Open LANE Dissertation FINAL.Pdf

The Pennsylvania State University The Graduate School College of Engineering

THE DEVELOPMENT OF A COMPREHENSIVE ANNULAR FLOW

MODELING PACKAGE FOR TWO-PHASE THREE-FIELD

TRANSIENT SAFETY ANALYSIS CODES

A Dissertation in Nuclear Engineering by Jeffrey W. Lane

Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

August 2009

The dissertation of Jeffrey W. Lane was reviewed and approved* by the following:

Lawrence E. Hochreiter Professor of Mechanical and Nuclear Engineering Dissertation Co-Advisor Co-Chair of Committee

Fan-Bill Cheung Professor of Mechanical and Nuclear Engineering Dissertation Co-Advisor Co-Chair of Committee

Kostadin Ivanov Distinguished Professor of Nuclear Engineering

John H. Mahaffy Associate Professor Emeritus of Nuclear Engineering

Gita Talmage Professor of Mechanical Engineering

Ali Borhan Professor of Chemical Engineering

David L. Aumiller, Jr. Fellow Engineer, Bechtel Marine Propulsion Corporation Special Member

Jack S. Brenizer Professor of Mechanical and Nuclear Engineering Program Chair of Nuclear Engineering

*Signatures are on file in the Graduate School.

Abstract

The annular two-phase flow regime is important to several applications and most notably the safety analysis of nuclear reactors. Such analyses require an accurate prediction of the phenomena associated with this regime, including the pressure gradient as well as the distribution of liquid and the interfield rate of exchange between the film and dispersed droplet fields. In general, the nuclear industry uses transient safety analysis codes, such as COBRA-TF, to predict phenomena of interest for various reactor accident scenarios and ensure the safe design of the system. COBRA-TF is a best-estimate thermal-hydraulic analysis tool developed for Light Water Reactors (LWR) and the primary feature of COBRA-TF is that it provides a three-field representation of two-phase flow (vapor/non-condensable gases, continuous liquid or films, and dispersed liquid or droplets). This representation is regarded as the most physically accurate approach for analyzing situations where liquid can coexist in both continuous and discrete forms, as is the case for annular-mist and counter-current flow situations, since substantial differences can exist in the velocity and flow direction for these two fields.

The prediction of annular flow situations requires a variety of constitutive relationships to describe the mass, momentum, and energy exchange that occurs between the flow fields and provide closure to the set of momentum equations. An initial assessment of the predictive capability of COBRA-TF indicated that the modeling package that was used in the baseline version of the code did not provide adequate predictions when a variety of annular flow experiments were simulated. As a result, the goal of the current study was to assemble a physically-based and self-consistent annular flow modeling package that is amenable to implementation in three-field analysis environments and accurately captures the variation in entrainment and interfacial drag within co-current and counter-current regimes over the pressure range of interest (atmospheric to 2000-psia). The constitutive relations available in the open- literature were assessed relative to the models employed in the modeling package that was applied in the baseline version of COBRA-TF. Where necessary, model upgrades were made in an effort to utilize the most appropriate models that are based on either the physics of the flow or developed from experimental data collected over the desired range of conditions. The models that were incorporated into the newly proposed modeling package were either based on those developed in previous studies or developed uniquely within the current study. It is important to

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note that the current study used COBRA-TF to provide the baseline modeling package as a means for comparison and as a vehicle for assessing the newly proposed modeling packages; however, the proposed packages are amenable to implementation into any other three-field analysis tool.

The proposed modeling packages for co-current and counter-current annular flow are outlined in Chapters 4 and 5, respectively. The co-current modeling package: 1) applies an interfacial shear model that explicitly accounts for the presence of interfacial waves, 2) idealizes the structure of the interface in a manner that is consistent with both the interfacial shear model and other visual observations, 3) includes three mechanistic-based entrainment rate models (roll wave stripping, Kelvin-Helmholtz lifting, and liquid bridge breakup) that calculate a theoretical entrainment rate for a single wave based on the physical structures and controlling phenomena as they are currently understood for each mechanism, and 4) provides a functional relationship between the actual and theoretical entrainment rates based on comparisons to experimental data to account for any deficiencies that exist in the theoretical model. This methodology improves the physical basis of the modeling package while simultaneously leveraging the available experimental data to ensure the modeling package is able to accurately reflect the experimental data.

Meanwhile, the three-field Counter-Current Flow Limitation (CCFL) model developed in the current study is based on an empirical model that has been shown to suitably correlate specific sets of data over a wide range of flow path dimensions and geometries. The resulting correlation provides a quantitative description of the experimentally determined flooding curve. The proposed model compares the flow conditions predicted by the code to the results of the user- specified CCFL correlation to determine if the standard set of momentum equations should be replaced with a newly developed set of CCFL momentum equations. The proposed model also provides appropriate entrainment rate models (pool and excess film) and necessary criterion to exit the model in a stable manner. In general, this approach provides flexibility to the code user and again leverages the available experimental data to improve the predictive capability of the code since a universal model has yet to be determined for this phenomenon. While not entirely mechanistic, this approach ensures the proper amount of liquid flow can penetrate these regions, which is preeminent to achieving accurate predictions of coolant and temperature distributions for Loss-of-Coolant Accident (LOCA) scenarios. Overall the development of this model is a unique aspect of the current study because of the explicit treatment of the entrained field, which

previously suggested models did not consider because they were aimed at two-field analysis environments.

The results of the current study indicate that the inclusion of these newly proposed modeling packages for both co-current and counter-current annular flow has provided increased accuracy in the predictions of phenomena that are of interest to reactor safety analyses. In particular, the mean relative error in entrained fraction was reduced from 20.2% (underprediction) to 4.5% (overprediction) and the mean relative error in axial pressure gradient was reduced from 108.2% to 7.6% (both overprediction) for co-current upward annular flow situations following the implementation of these packages into COBRA-TF and the code-to-data agreement of several different parameters within the counter-current flow regime was improved significantly. It was also shown that the proposed co-current annular modeling package: 1) provided reasonable estimates of a variety of more fundamental annular flow parameters such as wave spacing, velocity, and intermittency, and 2) was able to capture the general behavior within the developing flow region. Both these results provide confidence that the proposed modeling package reasonably reflects the underlying physics of the annular regime. Moreover, the current study is one of the few works that has examined the predictive capabilities of transient analysis codes within the developing, or non-equilibrium, annular flow region.

The methodology employed in the current study is not meant to provide a final solution to this complex problem; however, given the importance of these phenomena to the safety analysis of various reactor accident scenarios and the abundance of available experimental data, it would be inopportune not to employ this modeling methodology and improve the predictive capabilities of three-field transient analysis codes until a more viable approach is ascertained. Regardless, the current study has both provided a functional modeling package that has presently improved the predictive capabilities of three-field analysis tools and established a new baseline for future research and model development activities in this area.

Table of Contents

List of Figures ...... x List of Tables...... xix Nomenclature ...... xx Acknowledgements ...... xxv 1. Introduction ...... 1 1.1. Overview of Annular Flow...... 2 1.2. Comparison of Annular to Single-Phase Flow Situations...... 10 1.3. Importance of Annular Flow ...... 12 1.4. Scope and Objectives of the Current Study ...... 15 2. Background ...... 19 2.1. Transient Safety Analysis Codes...... 20 2.1.1. COBRA-TF Code Description...... 23 2.2. Definition of Annular Flow Regime Boundaries ...... 33 2.2.1. Baseline Pre-CHF Flow Regime Map in COBRA-TF...... 33 2.2.2. Co-Current Upward Annular Flow Regime Boundary ...... 37 2.2.3. Local Minimum in the Pressure Gradient...... 43 2.2.4. Conditions Required for Entrainment to Occur...... 45 2.2.5. Conclusions...... 50 2.3. Review of Works Focused on Interfacial Phenomena...... 52 2.3.1. Interfacial Structure...... 52 2.3.2. Interfacial Shear Stress Definition ...... 61 2.3.3. Current COBRA-TF Calculation of Interfacial Shear Stress...... 64 2.3.4. Interfacial Friction Factor...... 66 2.3.5. Droplet Drag ...... 80 2.3.6. Conclusions...... 84 2.4. Review of Works Focused on Entrainment Phenomena...... 86 2.4.1. Entrainment Mechanisms ...... 91

2.4.2. Entrainment and Deposition Rates...... 96 2.4.3. Entrained Droplet Size and Size Distribution...... 110 2.4.4. Conclusions...... 121 2.5. Review of Works Focused on Flooding and Flow Reversal...... 123 2.5.1. Counter-Current Flow Situations ...... 124 2.5.2. Mechanism of Flooding and Flow Reversal...... 126 2.5.3. Available CCFL Correlations ...... 130 2.5.4. Conclusions...... 138 2.6. Concerns with Currently Available Models...... 140 2.7. Motivation for the Current Study ...... 142 3. Methodology ...... 145 3.1. Experimental Data for Model Development and Assessment...... 145 3.1.1. Description of Desired Data ...... 146 3.1.2. Discussion of Entrained Fraction Measurement Techniques ...... 147 3.1.3. Co-Current Annular Flow Experiments ...... 149 3.1.4. Counter-Current Flow Limitation (CCFL) Experiments...... 155 3.2. Approach...... 157 3.2.1. Description of Input Parameters ...... 157 3.2.2. Optimization Scheme for Model Development ...... 160 3.2.3. Execution of Cases for Model Assessment ...... 165 3.2.4. Comparison of Results ...... 166 3.2.5. Sensitivity Studies...... 168 3.3. Conclusions ...... 170 4. Annular Flow Modeling Package ...... 172 4.1. Results Obtained using the Baseline Modeling Package in COBRA-TF ...... 172 4.2. Description of Proposed Modeling Package ...... 177 4.2.1. Preliminary Modifications...... 177 4.2.2. Regime Boundaries ...... 180 4.2.3. Interfacial Shear Stress...... 185

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4.2.4. Entrainment Rate...... 202 4.2.5. Droplet Drag ...... 232 4.2.6. Deposition Rate...... 232 4.2.7. Entraining Drop Size ...... 233 4.2.8. Wall Shear Stress ...... 234 4.2.9. Churn-Turbulent Regime...... 235 4.2.10. Conclusions on Proposed Modeling Package...... 236 4.3. Presentation and Assessment of Results ...... 239 4.3.1. Entrained Fraction...... 242 4.3.2. Pressure Gradient ...... 246 4.3.3. Qualitative Assessment of Developing Flow Parameters...... 247 4.3.4. Results of Cousins & Hewitt [67] Simulations...... 251 4.3.5. Qualitative Comparisons of Other Parameters ...... 253 4.4. Results of Sensitivity Studies ...... 266 4.5. Conclusions ...... 272 5. Counter-Current Flow Limitation Model ...... 273 5.1. Model Basis...... 274 5.2. Model Description ...... 276 5.3. Model Implementation...... 278 5.3.1. Activating the CCFL Model ...... 278 5.3.2. CCFL Momentum Equations...... 281 5.3.3. CCFL Entrainment Rate Models...... 286 5.3.4. Exiting the CCFL Model ...... 291 5.3.5. Assumptions...... 293 5.4. Required COBRA-TF Code Modifications...... 293 5.5. Verification Problem Description and Results ...... 294 5.6. Simulation of Dukler & Smith [43] Experiments...... 311 5.7. Conclusions on CCFL Model...... 316 6. Conclusions ...... 318

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7. Recommendations for Future Work ...... 320 8. References ...... 322 Appendix A: Subroutine Listing of Proposed Annular Flow Model ...... 330 Appendix B: Code-to-Data Comparisons of the Würtz [79] Experiments ...... 347 Appendix C: Qualitative Comparisons to the Disturbance Wave Characteristics Observed by Sawant et al. [57] ...... 353 Appendix D: Subroutine Listings for Proposed Counter-Current Flow Limitation (CCFL) Model ...... 358

List of Figures

Figure 1-1: Annular Two-Phase Flow [1]...... 1

Figure 1-2: Pressure Gradient as a function of Gas Flow Rate ranging from Counter- Current through Co-Current Annular Flow Conditions [4,5]...... 3

Figure 1-3: Summary of Direction and Qualitative Variation in Several Quantities for the Different Annular Flow Regimes...... 4

Figure 2-1: Schematic of Staggered Mesh Scheme used by COBRA-TF...... 25

Figure 2-2: Visual Representation of Pre-CHF Flow Regimes...... 34

Figure 2-3: Baseline COBRA-TF Cold Wall Flow Regime Map...... 35

Figure 2-4: Relationship between entrained fraction and gas velocity for a constant total liquid flow rate [3]...... 46

Figure 2-5: Qualitative schematic of conditions required for entrainment by the roll-wave mechanism [3]...... 48

Figure 2-6: Comparison of Droplet Drag Coefficient Expressions...... 83

Figure 2-7: Droplet Entrainment by the Kelvin-Helmholtz Lifting Mechanism [40]...... 92

Figure 2-8: Droplet Entrainment by the Boundary Layer Stripping Mechanism [74]...... 93

Figure 2-9: Droplet Entrainment by the Liquid Bridge Breakup Mechanism [3]...... 94

Figure 2-10: Possible Two-Phase Flow Situations [8]...... 123

Figure 2-11: Flooding and Flow Reversal [5]...... 125

Figure 4-1: COBRA-TF Predicted versus Experimentally Measured Axial Pressure Gradient using the Baseline Modeling Package...... 174

Figure 4-2: COBRA-TF Predicted versus Experimentally Measured Outlet Entrained Fraction using the Baseline Modeling Package...... 174

Figure 4-3: Comparison of the COBRA-TF Predicted and the Experimentally Measured Outlet Entrained Mass Flow Rate as a function of Flowing Quality for the Hewitt & Pulling [83] Experiments in a 12-foot test section using the Baseline Modeling Package...... 175

Figure 4-4: Comparison of the COBRA-TF Predicted and the Experimentally Measured Outlet Entrained Mass Flow Rate as a Function of Flowing Quality for the Yanai [64,84] Experiments using the Baseline Modeling Package...... 175

Figure 4-5: Comparison of the Experimentally Measured and the COBRA-TF Predicted Entrained Flow Rate as a function of Axial Elevation for the Cousins & Hewitt [67] Experiments using the Baseline Modeling Package...... 176

Figure 4-6: Comparison of the Experimentally Measured and the COBRA-TF Predicted Axial Pressure Gradient as a function of Axial Elevation for the Cousins & Hewitt [67] Experiments using the Baseline Modeling Package...... 176

Figure 4-7: Comparison of Film Thickness Calculations...... 180

Figure 4-8: Comparison of the Correlation for the Wave Height Standard Deviation and the specified Spline Fit as a function of Mean Film Thickness...... 192

Figure 4-9: Comparison of the Velocity Correlations as a function of Mean Dimensionless Film Thickness (in interfacial units)...... 195

Figure 4-10: Intermittency as a function of Dimensionless Film Thickness (in interfacial units)...... 199

Figure 4-11: Schematic of Idealized Interfacial Structure...... 203

Figure 4-12: Correlation between the Actual and Theoretical Entrainment Rates for the Roll Wave Stripping Mechanism as a function of Relative Wave Height...... 214

Figure 4-13: Comparison of Optimized and Correlated Results for the Proposed Functional Relationship between the Actual and Theoretical Entrainment Rates for the Roll Wave Stripping Mechanism...... 215

Figure 4-14: Disc Integration Method [95]...... 220

Figure 4-15: Streamlines in Two-Phase Flow with a Wavy Interface [5]...... 223

Figure 4-16: Pressure Gradient Normal to a Wavy Interface [5]...... 224

Figure 4-17: Normal Stresses and Coordinate Systems Utilized by Holowach [33,34] in the Interfacial Instability Calculation...... 227

Figure 4-18: Correlation between Maximum and Actual Entrainment Rate for the Kelvin- Helmholtz lifting mechanism as a function of Film Reynolds Number...... 230

Figure 4-19: Proposed COBRA-TF Pre-CHF Flow Regime Map...... 238

Figure 4-20: COBRA-TF Predicted versus Experimentally Measured Results for Outlet Flowing Quality using the Proposed Modeling Package...... 239

Figure 4-21: Comparison of the COBRA-TF Predicted Vapor Void Fraction as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet...... 240

Figure 4-22: Ratio of COBRA-TF Predicted Entrainment to Deposition Rates at the Outlet as a function of Measured Flowing Quality using the Proposed Modeling Package...... 241

Figure 4-23: COBRA-TF Predicted versus Experimentally Measured Outlet Entrained Fraction using the Proposed Modeling Package...... 242

Figure 4-24: Comparison of COBRA-TF Predicted Experimentally Measured Outlet Entrainment Mass Flow Rate as a Function of Flowing Quality for the Hewitt & Pulling [83] Experiments in the 12-foot Test Section using the Proposed Modeling Package...... 243

Figure 4-25: Comparison of COBRA-TF Predicted Experimentally Measured Outlet Entrainment Mass Flow Rate as a Function of Flowing Quality for the Hewitt & Pulling [83] Experiments in the 6-foot Test Section using the Proposed Modeling Package...... 243

Figure 4-26: Comparison of COBRA-TF Predicted Experimentally Measured Outlet Entrainment Mass Flow Rate as a Function of Flowing Quality for the Yanai [64,84] Experiments using the Proposed Modeling Package...... 244

Figure 4-27: COBRA-TF Predicted versus Experimentally Measured Results for Axial Pressure Gradient using the Proposed Modeling Package...... 246

Figure 4-28: Axial Distribution in the COBRA-TF Predicted Film Reynolds Number for the Singh [82] Experiments using the Proposed Modeling Package...... 248

Figure 4-29: Axial Distribution in the COBRA-TF Predicted Entrained Fraction for the Singh [82] Experiments using the Proposed Modeling Package...... 249

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Figure 4-30: Axial Distribution in the COBRA-TF Predicted Ratio of Entrainment to Deposition Rate for the Singh [82] Experiments using the Proposed Modeling Package...... 249

Figure 4-31: Axial Distribution in the COBRA-TF Predicted Film Reynolds Number for the Keeys et al. [81] Experiments using the Proposed Modeling Package...... 250

Figure 4-32: Axial Distribution in the COBRA-TF Predicted Entrained Fraction for the Keeys et al. [81] Experiments using the Proposed Modeling Package...... 250

Figure 4-33: Axial Distribution in the COBRA-TF Predicted Entrained Fraction for the Keeys et al. [81] Experiments using the Proposed Modeling Package...... 251

Figure 4-34: Experimentally Measured and COBRA-TF Predicted Entrained Flow Rate as a function of Axial Elevation for the Cousins & Hewitt [67] Experiments using the Proposed Modeling Package...... 252

Figure 4-35: Experimentally Measured and COBRA-TF Predicted Axial Pressure Gradient as a function of Axial Elevation for the Cousins & Hewitt [67] Experiments using the Proposed Modeling Package...... 252

Figure 4-36: Ratio of COBRA-TF Predicted Ratio of Film to Total Interfacial Drag at the Outlet as a function of Dimensionless Superficial Gas Velocity using the Proposed Modeling Package...... 254

Figure 4-37: Comparison of the COBRA-TF Predicted Ratio of the Mean Entrained to Mean Vapor Velocities as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet...... 254

Figure 4-38: Ratio of COBRA-TF Predicted Ratio of Deposition to Entrainment Momentum at the Outlet as a function of Dimensionless Superficial Gas Velocity using the Proposed Modeling Package...... 255

Figure 4-39: Comparison of the COBRA-TF Predicted Mean Vapor to Film Relative Velocity as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet...... 255

Figure 4-40: Comparison of the COBRA-TF Predicted Disturbance Wave to Base Film Velocities as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet...... 257

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Figure 4-41: Comparison of the COBRA-TF Predicted Ratio of Disturbance Wave Amplitude to Mean Film Thickness as a function of Film Reynolds Number at the Test Section Outlet...... 257

Figure 4-42: Comparison of the COBRA-TF Predicted Equivalent Angle of the Disturbance Waves as a function of Film Reynolds Number at the Test Section Outlet...... 258

Figure 4-43: Comparison of the COBRA-TF Predicted Velocity Enhancement Factor at the Disturbance Wave Crest as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet...... 260

Figure 4-44: Comparison of the COBRA-TF Predicted Intermittency as a function of Film Reynolds Number at the Test Section Outlet...... 260

Figure 4-45: Comparison of the COBRA-TF Predicted Ratio of the Mean to Critical Film Thickness as a function of Film Reynolds Number at the Test Section Outlet...... 261

Figure 4-46: Comparison of the COBRA-TF Predicted Disturbance Wave Frequency as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet. 261

Figure 4-47: Comparison of the COBRA-TF Predicted Disturbance Wave Spacing as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet. 262

Figure 4-48: Comparison of the COBRA-TF Predicted Number of Disturbance Waves per Computational Cell as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet...... 262

Figure 4-49: Comparison of the COBRA-TF Predicted Weber Number (based on Film Thickness) as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet...... 264

Figure 4-50: Comparison of the COBRA-TF Predicted Ratio of the Actual to Theoretical Entrainment Rate as a function of Film Reynolds Number at the Test Section Outlet...... 264

Figure 4-51: Comparison of the COBRA-TF Predicted Ripple Wavelength as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet...... 265

Figure 4-52: Sensitivity of COBRA-TF Predicted Entrained Fraction on Computational Node Height using the Proposed Modeling Package...... 267

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Figure 4-53: Sensitivity of COBRA-TF Predicted Axial Pressure Gradient on Computational Node Height using the Proposed Modeling Package...... 267

Figure 4-54: Sensitivity of COBRA-TF Predicted Entrained Fraction on Liquid Injection Area using the Proposed Modeling Package...... 268

Figure 4-55: Sensitivity of COBRA-TF Predicted Axial Pressure Gradient on Liquid Injection Area using the Proposed Modeling Package...... 268

Figure 4-56: Sensitivity of COBRA-TF Predicted Entrained Flow Rate on Inlet Entrained Friction using the Proposed Modeling Package...... 270

Figure 4-57: Sensitivity of COBRA-TF Predicted Axial Pressure Gradient on Inlet Entrained Friction using the Proposed Modeling Package...... 270

Figure 4-58: Sensitivity of COBRA-TF Predicted Entrained Flow Rate on Inlet Entrained Droplet Diameter using the Proposed Modeling Package...... 271

Figure 4-59: Sensitivity of COBRA-TF Predicted Axial Pressure Gradient on Inlet Entrained Droplet Diameter using the Proposed Modeling Package...... 271

Figure 5-1: Schematic of Pool Entrainment Mechanism [98]...... 288

Figure 5-2: Noding Diagram of CCFL Verification Problem...... 295

Figure 5-3: Comparison of COBRA-TF Predicted Results for the CCFL Verification Problem With and Without a Wallis-Type CCFL Boundary Condition Applied when the Vapor Flow Rate is Increased over the course of the Simulation...... 299

Figure 5-4: Comparison of COBRA-TF Predicted Results for the CCFL Verification Problem With and Without a Kutateladze-Type CCFL Boundary Condition Applied when the Vapor Flow Rate is Increased over the course of the Simulation...... 300

Figure 5-5: COBRA-TF Predicted Flow Regime as a function of Simulation Time for the CCFL Verification Problem when the Vapor Flow Rate is Increased over the course of the Simulation...... 302

Figure 5-6: COBRA-TF Predicted Liquid Mass Flow Rate in Node 3 as a function of Simulation Time for the CCFL Verification Problem when the Vapor Flow Rate is Increased over the course of the Simulation...... 303

Figure 5-7: COBRA-TF Predicted Entrainment Rate in Node 5 as a function of Simulation Time for the CCFL Verification Problem when the Vapor Flow Rate is Increased over the course of the Simulation...... 303

Figure 5-8: COBRA-TF Predicted Continuous Liquid Volume Fraction in Node 5 as a function of Simulation Time for the CCFL Verification Problem when the Vapor Flow Rate is Increased over the course of the Simulation...... 304

Figure 5-9: Comparison of COBRA-TF Predicted Flow Rates as a function of Simulation Time for the CCFL Verification Problem when No CCFL Boundary Condition is Applied and the Vapor Flow Rate is Increased over the course of the Simulation. 305

Figure 5-10: Comparison of COBRA-TF Predicted Flow Rates as a function of Simulation Time for the CCFL Verification Problem when a Wallis-Type CCFL Boundary Condition is Applied and the Vapor Flow Rate is Increased over the course of the Simulation...... 305

Figure 5-11: Comparison of COBRA-TF Predicted Flow Rates as a function of Simulation Time for the CCFL Verification Problem when a Kutateladze-Type CCFL Boundary Condition is Applied and the Vapor Flow Rate is Increased over the course of the Simulation...... 306

Figure 5-12: Comparison of COBRA-TF Predicted Results for the CCFL Verification Problem With and Without a Wallis-Type CCFL Boundary Condition Applied when the Vapor Flow Rate is Decreased over the course of the Simulation...... 307

Figure 5-13: Comparison of COBRA-TF Predicted Results for the CCFL Verification Problem With and Without a Kutateladze-Type CCFL Boundary Condition Applied when the Vapor Flow Rate is Decreased over the course of the Simulation...... 308

Figure 5-14: COBRA-TF Predicted Flow Regime as a function of Simulation Time for the CCFL Verification Problem when the Vapor Flow Rate is Decreased over the course of the Simulation...... 309

Figure 5-15: COBRA-TF Predicted Liquid Mass Flow Rate in Node 3 as a function of Simulation Time for the CCFL Verification Problem when the Vapor Flow Rate is Decreased over the course of the Simulation...... 310

Figure 5-16: Correlation of CCFL Data Collected by Dukler & Smith [43]...... 311

Figure 5-17: Comparison of COBRA-TF Predicted Liquid Downflow Rate With and Without a CCFL Boundary Condition Applied to the Experimental Results Collected by Dukler & Smith [43] for a Liquid Injection Rate of 500-lb m/hr...... 313

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Figure 5-18: Comparison of COBRA-TF Predicted Outlet Entrained Flow Rate With and Without a CCFL Boundary Condition Applied to the Experimental Results Collected by Dukler & Smith [43] for a Liquid Injection Rate of 500-lb m/hr...... 313

Figure 5-19: Comparison of COBRA-TF Predicted Liquid Film Upflow Rate With and Without a CCFL Boundary Condition Applied to the Experimental Results Collected by Dukler & Smith [43] for a Liquid Injection Rate of 500-lb m/hr...... 314

Figure 5-20: Comparison of COBRA-TF Predicted Axial Pressure Gradient With and Without a CCFL Boundary Condition Applied to the Experimental Results Collected by Dukler & Smith [43] for a Liquid Injection Rate of 500-lb m/hr...... 315

Figure 5-21: Comparison of COBRA-TF Predicted Mean Film Thickness With and Without a CCFL Boundary Condition Applied to the Experimental Results Collected by Dukler & Smith [43] for a Liquid Injection Rate of 500-lb m/hr...... 315

Figure B-1: Comparison of Experimentally Measured and COBRA-TF Predicted Outlet Entrained Flow Rate as a function of Flowing Quality for the Würtz [79] Experiments conducted in a 0.394-inch diameter tube at 435-psia...... 348

Figure B-2: Comparison of Experimentally Measured and COBRA-TF Predicted Axial Gradient as a function of Flowing Quality for the Würtz [79] Experiments conducted in a 0.394-inch diameter tube at 435-psia...... 348

Figure B-3: Comparison of Experimentally Measured and COBRA-TF Predicted Outlet Entrained Flow Rate as a function of Flowing Quality for the Würtz [79] Experiments conducted in a 0.394-inch diameter tube at 725-psia...... 349

Figure B-4: Comparison of Experimentally Measured and COBRA-TF Predicted Axial Gradient as a function of Flowing Quality for the Würtz [79] Experiments conducted in a 0.394-inch diameter tube at 725-psia...... 349

Figure B-5: Comparison of Experimentally Measured and COBRA-TF Predicted Outlet Entrained Flow Rate as a function of Flowing Quality for the Würtz [79] Experiments conducted in a 0.394-inch diameter tube at 1015-psia...... 350

Figure B-6: Comparison of Experimentally Measured and COBRA-TF Predicted Axial Gradient as a function of Flowing Quality for the Würtz [79] Experiments conducted in a 0.394-inch diameter tube at 1015-psia...... 350

Figure B-7: Comparison of Experimentally Measured and COBRA-TF Predicted Outlet Entrained Flow Rate as a function of Flowing Quality for the Würtz [79] Experiments conducted in a 0.394-inch diameter tube at 1305-psia...... 351

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Figure B-8: Comparison of Experimentally Measured and COBRA-TF Predicted Axial Gradient as a function of Flowing Quality for the Würtz [79] Experiments conducted in a 0.394-inch diameter tube at 1305-psia...... 351

Figure B-9: Comparison of Experimentally Measured and COBRA-TF Predicted Outlet Entrained Flow Rate as a function of Flowing Quality for the Würtz [79] Experiments conducted in a 0.787-inch diameter tube at 1015-psia...... 352

Figure B-10: Comparison of Experimentally Measured and COBRA-TF Predicted Axial Gradient as a function of Flowing Quality for the Würtz [79] Experiments conducted in a 0.787-inch diameter tube at 1015-psia...... 352

Figure C-1: Comparison of the COBRA-TF Predicted Disturbance Wave Velocity as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet. 354

Figure C-2: Comparison of the COBRA-TF Predicted Disturbance Wave Velocity as a function of Film Reynolds Number at the Test Section Outlet...... 354

Figure C-3: Comparison of the COBRA-TF Predicted Disturbance Wave Amplitude as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet. 355

Figure C-4: Comparison of the COBRA-TF Predicted Disturbance Wave Amplitude as a function of Film Reynolds Number at the Test Section Outlet...... 355

Figure C-5: Comparison of the COBRA-TF Predicted Disturbance Wave Spacing as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet. 356

Figure C-6: Comparison of the COBRA-TF Predicted Disturbance Wave Spacing as a function of Film Reynolds Number at the Test Section Outlet...... 356

Figure C-7: Comparison of the COBRA-TF Predicted Disturbance Wave Frequency as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet. 357

Figure C-8: Comparison of the COBRA-TF Predicted Disturbance Wave Frequency as a function of Film Reynolds Number at the Test Section Outlet...... 357

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List of Tables

Table 2-1: Summary of Conservation Equations used by COBRA-TF...... 27

Table 2-2: Summary of Disturbance Wave Characteristics obtained by Sawant et al. [57]...... 57

Table 3-1: Summary of Test Conditions for Available Co-Current Upward Annular Entrainment Experiments...... 154

Table 3-2: Summary of Boundary Conditions...... 158

Table 3-3: Summary of Modeling Parameters...... 160

Table 4-1: Results of the Statistical Analysis for Entrained Fraction in Co-Current Upward Annular Flow...... 245

Table 4-2: Results of the Statistical Analysis for Pressure Gradient in Co-Current Upward Annular Flow...... 247

Table 5-1: Summary of Results for CCFL Verification Problem...... 298

Table C-1: Comparison of Qualitative Trends in Disturbance Wave Characteristics for Results Predicted by Proposed Modeling Package and those Observed by Sawant et al. [57]...... 353

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Nomenclature

General

A area [L 2]

Ao correlating parameter (Bharathan & Wallis [11]) A′′′ interfacial area concentration density [L -1] B log law parameter

Bo correlating parameter (Bharathan & Wallis [11])

Bn set of correlating parameters C droplet concentration [M 1L-3] or geometry dependent CCFL correlating constant

cb parameter in log law velocity profile

Cd droplet drag coefficient

C p specific heat

Cw wave velocity or correlating parameter ( Kocamustafaogullari et al. [76])

Cµ mixture viscosity parameter D diameter of flow path or droplets [L 1] E entrained fraction f friction factor, frequency [T-1], or specified function f ′ derivative of specified function F correlating parameter (Henstock & Hanratty [27]) F′ force per unit length in axial direction g gravitational constant [L 1T-2] G mass flux [M1T-1L-2] or correlating parameter (Henstock & Hanratty [27])

h fg enthalpy of vaporization H thickness of non-circular flow path [L 1] or CCFL non-dimensional parameter i imaginary number isij flow regime identifier j superficial velocity [L 1T-1] k mass transfer coefficient [L1T-1] or wave number K COBRA-TF definition of drag coefficient

ks sand grain roughness L Length or Laplace capillary length m mass [M1], wave slope, or CCFL correlating constant n exponent for velocity profile or normal distance from interface N refers to number or normal stress p pressure P perimeter q′′ wall heat flux

r radial distance [L1] R ramping function or tube radius [L 1]

R1−4 defined parameters associated with two-zone interfacial shear model s distance along surface of the interface S interfield mass transfer flux [M 1T-1L-2] S′ interfield mass transfer flow rate [M 1T-1] 1 t E time scale for entrainment mechanism [T ]

tTB total dimensionless breakup time (Pilch & Erdman [85]) ∆t computational time step size [T 1] T Temperature u,v local velocity [L 1T-1] U bulk mean velocity [L 1T-1] U * friction velocity [L 1T-1] V volume [L 3] W field mass flow rate [M 1T1] or width of non-circular flow path [L 1] wght CCFL weighting parameter x optimized values y distance from surface in normal direction Y CCFL property group [T1L-1] z axial distance along flow direction [L 1] ∆z length of computational cell in axial or flow direction [L1]

Statistical Parameters

IA index of agreement ME mean error MFE mean fraction error

ρ xy cross-correlation coefficient N number of data points in set

Oi i-th datum in experimental set

Pi i-th datum in computed set O average of observed data P average of computed data

Err max maximum possible error

xxi

Greek Letters

α unsubscripted this is void fraction, with subscript this is volume fraction of the subscripted phase β CCFL correlation indicator δ film thickness [L1] δ mean film thickness [L1] ε amplitude [L1] εˆ non-dimensional roughness η efficiency or normalized distance within stripping layer κ von Kármán constant λ wavelength [L1] or wave spacing [L1] µ dynamic viscosity ν kinematic viscosity π pi, mathematical constant ρ density [M 1L-3] ∆ρ density difference (liquid – vapor) σ surface tension 1 σ δ film thickness standard deviation [L ] + σ δ dimensionless film thickness standard deviation τ shear stress ψ intermittency Π dimensionless functions used to correlate results Θ model function χ curvature of interface

Subscripts

∞ fully-developed or annular equilibrium 1-wave refers to a quantity calculated for a single wave 32 Sauter mean value AN annular regime B breakup bridge liquid bridging CCFL counter-current flow limitation cell related to computational cell CHF related to critical heat flux crit critical CT churn-turbulent d entrained or dispersed droplets D deposition D2U transition from downflow to upflow

xxii

DW disturbance wave duct flow path with a non-circular cross-section e dispersed or entrained droplet field ent entrainment E entraining exfilm excess film flow model exp experimental flood flooding FR flow reversal gc gas core region (vapor + dispersed droplet fields) GS gas superficial H hydraulic HH refers to Henstock & Hanratty [27] i interfacial I imaginary component incpt entrainment inception criterion j computational cell k experimental run or generic field/phase placeholder l continuous liquid / film field or liquid phase l,tot total liquid (continuous + entrained fields) lam laminar lim limiting dimension LBB liquid bridge breakup LS liquid superficial m mixture M total number of cases considered max maximum min minimum mod modified model theoretical model value n counter variable N total number of correlating parameters Γ phase change pool pool entrainment mechanism pred predicted value R real component RW ripple wave s surface or smooth wall value sat saturation SL stripping layer ST stable annular film regime sup entrainment suppression criterion tube flow path with a circular cross-section

xxiii

turb turbulent UN unstable annular film regime v vapor field or phase vl between vapor and continuous liquid fields ve between vapor and dispersed droplet fields w wall or wave x cross-sectional

Superscripts

i iterate number n explicit quantity n +1 implicit or new time quantity n~ +1 tentative new time quantity

Dimensionless Parameters

δ + dimensionless film thickness (in interfacial units) δ * dimensionless film thickness D* dimensionless diameter Fr Froude number F′′ non-dimensional force per unit area

H k generic CCFL parameter * jk dimensionless superficial velocity

Ku k Kutateladze number

N µ viscosity number p* non-dimensional pressure Re Reynolds number

Re s Friction Reynolds number Sc Schmidt number Sr Strouhal number * τ i,vl non-dimensional interfacial shear stress We Weber number u′,v′ dimensionless velocities n′, s′ dimensionless directions

xxiv

Acknowledgements

First I would like to thank my dissertation advisor and co-committee chair, Professor Lawrence Hochreiter, who unfortunately passed away during the completion of this work. I sincerely appreciate all of his time and support during my time as both an undergraduate and graduate student at Penn State. His knowledge of and passion for nuclear engineering were instrumental both in the completion of this work and my development as a nuclear engineer. I am thankful to of had the opportunity to study under him.

Next, I would like to thank my mentor at Bechtel Marine Propulsion Corporation, Dr. David Aumiller, Jr. I am forever grateful for his immeasurable time and support during the completion of this work. His expertise in thermal hydraulic code development and analysis tools was extremely helpful throughout this process and the opportunity to work along side him during the past several years has proved to be an invaluable experience.

I would also like to acknowledge the contributions of the other members of my committee: co- committee chair Professor Fan-Bill Cheung, Professor John Mahaffy, Professor Kostadin Ivanov, Professor Gita Talmage, and Professor Ali Borhan. Their contributions and review of this work have helped shape the finished product. Additionally, I would like to thank Dr. Evan Hurlburt, Dr. Larry Fore, and Dr. Jack Buchanan, all of Bechtel Marine Propulsion Corporation, for sharing their annular flow expertise and insights throughout the completion of this work.

Special thanks are due to my family, and in particular to my fiancée, Jessica Henry, to my parents, George and Diane, and to my brothers, Brandon and Stephen. Jessica provided unwavering support, understanding, and encouragement throughout this entire process and I would not have been able to complete this work without her. My parents have supported and encouraged me throughout my life and have provided me with opportunities to pursue my interests. I would not be where I am today without them.

Finally, this research was performed under appointment to the Rickover Graduate Fellowship Program sponsored by Naval Reactors Division of the U.S. Department of Energy. Without the financial support of this program this work would not have been possible.

xxv

1. Introduction

The annular two-phase flow regime in a vertical orientation consists of a gas core flowing in the center of a duct surrounded by a thin liquid film flowing along the walls of the duct. A visual representation of annular flow is provided in Figure 1-1. A high degree of mechanical non- equilibrium exists between the gas core and film regions in this regime, with the vapor core moving much faster than the liquid film. Random perturbations in the flow can cause a wavy interface to develop on the film and the faster moving vapor phase can shear liquid from the interface causing it to become entrained in the form of droplets in the gas core region. This situation is commonly referred to as the annular mist flow regime. The flow of the two-phases for annular or annular mist situations can be either co-current or counter-current depending on the respective phasic flow rates, geometric configuration, and other conditions.

Figure 1-1: Annular Two-Phase Flow [1].

An accurate prediction of the pressure gradient and entrained fraction within vertically oriented annular flow is largely dependent on the film thickness and the structure of the interface [2]. The film thickness dictates the impact of the gravitational force, while the interfacial structure determines the interfacial shear stress and the potential for the entrainment of droplets to occur.

The interfacial shear stress influences the film thickness and phasic mass flow rates, both which impact the pressure gradient, while entrainment significantly alters the mass, momentum, and energy exchange between the flow fields [3]. In general, the relative magnitude of the wall shear, interfacial shear, and gravitational force dictates whether co-current or counter-current conditions will exist for a given set of flow conditions.

This chapter provides a detailed description of the characteristics of the annular flow regime ranging from counter-current through co-current upward conditions. Then the importance of annular flow in several different applications is discussed. Finally, an overview of the scope and objectives of the current study are provided.

1.1. Overview of Annular Flow

The annular two-phase flow regime covers a wide range of conditions in two-phase flow, occurring at qualities just above a few percent. This section provides a brief overview of the characteristics of the different annular regimes as the flow moves from a situation of counter- current flow through flooding and into co-current upward annular flow as it is currently understood. Qualitative changes in parameters such as the interfacial structure, film thickness, interfacial shear stress, wall shear stress, entrainment and pressure gradient are discussed. This section also provides several definitions and attempts to clarify some terminology that is often used interchangeably within the open-literature. In order to maintain a level of consistency the definitions provided here will be used throughout the remainder of the current study.

One common method for depicting the variation in the behavior of the annular flow regimes is shown in Figure 1-2. It can be seen from this figure that the available data indicates that for fixed liquid flow the pressure gradient is non-monotonic with respect gas flow. The two portions of the curve depicted in this figure are obtained by taking pressure gradient measurements below (left half) and above (right half) the liquid injection location. An annotated approximation of this curve is provided in Figure 1-3. It is indicated on Figure 1-3 the region of existence for each of the annular flow regimes that will be described within this section and also summarizes the direction and qualitative variation in several different quantities within each regime. For

example, this figure contains a smaller schematic for each regime that indicates: 1) the direction of the interfacial shear stress and wall shear stress and 2) the direction of flow within the film both near the wall and near the interface. The arrows associated with the film flow direction have been sized to indicate the relative magnitude of the liquid velocity at the wall and the interface. Additionally, arrows are provided to show whether the magnitude of the interfacial and wall shear stresses increase or decrease through a given regime as the vapor flow rate is increased. Finally, a relative comparison between the magnitudes of the two shear stress quantities within a given regime is provided.

As shown in Figure 1-3 the curve presented in Figure 1-2 can be easily divided into four distinct regions. As the gas flow rate is increased from zero for a given liquid flow rate the flow moves from: 1) counter-current flow, 2) through the flooding and flow reversal points, 3) to a churn- annular regime, and 4) finally to co-current upward annular flow. The characteristics of each of these regions and the transition points will now be described individually.

Figure 1-2: Pressure Gradient as a function of Gas Flow Rate ranging from Counter-Current through Co-Current Annular Flow Conditions [4,5].

Counter-Current Flooding and Churn-Annular Co-Current Upward Flow Reversal

|τw| Region Governed by CCFL τw τi |τi|

τw τi |τw| < | τi|

|τw| Flow |τi| Reversal

|τw| >> | τi| τw = 0

Flooding τw τi Point τw τi Axial Pressure Gradient Axial Pressure

τw τi

|τw| |τw|

|τi| |τi|

|τw| |τw| < | τi| |τw| << | τi|

|τi|

|τw| > | τi|

Vapor Flow Rate Figure 1-3: Summary of Direction and Qualitative Variation in Several Quantities for the Different Annular Flow Regimes.

In counter-current flow, where vapor flow upwards and liquid flows downwards, the interfacial shear caused by the vapor upflow is not sufficient to cause net liquid flow in the upward direction and thus no liquid flows out of the top of the flow path or is present above the location where it is supplied. Therefore, the downward liquid mass flow rate equals the mass flow rate of the liquid that is supplied to, or penetrates into, the flow duct. The interface between the falling film and the vapor upflow in this region is relatively smooth, which suggests the interfacial shear stress is

small. As a result the pressure gradient is small, since the weight of the film is on the same order of magnitude as the wall shear force, and very little entrainment exists in this region.

As the vapor flow rate is increased and the flooding point is approached for a given liquid flow rate the interface becomes increasingly rougher, which increases the interfacial shear stress and pressure gradient. The increased interfacial shear reduces the magnitude of the downward film flow rate and decreases the wall shear stress to the point where it is the same order of magnitude as the interfacial shear stress. This last characteristic is unlike the flooding and churn-annular regions that will be discussed next where the wall shear stress is negligible compared to the interfacial shear stress [6]. The increased interfacial shear also allows for thicker films to be supported and as a result the film thickness and liquid volume fraction increase through this region. The results of Zabaras & Dukler [7] show that mean film thickness was largely unaffected by increasing gas velocities at low gas flow rates, but a sharp exponential type increase was observed in the measured mean film thickness just prior to the flooding point, which results in an increased vapor velocity and leads to the development of large stagnation waves developing on the interface, which many researchers [4,8,9] have hypothesized cause flooding to occur. Following this maximum the mean film thickness decreases as the waves break up and became entrained. Similarly, a sharp decrease in the wall shear stress was measured by Zabaras & Dukler [7] just prior to the flooding point and then the wall shear stress approached zero with further increases in gas flow, which is as expected since the interfacial shear begins to partly support the weight of the film and the mean velocity of the film approaches zero. However, until the flooding condition is reached the net liquid flow remains in the downward direction and very little entrainment occurs.

The second distinct region exists between the flooding and the flow reversal points. In this region the liquid mass flow rate is partitioned between upflow and downflow with the amount of liquid downflow decreasing with increasing vapor flow rate. In other words, the liquid film displays an “oscillatory” behavior with the waves being transported upwards while the film between the waves drains downwards. The flooding condition is defined as the point where the interfacial drag is sufficient to prevent some portion of the liquid from flowing downwards. Meanwhile the flow reversal condition is defined as the point where the interfacial drag is sufficient to prevent any liquid from penetrating below the point that it is supplied to the flow duct. At the flow

reversal point the net liquid flow is in the upward direction and therefore this condition represents to the boundary between counter-current and co-current flow. The difference between vapor flow rate needed for flooding and flow reversal is dependent upon the liquid flow rate, with smaller liquid flow rates requiring a smaller difference. [7,10]

It is important to note that for vapor flow rates equal to and just greater than those corresponding to the flow reversal condition, the liquid flow near the wall may still be moving locally downwards; therefore the wall shear stress at this point is not equal to zero. Zabaras et al. [10] determined that at the flow reversal point that the gravitational contribution to the total pressure drop (determined from the time-averaged film thickness measurement) exceeded the total measured pressure drop, which indicates a compensating force must be acting in the upward direction at the wall, meaning that the liquid flow adjacent to the wall must be in the downward direction. Additionally, the flow reversal point does not correspond to a “hanging film” condition, where the wall shear stress, interfacial shear stress, and body force (i.e. gravity) sum to zero, because some liquid upflow has already been observed following the flooding point. [7,10]

The pressure gradient increases drastically through the region between the flooding and flow reversal points. Bharathan & Wallis [11] suggest that the sudden and drastic change in the pressure gradient in this region may suggest that the transition between counter-current and co- current flow is not approached as the limit of a continuous process, but rather is the result of a marked instability. They suggest this instability can be caused either by the stagnation and reversal of wave motion or significant amounts of entrainment resulting from liquid bridge breakup [11]. The exact mechanism is dependent on the diameter and geometry of the flow duct, but these effects have not yet been quantified. Regardless, the result is an increased apparent roughness and interfacial shear and decreased wall shear through this region as the liquid flow transitions from net downflow to net upflow. Meanwhile, the film thickness can either increase or decrease depending on the amount of entrainment that occurs, which is governed by the flooding mechanism. Lastly, the entrainment that occurs in this region determines the distribution of liquid between the droplets and film at the onset of co-current flow.

The third distinct region of annular flow, which occurs at gas velocities immediately above those required for flow reversal, is highly complex with a large amount of interfacial wave activity

causing a considerable amount of liquid entrainment to occur [12]. This region is commonly referred to as the churn-annular regime, but it has also been referred to as the thick film, film- churning, liquid breakup, or upflow with recirculation regime. For the purpose of clarity and conciseness this region will be referred to as the churn-annular regime throughout the remainder of this work. In general the liquid motion in this region is oscillatory, but not in a periodic or regular manner [13]. Since the vapor flow rates associated with this region are greater than those required for flow reversal the interfacial shear is sufficient to transport the net liquid flow upwards. However, based on wall shear stress measurements it has been found that in the churn- annular regime liquid downflow can still occur locally near the wall [10]. This behavior requires that a recirculation effect exists within the film to satisfy conservation of mass. Hewitt et al. [12] used a dye tracing visualization technique to show that the film surface in this region is traversed by large waves that deposit a liquid film in their wake. This liquid film initially travels upwards, but then gradually decelerates and may reverse direction before the arrival of the next wave [12].

Previously this region has been thought to be representative of churn-turbulent flow, where interfacial waves could become large such that “bridging” could occur across the cross section of the flow duct. However, the “bridging” phenomenon is hypothesized to be highly dependent on the size and geometry of the flow path. This phenomenon may occur in smaller diameter flow ducts, but film thickness measurements of Zabaras et al. [10] and others in larger diameter tubes (≥ 2-inches) indicate that bridging did not occur within this region. Instead, Zabaras et al. [10] suggest that in this region the wall shear stress fluctuates about zero with alternating sign due to the recirculation. The film interface is characteristically unstable, which prohibits the presence of coherent roll waves that travel significant distances. Instead, large disturbance waves grow at the interface and then subsequently breakup by vapor undercutting causing large amounts of entrainment. These waves govern the transport of liquid in this regime [14]. Larger wave heights lead to more violent the flow oscillation and make it more likely that significant portions of the wave will become entrained [14]. In any event, provisions should be made to determine whether churn-turbulent (liquid bridging) or churn-annular (large disturbance wave, no bridging) flow conditions exist in both experiments and predictive tools because the two entrainment mechanisms are fundamentally different. Additionally, the experimental results indicate the size of droplets generated by the liquid bridge breakup mechanism tend to be larger than those generated by the wave undercutting mechanism associated with churn-annular flow [15].

As the vapor flow rate is increased the amplitude and number of the disturbance waves decreases, while the wave velocity increases [14]. As the vapor flow rate is increased, a point of balance between the measured pressure gradient and the gravitational contribution is reached. This corresponds to the point where the wall shear stress is exactly zero [7,10]. This point occurs somewhere between the flow reversal and the local minimum in the pressure gradient; however, the vapor velocity corresponding to this condition is a function of the liquid flow rate [7,10]. Also, as this point is approached the flow throughout the film moves towards becoming unidirectional upwards and therefore the amount of recirculation occurring within the film is reduced. The overall result of these behaviors is that the pressure gradient and entrained fraction have both been found to decrease through this region [16,17,18]. The physical basis for this behavior is that as the large dissipative effects associated with the recirculation within the film are reduced the corresponding pressure gradient decreases. Less recirculation also reduces the instability within the film leading to fewer and smaller interfacial waves, which makes entrainment less likely to occur and acts to further reduce the pressure gradient by reducing the interfacial drag area. The churn-annular regime is terminated by a local minimum with respect to increasing vapor flow being observed in the pressure gradient and entrained fraction; however, it should be noted here that there is some question as to the existence of local minimum at higher pressures or larger flow rates [16,17,18].

Flow over a wavy surface experiences large positive and negative pressure gradients because of the wave-induced expansion and contraction of the streamlines [5]. The large amplitude waves associated with both the churn-annular regime and at low gas velocities in the co-current upward regime can induce separated flow as the vapor phase passes the wave crest. This separated flow region can extend over most of the wave trough, as confirmed by the experimental results of Buckles et al. [19] and the numerical results of Henn & Sykes [20], with the reattachment point moving further downstream with increasing wave amplitude. It is important to understand that such a region does exist since the total drag force over the wave is related to the total energy dissipation in the flow and therefore it may to necessary to explicitly account for this effect in interfacial drag models.

The fourth and final distinct region of annular flow consists of unidirectional upflow throughout the entire film region and is referred to as the co-current upward annular regime. The

measurements of wall shear by Zabaras et al. [10] within this region showed the wall shear to fluctuate, but always be less than zero (i.e. point in the downward direction, thus corresponding to liquid upflow near the wall). As the gas velocity is increased through this regime the film interface becomes increasingly rougher as the relative velocity between the vapor and film increases. In turn this leads to an increase in both the interfacial shear and the pressure gradient. Also, the increasing interfacial shear leads to larger film velocities, decreasing film thicknesses, and a higher propensity for entrainment.

The co-current upward annular regime can be subdivided into three regions. At low gas velocities immediately following the transition from churn-annular flow, the film is relatively smooth. Experimental evidence indicates a critical vapor flow rate exists a for vapor flow rates less than this critical value where traveling roll waves do not exist and entrainment does not occur [3]. This critical condition is referred to as the entrainment inception criterion.

As the gas velocity is increased further, the onset of film instability causes an increase in pressure gradient due to the increased roughness caused by the development of large interfacial waves. This region is dominated by the upward propagation of roll waves that, unlike the churn-annular regime, are coherent over lengths of several tube diameters [14]. The primary mechanism for entrainment in this region is liquid being sheared off the tops of the wave crests, which supports the idea that the presence of roll waves is a prerequisite for entrainment to occur [3]. The experimental results indicate the droplet sizes generated by the roll wave mechanism tend to be smaller than those generated by either the liquid bridge breakup or wave undercut mechanisms described previously [15].

The entrained fraction and pressure gradient continually increase with increasing vapor flow rate through this second region that exists following the onset of entrainment. Increasing the vapor flow rate leads to increased interfacial shear, which causes more entrainment and decreases the amplitude of the roll waves [2]. Meanwhile both the increased interfacial shear and the increased amount of entrainment act to increase the pressure gradient. For the purpose of clarity and conciseness this region of co-current upward annular where entrainment can occur will be referred to as the unstable film regime throughout the remainder of the current study. As

previously mentioned, other works have also referred to this regime as the annular mist or annular dispersed flow regime.

The film thickness continually decreases with increasing vapor flow rate through the unstable film regime due to both increased interfacial shear and entrainment. Based on the results of their experiments Sawai et al. [14] suggest that for larger vapor flow rates a thin liquid film without interfacial waves can be observed. Ishii & Grolmes [3] postulate that the thinning of the liquid film by the high shear flow of the gas leads to the submergence of the liquid film in the gas boundary layer, which causes a substantial decrease in the interaction of the two phases and prevents both the presence of roll waves and further entrainment from occurring. This suggests that in adiabatic situations complete disappearance of the liquid film will never occur, and thus in heated situations the last remaining amount of liquid film is removed purely by evaporation as the “dryout” condition is approached. A critical film thickness required to support roll waves places an upper limit on the entrained fraction that is possible for a given adiabatic flow situation. This third region of co-current upward annular, where entrainment cannot occur due to the thinning of the liquid film, will be referred to as the stable film regime throughout the remainder of the current study.

1.2. Comparison of Annular to Single-Phase Flow Situations

Annular flow represents a unique flow regime because of the strong coupling of the vapor, liquid film, and entrained or discrete droplet fields. In annular flow the pressure gradient is enhanced relative to the single-phase case in a smooth pipe for the same vapor flow rate due to: 1) the waviness of the interface, 2) the dynamic interchange between the two liquid fields, and 3) the presence of droplets within the flow, which imposes additional loss mechanisms within the flow.

In annular flow a continuous exchange occurs between the entrained and continuous liquid fields due to entrainment and deposition and a significant momentum difference exists between the continuous liquid (low) and entrained (high) fields [21]. The entraining droplets are accelerated in the gas core region by the drag force exerted by the faster moving gas phase, which consumes energy and represents an additional loss that does not exist in single-phase flow [21,22].

Meanwhile, the depositing droplets have a much larger velocity than those being entrained and as a result a net momentum transfer occurs to the film that aids in driving it upwards; however, Fore & Dukler [23] propose the depositing droplets tend to lose their momentum at the wall rather than within the film region. In this case, instead of imparting momentum to the film, the depositing droplets represent an additional discrete loss mechanism that appears to increase the wall friction and accelerate the corrosion process [24,25]. If this is in fact the case then the droplet exchange process is non-conservative (i.e. losses by acceleration in entrainment and losses by impaction in deposition).

Regardless of whether the depositing droplets transfer their momentum to the film or at the wall it was determined by Lopes & Dukler [26] that the momentum exchange process can comprise a substantial portion of the total pressure gradient observed in annular flow; however, the available interfacial film friction factor correlations were either developed using experimental data with little or no entrainment present within the flow (Wallis [4]) or without explicitly considering this effect when reducing the experimental data (Henstock & Hanratty [27] and Whalley & Hewitt [28]). The results presented by Lopes & Dukler [26] suggest that this oversimplification may be one reason that a satisfactory interfacial friction factor correlation has yet to be found that is suitable over a wide range of conditions.

Once the droplets are accelerated by the vapor field after becoming entrained they still impose an additional drag component relative to single-phase flow. Even though the relative velocity between the vapor and droplets tends to be much less than that between the vapor and film regions this contribution can still be significant since the droplets represent a larger interfacial area than the liquid film. Therefore, the interfacial area tends to drive the magnitude of the droplet drag, whereas the magnitude of the drag force on the film is driven primarily by the velocity difference.

Additionally, previous works have shown that the presence of droplets can either suppress or augment the turbulence of the gas core region depending on the droplet size [22,26]. Experiments conducted by Tsuji et al. [29] using air-polystyrene, the latter of which has a density similar to water, varying both particle size and concentration. The results confirmed that turbulence acts upon smaller spheres, which consumes energy from the gas and thus suppresses

the turbulence, while the larger spheres were unaffected by the turbulent eddies, but rather acted in the same manner as a spacer grid enhancing the turbulence [29]. Independently Azzopardi et al. [30] made drop size measurements using a phase Doppler technique, which simultaneously measured the droplet velocity. The results indicated a narrower range of velocities for the larger droplets, which Azzopardi et al. [30] attribute smaller drops being more affected by the turbulence than larger droplets.

Additionally, the heat transfer in annular flow is also enhanced relative to the single phase situation due to: 1) the presence of droplets, which have a large surface-to-volume ratio for interfacial heat transfer to occur with the higher temperature vapor phase and 2) the continuous mass transfer of drops entraining, depositing, and re-entraining provides an improved means for transferring energy between the fields and the heated surface [23]. The evaporation of the droplets in the gas core acts to reduce the vapor temperatures, with the degree of reduction being dependent on the size and velocities of the droplets [17]. The increased turbulence caused by the presence of dispersed droplets also enhances the convective heat transfer to the vapor. As a result the characterization of dispersed droplets is important to the modeling and prediction of momentum and heat and mass transfer in annular two-phase flow [31].

1.3. Importance of Annular Flow

Situations involving two-phase annular flow can arise in several different fields. The examples provided here are related to reactor safety analysis within the nuclear industry and the extraction and transport of gas from underground wells, but other applications include film cooling in rocket engines and steam generator analysis.

Annular flow can exist in a loss-of-coolant accident (LOCA) scenario in a nuclear reactor during both the blowdown and reflood phases. In low flooding rate scenarios annular flow typically exists upstream of the dryout location. The depletion of the liquid film is governed by the evaporation and the net rate of entrainment. A mass balance on the liquid film yields [32]:

d(G ) 4  q ′′  l S S  (1-1) =  D − E −  dz DH  h fg 

If the rate of entrainment and evaporation exceeds the deposition rate then the film thickness will decrease and dryout will be approached. The disappearance of the liquid film results in the critical heat flux (CHF) being exceeded and leads to a significant deterioration in the heat transfer capability.

In modeling reflood scenarios, the quench front velocity and the liquid carryover are also impacted by the distribution of the liquid between entrainment and liquid film. A recently completed work by Holowach [33,34] indicated COBRA-TF [35] underpredicted the entrained fraction at low pressures and severely overpredicted the axial pressure gradient relative to experimental data. The excess liquid within the film region predicted by the simulations resulted in a more rapid quench front velocity and a shorter reflood transient time relative to the experimental data.

In the event that dryout does occur (low flooding rate scenarios) the entrainment generated in the annular regime represents the source of liquid that is carried to upper elevations [36]. The post- CHF regime that occurs is referred to as Dispersed Flow Film Boiling (DFFB) and the prediction of the Peak Cladding Temperature (PCT), which typically occurs within the region occupied by the DFFB regime, is heavily dependent on the amount of entrained liquid present [36]. The presence of droplets provides an increased cooling effect through the mechanisms described in Section 1-2 (evaporation and convective enhancement) as well as their potential to impact the heated surface. Given that the entrainment generated in the annular regime serves as a boundary condition to the DFFB regime an accurate annular flow model is a prerequisite for assessing the DFFB models in COBRA-TF. Furthermore, since the PCT ultimately determines whether core cooling can be maintained throughout the transient it directly impacts the allowable operating space of the design and makes an accurate prediction of this quantity essential to reactor safety analyses.

During a hot leg break in some Westinghouse designed commercial Pressurized Water Reactors (PWRs) two means exist for liquid to reach the reactor core and provide cooling. On one hand, a liquid spray injection system is activated in the upper plenum as part of the Emergency Core Cooling System (ECCS). Liquid from this system can penetrate through the upper tie plate and into the core to generate a top down quench; however, the penetration of ECCS water into the reactor core from above can be impeded by the flow of rising steam, which is generated at lower elevations within the core by the boiling of liquid from both the bottom-up reflood process and the liquid that has already penetrated downwards. Similarly, liquid that flows through the intact cold leg can flow down the downcomer to provide cooling water for bottom-up reflood, but this downward liquid flow can also be impeded by the flow of rising steam. In both cases if the rising steam flow rate is sufficiently large it can prevent any liquid from penetrating downwards. This condition is often referred to as the counter-current flow limitation (CCFL) and an accurate prediction this phenomenon is a prerequisite for determining the thermal-hydraulic response of the reactor core. Specifically this phenomenon impacts the resulting coolant and temperature distributions within the core and determines the effectiveness of the ECCS for safety analysis purposes. [36]

Situations involving annular flow are also common in the transport of gas from underground wells. The extraction of natural gas often occurs simultaneously with the extraction of some liquid (water, oil, condensate, etc.). The liquid flows partly as a film along the tube walls and partly as entrained droplets in the gas core. The optimization of the transport and gas separation process requires the distribution of the phases to be known [2]. Additionally, near the end of life of a gas well the gas production rate decreases substantially. As a result, a point is reached where the interfacial drag force between the gas and liquid may no longer be sufficient to carry all of the liquid to the surface, at which point the liquid begins to drain downwards (i.e. flow reversal). Eventually this will cause the liquid to accumulate and block the upflow of gas through the production lines and thus causing gas production to cease. This phenomenon in the gas industry is referred to as liquid loading. [17,37]

1.4. Scope and Objectives of the Current Study

Due the importance of the annular regime and the inadequacies in the predictive capabilities for the important parameters within this regime by two-phase, three-field transient safety analysis codes, such as COBRA-TF, the overall goal of the current study was the development of a comprehensive annular flow modeling package for these types of codes. The variation in interfacial drag, entrainment, and drop sizes ranging from co-current through counter-current conditions over a range of pressures (atmospheric to 2000-psia) has been considered. One unique aspect of the current study is the focus on developing a modeling package strongly based on the physical mechanisms causing the different phenomena of interest rather than relying on dimensionless parameters. The explicit consideration of several of the sub-regimes within annular flow through the application of regime specific models has also been emphasized.

The approach of developing a comprehensive modeling package for each annular regime is necessitated by the fact that it is not possible to isolate a single phenomenological model within transient analysis codes, especially for annular flow regimes where a strong coupling exists between the physical models and closure relationships. The predictions by transient analysis codes are result of the modeling package as a whole and implementing an advanced model for one parameter affects the other existing models. Upgrading one particular model may in fact adversely affect the overall predictive capability of the code depending upon how it interacts with the existing models. For example, in the case of entrainment and deposition rates compensating errors between the two models may yield the correct prediction of integral effect (i.e. the outlet entrained flow rate), but mispredict the local phenomena, thus yielding inaccurate estimates of the local exchange rate and the axial distribution of the entrained flow rate. Since it is desired to achieve the correct result for the correct reason the process of model replacement begins. Upgrading one of the models, such as entrainment rate, to more accurately reflect the physical mechanisms and improve the prediction of the local phenomenon, also changes the predicted deposition rate. This process can remove compensating errors that existed previously and can result in a worsened prediction of the integral effect of the two models. This necessitates further model replacement and development activities and is the reason that a comprehensive approach must be employed.

The scope of the current work can be subdivided into three main areas:

1) Co-Current Upward Annular Flow (both unstable and stable film regimes) 2) Churn-Annular/Churn-Turbulent Flow 3) Counter-Current Flow Limitation (CCFL)

The proposed modeling package includes improved definitions of regime boundaries and the explicit consideration additional sub-regimes within annular flow that were not considered previously by COBRA-TF. Doing this has led to the inclusion of models in the proposed modeling package to explicitly consider several other phenomena that were previously not considered by COBRA-TF, such as entrainment by the liquid bridge breakup mechanism and CCFL.

The proposed modeling package has been implemented and tested using the two-phase, three- field COBRA-TF transient safety analysis code; however, the proposed modeling package is amenable to implementation into any three-field analysis tool. Throughout the current study careful attention was paid to ensure the proposed modeling package:

a) provides an accurate physical representation of the situation of interest as it is currently understood using mechanistic-based models wherever possible, b) is continuous between the different regimes c) maintains a level of consistency in the calculation of parameters that appear in more than one model, d) is able to capture the changes in the phenomena under the different conditions of interest, e) is amenable to implementation into two-fluid, three-field safety analysis codes, such as COBRA-TF, that provide the ability to model vapor, continuous liquid, and entrained droplet fields separately, f) maintains the overall computational efficiency and numerical stability of the code, g) minimizes the noding sensitivity of the selected models, and h) wherever possible utilizes local parameters that can be readily and accurately calculated on a subchannel basis within the code.

It will be shown throughout Chapter 2 that the modeling package included in the baseline version of COBRA-TF that was used in the current study did not satisfy several of these criteria. The previously performed research conducted by Holowach [33,34], which focused only on the co- current upward annular regime, served as a basis for the current study.

The focus of the model development in the current work was for nuclear applications and therefore wherever experimental data was needed for either model development or verification activities an emphasis was placed on finding experiments conducted using vertically oriented test sections and using steam-water over a range of pressures of interest. Doing this increases the validity of the proposed modeling packages for reactor analysis applications. It will be seen in Chapter 3 that a substantial amount of data exists that meets these criteria for the co-current upward annular regime; however, considerably less data is available to characterize the churn- annular, co-current downward, and counter-current regimes and the data that does exist for these regimes consists primarily of air-water data at low pressures. Other work has shown models developed from air-water data tend to overpredict the interfacial drag for steam-water situations at similar conditions [28,38]. Despite this fact air-water based models have been traditionally used in transient safety analysis codes to model reactor scenarios.

As anticipated, the use of regime specific models that are either physics-based on developed from experimental data taken over the range of conditions of interest have improved the overall predictive capability of the code, as is shown in Chapters 4 and 5. Chapter 4 presents the proposed annular flow modeling package and associated results. In particular the implementation of a two-zone interfacial drag model and the development of mechanistic-based entrainment models for the a) roll wave stripping, b) bag breakup, and c) liquid bridge breakup mechanisms are discussed. In addition to improving the predictive capabilities in the co-current annular regime the work outlined in Chapter 4 was a prerequisite for the work associated with the transition to counter-current flow, which is outlined in Chapter 5. Since the CCFL phenomenon in primarily dependent on the accurate prediction of the interfacial drag between the vapor and liquid film fields the predictive capability of COBRA-TF for this quantity had to be addressed before a three-field counter-current flow limit (CCFL) model could be developed and implemented. A similar model was previously available for two-fluid analysis approaches, but prior to the current study no model had been developed for three-field analysis approaches. The

development, implementation, and associated results of this unique model are presented in Chapter 5. Finally, in Chapter 6 several conclusions on the proposed modeling package are provided and in Chapter 7 some recommendations for future work are offered.

2. Background

The annular flow regime is characterized by a strong coupling between the interfacial and interfield transfer of mass, momentum, and energy such that it presents a complex scenario to analyze in a mechanistic and self-consistent manner. The rate of exchange and the mechanism causing entrainment is dependent on the structure of the interface, which is unstable and constantly changing. The structure impacts the drag area and the potential for liquid to become entrained, but is also dependent on the respective flow rates of the liquid film and gas core regions.

The objectives of the literature review presented in this chapter are: 1) identify works that provide insight to the physical mechanisms occurring within the different annular regimes, 2) assess the applicability of the models currently available in the open-literature, and 3) determine the current state-of-the-art modeling techniques for annular flow phenomena. To meet these objectives a brief overview of the COBRA-TF code is first provided and then several of the previously conducted works that are the most relevant to the current study are reviewed within this chapter. In particular the models currently used by COBRA-TF are presented and then compared to others proposed in the open-literature to indentify the need for improvement. Wherever possible a focus has been placed on theoretical or mechanistic based models and works involving steam-water; however, it will be seen that few models currently available were developed in this manner or correlated using this type of data. Sections discussing the definition of flow regime boundaries, interfacial phenomena, entrainment phenomena, and flooding and flow reversal are provided. The discussion of available experimental data for model development and assessment will be postponed until Chapter 3. The survey and evaluation of the physical mechanisms and current models presented in this chapter identifies the need for the current study and serves as a basis for the proposed modeling package.

2.1. Transient Safety Analysis Codes

Thermal-hydraulic analysis codes are used within the nuclear industry to simulate different types of reactor transient and accident scenarios and predict the peak fuel and cladding temperatures observed in the reactor core. This result determines whether core cooling can be maintained for each scenario and thus impacts the allowable operating space for the design. Different types of codes exist and they differ in the level of detail and scenarios that to which each are applied.

The current work is focused on subchannel analysis codes, and more specifically those that provide the ability to model the liquid phase using two fields (i.e. continuous and discrete). The use of two liquid fields provides a more accurate means of representing the liquid phase over a wide range of two-phase situations and allows for a more natural transition between flow regimes. In annular flow situations, the velocity and direction of motion of droplets within the flow can be significantly different from that of the film region. For example, in counter-current flow liquid can flow downward along the wall as a film while also being carried upwards in the form of entrained droplets. In these situations the use of a single liquid field can not adequately describe the flow of this phase or the interaction of the liquid and vapor phases. Additionally, the impact of the entrained droplets on the physics of annular flow and reflood situations is well established in the literature and the explicit representation of this field is required for accurate analyses of these situations [36]. However, droplet distributions (both location and size) are not typically considered by these codes, but instead it is assumed that the droplets are uniformly distributed across the flow duct and can be characterized by a single size.

These codes solve the set of conservation equations (mass, momentum, and energy for each field) within each control volume based on the input (physical dimensions, boundary conditions, initial conditions, etc.) supplied by the user. The solution of the conservation equations for the three fields requires suitable constitutive models be defined to provide closure to the equation set by quantifying the mass, momentum, and energy exchange of a given field with: a) the other fields and b) any surfaces within the control volume. For example, the momentum equations for each field are coupled to one another through the interfacial and interfield terms (drag as well as heat and mass transfer). Momentum exchange due to mass transfer occurs to or from the vapor phase by evaporation or condensation of either liquid field. Momentum exchange due to mass transfer

also occurs between the two liquid fields during the entrainment and deposition processes. Meanwhile, the continuity equations are coupled to one another by the mass transfer that occurs to or from the liquid fields by both condensation/evaporation and entrainment/deposition. Relationships are also needed to describe the interfacial area between each field within each flow regime.

The selection of the appropriate set of closure relationships to be applied by the code in a given situation is dictated by the flow and heat transfer regime maps. The relationships required by COBRA-TF to simulate steam-water annular flow situations assuming adiabatic conditions include:

1) wall friction factor 2) interfacial friction factor between the continuous liquid and vapor fields 3) interfacial drag coefficient between the dispersed droplet and vapor fields 4) entrainment rate 5) deposition rate 6) entraining drop size 7) interfacial heat transfer coefficient between the continuous liquid and vapor fields 8) interfacial heat transfer coefficient between the continuous liquid and vapor fields

Constitutive relations for interfacial heat transfer are required for adiabatic steam-water situations since the pressure gradient continuously decreases the saturation temperature along the length of the flow path and therefore causes evaporation to occur. With the exception of the wall friction factor, models that are applied are typically are flow regime dependent. It should be noted that no model or empirical correlation is need for the calculation of film thickness in annular flow because this quantity can be calculated from a geometric relation between the flow path dimensions (input) and the calculated liquid volume fraction for the continuous liquid field (solution variable). However, doing this does assume that the liquid film is uniformly distributed around the periphery of the flow path, which is a reasonable assumption for circular flow paths [2,9], but is less reasonable in non-circular flow ducts with large aspect ratios where “cornering” can occur [39]. This same assumption applies to the calculation of interfacial area for the continuous liquid field in the annular regime.

In addition to the peak temperature predicted by the code during reactor safety analyses, other important parameters calculated by the code during the solution of the conservation equations that are used to characterize the flow include:

1) pressure gradient 2) flow distributions (i.e. quality and entrained fraction) 3) field velocities 4) void fractions and film thickness

Since each of these parameters impacts the resulting peak temperature and temperature distribution, assessing the ability of the code to predict these different parameters through code- to-data comparisons of experiments is a useful tool. The accurate prediction of these more fundamental parameters provides confidence the that code accurately reflects the physics of the flow situation and increases the confidence in the prediction of peak temperature since limited data of this type exists.

As with other two-phase flow regimes the most significant barrier to accurately predicting the behavior of this regime over the range of conditions of interest is the inability to isolate phenomena and to collect experimental data that supports the development of appropriate constitutive relationships for the various phenomena associated with annular flow. While it will be shown later in this chapter that many previous studies have been focused on predicting phenomena within the annular regime, most of the relationships that have been suggested are highly empirical, relying heavily on simple correlations of dimensionless parameters, and as a result are unreliable outside of the range of conditions from which they were developed. Lopes & Dukler [26] suggest one reason for this could be insufficient reduction of experimental data prior to correlating the results. While it is difficult to isolate individual effects, not explicitly accounting for the effects introduced by the dispersed droplet field can lead to double counting and the inclusion of compensating errors in the calculation when such correlations are used within three-field analysis codes. For example, interfacial friction factors are generally calculated directly from the total measured pressure gradient and empirically correlated. At best, the effect of gravity is removed and the effects of droplet drag and the momentum exchange by the entrainment/deposition process are lumped into the resulting correlation; however, three-field

analysis methods include explicit models to account for these contributions. Therefore, it is important that the level of data reduction be consistent with the fields considered in the analysis. For example, when developing a correlation to support two-field analysis tools, where the entrained droplet field is not considered explicitly, it is desired to include the effects of the droplet field implicitly within the correlation so they can still be accounted for by the code. Most of the correlations currently available fall into this category and can be referred to as “lumped parameter” correlations; however, such models cannot be applied to situations that explicitly consider three-fields. Additionally, even if it is desired to develop a correlation to support three- field analysis tools, most of the available experimental data does not contain sufficient measurements to support such data reduction efforts. In some cases assumptions are made to support such data reduction and provide a correlation that is applicable to three-field analysis methods; however, the accuracy of the assumptions is unknown.

2.1.1. COBRA-TF Code Description

COBRA-TF (CO olant Boiling in Rod Arrays - Two Fluid) is a best-estimate thermal-hydraulic analysis computer code used to provide estimates of reactor conditions for design basis accidents and anticipated transients in Light Water Reactors (LWR). It was originally developed at the Pacific Northwest Laboratory under the sponsorship of the United States Nuclear Regulatory Commission [35]. COBRA-TF uses a separated flow formulation to solve multiphase flow problems on a Eularian-Eularian (time and space) mesh using the time-averaged transient conservation equations for each field. Eularian time averaging allows for variables for a given computational cell to be continuous with time while space averaged conservation equations are used.

In addition to the conservation equations, COBRA-TF also contains a droplet interfacial area transport equation. This equation calculates the total droplet interfacial area in a computational cell considering the sources and sinks of interfacial area such as entrainment from, and deposition of the drops to, a liquid film (continuous liquid) as well as droplet coalescence and breakup. The interfacial area calculated by this equation is used in: 1) the energy equations to determine the

interfacial heat transfer between the vapor and entrained droplet fields and 2) the momentum equations to determine the mass transfer associated with the interfacial heat transfer.

2.1.1.1. Solution Scheme

COBRA-TF utilizes the finite-difference equations in semi-implicit form with a Eularian staggered mesh to increase the stability of the solution. In the staggered mesh approach the momentum cells are centered on the edges of the continuity and energy cells. Therefore the phasic flow rates and velocities are solved for at the edges of the continuity cell (which corresponds to the center of the momentum cell) and the state variables (density, pressure, enthalpy, and volume fractions) are obtained at the cell centers. An example of the implementation of a staggered mesh is shown in Figure 3-1, where the thicker black line represents the test section that is being modeled. The continuity and energy mesh is given on the left and the corresponding momentum mesh is given on the right. The continuity and energy mesh cells (denoted by subscript J) lie a distance of z/2 below the center of the correspondingly subscripted momentum cell (denoted by subscript j). Phantom cells are added to the continuity and energy mesh above and below the subchannel to allow for the specification of inlet and outlet state variables using either one of the available boundary conditions or by connection to another section. All subchannels specified in COBRA-TF must either have a boundary condition or a connection to another section specified at both the top and bottom of the subchannel.

Donor cell differencing is used to provide values of the convected parameters at the edges of the control volumes used to solve the mass and energy equations. This technique increases the stability of the solution while introducing numerical diffusion into the calculation. Meanwhile, linear averages are used to calculate values of the state variables (ρl , ρv ,α l ,α v ,α e ) when calculating the constitutive relationships that are required to solve the momentum equations. However, it is important to note that volume weighted averages are not used, but this does not present an issue in for the current work since uniform nodalization has been used for all cases analyzed.

Figure 2-1: Schematic of Staggered Mesh Scheme used by COBRA-TF.

In COBRA-TF a computational cell is characterized by: a) the axial cross-sectional area, b) the height and c) the number and widths of the lateral connections to adjacent cells. COBRA-TF assumes all sections are vertically oriented and the positive axial flow is vertically upward. COBRA-TF does not require a uniform mesh size; however, the size of the computational cells within the mesh should be made to be large enough to obtain a solution in a reasonable (user- defined) amount of time, but at the same time small enough to capture the important effects within the flow. Since a semi-implicit form is used the time step size is limited by the material Courant limit:

∆z ∆t < (2-1) U

COBRA-TF uses a mass flow formulation of the Courant number that reduces to the above equation for one-dimensional, single-phase, steady-state scenarios.

In the subchannel formulation used by COBRA-TF transverse or lateral flow is assumed to occur through “gaps”. This approach allows for a single transverse momentum formulation to be applied to all gaps regardless of orientation and therefore reduces the number of momentum equation types per field from three (x, y, and z) to two (lateral and transverse). This formulation provides a more flexible scheme for modeling irregular geometries than a full Cartesian geometry.

The representation of two-phase flow employed by COBRA-TF considers the following fields: 1) continuous vapor, 2) continuous liquid (for low void fraction flow and films), 3) entrained droplets (for dispersed flow), and 4) non-condensable gases. As previously described, the use of two liquid fields provides the most physically accurate representation of annular flow situations. The set of conservation equations considered by COBRA-TF to model these fields includes: four (4) continuity equations, three (3) momentum equations, and two (2) energy equations. As expected a continuity equation is written individually for each of the fields. However, COBRA- TF assumes that the vapor and non-condensable gas fields travel with the same velocity so the momentum equations for these two fields can be combined. Therefore, only three momentum equations are needed to completely describe the flow. This approach still allows for mechanical non-equilibrium to exist between both the vapor and liquid fields and the continuous liquid and entrained droplet fields.

COBRA-TF also assumes thermal equilibrium exists between both the vapor and non- condensable gas fields and the continuous liquid and entrained droplet fields. This assumption is reasonable since the vapor and non-condensable gas are well-mixed and thus these fields tend toward thermal equilibrium. Meanwhile the assumption is justified between the continuous liquid and droplet phases due to the large rate of mass transfer that typically exists between the two fields, which tends to bring them towards thermal equilibrium. As a result, the number of energy equations can be reduced by two while still allowing for thermal non-equilibrium to exist between the vapor/non-condensable gas and liquid phases. A summary of the conservation equations

considered by COBRA-TF is provided in Table 2-1 and a listing of the individual equations is provided in Reference [35].

Table 2-1: Summary of Conservation Equations used by COBRA-TF. Number of Type Fields Equations 1) vapor Continuity 2) non-condensable 4 (scalar) 3) continuous liquid 4) entrained droplets Momentum 1) vapor / non-condensable (axial and 3 2) continuous liquid transverse) 3) entrained droplets Energy 1) vapor / non-condensable 2 (scalar) 2) continuous liquid / entrained droplets

2.1.1.2. Interfacial Area Transport Equation

In addition to the conservation equation described above, COBRA-TF utilizes an interfacial area transport equation to calculate the interfacial area concentration, or density, for the dispersed field. This equation calculates the rate of change in the interfacial area concentration for a given cell by considering: 1) the net convective contribution to the interfacial area coming from flows into and out of the cell (both axial and transverse), 2) the net rate of interfacial area generated due to combined effect of the entrainment and deposition processes ( Ai′′,′E ), 3) the rate of change of interfacial area due to droplet breakup and coalescence ( Ai′′,′B ), and 4) the rate of change of interfacial area due to phase change ( Ai′′,′Γ ). The resulting governing equation is given as:

d d d d ()()A ′′′ + ∇ ⋅ A ′′′ U = ()A ′′′ + ()A ′′′ + ()A ′′′ (2-2) dt i,d i,d e dt i,E dt i,B dt i,Γ

It should be mentioned that when calculating the net contribution of entrainment/deposition, which is done using:

  6 S E′ S D′ ∆t Ai,′′′ E =  −  (2-3) ρl  DE Dd  Vcell

COBRA-TF assumes the depositing droplets have the mean diameter of those already present, or entrained, within the flow, Dd, which is calculated as:

6αe Dd = (2-4) Ai,′′′ d

while the entraining droplets are born with a diameter, DE, which is calculated using flow regime- specific empirical correlations. Currently the Tatterson et al. [40] correlation, which will be described in detail later, is used by COBRA-TF to calculate the size of entraining droplets in annular flow.

Equation (2-2) is solved explicitly in the POST3D subroutine following the solution of the conservation equations. The resulting interfacial area concentration that is calculated is used to determine the size of the entrained droplets and interfacial area of the dispersed phase at the next time step. The entrained drop size is given as Equation (2-4) and the interfacial area of the dispersed phase is calculated as:

6α e Ax ∆z Ai,d = Ai,′′′dVcell = (2-5) Dd

2.1.1.3. Relevant Subroutines

The solution routine in COBRA-TF is driven by the TRANS subroutine and consists of three steps: 1) pre-pass calculations, 2) solution over a single time step (outer iteration), and 3) post- pass calculations. This section briefly summarizes the calculations performed during each step.

The pre-pass calculations are controlled by the PREP3D subroutine. The calculations in this subroutine are based on known information from the previous time step and then the calculated quantities remain fixed throughout the time step. Calculated parameters include: boundary conditions, heat transfer coefficients (HCOOL) and the corresponding wall heat flux (HEAT), the surface temperatures from the solution of the conduction equations (TEMP), and the quench front location (QFRONT).

Next, the hydrodynamic state of the system is determined by solving the semi-implicit form of the discretized and linearized set of thermal-hydraulic conservation equations. This sequence of calculations is controlled by the TRANS subroutine while the XSCHEM subroutine linearizes the equations and the INTFR subroutine calculates: 1) interfacial and wall drag coefficients, 2) interfacial heat transfer coefficients, and 3) entrainment and deposition rates. The first expressions listed should be calculated on the momentum mesh, using averaged quantities for the void fraction and physical properties, while all the other expressions listed should be calculated on the on the continuity mesh using donored values for the flow rates. Once these values are known the momentum equations are solved by Gaussian elimination using currently known values to estimate of the flow at the end of the time step. These estimated phasic mass flow rates are determined as a function of the implicit pressure gradient and are used to calculate the corresponding velocities, which are needed to solve the continuity and energy equations. The continuity and energy equations are then linearized with respect to the independent variables, which produces a Jacobian matrix. Lastly, the pressure matrix is solved using either direct matrix inversion or iterative methods.

The sequence concludes with the post-pass calculations, which are controlled by the POST3D subroutine. If an acceptable solution is obtained then this routine unfolds the values of the independent variables from the system matrix, updates the mass flow rates and fluid densities,

and solves the droplet interfacial area transport equation. However, if an acceptable solution is not achieved, then this routine resets the data arrays to their state at the beginning of the time step and the iterations are repeated with a reduced time step size. A series of limits are imposed to restrict the time step size when fluid conditions are changing rapidly or increase the time step size for a slower transient. These include the courant limit, which was described earlier, pressure- temperature change limits, and a void fraction change limit. The user also specifies the minimum and maximum allowable time step sizes.

2.1.1.4. Time and Space Averaging of Variables

Ramps are used throughout the code to minimize discontinuities in both void fraction and time in an attempt to enhance the numerical stability of COBRA-TF while maintaining a reasonable computing time. For example, the computed values of the constitutive relationships for a given phase are ramped to zero as that phase is depleted within a given cell or, as will be discussed later, the variables computed for the churn-turbulent flow regime consist of a void fraction weighted ramp between those calculated in the large bubble and annular regimes. On the other hand, the time smoothing acts to under-relax terms such as heat transfer coefficients and interfacial drag coefficients to prevent these quantities from changing orders of magnitude over the course of a single time step. Such drastic changes can be caused by changes in the flow or heat transfer regime in a given cell.

The most common implementation for void fraction smoothing within the code consists of linear ramps. However, caution must be used with these ramps, especially those between flow regimes, because large transitions can occur. Depending on the parameter considered the change can be several orders of magnitude, which linear ramps do not handle well, especially if the parameter is decreasing. Additionally, the time smoothing routines used by COBRA-TF were originally, time step size dependent (i.e. rate of change differs depending on time step size), but these routines have since been updated to be time step size independent.

2.1.1.5. Boundary Conditions

COBRA-TF offers several different options for specifying boundary conditions. Brief descriptions of the boundary conditions that are relevant to the current study are provided here.

A Type 1 boundary condition allows for the pressure and enthalpy to be specified in any cell. For this type of boundary condition the momentum equations are then solved at the cell edges to determine the flow into or out of the cell. Essentially a Type 1 boundary condition acts as a mass, momentum, and energy source if flow is out of the cell, or a sink, if flow is into the cell. This type of boundary condition is commonly applied at an outlet. It should be noted that when this type of boundary condition is applied at an outlet then the void fraction and enthalpy that are specified are never considered in the solution unless downflow is occurring since donor cell differencing is used.

A Type 2 boundary condition allows for the mass flow rate, enthalpy, and void fraction to be specified at a momentum cell center. The type of boundary condition replaces the momentum equations at the specified interface with the user specified information. It should be noted that if the specified flow is a two-phase mixture it is assumed that the phases have the same velocity at the interface (i.e. homogeneous flow). A Type 2 boundary condition is commonly applied at an inlet.

The final type of boundary condition that is relevant to this work is a Type 4 boundary condition. This type of boundary condition allows for a mass and energy source to be specified in any cell and is most commonly used to simulate injection lines. This type of boundary condition requires the mass flow rate, enthalpy, and injection area to be specified.

In summary, Type 1 and Type 2 boundary conditions directly control the parameter of interest at the specified cell or flow connection. On the other hand, a Type 4 boundary condition can be thought of as a connection to the side of the specified cell. It should also be noted that for both Type 2 and Type 4 boundary conditions the specified liquid flow can be partitioned between the continuous and discrete liquid fields. If liquid is supplied to the discrete liquid field the initial droplet size must also be specified. This feature allows for more accurate modeling of the spray

injection and porous sinter techniques for introducing liquid into the test section, which provides a more accurate representation of the tests being modeled. Sensitivity studies on the initial droplet fraction, initial drop size, and size of injection area were performed as part of the current study and the results are discussed in Chapter 4.

2.1.1.6. Conclusions on COBRA-TF

As previously mentioned, the ability to model liquid in the form of both continuous and discrete forms make COBRA-TF well suited for calculating annular flow situations; however, the previous work conducted by Holowach [33,34] identified several concerns with the models currently used by the code. The remainder of this chapter is focused on examining the available models for the constitutive relationships outlined in Section 2.1. For each phenomena of interest the model or approach currently used by COBRA-TF for considering that phenomena is first presented and then the concerns the current treatment is discussed. Then other models or approaches that have been proposed in the open-literature for considering the given phenomena are discussed, with the goal being to identify models that were developed either: a) from a first- principles approach or b) using experimental data collected over the range of interest (i.e. steam- water at both low and high pressure). It is anticipated that this approach will lead to a more suitable modeling package that will improve the predictive capability of the COBRA-TF code within the annular regimes.

2.2. Definition of Annular Flow Regime Boundaries

As stated by Hewitt & Jayanti [41] the purpose of designating a specific flow regime is to enable models to be applied in that regime that reflect the distinctive characteristics of the regime. By dividing flows into specific regimes and applying specific models within each regime that are representative of the behavior of that regime it is anticipated that an improved phenomenological prediction is possible [41]. As previously mentioned several sub-regimes exist within annular flow that are not currently considered explicitly by COBRA-TF. Given the drastic differences in interfacial structure and entrainment mechanisms between counter-current, churn-annular, and unstable and stable co-current upward annular regimes it is believed that the explicit consideration and implementation of regime specific models into transient analysis codes, such as COBRA-TF, is warranted. However, utilizing this approach requires appropriate criteria for distinguishing regime boundaries to be specified.

This section first presents the flow regime map for pre-CHF conditions that is currently used by COBRA-TF. Then, previously conducted works that characterized: 1) the flow conditions associated with the local minimum in the pressure gradient, which serves as the boundary between the churn-annular and unstable film regimes, 2) the entrainment inception criterion, and 3) the critical film thickness required to support roll waves, which serves as the boundary between the unstable and stable film regimes, are discussed. An additional regime transition, which is often referred to as the counter-current flow limitation (CCFL), is needed between the counter-current and churn-annular or churn-turbulent regimes; however, the discussion of this regime transition will be postponed until Section 2.5.

2.2.1. Baseline Pre-CHF Flow Regime Map in COBRA-TF

COBRA-TF employs two flow regime maps: 1) pre-CHF, also referred to as normal or cold wall and 2) post-CHF, also referred to as hot wall. The latter is utilized if the surface temperature of any single structure within the computational cell exceeds:

TCHF Ts > minimum  (2-6) 705 3. o F

This relationship is assumed to express the demarcation condition between fully and partially wettable surfaces.

Following the selection of the appropriate flow regime map (i.e. pre-CHF or post-CHF) the two- phase flow pattern is selected based solely on the void fraction within the given cell. As previously mentioned the selected flow pattern dictates the constitutive relationships that are applied. The pre-CHF flow regimes in COBRA-TF include: small bubble (SB), small-to-large bubble (SLB), churn-turbulent, which includes unstable annular (UN), and stable annular (AN). A visual representation of these regimes is provided in Figure 2-2 and a schematic of the current selection logic in COBRA-TF for pre-CHF conditions is given in Figure 2-3.

Figure 2-2: Visual Representation of Pre-CHF Flow Regimes.

αv < ala False ala=max(0.8,1-alcrit)

True

Small Bubble Regime (SB) isij = 1

αv > alb False alb = 0.2

True

Large Bubble Regime (LB) Logic Note: this regime cannot exist individually

Small -to -Large Bubble Transition (SLB) isij = 3 Note: ramp between SB and LB values

αv > alsa False alsa = 0.5

True

Ann ular - Stable Film (ST) isij = 5 fi = Wallis

False αv < ala

True

Annular - Unstable Film (UN) Logic fi = max(5x Wallis, Henstock & Hanratty) Note: this regime cannot exist individually

True

Churn -Turbulent Regime (CT) isij = 5 Note: ramp between SLB and UN values

Figure 2-3: Baseline COBRA-TF Cold Wall Flow Regime Map.

The small-to-large bubble regime in COBRA-TF models the transition from bubbly to slug flow that occurs when the dispersed bubbles agglomerate to form larger bubbles, which are also referred to as slugs. In this regime COBRA-TF assumes that small bubbles occupy the portion of the vapor phase equal to the maximum void fraction associated with the small bubble regime (i.e. 20%). The remaining vapor is then assumed to be occupied by one or more large spherical bubbles. The implementation of this regime in this manner allows for a continuous transition from bubble to slug flow.

The churn-turbulent regime in COBRA-TF consists of a linear interpolation between the small-to- large bubble and the unstable annular film regimes. Meanwhile, the transition between the churn- turbulent and stable annular film regimes (αAN ) in COBRA-TF is currently based on the maximum of two criteria: 1) a liquid bridging criterion and 2) a stable liquid film criterion. The details of these criteria and some potential issues associated with this current definition of the flow regime transition will be discussed in the following section. Lastly, the unstable and stable annular regimes are assumed to consist of a thin, uniformly distributed liquid film on the wall with entrained spherical droplets of dispersed liquid flowing in a gas core region. These regimes are the focus of the current work.

It is important to note here that in the case of transition regimes (i.e. small-to-large bubble and churn-turbulent) the calculations performed by COBRA-TF are first done by assuming that the flow consists entirely of a given regime. Assuming a void fraction of 0.55, which corresponds to the churn-turbulent regime, COBRA-TF performs calculations for the small bubble, large bubble, and unstable film regimes assuming the flow consists entirely of each regime at the given set of flow conditions, regardless of whether a given regime could physically exist at those conditions. For example, it is unlikely that the small bubble regime could not exist on its own at this assumed void fraction condition, but since it could be anticipated that some small bubbles may still be present within the flow as this condition this approach provides a method to account for this contribution. While this approach may seem unphysical, it does allow for the void fraction and velocity conditions used in the calculations for each regime to be consistent with one another. Once the values for each regime are known COBRA-TF applies a void fraction based weighting function to partition the result from each regime and calculate and effective result for the given set of conditions. This approach is used to provide smooth transitions between different regimes.

In the case of the churn-turbulent regime calculation COBRA-TF first calculates the “effective” result for the small-to-large bubble transition regime by applying a void fraction weighted ramp between the values calculated individually for the small and large bubble regimes. Then, COBRA-TF calculates the effective result for the churn-turbulent regime by applying a second void fraction weighted ramp between the effective result from the small-to-large bubble transition regime and the value calculated for the unstable annular regime.

The wall shear stress is carried by the continuous liquid field for pre-CHF flow regimes, except at very high void fractions where a ramp is employed to provide a smooth transition to the vapor field carrying the wall drag. As expected, no wall drag component ever exists for the entrained field. Additionally, no unstable film regime exists within COBRA-TF for modeling transverse flows. Moreover, in the transverse direction the calculated stable film interfacial friction factor is multiplied by the number of lateral gaps associated with a given computational cell prior to being applied.

2.2.2. Co-Current Upward Annular Flow Regime Boundary

This section first reviews the annular flow regime boundary currently applied by COBRA-TF and then discusses several concerns with the current approach. Following this an alternative approach that has been proposed in the open-literature is examined.

2.2.2.1. Current Definition in COBRA-TF

As previously indicated the transition between the churn-turbulent and stable annular film regimes (αAN ) in COBRA-TF is currently based on the maximum of two criteria: 1) a liquid bridging criteria and 2) a stable liquid film criteria, which is given as:

α bridge α AN = max  (2-7) 1−α l,crit

This transition point is calculated on a cell-by-cell basis throughout the computational mesh. If the void fraction in a given computational cell exceeds the maximum of the two criteria for that cell then the flow within that cell is assumed to in the stable annular regime. Otherwise the flow is assumed to be churn-turbulent, where a linear ramp between the small-to-large bubble and unstable annular regimes is imposed.

The first criterion in Equation (2-7) represents the void fraction above which bridging across the flow duct cannot occur and corresponds to a void fraction of 80% in large hydraulic diameter geometries and 60% for small hydraulic diameter geometries as suggested by Holowach [42]. The second criterion in Equation (2-7) is based on the analysis presented by Richter [8] to determine the conditions required for entrainment to occur from the interface in terms of a critical liquid volume fraction for the continuous liquid field. Balancing the forces acting on the interface in the direction normal to the wall (pressure and surface tension) and assuming the wave has a semicircular shape yields:

1 2 σ ρvU v = (2-8) 2 ε w

Then by approximating the wave amplitude to be four times the mean film thickness (δ ≈ ε w ), as suggested by Wallis [4], applying the thin film approximation, which is given as:

D δ = H ()1− α (2-9) 4 l,crit and finally solving the resulting expression for the critical liquid volume fraction for the continuous liquid field yields:

0.2 σ α l,crit = 2 (2-10) ρv ()U v −U l DH

If the continuous liquid volume fraction exceeds this critical value it suggests that the disruptive force of the pressure gradient over the wave crest exceeds the stabilizing force of the surface tension such that the interfacial wave disintegrates (i.e. “unstable”) and liquid from the wave will be carried upwards with the vapor flow in the form of droplets. Therefore this expression describes the conditions for entrainment to occur, but it is not currently applied in this manner within COBRA-TF. This and other potential issues associated with using this definition for transition between churn-turbulent and stable film flow as well as the current treatment of the churn-turbulent regime in general will be discussed in the following section.

2.2.2.2. Concerns with Current Approach

The current churn-turbulent and annular flow regime logic in COBRA-TF poses several concerns. First, the critical liquid volume fraction can fluctuate within a given computational cell from one time step to the next due to the dependence on relative velocity. Therefore, the code can predict the flow is in the stable annular regime for one time step and then in the churn-turbulent regime in the next time step. This oscillation in flow regime, coupled with the bifurcation that currently exists in the interfacial friction factor that is applied in the stable and unstable annular regimes (see Sections 2.3.4.1 and 2.3.4.2), can cause significant differences in the predicted interfacial drag and is a potential source of instability within the code. Additionally, the critical liquid volume fraction, and hence the transition to annular flow, can vary throughout the computational mesh. This represents a potential inconsistency and a source of instability that is not possible for the current implementations of any other flow regime transitions in COBRA-TF.

Second, the only difference in the calculations performed for the stable and unstable annular regimes is in the interfacial friction factor. As expected the interfacial friction factor in COBRA- TF is enhanced in the unstable annular regime relative to the stable annular regime. For the stable annular regime the Wallis [4] interfacial friction factor correlation is applied while in the unstable annular the maximum of five times the Wallis [4] correlation and the Henstock & Hanratty [27] correlation is applied based on the suggestion of Dukler & Smith [43]. However, the application of this enhancement for the current flow regime logic yields in a discontinuity of at least a factor of five in the code calculated interfacial drag at the boundary between the stable and unstable film

regimes when the stable liquid film criterion is greater than the liquid bridging criterion. This bifurcation introduces a numerical instability that can prevent the code from achieving a steady- state result in some annular flow situations.

Third, since the stable liquid film criterion is derived from a force balance on a wave crest consisting of the disruptive force of the pressure gradient and the stabilizing force of the surface tension it can be assumed that the stable annular regime should correspond to annular flow where interfacial waves cannot be supported while the unstable annular regime should correspond to annular flow where interfacial waves exist. This is the physical basis for the enhanced interfacial friction factor used by COBRA-TF in the unstable film regime. However, if interfacial waves cannot be supported then the entrainment rate should be zero for the stable film regime, but the current implementation of this regime in COBRA-TF does not enforce this physically based limit.

Fourth, combining the unstable film regime with the churn-turbulent regime does not provide an accurate physical representation of the unstable film regime. The structure of the flow and the mechanisms for entrainment are drastically different between the two regimes. For example, in a true churn-turbulent regime, where liquid bridging can occur, the faster moving vapor acts to breakup this liquid bridge causing significant amounts of entrainment. Meanwhile, in an unstable annular regime, where liquid bridging cannot occur, entrainment occurs by the faster moving vapor shearing liquid from the crests of interfacial waves. Previous research has indicated that the droplet sizes that are created by the liquid bridge breakup mechanism tend to be larger than those generated by the roll wave mechanism present in unstable annular flow [15]. Also, since COBRA-TF currently does not explicitly consider the churn-turbulent regime it does not utilize an entrainment model based on the liquid bridge break-up or apply regime specific drop size correlations.

Lastly, it can be seen in Equation (2-10) that the critical liquid volume fraction criterion is inversely proportional to the relative velocity squared. Therefore, increases in this quantity increase the void fraction transition to the stable film regime and can cause COBRA-TF to invoke the churn-turbulent logic at very large void fractions (~0.995). The implications of this include:

1) the COBRA-TF results at large void fractions are still influenced by small and large bubble regime calculations since the churn-turbulent consists of a linear ramp that includes these quantities. It is unlikely that considering these contributions at such large void fraction conditions is physically reasonable.

2) the interfacial drag tends to be much larger in the bubble regimes than the annular regimes due to the increased interfacial area. This difference can be an order of magnitude or larger. A linear ramp, which more strongly considers the larger order of magnitude component, is employed by COBRA-TF in the churn-turbulent regime. This may explain the severe overprediction of the pressure gradient by COBRA-TF relative to the experimental data when several annular flow tests were modeled by Holowach [33,34]. One possible solution would be to utilize a logarithmic ramp in this regime, which more equally weights the contribution of the two components that differ by at least an order of magnitude. However, it should be noted that drag is numerically stabilizing within transient analysis codes and therefore reducing the drag in the churn-turbulent regime may increase the instability of the code within this regime. On the other hand, this instability may be physically accurate due to the oscillatory motion that characterizes the churn-turbulent regime.

3) a linear, void fraction weighted ramp is used in the churn-turbulent regime to suppress the calculated the entrainment and deposition rates since by definition no entrainment can exist in the bubble flow regimes.

These concerns have led to alternative flow regime transition criteria to be examined as part of the current study.

2.2.2.3. RELAP5-3D [44] Annular Flow Regime Transition Criteria

For the purposes of comparison the annular flow transition criteria currently employed by RELAP5-3D [44] were also examined. In co-current upflow the transition from slug flow (which RELAP5-3D assumes to include both the slug and churn flow regimes) to annular mist flow is

based on the minimum void fraction associated with either the Wallis [4] flow reversal criterion or the Kutateladze [45] onset of entrainment criterion. This criterion is based on the suggestion of Putney [46] and the former condition is given as:

* jv,crit = 0.1 (2-11) while the latter condition is given as:

Ku v,crit = 2.3 (2-12)

The critical values were suggested by McQuillan & Whalley [47] based on comparisons with experimental data in pipes ranging from 1 to 10.5 cm in diameter and covering a wide range of fluid conditions. Solving each expression for the corresponding critical value of the vapor void fraction and imposing restraints to prevent unphysical values from being obtained yields:

1 1   2 4   [gD H ∆ρ] []gσ∆ρ  α AN = max  ,5.0 min  1 2.3, 1 9.0,  (2-13)  2 2    ρv U v ρv U v 

The lower limit of 0.5 corresponds to the maximum void fraction at which bubbly-slug flow can exist and the upper limit of 0.9 corresponds to a void fraction above which annular flow must exist. A similar upper limit on the annular flow transition does not currently exist in COBRA-TF.

Applying both criteria together causes the Wallis [4] criterion to control the transition when the hydraulic diameter is less than a critical value, which is given as:

1  σ  2 DH ,crit = 10 24.   (2-14)  g∆ρ  while the Kutateladze [45] criterion controls the transition when the hydraulic diameter is greater than this critical value. This approach is consistent with several other findings on the mechanism

that controls CCFL that will be discussed later in this chapter and results in a transition boundary which is continuous in diameter. This approach warrants consideration for implementation into COBRA-TF.

2.2.3. Local Minimum in the Pressure Gradient

The conditions corresponding to the local minimum in the pressure gradient are widely accepted as the transition between churn-annular and co-current upward annular flow. Barbosa et al. [18] also observed a local minimum to occur in the entrained fraction at this same point. As will be described later in Section 2.3 the structure of the interface and the entrainment mechanism associated with each of these regimes are completely different and as a result it is important that the location of the local minimum be characterized to demarcate between the two regimes and allow for regime-specific models to be applied.

The physical basis for the existence of the local minima in the pressure gradient and entrained fraction can be explained as follows. As the gas flow rate is increased and the flow moves through the churn-annular regime the liquid flow within the film region becomes unidirectional upwards, which reduces the large dissipative effects associated with the recirculation behavior within the film and the highly oscillatory motion of the interfacial waves subsides [41]. This causes a decrease in both the pressure gradient and entrained fraction through the churn-annular regime. Then, as the gas velocity is further increased a critical gas velocity required to support upward propagating roll waves is exceeded and the unstable film regime within co-current upward annular flow is established. The pressure gradient and entrained fraction both increase through this regime due to the increasing presence of roll waves. [17]

It has been suggested that the local minimum corresponds to the point where liquid can no longer be flowing downward at any point within the film (i.e. wall shear stress equals zero); however, these two points do not necessarily correspond to one another as indicated by Hewitt & Hall- Taylor [5] and confirmed through experiments by Zabaras et al. [10] and Govan et al. [16]. Instead, Govan et al. [16], Barbosa et al. [18], and Hewitt & Govan [48], have all suggested the

location of the local minimum corresponds to a dimensionless superficial gas velocity condition of:

* jv,min ≈ 0.1 , and this represents to the condition where the inertial forces of the gas balance the hydrostatic forces of the flow. Meanwhile Van’t Westende et al. [17] suggest that the location of the local minimum in pressure gradient corresponds to a densimetric vapor Froude number of 1.0, which is defined as:

2 jv ρv Fr v = (2-15) gD H ∆ρ

On the other hand, Azzopardi [49] suggested the change in entrainment mechanism between the two regimes is responsible for this behavior. As a result it was proposed that this point could be identified using a critical Weber number criterion where Azzopardi [49] assumed that ratio of wave height to mean film thickness to be approximately 3.5. The resulting expression is given as:

ρ j 2ε ρ j 2 ( 5.3 δ ) We = v v w = v v = 25 (2-16) crit σ σ

Verbeek et al. [50] examined the effect of tube diameter and physical properties on the measured entrainment. It was and found that the local minimum shifted to higher gas velocities for larger tube diameters, lower surface tensions, and larger liquid viscosities. This viscosity effect was confirmed by Fore & Dukler [23] for the entrained fraction, but a corresponding shift in the location of the local minimum in the pressure gradient was not observed. Zabaras et al. [10] also confirmed the viscosity effect, but they did not measure the entrained fraction so no conclusion can be drawn about the behavior of this parameter from their experiments.

There is some question as to the existence of this behavior over a range of pressures and flow rates. The local minima in pressure gradient and entrained fraction has not been observed in

experiments conducted by Fore et al. [31,51] at higher pressure (250-psia), but was observed in the same experiments at lower pressure (50-psia). It can be shown that the superficial gas velocity decreases significantly as pressure increases and therefore the superficial gas velocity at the flooding point may not be sufficient to yield churn-annular flow at higher pressures. In these situations Jayanti & Hewitt [13] suggested a direct transition to co-current upward annular flow can occur and thus the churn-annular regime and corresponding local minimum does not exist.

2.2.4. Conditions Required for Entrainment to Occur

Researchers have suggested the conditions required for entrainment to occur in co-current upward annular flow correspond closely to the conditions required to support upward propagating roll waves [52]. Azzopardi [53] indicates that the presence of roll waves is a necessary, but not sufficient, condition for entrainment to occur; however, for the purposes of the current study it will be assumed that these two conditions coincide. Based on this idea it then becomes important to distinguish between conditions where interfacial waves can and cannot be supported to allow: 1) for the code calculated entrainment rate to be ramped to zero for situations where entrainment cannot occur and 2) for appropriate interfacial drag models to be applied based on the physical structure of the interface.

As mentioned previously it has been observed by several experimenters that two different conditions can arise within co-current upward annular flow where liquid cannot be entrained from the liquid film region. The first condition, which will be referred to as the entrainment inception criterion, corresponds closely to a critical gas velocity that must be exceeded before upward propagating roll waves will be observed on a given liquid film. Experimental data indicates that following the entrainment inception point the entrained fraction increases slowly at first before reaching a region of more linear growth with increasing gas velocity as shown in Figure 2-4 [3]. For convenience some researchers have simply defined the entrainment inception point using linear extrapolation from the linear growth region. This, coupled with sensitivity of detecting this point to the liquid injection method, tube diameter, and method of detection, has yielded vast differences in the results predicted by correlations currently available in the open-literature [3].

Following the onset of entrainment the liquid film thickness gradually decreases and a point may be reached at high gas velocities where the film becomes sufficiently thin such that it can no longer support roll waves and entrainment ceases to occur. This condition is the reason for the asymptotic behavior in the entrained fraction at high gas velocities as shown in Figure 2-4. This second condition can be thought of as a critical film Reynolds number or flow rate for entrainment and will be referred to in the current study as the entrainment suppression criterion. Based on this definition the critical liquid volume fraction criterion that is defined in Section 2.2.2.1 can be thought of as an entrainment suppression criterion even though it is not currently implemented within COBRA-TF in this manner.

Figure 2-4: Relationship between entrained fraction and gas velocity for a constant total liquid flow rate [3].

Several studies have been aimed at defining entrainment inception and suppression criteria, but most of the existing correlations are strictly empirical. This section provides brief reviews of several works that have been focused on characterizing these conditions and the resulting correlations that have been proposed.

Hewitt & Govan [48]

In this work the critical film Reynolds number for entrainment is calculated using the expression originally proposed by Owen & Hewitt (1987), which is given by Hewitt & Govan [48] to be:

 µ ρ   v l  (2-17) Re l,crit = exp  .5 8504 + .0 4249   µl ρv  It is obvious that this expression for the entrainment suppression criterion is highly empirical and, more importantly, the fact that this correlation is not dependent on surface tension brings the applicability of this expression to a wide range of fluid conditions into question. Azzopardi [25] indicates this correlation overpredicts the experimentally determined critical liquid Reynolds numbers for wave inception from various researchers.

Asali et al. [54]

Asali et al. [54] suggest the critical film Reynolds number for entrainment is 330 for vertical flows, decreasing slowly with increasing liquid viscosity. A similar result was obtained by Fore & Dukler [23], but neither provided a functional relationship for this quantity. Additionally, Asali et al. [54] hypothesize that only films thicker than the laminar sublayer will increase the interfacial friction factor above the analogous single phase value. Asali et al. [54] suggests that for entrainment to occur the film must exceed a dimensionless film thickness (in interfacial units) of 4, regardless of the vapor flow rate. A similar condition was proposed by Hurlburt et al. [55], except they suggest a value of 12 for the steam-water data they considered.

Ishii & Grolmes [3]

Ishii & Grolmes [3] developed a series of entrainment inception correlations from simple physical models for co-current annular flow. Models are presented for: 1) a low Reynolds number region

(Re l < 160 ), where entrainment is assumed to occur by the wave undercutting mechanism, 2) a

higher Reynolds number region (Re l > 1635 ), where entrainment is assumed to occur by the roll-wave mechanism, and 3) a transition region between these two regimes. No theoretical arguments were presented in support of the transition values suggested and separate correlations are presented for both low and high viscosity fluids. [3]

At film Reynolds numbers greater than 1635 Ishii & Grolmes [3] suggest the critical gas velocity needed for entrainment becomes nearly constant, which suggests the presence of a completely rough-turbulent film. This idea is consistent with the analogy to turbulent flow over rough surfaces where the friction factor becomes Reynolds number independent. For film Reynolds numbers between 160 and 1635 Ishii & Grolmes [3] suggest the presence of a transition regime where the critical gas velocity for entrainment becomes a function of the film Reynolds number. The magnitude of the film flow contributes to the momentum exchange between the phases in this regime. A schematic of these two regimes is given in Figure 2-5.

Figure 2-5: Qualitative schematic of conditions required for entrainment by the roll- wave mechanism [3].

Ishii & Grolmes [3] also proposed an entrainment suppression correlation and this criterion is based on a condition where the film thickness does not penetrate the gas boundary layer and thus

interaction between the two phases cannot occur to cause entrainment. The resulting correlation is given as:

3 3 3  δ +  2  ρ  4  µ  2  min   l   v  Re l,crit =       (2-18)  .0 347   ρv   µl  where:

+ δ min = 10 (2-19)

Later work by Azzopardi [25] indicates this expression underpredicts the experimentally determined critical liquid Reynolds numbers for wave inception from various researchers, but suggests more suitable agreement can be obtained by using a value of 30 for the minimum dimensionless film thickness (in interfacial units) in this expression.

Woodmansee & Hanratty [56]

Woodmansee & Hanratty [56] studied the critical conditions required for entrainment to occur from the interface for air-water flowing co-currently in a horizontal, non-circular duct (12-inch x 1-inch). It was observed that for thick liquid films the critical gas velocity for entrainment to occur was greater than that required to generate roll waves and both conditions were independent of liquid flow rate. Meanwhile, as the film thickness decreased the two critical gas velocities increased, becoming a strong function of the film thickness, but became indistinguishable from one another for very thin films (i.e. onset of atomization coincides with the onset of roll waves). A condition also existed for very thin films where atomization would not occur regardless of gas velocity. These observations are consistent with the models proposed by Ishii & Grolmes [3].

Woodmansee & Hanratty [56] also indicated that the critical conditions are not strongly dependent on the liquid viscosity, but do increase with increasing surface tension. However, it is suggested that the reason for the critical film thickness below which roll waves cannot be

supported is that viscous damping, caused by the presence of the wall, is so strong such that waves cannot be sustained if they are too long compared to the film thickness. In turn, as the film thickness decreases prior to reaching the critical film thickness the critical gas velocity must increase since the length of the waves decreases due to the viscous damping.

2.2.5. Conclusions

The appropriate definition of regime transition points for the different annular flow regimes is important to the current study to ensure the appropriate models are applied based on the physical structures present within the flow. A wide variety of criteria have been suggested in the open- literature for these transitions and the current implementation in COBRA-TF may not provide an accurate reflection of the physics of annular flow. Rewriting the flow regime selection logic to explicitly account for sub-regimes within annular flow and then incorporating regime-specific models into the modeling package should improve the overall predictive capability of the code. The regime transition criteria that will be included in the proposed modeling package include:

1) a transition between counter-current and churn-annular or churn-turbulent flow, which will be handled by an explicit Counter-Current Flow Limitation (CCFL) model that will be developed as part of the current study.

2) a transition between the churn-annular and unstable film regimes, which will governed by the maximum of the gas velocities associated with: a) the local minimum in the pressure gradient assuming this condition corresponds to a dimensionless superficial gas velocity of 1.0, and b) the entrainment inception criterion correlations proposed by Ishii & Grolmes [3].

3) a transition between the unstable and stable film regimes, which will be based on a criterion that is consistent with the interfacial drag model included in the proposed modeling package and serve as the entrainment suppression criterion.

4) a transition between churn-turbulent and annular flow, which will be based on the bridging criterion currently used by COBRA-TF. If the void fraction in the computational cell is less than this criterion then checks will then be made on the wave amplitude relative to the limiting dimension of the flow path to determine whether the conditions are representative of churn-annular or churn-turbulent flow.

These criteria and their implementation will be discussed in greater detail in Chapters 4.

2.3. Review of Works Focused on Interfacial Phenomena

The prediction of the pressure gradient in annular flow for a three-field analysis approach requires appropriately defined expressions for the interfacial shear stress or drag between both the vapor and continuous liquid fields and the vapor and dispersed droplet fields. The latter contribution can be quantified using a droplet drag coefficient while Bharathan & Wallis [11] suggest the former contribution consists primarily of form drag over interfacial waves that protrude into the gas core region. This has led to the correlation of interfacial friction factors in an analogous manner to wall friction factors for single-phase situations, likening the waviness of the interface to surface roughness; however, due to the empiricism associated with this approach and the significant differences in the interfacial structure between the different annular regimes, regime- specific interfacial friction factors may need to be applied. This section reviews some of the previously conducted works focused on characterizing the interfacial structure of the film region and quantifying the interfacial shear stresses that exist between the different fields in annular flow.

2.3.1. Interfacial Structure

The structure of the interface between the liquid film and the gas core region determines the apparent roughness and drag area, both of which impact the calculation of the interfacial shear stress between the two fields and the amount of entrainment that occurs. For gas flow rates greater than the local minimum in the pressure gradient the co-current upward annular regime exists and the liquid film is characterized by the presence of upward propagating roll waves that travel at a relatively constant velocity and remain coherent over an appreciable distance [2,14,25,53,57]. As described in Section 2.2.4, once the critical gas velocity required to support these roll waves is exceeded the amount of entrained liquid present within the flow increases with increasing gas velocity, up to the point where the film thickness becomes so thin that it can once again no longer support roll waves [3,48]. As this point the propensity for entraining liquid from the interface decreases to zero and the amount of entrained liquid within the flow asymptotically approaches a limiting value. In contrast, for gas flow rates less than the local minimum in the pressure gradient the churn-annular regime is assumed to exist where the liquid film is chaotic

with large, unstable waves that can move either upwards or downwards, but do not travel a significant distance before disappearing by coalescence or breakup [10,12,14,16,18,41]. In this regime the waves are broken up by vapor undercutting, thus causing large amounts of entrainment; however, the large drops that are generated tend to quickly redeposit on the liquid film [41].

Most models currently available in the open-literature do not explicitly model the interfacial waves due to their complex motion and constantly changing structure, but it is important to characterize these waves, both qualitatively and quantitatively, to gain a physical understanding of the phenomena associated with annular flow and to assist with the future development of physics-based models. Quantitative parameters of interest include wave velocity, amplitude, frequency, and spacing. Most attempts to characterize these properties have utilized data obtained with conductance probes, but other work has indicated that fluctuations in the instantaneous measurement of wall shear stress are related to fluctuations in the film thickness corresponding to the passage of disturbance waves [10,58]. Therefore this type of measurement can also be used to provide information on the wave frequency and possibly size. This also indicates that the wall shear stress may be enhanced in annular flow relative to the single-phase case due to the existence of interfacial waves, which would suggest it is not adequate to apply the same wall friction factor correlations in both cases as is currently done. This section reviews several works aimed at characterizing the interfacial structure and wave motion properties in the different regimes of annular flow. A detailed review of disturbance wave behavior (i.e. frequency and velocity) is provided by Azzopardi [25].

2.3.1.1. Co-Current Upward Annular Flow

This section reviews several works associated with characterizing the interfacial structure of co- current upward annular flow. Most works described here focused on the unstable film regime, but some did attempt to determine the conditions where roll waves could not be supported.

Belt et al. [2]

Belt et al. [2] conducted experiments in the co-current upward annular regime using air-water at atmospheric pressure in a vertically oriented, 2-inch diameter tube. The results indicted upward traveling roll waves were the dominant structures separated by smaller ripple waves. They found the roll waves were not uniformly distributed around the circumference of the tube, but the structures were coherent. It was determined that the time between waves was well fit by a gamma distribution, which suggests each roll wave is an individual and randomly distributed process, independent of neighboring roll waves. The measurements also indicted that the roll waves move with an approximately constant velocity and the average height of the waves is roughly four times the mean film thickness; however, this ratio was found to deviate by as much as 25% and was found to be a function of superficial velocities. In particular the wave heights tended to be smaller than four times the mean film thickness for large gas superficial velocities and larger than four times the mean film thickness for small gas superficial velocities. Finally, a critical liquid Reynolds number was found below which roll waves do not exist. Belt et al. [2] also indicated a sharp decrease in the measured wave velocity as this critical condition was approached, which confirms that roll waves tend to move faster than the mean velocity of the film; however, no measurement of entrainment was made to confirm or deny liquid was being entrained from the interface at the critical condition. [2]

Sawai et al. [14]

Sawai et al. [14] conducted air-water experiments at atmospheric pressure in a 1-inch diameter, vertically oriented acrylic resin tube to measure the pressure gradient and liquid holdup for various flow rate combinations. It was found that as the liquid flow rate decreased the wave velocity became much larger than the average liquid velocity, which contradicts the result obtained by Belt et al. [2] that the wave velocity decreased as the critical condition was approached. Sawai et al. [14] suggest that the frictional pressure gradient in the co-current upward annular regime is dominated by the propagation of waves rather than the steady liquid film flow. Additionally, similar to findings of Belt et al. [2], it was observed that for small liquid velocities and large vapor velocities a thin liquid film without interfacial waves existed; however,

again in these experiments no measurements were made to confirm or deny whether entrainment was or was not occurring. [14]

Okada et al. [59]

Okada et al. [59] identified two types of disturbance waves: 1) DW1 - two-dimensional waves that are coherent about the circumference of the flow path and 2) DW2 - three-dimensional wave structures. Coherence was quantified using the sharpness of the peak observed in the autospectral density signal from film thickness probes. Okada et al. [59] found that for a given tube diameter and liquid flow rate the overall frequency of waves (DW1+DW2) increased with increasing gas velocity; however, the frequency of coherent waves (DW1) increased while the frequency of non- coherent waves decreased with increasing gas velocity. Martin & Azzopardi (1985) observed the coherence also decreased with increasing tube diameter [25].

Woodmansee & Hanratty [56]

Woodmansee & Hanratty [56] took high speed motion pictures of air-water flowing co-currently in a horizontal, non-circular duct (12-inch x 1-inch) near the critical conditions for entrainment to occur. The structure of the interface was found to consist of large disturbance, or roll, waves that varied in length between three to twelve inches, but the ratio of length to height was always less than 3:1 (corresponds to a maximum wave slope of 18-degrees). Additionally, small wavelets, or ripples, were found to be superimposed on the roll wave structure. These ripples were described as being broad-crested or two-dimensional, which is in contrast to the previous suggestion of Taylor (1963) that these structures were three-dimensional in nature. Additionally, the ripple waves were observed to have much shorter lengths (roughly 0.03-0.06-inches), and lifetimes than the disturbance wave, but can have a height that is on the same order of the height of the disturbance wave. The presence of the ripple waves results an increased roughness on the tops of the roll waves that provide the frothy, or agitated, appearance, which can be one explanation for the enhancement of interfacial shear relative to stationary, wavy wall situations. This effect will be discussed in greater detail in Section 2.3.4.

Azzopardi [53]

Azzopardi [53] studied the properties of interfacial waves in co-current upward annular flow conditions using air-water in a vertically oriented 1.25-inch tube at 36-psia. The work concentrated on characterizing the dependency of wave frequency, velocity, and spacing on the liquid and vapor mass fluxes. The results are based on the analysis of film thickness measurements obtained using conductance probes located axially along the test section. Azzopardi [53] observed two distinct peaks in the frequency of peak height versus peak height plot corresponding to two distinct distributions: one for ripples and one for disturbance waves. Additionally, the wave frequency and velocity of the disturbance waves were found to increase with both vapor and liquid flow rates, while the wave spacing decreased since more waves were present. However, if the waves become too close they begin to coalesce and therefore an asymptotic limit exists for this spacing. A similar finding was obtained by Belt et al. [2]. It was also found that increased vapor flow rates leads to more uniform wave velocities and as a result the asymptotic limit for wave spacing decreases with increased vapor flow rates. Finally, wave frequency and spacing were found to decrease with distance along the tube due to waves coalescing, but the velocity of the waves only increased slightly. Other work has suggested the wave velocity is dependent on the wave amplitude, which does not appear to be consistent with this result since wave coalescence should lead to larger wave amplitudes assuming the characteristic angle is unchanged. [53]

Sawant et al. [57]

Sawant et al. [57] conducted air-water annular flow experiments in a 0.37-inch diameter, vertically oriented, stainless steel tube for three outlet pressure conditions (atmospheric, 58-psia, and 84-psia) to characterize the properties of disturbance waves. Disturbance wave velocity, frequency, amplitude, and spacing are estimated from the film thickness measurements using statistical analysis methods. Gas velocities were increased from near the churn-annular regime boundary to well within the annular regime, but conditions examined are limited to relatively low liquid Reynolds numbers (500-5700). The qualitative results of these experiments are summarized in Table 2-2 and in general are consistent with those of Azzopardi [53]; however,

Sawant et al. [57] only took measurements at a single point located roughly 8-ft (270 L/D) downstream of the liquid injection point. At this location the conditions were said to be representative of fully-developed, or equilibrium annular flow conditions and therefore no conclusions can be made regarding developing flow effects.

Table 2-2: Summary of Disturbance Wave Characteristics obtained by Sawant et al. [57]. Increasing Gas Increasing Liquid Increasing Wave Property Velocity Reynolds Number Pressure Velocity Increases Increases Increases Slightly Amplitude Decreases Increases Decreases Spacing Decreases No Dependence Decreases Frequency Increases Increases Increases

It should be noted that the wave velocity was found to increase linearly with gas velocity at lower gas velocity conditions, but approached an asymptotic limit at higher gas velocities. The experimental results also indicated the average spacing between two successive waves ranged between 2 and 10 inches, with wave spacing decreasing sharply with increasing gas velocity for low gas velocity conditions, but asymptotically approaching a limiting value at higher gas velocities. [57]

Finally, Sawant et al. [57] compared the wave frequency results from these experiments to the results predicted by available correlations and it was found these correlations failed to predict the pressure effect. As a result a new correlation is proposed by Sawant et al. [57] where the disturbance wave frequency is correlated based on a fit to their experimental data using the Strouhal number, liquid Reynolds number, and density ratio. This new correlation was found to predict the disturbance wave frequency data from these experiments, as well as those conducted by independently by Schadel (1988) and Willetts (1987), within ±25% deviation.

Hurlburt et al. [55]

Hurlburt et al. [55] developed correlations for the intermittency and average amplitude for both ripple and disturbance waves based on the high-resolution film thickness data of Fore & Dukler [23], which was taken using air-water at low-pressure in a 2-inch diameter tube, and Fore et al. [31, 51], which was taken using both steam-water and nitrogen-water at high pressure in a non- circular duct with a large aspect ratio. The correlations for these parameters were needed to support their development of a two-zone interfacial drag model, which will be discussed later in this chapter. The intermittency is defined as the fraction of time the surface is occupied by disturbance waves and provides a means for quantifying when a reduction in disturbance wave activity occurs. Hurlburt et al. [55] found that the intermittency correlated with the dimensionless film thickness (in interfacial units) with the intermittency asymptotically approaching a value of roughly 40% for large values of this quantity and approaching zero for dimensionless film thicknesses (in interfacial units) of roughly 12 for the steam-water data considered. This result is substantially larger than the value of 11% suggested by Kataoka et al. [60] for the ratio of the interfacial area of the wave crests to the interfacial area of the entire film.

Meanwhile, the amplitude of both types of waves was related to the standard deviation of the time-averaged film thickness measurements for each region. Relating these two parameters is physically meaningful since the standard deviation should approach zero for smooth films. The resulting steam-water correlation includes an offset, or critical mean film thickness, of 30-m (9.84x10 -5 ft), below which no disturbance waves were observed. The steam-water correlation also suggests that for thicker films the ratio of disturbance wave amplitude to mean film thickness reaches an asymptotic upper limit of roughly 1.4, with this ratio obviously being less for thinner films. This ratio is significantly less than that obtained by Belt et al. [2] and suggested by Wallis [4], but is consistent with the more recent findings of Rodríguez [61]. One explanation may be the significantly smaller hydraulic diameter flow duct used by Fore et al. [31,51] in their experiments relative to the 2-inch diameter tube used by Belt et al. [2]. The larger pressure gradients and increased gas velocities observed in smaller diameter flow ducts may be an indication that entrainment can occur from smaller amplitude waves in these situations, which prevents such large amplitude waves from being observed. Additionally, it should be noted that the suggestion of Wallis [4] is not based on direct measurement. Instead Wallis [4] used

comparisons to experimental pressure gradient data to infer an estimate of the equivalent roughness and therefore assumes stationary waves and neglects the highly dissipative nature of this region.

Henn & Sykes [20]

Henn & Sykes [20] used Large-Eddy Simulation (LES) to investigate fully-developed turbulent flow of liquid through a 2-D flow duct where the lower wall is sinusoidal. Since in can be anticipated that wavy surfaces will generate large, resolvable eddies, especially if flow separation occurs, this type of flow is well-suited to modeling using LES. A periodic boundary condition is applied to model the sinusoidal surface and a no-slip condition is imposed on the surfaces. The standard computational domain contains two wavelengths, which was determined to be sufficient from a sensitivity study performed by comparing these results to those obtained using one and four wavelengths.

The numerical results were compared to the experimental observations of such researchers as Buckles et al. [19] and Zilker & Hanratty who have performed extensive measurements of flow over wavy walls for Reynolds numbers around 10 4. Reasonable agreement was obtained for wave slopes between 0 and 0.628, where the wave slope is defined as:

ε m = w (2-20) λ

Most important to the current study is that Henn & Sykes [20] integrated the LES predicted surface pressures to calculate form drag as a function of wave slope and were able to correlate the results. It should be noted that when doing this Henn & Sykes [20] adjusted the calculated pressure results so that the LES and experimental results matched in the wave trough. Their results indicated the drag increased quadratically with slope for small amplitude waves with a somewhat slower increase for larger amplitudes. The resulting correlation for the non- dimensional drag force (per unit area) in the flow direction is given as:

2 Fz′′ = 12.0 (mπ ) for: m < 1.0 (2-21)

This expression could be equated with the code predicted drag force to provide an estimate the disturbance wave spacing. 2.3.1.2. Churn-Annular Flow

This section reviews several works associated with characterizing the interfacial structure of the churn-annular regime. As indicated previously, the churn-annular regime is a chaotic region that exists following the flow reversal point and prior to the establishment of co-current upward annular flow.

Hewitt et al. [12]

A dye tracing visualization technique showed in the churn-annular regime the film surface is traversed by large waves that deposit a liquid film in their wake. This liquid film initially travels upwards, but then gradually decelerates and may reverse direction before the arrival of the next wave. [12]

Moalem-Maron & Dukler [62]

Moalem-Maron & Dukler [62] hypothesized that in the region between the flow reversal point and the minimum in the pressure gradient that the liquid film thickness randomly oscillates to satisfy the two possible states for the system. They conducted a theoretical analysis and were able to show that in fact two-possible solutions to the system exist near the flow reversal point, corresponding to the liquid moving either upwards or downwards at the wall. This switching is suppressed with increasing vapor flow rate as the local minimum is approached and the flow throughout the film moves towards becoming unidirectional upwards. [62]

Zabaras et al. [10]

Experiments were conducted to confirm the hypothesis of Moalem-Maron & Dukler [62] by simultaneously measuring the magnitude and direction of the wall shear stress along with the pressure gradient and film thickness as a function of time as the gas flow rate was increased from the flow reversal point through the local minimum in the pressure gradient. As expected, the measurements confirmed that for gas flow rates between the flooding point and the local minimum in the pressure gradient the wall shear stress could be either positive to negative. Little or no correlation existed between the instantaneous magnitudes of the wall shear stress and the film thickness in this region. On the other hand, for gas flow rates greater than those corresponding to the local minimum in the pressure gradient, the wall shear stress was always negative, corresponding to upflow at the wall, and excellent correlation existed between the instantaneous magnitudes of the wall shear stress and film thickness. The fractional amount of time that the wall shear stress was negative was also determined and the results showed that this quantity increased from about 20% at the flow reversal condition to 100% near the location of the minimum in the pressure gradient. [10]

Finally, it was shown that the Wallis [4] interfacial friction factor correlation did an adequate job of predicting the data at higher gas flow rates corresponding to those well beyond the pressure gradient minimum, but severely underpredicted the data at the lower gas flow rates corresponding to those in the churn annular regime [10]. This result confirms the need for regime-specific interfacial friction factor correlations to be instituted in the proposed modeling package.

2.3.2. Interfacial Shear Stress Definition

An appropriately defined interfacial shear model is essential to the accurate prediction of annular flow situations. The resulting entrainment rate, which directly impacts the liquid flow distribution between the continuous and entrained fields, is strongly dependent on this parameter [33]. In turn this parameter also influences the distribution of liquid between the film and droplet fields, the resulting flow rates, and the pressure gradient. The most common definition of this quantity between the vapor and continuous liquid fields is given as:

f τ = i,v ρ ()U −U 2 (2-22) i,vl 2 v v i

The primary difficulty with this expression is quantitatively characterizing the liquid portion of the relative velocity term. It has been suggested in the open-literature that this velocity should be the mean velocity of the liquid film, the velocity of the waves, or the velocity of the interface, which can be taken to be twice the mean velocity of the liquid film if a constant shear stress profile throughout the film is assumed. COBRA-TF applies Equation (2-22) using the mean velocity of the continuous liquid field for the liquid portion of the relative velocity term.

The remaining discussion within this section is focused on various simplifying assumptions that are often applied to Equation (2-22) when reducing experimental data to develop interfacial friction factor correlations and the potential impact these assumptions have on the applicability of the resulting correlations. Since the film velocity is often not measured or known the most common approach assumes the wave or interface velocity component of this expression is negligibly small relative to the bulk velocity of the vapor phase. Meanwhile, experimental evidence indicates that the velocity of the disturbance waves can be as much as one order of magnitude greater than the mean velocity of the entire film [1,53]. Belt et al. [2] indicate based on their experiments that neglecting the liquid velocity component can cause the interfacial shear to be overpredicted by 10-17%.

Since the film thickness is typically small, and often not measured, it is also commonly assumed that the vapor velocity can be replaced with the superficial vapor velocity, which yields:

f τ = i,v ρ j 2 (2-23) i,vl 2 v v

Still others, such as Whalley & Hewitt [28], suggest that the calculation of interfacial shear should utilize an effective vapor core superficial velocity and density to consider the effect of the entrained droplets, which alters the inertia of the gas core region. In this case the interfacial shear stress is defined as:

f τ = i,v ρ j 2 (2-24) i,vl 2 gc gc

However, the calculation of the gas core density and superficial velocity typical assumes homogenous flow in this region, but Lopes & Dukler [26] presented experimental results showing the slip ratio between the vapor and the droplets to be on the order of 50% for gas Reynolds numbers between 5000 and 10000, while Fore & Dukler [24] suggest that droplets at the centerline travel, on average, at 80% of the local mean gas velocity. As a result a homogenous flow assumption may be invalid. Hurlburt et al. [55] tried using the gas core density when developing their two-zone interfacial shear model; however the resulting predictions were found to overestimate the experimental data and therefore was not used in their final model.

Regardless of the assumptions discussed within this section, it is obvious that the prediction of the interfacial shear stress is heavily dependent upon utilizing a suitably defined interfacial friction factor. A summary of the most relevant interfacial friction factor correlations that have been proposed in the open-literature will be provided in the following sections, but first it is important to understand the forces acting on an interfacial wave. The drag force is defined as the component of the resultant force exerted by a fluid in a body in the direction parallel to the motion of the fluid. In general the total drag force over the surface of a wave consists of the pressure variation, or form loss, that acts normal to the surface of the wave and the stress that acts along the surface of wave due to the difference in velocity of the fluids at the interface [19]. Therefore the total average drag force (per unit area) over the length of a wave is comprised of:

Fz′′ = Fp′′ + Fs′′ (2-25) where the average non-dimensional pressure loss (per unit area) over the length of the wave is:

1 ε wπ *  z   2πz   z  Fp′′ = − ∫ ps  sin  d  (2-26) λ 0  λ   λ   λ  and the non-dimensional pressure at the surface is given as:

−1 *  z  1 2  ps   = p(z)  ρU  (2-27)  λ  y=0 2 

Meanwhile, the average non-dimensional surface shear force (per unit area) over the length of the wave is:

1 *  z   z  Fs′′ = ∫τ i,vl  d  (2-28) 0  λ   λ  and the non-dimensional shear stress is given as:

−1 *  z  1 2  τ i,vl   = τ (z)  ρU  (2-29)  λ  y=0 2 

Recognizing that in reality two distinct forces act on a wave highlights that interfacial friction factor correlations effectively lump these contributions into this single parameter. Buckles et al. [19] integrated the surfaces stresses from their experiments of flow over a wavy wall and indicated that the surface shear component of the total force is negligible relative to the pressure component (roughly an order of magnitude smaller). A similar result was obtained by Henn & Sykes [20] when they integrated their LES predicted surface shear stress when simulating the experiments of Buckles et al. [19]. This result supports the theory of Bharathan & Wallis [11] that the interfacial shear stress consists primarily of form drag over the waves.

2.3.3. Current COBRA-TF Calculation of Interfacial Shear Stress

As alluded to in the previous section the definition of interfacial shear stress used by COBRA-TF is:

1 τ = f ρ U U (2-30) i,vl 2 i,v v vl vl

where:

U vl = U v −U l (2-31)

The form of the momentum equations applied by COBRA-TF employs an interfacial drag coefficient, which is defined in the axial direction for the interface between the vapor and the film as:

τ i,vl Ai,l 1 fi,v ρv U vl Ai,l K vl = = (2-32) U vl ∆z 2 ∆z and assuming the film is uniformly distributed around the periphery of the flow path the interfacial area between the continuous liquid and vapor phases can be obtained from simple geometric relationships and the resulting expression is given as:

4Ax ∆z α v + α e Ai,l = Ps ∆z α v + α e = (2-33) DH

The definition of the interfacial drag coefficient allows the force per unit length acting on the interface to be calculated as:

Fv′l = KvlU vl (2-34)

Finally, it should be mentioned that COBRA-TF currently applies the same interfacial friction factor correlations, which were developed primarily for co-current flow situations, for all annular flow regimes, including the counter-current, or falling film, regime. Other works have highlighted the inaccuracy of doing this [6,10,16] and as a result a primary focus of the current work was the identification and implementation of regime-specific interfacial friction factor models.

2.3.4. Interfacial Friction Factor

Most of the existing interfacial film friction factor correlations for annular flow, such as the one suggested by Wallis [4], are highly empirical and based on the analogy of friction in single-phase turbulent flow in rough pipes. However, measurements of the pressure gradient across a solid wavy surface having the similar structure (size and shape) to the waves expected in annular flow give much lower estimates of the pressure gradient than those measured in annular flow and an unrealistically large relative roughness is needed to match the measured pressure gradient [10,55]. Zabaras et al. [10] suggested the reason for the enhancement relative to the solid surface is due to dynamic characteristics of the interfacial waves. Hurlburt et al. [55] were able to quantify this hypothesis based on fits to experimental data. Their results suggest the ‘ cb’ parameter in the log law velocity profile for disturbance waves is 4.7 compared to the commonly used values of 0.3 and 0.8 for sand roughness or stationary wavy wall situations, respectively [55]. This difference suggests that the dynamic shape of the disturbance waves, the wave’s local small scale roughness, and developing flow effects near the wave result in much larger drag than is observed in flows with stationary wavy walls [55]. This finding may indicate that the value of four (4) for the ratio of wave amplitude to mean film thickness that is implied by the Wallis [4] correlation is in fact unrealistically large, as the data from several experiments seems to suggest [55,61], but such a large value was necessary to correlate the experimental data since the dynamics of the wave are not considered.

Meanwhile, some researchers have also suggested the presence of the liquid film and droplets causes the vapor velocity profile to deviate from the flattened profile characteristic of single- phase turbulent flow to more of a center peaked profile [18]. This difference is primarily a result of larger droplets tending toward the periphery, thus allowing the velocity to be larger in the center [17]. This change would result in the power-law coefficient ( n) of the velocity profile being less than the typical single-phase value of seven (n = 7) for smooth conduits and would correspond to a larger friction factor than an analogous single-phase flow situation since, according to Nunner’s results [63]:

1 f ≈ (2-35) 4n2

Also, a continuous exchange of mass, momentum, and energy occurs between the entrained and continuous liquid fields in annular flow due to entrainment and deposition. The work of Lopes & Dukler [26] and Fore & Dukler [23] attempted to isolate the contributions of gravity, interfacial shear, and droplet exchange to the total pressure gradient using momentum balances and the results from both works indicate that the momentum exchange process can comprise a substantial portion of the total pressure gradient observed in annular flow. The results of Fore & Dukler [23] indicated that droplet interchange accounted for as much as 20% of the overall pressure gradient in annular flow, while Lopes & Dukler [26] observed an even larger percentage, especially at larger liquid flow rates. The results of Lopes & Dukler [26] also indicate that the pressure loss due to gravity is negligible in all cases examined.

The inability to isolate the interfacial wave effect from the momentum exchange process may be one reason a suitable correlation covering a wide range of conditions has been unable to be defined [26]. For example, many of the interfacial friction factor correlations that are currently available, such as those of Wallis [4], Henstock & Hanratty [27], and Bharathan & Wallis [11], were developed using experimentally measured total pressure gradients, without explicitly accounting for the momentum transfer due to the entrainment and deposition of droplets [26]. Lopes & Dukler [26] show that when the interfacial friction factor is simply calculated using the total pressure gradient a unique relation to the relative film thickness does not exist, but rather a significant dependence liquid flow rate is displayed. Meanwhile, if the interfacial shear contribution to the experimental pressure gradient is isolated by correcting the experimental data for the effect of the momentum exchange due to entrainment/deposition, then the calculated interfacial friction factors are simply a function of relative film thickness, as suggested by Wallis [4]. Lopes & Dukler [26] suggest that Wallis [4] was able to obtain a similar result when developing his correlation only because a minimal amount of entrainment existed in the data that was considered and thus the momentum exchange component would have been negligibly small. Similarly, a unique relationship has been obtained for single phase wall friction factors because no momentum transfer due to mass transfer occurs at the wall.

It is more important for the purposes of the current study to recognize that not accounting for these effects when reducing and then correlating experimental data causes these effects to be implicitly included in the resulting correlation. This is desirable for two-fluid analysis

approaches because it provides a means for implicitly considering the contributions of the droplet field to the overall flow parameters; however, such a correlation presents an issue for three-field analysis approaches where separate droplet models have been included to explicitly account for these mechanisms. Therefore using such a correlation can cause droplet contributions to be double accounted for and lead to the introduction of compensating errors within the code. As a result caution must be exhibited with regards to this topic when assessing the applicability of available interfacial friction factors for use in three-field analysis environments.

Additionally, it will be seen in the following sections that a majority of the correlations that have been developed to date have used air-water data at low pressures. Meanwhile, the work by Hossfeld et al. [38] examined the extension of air-water based correlations to situations involving steam-water. The results indicted that the air-water based interfacial friction correlations considerably overpredict the steam-water data considered. A similar result was also found by Whalley & Hewitt [28] in their work. As a result, it is recommended that separate correlations be used for the analysis of air-water and steam-water situations. Hossfeld et al. [38] also suggest that this discrepancy may indicate that phase change phenomena may have a significant bearing on interfacial drag and may represent another effect that needs to be considered when performing data reduction involving steam-water.

Finally, for low gas flow rates the effect of gravity on the velocity profile within the liquid film becomes important. Under these conditions the churn-annular regime exists where downflow can occur near the wall despite to overall flow being upwards. This situation enhances the size of the interfacial waves and the apparent roughness of the film. Hurlburt et al. [55] indicate that in this region it may be necessary to explicitly account for the losses in the recirculation zone that exists following the expansion of the vapor phase in the region just past the wave crest and the length of which increases with the height of the wave. These ideas highlight the need for regime specific correlations to be developed and applied. [36,41]

The following section reviews several of the interfacial friction factor correlations that are currently available in the open literature. A focus is placed throughout this section on identifying regime-specific interfacial fraction factors since it is unlikely that a single correlation can account

of the changes in interfacial structure that exists between the different regimes. Sections related to stable films, unstable films, churn-annular flow, and counter-current flow are provided.

2.3.4.1. Stable Film Flow

This section focuses on interfacial friction factors that were developed from data that was taken under conditions where interfacial waves and entrainment were suppressed. In this regime COBRA-TF currently uses the interfacial film friction factor suggested by Wallis [4], which suggests the interfacial friction factor is a function of only the relative film thickness and is given as:

   δ  f i,ST = .0 005 1+ 300  ≈ .0 005 ()1+ 75 α l (2-36)  DH 

When the thin film approximation is applied then the latter form of the correlation is obtained. The 0.005 factor in front of the Wallis [4] correlation corresponds to the fully-rough turbulent regime where the friction factor is independent of the Reynolds number. Wallis [4] noted the similarity between his relationship and the one used to approximate the single-phase friction in rough tubes suggested by both Nikuradse and Moody, which is given as:

 k   s  f w = .0 005 1+ 75  (2-37)  DH 

The comparison of these two expressions suggests that the amplitude of the roll waves in annular flow is analogous to sand grain roughness in rough pipes and implies the ratio of wave amplitude to mean film thickness is roughly equal to four. The work of Belt et al. [2] discussed previously confirmed a value of four for this ratio is reasonable, but the more recent work by Hurlburt et al. [55] indicates such a value for this ratio may be too large.

Later work by Whalley & Hewitt [28] showed that the Wallis [4] correlation significantly overpredicted high pressure steam-water data, which is consistent with the observations of Hossfeld et al. [38]. Similarly, Fore et al. [63] compared the results predicted by the Wallis [4] correlation to several sets of experimental data and indicted was inaccurate over the entire range of conditions considered. Others have tried to replacing fully-rough turbulent friction factor (0.005) in Equation (2-37) with a Blasius-type relationship:

  1  .0 316  − 25.0 f s =  25.0  = .0 079 Re v (2-38) 4  Re v 

However, Fore et al. [63] indicate this inclusion does not provide a strong enough vapor Reynolds number dependence and actually performs slightly worse at the higher vapor Reynolds numbers. Meanwhile, Sugawara [64] suggested using a film, rather than vapor, Reynolds number should be used in this relationship.

Due to the inadequacy of the predictions Fore et al. [63] proposed a modified form of the Wallis [4] correlation based on the idea of transition roughness where it is suggested that for lower gas Reynolds numbers and thicker films the friction factor is enhanced and in this region a Reynolds number dependence exists, similar to single phase flow. Fore et al. [63] suggest a simple multiplication factor, the values of which were determined from a fit to the experimental data sets that were considered, to impart a stronger dependence than the simple inclusion of a Blasius-type relationship. The proposed correlation also includes an offset, which suggests some finite relative film thickness is required to generate interfacial waves that enhance the interfacial friction above the ‘smooth’ tube value, to improve the predictions for very thin films. The resulting correlation is given as [63]:

  17500  δ  f = .0 005 1+ 300 1+  − .0 0015  (2-39) i   D   Re gc  H  where:

(Gv + Ge )DH Re gc = (2-40) µv

The formulation of this correlation is somewhat contradictory since the correlation is still preceded by the fully-rough value of the friction factor (Reynolds number independent), while at the same time including a Reynolds number dependence to capture the effect of transition roughness. More importantly, this correlation is still highly empirical, relying entirely on dimensionless parameters to correlate experimental data. In any event, this correlation was able to predict all the data from the three different experiments considered to within 25% uncertainty, but significant differences still existed for thicker films present at lower gas velocity conditions [63]. Other correlations, such as the one suggested by Henstock & Hanratty [27], which is used by COBRA-TF for the unstable film regime, have done a better job of predicting the behavior of thicker films, but this correlation was developed considering only low-pressure air-water data.

2.3.4.2. Unstable Film Flow

This section focuses on interfacial friction factors for the unstable film flow regime where interfacial roll waves are the dominate structures, which act to increase the apparent roughness or drag area of the interface relative to the stable film regime. Most of the correlations that have been proposed in the open literature were developed for this regime. Currently for the unstable film regime COBRA-TF applies the maximum of: 1) five times the Wallis [4] prediction and 2) the result predicted the Henstock & Hanratty [27] correlation.

5 f i,ST f i,UN = max  (2-41)  fi,HH in the axial direction. The Henstock & Hanratty [27] correlation was developed using co-current and counter-current film flow data for unstable films and the multiplication factor associated with the Wallis [4] correlation comes from the difference between the pressure drop characteristics for

stable and unstable films that was observed by Dukler & Smith [43]. For vertical flow situations the Henstock & Hanratty [27] correlation is given as:

f   τ  i,HH  i,vl  = 1+1400 F 1− exp −  (2-42) f s   ρl gδ  where the following approximation was suggested:

3 2 τ i,vl (1+1400 F ) = (2-43) ρl gδ 13 2. FG and the friction factor for a smooth surface is calculated as:

− 20.0 f s = .0 046 Re v (2-44)

The parameter ‘G’ is a measure of the importance of gravity relative to interfacial drag and is given as [27]:

ρl gD H G = 2 (2-45) ρv µv f s

This expression is most important for situations involving vertical downflow. On the other hand, the parameter ‘F’ is similar to the Martinelli parameter and is used to characterize the flow conditions. This expression is given as [27]:

δ + µ ρ l v (2-46) F = 9.0 Re v µv ρl where the dimensionless film thickness is given as [27]:

4.0 + 5.0 5.2 9.0 5.2 δ = [( .0 707 Re l ) + ( .0 0379 Re l ) ] (2-47)

It should be noted that in addition to only considering air-water data when developing this correlation Henstock & Hanratty [27] also only used data from 1.24 and 1.36-inch diameter tubes. Asali et al. [54] later indicated that this correlation does not adequately predict the effect of pipe diameter on interfacial friction factor or the influence of film flow rate on film thickness. Meanwhile, Fore et al. [63] show that the Henstock & Hanratty [27] correlation adequately reflects the behavior of the data for thicker films at low pressure, but it severely overpredicts the friction factors for a newly collected set of nitrogen-water experimental data taken at 250-psia. Additionally, due to the discrepancies found in using air-water based correlations to predict steam-water conditions, Hossfeld et al. [38] proposed an alternative form of the Henstock & Hanratty [27] correlation based on steam-water data; however, the correlation is still highly empirical in nature and as a result it was decided not to pursue using this correlation in the current study.

Whalley & Hewitt [28]

Whalley & Hewitt [28] proposed a correlation for the interfacial friction factor developed considering both air-water and steam-water data. The correlation is given as:

1    3  f i  ρl δ  = 1+ 24   (2-48) f   ρ  D  s   v  H  where smooth friction factor is calculated using Equation (2-38), but the Reynolds number in this expression is replaced by an effective vapor core Reynolds number, which considers the effect of the entrained droplets. Due the wide range of data considered and the amenability to implementation within the structure of a transient analysis code Holowach [33,34] utilized this correlation in his work, and while it does represent an improvement over the Wallis [4]

correlation, Hewitt & Govan [48] suggest that this correlation is unsatisfactory because a large amount of scatter still exists.

Hurlburt et al. [55]

The interfacial friction factor correlations that have been discussed thus far have attempted to globally average the interfacial shear based on the mean vapor and film velocities and the mean film thickness, which is assumed to be related to wave amplitude. Meanwhile, as discussed previously, several researchers, such as Belt et al. [2], Sawai et al. [14], and Sawant et al. [57], have proposed that the liquid film consists of two distinct regions: 1) disturbance waves and 2) a base film substrate, which travel with significantly different velocities. In reality the disturbance waves travel with a faster velocity than the base film and an increase in vapor velocity is observed at the wave crest due to the reduction in flow area at this point. These parameters govern the local relative velocity between the two fields and an accurate estimate of this quantity is crucial to the prediction of interfacial shear stress, especially at higher pressures, where the smaller density ratio results in lower relative velocities, and in smaller flow paths, where the vapor velocity is greatly increased near the wave crests; however, none of the models discussed to this point explicitly account for these effects. [55]

As a result Hurlburt et al. [55] take a unique approach to calculating the interfacial shear stress in annular flow by proposing a model that calculates the shear in the disturbance wave and base film regions separately. Doing this allows for: 1) separate interfacial friction factors to be applied to each region, which uses a regional average, instead of a global average, estimate of the relative roughness and 2) a more precise estimate of the relative velocity between the vapor and liquid that exists in each region. Hurlburt et al. [55] estimate the relative contribution of each zone to the total interfacial shear stress using a wave intermittency, which, as mentioned previously, is defined as the fraction of time that disturbance waves are present on the surface and provides a quantitative relationship for predicting when a reduction in disturbance wave activity will occur. The loss of disturbance waves results in lower wall heat transfer rates and decreased transfer mixing within the film because this condition is also coupled with a loss in entrainment.

It is noted by Hurlburt et al. [55] that the shear predictions at low gas velocities near the transition to annular flow are consistently underpredicted. It is assumed that this occurs because a log law velocity profile is least accurate for very large waves that extend nearly to the centerline of the flow path. Also, as previously mentioned, a recirculation zone exists following the expansion of the vapor phase in the region just past the wave crest, which increases with the height of the wave, and this approach does not account for these losses.

The model proposed by Hurlburt et al. [55] requires estimates of relative roughness, vapor velocity, and liquid velocity for each zone (base film substrate and interfacial waves) as well as an estimate of the intermittency. All of these parameters can be obtained either from calculations performed by COBRA-TF or from correlations provided by Hurlburt et al. [55], which makes this model amenable to implementation. Based on the merits of this approach it has been decided to implement this model as part of the proposed modeling package, and therefore the discussion of the specific equations associated with this model will be postponed until Chapter 4, where the proposed modeling package is presented. However, it should be noted that a level of empiricism still exists within this model because slight differences in the correlating constants exist in the proposed forms of the wave amplitude and intermittency correlations for the different fluids considered (steam-water, air-water, nitrogen-water). Additionally, the values used for the ‘ cb’ parameter in the log law velocity profile are based on fits to experimental data.

2.2.3.3. Churn-Annular Flow

This section focuses on interfacial friction factors for the churn-annular regime. The works discussed in this section compared experimental results to several of the co-current upward annular flow correlations that were discussed in the previous section and unsatisfactory agreement existed. As a result specific correlations were developed for this regime. COBRA-TF does not currently contain an explicit model for the churn-annular regime and therefore a suitable interfacial friction factor for this regime will need to be selected for inclusion in the proposed modeling package. However, significantly fewer works have been conducted in this regime and more importantly the majority of these works have been conducted using air-water at low- pressures.

Bharathan & Wallis [11]

Bharathan & Wallis [11] conducted a series of air-water experiments in tubes with diameters ranging from 0.25 to 6-inches. It is suggested that when the dimensionless diameter, is much greater than one (i.e. D* >> 1) the interfacial friction factor should be independent of this parameter [11]. Based on this observation Bharathan & Wallis [11] proposed the following functional form for the interfacial friction factor correlation:

* Bo f i = .0 005 + Ao (δ ) (2-49) where the dimensionless film thickness is defined as:

1 − 2 *  σ  δ = δ   (2-50)  g∆ρ 

The correlating constants in this equation were determined from comparisons to their experimental data and were found to be:

07.9 − 56.0 + D* Ao = 10 (2-51) and:

74.4 B = 63.1 + (2-52) o D* where the dimensionless diameter is defined as:

1 − 2 *  σ  D = DH   (2-53)  g∆ρ 

This correlation is a function of the liquid film thickness, tube diameter, interfacial surface tension, and gravity. It can be seen when examining the correlation that for a given dimensionless film thickness the interfacial friction factor for a smaller tube is larger. Additionally, for large tube diameters both the correlating constants approach finite limits and the functional dependence on the tube diameter is removed. These observations are physically consistent since relative surface roughness decreases as the tube diameter increases and this behavior is expected to be asymptotic since a point will be reached where the gas core is largely unaffected by the presence of the liquid film. The ability to capture this effect is a unique aspect of the Bharathan & Wallis [11] correlation.

Zabaras & Dukler [7]

Zabaras & Dukler [7] calculated the interfacial shear stress from their experimental measurements of mean pressure gradient and film thickness. When performing this data reduction Zabaras & Dukler [7] neglected the interfacial velocity since it was not measured experimentally, which would result in an overprediction of the interfacial shear stress; however, the liquid velocity contribution should be relatively small within the churn-annular regime and so this approach may be reasonable. The calculated interfacial friction factors were then compared to values predicted by the Bharathan et al. [65] correlation, which is given as:

04.2   δ   04.2     f i = .0 005 + []24 ()1− α gc = .0 005 + 406   (2-54)   D   and was developed prior to the correlation proposed by Bharathan & Wallis [11] that was given previously. Zabaras & Dukler [7] showed that reasonable agreement was obtained between their experimental data and this correlation and highlights the need for regime-specific correlations to be applied.

Govan et al. [16]

Govan et al. [16] conducted a series of experiments involving churn-annular flow using air-water in a 1.26-inch diameter tube at 19-psia. The results of these experiments were then compared to the models utilized by TRAC-PF1/MOD1 and RELAP5. TRAC-PF1/MOD1 utilizes the Wallis [4] correlation while RELAP5 uses the correlation of Bharathan et al. [65], which was given as Equation (2-54). Govan et al. [16] calculate the interfacial shear for the experiments using the measurements of pressure gradient and total liquid fraction. When reducing their experimental data, Govan et al. [16] neglect the interfacial velocity, as was also done by Zabaras & Dukler [7], but also neglect entrainment since, unlike in the experiments by Zabaras & Dukler [7], the film thickness was not measured.. The comparison of experimental and predicted results presented by Govan et al. [16] show that much better agreement was obtained using the Bharathan et al. [65] correlation, which only slightly overpredicted the data, relative to the Wallis [4] correlation, which severely underpredicted the data [16]. These results further highlight the need for regime- specific interfacial friction factor correlations to be incorporated and applied within COBRA-TF as part of the current study.

2.3.3.4. Counter-Current Flow

This section focuses on interfacial friction factors for the counter-current regime. Similar to the churn-annular regime, the work discussed in this section compared experimental results to several of the correlations that were developed for co-current upward annular flow and unsatisfactory agreement existed. As a result specific correlations were developed for this regime. Currently COBRA-TF does not apply a separate interfacial friction correlation for the counter-current regime, but rather the same correlation is used for all annular flow situations. Selecting a more suitable interfacial friction faction for this regime for inclusion in the proposed modeling package has been shown to improve the predictive capability of the code.

Abe et al. [6]

Abe et al. [6] assess the predictive capability of TRAC-PF1 for counter-current flow through code-to-data comparisons of experiments conducted by Bharathan et al. [65]. Three different combinations of interfacial and wall friction factor correlations were tested, including:

1) fi = Wallis [4], fw = homogeneous co-current flow (simulates TRAC-PF1)

2) fi = Wallis [4], fw = 0.005

3) fi = Bharathan & Wallis [11], fw = 0.005

Abe et al. [6] show that all three sets of correlations overestimate the downward flow of water, underestimate the pressure gradient, and more importantly do not capture the overall qualitative trend of the data. Moreover, the results indicate that the interfacial factor is underpredicted by the Wallis [4] correlation and overpredicted by the Bharathan & Wallis [11] correlation. [6]

Abe et al. [6] suggest based on these findings that in counter-current flow the interfacial and wall friction factors cannot be determined independently, but rather must be determined simultaneously by considering a mutual relationship between the two shear stresses. Abe et al. [6] propose a simple analytical model that considers the radial variation in the shear stress and velocity profiles using an eddy viscosity. Ultimately the model utilizes the measured superficial velocities (liquid and vapor) and the measure pressure gradient and then iteratively solves for the corresponding values of interfacial and wall shear, void fraction, and eddy viscosity. Since this work focused on two-fluid codes the model assumes no entrainment; however, prior to reaching the flooding point a minimal amount of entrainment exists so this assumption may be reasonable and the resulting correlation may still be applicable to three-field analysis environments.

The results predicted by this newly proposed model showed favorable agreement with the counter-current flow data of Bharathan et al. [65] for liquid holdup, and Zabaras & Dukler [7] for wall shear stress. Some other important observations made by Abe et al. [6] include:

1) The wall friction factor predicted by the proposed model is much larger than that predicted by the current approaches that were tested. [6] 2) The results from the proposed model show the eddy viscosity to increase with increasing film thickness and decreasing tube diameter. This result is consistent with the qualitative observation made by several researchers that disturbance wave activity increases with relative film thickness. [6] 3) It was shown that at low gas velocities the liquid profile predicted by the proposed model was rather flat and the local velocity at the interface was nearly zero. At higher velocities the local velocity at the interface became significant and the velocity profile became peaked. [6]

Based on the results of their proposed analytical model Abe et al. [6] suggest new correlations for the interfacial and wall friction factors in counter-current flow:

* 27.1 * − 37.0 f i = .0 005 + (δ ) (D ) (2-55) and:

300 f w = (2-56) Re l where the standard definition of the liquid Reynolds number based on diameter is used. Abe et al. [6] state these correlations are valid for counter-current flow situations involving water downflow with dimensionless superficial vapor velocities between 0.11 and 0.79, dimensionless superficial liquid velocities between 0.0 and 0.15, and tube diameters between 1 and 6-inches. [6]

2.3.5. Droplet Drag

The characterization of dispersed droplets is important to the modeling and prediction of momentum and heat and mass transfer in annular two-phase flow [31]. The droplet drag model

currently applied by COBRA-TF is based on the work by Ishii () that assumes a single droplet in an infinite vapor medium is analogous to a single bubble in an infinite liquid field [35]. This same model is used for the dispersed flow regime present downstream of an annular mist regime. The droplet diameter is calculated using Equation (2-4) and the interfacial area of the dispersed droplets is calculated Equation (2-5). Meanwhile, the droplet interfacial area density is obtained from the solution of the droplet interfacial area transport equation as described in Section 2.1.1.2. The interfacial shear stress definition applied by COBRA-TF between the vapor and dispersed droplet fields is given as:

1 τ = C ρ U U (2-57) i,ve 8 d v ve ve where:

U ve = U v −U e (2-58)

As mentioned previously the form of the momentum equations applied by COBRA-TF utilizes an interfacial drag coefficient, which between the vapor and dispersed droplet fields is defined in the axial direction as:

τ  A   A   i,ve  i,d  1  i,d  K ve =    = Cd ρv U ve   (2-59)  U ve  ∆z  8  ∆z 

Substitution of Equation (2-5) into the above equation yields a final expression of:

1  6α A   e x  Kve = Cd ρv U ve   (2-60) 8  Dd 

Currently the droplet drag coefficient is calculated by as:

 45.0  Cd = max  24 75.0 (2-61)  ()0.1 + 1.0 Re d ,mod Re d ,mod where the modified Reynolds number is calculated as:

Dd ρv U v −U e Re d ,mod = , (2-62) µv,m using a mixture viscosity, which is defined as:

Cµ µv,m = µvα v , (2-63) with:

(µl + 4.0 µv ) Cµ = − 5.2 (2-64) µl + µv

It was suggested by Ishii () that using an appropriately defined mixture viscosity when calculating the Reynolds number allows for the drag coefficient of a single particle in a multi-particle system to be calculated using the same expression that has been obtained for of a single particle in an infinite medium. The mixture viscosity accounts for the increased resistance experienced by a particle moving within a multi-particle system relative to a single particle in an infinite medium because the particle must deform along with the fluid and neighboring particles. [35]

Lee & No [66] implemented a three-field model for annular flow based on the physical models of COBRA-TF was implemented into the one-dimensional components of TRAC-M/F90. The implementation of the model for co-current situations and the selection of the constitutive relationships are discussed. An important finding by Lee & No [66] came from a sensitivity study that was conducted and showed that the droplet diameter and droplet drag had a negligible

effect on the calculated pressure drop and entrained mass flow rate. In any event, Lee & No [66] used droplet drag coefficient suggested by Clift & Gauvin (1970), which is given as:

24 .0 687 42.0 Cd = ( 0.1 + 15.0 Re d ,mod )+ 4 − 16.1 (2-65) Re d ,mod 1+ 25.4 x10 Re d ,mod

It can be seen that the first term in this equation is similar to the expression for droplet drag coefficient that is currently used by COBRA-TF given in Equation (2-61); however, this model includes a second term, which represents the fully turbulent situation, and provides a smooth transition between the regimes. The applicable range of this correlation is for Reynolds numbers less than 3x10 5. A comparison of Equations (2-61) and (2-65) as a function of Reynolds number is provided in Figure 2-6. The features of Equation (2-65) represent an improvement over the currently used expression and as a result this expression will be utilized in the proposed modeling package presented in Chapter 4.

3 1× 10 Baseline COBRA-TFCurrent Model COBRA-IE – Eqn. (2-61) Model Clift & Gauvin Correlation Clift & Gauvin (1970) – Eqn. (2-65) 100

1 DropletDrag Coefficient (Cd)

0.1 3 4 5 6 0.1 1 10 100 1× 10 1× 10 1× 10 1× 10

Droplet Reynolds Number (Re) Figure 2-6: Comparison of Droplet Drag Coefficient Expressions.

2.3.6. Conclusions

The calculation of the pressure gradient in annular two-phase flow requires accurate predictions of both the interfacial and wall shear stresses. The modeling of interfacial drag is also of significant importance to the entrainment process since it dictates the flow rates, void fractions, and predicted flow regimes. Therefore, an appropriate interfacial drag model is a prerequisite for the prediction of entrainment phenomena.

The works outlined in this section have highlighted several issues. First, the inability to isolate the interfacial wave effect from the droplet drag and momentum exchange processes can cause these effects to be lumped into the calculated interfacial friction factors. This can limit the applicability of the available interfacial friction factor correlations for three-field modeling approaches where models are already included to explicitly consider the effects of droplets. Similarly, the inadequacy of the assumptions used in the data reduction process, such as neglecting the interface velocity, can lead to considerable errors.

Second, the interfacial structure is markedly different between the various annular flow regimes and as a result it is doubtful that the same phenomenological models are applicable in each regime. Meanwhile, with the exception of the model proposed by Hurlburt et al. [55], the interfacial friction factors that are currently available to quantify the interfacial shear in annular flow are highly empirical, relying primarily on dimensionless parameters, and extending these correlations beyond the range which they were developed, especially to different regimes, can lead to significant errors. Additionally, comparisons of available steam-water experimental data has indicated significant differences from the results predicted by air-water based interfacial fraction factor correlations that have been traditionally used by transient safety analysis codes within the nuclear industry. In response to these factors one of the primary objectives of the current study was to utilize regime-specific correlations based on steam-water data wherever possible; however, these types of models are not readily available in all cases, and more specifically for the churn-annular and counter-current flow regimes. Despite this fact it is believed that since regime-specific correlations were not utilized previously for these regimes the implementation of correlations based on air-water are a step in the right direction and will be

sufficient for the purposes of the current study since no steam-water correlations are available at this time.

Lastly, the current state-of-the-art for predicting interfacial shear in annular flow is the two-zone model that has been proposed by Hurlburt et al. [55]. This model reflects the physical structure of the flow by explicitly accounting for: 1) the increased vapor velocity seen by the wave crests, 2) the difference in velocities between the disturbance wave and base film substrate regions, and 3) the fraction of the surface occupied by disturbance waves, which is quantified using the intermittency. Overall this model provides a more accurate estimate of the interfacial shear, especially in small hydraulic diameter and high pressure situations.

The use of a shear model that is based on the physical structure of the flow then allows for entrainment rate models that are consistent with this structure to be developed. For example, coupling the intermittency and velocity of the disturbance waves predicted by the two-zone interfacial shear model proposed by Hurlburt et al. [55] with an estimate of wave frequency, which can be obtained using a correlation such as the one proposed by Sawant et al. [57], allows for the disturbance wave spacing and length of the wave to be estimated. Quantifying these parameters further characterizes the interfacial structure and aids in the development of a physics- based entrainment model. As a result of these advantages and the desire to utilize mechanistically based models the two-zone interfacial shear proposed by Hurlburt et al. [55] model will be implemented as part of the proposed modeling package that will be discussed in Chapter 4.

2.4. Review of Works Focused on Entrainment Phenomena

Entrainment, or atomization, is defined as mass transfer from the continuous liquid to the dispersed droplet field that is caused by the interfacial shear forces imparted by the vapor field on the interface. Entrainment can occur: 1) from a liquid film in both co-current and counter-current annular flow, 2) at the quench front during reflood, and 3) from falling films during top down reflood. The understanding of the physical phenomena leading to the entrainment of liquid from an interface by a gas flow is of considerable practical importance for the effective modeling of heat and mass transfer processes in two-phase flow [3]. The physics, and thus the amount and size of entrained droplets, varies for each situation. This work focuses primarily on the first mechanism listed. At this point it is important to distinguish between the entrainment rate, which refers to the rate at which this transfer occurs, from the from the entrained flow rate, which refers to the mass flow rate of the dispersed droplet field. These two notations are often used interchangeably within the open-literature, but in reality refer to two distinctly different quantities.

Contrary to entrainment, deposition or de-entrainment consists of mass transfer from the dispersed droplet to continuous liquid fields. It is typically assumed that deposition in annular flow occurs due to the turbulent diffusion within the gas core region that can impart a transverse velocity to entrained droplets and cause them to be deposited on the liquid film. More recently other theories have been proposed that suggest that describing droplet deposition as a diffusion- like process is an oversimplification because due to the large size of the droplets their motion is unaffected by the turbulence [26]. For example, Lopes & Dukler [26] have developed a deposition model based on droplet trajectory and inertia, where the momentum imparted to the drops is attributed to the atomization process. This approach was not pursued here due to the complexity of the proposed model, which does not make it amenable to implementation into COBRA-TF, but the results show remarkable agreement with the experimental data of Cousins & Hewitt [67]. Still other correlations have treated deposition and droplet coalescence in a manner similar to kinetic gas theory, where a mean free path between collisions and the probability of experiencing a collision after traveling some distance are quantified. This approach was utilized by Ambrosini et al. [68] when developing their drop size correlation. The droplet coalescence model currently used by COBRA-TF is designed to remove droplets when the packing fraction

becomes too large relative to the maximum theoretical packing fraction, which is calculated assuming a simple cubic close packed structure. Meanwhile, Jepson et al. [32] and Azzopardi et al. [69] both found the deposition mass transfer coefficient to decrease with deposition length. Jepson et al. [32] found this parameter to asymptote to a nearly constant value with length and suggests this is an indication that larger drops, which contain the majority of the liquid, preferentially deposit first due to direct impaction. Once the larger droplets have deposited then the smaller droplets deposit at a constant rate due to a diffusion-type method. It should be noted here that droplet deposition can also occur due to flow area changes or upon impact with structures within the flow path (i.e. spacer grids, tube banks) and COBRA-TF does contain models to account for these effects; however, deposition mechanisms of these types will not be discussed nor are they considered as part of the current study.

Previous works [70,71] have identified three different entrainment regions, including: 1) a geometry or injection method dependent entrance region, 2) a developing region and 3) a fully developed or annular flow equilibrium region. Both the entrained flow rate and the entrained fraction, which is defined as:

W W E = e = e (2-66) Wl,tot Wl +We have been found to vary significantly in the first two regions, but be relatively constant in the third. The manner in which the liquid is introduced into the flow duct affects the entrained fraction in, and the length of, the developing flow regions. For example, smooth injection through a porous sinter yields an initial entrained flow rate of zero, where the liquid film thins along the length of the flow path as liquid becomes entrained. However, at low liquid flow rates some finite distance, or critical length, may exist in this entrance region where no entrainment occurs from the interface as roll waves develop on the surface of the film [70]. This critical length has been found to decrease as the liquid flow rate is increased, and as a result some critical liquid flow rate may exist above which the critical length goes to zero [70]. On the other hand, if liquid is suddenly introduced as an axial jet, the initial entrained flow rate is greater than zero and the liquid film thickens along the length of the flow path. The experimental data for this method

of injection indicates a sharp decrease in the entrained flow rate just downstream of the injection location, which is most likely an indication that the deposition mechanism in this region is not controlled by diffusion, but rather is controlled by the initial droplet momentum. As a result of these differences, it generally requires a longer distance for annular flow equilibrium conditions to be attained for a porous sinter versus droplet injection situation; however, once annular flow equilibrium conditions are reached the film thickness remains constant and no difference should exist in the entrained fraction for a given set of flow rates regardless of the method which the liquid is introduced [5].

This concept of equilibrium in annular flow has been proposed, where the axial rate of change of the entrained fraction tends toward zero, corresponding to a situation where the entrainment and deposition rates are equal. However, even for adiabatic air-water situations it is unlikely that annular flow equilibrium conditions are ever attained for vertical systems since the axial pressure gradient along the flow path leads to a continual expansion of the vapor, which increases the vapor velocity and interfacial shear stress [5]. This is commonly referred to as the gas expansion effect and is most important: a) in smaller flow ducts [5] and b) at low pressures [72]. These to situations tend to exhibit the largest relative change in pressure over the length of the test section. In the case of steam-water conditions it is even less likely that annular flow equilibrium conditions will be attained since the pressure gradient also causes phase change and mass transfer even in adiabatic experiments due to the changing saturation temperature. Despite these concerns Hewitt & Hall-Taylor [5] still suggest the concept of annular flow equilibrium is still worth retaining since it represents a “potential value” to which the system is tending.

The amount of entrained liquid present in annular flow can be calculated in two ways. The first is through direct correlation of the entrained fraction. Meanwhile, the second requires individual predictions of the entrainment and deposition rates coupled with the continuity equations for the dispersed droplet and liquid film fields. Both the entrained fraction and the entrained liquid flow rate, present within the flow represents the integral effects of the two rate processes and represents a convenient parameter to correlate since it can be obtained directly from experimental measurement. Other advantages of entrained fraction correlations are that they provide information on the amount of droplet flow in terms of macroscopic variables and applying such a correlation within a two-field analysis environment allows the effects of entrained droplets to be

considered in some regards even though this field is not modeled explicitly [71]. The entrained fraction correlation proposed by Ishii & Mishima [72], which predicts annular flow equilibrium values, is currently the most widely used by two-field codes, such as RELAP5-3D [44], and is given as:

−7 25.1 25.0 E∞ = tanh [ 25.7 x10 We e Re l ] (2-67) where the effective Weber number for entrainment is defined as [72]:

1 ρ j 2 D   3 v v H  ∆ρ  We e =   (2-68) σ  ρv 

However, two issues exist with both this, and most other, entrained fraction correlations that have been proposed. First, the data used to develop them was primarily collected using air-water at low pressures and are not reliable outside of these conditions, particularly for steam-water applications. For example, work by Okawa et al. [73] has shown the accuracy of the Ishii & Mishima [72] correlation decreases for air-water situations in larger diameter tubes, while it underpredicts steam-water data for low entrained fractions and overpredicts the steam-water data for high entrained fractions. Second, the correlated parameter is typically measured for quasi- equilibrium, or fully-developed, flow conditions (i.e. several hundred L/D’s downstream of the liquid injection point) and therefore are not applicable in developing flow or transient situations. While some correction factors have been proposed to consider developing flow effects [68,72], they are specific to a given liquid injection technique. Regardless, these correlations are developed from pointwise measurements of entrained fraction and do not allow for the convection of drops created somewhere else to be considered.

These deficiencies highlight the desire to individually calculate the entrainment and deposition rates. This approach provides a more mechanistic calculation, representing the physical transport processes occurring at the interface, but also should be able to implicitly capture developing or transient flow effects, as well as the entrainment inception and suppression criterion, assuming the physics and parameters controlling the processes are accurately represented. However, the

difficulty with this approach arises from the inability to develop appropriate constitutive relationships for these rates. A mass balance on the dispersed droplet field assuming adiabatic conditions and no mass transfer due to phase change yields:

z We ()z =We + ()S E′ − S D′ dz (2-69) z=0 ∫0

It can be seen from this equation that even if the inlet condition and axial distribution of entrained mass flow rate are known the expression still contains two unknowns.

In the case of deposition this issue can be circumvented using a double extraction technique, where the liquid film through a porous sinter and then a new liquid film to be created on the surface. As long as the conditions of the film remain below that required for the onset of entrainment this newly established film is a result of only the deposition mechanism and subsequent extraction of this film allows for a correlation to be developed. A similar unidirectional experiment cannot be conducted to isolate the entrainment process since it is not possible to extract droplets without causing a significant disturbance to the flow [25]. Instead, most entrainment rate correlations that have been proposed were developed from deposition rate measurements within the annular flow equilibrium region by equating the two rates. However, it can be shown that the entrained flow rate within the annular flow equilibrium region is independent of the entrainment and deposition rates within this same region. Rewriting Equation (2-69) to independently consider developing and annular flow equilibrium regions yields:

z z W z > z =W + ∞ S ′ − S′ dz + S′ − S ′ dz (2-70) e ()∞ e z=0 ∫ ()()E D ∫ E D 0 z∞ and then simplifying based on the definition of annular flow equilibrium yields:

z∞ We,∞ = We + ()S E′ − S D′ dz (2-71) z=0 ∫0

Equation (2-71) indicates that the entrained flow rate in the annular flow equilibrium region is dependent only the entrained flow rate at the inlet and the entrainment and deposition rates within

the developing flow region, but the extension of the available entrainment and deposition rate correlations, which are based on annular flow equilibrium conditions, to developing flow situations has not been verified. Additionally, unlike the available deposition rate models, which are based on the assumption that this rate is controlled by a diffusion mechanism, most of the entrainment rate correlations currently available are not based on the physics of the mechanisms assumed to be causing entrainment, such as interfacial instability, but rather rely on dimensionless parameters. In general, the inadequacy of the correlations for entrainment and deposition rates currently represents severe limitation of annular flow modeling [48].

This section provides a review several works that are focused on characterizing entrainment and droplet phenomena in the regimes of interest. The amount and size of entrained droplets that are generated in annular flow is dependent on whether the flow is co-current upward, churn-annular, or counter-current because the entrainment mechanism and resulting drop sizes generated is different for each of these regimes. Entrainment mechanisms and their region of importance, models for entrainment and deposition rates, and correlations for entrained drop sizes are all discussed in the following sections.

2.4.1. Entrainment Mechanisms

Hydrodynamic and surface tension forces govern the motion and deformation of the interface and extreme deformation results in the breakup of a portion of the wave crest causing droplets to be formed [3]. In general entrainment occurs when the interfacial shear overcomes the restraining force of surface tension. The interfacial shear is proportional to the roughness of the interface and the driving force is the relative velocity between the vapor and film fields [72]. The experimental data of Hewitt & Whalley [28], Jepson et al. [32], and Ishii & Mishima [72] appear to confirm this mechanism since the entrained fraction was observed to increase with increasing interfacial shear and decreasing surface tension.

The interface in two-phase flow is extremely complex and can lead to a variety of different entrainment mechanisms. Ishii & Grolmes [3] indicate five entrainment mechanisms exist in annular flow, including: 1) bag breakup or wave undercut, 2) roll wave, 3) liquid bridge breakup,

4) bubble breakthrough , and 5) droplet impingement, but only the first three listed yield significant amounts of entrainment such that they should be modeled explicitly.

According to Tatterson et al. [40], and based on the observations of Woodmansee & Hanratty [56], a Kelvin-Helmholtz type instability (i.e. the imbalance of the suction force caused by the pressure variation over the wave crest and the component of the surface tension force that acts normal to the wall) is responsible for entrainment by the bag breakup mechanism. The high speed motion pictures of Woodmansee & Hanratty [56] indicate that for conditions near the onset of entrainment the mechanism for entrainment is the removal of small wavelets, or ripples, that are superimposed on the larger disturbance, or roll, wave structures. Taylor (1941) previously suspected these ripples were three-dimensional waves that would be “plucked” from the interface, but these experiments suggest the ripples are best described as broad-crested or two-dimensional. In any event, atomization occurs when one of these ripples is suddenly accelerated, moves toward the front of the roll wave, and is then lifted by the air stream with the base still attached to the liquid film. As the wavelet deforms the radius of curvature of the wavelet decreases and as a result the vertical component of the surface tension force increases, as shown in Figure 2-7(a). If the surface tension force is large enough it can inhibit further distortion of the wave crest, otherwise wave deformation will continue until a critical or unstable condition exists where the surface tension force acts entirely in the vertical direction, as shown in Figure 2-7(b). Further distortion of the wavelet beyond this point causes the surface tension force to decrease, as shown in Figure 2-7(c) and droplet entrainment to occur by the stretched ligament being blown into an arc and then rupturing in a number of places [56]. Woodmansee & Hanratty [56] observed as many as ten to twenty droplets can be created by this process. The drop size correlation developed by Tatterson et al. [40], which will be discussed later, is based on this mechanism. The remaining portion of the unstable wave that does not become entrained lies down and is swept into the disturbance wave.

Figure 2-7: Droplet Entrainment by the Kelvin-Helmholtz Lifting Mechanism [40].

On the other hand, at larger gas velocities the roll wave, or stripping, entrainment mechanism exists where atomization occurs by shearing of liquid off the wave crests of upward propagating roll waves due to the shear stress exerted by the faster moving vapor field on the liquid interface [40]. The two possible forces resisting the entrainment of droplets from an interface by this mechanism are the surface tension and internal viscous shear [74]. The entrainment inception correlations proposed by Ishii & Grolmes [3] were developed by equating the interfacial shear and surface tension forces; however, Fore [74] argues that since the disturbance waves have a large length to height ratio the radius of curvature of the interface is also large, which minimizes the effect of surface tension such that momentum required to break through the interface is small. As a result Fore [74] proposed an entrainment rate model where the internal shear stress is the primary force resisting droplet formation. Fore [74] suggests that for entrainment to occur the fluid at the crest of the wave must travel faster than the bulk average velocity of the wave and for this to occur the opposing force of internal shear stress becomes negligible at some point within the wave. The liquid located between the interface and the location of minimal viscous shear can become entrained and this region can be treated as a “boundary layer”. A visual depiction of the boundary layer stripping mechanism proposed by Fore [74] is provided in Figure 2-8.

Figure 2-8: Droplet Entrainment by the Boundary Layer Stripping Mechanism [74].

In situations of counter-current or churn-turbulent flow the situation is further complicated since near the flooding point the interface is extremely unstable and large amplitude waves appear. These waves can be either: a) carried upwards by the gas flow, b) buildup and eventually be broken up causing entrainment by a wave undercutting mechanism, or c) in smaller flow ducts extend to form a liquid bridge, as shown in Figure 2-9, where the faster moving gas flow then breaks this bridge and entrains large amounts of liquid. [3]

Figure 2-9: Droplet Entrainment by the Liquid Bridge Breakup Mechanism [3].

Another entrainment mechanism exists where droplets are created when bubbles, generated either at the heated surface or due to the frothy nature of the wave front, break through the interface. This mechanism was observed by Woodmansee & Hanratty [56], but is often a small contributor relative to the other mechanisms and therefore is not accounted for explicitly. Similarly, Nigmatulin et al. [75] has shown that some drops are created by the impingement of drops at the interface. Experiments were conducted with film flow rates below the entrainment inception criterion such that no entrainment was observed. Then, solid spheres were injected along the tube centerline and entrained droplets were observed following deposition of the sphere on the film [75]. This effect was also observed visually by Woodmansee & Hanratty [56] in their experiments following the deposition of a newly generated droplet. However, given this effect is small it is also not accounted for explicitly.

In addition to the different entrainment mechanisms there has also been some discussion related to the resulting drop sizes observed in annular flow. For example, Kataoka et al. [60] suggest

experimental evidence indicates that the drops present in annular dispersed flow are too small to be generated by the droplet disintegration mechanism based on critical Weber number criterion using the relative velocity between the gas and the drops. Kataoka et al. [60] go on to suggest that this observation indicates a majority of the droplets present in annular dispersed flow are generated at the time of entrainment and not during flight within the gas core region, which also implies that the governing parameter for droplet size is the relative velocity between the gas core and the liquid film. Meanwhile, the correlations of Azzopardi et al. [69], Lopes & Dukler [26], and Kocamustafaogullari et al. [76] suggest droplets are initially entrained with diameters greater than that allowable by the maximum stable drop size. Kocamustafaogullari et al. [76] suggest that the local droplet size measurements of Lopes & Dukler [26], which showed the average drop size to decrease towards the center of the tube, are an indication that the droplet size is not determined at the time of entrainment, but rather is controlled by the mechanics of gas-droplet interaction because as the droplet enters the stream it begins to oscillate and breakup. As a result the correlations proposed by each of these researchers is based on the interaction between the droplets and the gas phase turbulence assuming that droplet breakup occurs when the natural frequency of the oscillation of the drop is equal to the characteristic frequency of turbulence. These two suggestions appear to be contradictory, but in reality the difference could be attributed to a difference in entrainment mechanism. Azzopardi [15] suggests droplets generated by the wave undercutting mechanism produces larger drops then those generated by the roll wave mechanism. Therefore, droplet breakup may occur within the gas core for mechanisms such as wave undercut or liquid bridge disintegration if the initial drop sizes are larger than the maximum stable drop size.

In either case it is apparent that the amount and size of entrained droplets in annular flow is strongly dependent on whether the flow is co-current upward, churn-annular, churn-turbulent, or counter-current because the entrainment mechanism is different for each of these situations. Azzopardi along with several different colleagues [15,21,22,25,30,49,68,69], Ishii along with several different colleagues [3,60,71,72], as well as Dukler along with several different colleagues [24,26] have characterized the entrained drop sizes in annular flow. Section 2.4.3 will address this topic, but first the available models for calculating the entrainment and deposition rates will be discussed.

2.4.2. Entrainment and Deposition Rates

This section explores several entrainment and deposition rate models that have been proposed in the open-literature. Sections have been provided focused on deposition rate models (2.4.2.1), as well as entrainment rate models in co-current upward annular (2.4.2.2), churn-annular/churn- turbulent (2.4.2.3), and counter-current flow (2.4.2.4). Each section outlines the models currently used by COBRA-TF and then provides a brief review of several other models that have been more recently proposed in the open-literature as a means of exploring alternative approaches to improve the predictive capability of the code in these regimes. Summaries of these models, as well as others that have been proposed, has also been provided by Okawa et al. [73], Stevanovic & Studovic [77], and several others. 2.4.2.1. Deposition Models

As mentioned previously deposition in annular flow is typically assumed to occur due to the turbulent diffusion [48,64,67,70], but other theories have been proposed more recently based on droplet trajectory and inertia [26] as well as kinetic gas theory [68]. This section briefly describes the deposition rate model currently applied by COBRA-TF and some alternative models available in the open-literature.

2.4.2.1.1. Current COBRA-TF Deposition Model

COBRA-TF only models deposition in computational cells experiencing pre-CHF conditions (i.e. where the surface temperatures of all structures located within that computational cell are less than that given by Equation (2-6)). The current deposition rate model used by the code was suggested by Cousins et al. [70] and is based the diffusion mechanism. The model is given as:

S D′ = k DCA s (2-72) where the mean droplet concentration in the gas core is calculated as:

α ρ α ρ C = e l = e l (2-73) α e +α v 1−α l

The deposition mass transfer coefficient was suggested by Whalley et al. [78] to be a function of only surface tension and is given as:

 ft  k = []max ().3 0491 x10 12 σ .5 3054 12, .491 σ .0 8968   (2-74) D  sec 

This correlation considered steam-water data ranging between 20 and 2600-psia. In addition to impacting the distribution of liquid and the mass, momentum, and energy exchange between the two liquid fields, the calculated deposition rate also influences the rate of change of the interfacial area for the entrained droplet field, as shown in Equation (2-3).

2.4.2.1.2. Alternative Deposition Models

This section briefly describes some of the deposition rate models that have been proposed based on the diffusion mechanism, including those by Hewitt & Govan [48] and Sugawara [64], since these types of models are currently the most amenable to implementation into COBRA-TF and are therefore the most relevant to the current study.

Hewitt & Govan [48]

Hewitt & Govan [48] developed a deposition rate correlation considering a range of fluids including steam-water data ranging from 580 to 1600-psia. The deposition rate correlation is based on the diffusion mechanism even though Hewitt & Govan [48] suggest that the deposition rate may not be proportional to droplet concentration. The proposed correlation is given as:

 − 65.0  σ  C ρv  kD = .0 182 min  ,0.1    (2-75) ρv DH   3.0   where the droplet concentration is calculated assuming homogenous flow in the gas core. It was shown by Hewitt & Govan [48] that this correlation captured the overall trend, but a considerable amount of scatter still exists when these correlations are plotted against the experimental data. Later work by Okawa et al. [73] showed that using this correlation, along with the entrainment rate correlation proposed in the same work, provides reasonable agreement for a majority of the data they considered, but overpredicts the entrained fraction for air-water in small tubes and underpredicts the entrained fraction for steam-water at low gas flow rates. Meanwhile, Fore et al. [51] proposed that better agreement could be obtained with their low pressure air-water and high pressure nitrogen-water test data if the correlation was modified to be:

 − 65.0  σ  C ρv  kD = 10.0 min  ,0.1    (2-76) ρv DH   4.0   but again significant scatter still existed. In addition to the scatter, the primary drawback of either form of the correlation is that, as pointed out by Holowach [33,34], the inclusion of a hydraulic diameter dependence in a model that assumes droplet deposition occurs by a diffusion based mechanism is unclear and highlights the notion that in reality droplet deposition may not be governed by this mechanism.

Sugawara [64]

Sugawara [64] presents a droplet deposition model that is based on the turbulent diffusivity of entrained droplets in the vapor core using an analogy between heat and mass transfer, which yields:

ρ −2 k = 0.9 x10 −3U v Re − 2.0 Sc 3 (2-77) D v C v where the droplet concentration is calculated using Equation (2-73). This correlation was based on available air-water data at low pressures and steam-water data at high pressures.

2.4.2.2. Entrainment Models for Co-Current Upward Annular Flow

This section reviews several of the models that have been proposed to calculate entrainment rate within the co-current upward annular regime, including those by Würtz [79], which is currently used by COBRA-TF, as well as Sugawara [64], Holowach [33,34], and Fore [74].

2.4.2.2.1. Current COBRA-TF Entrainment Model

COBRA-TF currently calculates mass flux of entraining liquid from the liquid film in co-current annular flow using the Würtz [79] correlation, which is given as:

 k τ U µ  lbm   s i,vl  v l   (2-78) S E = 41.0    2   σ  σ  ft s 

The interfacial shear stress in this expression is calculated by COBRA-TF using Equation (2-30) and the second dimensionless parameter in this expression represents a dimensionless gas velocity. Meanwhile, an equivalent sand grain roughness expression, which represents the length scale for the entrainment force due to surface tension, was developed by Würtz [79] using a film flow model that applied Prandtl’s turbulent two-layer model for the velocity profile in the film and a turbulent, logarithmic profile for completely rough walls for the velocity profile in the gas core. The result was a general correlation between the equivalent sand grain roughness and the film thickness for film thicknesses less than 800-µm, which is given as:

3 −1 2 6 −2 3 9 −3 4 k s = 57.0 δ + ( 25.6 x10 ft )δ − ( 56.3 x10 ft )δ + ( .1 5739 x10 ft )δ (2-79)

It should be noted that, similar to the deposition rate, the entrainment rate calculated by COBRA- TF also influences the rate of change of the interfacial area for the entrained droplet field, as shown in Equation (2-3).

The correlation proposed by Würtz [79] was developed considering the low-pressure, air-water data collected by Whalley et al. [80] and the high-pressure, steam-water data collected by Würtz [79]. This correlation was developed by assuming annular flow equilibrium conditions existed in these experiments at the measurement location so that a deposition model could be used to calculate the “experimental entrainment rate” and then correlated against, where the result was Equation (2-78). The “experimental entrainment rate” was calculated using a diffusion based deposition model assuming: 1) homogenous flow existed in the gas core when calculating the droplet concentration, which according to the results presented by Fore & Dukler [24] and Lopes & Dukler [26] that were discussed in Section 2.3.2 may not be appropriate, and 2) the deposition mass transfer coefficient could be estimated using the correlation proposed by Whalley et al. [78] given previously as Equation (2-74). This approach couples the resulting entrainment model to the values of the deposition mass transfer coefficient predicted by this model and explains why COBRA-TF currently uses these two models in conjunction with one another. Additionally, the experimental interfacial shear stress, which was used in correlating the data, was calculated by Würtz [79] from the measured total pressure gradient by: a) estimating of the void fraction using the Bankoff-Jones formula, b) estimating the film thickness using this calculated void fraction, which neglects the entrained void fraction contribution, and c) correcting the total pressure gradient for the effects of gravity and acceleration, but neglecting the effects of momentum exchange due to droplet exchange and the gas expansion effect, which may not be valid based on the discussion in Sections 2.3.4 and 2.4, respectively.

The results of the correlation were then compared to the high-pressure, steam-water data collected by Keeys et al. [81] and Singh [82] with reasonable agreement; however, it should be noted that the model overestimated the data collected by Würtz [79] at low gas velocities (i.e. thick films and low qualities), but the same behavior was not observed when this correlation was used to

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predict the other data sets. Würtz [79] suggested this occurred because the actual film flow rate from these experiments was underestimated at these conditions. Additionally, this correlation does not directly consider pressure effects, through a density ratio, or the effect of mass flow rate. It is expected that these two variables have a significant effect on the entrainment phenomena. Later works by Sugawara [64], Holowach [33], and some preliminary results of this current work have shown this correlation to severely underpredict the low pressure stream-water data collected by Hewitt & Pulling [83].

2.4.2.2.2. Alternative Co-Current Upward Annular Flow Entrainment Model

As a result of the concerns discussed in the previous section several other entrainment rate models have been proposed for this regime. The model of Sugawara [64], which tried to improve the predictive capability of the Wϋrtz [79] correlation at lower pressures through of inclusion of several correction factors, will be discussed first, but the primary focus will be placed on the mechanistic-based models proposed by Holowach [33,34] and Fore [74], both of which served as the basis for the entrainment models developed in the current study.

Sugawara [64]

A droplet entrainment model based on the roll wave mechanism is presented using a force balance consisting of the disruptive force of the interfacial shear stress and the restraining force of the surface tension. This correlation uses the same dimensionless parameters as Würtz [79], but also includes a density ratio to account for the pressure effect. The correlation is given as:

4.0 τ λ U µ  ρ   lbm   i,vl  v l  l    (2-80) S E = .0 219      2   σ  σ  ρv   ft s  where:

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k for Re > 1x10 5 λ =  s v (2-81)  5 ks []2.136log 10 ()Re v − 68.9 for Re v ≤ 1x10

and ‘ks‘ is the hydrodynamic equivalent wave roughness that was suggested by Würtz [79]. The logarithmic correction at low Reynolds numbers accounts for the differences between results obtained in a series of low-pressure, steam-water experiments conducted by Hewitt & Pulling [83] and the original expression proposed by Würtz [79]. Comparisons indicated that the hydrodynamic equivalent wave height decreases with decreasing gas Reynolds number. Holowach [33] suggests the gas Reynolds number curve fit had to be applied as a means of accounting for gravity since it was neglected in the force balance; however, it was shown by Lopes & Dukler [26], as well as later in the current study, that the gravitational force is negligibly small relative to the surface tension and drag contributions.

Code-to-data comparisons of the liquid film and entrainment flow rates were made using the Film Dryout Analysis code in Subchannels (FIDAS), which provides a three-field representation of two-phase flow. The adiabatic annular flow tests conducted by Würtz [79], Keeys et al. [81], Hewitt & Pulling [83], and Yanai [84], were considered and the results were favorable [64]. However, later work by Okawa et al. [73] showed this correlation, coupled with the deposition rate correlation proposed in the same work, performed well for high pressure steam-water data, but in general performed very poorly for air-water conditions.

Holowach [33,34]

Holowach [33,34] developed a physical model for droplet entrainment in annular flow based on the recommendation of Woodmansee & Hanratty [56] that entrainment occurs due to a Kelvin- Helmholtz type instability. The proposed model is based on:

1) a stability analysis, which is used to calculate wavelength and wave velocity, 2) a shear flow model, which is used to calculate wave amplitude, and

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3) a force balance, which is used to calculate the maximum volume of liquid that can become entrained from a single wave crest.

Holowach [33,34] assumed a three-dimensional sine wave structure based on the observations of Woodmansee & Hanratty [56], that smaller ripple waves, and not the large disturbance waves, were the primary source of entrainment in annular flow. The stability analysis, which relates the normal stresses and the surface tension acting at the interface, was conducted using perturbation theory to determine the critical wavelength at which the interface becomes unstable for a given set of fluid conditions (e.g. pressure, mean film thickness, and field velocities). The slope of the interface was assumed to be small, which allows the fluid motion equations to be linearized and provided a simplified solution. Once the wavelength is known the corresponding wave velocity could be determined.

Holowach [33,34] then employed the shear flow model proposed by Ishii & Grolmes [3] to calculate the wave amplitude, which is given as:

f (N µ )µlU l ε w = (2-82) τ i,vl

However, instead of using the code calculated value for the continuous liquid field velocity in this expression, Holowach [33,34] used a value obtained by preserving the interfacial shear stress from both sides of the interface. The code calculated value, obtained using the Whalley & Hewitt [28] interfacial friction factor, was used for the vapor side and equated to the result predicted for the liquid side, which was obtained using the Hughmark interfacial friction factor that was given by Ishii & Grolmes [3]. While this approach is valid if the liquid velocity is unknown, employing this approach here results in inconsistent velocities for the continuous liquid field being used by the code and the entrainment model.

The model assumes the maximum amount of liquid that can be entrained from a wave crest occurs when the vector sum of drag force, the gravitational force, and the surface tension forces on the wave in the direction of the flow equals zero. The force balance was performed on a single

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wave crest to determine the point on the wave where the drag force would exceed the restraining forces of surface tension and gravity. Expressions were obtained for the effective drag area and the maximum volume of the wave crest that can be entrained in terms of a correlating dimension. The force balance could then be iteratively solved for this dimension. It should be noted that, while Holowach [33,34] assumed the sine wave structure to calculate the quantities of interest (i.e. frontal area of the wave, volume of the wave crest, etc.) it was assumed for the purposes of the force balance that the wave deformed to reflect a roll wave structure just prior to breaking up. This prevented the surface tension forces on the front and backsides of the wave from canceling out one another as occurs for a symmetric wave structure.

Holowach [33,34] used the drag coefficient proposed by Kataoka et al. [60] when calculating the drag force for the wave crest, but two issues exist with doing this. First, the underlying assumption applied by Kataoka et al. [60] when developing their drag coefficient expression is that in annular flow the interfacial drag force per unit volume acting on the dispersed droplets is approximately equal to the interfacial shear force per unit volume acting on the liquid film. It is not clear that this is a valid assumption. Second, using an independent expression for the drag coefficient does not preserve the shear force between the code calculated value, which Holowach [33,34] obtained using the Whalley & Hewitt [28] interfacial friction factor, and that used by the entrainment model.

The model considered the presence of a homogeneous gas and droplet mixture in the gas core region, which results in a larger effective gas core density and impacts the drag force on the wave crest. Holowach [33,34] indicates that since momentum transfer occurs from the droplets to the film during deposition, which can increase the subsequent amount of entrainment from the film, it was important that a gas core, rather than vapor, density be used.

Holowach [33,34] then proposed an expression for the maximum entrainment rate based on the typical scaling analysis approach which is given as:

VE ρl N w ∆z S E′ ,max = (2-83) t E λ

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where it is suggested that the time scale for entrainment can be estimated as:

λ t E = (2-84) U gc −U l and the number of waves in the control volume is given as:

P N = w (2-85) w λ Equation (2-85) assumes that these waves are uniformly distributed over the entire surface of the liquid film and that the wavelength calculated from the stability analysis that was performed in the direction of the flow also governs the spacing of the waves circumferentially around the flow path. The calculated wavelength was also used to relate the length of the control volume to the length of the computational node, as shown in Equation (2-83) to scale the calculated entrainment rate for a single wave.

Lastly, Holowach [33,34] compared the results predicted by this model results to the steam-water experimental data of Hewitt & Pulling [83] and Keeys et al. [81] to determine a correlation between the actual entrainment rate in terms of the maximum entrainment rate as predicted by this theoretical model. Corrections for pressure, viscosity number, and film Reynolds number were included in the final expression, which is given as:

7  ρ  8   Re −1000  ′ 67.1 3  l  ′  l  S E = .0 0311 Re l N µ   S E,max min  ,0.1  (2-86)  ρv    3000 −1000 

The linear ramp was instituted to transition the entrainment rate to zero as the film moved from turbulent to laminar and Holowach [33,34] indicated such a ramp was needed to compensate for the use of the shear flow model, which is invalid in this region. The results of this work were then validated using a wider set of experimental data, including that collected by Würtz [79] and Singh [82], and it was shown that substantial improvement was obtained in the code-to-data predictions of both entrainment mass flow rate and pressure gradient using this model.

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This model provides a more mechanistic calculation than previously suggested models and is amenable to implementation in transient two-phase flow codes. The drawback of the correlation suggested by Holowach [33,34] is that since the correction factor on the entrainment was correlated iteratively by comparing the measured and predicted entrained mass flow rate at the outlet and is implicitly coupled to all the other constitutive relations used by the code. In reality, the proposed correlation for the relationship between the maximum and theoretical entrainment rate reflects the integral effect of the entrainment and deposition rates over the entire axial length of the test section; however, the accuracy of the predictions axially along the test section (i.e. in the developing, or non-equilibrium, flow region) was not confirmed. This effect will be explored in the current study using the experimental data collected by Cousins & Hewitt [67].

Fore [74]

As mentioned previously Fore [74] proposed an entrainment rate model where the internal shear stress, rather than surface tension, is the primary force resisting droplet formation by the roll wave mechanism. As gas flows over the wave a boundary layer forms in the liquid adjacent to the interface marked by a large velocity gradient at the interface and a negligible one at some point within the wave. Fore [74] suggests the liquid located between the interface and the point of minimal viscous shear can be treated as a “boundary layer”. It is proposed that the “boundary layer” extends from a stagnation point located along the interface on the windward side of the wave to the peak of the wave, increasing in thickness with distance. All the liquid within this “boundary layer” moves faster than the bulk average velocity of the wave such that at the peak of the wave this liquid is free to leave the wave, generating droplets of a size related to the thickness of the boundary layer. Therefore, the existence of such a stagnation point on the windward side of the wave is a prerequisite for entrainment to occur by this mechanism.

Fore [74] provides models to: 1) determine the location of the stagnation point (i.e. where the local fluid velocity is equal to the bulk average wave velocity), 2) calculate the thickness of the stripping layer at the wave peak, and 3) calculate the velocity profile in this layer. From these calculations the mass flow rate of liquid that can leave the wave can be calculated. Fore [74] included factors to account for the spacing and asymmetry of the waves. The primary difficulty

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with the model is that it requires estimates of: 1) the length of the wave, 2) the bulk average velocity of the wave, 3) the interfacial shear stress, and 4) the time variation in both wave height and wall shear stress. When proposing this model Fore [74] obtained estimates of these parameters from the experiments conducted as part of the same work and also published in Fore & Dukler [23]; however, the generalization of this model requires estimates of these parameters. Such estimates were not readily available, but the more recently proposed two-zone interfacial shear model proposed by Hurlburt et al. [55] provides estimates of these parameters. As a result, the model proposed by Fore [74] should be considered for inclusion in the proposed modeling package.

2.4.2.3. Churn-Annular/ Churn-Turbulent Flow

COBRA-TF does not currently contain an explicit model to characterize entrainment generated by the liquid bridge breakup mechanism. Due to the difference controlling parameters and resulting drop sizes generated by this mechanism it is desired to include models to consider these phenomena in the proposed modeling package.

Liquid Bridge Breakup Model [85]

The entrainment rate due to liquid bridge breakup is given as:

S E′ ,LBB = ρlα l f Bη B Ax dz (2-87)

This model utilizes correlations that were recommended by Pilch & Erdman [85] for the breakup of isolated liquid drops that are suddenly exposed to a high-velocity flow field. The total dimensionless breakup time for this mechanism is based on a critical Weber number criterion, which refers to the minimum ratio of the disruptive hydrodynamic forces to the stabilizing surface tension force that is needed for acceleration induced fragmentation of a droplet to occur [85]. Meanwhile, the total breakup is referred to as the condition where no further fragmentation of

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droplets occurs and the correlation proposed by Pilch & Erdman [85] for the time to reach this state, assuming the flow field remains constant over this time scale, is given to be:

6(We −12 )− 25.0 for 12 ≤ We < 18  B B  25.0 tTB =  45.2 ()We B −12 for 18 ≤ We B ≤ 45 (2-88)  − 25.0 14 1. ()We B −12 for We B ≥ 45

The Weber number in this expression is based on the effective initial diameter of the continuous liquid phase and the velocities of the continuous vapor and liquid phases:

ρ D (U −U )2 We = v d ,max v l (2-89) B σ

The velocity of the continuous liquid field in this equation is neglected since the conditions near the flooding point where this mechanism is assumed to occur is characterized by oscillatory motion of the liquid phase. The drop size in this expression corresponds to the maximum stable drop size based on the Taylor instability criterion and is given as:

  σ  Dd ,max = max  .0 00328 ft ,min DH 2,  (2-90)   g∆ρ 

The breakup frequency is then given by:

(U v −U l ) ρv f B = (2-91) tTB Dd ,max ρl where again the liquid velocity in this expression is neglected. Meanwhile the breakup efficiency is assumed to be one minus a collision probability where the collision probability is the ratio of the actual to the maximum dispersed phase volume fraction. The maximum value of 0.52

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corresponds to a simple cubic close packed structure. The resulting expression for the breakup efficiency is given to be:

    1 α e 1 α e η B = 1−   = 1−   (2-92) α max α e + α v  52.0 α gc 

This model has been implemented as part of the proposed modeling package to explicitly calculate the entrainment rate by the liquid bridge breakup mechanism.

2.4.2.4. Counter-Current Flow

In a counter-current flow situation the entrainment rate is currently calculated by COBRA-TF as the difference between the actual film flow rate and the film flow rate associated with the critical liquid volume fraction. In other words, in a counter-current flow situation the film flow rate in excess of the critical value is assumed to become entrained. The resulting expression for the entrainment rate is:

S E′ ,exfilm = (α l − α l,crit )Wl (2-93) where critical liquid volume fraction in the expression is calculated using Equation (2-10).

Preliminary assessments conducted as part of the current work indicted that using this expression resulted in a severe overprediction of the entrainment present in this regime relative to experimental data collected by Dukler & Smith [43]. In reality a minimal amount of entrainment is observed for counter-current flow situations until the flooding point is reached [10,43]. Additionally, using this correlation, coupled with the severe overprediction in the interfacial drag between the vapor and film, resulted in an underprediction by COBRA-TF of the amount of liquid flow that could penetrate downwards as a film in counter-current flow situations relative to experimental data collected by Dukler & Smith [43]. The large entrainment rate, caused by the small value of the critical liquid volume fraction, resulted in nearly the entire liquid film

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becoming entrained and then transported upwards. As a result of these concerns an explicit three- field counter-current flow limitation (CCFL) model was developed and implemented as part of the current study. Included in this model was logic to deactivate this entrainment mechanism at elevations below the location wherever this newly developed model is applied.

2.4.3. Entrained Droplet Size and Size Distribution

The size and number of entrained droplets influences the interfacial area of this field and an accurate prediction of this quantity is needed because is dictates the interfacial drag of this field and enhanced heat transfer capability of this regime. Transient analysis codes, such as COBRA- TF, do not model individual drops, but rather rely on a characteristic drop size to represent distribution of drops sizes. Two common characteristic sizes include the volume mean diameter and the Sauter mean diameter. The volume median diameter is defined so that 50% of the droplets contained within the volume have diameters larger than the volume median diameter [40]. Meanwhile, the Sauter mean diameter corresponds to a diameter of a drop having the same volume-to-surface area ratio as the entire distribution and is given as:

6Vd Dd 32, = (2-94) Ai,d

Since the interfacial area and volume of liquid are important to the heat transfer predictions for dispersed flows it is desired to preserve these quantities and as a result COBRA-TF uses the Sauter mean diameter for the characteristic drop size.

Several difficulties exist in determining a characteristic dimension from experimental measurements and a brief discussion of this topic is provided here for completeness, but more detailed reviews of this subject, as well as the different drop size measurement techniques and their inherent issues, has been provided by Azzopardi [22,25], Fore & Dukler [24], and Fore et al. [31]. The primary difficulties include: 1) obtaining optical access to the gas core region without significantly disrupting the flow and 2) overcoming the statistical errors that can be introduced if:

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a) the sample size is not large enough, b) the measurement range of the technique used is insufficient, c) an inappropriate droplet size distribution is assumed, or d) droplets deviate from their assumed spherical shape, which tends to occur with increasing droplet size. In particular, Tatterson et al. [40] suggest based on their results that 90% of the dispersed liquid volume is carried by about 10% of the drops. Therefore, if several large drops are excluded, which can occur for a variety of reasons, this can result in the characteristic diameter being significantly underpredicted. Additionally, the laser diffraction measurement technique requires a drop size distribution to be assumed and typically the Rosin-Rammler function is used; however, the more recent measurements of Fore & Dukler [24] yielded a bimodal drop size distribution, which discounts the use single-mode functions.

More importantly, experiments measure the entrained drop size (i.e. size of droplets flowing in the gas core), while COBRA-TF and other transient analysis codes require a constitutive relation for the entraining drop size (i.e. size of droplets generated at the interface). Entrained drop size is influenced by droplet coalescence and breakup, while entraining drop size is governed by the entrainment mechanism. Since entraining droplet size is not measured, typically entrained drop size correlations are applied to predict entraining drop size; however, COBRA-TF includes explicit models for droplet coalescence and breakup that are then considered by the interfacial area transport equation for the dispersed field that was described in Section 2.1.1.2. Therefore, applying such correlations may cause these effects to be double accounted. Additionally, the resulting drop size correlations, like those provided for entrained fraction, are highly empirical and use macroscopic variables. These correlations provide static predictions of the drop size and, unlike the use of an interfacial area transport equation for the dispersed field, do not provide a means for convecting drops for one region to the next.

2.4.3.1. Experimental Observations

The purpose of this section is to briefly review some of the important qualitative findings of annular flow drop size experiments that have been conducted.

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Azzopardi [49]

Azzopardi [49] conducted a series of experiments to determine the effect of: a) vapor and liquid total flow rates, b) entrained flow rate or droplet concentration, c) liquid injection method (porous sinter and axial jet), and d) tube diameter (1.25 and 5-inches) on resulting drop sizes in co-current annular flow. The experiments were conducted at atmospheric pressure using air-water in vertically oriented test section with a length of roughly 13-feet. Film flow rates were measured by extracting the liquid film through a porous sinter and drop sizes were measured just downstream of the liquid film extraction using a laser-diffraction technique. [49]

The experimental results presented by Azzopardi [49] show the Sauter mean diameter to initially decrease for low liquid mass fluxes and then increase with both increasing liquid mass flux and decreasing vapor mass flux regardless of the tube diameter or method of liquid injection. The same qualitative trends were also observed by Jepson et al. [32], Azzopardi et al. [21] and Teixeira et al. [86]. The decrease in drop size with increasing gas velocity is consistent with the concept of a shear driven process creating the droplets. Also, the difference in behavior between low and high liquid mass flux is suggested by Azzopardi [49] to be a result of different mechanisms causing entrainment in these two regions.

For a given combination of phasic mass flow rates Azzopardi [49] found the Sauter mean diameter to be slightly smaller for the porous sinter method of liquid injection than the axial jet liquid injection. Meanwhile, the drop size was shown to increase slightly along the length of the test section. The former result suggests that the drop size is mainly determined at the time the droplet is generated, but the latter implies that some droplets do coalesce as they flow in the gas core region, causing larger droplets to remove smaller droplets, thus increasing the average size. Based on this finding Azzopardi [49] suggests that the drop size is dependent on the entrained liquid mass flux or droplet concentration instead of film thickness or film mass flow rate as has been suggested by other researchers [40,87].

Finally, it should be noted that Azzopardi [49] found the drop size to be independent of the tube diameter for a given vapor flow rate. Azzopardi [49] suggests that this is because entrainment is a localized process and the local curvature of the interface is much greater than the curvature of

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the tube such that the droplets are unaware of the tube size; however, this argument becomes increasingly inaccurate as the tube diameter decreases.

Fore & Dukler [24]

Fore & Dukler [24] used a laser-grating technique to obtain simultaneous measurements of drop size and velocity in co-current annular flow in a 2-inch diameter, vertically-oriented acrylic tube. Measurements were made using air and two liquids (water and a 50% glycerin-water mixture) to examine the effect of viscosity. The results showed the mean drop size to increase with both increasing liquid flow rate and increasing viscosity. Additionally, it was found the data collapsed reasonably well when Sauter mean diameter was plotted as a function of droplet concentration, which is consistent with the argument of Azzopardi [49].

Andreussi et al. [87]

Andreussi et al. [87] made measurements of drop size in co-current downward annular flow using air-water at an outlet pressure of 17.5-psia. A photographic technique, that allowed the simultaneous measurement of droplet size and velocity in a very small volume, was used in this work. The experimental data showed the volume median drop size to decrease with increasing gas velocity. The results are consistent in both tend and magnitude with those found in the co- current upward experiments by Cousins & Hewitt [67]. Meanwhile, the volume median drop diameters measured by Andreussi et al. [87] are about 60% less, and display the opposite trend (i.e. drop size increasing with gas velocity), than that suggested by the Tatterson et al. [40] correlation. Additionally, the results of Andreussi et al. [87] show the drop size to be a stronger function of gas and liquid flow rates than this correlation. Lastly, Andreussi et al. [87] found larger drops tended toward the center of the flow path, which contradicts the observation of Lopes & Dukler [26] from their co-current upward annular flow experiments. The differences in between co-current upflow and co-current downflow highlighted by these experiments are unavoidable because the film instability/flow reversal mechanism that can cause a breakdown of annular flow for upflow conditions are not possible for downflow conditions. The fact that

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gravity acts in the same direction as wave motion also leads to significant differences, such as smaller wave amplitudes and the delay in entrainment inception point to larger gas velocity conditions. [87]

2.4.3.2. Drop Size Correlations and Assessments

This section briefly examines several of the droplet size correlations that have been proposed and assessments of these correlations that have been conducted by various researchers. A more detailed review has been provided by Azzopardi [25]. It will be seen in this section that some conflicting suggestions have been made as to parameters that effect drop size and the applicable conditions for several of the correlations that have been proposed.

2.4.3.2.1. Current COBRA-TF Drop Size Correlation

The size of the entraining droplets is currently calculated in COBRA-TF using the expression developed by Tatterson et al. [40]. The analysis of Tatterson et al. [40] is based on the work of Taylor (1940), who studied the instability of a stationary viscous liquid of infinite depth interacting with a high velocity inviscid gas stream. Tatterson et al. [40] modified this approach to allow for a finite liquid depth and assumed that the droplets scale as the diameter of the fluid ligament removed from the wave crest, rather than the wavelength of the unstable wave. The correlation was then assumed to apply in annular dispersed flow situations, where a mobile interface exists, since the velocity of the vapor is much greater than that of the film. The proposed correlation for Sauter mean diameter, assuming an upper-limit log-normal distribution, is given as:

  5.0   D σ D = .0 0112  H  (2-95) E  f   s ρ U 2   2 v vl 

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where the friction factor in this expression is calculated using Equation (2-44). COBRA-TF then imposes a series of physical constraints on the predicted drop size so it does not become unphysically large.

DE = min [DE , DH 13.0, ft , Dlim ] (2-96)

Tatterson el al. [40] considered low pressure air-water data from three experiments, which used both tubes and non-circular ducts, as well as considering co-current upflow, co-current downflow, and horizontal flow situations. Tatterson et al. [40] concede that when the data are plotted using the expression given as Equation (2-95) some scatter does exist, but since each of the tests considered used different orientations, flow directions, geometries, and droplet measurement techniques, and because the error does not appear to be systematic in any one of these variables, the variability is assumed to be within experimental error. As a result they assume it was sufficient to determine the correlating constant by taking the arithmetic average of all the data considered.

The correlation proposed by Tatterson et al. [40] is nearly independent of liquid film flow rate and thus does not capture the increase in drop size that has been observed with increasing liquid flow rate [24,49]. Additionally, this correlation is dependent upon film thickness, which contradicts the observation of Azzopardi [49] that droplet size depends on the entrained, and not film, mass flow rate due to droplet coalescence . Also, Kataoka et al. [68] point out that the potential flow assumption used by Tatterson et al. [40] does not apply in the highly turbulent gas core region, and thus this correlation is not applicable over a wide range of gas Reynolds numbers.

2.4.3.2.2. Alternative Drop Size Correlations

Correlations for the characteristic drop size present in annular dispersed flow have also been proposed by several other researchers, including: Lopes & Dukler [26], Kataoka et al. [60], Azzopardi [49], Azzopardi et al. [30], Andreussi et al. [87], Kocamustafaogullari et al. [76], and

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Ambrosini et al. [68]. Most of these correlations were developed from data collected using air- water at low pressures (1-2 atm). Those correlations that are most applicable to the current study are briefly described within this section.

Lopes & Dukler [26]

Lopes & Dukler [26] concluded that the maximum droplet size in annular flow is mainly controlled by the pressure fluctuations of the turbulent flow around the droplet. They suggested a simple method for predicting drop size based on a critical Weber number of 0.194 using their data taken with air-water at atmospheric pressure.

Azzopardi [49], Azzopardi et al. [30], and Andreussi et al. [87]

The correlations of Azzopardi [49] and Azzopardi et al. [30] consist of two terms, where the first describes the effect of the vapor phase while the second term accounts for the effect of droplet coalescence using the entrained liquid concentration. Andreussi et al. [87] (co-current downflow data) suggested a correlation of a similar form; however, their second term contained film thickness and tube diameter. Andreussi et al. [87] reasoned that as the film thickness decreases smaller droplets are generated, which may be true since smaller amplitude waves and larger vapor velocities are generally associated with smaller film thicknesses; however, based on the qualitative observation that drop sizes increase with decreasing film thickness, coalescence must dominate in these situations. In any case, since COBRA-TF requires the entraining droplet size be specified and an explicit coalescence model exists within the code, it is not desired to account for this effect in this correlation and therefore these types of correlations are of limited use to the current study.

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Kocamustafaogullari et al. [76]

Kocamustafaogullari et al. [76] proposed a correlation for the maximum stable drop size is developed using a critical Weber number based on the balance of external and surface stresses. Two different surface stresses exist, namely viscous and dynamic stresses, but Kocamustafaogullari et al. [76] indicate the viscous stresses for a droplet are negligible in annular dispersed flow. Meanwhile, two dynamic stresses exist where the first is due to the local relative motion around the droplet and the second is due to the change in eddy velocity over the length of the droplet. Kocamustafaogullari et al. [76] argue that for annular dispersed flow, where the density of the dispersed phase is much greater than that of the continuous phase, the disruptive forces based on the local relative velocity is much larger than those generated by the turbulent eddies. Therefore, the disruptive force that governs droplet disintegration in a gas stream is the local relative velocity.

Based on these arguments Kocamustafaogullari et al. [76] propose an expression for a critical Weber number, which is given as:

ρ U 2 D We = v ve ,max d ,max (2-97) crit 2σ

Kocamustafaogullari et al. [76] use an expression suggested by Levich (1962) for the limiting local relative velocity that a fluid will flow around a particle suspended in it of a given size along with an expression given by Lopes & Dukler [26] for the energy dissipation rate. Additionally, the expression suggested by Ishii & Grolmes [3] was used to calculate the interfacial shear stress. Substituting these equations into Equation (2-97) and then correlating using the data collected by Lopes & Dukler [26] allowed Kocamustafaogullari et al. [76] to propose the following correlation for the maximum stable drop size:

4 1 15 4 15 Dd ,max 4 3  ρ µ   Re  − 15 − 5  v v   GS  = .2 609 Cw We GS     (2-98) DH  ρ l µl   Re LS 

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This correlation suggests that the maximum stable drop size decreases slightly with both increasing superficial gas velocity and increasing superficial liquid velocity. It also suggests the maximum stable drop size increases linearly with increasing tube diameter, which is inconsistent with the observations of Azzopardi [49]. Meanwhile, this correlation is independent of gas viscosity, but suggests the maximum stable drop size increases slightly with increasing liquid viscosity, which is consistent with the observations of Fore & Dukler [24]. Finally, as desired, this correlation does not consider the effects of droplet coalescence since there is no dependence on the entrained liquid flow rate.

The maximum stable droplet diameter was then related to the Sauter mean diameter by assuming the droplets follow an upper-limit log normal distribution and the resulting expression is:

65.0 D = D (2-99) d 32, 2.609 d ,max

These correlations show reasonable agreement with the same sets of data considered by Tatterson et al. [40] during the development of their correlation, but the comparisons with the data of Jepson et al. [32] are much less promising. The experimental data collected by Jepson et al. [32] shows a dependence on the liquid Reynolds number for the various fluids tests that is not captured by the proposed correlation.

2.4.3.2.3. Assessments of Drop Size Correlations

Assessments of drop size correlations to experimental data have been conducted by: Teixeira et al. [86], Azzopardi et al. [21,30], Kocamustafaogullari et al. [76], and Fore et al. [31]. With regards to the assessments that have been conducted it is important to recognize the differences in geometry and orientation of the flow paths that have been used in the experiments. In particular, Cousins & Hewitt [67] and Pogson et al. [88] used tubes while Tatterson (1975) and Wicks & Dukler (1966) used non-circular ducts. Meanwhile, Cousins & Hewitt [67] and Pogson et al. [88] took data for upflow conditions, Wicks & Dukler (1966) considered downflow situations, and Tatterson (1975) conducted experiments for horizontal flow.

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Teixeira et al. [86]

Teixeira et al. [86] compared the drop size correlations of Tatterson et al. [40], Azzopardi [49], Andreussi et al. [87], among others, with several sets of experimental data. The best agreement was obtained using Azzopardi [49], while Tatterson et al. [40] severely overpredicted the data and Andreussi et al. [87] underpredicted the data.

Azzopardi et al. [21]

Azzopardi et al. [21] compared several drop size correlations to the experimental data that was collected in this work. The comparisons with the correlation proposed by Tatterson et al. [40] were not favorable and indicate this correlation has too strong of a dependence on the vapor velocity. Additionally, the dependence on liquid film thickness, rather than entrained flow rate, in this correlation results in a decrease in drop size being predicted with increasing liquid flow rate, which contradicts the experimental data. Azzopardi et al. [21] suggests the latter disparity may be due to a difference in the type entrainment mechanisms between their data and the data used by Tatterson et al. [40] to develop their correlation. Azzopardi et al. [21] indicates that their data was taken in the region where the roll wave, or ligament breakup mechanism, is most important, while the data used in developing the Tatterson et al. [40] correlation was taken in the region where the wave undercutting, or bag breakup, mechanism is most important. Earlier work by Azzopardi [49] showed that the Tatterson et al. [40] correlation gave favorable predictions of the collected by Lopes (1985), which was taken at flow rates where the bag breakup mechanism would be expected.

Azzopardi et al. [30]

Azzopardi et al. [30] compared of the data collected by Jepson et al. [32], Azzopardi et al. [21], and Teixeira et al. [86] in 0.394, 0.787, and 1.26-inch diameter tubes, respectively, using various fluids with several correlations, including those proposed by Tatterson et al. [40] and Azzopardi [49]. The poor agreement resulted in the development of a new empirical correlation that applies

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to situations where the Weber number, based on wave height, is greater than 25 (suggested to be transition point between entrainment mechanisms and where the roll wave/ligament breakup regime exists) [30]. Following development this correlation was compared with the experimental data of Andreussi et al. [87] with reasonable agreement, but this correlation does a poor job of predicting the Azzopardi [49] data in a 0.125-m diameter tube. [30]

Kocamustafaogullari et al. [76]

Kocamustafaogullari et al. [76] compared the droplet sizes predicted by the correlation of Lopes & Dukler [26] to the experimental data of Jepson et al. [32], which examined property effects such as surface tension and fluid density ratios on drop sizes. The correlation of Lopes & Dukler [26] provided reasonable estimate of the data when the density ratio was similar to that of air- water at atmospheric pressure, but worsened as the density ratio moved away from this value.

Fore et al. [31]

Fore et al. [31] conducted experiments measuring drop size distribution and pressure gradient in a vertically oriented non-circular duct (4 x 0.2-inch) using nitrogen-water at 50 and 250-psia. Fore et al. [31] then made comparisons using this newly collected data and the previously collected data of Fore & Dukler [24], Cousins & Hewitt [67], and Wicks & Dukler (1966) with several previously published drop size correlations, including those proposed by Tatterson et al. [40], Kataoka et al. [60], and Kocamustafaogullari et al. [76]. Fore et al. [31] indicate that the correlation proposed by Tatterson et al. [40] predicts data of Fore & Dukler [24] reasonably well, but significantly underpredicts the newly collected data from their experiments. Meanwhile, they show that the correlation proposed by Kataoka et al. [60] does a reasonable job of predicting their data collected at 3.4-atm, but significant scatter does exist. More importantly, this correlation significantly underpredicts both the higher pressure data from this study (17-atm) and the higher viscosity data collected by Fore & Dukler [24] using air with a 50% glycerin-water mixture. These results lead Fore et al. [31] to suggest that the effects of gas density and viscosity are not properly accounted for by the Kataoka et al. [60] correlation.

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The maximum drop size correlation of Kocamustafaogullari et al. [76] was found to predict all data considered reasonably well, except for underpredicting all of the data from Wicks & Dukler (1966), which is consistent with the findings presented by Kocamustafaogullari et al. [76]. Meanwhile, the Sauter mean diameter correlation proposed by Kocamustafaogullari et al. [76] was found to underpredict the newly collected data obtained by Fore et al. [31] in this study. Fore et al. [31] suggest this difference is due to the assumption of a fixed type distribution and a change to the correlating coefficient in Equation (2-99) from 0.65 to 1.3 is suggested by Fore et al. [31] to provide a better fit to the data, but significant scatter still exists.

2.4.3.3. Conclusions on Applicability of Available Drop Size Correlations

From this review it is obvious that the differences in entrainment mechanism, test orientation, flow direction, geometry, and droplet measurement technique must all be considered when selecting an appropriate drop size correlation. As a result, the application of regime, or entrainment mechanism, specific drop size correlations was emphasized in the current study. However, it was also seen in this section that several inconsistencies and a large amount of scatter exists in the available correlations. For example, the drop sizes reported by Azzopardi and his colleagues [15,21,22,25,30,49,68,69] generally tend to be much smaller than those reported by Dukler and his colleagues [24,26]. Lopes & Dukler [26] suggest the difference is the calculation of the Sauter mean diameter is very sensitive to larger drops and the laser-diffraction measurement technique used by Azzopardi is unable to reliable detect drop with a diameter greater than 500-m. More importantly the available correlations were developed based on entrained, rather than entraining droplet size. These limitations, among others, will need to be addressed in future studies aimed at quantifying drop sizes within annular flow situations.

2.4.4. Conclusions

Accurate predictions of the creation (entrainment), existence (size and motion), and removal (deposition) of droplets is required to accurately predict annular flow phenomena [25]. First and foremost an accurate prediction of the distribution and interchange of liquid between the film and

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droplet fields is needed to calculate the occurrence and location of dryout. As described previously an applicable interfacial shear model is essential to the prediction of entrainment since it is postulated that this shear is strongly related to the entrainment process. Additionally, the size and number of the entrained droplets impacts the post-CHF heat transfer through the available interfacial area for de-superheating the vapor and their ability to potentially impact the heated surface. The presence of entrained droplets also increases both the momentum of the gas core region due to the increased density of the liquid and the pressure gradient for the flow through the additional drag that exists between the vapor and droplets.

This section has indicated several differences in the entrainment behavior for the different regimes of annular flow. Similar to interfacial shear, most models that have been developed are for air-water at low pressures and rely on dimensionless parameters rather than physical mechanisms. Additionally, it was seen that COBRA-TF does apply different entrainment models for co-current or counter-current flow, but it does not include separate models for churn-annular/ churn-turbulent flow. As a result, models for these phenomena had to be identified and will be included in the proposed modeling package. Lastly, it was seen that the area with the most uncertainty is the drop size correlations. Several concerns were highlighted and will need to be addressed in future studies.

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2.5. Review of Works Focused on Flooding and Flow Reversal

Counter-current flow consists of vapor, with or without entrained droplets, flowing upward while a liquid film flows downwards. As shown in Figure 2-10 a limited range of conditions (Quadrant 2) exist where counter-current flow can occur; however, as discussed previously it is an important phenomenon in reactor accident scenarios because it affects the flow rate of water that can reach the reactor core to provide cooling.

Figure 2-10: Possible Two-Phase Flow Situations [8].

Possible counter-current flow conditions can be summarized by saying: for a given vapor flow rate upwards through a flow path there exists a corresponding maximum flow rate of liquid that can penetrate downwards. The collection of these points defines the flooding curve, which is also referred to as the Counter-Current Flow Limitation (CCFL). A critical vapor flow rate then exists above which no liquid can penetrate downwards, which corresponds to the flow reversal point and the onset of co-current flow.

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The discussion in this section is focused on describing situations where CCFL can occur, the mechanisms of CCFL, the current COBRA-TF treatment of CCFL, and a brief summary of the available models and correlations for predicting this phenomenon.

2.5.1. Counter-Current Flow Situations

Essentially two different means exist for counter-current flow to occur with the difference between them being how the liquid is introduced to the flow duct. In the first situation liquid pools above a flow duct where vapor upflow exists. Starting from a condition of zero vapor flow the initial downward liquid flow rate is dictated by the dimensions of the flow duct and the water level above the flow duct since this is the driving head, but as much liquid as possible will penetrate downwards. The amount of liquid that can penetrate downwards will decrease as the vapor flow is increased until eventually the vapor flow rate becomes sufficient to prevent any liquid from penetrating. The flow pattern moves from bubbly, to slug, and finally to annular as the vapor flow is increased. This type of situation follows the flooding curve shown in Figure 2- 10 the entire way through the counter-current flow region (Quadrant 2) from Point A to the flow reversal point (Point B). This situation is representative of the conditions in a commercial Pressurized Water Reactor (PWR) following the initiation of the liquid spray injection system in the upper plenum. Liquid from this system collects above the core and tries to penetrate the flow paths as a falling film to provide additional heat transfer capability in the upper elevations and potentially create a top-down quench; however, vapor generated at lower elevations from both the reflood process and the boiling of liquid that has already penetrated propagates upwards and can prevent further liquid from penetrating into the flow duct. Due to the importance of this phenomenon it is necessary to characterize and be able to predict these conditions.

In the second situation liquid is introduced within the flow duct, which is more representative of liquid condensate entering a well during the extraction of gas and is of less interest for the analysis of reactor accident scenarios. Figure 2-11 provides a visual representation of this scenario. Experimentally this situation is simulated using porous sinter injection and the initial downward liquid flow rate for a zero vapor flow condition is dictated by the amount of liquid supplied to the flow duct. For this method of liquid introduction situations can exist at low vapor

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flow rates where the liquid flow is unaffected by the vapor flow. This situation occurs for any Point C on Figure 2-10 where the magnitude of the injected liquid flow rate is greater than zero, but less than that corresponding to Point A. At these low vapor flow rate conditions all the liquid introduced within the flow duct flows downward and either: 1) out the bottom of the flow duct, as shown in Figure 2-11, or 2) in the case of gas extraction wells, collects at the bottom and can block the flow of vapor causing gas production capability of the well to cease. Case 1 is depicted on Figure 2-10 as a propagation along a path of constant liquid flow as the upward gas flow rate is increased until eventually a critical value is reached where some liquid is prevented from flowing downward. This condition, noted as Point D on Figure 2-10, corresponds to the intersection with the flooding curve and as such is referred to as the flooding point.

Figure 2-11: Flooding and Flow Reversal [5].

A variety of phenomena accompanying the flooding point have been suggested in the literature including a sharp increase in pressure gradient and entrainment and enhanced chaotic wave motion of the interface [7]. It has been hypothesized that these behaviors were a result of liquid bridging within the tube; however, the results of Dukler & Smith [43] and Zabaras & Dukler [7] suggest that this is not the case for larger diameter flow paths (~> 2-inches). As the gas flow rate is increased further the amount of liquid downflow continues to decrease until the point were no

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more net liquid downflow occurs. Again this point is referred to as the flow reversal point. Between the flooding and flow reversal points the liquid is partitioned between downflow and upflow and this partition is governed by the flooding curve where the flow moves from some Point D to Point B on Figure 2-10.

2.5.2. Mechanism of Flooding and Flow Reversal

In general three mechanisms for flooding have been proposed and investigated in the open- literature. These include: 1) a wave transport mechanism, 2) a liquid bridging phenomenon, and 3) a droplet transport mechanism. This section will examine each mechanism and briefly describe some previously conducted works that support the existence of each mechanism. Finally, the work of Jayanti et al. [89] will be described in an attempt to resolve the discrepancies that appear to exist between these mechanisms.

2.5.2.1. Wave Transport Mechanism

The wave transport mechanism assumes that flooding occurs when the interfacial waves reverse direction and propagate upwards by the drag force exerted by the gas flow [9]. In this case the interfacial waves are assumed to flow downward for gas flow rates below the flooding point. Then as the gas flow rate is increased the wave velocity decreases until becoming stationary at the flooding point before finally propagating upwards and eventually causing flow reversal to occur as the gas flow rate is further increased [9]. The results of the experiments conducted by McQuillan et al. [9] and Govan et al. [16] support the existence of this mechanism

2.5.2.2. Liquid Bridging Mechanism

The liquid bridging is assumed to occur when the interfacial waves become unstable and grow to be sufficiently large that they form a liquid bridge across the flow path. The liquid bridge blocks the gas flow, which causes the liquid slugs to be broken and intermittently forced upwards by the

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faster moving gas flow [7]. Initially most researchers suggested that liquid bridging was the mechanism for flooding and analytical approaches for modeling the phenomenon involved estimating the gas flow rate where waves would become unstable and grow until bridging occurred. However, several investigators, including Dukler & Smith [43] and Zabaras & Dukler [7], later measured the mean and maximum film thickness in a series of flooding experiments and were able to conclude that no bridging occurred in the 2-inch diameter tubes they examined since the maximum film thickness did not exceeded 20% of the tube radius for any of the conditions tested.

Additionally, Zabaras & Dukler [7] used two conductance probes in their experiments to allow wave velocity and direction to be calculated. It was determined that for all cases the wave velocities, even through the flooding point, were in the downward direction. Obviously the waves traveled faster for the higher liquid flow rates, but the wave velocities themselves were rather insensitive to the increasing upward gas velocity until just prior to the flooding point. At this point the wave velocities began to decrease towards zero, but the wave reversal condition was not reached even though flooding had occurred. As a result Zabaras & Dukler [7] also discount the possibility that the wave transport mechanism was the mechanism of flooding in these experiments.

2.5.2.3. Droplet Transport Mechanism

The droplet transfer mechanism suggests that the waves on the interface become unstable and breakup due to interfacial shear at a lower gas velocity than that required to either: a) transport the entire wave upwards or b) prior to the wave growing large enough to bridge the flow path. It is hypothesized that this process yields an ample source of drops and the transport of droplets entrained in this manner is sufficient to cause flooding. This mechanism suggests CCFL is governed solely by the conditions required to transport droplets, rather than films, upwards. Based on the discussion in the previous section it is clear that the experiments of Dukler & Smith [43] and Zabaras & Dukler [7] support the existence of this mechanism.

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The gas producing industry has used a correlation developed by Turner et al. [37] based on this mechanism to estimate the minimum gas velocity needed to suspend the largest droplet within the gas core. Balancing the drag force imparted by the gas flow on a drop and the force of gravity allows the gas velocity to be correlated in terms of an unknown drop diameter that can be expressed in terms of a critical Weber number criterion. It is assumed that when the gas flow in the production line falls below the estimated minimum value that flow reversal, or liquid loading, will occur and gas production will cease [17].

2.5.2.4. Conclusions on CCFL Mechanism

From the discussion provided in the previous sections it appears that a discrepancy exists in the proposed flooding mechanisms. On one hand, the droplet transport mechanism implicitly assumes CCFL is governed solely by the upward transport of the dispersed phase while on the other hand the liquid bridging and wave transport mechanisms suggest CCFL is governed by the upward transport of the liquid film. However, given that a fluid will always takes the path of least resistance, a physical basis can be described for the occurrence of each mechanism.

For example, the wave transport mechanism requires the interfacial shear force to exceed the weight of the wave to move the wave upwards. Such a condition requires a large gas velocity and form drag to exist over the wave. These conditions are more likely to occur in smaller diameter tubes where: 1) circumferentially coherent waves are more likely to be formed and 2) the ratio of wave amplitude to diameter is the largest. These conditions provide the greatest percent reduction in flow area for the gas past the wave crest, which thereby creates the largest drag. It would be expected that flooding by this mechanism would exhibit a strong diameter dependence, which is consistent with the Wallis [4] form of the CCFL correlation that will be discussed in Section 2.5.3.3 and has been widely found to correlate flooding data in tubes smaller than 2- inches in diameter.

On the other hand, in larger diameter tubes circumferentially coherent waves are not as easily formed [2,59] meaning the entire gas phase is not forced to pass over a wave crest simultaneously. More importantly the percent reduction in flow area at the wave crest is

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significantly less in larger tubes. As a result the form drag would not be as great in larger diameter tube as it would be in smaller diameter tubes, which in turn requires a much larger gas velocity to transport a wave of the same amplitude upwards in larger diameter tube relative to a smaller diameter tube. It is anticipated that prior to reaching the gas velocity needed to transport the entire wave upwards, the wave would become unstable and a portion of the wave would become entrained. In this case it is anticipated that flooding would become primarily dependent on droplet transport rather than wave transport. Flooding by a wave instability and droplet transport mechanism should not exhibit a diameter dependence, but rather a surface tension effect should exist to consider the instability of the interface. This is consistent with the Kutateladze [45] form of the CCFL correlation that will be discussed in Section 2.5.3.3 and has been widely found to correlate flooding data in larger tubes. [89]

The situation is further complicated for non-tubular geometries such as rod bundles or non- circular flow ducts where the liquid may be non-uniformly distributed around the periphery of the flow duct [39]. Mishima & Nishihara [39] determined by experiment that in non-circular ducts with large aspect ratios most of the liquid film forms on the narrower side walls of the non- circular duct, while the film on the wider side walls is much thinner and unwetted patches could sometimes appear on this surface. A similar result was observed by Fore et al. [31,51]. Jayanti et al. [89] proposed that in these types of flow paths it would be unlikely that a coherent wave would be formed around the entire perimeter due to the large linear dimensions and therefore it would be more likely that flooding would occur by the droplet transport mechanism in these types of geometries. Similarly, Jayanti et al. [89] proposed that it is also unlikely that a coherent wave will be formed simultaneously on all adjacent rods in a rod bundle and therefore it would be more likely that flooding would also occur by the droplet transport mechanism in these types of geometries. This conclusion is consistent with the Kutateladze [45] form of the CCFL correlation being more commonly found to correlate results taken in both these types of geometries. [89]

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2.5.3. Available CCFL Correlations

CCFL correlations or models provide a means of quantifying the flooding curve. This section presents several of the physics-based and empirical correlations for predicting the phenomenon of flooding and flow reversal that have been suggested in the open-literature.

2.5.3.1. Current COBRA-TF Treatment of CCFL

The inability to predict the CCFL phenomena within a three-field analysis structure is an indication that the interfacial drag, and corresponding entrainment rate, is inaccurately predicted relative to experimental data. One means of accounting for this deficiency is to alter the code predicted interfacial shear stress between the vapor and continuous liquid fields based on the results predicted by CCFL correlation that more suitably represents the experimental data. In this approach if the code predicted liquid flow exceeds the allowable predicted using one CCFL correlation then the code predicted interfacial drag is artificially enhanced by a factor of:

Code predicted downflow for continuous liquid field W = l (2-100) Allowable liquid downflow predicted by CCFL correlatio n Wl,crit

This artificial enhancement of the interfacial drag serves to reduce the amount of liquid downflow; however, there is no experimental evidence to support the existence of a linear relationship between this ratio and the interfacial drag. This option is currently available in COBRA-TF, but was not activated at the time of the current study.

2.5.3.2. Physics-Based Correlations

Several attempts have been made to develop physically based models have been made by those such as Richter [8], McQuillan et al. [9], and Bharathan & Wallis [11], but since the flooding phenomenon is complex these approaches have been met with limited success. More

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importantly, the range of applicability of these models is limited since they must assume a given mechanism, but the conditions under which a given mechanism occurs has yet to be quantified.

Richter [8]

Richter [8] developed a flooding correlation based on force balances and a wave stability criterion that accounts for the contradiction between the work of Wallis [4] and Pushkin & Sorokin [90]. In general, the proposed correlation approximates the Wallis [4] solution for small diameter tubes and the Pushkina & Sorokin [90] solution for larger diameter tubes; however, this correlation does not account for differences in entrance conditions or allow them to be considered in any way.

McQuillan et al. [9]

As previously mentioned McQuillan et al. [9] suggested that flooding occurs due to interfacial waves reversing direction. As a result they suggest flooding could be modeled using a force balance where the weight of the falling film is balanced by the forces tending to cause the wave to propagate upwards. McQuillan et al. [9] apply Bernoulli’s theorem to obtain a simple expression for the pressure drop associated with the wave, which yields an expression that is analogous to a form loss due to a sudden contraction. The height of the wave was then related to the gas velocity by assuming horizontal equilibrium of the wave at the flooding condition, which is assumed to be governed by the competing effects of surface tension and pressure differences. The final expression relates the wave amplitude at the flooding point to the liquid superficial velocity in terms of surface tension, liquid density, the wall friction factor, tube diameter, and a proportionality constant. The proportionality constant is needed to consider the radius of curvature of the wave, which is unknown but affects the surface tension force. In general the model presented by McQuillan et al. [9] provided favorable agreement with experimental data for low liquid flow rates, but begin to overestimate the critical vapor velocity for larger liquid low rates. McQuillan et al. [9] state that their approach provides better agreement than the other theoretical models available at the time, but does not perform as well as empirical correlations.

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Bharathan & Wallis [11]

Bharathan & Wallis [11] pursue an idea originally proposed by Wallis that the maximum limiting fluxes lie on an envelope of constant liquid volume fraction curves and that end conditions cause the actual flooding curve to lie within the envelop. Force balances on the gas core, neglecting entrainment, and entire flow regions were used along with expressions for the wall and interfacial friction factors. Bharathan & Wallis [11] are able to show that two solutions exist for a given set of limiting fluxes corresponding to different void fractions. The two solutions correspond to smooth films, where wall shear stress is much greater than the interfacial shear stress, or rough films, where the interfacial shear stress is much greater than the wall shear stress. A limiting envelope of curves is created by plotting the limiting fluxes for constant void fraction values. Bharathan & Wallis [11] then suggest and show that the end conditions further limit fluxes and thus points fall within this envelop. While the analysis of Bharathan & Wallis [11] in an interesting approach that provides a method for bounding the flooding phenomenon, it does not provide any means of quantifying the effects of the end conditions. It also does not consider, and thus quantify, the effects of geometry differences. As a result this analysis approach was not pursued in the current study.

2.5.3.3. Empirical Correlations

As demonstrated in the previous section, the complexity of the flooding phenomenon has precluded a widely-applicable model from being determined to date. The most successful predictions have come from empirical correlations, such as those that have been suggested by Wallis [4], Kutateladze [45], and Pushkina & Sorokin [90]. This section will review these models and then discuss the implementation of a CCFL model into the two-phase, two-field RELAP5-3D analysis code. This model will serve as the basis for the development of a three-field CCFL within the current study.

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Wallis [4]

Wallis [4] suggested a correlation to predict the conditions for flow reversal using a dimensionless superficial velocity. This correlation is given as:

1 1 * 2 * 2 jv + mj l = C (2-101)

* * where m and C are empirical constants and jv and jl are the dimensionless gas and liquid fluxes, respectively. The dimensionless fluxes, which represent the ratio of inertial to hydrostatic forces, are given as:

1 2 * ρk jk jk = 1 , where: k = l or v (2-102) 2 []gD H ∆ρ

Meanwhile ‘C’ is said to be dependent upon the design of the ends of the test section and typically ranges between 0.7 and 1.0. In particular Wallis [4] suggests C = 0.725 for sharp-edged flanges. On the other hand, ‘ m’ considers mass transfer due to phase change and it has been found to be approximately unity for turbulent flow with no phase change (i.e. air-water). This correlation is based on experiments in small diameter tubes (0.5 to 2.0-inches) and neglects the effects of viscosity and surface tension which Bharathan & Wallis [11] suggest also merit consideration even in situations where flooding is governed by the wave transport mechanism because these parameters are postulated to influence wave amplitude.

Kutateladze [45] and Pushkina & Sorokin [90]

Kutateladze [45] suggests that the stability of a flow is dependent upon the stability of the individual structures that comprise the flow (e.g. droplets, films, bubbles) and indicates the stability of a given structure is dependent upon: a) the disruptive force of the dynamic heads of the phases in contact with the structure and b) the stabilizing force of the pressure caused by the surface tension. This leads to the introduction of the following dimensionless group:

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2 2 (ρU ) L Ku = 2 1 (2-103) σ

2 where (ρU )2 is the dynamic head of the phase in contact with the structure and L1 is the characteristic dimension of the structure. Kutateladze [45] indicates that the characteristic dimension can be written as:

1  σ  2 L1 ~   (2-104)  g()ρ1 − ρ2 

where ρ1 and ρ2 are the densities of the structure and the phase in contact with the structure, respectively. Introducing this expression yields a stability criterion that assumes the form of:

1 2 ρ2 U2 Ku = 1 (2-105) 4 []gσ ()ρ1 − ρ2

This parameter is often referred to as the Kutateladze number. It should be noted that the Kutateladze number can also be derived from the dimensionless superficial velocities by replacing the physical geometric parameter (i.e. the hydraulic diameter) with the Laplace capillary length constant given as Equation (2-104).

Assuming a droplet transport mechanism governs the flooding phenomenon it is desired to quantify the stability of the liquid film. Writing Equation (2-105) to consider this condition and assuming the actual vapor velocity can be replaced by the superficial vapor velocity, as is often done for annular flow situations since the film thickness is typically small, yields:

1 2 ρv jv Ku v = 1 (2-106) []gσ∆ρ 4

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This expression represents the ratio of the inertial forces of the vapor phase to the buoyancy and surface tension forces.

For stability the vapor velocity must be less than that corresponding to a critical value of the Kutateladze number and this criterion is based on comparisons to experimental data. The work of Pushkina & Sorokin [90], which considered experiments conducted using air-water at atmospheric pressure in tubes with diameters great than 6-inches, suggests the critical gas velocity needed to prevent any liquid from penetrating downwards independent of tube diameter and corresponds to:

Ku v, crit = 2.3 (2-107)

It has been proposed that a CCFL correlation can be written with the same form as the Wallis [4] correlation where the dimensionless superficial velocities are replaces with the Kutateladze number. This correlation is given as:

1 1 2 2 Ku v + mKu l = C (2-108) where:

1 2 ρv jk Ku k = 1 for: k = l or v (2-109) []gσ∆ρ 4

The discrepancy between the Wallis [4] and Kutateladze [45] correlations is that the Wallis [4] correlation suggests CCFL is dependent upon diameter, while the Kutateladze [45] correlation is independent of diameter and instead suggests a surface tension effect. As alluded to in Section 2.5.2.4 these differences are consistent with the type of mechanism that is assumed to be governing flooding in the different ranges of tube diameters considered when developing each correlation. This idea was first presented by Wallis & Makkenchery [91] and explored further by Jayanti et al. [89].

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Wallis & Makkenchery [91]

Wallis & Makkenchery [91] attempted to explain the apparent contradictions between the work of Wallis [4] and Pushkina & Sorokin [90] by suggesting the Kutateladze number is only appropriate when the characteristic dimension becomes unimportant. Wallis & Makkenchery [91] suggest that in larger tubes with thin liquid films the perturbation in the gas flow due to interfacial waves is confined within the boundary layer region of the gas core and therefore the result is independent of tube diameter. However, Wallis & Makkenchery [91] also point out that in very small tubes capillary effects can cause a liquid bridge to be formed by surface tension forces regardless of the phasic flow rates. This means that liquid downflow can be prevented even for a zero vapor flow condition the result also becomes diameter independent.

Jayanti et al. [89]

Jayanti et al. [89] attempted to quantify the hypothesis presented by Wallis & Makkenchery [91] using a Computational Fluid Dynamics (CFD) code to determine the force exerted on a standing wave by gas flowing over it. The analysis assumes that flooding corresponds to the point where upward transport of the waves can occur and that this point corresponds to the condition where the weight of the wave is balanced by the form drag due to the pressure variation over the wave. Jayanti et al. [89] correlated the form drag coefficient required to balance the weight of the wave from their computational results as a function of the reduction in cross-sectional area available for gas flow at the wave crest. The results indicate that as the tube diameter increases the drag force on a wave of given amplitude decreases substantially and therefore the corresponding critical gas velocity required to transport the wave upwards increases significantly.

Jayanti et al. [89] compared their results to those predicted by the Wallis [4] and Kutateladze [45] forms of the CCFL correlation. For smaller tube diameters their results compare well with the results predicted using the Wallis [4] form of the CCFL correlation and suggests this agreement indicates flooding must occur by the interfacial wave transport mechanism in these situations. However, for large tube diameters their results indicate that the required gas velocity obtained from their computational results for transporting waves upwards exceeds the gas velocity required

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to entrain and suspend droplets within the flow as predicted by the Kutateladze [45] form of the CCFL correlation. This result indicates that the wave becomes unstable and breakup at a gas velocity less than that required to transport the entire wave upwards. Despite these suggestions Jayanti et al. [89] did not propose any criterion to quantify the demarcation between conditions where each mechanism would be supposed to occur. [89]

Additionally, Jayanti et al. [89] found that the results of the Wallis [4] and Kutateladze [45] correlations agree well with one another for data taken at atmospheric pressure in tubes with a 2- inch diameter. Meanwhile, since the Kutateladze [45] correlation does not exhibit a tube diameter dependence the flooding velocities predicted by the Wallis [4] correlation are less than those of the Kutateladze [45] correlation for tube diameters less than 2-inches and greater than those of the Kutateladze [45] correlation for tube diameters greater than 2-inches. [89]

RELAP5-3D [44]

Upon examination of the two forms of the CCFL correlation that have been proposed in the open- literature (i.e. Wallis [4] and Kutateladze [45]) it becomes clear that their similarities make them amenable to the development of a single expression involving a weighting parameter. Then, depending on the value of the weighting parameter the new expression can retain the properties of either the Wallis [4] or Kutateladze [45] correlation, or if desired some combination of the two. This approach is employed in RELAP5-3D [44], where a new parameter is defined as:

1 2 jk ρ k H k = 1 , where: k = l or v (2-110) []g()wght ∆ρ 2 and the weighting parameter in this expression is given as:

(1−β ) β wght = DH L (2-111) where the Laplace capillary length in this expression is calculated as:

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σ L = (2-112) g()ρl − ρv

A generalized form of the CCFL correlation can now be written as:

1 1 2 2 H v + mH l = C (2-113)

In this formulation if β = 0 then the Wallis [4] form of the CCFL equation is obtained and if β = 1 then the Kutateladze [45] form of the CCFL equation is obtained. Using this formulation provides the user greater flexibility when specifying the form of the CCFL correlation and also represents a more efficient programming approach than coding each form of the correlation separately.

2.5.4. Conclusions

Given the importance of CCFL to the distribution of coolant and resulting temperature distributions for postulated reactor accident scenarios a large amount of experimental data has been collected for the specific configurations of a given reactor design. However, from this review it is clear that a variety mechanisms, each governed by a different phenomenon, have been shown to causing flooding for different scenarios, but more importantly the mechanism that occurs in a given situation has not yet been appropriately quantified. Additionally, the exact mechanism that occurs appears to be extremely sensitive to several effects where the most apparent of these are the diameter and geometry of the flow path, but some uncertainty also exists in the sensitivity of inlet and outlet conditions and several other factors that have not yet been resolved.

Currently the most suitable agreement to a specific set of experimental data tends to be provided by empirical flooding correlations, where correlating constants are used to quantitatively account for any unresolved effects. In particular, the Wallis [4] form of the CCFL correlation tends to be more applicable in smaller diameter tubes, where flooding is supposed to be governed by the

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reversal of direction of interfacial waves, while the Kutateladze [45] form of the CCFL correlation tends to be more applicable in larger diameter tubes and non-circular geometries where flooding is supposed to be governed by the wave instability and droplet transport mechanisms.

Therefore, even though a suitable mechanistic-based model for CCCL is unavailable at this time, it is still desired to leverage the available experimental data and apply an appropriately defined correlation as a means of improving the predictive capabilities of transient analysis codes. As a result a three-field CCFL model has been developed within the current study. Additionally, given that CCFL can occur at a variety of locations within a reactor, including both small (subchannel) and large (downcomer) dimensions and a range of geometries, it is desired to have the ability to apply either the Wallis [4] or Kutateladze [45] form of the CCFL correlation. Therefore, it was decided to extend the generic implementation from RELAP5-3D [44] to the more complex physics associated with a three-field analysis environment. The developed model includes appropriate entrainment rate models as well as unique tests for activating and de-activating the model.

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2.6. Concerns with Currently Available Models

The lack of appropriate constitutive relationships for interfacial drag, entrainment, and drop size is currently the most significant barrier to accurately predicting the behavior of the annular flow regime over a wide range of conditions. The review of available regime transition, interfacial shear, entrainment, drop size, and flooding models, and in particular those currently used by COBRA-TF, presented in this chapter has highlighted several concerns. First, a majority of the relationships that have been proposed are highly empirical and are based primarily on low pressure air-water data. These correlations have been shown to provide reasonable agreement over a limited range, but are not reliable beyond the range which they were developed. In particular, it was identified in previous works [28,38], and confirmed in the current study (see Section 4.1), that applying correlations for film interfacial drag based on air-water to steam-water situations results in a severe overprediction of the axial pressure gradient. Despite this fact these types of models are commonly used by transient safety analysis codes for modeling nuclear reactor scenarios. Additionally, while the use of three-field analysis approaches that model the liquid film and dispersed droplet regions separately do provide a more accurate representation of the physics of the flow, careful attention must be ensure that the experimental data used for model development was reduced to a level commensurate with flow fields considered by the code. For example, since the effects of droplet drag and momentum exchange have not been explicitly considered in most previous works the majority of available empirical models are not readily extendable to three-fluid analysis scenarios.

Additionally, the convenience of using empirical relationships has precluded most of the available models from explicitly accounting for the physical mechanisms controlling the phenomena of interest. For example, many researchers agree there is a clear presence of interfacial instability phenomena directly impacting the entrainment process in annular flow [33], but most of the available models do not directly consider this effect. Similarly, the interfacial drag in annular flow has been attributed to the roughness associated with the presence of disturbance waves and smaller amplitude ripple waves, but most models are correlated using the mean film thickness. Instead of considering these effects most of the current models instead rely heavily on dimensionless parameters since this approach does not require a complete understanding of the controlling mechanism. For some parameters, such a detailed level of

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understanding does not exist. Additionally, the use of dimensionless parameters tends to be more computationally efficient, but this approach limits the applicability of such models since it is less likely that they will be able to implicitly handle changes in the pressure and flow conditions. This may be one reason that single models for phenomena such as entrainment and film interfacial drag have not yet been found to be accurate over a wide range of conditions. Therefore, the current study aims to develop and utilize models that are based on the physical mechanisms associated with annular flow. Such models have been proposed by Hurlburt et al. [55] for interfacial drag and by Holowach [33,34] and Fore [74] for entrainment rate; however, the cited models are only appropriate for a portion of annular flow conditions. The current study integrates models similar to those proposed in References [33], [34], [55], and [74] into a comprehensive model covering the entire range of conditions for annular flow.

The ability to model liquid in the form of both continuous and discrete forms makes COBRA-TF will suited for calculating annular flow situations; however, the literature review and some preliminary analyses conducted as part of the current study have highlighted several concerns with the models used by the baseline version of the code. In addition to the use of air-water based correlations, the implementation of the churn-turbulent and annular flow regimes in the baseline version of COBRA-TF does not provide an accurate reflection of the physics of these regimes. Also, the baseline version does not explicitly consider the liquid bridging or wave undercut entrainment mechanisms nor does it apply different drop size correlations for the different annular flow regimes. Given the drastic differences in the structure of the interface, entrainment mechanism, and resulting drop sizes for the counter-current, churn-annular, unstable film, and stable film regimes it is believed that the explicit consideration of and the implementation of regime specific models into transient analysis codes for each of these regimes is warranted and therefore will be addressed as part of the current study.

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2.7. Motivation for the Current Study

This chapter has shown that an extensive amount of research has been conducted within the annular flow regime in an attempt to understand and quantify various phenomena of interest; however, the coupled nature of important phenomena in this regime (i.e. interfacial structure and entrainment) has made it difficult to experimentally isolate the effect of a single phenomenon to collect experimental data that supports the development of appropriate constitutive relationships for the interfacial and interfield exchange of mass, momentum, and energy. These complexities has made it more convenient, and computationally efficient, in previous studies to rely on simple correlations that are a functions of a few dimensionless parameters, rather than physical mechanisms, to quantify these phenomena. As a result of these limitations the predictive capability of reactor analysis codes has not been improved commensurate with the amount of research that has been conducted on the various physical phenomena. In response to this finding the current study leverages the existing physical knowledge and available experimental data to provide the desired level of accuracy in COBRA-TF.

Based on the findings of the literature review the current study was proposed in an effort to address the concerns outlined in the previous section and improve the predictive capabilities of these types of codes. However, similar to the difficulty in experimentally isolating a single phenomenon, the strong coupling that exists between the three fields in annular flow also makes it difficult to isolate the effect of a single parameter or correlation within a computational environment because the code prediction results from the individual predictions by each sub- model (i.e. drag, entrainment, deposition, etc.). This necessitates the comprehensive approach taken when developing the proposed modeling package in the current study. The logical progression for addressing the required models is:

1. regime boundaries or transitions 2. interfacial drag 3. entrainment rate 4. drop sizes generated 5. deposition rate 6. CCFL model

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The appropriate definition of flow regime transitions provides the foundation for the proposed modeling package. The accurate definition of these points is required because it outlines the conditions under which a given set of models will be applied and it is desired that these models be based on the physical structures that are expected to exist within the flow. Once regime definitions are established appropriate models can be selected to quantify the interfacial drag within each regime. The counter-current, churn-turbulent, churn-annular, and co-current upward regimes are characterized by markedly different interfacial structures; as such, it is doubtful that the same phenomenological model is applicable for all regimes. Much like the appropriate definitions for regime transitions are a prerequisite for the calculation of interfacial drag, an appropriate interfacial drag model is a prerequisite for the calculation of entrainment rate since a strong relationship exists between these two parameters. Once the interfacial shear model has been selected, the entrainment rate models can be addressed. The use of an interfacial shear model that is based on the physical structure of the flow then allows the results from the calculation of interfacial shear can then be used as input to a mechanistic entrainment model. Then, based on the entrainment mechanism appropriate drop size correlations can be selected. Next, a deposition rate model can be selected.

Lastly, the ability to assess the predictive capability of a code in the CCFL region, and then if necessary develop a three-field CCFL model, requires an accurate prediction of the interfacial drag within the annular regime because this quantity directly impacts the predicted field flow rates. Given the severe overprediction in the pressure gradient in the annular regime that was observed by both Holowach [33,34] and in the baseline results of the current study when using the baseline modeling package in COBRA-TF it was not feasible to consider the CCFL region prior to resolving the issues with the annular regime. Therefore, following the development of an improved annular flow modeling package in the first part of the work, as outlined above, the predictive capability of the code within the CCFL region was considered. The initial assessment revealed that even after resolving the predictive capability of the code in the annular regime a suitable prediction of the flow reversal phenomenon and CCFL region was not obtained. As a result a first-of-a-kind, three-field CCFL model was developed, implemented, and verified as part of the current work. This model leverages the available experimental data for this region while simultaneously improving the predictive capabilities of the code.

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In general, the focus of the current study is to improve the overall predictive capability of the code for annular flow situations by:

1. explicitly considering sub-regimes of annular flow 2. using more suitable definitions of regime boundaries 3. utilizing more appropriate closure relationships by incorporating regime-specific models for interfacial friction factor, entrainment mechanism, and drop size.

During the current study careful attention was paid to ensure the selected models provide an accurate physical representation of the phenomena of interest as they are currently understood and are based on steam-water data wherever possible to increase their validity for nuclear reactor analysis applications. An implied restraint exists on this study that the selected models must be amenable to implementation into the code and do not adversely impact the computational efficiency or numerical stability of the code in a significant manner. The current study used COBRA-TF to provide the baseline modeling package as a means for comparison and as a vehicle for assessing the newly proposed modeling packages, but the proposed package can be easily implemented in any three-field analysis tool. It is anticipated that this overall approach will lead to a more suitable modeling package that will improve the predictive capability of three- field analysis codes within the annular regimes. The remaining chapters will present the methodology used in the current study as well as the proposed modeling package and the associated results.

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3. Methodology

The literature review presented in the previous chapter highlighted that most of the available constitutive models available for the annular regime rely on simple correlations rather than mechanistic models. As a result, the focus of the current study is to develop a comprehensive modeling package that: 1) utilizes models that capture the physical phenomena as it is currently understood using a first-principles approach and 2) provides a self-consistent set of calculations, where parameters that appear in more than one model are calculated in the same manner within each model. Then, available experimental data is leveraged to develop a relationship between the theoretical and actual entrainment rates to account for any model deficiencies. Doing this ensures the modeling package is able to accurately reflect the experimental data while simultaneously improving the underlying physical basis of the code. This approach requires detailed experimental data covering the range of conditions of interest to be available. This section will summarize and assess the available experimental data to confirm a suitable database exists for both model development and assessment activities as well as to identify any deficiencies that exist. Then the modeling methodology employed in the current study will be discussed and finally the means for model assessment will be presented.

3.1. Experimental Data for Model Development and Assessment

This section reviews several of the annular flow experiments that have been conducted and discusses data that is readily available in the open literature. First, the desired data is described and then a review of several experiments that have been conducted is provided. Sections highlighting various experimental measurement techniques that have been applied are also provided as a means of determining which data sets are the most reliable for either model development or assessment.

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3.1.1. Description of Desired Data

The parameters of primary interest within the annular regime include: 1) the entrained fraction or film and entrainment flow rates, 2) film thickness, 3) pressure gradient, and 4) drop size. Code- to-data comparisons of the entrained fraction provide an assessment of the entrainment model, while the code-to-data comparisons of the pressure gradient provide an assessment of the interfacial drag model. Additionally, code-to-data comparisons of drop size are important to assessing the interfacial area available for heat and mass transfer. However, it will be seen in this section that for the data currently available in the open-literature a single experiment rarely measured the first three parameters simultaneously and most often never measured the fourth. Given that compensating errors can result in an accurate prediction of one parameter, but mispredict another parameter, it is desired to have the ability to verify the code predictions against as many parameters as possible. Such multi-tiered checks provide increased confidence that the underlying physics of the flow are being accurately captured by the code.

Numerous researchers have conducted annular flow experiments using several different working fluids in a variety of geometries and orientations. In these tests annular flow is typically established by flowing gas through the test section and then introducing the liquid into the test section peripherally using a porous sinter. This technique encourages the liquid to remain attached to the walls except that portion that becomes entrained by the vapor flow. Other experimenters have used annular slot or liquid jet injection techniques. The annular slot technique has been required to achieve larger liquid flow rates than are possible using a porous sinter. Meanwhile, the use of liquid jet injection causes the deposition process to be primarily responsible for driving the flow towards annular flow equilibrium, which is unlike porous sinter injection where the entrainment process is primarily responsible for driving the flow towards annular flow equilibrium. COBRA-TF provides the ability to specify the fraction of liquid initially entrained when defining a boundary condition, which is advantageous, and also makes it desirable to simulate experiments with different liquid injection techniques in the current study to ensure the code captures the differences in the physics associated with each technique.

Since the primary application of interest is nuclear reactors, a focus was placed on identifying available steam-water data over a range pressures, flows, and hydraulic diameters. Additionally,

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it was desired to use single-tube tests to eliminate the effects of turbulent mixing and void drift. It was also desired to use unheated tests to minimize the influence of mass transfer due to phase change on the results so as to isolate the mass transfer by entrainment and deposition; however, the interfacial heat transfer and associated mass transfer due to evaporation cannot be eliminated in steam-water experiments due to the pressure gradient, which reduces the saturation temperature along the test section. This complexity initially makes the use of air-water experimental data an attractive option, but given the substantial difference in the interfacial drag that has been observed for air versus steam in previous experiments [28,38] it was not desired to rely on air-water experimental data in the current work when the primary application is steam- water environments.

3.1.2. Discussion of Entrained Fraction Measurement Techniques

Two methods have been discussed in the open-literature to obtain measurements of the liquid film or dispersed droplet flow rates in annular flow experiments. The first consists of the extraction of the liquid film through a porous sinter. In this method the pressure differential across the porous sinter is varied and the flow that is extracted is passed through a condenser. Performing an energy balance on the condenser allows the extracted film flow rate to be calculated. When the differential pressure is too low incomplete film removal can occur and when the differential pressure is too high droplets from the gas core can be extracted along with the film itself; however, collecting data at several differential pressure conditions and then plotting the extracted liquid flow rate as a function of the extracted steam flow rate generates a “suction curve” that allows the condition of complete film removal to be determined. This method of measurement was used by: Hewitt & Pulling [83], Keeys et al. [81], Würtz [79], Singh [82], Yanai [64,84], Cousins et al. [70], Govan et al. [16], and Fore & Dukler [23]. Obviously this method measurement decreases in accuracy with increasing film flow rate because the enhancement of interfacial waves makes it more difficult to distinguish between the condition for complete film removal and the condition where droplets are extracted from the gas core. Würtz [79] cited this reason when describing the underestimation of the film flow rates at low gas velocities (i.e. thicker films) for his experiments relative to other experimental data. Also, as the film flow rate increases it may not be possible to generate a large enough differential pressure for the entire

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liquid film to be extracted or the length of the sinter may prohibit complete removal. Singh [82] attempted to study the effect of sinter length on the measured film flow rate using sinter lengths of ½, 1, and 2 inches; however, the trends in data are inconsistent (i.e. film flow rates were larger for 1-inch sinter than both the ½ and 2 inch sinters), which Singh [82] attributed to excessive oxidation/clogging in the 2-inch sinter since during the study this sinter was installed for over one month, while the other two were each installed for only about a week.

The second method involves the direct measurement of the entrained liquid mass flow using a Pitot-type sampling probe. For this method the probe is typically moved transversely across the flow path, taking measurements at several locations, and then the total mass flow rate of entrained droplets is calculated by integrating the measured local entrained liquid mass flux profile over the area of the gas core. This measurement technique was employed by Fore et al. [31,51], Barbosa et al. [18], and Andreussi et al. [87]. The primary difficulty with this technique is extracting the droplets without disturbing the flow. More importantly, this technique can introduce errors into the measured quantity since as the probe approaches the liquid film it can intercept interfacial waves, which are then wrongly contributed to the measurement of entrained flow rate. It is anticipated that the error introduced in this case is larger than the error introduced by the extraction of several droplets in the film removal technique. Additionally, as was mentioned in Section 2.4.3, the majority of the entrained liquid can be carried in several large droplets and thus missing one of these droplets can significantly impact the resulting measurement.

An additional concern that should be highlighted is that since the flow rate of only one of the fields is measured in the experiments the flow rate for the other liquid field and the entrained fraction are typically calculated using the measured quantity and the measured total liquid flow rate that was supplied to the test section [83]. This approach assumes the mass transfer due to phase change along the length of the test section is negligibly small, which becomes increasingly inaccurate as: 1) the relative change in pressure over the length of the test section increases and 2) the liquid and vapor are introduced with a larger degree of thermal non-equilibrium. The overall effect this has on the reported values is probably small and within the experimental uncertainty of the measurements, but it is important to recognize this effect does exist. In any event, the code- to-data predictions in the current study were made to the quantity that was actually measured in the experiment to minimize any bias that may exist in the calculated quantities.

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3.1.3. Co-Current Annular Flow Experiments

This section examines several unheated co-current annular flow experimental programs that have been conducted. These experiments primarily utilize steam-water and cover the desired range of conditions, but vary in the method used to introduce the liquid to the system and in the hydraulic length (L/D H) of the test sections. The data collected in these experimental programs corresponds primarily to the co-current upward annular regime, but some data was collected within the churn- annular/churn-turbulent regime.

Hewitt & Pulling [83]

Unheated experiments involving co-current upward steam-water annular flow were conducted in a vertically oriented 0.366-inch diameter tube for an outlet pressure range of 40 to 65-psia. A 5 2 single, fixed total mass flux of 2.19x10 lb m/ft -hr was tested and the flowing quality was varied over the range of 10% to 80% for each pressure condition tested. Both 6-ft and 12-ft test section lengths were used. Therefore, in addition to capturing any pressure effects at low pressure, these tests allowed for either: a) the existence of annular flow equilibrium conditions to be confirmed or b) developing flow effects to be captured. Metered flows of slightly superheated steam and slightly subcooled water were supplied to the test section. The water was injected using a 2-inch long porous sinter section and the film flow was extracted and measured at the test section outlet using a second porous sinter device. [83]

Keeys et al. [81]

Similar unheated experiments involving co-current upward steam-water annular flow were also conducted by Keeys et al. [81] at higher pressures (500 and 1000-psia). The test section was a 12-ft vertically oriented tube with a diameter of 0.496-inches. The total mass flux ranged from 6 2 1.0-2.0 x 10 lb m/ft -hr with flowing qualities between 15% and 75%. A total of twenty-one data points were collected. The methods of injection and collection were the same in this test as was described for the Hewitt & Pulling [83] tests. [81]

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Würtz [79]

A series of heated and unheated co-current upward steam-water annular flow experiments were conducted at the Ris ø National Laboratory. Over 250 experiments were performed between 435- psia and 1305-psia in one annular and two tubular test sections, each measuring 29.5-feet in length. The tests of primary interest to the current study are the unheated tests in the tubular test sections, which includes: 1) 81 experimental data points collected in a 0.394-inch diameter test section covering the entire range of pressures listed above with mass fluxes in the range of 0.37- 6 2 2.21 x 10 lb m/ft -hr and 2) 21 experimental data points collected in a 0.787-inch diameter test 6 2 section at 1015-psia with mass fluxes between 0.37-1.47 x 10 lb m/ft -hr. Comparing the results from these tests allows the influence of tube diameter to be assessed. [79]

Unlike the other tests discussed to this point Würtz [79] did not establish annular flow at the inlet, but rather used an electrically heated steam generator (i.e. pre-heater) to provide a two-phase mixture with a desired quality to the test section. The film flow rate was measured at the test section outlet by extracting the film through a porous wall section in a similar manner as the other tests discussed. The pressure gradient was also measured in these tests and the mean value over the last meter of the test section is reported in Reference [79]. The simultaneous measurement of pressure gradient and entrained fraction provides the means to simultaneously assess the accuracy of the models associated with both entrainment and interfacial drag, which neither the experiments conducted by Hewitt & Pulling [83] or Keeys et al. [81] is able to provide. [79]

Singh [82]

Unheated co-current upward annular flow experiments were conducted using steam-water at 1000 and 1200-psia in a 0.493-inch diameter tube with a length of about 7.5-feet. This set of experiments is unique since the liquid was introduced into the mixer as droplets using spray injector ( E ≈ 0.0 ) instead of as a film using a porous sinter ( E ≈ 0.1 ) as was done in the z=0 z=0 other experiments that have been considered. Therefore, it is anticipated that the tests conducted by Singh [82] provide a better assessment of the deposition model than the others that have been

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discussed previously since annular flow equilibrium is approached for this situation by depositing more droplets than are being entrained. It should also be noted that due to the short length of this test section (~183 L/D H) relative to the other tests that were discussed it is possible that annular flow equilibrium did not exist at the outlet. As a result these tests also provide useful data to assess developing flow effects. [82]

In these tests separate liquid and vapor streams were introduced to a mixer just prior to the test section inlet and a preheater was used to superheat the steam by about 10°C. Meanwhile, a separate preheater was used to heat the water to a few degrees below the saturation temperature and a porous sinter was used to measure the film flow rate at the outlet. These experiments also measured the total pressure gradient over roughly the entire test section (inlet to 7-ft). Given the oxidation/clogging issue described previously only the data collected using the ½-inch sinter, which Singh [82] indicates are the most reliable, were considered as part of the current study. This data consists of 11 data points collected at 1000-psia and 3 data points collected at 1200-psia 6 with total mass fluxes ranging between 0.2-0.7 x 10 lb m/hr-ft and qualities ranging between 0.3 and 0.92. [82]

Yanai [64,84]

Unheated, co-current upward steam-water annular flow experiments were conducted with an outlet pressure of 49.3-psia in a vertically oriented tube with a diameter of 0.472-inches and a 5 2 length of approximately 7.55-feet. Total mass fluxes of 1.03, 1.54, and 2.06 x 10 lb m/ft -hr were tested with various flowing qualities ranging between 10% and 80%. Therefore, unlike the Hewitt & Pulling [83] tests, these tests capture any flow rate effects at low pressure. Additionally, the liquid was introduced in these tests using a 0.0236 diameter multi-nozzle mixer. The results of a sensitivity study conducted by Sugawara [64] using the three-field Film Dryout Analysis in Subchannels (FIDAS) code suggests that an initial entrained fraction of 50% ( E ≈ 5.0 ) yields the best agreement with the experimental data for this type of liquid z=0 injection. Therefore this value will be used initially in the current study; however a similar type of sensitivity study will be conducted as part of the current study and the results of this are

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presented in Chapter 4. Meanwhile the film flow was again extracted and measured at the test section outlet using a porous sinter device, similar to the other tests discussed.

Cousins et al. [70]

All of the experiments discussed to this point only provided a pointwise measurement of entrained fraction at the outlet and it is often assumed annular flow equilibrium was attained by this point, but a limited amount of evidence is available to support this theory. Regardless, as discussed in Section 2.4, even if annular flow equilibrium was attained, the measured entrained fraction within the region where annular flow equilibrium exists is only dependent on the behavior within the developing flow region. Therefore, it is desired to obtain a set of experimental data that includes measurements of the entrained fraction as a function of axial position to make code-to-data predictions for the purposes of model assessment within the developing flow region, but the available data of this type is extremely limited. One of the few available data sets was collected by Cousins et al. [70], which measured the entrained fraction and pressure gradient as a function of axial position in a 0.375-inch diameter test section; however, these experiments used air-water at low pressure (20-psia outlet). Since these experiments captured developing flow effects the consideration of these experiments in the current study will provide increased confidence that the an accurate prediction of the integral effect (i.e. outlet entrained fraction) by the code is achieved for the correct physical reason. However, these comparisons are complicated due to the differences in interfacial drag for air- water compared to steam-water.

3.1.3.1. Conclusions on Available Co-Current Annular Flow Data

Several unheated co-current vertical annular flow experiments involving steam-water covering the range of conditions of interest are identified and evaluated in the previous section. A summary of test conditions is provided in Table 3-1. A range of pressures is represented and the range of tube diameters covers the range of hydraulic diameters of subchannels typically employed in commercial reactor applications. Also, as mentioned previously the inclusion of the

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data collected by Singh [82] and Yanai [64,84] allows for the effects of developing flow and liquid injection method to be assessed. However, no low-pressure steam-water pressure gradient data and no steam-water film thickness or drop size measurements at any pressure were obtained in these experiments.

The last row of Table 3-1 indicates which data sets were used for the purposes of model development in the current study and the remaining sets were then used for model assessment. Careful attention was paid when selecting the subset of the data from these tests that would be used for model development to ensure that it adequately covered the ranges of interest, but at the same time it was desired to leave some gaps to ensure that finished model was extendable to other conditions. For example, the data collected by Würtz [79] at 1015-psia in the 0.394-inch diameter test section was purposely excluded to assess whether the proposed model could handle these cases, which should roughly be an interpolation of the data considered at the other pressures in this same test section. Similarly, the data from several of the pressure conditions tested by Hewitt & Pulling [83] in their longer test section were excluded for the same reason. Additionally, the data collected by Würtz [79] in the 0.787-inch diameter test section was excluded to assess the ability of the proposed model to be extended to larger tube diameter situations. Meanwhile, the data collected by Yanai [64,84] was excluded to assess the extension of the model to a wider range of flow rates at low pressure. Also, excluding the data collected by Singh [82], Yanai [64,84], and Hewitt & Pulling [83] from the shorter test section allowed the ability of the proposed model in handling situations where non-equilibrium conditions at the outlet were supposed to exist to be assessed. Finally, excluding the data collected by Singh [82] and Yanai [64,84] also allowed for the ability of the model to handle cases with different injection types since all the cases considered for model development used the porous sinter liquid injection technique. The ability of the proposed model to handle these situations without explicitly including them in the model development process provides confidence that the model is able to capture the physical mechanisms causing entrainment within the flow.

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Table 3-1: Summary of Test Conditions for Available Co-Current Upward Annular Entrainment Experiments. Hewitt & Yanai Singh Keeys et al. Würtz Test Program Pulling [83] [64,84] [82] [81] [79] Hydraulic Diameter 0.366 0.472 0.493 0.496 0.394 0.787 (inches) Test Section Length 12.0 6.0 7.546 7.5 12.0 29.53 (feet)

L/D H ~393 ~197 ~191 ~183 ~288 ~898 ~450

35, 40, 45, 40, 50, 435, 725, Pressure (psia) 50, 55, & 50 1000 & 1200 500 & 1000 1015 & 65 1015, & 1305 65 Total Mass Flux 1.03, 1.54, & -5 2 2.19 2.0-7.0 9.68-20.4 3.69-22.1 3.69-14.8 (10 lb m/ft -hr) 2.06

Mass Flow 125, 187, & 160 265 - 930 1345 - 2750 300 - 1900 1250 - 5000 Rate (lb/hr) 250 Dimensionless Superficial Vapor 3.1-20.9 3.0-12.0 3.8-11.8 4.9-16.8 1.6-29.0 1.4-11.3 Velocity Flow Quality 0.10-0.80 0.10-0.80 0.30-0.93 0.14-0.69 0.10-0.60 0.20-0.70

E x x X x x x x

dP/dz - - - x - x x

δ ------Measured Measured Parameters Parameters

Dd ------

Liquid Spray/ Porous Sinter Multi-Nozzle Porous Sinter Porous Sinter Injection Type Droplet

Number of Usable 70 (for E) 70 38 21 14 21 20 Data Points 79 (for dP/dz) Data collected using 1-inch Runs Excluded N/A N/A N/A N/A 234 & 240 N/A and 2-inch sinters Yes, but only data at Yes, but 435, 725, & only Yes, but not Used for Model 1305-psia data at 35, No No No runs 4, 5, 6, No Development? with outlet 50, and 65- and 10 flowing psia qualities > 20% Würtz [79] cases that were excluded from Model Development: 234-265 (1015-pisa) as well as 201, 207, 208, 209, 214, 219, 220, 226, 227, 228, 268, 269, 274, 275, 276, 277, and 278.

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3.1.4. Counter-Current Flow Limitation (CCFL) Experiments

A variety of Counter-Current Flow Limitation (CCFL) experiments have been conducted over a range of hydraulic diameters and in various configurations; however, most of these experiments have been conducted using air-water at low pressure conditions. Additionally, since all of the governing parameters have not yet been resolved, the data from each of these experiments is specific to the test configuration. Experiments have been conducted to either determine the mechanism of CCFL or support model development efforts by: Wallis [4], Zabaras et al. [7], Richter [8], McQuillan & Whalley [9], Bharathan & Wallis [11], Govan et al. [16], Dukler & Smith [43], Bharathan et al. [65], Pushkina & Sorokin [90], and Wallis & Makkenchery [91], among others. For the purposes of the current study it was decided to focus on the experiments conducted by Dukler & Smith [43].

Dukler & Smith [43]

Low-pressure, air-water experiments were conducted to measure pertinent flow rates, pressure gradients, and liquid film thicknesses (conductance probes) within the flooding region. Liquid injection rates of 100, 250, 500, and 1000-lb m/hr (corresponding to liquid Reynolds numbers of

310, 776, 1552, and 3105) were tested. Air flow rates ranged from 136 to 330-lb m/hr for the pressure gradient data and from 0 to 280-lb m/hr for the film thickness data.

The test section was a Plexiglas tube 13-feet long and 2-inches in diameter with a 5-foot long calming section for the incoming air located below the inlet. The inlet of the test section was tapered from 2-inches to 5-inches, which created an expansion nozzle and allowed the falling film to exit the test section with minimal interaction with the incoming air flow. Doing this minimized the entrainment generated in the inlet region so that the only entrained droplets generated within the test section would be measured at the outlet.

The system consisted of a porous sinter for supplying liquid to the test section, four measurement stations (film thickness and pressure gradient), inlet section for measuring the liquid film downflow, and exit section for measuring the film and entrained liquid flows out of the top of the

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test section. For the pressure gradient measurements two measurement stations were placed above the liquid entrance and the other two were placed below. Meanwhile, for the film thickness measurements only one measurement station was placed above the liquid entrance and the other three were placed below. Time traces of these parameters were recorded and then the data was processed to determine time-averaged mean values for these parameters.

3.1.4.1. Conclusions on Available CCFL Data

Given the nature of the CCFL model that will be developed in the current study it is not desired to simulate a range of tests as will be done for the other regimes, but rather it was necessary to develop a simple verification problem, which will be described along with the proposed model in Chapter 7, to ensure the appropriate implementation of the model. Once the model was verified it was desired to simulate a single set of experiments to make code-to-data predictions both with and without the newly developed CCFL model activated to show the improvement in the predictive capability of the code. Since the experiments conducted by Dukler & Smith [43] were the best instrumented of those experiments available they were selected to be used in the current study to make these comparisons.

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3.2. Approach

As mentioned previously, the current study uses the predictions of various annular flow experiments for model development and assessment. The following sections describe the various aspects associated with the analysis and assessment of the experiments considered in the current study. The procedure of model development and assessment is described in three parts: 1) methodology and development of input, 2) execution of cases, and 3) post-processing, including error analysis and sensitivity studies.

3.2.1. Description of Input Parameters

The section provides a general overview of how the experiments were simulated using COBRA- TF in the current work. A summary of the boundary conditions that were applied when simulating these tests is provided in Table 3-2. A mass source boundary condition (Type 4) was used to introduce the liquid into the system when simulating these tests because this represents a better approach than specifying two-phase flow conditions using a Type 2 boundary condition, which assumes the flow is both homogeneous and in thermodynamic equilibrium. Using a Type 2 boundary conditions also requires the inlet vapor void fraction, which was not measured in these tests, to be specified. Specifying the flow rate and injection area using a mass source boundary condition allows the fluid to be introduced with a more appropriate velocity, assuming the liquid injection area is known; however, for some of the experiments used in the current study the liquid injection area was not specified in the experimental data report and as a result various assumptions were applied to estimate this quantity. Additionally, for several of the experiments injected two-phase mixtures that were not in thermal equilibrium to the test section and the use of a mass source boundary condition allows the appropriate enthalpies of each phase to be specified. Finally, the use of a mass source boundary condition also provides the ability to specify an entrance length, where the liquid is not introduced into the test section until several nodes downstream of the inlet where the vapor flow is introduced, which is more representative of the actual test section configuration. While the effect of doing this may be minimal, it does provide a more accurate representation of the test conditions.

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Table 3-2: Summary of Boundary Conditions. Location Type Parameters Specified Experimentally measured inlet vapor mass flow rate and Inlet 2 enthalpy. Outlet 1 Experimentally measured system pressure. Liquid Experimentally measured liquid mass flow rate and 4 Injection enthalpy. Applied several nodes downstream of the inlet.

As alluded to previously, various assumptions were applied in the current study to calculate the liquid injection area associated with the mass source boundary condition based on the amount of information provided in the report for each experiment for this parameter. For example, no information was provided regarding the porosity of the sinters used and the length of the sinter was only provided by Hewitt & Pulling [83]. A brief description of the basis for the sizing of the liquid injection area for the tests simulated in the current study, along with other details of the input parameters used, is contained in Table 3-3. Given the large amount of uncertainty associated with this parameter a sensitivity study on this input parameter was performed as part of the current study and the results are presented in Chapter 4.

It can also be seen in Table 3-3 that computational node heights between 3 and 4-inches were used in the current study. These node sizes were selected because they provided a reasonable level of axial detail within the simulations while maintaining a reasonable computational time for the code; however, during the current study in was desired limit the sensitivity of the proposed modeling package on computational node height. As result, it was necessary to check the sensitivity of the predictive capability of the proposed modeling package to this parameter. The results of this analysis are also presented in Chapter 4.

It should also be noted that the measured outlet flowing quality and total mass flow rate was used to determine the phasic mass flow rates at the inlet that were supplied as boundary conditions to COBRA-TF when simulating the experiments conducted by Hewitt & Pulling [83], Yanai [64,84], Singh [82], and Würtz [79] . This approach, which had to be used for these tests since it was the only information that was provided, assumes the mass transfer due to phase change along the length of the test section is negligibly small. As mentioned previously, this assumption

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becomes increasingly inaccurate as: 1) the relative change in pressure over the length of the test section increases and 2) the fluid is introduced with a larger degree of thermal non-equilibrium.

All of the simulations were initialized with pure vapor in the test section. The simulation would then be run for a total of 10-seconds, with the liquid flow rate being linearly “ramped on” over the course of the first second of the simulation. This total time period was determined to be sufficient for achieving a steady-state solution for all of the cases examined,. This method was applied when simulating the experiments conducted by Keeys et al. [81] and Singh [82], where single combinations of pressure, total flow, and flowing quality were tested; however, in each of the other experiments considered a range of flow qualities were tested for each pressure / total flow rate combination. Therefore, when simulating these tests a single input deck would be used with forcing functions being applied to changes the flowing quality over the course of the simulation. In these cases, after the first 10-seconds of the simulation the phasic flow rates would be linearly varied over a simulation time period of 1-second to achieve the next desired flowing quality. The simulation would then be run for another 9-seconds to allow steady-state conditions to be obtained. This process would be repeated until all flowing quality conditions were tested for a given pressure / total flow rate condition. The hysteresis effect of simulating the experimental cases in this manner as opposed to individually will also be examined as part of the sensitivities study.

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Table 3-3: Summary of Modeling Parameters. Hewitt & Yanai Würtz Singh Test Program Keeys et al. [81] Pulling [83] [64,84] [79] [82] Computational Cell 3.0 3.937 3.937 3.0 3.0 Length (inches)

Entrance Length (feet) 1.0 0.656 1.64 0.5 1.0

Diameter of single Assumed area Assumed axial Assumed area to be 2-inch long nozzle was 0.0236- to be equal to jet has an area equal to the flowing sinter that is inches, Assumed Injection Area the cross- equal to 10% of quality multiplied by assumed to have multi-injector sectional area the total flow the cross-sectional a 50% porosity consisted of 20 of the tube area area of the tube nozzles. Inlet Entrained 0.0 0.5 0.0 1.0 0.0 Fraction Condition Information Information Condition Mass Source Boundary Boundary Mass Source Inlet Drop - 0.005 - 0.005 - Diameter (feet) Simulated a given pressure/total flow condition using a single input deck. Step decreases in flowing Notes Each case simulated individually. quality were specified using forcing function tables for both vapor and liquid mass flow rate. Note: Holowach [33,34] used 6-inch nodes and a two-phase inlet boundary condition when simulating the tests conducted by Hewitt & Pulling [83], Keeys et al. [82], Singh [81], and Würtz [79] in his work.

3.2.2. Optimization Scheme for Model Development

The methodology employed in the current study when developing the proposed co-current upward annular flow package was to utilize or develop theoretical models related to the structures present within the flow and mechanisms causing the phenomena of interest and then a relationship between the actual and theoretical entrainment rates was developed based on available experimental data to account for any model deficiencies. As mentioned previously this approach ensures the modeling package is able to accurately reflect the experimental data while simultaneously improving the underlying physical basis of the code. Additionally, the development of such a relationship also provides insight into the processes that are not being accurately captured and lays the foundation for future model development activities. This approach was applied to the code calculated entrainment rate in the current study since it is hypothesized that this is the single most important parameter to achieving an accurate prediction of annular flow because it governs the distribution of liquid between the film and entrained fields.

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Additionally, the constitutive relationships currently available for this parameter have the highest uncertainty of any of the relationships required for predicting annular flow behavior.

For the purposes of the current study the optimized value of the multiplier was defined as the value that minimized the relative error in the entrained fraction when applied uniformly along the test section to the code calculated entrainment rate in each computational cell.

S′ = x S′ (3-1) ( E ) j k ( E,Model ) j where the subscript ‘ k’ refers to the experimental case and ‘ j’ refers a given computational cell.

The optimized value of the multiplier was obtained individually for each of the model development cases. This subset was outlined in Table 3-1 and consisted of 83 steam-water experimental cases out of a total of 263 steam-water experimental cases considered in the current study. A driver program was written in MATLAB to determine the optimized value of the multiplier for each case using the secant method. The automation of this process using a driver program allowed more experimental cases to be considered in the current study while simultaneously limiting the chance for human error. Automating this process also allowed more model combinations to be tested since the overall execution of the simulations was drastically simplified and more importantly allowed the focus to be placed on improving code-to-data comparisons and examining data rather than running cases. The driver program was designed to:

1) read in the test section geometry, test conditions, and experimental data, 2) generate the input deck, 3) select a value for the multiplier to apply to the code calculated entrainment rate in each computational cell, 4) execute COBRA-TF using the specified value of the multiplier, 5) calculate the relative error between the experimental and predicted entrained fraction, 6) if the convergence criterion was not satisfied the secant method was applied to calculate the next value for the multiplier to be tested 7) repeat steps 4 through 6 until convergence was achieved.

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The secant method is derived from the Newton-Raphson Method for solving non-linear equations by approximating using a first-order backwards finite-difference relationship.

i i+1 i f (xk ) xk = xk − i (3-2) f ′()xk where:

f xi − f xi−1 ′ i ( k ) ( k ) f ()xk = i i−1 (3-3) xk − xk

Substitution yields the recursive formula:

i i i−1 i+1 i f (xk )(xk − xk ) xk = xk − i i−1 (3-4) f ()()xk − f xk

The independent variable in Equation (3-4) is the value of the multiplier and the dependent variable is the relative error in the entrained fraction, where the relative error is defined as:

i i Ek,exp − E(xk ) f ()xk = (3-5) Ek,exp

The secant method requires two pairs of independent parameters and associated errors to initiate the algorithm, but unlike the bisection method, it is not required that these points bracket the solution. Unlike the bisection method, the secant method may or may not converge, but when it does converge it does so faster than the bisection method. Convergence was assumed to have occurred when the absolute relative error in the entrained fraction was less than 1%.

Once the optimized value of the multiplier for each case was determined a multi-variable regression analysis was used to correlate these results to determine a relationship between the actual and theoretical entrainment rate in terms of COBRA-TF predicted values at the test section

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outlet. In general, the goal of regression analysis is to determine the functional relationship between a response variable and set of predictor variables. While the dependent variable in this case (i.e. outlet entrained fraction) is subject to experimental uncertainty, the error in the COBRA-TF prediction is assumed to depend solely on the predicted values of the multiplier. Therefore, given the set of optimized multipliers:

(x1 , x2 ,..., xM ) (3-6) a model function can be selected:

Θ = f (Π1 ,Π 2 ,..., Π N ) where: M >> N (3-7) and the dependence of selected dimensionless parameters on the response value can be determined. The ‘regress’ function in MATLAB does this by minimizing the sum of the squares of the deviations of the data from the model function (i.e. least-squares fit). It was assumed in the current study that the correction factor could be correlated using:

N Bn Θ = B0 ∑Π n (3-8) n=1

This non-linear model function can be transformed into a domain in which this function is linear, which allows the standard linear regression approach to be applied. Taking the natural logarithm of Equation (3-8) yields:

ln ()()xk = ln B0 + ∑ Bn ln ()Π k,n for: k = 2,1 ,..., M (3-9) n=1

This expression can be rewritten in matrix form as:

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ln (B0 )  ln ()x1  1 ln ()()()Π 1,1 ln Π 2,1 L ln Π ,1 N       B1 ln ()x 1 ln ()()()Π ln Π L ln Π    2  =  1,2 2,2 ,2 N  B  (3-10)  M  M M M O M  2      M  ln x 1 ln Π ln Π L ln Π  ()M   ()()()M 1, M 2, M ,N    BN  or:

X = ΠB (3-11)

It is then desired to solve this expression for the vector of partial regression coefficients by:

B = [Π]−1 X (3-12) using MATLAB. Once the coefficients are known a functional relationship between the actual and theoretical entrainment rate is available.

As mentioned previously the values of parameters correlated against (Π) for this analysis corresponded to values predicted by COBRA-TF at the test section outlet. Therefore, the extension of this correlation to local conditions required several iterations to arrive at the final result because certain parameters, such as film Reynolds number, film thickness, etc., can vary substantially over the axial length of the test section.. In other words, a set of optimized values would be obtained and correlated and then the resulting correlation would be applied in COBRA- TF as:

S′ = x S′ Θ (3-13) ( E ) j k ( E,Model )j such that a new set of optimized values could then be obtained and recorrelated. Small variations were observed in the second iteration of this process, but by the third iteration the newly determined optimized values for the multiplier were typically very close to 1.0, thus indicating that a converged solution was obtained.

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3.2.3. Execution of Cases for Model Assessment

Following the development of the new entrainment rate model it was validated using a much larger experimental data base. To conduct this assessment a series of PERL scripts were written to:

1) read in the test section geometry, test conditions, and experimental data, 2) generate the input decks, 3) execute the code, 4) store input and output files, 5) strip the desired information from the output files at the appropriate axial elevations, 6) perform any necessary data reduction and generate output files that compare the experimental and COBRA-TF predicted values, 7) generate desired plotting scripts and create desired plots using the AptPlot software package.

Again, automating this process allowed more experimental cases to be considered in the current study while simultaneously limiting the chance for human error. The automation also provided a higher level of quality assurance and repeatability between the cases run with the original and proposed model sets.

For the purposes of the current study 171 steam-water experimental cases were examined in addition to the 83 steam-water experimental cases that were used for model development (263 total steam-water data points). Additionally, a total of 55 air-water experimental data points were examined from the tests conducted by Cousins & Hewitt [67]. Finally, while not examined as part of the current study, it should be noted that an additional 230 experimental cases conducted using air-water at low pressure as well as 252 experimental cases conducted using nitrogen-water at both low and high pressure provided by Fore et al. [51]. This data can be used for additional assessment of the proposed model set.

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3.2.4. Comparison of Results

As previously mentioned the desired quantities in annular flow for which to make code-to-data comparisons are: 1) entrained fraction, 2) pressure gradient, 3) film thickness, and 4) drop size. Since the entrained fraction was not measured directly, but was instead inferred from measurements of film or entrained flow rate assuming no mass transfer due to phase change, it was desired to make direct comparisons to the measured quantity; however, comparisons with the entrained fraction are still made since it provides a means for normalizing the results.

The results obtained using the original and proposed modeling packages were compared to one another on the basis of an error analysis. Based on the suggestion of the team of developers for the United State Nuclear Regulatory Commission’s (USNRC) Automated Code Assessment Program (ACAP), the Figures of Merit (FOM) that provide the best means of quantifying the agreement between experimental and predicted values of pointwise data include: 1) Index of

Agreement (IA), 2) Mean Fractional Error (MFE), and 3) Cross-Correlation Coefficient ( ρxy ). These metrics are calculated as:

N 2 ∑[]()Oi − O − ()Pi − P i=1 IA = 0.1 − N , (3-14) 2 ∑[]Oi − O + Pi − P i=1

 N  1 (Oi − Pi ) MFE = ∑ 2/1  , (3-15) N  i=1 [( Oi + Pi ]2/)  and:

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 N   (O −O )( P − P)   ∑ i i  ρ = i=1 xy  2/1  (3-16)  N  N    2  2   ∑(Oi −O ) ∑(Pi − P)    i=1  i=1  

It is indicated by ACAP team of developers that the MFE and ρxy metrics capture the distinct features of bias and correlation of the data. In addition to these metrics, the standard definition of mean relative error, given as:

 N  1 (Oi − Pi ) MRE = ∑  (3-17) N  i=1 Oi  and absolute mean relative error, given as:

N 1  Oi − Pi  MRE = ∑  (3-18) N  i=1 Oi  were used as additional means of quantifying the results. The error analysis was conducted in the current study using the MATLAB software package due to the powerful calculation capabilities of this program for matrix data structures.

It should be noted that a physical bound exists on the calculation of error for the entrained fraction or entrained mass flow rate when the code overpredicts these values since these values cannot exceed one or the total liquid flow rate, respectively. The maximum error for overprediction is therefore calculated as:

W W ( l,tot − e )exp ( 0.1 − Eexp ) Err max = ≈ (3-19) We Eexp

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In addition to making code-to-data comparisons of the standard annular flow quantities described above it was also desired to compare several other available parameters to other results that have been presented in the open-literature or to provide additional insight into annular flow processes. These comparisons were made both at the outlet and axially along the flow path. For example, the ratio of entrainment rate to deposition rate along the test section was calculated to determine if annular flow equilibrium conditions had been predicted to exist. Also, the force per unit length acting on the film to the total interfacial force per unit length, which is:

K U vl vl (3-20) KvlUvl + Kve Uve and the ratio of momentum transfer by deposition to entrainment, which is:

S U D e (3-21) SEUl were calculated to determine the relative impact of these parameters. Additionally, disturbance wave height, length, slope, velocity, and intermittency were calculated for the purposes of comparison to values for these quantities that have been observed by other researchers.

3.2.5. Sensitivity Studies

Based on several of the concerns outlined in Section 3.2.1 a series of sensitivity studies were performed as part of the current study. The parameters of interest include: computational node height, liquid injection area (i.e. area associated with mass source boundary condition), initial entrained fraction, initial drop size, and hysteresis effects. The computational node size sensitivity was assessed by increasing the length of the computational nodes by a factor of two, which reduced the total number of computational nodes by a factor of two, for all of the cases considered and then comparing the results. Meanwhile, the sensitivity of the results to the liquid injection area was assessed by reducing the area used to simulate each cases considered by a

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factor of two. Next, the sensitivity of the results to the initial entrained fraction and initial drop size were assessed by varying these parameters for the series of experimental runs conducted by

Yanai [64,84] at a total mass flow rate of 250-lb m/hr. Lastly, the hysteresis effect of simulating the cases using a forcing function to induce step changes in flowing quality as a function of simulation time versus simulating each case individually was also assessed. The results of these sensitivity studies are presented in Chapter 4.

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3.3. Conclusions

The experiments outlined in this chapter represent a suitable database for both model development and assessment as part of the current study, but several gaps do exist. A sufficient amount of steam-water covering the range of interest exists in the co-current upward annular regime for entrained fraction, but less data covering a smaller range exists for the pressure gradient and film thickness. Most importantly, no pressure gradient measurements were made in any of the steam-water experiments conducted at low pressure. Furthermore, no steam-water data is currently available for drop size or within the CCFL region and the availability of data within the churn-annular/churn-turbulent region is severely limited. It was mentioned in Section 3.2.3. that a wide range of annular flow data that was obtained using both air-water and nitrogen-water is readily available that simultaneously measures all of the parameters of interest, including drop size; however, this data was not used in the current study due to the documented differences in interfacial drag for steam-water and other working fluids. The inability to isolate the effect of a single phenomenon when simulating annular flow conditions would make it difficult to draw any conclusions relative to the ability of the code to predict these other parameters. Given that the primary application was for steam-water environments it was desired to focus on this type of data in the current work, but this other data may be useful for future work

The use of automation schemes in various aspects of the current study, including optimization, execution, post-processing, and assessment, allowed a larger number of cases to be considered, reduced the possibility for human error, and increased the repeatability of the results. However, the most important aspect of the automation schemes was that they allowed the time and effort to be focused on the evaluation of results rather than the execution of post-processing of the cases simulated. This allowed for a variety of different model combinations to be tested and ensure the appropriate behavior of the annular regime, as it is currently understood based on previously conducted experimental studies, was being captured by the proposed modeling package.

Lastly, the methodology employed in the current study provides a means of implementing models that are based on the physical mechanisms causing entrainment, as they are currently understood, while simultaneously leveraging the available experimental data to ultimately improve the predictive capability of the code. Ideally, the functional relationship between the actual and

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theoretical entrainment rates would be equal to one; however, it is unlikely this will occur due to the variety of models, each developed in separate, unrelated studies, that must come together to make such a prediction. Regardless, it is likely that the correlating parameters determined in the current study will provide insight on which correlations contained within the model may not be accurately capturing some behavior relative to the experimental data. This methodology is not meant to provide a final solution to this complex problem, but rather provide a means for improving the predictive capability until more detailed data that is able to support further model development, or provides the ability to develop a consistent set of models from a single experimental study, becomes available.

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4. Annular Flow Modeling Package

This chapter presents the proposed annular flow modeling package that has been developed during the current study. First, the results obtained using the modeling package in the baseline version of COBRA-TF are presented. Then, the constitutive relationships and physical models included in the proposed modeling package are discussed. Lastly the results obtained using this newly proposed modeling package once it was implemented in COBRA-TF are presented. It will be seen that a significant improvement in the predictive capability of the code within the annular regime was obtained when the newly proposed modeling package was employed.

4.1. Results Obtained using the Baseline Modeling Package in COBRA-TF

As previously alluded to, the current study was proposed because the predictions obtained using the modeling package in the baseline version of COBRA-TF did not provide reasonable agreement with the experimental data. For example, Figure 4-1 shows the code-to-data comparison of the axial pressure gradient from annular flow tests conducted by Singh [82] and Würtz [79] (113 total data points) while Figure 4-2 shows the code-to-data comparison of the outlet entrained fraction for all of the steam-water cases that were considered in the current study (254 total data points). The mean relative error in the pressure gradient for the baseline modeling package was 108% (overprediction) while the mean relative error in the outlet entrained fraction was 20% (underprediction). More importantly, it can be seen in Figure 4-2 that the worst predictions were obtained for the low pressure data that was considered. Reasonable agreement was obtained for the higher pressure cases considered, which is to be expected since a majority of the cases that were simulated in the current study corresponded to data that was used to develop the entrainment rate model that was used by the baseline version of COBRA-TF.

Figures 4-3 and 4-4 provide a more detailed comparison of the outlet entrained flow rate as a function of flowing quality for the low pressure tests conducted by Hewitt & Pulling [83] and Yanai [64,84], respectively. As indicated on the figures, the symbols correspond to the COBRA- TF predicted values while the lines correspond to the experimental data. The tests conducted by

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Hewitt & Pulling [83] were run at a single total mass flow rate (160-lb m/hr) with various outlet pressures while the tests conducted by Yanai [64,84] were run at a single outlet pressure (50-psia) with various total mass flow rates. As highlighted previously, these figures show that COBRA- TF severely underpredicted a majority of the experimental data from these tests, but the code did capture the overall trends observed in the experimental data.

Lastly, Figures 4-5 and 4-6 show the code-to-data predictions of entrained flow rate and pressure, respectively, as a function of pressure for the low pressure, air-water experiments conducted by Cousins & Hewitt [67]. These tests were run at a fixed air mass flow rate and fixed inlet pressure while varying the liquid mass flow rate. The results presented in Figures 4-5 and 4-6 correspond to cases where the air flow rate was 40-lb m/hr and the inlet pressure was 40-psia. Unlike Figures 4-3 and 4-4, the symbols in the Figures 4-5 and 4-6 correspond to the experimental data while the lines correspond to the COBRA-TF predicted values. The results shown in Figure 4-5 for these tests display an overprediction in the entrained fraction and also highlight the existence of numerical instabilities in the COBRA-TF solution within the annular regime for the cases with lower liquid flow rates. These conditions correspond to vapor void fractions near the transition between the unstable and stable film regimes and the reasons for this behavior in this region were outlined in Section 2.2.2.2. Meanwhile, since a fixed inlet pressure was used when conducting these tests the measured outlet pressure had to be supplied as a boundary condition to COBRA- TF to simulate these experiments. As a result the difference in the predicted and measured quantities is evident at the lower axial elevations, as seen in Figure 4-6. It can be seen that an overprediction exists, but it is not as severe as was observed for the steam-water cases. This improved prediction for air-water cases is to be expected since the interfacial drag model in the baseline version of COBRA-TF was developed using low pressure air-water experimental data.

The results presented within this section provide the baseline for the current study. During the course of the current study it was desired to remove the biases in the COBRA-TF predictions to improve the predictive capability of the code within the annular regime. The remainder of this chapter will present the proposed model and the associated results. A comparison between the results obtained using the baseline and newly proposed modeling package in Section 4.3 will reveal the ability of the proposed modeling package to address the concerns highlighted by the results presented in this section.

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Figure 4-1: COBRA-TF Predicted versus Experimentally Measured Axial Pressure Gradient using the Baseline Modeling Package.

Figure 4-2: COBRA-TF Predicted versus Experimentally Measured Outlet Entrained Fraction using the Baseline Modeling Package.

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Experimental Data Total Liquid Flow Rate

COBRA-TF Prediction

Total Liquid Flow Rate

Experimental Data COBRA-TF Prediction

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Total Liquid Flow Rate Pinlet = 40-psia

Wair = 40-lb m/hr

COBRA-TF Prediction

Experimental Data

Pinlet = 40-psia

Wair = 40-lb m/hr COBRA-TF Prediction

Total Liquid Flow Rate

Experimental Data

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4.2. Description of Proposed Modeling Package

The literature review presented in Chapter 2 revealed several models for predicting various phenomena of interest in annular flow that were more suitable than those used by the annular flow modeling package in the baseline version of COBRA-TF. Models for additional phenomena within annular flow that were previously not considered by COBRA-TF were also identified. Then, Section 2.7. outlined the desirable approach for developing an annular flow modeling package. Using this approach the following sections provide details on the constitutive relationships that were incorporated into a comprehensive annular flow modeling package that was developed within the current study.

4.2.1. Preliminary Modifications

The modifications described in this section were made to enhance the ability of COBRA-TF to model various flow paths and provide a more accurate calculation of several annular flow quantities. First, based on the suggestion of Holowach [42], the ability to specify channel- specific flow regime transitions depending on the hydraulic diameter of the flow path being modeled was implemented into the code. The most significant impact of this change to the current work is that for small hydraulic diameter flow paths the vapor void fraction corresponding the liquid bridging criterion is reduced from 0.8 to 0.6. As will be discussed in the next section this criterion will serve as the transition between churn-turbulent and unstable annular flow in the flow regime map for the proposed model set. Concurrent with this change the ability to specify whether the flow path is circular or non-circular, which impacts the calculated wall friction factor [93], was also added to COBRA-TF. Given the range of hydraulic diameters and flow path geometries that have been examined in various annular flow experimental programs it was desired to implement these to capabilities into the code as part of the current work.

Additionally it was found that the surface areas associated with the calculation of the entrainment and deposition rates in the baseline modeling package corresponded to the surface area of the flow path rather than the interfacial area of the liquid film. The flow path surface area was most likely used since the entrainment and deposition rate models were developed using this quantity

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because the measurements required to calculate the interfacial area of the film were not available. However, since the mass transfer actually occurs at the interface, and not the wall, the entrainment and deposition rate calculations within COBRA-TF were updated to use the interfacial area of the film.

Lastly, as mentioned previously a constitutive relationship is not used by COBRA-TF to calculate the film thickness, but rather this quantity is determined using the code calculated value of the continuous liquid volume fraction by geometrically relating this quantity to the flow path assuming the film is uniformly distributed around the periphery. Two methods exist for deriving a relationship between the relative film thickness (δ DH ) and continuous liquid volume fraction

(α l ) in annular flow for tubular geometries. The first method, which is currently employed by COBRA-TF, is based on the definition of volume fraction which is:

2 2 2 2 2 A DH − Dgc DH − (DH − 2δ ) δ  δ  l   α l = = 2 = 2 = 4 + 4  (4-1) Ax DH DH D  D 

Then by applying the thin film approximation (i.e. δ << D ) the last term in this equation can be neglected and rearranging the remaining expression yields a relationship between the relative film thickness and continuous liquid volume fraction, which is given as:

δ α = l (4-2) DH 4

Meanwhile, an alternative approach for deriving the same type of relationship is:

1 δ = (D − D ) (4-3) 2 H gc

Rearranging this expression so it can be written in terms of the relative film thickness leaves:

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δ 1  Dgc  = 1−  (4-4) DH 2  DH 

Then, the definition for the gas core void fraction, which is:

2 Agc Dgc α gc = 1−α l = = 2 (4-5) Ax DH can be applied to Equation (4-4) to simplify this expression. Solving Equation (4-5) for the ratio of diameters as a function of the continuous liquid volume fraction and substituting this result into Equation (4-4) leaves:

δ 1 = [1− ()1−α l ] (4-6) DH 2

The derivation of this expression does not require the thin film approximation and thus is exact. A comparison of the results predicted by Equations (4-2) and (4-6) is provided in Figure 4-7. Slight differences do exist, but overall the results predicted by these two expressions are similar in the continuous liquid volume fraction range where annular flow can exist in COBRA-TF assume no entrainment (i.e. α l < 4.0 for small hydraulic diameters, α l < 2.0 for large hydraulic diameters). In any event, the calculation of film thickness in COBRA-TF was modified to use the exact expression given by Equation (4-6) rather than the thin film approximation expression given by Equation (4-2).

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0.15 Thin Thin Film Film ApproximationApproximation - Equation (4-2)

Actual Exact Expression Film Thickness - Equation (4-6)

0.1

0.05 Relative FilmRelative Thickness

0 0 0.1 0.2 0.3 0.4

Continuous Liquid Void Fraction Figure 4-7: Comparison of Film Thickness Calculations.

4.2.2. Regime Boundaries

After reviewing the available literature and examining the pre-CHF flow regime map in the baseline version of COBRA-TF (see Figure 2-2) it was determined that the logic for the churn- turbulent and unstable annular regimes as it existed previously did not provide an accurate physical representation of these regimes. Given that the definition of the regime boundaries provides the foundation for the modeling package, a significant rewrite of the flow regime selection logic within the annular regime was required. As discussed in Section 2.2.5., the definition of an annular flow regime map requires criterion to describe:

1) the demarcation between the churn-turbulent and unstable film regimes 2) the entrainment inception point, 3) the entrainment suppression point, which also serves as the demarcation between the stable and unstable film regimes in annular flow, and 4) the transition between counter-current and churn-annular/churn-turbulent flow.

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This last item will be addressed using an explicit Counter-Current Flow Limitation (CCFL) model that was developed as part of the current study and is presented in Chapter 5. In addition to the items outlined above, a criterion is needed to distinguish between the occurrence of two entrainment mechanisms that exist within the unstable annular flow regime (i.e. roll wave stripping and Kelvin-Helmholtz lifting). This change in mechanism may also correspond to the transition between churn-annular and co-current upward annular flow, but not enough information is available at this time to adequately assess the validity of this notion. In general, it was described in Chapters 1 and 2 that these transition points correspond to changes in the interfacial structure and therefore defining the flow regime map in this manner allows models that are consistent with the expected structure to be applied.

First, the demarcation between the churn-turbulent and unstable film regimes is defined within the proposed modeling package using the liquid bridging criterion. This value corresponds to a vapor void fraction of 0.8 or 0.6 for large or small hydraulic diameter channels, respectively. Within the churn-turbulent regime a void fraction weighted, logarithmic ramp is used to calculate effective interfacial drag, entrainment rate, and entraining drop size for this regime. The treatment of the churn-turbulent regime will be discussed in more detail in Section 4.2.9.

Second, the model proposed by Ishii & Grolmes [3] was selected to describe the entrainment inception point. For low viscosity fluids (N µ < 15/1 ) this model is given as [3]:

− 1  5.1 Re 2 for Re < 160  l l σ ρ  1 U ≥ l 11 78. N 8.0 Re − 3 for 160 ≤ Re ≤ 1635 (4-7) v,crit α µ ρ µ l l v l v  8.0 N µ for Re l > 1635 where the viscosity and film Reynolds numbers in this expression are defined as

µl N µ = (4-8) σ ρ σ l g∆ρ

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and:

ρlα lU l DH Re l = , (4-9) µl respectively. If the mean vapor field velocity calculated by COBRA-TF is less than two times the critical value predicted by Equation (4-7) then a suppression factor, which is calculated as:

   U v −U v,crit Rincpt = max  ,0.0 min  ,0.1  (4-10)   U v,crit  is applied to the calculated entrainment rate. This is done to provide a smooth transition into and out of conditions where entrainment can and cannot occur.

Third, the two-zone interfacial shear model proposed by Hurlburt et al. [55] discussed in Section 2.2.3. was incorporated into the proposed modeling package. The correlation for intermittency proposed by Hurlburt et al. [55] includes a critical value of the mean dimensionless film thickness (in interfacial units), which based on the definition of intermittency corresponds to the onset of disturbance waves. A critical value of:

+ δ crit =12 (4-11) is suggested by Hurlburt et al. [55], where the mean dimensionless film thickness (in interfacial units) is defined as:

δρ U * δ + = l l , (4-12) µl and the friction velocity for the film is defined as:

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* τ i,vl U l = , (4-13) ρl

Given that other works have suggested that the presence of interfacial waves is a prerequisite for entrainment to occur this criterion is applied as the entrainment suppression point and the demarcation between the unstable and stable film regimes in the proposed modeling package. Using this criterion also ensures consistency between the flow regime map and the interfacial shear model that is used in the modeling package. To ensure the criterion proposed by Hurlburt et al. [55] is in reasonable agreement with other experimentally obtained conditions, the correlation proposed by Asali et al. [54], which is given as:

+ 6.0 δ = 34.0 Re l (4-14) can be used to calculate the corresponding critical film Reynolds number. Solving Equation (4- 14) yields:

Re l,crit = 380 (4-15)

This result is similar to other experimentally obtained results discussed in Section 2.2.4. and provides confidence that the criterion given by Equation (4-11) is reasonable. Similar to the entrainment inception criterion, if the mean dimensionless film thickness (in interfacial units) calculated by COBRA-TF is less than two times the critical value given by Equation (4-11) then a suppression factor, which is calculated as:

  δ + − δ +  R = max  ,0.0 min  ,0.1 crit  (4-16) sup  +    δ crit  is applied to the calculated entrainment rate.

At this point, the entrainment rate in the unstable film regime can be calculated as:

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S E′ ,UN = Rincpt Rsup S E′ ,UN , pred (4-17)

Lastly, within the unstable regime two different entrainment mechanisms are hypothesized to exist. At lower gas velocities near the onset of entrainment it is supposed that entrainment occurs by a Kelvin-Helmholtz lifting mechanism while at larger gas velocity conditions well away from the onset condition entrainment is supposed to occur by a roll wave stripping mechanism. Entrainment rate models for each mechanism have been developed within the current study and will be presented in Section 4.2.4. Azzopardi [49] suggested a Weber number (based on film thickness) describes the transition between these two mechanisms and the proposed relationship was given previously as Equation (2-16).

Based on results obtained from COBRA-TF predictions of the experiments conducted by Hewitt & Pulling [83] it was determined that the local maximum in the entrained flow rate versus flowing quality observed in the experimental data (see Figure 4-3) corresponds to the critical Weber number proposed by Azzopardi [49]. As a result when the Weber number is less than this critical value (i.e. flowing quality is less than that corresponding to the local maximum) entrainment is supposed to occur by the Kelvin-Helmholtz lifting mechanism and conversely when the Weber number is greater than this critical value (i.e. flowing quality is greater than that corresponding to the local maximum) entrainment is supposed to occur by the roll wave stripping mechanism. As a result a Weber number weighted linear ramp is instituted in the proposed model set to provide a smooth transition in the calculated entrainment rate and entrained drop size for these two mechanisms, which is given as:

  We crit − 20  RUN = max  ,0.0 min  ,0.1  (4-18)   30 − 20  such that:

S E′ ,UN = S E′ ,RW ( 0.1 − RUN ) + S E′ ,DW RUN (4-19) and:

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DE,UN = DE,RW ( 0.1 − RUN ) + DE,DW RUN (4-20)

It has been suggested that this change in entrainment mechanism also corresponds to the local minimum in the pressure gradient and therefore the transition between churn-annular and co- current upward annular flow; however, Hewitt & Pulling [83] did not take pressure gradient measurements in their study and therefore it is not possible to determine if the local maximum in entrained flow rate corresponds to the local minimum in the pressure gradient. In the proposed modeling package it is assumed that these two conditions do coincide, but it would be desired to obtain corresponding measurements of pressure gradient and entrained flow rate in a future low pressure, steam-water annular flow experiment to assess validity of using this criterion as the transition between churn-annular and co-current upward annular flow.

Additionally, based on the COBRA-TF predictions of the other annular flow tests simulated in the current study only a few of the higher pressure cases considered reached Weber numbers below the critical value proposed by Azzopardi [49]. Instead it appears that at higher pressure the flow becomes churn-turbulent prior to reaching these conditions, which may be an indication that the churn-annular regime does not exist at higher pressures. Assuming the local maximum in entrained fraction does correspond to the local minimum in pressure gradient, such an observation would be consistent with the non-existence of the local minimum in the pressure gradient at higher pressures that was observed by Fore et al. [51].

4.2.3. Interfacial Shear Stress

Based on the significant difference in behavior between the wave and base film substrate regions that was observed by Sawant et al. [57] and Belt et al. [2] it is unlikely that a model that does not explicitly consider these two regions can accurately capture the average behavior of the film in annular flow. As a result is was desired to incorporate the two-zone interfacial shear model proposed by Hurlburt et al. [55] into the proposed modeling package. In addition to providing a calculation of interfacial shear stress that reflects the physical structure of the flow, the two-zone model also provides estimates of several other parameters, such as wave amplitude, wave

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velocity, and intermittency, which can be used as input to a mechanistic-based entrainment model.

Hurlburt et al. [55] use the log law with a roughness offset to describe the velocity profile of the vapor field within each zone. Azzopardi [25] indicates the use of this profile is possible based on the results presented by Owen & Hewitt (1987) because the presence of droplets suppresses the turbulence in the gas core. The log law profile is given as:

U 1  yU *   k  (4-21) * = ln   + B − ∆Bk U k κ  ν v  where k = b,w , refers to the base film and wave zones, respectively. Meanwhile, the log law parameter is:

B = 0.5 , (4-22) the von Kármán constant is:

κ = 41.0 , (4-23) and the velocity offset, which is a function of roughness, is approximated for annular flow situations as:

 ε U *   ε U *  1  k k  1  k k  ∆Bk = ln 1+ cB,k  ≈ ln cB,k  (4-24) κ  ν v  κ  ν v 

The parameter ‘ cB,k ’ varies with roughness type and Hurlburt et al. [55] suggest that:

cB,w = 7.4 (4-25) and:

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cB,b = 8.0 (4-26)

These values were obtained by Hurlburt et al. [55] from a fit to the experimental pressure gradient data collected by both Fore & Dukler [23] and Fore et al. [31,51]. As previously mentioned, the larger value for the disturbance wave region is consistent with the highly dissipative nature of this region [55].

Integrating the log law profile between centerline and the top of the roughness element in each zone:

y y U  1 wall − k  1  yU *    v  k  B dy B (4-27) * =  ∫  ln   +   − ∆ k U  A κ ν   k  x,k y0   v    and then invoking the definition of the friction velocity for the vapor phase, but neglecting the liquid velocity component:

2 2 * τ i,k fi,k ()U v −U k fi,kU v U k = = ≈ (4-28) ρv 2 2 allows expressions for the interfacial friction factors for each zone to be determined as:

2 2 U *   1 ywall − yk  1  yU *     f ≈ 2 k  = 2 ln  k  + B dy − ∆B  (4-29) i,k    ∫      k  U v  Ax,k κ ν v      y0      

It can be seen that the bounds of integration are geometry dependent and therefore separate expressions are obtained for circular and planar geometries. The integral for circular geometry is:

R−()ε +δ U  1 k min  1  ()R − r U *    v r  k  B dr B (4-30) * =  2 ∫ 2π  ln   +   − ∆ k U k π[]R − ()ε k + δ min 0 κ  ν v   

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and the resulting expression the interfacial friction factor is:

  2   58.0 f ()c ,εˆ =   (4-31) i,k,tube B,k k  ln εˆ 1 ()εˆ +1  − k − ln c + 05.1 + k  2 B,k ˆ   ()εˆk −1 2 ()ε k −1  where the non-dimensional roughness is defined as:

ε k εˆk = (4-32) R − δmin

Meanwhile, the integral for planar geometry is:

 H  −()ε k +δ min U  1 2  1  yU *    v  k  (4-33) * =  ∫  ln   + Bdy  − ∆Bk U H κ ν  k  − ()ε + δ 0   v     2 k min  and the resulting expression for the interfacial friction factor is:

  2    58.0  f i,k,duct ()cB,k ,εˆk = (4-34)  ln εˆk   − ln cB,k + 05.1   εˆk −1  where the non-dimensional roughness is defined as:

ε εˆ = k (4-35) k H −δ 2 min

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As discussed in Section 4.2.1. the ability for the user to specify the geometry (i.e. circular or non- circular) on a subchannel basis was added to COBRA-TF during the current study. As a result it is possible to utilize the appropriate form of the friction factor correlation based on the geometry of the flow path being modeled.

A set of limits were imposed on the calculated interfacial friction factor to ensure physically reasonable values were obtained. These consisted of a lower limit equal to the smooth tube, fully-turbulent value of 0.005 and an upper limit equal to five times the value obtained using the Whalley & Hewitt [28] correlation, which was given as Equation (2-48). These limits were primarily needed to maintain physically reasonable values and improve numerical stability during transient situations. These limits were not typically invoked in the steady-state solutions for the tests analyzed in the current study.

At this point it should be mentioned that for the purposes of their work Hurlburt et al. [55] assumed the demarcation between the wave and base film zones occurred at a value equal to 110% of the mean film thickness. They used this definition when analyzing the experimental data to correlate the zonal average peak height and the intermittency. Hurlburt et al. [55] also define the minimum thickness of the film substrate as one-half the mean film thickness:

δ min ≅ 5.0 δ (4-36) such that the average zonal peak height is calculated as:

δ k = δ min + ε k (4-37)

Hurlburt et al. [55] suggest that the average zonal roughness, ‘ εk’, is related to the standard deviation in the film thickness and the resulting expressions for the wave and base film zones are given as:

ε w = 1.2 σ δ (4-38)

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and:

ε b = 6.0 σ δ (4-39)

Hurlburt et al. [55] propose that the standard deviation for the steam-water data considered can be adequately described by:

σ δ = 65.0 (δ − δ crit ) (4-40) where:

δcrit = 30 µm (4-41) and the applicable range of Equation (4-38) based on the experimental data used to develop the correlation is given by Hurlburt et al. [55] to be:

50 µm < δ < 650 µm (4-42)

The existence of an offset or critical value in Equation (4-40) suggests that a minimum film thickness is required for interfacial waves to exist and for film thicknesses below this value can be treated as a “smooth” film. This is consistent with the observations of Asali et al. [54]. However, there is a potential for an inconsistency to exist between the critical values given by Equations (4-11) and (4-41). Examples of this were observed in the analysis of the Hewitt & Pulling [83] experiments during the current study. Situations occurred where the COBRA-TF predicted mean film thickness was less than the critical value suggested by Equation (4-41) and therefore the calculated disturbance wave amplitude is zero; however, simultaneously the mean dimensionless film thickness calculated using Equation (4-12) was greater than the critical value suggested by Equation (4-11). As a result the calculated intermittency was non-zero, which suggests interfacial waves should exist. Since the presence of interfacial waves is a prerequisite for entrainment to occur, and since a non-zero entrained flow rate was measured experimentally for the concerned cases, it is desired that the calculated entrainment rate also be non-zero;

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however, as the wave amplitude approaches zero the entrainment rate calculated by the proposed models developed in the current study also inherently approach zero.

One way to alleviate this inconsistency is to use an alternative functional relationship that was provided by Hurlburt et al. [55], which correlated the dimensionless wave standard deviation and dimensionless film thickness (in interfacial units). This expression is given as:

+ + + 9.0 σ δ = (δ −δ crit ) , (4-43) but using this form of the correlation requires the total interfacial shear stress to be known a priori; however, this is the desired solution variable. It was decided not to implement an iterative solution scheme for this parameter since it would encompass essentially the entire two-zone interfacial shear model and would likely impair the computational efficiency of the code.

Instead, it was decided to modify the explicit form of the functional relationship given as Equation (4-40) such that it would not rigidly impose the critical film thickness criterion given by Equation (4-41) for very thin films, but rather provide a smooth transition to zero as the mean film thickness approached zero. This approach was deemed appropriate since a separate ramp, which was given as Equation (4-16) and was defined in such a manner to be consistent with the critical mean dimensionless film thickness (in interfacial units) that was given as Equation (4-11), was applied to the calculated entrainment rate in the proposed modeling package to provide a smooth transition into and out of conditions where entrainment can and cannot occur based on the amount of disturbance wave activity (i.e. intermittency). Additionally, evidence was provided by Hurlburt et al. [55] that the same critical thickness, given by Equation (4-41), does not apply for the different fluid types considered (e.g. steam-water, nitrogen-water, and air-water); however, the variation in this parameter has not yet been quantified.

A modified form the Equation (4-40), which retains the properties of the original correlation within the applicable range of the correlation, can be obtained by applying a spline fit that forces the wave height to zero as the film thickness goes to zero. A polynomial spline fit was used so that an additional restriction could be imposed to ensure the spline fit has the same derivative as

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the correlation at the upper limit of the range where the spline fit was imposed. The spline fit was applied over the region of mean film thickness ranging from zero to two times the critical mean film thickness suggested by Equation (4-41). The resulting expression is:

(δ − δ ) if δ > 2δ  crit crit 2 (4-44) σ δ ≈ 65.0  ()δ  if δ ≤ 2δ crit 4δ crit

A comparison between the correlation for the wave height standard deviation, given as Equation (4-41), and the imposed spline fit is provided in Figure (4-8).

Correlation - Equation (4-40) 40 Spline Fit - Equation (4-44)

20 Wave Height Standard Standard Height Deviation Wave

0 0 20 40 60 80 100 120 Mean Film Thickness ( µm) Figure 4-8: Comparison of the Correlation for the Wave Height Standard Deviation and the specified Spline Fit as a function of Mean Film Thickness.

At this point the zonal interfacial friction factors can be calculated. Next, the zonal average interfacial shear stress can be defined as:

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2 f ρ  A  i,k v  x,gc  (4-45) τ i,k =  U v −U k  2  Ax,k 

In this expression the bulk mean velocity of the vapor field is multiplied by an area ratio to provide a more accurate estimate of the vapor velocity at the “wave” crests within each zone. For a tubular geometry this ratio is given as:

2 Ax,gc (R − δ ) = 2 (4-46) Ax,k ()R − δ k and for a non-circular geometry this ratio is given as:

Ax,gc (H − 2δ )(W − 2δ ) = (4-47) Ax,k (H − 2δ k )(W − 2δk )

It can be seen from Equation (4-43) that the calculation of the zonal average shear stress requires the base film and wave zonal average velocities to be estimated. Hurlburt et al. [55] suggest these quantities can be calculated using the laminar correlation of Asali et al. [54], which is given as:

+ 6.0 δ k = 34.0 Re l,k (4-48) and the turbulent correlation Henstock & Hanratty [27], which is given as:

+ 9.0 δ k = .0 0379 Re l,k (4-49) where the zonal average dimensionless film thickness (in interfacial units) in these expressions is defined as:

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* + δ k ρlU l,k δ k = (4-50) µl and the corresponding zonal friction velocity is defined as:

* τ i,k U l,k = (4-51) ρl

Assuming the thin film Reynolds number approximation, which is given as:

4U kδ k Re l,k = (4-52) µl is valid then Equations (4-48) and (4-49) can be rearranged to solve for the zonal average velocity. Hurlburt et al. [55] suggest the wave region must be turbulent, but allow the base film region to be either laminar or turbulent. A hybrid form of the velocity correlations, that maintains the characteristics of the individual laminar and turbulent correlations in the respective regions, can be obtained as:

− 1 2 −2 1 −2 2  3 9  U = U *  5.1 ()δ + + 5.9 ()δ +  (4-53) k l,k [ k ] [ k ] 

A comparison of this hybrid form of the correlation, Equation (4-53), to the laminar, Equation (4- 48), and turbulent, Equation (4-49), forms is provided in Figure 4-9.

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60 Hybrid - Equation (4-53) Turbulent - Equation (4-49) Laminar - Equation (4-48)

20 Normalized Velocity (U/U*)

0 0 50 100 150 200

Mean Dimensionless Film Thickness (in interfacial units) Figure 4-9: Comparison of the Velocity Correlations as a function of Mean Dimensionless Film Thickness (in interfacial units).

It was found in the current study that in order to maintain the numerical stability of the code and achieve a converged solution at smaller mean dimensionless film thicknesses (in interfacial units) it was necessary to use the hybrid form of the correlation to estimate the zonal average velocities for both the base film and wave zone regions. Substituting Equation (4-53) into Equation (4-45) yields an expression for the zonal average interfacial shear stress, which is given as:

  2 −2     3  f ρ A τ  δ k τ i,k ρl τ = i,k v  x,gc U − i,k  5.1    + (4-54) i,k 2  A v ρ   µ    x,k l   l     

1 2 − 2 1 −2      9     δ k τ i,k ρl    5.9     µ      l      

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Solving the above equation requires an iterative solution procedure since the zonal average velocity defined by Equation (4-53) is a function of the zonal average interfacial shear; however, if one approximates the interfacial shear stress in the dimensionless film thickness components of Equation (4-54) by neglecting the liquid velocity contribution then an explicit solution can be obtained. Since the magnitude of these terms is relatively small compared to the other terms in the equation the effect of doing is negligible. The resulting expression is:

  2 −2    2  3    f ρ  A     i,k v  x,gc    δ k U v ρl    2  A  f i,k ρv Ax,gc τ i,k    x,k    τ ≈  U −  5.1 + (4-55) i,k 2  A v ρ   µ   x,k l   l                 

1 2 − 2 1 −2     2  9     f ρ  A     i,k v  x,gc     δ k U v ρl     2  A     x,k      5.9    µ      l                  

One can then define the following parameters:

A U = x,gc U , (4-56) v k v Ax,k

f ρ R = i,k v , (4-57) 1 2 and:

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δ R U 2 ρ k 1 ( v k ) l R2 = , (4-58) µl which allows Equation (4-55) to be rewritten as:

2  − 1  τ 2 −2 1 −2 2 τ ≈ R U − i,k 5.1 ()R 3 + 5.9 ()R 9  (4-59) i,k 1  v k {[]2 []2 }   ρl 

Finally, this expression can be solved explicitly for the average zonal interfacial shear stress:

2 (U ) 4 v  3 3  τ ≈ k ρ R R ρ R + R − 2 5.1 5.9 R ρ R R (4-60) i,k 2  l 1 3[]l 3 4 ( )( ) ()2 ()l 1 2  []ρl R3 − R4   where:

10 2 9 2 R3 = [( 5.1 ) (R2 ) + ( 5.9 ) ] (4-61) and:

4 2 2 3 R4 = ( 5.1 ) ( 5.9 ) R1 (R2 ) (4-62)

Once the zonal average interfacial shear is known the corresponding zonal average velocity can be calculated using Equation (4-45). To ensure physically reasonable results are obtained for the predicted zonal average velocities a series of restrictions were imposed. For the wave region it was required that:

U l < CDW < U v (4-63) and for the base film region it was required that:

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0 < U b < U l (4-64)

Then, based on the definition of intermittency (i.e. fraction of time that disturbance waves are present on the surface), that the average total shear stress can be defined as:

τ i,vl = τ i,wψ +τ i,b (1−ψ ) (4-65)

This expression contains two unknowns, and therefore the calculation of the average total interfacial shear stress requires an independent correlation for intermittency. Hurlburt et al. [55] propose such a relationship based on the scaling of single-phase turbulent kinetic energy in the near wall region. The resulting correlation for the steam-water data considered is given as:

 +   + 2   +   δ crit 4  (δ − )  (δ crit − 2)  ψ ≈ 38.0 1− + exp −  − exp − (4-66)  + +    δ δ   4   4  where the critical and mean dimensionless film thicknesses (in interfacial units) are defined as Equations (4-11) and (4-12), respectively. Figure 4-10 shows the asymptotic behavior of the intermittency correlation as a function of dimensionless film thickness (in interfacial units).

Similar trends were observed for the experimental data collected with other fluids; however, slight differences exist in the leading coefficient in the correlation and the critical dimensionless film thickness (in wall units). The difference in the dimensionless film thickness (in wall units) required for the onset of disturbance waves for the different fluids indicates that a surface tension or viscosity effect may exist, but this relationship has not yet been quantified. Since again the primary focus of the current work is for nuclear reactor applications the steam-water form of the correlation given as Equation (4-66) was used in the proposed modeling package.

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0.4

0.3

0.2 Intermittency 0.1

0 0 50 100 150 200

Dimensionless Film Thickness (in interfacial units) Figure 4-10: Intermittency as a function of Dimensionless Film Thickness (in interfacial units).

Upon examining the intermittency correlation given above one can notice that this correlation is a function of the total average interfacial shear stress through the friction velocity. Meanwhile, the total average interfacial shear stress is also a function of intermittency and as a result this correlation cannot be solved explicitly. Although explicit forms of the zonal interfacial shear stress equations could be obtained with a simple assumption, no such simplification is possible here and therefore implementing this model into COBRA-TF requires Equations (4-65) and (4- 66) to be solved iteratively using a numerical method. Since the solution interval can be bounded by the shear stress values calculated for each zone the bisection method can be applied. The difference in the intermittency calculated by Equations (4-65) and (4-66) can be determined and compared to a convergence criterion.

If the calculated intermittency is equal to zero then as indicated in the previous section the film is assumed to be in the stable film regime and a “one-zone model”, which is given as:

f ρ τ = i,b v ()U −U 2 (4-67) i,vl 2 v l should be applied to calculate the total interfacial shear stress.

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At this point the total average interfacial shear stress is known and the calculations associated with the proposed interfacial shear model are complete; however, the form of the momentum equations used by COBRA-TF requires the corresponding interfacial drag coefficient, which is calculated as:

τ i,vl Ai,l K vl = (4-68) U v −U l ∆x where:

4 α v + α e Ax,tot ∆x Ai,l = Pw α v + α e ∆x = (4-69) DH

The calculation of the interfacial area of the film assumes the film is uniformly distributed around the periphery of the flow path. It is important to note that in Equation (4-68) an absolute value is applied to the relative velocity quantity so that the calculated coefficient is always positive. The appropriate sign for the shear stress is applied later during the calculations conducted in the XSCHEM subroutine, where the drag force per unit length is calculated as:

F d = K ()U − U (4-70) ∆z vl v l

This quantity, equal and opposite, is applied to both the continuous liquid and vapor field momentum equations.

Meanwhile, several other constitutive relationships calculated in INTFR require an the interfacial friction factor. A total effective interfacial friction factor for the two-zone model can be calculated as:

2τ i,vl f i,eff = 2 (4-71) ρv ()U v −U l

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In summary, the input quantities required for the interfacial drag model includes:

1) continuous liquid and vapor field average velocities (U l ,U v )

2) vapor void fraction (α v )

3) liquid and vapor densities and viscosities (ρv , ρ l , µv , µl )

4) continuous liquid volume fraction or mean film thickness (α liq ,δ )

5) hydraulic diameter (DH ) 6) flow path geometry (circular or non-circular)

7) area of flow path (Ax,tot ), only if non-circular geometry

Each of these quantities is readily available within COBRA-TF or any other three-field analysis code. Meanwhile, the output required from the interfacial shear model for COBRA-TF includes:

1) effective drag coefficient (K vl )

2) effective interfacial friction factor ( fi,eff )

Finally, the output required from the two-zone interfacial shear model to be provided as input for the entrainment model, which will be discussed in the next section, includes:

1) disturbance wave height (ε w )

2) wave zone interfacial shear stress (τ i,w )

3) disturbance wave velocity (CDW ) 4) intermittency (ψ )

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4.2.4. Entrainment Rate

The proposed modeling package includes explicit models for three different entrainment mechanisms: 1) roll wave stripping, 2) Kelvin-Helmholtz lifting, and 3) liquid bridge breakup. The proposed modeling package assumes that within the unstable annular regime entrainment can occur by either the roll wave stripping or Kelvin-Helmholtz lifting mechanism based on the calculated Weber number (based on film thickness) as explained in Section 4.2.1. Meanwhile, in the churn-turbulent regime the calculated entrainment rate for the unstable film regime is augmented by the liquid bridge breakup mechanism.

Within this section the idealization of the interfacial structure is first outlined. Next, the proposed entrainment rate models are is presented. The models used for each mechanism are based on previously conducted work, but have been modified to use quantities that are consistent with the rest of the proposed modeling package. For each mechanism, the theoretical basis for the model is first presented then, if necessary, a relationship between the theoretical and actual entrainment rates is provided. As described in Section 3.2.2., such a relationship accounts for any deficiencies in the proposed theoretical model based on comparisons to experimental data.

4.2.4.1. Disturbance Wave Frequency and Spacing

The calculation of entrainment rate from a single wave using mechanistic based models requires the structure of the interface to be idealized to allow for the length and spacing of the disturbance waves to be estimated. These quantities are needed to calculate the number of waves residing in a given computational cell, which scales the calculated entrainment rate. In reality the interface is highly irregular and chaotic, but an idealized structure should approximate the average behavior. A schematic of the resulting idealized interfacial structure described within this section is given in Figure 4-11.

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Figure 4-11: Schematic of Idealized Interfacial Structure.

In order to maintain consistency with the two-zone interfacial drag model, several parameters calculated by that model were used as input for characterizing the structure of the interface and within the proposed entrainment rate models. For example, the previously calculated disturbance wave height (ε w ), wave velocity (CDW ), and intermittency (ψ ) are coupled with the wave

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frequency correlation proposed by Sawant et al. [57] to quantify the idealized interfacial structure. Sawant et al. [57] correlated their experimental results from air-water experiments in a vertically-oriented, 0.37-inch diameter tube at low pressures (atmospheric, 58-psia, and 84-psia) using the Strouhal number. The resulting correlation is given as:

− 64.0 f D  ρ  DW H 27.0  l  Sr = = .0 086 Re l,tot   (4-72) jv  ρv 

While this correlation did considered conditions ranging from the near the churn-annular regime boundary to well within the annular regime, only relatively low liquid Reynolds numbers (500- 5700) were tested. Additionally, this correlation is based on macroscopic, rather than local parameters, which is highlighted by the use of a total liquid Reynolds number and superficial gas velocity in the correlation. Therefore, while the hydraulic diameter is representative of the conditions of interest for the current study, the consideration of only low-pressure, low liquid Reynolds number conditions using air-water causes extension of this correlation to nuclear reactor applications to be questioned; however, a more suitable correlation is not available at this time and as a result this correlation was applied in the proposed modeling package to estimate the disturbance wave frequency.

For the purposes of the current study it is assumed that the disturbance waves are coherent ring wave structures uniformly distributed around the periphery of the flow duct that move with a constant velocity. As indicated by Azzopardi [25] the uniform distribution becomes increasingly accurate with decreasing hydraulic diameter and therefore the assumption is deemed appropriate given the relatively small diameters of the flow paths associated with nuclear reactor applications. A separate factor could be included to consider the potential asymmetry of the waves, which would act to reduce the calculated entrainment rate, but since a functional relationship for quantifying asymmetry is not available such an effect was not accounted for explicitly in the proposed model. Rather it was decided to allow this effect, if it does exist, to be captured implicitly within the functional relationship between the actual and theoretical entrainment rates.

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Meanwhile, the constant wave velocity assumption is justified based on the results of Belt et al. [2], Azzopardi [53], and Sawant et al. [57] and allows the effective wavelength of the disturbance waves, or the wave spacing, to be calculated as:

CDW λDW = (4-73) f DW where again the disturbance wave velocity in this expression is estimated as part of the two-zone interfacial drag model calculations. The effective wavelength calculated by Equation (4-71) relates the length of the control volume encompassing a single wave to the height of the computational cell. This determines the number of waves per computational cell and directly impacts the resulting entrainment rate calculated for that computational cell. The number of waves per computational cell is calculated as:

dz N DW = (4-74) λDW

Next, the constant velocity assumption and the definition of intermittency allow the length of the disturbance waves to be calculated as:

λ L = DW (4-75) DW ψ

This quantity is needed to: 1) calculate the length of the stripping boundary layer for the roll wave stripping entrainment mechanism and 2) scale the number of ripple waves that reside on the surface of a single disturbance wave when calculating the entrainment rate by the Kelvin- Helmholtz lifting mechanism.

Two restrictions are imposed on the length of disturbance waves estimated using Equation (4-73). First, the experimental observations of Hurlburt & Newell [94] indicate the maximum surface angle of disturbance waves in horizontal air-water annular flow was 5-degrees, with the surface

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angles of most waves being much less than this (average value of 1.5-degrees). Based on this observation the minimum allowable length of disturbance waves given as:

ε L = w (4-76) DW ,min tan ()50

If the length of the disturbance waves calculated by Equation (4-75) is less than the allowable value calculated by Equation (4-76) then the length of the disturbance waves is set to the value calculated by Equation (4-76).

Second, as previously mentioned, a direct relationship exists between the number of waves that are supposed to reside within a given computational cell and the calculated entrainment rate for that cell. Upon examining Equations (4-73) and (4-74) it can be seen that the number of waves per computational cell is directly related to the calculated frequency. Since the frequency correlation used in the current study was not developed from experimental data that is representative of conditions experienced within nuclear reactors, it is desired to impose a restriction on the minimum allowable disturbance wave spacing to ensure a physically reasonable value is maintained when the wave frequency correlation is extended beyond the range of conditions from which it was developed. A minimum spacing of 0.5-inches, which is significantly less than the minimum value of 2-inches observed by Sawant et al. [57], was imposed within the proposed model. Applying this type of limit precludes a sensitivity to computational node height from being built into the proposed model. If the disturbance wave spacing calculated using Equation (4-73) is less than this value then the calculated wave frequency is adjusted accordingly.

Now that the interfacial structure has been idealized, individual, mechanistic-based models can be developed to calculate the entrainment rate by each mechanism.

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4.2.4.2. Roll Wave Stripping Mechanism

As described previously, for larger gas velocities, well away from the entrainment inception point, entrainment occurs by the roll wave stripping mechanism where droplets are sheared from the interface at the roll wave crests by the faster moving vapor field. Two primary forces exist to resist entrainment by this mechanism: 1) surface tension and 2) internal viscous shear. Fore [74] suggests that the effect of surface tension is minimized due to the large radius of curvature of the disturbance wave surface. As a result Fore [74] proposed an entrainment rate model for the roll wave stripping mechanism where the internal viscous shear is the primary force resisting droplet formation.

Fore [74] suggests that for entrainment to occur the local velocity of the fluid at the crest of the wave must be greater than the bulk average velocity of the wave (CDW ). If this is true, a point must exist within the wave where the local velocity is equal to the bulk average velocity and at this point the opposing force of internal viscous shear becomes negligible. The location of this point on the windward side of the disturbance wave interface can be thought of as a stagnation point and therefore the existence of such a point is a prerequisite for entrainment to occur by this mechanism. Since a large velocity gradient exists at the interface and a much reduced gradient exists at the inner edge of this layer Fore [74] also suggests that this region near the surface of the wave, which will be referred to as the stripping layer, can be treated as a “boundary layer” and the velocity profile in this region can be approximated using a cubic polynomial. The liquid contained within this stripping layer is free to leave the disturbance wave at the peak, generating droplets of a size related to the thickness of the boundary layer.

For the purposes of the current study it was assumed as a first order approximation that the stagnation point (i.e. the point where the local fluid velocity is equal to the bulk average wave velocity) occurs a distance from the wall that corresponds to minimum film thickness, which is defined as Equation (4-36). This approximation is justified since by definition in the two-zone interfacial shear model the liquid contained within the base film substrate moves with a velocity less than, and the wave moves with a velocity greater than, that of the bulk average velocity of the liquid film. It is also assumed that the stagnation region extends over the entire length of the

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disturbance wave (LDW ) and the thickness of the stripping layer is equal to the disturbance wave amplitude (ε w ). To calculate the flow rate within the stripping layer at the peak of the disturbance wave it is appropriate to position the coordinate system at the location of the stagnation point on the interface, as shown in Figure 2-8, such that the coordinate system moves with the bulk average velocity of the disturbance wave. Writing the Navier-Stokes equation in the tangential direction along the interface of the wave neglecting the tangential viscous term yields:

2 ∂u ∂u 1 ∂p µl ∂ u u + v = − − g + 2 (4-77) ∂s ∂n ρ l ∂s ρl ∂n

Fore [74] non-dimensionalized this expression using the hydraulic diameter for the length scale and the friction velocity for the velocity scale. The dimensionless directions are defined as:

s s′ = (4-78) DH and:

n n′ = (4-79) DH

Meanwhile, the friction velocity is defined as:

τ U * = i (4-80) ρl

The interfacial shear stress in this expression is calculated as one-half the zonal average interfacial shear stress for the disturbance waves that was calculated by the two-zone model (i.e.

τ i = 5.0 τ i,w ). This factor is included since the quantity calculated by the two-zone model

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includes both skin friction and form drag; however for the calculation of entrainment rate by the roll wave stripping mechanism it is desired to only include the effect of skin friction. Similar to Fore [74], the current study assumes as a first-order approximation that skin friction and form drag each account for 50% of the total drag force. Buckles et al. [19] found the skin friction accounted for only 10% of the total interfacial drag; however, much larger wave slopes (~20%) were examined in their work than what is typically associated with disturbance waves (<5%).

The dimensionless velocities can then defined as:

u u′ = (4-81) U * and:

v v′ = (4-82) U *

Lastly, a friction Reynolds number can be defined as:

* ρlU DH Re s = (4-83) µl and a Froude number can defined, assuming a constant pressure gradient along the length of the disturbance wave, as:

* 2 * 2 2 (U ) (U ) Fr = ≈ (4-84)   gD  1 ∂p  H DH  g +   ρl ∂s  where as a first-order approximation for the purposes of the current work the pressure gradient contribution is neglected. Substituting Equations (4-76) through (4-82) into Equation (4-75)

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yields a dimensionless form of the Navier-Stokes equation in the stripping layer, which is given as:

∂u′ ∂u′ 1 1 ∂ 2u′ ′ ′ u + v = − 2 + 2 (4-85) ∂s′ ∂n′ Fr Re s ∂n′

By definition, at the inner edge of the stripping layer (δ SL ):

u′ = 0 (4-86) and:

∂u′ = 0 (4-87) ∂n′ in the moving coordinate system, which allows Equation (4-85) to be written at this location as:

∂ 2u′ Fr 2 2 = (4-88) ∂n′ Re s

Meanwhile, at the surface of the disturbance wave the assumed skin friction component of the interfacial shear stress is related to Newton’s Law of Viscosity:

∂u 5.0 τ i,w = −µl (4-89) ∂n n=0

Assuming the interfacial shear stress is constant along the length of the disturbance wave then a dimensionless form of Equation (4-89) can be written as:

∂u′ = −Re s (4-90) ∂n′ n′=0

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Fore [74] tested cases using both a constant interfacial shear stress and allowing the interfacial shear stress to increase linearly between the stagnation point and the wave peak; however, the difference in entrainment rate predicted for the two cases did not differ by more than 15% and therefore assuming a constant interfacial shear stress along the length of the wave was deemed appropriate for the purposes of the current study.

Fore [74] then applied the Pohlhausen boundary layer approach to specify a cubic polynomial approximation to the velocity profile in the stripping layer based on these boundary conditions. The resulting profile is given as:

δ ′2 Re  1 3  δ ′Re u′ = s  − η 2 +η 3  − s ()−1+ 3η − 3η 2 +η 3 (4-91) 3 Fr 2  2 2  3 where the dimensionless thickness of the stripping layer is defined as:

δ δ ′ = SL (4-92) DH and the normalized distance within the stripping layer is defined as:

n′ n η = = (4-93) δ ′ δ SL

As previously mentioned, it was assumed within the current work that the thickness of the stripping layer at the wave peak could be estimated as the disturbance wave height calculated within the two-zone interfacial shear model (i.e. δ SL = ε w ).

Using this information the mass flow rate within this layer at the peak can be calculated. By definition the mass flow rate is given as:

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S E′ ,DW 1, −wave = ∫∫ ρludA x (4-94)

In non-dimensional form, Equation (4-94) is given as:

δ ′ S′ = ρ U * A u′dn′ (4-95) E,DW 1, −wave l δSL ∫ 0

Then, rearranging Equation (4-93) allows Equation (4-95) to be written in normalized coordinates as:

1 S′ = ρ U * A δ ′ u′dη (4-96) E,DW 1, −wave l δSL ∫ 0

Substituting the specified velocity profile, which was given as Equation (4-91) and conducting the integration yields a final expression for the mass flow rate of liquid leaving a single disturbance wave, which is given as:

 δ ′3  * Re s  2  S E′ DW wave = ρ U A + δ ′ (4-97) , 1, − l δSL  2  12  Fr  where:

π 2 2 Aδ = {[]DH − ()δ min + ε w − δ SL − []DH − ()δ min + ε w } (4-98) SL 4 for circular geometries and:

A = [H − 2(δ + ε − δ )][W − 2(δ + ε − δ )] (4-99) δSL min w SL min w SL

− [H − 2(δ min + ε w )][W − 2(δ min + ε w )]

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for non-circular geometries.

The resulting theoretical entrainment rate for a given computational cell is then obtained by multiplying the result obtained using Equation (4-97) by the number of disturbance waves that reside within the computational cell, which is estimated using Equation (4-74). The resulting theoretical entrainment rate for the roll wave mechanism is therefore given as:

S E′ ,DW ,Model = N DW S E′ ,DW 1, −wave (4-100)

Next, a functional relationship (Θ DW ) between the actual entrainment rate and the theoretical entrainment rate could be determined such that the actual entrainment rate is given as:

S E′ ,DW = Θ DW S E′ ,DW ,Model (4-101)

As outlined in Section 3.2.2. such a relationship was determined using a subset of the experimental data available for the co-current upward annular regime. Based on the analysis conducted in the current study using COBRA-TF, which considered 67 experimental data points where the roll wave stripping mechanism is supposed to exist based on the criterion proposed by Azzopardi [49] and given previously as Equation (2-16). As shown in Table 3-1, this subset of experimental data encompassed a range of pressures and flow rate, but left sufficient gaps to assess the model against a much larger set of data covering a wide range of conditions. Based on these results, the proposed functional relationship between the actual and theoretical entrainment rates determined to be :

− .1 0556 .0 8014  ε   ρ  −6  w   l  ΘDW = .5 7223 x10     (4-102)  DH   ρv 

This relationship indicates an inverse relationship with a normalized disturbance wave height exists and suggests a pressure dependence is not being completely captured by the proposed entrainment model.

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The dependence of the above relationship on the relative disturbance wave height is shown in Figure 4-12 and a general comparison between the optimized and correlated results is provided in Figure 4-13. It can be seen in Figure 4-13 the range of values for the relationship between the actual and theoretical entrainment rates is roughly 10 -3 to 1. Smaller values of this relationship indicate a severe overprediction in the entrainment rate by the theoretical model relative to the experimental data. Examining Equation (4-102) it can be seen that the values predicted by this relationship decrease with increasing pressure and increasing relative disturbance wave height. In order to ensure a reasonable value for this multiplier is obtained a maximum value of ten is imposed within the proposed modeling package.

Figure 4-12: Correlation between the Actual and Theoretical Entrainment Rates for the Roll Wave Stripping Mechanism as a function of Relative Wave Height.

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Figure 4-13: Comparison of Optimized and Correlated Results for the Proposed Functional Relationship between the Actual and Theoretical Entrainment Rates for the Roll Wave Stripping Mechanism.

In summary, the entrainment rate model outlined in this section for the roll wave stripping mechanism represents an improvement over the other models currently available because it provides a mechanistic-based calculation of the phenomenon as it is currently understood, but simultaneously leveraging the available experimental data to improve the predictive capabilities of three-field analysis tools. The significant improvement in the predictive capability of the COBRA-TF resulting from the inclusion of this model is shown in Section 4.3. Potential improvements to this model could address the following assumptions:

1) The assumption that the disturbance wave frequency correlation, which is based on data collected using air-water at low pressures and low liquid Reynolds numbers, is applicable to conditions experienced in nuclear reactor applications.

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2) The assumption that the cubic velocity profile decribes the liquid velocity profile within the stripping region. This assumption influences the calculated mass flow rate of liquid leaving the wave. 3) The assumptions that: a) the location of the stagnation point corresponds to the minimum film thickness given by Equation (4-36), b) the stripping region extends along the entire disturbance wave length, and c) the thickness of the stripping region at the peak of the disturbance wave is equal to the height of the disturbance wave. 4) The assumption that the disturbance wave is a coherent, ring wave structure that is uniformly distributed around periphery of flow duct.

This working model provides a starting point for future studies where the aforementioned limitations could be addressed.

4.2.4.3. Kelvin-Helmholtz Lifting Mechanism

As described previously the visual observations of Woodmansee & Hanratty [56] indicate that near the entrainment inception point (i.e. low gas velocity conditions) atomization occurs by the removal of small wavelets, or ripples, that are superimposed on the larger disturbance, or roll, wave structures. Atomization occurs when one of these ripples is suddenly accelerated and moves toward the front of the roll wave, at which point it is lifted by the gas stream with the base still attached to the liquid film, as shown in Figure 2-7. The stretched ligament is then blown into an arc by the faster moving gas phase until the ligament ruptures in a number of places, thereby creating several droplets. Woodmansee & Hanratty [56] proposed that atomization by this mechanism occurs due a Kelvin-Helmholtz type instability, but this mechanism has also been referred to as the bag breakup mechanism based on the shape of the liquid ligament just prior to rupture.

Following the typical scaling analysis approach, the general time scale for entrainment can be estimated as the mass of liquid that becomes entrained divided by the rate at will the liquid becomes entrained, which is given as:

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mE ρlVE t E = = (4-103) S E′ S E′

Solving this expression for the entrainment rate yields:

ρlVE S E′ = (4-104) t E

The volume in this expression refers to the total volume that becomes entrained. Since the entrainment rate model that is developed within this section for the Kelvin-Helmholtz lifting mechanism calculates the volume for a single ripple wave, this value must be scaled by the number of waves contained within the computational cell. Therefore, the total volume that becomes entrained is:

VE = VE,RW N RW (4-105) and Equation (4-104) can be rewritten for the Kelvin-Helmholtz lifting mechanism as:

ρlVE,RW N RW S E′ ,RW ,model = (4-106) t E,RW

It can be seen from this equation that the calculation of the entrainment rate for the Kelvin- Helmholtz lifting mechanism in a given computational cell using this approach requires estimates of:

1) the number of ripple waves within the computational cell, 2) the time scale associated with entrainment, and 3) the volume of liquid that becomes entrained from a single wave.

First, the number of ripple waves that reside within a computational cell can be estimated as:

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67B 8  dz  L  P  N =   DW  i,DW  (4-107) RW  λ  λ  λ  12DW 3 RW 142RW 43 A C where:

Term A ≡ number of disturbance waves per computational cell, Term B ≡ number of ripple waves located along the axial length of the disturbance wave, Term C ≡ number of ripple wave located circumferentially around the interface, and the distance around the periphery of the interface at the crest of the disturbance wave is calculated as:

Pi,DW = π [DH − 2(δ min + ε w )] (4-108) for circular geometries and as:

Pi,DW = 2{[H − 2(δ min + ε w )]+ [W − 2(δ min + ε w )]} (4-109) for non-circular geometries.

The disturbance wavelength (i.e. spacing) and length of the disturbance waves are calculated using Equations (4-73) and (4-74), respectively, while an estimate of the wavelength of the ripple waves that reside on the tops of the disturbance waves will be obtained using a stability analysis that will be outlined later in this section. It should be noted that Equation (4-107) assumes: 1) the spacing of the ripple waves calculated by the stability analysis, which is performed in the axial direction, also governs the spacing of the ripple waves around the periphery of the flow path, 2) the ripple waves are equally spaced along the entire length of the disturbance wave, and 3) the ripple waves are uniformly spaced around the periphery of the interface. Woodmansee & Hanratty [56] observed the waves responsible for entrainment by this mechanism primarily reside

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near the crests of the disturbance roll waves; however, the fraction of length occupied by ripple waves has not yet been quantified experimentally so the uniform axial distribution assumption must be applied. As a result the estimate of the entrainment rate obtained using Equation (4-106) represents the maximum possible entrainment rate for the computational cell, such that:

S E′ ,RW ,model = S E′ ,RW ,max (4-110)

Then, similar to the approach used to develop the roll wave stripping entrainment model presented in the previous section, a functional relationship between the maximum and actual entrainment rates can be developed based on comparisons to experimental data, which would implicitly capture the fraction of disturbance wave length actually occupied by ripple waves and allow the actual entrainment rate to be calculated as:

S E′ ,RW = Θ RW SE′ ,RW ,Model (4-111)

The proposed relationship for the Kelvin-Helmholtz lifting mechanism will be presented later in this section.

Next, the time scale for entrainment can be estimated as the transit time required for the vapor to pass over a ripple wave, which can be calculated as:

λ t = RW (4-112) E,RW U − C v w RW

Again, an estimate of the wavelength of the ripple waves that reside on the tops of the disturbance waves, as well as the velocity of the ripple waves, will be obtained using a stability analysis that will be outlined later in this section. Meanwhile, the vapor velocity in this expression corresponds to the enhanced velocity that exists at the crest of the disturbance wave that was estimated as part of the two-zone interfacial shear model calculations using Equation (4-56).

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Third, estimating the volume that becomes entrained requires a geometry for the interfacial waves to be assumed and the height and spacing of the waves be known. For the purposes of the current work a three-dimensional sinusoidal wave structure is assumed, which can be described using:

   2πz  y = ε RW cos   (4-113)  λRW 

The three-dimensional structure is consistent with the observations of Taylor (1940); however, Woodmansee & Hanratty [56] suggest the wave structure to be more broad-crested (i.e. two- dimensional). Since it is unlikely that these smaller ripple waves are coherent ring wave structures it was decided for the purposes of the current study to quantify the volume of liquid contained in a single wavelet assuming a three-dimensional structure. The use of a sinusoidal function provides a simplified means of quantification, allowing a volume integration to be performed about the y-axis using the disc method of integration, as depicted in Figure 4-14, to determine the volume of liquid contained in a single ripple wave.

Figure 4-14: Disc Integration Method [95].

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Solving Equation (4-113) in terms of ‘ z’ yields:

  λRW −1 y z = cos   (4-114) 2π  ε RW 

In integrating a function by rotation about the vertical axis, the following general formula may be used [95]:

d V = π ∫[]R()y 2 dy (4-115) c

It was assumed that the entire volume of liquid calculated by Equation (4-115) becomes entrained. In reality only some fraction of this volume becomes entrained while the remaining portion of the unstable wave lies down and is swept into the disturbance wave [56]. This assumption further substantiates the notion that the entrainment rate predicted by Equation (4- 106) in the proposed model represents the maximum possible entrainment rate for the cell. Writing the bounds of integration in Equation (4-115) to reflect this assumption and substituting Equation (4-114) yields:

2 ε RW λ   RW −1  y  VE,RW = π ∫  cos   dy (4-116) 0  2π  ε RW 

Carrying out this integration yields a final expression for the maximum volume of liquid that becomes entrained from a single ripple wave by the Kelvin-Helmholtz lifting mechanism, which is given as:

ε λ2 (π − 2) V = RW RW (4-117) RW 4π

The amplitude of the ripple waves can be estimated using the shear flow model proposed by Ishii & Grolmes [3], which is given as:

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µlCDW ε RW = f ()N µ (4-118) τ i,w

Since the hydrodynamics inside of the wave crest can be described in terms of the viscous and surface forces, Ishii & Grolmes [3] used the viscosity number, which was defined previously as Equation (4-8), to correlate a variety of experimental data covering a range of geometries, fluids, and flow conditions; however, no steam-water data were considered. The resulting correlations indicate that for low viscosity numbers (< 1/15), which describes the values for steam-water over the pressure range of interest, the motion of the liquid in the wave is dominated by the surface tension effects [3], which is consistent with the Kelvin-Helmholtz lifting mechanism, as depicted in Figure 2-7. The correlation given by Ishii & Grolmes [3] for the low viscosity number region is given as:

1 f ()N µ = 8.0 (4-119) 3[]11 78. ()N µ

The use of Equations (4-8) and (4-119) in Equation (4-118) provides an estimate of the ripple wave amplitude for entrainment by the Kelvin-Helmholtz lifting mechanism.

Lastly, as previously alluded to, estimating the entrainment rate by the Kelvin-Helmholtz lifting mechanism using this approach requires estimates of the ripple wavelength, or spacing, as well as the ripple wave velocity. Estimates of both these quantities can be obtained from a stability analysis performed on the surface of a disturbance wave. In the case of two inviscid fluids with different densities that travel with mean velocities of U G and U L and are separated by a vertical interface on which interfacial waves reside the amount of distortion of the streamlines decreases with increasing distance from the interface, as shown in Figure 4-15.

This type of interfacial instability for co-current upward flow of a gas and liquid phase is discussed in detail by Hewitt & Hall-Taylor [5]. Hewitt & Hall-Taylor [5] explain that as the gas flows around the liquid wave, centrifugal forces are set up and these forces must be balanced by a

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pressure gradient in the direction normal to the streamline. At any position, the equation for the pressure gradient normal to the streamline is given as:

∂p − = ρ χ()U − C 2 (4-120) ∂η g G w

The gas velocity can be assumed to be nearly constant with increasing distance from the interface and since the curvature of the streamlines decreases with increasing distance from the interface, as shown in Figure 4-15, the magnitude of the pressure gradient therefore decreases with increasing distance from the interface. Ultimately the pressure gradient will decrease to zero at an infinite distance from the interface since the pressure and velocity fields will be undisturbed. The qualitative variation in pressure gradient normal to the streamline as a function of distance from the interface is shown in Figure 4-16. [33,34]

Figure 4-15: Streamlines in Two-Phase Flow with a Wavy Interface [5].

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Figure 4-16: Pressure Gradient Normal to a Wavy Interface [5].

The pressure at the interface is found by integrating Equation (4-120) from the interface to infinity. Holowach [33,34] explains that when examining the gas side of the interface it is deduced from the results of this integration that for surfaces of positive curvature (i.e. wave crests) the pressure at the interface will be less than the undisturbed pressure at infinity. Conversely, for surfaces of negative curvature (i.e. wave troughs), the pressure at the interface is greater than that at infinity. In other words, the gas flowing past the wavy interface generates an increased pressure over the troughs and suction over the crests [5], thereby resulting in the gas exerting a periodic normal stress that is 180-degrees out of phase with the wave displacement. Therefore, over a complete wavelength the gas does not exert a net force on the interface, but the pressure distribution does give rise to local forces that can cause distortion of the interface [5].

On the other hand, when examining the liquid side of the interface, it can be seen from Figure 4- 15 that the curvature of the streamlines is such that the liquid exerts an outward pressure at the crests and an inward suction at the troughs, and thereby further assists the gas phase by distorting the interface. The pressure and the resulting suction force at the wave crest are proportional to the product of the gas density and the square of the relative velocity between the gas and wave. [33,34]

For a given condition in which the wave velocity is complex (CRW = CRW ,R + iC RW ,I ) the ripple wave height at the interface of the disturbance wave can be described using:

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kC RW , I t ε (y) = ε we cos [k(z − CRW ,Rt)] (4-121)

where ‘εw’ is the disturbance wave amplitude and the ‘ k’ is the wave number, which is given as:

2π k = (4-122) λRW

For positive values of the imaginary component of the wave velocity the wave amplitude will increase exponentially with time while for negative values of the imaginary component of the wave velocity the wave amplitude will decrease exponentially with time. When the imaginary component of the wave velocity is equal to zero, the wave amplitude will remain constant. Therefore, the problem of interfacial stability, which is associated with droplet entrainment by the Kelvin-Helmholtz mechanism, can be reduced to determining the conditions that cause the imaginary component of the wave velocity to be greater than zero.

Holowach [33,34] obtains an expression for the critical wavelength at which the interface becomes unstable for a given set of fluid conditions by relating the normal stresses and the surface tension forces acting on the interface using perturbation theory. The effect of gravity was not included in the analysis since the situation of interest (i.e. nuclear reactor applications) is primarily concerned with vertical annular flow. Holowach [33,34] applies axisymmetric, laminar, and inviscid flow assumptions as well as assuming the slope of the interface to be small. This last assumption allowed the fluid motion equations to be linearized and provided a simplified solution that could be applied to the analysis of the entrainment phenomenon. Meanwhile, the inviscid assumption is justified by Holowach [33,34] based on the results of a separate scaling analysis.

Following the approach of Hewitt & Hall-Taylor [5], Holowach [33,34] conducted separate analyses for the gas core and liquid film region; however, Hewitt & Hall-Taylor [5] assumed the wavelength was considerably greater than the film thickness, which is true for disturbance waves and greatly simplifies the analysis, but is inconsistent with the observations of both Woodmansee & Hanratty [56] and Cohen & Hanratty [96] that the primary means of droplet entrainment is

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ripple waves that have a wavelength that is on the same order of magnitude as the film thickness. As a result Holowach [33,34] eliminated this assumption in his analysis.

Based on the observations of Woodmansee & Hanratty [56], the analysis conducted by Holowach [33,34] was modified within the current study to be applied at the surface of the disturbance waves rather than at the surface corresponding to the mean film thickness since this is where the ripple waves responsible for entrainment by the Kelvin-Helmholtz lifting mechanism actually reside. The disturbance wave velocity and amplitude applied in the stability analysis are consistent with those are calculated by the two-zone interfacial shear model. Additionally, Holowach [33,34] suggested it was more appropriate to use gas core values (density or velocity), rather than mean vapor values; however, this suggestion is inconsistent with the findings of Hurlburt et al. [55]. Therefore in the current work the vapor density and the enhanced vapor velocity, which was calculated within the two-zone model, was used instead of effective gas core values.

Accounting for the modifications listed above the resulting expressions for the normal stresses exerted by each phase on the interface based on the results presented by Holowach [33,34] are:

2 N = N − kε ρ U − C eikz (4-123) i,v i,v w v ( v w RW ) for the vapor and as:

2 [cosh (kε w )−1] ikz Ni,l = Ni,l − ε w ρl ()CDW − CRW e (4-124) sinh ()kε w for the liquid film. A diagram provided by Holowach [33,34] of the normal stresses at the interface, and the requisite coordinate systems used to conduct the separate liquid film and vapor analyses is shown in Figure 4-17.

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Figure 4-17: Normal Stresses and Coordinate Systems Utilized by Holowach [33,34] in the Interfacial Instability Calculation

Accounting for the different coordinate systems (r = 0 at the centerline of the flow path for the vapor analysis, and y = 0 at the wall for the liquid film analysis), a balance of the vapor normal stress, liquid normal stress, and surface tension force is given by Holowach [33,34] as:

Ni,l − N i,v = σχ (4-125) where the curvature of the interface that is expressed as:

2 ikz χ = k ε we (4-126)

For a stable interface condition, the stress of the surface tension exactly balances the effects of the normal stresses. Conversely, the wave will grow in amplitude when the sum of the local liquid and gas normal stresses exceeds the surface tension stress, leading to the Kelvin-Helmholtz

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instability, where the surface tension stress is no longer capable of retaining the wave configuration. [33,34]

Holowach [33,34] determined the stability condition by solving Equation (4-125). Substituting Equations (4-123), (4-124), and (4-126) into Equation (4-125), canceling the exponential and perturbed wave amplitude terms, combining the wave number terms, and canceling the average normal stress exerted by the gas and liquid phases at the interface yields the characteristic equation for ripple wave motion, which is given as:

2 [cosh (kε ) −1] 2 ρ C − C w + ρ U − C = σk (4-127) l ()DW RW v ()v w RW sinh ()kε w

This result, which applies to upward, co-current annular flow where U v > U l , is a quadratic equation that can be solved for the real and imaginary parts of the ripple wave velocity. Based on the results given by Holowach [33,34] the real and imaginary components of this expression are:

[cosh (kε )−1] w ρ C + ρ U l DW v v w sinh ()kε w CRW ,R = (4-128) []cosh ()kε w −1 ρl + ρv sinh ()kε w and:

1 2   []cosh ()kε −1  []cosh ()kε −1 2  σk w ρ + ρ  − w ρ ρ ()U − C   sinh kε l v  sinh kε l v v w DW   ()w  ()w  (4-129) CRW ,I = ± []cosh ()kε w −1 ρ l + ρv sinh ()kε w

As previously mentioned, instability occurs by definition when the imaginary component of the ripple wave velocity is greater than zero and therefore the critical ripple wavelength can be obtained by solving:

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 [cosh (kε ) −1]  [cosh (kε )−1] 2 k w  w U C (4-130) σ  ρl + ρ v  − ρl ρv ()v w − DW = 0  sinh ()kε w  sinh ()kε w

Direct solution for the critical ripple wavelength in Equation (4-130) is not possible, but instead a numerical method must be applied. Since it is not possible to easily bound the solution variable a secant method can be applied to determine this wavelength. Once the critical ripple wavelength is known the ripple wave velocity can be obtained using Equation (4-128). [33,34]

At this point estimates for all of the quantities needed to calculate the maximum entrainment rate for a given computational cell using Equation (4-106) have been obtained. The next step is to develop a functional relationship between the actual and maximum entrainment rates for the Kelvin-Helmholtz lifting mechanism based on comparisons to experimental data. As outlined in Section 3.2.2. such a relationship was determined using a subset of the experimental data available for the co-current upward annular regime. In the case of the Kelvin-Helmholtz lifting mechanism this subset consisted of 36 experimental data points where the COBRA-TF predicted Weber number (based on film thickness) was less than 25, as suggested by Azzopardi [49]; however, it should be noted all of the experimental data points that satisfied this condition corresponded to the low pressure experiments conducted by Hewitt & Pulling [83]. Weber numbers (based on film thickness) this low were not predicted by COBRA-TF for any of the higher pressure cases considered. Possible explanations for this were offered in Section 4.2.2. In any event, the proposed functional relationship between the actual and maximum entrainment rates for the Kelvin-Helmholtz lifting mechanism is:

  −4   Re l −1750  ΘRW = 3.2 x10 min  ,0.1 max  ,0.0  (4-131)    6000 −1750 

Figure 4-18 provides a comparison between the optimized multipliers and the correlated result obtained using Equation (4-131) as a function of film Reynolds number. It can be reasoned that the behavior of the relationship between maximum and actual entrainment rate at low film Reynolds numbers occurs because the shear flow model used to calculate ripple wave amplitude

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is no longer applicable. In addition this range of film Reynolds numbers approximately corresponds to the transition from turbulent to laminar flow.

1.E-02

Optimized Values 1.E-03 Correlated Result

1.E-04

1.E-05 Ratio of Actual to Maximum Entrainment Rate 1.E-06 0 2000 4000 6000 8000 10000 12000 14000 Film Reynolds Number

Figure 4-18: Correlation between Maximum and Actual Entrainment Rate for the Kelvin- Helmholtz lifting mechanism as a function of Film Reynolds Number.

In summary, the proposed entrainment rate model for the Kelvin-Helmholtz mechanism represents an improvement over currently available models since it is based on the physical phenomena supposed to govern entrainment by this mechanism as it is currently understood. It will be shown in Section 4.3 that the inclusion of this model significantly improves the predictive capability of COBRA-TF. Potential improvements to this model could address the following assumptions:

1) The assumption that the entire volume of the ripple wave becomes entrained. 2) The estimated time scale for entrainment.

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3) The assumption that ripple waves are uniformly distributed along the entire length of the disturbance wave. 4) The assumption that the wavelength calculated from the stability analysis, which was performed in the axial direction, also governs the spacing of the ripple waves in the peripheral direction. 5) The assumption that the functional relationship presented as Equation (4-131), which was developed only on the basis of low pressure experimental data, can be extended to higher pressures.

Again this working model provides a starting point for future studies where the aforementioned limitations could be addressed.

4.2.4.4. Liquid Bridge Breakup Mechanism

As previously mentioned the churn-turbulent regime in COBRA-TF consists of linear ramp between the annular and small-to-large bubble flow regimes. Since entrainment cannot exist in bubbly flow, the logic in the baseline version of COBRA-TF linearly ramped the entrainment rate calculated for the annular regime from zero to the calculated value with increasing void fraction through the churn-turbulent regime. In reality the roll wave stripping entrainment rate is augmented by the liquid bridge breakup mechanism in the churn-turbulent regime. The entrainment rate model that has been included in the proposed modeling package was presented previously within Chapter 2 as Equations (2-87) through (2-92). Unlike for the models presented for the roll wave stripping and Kelvin-Helmholtz lifting mechanisms, the comparisons to experimental data indicated that a relationship between the actual and theoretical entrainment rates is not needed for the liquid bridge breakup model. Incorporating this model into the proposed modeling package allows for the explicit consideration for entrainment occurring by this mechanism, which was not previously considered by the code.

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4.2.5. Droplet Drag

The proposed modeling package utilizes the droplet drag coefficient suggested by Clift & Gauvin (1970) and was given previously as Equations (2-65) and (2-62). While the actual difference in the values predicted by this and the previous correlation is negligible, as shown in Figure 2-6, the inclusion of this model still represents an improvement over the previous model for two reasons. First, Equation (2-65) provides a transition between drag regimes that has a continuous first derivative, which was not true for the previously used expression. Second, this expression captures a shedding effect near the transition point, which acts to reduce the drag coefficient in this region and was not captured by the previously used expression. Therefore, this expression more accurately reflects the physics of the flow. While these are only minor improvements that will most likely have a negligible effect on the calculated results the modification was still worthwhile for the reasons outlined above.

4.2.6. Deposition Rate

It was decided to continue using the deposition rate model from the baseline modeling package within COBRA-TF in the proposed modeling package. This diffusion-based model proposed by Cousins et al. [70] uses the correlation for the deposition mass transfer coefficient that was proposed by Whalley et al. [78], which is a function of surface tension only and was developed considering steam-water data over the range of pressures of interest to the current study. These expressions were given previously as Equations (2-72) and (2-74), respectively. However, as outlined in Section 4.2.1., the deposition rate model given as Equation (2-72) was updated to utilize the interfacial area of the film, instead of the surface area of the flow duct, since it provides a more accurate physical representation of the point where droplet deposition actually occurs. The updated form of this model is given as:

S D′ = k DCA i,l (4-132)

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where the interfacial area of the film in this expression is calculated using Equation (4-69). Given the amount of scatter that exists in the other available deposition rate models and the lack of understanding in the exact mechanisms causing deposition to occur in annular-dispersed flow, it was decided that the continued use of this model provided the best available option at this time.

4.2.7. Entraining Drop Size

The inclusion of the mechanism-specific entrainment rate models in the proposed modeling package also allows for the mechanism-specific entraining droplet size correlations to be applied. The difference in measured drop sizes within the different regions where each mechanism is supposed to exist, is well documented in the open-literature. As a result the ability to specify mechanism-specific drop size correlations provides a more realistic representation of annular flow. This section outlines the correlations for Sauter mean diameter that were coupled with each entrainment rate model in the proposed modeling package.

First, based on the suggestion of Azzopardi et al. [21] and Azzopardi [49] the Tatterson et al. [40] correlation, which was given as Equation (2-95), is used to predict the entraining droplet size for the Kelvin-Helmholtz lifting mechanism in the proposed modeling package. Meanwhile, Fore et al. [31,51] indicate that the most suitable agreement with their experimental data, which was taken under conditions that correspond to the existence of the roll wave stripping mechanism, was obtained using the drop size correlation that was proposed by Kocamustafaogullari et al. [76]. Based on this suggestion the correlation proposed by proposed by Kocamustafaogullari et al. [76], given as Equations (2-98) and (2-99), is used in the proposed modeling package when entrainment is supposed to occur this mechanism. Lastly, the entraining drop size for the liquid bridge breakup mechanism is governed in the proposed modeling package by a critical Weber number criterion, which is consistent with this mechanism for entrainment. The correlation used in the current study is given as:

5.6 σ DE,LBB = .0 06147 2 (4-133) ρv ()U v −U l

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A similar set of restrictions as was given previously as Equation (2-96) is imposed to prevent unphysically large values from being obtained.

4.2.8. Wall Shear Stress

The previously conducted work by Holowach [33,34] provided code-to-data comparisons of the single-phase liquid pressure gradient using COBRA-TF and a series of tests conducted at the University of California - Berkley [97]. Reasonable agreement was obtained which provides confidence that the wall friction factor used by COBRA-TF is adequate. As a result the wall friction factor correlation used in the modeling package in the baseline version of COBRA-TF, which is given as:

 f w,lam if Re l < 2000  f w =  f w,turb if Re l > 5000 (4-134)  ()1.0 - R w f w,lam + R w f w,turb otherwise was applied in the proposed modeling package, where:

64 f w,lam = , (4-135) φ Re l

1 − 3 f w,turb = 55.0 [ 01.0 + (φ Re l ) ], (4-136)

Re − 2000 R = l , (4-137) w 5000 − 2000 and where φ is a geometry specific factor that has a value of one for circular flow paths.

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However, it should be noted that Fore & Dukler [23] used their experimental data from vertical annular flow experiments in a 2-inch diameter tube with a momentum balance analysis to determine the mechanism of momentum transfer from depositing droplets. It is typically assumed that the momentum carried by depositing droplets is transferred to the liquid film. In this situation two distinct forces act to drive the liquid film upwards: 1) the interfacial friction between the liquid film and the vapor and 2) the momentum transferred by the droplets as they decelerate in the liquid film. However, based on their results Fore & Dukler [23] suggest that a droplet that has a diameter that is comparable to the mean film thickness would likely pass through the liquid film and effectively decelerate at the wall. This would result in a local, impulsive increase in the wall shear stress at the point of impact, such that the total wall shear is comprised of both a continuous stress exerted by the mean film flow and discrete stresses exerted by depositing droplets. If this is the case the depositing droplet would effectively transfer zero momentum to the liquid film and then rather than aiding in driving the liquid film upwards, the deposition process acts as an additional loss mechanism. Similarly, previously conducted works have indicated an enhancement in the measured wall shear stress corresponding the passing of a disturbance wave [10,58]. Neither of these effects were considered explicitly in the current study, but these effects on wall shear enhancement in annular flow may warrant considerations in future studies.

4.2.9. Churn-Turbulent Regime

As mentioned previously, within the churn-turbulent regime a void fraction weighted, logarithmic ramp is used to calculate effective values of:

1) the interfacial drag using the calculated values from the small-to-large bubble transition regime and the two-zone model

2) the interfacial heat transfer coefficients using the calculated values from the small-to- large bubble transition regime and the annular regime

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3) the entrainment rate and entraining drop size using the calculated values from the liquid bridging mechanism and either the roll wave stripping or Kelvin-Helmholtz lifting mechanism (whichever is determined to exist in the unstable annular regime).

The ramping function is given as:

  α − 5.0  R   v  (4-138) CT = max ,0.0 min  ,0.1    α bridge − 5.0  where again the liquid bridging criterion corresponds to a vapor void fraction of 0.8 or 0.6 for large or small hydraulic diameter channels, respectively. Then the desired quantities are calculated as:

xCT = exp [ln (xSLB )( 0.1 − RCT )+ ln (xUN )RCT ] (4-139)

In general the churn-turbulent regime is highly chaotic and applying a ramping function in this manner within this region provides a smooth transition between flow regimes where the dominate continuous field or phase switches from liquid to vapor.

4.2.10. Conclusions on Proposed Modeling Package

The proposed annular flow modeling package outlined in this section represents an improvement of the modeling package used in the baseline version of COBRA-TF for several reasons. First, the pre-CHF flow regime map associated with the proposed modeling package, which is shown in Figure 4-19, provides a more realistic representation of the annular and churn-turbulent regimes relative to the original pre-CHF flow regime selection logic that was presented previously in Figure 2-3. Second, the models incorporated in the proposed modeling package explicitly consider the interfacial structure and governing phenomena as they are currently understood. It will be shown in the following sections that the use of mechanistic-based models, rather than simple correlations, allows the applicable range of the models to be extended. Third, wherever

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empirical models are still be applied within the modeling package, these models are now based on steam-water, rather than air-water, data. Fourth, the proposed model provides a self-consistent set of calculations, where parameters that appear in more than one model are calculated in the same manner within each model. Lastly, the proposed modeling package includes entrainment rate models for three different mechanisms and mechanism-specific correlations for predicting the corresponding entraining drop size. Previously a single set of empirical correlations for entrainment rate and entraining drop size were applied by COBRA-TF for all annular and churn- turbulent flow situations.

It will be shown in the remainder of this chapter that implementing the proposed annular flow modeling package into COBRA-TF significantly improved the predictive capabilities of the code within the annular and churn-turbulent regimes relative the to results presented previously in Section 4.1. It will be also shown in Section 4.3.4. that the comprehensive approach used to develop the proposed modeling package in the current study has also eliminated some numerical instabilities that existed previously within the code. A listing of the FORTRAN source code containing the proposed modeling package that was outlined in this section is given in Appendix A. This subroutine was implemented into COBRA-TF in the current study by replacing the baseline modeling package with a call to this subroutine from within the INTFR subroutine; however, this portion of code can readily be implemented into other two-phase, three-field transient analysis code.

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False αv < ala ala=bridging criterion

True

Small Bubble Regime (SB) isij = 1

αv > alb False alb = 0.2

True

Large Bubble Regime (LB) Note: this regime cannot exist individually

Small -to -Large Bubble Transition (SLB) isij = 3 Note: ramp between SB and LB values

αv > alsa False alsa = 0.5

True

Annular - Stable Film (ST) isij = 5 fi = One-Zone Model, S E = 0

ψ > 0 False Intermittency

True

Annular - Unstable Film (UN) isij = 4 fi = Two-Zone Model, S E = Roll Wave or Lifting

αv < ala False ala=bridging criterion

True

Churn -Turbulent Regime (CT) isij = 3 fi = Ramp between SLB and UN values SE = Liquid Bridge Breakup augmented by UN Regime Value

Figure 4-19: Proposed COBRA-TF Pre-CHF Flow Regime Map.

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4.3. Presentation and Assessment of Results

This section presents the COBRA-TF predicted results using the newly proposed annular flow modeling package outlined in the previous section. The foundation of this assessment is quantitative comparisons between of the predictive capability of the code using both the baseline and proposed modeling packages, but qualitative comparisons of various parameters are also provided. Qualitative comparisons are made both axially along the test section for a given test, which allows the developing flow effects to be observed, as well as between the various tests considered at the outlet of the test section. This type of information provides confidence the proposed model adequately captures the overall trends of the experimental data.

The results for COBRA-TF predicted versus experimentally measured flowing quality for the 263 steam-water experimental cases considered in the current study are shown in Figure 4-20. These results provide confidence that the input parameters for the simulations are correct and that a stable, steady-state solution was obtained.

Figure 4-20: COBRA-TF Predicted versus Experimentally Measured Results for Outlet Flowing Quality using the Proposed Modeling Package.

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Meanwhile, the results presented in Figure 4-21 for COBRA-TF predicted vapor void fraction at the outlet of the test section indicate that COBRA-TF suggests 34 experimental cases considered correspond to churn-turbulent flow conditions. All of these cases correspond experiments conducted at higher pressures. As indicated on the flow regime map presented as Figure 4-19, the liquid bridging criterion serves as the transition between the annular and churn-turbulent regimes in the proposed modeling package and in the case of large hydraulic diameter flow paths this criterion corresponds to a vapor void fraction value of 0.8. As expected the churn-turbulent cases predicted by COBRA-TF occur at lower dimensionless superficial gas velocity conditions.

Annular Regime

Churn-Turbulent Regime

Figure 4-21: Comparison of the COBRA-TF Predicted Vapor Void Fraction as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet.

Lastly, Figure 4-22 shows the ratio of entrainment to deposition rate predicted by COBRA-TF at the outlet of the test section. If annular flow equilibrium conditions were predicted to exist at the outlet of the test section in the simulations then this ratio should be approximately one. This figure highlights that annular flow equilibrium conditions are obtained for a majority of the cases considered, but most notably the experiments conducted by Keeys et al. [81] and Singh [82] are

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not predicted by COBRA-TF to reach annular flow equilibrium. Such a result is reasonable given the shorter hydraulic lengths for these tests relative to hydraulic lengths used in the other experiments considered in the current study. These results also display the fundamental difference in how annular flow equilibrium is approached for the porous sinter (Keeys et al. [81]) and droplet injection (Singh [82]) techniques. In the case of a porous sinter the entrainment process is primarily responsible for driving the flow towards annular flow equilibrium such that in the developing flow region the entrainment rate exceeds the deposition rate. Meanwhile the converse is true in the case of droplet injection where the deposition process is primarily responsible for driving the flow towards annular flow equilibrium.

Figure 4-22: Ratio of COBRA-TF Predicted Entrainment to Deposition Rates at the Outlet as a function of Measured Flowing Quality using the Proposed Modeling Package.

The following sections present the results for entrained fraction, axial pressure gradient, and several other parameters obtained using the newly proposed annular flow modeling package. In addition to the results in these sections, detailed results obtained from the simulation of the tests conducted by Würtz [79] are provided in Appendix B.

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4.3.1. Entrained Fraction

The results obtained for entrained fraction or entrained flow rate using the proposed model are presented in this section in Figures 4-23 through 4-26. Meanwhile the statistical results comparing the error in the entrained fraction for when the baseline and proposed modeling packages were applied is provided in Table 4-1.

Comparing the results presented in Figure 4-23 to those presented previously in Figure 4-2 provides a visual representation of the improvement in the predictive capability of this parameter by the code using the proposed modeling package. Comparing Figures 4-24 and 4-26 to Figures 4-3 and 4-4, respectively, further highlights the substantial improvement that is achieved in the entrainment predictions at low pressures using the proposed modeling package. The corresponding regions where each entrainment mechanism is supposed to occur is also indicted on Figure 4-24.

Figure 4-23: COBRA-TF Predicted versus Experimentally Measured Outlet Entrained Fraction using the Proposed Modeling Package.

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Total Liquid Entrainment Flow Rate supposed to occur by Kelvin- Experimental Data Helmholtz Lifting Mechanism COBRA-TF Prediction

We ≈ 25 Entrainment supposed to occur by Roll Wave Stripping Mechanism

Total Liquid Flow Rate Experimental Data

COBRA-TF Prediction

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Total Liquid Flow Rate

Experimental Data COBRA-TF Prediction

The statistical results indicate that the mean relative error in the predicted outlet entrained fraction is reduced from 20.2% (underprediction) to 4.5% (overprediction) using the proposed modeling package. As illustrated in the figures the most substantial improvement in the code-to-data comparisons was obtained for the low pressure data of Hewitt & Pulling [83] and Yanai [64,84]. Meanwhile, a slight increase occurred in the mean relative error for the predictions of the tests performed by Keeys et al. [81] and Singh [82], as well as by Würtz [79] in the smaller diameter test section. This increase is reasonable given that the entrainment model used in the baseline version of COBRA-TF was developed by Würtz [79] using the data collected in the same set of experiments and encompassed the range of conditions explored by both Keeys et al. [81] and Singh [82]. However, using the proposed modeling package decreased the observed error for the tests conducted by Würtz [79] in the larger diameter test section from 9.3% (underprediction) to 1.3% (overprediction). This result is extremely positive since the data from these tests was not used to develop the proposed entrainment rate models and the hydraulic diameter of these tests is much larger than that of any of the tests used for the model development. In general the results

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presented in this section indicated that the proposed modeling package has: 1) enhanced the predictive capability of the code for entrainment phenomena, 2) eliminated the bias that existed at low pressures, and 3) provides a mechanistic-based calculation of entrainment phenomena by a variety of mechanisms

Table 4-1: Results of the Statistical Analysis for Entrained Fraction in Co-Current Upward Annular Flow. Hewitt & Yanai Keeys et Singh Würtz Test Program Total Pulling [83] [64,84] al. [81] [82] [79]

L/D H ~393 ~197 ~191 ~288 ~195 ~898 ~450 -

Low High High Test Low Pressure, Pressure, Pressure, Pressure, High Pressure - Conditions Low Mass Flux Very Low High Mass Low Mass Mass Flux Flux Flux

Total Number 70 38 21 21 14 70 20 254 of Data Points Test Parameters Multi- Porous Porous Porous Liquid Porous Sinter Droplet Nozzle Sinter Sinter Sinter - Injection Type Injection Injection Injection Injection Injection Injection Yes, but Yes, but only Used for only data data at 83 Model No No Yes No at 435, No 35, (35+17+31) Development 725, 50, and and 1305 65 Baseline Mean 34.2 58.1 40.7 0.05 -37.3 0.8 9.3 20.2 Model Set Relative Proposed Error (%) -11.2 6.3 21.0 6.0 -37.2 -8.8 -1.6 -4.5 Model Set Absolute Baseline 42.3 58.1 44.9 4.5 40.5 25.2 14.2 34.6 Mean Model Set Relative Proposed 18.0 10.6 24.1 11.0 42.7 24.4 8.8 19.2 Error (%) Model Set Baseline Cross- 0.767 0.855 0.725 0.875 0.675 0.847 0.881 0.789 Model Set Correlation Proposed Coefficient 0.949 0.963 0.937 0.682 0.591 0.883 0.890 0.905 Model Set Baseline 0.709 0.548 0.691 0.921 0.715 0.909 0.891 0.828 Index of Model Set Agreement Proposed 0.972 0.969 0.938 0.736 0.680 0.938 0.942 0.951 Model Set Baseline Mean 30.3 46.9 31.0 0.06 -16.3 5.8 7.9 19.1 Model Set Fractional Proposed Error (%) -2.4 4.9 12.6 5.5 -15.0 -0.5 -0.4 0.6 Model Set Note: With the definitions given as Equations (3-15) and (3-17) a negative value in the mean relative or mean fractional error corresponds to an overprediction.

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4.3.2. Pressure Gradient

The results obtained by COBRA-TF for the axial pressure gradient using the proposed modeling package are presented in Figure 4-27. Comparing these results to those presented previously in Figure 4-1 that were obtained using the baseline modeling package highlights the significant improvement that has been achieved in the predictive capability of the code for this quantity. The results of the statistical analysis for this parameter is provided in Table 4-2. Overall, applying the proposed modeling package reduced the mean relative error from 108.2% (overprediction) to 7.6% (overprediction) for the 113 experimental cases considered in the current study. The data considered does cover a range of hydraulic diameters and hydraulic lengths (L/D H); however, as mentioned previously no experimental pressure gradient data was available for low pressure conditions and thus could not be used for comparisons in the current study. It is important to note that these results were obtained by implementing the two-zone interfacial shear model as it was originally proposed by Hurlburt et al. [55] without recorrelation or optimization. The only slight modifications that were needed were related to numerical stability.

Figure 4-27: COBRA-TF Predicted versus Experimentally Measured Results for Axial Pressure Gradient using the Proposed Modeling Package.

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Table 4-2: Results of the Statistical Analysis for Pressure Gradient in Co-Current Upward Annular Flow. Singh Würtz Test Program Total [82] [79]

L/D H ~195 ~898 ~450 - High Pressure, Test Conditions Low Mass High Pressure - Flux Test Total Number Parameters 14 79 20 113 of Data Points

Porous Porous Liquid Droplet Sinter Sinter - Injection Type Injection Injection Injection Baseline Mean -111.6 -112.4 -89.6 -108.2 Model Set Relative Proposed Error (%) 16.8 -14.6 3.3 -7.6 Model Set Absolute Baseline 111.5 112.4 89.6 108.2 Mean Model Set Relative Proposed 16.8 15.5 4.9 13.8 Error (%) Model Set Baseline Cross- 0.955 0.973 0.969 0.974 Model Set Correlation Proposed Coefficient 0.960 0.987 0.997 0.988 Model Set Baseline 0.482 0.602 0.640 0.645 Index of Model Set Agreement Proposed 0.915 0.984 0.998 0.988 Model Set Baseline Mean -56.9 -92.4 -40.8 -78.8 Model Set Fractional Proposed Error (%) 11.6 -12.7 1.6 -7.2 Model Set Note: With the definitions given as Equations (3-15) and (3-17) a negative value in the mean relative or mean fractional error corresponds to an overprediction.

4.3.3. Qualitative Assessment of Developing Flow Parameters

As mentioned previously it is useful to examine the axial variation of a variety of parameters to examine the relative change over the length of the test section and ensure the expected trends are captured. Given the different liquid injection techniques the behavior within the developing region should be fundamentally different within the developing flow region for the tests conducted by Keeys et al. [81] and Singh [82]. Figures 4-28, 4-29, and 4-30 provide the axial variation in the COBRA-TF predictions of film Reynolds number, entrained fraction, and ratio of

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entrainment-to-deposition rate, respectively, for the tests conducted by Singh [82]. In these cases the predicted film Reynolds number increases with axial length as the film thickens due to deposition. Correspondingly the entrained fraction decreases from 1.0 with increasing length. These parameters appear to asymptote to annular flow equilibrium conditions, but do not reach a constant value, which further highlights that annular flow equilibrium conditions were not predicted by COBRA-TF at the outlet for these tests. It should also be noted that the predicted film Reynolds number can increase by roughly an order of magnitude over the length of the test section for these tests. Figure 4-30 highlights that in droplet injection situations annular flow equilibrium is approached by depositing more droplets that are entrained. It can also be inferred from this figure that the proposed model does capture an entrance length effect where the film must become sufficiently thick before entrainment can occur. Each of the trends described here is as expected and desired. Meanwhile, Figures 4-31, 4-32, and 4-33 provide the axial variation in the COBRA-TF predictions of film Reynolds number, entrained fraction, and ratio of entrainment-to-deposition rate, respectively, for the tests conducted by Keeys et al. [81]. As expected the general trends for these tests, which used porous sinter injection, are opposite of those predicted for the droplet injection tests conducted by Singh [82].

Figure 4-28: Axial Distribution in the COBRA-TF Predicted Film Reynolds Number for the Singh [82] Experiments using the Proposed Modeling Package.

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Figure 4-29: Axial Distribution in the COBRA-TF Predicted Entrained Fraction for the Singh [82] Experiments using the Proposed Modeling Package.

Figure 4-30: Axial Distribution in the COBRA-TF Predicted Ratio of Entrainment to Deposition Rate for the Singh [82] Experiments using the Proposed Modeling Package.

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Figure 4-31: Axial Distribution in the COBRA-TF Predicted Film Reynolds Number for the Keeys et al. [81] Experiments using the Proposed Modeling Package.

Figure 4-32: Axial Distribution in the COBRA-TF Predicted Entrained Fraction for the Keeys et al. [81] Experiments using the Proposed Modeling Package.

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Figure 4-33: Axial Distribution in the COBRA-TF Predicted Entrained Fraction for the Keeys et al. [81] Experiments using the Proposed Modeling Package.

4.3.4. Results of Cousins & Hewitt [67] Simulations

Figures 4-34 and 4-35 show the experimental and predicted results for entrained flow rate and pressure, respectively, for the series of tests conducted by Cousins & Hewitt [67] at an inlet pressure of 40-psia. It can be seen by comparing these results with those presented previously in Figure 4-5 and 4-6 that a significant improvement in the predictive capabilities of both quantities is obtained using the proposed modeling package. In particular, the entrainment rate and inlet pressure are both slightly underpredicted using the proposed modeling package, whereas both were significantly overpredicted using the baseline modeling package (note difference in axial scale between Figures 4-5 and 4-34). Additionally, the proposed model captures the general trends in experimental data for entrained flow rate. Specifically, the upward turn in entrained flow rate at the upper elevations for the larger liquid flow cases, which is caused by the gas expansion effect, is captured by the proposed model. Lastly, the numerical instabilities that were observed in Figure 4-5 using the baseline modeling package no longer exist once the proposed modeling package was implemented.

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Total Liquid Flow Rate Experimental Data

COBRA-TF Prediction

Pinlet = 40-psia

Wair = 40-lb m/hr

Figure 4-34: Experimentally Measured and COBRA-TF Predicted Entrained Flow Rate as a function of Axial Elevation for the Cousins & Hewitt [67] Experiments using the Proposed Modeling Package.

Pinlet = 40-psia

Wair = 40-lb m/hr COBRA-TF Prediction

Total Liquid Flow Rate

Experimental Data

Figure 4-35: Experimentally Measured and COBRA-TF Predicted Axial Pressure Gradient as a function of Axial Elevation for the Cousins & Hewitt [67] Experiments using the Proposed Modeling Package.

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4.3.5. Qualitative Comparisons of Other Parameters

It is also worthwhile to make comparisons of the values predicted by the proposed modeling package for various quantities, such as wave slope, wave spacing, and velocity ratios, to values that have been published previously in the open-literature. Such comparisons are provided in this section. First, Figure 4-36 shows the ratio of film-to-total interfacial drag predicted by COBRA- TF at the test section outlet for the experimental cases considered. This figures shows the film interfacial drag accounts for roughly 75% or greater of the total interfacial drag and based on the COBRA-TF predictions the droplet drag tends to be more significant for the lower pressure conditions. It can be seen in Figure 4-37, which presents the COBRA-TF predicted ratio of entrained-to-vapor field mean velocities, that the primary reason for the greater importance of droplet drag a low pressures is that COBRA-TF predicts the entrained field to travel at roughly 70% of the vapor velocity, whereas at higher pressure the flow tends to be closer to mechanical equilibrium such that the effect of droplet drag is reduced. The COBRA-TF predicted ratio of entrained-to-vapor velocities at low pressure is reasonable based on the experimental observations of both Fore & Dukler [23] and Lopes & Dukler [26]. These researchers suggested values of 80% and 50%, respectively, for air-water annular flow at low pressure,. Meanwhile, it can also be seen in Figures 4-36 and 4-37 that COBRA-TF predicts the entrained droplets to be moving faster than the mean vapor velocity for some cases, typically at lower gas dimensionless velocities, and as a result the corresponding ratio of film-to-total interfacial drag is predicted to be greater than one for these cases. The reason for this behavior is unknown at this time and will warrant future investigation.

Next, Figure 4-38 shows the COBRA-TF predicted ratio of momentum for the depositing and entraining droplets as a function of film Reynolds number. Since the majority of the cases considered in the current study are near annular flow equilibrium conditions at the outlet, such that the deposition and entrainment rates are approximately equal (see Figure 4-22), the results presented in Figure 4-38 indicate that the ratio of entrained-to-film field velocities decreases with increasing film Reynolds number. Meanwhile, Figure 4-39 shows that the COBRA-TF predicted relative velocity between the vapor and film fields increases with increasing dimensionless superficial gas velocity, which is to be expected, but the relative velocity also tends to be about four times larger for the lower pressure cases compared to the higher pressure cases considered.

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Figure 4-36: Ratio of COBRA-TF Predicted Ratio of Film to Total Interfacial Drag at the Outlet as a function of Dimensionless Superficial Gas Velocity using the Proposed Modeling Package.

Figure 4-37: Comparison of the COBRA-TF Predicted Ratio of the Mean Entrained to Mean Vapor Velocities as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet.

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Figure 4-38: Ratio of COBRA-TF Predicted Ratio of Deposition to Entrainment Momentum at the Outlet as a function of Dimensionless Superficial Gas Velocity using the Proposed Modeling Package.

Figure 4-39: Comparison of the COBRA-TF Predicted Mean Vapor to Film Relative Velocity as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet.

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Next, Figure 4-40 shows that the two-zone interfacial shear model, proposed by Hurlburt et al. [55] and incorporated into the proposed modeling package, predicts the disturbance waves to travel roughly two to three times faster than the base film substrate. This is consistent with other observations that have been made in experimental studies. Meanwhile, Figure 4-41 shows the ratio of wave amplitude to mean film thickness that is predicted by a correlation proposed by Hurlburt et al. [55] within the two-zone interfacial shear model. As expected this ratio increases with film Reynolds number and the predicted values for the cases considered in the current study do approach the asymptotic upper limit of 1.36 that exists in the correlation proposed by Hurlburt et al. [55].

Then, Figure 4-42 shows the equivalent angle of disturbance waves as a function of film Reynolds number predicted by the COBRA-TF using the proposed modeling package. The equivalent angle of is calculated as:

 ε  −1  w  θ DW = tan   (4-141)  LD 

The values shown in Figure 4-42 are consistent with those observed by Hurlburt & Newell [94] in horizontal annular flow. Additionally, it should be mentioned that the upper limit of 5-degrees included in the modeling package is not activated for any of the steady-state results.

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Figure 4-40: Comparison of the COBRA-TF Predicted Disturbance Wave to Base Film Velocities as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet.

Figure 4-41: Comparison of the COBRA-TF Predicted Ratio of Disturbance Wave Amplitude to Mean Film Thickness as a function of Film Reynolds Number at the Test Section Outlet.

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Figure 4-42: Comparison of the COBRA-TF Predicted Equivalent Angle of the Disturbance Waves as a function of Film Reynolds Number at the Test Section Outlet.

Figure 4-43 shows the vapor velocity enhancement factor at the crests of the disturbance waves as a function of dimensionless superficial gas velocity. This factor is calculated within the two-zone interfacial shear model using Equation (4-56) to provide a more realistic estimate of the vapor velocity at this location assuming the disturbance waves are coherent, ring wave structures. Factors as large as 1.35 were observed for the cases examined in the current study, with these values decreasing with increasing dimensionless superficial gas velocities because the film thickness also tends to decrease under these same conditions and this reduces the disturbance wave amplitude. However, it should be noted that in the proposed modeling package a lower limit of one was imposed on the calculated value for this quantity and it can be seen in Figure 4- 43 that this limit is activated in the steady-state solutions obtained for several of the cases considered in the current study and especially for those corresponding to lower pressure situations. The activation of this is an indication that the disturbance wave heights predicted by the correlations within the two-zone interfacial shear model are less than the mean film thickness. In reality it is not possible for both the base film and disturbance wave heights to be less than the

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mean film thickness. This is an inconsistency within these correlations that should be addressed in the future.

Another inconsistency in the calculations included in the proposed model is highlighted by the results presented in Figures 4-44 and 4-45. Figure 4-44 shows the intermittency predicted by COBRA-TF using the two-zone interfacial shear model as a function of film Reynolds number. Intermittencies observed in cases examined in the current study range from 15% up to the asymptotic upper limit of Equation (4-66), which is 38%. Since all of the observed intermittencies are greater than zero then all of the mean dimensionless film thicknesses (in interfacial units) predicted by COBRA-TF using Equation (4-12) must be greater than the critical value for this quantity, which was given in Equation (4-11) to be twelve. However, as shown in Figure 4-45, the COBRA-TF mean film thicknesses were not all greater than the critical value suggested by Equation (4-41). This inconsistency in the two critical values was explained previously in Section 4.2.3. and was the reason for the imposed spline fit, given in Equation (4- 44), for film thicknesses close to the critical value suggested by Equation (4-41).

Meanwhile, Figure 4-46 shows the disturbance wave frequency predicted using the correlation proposed by Sawant et al. [57] within the proposed modeling package. As previously mentioned this correlation was developed from low pressure, air-water data at relatively low liquid Reynolds numbers. Therefore, the extension of this correlation to higher pressure steam-water conditions that are representative of nuclear reactor conditions is questionable. It can be seen in Figure 4-46, that the predicted wave frequency is significantly less for the higher pressure cases than the lower pressure cases at the same film Reynolds number condition. As a result, the disturbance wave spacing, which is shown in Figure 4-47, is also significantly less for the higher pressure cases. In turn, a larger number of waves are predicted to exist per computational cell for the higher pressure cases, as shown in Figure 4-48. In general, the disturbance wave spacing predicted by the proposed modeling package ranges from 0.36 to 9.6-inches, which is in reasonable agreement with the disturbance wave spacing of 2 to 10-inches observed by Sawant et al. [57], but the smaller spacings for the higher pressure cases causes upwards of ten disturbance waves to be predicted to exist within a single, 4-inch tall computational cell. This value is slightly higher than expected. Additionally, the imposed upper limit of ten disturbance waves per computational cell is invoked in the steady-state solutions for some of the higher pressure cases considered.

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Figure 4-43: Comparison of the COBRA-TF Predicted Velocity Enhancement Factor at the Disturbance Wave Crest as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet.

Figure 4-44: Comparison of the COBRA-TF Predicted Intermittency as a function of Film Reynolds Number at the Test Section Outlet.

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Figure 4-45: Comparison of the COBRA-TF Predicted Ratio of the Mean to Critical Film Thickness as a function of Film Reynolds Number at the Test Section Outlet.

Figure 4-46: Comparison of the COBRA-TF Predicted Disturbance Wave Frequency as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet.

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Figure 4-47: Comparison of the COBRA-TF Predicted Disturbance Wave Spacing as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet.

Figure 4-48: Comparison of the COBRA-TF Predicted Number of Disturbance Waves per Computational Cell as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet.

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Next, Figure 4-49 shows the COBRA-TF predicted Weber number (based on film thickness), which is calculated using Equation (2-16), at the outlet of the test section in the simulations. It can be seen on this figure that a majority of the experimental cases considered corresponded to conditions where entrainment was supposed to occur by the roll wave stripping mechanism, but several of the lower pressure cases and three high pressure cases corresponded to the region where the Kelvin-Helmholtz lifting mechanism is supposed to cause entrainment. Meanwhile, Figure 4-50 shows the values for the functional relationship between the actual and theoretical entrainment rates for the disturbance wave entrainment mechanism predicted by Equation (4-102) for the cases considered in the current study. It can be seen in this figure that much smaller values for this relationship, which correspond to an overprediction by the theoretical model, are predicted for the higher pressure cases relative to the lower pressure cases, which is consistent with the postulated breakdown of the wave frequency correlation used in the proposed modeling package when it is extended to higher pressure situations. Lastly, Figure 4-51 shows the predicted values for the critical ripple wavelength obtained from the stability analysis associated with the proposed entrainment rate model for the Kelvin-Helmholtz lifting mechanism. The predicted values shown in this figure are slightly less than the ripple wavelengths of 0.03 to 0.06- inches observed by Woodmansee & Hanratty [56], but these results are in reasonable agreement.

In general, the results presented in this section indicate the proposed modeling package provides physically reasonable estimates for a variety of quantities of interest within annular flow; however, these results also highlighted some areas of concern within the proposed modeling package that can be addressed in future works. In addition to the comparisons provided in this section, qualitative comparisons between the trends in disturbance wave characteristics (i.e. velocity, amplitude, spacing, and frequency) observed by Sawant et al. [57] and those predicted by the proposed modeling package are provided in Appendix C. Overall the results shown in this section and in Appendix C indicate the proposed modeling package has improved the underlying physics of the annular flow calculations within COBRA-TF and allows the code to provide estimates of additional quantities of interest within this regime (i.e. intermittency, frequency, etc.) that were previously unavailable when the baseline modeling package was applied.

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Entrainment supposed to occur by Roll Wave Stripping Mechanism

Entrainment supposed to occur by Kelvin-Helmholtz Lifting Mechanism

Figure 4-49: Comparison of the COBRA-TF Predicted Weber Number (based on Film Thickness) as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet.

Figure 4-50: Comparison of the COBRA-TF Predicted Ratio of the Actual to Theoretical Entrainment Rate as a function of Film Reynolds Number at the Test Section Outlet.

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Figure 4-51: Comparison of the COBRA-TF Predicted Ripple Wavelength as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet.

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4.4. Results of Sensitivity Studies

The results presented in the previous sections were obtained using the modeling methodology outlined in Table 3.3; however, it is important to ensure the proposed modeling package is not extremely sensitive to this particular modeling methodology since most of the parameters outlined in Table 3.3 are user-specified. Therefore, a series of sensitivity studies were conducted to verify that the predictive capability of the proposed modeling package is not adversely affected when various values for the parameters specified in Table 3.3 are applied. In particular, it was desired to examine the sensitivity of the proposed modeling package to computational node height, liquid injection area, injected entrained fraction, and injected drop size.

Figures 4-52 and 4-53 show the results of the computational node height sensitivity study for entrained fraction and axial pressure gradient, respectively. This study was performed by doubling the heights of the computational nodes used in the simulations. In these figures the open symbols correspond to the results obtained using the original noding and the solid symbols correspond to the results obtained when the computational node heights are doubled. Therefore, if the solid symbols lay within the open symbols then there is no effect on the predicted result. Based on this it can be seen in Figures 4-52 and 4-53 that the proposed modeling package shows very little sensitivity to the computational node height used in the simulations.

Similarly, Figures 4-54 and 4-55 show the results of the liquid injection area sensitivity study for entrained fraction and axial pressure gradient, respectively. This study was performed by reducing the liquid injection area associated with the mass source boundary condition by a factor of two for each experimental case considered. The open symbols correspond to the original predictions and then the solid symbols correspond to the results obtained when the liquid injection area was reduced by a factor of two. These results indicate that the proposed modeling package also shows very little sensitivity to the liquid injection area.

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Open Symbols ≡ Original Node Heights (3 to 4-inches)

Solid Symbols ≡ Node Heights Doubled (6 to 8-inches)

Figure 4-52: Sensitivity of COBRA-TF Predicted Entrained Fraction on Computational Node Height using the Proposed Modeling Package.

Open Symbols ≡ Original Node Heights (3 to 4-inches)

Solid Symbols ≡ Node Heights Doubled (6 to 8-inches)

Figure 4-53: Sensitivity of COBRA-TF Predicted Axial Pressure Gradient on Computational Node Height using the Proposed Modeling Package.

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Open Symbols ≡ Original Liquid Injection Areas

Solid Symbols ≡ Liquid Injection Areas reduced by a factor of 2

Figure 4-54: Sensitivity of COBRA-TF Predicted Entrained Fraction on Liquid Injection Area using the Proposed Modeling Package.

Open Symbols ≡ Original Liquid Injection Areas

Solid Symbols ≡ Liquid Injection Areas reduced by a factor of 2

Figure 4-55: Sensitivity of COBRA-TF Predicted Axial Pressure Gradient on Liquid Injection Area using the Proposed Modeling Package.

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Next, the sensitivity of the proposed modeling package to the inlet entrained fraction and inlet entrained drop size was examined using the series of tests conducted by Yanai [64,84] at a total flow rate of 250-lb m/hr. These tests were selected since the used a multi-nozzle injector, which presents the largest uncertainty in the inlet entrained fraction. The results of these studies are presented in Figures 4-56 through 4-59 and show very little sensitivity to these two parameters.

Lastly, although not shown explicitly, it should be noted that no hysteresis effect was observed when comparing results of simulations that were run: a) individually (i.e. a single pressure, total flow, and flowing quality combination) versus b) in series (i.e. a single input deck for a given pressure and total flow condition where the flowing quality was decreased in step increments over the course of the simulation.

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Figure 4-56: Sensitivity of COBRA-TF Predicted Entrained Flow Rate on Inlet Entrained Friction using the Proposed Modeling Package.

Figure 4-57: Sensitivity of COBRA-TF Predicted Axial Pressure Gradient on Inlet Entrained Friction using the Proposed Modeling Package.

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Figure 4-58: Sensitivity of COBRA-TF Predicted Entrained Flow Rate on Inlet Entrained Droplet Diameter using the Proposed Modeling Package.

Figure 4-59: Sensitivity of COBRA-TF Predicted Axial Pressure Gradient on Inlet Entrained Droplet Diameter using the Proposed Modeling Package.

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4.5. Conclusions

The proposed annular flow entrainment model that has been outlined in this chapter has addressed the primary concerns with the modeling package that was used in the baseline version of COBRA-TF. As mentioned previously, the proposed modeling package considers the interfacial structure and relevant mechanisms that are supposed to govern annular phenomena as they are currently understood while not significantly impacting the computational efficiency of the code. Additionally, the proposed modeling package maintains the numerical stability of the code by integrating several models that were developed previously in separate, unrelated studies in a self- consistent manner and ensuring the models are continuous at regime boundaries. The proposed modeling package is also more suitable for nuclear reactor applications than the modeling package used in the baseline version of the code, since it is based primarily on steam-water data. Also, the proposed modeling package was shown to provide reasonable agreement with the available experimental data within the developing, or non-equilibrium, annular flow region. Lastly, the methodology of incorporating mechanistic-based models into the proposed modeling package in an attempt to improve the underlying physical basis of the code, while simultaneously leveraging the available experimental data to ensure the modeling package is able to accurately reflect the experimental data, has both presently improved the predictive capability of COBRA- TF as well as provided a basis for future model development activities in this area. For example, future work can be aimed at addressing the weaknesses of the modeling package that were highlighted in Sections 4.2.4.2. and 4.2.4.3. Such work can include both experimental studies, that provide detailed information on annular flow characteristics over the range of conditions applicable to nuclear reactor applications, as well as theoretical studies, that provide models based on the underlying physics of annular flow.

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5. Counter-Current Flow Limitation Model

As highlighted in Sections 2.2.5. and 4.2.2., an appropriate transition criterion between the counter-current and co-current upward annular flow regimes is required to complete the proposed annular flow modeling package that was outlined in the previous chapter. The transition to annular flow is referred to as the flow reversal point; however due to the importance of the counter-current flow regime to nuclear reactor safety analyses (see Section 1.3) it was desired to provide a model that not only defines this transition, but also captures the characteristics of the counter-current flow regime. Previous works have been unable to ascertain a mechanistic-based model that is applicable over a wide range of conditions and flow path geometries, but it has been shown that the maximum amount of downward liquid penetration for a given vapor upflow condition within the counter-current regime can be described by an appropriately defined Counter-Current Flow Limitation (CCFL) correlation. This correlation, also referred to as a flooding curve, implicitly captures the associated flooding and flow reversal criteria and governs the partition between liquid upflow and downflow within the counter-current annular flow regime.

This chapter describes the development a three-field CCFL model to address the counter-current flow regime that can be implemented into three-field analysis tools that provide the ability model the liquid using both continuous and dispersed liquid fields. The development of this model is a unique aspect of the current study because of the explicit treatment of the entrained field, which previously suggested models did not consider because they were aimed at two-field analysis environments. This chapter provides a basis for this new model, outlines the derivation of the necessary equations for the model, and discusses the implementation of the model in COBRA-TF as well as the limitations of the model. A generic verification problem is developed and the results are presented to confirm the model was implemented correctly in COBRA-TF and is behaving in the desired manner. Next, the CCFL experiments conducted by Dukler & Smith [43] were simulated using COBRA-TF and some of these results are presented comparing the predictive capability of the code with and without the newly proposed modeling being applied.

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5.1. Model Basis

The review of previously conducted work focused on the flooding and flow reversal phenomena that was presented in Section 2.4. indicated that an adequate deterministic, physics-based model for predicting the behavior within the CCFL region over the desired range of hydraulic diameters and geometric configurations has yet to be resolved. In general, the partition between liquid upflow and liquid downflow within the counter-current regime is dependent upon the balance of the gravitational and the wall and interfacial shear forces [6]. The quantity with both the largest uncertainty and the greatest influence on this prediction is the interfacial drag between the vapor and continuous liquid fields. As a result, the inability of the code to predict the proper amount of liquid to penetrate these regions is a consequence of the code predicted interfacial drag being either too low or too high relative to that suggested by the available experimental data taken at these same locations. Given the severe overprediction of the pressure gradient by COBRA-TF when the baseline modeling package was applied (see Figure 4-1) it was obvious that an accurate prediction of this quantity did not exist. As a result the proposed annular flow modeling presented in Chapter 4 was developed in the current study to address these concerns. An accurate prediction of the interfacial drag, which this newly proposed modeling package provides (see Figure 4-27), was a prerequisite for the assessment of the counter-current flow regime and the development of an explicit CCFL model.

In nuclear reactor applications CCFL typically does not occur within the core itself, but rather at discrete locations within the system where gravity-driven drainage of liquid can be limited by the upward flowing vapor. These locations are typically associated with reactor internal structures where flow area expansion occurs because the vapor velocity is the largest in the constricted area just below the expansion, and thus corresponds to the limiting location for liquid downflow. Examples of such configurations within commercial reactor cooling systems include the top of the upper core tie plate, the top of the downcomer annulus, and steam generator support plates. Typically full-size or scaled experiments have been conducted to obtain CCFL data for these specific regions/configurations and this data has then been found to be correlated reasonably well using either the Wallis [4] or Kutateladze [45] form of the flooding correlation once appropriate values for the correlating constants ‘m’ and ‘C’ are determined. The resulting correlation provides a quantitative description of the experimentally determined flooding curve in this region

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and by definition provides a prediction of the maximum allowable liquid downflow for a given vapor upflow condition. The ability for a transient analysis code to leverage this experimental data by applying an appropriately defined CCFL correlation in regions where CCFL may occur is advantageous. While not entirely mechanistic, this approach ensures the proper amount of liquid flow can penetrate these regions, which is preeminent to achieving accurate predictions of coolant and temperature distributions for LOCA scenarios.

The inclusion of a CCFL correlation to govern the downward flow of liquid within the counter- current flow regime replaces the code calculated interfacial drag with a more appropriate result based on experimental data. In general it is desired to incorporate physically-based mechanistic models into transient analysis codes rather then relying on empirical correlations; however, at this time relying on a completely mechanistic approach to determine the CCFL point is unrealistic due to the unresolved geometric dependencies of this phenomenon. Moreover, a wide variety of geometric configurations precludes the development of a universal CCFL model with the available experimental data and techniques. The treatment of CCFL that is outlined in this chapter precludes the need for the diameter and geometric dependencies to be resolved at this time while still leveraging the experimental data that is readily available.

It has been suggested by Bharathan & Wallis [11] that the CCFL phenomenon is the result of a marked instability in the flow and therefore the interfacial drag models developed for “normal” annular flow situations may no be able to capture the behavior in this region. Meanwhile, Hurlburt et al. [55] indicate that their two-zone interfacial shear model becomes increasingly inaccurate as the vapor velocity decreases because the model does not explicitly consider the losses associated with the recirculation zone that exists on the backside of the larger interfacial waves. This effect acts to significantly enhance the pressure gradient near the flow reversal point.

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5.2. Model Description

Based on the findings of previous studies it is desired to have the ability to provide a model that allows the user to apply either the Wallis [4] or Kutateladze [45] form of the CCFL correlation with the appropriate values for the correlating constants. Both of these provisions make a CCFL model of this type more flexible. It was shown in Section 2.4.2. that the two forms of this correlation can be reduced to a single correlation if a weighting parameter is introduced. This technique lends itself to more efficient programming, thus limiting the opportunity for error. Additionally, since the CCFL phenomenon typically occurs at discrete locations within the reactor system it was decided for the purposes of the current work that the implementation should treat the CCFL model as a “boundary condition”. Doing this allows the user to specify: 1) the computational cell where the model is applied, 2) the values of the correlating constants, and 3) a flag ( β) that specifies the desired form of the CCFL correlation to apply ( β = 0 for the Wallis [4] form and β = 1 for the Kutateladze [45] form).

When CCFL conditions are determined to exist in a specified cell where the model is applied (i.e. liquid downflow rate exceeds the allowable amount dictated by an appropriately defined CCFL correlation) then the standard set of momentum equations normally considered by the code is modified to include the user-specified CCFL correlation. A similar approach has been used previously in RELAP5-3D; however, the unique aspect of the model developed in the current study is the implementation of the CCFL model into code that considers both discrete and continuous liquid fields separately. RELAP5-3D is a two-field analysis tool and therefore extending this approach to three-field analysis environments requires the development of new methods surrounding the treatment of the excess liquid relative to the allowable liquid downflow that collects in the region above where the CCFL boundary condition is applied. As a result, specific entrainment rate models had to be included in the proposed model.

In the proposed model a linearized form of the CCFL correlation replaces the continuous liquid field momentum equation. Then, the vapor field momentum equation is replaced by a total momentum equation, which is obtained by summing the three standard field momentum equations. Finally, the standard entrained liquid momentum equation is used. Using the total

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momentum equation ensures that the wall drag, gravity, and pressure terms will still be satisfied, but eliminating both the continuous liquid and vapor momentum equations prevents the code calculated interfacial drag between these two fields from impacting the calculation. Instead, a more reasonable value for this quantity based on experimental data is implicitly captured by the CCFL correlation. Concurrently, this approach maintains the code calculated values for: 1) the interfacial drag between the dispersed droplet and vapor fields, 2) the entrainment rate, and 3) deposition rate since the droplet momentum equation is still considered. The next section outlines the proposed model in its entirety.

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5.3. Model Implementation

This section describes the various aspects of the three-field CCFL model that was developed in the current study. Sections discussing the activation of the model, the derivation of the necessary momentum equations, the proposed entrainment rate models, exiting the model, and model limitations are provided.

5.3.1. Activating the CCFL Model

Within a given computational cell where a CCFL boundary condition is applied the flow conditions are tested explicitly during each time step. First, it must be determined if counter- current flow exists. In COBRA-TF the calculated mass flow rates for each field have direction associated with them (i.e. downflow corresponds to a negative value for mass flow rate). Therefore, counter-current flow conditions can be determined to exist if the following two criteria are satisfied:

n n a) Wv Wl < 0 , (5-1) which if satisfied indicates the continuous liquid and vapor fields are moving in opposite directions, and

n b) Wl < 0 (5-2) which if satisfied indicates the continuous liquid field is moving downwards.

If counter-current flow is found to exist in the cell of interest, the maximum allowable liquid downflow rate for the continuous liquid phase is calculated using the user-specified form of the CCFL correlation and correlating constants. A general expression for this quantity can be determined by solving the generic form of the CCFL correlation, which is given as:

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1 1 2 2 H v + mH l = C (5-3) for the liquid component, which yields:

1 2 H l = (C − H v ) (5-4) m 2

Again, the dimensionless parameters in this expression are defined as:

1 2 jk ρ k H k = 1 , for: k = l or v (5-5) 2 []g ⋅ wght ⋅ ()ρl − ρv and the weighting parameter in this expression is given as:

(1−β ) β wght = DH L (5-6)

The second length scale in this expression is defined in Equation (2-112). If a value of β = 0 is applied to the weighting parameter then the Wallis [4] form of the CCFL correlation is obtained while if a value of β = 1 is applied to the weighting parameter then the Kutateladze [45] form of the CCFL correlation is obtained.

Since the solution variable in COBRA-TF is the phasic mass flow rate it is desired to rewrite the dimensionless parameters (Hk) in terms of this variable rather than the superficial velocity. The superficial velocity is defined as:

Wk jk = , for: k = l or v (5-7) ρ k Atot

Based on this definition and the form of the dimensionless parameter defined in Equation (5-5) it is advantageous to define the following parameter:

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1 1 Yk = , where: k = l or v (5-8) Ax g ⋅ wght ⋅ ρ k ()ρl − ρv such that:

H k = YkWk , for: k = l or v (5-9)

Inserting the above relationship into Equation (5-4) and solving for the continuous liquid field mass flow rate yields:

1 2 W C Y W (5-10) l,crit = 2 ( − v v ) m Yl

This expression governs the magnitude of the maximum allowable downward mass flow rate of liquid for a given vapor upflow condition. It should be noted that the implementation of this expression in a three-field computational environment assumes that liquid only flows downwards in the continuous liquid field (i.e. no falling drops). This assumption is reasonable, given that as the vapor velocity decreases the propensity for entrainment to occur also decreases because, as discussed in Section 2.4.1. and 2.4.2.,the entrainment rate is primarily dependent on the interfacial shear stress. Also, previously conducted works indicate that the falling films present in counter-current flow situations are relatively smooth and a minimal amount of entrainment exists until the flooding point is reached [10,43].

Also, if:

C − YvWv ≤ 0 (5-11) in Equation (5-10) then:

Wl,crit = 0 , (5-12)

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meaning that the CCFL correlation indicates no liquid can penetrate downwards. Therefore, Equation (5-11) governs the flow reversal point and a restriction must be placed on Equation (5- 10) to prevent a negative value from being calculated for the quantity given as Equation (5-11). In addition, Equation (5-10) as it is written determines the magnitude of liquid mass flow. The direction of the flow is implied to be downward, but to be consistent with COBRA-TF a negative sign must be inserted in this expression to explicitly denote that this is a downflow quantity. Finally, the test to determine if CCFL conditions exist at a given time step in the computational cell of interest is done using explicit quantities calculated during the previous time step, which are denoted by the superscript ‘ n’. The inclusion of these different considerations yields a final expression for the critical, or maximum allowable, continuous liquid field mass flow rate in the downward direction, which is given as:

1 2 W n C Y nW n (5-13) l,crit = − 2 n [max ( ,0.0 − v v )] m Yl

If the code predicted downward mass flow rate for the continuous liquid field exceeds the critical value suggested by this expression in a computational cell where a CCFL boundary condition is applied then the following expression will be satisfied:

n n Wl,crit >Wl (5-14) and the CCFL model should be invoked. At first glance this expression given as Equation (5-14) may appear to be reversed; however, since both quantities in this expression are negative it takes this form.

5.3.2. CCFL Momentum Equations

When the CCFL model is invoked then the standard set of momentum equations is replaced with an alternative set of equations that more accurately governs the counter-current flow regime based on available experimental data. A three-field computational environment requires three

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momentum equations and the standard set of momentum equations includes equations for: 1) the continuous liquid field, 2) a combined equation for the vapor/non-condensable gas mixture since the two fields are assumed to move at the same velocity, 3) the entrained liquid field. The generic forms of these three equations are given as:

n~+1 n n n n n~+1 n n~+1 Wl = A1 + B1 ∆P + C1 Wl + D1 Wv , (5-15)

n~+1 n n n n n~+1 n n~+1 n n~+1 Wv = A2 + B2 ∆P + C2 Wl + D2 Wv + E2We , (5-16) and:

~n+1 n n n n n~+1 n ~n +1 We = A3 + B3 ∆P + D3 Wv + E3 We (5-17)

The corresponding matrix form of these equations is:

 n n  n~+1   n n n  C1 −1 D1 0 Wl − A1 − B1 ∆P  n n n  n~+1   n n n   C2 D2 −1 E2 Wv  = − A2 − B2 ∆P  (5-18)  n n  n~+1   n n n   0 D3 E3 −1We  − A3 − B3 ∆P 

In these expressions the ‘n’ superscript indicates an explicit quantity and the ‘ n~ + 1 ’ superscript indicates a tentative new time quantity. The simultaneous solution of these equations by Gaussian elimination yields tentative new time values for the field flow rates that are then substituted into the Jacobian that is used to solve the continuity and energy equations and determine the state variables (pressure, enthalpy, void fractions). Since the Jacobian considers a pressure derivative term the tentative new time values obtained from the solution of Equation (5- 18) can be updated following the solution of the Jacobian based upon the calculated pressure gradient as:

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 n+1   n~+1   n  Wl Wl Wl  n+1   n~+1  ∂  n  n+1 Wv  = Wv  + Wv ∆P (5-19) ~ ∂P  n+1   n +1   n  We  We  We  where the derivative of the flow rates with respect to pressure gradient can be calculated from Equation (5-18) as:

 n   n  Wl B1 ∂  n  n −1  n  Wv  = −[]LHS B2  (5-20) ∂P  n   n  We  B3 

On the other hand, as alluded to previously, when the CCFL model is invoked it is proposed that: 1) the continuous liquid field momentum equation should be replaced by a mixture, or total, momentum equation, 2) the vapor/non-condensable gas momentum equation should be replaced by governing equation for the CCFL phenomenon, and 3) the momentum equation for the entrained field should be considered. The momentum equation for the entrained field is unchanged from what is applied in the standard momentum equation set and given previously as Equation (5-17). Meanwhile, the total momentum equation is obtained by summing the three individual momentum equations given by Equations (5-15) through (5-17), which yields:

n~+1 n~+1 n~+1 n n n n n n n Wl +Wv +We = (A1 + A2 + A3 )+ (B1 + B2 + B3 )∆P (5-21)

n n n~+1 n n n n~+1 n n n~+1 + (C1 + C2 )Wl + (D1 + D2 + D3 )Wv + (E2 + E3 )We

Lastly, the generic form of the CCFL correlation given by Equation (5-3) can be expanded and written in terms of field mass flow rates using Equation (5-9). Doing this, and again inserting a negative sign to consider the flow is in the downward direction, yields:

1 2 Yl Wl = − (C + Yv Wv − 2C Yv Wv ) (5-22) m2

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It is important to note that this expression is only valid if the calculated liquid flow rate is less than or equal to zero. This requires a provision be included in the proposed model to protect against situations where “velocity flip-flop” is predicted to occur during a given computational time step (i.e. the continuous liquid field changes from moving downwards to upwards). Such a situation inherently occurs in this region at the flow reversal point. In the event that a positive liquid film flow rate is calculated for a given time step when the CCFL model is activated, the time step is failed and the code backs up and repeats the time step with the same computational time step size, but forces the calculated liquid flow rate to be equal to zero. This prevents “velocity flip-flop” and allows for the appropriate flow direction to be evaluated at the next time step.

It is desired to evaluate Equation (5-22) using quantities based on the previous time step and solve for the flow rates implicitly at the new time step, such that:

n n+1 1 2 n n+1 n n+1 Yl Wl = − [C + Yv Wv − 2C Yv Wv ] (5-23) m 2

Incorporating Equation (5-23) into the matrix construct to solve for the field mass flow rates requires the last term in this expression to be linearized. Linearization involves approximating the output of a given function based on the known value and slope of the function when evaluated at a nearby point. Such an expression is described mathematically by:

f (xt+∆t ) ≈ f (xt )+ f ′(xt )(xt+∆t − xt ) (5-24)

This relationship is a first-order linearization that stems from a Taylor series expansion of a function about a known point neglecting higher-order terms. Linearizing a given function requires that the evaluation points are “close” to one another and the function in continuous over the interval of interest. Both criteria are satisfied in this case since the time step sizes are generally small so the mass flow rate does not change drastically between time steps. In this case the function that needs to be linearized is:

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f (x) = x (5-25) and it therefore follows that:

1 f ′()x = (5-26) 2 x

The resulting linearized relationship in terms of explicit and implicit vapor mass flow rates is given as:

n+1 n 1 n+1 n Wv ≈ Wv + (Wv −Wv ) (5-27) n 2 Wv

n+1 n It can be seen that for a steady-state solution (defined by Wv = Wv ) the linearized form yields the exact result, as desired. Substitution of this expression into Equation (5-23) yields:

   n n+1 1  2 n n+1 n  n 1 n+1 n  Yl Wl = − C + Yv Wv − 2C Yv Wv + ()Wv − Wv (5-28) m 2   n    2 Wv 

Expanding and then simplifying Equation (5-28) yields a final expression of:

 C  ()m2Y n W n+1 + 1− Y nW n+1 = −C C − H n (5-29) l l  n  v v ()v  Hv 

As a means of simplification several new parameters can be defined as:

n ACCFL = C(C − H v ), (5-30)

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2 n CCCFL = m Yl , (5-31) and:   n  C  DCCFL = Yv 1− (5-32)  n   H v 

In summary, the matrix form of the CCFL momentum equation set, which consists of Equations (5-17), (5-21), and (5-28), is given as:

 n~+1   CCCFL DCCFL 0  Wl   n~+1  ()()()C1 + C2 −1 D1+D 2 +D 3 −1 E 2 +E 3−1 Wv  (5-33)   n~+1   0 D3 E3 −1 We 

 − ACCFL    = − ()()A1 +A2 + A3 − B1+B 2 +B 3 ∆P    − A3 − B3∆P 

By design, the matrix structure of Equation (5-33) is the same as that produced by the standard set of momentum equations that was given previously as Equation (5-18). As a result the existing Gaussian elimination matrix solver included in COBRA-TF, which is specific to this structure, does not have to altered to apply the proposed CCFL model. This solver is contained within the solve_vel.f90 subroutine and provides estimates of both the tentative new time field flow rates and the derivatives of the field flow rates with respect to pressure gradient.

5.3.3. CCFL Entrainment Rate Models

The unique aspect of developing an explicit CCFL model for three-field analysis tools is the treatment of the entrained liquid field. For example, the governing equation for CCFL given by Equation (5-3) dictates the magnitude of the total liquid downflow, but does not provide any information on the distribution between the film and droplet fields. It was mentioned previously

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the proposed model assumes liquid can only penetrate downwards in the form of a liquid film and this assumption is justified based on experimental observations; however, similar visual observations also indicate the presence of entrainment at the CCFL location such that some of the liquid that does not penetrate downwards in the form of a falling film is swept upwards away from this location in the form of liquid droplets. Therefore, an entrainment rate model must be implemented to account for this mechanism. Such an entrainment mechanism should also improve the numerical stability of the model because it provides a means for removing liquid from the CCFL location instead of allowing it to pool up. If the liquid that was not allowed to penetrate downwards simply collected in this region it could cause a “slugging” behavior in the predictions that could lead to the code to predicting conditions where the CCFL model is continuously activated and deactivated and thus prevent a stable solution from being obtained.

Two different entrainment mechanisms are postulated for CCFL situations. If a large area expansion occurs at the CCFL location it can be anticipated that a pool entrainment type mechanism will exist. A pool entrainment mechanism consists of vapor flowing through a stagnant liquid pool, causing bubble bursting, splashing, and foaming at the pool surface which creates droplets that are carried away by the vapor as depicted in Figure 5-1 [98]. Some portion of these droplets fall back to the surface by gravity or are deposited further downstream of the pool location. The flow within the bubbling pool can be described as churn-turbulent where droplets are created by a momentum exchange mechanism between the faster moving vapor and the stagnant liquid [98]. An example of such a situation in a commercial nuclear reactor is the region where flow exits the core through the upper tie plate and into the upper plenum.

The pool entrainment model applied in the newly proposed CCFL model is from Kataoka & Ishii [98]. This semi-empirical model is based on a simple physical model and comparisons to available experimental data is given as:

−3 S E′ , pool = 84.4 x10 U vα v ∆ρAx, pool (5-34) where the area in this expression is the surface area of the pool.

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Figure 5-1: Schematic of Pool Entrainment Mechanism [98].

On the other hand, if no area change exists at the CCFL location, similar to situations within gas extraction wells where liquid is injected within the flow path, the liquid does not substantially accumulate at the liquid injection point such that it would not create an inverted pool. Therefore, the application of a pool entrainment model in this situation would be inappropriate. Rather, it can be anticipated that the entrainment rate for this situation is governed by an excess film flow model. Such a model existed in the modeling package applied in the baseline version of COBRA-TF and was given previously as Equation (2-93). This model suggests that liquid flowing in excess of the critical liquid volume fraction value predicted by Equation (2-10) becomes entrained; however, the implementation of this model has been altered in the proposed annular flow modeling package relative to the previous implementation. In particular, entrainment by this mechanism is deactivated in the proposed modeling package for computational cells that reside in the same channel and are located at elevations below the point where a CCFL boundary condition is applied when counter-current flow exists in the computational cell, but the liquid flow is less than the critical value that is predicted by the same form of the CCFL correlation that is applied in that channel. This change is in response to the visual observations of several researchers that the film is relatively smooth and a minimal amount of entrainment occurs in counter-current flow situations until the flooding point is reached [10,43].

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Applying the excess film model in the proposed modeling package to calculate the entrainment rate in the computational cell located above the point where the CCFL boundary condition is applied has several advantages. The large entrainment rate calculated by this model, caused by the small value of the critical liquid volume fraction predicted by Equation (2-10), results in a significant portion of the liquid becoming entrained such that it can be transported upwards as droplets rather than simply collecting at the CCFL location. This behavior is both physically consistent with experimental observations when liquid is injected within the flow path and it is numerically stabilizing within the code.

The entrainment rate for both mechanisms is calculated using explicit quantities within the INTFR subroutine in the continuity cell located above the location where the CCFL boundary condition is applied. Then, the calculated entrainment rate for the two individual mechanisms is scaled by the area ratio that exists between the momentum area associated with the CCFL location (Ax, j ) and area of the continuity cell located directly above the CCFL location

(A x,J = j+1 ). Doing this yields an expression that applies the calculation for the expected mechanism in the desired situations and the resulting expression is given as:

  A   A  S′ R S′  x,J = j+1  S′  x,J = j+1  (5-35) E,CCFL = CCFL E,exfilm   + E, pool 1−    Ax, j   Ax, j  where:

   W   W  R = 1− max  ,0.0 l  min  ,0.1 v  (5-36) CCFL   W   W    l, flood   v,FR 

The ramping function given by Equation (5-36) increases the calculated entrainment rate as the flow approaches the flow reversal point. The two parameters used to normalize this expression are calculated as:

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C 2 Wl, flood = − 2 (5-37) m Yl which is obtained by setting the vapor flow rate to zero in the CCFL correlation, and:

C 2 Wv,FR = (5-38) Yv which is obtained by setting the liquid flow rate to zero in the CCFL correlation.

An additional ramp is then applied to the calculated entrainment rate to consider the transition from counter-current, where the above models should be applied, to co-current flow, where the entrainment rate models described in Chapter 4 should be applied. Therefore, the finalized expression for the calculated entrainment rate in the continuity cell located above the momentum cell where a CCFL boundary condition is applied is given as:

S = S R + S 1− R (5-39) ( E, pred )J = j+1 E,CCFL D2U E,UN ( D2U ) where the ramping function in this expression is given as:

  W − 2W    v v,FR  (5-40) RD2U = max ,0.0 min  ,0.1    3Wv,FR 

This vapor flow rate weighted ramp provides a smooth transition in the calculated entrainment rate as the flow moves from a counter-current to co-current upward situation.

The stability of this method for considering the entrainment phenomena in CCFL situations will be discussed in the verification discussion of this work presented in Section 5.5. As described previously, a diffusion-based model is used to calculate the deposition rate in COBRA-TF. As a result the deposition rate is dependent upon the droplet concentration, or entrained mass flow rate.

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Therefore, an increase in the entrainment rate leads to a corresponding increase in the deposition rate. The net result could be a recirculation effect, where liquid that becomes entrained in the cell located above the CCFL location then being de-entrained in the cell located at the next axial level and creating a falling film that flows back towards the CCFL location. While this behavior may be physically consistent with the situation of interest, it could lead to numerical stability issues.

5.3.4. Exiting the CCFL Model

As mentioned previously, applying the newly proposed set of momentum equations for CCFL situations replaces the code calculated interfacial drag coefficient with a more appropriate value based on experimental data. Therefore, following the solution of the CCFL momentum equations an effective interfacial drag coefficient can be back-calculated using the momentum equation for the continuous liquid field, which can be rewritten from Equation (5-15) in terms of implicit and explicit quantities as:

n n+1 n n+1 n n n+1 (C1 −1)Wl + D1 Wv = −A1 − B1 ∆P (5-41) where in this expression:

n+1 n+1 n+1 ∆P = [PJ −1 − PJ ], (5-42)

  n dt K vl dz C1 = −  + K w,l dz  , (5-43) dz  5.0 Ax, j []()()α l ρl J + α l ρl J +1  and:

dt  K dz  D n =  vl  (5-44) 1 dz n n  5.0 Ax, j []()()α v ρv J + α v ρv J +1 

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Following the substitution of these expressions, Equation (5-41) can be solved for the interfacial drag coefficient as:

An B n  1  1 + 1 ∆P n+1 −  K + W n+1 dt dt  w,l dt  l K = (5-45) vl,CCFL W n+1 W n+1 l − v 5.0 A n n x, j−1 []()()α l ρl J −1 + α l ρl J 5.0 Ax, j−1 []()()α v ρv J −1 + α v ρv J

This calculation is performed within POST3D at the j+1 level relative to where the CCFL boundary condition is applied following the update of the flow rate and pressure quantities at the new time step. The explicit values in this equation, along with the code predicted interfacial drag value, are stored following their calculation in XSCHEM. If code calculated interfacial drag, which is calculated on an explicit basis, exceeds this back-calculated value, which is calculated on an implicit basis, then the time step is failed and the code backs-up and repeats the calculations using the standard set of momentum equations and the code predicted interfacial drag value. This criteria is used to determine when the CCFL model should be deactivated following a flow reversal scenario.

Also within the POST3D subroutine, calculations are performed on an implicit basis at the j+2 level relative to where the CCFL boundary condition is applied to determine if the calculated magnitude of liquid downflow exceeds the maximum liquid flow available to penetrate downwards. The magnitude of the maximum available liquid flow is calculated as:

  α v,J +1 Wl,max = Wl,inj ,J +1 − min (),0.0 Wl, j+1   (5-46)  5.0 ()α v,J +1 + α v,J +2  where the two components of this expression correspond to the injected liquid flow specified by a mass source boundary condition in the computation cell of interest and the liquid flow that is flowing downwards into the computation cell of interest from above. The second term in this expression accounts for the contribution generated by the recirculation effect that was described in the previous section. If the magnitude of the code predicted downward liquid flow rate

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exceeds this value then the time step is failed and the code backs up and repeats the calculations using the standard set of momentum equations. This criteria is needed to determine when the CCFL model should be deactivated for vapor flow rates less than those associated with the flooding point. If such a condition exists where the code predicts more liquid should flow downward than is available then the flow regime identifier is assigned a value of 8. It should also be noted that the flow rate predicted by Equation (5-46) is used by the excess film flow model that was described in the previous section to calculated the entrainment rate by this mechanism.

5.3.5. Assumptions

This section restates the assumptions that used in the development and implementation of the CCFL model. First, the implementation: a) allows for only CCFL boundary condition to be applied per channel and b) does not allow for a CCFL boundary condition to be applied in the last node of a section. Next, the proposed modeling package implicitly assumes that liquid only penetrates downwards in the continuous liquid field (i.e. no falling drops). Lastly, the model is only activated if the code predicts too much liquid downflow relative to amount allowed by the CCFL correlation that is specified; however, once the model is activated the model remains activated until either: a) the code predicted interfacial drag exceeds the effective value calculated by Equation (5-45) for the CCFL region or b) the magnitude of liquid downflow predicted by the model exceeds the available liquid flow that is calculated by Equation (5-45).

5.4. Required COBRA-TF Code Modifications

Unlike the implementation of the co-current annular flow modeling package shown in Appendix A, it was not possible to contain the implementation of the newly proposed CCFL model to a single, independent subroutine. To support the newly proposed CCFL model logic had to be incorporated into a variety of subroutines. A listing of the logic that had to be added to these different subroutines within COBRA-TF to support this model is provided in Appendix D.

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5.5. Verification Problem Description and Results

A simple verification problem was created to confirm the correct implementation of the newly proposed CCFL model in COBRA-TF and compare the predictive capability of code with and without this model. The problem designed for this purpose consisted of a 2-inch diameter tube and used steam-water at 250-psia as the working fluid with 4-inch tall computational nodes. A noding diagram indicating the applied boundary conditions is provided in Figure 5-2. The centers of the computational cells corresponding the continuity and energy mesh are denoted by ‘J’ and the centers of the computational cells corresponding to the momentum mesh (not shown) are denoted by ‘j’.

A pressure boundary condition (Type 2) was applied at the outlet and the newly defined CCFL boundary condition (Type 7) was applied to the momentum cell designated by j = 4. Two mass source boundary conditions were applied to supply vapor and liquid to the test section in continuity cells J = 3 and J = 5, respectively. A constant liquid injection rate of 0.35-lbm/sec was supplied to the test section in the form of a liquid film throughout the simulation. Meanwhile, the vapor injection rate was step changed monotonically every 10-seconds throughout the simulation. Two versions of the verification problem were executed: first by increasing and then by decreasing the vapor flow rate over the course of the simulation. This was done to ensure the model was applied under the appropriate conditions in either case and could be entered and exited in a stable manner when approach from either direction.

A large computational cell, with a cross-sectional area 500 times larger than that associated with a 2-diameter tube, was attached to the bottom of the test section with a no-flow boundary condition applied at the bottom of this cell. Such a computational cell provided a location for the downward penetrating liquid to collect without significantly affecting the upward flowing vapor; however, it should be noted that the inclusion of such a node does create a feedback mechanism within the problem. Since the problem is initialized to be vapor-solid the downward flow of liquid into this large computational cell forces vapor flow out of this node. As a result, even for the condition where the injected vapor flow rate is zero, a small, non-zero vapor flow rate still exists within the system. More importantly, if a slug of liquid flows downward this creates a

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corresponding slug of vapor flow that propagates upwards. This feedback mechanism can be a source of instability in the solution that can prevent a converged solution from being obtained.

Pressure BC (Type 2)

j = 6

J = 6

j = 5

Liquid Mass Source BC J = 5 (Type 4)

CCFL BC j = 4 (Type 7)

J = 4

j = 3

Vapor Mass Source BC J = 3 (Type 4)

j = 2

J = 2

j = 1

No Flow BC attached to the bottom of a large node where the downward draining liquid can collect

Figure 5-2: Noding Diagram of CCFL Verification Problem.

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This verification problem was run with three different channels. The first channel did not have a CCFL boundary condition applied to provide a baseline for comparison. These baseline results were obtained using the newly proposed annular modeling package outlined in Chapter 4. The other two channels had Wallis-type and Kutateladze-type CCFL boundary conditions applied with appropriate values of the correlating constants for each form of the CCFL correlation. The selected values were based on those suggested within previously conducted works. The Wallis- type CCFL boundary condition was applied with:

β = 0.0 C = .0 700 (5-47) m = 0.1

At 250-psia and using these specified values of the CCFL constants the values for relevant parameters within the model are:

wght = DH = 2 − inches sec Yv = .3 689 (5-48) lb m sec Yl = .0 371 lb m

Meanwhile, the Kutateladze-type CCFL boundary condition was applied with:

β = 1.0 C = 2.3 = .1 789 (5-49) m = 0.1

At 250-psia and using these specified values of the CCFL constants the values for relevant parameters within the model are:

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wght = L = .6 869 x10 −3 ft sec Yv = 18 17. (5-50) lb m sec Yl = .1 828 lb m

Based on these values and the specified liquid injection rate of 0.35-lb m/sec, Equation (5-10) indicates that the liquid downflow should first be impeded at vapor flow rates of 0.031-lb m/sec and 0.054-lb m/sec for the Wallis-type and Kutateladze-type forms of the CCFL correlations, respectively. For vapor flow rates less than these values the predicted downward liquid flow rate should be equal to the injected flow rate. The reason for the difference between the injected flow rate and the flow rates reported in Table 5-1 at time zero is that the reported flow rates are associated with the momentum cell whereas the injected flow rate is applied to the continuity cell. The two rates are different due to donor cell differencing. Based on the injected vapor flow rate as a function of time that was applied in the verification problem (see Table 5-1) this should result in the CCFL model being activated roughly 30-seconds into the simulation for the channel where the Wallis-type CCFL correlation is applied and roughly 50-seconds into the simulation for the channel where the Kutateladze-type CCFL correlation is applied. It can be seen in Table 5-1 (as well as in Figure 5-6 that will be presented later) that this condition occurred at the desired times in the simulations and a times earlier than this in the simulation the downward liquid flow was not impeded, but rather constant and equal to the injected flow rate.

Meanwhile, based on these values given above and using Equation (5-38), flow reversal or the point of zero liquid penetration should occur at vapor flow rates of 0.133- lb m/sec and 0.176- lb m/sec for the Wallis-type and Kutateladze-type forms of the CCFL correlations, respectively. Based on the injected vapor flow rate as a function of time that was applied in the verification problem (again see Table 5-1) this should result in the downward penetration of liquid ceasing roughly 110-seconds into the simulation for the channel where the Wallis-type CCFL correlation is applied and roughly 150-seconds into the simulation for the channel where the Kutateladze- type CCFL correlation is applied. Again it can be seen in Table 5-1 (as well as in Figure 5-6 that will be presented later) that this condition occurred at the desired times in the simulations.

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Table 5-1: Summary of Results for CCFL Verification Problem. Wallis-Type CCFL BC Kutateladze-Type BC with C=0.700, m=1.0 with C=1.789, m=1.0 Injected Maximum Maximum Simulation Vapor COBRA-TF COBRA-TF COBRA-TF COBRA-TF Allowable Allowable Time (sec) Flow Predicted Predicted Predicted Predicted Downward Downward Rate Vapor Liquid Vapor Liquid Liquid Liquid Momentum Momentum Momentum Momentum Momentum Momentum Flow Rate Flow Rate Flow Rate Flow Rate Flow Rate Flow Rate 0-10 0.000 0.003 -0.285 -0.953 0.003 -0.285 -1.324 10-20 0.013 0.015 -0.287 -0.582 0.015 -0.287 -0.878 20-30 0.025 0.027 -0.288 -0.398 0.027 -0.288 -0.648 30-40 0.035 0.035 -0.308 -0.313 0.037 -0.290 -0.514 40-50 0.050 0.048 -0.206 -0.210 0.052 -0.292 -0.365 50-60 0.060 0.056 -0.163 -0.162 0.062 -0.290 -0.290 60-70 0.080 0.068 -0.108 -0.107 0.079 -0.189 -0.191 70-80 0.100 0.082 -0.061 -0.061 0.094 -0.128 -0.127 80-90 0.120 0.094 -0.032 -0.033 0.104 -0.094 -0.094 90-100 0.130 0.105 -0.016 -0.016 0.109 -0.080 -0.080 100-110 0.140 0.115 -0.006 -0.006 0.114 -0.066 -0.067 110-120 0.160 0.137 0.000 0.000 0.137 -0.025 -0.024 120-130 0.170 0.148 0.000 0.000 0.147 -0.013 -0.013 130-140 0.180 0.158 0.000 0.000 0.158 -0.005 -0.005 140-150 0.190 0.170 0.000 0.000 0.170 -0.001 -0.001 150-160 0.200 0.181 0.000 0.000 0.181 0.000 0.000 160-170 0.225 0.206 0.000 0.000 0.206 0.000 0.000 170-180 0.250 0.234 0.000 0.000 0.235 0.000 0.000 180-190 0.275 0.268 0.000 0.000 0.269 0.000 0.000 190-200 0.300 0.293 0.000 0.000 0.294 0.000 0.000 200-210 0.325 0.316 0.000 0.000 0.319 0.000 0.000 210-220 0.350 0.340 0.000 0.000 0.345 0.000 0.000 220-230 0.375 0.364 0.000 0.000 0.369 0.000 0.000 230-240 0.400 0.388 0.000 0.000 0.394 0.000 0.000 240-250 0.500 0.485 0.000 0.000 0.490 0.000 0.000 250-260 0.600 0.582 0.000 0.000 0.588 0.000 0.000 260-270 0.750 0.731 0.000 0.000 0.734 0.000 0.000 270-280 1.000 0.981 0.000 0.000 0.981 0.000 0.000 280-290 1.500 1.482 0.000 0.000 1.482 0.000 0.000 290-300 2.000 1.982 0.000 0.000 1.982 0.000 0.000 Notes: All flow rates listed in this table are given in (lb m/s) and the maximum flow rates are calculated based on the system pressure (250-psia) and not the local pressure at the CCFL location.

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The COBRA-TF predicted results of this verification problem when the Wallis-type and Kutateladze-type CCFL boundary conditions are applied are compared to the baseline results in Figures 5-3 and 5-4, respectively. The results shown in these figures were obtained by averaging the COBRA-TF predicted vapor and liquid flow rates over the last two-seconds of the simulation time for a given injected vapor flow rate condition. It can be seen that for vapor flow rates below the flooding condition the downward liquid flow is independent of the vapor flow. Then, as the vapor flow rate is increased further the flooding condition is reached and the allowable liquid downflow is governed by the specified CCFL correlation. Finally, at larger vapor flow rates the flow reversal condition is exceeded and the downward penetration of liquid is prevented.

Results obtained by increasing the vapor flow rate throughout the simulation

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Results obtained by increasing the vapor flow rate throughout the simulation

Meanwhile, Figures 5-5, 5-6, 5-7, and 5-8 show the COBRA-TF predicted flow regime, liquid flow rate, entrainment rate, and continuous liquid volume fraction as a function of simulation time for the three different channels. The former two parameters are plotted for the computational node where the CCFL boundary condition is applied (j = 4) while the latter two parameters are plotted for the computational node positioned above this location (J = 5). It should be noted that the results shown in these figures contain the COBRA-TF values predicted at every single computational time step throughout the simulation. This was done to capture any numerical instabilities that may have existed.

In Figure 5-5, a value of ‘4’ for the flow regime indentifier corresponds to the unstable annular regime, where the intermittency is predicted to be greater than zero, a value of ‘5’ corresponds to the stable annular regime, where the intermittency is zero, and a value of ‘7’ corresponds to situations where the newly proposed CCFL model is activated. This figure shows that the CCFL model is first activated at the desired times within the simulation for the two channels where a

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CCFL boundary condition was applied. Meanwhile, both channels do not exit the CCFL model until roughly 180-seconds even though flow reversal was predicted to have occurred at 110- seconds and 150-seconds in the channels where the Wallis-type and Kutateladze-type CCFL boundary conditions were applied, respectively. As described in Section 5.3.4., once the CCFL model is activated it remains activated until the code calculated drag exceeds the effective value for the CCFL model that is back-calculated using Equation (5-45). As a result, the explicit CCFL model must still be used for vapor velocities larger than those associated with flow reversal since the code predicted interfacial drag is not sufficient to prevent liquid downflow. In this particular problem the code calculated drag interfacial drag does not exceed the effective value associated with the CCFL model until 180-seconds into the simulation.

On the other hand, Figure 5-6 shows that the flow reversal, or the point of zero liquid penetration, also occurs at the desired times in the simulation for the two channels where the CCFL boundary condition was applied. The points where liquid downflow is first impeded can also be seen on this figure. Figure 5-6 also shows that, in general, the baseline version of COBRA-TF, which used the newly proposed annular flow modeling package, provided a reasonable prediction of the allowable downward liquid flow rate, but for this specific problem yielded too much liquid downflow for conditions near both the flooding and flow reversal points relative to the results obtained when the CCFL model was applied. It should also be noted that based on the results shown in this figure it can be suggested that the solution is much more stable when the CCFL model is imposed relative to the results obtained using the baseline version of the code.

Next, Figure 5-7 shows the entrainment rate increases through the first part of the simulation as the vapor flow rate is increased and the flow moves through the CCFL region. This increase is governed primarily by Equation (5-36) in the proposed CCFL model. As desired, the entrainment rate is zero in the two channels where the CCFL boundary condition is applied prior to the CCFL model being activated. The peak entrainment rate predicted by the excess film flow model is slightly less than the injected liquid flow rate of 0.35-lbm/sec. Then, in the second part of the simulation, once the vapor flow rate is greater than two times the value associated with the flow reversal point, the predicted entrainment rate decreases as the flow transitions to toward co- current upward and the entrainment models associated with this regime. In general, this decrease occurs because the entrainment rate associated with the co-current annular regime tends to be

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much less than that associated with the CCFL region as the amplitude of interfacial waves subsides and the entrainment mechanism changes from wave undercutting to roll wave stripping. The weighting of the entrainment rates calculated for the two mechanisms is governed by Equation (5-40) in the proposed model.

Figure 5-8 shows that, as expected, the COBRA-TF predicted continuous liquid volume fraction first increases and then decreases with increasing vapor flow rate while the CCFL model is invoked. The initial increase occurs due to the small entrainment rate and prevention of liquid downflow that occurs near the flooding point, but then as the entrainment rate increases as the flow reversal point is approached the continuous liquid volume fraction decreases because a larger portion of the liquid that can no longer penetrate downwards is swept away rather than collecting in this region.

Figure 5-5: COBRA-TF Predicted Flow Regime as a function of Simulation Time for the CCFL Verification Problem when the Vapor Flow Rate is Increased over the course of the Simulation.

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Figure 5-7: COBRA-TF Predicted Entrainment Rate in Node 5 as a function of Simulation Time for the CCFL Verification Problem when the Vapor Flow Rate is Increased over the course of the Simulation.

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Meanwhile, Figures 5-9, 5-10, and 5-11 show various flow rates predicted by COBRA-TF at locations above and below the point where the CCFL boundary condition was applied (j =4) for each of the cases simulated in the verification problem. Figure 5-9, which shows the COBRA-TF predicted results for the baseline case where no CCFL boundary condition was applied, indicates that for low vapor flow rates that exist early on in the simulation the downward flowing liquid is predicted to be distributed between the continuous liquid and dispersed droplet fields. This result contradicts available experimental data [10,43]. Additionally, a large amount of instability exists in the solution and prevents a converged solution from being obtained for some conditions, particularly near the flooding and flow reversal conditions. Meanwhile, Figures 5-10 and 5-11, which show the COBRA-TF predicted results when Wallis-type and Kutateladze-type CCFL boundary conditions are applied, indicate that as expected the downward flowing liquid is contained entirely within the continuous liquid field. These results are also much more stable than those predicted without a CCFL boundary condition being applied.

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It can also be seen in Figures 5-9, 5-10, and 5-11 that, as anticipated, a recirculation effect is predicted by the code where once entrainment is predicted to exist and travel upwards. It can be seen that a downward liquid flow rate is predicted for the cell located above the CCFL boundary condition immediately following the activation of the CCFL; however, this effect does not appear to impact the numerical stability of the code, most likely because this contribution is considered in the flow rate supplied to the excess film flow entrainment model as shown in Equation (4-46). As the vapor flow rate is increased further the continuous liquid flow rate in this computational cell eventually reverses direction and propagates upwards.

All of the results that have been shown to this point for the verification problem correspond to those obtained when the vapor flow rate was increased throughout the simulation; however, as previously mentioned it was desired to also run the same problem, but decrease the vapor flow rate throughout the simulation, to ensure the model could be entered and exited in a stable manner

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from both directions. The COBRA-TF predicted results when the verification problem was run in this manner are shown in Figures 5-12 and 5-13 for the Wallis-type and Kutateladze-type CCFL boundary conditions, respectively. Again, the results shown in these figures were obtained by averaging the COBRA-TF predicted vapor and liquid flow rates over the last two-seconds of the simulation time for a given injected vapor flow rate condition. The results shown in these figures exhibit the desired behavior for the newly proposed CCFL model.

Results obtained by decreasing the vapor flow rate throughout the simulation

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Results obtained by decreasing the vapor flow rate throughout the simulation

Meanwhile, Figures 5-14 and 5-15 show the COBRA-TF predicted flow regime and liquid flow rate in the computational node where the CCFL boundary condition is applied (j = 4). Again, the results shown in these figures contain the COBRA-TF values predicted at every single computational time step throughout the simulation. It can be seen in Figure 5-14 that the CCFL model is invoked roughly 120-seconds into the simulation when the code predicts liquid can penetrate downward, as shown in Figure 5-15, but the CCFL equation indicates no liquid should penetrate downwards. As expected, but not shown here explicitly, the code interfacial drag is less than the effective value predicted by the CCFL model at this point. It should also be noted that the time associated with this condition is consistent with the results obtained for the verification problem when it was run with increasing vapor flow throughout the simulation where the CCFL model was deactivated at 180-seconds (300-seconds – 180-seconds = 120-seconds).

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Similarly, the CCFL model is deactivated when the vapor flow rate is decreased throughout the simulation at roughly 260-seconds and 270-seconds when the Wallis-type and Kutateladze-type forms of the CCFL correlations are applied, respectively. The model is deactivated when the downflow predicted by the CCFL correlation exceeds the available liquid flow, which is designated by a value of ‘8’ being predicted for the flow regime identifier. These results are also consistent with the simulation times when the model was first activated in the simulations run with increasing vapor flow rates.

Figure 5-14: COBRA-TF Predicted Flow Regime as a function of Simulation Time for the CCFL Verification Problem when the Vapor Flow Rate is Decreased over the course of the Simulation.

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Overall, the results shown in this section indicate the model is: a) activated under the appropriate conditions, b) provides an accurate prediction based on the CCFL correlation that is specified, and c) is entered and exited in a stable manner when the CCFL region is approached from either direction (i.e. increasing or decreasing vapor flow rate for a constant liquid flow rate). These results provide confidence that the proposed model has been implemented correctly within COBRA-TF and is behaving in the desired manner. The next section provides additional results obtained using this newly proposed model when the experiments conducted by Dukler & Smith [43] were simulated with COBRA-TF.

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5.6. Simulation of Dukler & Smith [43] Experiments

As previously described, low-pressure air-water experiments were conducted by Dukler & Smith [43] to measure pertinent flow rates, pressure gradients, and liquid film thicknesses (conductance probes) within the flooding region. Liquid injection rates of 100, 250, 500, and 1000-lb m/hr (corresponding to liquid Reynolds numbers of 310, 776, 1552, and 3105) were tested, but only the experimental and COBRA-TF predictions for the 500-lb m/hr case is presented in the current study. Using the experimental data collected by Dukler & Smith [43] for measured liquid downflow as a function of vapor upflow one can calculate the corresponding dimensionless gas velocities and allow the appropriate values for the CCFL correlating constants (‘ C’ and ‘ m’) to be determined. The experimental data and resulting curve fit using the Wallis [4] form of the CCFL correlation are presented in Figure 5-16. The resulting experiment-specific governing equation for the CCFL region is found to be:

* * jg + .0 9375 jl = .0 8841 (5-51)

Figure 5-16: Correlation of CCFL Data Collected by Dukler & Smith [43].

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This information allows for a CCFL boundary condition to be defined and the newly proposed CCFL model to be applied. The following figures compare the experimental results published by

Dukler & Smith [43] for a liquid injection rate of 500-lb m/hr to the COBRA-TF predictions that were obtained both with and without a CCFL boundary condition applied when this same set of experiments was simulated by the code. A model with the same types of boundary conditions that were applied in the verification problem (see Figure 5-2) was created to simulate these experiments, except the overall test section length was much longer (~13-feet). The porous sinter used for liquid injection in these experiments was located roughly at the center of this axial length, with the flow rates being measured at the inlet and outlet of the test section, the pressure gradient being measured over a distance of 68-inches starting just above the liquid injection location, and the film thickness being measured 52-inches below the liquid injection location. As a result of these extended distances, the experimental data and COBRA-TF predictions reflect the integral effect of both the counter-current and co-current annular flow regimes. Additionally, these experiments were conducted using air-water, while the proposed annular flow modeling package was optimized for steam-water situations. This are important factors to consider when making code-to-data comparisons of these experiments.

Figure 5-17 provides a comparison of the experimental and COBRA-TF predicted liquid downflow rates. It can be seen that significant improvement with the experimental data was obtained following the implementation of an appropriately defined CCFL boundary condition. The remaining difference between the experimental data and COBRA-TF predictions is a result of deviations in the curve-fit, defined by Equation (5-51), from the experimental data. Meanwhile, Figures 5-18 and 5-19 compare the experimental and COBRA-TF predicted entrained and film upflow rates at the test section outlet. It can be seen in these figures that in both cases COBRA-TF overpredicts the entrained flow rate relative to the experimental data, but applying the CCFL model significantly reduced the magnitude of overprediction relative to the baseline results. Correspondingly, COBRA-TF drastically underpredicts the film upflow rate at the outlet relative to the experimental data. Again, since these measurements and predictions are made well above the liquid injection location they reflect an integral effect of CCFL and the annular flow.

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Lastly, Figures 5-20 and 5-21 show the experimentally measured and COBRA-TF predicted axial pressure gradient above the liquid injection location and mean film thickness below the liquid injection location, respectively. It can be seen in Figure 5-20 that applying a CCFL boundary condition caused the COBRA-TF prediction of the pressure gradient increase to become more aligned with the experimental data; however, the predicted rate of increase with increasing air flow rate was not as large as was observed in the experiment. Similarly, it can be seen in Figure 5-21 that applying a CCFL boundary condition caused the COBRA-TF prediction of mean film thickness below the liquid injection location to be in much better agreement with the experimental data; however, neither prediction by COBRA-TF was able to capture the increase in this quantity at the flooding point that was observed in the experimental data.

Overall, the results shown in this section indicate that the inclusion of the newly proposed three- field CCFL model in COBRA-TF has improved the predictive capability of the code when simulating these experiments with an appropriately defined CCFL boundary condition; however, room for improvement does still exist.

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5.7. Conclusions on CCFL Model

Throughout the current study a focus was placed on utilizing regime specific models to provide a more accurate representation of the situation of interest. Given that COBRA-TF is well suited for calculating counter-current flow situations since it models the continuous and discrete liquid fields separately, it was desired to include an explicit CCFL model to improve the predictive capabilities of the code in this region relative to available experimental data. Prior to the current study no CCFL model was available for use in three-field analysis environments and an accurate prediction of this phenomenon was not achieved.

Ideally the flooding curve would be inherently predicted by the interfacial drag models within the code; however, given the suggestion that CCFL is not approached as the limit of a continuous process, but rather is a result of a marked instability [11], it may be reasonable to assume that the same phenomenological models that are applicable in the co-current annular regime may be unable to capture this effect. This idea is further supported by the inability of the code to accurately predict this phenomena even after the proposed annular flow modeling package was implemented. As a result a separate criterion must be applied.

The newly proposed three-field CCFL model that was developed in the current study compares the flow conditions predicted by the code in the computational cell where the model is applied to the results of a user-specified form of the CCFL correlation. This provides flexibility to the user since the unresolved flow path diameter and geometric dependencies of this phenomenon have precluded a universal CCFL model from being determined. In the proposed model, if the code is allowing too much liquid downflow relative to this correlation then the normal set of momentum equations is replaced with a set of CCFL momentum equations that was developed and presented in this chapter. The proposed model also provides appropriate entrainment rate models and necessary criterion to enter and exit the model in a stable manner. The implementation of the model was verified and was shown to provide increased stability in the code predictions relative to the results that were obtained without the newly proposed model. Following the implementation of the proposed model substantial improvement in the code-to-data agreement of the Dukler & Smith [43] experiments was obtained.

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Given the current state of CCFL knowledge and available predictive models the approach outlined in this chapter currently represents the most viable method for predicting this phenomenon. This methodology is not intended to be a final resolution to predicting the CCFL phenomenon in transient safety analysis codes, but due to the importance of this phenomenon to accurately predicting other parameters of interest and the abundance of location-specific experimental data that exists, it would be inopportune not to employ this modeling methodology and leverage the available experimental data to improve the predictive capabilities of such codes until a more mechanistic approach is ascertained.

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6. Conclusions

The current study has proposed a functional modeling package for predicting annular flow situations over a wide range of conditions within a three-field analysis environment. Chapter 2 presented a detailed literature review that provided insight on the physical mechanisms governing various phenomena in annular flow and highlighted currently available models for the constitutive relationships that are required to predict the annular regime using a three-field analysis tool. This process revealed an extensive amount of research that has been conducted within the co-current and counter-current annular flow regimes, but the complex nature of the interfacial structure and entrainment phenomena has made it more convenient, and computationally efficient, to rely on simple correlations that are a functions of a few dimensionless parameters, rather than physical mechanisms, to quantify these phenomena; however, it was shown in Sections 4.1. and 5.6., that when such models are applied they do not accurately reflect the experimental data over a wide range of conditions.

Based on these results this study aimed to leverage the existing physical knowledge and experimental data to provide a functional modeling package that is self-consistent and provides the desired level of accuracy for three-field analysis tool such as COBRA-TF. Wherever possible the proposed modeling package incorporates models that are based on the physical structures and phenomena governing the flow as it is currently understood. The result is a new set of modeling packages that cover the annular flow regime ranging from counter-current to co-current situations. The current study used COBRA-TF as the vehicle for assessing the newly proposed modeling packages, but these packages can be easily implemented in any other three-field analysis tool; however, it should be noted that the co-current annular package must be implemented as a whole, rather than in a piece-wise manner, because the methodology applied in the current study causes uncertainties in each individual model to be implicitly captured in the functional relationship between the actual and theoretical entrainment rates.

In general, it was shown in the current study that the inclusion of the proposed modeling packages for both co-current and counter-current annular flow has provided increased accuracy in the predictive capabilities of COBRA-TF for situations of interest to reactor safety analyses. In particular, the implementation of these packages into COBRA-TF reduced the mean relative error

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from 20.2% (underprediction) to 4.5% (overprediction) in entrained fraction and from 108.2% to 7.6% (both overprediction) in axial pressure gradient for co-current upward annular flow situations and significantly improved the code-to-data agreement of several different parameters within the counter-current flow regime. It was also shown that the proposed co-current annular modeling package provided reasonable estimates of a variety of parameters that are important in annular flow and was able to capture the general behavior within the developing flow region. Both these results provide confidence that the proposed modeling package reasonably reflects the underlying physics of the annular regime. Moreover, the current study is one of the few works that has examined the predictive capabilities of transient analysis codes within the developing, or non-equilibrium, annular flow region.

The methodology employed in the current study provides a means of implementing models that are based on the physical mechanisms, as they are currently understood, while simultaneously leveraging the available experimental data to improve the predictive capability of the code. Ultimately it is desired to develop a modeling package that is void of empirical correlations and does not require functional relationships to predict the experimental data, but in the meantime a need does exist within the nuclear industry to provide analysis tools that are able to provide accurate predictions of annular flow phenomena. Therefore, this methodology is not meant to provide a final solution to this complex problem, but rather provide a means for improving the predictive capability until more detailed data are available to support further model development, or provides the ability to develop a consistent set of models from a single experimental study. The newly proposed modeling packages developed in the current study provides both a bridge towards this ultimate goal and a new baseline for future research to be conducted to continue enhancing the predictive capabilities of three-field analysis tools within the annular regime.

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7. Recommendations for Future Work

The current study has highlighted several areas of future work that need to be addressed to support the continued development of annular flow models and improve the predictive capabilities of three-field transient safety analysis codes. Such work should include both the collection of detailed experimental data under representative conditions for nuclear reactor applications as well as theoretical investigations that attempt to model annular flow phenomena using a first-principles approach.

First, no experimental pressure gradient data are readily available for the annular flow at low pressures. Such data are needed to: 1) assess the ability of the proposed modeling package to predict this quantity under these conditions, and 2) determine if the entrainment mechanism transition proposed by Azzopardi [49] also corresponds to the transition between the churn- annular and co-current upward annular regimes.

Second, steam-water data are also needed over the entire range of pressures within the churn- annular regime as well as for entrained drop size and mean film thickness. Such data is readily available for air-water and nitrogen-water, but the difference in interfacial drag between these and steam-water make comparisons difficult. As a result, no comparisons of the film thickness or drop size predicted by the proposed modeling package were made in the current study. The simultaneous measurement of entrained fraction, pressure gradient, film thickness, and drop size for steam-water annular flow over a range of pressures would provide the data necessary to assess the predictive capability of a transient safety analysis code using a consistent set of experimental data.

Third, the measurement of other important annular flow parameters, such as wave amplitude, frequency, velocity, intermittency, etc. , in future experimental studies are desired to further ensure the underlying physics of the annular regime is being accurately resolved. For example, the disturbance wave frequency correlation employed in the proposed modeling package was developed using low pressure air-water data and, as shown in Appendix C, the extension of this correlation to high pressure steam-water environments may be inappropriate. The prediction of this quantity strongly influences the resulting entrainment rate predicted by the proposed annular

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flow modeling package because it governs the number of disturbance waves that reside in a given computational cell. The development of a more appropriate correlation for this quantity would certainly improve the proposed modeling package and likely reduce the influence that the functional relationship between the actual and theoretical entrainment rates has on the predication of this quantity. Similarly, several assumptions were applied during the development of the roll wave stripping and Kelvin-Helmholtz lifting entrainment rate models that were proposed in the current study. The assessment of theses assumptions and the quantification of the assumed quantity in future experiments would also likely reduce the influence of the functional relationship.

Lastly, further investigation into a more appropriate deposition model and the explicit consideration of both a) the wall shear enhancement in annular flow and b) the losses associated with the recirculation zone that exists on the backside of interfacial waves may be warranted in future studies.

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8. References

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66. Lee, S., and No, H.: “Assessment of an Entrainment Model in Annular-Mist Flow for a Three-Field TRAC-M,” Nuclear Engineering and Design , Vol. 237, pp. 441-450 (2007). 67. Cousins, L., and Hewitt, G.: “Liquid Phase Mass Transfer in Annular Two-Phase Flow: Droplet Deposition and Liquid Entrainment,” AERE-R-5657 , Harwell, England (1968). 68. Ambrosini, W., Andreussi, P., and Azzopardi, B.: “A Physically Based Correlation for Drop Size in Annular Flow,” International Journal of Heat and Mass Transfer , Vol. 17, No. 4, pp. 497-507 (1991). 69. Azzopardi, B., Freeman, G., and King, D.: “Drop Sizes and Deposition in Annular Two- Phase Flow,” AERE-R-9634 , Harwell, England (February 1980). 70. Cousins, L., Denton, W., and Hewitt, G.: “Liquid Mass Transfer in Annular Two-Phase Flow,” Symposium on Two-Phase Flow , Exeter, England, Paper C4 (1965). 71. Kataoka, I., and Ishii, M.: “Mechanism and Correlation of Droplet Entrainment and Deposition in Annular Two-Phase Flow,” NUREG/CR-2885, ANL-82-44 , Argonne National Laboratory, Argonne, IL (July 1982). 72. Ishii, M., and Mishima, K.: “Droplet Entrainment Correlation in Annular Two-Phase Flow,” International Journal of Heat and Mass Transfer , Vol. 23, No. 10, pp. 1835-1846 (1989). 73. Okawa, T., Kitahara, T., Yoshida, K., Matsumoto, T., and Kataoka, I., “New Entrainment Rate Correlation in Annular Two-Phase Flow Applicable to Wide Range of Flow Conditions,” International Journal of Heat and Mass Transfer , Vol. 45, pp. 87-98 (2002). 74. Fore, L.: “Droplet Entrainment in Vertical Gas-Liquid Annular Flow,” Ph.D. Dissertation, University of Houston (1993). 75. Nigmatulin, R., et al. : “Entrainment and Deposition Rates in a Dispersed-Film Flow,” International Journal of Multiphase Flow , Vol. 22, No. 1, pp. 19-30 (1996). 76. Kocamustafaogullari, G., Smits, S., and Razi, J.; “Maximum and Mean Droplet Sizes in Annular Two-Phase Flow,” International Journal of Heat and Mass Transfer , Vol. 37, No. 6, pp. 955-965, (1994). 77. Stevanovic, V., and Studovic, M.: “A Simple Model for Vertical Annular and Horizontal Stratified Two-Phase Flows with Liquid Entrainment and Phase Transitions: One- Dimensional Steady State Conditions,” Nuclear Engineering and Design , Vol. 154, pp. 357- 379 (1995).

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78. Whalley, P., Hutchinson, P., and Hewitt, G.: “The Calculation of Critical Heat Flux in Forced Convection Boiling,” 5th International Heat Transfer Conference , Tokyo, Japan (September 1974). 79. Würtz, J.: “An Experimental and Theoretical Investigation of Annular Steam-Water Flow in Tubes and Annuli at 30 to 90 bar,” RISO REPORT NO. 372 , RISO National Laboratory, Roskilde, Denmark (April 1978). 80. Whalley, P., Hewitt, G., and Hutchinson, P.: “Experimental Wave and Entrainment Measurements in Vertical Annular Two-Phase Flow,” AERE-R-7521 , Harwell, England (1973). 81. Keeys, R., Ralph, J., and Roberts, D.: “Liquid Entrainment in Adiabatic Steam-Water Flow at 500 and 1000 psia,” AERE-R-6293 , Harwell, England (1970). 82. Singh, K.: “Liquid Film Flow-Rate Measurement at Elevated Pressures,” M. S. Thesis, University of Windsor, Windsor, Ontario, Canada (1967). 83. Hewitt, G., and Pulling, D.: “Liquid Entrainment in Adiabatic Steam-Water Flow,” AERE-R- 5374 , Harwell, England (1969). 84. Yanai, M.: “Study on boiling heat transfer in a channel”, Ph.D. Thesis, Kyoto University (1971). 85. Pilch, M., and Erdman, C.: “Use of Breakup Time Data and Velocity History Data to Predict the Maximum Size of Stable Fragments for Acceleration-Induced Breakup of a Liquid Drop,” International Journal of Multiphase Flow , Vol. 13, pp. 741-757 (1987). 86. Teixeira, J., Azzopardi, B., and Bott, T.; “The Effect of Inserts on Drop Sizes in Vertical Annular Flow,” 2nd UK National Conference on Heat Transfer , University of Strathclyde (1988). 87. Andreussi, P., Romano, G. and Zanelli, S.: “Drop Size Distribution in Annular Mist Flows,” 1st International Conference on Liquid Atomization and Spray System, Tokyo, Japan (August 1978). 88. Pogson, J., Roberts, J., and Waibler, P.: “An Investigation of the Liquid Disintegration in Annular Mist Flows,” Transactions of ASME, Journal of Heat Transfer , Vol. 92, pp. 651-658 (1970). 89. Jayanti, S., Tokarz, A., and Hewitt, G.: “Theoretical Investigation of the Diameter Effect on Flooding in Countercurrent Flow,” International Journal of Multiphase Flow , Vol. 22, pp. 307-324 (1996).

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90. Pushkina, O., and Sorokin, Y.: “Breakdown of Liquid Film Motion in Vertical Tubes,” Heat Transfer – Soviet Research , Vol. 1, No. 5, pp. 56-64, (1969). 91. Wallis, G., and Makkenchery, S.: “The Hanging Film Phenomenon in Vertical Annular Two- Phase Flow,” Journal of Fluids Engineering, Trans. of ASME , pp. 297-298 (September 1974). 92. Okawa, T., Kotani, A., and Kataoka, I.: “Experiments for Liquid Phase Mass Transfer Rate in Annular Regime for a Small Vertical Tube,” International Journal of Heat and Mass Transfer, Vol. 48, pp. 585-598 (2000). 93. White, F.: Fluid Mechanics – 2 nd Edition , McGraw Hill Inc, New York, (1986). 94. Hurlburt, E., and Newell, T.: “Optical Measurement of Liquid Film Thickness and Wave Velocity in Liquid Film Flows,” Experiments in Fluids , Vol. 21, pp. 357-362 (1996). 95. Larson, R., and Hostetler, R.: Calculus with Analytic Geometry – Third Edition, D.C. Heath and Company, Lexington, Massachusetts (1986). 96. Cohen, L., and Hanratty, T.: “Generation of Waves in the Concurrent Flow of Air and a Liquid,” AIChE Journal, Vol. 11, pp. 138-144 (1965). 97. Seban, R. A., et al. : “UC-B Reflood Program: Experimental Data Report,” EPRI NP-743, University of California at Berkeley, (April 1978). 98. Kataoka, I, and Ishii, M.: “Mechanistic Modeling and Correlations for Pool-Entrainment Phenomenon,” NUREG/CR-3304 , ANL-83-37 , Argonne National Laboratory, Argonne, IL (April 1983).

329

Appendix A: Subroutine Listing of Proposed Annular Flow Model

MODULE ENTRRATE

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! This module contains subroutines to calculate the droplet entrainment rate ! from a film in co-current annular flow. The model is was developed in the ! dissertation of Lane (Pennsylvania State University, 2009) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE CONTAINS

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

FUNCTION CircArea(dhydj,off) REAL :: CircArea,dhydj,off CircArea = 0.25*(4.*atan(1.))*((dhydj-2.0*off)**2) END FUNCTION CircArea

FUNCTION NonCircArea(H,W,off) REAL :: NonCircArea,H,W,off NonCircArea = (H-2.0*off)*(W-2.0*off) END FUNCTION NonCircArea

FUNCTION CircWP(dhydj,off) REAL :: CircWP,dhydj,off CircWP = (4.*atan(1.))*(dhydj-2.0*off) END FUNCTION CircWP

FUNCTION NonCircWP(H,W,off) REAL :: NonCircWP,H,W,off NonCircWP = 2.0*((H-2.0*off)+(W-2.0*off)) END FUNCTION NonCircWP

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Annular Flow Modeling Package Developed by Lane, 2009 ! (returns sde,xkic,fi,se,dsaut,isijl,liqbrg) SUBROUTINE ANENTR(i,j,delt,press,rll,rvl,visl,visv,sigma,dx,dhydj, & gpth,acontl,chantype,hbar,Uv,av,Ul,aliq,Ue,ae,xkw, & sde,xkic,fi,se,dsaut,isijl,liqbrg) ! Film Interfacial Drag based on Two-Zone Model of Hurlburt et al. (2006) ! Droplet Drag Coefficient calculated using the model of Clift & Gauvin (1970) ! Deposition Rate calculated using the model of Cousins et al. with Whalley (1974) ! Entrainment Rate and corresponding Entraining Drop Size calculated ! allowing three different mechanisms:

! 1) Roll Wave Stripping - shear driven process, base model developed ! by Fore (1993) with correction factor developed by Lane (2009), ! wave dimensions consistent with those predicted by Two-Zone Model, ! drop Size calculated using Kocamustafaogullari et al. (1994) ! 2) Bag Breakup - "plucking" of small wavelets residing on tops of ! disturbance waves, uses Holowach (2002) Kelvin-Helmholtz stability ! analysis to calculate wavelength with Ishii & Grolmes expression ! for wave height, drop size calculated using Tatterson (1980) ! 3) Liquid Bridge Breakup - uses the model based on the work of Pilch & ! Erdman (1985)

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

use Model_Options, only : entr_print,use_liqbreak,jwldev,corrrip

IMPLICIT NONE

! Passed INTEGER :: i,j,chantype ! Input REAL :: delt,press,rll,rvl,visl,visv,sigma,dx,dhydj,gpth ! Input REAL :: acontl,hbar,Uv,av,Ul,aliq,Ue,ae,xkw ! Input

330

REAL :: sde,xkic,fi,se,dsaut ! Output INTEGER :: isijl ! Output LOGICAL :: liqbrg ! Output ! Local REAL :: gc,pi,noncirc,frct,icut,diff,height,Refilm,tiw,hplus,hcrit REAL :: fiw,Cw,hwave,Utip,fib,Ub,L,b,wlh,crw,ahgt,ehatb,ehatw,Recrit,hpc REAL :: intm,titot,lamda,a,c,Vol,theta,Fdrag,Fgrav,Fsurf,Fbal,H,W REAL :: secfc,freq,ScaleFact REAL :: we,ubnd,lbnd,serip,sedis,ramp,seoff,dsaut_tat REAL :: tauw,blhgt,Lentr,dele INTEGER :: iter LOGICAL :: ripple

gc = 32.2 pi = 4.*atan(1.)

! Initialize Variables CALL Initialize(ehatb,ehatw,tauw,tiw,blhgt,Lentr,dele,a,b,c,L,lamda,wlh, & Fdrag,Fgrav,Fsurf,Fbal,Vol,height,ahgt,Utip,Cw,crw,Ub, & fiw,fib,titot,intm,secfc,ScaleFact,fi,dsaut,xkic,se,iter)

! Fraction of mean film thickness that resides as base film substrate frct = 0.5 ! Fraction of mean film thickness that distinguishes disturbance wave region icut = 1.1 diff = icut - frct

! Approximate Width of rectangular channel by H = dhydj/2 noncirc = acontl/(0.5*dhydj) if (chantype>0) then ! if circular channel set width to zero noncirc = 0.0 endif

! Film Reynolds Number Refilm = rll*abs(Ul)*aliq*dhydj/visl Refilm = max(Refilm,1.) ! Consistent with Two-Zone model, but limit is not used actively within the model Recrit = 317.

! Two Zone Model proposed by Hurlburt et al. (2006) ! (returns isijl,titot,intm,xkic,fi,fiw,fib,Cw,Ub,Utip,height,Refilm, ! tiw,hplus,hcrit,ehatb,ehatw,Recrit) CALL TwoZone(i,j,delt,rll,rvl,visl,visv,dhydj,noncirc,acontl,hbar,frct, & Uv,av,Ul,aliq,Ue,ae,Refilm,isijl,titot,intm,xkic,fi,fiw, & fib,Cw,Ub,Utip,height,tiw,hplus,hcrit,ehatb,ehatw,hpc)

! Test if liquid bridging is occuring hwave = height + frct*hbar liqbrg = .false. if (noncirc > 0.) then H = 0.5*dhydj W = noncirc if ((H-2.0*hwave) <= 0.) liqbrg = .true. else if ((dhydj-2.0*hwave) <= 0.) liqbrg = .true. endif

! Calculate Deposition Rate using Cousins with Whalley (returns sde) CALL CalcDep(0,rll,rvl,sigma,dhydj,dx,acontl,aliq,ae,sde) ! Calculate Entraining Drop Size (returns dsaut) CALL DropSize(rll,rvl,visl,visv,sigma,dhydj,gpth,hbar,Uv,av,Ul,aliq,Ue,ae,dsaut)

! Check for co-current flow with vapor moving faster than the liquid if ((abs(Uv)>abs(Ul)).and.((Uv*Ul)>=0.0).and.(.not. liqbrg)) then

! Calculate disturbance wave spacing (returns freq,lamda,L) CALL WaveSpace(rll,rvl,visl,dhydj,Ul,aliq,Ue,ae,Uv,av,Cw,dx,height, & intm,freq,lamda,L)

! Calc. Eff Weber number to determine entraiment mechanism (Lift or Shear) we = rvl*(abs(Uv)*av)**2*(3.5*hbar)/(gc*sigma) we = max(we,1.) ! Transition value is 25 (Azzopardi), so smooth over 20-30 range ubnd = 30.

331

lbnd = 20.

! Roll wave entrainment mechanism calculations (we > lbnd) ripple = .true. if (we > lbnd) ripple = .false. if (jwldev) ripple = corrrip if (.not. ripple) then ! Calculate wall shear stress tauw = xkw*rll*aliq*abs(Ul)*dhydj/4. ! Calculate entrainment rate by boundary layer stripping mechanism CALL ENTRSTRIP(hplus,hpc,press,rvl,rll,visl,sigma,dhydj,noncirc,frct, & hbar,L,dx,lamda,height,Cw,tauw,tiw,Uv,av,Ul,aliq,Ue,ae, & Refilm,titot,blhgt,Lentr,dele,ScaleFact,se,secfc) if (we < ubnd) sedis = se endif

! Kelvin-Helmholtz entrainment mechanism calculations (we < ubnd) ripple = .false. if (we < ubnd) ripple = .true. if (jwldev) ripple = corrrip if (ripple) then ! Taterson (1980) drop size correlation dsaut_tat = 0.0112*sqrt(dhydj)/sqrt(0.023*rvl*(abs(Uv-Ul)+0.001)**2/ & (sigma*gc*(rvl*abs(max(Uv,0.001))*av*dhydj/visv)**0.2)) dsaut_tat = min(dsaut_tat,dhydj-2.*hbar,gpth-2.*hbar) dsaut_tat = max(dsaut_tat,1.e-8) if (we < lbnd) dsaut = dsaut_tat

! Stability analysis to determine critical wavelength (returns wlh,crw,ahgt) CALL Stability(rll,rvl,visl,sigma,Utip,Cw,hwave,height,L,fiw,wlh,crw,ahgt) ! Assume entire volume of 3-d ripple wave becomes entrained Vol=ahgt*(wlh**2)*(pi-2.0)/(4.0*pi) ! Calculate Entrainment Rate (returns se) CALL CalcSe(Utip,crw,dhydj,noncirc,dx,hwave,lamda,L,wlh,Vol,rll,rvl,visl, & visv,sigma,Uv,av,Ul,aliq,Ue,ae,Refilm,ripple,ScaleFact,se,secfc) if (we > lbnd) serip = se endif

! If necessary, ramp between two results if ((we > lbnd).and.(we < ubnd)) then ramp = (we - lbnd)/(ubnd - lbnd) ramp = max(0.,min(1.,(3.-2.*ramp)*ramp**2)) if (jwldev) then ramp = 1.0 if (corrrip == .true.) ramp = 0.0 endif ! Effective entrainment rate and drop size se = serip*(1.-ramp) + sedis*ramp dsaut = dsaut_tat*(1.-ramp) + dsaut*ramp dsaut = max(dsaut,1.e-8)

endif

elseif (liqbrg .and. use_liqbreak) then

! Liquid Bridge Breakup Model (returns se and dsaut) CALL LBBEntr(rll,rvl,sigma,dhydj,abs(Uv-Ul),av,ae,acontl,dx,gpth,se,dsaut)

endif

! Entrainment inception and suppression criterion CALL ENTRRAMPS(rll,rvl,visl,visv,sigma,dhydj,Refilm,Uv,av,Ul,aliq, & hplus,hpc,seoff) se = se*seoff

RETURN

END SUBROUTINE ANENTR

332

@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ ! ! TWO-ZONE INTERFACIAL SHEAR SUBROUTINES ! !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Calculate Zonal Interfacial Shear Stress SUBROUTINE TwoZone(i,j,delt,rll,rvl,visl,visv,dhydj,noncirc,acontl, & hbar,frct,Uv,av,Ul,aliq,Ue,ae,Refilm,isijl,titot,intm,xkic, & fi,fiw,fib,Cw,Ub,Utip,height,tiw,hplus,hcrit,ehatb,ehatw,hpc) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

use Model_Options, only : entr_print use contrlr, only : timet

IMPLICIT NONE

! Passed REAL :: delt,rll,rvl,visl,visv,dhydj,noncirc ! Input REAL :: acontl,hbar,frct,Uv,av,Ul,aliq,Ue,ae,Refilm ! Input INTEGER :: i,j,isijl ! Input/Output REAL :: titot,intm,xkic,fi,fiw,fib,Cw,Ub,Utip ! Output REAL :: height,tiw,hplus,hcrit,ehatw,ehatb,hpc ! Output

! Local REAL :: m_to_ft,WP,hwave,stlc,intlc,stdev,hbase,tib,Utb REAL :: H,W,urvl,bm,wm,fs,fact,dmy,fi_2zone,fi_whal

! smooth tube friction factor fs = 0.005 ! Conversion Factor m_to_ft = 3.281

! Steam-Water Values hcrit = (30.0e-6)*m_to_ft ! Critical film thickness stlc = 0.65 ! Leading coefficient in stdev correlation hpc = 12.0 ! Critical hplus (non-dim hbar in interfacial units) intlc = 0.38 ! Leading coefficient in intermittency correlation

! Multiplier on stdev to calculate wave height in base film zone bm = 0.6 ! Multiplier on stdev to calculate wave height in wave zone wm = 2.1

! Explicit form of stdev correlation stdev = stlc*(hbar-hcrit) ! Polynomial spline to ramp this value to zero as hbar approaches zero if (hbar < (2.*hcrit)) stdev = stlc*hbar**2/(4.*hcrit)

! Relative Velocity urvl = abs(Uv-Ul)+0.001 ! Perimeter of Gas Core region WP = CircWP(dhydj,hbar) if (noncirc > 0.) then H = 0.5*dhydj W = noncirc WP = NonCircWP(H,W,hbar) endif

! Assumes unstable film exists isijl = 4 ! Base Film Zone (returns fib,Ub,hbase,tib,Utb,ehatb) CALL ZonalCalcs(rll,rvl,visl,visv,dhydj,noncirc,hbar,hcrit,frct, & Uv,av,Ue,ae,Ul,aliq,stdev,bm,fib,Ub,hbase,tib,Utb,ehatb)

! Wave Zone (returns fiw,Cw,hwave,tiw,Utip,ehatw) CALL ZonalCalcs(rll,rvl,visl,visv,dhydj,noncirc,hbar,hcrit,frct, & Uv,av,Ue,ae,Ul,aliq,stdev,wm,fiw,Cw,hwave,tiw,Utip,ehatw) ! Restrict wave zone shear stress to be greater than 1.1x base zone shear stress if (tiw < (1.1*tib)) then tiw = 1.1*tib

333

! assumes correct relative velocity fiw = 2.*tiw/(rvl*(Utip-Cw)**2) CALL WhalleyFi(Uv,av,rvl,visv,Ue,ae,rll,hwave,dhydj,fi_whal) if ((fiw < fs) .or. (fiw > 5.*fi_whal)) then if (fiw < fs) fiw = fs if (fiw > 5.*fi_whal) fiw = 5.*fi_whal Cw = Utip - (2.*tiw/(fiw*rvl))**0.5 Cw = min(Uv,max(Cw,Ul)) endif endif

! Average height of disturbance waves height = hwave-frct*hbar

! Iterative Solution of Int and Total Int. Shear Stress (returns intm & titot) CALL CalcInt(i,j,delt,rll,visl,hbar,hpc,intlc,tiw,tib,intm,titot,hplus)

! Equivalent or average interfacial friction factor (needed by intfr) fi = titot/(rvl*urvl**2) fi_2zone = fi

! Whalley interfacial friction factor correlation CALL WhalleyFi(Uv,av,rvl,visv,Ue,ae,rll,hbar,dhydj,fi_whal) ! Restrict equivialent friction factor fi = min(5.*fi_whal/2.,max(fi,fs/2.)) if (fi /= fi_2zone) then ! Total shear stress titot = rvl*fi*(urvl**2) ! Dimensionless film thickness (in interfacial units) hplus = hbar*(titot*rll)**0.5/visl ! Intermittency CALL TwoEqns(titot,hbar,hpc,intlc,rll,visl,0.,1.,dmy,intm) ! Recalculate zonal shears assuming ratio of two is unchanged fact = tib/tiw tiw = titot/(intm*(1.-fact) + fact) tib = fact*tiw ! Recalculate zonal friction factors assuming velocities are unchanged ! Wave Zone fiw = 2.*tiw/(rvl*(Utip-Cw)**2) CALL WhalleyFi(Uv,av,rvl,visv,Ue,ae,rll,hwave,dhydj,fi_whal) ! If this violates limits then alter zonal velcoity if ((fiw < fs) .or. (fiw > 5.*fi_whal)) then if (fiw < fs) fiw = fs if (fiw > 5.*fi_whal) fiw = 5.*fi_whal Cw = Utip - (2.*tiw/(fiw*rvl))**0.5 Cw = min(Uv,max(Cw,Ul)) endif ! Base Film Zone fib = 2.*tib/(rvl*(Utb-Ub)**2) CALL WhalleyFi(Uv,av,rvl,visv,Ue,ae,rll,hbase,dhydj,fi_whal) ! If this violates limits then alter zonal velcoity if ((fib < fs) .or. (fib > 5.*fi_whal)) then if (fib < fs) fib = fs if (fib > 5.*fi_whal) fib = 5.*fi_whal Ub = Utb - (2.*tib/(fib*rvl))**0.5 Ub = max(0.001,min(Ub,Ul)) endif endif

! Corresponding Drag Coeffeficient (also needed by intfr) xkic = titot*WP/urvl

! Stable film criteria (should be the same) if ((intm == 0.) .or. (hplus < hpc)) isijl = 5

RETURN

END SUBROUTINE TwoZone

334

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Calculate Zonal Interfacial Shear Stress SUBROUTINE ZonalCalcs(rll,rvl,visl,visv,dhydj,noncirc,hbar,hcrit,frct, & Uv,av,Ue,ae,Ul,aliq,stdev,mult,fi,Vel,hcrest,ti,Utip,ehat) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE

! Passed REAL :: rll,rvl,visl,visv,dhydj,noncirc,hbar,hcrit ! Input REAL :: frct,Uv,av,Ue,ae,Ul,aliq,stdev,mult ! Input REAL :: fi,Vel,hcrest,ti,Utip,ehat ! Output

! Local REAL :: hmin,eps,awc,H,W,R1,R2

hmin = frct*hbar ! thickness of base film substrate eps = mult*stdev ! zonal roughness

! Calculate interfacial friction factor CALL FricFact(mult,eps,dhydj,hmin,noncirc,Uv,av,rvl,visv,Ue,ae,rll,ehat,fi) ! Total height of wave hcrest = hmin+eps ! Estimate mean vapor velocity at tip of wave awc = CircArea(dhydj,hcrest)/CircArea(dhydj,hbar) if (noncirc > 0.) then H = 0.5*dhydj W = noncirc awc = NonCircArea(H,W,hcrest)/NonCircArea(H,W,hbar) endif ! Restrict velocity enhancement factor awc = max(0.7,min(awc,1.0)) ! Calculate effective velocity at crest Utip = Uv/awc ! Define Parameters R1 = 0.5*fi*rvl R2 = (hcrest*(R1*(Utip**2)*rll)**0.5)/visl ! Calculate Zonal Interfacial Shear Stress (lbm/ft-s2) CALL ShearStress(R1,R2,Utip,rll,ti) ! Calculate Zonal Velocity Vel = Utip - (2.*ti/(fi*rvl))**0.5 ! Impose physical restrictions on calculated velocity if (mult > 1.) then Vel = min(Uv,max(Vel,Ul)) ! Wave Zone else Vel = max(0.001,min(Vel,Ul)) ! Base Film Zone endif ! Calculate interfacial shear stress assuming correct fi and Utip values ti = 0.5*fi*rvl*(Utip-Vel)**2

RETURN

END SUBROUTINE ZonalCalcs

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Calculates interfacial friction factor based on log law SUBROUTINE FricFact(mult,eps,dhydj,hmin,noncirc,Uv,av,rvl,visv,Ue,ae,rll,ehat,fi) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE

! Passed REAL :: mult,eps,dhydj,hmin,noncirc,Uv,av,rvl,visv,Ue,ae,rll ! Input REAL :: ehat,fi ! Output ! Local REAL :: uplim,cb,H,W,fs,fi_whal

! Note: mult = 2.1 for wave zone, 0.6 for base film zone

! Impose upper limit on relative roughness values uplim = 0.1143 ! Base film Zone (0.4*0.6/2.1) if (mult > 1.0) uplim = 0.4 ! Wave Zone

! Log-law parameter (varies between zones)

335

cb = 0.8 if (mult > 1.0) cb = 4.7

if (noncirc > 0.) then ! Relative roughness and zonal interfacial fi for non-circular geometry H = 0.5*dhydj W = noncirc ehat = eps/(0.5*H-hmin) if (ehat > uplim) then ehat = uplim eps = ehat*(0.5*H-hmin) endif fi = (0.58/(log(ehat)/(ehat-1.0)-log(cb)+1.05))**2 else ! Relative roughness and zonal interfacial fi for circular geometry ehat = eps/(0.5*dhydj-hmin) if (ehat > uplim) then ehat = uplim eps = ehat*(0.5*dhydj-hmin) endif fi = (0.58/(-log(ehat)/((ehat-1.0)**2)-log(cb)+1.05+0.5*(ehat+1.0) & /(ehat-1.0)))**2 endif

! Impose restrictions on interfacial friction factor fs = 0.005 CALL WhalleyFi(Uv,av,rvl,visv,Ue,ae,rll,eps+hmin,dhydj,fi_whal) fi = min(5.*fi_whal,max(fi,fs))

RETURN

END SUBROUTINE FricFact

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Whalley Interfacial Friction Factor Correlation SUBROUTINE WhalleyFi(Uv,av,rvl,visv,Ue,ae,rll,thick,dhydj,fi) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE

! Passed REAL :: Uv,av,rvl,visv,Ue,ae,rll,thick,dhydj ! Input REAL :: fi ! Output ! Local REAL :: regc,Ugc,rhogc

! Gas core velocity (momentum over density) Ugc = (abs(Uv)*av*rvl + abs(Ue)*ae*rll)/(av*rvl + ae*rll) ! Gas core density rhogc = (av*rvl + ae*rll)/(av+ae) ! Gas core Reynolds Number regc = max(1.,Ugc*rhogc*dhydj/visv) ! Whalley interfacial friction factor correlation fi = 0.079*(regc**-0.25)*(1.+24.*thick/dhydj*((rll/rvl)**0.333))

RETURN

END SUBROUTINE WhalleyFi

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Calculates interfacial shear stress in lbm/ft-s2 SUBROUTINE ShearStress(R1,R2,Utip,rll,taui) ! Explicit form, neglects wave velocity in hplus portion ! Allows film to be either laminar (Asali) or turbulent (Henstock & Hanratty) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE

! Passed REAL :: R1,R2,Utip,rll ! Input REAL :: taui ! Output ! Local REAL :: term0,term1,term2,term3,numer,denom

336

REAL :: G1S,G9S,SF,FT,OF

G1S = 1.5**2 G9S = 9.5**2 SF = 17./15. FT = 4./3. OF = 1./15.

term0 = rll*(G9S + G1S*(R2**SF)) term1 = G1S*G9S*R1*(R2**FT) denom = (term0 - term1)**2 term2 = R1*term0*(term0 + term1) term3 = (4.0*G1S*G9S*((R1*term0)**3)*((R2**OF)**20.))**0.5 numer = (Utip**2)*(term2-term3) taui = max(numer/denom,0.0)

RETURN

END SUBROUTINE ShearStress

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Bisection Method to solve for Intermittency and Total Int. Shear Stress SUBROUTINE CalcInt(i,j,delt,rll,visl,hbar,hpc,intlc,tiw,tib,intm,titot,hplus) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE

! Passed INTEGER :: i,j ! Input REAL :: delt,rll,visl,hbar,hpc,intlc,tiw,tib ! Input REAL :: intm,titot,hplus ! Output ! Local REAL :: ps0,ps1,ps2,fps0,fps1,fps2,intm0,intm1 INTEGER :: psct

! Lower bound ps0 = tib CALL TwoEqns(ps0,hbar,hpc,intlc,rll,visl,tib,tiw,fps0,intm0) ! Upper bound ps1 = tiw CALL TwoEqns(ps1,hbar,hpc,intlc,rll,visl,tib,tiw,fps1,intm1)

! Check to make sure root exists on interval if ((fps0*fps1) >= 0.0) then intm = 0.0 titot = tib else do psct=1,100 ps2=(ps1+ps0)/2. CALL TwoEqns(ps2,hbar,hpc,intlc,rll,visl,tib,tiw,fps2,intm) intm = min(0.38,max(0.0,intm)) if ((ps2-ps0) < (1.e-8)) exit if ((fps0*fps2) > 0.0) then ps0 = ps2 else ps1 = ps2 endif enddo endif

! Calculate total shear stress titot = intm*tiw+(1.-intm)*tib ! Calculate dimensionless film thickness (in interfacial units) hplus = hbar*(titot*rll)**0.5/visl

RETURN

END SUBROUTINE CalcInt

337

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Solves the two expressions for Intermittency using a given value ! Total Int. Shear Stress. Returns difference of two results. SUBROUTINE TwoEqns(guess,hbar,hpc,intlc,rll,visl,tib,tiw,result,psi2) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE

! Passed REAL :: guess,hbar,hpc,intlc,rll,visl,tib,tiw ! Input REAL :: result,psi2 ! Output ! Local REAL :: psi1,x,term1,term2,term3

! Equation 1 (Definition of intermittency) psi1 = (guess-tib)/(tiw-tib)

! Equation 2 (Intermittency correlation x = hbar*(guess*rll)**0.5/visl term1 = hpc/x term2 = exp(-(x-2.0)/4.0) term3 = exp(-(hpc-2.0)/4.0) psi2 = intlc*(1.0-term1+(4.0/x)*(term2-term3)) if (x < hpc) psi2 = 0.0

! Difference in two results result = psi1 - psi2

RETURN

END SUBROUTINE TwoEqns

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Calculates the spacing between disturbance waves and based on the ! frequency correlation suggested by Sawant et al. (2007) SUBROUTINE WaveSpace(rll,rvl,visl,dhydj,Ul,aliq,Ue,ae,Uv,av,Cw,dx,height, & intm,freq,lamda,L) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE

! Passed REAL :: rll,rvl,visl,dhydj,Ul,aliq,Ue,ae,Uv,av ! Input REAL :: Cw,dx,height,intm ! Input REAL :: freq,lamda,L ! Output ! Local REAL :: pi,reliq,maxangle,minL,minspace

pi = 4.*atan(1.)

! Total liquid Reynolds number reliq = rll*(abs(Ul)*aliq+abs(Ue)*ae)*dhydj/visl ! Wave frequency freq = 0.086*(reliq**0.27)*((rll/rvl)**-0.64)*abs(Uv)*av/dhydj ! Relate frequency to wave length assuming constant wave velocity lamda = Cw/freq ! Relate the intermittency to the length of the disturbance wave region assuming ! a constant wave velocity L = lamda*intm

! Impose restriction of maximum angle of wave based on Hurlburt & Newell (1996) maxangle = 5.*pi/180. ! since height is fixed this translates to a minimum allowable length of the wave minL = height/tan(maxangle) if (L < minL) then L = height/tan(maxangle) lamda = L/intm freq = Cw/lamda endif

! Also limit minimum disturbance wave spacing minspace = 0.5/12. if (lamda < minspace) then lamda = minspace

338

L = lamda*intm freq = Cw/lamda endif

RETURN

END SUBROUTINE WaveSpace

!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ ! ! ROLL-WAVE ENTRAINMENT RATE SUBROUTINES ! !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Entrainment rate based on boundary layer stripping model of Fore (1993) SUBROUTINE ENTRSTRIP(hplus,hpc,pll,rvl,rll,visl,sigma,dhydj,noncirc, & frct,hbar,L,dx,lamda,height,Cw,tauw,taui,Uv,av,Ul,aliq,Ue,ae, & Refilm,titot,blhgt,Lentr,dele,ScaleFact,se,secfc) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE

! Passed REAL :: hplus,hpc,pll REAL :: rvl,rll,visl,sigma,dhydj,noncirc,frct,hbar,L,dx,lamda ! Input REAL :: height,Cw,tauw,taui,Uv,av,Ul,aliq,Ue,ae,Refilm,titot ! Input REAL :: blhgt,Lentr,dele,ScaleFact,se,secfc ! Output ! Local REAL :: g,sknfrc,ustr,nul,Res,Fr2

! Note: tauw and taui need to be in lbm/ft-s2

g = 32.2

! Set stripping length equal to length of wave Lentr = L

if (Lentr == 0.0) then dele = 0. se = 0. else ! Assume 50/50 split in form drag and skin friction sknfrc = 0.5*taui ! Friction velocity ustr = sqrt(sknfrc/rll) ! Kinematic viscosity nul = visl/rll ! Reynolds number Res = ustr*dhydj/nul ! Froude number squared Fr2 = ustr**2/(g*dhydj) ! Set stripping thickness equal to height of wave dele = height ! Calculates the entrainment rate CALL SERAT(rvl,rll,visl,frct,hbar,dhydj,noncirc,dx,lamda,height,dele,ustr, & Res,Fr2,Uv,av,Ul,aliq,Ue,ae,ScaleFact,se,secfc) endif

RETURN

END SUBROUTINE ENTRSTRIP

339

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Calculates the entrainment rate by the BL stripping mechanism SUBROUTINE SERAT(rvl,rll,visl,frct,hbar,dhydj,noncirc,dx,lamda,height,dele,ustr, & Res,Fr2,Uv,av,Ul,aliq,Ue,ae,ScaleFact,se,secfc) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

use Model_Options, only : jwldev use dmpctrl, only : corrfact

IMPLICIT NONE

! Passed REAL :: rvl,rll,visl,frct,hbar,dhydj,noncirc,dx,lamda,height ! Input REAL :: dele,ustr,Res,Fr2,Uv,av,Ul,aliq,Ue,ae ! Input REAL :: se,ScaleFact,secfc ! Output ! Local REAL :: hmin,hwave,aentrw,H,W,thk,intgrl,semax,g,gc,rhorat

g = 32.2 gc = 32.2

! Geometric Parameters hmin = frct*hbar hwave = hmin+height

! Area of striping layer, needed to compute mass flow rate aentrw = CircArea(dhydj,hwave-dele)-CircArea(dhydj,hwave) if (noncirc > 0.) then H = (0.5*dhydj) W = noncirc aentrw = NonCircArea(H,W,hwave-dele) - NonCircArea(H,W,hwave) endif

! dimensionless thickness of stripping region thk = dele/dhydj ! Integral of cubic velocity profile in stripping region intgrl = ((thk**2)/Fr2+thk)*Res/12. ! Number of disturbance waves per computational cell ScaleFact = dx/lamda ! Calculate maximum entrainment rate semax = rll*ustr*aentrw*intgrl*ScaleFact ! Density Ratio rhorat = rll/rvl ! Correction factor based on comparisons to experimental data secfc = (5.7223e-06)*thk**(-1.0556)*rhorat**(0.8014) secfc = min(secfc,10.) ! Logic for Model Development version of the code if (jwldev) secfc = corrfact ! Actual entrainment rate se = secfc*semax

RETURN

END SUBROUTINE SERAT

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Kocamustafaogullari et al. (1994) Drop Diameter Correlation SUBROUTINE DropSize(rll,rvl,visl,visv,sigma,dhydj,gpth,hbar,Uv,av, & Ul,aliq,Ue,ae,dsaut) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE

! Passed REAL :: rll,rvl,visl,visv,sigma,dhydj,gpth ! Input REAL :: hbar,Uv,av,Ul,aliq,Ue,ae ! Input REAL :: dsaut ! Output ! Local REAL :: gc,g,nuh,cwh,Wed,regas,reliq

gc = 32.2 g = 32.2

! viscosity number (dim'less) nuh = visl/sqrt(gc*rll*sigma*sqrt(gc*sigma/(g*(rll-rvl))))

340

! accounts for effect of surface tension on internal flow (dim'less) cwh = 1./(35.34*nuh**0.8) ! Correlating Constant and Drag Coeff dsaut = 1.3*dhydj*(cwh**(-4./15.)) ! Weber number Wed = rvl*((max(Uv,0.001)*av)**2)*dhydj/(gc*sigma) Wed = max(Wed,1.) dsaut = dsaut*(Wed)**(-0.6) ! Reynolds numbers regas=rvl*max(abs(Uv),0.001)*av*dhydj/visv regas = max(regas, 1.) reliq = rll*(max(abs(Ul),0.001)*aliq+max(abs(Ue),0.001)*ae)*dhydj/visl reliq = max(reliq,1.) dsaut = dsaut*(((regas**4)/reliq)**(1./15.)) ! Density and viscosity ratios dsaut = dsaut*(((rvl/rll)*(visv/visl))**(4./15.)) ! Physical constraints dsaut = min(dsaut,dhydj-2.*hbar,gpth-2.*hbar) dsaut = max(dsaut,1.e-8)

RETURN

END SUBROUTINE DropSize

!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ ! ! BAG-BREAKUP ENTRAINMENT RATE SUBROUTINES ! !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Stability analysis used to calculate the critical wavelength and height ! of the ripple waves that reside on the top of disturbance waves and ! are assumed to be the source of entrainment SUBROUTINE Stability(rll,rvl,visl,sigma,Utip,Cw,hwave,height,L,fiw,wlh,crw,ahgt) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE

! Passed REAL :: sigma,rvl,rll,visl,Utip,Cw,hwave,height,L,fiw ! Input REAL :: wlh,crw,ahgt ! Output ! Local REAL :: pi,gc,g,Urel,ti,wlh1,ps0,ps1,dmvps0,dmvps1 REAL :: fps0,fps1,nuh,cwh INTEGER :: psct

pi = 4.*atan(1.) gc = 32.2 g = 32.2

! Relative velocity between gas core and disturbance wave (ft/sec) Urel = max(abs(Utip-Cw),0.001) ! Interfacial shear stress for disturbance wave (lbm/ft-s2) ti = 0.5*rvl*fiw*Urel**2

! Initialize kelvin helmholtz wavelength (ft) wlh=2.*pi/((Urel**2)*rvl/(gc*sigma)) wlh1=wlh

! Secant method to solve for critical wavelength (wlh in ft) by setting imaginary ! component of stability analysis result equal to zero ps0=wlh*0.8 if ((2.*pi*hwave/ps0) > 5.) then dmvps0=1. else dmvps0=(cosh(2.*pi*hwave/ps0)-1.)/sinh(2.*pi*hwave/ps0) endif fps0=rvl*rll*dmvps0*(Urel**2) - gc*sigma*2.*pi/ps0*(rll*dmvps0 + rvl)

ps1=wlh*1.2

341

do psct=1,100 if ((2.*pi*hwave/ps1) > 5.) then dmvps1=1. else dmvps1=(cosh(2.*pi*hwave/ps1)-1.)/sinh(2.*pi*hwave/ps1) endif fps1=rvl*rll*dmvps1*(Urel**2) - gc*sigma*2.*pi/ps1*(rll*dmvps1 + rvl)

wlh=ps1-fps1*(ps1-ps0)/(fps1-fps0) if (abs(wlh-ps1) < 1.e-15) exit

ps0=ps1 fps0=fps1 ps1=wlh

enddo ! End of secant method

! Finalize kelvin helmholtz wavelength (ft) wlh=min(5.*wlh1,wlh,L)

! Compute ripple wave velocity (crw,ft/s) from real component if ((2.*pi*hwave/wlh) > 5.) then dmvps1=1. else dmvps1=(cosh(2.*pi*hwave/wlh)-1.)/sinh(2.*pi*hwave/wlh) endif crw=(dmvps1*rll*Cw + rvl*Utip)/(dmvps1*rll + rvl) ! Limit ripple wave velocity so it must be greater than disturbance wave velocity crw = min(Utip,max(crw,Cw))

! Viscosity number (dim'less) nuh=visl/sqrt(gc*rll*sigma*sqrt(gc*sigma/(g*(rll-rvl)))) ! Parameter that accounts for effect of surf. tension on internal flow (dim'less) cwh=1./(35.34*(nuh**0.8)) ! Ripple wave amplitude (ft) using shear flow model of Ishii & Grolmes ahgt=cwh*visl*crw/ti

RETURN

END SUBROUTINE Stability

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Apply correction factor to convert maximum to actual entrainment rate SUBROUTINE CalcSe(Utip,crw,dhydj,noncirc,dx,hwave,lamda,L,wlh,Vol,rll, & rvl,visl,visv,sigma,Uv,av,Ul,aliq,Ue,ae,Refilm, & ripple,ScaleFact,se,secfc) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

use Model_Options, only : jwldev use dmpctrl, only : corrfact

IMPLICIT NONE

! Passed REAL :: Utip,crw,dhydj,noncirc,dx,hwave,lamda,L,wlh,Vol,rll ! Must have REAL :: rvl,sigma,Uv,av,Ul,aliq,Ue,ae,Refilm,visl,visv ! To correlate REAL :: ScaleFact,se,secfc ! Output LOGICAL :: ripple ! Local REAL :: gc,g,trace,H,W,period,semax,we,filmthick,nuh REAL :: jg,jltot,jtot,jgstr,denrat,Fr,Weent,comp,compl,qual

gc = 32.2 g = 32.2

! Perimeter of Film trace = CircWP(dhydj,hwave) if (noncirc > 0.) then H = (0.5*dhydj) W = noncirc trace = NonCircWP(H,W,hwave) endif

342

! Number of Ripple Waves per computational cell ! # of DW, # of RW (axially), # of RW (circumfrentialy) ScaleFact = (dx/lamda)*(L/wlh)*(trace/wlh) ! Estimate Time Scale for Entrainment period = max(wlh,1.e-12)/max(abs(Utip-crw),0.001) ! Calculate Maximum Entrainment Rate semax = (Vol*rll/period)*ScaleFact ! Correction Factor Based on Comparisons to Experimental Data secfc = 2.3E-4 if (Refilm < 6000.) secfc = secfc*(Refilm - 1750.)/(6000. - 1750.) if (Refilm < 1750.) secfc = 0. ! Flag to specify model development version if (jwldev) secfc = corrfact ! Actual entrainment rate se = secfc*semax

RETURN

END SUBROUTINE CalcSe

!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ ! ! LIQUID BRIDGE BREAKUP ENTRAINMENT RATE SUBROUTINES ! !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Liquid Bridge Breakup Model SUBROUTINE LBBEntr(rll,rvl,sigma,dhydj,uvjl,all,ael,acontl,dx,gpth,se,dsaut) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE

! Passed REAL :: rll,rvl,sigma,dhydj,uvjl,all,ael,acontl,dx,gpth ! Input REAL :: se,dsaut ! Output ! Local REAL :: gc,g,wecrit,ddtay,we_lbb,tdbt,ramplbb,freq,efflbb,fb

gc = 32.2 g = 32.2

se = 0.0

! critical value for breakup wecrit = 12. ! equivalent liquid bridge drop size ddtay = max(0.00328,min(dhydj,gpth,2.*sqrt(gc*sigma/(g*(rll-rvl))))) ! weber number we_lbb = rvl*ddtay*uvjl**2/(sigma*gc) ! total dimensionless breakup time, if we is not > wecrit then se is not ! calculated for this mechanism if (we_lbb > wecrit) then if (we_lbb < 18.) then tdbt = 6.*(we_lbb - wecrit)**-0.25 elseif (we_lbb < 45.1) then tdbt = 2.45*(we_lbb - wecrit)**0.25 else tdbt = 14.1*(we_lbb - wecrit)**-0.25 endif ! chance of breakup increases with increased presence of vapor ramplbb = max(0.,(all-0.1)/0.9) ! breakup frequency freq = ramplbb*abs(uvjl)*sqrt(rvl/rll)/(tdbt*ddtay) ! breakup efficiency efflbb = 1. - (ael/max(0.1,all+ael))/0.52 ! use max se of this and annular calc se = rll*(1.-all-ael)*freq*efflbb*acontl*dx endif

343

! Drop size, from Flow3D dsaut = 0.06147*6.5*sigma*gc/(rvl*uvjl**2) dsaut = min(dsaut,dhydj,0.13,gpth) dsaut = max(dsaut,1.e-8)

RETURN

END SUBROUTINE LBBEntr

!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ ! ! GENERAL SUBROUTINES ! !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ !@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Ramp entrainment rate to zero as inception or suppresion criteria ! are approached SUBROUTINE ENTRRAMPS(rll,rvl,visl,visv,sigma,dhydj,Refilm,Uv,av,Ul,aliq, & hplus,hpc,seoff) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE

! Passed REAL :: rll,rvl,visl,visv,sigma,dhydj,Refilm ! Input REAL :: Uv,av,Ul,aliq,hplus,hpc ! Input REAL :: seoff ! Output ! Local REAL :: Remin,hmm,nuh,ugcrit,g,gc,fact,sesup,seinc

g = 32.2 gc = 32.2

! Ishii & Grolmes Entrainment Inception Criterion (Critical Vapor Velocity) hmm = ((rvl/rll)**0.5)*(visl*av/(32.2*sigma)) hmm = max(hmm,1.e-8) nuh=visl/sqrt(gc*rll*sigma*sqrt(gc*sigma/(g*(rll-rvl)))) if (Refilm > 1635.) then ugcrit = nuh**0.8/hmm elseif (Refilm < 160.) then ugcrit = 1.5*(Refilm**-0.5)/hmm else ugcrit = 11.78*(nuh**0.8)*(Refilm**-0.333)/hmm endif

seinc = 1.0 if (abs(Uv) < (2.*ugcrit)) then seinc = (abs(Uv) - ugcrit)/ugcrit seinc = (3.-2.*seinc)*seinc**2 if (abs(Uv) < ugcrit) seinc = 0.0 endif

! Critical Film Condition Consistent with Two-Zone Model (Hurlburt et al. ) sesup = 1.0 if (hplus < (2.*hpc)) then sesup = (hplus - hpc)/hpc sesup = (3.-2.*sesup)*sesup**2 if (hplus < hpc) sesup = 0.0 endif

! Deactivate Ishii & Grolmes Criteria ! seoff = min(sesup,seinc) seoff = sesup

RETURN

END SUBROUTINE ENTRRAMPS

344

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Deposition rate based on empirical diffusion model of cousins SUBROUTINE CalcDep(def,rll,rvl,sigma,dhydj,dx,acontl,aliq,ae,sde) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE

INTEGER :: def REAL :: rll,rvl,sigma,dhydj,dx,acontl,aliq,ae,sde REAL :: aintfc,conc,xkd,gc

gc = 32.2 aintfc = 4.0*acontl*sqrt(1.-aliq)*dx/dhydj conc = ae*rll/(1.-aliq)

! Mass tranfer coefficient suggested by Whalley (Original Model) xkd = max(12.4911*sigma**0.8968, 3.0491e12*sigma**5.3054) ! Mass tranfer coefficient suggested by Hewitt & Govan (1990) if (def>0) xkd=0.182*sqrt(gc*sigma/(rvl*dhydj))*min(1.0,((conc/rvl)/0.3)**-0.65) sde = sde + xkd*conc*aintfc

RETURN

END SUBROUTINE CalcDep

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! ! Initialize Important Variables SUBROUTINE Initialize(ehatb,ehatw,tauw,tiw,blhgt,Lentr,dele,a,b,c,L,lamda,wlh,Fdrag,Fgrav,Fsurf,fbal ,Vol,height,ahgt,Utip,Cw,crw,Ub,fiw,fib,titot,intm,secfc,ScaleFact,fi,dsaut,xkic,se,iter) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

IMPLICIT NONE

REAL :: ehatb,ehatw,tauw,tiw,blhgt,Lentr,dele REAL :: a,b,c,L,lamda,wlh,Fdrag,Fgrav,Fsurf,Vol,fbal,height,ahgt REAL :: Utip,Cw,crw,Ub,fiw,fib,titot,intm,secfc REAL :: ScaleFact,fi,dsaut,xkic,se INTEGER :: iter

ehatb = 0.0 ehatw = 0.0 tauw = 0.0 tiw = 0.0 blhgt = 0.0 Lentr = 0.0 dele = 0.0

a = 0.0 b = 0.0 c = 0.0 L = 0.0 lamda = 0.0 wlh = 0.0

Fdrag = 0.0 Fgrav = 0.0 Fsurf = 0.0 fbal = 0.0

Vol = 0.0 height = 0.0 ahgt = 0.0

Utip = 0.0 Cw = 0.0 crw = 0.0 Ub = 0.0

fiw = 0.0 fib = 0.0 titot = 0.0 intm = 0.0

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secfc = 0.0 ScaleFact = 0.0 fi = 0.0 dsaut = 1. xkic = 0.0 se = 0.0

iter = 0

RETURN

END SUBROUTINE Initialize

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! END MODULE ENTRRATE !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

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Appendix B: Code-to-Data Comparisons of the Würtz [79] Experiments

Contained within this appendix is a comparison of the experimentally measured and COBRA-TF predicted results for the Würtz [79] experiments using the proposed annular flow modeling package outlined in Chapter 4. As described in Chapter 3, Würtz [79] conducted steam-water annular flow experiments in both 0.394-inch and 0.787-inch diameter tubes while measuring outlet film flow rate and axial pressure gradient over the last 3.28-feet of the 29.53-foot long test section. The results from the 0.394-inch tube tests, which were conducted at 435, 725, 1015, and 1305-psia, are presented in Figures B-1 through B-8 and the results from the 0.787-inch tube tests, which were conducted only at 1015-psia, are presented in Figures B-9 and B-10. On these figures the experimental data are shown as solid lines while the symbols represent the COBRA- TF predictions. The legend on each figure lists the corresponding total mass flow rate for each test, which remained constant for a given test, while the flowing quality was varied. The additional line shown for each experiment on the entrained flow rate figures represents the total liquid mass flow rate, which decreases for a given test as the flowing quality increases.

From these results it can be seen that the proposed model set provides reasonable agreement with the experimental data taken by Würtz [79]. The model set is able to capture the overall trends observed in the experimental data. The most significant deviations in the predicted entrained mass flow rate occur near the transition between annular and churn-turbulent flow for the larger total mass flow rates tested (e.g. see Figure B-5 around a flowing quality of 20-25%). Additionally, as the system pressure increases an overprediction occurs in the axial pressure gradient for the larger total mass flow rates tested in smaller diameter tube (e.g. see Figures B-6 and B-8); however, the same overprediction is not seen in the larger diameter tube (e.g. see Figure B-10). Without corresponding film thickness measurements to verify the code predictions of this quantity is difficult to ascertain the reason for the deviations in the predicted axial pressure gradient.

347

348

349

350

351

352

Appendix C: Qualitative Comparisons to the Disturbance

Wave Characteristics Observed by Sawant et al. [57]

To further assess the capability of the proposed annular flow modeling package to capture the underlying physics of the flow, qualitative comparisons can be made between the trends in disturbance wave characteristics that were observed by Sawant et al. [57] and the results predicted by the proposed modeling package. The vertical annular flow experiments conducted by Sawant et al. [57] used air-water in a 0.37-inch diameter tube for three low pressure outlet conditions (atmospheric, 58-psia, and 84-psia) and relatively low liquid Reynolds numbers (500- 5700). The qualitative trends observed in the experimental data, which was collected roughly 8-ft (270 L/D) from the liquid injection location, were summarized in Table 2-2 and this experimental data was used to provide a disturbance wave frequency correlation that was applied in the proposed modeling package (see Equation (4-72)). Figures C-1 through C-8 present the disturbance wave velocities, amplitudes, spacings, and frequencies predicted by the proposed modeling package for the steam-water data examined in the current study as a function of both dimensionless superficial gas velocity and film Reynolds number. Sawant et al. [57] compared their results to the total liquid Reynolds number because they did not make entrained fraction or film thickness measurements in their experiments; however, the film Reynolds number is a more realistic quantity to compare to if it is available. In general, the same trends observed in the these experiments were predicted by the proposed modeling package; however, some differences do exist. In particular, the difference in trend between the experimental and predicted wave frequency may indicate an issue with the extension of this correlation to higher pressures and larger liquid Reynolds numbers and certainly warrants further investigation in future studies given the importance of this correlation to the resulting predictions by the modeling package.

Table C-1: Comparison of Qualitative Trends in Disturbance Wave Characteristics for Results Predicted by Proposed Modeling Package and those Observed by Sawant et al. [57]. Increasing Gas Velocity Increasing Liquid Reynolds Number Increasing Pressure Wave Sawant et Sawant et Sawant et Property Current Study Current Study Current Study al. [57] al. [57] al. [57] Velocity ↑ No Trend ↑ No Trend ↑ (Slightly) Same Trend Amplitude ↓ Same Trend ↑ Same Trend ↓ Opposite Trend Spacing ↓ Same Trend No Dependence Same Trend ↓ Same Trend Frequency ↑ Opposite Trend ↑ No Trend ↑ Opposite Trend

353

Figure C-1: Comparison of the COBRA-TF Predicted Disturbance Wave Velocity as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet.

Figure C-2: Comparison of the COBRA-TF Predicted Disturbance Wave Velocity as a function of Film Reynolds Number at the Test Section Outlet.

354

Figure C-3: Comparison of the COBRA-TF Predicted Disturbance Wave Amplitude as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet.

Figure C-4: Comparison of the COBRA-TF Predicted Disturbance Wave Amplitude as a function of Film Reynolds Number at the Test Section Outlet.

355

Figure C-5: Comparison of the COBRA-TF Predicted Disturbance Wave Spacing as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet.

Figure C-6: Comparison of the COBRA-TF Predicted Disturbance Wave Spacing as a function of Film Reynolds Number at the Test Section Outlet.

356

Figure C-7: Comparison of the COBRA-TF Predicted Disturbance Wave Frequency as a function of Dimensionless Superficial Vapor Velocity at the Test Section Outlet.

Figure C-8: Comparison of the COBRA-TF Predicted Disturbance Wave Frequency as a function of Film Reynolds Number at the Test Section Outlet.

357

Appendix D: Subroutine Listings for Proposed Counter- Current Flow Limitation (CCFL) Model

1) module.f to declare the variables associated with the new CCFL model.

2) setin13.f to read in the user-specified information associated with the new CCFL boundary condition from the input deck and ensure physically reasonable values are provided.

3) intfr.f to: a) deactivate the baseline counter-current flow entrainment model (excess film flow) in nodes located at lower elevations in the same channel where a CCFL boundary condition is applied when counter-current flow conditions exist, b) determine if the CCFL model should be activated in a given node where a CCFL boundary condition is applied, and c) calculate the appropriate entrainment rate in the node located above the point where the CCFL BC is applied. All of these calculations are done on an explicit basis. It should also be mentioned that when the CCFL model is invoked the flow regime identifier is assigned a value of 7.

4) xschem.f to calculate and solve the set of CCFL momentum equations, instead of the standard set of momentum equations, when it is determined that the CCFL model should be activated.

5) post3d.f to use the resulting values for the current time step and back-calculate an effective interfacial drag when the CCFL model is invoked as well as determine if: a) velocity flip-flop occured, b) the code predicted interfacial drag exceeds the back- calculated CCFL value such that the CCFL model should be exited (for increasing vapor

358

flow conditions), or c) the calculated liquid downflow exceeds the available liquid flow such that the CCFL model should be exited (for decreasing vapor flow conditions). All of these calculations are performed on an implicit basis.

6) trans.F and timstp.f to support the backup logic for a failed time step.

7) dumpit.f and restrt.f to store the CCFL boundary condition values to support the restart capability of the code.

A listing of the logic that had to be added to these different subroutines within COBRA-TF to support this model is provided in the remainder of this Appendix.

Note: it is not shown with this Appendix, but logic was also added to trans.F and timstp.f to support the backup logic for a failed time step as well as dumpit.f and restrt.f to store the CCFL boundary condition values to support the restart capability of the code

359

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Within module.f a module was added to declare the variables associated with the new CCFL model. Note : ‘mflddim’ sets maximum number of CCFL boundary conditions that can be applied. This vaule was set to 20 in respar.h.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

. . Existing logic... .

module ccfl include "respar.h" integer msp,mspx ! new CCFL model variables real :: ccfl_incpt(mflddim),ccfl_slope(mflddim), $ ccfl_beta(mflddim),ccfl_flcrit(mflddim), $ chiv(mflddim),chil(mflddim), $ ccfl_a(mflddim),ccfl_b(mflddim), $ ccfl_fldpt(mflddim),ccfl_FRpt(mflddim), $ ccfl_stor(mflddim,5),compem(20), $ ccfl_flcritn(mflddim),ccfl_avail(mflddim) real :: wght,caplgth,denom,liqprop,vapprop,seccfl real :: ccfl_xk,flood logical :: usenorm(mflddim) = .false. logical :: failedhere(mflddim) = .false. integer :: ccfl_bcnum(mcdim,2) = 0 logical :: ccfl_fail,ccfl_flg,no_secc,ccfljp1 end module ccfl

. . Existing logic... .

360

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Within setin13.f logic was added to read in the user-specified information associated with the new CCFL boundary condition, which was assigned a value of 7, from the input deck and ensure physically reasonable values are provided.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

. . Existing logic... .

elseif (isp == 7) then

! JWL - 08/15/08 - read in CCFL boundary condition values

!! Update counter varaible msp = msp + 1

!! Store boundary condition number for later use ccfl_bcnum(ibound(1, n),1) = n !! Store counter value for later use ccfl_bcnum(ibound(1, n),2) = msp

!! Check for valid input values if ((pvalue(n) <= 0.) .or. (hvalue(n) <= 0.) .or. & (xvalue(n) < 0.) .or. (xvalue(n) > 1.)) then write (i3, *) write (i3, *) 'CCFL BC values invalid ',ibound(1, n),' ', & ibound(2, n) linperr = .true. write (i3, *) endif

!! Ensure both slope and intercept are positive values ccfl_incpt(msp) = max(0.,pvalue(n)) ccfl_slope(msp) = max(0.,hvalue(n)) !! Logic written such that default is Wallis form of CCFL correlation ccfl_beta(msp) = min(1.,max(0.,xvalue(n))) ccfl_flcrit(msp) = -(ccfl_incpt(msp)/ccfl_slope(msp))**2

endif

. . Existing logic... .

361

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Within intfr.f logic was added to: a) deactivate the default counter-current flow entrainment model (excess film flow) in nodes located at lower elevations in the same channel where a CCFL boundary condition is applied when counter-current flow conditions exist, b) determine if the CCFL model should be activated in a given node where a CCFL boundary condition is applied, and c) calculate the appropriate entrainment rate in the node located above the point where the CCFL BC is applied. All of these calculations are done on an explicit basis. It should also be mentioned that when the CCFL model is invoked the flow regime identifier is assigned a value of 7.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

. . Existing logic... .

use ccfl use ENTRRATE

. . Existing logic... .

!!!!start channel loop here...

do ic = 1, ichn

i = ncsec(ic, isect) appp = amom(i, j) dhydj = dhyd(i, j) acontl = acont(i, j) wperim = 4.0*appp/dhydj rlp = 0.5*(rl(i, j) + rl(i, jp1)) rstp = 0.5*(rv(i, j) + rv(i, jp1)) rvp = rstp + 0.5*(rmgas(i, j) + rmgas(i, jp1)) alp = 0.5*(al(i, j) + al(i, jp1)) aep = 0.5*(ae(i, j) + ae(i, jp1)) aliqp = 0.5*(aliq(i, j) + aliq(i, jp1)) all = al(i, j) aliql = aliq(i, j) ael = ae(i, j) allp = al(i, jp1) rll = rl(i, j) rvl = rv(i, j)+rmgas(i,j) feml = fem(i, j) flml = flm(i, j) fgml = fgm(i, j) femml = fem(i, jm1) flmml = flm(i, jm1) fgmml = fgm(i, jm1) isp = iaspec(i, j) isijp = isij(i, jp1) ispm = iaspec(i, jm1)

. . Existing logic... .

! JWL - 08/15/08 - Start of insertion of new CCFL Model

no_secc = .false.

! JWL - 08/15/08 - The following section of code is used to determine if the ! cell is located at a lower elevation in a channel where a ! CCFL BC is applied. If so, and if counter-current flow ! conditions exist in the cell, then the vapor flow rate required to ! impede liquid downflow is calculated based on the continuous liquid ! flow rate and the parameters associated with the CCFL correlation ! that is applied in that same channel. If the vapor flow rate is less ! than the value calculated then the entrainment rate for the cell is ! set set to zero. ! ! ccfl_bcnum(i,1) stores the BC counter value, done in setin13 ! if this value is non-zero then a CCFL BC is applied in this channel

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! ccfl_bcnum(i,2) stores the CCFL BC counter value, done in setin13 ! this is needed to access array values that correspond to CCFL BC for ! the channel of interest ! ispec(ccfl_bcnum(i,1)) should contain 7 (CCFL BC type number) ! if ccfl_bcnum(i,1) is non-zero ! ibound(2,ccfl_bcnum(i,1)) contains the axial node number where the ! CCFL BC is applied in this channel

if (ccfl_bcnum(i,1) > 0) then if ( (fgml > 0.) .and. (flmml <= 0.) .and. & & (j <= (ibound(2,ccfl_bcnum(i,1)) + 1)) ) then msp = ccfl_bcnum(i,2) ! Calculate length scale for Kutateladze correlation caplgth = sqrt(sigma/max(rll-rvl,1.e-8)) ! Calculate weighting factor to determine which CCFL correlation is applied wght = dhydj**(1.-ccfl_beta(msp))*caplgth**ccfl_beta(msp) ! Property groups denom = acontl*sqrt(gc*wght*max(rll-rvl,1.e-8)) vapprop = 1./(sqrt(rvl)*denom) liqprop = 1./(sqrt(rll)*denom) ! Vapor flow rate where liquid downflow should be impeded flood = (max(0.,ccfl_incpt(msp)-ccfl_slope(msp)* & & sqrt(abs(flmml)*liqprop)))**2/vapprop if (fgml < flood) no_secc = .true. no_secc = .true. endif endif

ccfl_flg = .false.

! JWL - 08/15/08 - The following section of code contains the CCFL model. Tests are ! done to determine if a CCFL BC is applied in the cell and then tests ! are done to determine if the CCFL model should be invoked. ! ! Determine if: 1) counter-current flow exist in this cell (explicit) ! 2) if too much liquid flows downward (explicit) ! 3) if CCFL conditions existed in the previous time step ! 4) if the normal mom. eqns. should be applied (implicit) ! Note: the implicit tests are performed in post3d

! Test to determine if a CCFL BC is applied in this cell if (isp == 7) then ! BC counter variable msp = ccfl_bcnum(i,2) ! Calculate length scale for Kutateladze correlation caplgth = sqrt(sigma/max(rlp-rvp,1.e-8)) ! Calculate weighting factor to determine which CCFL correlation is applied wght = dhydj**(1.-ccfl_beta(msp)) * caplgth**ccfl_beta(msp) ! Property groups denom = appp*sqrt(gc*wght*max(rlp-rvp,1.e-8)) chiv(msp) = 1./(sqrt(rvp)*denom) chil(msp) = 1./(sqrt(rlp)*denom) ! Maximum allowable liquid downflow old_flcrit = ccfl_flcrit(msp) if (fgml >= 0.) then ccfl_flcrit(msp) = -(1./(ccfl_slope(msp)**2*chil(msp)))* & & (max(0.,ccfl_incpt(msp)-sqrt(chiv(msp)*fgml)))**2 else ccfl_flcrit(msp) = 0. endif ccfl_flcrit(msp) = sm_fact*old_flcrit + & & (1.-sm_fact)*ccfl_flcrit(msp)

! liquid flow corresponding to zero vapor flow condition ccfl_fldpt(msp) = -(ccfl_incpt(msp)/ccfl_slope(msp))**2/ & & chil(msp) ! vapor flow corresponding to zero liquid flow condition ccfl_FRpt(msp) = (ccfl_incpt(msp))**2/chiv(msp)

! Test flow conditions ! Note: zero liquid flow condition is included in test on liquid flow rate ! since this flow rate is set to zero during the previous time step ! if flow reversal has occured. Also, a less than or equal used when ! comparing the actual and critical liquid flow rates to again consider ! flow reversal situations where flm and fl_crit are both zero and the

363

! CCFL model should still be invoked to prevent liquid downflow cond1 = (fgml > 0.) .and. (flml <= 0.) cond2 = (flml <= (ccfl_flcrit(msp)+0.005)) cond3 = (isijn(i,j)==7).and.(flml<0.) cond4 = .not. usenorm(msp)

! Test to determine if CCFL model should be invoked if (cond1.and.(cond2.or.cond3).and.cond4) then ! Set identifiers ccfl_flg = .true. no_secc = .true. ! If CCFL model is invoked it is assumed annular flow exists goto 50 endif endif

! JWL - 08/15/08 - End of insertion of CCFL Model

. . Existing logic... .

50 continue

!!!! annular flow logic

!!!! JWL - 07/18/07 - Never Reset Relative Velocity to Appropriate Value after Large !!!! Bubble calculation, which may use bubble rise velocity urvl = urvlan

! ft = aliql*dhydj*0.25 ! JWL - 07/02/08 - replaced thin film approx with exact expression aliqlf = aliql ft = 0.5*dhydj*(1.-sqrt(1.-aliqlf))

!!!! film interfacial area alve = alp + aep aintfc = 4.0*appp*sqrt(alve)*dx/dhyd(i, j)

use_logCT = .true.

! Use model set developed by Lane (2009) CALL ANENTR(i,j,delt,pll,rlp,rvp,visl,visv,sigma,dx,dhydj, & gpth,appp,chan_type(i),ft,uvj(i),alp,ulj(i),aliqp, & uej(i),aep,xkwltp,sde,xkic,fi,se,dsaut,isijl,liqbrg) seun = se xkiun = xkic

! JWL - 07/17/08 - Added Liquid Bridge Breakup Model to calculate entrainment rate in the ! churn-turbulent regime se_brig = 0. if ((alp<=ala_brig).and.use_liqbreak.and.(dhydj<.083)) then CALL LBBEntr(rlp,rvp,sigma,dhydj,uvjl,alp,aep,appp,dx, & gpth,se_brig,dsaut_brig) fa = max(0.,min(1.,(alp-alsa)/(ala_brig-alsa))) if (use_logCT) then se=exp(log(max(se_brig,1e-8))*(1.-fa)+log(max(se,1e-8))*fa) dsaut = exp(log(max(dsaut_brig,1e-8))*(1.-fa) + & log(max(dsaut,1e-8))*fa) else se = se_brig*(1.-fa) + se*fa dsaut = dsaut_brig*(1.-fa) + dsaut*fa endif sect = se endif seann = se

! JWL - 04/13/09 - Churn-Turbulent Logic fa = 1. if ((alp <= ala_brig)) then isijl = 3

364

fa = max(0.,min(1.,(alp - alsa)/(ala_brig - alsa))) fa = (3.-2.*fa)*fa**2

if (ccfl_flg) then fa = 1. xkilb = 0. endif

fam = 1.0 - fa ! JWL - 07/23/08 - logrithmic smoothing of xkic in the churn-turbulent regime if (use_logCT) then xkic = exp(fa*log(max(xkic,1.e-8))+ & fam*log(max(xkilb,1.e-8))) else xkic = fa*xkic + fam*xkilb endif xkict = xkic endif

! Set CCFL Flow regime identifier if (ccfl_flg) isijl = 7

!!!! film entrainment source

!!!! ... find maximum liquid flow in cell

wlinjt = 0.0 if (isp == 4) then minje = minje + 1 wlinjt = wlinj(minje) if (wlinjt < 0.0) wlinjt = 0.0 endif if (isp == 5) then mske = msk + 1 wlinjt = wlsink(mske)*aliqs(mske)*rlsink(mske)/(aliql*rll) if (wlinjt < 0.0) wlinjt = 0.0 endif

! JWL added to make sure more liquid is not extracted than exists if (ispm == 7) ccfl_avail(ccfl_bcnum(i,2)) = wlinjt

if (flml < 0.0) wlinjt = wlinjt - flml*aliq(i, jp1)/aliqp

!!!! ... entrainment for countercurrent flow !!!! based on critical film thickness

flmse = wlinjt + wlmse aliqlm = 0.5*(aliql + aliq(i, jm1)) jstar = jm1 if (flmml < 0.0) jstar = j flmse = max(flmse, abs(flmml)*aliq(i, jstar)/aliqlm) uvrel = max(vectur, (uvjm(i) - uljm(i))*(1.0 - aliqp)/(1.0 - & 2.5*aliqp)) secc = (aliql - 2.0*sigma*gc/(dhydj*rvp*uvrel**2))*flmse/aliql secc = max(0.0, secc)

! JWL CCFL MODEL if (no_secc) secc = 0. if ((flmml<=0.) .and.(fgml>0.)) se=secc

! JWL - 04/13/09 - Entrainment Rate for node above CCFL BC if (ispm == 7) then sepool = 0. sexflm = 0. seccfl = 0. dccfl = 1.e-8 msp = ccfl_bcnum(i,2) if ( (isij(i,jm1) == 7).or.(fgmml > ccfl_FRpt(msp)) ) then scaling = min(1.,acontl/amom(i,jm1))

! Ishii Pool Entrainment Model (occurs if area change) sepool = 4.84e-03*abs(uvjm(i))*all*(rll-rvl)*amom(i,jm1) sepool = max(0.,sepool)

365

sepool = sepool*(1.-scaling)

! Excess Film Model (occurs if no area change) uvrel = (uvjm(i) - uljm(i))**2 sexflm = aliql - 2.0*sigma*gc/(dhydj*rvl*uvrel) sexflm = max(0.,sexflm*flmse/aliql) sexflm = sexflm*scaling

! Add two area weighted components seccfl = sexflm + sepool

! Apply liquid flow rate based ramp (relative to flooding pt) if (flmml < 0.) then seccfl=seccfl*max(0.,min(1.,1.-sqrt(flmml/ccfl_fldpt(msp)))) endif

if (fgmml > 0.) then seccfl = seccfl*max(0.,min(1.,sqrt(fgmml/ccfl_FRpt(msp)))) endif

! Size of Entraining droplets dccfl = 0.06147*6.5*sigma*32.2/(rvp*uvjm(i)**2) dccfl = min(dccfl,dhydj,0.13,gpth) dccfl = max(dccfl,1.e-8) endif

! Apply vapor flow rate based ramp (relative to flow reversal pt) facc=max(0.,min(1.,(fgmml-2.*ccfl_FRpt(msp))/ & & (3.*ccfl_FRpt(msp))))

! Calculate entrainment rate for node above ccfl se = seccfl*(1.-facc) + seann*facc dsaut = dccfl*(1.-facc) + dsaut*facc

endif

. . Existing logic... .

366

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! within xschem.f logic was added to to calculate and solve the set of CCFL momentum equations, instead of the standard set of momentum equations, when it is determined that the CCFL model should be activated. This logic was located immdediately following solution of normal momentum equations

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

. . Existing logic... .

! JWL - 08/15/08 - Start of insertion of CCFL Model

isp = iaspec(i,j)

! Check if CCFL BC is applied in the cell being considered if (isp == 7) then

! Update counter for CCFL BCs mspx = ccfl_bcnum(i,2) ! if (isij(i,j) == 7) then

! Store several quantities that were calculated for the normal ! liquid momentum equation so that an effective vap-liq ! interfacial drag coeff for CCFL conditions can be ! back-calculated later in post3d subroutine ccfl_stor(mspx,1) = atmp(1)/delt ccfl_stor(mspx,2) = btmp(1)/delt ccfl_stor(mspx,3) = xkwlx(i) + 1./delt ccfl_stor(mspx,4) = alrl ccfl_stor(mspx,5) = avrv

! Begin calculating momentum equations for CCFL Model c c ...total momentum equation (replaces vapor field momentum eqn.) c ! explicit contribution ! neg. sign inserted implicitly by solve_vel routine atmp(2) = atmp(1) + atmp(2) + atmp(3) ! pressure gradient contribution ! neg. sign inserted implicitly by solve_vel routine btmp(2) = btmp(1) + btmp(2) + btmp(3) ! cont. liquid field contribution ctmp(2) = ctmp(1) + ctmp(2) - 1. ! vapor field contribution dtmp(2) = dtmp(1) + dtmp(2) + dtmp(3) - 1. ! entrained liquid field contribution etmp(2) = etmp(2) + etmp(3) - 1. c c ...CCFL correlation (replaces continuous liquid field momentum eqn.) c ! explicit contribution ! neg. sign inserted implicitly by solve_vel routine atmp(1) = ccfl_incpt(mspx)*(ccfl_incpt(mspx) - & & sqrt(chiv(mspx)*fgm(i,j))) ! pressure gradient contribution ! no pressure dependence btmp(1) = 0. ! cont. liquid field contribution ctmp(1) = (ccfl_slope(mspx))**2*chil(mspx) ! vapor field contribution dtmp(1) = chiv(mspx)*(1.-ccfl_incpt(mspx) & & /sqrt(chiv(mspx)*fgm(i,j))) ! entrained liquid field contribution ! not considerd by correlation, assumes only liquid that ! can penetrate downwards is in the form of a liquid film etmp(1) = 0

! If the vapor flow rate is greater than or equal to ! this value then the liquid flow rate is set to zero if ( (fgm(i,j) >= ccfl_FRpt(mspx)) .or. & & (.not. usenorm(mspx) .and. &

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& failedhere(mspx))) then atmp(1) = 0. ctmp(1) = 1. dtmp(1) = 0. failedhere(mspx) = .false. endif c c ...entrained liquid field momentum eqn. c ! Most terms unchanged ctmp(3) = 0. ! entrained liquid field contribution ! minus 1 not included implicitly in solve_vel routine etmp(3) = etmp(3) - 1.

! Solution of CCFL Momentum Equations ctmp(1) = ctmp(1) + 1. dtmp(2) = dtmp(2) + 1. etmp(3) = etmp(3) + 1. call solve_vel(atmp,btmp,ctmp,dtmp,etmp,deltap, & & mdot,dmdp)

endif endif

! JWL - 08/15/08 - End of insertion of CCFL Model

. . Existing logic... .

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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! within post3d.f logic was added to use the resulting values for the current time step and back-calculate an effective interfacial drag when the CCFL model is invoked as well as determine if: a) velocity flip-flop occured, b) the code predicted interfacial drag exceeds the back-calculated CCFL value such that the CCFL model should be exited (for increasing vapor flow conditions), or c) the calculated liquid downflow exceeds the available liquid flow such that the CCFL model should be exited (for decreasing vapor flow conditions). All of these calculations are performed on an implicit basis.

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

. . Existing logic... .

! JWL - 08/15/08 - added to support new CCFL Model

jm2 = jm1 - 1 if (j>2) then

if (iaspec(i, jm2) == 7) then

msp = ccfl_bcnum(i,2)

! Must go up two levels so the flow rate above ccfl node is known ! put limit that downflow through CCFL node cannot exceed this value wlinjt = ccfl_avail(msp) aliqp = 0.5*(aliq(i,j)+aliq(i,jm1)) if (flm(i,jm1) < 0.) wlinjt = wlinjt - & & flm(i,jm1)*aliq(i,j)/aliqp

if (usenorm(msp)) then if ( flm(i,jm2) < 0. ) then if (failedhere(msp)) then isij(i,jm2) = 8 else usenorm(msp) = .false. ccfl_fail = .true. if ((msp==what).and. ccfl_print) & & write(88,*) timet,'need to use ccfl not norm' endif endif endif

if (isij(i, jm2) == 7) then

ccfl_xk = (ccfl_stor(msp,1) + ccfl_stor(msp,2)* & & (p(i,jm2) - p(i,jm1))/ri144-ccfl_stor(msp,3)* & & flm(i,jm2) ) / & & (flm(i,jm2)/ccfl_stor(msp,4) - fgm(i,jm2)/ & & ccfl_stor(msp,5))

if ((flm(i,jm2) < 0.) .and. & & (abs(flm(i,jm2)) > wlinjt)) then usenorm(msp) = .true. failedhere(msp) = .true. ccfl_fail = .true. if ((msp==what).and. ccfl_print) & & write(88,*) timet,'exceeds available'

elseif ( (flm(i,jm2) > 0.) .and. & & (1.1*ccfl_xk > xk(i,jm2)) ) then usenorm(msp) = .false. failedhere(msp) = .true. ccfl_fail = .true. if ((msp==what).and. ccfl_print) & & write(88,*) timet,'velocity flip flop'

elseif ((flm(i,jm2) >= 0.) .and. & & (1.1*ccfl_xk < xk(i,jm2))) then usenorm(msp) = .true. failedhere(msp) = .false. ccfl_fail = .true.

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if ((msp==what).and. ccfl_print) & & write(88,*) timet,'code drag exceeds ccfl'

else xk(i,jm2) = ccfl_xk if ((msp==what).and. ccfl_print) & & write(88,*) timet,'ccfl solution used'

endif

endif endif

! JWL - 08/15/08 - end insertion to support new CCFL Model

. . Existing logic... .

370

VITA Jeffrey W. Lane

Jeffrey William Lane was born on August 11, 1981 in Harrisburg, Pennsylvania, to George and Diane Lane. He graduated from Mechanicsburg Area Senior High School in 2000. He then attended the Pennsylvania State University, where he graduated with Bachelor of Science degrees in both Mechanical Engineering and Nuclear Engineering in December 2004. As an undergraduate student Jeffrey was a member of the Schreyer’s Honors College and for his honors thesis research he performed experiments to characterize the flame structure and determine the lean blowout (LBO) limits of a swirl-stabilized, dump combustor operating on natural gas. This research was conducted under the supervision of Dr. Domenic Santavicca and the sponsorship of the United States Department of Energy University Turbine System Research Program.

Following graduation, Jeffrey continued his academic career as a graduate student at the Pennsylvania State University studying nuclear engineering. During this time he worked as a research assistant in the Thermal-Hydraulic Analysis Group for Dr. L. E. Hochreiter and served as a teaching assistant for a senior-level reactor thermal-hydraulics course. He graduated with a Masters of Science degree in Nuclear Engineering in May 2006, where his thesis was titled: “Performance Assessment of the Two-Phase Pump Head Degradation Model in the RELAP5-3D Transient Safety Analysis Code ”. Jeffrey then pursued a doctoral degree, also at Penn State in nuclear engineering, studying with Dr. Hochreiter under appointment to the Rickover Graduate Fellowship Program sponsored by Naval Reactors Division of the U.S. Department of Energy.

His research interests include thermal-hydraulics and two-phase flow as applied to transient safety analysis codes. In particular he is interested in annular flow, dispersed flow film boiling, and counter-current flow limitation (CCFL) phenomena. His work has been presented at the American Nuclear Society (ANS), International RELAP Users Group (IRUG), and NURETH-12 conferences and he has published in the Nuclear Engineering and Design Journal. Jeffrey is a member of the American Nuclear Society (ANS) and enjoys playing basketball and golfing.