University of Nevada, Reno a Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philo

University of Nevada, Reno

Computational Investigation of Cavitation Performance and Heat Transfer in Cryogenic Centrifugal Pumps with Helical Inducers

A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering

Enver Sinan Karakas

Dr. Cahit A. Evrensel/Dissertation Advisor

May, 2019

THE GRADUATE SCHOOL

We recommend that the dissertation prepared under our supervision by

Entitled

be accepted in partial fulfillment of the requirements for the degree of

, Advisor

, Committee Member

, Comm ittee Member

, Committee Member

, Graduate School Representative

David W. Zeh, Ph.D., Dean, Graduate School

i. ABSTRACT

In this dissertation, heat transfer and cooling mechanism of a cryogenic submerged pump motor, and cavitation behavior of a cryogenic pump are investigated. Cryogenic pumps for Liquefied Natural Gas (LNG) applications are unique in design with respect other industrial pump applications. The two distinct features of the cryogenic pumps are:

(1) pump motor is submerged to process liquid for improved cooling, and (2) a helical style inducer is utilized at the suction side of the pump to enhance the cavitation performance.

The first part of this dissertation investigates the cooling of the submerged induction motor considering normal operating conditions of a typical cryogenic pump to understand the effect of the motor rotor geometry. In this effort, the flow structure across the motor annulus, which is commonly known as “motor air gap”, is investigated in the presence of realistic rotor geometries commonly found in practical applications. The purpose of this study is to observe and understand the fluid dynamics of Taylor-Couette-Poiseuille (TCP) flow and its significance to the cooling performance of the submerged motor, considering various configurations of the rotor surface. Computational Fluid Dynamics (CFD) simulations are conducted to determine the flow characteristic across the “motor air gap” and temperature distribution at the stator section of the motor. CFD simulations results are compared to temperature and flow calculations according to previously published TCP flow equations and dimensionless correlations. The entrance region is explored for each rotor geometry and its impact to overall cooling performance is discussed. It is concluded that while the rotor surface has modest effect to heat transfer, TCP flow characteristics differ for each geometric configuration. In addition, TCP flow and heat transfer

correlations can predict the temperature profile, if the similarity between dimensionless parameters are obtained.

In the second part of this dissertation, the cavitation performance of the cryogenic pump in terms of Net Positive Suction Head (NPSH) is investigated considering two different style helical inducers. The inducers utilized in cryogenic industrial pumps have open shroud design with a constant tip diameter. Due to the open shroud design, inducers tip clearance has an important role in cavitation performance due to possible back leakage through the gap. The effect of the tip clearance is investigated to understand the flow structure at the inducer tip location and its impact to cavitation performance. A constant pitch inducer and a variable pitch inducer are considered in the study of the tip clearance effect. CFD simulations are performed for various tip clearances. CFD simulation results are compared to actual test data to validate the simulation predictions. The objective of this research is not only to determine the tip clearance effect to the cavitation performance, but also to understand how the tip clearance can change the cavitation performance of different style inducers. A transport-equation based cavitation model based on Rayleigh-Plesset’s bubble dynamics is implemented in the CFD code to explore the bubble dynamics and cavitation at the inducer and impeller eye sections. It is concluded that tip clearance plays an important role in cavitation performance and sensitivity to tip clearance in terms of cavitation performance is different for each style inducer.

The third and last part of this dissertation is concerned with the review of four commonly used transport-equation based cavitation models. Comparison of these models and assessment of their applicability to predict cavitation performance of a cryogenic

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industrial pump with a variable pitch helical inducer are discussed in detail. To this aim, experimental results are compared to simulation findings to validate accuracy and assumptions of each model. Lastly, modifications to original empirical constants are proposed to obtain agreement with the actual cavitation and bubble dynamics observed during experimental testing. Two new sets of empirical constants are proposed to improve the prediction of cavitation in cryogenic pumps. Results show that with the new empirical constants, cavitation performance of a cryogenic pump with a helical inducer can be determined. It is shown that including the concentration of undissolved non-condensable gases and the turbulent pressure fluctuations in the models had no impact to the cavitation performance for this particular case.

ii. ACKNOWLEDGMENTS

I would like to thank my family members for their great support. I feel grateful to have such an understanding and supportive family. Especially, my wife, Beril Karakas, has been very caring and kind from the beginning to the end of this journey. My parents and my only sister have given me strength to pursue my degree.

I would also like to thank my advisor and committee members. Especially, Dr.

Aureli has been extraordinarily supportive, and assisted in every matter and helped me in my advancement. I would have not succeeded and finished my degree without the support and assistance of my advisor and committee members.

I want to thank all of my friends and colleagues at Turbomachinery Laboratory, who have helped me during this study. I would especially like to thank Dr. Tokgoz and

Burak Gulsacan for their contributions and sharing their ideas in this project.

Finally, I would like to thank Elliott Group – Cryodynamics Products and Ebara

Corporation, Japan for funding this research.

Table of Contents i. ABSTRACT ...... i ii. ACKNOWLEDGMENTS ...... iv List of Tables ...... vii List of Figures ...... viii CHAPTER 1. INTRODUCTION: ...... 1 CHAPTER 2. FLOW AND HEAT TRANSFER THROUGH THE AIR GAP OF A SUBMERGED MOTOR PUMP: EFFECT OF GROOVES AT THE ROTOR SURFACE .. 9 Abstract ...... 9 1. Introduction ...... 10 2. Problem Statement ...... 18 2.1 Induction Motor Configurations and Cooling Mechanism ...... 18 2.2 Formulation of the Cooling Mechanism ...... 19 2.3 Flow Structure and Convective Heat Transfer: ...... 22 2.4 Correlation and Formulation of Heat Transfer Coefficient: ...... 25 3. CFD Analyses and Results ...... 29 4. Conclusions ...... 43 5. Acknowledgment ...... 45 References ...... 45 CHAPTER 3. CAVITATION PERFORMANCE OF CONSTANT AND VARIABLE PITCH HELICAL INDUCERS FOR CENTRIFUGAL PUMPS: EFFECT OF INDUCER TIP CLEARANCE...... 51 Abstract ...... 51 1. Introduction ...... 52 2. Pump and Inducer Specifications ...... 58 3. Numerical Simulation ...... 60 3.1 Computational Fluid Dynamics (CFD) ...... 61 3.2 Cavitation Parameters ...... 62 3.3 Cavitation Model ...... 63 3.4 Geometric Model and Grid ...... 65 4. Experimental Setup ...... 67 5. Results ...... 70 6. Conclusions ...... 88

7. Acknowledgment ...... 90 References ...... 91 CHAPTER 4. COMPARISON AND APPLICATION OF TRANSPORT-EQUATION BASED CAVITATION MODELS TO INDUSTRIAL PUMPS WITH INDUCERS ...... 95 Abstract ...... 95 1. Introduction ...... 96 2. Bubble Dynamics and Cavitation Models ...... 102 3. Subject Pump: Description of Assembly and Experimental Setup ...... 108 3.1 Pump Assembly ...... 108 3.2 Experimental Setup ...... 110 4. Numerical Simulation ...... 111 4.1 Computational Fluid Dynamics (CFD) Implementation ...... 111 4.2 Pump Cavitation Performance Parameters...... 113 4.3 Geometric Model and Grid ...... 114 5. Results ...... 116 6. Conclusions ...... 128 7. Acknowledgment ...... 130 References ...... 130 CHAPTER 5. CONCLUSIONS AND FUTURE WORK ...... 136 References ...... 143

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List of Tables CHAPTER 2. FLOW AND HEAT TRANSFER THROUGH THE AIR GAP OF A SUBMERGED MOTOR PUMP: EFFECT OF GROOVES AT THE ROTOR SURFACE Table-1 Fluid composition in mole fraction...... 19 Table-2 Inlet Conditions with corresponding flow rate for a given operating cases [34]...... 21 Table-3 Cargo Pump Operating Details [34]...... 21 Table-4 Correlations for Nusselt number as a function of dimensionless parameters – smooth annulus...... 27 Table-5 Correlations for Nusselt number as a function of dimensionless parameters – slotted rotor or stator (axial grooves)...... 27

Table-6 Stator Surface Temperature at exit of the annulus, Ts...... 34 Table-7 Calculation results of discretization error according to [37], please refer to [37] for description of each parameter used to determine the error involved with the grid size...... 35 CHAPTER 3. CAVITATION PERFORMANCE OF CONSTANT AND VARIABLE PITCH HELICAL INDUCERS FOR CENTRIFUGAL PUMPS: EFFECT OF INDUCER TIP CLEARANCE Table-1 Pump Performance Data and Operational Specifications (Full Scale)...... 58 Table-2 Geometric parameters and specification of the inducers (Scaled).Geometric parameters and specification of the inducers (Scaled)...... 60 Table-3 Calculation results of discretization error according to [30]...... 62 Table-4 Cavitation model constant parameter values [21]...... 65 CHAPTER 4. COMPARISON AND APPLICATION OF TRANSPORT-EQUATION BASED CAVITATION MODELS TO INDUSTRIAL PUMPS WITH INDUCERS Table-1 Pump Performance Data and Specifications (Full Scale)...... 109 Table-2 Geometric parameters and specification of the inducers (Scaled)...... 110 Table-3 Calculation results of discretization error according to [34]. Refer to [34] for description and calculation of each parameter...... 113

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List of Figures

CHAPTER 2. FLOW AND HEAT TRANSFER THROUGH THE AIR GAP OF A SUBMERGED MOTOR PUMP: EFFECT OF GROOVES AT THE ROTOR SURFACE Figure-1 Rotor surface configurations commonly utilized in cryogenic pump motors. (a) Smooth, (b) axial (non-skewed) grooves, (c) skewed grooves. Courtesy of Elliott Cryodynamics. .... 17 Figure-2 Rotor surface configurations in analyses (a) Smooth, (b) axial (non-skewed) grooves, (c) skewed grooves 5 degrees (ccw), (d) skewed grooves 5 degrees reversed (cw), (e) skewed grooves 10 degrees (ccw), (f) skewed grooves 10 degrees reversed (ccw)...... 17 Figure-3 Coolant flow through the motor, Elliott Cryodynamics’ shipboard cargo pump...... 18 Figure-4 Simplified model for Taylor-Couette-Poiseuille flow with constant heat flux at stator. 20 Figure-5 Mesh quality shown at (a) inlet region, (b) Isometric view of the mesh at annulus, applicable to geometries with and without grooves, and isometric view of the groove region...... 31 Figure-6 Convergence Plot: residual RMS values for fluid velocity, pressure and heat transfer are plotted, along with average temperature at stator surface, against time step...... 34

Figure-7 Stator Surface Temperature (Ts) as a function of axial location for smooth rotor...... 36

Figure-8 Stator Surface Temperature (Ts) as a function of axial location for axial (non-skewed) grooves...... 37

Figure-9 Typical Stator Heat Transfer Coefficient and Stator Surface Temperature (Ts) as a function of axial location. 5° CW Skewed Grooves case is shown...... 37 Figure-10 Temperature comparison between CFD simulations – averaged data...... 39 Figure-11 Typical Temperature distribution at stator surface, 5° CW Skewed Grooves case is shown...... 39 Figure-12 Typical flow structure with Taylor vortices and entrance region, 5° CW Skewed Grooves case is shown...... 40 Figure-13 Taylor vortices along XY Plane. (a) Smooth case no axial flow, (b) smooth case with axial flow, (c) axial (non-skewed) grooves with axial flow, (d) skewed 5° CCW grooves with axial flow, (e) skewed 10° CCW grooves with axial flow, (f) skewed 5° CW grooves, (g) skewed 10° CW grooves...... 41 Figure-14 Entrance length vs skew angle...... 42

CHAPTER 3. CAVITATION PERFORMANCE OF CONSTANT AND VARIABLE PITCH HELICAL INDUCERS FOR CENTRIFUGAL PUMPS: EFFECT OF INDUCER TIP CLEARANCE Figure-1 Typical configuration of a cryogenic LNG in-tank pump by Elliott Group Cryogenic Products [1]...... 53 Figure-2 Helical style axial inducer geometries, (a) variable pitch, (b) constant pitch inducer. ... 59 Figure-3 Pump assembly consists of inducer, impeller and the diffuser vane used in simulations...... 65 Figure-4 Mesh quality of the pump hydraulic components...... 66 Figure-5 Mesh quality at the inducer tip clearance, Mesh quality at the inducer tip clearance, (a) =0.175mm (b) =0.35mm, (c) =0.70mm, (d) =1.05mm, (e) =1.40mm...... 67

Figure𝜹𝜹 -6 Schematic view𝜹𝜹 of the test facility.𝜹𝜹 ...... 𝜹𝜹 ...... 𝜹𝜹 ...... 68 Figure-7 Pump test section of the test facility...... 69 Figure-8 Pump performance curve...... 70 Figure-9 The total pressure distribution in meridional view for variable pitch inducer.Typical pressure distribution is shown for complete pump assembly...... 73 Figure-10 Axial velocity and backflow leakage of variable pitch inducer...... 74 Figure-11 Axial velocity and backflow leakage of constant pitch inducer ...... 75 Figure-12 The total pressure distribution in meridional view for constant pitch inducer for δ = 0.35 mm. Typical pressure distribution is shown for complete pump assembly...... 73 Figure-13 Pressure distribution on the blade surfaces for variable pitch inducer...... 75 Figure-14 Pressure distribution on the blade surfaces for constant pitch inducer...... 76 Figure-15 Head drop curve, cavitation number vs. head coefficient for variable pitch inducer. .. 77 Figure-16a Vapor volume fraction at the pump assembly for variable pitch inducer at inlet pressure “a”...... 79 Figure-16b Vapor volume fraction at the pump assembly for variable pitch inducer at inlet pressure “b”...... 79 Figure-16c Vapor volume fraction at the pump assembly for variable pitch inducer at inlet pressure “c”...... 80 Figure-16d Vapor volume fraction at the pump assembly for variable pitch inducer at inlet pressure “d”...... 81 Figure-17 Head drop curve, cavitation number vs. head coefficient for constant pitch inducer. . 83

Figure-18a Vapor volume fraction at the pump assembly for constant pitch inducer at inlet pressure “a”...... 84 Figure-18b Vapor volume fraction at the pump assembly for constant pitch inducer at inlet pressure “b”...... 84 Figure-18c Vapor volume fraction at the pump assembly for constant pitch inducer at inlet pressure “c”...... 85 Figure-18d Vapor volume fraction at the pump assembly for constant pitch inducer at inlet pressure “d”...... 85 Figure-19 Head Drop Curve for variable pitch inducer, δ = 0.35 mm...... 86 Figure-20 Cavitation and vapor formation for variable pitch inducer, (a) test versus corresponding (b) CFD simulation results are shown, δ = 0.35 mm...... 87 Figure-21 Relationship between the cavitation performance and the tip clearance...... 88

CHAPTER 4. COMPARISON AND APPLICATION OF TRANSPORT-EQUATION BASED CAVITATION MODELS TO INDUSTRIAL PUMPS WITH INDUCERS Figure-1 Typical LNG Storage tanks and illustration of Minimum Liquid Height with In-tank Cryogenic Pump...... 97 Figure-2 Pump assembly used in the simulations...... 109 Figure-3 Grid structure of the pump hydraulic components shown in exploded view for clarity...... 111 Figure-4 Experimental test facility...... 115 Figure-5 Pump cavitation performance curve based on head drop...... 118 Figure-6 Vapor Volume Fraction (a) and vapor volume (b) for each cavitation model, Pin=10000 Pa...... 120 Figure-7 Vapor Volume Fraction (a) and vapor volume (b) for each cavitation model, Pin=9000 Pa...... 121 Figure-8 Vapor Volume Fraction (a) and vapor volume (b) for each cavitation model, Pin=8000 Pa...... 123 Figure-9 Vapor Volume Fraction (a) and vapor volume (b) for each cavitation model, Pin=7000 Pa...... 124 Figure-10 Vapor Volume Fraction at Inducer Blade, Front View...... 126 Figure-11 Vapor Volume Fraction at leading edge of impeller blades, Front View...... 127 Figure-12 Vapor Volume at Pin=8000 Pa, (a) test, (b) CFD simulations...... 128

CHAPTER 1. INTRODUCTION:

Cryogenic industrial pumps are commonly used in Liquefied Natural Gas (LNG) for transfer (send out) and storage applications. The in-tank style cryogenic pumps are vertically suspended. They are installed inside a storage tank via retraction system to locate the pump into close proximity of the tank bottom [1]. The storage tanks are often in excess of 30 meters in height, making it impossible to have an external pump motor [2]. In order to overcome this problem a submerged motor design is utilized at the in-tank cryogenic pumps. Another advantage of employing a submerged motor design is due to hazardous area requirements. LNG (methane, propane, ethane and butane) is highly flammable and explosive. Therefore each electrical equipment or device must be certified according to the applicable fire and hazardous area codes [3]. With the submerged motor design, due to lack of oxygen in the liquid, no hazardous area certification is necessary for the electrical induction motors [3].

Another distinct design feature of a cryogenic pump is the implementation of the inducer at the downstream of the first stage impeller. This hydraulic component, assists the first stage impeller by increasing the suction pressure to prevent or delay early cavitation.

The need for an inducer is mainly due to the uniqueness of the in-tank pump applications.

Storage tanks for LNG applications cannot withstand internal pressure due to their construction [2]. The suction pressure to the pump is governed by the liquid level at the storage tank. A typical configuration of an in-tank pump is shown in Figure-1, Chapter 3.

Inducer design plays a very important role in reducing the minimum liquid level requirement at the tank at which the pump can be safely operated without cavitation.

In this dissertation, focus is given to these unique design features of the cryogenic pumps to better understand and present the theory behind inducer technology in terms of cavitation performance and, cooling mechanism and temperature profile of submerged motor pump with various rotor geometries.

Submerged induction motor of a cryogenic pump consists of a rotor (inner rotational cylinder) and a stator (outer stationary cylinder) which are concentric to each other. The fluid flow between two concentric cylinders without axial flow are well documented [4-11]. However, applications with axial flow component, such as in induction motors and applications with similar geometries, thermal and fluid flow characteristic are far less understood. The configuration of the cooling at induction motor is very similar to

Taylor-Couette-Poiseuille (TCP) flow, which describes the flow between rotating concentric cylinders in the presence of Taylor Vortices with axial flow. Therefore, the TCP flow model is adopted to study the cooling in induction motor. Most of the TCP flow studies are experimental that rely on empirical formulations for specific applications [8,

12-14]. However, a complete theory encompassing the generality of all cases seems lacking. In this dissertation, temperature profile at the stator and fluid flow along the

“motor air gap” are studied for six different configurations considering three different style rotors commonly used in cryogenic motors. The rotor geometry with smooth surface, longitudinal straight slots (non-skewed), and skewed slots with various skewing angles are considered. While the skewing is applied to improve electrical and magnetic properties of the induction motor, to best of the author’s knowledge the effect of the skewing to the heat transfer and cooling have not been investigated before. In order to fill this knowledge gap, fluid flow and heat transfer simulations are performed via CFD (Computational Fluid

Dynamics) to understand the effect of the rotor surface. In addition, hydrodynamic and thermal entrance region of motor annulus for each geometric case are investigated to determine the impact to the fluid flow and the cooling mechanism. Lastly, simulation results are compared to previously published TCP flow and heat transfer correlations defined in terms of dimensionless parameters. It is concluded that different geometries of the rotor has minimal impact to stator temperature profile. Therefore, previously published correlations of flow and heat transfer relations for non-skewed case are applicable to skewed rotor configurations, provided that relevant dimensionless parameters are matched.

In addition, stator temperature profile along the axial length can be predicted without necessity of complex CFD simulations by utilizing the most appropriate correlation under the condition of comparable values of dimensionless parameters and similar heating configuration. On the other hand, this study concludes that particular geometric configuration has a major effect on Taylor vortices and entrance region. This study contributes to understanding the effect of fluid structure and Taylor vortices to assist in the design of more stable and safe induction motors and generators.

The next two research subjects involve cavitation behavior, the prediction of cavitation and suction performance in cryogenic pumps with the helical style inducer.

Inducer technology was first initiated by NASA in 1970s at rocket turbopump applications

[15-17]. Rocket turbopumps used in space shuttles operate under very low suction pressure thus prone to cavitate [15-17]. The inducer geometry at turbopumps are much more aggressive with respect to helical style inducers employed in cryogenic industrial pumps.

Helical inducers at cryogenic pumps are two bladed with constant hub and tip diameters, and can have either variable or constant pitch design. Turbopump inducers can have more

than four blades with variable hub and tip diameter with a progressive pitch. Cavitation performance of the turbopump inducers are studied considering various blade shapes, number of blades, hub geometry, solidity (blade vs. fluid volume), leading and trailing edge of inducer blades to determine the effect of each geometric parameter to the cavitating

(two-phase) and non-cavitating performance [17-25]. The main purpose of all these studies are to enhance the cavitation performance by preventing or delaying the cavitation inception. Since the all inducers have open shroud design (no closed shroud at outer diameter section), the tip clearance plays an important role in cavitation performance. The pressure increase across the inducer results in a back leakage through the gap between the inducer tip and the related bore diameter. This back leakage creates a vortex at the inducer tip due to being in the opposite direction of the main meridional flow component. The back leakage vortex often causes early cavitation in inducer section and can be detrimental to the overall performance of the pump [26, 27]. Effect of the tip clearance at turbopump applications have studied for various turbopump inducers. Most of these studies include investigation of cavitation and bubble dynamics at turbopump inducer blades, tip clearance and hub sections. There is very limited study investigating the turbopump inducer’s cavitation performance considering the first stage impeller and vapor formation at impeller eye section [28]. The cavitation performance of helical style inducers has not been studied considering the effect of tip clearance for constant and variable pitch inducers. In order to identify and understand the cavitation behavior for various tip clearance configurations, vapor formation and bubble dynamics are investigated considering a pump assembly with inducer, impeller and a diffuser vane. The main purpose of this study is not only to determine the effect of the tip clearance to cavitating and non-cavitating performance but

also to understand how the tip clearance can change the cavitation performance of different style inducers. To this aim, CFD simulations are performed to predict the cavitation performance of the cryogenic pump in terms of Net Positive Suction Head (NPSH).

Experimental results are compared to the simulations to validate the assumptions and boundary conditions employed in the CFD code. Inducer tip clearance versus cavitation performance of the pump assembly with each style inducer are reported in this study. It is concluded that the sensitivity to the tip clearance differs between each style inducer in terms of cavitation performance. The cavitation performance of the inducer with variable

(progressive) pitch exhibit more dependency to the tip clearance with respect to the constant pitch inducer. In addition, it is observed that pump cavitation breakdown and critical cavitation point have not been reached without any formation of vapor at the impeller eye section. This finding emphasizes the importance of analyzing and studying the inducer’s cavitation performance as a pump assembly rather than an individual component.

At the last research subject, transport-based cavitation models are reviewed in details. Prediction of pump performance under cavitation conditions is challenging due to the complexity of three-dimensional fluid flow along with the variation in density.

Determination of vaporization and condensation poses difficulty due to phase change. The fundamental approach in modeling the cavitation is to determine the fluid phase (amount of vapor and liquid) as a function time and location. Different approaches are used to predict the density field. The most appropriate cavitation models used in the field of turbomachinery are the transport-equation based cavitation models [29-40]. Transport- equation based cavitation models are formulated based on source/sink terms. The sink and

source terms for vaporization and condensation are specified in terms of mass transfer rate.

This cavitation models are widely used in CFD simulations. Transport-equation based cavitation models are proposed in determination of cavitation behavior at turbomachinery and aerospace and complex fluid flow problems under two-phase flow condition

[29,30,36,39,40]. These models require determination of sink and source term constants empirically. According to the application and characteristics of the two phase flow, adjustment may need to be done to those empirical constants to improve the accuracy of the predictions for both vaporization and condensation cases. The main objective of this study is to determine the most suitable and accurate cavitation model for predicting the cavitation performance of a cryogenic pump with a helical inducer. Four commonly used cavitation models are selected for that purpose. Initially the distinctions between each model is reviewed. Then, CFD simulations are conducted by implementing each cavitation model to the CFD code, and results are compared to experimental tests to validate the assumptions and the accuracy of CFD simulations. Cavitation performance in terms of

NPSHr is calculated for both experimental tests and simulation results. Adjustments were made to the empirical constants for two of the cavitation models to obtain close approximation of vapor formation and condensation at the pump assembly based on experiments. According to the author, there has been no research study found that deals with comparison of cavitation models to determine their applicability and accuracy for helical style inducers and impellers for cryogenic pumps. In order to address this knowledge gap, CFD simulations are performed considering the complete pump assembly.

The results indicated that empirical constant needed to be modified based on vaporization and condensation for two of the models. Including volume of non-condensable gas (NCG),

and turbulent pressure fluctuations have no impact to the results, but increased the computing time considerably for this particular application. Therefore, it is recommended to not include NCG volume and effect of the turbulent pressure fluctuations to determination of cavitation performance for the cryogenic pump. It is found that the most appropriate models to determine the NPSH performance of the pump assembly with a helical inducer are ZGB (Zwart-Gerber-Belamri) [29], and SS (Sauer-Schnerr)[31] models considering the computing time, convergence level and agreement to test data based on visual comparison of vapor volume. Overall each cavitation model was within %3 of each other in terms of predicting NPSH performance. The CFD results were within 9% of the experimental tests, suggested to be acceptable considering the complexity of two phase simulations and minor difference in actual test setup and actual pump assembly model.

The rest of dissertation is organized as follows: Chapter 2 is the study of “Flow and

Heat Transfer through the Air Gap of a Submerged Motor Pump: Effect of Grooves at the

Rotor Surface”. This work is submitted to Heat and Mass Transfer Journal of Springer

Journals and is currently under review. Chapter 3 covers the study of “Cavitation

Performance of Constant and Variable Pitch Inducers for Centrifugal Pumps: Effect of

Inducer Tip Clearance”. This paper includes the details of inducer types, CFD simulations, experimental testing and the pump assembly. It is submitted to ASME’s Journal of Fluids

Engineering and at the review phase under editorial office of ASME. Chapter 4 includes the research related to “Comparison and Application of Transport-Equation Based

Cavitation Models to Industrial Pumps with Inducers”. This work is also under review by

Mathematical Problems in Engineering by Hindawi. The experimental work detailed in the studies outline in Chapters 3 and 4 have been conducted by Mr. Hiroyoshi Watanabe, Ph.

D., of Ebara Corporation, Japan. Nehir Tokgoz, Ph. D., assisted in implementation of cavitation models to CFD Code and comparison of cavitation models in the study of cavitation models discussed in Chapter 4. Finally, conclusions and future work of the dissertation involving all the related research are given in Chapter 5.

CHAPTER 2. FLOW AND HEAT TRANSFER THROUGH THE AIR GAP OF A SUBMERGED MOTOR PUMP: EFFECT OF GROOVES AT THE ROTOR SURFACE

Enver S. Karakasa,b, Matteo Aurelia, Cahit A. Evrensela aMechanical Engineering Department, University of Nevada, Reno 1664 N. Virginia Street, Reno, NV 89557-0312, USA bElliott Group, Cryodynamic Products 350 Salomon Circle, Sparks, NV 89434, USA

Abstract

In this paper, flow and convection heat transfer at the motor annulus (air gap) of a cryogenic pump with a submerged induction motor are studied. Flow structure and heat transfer for various geometries of inner cylinder (rotor), such as rotor with smooth surface, straight (non-skewed) slots, and skewed slots, are investigated. Heat transfer and cooling flow structure at the motor annulus of a cryogenic centrifugal pump are studied with emphasis on the effect of the rotor surface geometry. The purpose of this study is to better understand the fluid dynamics of Taylor-Couette-Poiseuille (TCP) flow with different rotor surface configurations and its significance on the heat transfer. To this aim, Computational

Fluid Dynamics (CFD) analyses are performed and the flow structure and temperature distribution are compared for different geometries of actual motor rotors. The entrance region is explored for each rotor configuration and its impact to heat transfer is discussed.

Finally, previously published TCP flow equations and dimensionless correlations are

compared to CFD results and literature data for each geometry of the rotor and the flow regime.

Keywords: Taylor-Couette-Poiseuille Flow, Taylor Vortices, Motor Air Gap, Cryogenic

Pump, Computational Fluid Dynamics, Entrance Region

1. Introduction

Most cryogenic centrifugal pumps for liquefied natural gas (LNG) applications have a submerged motor design to efficiently cool the motor assembly with the help of cryogenic process fluid. The main cooling mechanism is achieved by forced convection through the motor annulus at the motor cavity. Each motor and its cooling system are designed to minimize heat generation to prevent vapor formation at the motor cavity which can cause unstable operation. This cooling arrangement is applicable to all induction motors and generators used in turbomachinery applications. The configuration of the cooling is very similar to Taylor-Couette-Poiseuille (TCP) flow, which describes the flow between rotating concentric cylinders. Therefore, the TCP flow model is adopted in this paper to study the cooling in induction motors and generators.

Heat and mass transfer in the gap between concentric cylinders have been studied with or without axial flow to better understand the effect of rotating inner cylinder (rotor) and stationary outer wall (stator) on the cooling of the motors. As regards to closed concentric cylinders separated by a small gap without axial flow (Taylor-Couette flow, or

TC), the dynamics of flow is well documented and studied [1, 2, 5, 6, 12, 16, 26]. A few geometric configurations of motor annulus, such as very large gap with small radial aspect ratios ( <0.5) and rotating external cylinder case, have not been studied in great detail.

𝜂𝜂

The case of open cylinders with axial through flow (Taylor-Couette-Poiseuille, or TCP) is far less understood mostly due to entrance region of axial flow and its effect on Taylor vortices and overall flow structure. Most of the reported studies are experimental and correlations between flow and heat transfer are explored for specific geometries of the motor annulus. However, a complete theory encompassing the generality of all cases seems lacking at this time. Therefore, it is crucial to understand and determine the characteristics of TCP flow at the motor annulus and its impact on heat transfer, since insufficient cooling and resulting elevated temperatures often reduce the efficiency of motors and lifetime of the application [3, 18].

The motor annulus geometry with smooth surfaces with TC flow has been studied extensively over the years. Becker and Kaye [1] studied the flow at the motor annulus with the rotating rotor, to determine the effect of Taylor vortices and their influence on heat transfer across the annulus. They reported the correlation between Nusselt number and

Taylor number for different flow regimes, such as laminar and turbulent flow with and without presence of Taylor vortices. Molki et al. [2] investigated the laminar flow across an annulus with a rotating inner cylinder. Their study covered convective heat and mass transfer of fluids with Taylor number less than 500. Howey et al.[3], explored the convective heat transfer and air-gap flow in rotating electrical machines with concentric cylindrical and disk geometries. They investigated the convective heat transfer for both turbulent and laminar flow with and without Taylor vortices and for different air gap sizes and rotational speeds. They concluded that the heat transfer and flow structure would vary for all types of electrical machines, and it is primarily affected by the rotational speed, and the size of the motor annulus. Guo and Zang [4], focused on the effect of the natural and

forced convection in a vertical rotating cylinder. They investigated the effect of mixed convection with the rotating Reynolds number (Re ). They suggested a dimensionless

t parameter group to represent the relative importance of buoyance and inertial forces. Tzeng

[5] investigated the local heat transfer of a coaxial cylinder, with its inner cylinder rotating and the outer one stationary. The local heat transfer rate on the wall is correlated and compared to previously published correlations by other researchers. An empirical relationship between the parameters of centrifugal force, buoyancy force and heat transfer coefficient is established by Tzeng [5]. Jeng et al. [6] utilized longitudinal ribs on the inner rotating cylinder to enhance the heat transfer at an annulus with TCP flow. Three configurations of inner cylinder with smooth, with longitudinal ribs and longitudinal ribs with cavities were considered in their study. They published the change in the average

Nusselt number with respect to dimensionless pumping power (~Re3) for each configuration. They experimentally showed that mounting longitudinal ribs on the rotating inner cylinder enhances the heat transfer especially for laminar flow regime. Fenot et al.

[7] reviewed heat transfer between concentric cylinders with or without axial flow. They compared the correlations published by researchers on Taylor vortices (Taylor number) and Nusselt number for different flow regimes and different configurations for concentric cylinders, such as heated or cooled surface of rotors and stators. They compared models and correlations established in different studies for rotors with and without slots. They concluded that the TCP flow structure and its heat transfer aspect are not well understood and wide range of investigation remains open for exploration, for both smooth and slotted cylindrical gaps. Hayase et al. [8] studied the convection heat transfer of two coaxial cylinders with periodically embedded cavities. The inner rotating cylinder has an axially

grooved surface resulting in twelve circumferentially periodic cavities embedded in it.

Such geometry is common in electric motors and generators. Their study reveals shear induced recirculation flow in the cavities and its interaction with Taylor vortices in the annular space to enhance the heat transfer. Seghir-Ouali et al. [9] reported an experimental identification technique for the convective heat transfer coefficient inside a rotating cylinder with an axial flow rate. They proposed correlations between Nusselt number and axial and rotational Reynolds numbers. Tachibana and Fukui [12], considered heat transfer of the rotational and axial flow between two concentric cylinders and published correlations of heat transfer coefficient with respect to Taylor numbers for different flow velocities and regimes with and without axial flow. They implemented four different test apparatus for different cooling configurations and geometries of cylinders. They also considered a rotating cylinder with slots in the axial direction and published an empirical formulation of Nusselt number as a function of Reynolds number. K.S. Ball et al. [16] considered the buoyancy effect in a vertical annulus with a rotating inner cylinder.

Experiments were conducted with mixed convection flows within a vertical annulus to determine heat transfer rates. Flow visualization studies were performed by injecting small amount of smoke in the annulus and the flow structure was compared for different Grashof number for three different vertical annulus configurations. They concluded that for certain values of densiometric Froude number (Gr/Re2), the buoyancy force can dominate the flow, which strongly resembles the natural convection case. Murata and Iwamoto [17] numerically simulated and compared the heat and fluid flow in cylindrical annular flow passages to a conical flow passage with through flow and inner wall rotation. While keeping the axial Reynolds number (Rea) at 1000, the Taylor number was varied by

increasing the rotational speed of the inner cylinder for both cylindrical and conical case.

With the increase of rotational speed, much more complex Taylor vortices were observed for conical cylinders. Staton and Cavagnino [18] developed formulations used to predict convective cooling and flow in electric machines with motor annulus (air gap). They provide guidelines for choosing suitable thermal and flow network formulations. Their study brings together useful heat transfer and flow formulations, dimensionless correlations that can be applied to thermal analysis of electrical machines. Leclercq et al.

[19] examined the characteristics of TCP flow for eccentric cylinders with fixed outer cylinder. Their study includes numerical analyses of wide and relatively small air gap with different eccentricities. They report critical wavenumber and azimuthal Reynolds number at which the Taylor vortices are unstable. They conclude that the eccentricity between two cylinders, regardless of the axial flow rate, helps stabilization of Taylor vortices. This effect becomes even more important for higher values of eccentricity. Bouafia et al. [20] performed experimental studies on TCP flow to determine the correlation between Nusselt number and Taylor number for concentric cylinders with rotating inner cylinder. They explored smooth and slotted fixed outer cylinder (stator) to determine the effect of slots.

Their dimensionless correlations take both the tangential and the axial Reynolds number into account. CFD simulations were carried out for scraped surface heat exchangers that have two concentric cylinders with TC flow by Pawar and Thorat [21]. The Reynolds stress model (RSM) and k–ε model were used for Taylor vortex flow (Ta>300). The main purpose of their work was to analyze the effect of rotating scrapper on the existing flow patterns in simple annular flow using CFD analysis. It was concluded that the Couette flow no longer exists in the heat exchanger annulus because of the rotating blades which formed vortices

even at low Taylor number (Ta~5). They illustrated that the appearance of vortical flow was observed earlier in the presence of blades. They also compared the turbulence models used in the CFD simulations and concluded that RSM model works better than the k–ε model to characterize the vortices in the annular flow. Lancial et al. [22] investigated the

TCP flow in an annular channel of longitudinal slotted rotating inner cylinder, corresponding to a salient pole hydrogenerator. The main objective of their study was to improve the understanding of flow and thermal phenomena in electrical machines using a simplified scale model. Numerical analyses were performed and results were compared to experimental data to validate the scale hydrogenerator model. With the validation of their model, a parametric analysis was performed to investigate the main flow regimes to derive expressions for the Nusselt Number and heat transfer coefficient. The heat transfer coefficient is affected by the presence of vortices in the notch region and promotes more efficient cooling at the leading edge as compared to the trailing edge.

In this paper, the temperature distribution at the stator surface and TCP flow structure at the annulus of a liquid submerged cryogenic induction motor are investigated in the presence of realistic rotor geometries commonly found in practical applications.

Common geometries of the cryogenic pump motor rotor are shown in Figure-1. Specially, induction motors are often skewed primarily to prevent cogging condition during which, rotor magnetic locking may occur due to the possibility of strong coupling between rotor and stator. Other reasons for skewing include improving the starting torque and enhancing magnetic coupling of stator and rotor fluxes. While skewing is implemented in induction motors and generators to improve electrical, magnetic and torque characteristics, to the best of the authors’ knowledge, there has been no research conducted on the impact of skew

geometry and its configuration on the cooling performance of induction motors and generators. Therefore, to fill this knowledge gap, cooling flow and convection heat transfer under turbulent flow regime are investigated for six different surface configurations of the inner cylinder (rotor). Those configurations are: smooth surface, longitudinal straight slots

(non-skewed), and skewed slots in counter clockwise direction and clockwise direction with 5 and 10 degrees of skewing. The fluid models for each geometric configuration are shown in Figure-2. In addition, entrance region of each geometry for annulus is studied to better understand its impact on the overall heat transfer and flow. It is found that different geometries have modest effect on the temperature profiles and heat transfer characteristics, thus suggesting that existing correlations published in the literature for non-skewed geometries are applicable to skewed geometries, provided that relevant nondimensional parameters are matched. On the other hand, the particular geometric configuration has a major effect on the different flow structures developed in the system, including Taylor vortices and entrance length. This paper thus contributes to the understanding these effects which in turn can aid in the design of more efficient, stable, and safe induction motors and generators.

The rest of the paper is organized as follows. Section 2 includes the description of the physical problem, introduces the relevant nondimensional parameters and correlations, and presents the mechanisms for heat transfer. Section 3 reports the results of CFD analyses for different geometries and includes comparison with published correlations. Conclusions are reported in Section 4.

Axial (Non-skewed) groove Skewed groove

(a) (b) (c) Figure-1 Rotor surface configurations commonly utilized in cryogenic pump motors. (a) Smooth, (b) axial (non-skewed) grooves, (c) skewed grooves. Courtesy of Elliott Cryodynamics.

(a) (b) (c)

(d) (e) (f) Figure-2 Rotor surface configurations in analyses (a) Smooth, (b) axial (non-skewed) grooves, (c) skewed grooves 5 degrees (ccw), (d) skewed grooves 5 degrees reversed (cw), (e) skewed grooves 10 degrees (ccw), (f) skewed grooves 10 degrees reversed (ccw).

2. Problem Statement

2.1 Induction Motor Configurations and Cooling Mechanism

Heat generation in the motor cavity is mainly due to power losses of the induction motor. A typical cooling scheme of a submerged motor pump, which is the representative system focus of this paper, is given in Figure-3.

Motor Housing Exhaust, Back to Pump Suction

Stator

Motor Coolant Flow

Motor Bypass Holes

Pump Suction

𝜔𝜔 Figure-3 Coolant flow through the motor, Elliott Cryodynamics’ shipboard cargo pump.

Note that, in this application, the motor is cooled with the help of pumped liquid and through coolant holes as shown in Figure-3. Therefore, the coolant flow rate through annulus varies with the total discharge flow of the pump. The cooling fluid considered in this paper is methane with a mixture of ethane and nitrogen, as this fluid is readily available commercially. The fluid composition is reported in Table-1.

Table-1 Fluid composition in mole fraction.

Fluid Composition Methane: 98.94%

Ethane: 0.92%

Nitrogen: 0.14%

2.2 Formulation of the Cooling Mechanism

Figure-4 illustrates the simplified TCP flow model used in the analyses. The inner cylinder is the rotor, rotating with a speed of 1800 RPM, which is the common rotational speed for 60 Hz 4 pole induction motors. The outer cylinder is the stator, which is stationary and is heated due to power losses. Such losses are modeled as constant heat flux over the inner surface of the stator as shown in Figure-4. This is because there is constant heat generation from the stator section of the motor. The cylinders are vertically suspended and coolant flow ( ) is imposed in the axial direction between the rotor and the stator.

𝑚𝑚̇ For this specific pump motor, the stator has a length of 0.584 m ( ), with 0.357 m inside diameter ( ). The motor rotor is sleeved to the shaft with same𝐿𝐿 length (0.584 m) with

𝐷𝐷2

0.351 m outer diameter ( ). The resulting radial gap aspect ratio, = / , is 0.986.

1 1 2 The axial cylindrical gap 𝐷𝐷ratio, = / , is 195, where = ( 𝜂𝜂)/2𝐷𝐷=3𝐷𝐷 mm.

𝛤𝛤 𝐿𝐿 𝛿𝛿 𝛿𝛿 𝐷𝐷2 − 𝐷𝐷1

y x

Figure-4 Simplified model for Taylor-Couette-Poiseuille flow with constant heat flux at stator.

Based on the pump operating condition, the inlet conditions along with the coolant

flow rate at the motor are known and given in Table-2. Fluid thermodynamic and transport

properties are calculated by using published properties by the National Institute of

Standards and Technology (NIST) [23], based on the known fluid composition. For Elliott

Cryodynamics’ shipboard cargo pump shown in Figure-3, power loss due to the

inefficiency of the motor at pump rated flow is 24.8 kW as specified by the manufacturer.

Specifically, rated flow, input power, and the motor efficiency values are published in

Elliott Cryodynamics’ catalogs for pump model 12EC-24 and given in Table-3 [34].

Table-2 Inlet Conditions with corresponding flow rate for a given operating cases [34].

Inlet Temperature (Tm,i) -161.4 °C Motor Inlet Pressure (Pin) 5.09 bar (gauge) Coolant Mass Flow Rate ( ) 1.185 kg/s

𝒎𝒎̇

Table-3 Cargo Pump Operating Details [34].

Rated Capacity (Volumetric Flow Rate): 2040 m3/hr Pump Differential Rated Head: 130.6 m Pump Operating Speed: 1800 RPM Motor Input Power at Rated Capacity ( ): 462.2 kW

Motor Efficiency ( ): 𝑷𝑷𝒎𝒎 94.6% 𝐄𝐄𝐄𝐄𝐄𝐄

Based on the known input heat generation, temperature and pressure at the motor outlet ( , ), and surface temperature along the stator ( ) are calculated by

𝑜𝑜𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜𝑜𝑜 𝑠𝑠 determining𝑇𝑇 dimensionless𝑃𝑃 parameters, such as Nusselt number, Taylor𝑇𝑇 number, Grashof number and Reynolds number, and convection heat transfer coefficient. It should be noted that the stator surface temperature is crucial for the electrical motor applications since the highest temperature is expected to occur at this location. On the other hand, excess temperature can result in motor failure, lack of overall motor performance and instability

in pump operation. Therefore, the focus of this study is to calculate and compare the stator surface temperatures for each motor rotor geometry by using published correlations of dimensionless parameters and CFD simulations.

The details of the calculations are given in the next section.

2.3 Flow Structure and Convective Heat Transfer:

The inefficiency (power losses) of the motor is the heat source, and the heat flux from the stator is calculated as follows:

= (1 Eff) (1)

𝑞𝑞 − 𝑃𝑃𝑚𝑚 = (2)

𝐴𝐴𝑠𝑠 𝜋𝜋𝐷𝐷2𝐿𝐿 = (3) ′′ 𝑞𝑞 𝑞𝑞 𝐴𝐴𝑠𝑠 where is the heat power due to the inefficiency of the motor, Eff is the efficiency of the motor, 𝑞𝑞 is the surface area of the stator, and is the total input power to the motor. The

𝑠𝑠 𝑚𝑚 heat power𝐴𝐴 is applied to the inner surface of the𝑃𝑃 stator where the area is defined by equation

2. Heat flux ( ) is calculated based on the heat power applied on the stator surface, as in ′′ equation 3. It𝑞𝑞 is assumed that the outer and inner surfaces and ends of the stator and rotor are completely insulated (adiabatic) and heat transfer occurs in radial direction from heated stator towards the shaft rotor. This represents the worst-case scenario experienced by the fluid travelling along the motor annulus.

The most important parameter used in characterization of the cylindrical gap flow is the rotational speed of the rotor, . The Reynolds number for the tangential (rotational) direction, Re , can be defined by using𝜔𝜔 the rotational speed

𝑡𝑡

Re = (4) 2 1 h t 𝜌𝜌 𝜔𝜔 𝐷𝐷 𝐷𝐷 𝜇𝜇 where is the density of the liquid, is the rotational speed, is the hydraulic diameter

h and 𝜌𝜌is the dynamic viscosity. With𝜔𝜔 the addition of axial 𝐷𝐷flow to TC flow, the axial

Reynolds𝜇𝜇 number, Re , should also be defined to determine if the flow is in laminar or

𝑎𝑎 turbulent regime

Re = (5) a h a 𝜌𝜌 𝑉𝑉 𝐷𝐷 𝜇𝜇 Both Reynolds numbers are based on the hydraulic diameter, , which is defined

h for configurations with and without slots as follows: 𝐷𝐷

= 4 (6) 𝐴𝐴 𝐷𝐷h 𝑃𝑃 where is the effective flow area across the motor annulus and is the total perimeter of the annulus.𝐴𝐴 The Reynolds number based on effective velocity𝑃𝑃 can be calculated as follows:

Re = (7) eff h eff 𝜌𝜌 𝑉𝑉 𝐷𝐷 𝜇𝜇 with defined as:

𝑉𝑉eff

= + ( ) (8) 2 2 𝜔𝜔𝐷𝐷1 2 𝑉𝑉eff �𝑉𝑉a 𝛼𝛼 The calculation of effective velocity, , was first suggested by Gazley [33]. In

eff Eq. 8, is a coefficient that defines the significance𝑉𝑉 of axial velocity component over tangential.𝛼𝛼 The empirical value of = 0.25 is widely used for small gaps ( >0.2), suggesting that the mean tangential velocity𝛼𝛼 has a smaller effect on heat transfer than𝜂𝜂 the axial velocity [12, 20, 28, 30, 31, 33].

For the TCP flow, the Reynolds number itself is not sufficient to identify the complete characteristics of the flow and to determine whether there is presence of Taylor vortices. Therefore, the Taylor number must be calculated to determine the effect of the centrifugal forces, and confirm if Taylor vortices are present around the rotor. The Taylor number can be interpreted as the square ratio between the centrifugal force and viscous force. It is defined for two concentric cylinders in rotation as follows [11]:

Ta = (9) 2 22 3 𝜌𝜌 𝜔𝜔 𝐷𝐷1𝛿𝛿 2 𝜇𝜇 The actual geometry of the slots is not considered for the calculation of Taylor number, where only the motor annulus gap, , appears [12]. Once a certain rotational speed is exceeded for a given air gap, the Taylor vortices𝛿𝛿 start to occur between the outer cylinder and the inner cylinder. They appear as counter-rotating pairs. Kaye and Elgar [26] were the first to report the transition of flow regimes for a wide range of axial Reynolds number and

Taylor number. For values of Taylor number approximately around 1800 or greater, Taylor

vortices can be observed under zero axial flow condition ( = 0) [26]. With the increase

a in axial flow, the occurrence of Taylor vortices can be delayed𝑉𝑉 [26].

Since the motor assembly is vertically suspended, the effect of buoyancy must be included. The rotational parameter, Gr/Re , describes the relative importance of 2 buoyancy and rotational effects [16] where𝜎𝜎 ≡ the Grashof number (Gr) is defined as follows:

Gr = (10) 3 𝜌𝜌𝛿𝛿 𝛽𝛽𝛽𝛽∆𝑇𝑇 𝜇𝜇 Here, is the temperature differential across the gap, is the thermal coefficient of the volumetric∆𝑇𝑇 expansion. For values of greater than 10, the𝛽𝛽 buoyancy forces dominate the flow, and in the range 0.01< <10, both𝜎𝜎 buoyancy and the rotational forces affect the flow characteristics. For values of𝜎𝜎 below 0.01, the effect of buoyancy is negligible and there is no distortion of the Taylor vortices𝜎𝜎 [16, 24, 25].

2.4 Correlation and Formulation of Heat Transfer Coefficient:

The correlation for Nusselt number is established often by experiments for applications with certain geometry, working fluid, rotational speed and flow characteristics

(Re number). The most widely used correlation for Nusselt number is a function of

Reynolds number, Prandtl number (Pr) and Taylor number. The Prandtl number is the ratio of momentum diffusivity to thermal diffusivity. It is a function of specific heat , viscosity

𝑝𝑝 and the thermal conductivity of the fluid, and is defined as: 𝑐𝑐

𝜇𝜇 𝑘𝑘 Pr = (11) 𝑐𝑐𝑝𝑝 𝜇𝜇 𝑘𝑘

The following general equation can be written to define the relationship between the Nusselt number and the other dimensionless parameters for TCP flow applications:

Nu = Ta Re Pr (12) 𝑎𝑎 𝑏𝑏 𝑐𝑐 𝑁𝑁 , , , and are the empirical constants and vary based on fluid, geometry and rotor- stator𝑁𝑁 𝑎𝑎 𝑏𝑏heating𝑐𝑐 configuration. If the axial flow is not dominant, and relatively very small with respect to rotational component (Ta Re ), the correlation can be simplified as

a follows: ≫

Nu = Ta Pr (13) 𝑎𝑎 𝑐𝑐 𝑁𝑁 Table-4 reports a list of empirical correlations published in the literature for smooth rotor and stator. This list also has additional information regarding the dimensionless parameters calculated based on the geometry of the annulus, and range of Reynolds number and Taylor number for validity.

As mentioned before, the surfaces of rotors and stators often have axial slots in electrical motor applications. The correlations for Nusselt number with slotted rotor or stator configurations are reported in Table-5 [12, 20, 30, 31]. The particular configuration of slots and area ratios ( ) of slots (grooves) to annulus area are listed for comparison purposes of each geometry.𝐸𝐸 It is interesting to observe that, according to the studies by

Tachibana and Fukui [12], axial slots do not have any impact on the heat transfer and the same set of equations and correlations are reported by them for both smooth and axial grooved geometries.

Table-4 Correlations for Nusselt number as a function of dimensionless parameters – smooth annulus.

= = Configuration Correlation 𝟏𝟏 𝒂𝒂 𝜞𝜞 𝑳𝑳� 𝑫𝑫 𝐑𝐑𝐑𝐑 4 𝐓𝐓𝐓𝐓 . 98.4 𝜹𝜹 𝜼𝜼 0.956� 𝑫𝑫𝟐𝟐 1.1x10 - 1800 - Heated Rotor Nu = 0.046Re [20] 4 6 0 7 3.1x10 4x10 Cooled Stator Re = (Re + 0eff.25Re ) . 2 2 0 5 4.72, 6.25 0.221, 380 - 4,200 71 – 3,415 Heated Stator eff a . t . / Nu = 0.015 1 + 2.3 0 45 Re Pr [12] 0.882 Cooled Rotor ℎ 2 0 8 1 3 𝐷𝐷 𝐷𝐷 eff � � � 1� Re = , 𝐿𝐿 𝐷𝐷= ( + 0.25( ) ) . 𝜌𝜌 𝑉𝑉𝑒𝑒ff 𝐷𝐷ℎ 2 2 0 5 5 7 eff 𝑉𝑉eff. 𝑉𝑉. a 𝜔𝜔𝐷𝐷1 13.3 0.869 1.7x10 - 6x10 - Insulated Stator Nu𝜇𝜇= 0.023Re Pr [27] 5 10 0 8 0 25 13.7x10 12x10 Heated Rotor a 12 0.571 1.5x104 - 4.87x109- Heated Stator Nu = 0.023Re . Pr . (1 + ) . [28] 5 9 0 8 0 5 2 0 87 6.5x10 8.65x10 Insulated Rotor 1 eff 2 = 𝛽𝛽 2 𝐷𝐷ℎ 𝜔𝜔𝐷𝐷1 4 7 � . . 50 0.889 <1.12x10 <7.9x10 Heated Rotor Nu𝛽𝛽= 6.�137𝜋𝜋Re� � �Ta𝐷𝐷1� � Pr � 𝑉𝑉 𝑎𝑎 [�29] 0 77 0 127 1� Cooled Stator a 3 233 0.986 17,766 1.24x109 Heated Stator PRESENT STUDY Insulated Rotor (Most Similar to [28])

Table-5 Correlations for Nusselt number as a function of dimensionless parameters – slotted rotor or stator (axial grooves).

For an internal flow, convective heat transfer problem, the heat transfer coefficient can be calculated by knowing the Nusselt number as [13]:

Nu = (14)

ℎ 𝑘𝑘 𝐷𝐷ℎ where is the thermal conductivity that varies based on the temperature and the pressure of the 𝑘𝑘fluid. By using an energy balance, the temperature of the bulk fluid at the motor annulus outlet, , can be determined as:

𝑇𝑇𝑚𝑚 𝑜𝑜 = , , (15)

𝑞𝑞 𝑚𝑚̇ 𝑐𝑐𝑝𝑝�𝑇𝑇𝑚𝑚 𝑜𝑜 − 𝑇𝑇𝑚𝑚 𝑖𝑖� where , is the inlet temperature of the fluid that is listed in Table-2 in the previous

𝑚𝑚 𝑖𝑖 section.𝑇𝑇 The surface temperature of the stator is calculated based on Newton’s cooling law as:

= ( ) (16)

𝑞𝑞 ℎ𝐴𝐴𝑠𝑠 𝑇𝑇𝑠𝑠 − 𝑇𝑇𝑚𝑚 The surface temperature ( ) and the fluid temperature ( ) can be calculated by

𝑠𝑠 𝑚𝑚 using the equations for heat power𝑇𝑇 ( ). For constant heat flux, the𝑇𝑇 rate of change in fluid temperature with the axial location can𝑞𝑞 be written as:

= (17) ′′ 𝑑𝑑𝑇𝑇𝑚𝑚 𝑞𝑞 𝑃𝑃 𝑑𝑑𝑑𝑑 𝑚𝑚̇ 𝑐𝑐𝑝𝑝 By integrating from = 0, the fluid temperature can be calculated at a given axial location by following equation:𝑥𝑥

( ) = + (18) , ′′ 𝑞𝑞 𝑃𝑃 𝑇𝑇𝑚𝑚 𝑥𝑥 𝑇𝑇𝑚𝑚 𝑖𝑖 𝑥𝑥 𝑚𝑚̇ 𝑐𝑐𝑝𝑝 To determine the impact of buoyancy over rotational forces, the rotational parameter is calculated. It is found to be less than 0.01, which indicates that the buoyancy forces can 𝜎𝜎be neglected for this application [16, 24, 25].

3. CFD Analyses and Results

CFD analyses are performed to determine the temperature distribution at the stator surface and the flow structure for each rotor configuration under steady state condition.

ANSYS CFX software is used for the CFD simulations, and six different rotor surface profiles are analyzed. As in any other CFD tool, ANSYS CFX is based on the three fundamental physical principles of fluid dynamics, which are; mass and energy are conserved, and Newton’s 2nd law is satisfied (momentum conservation). The mass conservation and momentum equation are defined, respectively, as follows:

+ ( ) = 0 (19) 𝜕𝜕𝜕𝜕 ∇ ∙ 𝜌𝜌𝑼𝑼 ( ) 𝜕𝜕𝜕𝜕 + ( ) = + + (20) 𝜕𝜕 𝜌𝜌𝑼𝑼 ∇ ∙ 𝜌𝜌𝑼𝑼⨂𝑼𝑼 −∇𝑝𝑝 ∇ ∙ 𝜏𝜏 𝜌𝜌𝒈𝒈 𝜕𝜕𝜕𝜕 where, is the vector of velocity ( , , ), is the static pressure, is the stress tensor,

𝒙𝒙 𝒚𝒚 𝒛𝒛 is the𝑼𝑼 gravitational body force. 𝑼𝑼The effect𝑝𝑝 of the buoyancy is𝜏𝜏 neglected as it was explained𝜌𝜌𝒈𝒈 in the previous section. Therefore, the third term at the right-hand side of equation 2 can be ignored since the gravity is not important.

The CFX solver applies the conservation laws to each control volume. This is fundamentally done by integrating partial differential equations over all the control volumes in the region of interest. Then, these integral equations are converted to algebraic equations by generating a set of approximations for the terms in the integral equation. After that, an iterative solution is conducted to solve each algebraic equation [36].

For each rotor configuration, mass flow rate with a fluid temperature of -161.4 °C is defined at the inlet, while constant heat flux is applied to the outer (stator) surface.

Saturation and superheat tables are utilized in ANSYS to define the fluid’s thermodynamic and transport properties as a function of static temperature and pressure along the air gap.

No slip boundary condition is defined with an average surface roughness (RMS) of 250 micro inches (6.35 μm) for both the stator and rotor wall. The surface roughness is based on manufacturing requirements of both components. The inner cylinder rotates at 1800

RPM in the counter clockwise (CCW) direction as viewed from the inlet section. Adiabatic condition is assumed at inner cylinder (rotor) to simulate the worst (highest) temperature condition at the stator wall. To better capture the fluid circulation and the effects of the near wall region, eddy viscosity model with k-ω SST (shear stress transport model, see

[22]) option is defined as the turbulence model. This selection is supported by the findings of Lancial et al. [22] where several different turbulence models for analyzing TCP flow and corresponding heat transfer mechanisms are compared.

Periodic Symmetry

Annulus Groove Region Region

z Annulus Wall (a) Inlet Region Mesh Quality.

y x Annulus Region z Groove Groove Wall Periodic Region Symmetry Face

(b) Isometric View of Region Mesh

Figure-5 Mesh quality shown at (a) inlet region, (b) Isometric view of the mesh at annulus, applicable to geometries with and without grooves, and isometric view of the groove region.

Manual map meshing is utilized for each surface to achieve uniform element size

(0.5 mm) and shape with small aspect ratio. The near wall surfaces and grooves are meshed much more finely with an element size of 0.02 mm and with a growth rate of 1.2 to better capture the wall effect to fluid circulation and its impact to heat transfer. Figure-5 shows the mesh quality for annulus with and without grooves. Identical mesh quality and mesh refinement are used for all configurations. To assess if convergence is reached, residual

RMS (root mean square) of velocity based on momentum, pressure based on mass and heat transfer based on energy equations are plotted.

Residual level is a measure of convergence utilized by ANSYS CFX, which can be directly related to whether each equation (mass, momentum and energy) is solved accurately. The target residual RMS value of 1e-5 is throughout the analysis to reach convergence. In addition, the temperature at stator is also monitored during the simulations to ensure that the stator temperature has reached a steady solution. Since the RMS value of

1e-5 can still lead to a temperature variation up to 2 degrees in the solution (approximately

1% error in temperature), simulations were run until a steady stator temperature condition is obtained. Figure-6 shows the RMS residual values and the stator temperature convergence as a function of accumulated time steps.

1.00E+00 -157.0 -157.23 -157.5 1.00E-01 -158.0 1.00E-02 RMS Pressure-Mass RMS Fluid Velocity-Momentum -158.5 Heat Transfer-Energy 1.00E-03 -159.0

Stator Temperature C] ° -159.5 [

1.00E-04 - avg s T -160.0 1.00E-05 -160.5

VariableResidual Value(RMS) 1.00E-06 -161.0

1.00E-07 -161.5 0 2000 4000 6000 8000 10000 12000 14000 Accumulated Time Step

Figure-6 Convergence Plot: residual RMS values for fluid velocity, pressure and heat transfer are plotted, along with average temperature at stator surface, against time step.

The Grid Convergence Method described by Celik et al. [37] is used to estimate the discretization error. The error estimates are approximated in terms of fine-grid convergence index (GCI). Three different size grids are considered in the determination of GCI error.

The total number of elements, grid size, grid refinement factor between each grid, and error estimate along with the apparent order p are given in Table-6. According to the calculated stator temperature along the axial flow direction for three different selected grid size, GCI error is determined as 0.07% for the finest grid used in the simulations.

Table-6 Calculation results of discretization error according to [37], please refer to [37] for description of each parameter used to determine the error involved with the grid size.

Number of Elements (N): N1= 22,100,000 N2= 13,200,000

N3= 10,500,000

Grid Refinement (r): r21=1.953, r32= 1.728

Stator Temperature at Exit ( ): Ts1= -153.51°C, Ts2=-153.68 °C, Ts3=-154.06 °C

𝒔𝒔 Apparent order ( ): 𝑻𝑻 1.683

21 Extrapolated Value𝒑𝒑 of Stator Tsext = -153.42 °C

Temperature (Tsext):

21 Relative error (ea): ea = 0.11%

21 Extrapolated relative error (eext): eext = 0.05%

21 Grid Convergence Index (GCI): GCIfine = 0.07%

The results of CFD analyses are compared to the results by correlations discussed in section 2.4. The stator temperatures as a function of the axial location ( ) are given in

Figures-7 and 8, for smooth and axial (non-skewed) groove configurations,𝑥𝑥 respectively.

In addition, predicted values of stator temperature at the exit for each configuration are given in Table-7. As expected, consistently with Eq. 18, the fluid reaches its highest temperature at the stator surface at the exit location as it is continuously heated from the inlet to the outlet of the annulus. This location is thus critical to determine whether a phase change is likely at the annulus. If the temperature at this location along with the pressure

is below the saturation point of the liquid, vapor will start to form. It is important to ensure that vapor formation is prevented for stable and efficient pump operation.

Table-7 Stator Surface Temperature at exit of the annulus, .

𝑠𝑠 Correlation Rotor Configuration Stator Surface Temperature𝑇𝑇 at Exit, [°C]

Bouafia [20] Smooth -150.81 𝑻𝑻𝒔𝒔 Tachibana [12] Smooth -152.14 Childs [27] Smooth -138.13 Kuzay [28] Smooth -153.24 Aubert [29] Smooth -149.15 Present Study Smooth -153.39

Bouafia [20] Axial Grooves -154.55 Tachibana [12] Axial Grooves -152.14 Yanagida [30] Axial Grooves -149.31 Fenot [31] Axial Grooves >-120 Present Study Axial Grooves -153.48

Present Study 5° Skewed Grooves, CCW -153.30 Present Study 10° Skewed Grooves, CCW -153.43 Present Study 5° Skewed Grooves, CW -153.49 Present Study 10° Skewed Grooves, CW -153.51

It is observed that the CFD simulation results exhibit small oscillations of temperature along the axial location once the Taylor vortices are formed. This temperature behavior agrees with the formation and shape of the velocity profile of the Taylor vortices.

Similar oscillations are also noted for the heat transfer coefficient at the heated stator wall.

Figure-9 shows the typical heat transfer coefficient at the heated stator wall as a function of axial location. Once the Taylor vortices are formed, fluctuation in heat transfer coefficient occurs. This fluctuation can be explained by observing that each coupled vortex can create a stall in flow at certain locations of the wall surfaces, causing a sudden increase in heat transfer and temperature. It should be noted that it may not be possible to observe this behavior in experimental studies as the temperature measurements are not recorded and monitored in close enough proximity to resolve each coupled vortex.

-145 Bouafia [20] -147 Tachibana [12] -149 Kuzay [28] -151 Aubert [29] CFD, Smooth Rotor C] -153 ° [ s

T -155 -157 -159 -161 -163 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Axial Location, x [m]

Figure-7 Stator Surface Temperature (Ts) as a function of axial location for smooth rotor.

-145 Bouafia [20] -147 Tachibana [12]

-149 Yanagida [30]

-151 CFD, Axial (non-skewed) Grooves C] °

[ -153 S T -155 -157 -159 -161 -163 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Axial Location, x [m]

Figure-8 Stator Surface Temperature (Ts) as a function of axial location for axial (non- skewed) grooves.

60000 -153.49 -153 h Thermal Entrance -154 Region 50000 -155

40000 -156 K] - 2 -157 30000 C] [W/m °

-158 [ s T 20000 -159

-160 h StatorWall Heat Transfer Coefficient, 10000 Ts -161

0 -162 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Axial Location, x [m]

Figure-9 Typical Stator Heat Transfer Coefficient and Stator Surface Temperature (Ts) as a function of axial location. 5° CW Skewed Grooves case is shown.

According to Figure-7, the present CFD simulations for smooth rotor configuration shows the closest temperature correlations with the formulas by Kuzay [28]. The difference in stator surface temperature between Kuzay’s correlation [28] and the present CFD analysis is found to be less than 1%. It should be noted that Kuzay’s experimental study

[28] has the most similar Reynolds and Taylor numbers, as well as the same stator/rotor heating configuration, to those reported in this paper for the Elliott Cryodynamics’ pump motor.

As shown in Figure-8, the temperature results of correlations are generally in good agreement with the CFD analyses for the rotor with axial grooves. Somewhat larger discrepancies are obtained between the present results and the correlation by Fenot [31]. It should be however noted that the rotor configuration by Fenot [31] has 4 very large grooves, while the number of rotor grooves for Elliott Cryodynamics’ pump motor is 40 and they are relatively small. In addition, ratio for Fenot’s configuration is more than two times greater than any other geometry. These𝛤𝛤 geometric discrepancies can significantly alter the results. Due to this dissimilarity, the correlations by Fenot [31], is thus not included in Figure-8. Elliott Cryodynamics’ motor configuration is most similar to the one reported by Bouafia [20]. The difference in stator temperature between the CFD results and the predictions using Bouafia’s correlation is less than 1%.

Both the published heat transfer correlations and CFD simulation results show that the temperature at the stator surface is very similar for all cases with less than 0.5% difference. The temperature at the stator as a function of the axial location is given in

Figure-10 for all cases. A typical temperature distribution of the stator surface is given in

Figure-11. For skewed axial grooves, the temperature seems to be slightly lower at the stator surface for skewing in the CCW direction when compared to skewing in the CW direction.

-150 Smooth (No Grooves) -152 Axial Grooves (Non-skewed) 5 deg CCW Skewed Grooves 10 deg CCW Skewed Grooves -154 5 deg CW Skewed Grooves C] ° [ 10 deg CW Skewed Grooves s

T -156

-158

-160

-162 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Axial Location, x [m]

Figure-10 Temperature comparison between CFD simulations – averaged data.

y x

Figure-11 Typical Temperature distribution at stator surface, 5° CW Skewed Grooves case is shown.

The flow structure is examined for each case. Taylor vortices are apparent from the

end of the entrance region to the exit of the annulus as seen in Figure-12. Taylor vortices

for each case are illustrated in Figure-13 for comparison purposes. Simulations with axial

flow demonstrate that the Taylor vortices do not have the exact size of the annulus gap and

the center of the vortices does not coincide with the mid-point of the annulus. This suggests

that the flow in the axial direction can stretch each vortex in the axial direction and distort

their shape.

y x

Figure-12 Typical flow velocity contour at motor annulus with Taylor vortices and entrance region for 5° CW Skewed Grooves case.

As the vortices increase in length, the center of each vortex is also shifted towards

the rotor or stator side depending on the rotational direction of the vortex. If the skewing

and the rotational speed are in the same direction, the vortex width is relatively larger. This

suggests that not only the axial flow but also the groove configuration can impact the vortex

shape. It should be noted that the skewed and axial groove configurations have identical

Reynolds and Taylor numbers. However, flow structure differs slightly based on the vortex width, also with very slight (<1 °C) impact to the temperature distribution at stator surface.

In order to validate the influence of the axial flow on the Taylor vortices, a CFD simulation with no axial flow is also performed for smooth annulus case. Figure-13 (a) is showing the

Taylor vortices with no axial flow.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

Figure-13 Taylor vortices along XY Plane. (a) Smooth case no axial flow, (b) smooth case with axial flow, (c) axial (non-skewed) grooves with axial flow, (d) skewed 5° CCW grooves with axial flow, (e) skewed 10° CCW grooves with axial flow, (f) skewed 5° CW grooves with axial flow, (g) skewed 10° CW grooves with axial flow.

The hydrodynamic entrance length is determined from the simulations as the length with no vortex development is observed. For fully turbulent internal flow in non-rotating pipes circular or non-circular shapes, the entrance length (both thermal and hydrodynamic) can be estimated as 4.4 (Re)1/6 [35]. Accordingly, at a distance of 22 from the inlet,

ℎ h the heat transfer coefficient𝐷𝐷 should approach its asymptotic value [32]. It𝐷𝐷 is expected that,

for a rotating inner cylinder and in the presence of geometric differences of the rotor, the entrance length may vary. Furthermore, the entrance region may extend or decrease depending on the significance of tangential velocity over axial velocity.

] 27 h

D Grooved Case 25 Smooth Case 23

Entrance Region Length [ Length Region Entrance 17

15 -15 -10 -5 0 5 10 15 Skew Angle [°]

Figure-14 Entrance length vs skew angle.

The relation between the skew angle and the entrance length region is shown in

Figure-14. The smooth case displays a longer entrance region than the axial groove case, mostly due to the axial velocity being larger for this configuration. In addition, the smooth case has slightly smaller flow area with identical cooling mass flow rate. This results in a larger Rea for smooth rotor configuration (See Tables 4 and 5). The rotor with 10 degrees of skewing in the CCW direction has the shortest entrance region. This effect is mainly due to the absolute tangential velocity. For grooves skewed in the same direction of the rotational speed, the tangential velocity is relatively larger, hence the development of

vortices will occur much earlier. The opposite is observed for the rotor configurations with grooves skewed in the CW direction. The entrance region will be relatively longer due to smaller absolute tangential velocity of the fluid. For skewed grooves in the CCW direction, the position of each groove is 5 or 10 degree off the normal plane in the same direction of the inner wall rotation. This should result in slightly reduced shear stress at the inner wall, consequently reducing the entrance region. This observation partially contributes to explain the shape of the Taylor vortices for each case shown in Figure-13.

4. Conclusions

The convection heat transfer and cooling of a submerged motor pump are investigated under normal operating conditions to determine the temperature distribution at stator surface and the velocity profile for different configuration of rotor geometry. Rotor with a smooth surface, straight axial grooves, and various skewed grooves are studied by performing CFD simulations to identify differences in flow structure and temperature distribution.

The temperature at the stator surface is calculated by using published correlation between dimensionless parameters (Re, Nu, and Ta), for the different geometry of the

Elliott Cryodynamics’ pump submerged motor and compared against results of detailed

CFD simulations incorporating the precise geometry of the annulus. It is observed that the temperature profile is very similar among different geometries, and thus it can be concluded that the rotor configuration does not seem to play a critical role in the temperature distribution at stator surface. This conclusion is in strong agreement with the

findings from Tachibana and Fukui [12]. In the cases investigated, the exit temperature at the stator surface for each geometry is still low enough to prevent vapor formation.

As expected, the correlations with similar geometry ( , ) and dimensionless parameters (Re, Ta) have given the closest temperature results to𝛤𝛤 those𝜂𝜂 in CFD analysis.

This suggests that published correlations can be utilized for the calculation of temperature distribution even in the presence of skewed grooves on the rotor, as long as similar geometries and dimensionless parameters are retained. On the other hand, it is observed that even for identical Reynolds and Taylor numbers, groove geometry does impact the flow structure. The impact on flow structure seems however slight and has a modest effect on stator temperature, at least for the cases studied in this paper. Fluctuations in the stator temperature and heat transfer coefficient are observed along the axial direction once the

Taylor vortices are formed. These fluctuations have modest impact on the overall heat transfer for the cases studied in this paper. However, for relatively larger annulus gaps and/or different geometries, the magnitude of the fluctuations might be significant enough to impact the overall temperature distribution.

The flow structure for each surface profile demonstrates the presence of Taylor vortices. The axial flow and geometry of the rotor can affect the width and the origin of each vortex. Furthermore, each axial groove configuration has a noticeable and distinct impact on vortex shape, even though identical dimensionless parameters exist for each geometric configuration.

Finally, the extent of the entrance region for each surface profile is examined and it is concluded that there is a correlation between the entrance region length and the vortex

shape (width). The configuration with the narrowest vortex is the 10 degrees CW skewed case, which also has the longest hydrodynamic entrance region. The reason for this correlation seems to be associated with the magnitude of the tangential velocity, which is correlated to the development of the vortices. It is observed that temperature depends slightly on the vortex shape and length (within 1 °C). The vortices that are stretched and off center due to axial flow, exhibit relatively worse cooling characteristics at the stator surface. Further research should be devoted to understanding the correlation between the structure of the Taylor vortices and possible cooling improvements in induction motor applications.

5. Acknowledgment

The authors would like to thank Elliott Group, Cryodynamic Products - Ebara

Corporation for their support.

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CHAPTER 3. CAVITATION PERFORMANCE OF CONSTANT AND VARIABLE PITCH HELICAL INDUCERS FOR CENTRIFUGAL PUMPS: EFFECT OF INDUCER TIP CLEARANCE

Enver S. Karakasa,b, Hiroyoshi Watanabec, Matteo Aurelia, Cahit A. Evrensela

aMechanical Engineering Department, University of Nevada, Reno 1664 N. Virginia Street, Reno, NV 89557-0312, USA bElliott Group, Cryodynamic Products, Ebara Corporation 350 Salomon Circle, Sparks, NV 89434, USA cEbara Fluid Machinery & Systems Company, Ebara Corporation 11-1 Haneda Asahi-cho, Ohta-ku, Tokyo, Japan

Abstract

In this paper, the effect of the inducer tip clearance is studied to understand its impact on the cavitating and non-cavitating performance of centrifugal pumps. Helical inducers with constant pitch and with variable (progressive) pitch are considered.

Computational Fluid Dynamics (CFD) simulations of a single stage pump are conducted on each inducer type to determine the cavitating (two-phase) and non-cavitating (single- phase) performance for varying inducer tip clearance. The Rayleigh-Plesset cavitation model is used to understand the bubble dynamics under the assumptions of single fluid undergoing no thermal energy transfer between each phase. Experimental tests are conducted on a pump with variable pitch inducer to determine the true performance in cavitating and non-cavitating operating conditions. Experimental results are compared to

the simulations to validate the accuracy of the proposed numerical modelling. Net Positive

Suction Head (NPSH) with 3% differential head drop is used as a criterion to identify the true cavitation performance of each inducer configuration. It is found that, as the inducer tip clearance increases, excessive back leakage and larger vortex recirculation occur at the tip location. These result in pressure loss within the inducer and, consequently, degrades the cavitation performance. In addition, the change in cavitation performance with tip clearance is much more evident for variable pitch inducer geometries as compared to the constant pitch case. Furthermore, the impact on the non-cavitating performance of inducer tip clearance is found to be minimal.

Keywords: Inducer, Cavitation, Tip Clearance, Net Positive Suction Head, Computational

Fluid Dynamics

1. Introduction

In cryogenic application, in-tank centrifugal pumps are installed inside a Liquefied

Natural Gas (LNG) storage tank in a vertical pump column to transfer the fluid and often empty the tank to its lowest liquid level possible. In order to achieve the minimum liquid height during draining of the tank, each pump is fitted with an inducer which is fundamentally an axial flow impeller with high specific speed. Typical configuration of an in-tank pump is shown in Figure-1. A helical inducer is installed at the pump suction side in the lowest possible position, to reduce the required suction head to operate the pump safely [1]. This required suction head is also known as the NPSHr (Net Positive Suction

Head required).

Figure-1 Typical configuration of a cryogenic LNG in-tank pump by Elliott Group Cryogenic Products [1].

Inducers have been used in many industrial pump applications for which the inlet pressure is relatively low and the pump may suffer from cavitation if the inlet pressure drops below the saturation pressure of the pumped fluid. Inducer technology was initiated by NASA to improve the suction pressure of rocket turbopumps in the 1970s [2-4]. Rocket turbopumps have very little inlet pressure to draw fluid into the pump internals, which are thus prone to cavitate. The formation and types of cavitation in centrifugal pumps have been studied by Japikse et al. [5]. The cavitation can occur in different forms and is highly detrimental to the pumps directly and indirectly. Cavitation erosion can destroy inducer

and impeller blades due to vapor formation which can be considered as direct cavitation damage. Indirect failure modes are often due to high vibration and noise. Vapor formation cannot be tolerated in the pump internals especially at locations where a hydrodynamic fluid film is required. The locations that are vulnerable to cavitation are mainly pump shaft support locations such as shaft bushings, hydrodynamic or hydrostatic bearings, and rolling element ball bearings. The cavitation performance, occurrence and prevention of cavitation in pumps have been studied since the 1960s. Japikse in [6] summarized the design methodology, flow observations, design practices of inducers in the review of inducer design for industrial and turbopump applications. In this review, a large number of studies related to design guidelines of inducers can be found.

Cavitation performance of centrifugal pumps with helical inducers was investigated by Sutton in 1964 [7]. Therein, a relationship is suggested between the flow coefficient and the blade angle for the optimum suction performance. Acosta [8] investigated the effect of the inducer tip clearance on cavitation performance of helical inducers with 9° blade angle.

In this study, experiments were conducted to determine not only the effect of the inducer tip clearance but also of the solidity of the inducer (blade versus fluid volume) on the cavitation performance. It was found that increasing the tip clearance tends to reduce the hydraulic performance under non-cavitating condition. Regarding cavitation performance, the smallest tip clearance gave the best performance. This study also included information related to various modes of cavitating flows and their development as a function of flow and cavitation coefficients.

Due to difficulty, time and expense involved in testing pumps under two phase conditions, numerical methods have been developed to verify and enhance NPSHr performance of centrifugal pumps. Bakir et al. [9] developed a cavitation model implemented in a Computational Fluid Dynamics (CFD) package to simulate extensive cavitation in an inducer. Experimental and numerical solutions are presented and show reasonably good agreement. Their validation of the model was satisfactory for the head drop curves and the cavitation formation’s size and location at various flow rates on the inducer. Watanabe and Ichiki [10] designed two different style inducers for an industrial cryogenic pump by using three-dimensional inverse design methodology. The flow characteristics and suction performance of the new inducers were initially verified via two- phase CFD simulations of the complete pump assembly, including the existing impeller and diffuser vanes besides the newly designed inducers. The Rayleigh-Plesset bubble dynamics-based cavitation model [27] was implemented in their CFD study. The inducer designs were confirmed by water testing in two-phase flow condition. Head drop curves of the simulation and the testing show good agreements confirming the validity of their design methodology for cavitation performance. Another method used to determine the NPSHr performance is to investigate and calculate the blade loading of hydraulic components at the suction side of the turbomachinery. Inception and growth of the cavitation are highly related to the pressure distribution on the blade surfaces. Therefore, it is essential to control the pressure distribution (blade loading) on the blade surfaces [10,11].

Cavitation inception and formation depend on many design features of the inducer such as leading and trailing blade geometry (incidence angle, blade thickness, and blade shape, etc.), blade count, as well as the pre-existing flow characteristics downstream and

upstream of the inducer [12-15]. Leading edge shape has been shown to have an influence in cavitation performance [12-14]. A certain increase in slope towards an optimal value of the leading edges can enhance the cavitation performance at high flow rates and weaken pressure fluctuations. In addition, sharpening at the leading edge can have a slight influence on the cavitation performance as well [13]. It should be noted, however, that all the geometric considerations and design features aiming at improving the cavitation performance may negatively impact the non-cavitating performance of the inducer or even the overall pump performance. As an example, inducer blade count not only influences the

NPSH performance but also the overall head (pressure) of the pump. Therefore, it is essential to identify the appropriate number of blades when designing inducers [15].

The effect of the tip clearance has been investigated for turbopump inducers in [16-

19], where the non-cavitating and cavitation performance of turbopump inducers with various tip clearances are studied to better understand the flow structure and its impact to inducer and pump overall performance. Tip clearance has been shown to influence the internal flow structure due to back flow through the tip clearance, which results in back flow leakage vortices. Along the direction of the flow, pressure increases with the increase in fluid’s kinetic energy. The fluid’s overall pressure increases which eventually results in back flow at the tip clearance. This back flow and associated vortices can ultimately effect cavitation and the formation of bubbles, and consequently impact the overall performance

[20]. In addition, back flow can reduce the head increase under non-cavitation operating conditions as well. This can result in flow recirculation at the tip, loss of efficiency, and head loss. Therefore, tip clearance is an important design parameter in inducers. An optimum clearance shall be utilized for each application [6, 20].

While variable and constant pitch helical inducers are widely used in cryogenic in- tank pump applications, to best of the authors’ knowledge, there has been no research conducted on the impact of pitch style along with tip clearance to the non-cavitating and cavitating performance. Therefore, to fill this knowledge gap, this paper investigates, for both cavitating and non-cavitating conditions, the effect of tip clearance on pump performance for (i) an inducer with a progressive (variable) pitch and (ii) an inducer with a constant pitch design. CFD simulations are performed for the entire pump assembly, including two-bladed helical inducer with impeller and diffuser vane hydraulic components. Vapor formation and vapor volume fraction are investigated not only at the inducer section but also at the impeller suction region to better understand the cavitation inception, vapor propagation and cavitation breakdown point under different inlet pressure condition. Experiments are conducted to validate CFD simulation results in terms of

NPSH3 performance and vapor formation by comparing test data and actual cavitation images to the CFD simulation results. The change in cavitating and non-cavitating performance as a function of the inducer tip clearance is presented for each style inducer.

The rest of the paper is organized as follows. Section 2 includes specifications and technical information of the industrial pump, hydraulic components, and geometric details of each inducer. Section 3 covers the details of the CFD simulations. Section 4 describes the experimental test facility and the test procedure for the pump assembly with the inducer.

Section 5 reports the experimental and CFD simulation results. Conclusions are reported in Section 6.

2. Pump and Inducer Specifications

The overall specifications and main characteristics of the industrial pump investigated in this study are listed in Table-1. As indicated in the introduction section, this pump is used in an LNG storage tank and installed into a vertical pump column inside the tank. These pumps are known as retractable in-tank pumps. A typical cross section of the in-tank pump is given in Figure-1. The pump is furnished with a helical inducer to delay the cavitation inception so that it can operate at very low liquid levels, hence, with minimum suction pressure. It should be noted that LNG tanks are not pressurized, therefore, suction pressure is limited to the static height of the liquid inside the tank [29].

Table-1 Pump Performance Data and Operational Specifications (Full Scale).

Design Flow: 695 m3/hr

Pump Differential Head: 270 m

Pumping Fluid: LNG (98% Methane and 2% Propane in Molar Basis)

Fluid Temperature: -165 °C

Fluid Density: 438.6 kg/m3

Rotational Speed: 3000 RPM

The inducer design is crucial for in-tank pumps, as it ultimately determines the minimum liquid level in the tank required for stable operation. This liquid level is known as the Net Positive Suction Head required (NPSHr) for each in-tank pump. The goal is thus to minimize the NPSHr as much as possible, to maximize the height of usable liquid inside the tank. Considering the overall foot print of an LNG tank, reducing the NPSHr by

as little as few centimeters can result in substantial improvements in usable amount of fluid volume. For example, each LNG storage tank recently constructed in Corpus Christi, TX

Liquefaction Terminal has 1.6 million cubic meter capacity with base diameter of approximately 200 meters [29].

In-tank pumps utilize helical style axial inducers with variable or constant pitch construction. The axial inducers studied in this paper are shown in Figure-2. The geometric parameters of the inducers are given in Table-2. It should be noted that the pump is scaled down by 150/236 due to the limitations of the testing facility, and water at 25°C is used due to hazardous nature of LNG, see also Section 4. Geometric parameters are reported in

Table-2 for the experimental setup and simulations. Both inducers are two-bladed, with blades 180° apart in angular position. The main differences between the inducers are the pitch and the blade chord length.

(a) (b)

Figure-2 Helical style axial inducer geometries, (a) variable pitch, (b) constant pitch inducer.

Table-2 Geometric parameters and specification of the inducers (Scaled).Geometric parameters and specification of the inducers (Scaled).

Constant Pitch Inducer Variable Pitch Inducer

Rotational Speed: 1700 RPM 1700 RPM

Design Flow Coefficient ( ): 0.136 0.136

Blade Count (n): 𝚽𝚽 2 2

Tip Diameter (Dt): 150 mm 150 mm

Hub Diameter (d): 45 mm 45 mm

Tip Clearance ( ): 0.175 mm, 0.35 mm, 0.70 0.175 mm, 0.35 mm, 0.70 mm, 1.05 mm, 1.4 mm mm, 1.05 mm, 1.4 mm 𝜹𝜹 Dimensionless Tip Clearance 0.0012, 0.0023, 0.0047, 0.0012, 0.0023, 0.0047, ( = / ): 0.007, 0.0092 0.007, 0.0092

Pitch𝝀𝝀 𝜹𝜹 (P𝑫𝑫):𝒕𝒕 2.78 mm/10° rotation Variable Blade Chord Length (c): 70.4 mm 81.8 mm

3. Numerical Simulation

Initially, the non-cavitating performance of the pump is modelled to verify and validate the accuracy of the CFD code, geometrical model and the grid. Pump simulations are performed for different flow rates in non-cavitating conditions. Later, the cavitation performance is studied for a given flow rate in different cavitation conditions. Details of the CFD code, cavitation parameters and model, and grid geometry of the complete pump assembly including inducer, impeller and the diffuser vane are discussed in this section.

3.1 Computational Fluid Dynamics (CFD)

ANSYS CFX 19.1 turbomachinery computational code is utilized in simulating the pump assembly with and without cavitation. Since the flow is turbulent, Reynolds- averaged Navier Stokes equations (RANS) are solved along with an appropriate turbulence model. According to the findings by Mani et al. [28], - turbulence model is the most appropriate turbulence model for rocket turbopump inducer𝑘𝑘 𝜀𝜀 applications. Therefore, - turbulence model is used in the present study. 𝑘𝑘 𝜀𝜀

The Grid Convergence Method (GCI) method described by Celik et al. [30] is used to estimate the discretization error. The error estimates are approximated in terms of fine- grid convergence index (GCI index). Three different size grids are considered in the determination of GCI error for the finest grid, which is ultimately the grid used in the simulations. Total number of elements, grid size, grid refinement factor between each grid, and error estimate along with the apparent order p are reported in Table-3. According to the pump differential head calculations and NPSH estimations of three selected grids, the

GCI error is determined as 0.55%. This error margin is shown at cavitation performance versus tip clearance plot to justify its acceptance.

For each configuration, the inlet pressure is defined with an outlet mass flow rate as the boundary condition. Since there is no change in fluid temperature, calculations are performed based on isothermal condition (25 °C). Simulations are run under the steady state condition for both cavitating and non-cavitating cases until a desired residual level of convergence is reached. The target residual RMS value of 10-5 is achieved throughout the

analysis. This convergence level exceeds the convergence value suggested by ANSYS

CFX for steady state calculations [22].

Table-3 Calculation results of discretization error according to [30].

Number of Elements (N): N1= 13,500,000 N2= 6,800,000, N3= 4,700,000

Grid Refinement (r): r21=1.533, r32= 1.651

Pump Differential Head ( ): H1= 19.375 m, H2=19.128 m, H3=19.042 m

Apparent order ( ): 𝑯𝑯 3.189

21 Extrapolated Value𝒑𝒑 of Head Hext = 19.46 (Hext):

21 Relative error (ea): ea = 1.27%

21 Extrapolated relative error eext = 0.44% (eext):

21 Grid Convergence Index(GCI): GCIfine = 0.55%

3.2 Cavitation Parameters

Cavitation is defined as the formation of vapor bubbles in low pressure region within a flow. The formation of the vapor bubbles starts when the static pressure in the liquid reaches to the vapor pressure, , of the liquid for a given temperature. The cavitation

𝑣𝑣 inception and the tendency to cavitation𝑃𝑃 is defined in a non-dimensionalized form as follows [20]:

( ) = (1) 1 1 𝑣𝑣 𝑃𝑃2 − 𝑃𝑃 𝜎𝜎 2 𝜌𝜌𝑈𝑈

where is the cavitation number, is the suction pressure, is the reference velocity,

1 which is𝜎𝜎 the inducer tip speed and can𝑃𝑃 be defined as /2 where𝑈𝑈 is the rotational speed

𝑡𝑡 and is the inducer tip diameter. It should be noted𝜔𝜔 that,𝐷𝐷 for any flow𝜔𝜔 rate with or without

𝑡𝑡 presence𝐷𝐷 of vapor, there is a corresponding cavitation number. In the context of centrifugal pumps, there are three distinct cavitation numbers that can be related to the pump’s cavitation characteristics [20]. The first one is the cavitation inception number, , which

𝑖𝑖 corresponds to the initial formation of vapor bubbles within the pump. The second𝜎𝜎 one is the critical cavitation number, , at which the pump’s differential pressure drops by 3%

𝑐𝑐 with decrease in suction pressure.𝜎𝜎 Further reduction in suction pressure results in major differential pressure loss at the pump and this corresponds to the breakdown cavitation number, . In industrial pump applications, the Net Positive Suction Head requirement

𝑏𝑏 (NPSHr)𝜎𝜎 of the pump is defined as the suction head that corresponds to the critical cavitation number. It is often referred as NPSH3 which implies that pump head loss is 3%

[23]. In the present study, the focus is on the NPSH3 (critical cavitation number) performance for each configuration, as this is the cavitation performance specification accepted by many industrial norms and guidelines (ANSI, ISO and API) [23-25].

3.3 Cavitation Model

In ANSYS CFX, cavitation is defined as a multiphase homogenous model. It assumes that the vapor velocity field is same as that of the liquid. This approach, also known as the Equal-Velocity-Equal-Temperature (EVET) approach [9, 16] is commonly used to model cavitation.

The ZGB (Zwart-Gerber-Belamri) transport-based cavitation equation is implemented in ANSYS CFX simulate the cavitation. This equation is used to determine the mass transfer rate for controlling vapor generation and condensation [21, 27].

3 2 | | = sgn( ) (2) 3 𝛼𝛼𝑣𝑣𝜌𝜌𝑣𝑣 𝑃𝑃𝑣𝑣 − 𝑃𝑃 𝑚𝑚̇ 𝑓𝑓 � 𝑃𝑃𝑣𝑣 − 𝑃𝑃 𝑅𝑅𝑏𝑏 𝜌𝜌𝑙𝑙 where is the bubble radius, and are density of the vapor and liquid, respectively,

𝑏𝑏 𝑣𝑣 𝑙𝑙 is an𝑅𝑅 empirical constant with𝜌𝜌 distinct𝜌𝜌 values for condensation ( ) and vaporization

𝑐𝑐𝑐𝑐𝑐𝑐 (𝑓𝑓 ) since phase change rate is not the same for each case. Vaporization𝑓𝑓 often occurs at

𝑣𝑣𝑣𝑣𝑣𝑣 a𝑓𝑓 much faster rate as compared to condensation ( > ). Equation (2) for mass

𝑐𝑐𝑐𝑐𝑐𝑐 𝑣𝑣𝑣𝑣𝑣𝑣 transfer rate must be further modified to better represent𝑓𝑓 the𝑓𝑓 vaporization case since the increase in vapor volume fraction should cause a decrease in density in the bubble region.

Therefore, the vapor volume fraction, , in equation (2) is replaced with (1 ) for

𝑣𝑣 𝑏𝑏 𝑣𝑣 vaporization case, where is the volume𝛼𝛼 fraction of the bubble region. Finally,𝛼𝛼 − the𝛼𝛼 mass

𝑏𝑏 transfer rates for condensation𝛼𝛼 and vaporization can be rewritten as follows, respectively

[21]:

3 2 ( ) = if > (3) 3 𝛼𝛼𝑣𝑣𝜌𝜌𝑣𝑣 𝑃𝑃 − 𝑃𝑃𝑣𝑣 𝑚𝑚̇ 𝑐𝑐𝑐𝑐𝑐𝑐 −𝑓𝑓𝑐𝑐𝑐𝑐𝑐𝑐 � 𝑃𝑃 𝑃𝑃𝑣𝑣 𝑅𝑅𝑏𝑏 𝜌𝜌𝑙𝑙 3 (1 ) 2 ( ) = if > (4) 3 𝛼𝛼𝑏𝑏 − 𝛼𝛼𝑣𝑣 𝜌𝜌𝑣𝑣 𝑃𝑃𝑣𝑣 − 𝑃𝑃 𝑚𝑚̇ 𝑣𝑣𝑣𝑣𝑣𝑣 𝑓𝑓𝑣𝑣𝑣𝑣𝑣𝑣 � 𝑃𝑃𝑣𝑣 𝑃𝑃 𝑅𝑅𝑏𝑏 𝜌𝜌𝑙𝑙 Table-4 reports the values of constants used in the mass transfer rate equations implemented in the CFD simulations.

Table-4 Cavitation model constant parameter values [21].

Bubble Radius ( ): 1µm

𝒃𝒃 Bubble Region Volume𝑹𝑹 Fraction ( ): 5e-4

𝒃𝒃 Condensation Factor ( ): 𝜶𝜶 0.01

𝒄𝒄𝒄𝒄𝒄𝒄 Vaporization Factor ( 𝒇𝒇 ): 50

𝒗𝒗𝒗𝒗𝒗𝒗 𝒇𝒇

3.4 Geometric Model and Grid

The three-dimensional view and a side view of the geometric model are shown in

Figure-3. This model consists of a helical inducer, a radial type closed impeller and an axial diffuser vane. ANSYS BladeGen and TurboGrid software are used to create the grid for each hydraulic component.

Diffuser Impeller Vane

y y Inducer z x z x

Figure-3 Pump assembly consists of inducer, impeller and the diffuser vane used in simulations.

Initially, the shroud, hub, and vane geometry for each part are defined separately in

BladeGen. Then, the grid for each component is created, and finally each part is aligned and joined in the ANSYS CFX’s preprocessing step. The quality of the grid highly impacts the convergence level, hence, the accuracy of the results. Therefore, mesh refinement at areas of interest, a reasonable expansion rate (geometric size increase between maximum and minimum element sizes), and global mesh size are adjusted to achieve an acceptable convergence and computational time. GCI index is also calculated to determine the discretization error (See Table-3). The final grid of the hydraulic components is shown in

Figure-4. The blade tip, leading and trailing edge, vane root locations for each hydraulic component are meshed with refinement (finer meshing with smaller element size) to accurately capture the impact of the vane shape and geometric discontinuities.

Inducer y Impeller z x Diffuser Vane Figure-4 Mesh quality of the pump hydraulic components.

The total number of elements is approximately 13.5 million for the complete assembly, and 3.4 million elements are used to model the inducer section along with the gap between the inducer blade and the shroud. A closer view of the inducer tip location is given in Figure-5 to show the mesh quality at the tip clearance for each configuration of the inducer.

𝛿𝛿

(d) (e)

Figure-5 Mesh quality at the inducer tip clearance, Mesh quality at the inducer tip clearance, (a) =0.175mm (b) =0.35mm, (c) =0.70mm, (d) =1.05mm, (e) =1.40mm.

𝛿𝛿 𝛿𝛿 𝛿𝛿 𝛿𝛿 𝛿𝛿 4. Experimental Setup

Experiments are conducted in the Ebara Corporation’s Research and Development

Test Facility located in Futtsu Plant, Japan to determine the cavitating and non-cavitating

performance of the pump assembly. This test facility is a closed loop system which consists of a pressure control tank fitted with a vacuum pump, control and instrumentation devices along with the pump to be tested that is coupled to a driver (AC motor) [10].

Pump

Vacuum Pump Test Torque Section Meter Motor

TOP VIEW

Pump Test Section Torque Meter Motor

Pressure Control Tank

SIDE VIEW

Figure-6 Schematic view of the test facility.

The schematic view of the facility is shown in Figure-6. Water is used as the test fluid since LNG is not suitable for indoor testing due to its hazardous and flammable nature.

Pump hydraulics are scaled down by 0.64 for testing purposes due to the size limitations of piping, pressure tank and power rating of the motor driver. Scaled down inducer, impeller and the diffuser vane are installed in to the pump test section with an outlet guide vane at the downstream of the diffuser vane in order to reduce the angular momentum of

the fluid and to guide the flow to downstream piping smoothly. The cross-sectional view of the test pump is shown in Figure-7. Pressure and temperature transmitters are utilized to measure the suction and discharge pressure and temperature of the pump test section. A torque meter is coupled to the pump shaft to measure the torque input to determine the pump’s efficiency based on the mechanical input power to the pump shaft and the hydraulic output power.

Diffuser Vane

Inducer Outlet Guide Vane

Shaft

Impeller

Figure-7 Pump test section of the test facility.

The inducer section utilizes a clear transparent casing for viewing purposes to capture cavitation images under operating condition. A data acquisition system is utilized to monitor and record the flow rate, pressure and temperature at various locations, torque and rotational speed of the pump shaft. Tests are conducted at a constant rotational speed of 1700 rev/min under varying suction pressure and flow rate. Water temperature is kept constant at 25°C (ambient) while the pressure is adjusted with the vacuum pump attached

to the pressure tank. Tests are conducted on a variable pitch inducer with =0.35mm tip clearance only. 𝛿𝛿

5. Results

Both simulations and experiments are initially conducted on the non-cavitating performance of the pump assembly for flow rates between the minimum (20 m3/hr) and maximum flow rate (140 m3/hr) of the pump. Figure-8 shows the pump performance curve for the original pump configuration ( =0.35mm). This figure also demonstrates the agreement of the CFD simulations to𝛿𝛿 test data for the non-cavitation performance.

Additionally, non-cavitating results of the simulations are used as initial conditions for cavitation cases.

30 180 CFX Simulation Head, 0.175mm, 19.396 CFX Simulation Head, 0.35mm, 19.375 CFX Simulation Head, 0.70 mm, 19.281 25 CFX Simulation Head, 1.05 mm, 19.069 150 CFX Simulation Head, 1.40 mm, 18.943

20 120

15 90 Head[m]

∆ 10 Test Head Data, 0.35 mm 60 CFX Simulation Head, 0.175mm CFX Simulation Head, 0.35mm CFX Simulation Head, 0.70 mm 5 CFX Simulation Head, 1.05 mm 30 CFX Simulation Head, 1.40 mm Test Efficiency, 0.35 mm CFX Simulation Eff, 0.35 mm Efficiency [%] 0 0 0 20 40 60 80 100 120 140 160 Flow Rate [m3/hr]

Figure-8 Pump performance curve.

At relatively high flow rates (100 m3/hr and greater), a slight discrepancy is noted between experimental and simulation results. This discrepancy is due to the recirculation and backflow leakage at the test assembly. Specifically, impeller back leakage around the shrouds and recirculation around pump shaft at the discharge section towards the suction area are not included in simulations. As the pump flow rate increases, the leakage becomes more significant, which causes a greater discrepancy between test and simulations. At the pump rated duty point of 105.6 m3/hr, the difference between the CFD and the test results for variable pitch inducer with 0.35 mm tip clearance is within 3%. With the increase in the tip clearance, the overall pump differential head drops by 2% between the original tip clearance of =0.35 mm and the largest tip clearance of =1.75 mm. The differential head for each tip clearance𝛿𝛿 of the variable pitch inducer is also𝛿𝛿 shown in Figure-8. The total pressure distribution in the meridional view at the pump section is shown in Figure-9 with the variable pitch inducer at rated duty point (105.6 m3/hr).

Diffuser Vane

Impeller Inducer δ = 0.35 mm, λ = 0.0023

Figure-9 The total pressure distribution in meridional view for variable pitch inducer. Typical pressure distribution is shown for complete pump assembly.

The 2D meridional view displays the average of the values of any parameter

(pressure, velocity, etc.) along the circumference. The velocity vectors at the inducer section are examined to better understand and observe the significance of tip clearance on backflow leakage. Figures 10 and 11 show the velocity vectors in the axial direction and the vortex at the tip area due to backflow leakage at the inducer tip. With the increase in tip clearance, , the size of the vortex increases. This is mainly due to the amount of backflow. The𝛿𝛿 meridional pressure distribution of the pump assembly with constant pitch inducer is shown in Figure-12.

δ = 0.175 mm

δ = 1.05 mm

δ = 0.35 mm

δ = 1.40 mm

δ = 0.70 mm

z x

Figure-10 Axial velocity and backflow leakage of variable pitch inducer.

δ = 0.175 mm

δ = 1.05 mm

δ = 0.35 mm

δ = 1.40 mm

δ = 0.70 mm

z x

Figure-11 Axial velocity and backflow leakage of constant pitch inducer.

Diffuser Vane

Inducer Impeller

δ = 0.35 mm, λ = 0.0023

Figure-12 The total pressure distribution in meridional view for constant pitch inducer. Typical pressure distribution is shown for complete pump assembly.

For both inducer styles, an increase in tip clearance results in reduction of pressure differential. This is primarily due to the leakage through the inducer tip. However, the reduction of differential pressure at the inducer location is not significant enough to change the pump’s overall differential head. The increase in tip clearance changes the overall head by less than 2.3% for the variable pitch inducer, as per the pump performance curve given in Figure-8. For the constant pitch inducer, the change in pump differential pressure is less than 1.5%. This is mainly due to the magnitude of head generation by the inducer versus the other hydraulic components. Under non-cavitating operation, the variable pitch inducer head generation is around 13% of the total head generation of the complete pump assembly.

For the constant pitch inducer this is around 11%. Therefore, the change in inducer’s head does not necessarily have a considerable impact on the total head increase for the pump assembly. However, the differential head increase between the variable pitch and the constant pitch inducer is approximately 8% for a given tip clearance. The variable pitch inducer produces more head for a given tip clearance and flow rate due to its blade geometry. The progressive pitch design allows the fluid volume to decrease along the chord length, subsequently increasing the pressure while the fluid is pushed in the axial direction.

For constant pitch inducer, the pressure increase is only achieved by the increase in kinetic energy of the fluid by the rotational speed (shaft) and there is no change in fluid volume along the chord length.

As the backflow leakage increases with the increase in the tip clearance, a flow component in the opposite direction to the meridional axial flow is observed. This eventually results in vorticity at the inducer tip location. With increase in tip clearance, the size of the vortex also increases for both inducer configurations. The tip vortex is

detrimental to the pump performance: the vortex acts in the opposite direction of the axial flow in the pump, it can result in flow restriction or flow recirculation within the inducer.

The pressure distribution at the inducer blades is examined for all inducer configurations and tip clearances. The main purpose of this analysis is to identify where the pressure is decreasing and thus predict where the blade cavitation will most probably occur. In addition, the pressure distribution at the blade also indicates the blade loading and is related to the pressure increase along the inducer. Figure-13 shows the pressure distribution at the inducer blade surfaces for variable pitch inducer.

δ = 0.175 mm δ = 0.35 mm δ = 0.70 mm

Max Pressure Area

Low Pressure Area

z x δ = 1.05mm δ = 1.40 mm

Figure-13 Pressure distribution on the blade surfaces for variable pitch inducer.

With increasing tip clearance, the pressure between each blade decreases due to backflow leakage. This can be observed from the maximum pressure at the front side of

the inducer blade, which is close to 1.3x105 Pa. For the tip clearances of = 0.70 mm, 1.05

mm and 1.40 mm, the effective area of maximum pressure is considerably𝛿𝛿 decreased. There

are areas with relatively lower pressure at the front face between the two blades for =

1.05 mm and 1.40 mm tip clearance cases. These low-pressure areas are not evident 𝛿𝛿for

smaller tip clearances. In addition, at the leading edge of the blade, the pressure at the tip

section increases with increasing tip clearance. For all cases, the lowest pressure occurs at

the leading edge of the inducer tip where the fluid velocity would be high due to rotational

speed. This location is where the blade cavitation will most likely initiate.

δ = 0.175 mm δ = 0.35 mm δ = 0.70 mm

Low Pressure y Area z x δ = 1.05mm δ = 1.40 mm

Figure-14 Pressure distribution on the blade surfaces for constant pitch inducer.

Figure-14 shows the pressure distribution at the blades for the constant pitch

inducer. The only obvious difference between each tip clearance is at the front side where

the lowest pressure is calculated. The effective area of the low-pressure section increases

with the increase in tip clearance. Besides that location, the rest of the pressure distribution is fairly similar for each case despite the increase in tip clearance. This is probably due to the characteristics of the pressure differential across the constant pitch inducer. The pressure distribution difference between each tip configuration is insignificant, hence the back-leakage effect and pressure loss with the tip clearance increase is minimal for constant pitch inducer.

The head coefficient, (= 2 / ), as a function of cavitation number, , is 2 plotted in Figure-15 for the variable𝜓𝜓 pitch𝑔𝑔𝑔𝑔 𝑈𝑈 inducer for each tip clearance case. As the𝜎𝜎 tip clearance increases, the cavitation number increases, which suggests that increasing tip clearance results in early cavitation and higher NPSHr levels.

1.10 c b a d 1.05

1.00 3% Head Drop Line

0.95

Head Coefficient = / , ,

𝜓𝜓 0.90 λ λ=0.0012𝛿𝛿 𝐷𝐷𝑡𝑡 λ=0.0023 0.85 λ=0.0047 λ=0.007 λ=0.0092 0.80 0.25 0.50 0.75 1.00 1.25 1.50 σ, Cavitation Number

Figure-15 Head drop curve, cavitation number vs. head coefficient for variable pitch inducer.

The difference in cavitation number between the smallest and the largest gap is calculated as 11% for variable pitch inducer. Vapor formation and cavitation inception is visually examined for four different cavitation numbers, 0.9654, 0.8152, 0.6649 and

0.5147, identified as cases “a”, “b”, “c” and “d” in Figure-15. These cavitation numbers correspond to suction pressures of 10000, 9000, 8000 and 7000 Pa, respectively. Figure-

16 shows the vapor volume fraction for each cavitation number (“a”, “b”, “c” and “d” points in Figure 15) for the variable pitch inducer. The cavitation initiates at the tip of the leading-edge blade of the inducer and propagates with the reduction in suction pressure.

Cavitation point “a” in Figure-15, is the cavitation inception for the variable pitch inducer. For this cavitation number, the pump differential head (or head coefficient, ) has not yet dropped by 3% (NPSHa>NPSH3). Vapor formation is noted at this point, but𝜓𝜓 the vapor volume fraction is not significant enough to cause any degradation of the pump differential head. Vapor forms initially at the tip of the leading edge of the inducer blade where the velocity is high due to rotational speed and pressure is low due to backflow leakage. As expected, the vapor volume for the largest tip clearance, δ = 1.40 mm, is greater with respect to the smaller tip clearance configurations. This can be seen by examining vapor volume fraction distribution on the inducer leading and trailing edge tip locations for each configuration at Figure-17 (a). For tip clearances δ = 1.40 and 1.05 mm, the vapor has moved from the leading to the trailing edge of the vane across the tip region due to leak and pressure losses through the clearance. The same propagation of the vapor formation is not evident for other tip clearances (δ = 0.175, 0.35 and 0.75 mm), since the inducer is able to maintain the pressure between leading and trailing blades at cavitation point “a”.

z x δ = 0.175 mm δ = 0.35mm

δ = 1.05 mm δ = 1.40 mm δ = 0.75 mm Figure-16a Vapor volume fraction at the pump assembly for variable pitch inducer at inlet pressure “a”.

z x δ = 0.175 mm δ = 0.35 mm

δ = 0.70 mm δ = 1.05 mm δ = 1.40 mm

Figure-16b Vapor volume fraction at the pump assembly for variable pitch inducer at inlet pressure “b”.

z x δ = 0.175 mm δ = 0.35 mm

δ = 0.70 mm δ δ = 1.05 mm = 1.40 mm Figure-16c Vapor volume fraction at the pump assembly for variable pitch inducer at inlet pressure “c”.

z x δ = 0.175 mm δ = 0.35 mm

δ = 0.70 mm δ = 1.05 mm δ = 1.40 mm Figure-16d Vapor volume fraction at the pump assembly for variable pitch inducer at inlet pressure “d”.

Once the suction pressure drops to point “b” in Figure-15, the vapor formation extends to the impeller eye section for δ = 1.40 and 1.05 mm, with a reduction in pump discharge pressure. For δ = 0.175 and 0.35 mm, the vapor formation and the pump discharge pressure show no change, regardless of the suction pressure (cavitation number) reduction. There is only an insignificant increase at the leading-edge vapor volume for this case. The smaller tip clearance allows the inducer to maintain the pressure at the trailing sections since the backflow leakage is relatively less pronounced for δ = 0.175 and 0.35 mm with respect to other cases at cavitation.

For the cavitation number point “c”, the pump differential head further drops for δ

= 1.40 and 1.05 mm cases, and the vapor at the impeller eye further propagates. This vapor formation at the inducer and the impeller eye section results in 2% pump differential head drop for δ = 1.40 and 1.05 mm cases. For δ = 0.70 mm, the pump differential head drops, and vapor extends to the impeller eye and the trailing section of the inducer. For the smaller tip clearance cases, the inducer is still able to prevent pressure losses at the trailing section.

While the vapor volume increases at the leading edge of the inducer, there is only very small amount of vapor formation at the eye section of the impeller for δ = 0.35 mm. This can be explained with the prevention of pressure losses and leakage through a tighter gap.

The reduction of suction pressure to point “d” results in significant pump differential head drop for δ = 1.40 and 1.05 mm, which can cause cavitation damage if the pump is operated at this cavitation number continuously. The head drop beyond 3% with a sudden decrease in head is identified as the cavitation breakdown [20]. Figure-16 (d) shows that the high vapor volume fraction at the inducer blades and flow passages, and at the eye

section of the impeller, is blocking the fluid flow and causing the pump to lose prime for δ

= 1.40 and 1.05 mm cases. The vapor formation for case δ = 0.70 mm at point “d” also results in head drop by 3%. The blockage at the inducer and impeller eye is not as significant as the larger tip clearance cases. However, if the pump suction pressure is further reduced, this tip clearance will also reach the breakdown point, as indicated in the head drop curve in Figure 15. For δ = 0.70 and 0.35 mm, there is no blockage at the eye but vapor formation is observed. The vapor volume fraction for the smallest gap is not large enough to cause any considerable pump head drop. Therefore, at point “d”, a pump with the smallest inducer tip clearance can be operated safely without any concern of cavitation.

The head drop curve for the constant pitch inducer is shown in Figure-17 with the cavitation numbers corresponding to NPSH3. The head coefficient exhibits a sudden drop for the constant pitch inducer as opposed to a more gradual change seen with the variable pitch inducer. The difference between each case can be explained by examining the vapor volume fraction at the impeller eye. There is no vapor formation at the eye section of the impeller prior to point “d” for the constant pitch inducer. The vapor forms mostly at the inducer section prior to point “d”, where vapor propagation towards the eye section is only evident at point “d” for the constant pitch inducer. The vapor formation behaves differently for the variable pitch inducer case, as the vapor starts to develop gradually at the impeller eye. This behavior is highly dependent on the inducer blade geometry and the style of each inducer. For constant pitch inducer, the pressure increase across the inducer is relatively modest which also reduces the amount of backflow leakage.

1.10

c b a 1.05 d

1.00 3% Head Drop Line

0.95

= / 0.90 λ=0.0012 Head Coefficient λ 𝛿𝛿 𝐷𝐷𝑡𝑡 , λ=0.0023 𝜓𝜓 0.85 λ=0.0047 λ=0.007 λ=0.0092 0.80 0.25 0.50 0.75 1.00 1.25 1.50 σ , Cavitation Number

Figure-17 Head drop curve, cavitation number vs. head coefficient for constant pitch inducer.

It is important to observe that head drop and cavitation performance for each case

are affected differently whether the inducer is considered by itself on in the context of the

complete pump assembly. Without the pump assembly (excluding impeller and diffuser

sections), the head drop and the NPSH3 value for the inducer alone and the complete pump

assembly including all the hydraulic components will differ considerably as the constant

pitch inducer exhibit vapor formation mostly at the inducer section. In addition, head drop

by %3 (NPSH3) for the pump assembly occurs when vapor formation propagates to

impeller eye section. In other words, a large amount of vapor formation at the inducer does

not necessarily result in 3% or more head drop in the pump assembly. However, the large

amount of vapor formation at the inducer section reduces inducer’s differential head

significantly.

z x

δ = 0.175 mm δ = 0.35 mm

δ = 0.70 mm δ = 1.05 mm δ = 1.40 mm Figure-18a Vapor volume fraction at the pump assembly for constant pitch inducer at inlet pressure “a”.

z x δ = 0.175 mm δ = 0.35 mm

δ δ = 0.70 mm δ = 1.05 mm = 1.40 mm

Figure-18b Vapor volume fraction at the pump assembly for constant pitch inducer at inlet pressure “b”.

z x δ = 0.175 mm δ = 0.35mm

δ = 0.70 mm δ = 1.05 mm δ = 1.40 mm Figure-18c Vapor volume fraction at the pump assembly for constant pitch inducer at inlet pressure “c”.

z x

δ = 0.175 mm δ = 0.35 mm

δ = 0.70 mm δ = 1.05 mm δ = 1.40 mm Figure-18d Vapor volume fraction at the pump assembly for constant pitch inducer at inlet pressure “d”.

The experimental results of the cavitation performance for the variable pitch inducer with δ = 0.35 mm are compared to the simulation results to validate the assumptions and boundary conditions used in simulations. Figure-19 shows the head drop curve as a function of the NPSHa and the pump differential head. The difference between the test and simulation results is 9% in NPSH3 value. Considering the test instrumentations’ and simulation accuracy, this is an acceptable deviation for multiphase testing and analysis. The cavitation images of the variable pitch inducer are reported in

Figure-20, which demonstrates the cavitation inception, vortex tip cavitation and cavitation at the inducer blades. The vapor formation results of CFD simulations and test results show good agreement in terms of vapor volume fraction according to Figure-20. Visual comparison between the experiments and the simulations further confirms the accuracy of the prediction by CFD simulations.

NPSH3cfd = 0.35 m 20

NPSH3test = 0.38 m 10 Head [m] CFD Head NPSH3 CFD 5 Test Head NPSH3 Test 0 0.0 0.5 1.0 1.5 2.0 NPSH [m]

Figure-19 Head drop curve for variable pitch inducer, δ = 0.35 mm.

(a)

(b)

Figure-20 Cavitation and vapor formation for variable pitch inducer, (a) test versus corresponding (b) CFD simulation results are shown, δ = 0.35 mm.

Lastly, cavitation performance is reported as a function of the dimensionless tip clearance parameter, = / , in Figure-21. This figure demonstrates the sensitivity of

𝑡𝑡 the cavitation performance𝜆𝜆 𝛿𝛿 𝐷𝐷on the inducer tip clearance for each style inducer. The cavitation performance exhibits an approximately linear trend with increase in tip clearance for constant pitch inducer. Towards larger tip clearances, the cavitation number approaches a constant value for both inducer geometries. With enlarged tip clearances, the variable pitch inducer performs worse than the constant pitch inducer. This suggests that the inducer tip clearance becomes a critical design parameter for an inducer with relatively higher

differential pressure. For turbopump inducers, with variable hub diameter and more aggressive pitch, a greater impact to the cavitation number is expected to occur, as compared to the inducers studied in this paper. This is due to relatively higher differential pressure across the turbopump inducers [2-4, 6]. Figure-21 suggests that inducer tip clearance is a critical parameter and should be considered in the enhancement of the cavitation performance for inducer types with relatively higher differential pressure.

Cavitation Performance vs Tip Clearance (Non-dimensional) 0.55

0.53

σ 0.50 Discretization Error Bar

0.48 Constant Pitch Inducer Variable Pitch Inducer

0.45 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 λ δ = /Dt

Figure-21 Relationship between the cavitation performance and the tip clearance.

6. Conclusions

Helical style constant and variable pitch inducers under non-cavitating and cavitation operating conditions are investigated for various inducer tip clearances. CFD simulations are performed for all inducer styles for various tip clearances under single phase and cavitation (two-phase) condition. An experimental test is conducted for a

variable pitch inducer with tip clearance of δ = 0.35 mm under cavitation and non- cavitation conditions. The conclusions of this study are reported below:

• CFD simulations and experimental results are within 9% for the cavitation

performance predictions for NPSH3 value, while the discrepancy for non-cavitation

pump performance is 5%. The difference between tests and simulations results are

acceptable, considering the complexity and accuracy involved in testing and

simulations.

• According to the simulations results with different inducer tip clearances, excess

tip clearance will result in cavitation performance drop by 11% for the variable

pitch inducer, while this drop is 4.5% for the constant pitch inducer. Cavitation

performance of the variable pitch inducer is more sensitive to the changes to the

inducer tip clearance. The head rise across the variable pitch inducer is greater due

to the blade geometry and progressive pitch. The reduction in fluid volume between

blades along the meridional direction results in greater differential pressure at the

variable pitch inducer. However, this results in greater backflow leakage and

vorticity at the tip section. Therefore, sensitivity to the tip clearance is more

pronounced for the variable pitch inducer.

• There is a good agreement between the blade cavitation location per two-phase

simulations and the low-pressure regions according to single phase simulation

results. It is possible to approximate or possibly control the blade cavitation and its

initiation by examining or adjusting the blade loading (pressure).

• The simulations include the complete pump assembly (inducer, impeller and the

diffuser vane) since NPSH3 cavitation performance is defined as the NPSHa value

when pump’s differential head is dropped by 3%. While the pressure differential

can reduce by more than 3% at the inducer section, the pump can still be able to

maintain its differential pressure over NPSH3 limit. Large cavitation volumes at

inducer locations do not necessary result in the pump head drop to NPSH3 level,

unless the vapor formation propagates to the impeller eye section. Therefore, all the

pump hydraulic components should be included and considered in determination of

true NPSH3 cavitation performance of a pump.

• Increase in tip clearance results in excess back leakage and larger vortex

recirculation at tip location, which produces pressure loss within the inducer and,

consequently, degraded cavitation performance. The impact on the non-cavitating

performance is minimal. This is due to the percentage of the head generation of the

inducer being too small as compared to the head generation of the pump assembly.

7. Acknowledgment

The work was supported by Elliott Group Cryodynamics Product, Ebara

Corporation.

References

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[3] Scheer, D. D., Huppert, M. C., Viteri, F., Farquhar, J. “Liquid rocket engine axial-flow turbopumps”, NASA Technical Report, SP-8125, April 1978.

[4] Kovich G., “Cavitation performance of 84 deg helical inducer in water and hydrogen”,

NASA Technical Report, TN-D-7016, December 1970.

[5] Japikse D., Marscher W., Furst, R., “Centrifugal Pump Design and Performance”,

Concepts ETI, Inc. Wilder, VT 1997.

[6] Japikse, D. “Overview of Industrial and Rocket Turbopump Inducer Design”, Fourth

International Symposium on Cavitation, California Institute of Technology, Pasadena,

CA, 2001.

[7] Sutton, M., “Improving the Cavitation Performance of Centrifugal Pumps with Helical

Inducers”, BHR Group’s Technical Report, TN814, 1964.

[8] Acosta A. J., “An Experimental Study of Cavitating Inducers”, Proceedings of Second

Office of Naval Research Symposium on Naval Hydrodynamics, p533-557, ACR-38,

1958.

[9] Bakir, F., Rey, R., Gerber, A. G., Belmari, T., Hutchinson, B., “Numerical and

Experimental Investigations of the Cavitating Behavior of an Inducer” International

Journal of Rotating Machinery, 10:15-25, 2004.

[10] Watanabe, H., Ichiki, I., “Development of Cryogenic Pump Hydrodynamics Using

Inverse Design Method and CFD”, ASME Middle East Mechanical Exposition, 2007.

[11] Ashihara, K., Goto, A., “Effects of Blade Loading on Pump Inducer Performance and

Flow Fields”, ASME 2002 Joint U.S.-European Fluids Engineering Division Conference,

DOI: 10.1115/FEDSM2002-31201.

[12] Horiguchi, H., Watanabe, S., Tsujimoto, Y., Aoki, M., “Theoretical Analysis of

Cavitation in Inducers with Unequal Blades with Alternate Leading Edge Cutback: Part I

– Analytical Methods and the Results for Smaller Amounto of Cutback”, ASME Journal of Fluids Engineering, Vol. 122, 2000.

[13] Horiguchi, H., Watanabe, S., Tsujimoto, Y., “Theoretical Analysis of Cavitation in

Inducers with Unequal Blades with Alternate Leading Edge Cutback: Part II – Effects of the Amount of Cutback”, ASME Journal of Fluids Engineering, Vol. 122, 2000.

[14] Bakir, F., Koudiri, S., Noguera, R., Rey, R., “Experimental Analysis of an Axial

Inducer Influence of the Shape of the Blade Leading Edge on the Performances in

Cavitating Regimes”, ASME Journal of Fluids Engineering, Vol. 125, 2003.

[15] Guo, X., Zhu, Z., Cui, B., Shi, G., “Effects of the Number of Inducer Blades on the

Anti-cavitation Characteristics and External Performance of a Centrifugal Pump”, Journal of Mechanical Science and Technology, Vol. 30, 2016. DOI 10.1007/s12206-016-0510-1.

[16] Campos-Amezcua, R., Khelladi, S., Mazur-Czerwiec, Z., Bakir, F., Campos-

Amezcua, A., Rey, R., “Numerical and Experimental Study of Cavitating Flow Through an Axial Inducer Considering Tip Clearance”, Journal of Power and Energy, Vol. 227 (8),

2013.

[17] Fu, X., Yuan, J., Yuan, S., Pace, G., d’Agostino, L., “Effect of Tip Clearance on the

Internal Flow and Hydraulic Performance of a Three-Bladed Inducer”, International

Journal of Rotating Machinery, Vol. 2017, Article ID 2329591, 2017.

[18] Kim, C., Kim, S., Choi, C-H., Baek, J., “Effects of Inducer Tip Clearance on the

Performance and Flow Characteristics of a Pump in a Turbopump”, Journal of Power and

Energy, Vol. 231(5), 2017.

[19] Hong, S-S., Kim, J-S., Choi, C-H., Kim, J., “Effect of Tip Clearance on the Cavitation

Performance of a Turbopump Inducer”, Journal of Propulsion and Power, Vol. 22, No.1,

2006.

[20] Brennen, C. E., “Hydrodynamics of Pumps”, Concepts ETI, Inc, Norwich Vermont,

USA, Oxford University Press, Oxford OX3 6DP, England, 1994.

[21] Zwart, P., Gerber, A., Belamri, T., “A two-phase flow model for predicting cavitation dynamics,” 5th International Conference on Multiphase Flow, Yokohama, Japan, 2004.

[22] ANSYS CFX Release 19.1, User Manual - Basic Solver Theory.

[23] ANSI/IHI 9.6.1-2012, “Rotordynamic Pumps Guideline for NPSH Margin” Hydraulic

Institute. ISBN: 19357621102.

[24] ISO 13709:2009, “Centrifugal Pumps for Petroleum Petrochemical and Natural Gas

Industries”, 2009.

[25] API 610 11th Edition, “Standard for Centrifugal Pumps for Petroleum, Petrochemical and Natural Gas Industries”

[26] Rayleigh, L., “Pressure Developed in a Liquid During the Collapse of a Spherical

Cavity” Philosophical Magazine, Vol. 34, 1917.

[27] Plesset, M. S., “The Dynamics of Cavitation Bubbles”, Journal of Applied Mechanics,

Vol. 16, 1949.

[28] Mani, K. V., Cervone, A., Hickey, J-P., “Turbulence Modeling of Cavitating Flows in

Liquid Rocket Turbopumps”, Journal of Fluids Engineering, Vol. 139, 2017.

[29] https://www.bechtel.com/projects/corpus-christi-liquefaction-project/

[30] Celik, I.B., Ghia, U., Roache, P. J., Freitas, C. J., Coleman, H., Raad, P. E., “Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications”,

Journal of Fluids Engineering, Vol. 130, 2008.

CHAPTER 4. COMPARISON AND APPLICATION OF TRANSPORT-EQUATION BASED CAVITATION MODELS TO INDUSTRIAL PUMPS WITH INDUCERS

Enver S. Karakasa,b, Nehir Tokgoza,c, Hiroyoshi Watanabed, Matteo Aurelia, Cahit A. Evrensel a aMechanical Engineering Department, University of Nevada, Reno 1664 N. Virginia Street, Reno, NV 89557-0312, USA bElliott Group, Cryodynamic Products, Ebara Corporation 350 Salomon Circle, Sparks, NV 89434, USA cFaculty of Engineering, Osmaniye Korkut Ata University Department of Energy Systems Engineering, 80000 Osmaniye, Turkey dEbara Fluid Machinery & Systems Company, Ebara Corporation 11-1 Haneda Asahi-cho, Ohta-ku, Tokyo, Japan

Abstract

This paper investigates and compares four commonly utilized flow transport- equation based cavitation models (TEM) and their applicability to prediction of cavitation performance and bubble dynamics of an industrial centrifugal pump with a helical inducer.

The main purpose of this study is to identify the most appropriate cavitation model for predicting the cavitation performance of centrifugal pumps with inducers. Each cavitation model is reviewed in detail and the uniqueness of each model is outlined. These cavitation models are incorporated in a Computational Fluid Dynamics (CFD) code to study the vaporization and condensation transport rate of the fluid. Experimental tests are conducted on the pump to determine the true cavitation performance in terms of Net Positive Suction

Head (NPSH). Experimental results are compared to simulation results with different cavitation models to validate accuracy and assumptions of each model. Lastly, bubble formation, cavitation inception and bubble growth of each cavitation model and the experimental results are compared. Modifications to empirical constants for each cavitation model are suggested, to obtain agreement with the actual cavitation and bubble dynamics observed. Two new sets of empirical constants are advised for two of the models to improve the accuracy of predictions in determination of cavitation performance. Results show that with the new suggested empirical constants, the cavitation performance of an industrial pump with a helical inducer can be better predicted.

Keywords: Inducer, Cavitation, Net Positive Suction Head, Computational Fluid

Dynamics, Centrifugal Pump

1. Introduction

Cavitation behavior of a centrifugal pump is an important performance characteristic due to the requirement of safe pump operation in the event of low suction pressure or during emptying a storage tank or a container. In order to achieve acceptable performance and reduce the required Net Positive Suction Head (NPSHr), pumps often utilize an inducer to delay the cavitation or enhance the suction performance [1-3]. Inducer technology was first initiated by NASA due to lack of suction pressure in rocket turbopump applications. [1-3].

In industrial in-tank pump applications, the NPSHr determines the non-useable height of a liquid inside a storage tank. The non-usable liquid height is mainly the liquid level that is left in the tank and cannot be used due to NPSHr limitation. Below this liquid

level, the pump will suffer from cavitation due to lack of suction pressure. Since the construction of the storage tank limits the available suction pressure to the pump, an inducer is utilized in these applications to increase the suction pressure to the pump in order to reduce the minimum required liquid level. A typical in-tank pump application for hydrocarbon applications (Liquified Natural Gas) is given in Figure-1 demonstrating the non-usable liquid level for clarification. This liquid level corresponds to the NPSHr of the pump. The estimation of the NPSHr is critical in design and development of the inducers.

Pump Discharge

LNG Storage Tank Pump Motor Tank Discharge Column

In-tank Cryogenic Pump Shaft

Minimum Liquid Level ~ NPSHr Diffuser

Impeller TANK BOTTOM Helical Inducer

TANK BOTTOM

Figure-1 Typical LNG Storage tanks and illustration of Minimum Liquid Height with In- tank Cryogenic Pump.

Prediction of pump performance under cavitation conditions is challenging due to the complexity of three-dimensional fluid flow along with the variation in density.

Determination of vaporization and condensation, which requires modeling the physics of cavitation, poses difficulty due to phase change. The fundamental approach in modeling the cavitation is to determine the fluid phase (amount of vapor and liquid) as a function of time and location. Different approaches are used to predict the density field. One way to determine the density field is to solve the energy equation and obtain the density by employing equations of state or from saturation tables for a specific fluid [4,5]. This method involves a barotropic equation of state to express the mixture density as a function of local pressure. Another way to model the cavitation is by determining the liquid or the vapor amount either in terms of volume or mass, and then employing transport equations with source/sink terms in order to determine condensation and vaporization rate [7, 8, 10,

14]. The main challenge in this method is to determine the mass transfer rate for condensation and vaporization. Usually, the mass transfer rate is derived from Rayleigh-

Plesset’s model which is based on the dynamics of cavitation bubbles [6]. This method is more accurate than using barotropic equation of state as it takes the transient dynamics of the cavitation into account. Using equations of state assumes that, as the flow condition varies, the fluid instantly reaches its thermodynamics equilibrium [8]. This is arguably an over simplification of cavitation, since vaporization and condensation occur in a more complex physical process. The initiation of the vapor (bubbles) may occur in nuclei and grow, and then move into an area of higher pressure and collapse all within few thousands of a second within a turbomachine [9]. Therefore, cavitation models based on source/sink terms are reviewed in this paper.

Zwart et al. [7] proposed a Rayleigh Plesset based cavitation model based on the multiphase flow equations with mass transfer defining sink/source terms for vaporization and condensation mass transfer rate, respectively. This cavitation model is widely used in computational fluid dynamics analysis. It has been confirmed to provide reasonable predictions of cavitation dynamics around hydrofoils, along pump inducers and for simple venturi geometry [7]. Transport cavitation models based upon Rayleigh Plessets bubble dynamics are proposed by Uttukar et al. [8], Schnerr and Sauer [10], Singhal et. al [11],

Tsuda et al. [12], Senocak and Shyy [13,14], Jiang et al. [15], Kunz [16], Merkle et al. [17] and Kubota et al. [23]. Each of these cavitation models are more commonly used in research studies employing equations of state. However, transport-based cavitation models require determination of sink and source term constants empirically [8,11-14]. An adjustment to these empirical constants may be needed based on the rate of the phase- change for each application. Kinzel et al. [21] focused on reformulation of Rayleigh-Plesset based cavitation models to take the collapse source term and slip on bubble transport into account to enhance the predictions of numerical approximations. Kinzel suggested that gas expansion and contraction should be considered besides mass transfer rate of condensation and vaporization terms, as these mass transfer terms are not fully consistent with the physical phenomenon of cavitation [21].

With growing research attention to multiphase flow, Computational Fluid

Dynamics (CFD) is more frequently employed to solve the separate continuity equations for liquid and gas phase fields. Due to the difficulty, time and expense involved in testing pumps under two phase conditions, CFD methods have been developed to verify and enhance NPSHr performance of the centrifugal pumps so it is now possible to predict

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cavitation performance of turbomachinery with simulation methodologies. For example,

Bakir et al. [18,19] conducted numerical simulations via CFD and experimental investigation of the cavitation behavior of inducers. Effects of the inducer tip clearance and tip blade angle around the leading edge is studied in their research [19, 20]. CFD predictions show good agreement with the experimental test results, indicating that tip clearance and leading-edge angle can impact the suction performance. Tani et al. [22] studied the cavitation instabilities in an inducer by conducting steady state and unsteady

CFD simulations to better understand the interaction between the tip vortex and the blade for two different flow coefficients. The steady state CFD simulations in [22] showed no evidence of such an interaction. Unsteady CFD simulations showed that bubble collapse in a cavity has a strong influence on cavitation instability. The effect of flow rate (flow coefficient) to cavitation instability and its impact to back flow structure on rational cavitation in inducer is also investigated [22].

Watanabe [24] and Ashihara et al. [25], performed CFD simulations using three- dimensional inverse design methodology to design two different style inducers for an industrial cryogenic pump. The flow characteristics and suction performance of the new inducers were initially verified via two phase CFD simulations of the complete pump assembly that include all hydraulic components such as impeller and the diffuser vane.

Their cavitation model was based on Rayleigh-Plesset’s bubble dynamics with appropriate mass transfer rate terms for vaporization and condensation. The performance prediction of each inducer design was confirmed by water testing in two phase flow. Cavitation performance curves of the simulation and the testing show good agreement confirming the

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design methodology of three-dimensional inverse method for cavitation performance

[24,25].

The main purpose of this study is to identify the most appropriate and accurate cavitation model for predicting the cavitation performance of centrifugal pumps with a helical inducer. For that reason, four commonly used Rayleigh Plesset transport-based cavitation models are reviewed and utilized in to ANSYS CFX CFD code to predict cavitation performance. The distinction of each model and their applicability to cavitation in industrial pumps are discussed. Cavitation performance in terms of NPSHr is determined by simulations using each cavitation model and the results are compared to the experimental tests. Two of the selected models were modified in order to obtain better prediction in bubble dynamics and cavitation behavior in centrifugal pump. The main changes to the existing models are to the empirical constants suggested for applications with hydrofoil or simple shapes such as an orifice. To best of authors’ knowledge, there has been no research involving confirmation of the selected cavitation models and their applicability with original empirical constants to helical inducers considering complete pump assembly. In this study, new set of empirical constants are recommended for two of the selected models based on the comparisons of simulations to actual test data. In order to further validate and compare each model and the empirical constants, bubble dynamics and vapor formation in terms of vapor volume fraction are reviewed against the high-speed imaging of the experimental results.

The outline of the paper is as follows: Section 2 discusses the cavitation bubble dynamics and details of the cavitation models along with the numerical method and

102

assumptions. Section 3 describes the pump assembly with helical inducer. Section 4 covers

the numerical simulations and details including geometric model and CFD grid, boundary

conditions, discretization error calculation, convergence criteria. Section 5 describes the

details of the experimental setup and testing procedure. Section 6 reviews the results from

CFD simulations and experimental tests. Finally, conclusions are given in Section 7.

2. Bubble Dynamics and Cavitation Models

A liquid at constant temperature goes through vapor formation, if its pressure falls

below the saturation pressure. This process vapor formation is called cavitation. One of

the most extensively used model for cavitation is the spherical bubble model which is most

relevant to those forms of bubble cavitation in which nuclei grow to visible size when they

encounter a region of low pressure and collapse when they are subjected to higher pressure

[26].

Almost all cavitation models based on spherical bubble are based on the Rayleigh-

Plesset equation [6], which defines the relationship between the bubble radius, , and the

𝑏𝑏 pressure around the bubble. The simplified expression of the time rate of change𝑅𝑅 in bubble

radius is defined by [6]:

2 ( ) = (1) 3 𝑑𝑑𝑅𝑅𝑏𝑏 𝑃𝑃 − 𝑃𝑃𝑣𝑣 � 𝑑𝑑𝑑𝑑 𝜌𝜌𝑙𝑙 where is the reference pressure away from the bubble, is the vapor pressure for a

𝑣𝑣 given temperature𝑃𝑃 of the fluid, and is the liquid density𝑃𝑃 at the given temperature.

𝜌𝜌𝑙𝑙

103

Assuming all the bubbles have a spherical shape, the vapor volume fraction, , can be

𝑣𝑣 defined as a function bubble number density N (number of bubbles per unit volume)𝛼𝛼 as:

4 = (2) 3 3 𝛼𝛼𝑣𝑣 𝑁𝑁 � 𝜋𝜋𝑅𝑅𝑏𝑏 � Then, the mass transfer rate of a bubble can be expressed as follows:

2 ( ) = 4 (3) 3 𝑑𝑑𝑚𝑚𝑏𝑏 2 𝑃𝑃 − 𝑃𝑃𝑣𝑣 𝜋𝜋𝑅𝑅𝑏𝑏 𝜌𝜌𝑣𝑣� 𝑑𝑑𝑑𝑑 𝜌𝜌𝑙𝑙 Within the approach of Equal-Velocity-Equal-Temperature (EVET), which assumes that the vapor velocity field is same as the liquid [18, 27], interphase mass transfer rate is defined by Zwart et al. for condensation and vaporization by considering the volume fractions of vapor and the nucleation site as follows [7]:

3 2 ( ) = if > (4) 3 𝛼𝛼𝑣𝑣𝜌𝜌𝑣𝑣 𝑃𝑃 − 𝑃𝑃𝑣𝑣 𝑚𝑚̇ con −𝐶𝐶con � 𝑃𝑃 𝑃𝑃𝑣𝑣 𝑅𝑅𝑏𝑏 𝜌𝜌𝑙𝑙 3 (1 ) 2 ( ) = if > (5) 3 𝛼𝛼𝑏𝑏 − 𝛼𝛼𝑣𝑣 𝜌𝜌𝑣𝑣 𝑃𝑃𝑣𝑣 − 𝑃𝑃 𝑚𝑚̇ vap 𝐶𝐶vap � 𝑃𝑃𝑣𝑣 𝑃𝑃 𝑅𝑅𝑏𝑏 𝜌𝜌𝑙𝑙 where the empirical constants for condensation ( ) and vaporization ( ) have

con vap different values, since phase change rate is not the s𝐶𝐶ame for each case, is 𝐶𝐶the volume

𝑣𝑣 fraction of the vapor, and is the volume fraction of the nucleation site.𝛼𝛼 Equations (4)

𝑏𝑏 and (5) are widely used 𝛼𝛼in cavitation simulations and proved to predict cavitation performance of inducers with empirical constants, = 0.01, = 50, and = 5x10 −4 𝐶𝐶con 𝐶𝐶vap 𝛼𝛼𝑏𝑏

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with typical radius for the nuclei of = 10 m for water at ambient temperature −6 𝑏𝑏 [7,27,28]. This model is known as ZGB cavitation𝑅𝑅 model.

Singhal et al. cavitation model [11] is known as the “full cavitation model” (FC) which accounts for all first order effects such as phase change, turbulent pressure fluctuations and non-condensable gases (NCG). In this cavitation model, unlike ZGB model, bubble diameter is defined by the balance between aerodynamic drag and surface tension forces. The following correlation between the aerodynamic drag and surface tension is used to limit the size of the bubble diameter:

0.061 We = (6) 2 𝑆𝑆 𝑅𝑅𝑏𝑏 2 𝜌𝜌𝑙𝑙 𝑣𝑣𝑟𝑟𝑟𝑟𝑟𝑟 where We is Weber number which defines the relative importance of fluid’s inertia to surface tension, is the surface tension and is the relative velocity between the liquid

𝑟𝑟𝑟𝑟𝑟𝑟 and the vapor phase.𝑆𝑆 In addition to , mixture𝑣𝑣 density is included in the FC model to

𝑏𝑏 take the mixture’s continuity equation𝑅𝑅 into account along with liquid and vapor continuity equations. In contest, only bubble density number and mass change rate of a single bubble are considered in the ZGB model to define the expression for net phase change. Another term that is included in the FC model is the density of NCG. NCG can impact the cavitation inception and bubble dynamics [26,29]. Singhal et al. [11] obtained the mass transfer rates of the FC model as follows:

2 ( ) = 1 if > (7) 3 √𝑘𝑘 𝑃𝑃 − 𝑃𝑃𝑣𝑣 𝑚𝑚̇ con −𝐶𝐶𝑐𝑐 𝜌𝜌𝑙𝑙𝜌𝜌𝑣𝑣� � − 𝑓𝑓𝑣𝑣 − 𝑓𝑓𝑔𝑔� 𝑃𝑃 𝑃𝑃𝑣𝑣 𝑆𝑆 𝜌𝜌𝑙𝑙

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2 ( ) = if > (8) 3 √𝑘𝑘 𝑃𝑃𝑣𝑣 − 𝑃𝑃 𝑚𝑚̇ vap 𝐶𝐶𝑒𝑒 𝜌𝜌𝑙𝑙𝜌𝜌𝑙𝑙� 𝑓𝑓𝑣𝑣 𝑃𝑃𝑣𝑣 𝑃𝑃 𝑆𝑆 𝜌𝜌𝑙𝑙

where and are the empirical constants validated to be 0.02 and 0.01 for cavitation

𝑒𝑒 𝑐𝑐 flow across𝐶𝐶 a sharp𝐶𝐶 -edged orifice and flow over a hydrofoil [11], is the local turbulent kinetic energy, and are the mass fraction of the vapor and NCG,𝑘𝑘 respectively. The

𝑣𝑣 𝑔𝑔 FC model also𝑓𝑓 considers𝑓𝑓 the effect of turbulence on cavitation. Singhal et al. [11] incorporates the effect of turbulence by estimating turbulent pressure fluctuations as:

= 0.39 (9)

𝑃𝑃turb 𝜌𝜌𝜌𝜌 In the model, the saturation pressure is corrected by including the local values of the turbulent pressure fluctuations with a threshold to phase change pressure as:

= ( + 0.5 ) (10)

𝑃𝑃𝑣𝑣 𝑃𝑃sat 𝑃𝑃turb Schnerr and Sauerr [30] defined the mass transfer rate considering the mixture density term as:

= (11) 𝑑𝑑𝑚𝑚𝑏𝑏 𝜌𝜌𝑣𝑣 𝜌𝜌𝑙𝑙 𝑑𝑑𝛼𝛼𝑣𝑣 𝑑𝑑𝑑𝑑 𝜌𝜌 𝑑𝑑𝑑𝑑 where, is the density of the mixture. This is a similar approach to the FC model. Unlike the previous𝜌𝜌 models, Schnerr and Sauerr define the vapor volume fraction , by

𝑣𝑣 connecting the vapor volume fraction to the number of bubbles per volume of liquid:𝛼𝛼

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4 4

3 3 = = = (12) + 4 3 4 3 𝑣𝑣 𝑛𝑛0 𝑉𝑉𝑙𝑙 � 𝜋𝜋𝑅𝑅𝑏𝑏 +� 𝑛𝑛 0 � 𝜋𝜋𝑅𝑅𝑏𝑏 +� 1 𝑣𝑣 𝑉𝑉 3 3 𝛼𝛼 𝑣𝑣 𝑙𝑙 3 3 𝑉𝑉 𝑉𝑉 𝑛𝑛0𝑉𝑉𝑙𝑙 � 𝜋𝜋𝑅𝑅𝑏𝑏 � 𝑉𝑉𝑙𝑙 𝑛𝑛0 � 𝜋𝜋𝑅𝑅𝑏𝑏 � where and are the volume of vapor and liquid at the area of interest, is the number

𝑣𝑣 𝑙𝑙 0 of bubbles𝑉𝑉 per𝑉𝑉 volume of liquid. By using equation (10), the bubble radius𝑛𝑛 is defined by

[30] as follows:

3 1 / = (13) 1 4 1 3 𝛼𝛼𝑣𝑣 𝑅𝑅𝑏𝑏 � � − 𝛼𝛼𝑣𝑣 𝜋𝜋 𝑛𝑛0 Schnerr and Sauerr’s mass transfer rate for condensation and vaporization can be expressed as:

3 2 ( ) = (1 ) if > (14) 3 𝜌𝜌𝑙𝑙𝜌𝜌𝑣𝑣 𝑃𝑃 − 𝑃𝑃𝑣𝑣 𝑚𝑚̇ con 𝛼𝛼𝑣𝑣 − 𝛼𝛼𝑣𝑣 � 𝑃𝑃 𝑃𝑃𝑣𝑣 𝜌𝜌 𝑅𝑅𝑏𝑏 𝜌𝜌𝑙𝑙 3 2 ( ) = (1 ) if > (15) 3 𝜌𝜌𝑙𝑙𝜌𝜌𝑣𝑣 𝑃𝑃𝑣𝑣 − 𝑃𝑃 𝑚𝑚̇ vap 𝛼𝛼𝑣𝑣 − 𝛼𝛼𝑣𝑣 � 𝑃𝑃𝑣𝑣 𝑃𝑃 𝜌𝜌 𝑅𝑅𝑏𝑏 𝜌𝜌𝑙𝑙 The major difference in Schnerr and Saurrer’s (SS) cavitation model is that it does not rely on distinct empirical constants for vaporization and condensation. Interestingly, mass transfer rate is a function of (1 ), which is equal to zero, for the maximum

𝑣𝑣 𝑣𝑣 and minimum values of (0,1) and𝛼𝛼 reaches− 𝛼𝛼 its maximum for intermediate values. The

𝑣𝑣 number of bubbles per volume𝛼𝛼 of liquid must be defined in SS cavitation model. This number is assumed to be constant and therefore no additional bubbles are destroyed or created.

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Tsuda et al. [12] used the relationship between critical bubble radius and liquid

𝑐𝑐 pressure based on the homogeneous nucleation theory described by Brennen 𝑅𝑅[31]:

2 = (16) ( ) 𝑆𝑆 𝑅𝑅𝑐𝑐 𝑃𝑃sat − 𝑃𝑃 It is assumed that NCG within the working fluid are dissolved. Equation (16) can be further simplified observing that most cavitation flows have high Reynolds number.

Therefore, the flow shear stress is dominant compare to the surface tension, and the effect of the surface tension can be neglected:

1 = (17) ( ) 𝑅𝑅𝑐𝑐 𝑃𝑃sat − 𝑃𝑃 Tsuda et al. [12] implemented this definition of critical radius to the term of

𝑏𝑏 equation (1), to define the mass transfer rate without the term. Therefore, this𝑅𝑅 model is

𝑏𝑏 often referred as “reduced critical radius model” (RCR).𝑅𝑅 In addition, unlike the ZGB and

FC cavitation models, the RCR model defines mass transfer rates as a function of vapor mass fraction instead of volume fraction:

2 ( ) = ( ) if > (18) 3 𝑃𝑃 − 𝑃𝑃𝑣𝑣 𝑚𝑚̇ con −𝐶𝐶𝑐𝑐 𝑃𝑃 − 𝑃𝑃sat 𝑓𝑓𝑣𝑣 𝜌𝜌𝑙𝑙� 𝑃𝑃 𝑃𝑃𝑣𝑣 𝜌𝜌𝑙𝑙 2 ( ) = ( ) (1 ) if > (19) 3 𝑃𝑃 − 𝑃𝑃𝑣𝑣 𝑚𝑚̇ vap −𝐶𝐶𝑒𝑒 𝑃𝑃 − 𝑃𝑃sat − 𝑓𝑓𝑣𝑣 𝜌𝜌𝑣𝑣� 𝑃𝑃𝑣𝑣 𝑃𝑃 𝜌𝜌𝑙𝑙 Here, and are the empirical constants, which found to be 0.3 and 0.6 in

𝑒𝑒 𝑐𝑐 cavitation simulations𝐶𝐶 𝐶𝐶of a blunt body and two hydrofoils [12]. Another distinction of the

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RCR model is that the latent heat transfer, which is important for applications with cryogenic fluids, is used in the determination of . Specifically, the RCR model defines

𝑣𝑣 the pressure change caused by the latent heat as: 𝑃𝑃

= (20) (1 ) 𝛼𝛼 𝜌𝜌𝑙𝑙𝐿𝐿 ∆𝑃𝑃𝑙𝑙 𝐺𝐺sat − 𝛼𝛼 𝜌𝜌𝑣𝑣 𝐶𝐶𝐶𝐶𝑙𝑙 where is the latent heat of vaporization, is the specific heat and is the gradient

sat of saturation𝐿𝐿 curve for the working fluid.𝐶𝐶 Similar𝐶𝐶 to the FC model,𝐺𝐺 RCR model also considers the effect of turbulence on cavitation and finally defines the vapor pressure as follows:

= + 0.195 (21) , (1 ) 𝛼𝛼 𝜌𝜌𝑙𝑙𝐿𝐿 𝑃𝑃𝑣𝑣 𝑃𝑃sat ref − 𝐺𝐺sat 𝜌𝜌𝜌𝜌 − 𝛼𝛼 𝜌𝜌𝑣𝑣 𝐶𝐶𝐶𝐶𝑙𝑙 In equation (21) , is the saturation pressure of the fluid at the inlet (initial

sat ref condition). Note that for𝑃𝑃 cavitation problems under isothermal conditions, , is equal

sat ref to . 𝑃𝑃

𝑃𝑃sat 3. Subject Pump: Description of Assembly and Experimental Setup

3.1 Pump Assembly

The pump studied in this paper is a retractable type in-tank pump which is installed in to a LNG tank column via a retraction system. The performance specifications of the pump are listed in Table-1.

The in-tank pump for cryogenic applications often utilizes a helical inducer. It is a variable pitch, two-bladed inducer, with blades 180° apart in angular position. The main

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hydraulic components of the pump assembly are the inducer, closed type impeller and an axial diffuser vane. The 3D model of the hydraulic components is shown in Figure-2. The geometric parameters of the inducers for the present study are reported in Table-2. The pump hydraulic components are scaled down by 150/236 due to the limitations of the testing facility. Parameters in Table-2 are for scaled down configuration, represents the experimental setup and will be used in the simulations.

Table-1 Pump Performance Data and Specifications (Full Scale).

Design Flow: 695 m3/hr

Pump Differential Head: 270 m

Pumping Fluid: LNG (98% Methane and 2% Propane in Molar Basis)

Fluid Temperature: -165 °C

Fluid Density: 438.6 kg/m3

Rotational Speed: 3000 RPM

Diffuser Impeller Vane

Inducer

z x

Figure-2 Pump assembly used in the simulations.

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Table-2 Geometric parameters and specification of the inducers (Scaled).

Rotational Speed: 1700 RPM Design Flow Coefficient ( ): 0.136

Blade Count: 𝝋𝝋 2 Tip Diameter (D): 150 mm Hub Diameter: 45 mm Pitch (P): Variable Blade Chord Length (c): 81.8 mm

3.2 Experimental Setup

The experimental test facility is shown in Figure-3. It is a closed loop system with water at ambient temperature. It consists of a pressure control tank fitted with a vacuum pump to regulate the suction pressure of the pump to a desired level. The loop is furnished with various control and instrumentation devices to monitor and record, pressure, temperature and pump flow rate. As discussed, pump hydraulics are scaled down by 0.64 for testing purposes due to the size limitations and power rating of the motor driver. The inducer, impeller and the diffuser vane are installed in to the pump test section. A clear transparent casing is utilized at inducer section in order to capture high speed camera images of the inducer under cavitation conditions. Tests are conducted at constant rotational speed of 1700 rev/min and at the pump rated flow rate. During testing, water temperature is kept constant at 25°C and the suction pressure to the pump is adjusted with the vacuum pump attached to the pressure control tank.

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Pump Test Section

Pressure Control Tank

Torque Meter

Motor

Figure-3 Experimental test facility.

4. Numerical Simulation

Cavitation performance, bubble dynamics, vapor formation in the centrifugal pump with helical variable pitch inducer is studied for a given flow rate ( ) for different cavitation conditions, identified by cavitation number ( ). The details of𝜑𝜑 the CFD code, cavitation performance parameters, turbulence model, 𝜎𝜎grid geometry, estimation of the discretization error are reported in this section.

4.1 Computational Fluid Dynamics (CFD) Implementation

The software package ANSYS CFX version 19.1 is used to simulate the pump assembly under cavitation conditions. Reynolds-averaged Navier Stokes equations

(RANS) are solved along with an appropriate turbulence model via CFX’s iterative solver.

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- turbulence model is adopted to the CFD code according to findings of Mani et al [32] for𝑘𝑘 𝜀𝜀 rocket turbopump inducer applications.

Inlet pressure is defined and varied for each NPSH case, with an outlet mass flow rate as the boundary condition. Fluid temperature is kept constant at 25 °C since cavitation is assumed to be under isothermal condition. Simulations are run under the steady state condition. The target residual RMS value of 10-5 for mass, momentum and energy are obtained throughout the analyses. To improve the convergence level and reduce the number of iterations for cavitation cases, simulation under non-cavitating condition is first solved.

The solution of non-cavitating condition is defined as the initial conditions for cavitation cases. This not only reduces the number of iterations but also improves the convergence level. Cavitation models are defined along with the appropriate empirical constants and fluid properties by using CFX’s CEL (CFX Expression Language) option.

Regardless of available experimental test data, CFD discretization error is estimated by calculating the grid convergence index (GCI). The methodology of GCI index calculation and procedure are described by Celik et al [34]. Three different size grids are selected with refinement factors of r21 = 1.53 and r32 = 1.65. The finest grid is the grid used at the simulations with 13.5 million nodes. Information related to each grid such as total number of elements, grid size, grid refinement factor between each grid, and error estimate along with the apparent order p is given in Table-3. According to the NPSH estimations of three selected grids, GCI index error is determined as 0.4% for the grid (fine grid) utilized at the simulations.

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Table-3 Calculation results of discretization error according to [34]. Refer to [34] for description and calculation of each parameter.

Number of Elements (N): N1= 13,500,000 N2= 6,800,000 N3= 4,700,000

Grid Refinement (r): r21=1.533, r32= 1.651

Cavitation Number (σ): σ 1= 0.477 σ 2=0.4798 σ 3=0.4905 Apparent order ( ): 2.498

21 Extrapolated Value𝒑𝒑 of σ (σ ext): σ ext = 0.4755 21 Relative error (ea): ea = 0.59%

21 Extrapolated relative error (eext): eext = 0.31%

21 Grid Convergence Index (GCI): GCIfine = 0.4%

4.2 Pump Cavitation Performance Parameters

In order to better represent, understand and compare pump performance for cavitating and non-cavitating cases, three main dimensionless parameters are utilized.

Head and flow coefficients are the most commonly used dimensionless parameters in the investigation of pump performance for both cavitating and non-cavitating conditions. Head

and flow coefficients are expressed as [33]:

𝜓𝜓 𝜑𝜑 2 = (22) 𝑔𝑔𝑔𝑔 𝜓𝜓 2 𝑈𝑈 = (23) 𝑐𝑐𝑚𝑚 𝜑𝜑 𝑈𝑈

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where is the gravitational acceleration, is the head, and is the reference velocity, which is𝑔𝑔 the inducer tip speed /2, where𝐻𝐻 is the rotational𝑈𝑈 speed and is the inducer diameter. In equation (23), 𝜔𝜔is𝜔𝜔 the meridional𝜔𝜔 velocity of the fluid. 𝐷𝐷

𝑐𝑐𝑚𝑚 The third parameter is the cavitation number used to determine the cavitation characteristics defined as [26]:

( ) = (24) 1 1 𝑣𝑣 𝑃𝑃2 − 𝑃𝑃 𝜎𝜎 2 𝜌𝜌𝑈𝑈 where is the suction pressure, is the vapor pressure for a given temperature. The most

1 𝑣𝑣 important𝑃𝑃 cavitation number in 𝑃𝑃pump applications is the critical cavitation number, ,

𝑐𝑐 which corresponds to 3% pump differential head (pressure) drop [26]. Critical cavitation𝜎𝜎 number determines the NSPHr of a pump. Below this required liquid level or corresponding suction head (pressure), pump is not allowed to be operated due to possibility of cavitation damage. This critical NPSH level is also known as NPSH3 [26].

4.3 Geometric Model and Grid

The geometric model of the pump assembly is shown in Figure-2. The geometric model consists of a variable pitch helical inducer, a radial type closed impeller and an axial diffuser vane. The grid for each hydraulic component is separately created via ANSYS

Blade Gen and ANSYS Turbo grid software. Blade, hub and the shroud, if applicable, of each hydraulic component is initially mapped and then meshed to create a structured grid.

The mesh refinement at area of interest with a reasonable expansion rate and global mesh size are adjusted to improve the accuracy of the predictions, while the computational time

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is considered. Once the grid of each component is created, each part is aligned and joined in CFX’s preprocessing step. The exploded view of the grid for each component is shown in Figure-4.

Figure-4 Grid structure of the pump hydraulic components shown in exploded view for clarity.

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Locations with discontinuities such as blade tip, leading and trailing edges, vane root locations for each hydraulic component are meshed with refinement (finer meshing with smaller element size). Mesh refinement at inducer tip to shroud clearance is also applied to capture the backflow leakage and related vortices. Grid convergence index calculations are also carried out as explained in Section 4.1.

5. Results

Initially, the NPSH3 performance is reviewed for each model and the results are compared to the test results. The empirical constants for cavitation models for the FC and the RCR models are revised due to lack of convergence and unrealistic results with the empirical constants determined for specific applications outlined at their studies. For this application, due to variations and discrepancies in vaporization and condensation rates related to pump’s geometry and boundary (running) conditions, the following empirical constants are utilized for the FC and RCR cavitation models:

For “full cavitation model” (FC):

= 0.1 = 1

𝐶𝐶𝑐𝑐 𝐶𝐶𝑒𝑒 For “reduced critical radius cavitation model” (RCR):

= 0.4 = 40

𝐶𝐶𝑐𝑐 𝐶𝐶𝑒𝑒 The above constants are determined based on the simulation results of NPSH3 performance of the pump assembly and the agreement between each model and the NPSH3 test data. Head drop curve as a function of head and cavitation coefficient is plotted for each model. In order to construct head drop curves, head coefficient is monitored while

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inlet pressure (NPSH) is reduced gradually. Figure-5 is the head drop curve for simulations with various cavitation models and the experiment test. NPSH3 performance based on simulation results are very similar along different models. Critical cavitation number at 3% head drop varies between 0.465 and 0.51 according to the simulation results. Critical cavitation number is calculated as = 0.51, = 0.504, = 0.492 and =

ZGB SS FC RCR 0.465. The agreement between the first𝜎𝜎 three cavitation𝜎𝜎 models is𝜎𝜎 within 3.5% considering𝜎𝜎 the NPSH3 performance prediction.

Actual test results exhibit much early cavitation inception compare to any simulation result (greater ). The main reason is due to the leakage around the impeller front and back shroud, and𝜎𝜎 also the backflow through the shaft bushing. The effect of the leakage can also be observed at the non-cavitating performance of the pump according to

Figure-5. The head coefficient is 1.03 while testing under non-cavitating condition with high suction pressure (NPSH or ). Under the same pump operating conditions, head coefficient is predicted to be 1.06𝜎𝜎 at simulations at the absence of cavitation. As it was mentioned at Section 4.3, only the inducer tip clearance is modelled at the simulations to capture the effect of backflow vortex cavitation. Back leakage through impeller shrouds and backflow at shaft bushing are not included in the simulations. Regardless, the difference in head coefficient is reported to be less than 3% between the simulation and test results for non-cavitation case. For cavitation case this is around 9% which is acceptable considering the complexity of multiphase simulations in rotating machinery.

Vapor formation and vapor volume fraction at certain inlet conditions ( ) are reviewed for each cavitation model to observe the cavitation dynamics within the 𝜎𝜎pump

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hydraulic components. These are identified as “a”, “b”, “c” and “d” cavitation points in

Figure-5, which corresponds to pump suction pressure of 10000, 9000, 8000 and 7000 Pa gauge, respectively.

1.2

1.1 (d) (c) (b) (a) 3% Head Drop Line CFD 1.0 3% Head Drop Line Test 𝜓𝜓 0.9

0.8 ZGB 0.7 SS HeadCoeffieint, FC 0.6 RCR Test Data 0.5

0.4 0.00 0.25 0.50 0.75 1.00 1.25 1.50 Cavitation Number, σ

Figure-5 Pump cavitation performance curve based on head drop.

At cavitation point “a”, according to Figure-6, pump inlet pressure is high enough to prevent formation of any significant amount of vapor at the downstream of the inducer.

The vapor formation is mostly at the inducer tip section and front side (suction side) of the inducer blades. Differential pressure increase across the inducer prevents the vapor to propagate further down the inducer blades. There is evidence of tip vortex and back flow vortex cavitation due to formation of vapor along the tip clearance section of the inducer for each model. The vaporization at the front blades are more pronounced with the RCR and the FC cavitation models. While the vapor generation is greater at the front section of

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the inducer for the RCR cavitation model, there is no propagation of the vapor observed toward the back region of the inducer blades. In addition, according to Figure-11, there is slight vapor formation at the shroud side of the impeller front blades at the FC and the RCR models. This can be due to consideration of turbulence pressure distribution at these models and the correction to fluid vapor pressure. It should be noted that at the leading edge of the impeller blades adjacent the to shroud, overall fluid velocity is relatively higher due to the rotational component of the fluid velocity and normal distance to the center of rotation. This higher velocity region can result in relatively earlier inception of cavitation.

The ZGB and SS cavitation models exhibit similar vapor formation and volume fraction results. Overall, the vapor volume is not large enough to impact the pumps performance and no head loss is reported at point “a”.

According to Figure-7, at point “b” when pressure is reduced to 9000 Pa, vapor starts to propagate towards the center of rotation in radial direction at the inducer blades indicating that the rotational component is no longer the dominant factor, with the reduction of inlet pressure, resultant static pressure towards the center starts to fall below the saturation pressure. This is valid for all cavitation models regardless of the sensitivity to vapor pressure correction at the RCR and the FC models. There is also slight increase at impeller leading edge vapor formation for the RCR and FC models with the reduction in inlet pressure per Figure-11. The ZGB and SS cavitation models maintain the consistency of vapor formation, for both models exhibit very comparable results. According to Figure-

5 pump performance at point “b” does not show any signs of head drop or pressure loss due to vapor formation.

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ZGB SS

FC RCR

ZGB SS

FC RCR

Figure-6 Vapor Volume Fraction (a) and vapor volume (b) for each cavitation model, Pin=10000 Pa.

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ZGB SS

FC RCR

ZGB SS

FC RCR

Figure-7 Vapor Volume Fraction (a) and vapor volume (b) for each cavitation model, Pin=9000 Pa.

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At point “c”, according to Figure-5, a slight reduction to the pump head is noted for the FC and the RCR cavitation models. Per Figure-8, both models exhibit large amount of vapor formation at the inducer section with respect to the other two cavitation models. The vaporization has extended to the shaft (hub) section of the inducer for FC and RCR cavitation models. However, the reduction at head (head coefficient) has still not reached the 3% drop for any of the cavitation models. At the inducer blade surfaces, similar to point

“a” and point “b” suction conditions, the RCR cavitation model exhibit the highest vapor formation in terms of surface area. However, according to Figure-10, the concentration of the vapor is much less compared to rest of the models, suggesting the vapor formation is distributed more evenly at the surface of the inducer blades. With the SS cavitation model, the vapor is more concentrated at the inducer blades. Per Figure-11, the vapor volume fraction is very similar to previous point “b” case for all cavitation models. It can be concluded that the majority of the phase change still occurred at the inducer section, and pump hydraulics are able to maintain the pump discharge pressure.

With further reduction to inlet pressure (Pin=7000 Pa), reduction in head coefficient suggests that the pump has reached its critical cavitation number. Operating the pump under this condition can result in cavitation damage. Head drop is equal to or greater than

3% at this point. The vapor propagated to the leading edge of the impeller blades and started to block the meridional flow. There is also significant amount of vapor formation at the inducer blades, and vapor formation and propagation can be considered very similar between each case.

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ZGB SS

FC RCR

ZGB SS

FC RCR

Figure-8 Vapor Volume Fraction (a) and vapor volume (b) for each cavitation model, Pin=8000 Pa.

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ZGB SS

FC RCR

ZGB SS

FC RCR

Figure-9 Vapor Volume Fraction (a) and vapor volume (b) for each cavitation model, Pin=7000 Pa.

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It should be noted that with the original empirical constants suggested by Tsuda et al. [12] and Singhal et al. [11], cavitation inception and cavitation critical points have shown significant discrepancies in NPSH3 performance of the pump. At suction pressure of Pin =

6000 Pa, the FC model with suggested empirical constants of = 0.02 and = 0.01

𝑒𝑒 𝑐𝑐 shows no signs of phase change for this particular study of cavitation𝐶𝐶 dynamics. In𝐶𝐶 addition, the RCR model also required changes to the empirical constants to predict reasonable results. As Tsuda et al. [12] suggested, adjustment to empirical constants is necessary based on the physics of the cavitation problem.

Lastly, cavitation high speed camera images at cavitation inception are compared to the vapor formation plots of each model. Figure-12 shows the vapor volume at the inducer section for Pin=8000 Pa. The SS and ZGB cavitation models closely approximate the vapor volume and formation within the inducer section. RCR model indicates slightly larger vapor formation at the inducer section with respect to the experimental test. The FC model overestimated the vapor formation at the inducer section. However, simulation results at Pin=8000 Pa show that the difference in calculated pump differential head is within less than 1% of each cavitation model according to point “c” of Figure-5. The reported difference in vapor volume seen in Figure-12 does not influence the overall pump differential head, thus the NPSH performance. This proves that without any vapor propagation towards the eye section of the impeller or at the leading edge of impeller blades, pump differential head can be maintained without 3% loss. This finding emphasizes

126

the importance of modeling the pump assembly as a complete unit, rather than analyzing the inducer section only.

ZGB SS FC RCR

Figure-10 Vapor Volume Fraction at Inducer Blade, Front View.

127

ZGB SS FC RCR

Figure-11 Vapor Volume Fraction at leading edge of impeller blades, Front View.

128

ZGB SS

(a)

RCR FC

(b)

Figure-12 Vapor Volume at Pin=8000 Pa, (a) test, (b) CFD simulations.

6. Conclusions

Four commonly utilized flow transport-equation based cavitation models (TEM) are reviewed and implemented in this study. The details and differences of each model are outlined, and the cavitation performance prediction of each model is compared to actual test data. The conclusions of this study are summarized below:

129

Transport based cavitation models excluding SS model may require modifications to empirical constants as each cavitation problem may exhibit different cavitation inception, bubble dynamics and vapor formation. For the cavitation study of an industrial pump with a helical inducer, empirical constants of = 0.1 and = 1 are suggested for FC model

𝑐𝑐 𝑒𝑒 to better estimate the NPSH3 performance.𝐶𝐶 Also, 𝐶𝐶RCR cavitation model empirical constants are modified to = 0.4 and = 40 in order to obtain comparable results to

𝑐𝑐 𝑒𝑒 the actual experiments and predictions𝐶𝐶 by𝐶𝐶 other cavitation models. For any application with cavitation, the empirical constants may need to be adjusted as necessary.

For this particular cavitation study, inclusion of effects of NCG (=15 ppm) and turbulent pressure fluctuations do not alter or provide more accurate predictions of the cavitation performance in terms of NPSH3. There was, however, significant impact to computing time. Therefore, including NCG, turbulent pressure fluctuations, thermal effects, and etc. must be considered for problems where these parameters may or may not play an important factor to the overall results. The most appropriate models for the current particular application are the ZGB and SS models considering the computing time, convergence level and agreement to test data based on visual comparison. Since the SS model does not utilize any empirical constants, there were no changes or adjustments made to this model. The ZGB model with the original empirical constants advised by Zwart et al. [7] proved to predict NPSH3 performance and vapor formation closely to the test data.

Overall cavitation performance results (NPSH3) for each cavitation model was within 3% regardless of the differences in definition of mass transfer rate. All cavitation

130

models provide reasonable predictions with respect to experiments in terms of NPSH performance after the empirical constants are adjusted.

Without presence of any vapor formation towards the eye section of the impeller or at the leading edge of impeller blades, pump differential head can be maintained without

3% loss. This finding emphasizes the importance of modeling the pump assembly as a complete unit, rather than analyzing the inducer section only.

7. Acknowledgment

This work was supported by Elliott Group Cryodynamic Products, Ebara

Corporation.

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SP-8052, May 1971.

[2] Scheer, D. D., Huppert, M. C., Viteri, F., Farquhar, J. “Liquid rocket engine axial-flow turbopumps”, NASA Technical Report, SP-8125, April 1978.

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624, 2002.

132

[12] Tsuda, S., Tani, N., Yamanishi, N., “Development and Validation of a Reduced

Critical Radius Model for Cryogenic Cavitation”, ASME Journal of Fluids Engineering,

Vol.134, 2012.

[13] Senocak, I., Shyy, W., “Numerical Solution of Turbulent Flows with Sheet

Cavitation”, Proceedings of the 4th International Symposium on Cavitation, CAV 2001.

[14] Senocak, I., Shyy, W. “Evaluation of Cavitation Models for Navier-Stokes

Computations”, ASME Joint U.S.-European Fluids Engineering Division Conference,

2002.

[15] Jian, W., Yong, W., Houlin, L., Qiaorui, S., Dular, M., “Rotating Corrected-Based

Cavitation Model for a Centrifugal Pump”, ASME Journal of Fluids Engineering, Vol.140,

2018.

[16] Kunz, R. F., Boger, D. A., Stinebring D. R., Thomas, S. C., Jules, W. L., Gibeling H.

J., Venkateswaran, S., Govindan, T. R., “A Preconditioned Navier Stokes Method for Two- phase Flows with Application to Cavitation Prediction”, Computers & Fluids, Vol. 29, pp

849-875, 2000.

[17] Merkle, C. L., Feng, J., Buelow, P. E. O., “Computational Modeling of Dynamics of

Sheet Cavitation”, Proceedings of the 3rd International Symposium on Cavitation, France,

1998.

[18] Bakir, F., Rey, R., Gerber, A. G., Belamri, T., Hutchison, B., “Numerical and

Experimental Investigation of the Cavitating Behavior of an Inducer”, International Journal of Rotating Machinery, Vol. 10, pp 15-25, 2004.

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[19] Mejri, I., Bakir, F., Rey, R., Belamri, T., “Comparison of Computational Results

Obtained from a Homogeneous Cavitation Model with Experimental Investigation of

Three Inducers”, ASME Journal of Fluids Engineering, Vol.128, pp 1308-1323, 2006.

[20] Campos-Amezcua, R., Khelladi, S., Mazur-Czerwiec, Z., Bakir, F., Campos-

Amezcua, A., Rey, R., “Numerical and Experimental Study of Cavitating Flow Through an Axial Inducer Considering Tip Clearance”, Journal of Power and Energy, Vol. 227 (8),

2013.

[21] Kinzel, M. P., Lindau J. W., Kunz R. F., “An Assessment of Computational Fluid

Dynamics Cavitation Models Using Bubble Growth Theory and Bubble Transport

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[22] Tani, N, Yamanishi, N., Tsujimoto Y., “Influence of Flow Coefficient and Flow

Structure on Rotational Cavitation in Inducer”, ASME Journal of Fluids Engineering, Vol.

134, 2012.

[23] Kubota, A., Kato, H., Yamagachi, H., “A New Modelling of Cavitation Flows: A

Numerical Study of Unsteady Cavitation on a Hydrofoil Section”, Journal of Fluids

Mechanics Vol 240, pp 59-96, 1992.

[24] Watanabe, H., Ichiki, I., “Development of Cryogenic Pump Hydrodynamics Using

Inverse Design Method and CFD”, ASME Middle East Mechanical Exposition, 2007.

[25] Ashihara, K., Goto, A., “Effects of Blade Loading on Pump Inducer Performance and

Flow Fields”, ASME 2002 Joint U.S.-European Fluids Engineering Division Conference,

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[26] Brennen, C. E., “Hydrodynamics of Pumps”, Concepts ETI, Inc, Norwich Vermont,

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Amezcua, A., Rey, R., “Numerical and Experimental Study of Cavitating Flow Through an Axial Inducer Considering Tip Clearance”, Journal of Power and Energy, Vol. 227 (8),

2013.

[28] Karakas, E., “Computational Investigation of Cavitation Performance and Heat

Transfer in Cryogenic Centrifugal Pumps with Helical Inducers”, PhD Dissertation,

University of Nevada, Reno, 2019.

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[30] Schnerr, G. H., Sauer, J., “Physical and Numerical Modeling of Unsteady Cavitation

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136

CHAPTER 5. CONCLUSIONS AND FUTURE WORK

Two unique features of submerged motor cryogenic pumps with a helical inducer are reviewed and discussed in this dissertation. At first, the cooling and fluid flow along the air gap of the submerged motor are studied considering realistic geometries of rotor surface. CFD simulations are conducted to calculate the temperature profile along the stator surface and to observe the fluid flow along the air gap. Simulation results are compared to previously published temperature and flow correlations based on dimensionless parameters. Here are the main outcome and findings of this research:

• It is observed that the temperature profile is very similar among different

geometries, and thus it can be concluded that the rotor configuration does not seem

to play a critical role in the temperature distribution at stator surface. This

conclusion is in strong agreement with the findings from Tachibana and Fukui [8].

• In the cases investigated, the exit temperature at stator surface for each geometry

is still low enough to prevent vapor formation.

• As expected, the correlations with similar geometry ( , ) and dimensionless

parameters (Re, Ta) have given the closest temperature𝛤𝛤 results𝜂𝜂 to those in CFD

analysis. This suggests that published correlations can be utilized for the

calculation of temperature distribution even in the presence of skewed grooves on

the rotor, as long as similar geometries and dimensionless parameters are retained.

On the other hand, it is observed that even for identical Reynolds and Taylor

numbers, groove geometry does impact the flow structure. The impact on flow

137

structure seems however slight and has modest effect on stator temperature, at least

for the cases studied in this paper.

• Fluctuations in the stator temperature and heat transfer coefficient are observed

along the axial direction once the Taylor vortices are formed. These fluctuations

have modest impact on the overall heat transfer for the cases studied in this paper.

However, for relatively larger annulus gaps and/or different geometries, the

magnitude of the fluctuations might be significant enough to impact the overall

temperature distribution.

• The flow structure for each surface profile demonstrate the presence of Taylor

vortices. The axial flow and geometry of the rotor can affect the width and the

origin of each vortex. Furthermore, each axial groove configuration has noticeable

and distinct impact on vortex shape, even though identical dimensionless

parameters exist for each geometric configuration.

• Finally, the extent of the entrance region for each surface profile is examined and

it is concluded that there is a correlation between the entrance region length and

the vortex shape (width). The configuration with the narrowest vortex is the 10

degrees CW skewed case, which also has the longest hydrodynamic entrance

region. The reason for this correlation seems to be associated to the magnitude of

the tangential velocity, which is correlated to development of the vortices. It is

observed that temperature depends slightly on the vortex shape and length (within

1 °C). The vortices that are stretched and off center due to axial flow, exhibit

relatively worse cooling characteristics at the stator surface.

138

Further research should be conducted to better understand the correlation between the structure of the Taylor vortices and possible cooling improvements in induction motor applications. In addition, the effect of the Taylor vortices to rotordynamics of the pump motor section can be studied to understand how the rotordynamic coefficients are impacted by the vortices. Experimental verification of this study in cryogenic will take place to further validate the conclusions and results. To this point, there is still no established correlation that can be implemented to predict the heat transfer coefficients for any annuli and heating configuration. Further research can be devoted to better understand the effect of surface geometries to the cooling and heating of induction motors considering two phase flow or the heat transfer and cooling characteristics in the presence of vapor or vapor formation.

The second study involves cavitation performance of the pump assembly with a helical inducer and the effect of the tip clearance. CFD simulations are performed and results are validated via experimental testing of the pump assembly. The summary of the conclusions of this study is as follows:

• CFD simulations and experimental results are within 9% for the cavitation

performance predictions for NPSH3 value, while the discrepancy for non-cavitation

pump performance is 5%. The difference between tests and simulations results are

acceptable, considering the complexity and accuracy involved in testing and

simulations.

• According to the simulations results with different inducer tip clearances, excess

tip clearance will result in cavitation performance drop by 11% for the variable

139

pitch inducer, while this drop is 4.5% for the constant pitch inducer. Cavitation

performance of the variable pitch inducer is more sensitive to the changes to the

inducer tip clearance. The head rise across the variable pitch inducer is greater due

to the blade geometry and progressive pitch. The reduction in fluid volume between

blades along the meridional direction results in greater differential pressure at the

variable pitch inducer. However, this results in greater backflow leakage and

vorticity at the tip section. Therefore, sensitivity to the tip clearance is more

pronounced for the variable pitch inducer.

• There is a good agreement between the blade cavitation location per two-phase

simulations and the low-pressure regions according to single phase simulation

results. It is possible to approximate or possibly control the blade cavitation and its

initiation by examining or adjusting the blade loading (pressure).

• The simulations include the complete pump assembly (inducer, impeller and the

diffuser vane) since NPSH3 cavitation performance is defined as the NPSHa value

when pump’s first stage differential head is dropped by 3%. While the inducer’s

pressure differential can reduce by more than 3% at the inducer section only, the

pump can still be able to maintain its differential pressure over NPSH3 limit. Large

cavitation volumes at inducer locations do not necessary result in the pump head

drop to NPSH3 level, unless the vapor formation propagates to the impeller eye

section. Therefore, all the pump hydraulic components should be included and

considered in determination of true NPSH3 cavitation performance of a pump.

140

• Increase in tip clearance results in excess back leakage and larger vortex

recirculation at tip location, which produces pressure loss within the inducer and,

consequently, degraded cavitation performance. The impact on the non-cavitating

performance is minimal. This is due to the percentage of the head generation of the

inducer being too small as compared to the head generation of the pump assembly.

The effect of the tip clearance to the cavitation performance is compared for two style inducers for a given blade geometry and the diameter. A generalized dimensionless correlation, such as cavitation number as a function of dimensionless

(λ=δ/ Dt) can be established for helical style inducers by analyzing different diameter inducers with different blade geometry. This future study can help researchers and engineers in determination of the optimized tip clearance for any helical inducer application. In addition, rotordynamics aspect of the inducer should be studied and considered in determination of the tip clearance, since minimizing the back leakage via utilization of very small tip clearances may not be possible due to possible radial displacement of the inducer component under cavitating and non-cavitating operating condition. Experimental studies and simulations considering fluid and solid interaction under multiphase flow condition can be performed for this effort. Tests and simulations are performed by using water at ambient temperature at this study. Further research can be done to explore the bubble dynamics and cavitation performance of different fluids.

The cavitation models are reviewed at the final research subject related to comparison of transport-equation based cavitation models. At this study, four commonly used transport-equation based cavitation models are reviewed and implemented to the CFD

141

code to predict the NPSH performance of the cryogenic pump. The following is the summary of the conclusions related to this study:

• Transport based cavitation models excluding SS model [31] may require

modifications to empirical constants as each cavitation problem may exhibit

different cavitation inception, bubble dynamics and vapor formation. For the

cavitation study of an industrial pump with a helical inducer, new set of empirical

constants are suggested for FC and RCR models [32,33] to better estimate the

NPSH3 performance. For any application with cavitation, the empirical constants

may need to be adjusted as necessary.

• For this particular cavitation study, inclusion of effects of NCG (=15 ppm) and

turbulent pressure fluctuations do not alter or provide more accurate predictions of

the cavitation performance in terms of NPSH3. There was, however, significant

impact to computing time. Therefore, including NCG, turbulent pressure

fluctuations, thermal effects, and etc. must be considered for problems where these

parameters may play an important factor to the overall results.

• The most appropriate models for prediction of the cavitation performance of a

pump assembly are ZGB and SS models [29, 31] considering the computing time,

convergence level and agreement to test data based on visual comparison. Since SS

model does not utilize any empirical constants, there were no changes or

adjustments made to this model. ZGB model with the original empirical constants

advised by Zwart et al [29] proved to predict NPSH3 performance and vapor

formation closely to the test data.

142

• Overall cavitation performance results (NPSH3) for each cavitation model was

within 3% regardless of the differences in definition of mass transfer rate. All

cavitation models provide reasonable predictions with respect to experiments in

terms of NPSH performance after the empirical constants are adjusted.

Simulations and tests are performed considering water at constant temperature.

Further research can be done to determine the effect of the thermal changes and sensitivity of each model to thermal effects and fluids. The effect of the turbulence model was not explored at any of the studies. To the author’s knowledge, turbulence models under two- phase flow in turbomachinery is not widely researched. More studies can be conducted to understand the effect of turbulence model to the cavitation behavior in turbomachinery.

The focus of this study was to review and utilize transport-equation based cavitation models. A review and study of models employing equations of state can be performed in order to establish a comparison to transport-equation based models. One of the disadvantages of transport-based cavitation models are the empirical constants as they may need adjustment or rely on a constant that is empirically determined for a specific case. It should be noted even though SS model is free of empirical constants for condensation and vaporization mass transfer rate calculations, it still assumes that there are constant number of bubbles when the vaporization occurs. Therefore, each models accuracy is highly depending on constants which may not necessarily represent or can be used in modelling all cavitation cases. Further research should be devoted to establishing a cavitation model without utilization of any empirical constants.

143

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624, 2002.

[33] Tsuda, S., Tani, N., Yamanishi, N., “Development and Validation of a Reduced

Critical Radius Model for Cryogenic Cavitation”, ASME Journal of Fluids Engineering,

Vol.134, 2012.

[34] Senocak, I., Shyy, W., “Numerical Solution of Turbulent Flows with Sheet

Cavitation”, Proceedings of the 4th International Symposium on Cavitation, CAV 2001.

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Computations”, ASME Joint U.S.-European Fluids Engineering Division Conference,

2002.

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Cavitation Model for a Centrifugal Pump”, ASME Journal of Fluids Engineering, Vol.140,

2018.

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148

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