An Investigation of Steam-Explosion Loadings with SIMMER-II

LA--10639-MS DE90 00 9219

W. R. Bohl

'-...;'•- ;-Ofi :,F T

Los Alamos National Laboratory L( Los Alamos,New Mexico 87545 CONTENTS

LIST OF TABLES i x LIST OF FIGURES x FOREWORD , xi x ACKNOWLEDGMENTS xx i i ABSTRACT xxv I. SUMMARY OF THE REPORT 1 A. Introduction 1 B. Modification of SIMMER-II 2 C. Calibration to SNL Steam-Explosion Experiments 4 D. Calibration to Los Alamos Shallow-Pool Experiments 7 E. SIMMER-II Reactor Case Calculative Results 11 F. Containment Failure Probabilities 16 G. Conclusions and Recommendations 19 II. SIMMER-II MODIFICATIONS FOR THE MOLTEN-CORE/COOLANT INTERACTION PROGRAM 23 A. Introduction , 23 B. Summary of Modified Equations 25 1. AEOS Modifications 26 2. Heat-Transfer Modifications 29 3. Vaporization/Condensation Modifications 30 C. Discussion of the AEC" Modifications 33 1. Review of Difficulties in the Version 10 SIMMER-II Formulation 33 2. Examination of Proposed Solutions to AEOS Problems 35 3. The Vapor Heat-Capacity Modification 36 4. Modification of the Gas "Constant" at High Pressures 42 5. Definition of the Vapor Temperature for Single-Phase Cells and Other Modifications 43 6. Discussion of Results 46 7. Suggestions for Further Improvement 59 8. Conclusions 64 D. Modification of SIMMER-II Liquid-Liquid Heat Transfer for Water 64 E. Vaporization/Condensation Model Changes 71 F. Miscellaneous Corrections for the Molten-Core/Coolant-Interact ion Program 85 III. LOWER HEAD MODEL FOR SIMMER-II 87 A. Introduction 87 B. Input 87 C. Head Failure 90 D. Lower Head Motion 90 E. Boundary Conditions and Edits 93 F. Correction Set 98 G. Sample Problem 98 IV. CORRELATION TO SNL STEAM-EXPLOSION EXPERIMENTS 107 V. ANALYSIS OF LOS ALAMOS EXPERIMENTAL DATA FOR SHALLOW POOLS 119 A. Introduction 119 B. Experimental Correlation 119 C. Hypothetical Cases with NonunifoTtn Interfaces 130 D. Experimental Comparison with a Pressure Ratio of 50:1 150 E. Experimental Comparison with a Nonuniform Initial Interface .. 156 F. Conclusions 153 VI. ANALYSIS OF STEAM EXPLOSIONS WITH SIMMER-II 169 A. Case 1 - 20% Premixed 169 1. Investigation of Water Surface Area 173 2. Comparison with the ZIP Study Using the New Models 173 3. Case 1 Update 181 B. Case 2 - Conservative Premixing, Explosion, and Expansion .... 195 1. Explosion After 1 s of Mixing (Scoping Calculation) 197 2. Explosion after 0.7 s of Mixing (Second Scoping Calculation) 209 3. Case 2 - Final , 210 C. Case 3 - 75% Premixed 227 D. Case 4 - Simulation of an Incoherent Explosion 235 E. Case 5 - Upper Bound 238 F. Summary of SIMMER-II Steam-Explosion Cases 246 VII. PROBABILITY OF CONTAINMENT FAILURE 249 VIII. RESEARCH PRIORITIES ON STEAM EXPLOSIONS 265 APPENDIX A THE SIMMER-II MANUAL AEOS TREATMENT 273 I. MATERIAL TEMPERATURES 273 11. TWO-PHASE FLOW PRESSURES 280 III. SINGLE-PHASE FLOW STATES 284 IV. LIQUID MICROSCOPIC DENSITIES 287 V. AEOS INPUT 288 VI. DETERMINATION OF COMPONENT TEMPERATURES 297 VII. DETAILS IN THE EVALUATION OF SINGLE-PHASE LIQUID STATES 301 A. Sing]e-Component Systems 301 B. Multicomponent System 305 APPENDIX B CFS NODE /VAPOR/WRBAEOS (EOS CORRECTION SET) 311 APPENDIX C REVISED SIMMER-II INPUT DESCRIPTION FOR THE AEOS 321 APPENDIX D AEOS SIMULATION PROGRAM 327 APPENDIX E SUGGESTED VALUES FOR THE WATER AEOS INPUT 337 APPENDIX F STANDARD SIMMER-II LIQUID-LIQUID HEAT-TRANSFER MODEL 339 APPENDIX G CORRECTION SET FOR MODIFIED LIQUID-LIQUID HEAT TRANSFER 349 APPENDIX H VAPORIZATION/CONDENSATION CORRECTION SET , 353 APPENDIX I SIMMER-II MANUAL TREATMENT OF THE VAPORIZATION/CONDENSATION MODEL 359 APPENDIX J MISCELLANEOUS CORRECTIONS 373 APPENDIX K SUMMARY DESCRIPTION OF THE HEAD-FAILURE MODEL SUBROUTINE 377 APPENDIX L DEFINITIONS OF VARIABLES IN THE PLUGW CORRECTION SET 379 APPENDIX M LOWER HEAD DYNAMIC ANALYSIS 381 I. INTRODUCTION AND SUMMARY OF RESULTS OF ADINA STUDY 381 II. THE FAILURE MODEL AND CRITERIA 383 III. BASIS FOR SIMPLE MODEL TO ESTIMATE REQUIRED FAILURE IMPULSE (COLUMN 1 OF TABLE M-II) 384 IV. A SDOF MODEL THAT APPROXIMATES FAILURE TIMES 385 V. CONCLUSION 391 APPENDIX N CONSIDERATIONS REGARDING THE RESULTS FROM THE SDOF LOWER HEAD FAILURE MODEL 393 APPENDIX 0 A LIMITED REVIEW OF SNL STEAM-EXPLOSION EXPERIMENTS 395 APPENDIX P SIMMER-II INPUT FOR THE ANALYSIS OF THE MD-19 EXPERIMENT USING A UNIFORM MIXING ZONE 403 APPENDIX Q COMPARISON OF THREE-FIELD AND SIMMER-II CALCULATIONS OF COARSE PREMIXING 407

VI 1 APPENDIX R SIMMER-11 INPUT FOR THE ANALYSIS OF THE MD-19 EXPERIMENT USING A NONUNIFORM MIXING ZONE 433 APPENDIX S EFFECTS OF CHANGING THE RADIAL AND AXIAL CONSTRAINTS IN THE SIMULATION OF MD-19 441 APPENDIX T EXPLOSION CALCULATION STARTING WITH A STANDARD SIMMER-11 PREMIXED CONFIGURATION 445 APPENDIX U SIMMER-II PREMIXING WITH HIGH STEAM PRODUCTION RATES 447 APPENDIX V DETAILS ON THE UPPER BOUND SIMMER-11 STE.AM-EXPLOSION CALCULATION 453 REFERENCES 463 LIST OK TABLES

3 I. HIGHEST F ZERO WITH P(- = 171.4 kg/ir. AND ec; = 3.197 MJ/kg 52 11. AEOS/STEAM-TABLE COMPARISON FOR THE 22~MPa ISOBAR 53 III. AEOS/STEAM-TABLE COMPARISON FOR THE 50-MPa ISOBAR 54 IV. AEOS/STEAM-TABLE COMPARISON FOR THE l(H)-MPa ISOBAR 54

V. AEOS/STEAM-TABLE COMPARISON FOR THE 22-MPa ISOBAR IF a(J M IS SET TO 75 K 55 VI. SESAME RESULTS FOR FIG. 24 61 VII. LOWER HEAD MODELING INPUT 89 VIII. PLUG DATA FOR BOTTOM/TOP HEAD CALCULATION 95 IX. SAMPLE OUTPUT EDIT BEFORE LOWER HEAD FAILURE 96 X. SAMPLE OUTPUT EDIT AFTER LOWER HEAD FAILURE 97 XI. DEFINITION OF THE PXPLOT ARRAY 98 XII. CORRECTION SET FOR THE SIMMER-11 LOWER HEAD MODEL 99 XIII. INITIAL CALCULATIONS FOR FITTING TEST MD-19 112 XIV. SIMMER-II INPUT FOR THE REFERENCE CALCULATION 131 XV. FLUID KINETIC ENERGY PARTITION 192 XVI. SUMMARY OF STEAM-EXPLOSION CASES 247 XVII. DEFINITION OF PROBABILITY SPLIT LEVELS 249 XVIII. FEATURES THAT COULD BE USED IN SIMMER-II PARAMETERS OF STEAM EXPLOSION/EXPANSIONS 268 XIX. PRESENTLY INVESTIGATED IMPROVEMENTS COMPARING THE INITIALLY PROGR.AMMED THREE-FIELD FLUID-DYNAMICS ALGORITHM AND SIMMER-11 271 E-1. WATER AEOS INPUT 337 M-I. PRESSURE TRANSIENTS USED IN LOWER HEAD DYNAMIC ANALYSIS 381 M-II. SUMMARY OF LOWER HEAD FAILURE STUDY FOR A GENERIC PWR VESSEL 383 M-111. COMPARISON OF SDOF MODEL AND ADINA RESULTS 390 0-1. INTERMEDIATE-SCALE FITS TESTS 397 Q-I. SIMMER-II INPUT FOR THE MD-19 PREMIXING CALCULATION 414 LIST OF FIGURES

la. SNL reported water chamber base pressures in experiment MD-19 6 lb. Calculated water chamber base pressures in experiment MD-19 6 2. Shallow-pool experimental apparatus 7 3a. P4 transducer pressures reported for the shallow-pool experiments .... 9 3b. Calculated P4 pressures in the shailow-pool experiments 9 4. PWR structural representation for SIMMER-II steam-explosion calculations 12 5. Integrated upper head loading for case 1 13 6. Integrated upper head loading for case 2 14 7. Integrated upper head loading for case 3 15 8. Integrated upper head loading for case 4 16 9. Integrated upper head loading for case 5 < 17 10. Sample problem results 26 11. Triple-valued "F" function for water vapor 34 12. Temperature-averaged vapor heat capacities for water 37

13. SESAME/SIMMER-11 comparison plot for fitting w 44 14. "F" function for the revised AEOS 47 15. Steam-table comparison with saturated density and temperature inputs , 48 16. Saturation pressures with internal energy and density input into SIMMER- II 49 17. Internal energies of vapor with density and temperature input along the 22-MPa isobar 50 18. Derivative of (Eq. 25) as a function of &Q ^, with TQ just above 1"crt for water 51 19. Remaining triple-valued nature of the "F" function near the critical point 52 20. A typical EOS comparison at low steam densities 56 21. Additional comparison of EOS data at low density 57 22. EOS comparison at high temperature 58

3 23. A SESAME/SIMMER-II comparison at PQ = 316 kg/m 59 24. Pressure as a function of liquid internal energy taken from the density and temperature describing a ficticious saturation curve above T^ . .. 60 25. An example showing that p > p* is not permitted 63 26. Pressure in the sample problem with new heat transfer 69 27. Pressure in the sample problem with original heat transfer 69 28. Water temperature in the sample problem with new heat transfer 70 29. Water temperature in the sample problem with original heat transfer .. 71 30. Pressure in the sample problem with new revised heat transfer 72 31. Water temperature in the sample problem with new revised heat transfer 72 32. Mesh and structure geometry for the quasi-mechanistic reactor steam-explosion problem 88 33. Geometric indices for a radial node in the SIMMER-II moving lower head model . , , 92 34. SIMMER-II treatment of the fluid dynamics at the lower head interface 93 35. EXO-FITS apparatus 108 36. Typical instrumented water chamber 109 37. Initial configuration for the SIMMER-II representation of test MD-19 110 38. Base pressure 6,f the water chamber in experiment MD-19 Ill 39. Best fit for a uniform interaction zone (water-droplet size 75 ^m; heat-transfer multiplier 1.7) 113 40. Initial thermite distribution for the three-field calculation compared with the reported experimental results 114 41. Sensitivity of the peak pressure to the heat-transfer multiplier for a 0. 3-mm fuel droplet and a O.O75-mm water droplet 115 42. Best fit starting from a nonuniform interaction zone with a 0.3-mm fuel droplet and a 0.075-mm water droplet 115 43. Water phase pressure for experiment MD-19 , 116 44. Pressure at location (r,z) = (0.27 m, 0.30 m) measured from the bottom of the Lucite container 117 45. Experimental setup for the shallow-pool tests 120 46. SIMMER-II mesh using equal area radial nodes and a pressure vs time bottom boundary condition 122 47. Experimental pressure recorded at location Pi 123 48. PI recorded pressures over the time range of interest for SIMMER-II calculations 49. Experimental P4 pressure trace 124 50. Calculated pressure for comparison at the P4 transducer 124 51. Liquid-volume-fraction contour plots produced by SIMMER-II 126 52. Contour plots with time-step control revisions 128 53. Revised calculation for comparison with the P4 transducer 130 54. Liquid-volume-fraction contour plots. Concave initial conditions .... 135 55. Pressure trace at the center of the upper endplate for the concave case 137 56. Liquid-volume-fraction contour plots. Convex initial conditions 138 57. Pressure trace at the center of the endplate foT the convex case 140 58. Liquid-volume fractions for the plate obstruction case 141 59. Pressure trace at the center of the endplate with a permanent obstruction 143 60. Reference case pressure integral 143 61. Concave case pressure integral 144 62. Convex case pressure integral =. 144 63. Plate obstruction case pressure integral 145 64. Center of tube pressure trace for the scaled-up case 145 65. Liquid-volume fractions for the scaled-up case 146 66. Upscaled case pressure integral 150 67. Experimental pressures recorded at 'location PI for the 50:1 pressure ratio test 151 68. PI record over the time range of interest for SIMMER-II calculations in the 50:1 pressure ratio test 151 69. Results at location P4 for the 50=1 pressure ratio test 152 70. Experimental results in the 50:1 pressure ratio test 153 71. Pressure integral in che 50:1 pressure ratio case 153 72. Contour plots for the 50:1 pressure ratio case 154 73. SIMMER-II representation of the experimental configuration with a plate , „. 157 74. PI pressure in test 16 with a stationary plate , 158 75. PI pressure in test 17 with a stationary plate 159 76. PI pressure in test 18 with a stationary plate 159 77. PI pressure in test 19 with a stationary plate 160 78. P4 pressure in test 16 with a stationary plate 160 79. P4 pressure in test 17 with a stationary plate 161

XI 1 80. P4 pressure in test 18 with a stationary plate 161 81. P4 pressure in test 19 with a stationary plate 162 82. Calculated P4 pressures for test 19 162 83. Calculated pressure integral, test 19 163 84. SIMMER-II liquid-volume-fraction plots for comparison with the experimental plate case 164 85. SIMMER-II results for an idealized expansion in a multinode envi ronmer.t 167 86. Configurat ion model for case 1 170 87. Initial core material distribution - 171 88. Initial water dist ibution 172 89. Liquid-volume-fraction plots for comparison with the ZIP study 175 90. Pressure at the inlet plenum bottom for comparison with the ZIP study 177 91. Pressure at the downcomer top for comparison with the ZIP study 177 92. Pressure at the top of the vessel for comparison with the ZIP study .. 178 93. Pressure on the head at 30° to the vertical for comparison with the ZIP study 178 94. Pressure on the head at 70° to the vertical for comparison with the ZIP study 179 95. Force on the head for comparison with the ZIP study 179 96. Total fluid kinetic energy (KE) for comparison with the ZIP study .... 180 97. Upward fluid kinetic energy for comparison with the ZIP study 180 98. Kinetic energy of materials not in the core and above core regions for comparison with the ZIP study 181 99. Pressure differential across the lower head in case 1 (update) 182 100. Lower head displacement before failure in case 1 (update) 183 101. Liquid-volume-fraction plots for case 1 (update) 184 102. Pressure at the inlet plenum bottom for case 1 (update) 189 103. Pressure at the top of the downcomer for case 1 (update) 189 104. Pressure at the top of the vessel for case 1 (update) 190 105. Pressure on the head at 30° to the vertical for case 1 (update) 190 106. Pressure on the head at 70° to the vertical for case 1 (update) 191 107. Force on the head for case 1 (update) 191 108. Total fluid (SINWER-II liquid and vapor velocity fields) kinetic energy for case 1 (update) 192

xm 109. Upward fluid kinetic energy for case 1 (update) 193 110. Downward fluid ki .etic energy for case 1 (update) 193 111. Downcomer and inlet pipe kinetic energy for case 1 (update) 19-1 112. Outlet pipe kinetic energy for case 1 (update) 194 113. Initial configuration for the SIMMER-II premixing calculation in case 2 196 114. Initial conditions, SIMMER-II conservative premixing calculation (case 2) 198 115. Water-fuel contact, SlMMER-II conservative premixing calculation (case 2) 198 116. Fuel contact with the support forging, SIMMER-II conservative premixing calculation (case 2) (time = 500 ms) 199 117. Initial contact of fuel with the lower head, DIMMER-11 conservative premixing calculation (case 2) (time = 700 ms) 199 118. Beginning of fuel pool breakup, SIMMER-II conservative premixing calculation (case 2) 200 119. Two-phase fuel pool formed, SIMMER-II conservative premixing calculation (case 2) > 200 120. Initial expansion instability, fluidized explosion conditions 201 121. Instability development, fluidized explosion conditions 202 122. Venting beginning under fluidized explosion conditions 202 123. Configuration just before impact, fluidized explosion conditions 203 124. Material impact beginning, fluidized explosion conditions 203 125. Material impact complete and rebound beginning, fluidized explosion cond i t i ons 204 126. Final configuration, fluidized explosion conditions 204 127. Inlet plenum pressure, fluidized explosion conditions ,. 205 128. Pressure at the top of the downcomer, fluidized explosion condi t ions 205 129. Pressure at the top of the vessel, fluidized explosion conditions .... 206 130. Head pressure at 30° to the vertical, fluidized explosion condi t ions 206 131. Head pressure at 70° to the vertical, fluioized explosion condi t ions 207 132. Force on the head, fluidized explosion conditions 207 133. Total fluid kinetic energy, fluidized explosion conditions 208

XIV 134. Upward fluid kinetic energy, fluidized explosion conditions 208 135. Miscellaneous fluid kinetic energy, fluidized explosion conditions ... 209 136. Corium below the bottom of the initial corium pool, SIMMER-II conservative premixing calculation (case 2) 210 137= Liquid-volume fractions, explosion at 0.7 s 211 138. Inlet plenum pressure, explosion at 0.7 s 214 139. Pressure at the top of the downcomer, explosion at 0.7 s 214 140. Pressure at the top of the vessel, explosion at 0.7 s 215 141. Head pressure at 30° to the vertical, explosion at 0.7 s 215 142. Head pressure at 70° to the vertical, explosion at 0.7 s , 216 143. Force on the head, explosion at 0.7 s 216 144. Tctal fluid kinetic energy, explosion at 0.7 s , 217 145. Upward fluid kinetic energy, explosion at 0.7 s 217 146. Miscellaneous fluid kinetic energy, explosion at 0.7 s 218 147. Liquid-volume fractions during the '. ~ mansion process, case 2 (final) 219 148. Inlet plenum pressure, case 2 (final) ,... 221 149. Pressure at the top of the downcomer, case 2 (final) 222 150. Pressure at the top of the vessel, case 2 (final) 222 151. Head pressure at 30° to the vertical, case 2 (final) , 223 152. Head pressure at 70° to the vertical, case 2 (final) 223 153. Force on the head, case 2 (final) , 224 154. Upward fluid kinetic energy, case 2 (final) 225 155. Downcomer fluid kinetic energy, case 2 (final) 225 156. Downward and outlet pipe fluid kinetic energy, case 2 (final) 226 157. Total fluid kinetic energy, case 2 (final) 226 158. Configuration model, case 3 228 159. Initial expansion of fuel in case 3 229 160. Head loading and reflection in case 3 230 161. Pressure at the inlet plenum bottom for case 3 231 162. Pressure at the top of the downcomer for case 3 231 163. Pressure at the top of the vessel for case 3 232 164. Pressure on the head at 30° to the vertical for case 3 232 165. Pressure on the head at 70° to the vertical for case 3 233 166. Force en the head for case 3 233 167. Total fluid kinetic energy, case 3 234

XV 168. Upward fluid kinetic energy, case 3 234 169. Miscellaneous fluid kinetic energy, case 3 235 170. Conf igurat ion model, case 4 237 171. Driving pressure for lower head failure, case 4 238 172. I -quid-volume-fraciion plots for post-explosion/expansion, case 4 .... 239 173. Inlet plenum pressure, case 4 242 174. Pressure at the top of the downcomer, case 4 ., 242 175. Pressure at the tcp of the vessel, case 4 243 176. Head pressure at 30° to the vertical, case 4 243 177. Head pressure at 70° to the vertical, case 4 244 178. Force on the head, case 4 244 179. Total fluid kinetic energy, case 4 245 180. Upward fluid kinetic energy, case 4 245 151. Miscellaneous fluid kinetic energy, case 4 ,, ,, = . = = = ..,.. 246 182. Diagram of an accident's progression focused on the alpha raode of 251 direct containment failure by a steam explosion 251 133 Visualization of the state resulting from a lailure of the core barrel prior to penetration of a coherent molten mass through the below-core structure 253 184. Melt flow into the lower plenum by sideways penetration of the core barrel 254 185. Proposed SEALS experiment configuration 270 A-l. Pressure-volume relation for the single-phase EOS 303 F-l. Physical representation of liquid-liquid interaction 340 F-2. Collision configuration for liquid m and liquid k 342 F-3. Assumed contact interval 344 F-4. Heat-transfer treatment during contact 346 M-l. Case 6- Regions used to approximate the spatial variation of the Table N'-l SIhiMER transient with four separate pressure-time histories 382 M-2. Failure initiation region from ADINA parameter studies , 386 M-3. "Expanding vessel" mode 388 0-1. Effect of pressure on explosion yield 396

0-2. FITS containment chamber ; 398 0-1- Initial volume fractions at the start of the three-field calculation AQS Q-2. Three-field calculation following impact with water 409

xvi Q-3. Three-field calculation showing initial total removal of water before the advancing melt - ... 410 Q-4. Three-field calculation after the melt density is reduced sufficiently to stop total water vaporization (removal) at the melt front 411 Q-5. Three-field calculation when the container bottom is being approached 412 Q-6. Three-field calculation at the time of a presumed steam explosion .... 413 Q-7. Initial volume fractions for the standard SIMMER-II calculation 419 0-8. Melt-water contact with standard SIMMER-II modeling 420 0-9. Melt penetration of water with standard SIVWER-II modeling 421 0-10. Start of low vapor production with standard SIMMER-II modeling 422 0-11- Results with standard SIMMER-II modeling as the container bottom is being approached . 423 0-12. Results at the time of the steam explosion with standard SIMMER-II mode 1 i ng . 424 Q-13. Initial conditions with revised SIMMER-II heat transfer 427 0-14. Melt-water contact with revised SlhMR-II modeling 428 Q-15. Initially strong vapor production with the revised SIMMER-II mode ling , 429 0-1^. Beginning of melt breakup with revised SIMMER-II modeling 430 Q-17. Melt breakup with revised SIMMER-II modeling 431 Q-18. Configuration at end of calculation for comparison with Figs. Q-6 and Q-12 432 S-l. Results starting from a nonur.iform interaction zone with a rigid-wall radial constraint > . 441 S-2. Initial conditions for the case with axial constraint 443 S-3. Pressures below the slug in the axially/radially constrained case .... 443 T-l. Explosion results starting with standard SIMMER-II calculated premixing , 445 U-l. Initial conditions, high steam production premixing case (t ime = 0.0 ms) , 448 U-?. Initial flow, high steam production premixing case (time = 100 ms) ... 448 U-3. Initial mixing, high steam production premixing case (t ime = 200 ms ) . 449 U-4. Premixture formed above support forging, high steam production premixing case (time = 300 ms) 449

xvn U-5. Counterflow beginning, high steam production premixing case (t ime = 400 ms) 450 U-6. Pool breakthrough close, high steam production premixing case (t ime = 500 ms ) 450 U-7. Conum pool fluidization beginning, high steam production premixing case (time = 600 ms) 453 U-8. Fuel spray reaching the top of the vessel, high steam production premixing case (time = 700 ms) 151 'J-9. Finai conditions, high steam production premixing case (t ime = 1000 ms) 452 V-l. Geometric setup for bounding case illustration 454 V-2. Comparison of an isentrope from the SESAME LOS with an adjusted SIhMER-II AEOS using the same input densities and internal energies .. 456 V-3. Pressure at the bottom of the inlet plenum for case 5 457 V-4. Pressure at node (11,60), where the head curvature begins for case 5 , 457 V-5. Pressure at node (6,65), the middle of the head curvature region for case 5 458 V-6. Pressure at the center of the head for case 5 458 V-7. Integrated force on the head for case 5 459 V-8. Liquid-volume fractions for case 5 460 V-9, Upward fluid kinetic energy for case 5 461 V-10. Downward fluid kinetic energy for case 5 ...., 461 V-ll. Downcomer and inlet pipe fluid kinetic energy for case 5 462 V-12. Outlet pipe fluid kinetic energy for case 5 462

XVI 1 1 FOREWORD

The mixing of a high-temperature melt with water is known to be capable of producing extremely rapid boiling. As a consequence of the inherent inertial constraints of the system, a conversion of the thermal energy of the melt into mechanical work can occur on an explosive time scale. This process is known as a steam explosion. A quantitative evaluation of the risk posed by steam explosions in severe reactor accidents was first reported in the 1975 Reactor Safety Study (WASH-1400).* The WASH-1400 Teport adopted a range of 10"1 to 10"4 for the conditional probability of containment failure given core melt. These numbers were the 5th and 95th percentiles of a skew lognormal distribution whose median was 10~2. After the Three Mile Island accident, increased emphasis was placed on steam-explosion research and analysis of potential steam-explosion consequences. In general, the WASH-1400 analysis was assessed as too pessimistic, particularly by the US Nuclear Regulatory Commission (NRC) Staff in NUREG-0850.** However, many uncertainties exist regarding the details of a postulated accident's progression and the steam-explosion phenomenology itself. These uncertainties were highlighted in a 1983 Monte Carlo study by Berman et al. Because of the difficulties in resolving core melt details, the question arises as to whether inherent processes will limit expected steam-explosion energetics so that the risk posed is negligible. The initial purpose of this study was to determine the maximum expected loads a steam-explosion might place on the upper head of a pressurized water reactor using a detailed hydrodynamics calculation of the postexplosion material expansion irrespective of any steam- explosion magnitude. If existing dissipative mechanisms could be found to be sufficient, for example, from slug breakup or from failure of the lower vessel head leading to venting and downward momentum transfer, the uncertainties in the

*W. A. Carbiener, P. Cybulskis, R. S. Denning, J. M. Genco, N. E. Miller, R. L, Ritzmau, R. Simon, and R. 0. Wooton "Physical Processes in Reactor Meltdown Accidents," App. VIII to "Reactor Safety Study," US Atomic Energy Commission report WASH-1400, NUREG-75/014 (October 1975). **NRC Staff, "Preliminary Assessment of Core Melt Accidents at the Zion and Indian Point Nuclear Power Plants and Strategies for Mitigating Their Effects," US Nuclear Regulatory Commission report NUREG-0850 (November 1981). *M. Berman, D. V. Swenson, and A. J. Wickett, "An Uncertainty Study of PWR Steam Explosions," Sandia National Laboratories report SAND83-1438, NUREG/CR-3369 (May 1984).

XI X steam-explosion process itself would be irrelevant to the issue of direct containment failure. This study was performed in 1984-85. The conclusion reached was that a measure of subjectivity is required to establish a computer model, to set up bounding cases, and to interpret the meaning of the resulting calculations. Although an updated study could improve on the accuracy of the details, this conclusion still appears valid. If pessimistic but physically possible assumptions are used, mechanistic analysis suggests that thermodynamic limits may be reducible by an order of magnitude. Nevertheless, we cannot completely dismiss consideration of the consequences of worst-case scenarios if the inertia of nearly an entire semi-molten core is postulated as present. To communicate some ideas regarding the likelihood of containment failure resulting from a steam explosion, a probabilistic approach was adopted. The approach is not as sophisticated as has been adopted by other authors in this field, such as by Theofanous et al.* Further, as the Challenger accident shows, sometimes low-probability estimates may be unrealistic. However, a probabilistic judgment was requested by NRC, and the approach used should give the reader a feeling on how computer code input needs to be biased to end-of-spectrum values to suggest containment failure. The WASH-1400 estimates for the conditional probability of containment failure from a steam explosion given core melt do appear to be high. On the other hand, the nuclear industry arguments dismissing all concerns based on what arc and what are not credible physical processes do not appear fully supportable at this time. To reduce the subjectivity required, some research activities are proposed. The highest priority suggestion is to obtain a better understanding of core melt phenomenology. Regarding steam explosions themselves, the most important issue may be addressing scaling questions. Steam explosions have been speculated to scale nonlinearly, and the existing database currently must be extrapolated by about 3 orders of magnitude when discussing severe accidents. The accident at Chernobyl has resulted in a desire for increased confidence that the consequences of severe accidents are known. Although some residual risk undoubtedly will always remain, improvements in the current level of understanding seem possible.

*T. G. Theofanous, M. Abolfadl, H. Amarasooriya, G. Lucas, B. Najafi, and E. Rumble, "On the Probabilistic Aspects of Steam Explosion Induced Containment Failure," US Nuclear Regulatory Commission report NUREG/CR-5030 (January 1989).

xx Some preliminary results of this study were published in NUREG-1116, the report of the NRC-sponsored Steam-Explosion Review Group.* Changes in the conclusions made to the NUREG-1116 presentation are noted where appropriate. The NUREG-1116 report also contains a bibliography for those wishing to pursue this subject further.

"Steam-Explosion Review Group (SERG), "A Review of the Current Understanding of the Potential for Containment Failure Arising from In-Vessel Steam Explosions," US Nuclear Regulatory Commission report NUREG-1116 (February 1985). ACKNOWLEDGMENTS This program, evaluating the consequences of steam explosions with SIMMER-II, would not have existed without the promotion and support of Charles Kelber, US Nuclear Regulatory Commission. Marshall Herman and his staff at Sandia National Laboratories were a significant help in furnishing experimental information and in providing suggestions and constructive criticism for the project. Appreciation is also extended to the numerous people at Los Alamos performing the word processing, editing, and graphics work necessary to prepare this report.

XXT 1 1 ABSTRACT

The purpose of this work was to provide a reasonable estimate of the maximum loads that might be expected at the upper head of a pressurized water reactor following an in-vessel steam explosion. These loads were determined by parametric cases using a specially modified and calibrated version of the SIMMER-II computer code. Using the determined range of loads, the alpha-mode containment failure probability was to be estimated using engineering judgment. In this context, an alpha-mode failure is defined as resulting from a missile, produced by a steam explosion, and assuming core melt has occurred. The SIMMER-II code first was upgraded for this assessment. An improved equation of state was implemented to treat the corium-water system, and appropriate nonequilibrium heut-transfer models were formulated to treat the low thermal conductivity of liquid water. Model parameters then were calibrated to Sandia National Laboratories (SNL) experimental data. Both pre-explosion coarse-mixing calculations and explosion analyses were performed. These analyses suggested parameters that would envelop the real behavior, although a unique fit to the data was not obtainable. To investigate a basis for judgment!! on the ability of SIMMER-II to calculate post-explosion slug breakup, ;i series of shallow-pool acceleration experiments was conducted and then analyzed using SIMMER-II. Correct global behavior was calculated, and the exercise suggested code parameters that should be conservative for -adulating the hypothetical reactor case. Finally, a lower-vessel head failure and motion model was developed and added to SIMMER-II. Only energetic steam explosions are a concern in evaluating containment failure. These energetic explosions could cause a rupture in the lower-vessel head area that could be expected to mitigate upwardly directed kinetic energy significantly. Five major types of parametric steam explosion cases were analyzed. First, several cases were calculated similarly to those in a 1980-Zion/Indian Point SIMMER-II study in which 20% of the total corium was mixed with water. These indicated a substantial reduction in the likelihood of energetic missile production using the code modifications, particularly with the lower-head failure model. Second, more mechanistic (but arguably conservative) cases were calculated starting from initially separated corium and water. The maximum unwardly diiected fluid kinetic energy achieved in these cases was about 500 MJ at t"ue time of head impact, with a peak upper-head force of 0.81 GN. Third, worst-case assumptions from a 1983 probabilistic SNL study were used; 94 000 kg of corium was postulated as mixing with 20 000 kg of vater. Heat transfer from solid-particulate corium to steam was limited, and a diffuse spray led to a peak upper-head force of 0.78 GN. Fourth, the calculated coarse-mixing situation from the second configuration was homogenized and used with degraded heat transfer to simulate a multiple-explosion environment. In this situation, lower-head failure was not calculated to occur, and the peak upper-head force wac 1.52 GN. Fifth, the premixture from the third configuration was used assuming instantaneous corium/water thermal equilibration along with additional assumptions to insure the expansion was calculated in a conservative fashion. The peak upper-head force WI».s now 12.4 GN. A rough analysis suggests that for the vessel head to become a large missile, a loading of approximately 1 GN is required under two-phase impact situations. After dismissing the fifth situation as impossible, only the multiple-explosion simulation exceeded this threshold. However, true multiple explosions were not calculated. The initial conditions were idealized (for example, with uniform material distributions radially), dissipation of energy in the upper-core structure was neglected (with only 900 MJ of upwardly directed fluid kinetic energy, this could be significant), and the tiic dependence of the nonuniform upper-head loading suggested formation of multiple missiles from the head apex that could be stopped by the missile shields. To quantify alpha-moHt .'ailure probabilities, an accident progression diagram was p spared and evaluated using an approach from a study cf Clinch Jiver Breeder Reactor core-disruptive accident energetics by Theofanous and Bell. The technology base, the SIMMER-II case? &nd the known SIMMER-II biases influenced the judgments made. E-oause of the edge-of-spectrum character of both the calculative assumptions leading to a large-scale steam explosion and the assumptions required to obtain significant head loading, this study suggests an upper limit for the containment failure probability given core melt of 0.01 if the vessel upper head and bolts have not been significantly degraded by high-temperature accident conditions. Some proposed research priorities were formulated that would allow increassd confidence on steam-explosion issues. In order of priority these were (a) increased knowledge, investigation, and modeling of severe-accident meltdown phenomena, (b) large-°:ale steam-explo' ion experiments with an associated modeling and analysis program, and (c) smaller scale experiments to address specific mechanisms. Increasing knowledge should reduce steam-explosion energetics uncertainties. However, a tolerable level of residual uncertainty needs to be defined to reduce the tendency for steam-explosion issues to remain open-ended.

xxvi AN INVESTIGATION OF STEAM-EXPLOSION LOADINGS WITH SIMMER-II

W. R. Bohl

Contributors

C. R. Bell, T. A. Butler, L. M. Hull, J. G. Bennett, and P. J. Blewett

I. SUMMARY OF THE REPORT

A. Introduction The purpose of this work was to provide a reasonable estimate of the maximum loads that might be expected at the upper head of a pressurized water reactor (PWR) following an in-vessel steam explosion. Using the determined range of loads, judgments were to be made on the potential for containment failure by missile production resulting from a steam explosion, commonly called alpha-mode failure, suming core melt had occurred. 'Hie SIMMER-II code was used to calculate the estimated loads. SIMMER-II is a multiphase material relocation code originally developed for fast reactor disrupted core accidents. A brief summary description of SIMMER-II is given in Chap. II. A previous study of this type, herein called the ZIP study, was performed in 1980. To provide some basic SIMMER-II capabilities lacking in the 1980 study, a brief program of code modification was undertaken as part of this work. These modifications are discussed in Sec. I.B. To calibrate the revised modeling to available steam-explosion experimental data, Sandia National Labora- tories' (SNL) intermediate-scale thermite-water experiments were assessed, and one particular well-characterized experiment, MD-19, was calculated with SIMMER-II. This discussion is in Sec. I.C. The adequacy of the SIMMER-II postexplosion/expansion dynamics was investigated by comparing calculations with scaled shallow-pool experiments. The experiments and the analysis performed are given in Sec. I.D. As the study proceeded, numerous scoping calculations on a full-scale PWR representation were performed; the results of these calculations are in Sec. I.E. These reactor results were integrated into an overall scheme for judging the probability of containment failure, presented in Sec, I.F. Finally, conclusions and recommendations from this study are given in Sec. I.G. To keep the volume of this summary chapter from being excessive, most of the above outlined sections present summary material only. Details are given in subsequent chapters of this report.

B. Modification of SIMMER-II The SIMMER-II code was modified to better represent a corium-water system, and to implement a lower head failure and motion model suggested by the ZIP study. The details of these changes are given in Chaps. II and III and Apps. A to N. The four major areas of code modification for treating the corium-water system are an improved equation of state (EOS), a more appropriate treatment of nonequilibrium heat transfer to liquid water, revised assumptions in the vaporization/condensation model regarding energies at which watsr is vaporized and condensed, and miscellaneous changes. The most significant EOS modification in terms of accuracy was to make the infinitely dilute, water-vapor's heat capacity temperature dependent, which better represents the increasing degrees of freedom of the water molecule and the hydrogen-oxygen system at high temperatures. For 3 000 K infinitely dilute steam, the new heat capacity relationship gives 1.62 times the value for infinitely dilute 400 K steam. The major modification to the EOS in terms of coding was to insert a relaxation constant so that as steam becomes more superheated, its internal energy becomes closer to the value for infinitely dilute water vapor. This is most important for stability because it limits the triple-valued nature of the vapor temperature as a function of density. It is «lso important for accuracy, re- moving a spurious pressure dependence for superheated steam, as well as allowing the inclusion of the ~83O-kJ/kg energy difference that appears between steam at the critical point and infinitely dilute steam at the critical temperature. A fitting procedure was applied to the new EOS relationships to compare them with both the steam tables^ and the Los Alamos SESAME tables. Agreement is within ±20% over the range of interest and generally is much better. Given the phenomenological uncertainties in steam explosions, these results exceed the required accuracy. If further improvements to the EOS are desired, the most cost-effective revision would be a better algorithm for the gas parameter, R, both near the critical point and at high steam densities and temperatures. The SIMMER-II heat-transfer problem with the corium-water system relates to the low thermal conductivity of water. As a consequence of model development based on liquid sodium, the standard SIMMER-II code transfers heat to bulk liquid, which then vaporizes. For water, both experiments and theoretical considerations support the view that steam-explosion liquid-liquid heat transfer occurs at the water surface. Some heat then is conducted into the water; the remainder causes vaporization. Implementation of this revised concept allows simulation of steam explosions in water-rich systems with nonequilibrium vaporization. The radiation heat transfer occurring during the coarse premixing phase of a steam explosion was not Tiodeled explicitly. Bounding cases that envelop a variety of heat-transfer modes and magnitudes were evaluated instead. Modeling a film-boiling flow regime with radiation and film layer conduction in these chaotic environments is beyond the scope of the SIMMER-II formalism at this time and would add only marginal improvement to the overall treatment by itself. The lack of physics for fuel-steam breakup and water interpenetration of the fuel are probably more important deficiencies in the SIMMER-II code. We found that inclusion of water subcooling into an effective heat of vaporization was necessary to obtain reasonable results from the revised heat- transler model. Past experience suggested that the complementary process, putting vapor superheat into the effective condensation energy, also would be desirable. Although conceptually straightforward, these changes did result ia somewhat complex coding modifications. The miscellaneous changes involved programming the liquid-water's thermal conductivity as a function of temper ture, changing the maximum water-droplet size algorithm, inserting an additional time-step control for the vapor-energy work term, and correcting of two minor FORTRAN coding errors. The necessity to tieat structural effects can be introduced with the observation that only energetic steam explosions are a concern in evaluating containment failure. These energetic explosions could cause a rupture in the lower vessel's head area, which could be expected to mitigate upwardly directed kinetic energy significantly. Finite-element calculations indicated that vessel failure could be expected at the radius where the outermost vessel penetrations were located. A circumferential split was expected to proceed around the head at this radius and to leave the inner portion of the head as a free body. A simple, single-degree-of- freedom spring-mass model was correlated to these finite-element results. This model accepts the average pressure loading over the failure region and furnishes the time and downward head velocity when the head disengages from the vessel. Motion of the head segment into the lower vessel cavity was treated by a modification of the SBiMER-II plug ejection model. The failed portion of the lower head becomes a free body whose acceleration is computed from its mass and the integrated forces acting on it. A two-dimensional representation of the cavity was used in this study; the volume available to the venting materials assumed the below core cavity was half filled with water.

C. Calibration to SNL Steam-Explosion Experiments The selection of SIMdER-II input parameters for reactor steam-explosion analysis can best be accomplished with reference to the experimental database. Two difficulties exist. First, any SIMMER-II model will be simplistic. The ability to propagate a mechanistic fragmentation (detonation) wave leaving behind a distribution of particle sizes doss not exist in the code. In fact, a generally agreed upon, theoretical fragmentation mechanism does not exist for steam explosions. Second, the reported experiments have random aspects. Later experimental results have tended to contradict earlier experiments and especially contradict early simplistic theories. Also, the experimental information is sufficiently ambiguous that any unique simulation is impossible. Because of these problems, one typical experiment was selected for computational simulation using a simple SIMMER-II representation. Details for this calibration as well as a brief review cf SNL steam-sxplosion data ate given in Chup. IV and Apps. 0 to T. The experiment s-lected was SNL test MD-19, in which 5.11 kg of iron- ilumina thermite was dropped intc 224 kg of amMent temperature water contained in a square Lucite box. Initially, a uniform interaction zone of a size reported by the experimentalists possessing a thermite/steam/water volume ratio of 0.04/0.48/0.48 was used. A two-phase, equal volume fraction steam/water chimney was assumed to exist above the interaction zone. The bottom of the box was assumed to be rigid and the sides and top to be at constant pressure. These lateral boundary conditions assume the Lucite has no strength. Alternative boundary conditions are discussed in App. S. Best results in the SIMdER-II simulation were achieved by having a water- droplet size at a fractional multiple of the fuel-droplet size. This relationship was considered reasonable because the surface tension of water is about an order of magnitude lower than that of iron-alumina thermite or corium. However, the fit was less than satisfactory, with a spurious high-pressure tail resulting from bulk heating of water in the interaction zone. A second fit then was made using a previously calculated fuel/water distribution based on a three-velocity-field film-boiling model. Here the outline of the calculated mixing zone was correlated to the vague pre-explosion, experimental configuration. The node with the peak thermite density had a thermite/ steam/water volume fraction of 0.27/0.37/0.36. The best fit achieved was with a fuel particle diameter of 300 ym, a water-droplet diameter of 75 ina, and a liquid-liquid heat-transfer multiplier of 0.2. (Note that the new liquid-liquid heat-transfer model was used with the 0.2 multiplier. This multiplier only acts on heat transfer to the water's surface.) A comparison with the base pressure transducer is shown in Figs, la and lb. The final exercise using SIKWER-II was to perform some pre-explosion coarse-mixing calculations, with thermite and water initially separated. Although the functional forms of some of the physics modeled in SINWER-II are less than desired, obtaining an appreciation of the qualitative errors involved was desirable. These results are used in Sec. I.E in running bounding cases for the problem of reactor meltdown. The first calculation assumed a uniform 15-mm droplet diameter with the nominal (unmodified) SIKMER-II heat-transfer model. Steam production and the extent of thermite dispersion were underestimated. This calculation was rerun with the heat-transfer models oi Sec. I.B. Excessive radial thermite dispersal resulted, and the interaction zone did not reach the bottom of the Lucite container in the required time. The configuration established with the first premising calculation was exploded with the same models and parameters used in the fit shown in Figs, la and lb to see the effect cf d.'fferezit premixing on the characteristics of the explosion. The peak pressure increased to 20.8 MPa; there was a longer time at high pressure, but the width of the pulse at 8 MPa was about the same. These results are consistent with what would be expected with less local vapor volume for early expansion and therefore pressure reduction. Although the calibration exercise was unable to provide a strongly supported, unique set of calibration parameters, it did provide some confidence that the SIKMER-II treatment was qualitatively reasonable. 200

CO < 100 • m

0.00200 0.0040O TIME (a)

Fig. la. SNL reported water chamber base pressures in experiment MD-19.

. • i • i i i .;—i 1.0 2.0 TIME (MS) Fig. lb. Calculated water chamber base pressures in experiment MD-19. Also, we obtained knowledge about the choices of parameters that would envelop the real behavior. As a caveat, details from other fully instrumented test series (FITS) experiments, such as those with double explosions or those producing significantly more efiicient explosions because of rigid walls, cannot be modeled mechanistically with SIMMER-II. Parametric studies coupled with additional model development are required to provide high confidence that consequences of all possible steam-explosion phenomena are included in any study.

P. Calibration to Los Alamos Shallow-Pool Experiments To investigate a basis to judge the ability of SIMMER-II to calculate postexplosion slug breakup, several series of shallow-pool acceleration experiments were conducted and then analyzed with SIMMER-II. Details of these experiments and their analysis are provided in Chap. V. A schematic of the test apparatus is shown in Fig. 2. The depth of the

1/4-mil water Mylar diaphragm 4-mil __ p. Mylar diaphragm

diaphragm nitrogen cutter

Fig. 2. Shallow-pool experimental apparatus. pool and the height of the free space above the pool are scaled using the actual reactor vessel's dimensions and the total amount of fuel available. As indicated in the schematic, a very thin (0,006-mm) diaphragm supports the pool in a 102-mm-i.d. Plexiglas tube. The pool is 50 mm deep, and there are 185 mm of free space above the pool. The bottom of the tube is separated from the nitrogen driver gas by a 0.1-mni Mylar diaphragm. The experiment is started by cutting the lower diaphragm; the pressure rises in the bottom tube and ruptures the thin diaphragm that supports the water. The increased pressure then induces the motion of the water that is of interest. Three sets of data are collected during the test: the pressure history just under the top diaphragm (PI), the pressure history at the endplate (P4), and a high-speed movie. Although some random behavior was observed, the results of these tests were useful. Three types of SIMMER-II calculations were performed, simulating both real and hypothetical experiments in this geometry. These were (1) calibration cases relating to one experimental test, (2) hypothetical cases with postulated curvature in the thin Mylar diaphragm supporting the pool, and (3) calculative comparisons with other shallow-pool experiments using the calibrated parameters. For the calibration cases, the initial gas pressure of the nitrogen driver was 0.56 MPa, and the evacuated space started at ~5.6 kPa. Consequently, these tests are said to have a 100:1 pressure ratio. The high-speed movies indicated progressive slug breakup as a function of distance, with complete breakthrough occurring at about the time of endplate impact. A reasonable simulation of the experimental pressure trace was obtainable with SIMMER-II by using a small drop diameter, 100 ym, and consequently maintaining close coupling between the liquid and vapor fields. Also, the value of a , the vapor volume fraction where the two-phase to single-phase transition occurs, was adjusted to 0.025 to attempt to maintain two-phase pressures as long as possible as the water impacted the endplate. The best comparison with experimental data is shown in Figs. 3a and 3b. Two types of hypothetical situations were examined. First, the inner one- half of the lower water's interface area was raised or lowered 1 cm to evaluate the effects of surface shapes on pool breakup. Second, one of these nonuniform situations was scaled up by a factor of 40 to evaluate the scalability of the treatment. The nonuniform interface resulted in an instability growth that def- initely made impact appear to be more of a two-phase spray, but actual venting of the gas through the water was not achieved. A further calculation was 20.0-1 PRESSURE RESPONSE FOR TEST SP7014

16.0-

CO 12.0-

8.0- CL

4.0-

0.0 0.0 1.0 2.0 3.0 4.0 TIME (ms)

Fig. 3a. P4 transducer pressures reported for the shallow-pool experiments. 20 0-1 PRESSURE RESPONSE FOR TEST SP7014

16.0-

12.0-

a. in m to 8.0-

4.0-

0.0 • • • i • • • • i • • • • i • • • • i • • • • i • • • • i • • • • i • • • ^ 0 0 10 2 0 3 0 4.0 5.C 6.0 7.0 8 0 TIME (ms) Fig. 3b. Calculated P4 pressures in the shallow-pool experiments. performed with half of the distorted lower interface actually blocked by a rigid disk to assess an experimental concept for testing the effects of interface distortion. The results suggested that the disk did not change the qualitative difference of the pool's behavior significantly. Therefore, a distorted interface experiment was planned. Increasing the scale (and driving pressure) did produce the expected increase in peak impact pressure by a factor of 40, but the upscaled case resulted in a more coherent impact. The two remaining calculations made for comparison with experiments were (a) one with a 50"-l pressure ratio and (b) one with a depressed lower pool interface by a central disk. The calculated impact pressure for the 50:1 case, which was obtained with the "calibrated" parameters, had peaks more than twice the experimental values. The character of the 50=1 calculation was much like that at 100-'l. Although slug breakup in the high-speed movie of the 50:1 experiment was similar to that for the 100:1, a significantly more diffuse impact was implied by the pressure trace. With a disk to lower the center of the lowtr water interface by 3 cm, both the experimental data and the calculation showed gas breakthrough before impact. However, now the calculative impact was significantly more diffuse than that measured. The following conclusions were reached in this brief SIMMER-II examination of shallow-pool dynamics. (a) The SIMMER-II simulation of shallow-pool breakup provides generally correct global behavior in terms of diffuse vs slug impact and overall impulse delivered by the fluid to the upper closure. It cannot, and probably need not, match the very short-lived pressure spikes measured in the experiments. (b) With an initially uniform pool, Lhe use of small drops in SIMMER-II probably would exaggerate upper head loadings in the reactor case (be conservat ive). (c) Even with exaggerated liquid/vapor coupling, the lack of a model for turbulent mixing leads SIMMER-II to overly accelerate slug breakup in the case where small nodes are used to simulate a nonuniform interface in this experiment. (d) Because the node size at reactor scale is much larger than the turbulent mixing length, exclusion of a turbulent mix ing model appears justified. Also, the nonlinear scaling of the SIMMER-II liquid/vapor drag relationship was a major contributor to increasing the impact coherence of the

10 upscaled calculation. (e) Because of unresolved scaling questions, further analysis of these experiments was judged less cost effective than parametric calculations involving reactor meltdown sequences.

E. SIMMER-II Reactor Case Calculative Results The range of uncertainty in the expected conditions resulting in a steam explosion during a core meltdown is large. If all possible initial conditions and modeling uncertainties were considered, a complete parametric study of steam explosions and their effects would be an enormous effort. The effort involved in even one SIMMER-II reactor calculation is appreciable. With the limited scope of this study, the limited understanding of steam explosions that exists, and the limited extent to which this understanding currently has been programmed into SIMMER-II, a massive parametric study was not appropriate. Rather, five scoping calculations were performed with SIMMER-II to address the upper bound question. Details of these calculations, as well as some additional results that help to integrate these cases into the context of a meltdown accident sequence are given in Chap. VI and Apps. U and V. The results are summarized in this section. The reactor's structural representation for these cases is shown in Fig. 4. The xn-vessel configuration is the same as in the previous ZIP study. A movable lower structure has been added for the lower head failure model. The five SIMMER-II cases can be summarized as follows. In case 1, we used the sane SIMMER-II corium and water geometry as in cases 1 and 2 of the 1980 ZIP study. This was a premixture including 20% of the total corium below a single-phase molten corium pool. The corium/steam/water volume fractions were 0.50/0.25/0.25. The fuel and water were assumed to be fragmented to 300-^m-diam globules within the premixture. In addition, 35 000 kg of steel particles were added between axial node 46 and node 60 to represent the mass, but not the strength, of the upper core structure for this case. The new SIMMER-II lower head failure, EOS, and heat-transfer models were used. The results of the calculation indicated a substantial reduction (compared with the ZIP study) in the likelihood of energetic missile production. The new models eliminated the spurious high-interaction-zone, single-phase pressures previously obtained. Lower head failure was calculated to occur at 3.5 ms, and about 57% of the kinetic energy was directed downward as a result. Upwardly

11 OUTLET PIPING

SOLID STRUCTURE

12.54 m INLET PIPING OOWNCOMER

CORE SUPPORT FORGING

MOVABLE STRUCTURE

••OUTLET TO KEYWAY

Fig. 4. PWR structural representation for SINWER-II steam-explosion calculations. directed kinetic energy at head impact was ~65O MJ. With the assistance of the downward venting of core material following lower head failure, the peak force on the head was reduced from 2.6 GN (giganewtons) to 1.0 GN. A plot of the integrated upper head loading is shown in Fig. 5. This head loading is near the threshold for failure if the head is near the normal operating temperature. In case 2, we used more mechanistic but arguably conservative initial conditions for a large-scale coherent steam explosion. Starting with a completely molten core at 3 100 K, a 1.85-m-diameter corium stream was allowed to pour into 18 000 kg of water in the lower plenum. Standard SINMER-II heat transfer was used with 20-mm-diameter, prefragmented corium globules. Based on the results in Sec. I.C, these assumptions should underestimate steam production and thereby permit extensive mixing. Mixing occurred around the edges of the corium stream, and was reasonably extensive because of the large mesh size used (greater than 10 cm). An explosion of the mixed corium and water was triggered 0.7 s after initiation of the pour, when 40 000 kg of corium had entered the lower plenum, and corium flow reversal was starting as a result of vapor production and plenum pressurization. This time was chosen because the mixing configuration appeared

— 12 0 -a FORCE ON HEAD 11.0-1 io o-; 90^ BOr r ro-i 6.o-; O 5.0-: Li. 4.0^

2.o-:

IO-;

oo '•• I • ' • 'I ' ' ' ' I ' ' ' 'T'' ' 'I ' ' ' ' I ' ' ' ' I ' ''' I ' ' ' 'I ' • ' • I 000 001 0 02 003 0 04 0 05 006 007 008 009 010 TIME (s)

Fig. 5. Integrated upper head loading for case 1.

13 to have the potential to produce maximum head loads. Examining details of timing sensitivity was beyond the scope of this study, but an explosion after 1 s of mixing is discussed in Chap. VI. The SNL-calibrated explosion parameters were used with the new SIMMER-II models. Steel particles were included to represent the upper internal structure. About 62% of the kinetic energy was directed downward. Upwardly directed kinetic energy at head impact was ~500 MJ. The peak upper head force was 0,81 GN, as shown in Fig. 6. In case 3, we used worst-case initial conditions as inferred from a recent SNL study.6 Here 75% of the core (94 000 kg) was assumed to mix with 20 000 kg of water. To limit the height of the premixture to that of the original corium pool, a 19% initial steam volume fraction was assumed. The corium globule size was assumed to be finely fragmented to 100 ^m in diameter. Although the new SIMMER-II models were used, no heat transfer from frozen corium to steam was calculated, limiting steam temperatures during the expansion to 922 K. No lower head failure was assumed, and the remaining core material and upper structure were assumed not to inhibit expansion. The upwardly directed kinetic energy was 1 830 MJ. However, as a consequence of the diffuse nature of the expansion, the peak upper head force was only 0.78 GN, as shown in Fig. 7. This result shows

10 0- FORCE ON HEAD 9 0-1

8.0 -I

7.0 -I

6.0-i u u 50-1 ££ O 40-1

3 0-i

20-1

10H

00 070 071 072 073 OH 075 076 077 078 079 080 0.81 TIME (s)

Fig. 6. Integrated upper head loading for case 2.

14 "few-, FORCE ON HEAD 8.0-

7.0-j

6.0 -

5.0- u •X. 4.0- o 30^

1.0-

0.0 0.00 0.01 0.02 003 0.04 0.05 TIME (s)

Fig. 7. Integrated upper head loading for case 3. that a complex relationship exists between the explosion and its delivered loads to the system. Simple correlations are not reliable. In case 4, we took the initial conditions from case 2 but homogenized the 40 000 kg of corium and 10 000 kg of water in the lower plenum. A slow "explosion" then was simulated in which droplet sizes were increased an order of magnitude over the SNL correlated values. This reduced the heat-transfer rats by 2 orders of magnitude. The idea was to simulate an incoherent, multiple explosion environment that could be more representative of the reactor situation. In case 4, the lower plenum was not calculated to fail. Upwardly directed kinetic energy at head impact was ~760 MJ. The peak upper head force was 1.52 GN, as shown in Fig. 8. This case had a significantly more coherent upper head impact than cases 1, 2, or 3, even though the explosion was relatively benign. Again, the complexity of relating explosion magnitudes anc characteristics to loads on the head is evident. Here the most benign explosion produced the largest challenge to the head.

15 FORCE ON HEAD 14 0-

120-J

10 0:

eog LL. 60 J

4 0

2.0

oo '"''' 000 001 002 0.03 004 005 0.06 0.07 0.08 0.09 010 Oil TIME (s)

Fig. 8. Integrated upper head loading for case 4.

In case 5 we used the case 3 premixture but with corium and water thermally equilibrated before the expansion. The Los Alamos SESAME tables were used to give a starting pressure of 900 MPa. Refitted parameters for the SIMMER-II EOS were derived to obtain these initial conditions. The remaining 25% of the core was placed on top of the premixed region. Both lower head failure and steel particles (for structure) were included in the calculation. The maximum upwardly directed kinetic energy was 7 250 MJ. The peak upper head force was 12.4 GN, which was obtained just before the calculation went unstable. The force plot is shown in Fig. 9. This result confirmed clearly that our calculative approach would produce the expected disastrous results for the upper limit assumptions and idealized physics.

F^ Containment Failure Probabilities To obtain an estimate for the probability of containment failure from an in-vessel steam explosion, given core melt, we must (a) judge how likely the initial conditions for a large-scale steam explosion are, (b) estimate the in-vessel energetics and loadings produced by the spectrum of possible steam explosions, and (c) evaluate the consequences of such loadings. A brief study

16 16.0 -| - FORCE ON HEAD 14.0-|

12.0-^

10.0-

8.0 o u. 6.0-

4.0^

2.0-

0.0 1 • • ' ' i I ' ' ' ' I ' ' ' ' I ' ' ' ' f' ' ' ' I ' ' ' ' I 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 TIME (s)

Fig. 9. Integrated upper head loading for case S. attempting to make such judgments is presented in Chap. VII. Here we summarize some conclusions from that study. First, although the best estimate probability is judged to be low (between 10~" and 10 ), this is simply a guess. Currently we lack the technology to construct a reliable probability density function for core melt consequences that would portray a best estimate or most probable value. Indeed, performing any "best estimate" mechanistic CQTB meltdown calculation, with models representing consensus phenomenology, was not possible because consensus phenomenology did not exist. The SIMMER-II calculations performed were for addressing the upper bound question. Second, any large-scale steam explosion or sequence of explosions leading to a containment challenge apparently must involve a sustained supercritical pressure. Although the industry degraded core rulemaking program (IDCOR) asserts the maximum meaningful pressure during the expansion phase of an explosion is about half the thermodynamic critical pressure, steam-explosion experiments exist with expansion phase pressures greater than 34 MPa sustained

17 ft * for more than 1 s. ' Thus, containment from large-scale steam explosions cannot be ruled out on the basis of limited pressures. Third, with the dissipation calculated by the SIMMER-II model, to obtain sufficient energy a large-scale steam explosion apparently must involve efficient energy transfer from a considerable fraction of a molten core, 20% or more, in a few tens of milliseconds. This leads to such requirements as forming a large molten corium pool, having a coherent pour, not obtaining early triggering of an explosion, and limiting steam generation so as to permit the extensive premixing of corium and water. All of these conditions are unlikely, but not outside the spectrum of reason. For example, triggering could be delayed by saturated water or elevated ambieit pressures, or the commonly quoted steady- state fluidization arguments, which limit mixing, could be defeated by the inertial effects involved in a large corium pour. Because of the edge-of-spectrum character of these phenomena, a 0.1 probability was judged to be a proper upper limit to associate with obtaining initial conditions resulting in sufficiently energetic steam explosions. Further explanation as to the meaning of this probability value is in Chap. VII. Fourth, obtaining the extreme conditions presented in case 5 appears to be impossible. Mixing 75% of the core requires water to rise through a molten corium pool in a film-boiling regime because of the necessity to accommodate the vapor volume produced. In any meaningful calculation attempting to attain this premixed configuration, Satisfaction of fluidization requirements resulting in corium dispersal would be assured. The assumptions of instantaneous temperature equilibration and pessimistic EOS parameters are also unreasonable. Case 5 merely demonstrates that if arbitrary mixing is allowed, and the explosion efficiency is allowed to approach the thermodynamic limit, SIMMER-II obtains the expected unacceptable result. Fifth, if the vessel head is to become a large missile, an integrated threshold head loading of approximately 1 GN is required if the head and bolts exist at operating temperature. Case 2, run with edge-of-spectrum initial conditions for a single large explosion, was below this level. Case 1 produced I GN, although its postulated prefixing as well as its explosion/expansion sequence involved edge-of-spectrum coisiderations. Case 4 achieved more than 1

'Information provided by K. H. Wohletz, Group ESS-1, Los Alamos National Laboratory, November 1984.

18 GN. However, true multiple explosions are not calculated. The initial conditions are idealized (for example, with uniform material distributions radially); dissipation of energy in the upper core structure is neglected (with 900 MJ of fluid kinetic energy, this could be significant); and the time dependence of the nonuniform upper head loading suggests formation of multiple missiles from the head apex, which could be stopped by the missile shields. In brief, case 4 is an edge-of-spectrum expansion calculation. Because 1 GN is exceeded only with edge-of-spectrum calculations, if the head and bolts exist at operating temperature, a 0.1 probability is judged appropriate to assign to vessel loadings of concern, given initial conditions resulting in energetic steam explosions. Sixth, any attempt to establish a simple alpha-mode failure threshold condition is fuzzy. An impulse limit is useful only if the impact involves single- phase liquid. Case 3 shows that high energies do not necessarily give unacceptable loads, if a highly two-phase spray is the impacting fluid. The 1-GN force limit is fuzzy because of the influence of the spatial loading distribution, the temperature of the vessel, and the response of the containment to any missile. Head loadings depend on the assumptions made in the explosion/expansion calculations. In this study, the presence of steel particles tended to concentrate loads, while lower head failure made upwardly directed impact loads less severe. Thus, consistent analysis must be performed to link initial conditions, mixing characteristics, explosion characteristics, expansion behavior, head-failure characteristics, and missile dynamics.

G. Conclusions and Recommendations This study has resulted in significant progress over the ZIP analyses. The SIMMER-II water EOS and heat-transfer mechanics are more credible. The significance of lower head failure can be evaluated consistently and shown to result in considerable mitigation for the constrained (by the corium pool) reactor meltdown case. The initial conditions used for a steam explosion can be related to a mixing calculation. Although SIMMER-II models can only parametrize the physics of steam explosions, ne correlated models now can simulate better the pressure-pulse timing for water-rich systems. Further, parameters can be chosen to provide a caiculative simulation of shallow-pool behavior. Using steel particles for the upper structure reduces fluid breakup and dispersal, which were nonconservative in the ZIP analyses.

19 Because of the edge-of-spectrum character of both the calculative assumptions leading to a large-scale stream explosion and those required to obtain significant head loading, the upper limit for the containment failure probability given core melt is estimated as 10 if the upper vessel head and bolts exist near ~550°F. This probability must be increased for higher temperatures. The importance of the high loadings in case 4, which were obtained from slow heat transfer simulating incoherent explosions, cannot be ignored. There is no method to quantify a true best-estimate probability, but a qualitative judgment suggests that values of 10 to 10 might be reasonable. Possible future research to obtain increased confidence on the steam-explosion issue is discussed in Chap. VIII. Beyond an extension of this study to better quantify the consequences of the calculated upper head loadings, the proposed priorities for additional research on the alpha-mode failure issue are (1) meltdown process modeling; (2) large-scale steam-explosion experiments with associated model development for test analysis, test interpretation, and reactor application; and (3) smaller scale experiments to address specific mechanisms. The reasons for this order of priority are as follows: 1. Understanding meltdown behavior would help place the alpha-mode of containment failure in its proper perspective among other safety issues. 2. Both the state of any corium pool before contact with water in the lower plenum and the mode of contact depend on the meltdown sequence, which is uncertain. 3. The state of the vessel's internal structures as well as the temperature of the vessel's head and head bolts depend on heat transfer during the core heat up (meltdown) phase of an accident. 4. Both SIhMER-II calculations* and the theories of Corradini, Fauske, and Theofanous lead to the conclusion that large-scale explosions will possess characteristics that differ from the smaller scale FITS tests. These differences should be observable with current technology, and their experimental characterization would help to resolve some important uncertainties.

*This is shown when the calculations for test MD-19 are compared with case 2 of Sec. VI. Further discussion is in Chap. VIII on research priorities.

20 5. Ultimately, increasing scientific knowledge, not performing parametric studies, will reduce uncertainty in the vapor-explosion field.

Finally, the tolerable level of residual uncertainty needs to be determined. Otherwise, steam-explosion issues will tend to remain open-ended.

21 II. SIMMER-11 MODIFICATIONS FOR THE MOLTEN-CORE/COOLANT-INTERACTION PROGRAM

A. Introduction The SIMMER-II code was used for the numerical calculations performed in this study. The SIMMER acronym stands for an S_ , Implicit, Multifield, Multicomponent, Eulerian Recriticality code. The code was designed to calculate the multiphase motion of liquid metal fast breeder reactor (LMFBR) core materials during a core disruptive accident. The capabilities of SIMMER-II to handle structure, rapid heat transfer, and nonequilibrium vaporization suggest that the code could be useful for calculating the consequences of steam explosions in water reactor degraded core accidents. The first modeling of steam explosions with SIMMER-II was reported in the Los Alamos analyses supporting the US Nuclear Regulatory Commission (NBC) Zion/Indian Point Project, the ZIP study. The SIMMER-II code used in this study, Version 10, has only minor changes from the SIMMER-II code used in the ZIP study in areas relevant to analyzing steam explosions. Problems observed in the ZIP study suggested modifications. First, in the ZIP study, the EOS had a spurious dependence on pressure for superheated steam near the critical pressure. A triple-valued function had to be inverted. Also, some steam pressures were observed to be high (by almost a factor of 2 at some temperatures and densities), whereas the internal energy of superheated steam was generally too low (achieving only 70% of its required value at 3 000 K). Second, heat transfer to liquid water was through a bulk heating process that was incompatible with water's low thermal conductivity. This mechanism could not be used to easily simulate a steam explosion in any experimentally observed water-rich system. Third, modification of the corium/water heat-transfer process to obtain surface vaporization required a change to the vaporization/condensation model, with the liquid subcooling included in an effective heat of vaporization to avoid spurious cooling of the bulk water. Fourth, the high pressures observed in any steam explosion sufficiently energetic to cause containment failure suggest including a lower head failure and motion model. Implementation of such structure modifications are given in Chap. III. The above problems led to the modifications described in this chapter. These improvements are supplemented by some miscellaneous changes developed as the work progressed and are described in Sec. II.F. Overall, the modifications do correct many previous problems and ailow a better assessment to be made of

23 the question of upper bound energetics. However, the physics involved in steam explosions is still incompletely understood, and, even with the indicated modifications, a best-estimate calculation is not possible using the SIMMER-II code. In Chap. II, a summary of the modified equations for the analytic equation of state (AEOS), the corium/water heat transfer, and the vaporization/condensation model is given in Sec. II.B. A detailed description of the AEOS modifications as well as a comparison with steam-table data and the Los Alamos SESAME library are given in Sec. II.C. Details of the heat-transfer modifications are in Sec. II.D, and Sec. II.E illustrates the implementation of the vaporization/condensation model changes. In the AEOS area, pressures can be calculated to within ±20% of steam-table or SESAME EOS values over the range of interest for water/steam and are usually much better. This agreement is considered acceptable because the amount of molten corium available, its heat content, and the degree to which mixing has occurred are more uncertain. The calculation of pressure for high-temperature liquid water results in larger discrepancies, but with steam procaction from nonequilibrium heating of liquid water, this regime generally is avoided. The improved AEOS algorithm has three coupled fitting parameters important for obtaining near-critical and supercritical steam pressures. This study suggests one set that is a qualitative compromise between accuracy and stability. The revised heat-transfer algorithm simply modifies the liquid-liquid heat- transfer expression so that energy is transferred to the water's surface, presumed to exist at the temperature of saturated water. Bulk water heating is now based only on the difference between the liquid-water temperature and the saturation temperature. The energy discrepancy between heat transferred to and removed from the water/steam interface vaporizes water (the water subcooling is included within the effective heat of vaporization). The primary changes to the vaporization/condensation model were to remove during vaporization liquid from the liquid field at its true energy, not the condensate energy, and to remove during condensation vapor from the vapor field at the mass-average vapor energy, not at the saturation value. The condensation modification was not truly required for this program, but it corrects some long-standing SIMMER-II problems, and introduces desirable symmetry to the formulation.

24 The miscellaneous corrections involve changes in the liquid thermal conductivity for water, the water-droplet size, boundary cell pressures, a time-step control based on the work term in the vapor-energy equation, and a minimum for an internal EOS paraneter. The AEOS modifications stand alone. The vaporization/condensation changes require the AEOS modifications. Both of these correction sets are suitable to be incorporated into Version 10 of SIMMER-II. This is particularly true if future SIMMER-II studies are to consider near-critical pressures, for example, -22 MPa for water. The heat-transfer modifications require the vaporization/- condensation changes and are limited to the assumption that water is liquid energy component 3. The miscellaneous corrections are independent of the other modil ications and can be inserted into Version 10 of SIMMER-II also, although generalization beyond the water database needs to be performed. The main numerical difficulty encountered was the triple-valued nature of two of the implicit functions. Although the improvements significantly reduced the severity of these problems relative to Version 10 of SIMMER-II, an obvious modification that would eliminate these problems was not apparent. One difficulty involves the calcula;ion of the vapor temperature in the AEOS corrections. The second comes from the expression for the vaporization/condensation rate. The method of treatment in Version 10 of SIMMER-II is awkward, ineffi- cient, and ineffective, because the true source of the problems was not recognized. In these corrections a bisection iteration procedure was developed to stop the Newton-Raphson algorithm from moving away from a domain where a solution must occur. As an initial example problem, one of the previous ZIP study SNL experimental simulations was run. Calculated pressures near the tank wall are shown in Fig. 10. Although an exercise in fitting was concluded to be required, the code did execute and produce plausible answers (given the input) with no indication of trouble.

B. Summary of Modified Equations Because the detailed discussions of code modifications in Sees. II.C, D, and E are rather extensive, a reference summary of the basic equations modified has been included. This section also illustrates the level of sophistication of the coding changes made for this program. Except where indicated, the terminology is that of the SIMMER-II manual.10 For convenience, Apps. A, F,

25 30

0 G.000 0.002 0.004 0.006 TIME (s)

Fig. 10. Sample problem results. and I provide the SIKWER-II, Version 10 treatments for the AEOS, liquid-liquid heat transfer, and vaporization/condensation. These appendices also contain definitions for the terminology.

The three separate areas addressed here are (1) AEOS modifications, (2) liquid-to-water heat-transfer and steam production modifications, and (3) vaporization/condensation model modifications.

1. AEOS Modifications, First, the vapor's heat capacity for infinitely dilute steam was made temperature dependent. A linear relationship seemed

26 satisfactory from the data available and the temperature range required. The expression used is

cvGM= aM+ bM (TG " 273"16) • (1) where

a^ = an input parameter (1 346 J/kg-K for water) and

b|yj = an input parameter (0.3302 J/kg-K for water).

Second, the nonlinear increase in the vapor's heat capacity as vapor becomes more superheated was modeled. A relaxation formula based on the temperature difference from the saturation line was used. The modified heat capacity, cvG M' was *ken substituted into the implicit SIMMER-1I expression for the vapor temperature. The expressions are

eG " I xm eG,M

& Xm cvG,M

where

eG,M = eVap,M " cvG,M TSat,M •

* + C c T * in -*g.*gMM + CLiq,LiqM " cvLM TMelt,M "" eG,M,old *.•, * »« 10. ITG-TSal,M,+(lM

eG,M,old ~ eVap,M " cvG,M TSat,M '

/3^ = an input parameter (~50 K for water), and

Ji. = an input parameter (~5 x 104 J/kg for water).

27 Third, at high steam densities and supercritical temperatures, the gas parameter, Rw, used to evaluate the steam pressure, increases significantly over the value for infinitely dilute steam. A simple formula increasing Rw under these conditions was added. The expressions combining this addition with the changes in R^ near the critical temperature (already in Version 10 of SIMMER-II) result in

^n RM = gM *n RcrtiM + (1 - gM) ^n [RMi

for PGm TG > <>Gm Vert ' and

-tn RM = gM 4n fM + (1 - gM) *n [R^ (1

for (pGm Vo * PGm TG

where

pGm TG - M " (pgm T

= an input parameter (-0.5 for water),

gM= 1 for TG <

a^ = an input parameter (~200 K used for water),

P R Crt,M Crt,M PCrt,M TCrt,M

ro uc (Pgm T0>o = the Pproduc^ t oof the vapor density and temperature at T Sat ,M =0-95 TCrt,M '

c = in ut value of heat RMi = cvGM,i (7M " • vGM,i P capacity, and

28 fw a» nominal fit for Rj^ when TQ <

Fourth, for consistency, these changes require a consistent modification of the coding defining the initial temperature for single-phase cells. The appropriate implicit formula is

NMAT-1 cvGM TSat,M/RM Tr = 0 (4) NMAT-1 G ' E £vMc vGM/RM m=l

where

p^y, = liquid vapor pressure at Tca^ u (fixed) ,

T<*4oat.., uai = TunT_ - superhea* t (fixed) , but

cvGM and RM dePend on TG •

2. Heat-Transfer Modifications. The normal model for liquid-liquid heat transfer between water and any other liquid component is turned off whenever the water is colder than the indicated liquid components, and the vaporization/condensation model is used. Heat transfer from water to colder liquids is still limited by the thermal conductivity of water. Heat from the hotter liquid components then is used to vaporize steam from the water surface according to

5 O k tt tt (r + r ) 45 Lm Lm L3 pm p3 , . 2fT tr + ID3 J — n . u ) 3 3 pm '"pS^c^Sat ,3 pm b rpm rp3

= min (?Lm cvLm/At'

29 (T Sat.3 pra

where

lf •P3 'P3 pra • rc * rpra if rpm < rp3 • c I is the heat capacity of liquid energy component m,

. is the thermal conductivity of liquid energy component m, and T is the input heat-transfer multiplier.

This means Eq. (5) defines an effective heat-transfer coefficient for heat transfer from water vapor to the hotter liquids. This coefficient, E^Q^, can H (T The s ecial cases are be inferred from qHLmG3 = LmG3 Sat,3 " "W* P H L3G3 0: H LmG3 if T Lm TSat,3: and HLmG3 if sin8le Phase'

3. Vaporization/Condensation Modifications. The revised equations for this model now are

Vapor-Energy Equation +1 NMAT-1 £ condensation P - rr m=l m GLm , vaporization

_ NMAT-1 2 ie+1 SkJ m=l k=l

NMAT-1 NCLE .m Lm m-1 m=l

Liquid-Energy Equat ions

e«+l .condens sn+l _n+l Con.M P e P e + LtxTv rr/c+l Lm Lm = Lm Lm ^ m GLm 1 •K • vaporiz 30 f At

m = 1,2,3,4 ;

Vapor-Continuity Equation

NMAT-1 NMAT-1 xfflffl rg(1 -j x+m ) PX[ (1 - xm) P[m ; (8) m=l m=l

Liquid-Continuity Equations

Structure-Energy Equations

NMAT-1 eSk + U I ilSkGm • k-1.2 ; (10) m=l

AEOS

NMAT eG 2 ^ Le + c II m=l -T* * RM 'G = PM TSat,M

Vapor-Energy Heat-Transfer Coefficients

(12) cpG/HGSk^ "

31 where

^ H LGm (

"GSk ^Sat.M ' lG

K+1 HSGk

= mass fraction of material M at iteration ie+1 ,

x = 1 is the nominal case,

x =« 0 if all of energy component m is vaporizing , and

the qjjLmG3 are given by the heat-transfer modification formalism; the remaining terms are defined by the SIVMER-II manual. Adding the vapor, liquid, and structure energies, Eqs. (6), (7), and (10), and setting

NCLE 2 5n+l n+1 sn _n . r fgn+l ,n+l sn _n \ . r 5 r.n+1 _n \ _ n PG CG " PG eG + ^ ^pLm eLm " pLm cLmJ + ^ pSk ^cSk " cSk) = ° m=l k=l

for conservation of energy, we find (for the nominal case where the x is 1)

2 + i 6m3 r k=l rGLm u

where

NCLE r*+1 k=l

32 6 2 is ^Qe Knoneckcr delta , and

" eCon,M if

C. Discussion of the AEOS Modifications The SIMMER-II AEOS and its usage is described in the SIMMER-II nr.nual, Rev. 2, in Eqs. (111-31-71), Eqs. (IV-2-33), Eqs. (IV-35-45), and in App. B. This material is included in App. A of this report for reference. This section describes the details of how the SIMMER-II AEOS was modified for this steam-explosion study,

1. Review of Difficulties in the Version 10 SIMMER-II Formulation. The basic AEOS problem is determining vapor stales for near-critical and above-critical pressures. From App. .A, the "F" function, Eq. (A-74), whose zero defines the vapor temperature, is triple-valued in this region. This is a consequence of the pressure dependence in h^ u, the heat of vaporization, which in the AEOS formulation is a component of all vapor states. The rapid decrease of h^ ^ near the critical pressure can lead to three possible values of TQ for one value of tQ unless the formalism is modified. An example of F as a function of TQ is given in Fig. 11. Determination of a solution with a pure Newton-Raphson procedure sometimes will not occur.

Besides the problems with pressure dependence when Tgaj.~ TQT^, the vapor energy shown in Fig. 11 is anomalously low. The SESAME tables gave a near-critical pressure of 24 MPa for steam at 9Q •> 15.5 kg/m3 and TQ - 3 100 K, but the required vapor energy was 10.0 MJ/kg, when normalized to the SIMMER-II zero energy reference point. SIMMER-II only required 64% of this vapor energy level. Another vapor state problem is that the pas "constant," Rw, which gives the partial pressure of a component from pm - PQ^ R™ TQ, depends on pressure and temperature, but the pressure dependence is only effective when p < p^.^ ^, and temperature dependence only occurs when TQ > T^ ^ «. This form has insufficient flexibility to represent all vapor states. Definition of a liquid state also involves some problems. First, there is an inconsistency between what SIMMER-II regards as a microscopic liquid density and the equilibrium vapor density produced by the vaporization/condensation

33 30

3 /0Q=15.5 kg/m

e^=6.425 x 106 J/kg 20

-5 WATER VAPOR

eG-e* -10

-15 I I I I 3010 3030 3050 3070 3090 3110 3130 3150 VAPOR TEMPERATURE (K)

Fig. 11. Triple-valued "F" function for water vapor.

)del whenever T. > L i, In this region the microscopic liquid density is a > Dnstant, f £1^ j_. but the equilibrium vapor density will follow a pseudosatura- ion line extending beyond the critical condition. A second problem is the in- ppropriate dependence of pressures in the liquid regime on densities through ae use of a constant compressibility for each material taken from the input Dnic velocities. Third, there are inaccuracies in the liquids' internal energies due to the numerical solution scheme. Fourth, the liquid state formulation is intimately tied to the hydrodynamics formalism. Extraction and replacement by a better procedure would involve widespread changes in methods. Finally, the SIMMER-II AEOS regards all material components as completely immiscible- For most LMFBR applications' this appears adequate, but some problems can arise in experimental simulations when this may not be true.

2. Examination of Proposed Solutions to AEOS Problems. Formulating and writing a new, thermodynamically consistent EOS for SIMMER-II is beyond the scope of this program. Direct use of the SESAME EOS is also not feasible. The necessary symbiosis between the SIMMER-II use of a sudden conversion from a two-phase state to a single-phase liquid state based on a particular void frac-

tion, aQ, and the SESAME tables has not been developed. The need for a

Tga^. > T^r^ would have to be eliminated from the SIMMER-II code because such artificial quantities would not be available from SESAME. The Van der Waals loops in the SESAME representation mean that with a single material the density can be triple-valued as a function of pressure. Thus there would be regimes where 3p/3Prj is small, causing the present pressure iteration to have difficulties. Finally, pieces of the AEOS exist in many places in the coding. Removal of such implied AEOS assumptions is a significant task at the present state of code maturity. The limited scope of this program permitted only minimal coding modifications. Because of the characteristics of the steam-expiosion problem, correcting the vapor energy away from the saturation line was judged to be the most important modification. The procedure was to define an effective variable vapor heat capacity that will relax the vapor's internal energy to the infinitely dilute value away from the saturation line. The relaxation rate depends on TQ - To . , not the absolute value of the pressure. This algorithm corrects the triple-valued nature of the "F" function shown in Fig. 11 at high

values of TQ and reduces the triple-valued extent of "F" when both TQ and Tga^. are near TQ^. The gas "constant" calculation was revised by adding another term at high pressure. This allows R*. to increase above c.^ju (7 - 1) at high steam densities and pressures, as it should. For single-phase cells, the definition of the vapor temperature was changed to be consistent with the variable heat capacity. This permits the single-phase/two-phase transition to be performed consistently. The other problems associated with defining the liquid

35 state listed in Sec. II.C.I were ignored. When the new heat-transfer algorithm is used, single-phase liquid states with Tj > ^Tt should generally be avoided.

3. The Vapor Heat-Capacity Modification. The first and most significant modification was a change in the vapor's heat capacity. An initial correction was formulated by examining the value of

c c dT dT 14 vG " ^T T vo ) / (/T T ) > ( ) T=TMelt T~TMelt

where cyo is the vapor's heat capacity at low pressure from the steam tables. The averaged c^ is plotted as a function of TQ in Fig. 12. The linear relationship above 7rrt suggested that

vGM ' aM+ bM

would be sufficient for SIMMER-II. For water, aM - 1346 J/(kg«K) and 2 bM - 0.3302 J/Ug'K ). Another part of the energy discrepancy was a consequence of the low energies of saturated vapor at near-critical pressures. Away from the saturation line these energies increase rapidly. To simulate this effect and to remove the pressure dependence as vapor becomes ircreasingly superheated, the SIMMER-II formula for the vapor's internal energy was examined. This formula is

eG 'I xm((eCon,M + hJgM PAvM> + cvGM(TG " TSat,M>) ' m

The goal was to have eQ agree with steam-table values at least in the infinitely

dilute limit, where TSat>M-0. In this limit eConM - eLiqM - cvLMTMeltM, h h d A ° B i l T h in the infinitely dilute limit

eG = I xmU°vSM " SLM^Melt.M + hfg,M +

36 1.8

1.7 p en

STEAM TABLE VALUES 1.6 SEE Eq. 04) H LJJ X o u. o W £L 1.5 CO

EXTRAPOLATED LINEAR RELATIONSHIP

I 400 800 1200 TEMPERATURE (°C)

Fig. 12. Temperature-averaged vapor heat capacities for water.

37 In Eq. (17) the values of cyci(j and hr- w are irrelevant because e^- ^ must be added for normalization to compare with the steam tables. The value of c^^ is given by Eq. (15). A least-squares fit to h^gM = h^gM(l - TSat >M/TCrt >M)*M was performed, giving h^ ^ = 3.09 MJ/kg. The values of T^^ y| = 273 K and cvLM = 4217 J/(kg*K), the heat capacity of water at low temperatures, do not allow for much adjustment. To fit the steam tables, a material-dependent con- tant was added to Eq. (17). This means the infinitely dilute vapor energy for

aterial M, CQ m ™, is defined by

eG,~,M = (cvSM " cvLM)TMelt,M + hfg,M + h^g,M + cvGMTG " *M -

where 5^ is an input parameter, with a negative sign so that input is positive.

To get ec <» |yj to agree with the steam tables at 1 573 K and low pressure,

$M = 0.05 MJ/kg. A modified heat capacity for vapor, CyQ^i, is now defined to force eg w to eG m M ^or a^ vapor conditions far from the saturation line of component M. By defining

eVap,M - eCon,M + h^g,M " PAvM

the requirement can be stated in mathematical terms as

eG,M = (1 " fM^eVap,M + cvGM(TG " ^at.M^ + fMcG,»,M (20)

• eVap,M + cvGM(TG " TSat,M) •

where f^ = 0 when TG = TSatfM and f£ -• 1 as TG •» - .

Eq. (20) can be solved for c^u to obtain

( G.'.M eVap,M) cw,M(TG cvGM - cvGM HF

38 One simple form for f/i is a rational function. Therefore, f^ was defined by

TG - TSat,M

where 0^ is an input relaxation constant. This gives

(eG,»,M ' eVap,M) " cvGM(TG " TSat,M) * TT,

In using Eq. (23), two further adjustments were found to be desirable. First, if we are to solve eG>M = eVapM + C^CTQ - TSat>M) for TG, CvGM must be continuous for all Tea* u- This is best accomplished by writing the denominator of c Eq. (23) as ITG - Toaj. wl + 0^. Second, we want CAQ^ > vGM" Therefore, Eq. (23) is written as

,. (eG,»,M " eVap.M) ' cvGM(TG " .

The main problem in the implementation of Eq. (24) is that a new "F" function is now required, or

eG ~ I xm CG,M F T'rG. - 0 , (25) xm cvGM

where eGM = eVapM - CyGM

The Newton-Raphson procedure to extract TQ requires that we evaluate dF/dT^, or dF

The easiest method discovered so far to evaluate the derivatives is to see that

h^g,M + cLiq,M " cvLM TMelt,M " eG,M,old SOM •

where eGMold - eVapM

In other words, C^QJ is defined in terms of temperature and the vapor's true heat capacity. Because the derivative

deG,M,old dTG

is given by Eq. (A-7o) with the addition of the term

' and

because C^Q^ is given by Eq. (15), we can derive (for CyQ^ >

TSat,M dT V ) - G TG-TSat.il+ TG "

for TG > TSatM , and

40 dc T vGM u [. , Sat.M ^ ——— = bM[l + - -——— J dTG TSat,M " TG + %

cvGM - cvGM 2(cvvGM (28) TSat,M ' TG +

for TSat,M > TG •

where

" T Sat,M TERM = cvLM - c -) + 1) rCrt,M " TSat,M

for TSat,M < 3 TCrt.M ' and

£ T 2T M( M " Sat,M j TERM cvLM " cvGM 1) 2 TCrt,M " TSat,M

for TSat,M > 3 TCrt,M •

dcvGM Obviously, for the case of vGM M

dcvGM . . . . . Oncn e ——— is obtained , dT

deG,M _ dTSat,M dcvGM (29)

where TERM is defined as above. This completes the equations necessary to

41 perform the iteration, Eq. (A-75). The "F" function is calculated by Eq. (25), the derivative of "F" is given by Eq. (26) and

Sat>M is taken from Eq. (A-79). dTG

4. Modification of the Gas "Constant" at High Pressures. The second modification comes from the desire to calculatethe vapor state variables in the high-temperature and high-pressure regime, for example hundreds of megapascals. Both the Hougen, Watson, and Ragatz chart and the steam tables suggest that the effective gas constant is higher than the infinitely dilute value under these conditions. The expedient modification for TQ > TQ.^ was to change the relaxation equation for the gas constant, Eq. (A-24), to be

RM - gM) -in [R .30) Mi

for PGm TG ? ^Gm TG^Crt • and> for continuity,

TGG)0 TGG)Crt

pGm TG -

TGG)0

and 0^ is an input constant. The two derivatives required for general transient operations are

3L d IG^M 9PGm ^ RM 9TG

42 These derivatives involve only minor changes from the procedures used to obtain Eq. (A-49) and Eq. (A-80) for Eq. (30). For Eq. (31) the expressions are

pGm 3RM pGm 3fM *MpGmT0(1 " 8 +

RM3TG RM 3pGm 8MGTG " TCrt,M+aG,M

PGmTG - where ^ = 1i +

These expressions have been simplified through the definition of f^, which allows

p T 9f J Gm G M fan ^A G ^M (34) = fM 3(pGmTG) fM 3pGm fM 3TG

The 0ii was selected as 0.5 for water by fitting SESAME data at 3 000 K shown in Fig. 13. The fit was for the case of &Q ^ = 200 K. This choice is explained in Sec. II.C.6.

5. Definition of the Vapor Temperature for Single-Phase Cells and Other Modifications, The third modification was in the definition of a vapor temperature for single-phase cells. Equation (A-40) now becomes an implicit e- quation for a function, G, defined by

NMAT-1 cvGM TSat,M/RM

m=l

43 = 3000 K 600

50 100 150 200 DENSITY (kg/m3)

Fig. 13. SESAME/SIkrtlER-II comparison plot for fitting

44 +1 n n For the Newton-Raphson procedure, Tg = TQ - G /(dG/dTG) . The derivative we need to evaluate is

NMAT-1

dG m=l dTG NMAT-1

m=l

NMAT-1 dc' I (^/Ifc - cvGM(5J?)/R2)} G(36) NMAT-1

m=l

For this situation, p"^ and consequently To.* « are constants. This simplifies

the evaluation of dCyQ^/dTQ from Eq. (28). The other derivative, dR^/dTG, can be evaluated from Eq. (33). This rather long iteration is only performed upon beginning a problem. During a transient, values of the vapor densities and vapor temperature from a previous time step are used to define c^w and R^ in the application of Eq. (35). However, when a single-phase cell becomes two-phase, only the previous time step TQ is used to define cA™ and R^ for use in Eq. (35). A separate iteration involving a function G', defined by

G< - *vM " pGmRMTG - ° ' (37)

is used to make Rw consistent with an updated P~ .

n n The iteration is defined by P^TQ = P^TQ - G' /(dG'/d(PGmTG) ) . Because TG is constant, the required derivative of R^ can be found using Eq. (32) in the one case that is more complex than that described in the SIMMER-II manual. Updates

45 have been included to cover all branches of the R^ evaluation. The logic implemented can be pursued by examining the FORTRAN coding. Four other minor modifications are included in the correction set for the EOS changes, WRBAEOS, (see App. B). These are as follows: (a) A bisection algorithm was superimposed on the Newton-Raphson iteration to so.ve Eq. (25). This prevents small values of dF/dTg from causing an extrapolation to a TQ outside a domain where a solution is known to exist from previous iterations. (b) The gas "constant" calculation was updated in the remaining input options to be consistent with the new formulation. The main change was to revise the solution of Eq. (37) to consistently initialize two-phase cells with the input option IINP = 1. (c) The iteration to solve Eq. (25) was removed for single-phase cells. It was replaced by a once-<;hrough pass to define the gas constant. The iteration is not required, and, as is shown in Sec. II.C.7, if

6. Discussion of Results. Now that the modifications to the AEOS have been described, we will discuss how well the AEOS works. First, convergence of the "F" function, Eq. (25), around the critical pressure at ~3 000 K is no longer a problem. Figure 14 shows a plot of a typical case. The pressure at the converged temperature (3 029 K) is 22.1 MPa. The SESAME tables give 25.2 MPa at this density (16.6 kg/m ) and temperature. The SESAME vapor energy is predicted to be 9.73 MJ/kg. Both the type of convergence and values obtained by SIMMER-II are at least reasonable for this case. We also made a comparison along the liquid and vapor saturation lines. The internal energies calculated for density and temperature input are compared with the steam-table values in Fig. 15. There is some distortion in SIMMEk-II because of the functional form assumed for the liquid energy at temperatures above two-thirds of the critical temperature (431.5 K or 158.4°C). A possible correction is to subtract one-half the internal energy of vaporization rather than one-half the enthalpy of vaporization in the formula for the liquid's saturation

46 v I ! 3 \ eQ=9.6 R 10® J/kg PQ=16.5 kg/m 20 " \ WATER VAPOR 15

10 - \

5 — - \

0 \ \

-5

• -10

-15 —

-20 I I i i 301C 3020 3030 3040 3050 VAPOR TEMPERATURE (K)

Fig. 14. "F" function for the revised AEOS.

temperature (Eq. A-6), In any case, if the saturated vapor densities and internal energies are input, the resulting plot of pressure as a function of temperature is quite reasonable, as is shown in Fig. 16. As the situation exists, an acceptable value for 0^, the relaxation constant in Eq. (22), is 50 ¥ With 0^ = 50 K, Fig. 17 shows a comparison of energy along the near-critical 22-MPa isobar above the critical temperature. The agreement is reasonable. To get a value for the gas "constant" relaxation

parameter, a^ ^, the worst case for Tgat > T^rt was selected. Values representing the critical density ard internal energy for wat;r were inserted 7 into the AEOS, and a tiial temperature of T^ = TCrt + 10" K was used. With % = 50 K and ty = 0,5, a plot of dF/dTg as a function of TQ is given in

47 ^ 3.0 STEAM-TABLE VAPOR SATURATION ENERGY

• = SIMMER- II

STEAM-TABLE LIQUID SATURATION ENERGY

I I I w0 100 200 300 400 TEMPERATURE (°C)

Fig. IS. Steam-table comparison with saturated density and temperature inputs.

Fig. 18. A value of 200 K (approximately 1 order of magnitude above that used in the ZIP study)1 appears to give an acceptably negative derivative. Unfortunately, F can still be triple-valued if f^ in Eq. (20) is sufficiently small, and we are close to the critical pressure. A plot of the "F"

function (aGM = 200 K, 0M = 50 K, and ^ - 0.5) for such a case is shown in Fig. 19. The idea was to produce saturated conditions at 20 MPa. The high SIMMER-II result of 22.42 MPa is 13% more than the low SIMMER-II result or 19.83 MPa. As might be expected, this triple-valued behavior becomes worse if 0M is increased, in other words, the approach to CQ „ ^ is slowed. Values for

the highest zero of F as a function of flM are shown in Table I. A lower value

48 STEAM-TABLE 10.0 SATURATION LINE

LU DC CO 1.0 CO = SIMMER-II LU DC

0.1 1 100 200 300 400 TEMPERATURE (°C)

Fig. 16. Saturation pressures with internal energy and input into SIIMER-II. of 0^| will reduce the problem; however, with &Q U of 200 K, the steam pressures above the critical temperature are already too low. Tables II-IV contain calculated pressures for 22, SO, and 100 MPa steam-table isobars, with density and temperature input, and ag ^ of 200 K, and a ^ of 0.5. In these tables, all mass is assumed to be vapor, and the calculation is only performed for a density less than ~500 kg/m . Higher steam densities are not anticipated in the molten- core/coolant-interaction problem. The maximum deviation is about 15% below the desired result. If 0M is reduced, the SHMiR-II internal energies necessary to achieve these temperatures will be increased, in other words, the points in Fig. 17 will be raised. Consequently, we do not wish to reduce 0^. A reduced

49 5.0 STEAM-TABLE CURVE>

4.5 H

a:o in Hi 4.0 •=SIMMER- n <

"" 3.5

3.0 I 1 400 600 800 1000 1200 TEMPERATURE (°C)

Fig. I" Internal energies of vapor with density and temperature input along the 22-MPa isobar.

50 100 200 300

Fig. 18. Derivative of Eq. (25) as a function of &Q ^, with TQ just above Tp ^ for water. aG M wwill improve the pressure relationships, as shown in Table V, giving the

calculated 22-MPa isobar with w = 75 K. Now the maximum error is only 4%. However, lowering EQ « increases instability when TQ ~ Tr j, as implied by Fig. 18. Formulation of a multivariate least-squares fitting procedure to find the best values for &Q ^, (3^, and 0^ seems desirable. Then, we could determine if the procedure is sufficiently stable, or if reformulation is required. In the meantime, the argument can be made that current values are acceptable for reasonably low-density steam, given the other uncertainties (see Fig. 20).

51 I i i i 20 •Q=3.1973 MJ/kg 3 15 PQ = 171.4 kg/m _

10 —

5 - \ —

u \

-5

-10 —

-15 — \- i I I I 630 650 370 690 TEMPERATURE (K) Fig. 19. Remaining triple-valued nature of the "F" function near the critical point.

TABLE I

3 F ZERO WITH PG - 171.4 kg/m AND eG « 3.197 MJ/kg Relaxation Temperature Pressure from (EQ. 22)

p(MPa) 50. 22.42 75. 23. 86 100. 25.01 150. 26.77 200. 28.08

52 TABLE II

AEOS/STEAM-TABLE COMPARISON FOR THE 22-MPa ISOBAR

Density (kg/m3) Temperature (K) AEOS Pressure (MPa)

207.3 648. 21.1 121.2 673. 19.4 101.4 698. 19.2 90.08 723. 19.0 76.25 773. 18.9 67.50 823. 18.9 61.20 873. 18.9 56.32 923. 19.0 52.38 973. 19.1 46.28 1073. 19.3 41.68 1173. 19.5 38.02 1273. 19.6 32.45 1473. 19.9 30.25 1573. 20.0

The greatest errors in a two-phase cell with the proposed AEOS formulation are with high steam densities as suggested by Fig. 13. Unfortunately, the only real comparison available is with SESAME, and as Fig. 21 shows, some deviation of SESAME from steam-table values occurs even at intermediate densities. Of course, deviation of SESAME tables from the AEOS is worse at higher densities and temperatures (Fig. 22). However, in the molten-core/coolant-interaction problem, cases with true high-density H20 can be bounded by running a parametric where pressures are obtained by heating liquid water rather than compressing steam. For T^ > ^Crt' *^e microscopic de^uity of liquid water is assumed to be p CTl' This is ~317 kg/m . Rapidly heated water will only go two-phase and consequently will be able to vaporize when its effective macroscopic density is less than this density. A plot of pressure as a function of temperature at a density of 316 kg/m is shown in Fig. 23. In the single-phase liquid regime, applying a low single-phase compressibility and adding a fictitious p

53 TABLE III

AEOS/STEAM-TABLE COMPARISON FOR THE SO-MPa ISOBAR

Density (kg/m3) Temperature (K) AEOS Pressure (MPa)

577.7 673.2 51.9 498. 3 698.2 54.7 .102. r> 723.2 52.3 256.9 773.2 44.5 195.4 823.2 44.4 163.6 873.2 44.3 143.6 923.2 44.4 129. 4 973.2 44.6 110.2 1073.2 45.1 97.25 1173.2 45.8 87.63 1273.2 46.4 80.03 1373.2 47.0 73.74 1473.2 47.4 68.42 1573.2 47.8

TABLE IV

AEOS/STEAM-TABLE COMPARISON FOR THE 100-MPa ISOBAR

Density (kg/m3) Temperature (K) AEOS Pressure (MPa)

529.0 773. 90.4 445.2 823. 94.2 374.4 873. 94.4 321.0 923. 93.8 282.0 973. 93.6 230.5 1073. 94.5 175. 7 1273. 98.7 133.9 1573. 104.8

54 TABLE V

AEOS/STEAM-TABLE COMPARISON FOR THE 22-MPa ISOBAR IF aGM IS SET TO 75 K

Density (kg/m3) Temperature (K) AEOS Pressure (MPa)

207.3 648. 21.2 121.2 673. 21.9 101.4 698. 22.1 90. 08 723. 22.1 76. 25 773. 21.8 67. 50 823. 21.6 61. 20 873. 21.4 56. 32 923. 21.3 52. 38 973. 21.2 46. 28 1073. 21.1 41. 68 1173. 21.1 38. 02 1273. 21.1 32. 45 1473. 21.1 30. 25 1573. 21.1

increase with density and low vapor pressure upon entering this regime. It is

true that pressures along a fictitious equilibrium Tca* line above Tp i, which goes deeply into the single-phase regime, are comparable with SESAME values (see Fig. 24). At e = 3.75 MJ (T £ 1 175 K) the maximum error is 21.5%. However, a single-phase cell in SIMMER-II could have the pressures indicated in Fig. 24 for much lower water densities. At most, water only has to occupy a volume fraction

equal to aQ in the single-phase regime when other liquids are present. For example, 1 000 K water in a single-phase cell could produce a pressure in excess of 2.87 GPa if more than 41 kg/m exists. The liquid in the remaining volume is assumed to be compressed to the same pressure in this example. (As another interesting observation, the SESAME curve in Fig. 24 is also somewhat strange. For any interested observer, the values used to generate the curve are given in Table VI. Note that the water's liquidus energy, 0.9043 MJ/kg, must be added to these values for comparison with the SIMMER-II AEOS.)

55 1 I I I I ! ^ / • /" V T=1 573.2 K /' >/ 90- SESAME^

/ , 70 - / y Hi */\ // \ STEAM TABLES CO CO 50 5U /7S = SIMMER- H /, r //

30 -

I I I I I 20 40 60 80 100 120 VAPOR DENSITY (kg/m3)

Fig. 20. A typical EOS comparison at low steam densities.

In conclusion, for H20 the revised AEOS produces pressures in the vapor field that are high for saturated water below the critical temperature (because of the low internal energy for the liquid water), perhaps ~15% high near the critical temperature (because of the triple-valued characteristic of the formulation), up to 15% low on isobars above the critical pressure when compared with steam tables (because of the value of aQ M chosen to increase stability), a little high at high pressure and temperature but low density as shown by Table IV, and perhaps as much as 20% low (for practical problems) at high vapor densities and high pressures. Greater vapor-field accuracy can be achieved in the future if instability problems do not affect practical SIMMER-II runs. Single-phase cells with supercritical liquid water still may possess unrealistically high pressures.

Appendix D gives the program to produce the results displayed in this section. Appendix E gives the proposed fitting parameters for water, corresponding to the input description of App. C.

56 5.6 o CLm 5.4 < STEAM TABLES

SESAME

CL O 5.0 T= 1573.2 K CL

20 40 60 80 100 120 VAPOR DENSITY (kg/m3)

Fig. 21. Additional comparison of EOS data at low density.

57 o 9.8 1 1 1 DC UJ T = 3000 K /SIMMER-n LLJ _j 9.4 l§90 o /SESAME a. 8.6- < 1 1 1 1 50 100 150 200 250 VAPOR DENSITY (kg/m^)

Fig. 22. EOS comparison at high temperature.

58 a. * 100 - 2UJ cc (0 (0 UJ IT 0.

800 1200 1600 2000 TEMPERATUFtc (K)

3 Fig. 23. A SESAME/SIMMER-II comparison at PQ = 316 kg/m .

7. Suggestions for Further Improvement. Except for the potcktial of additional error corrections, no further AEOS improvements were possible within the scope of this molten-core/coolant-interaction program. However, for future reference, we are providing a discussion of some items that deserve additional attention. These items are as follows: (a) A multivariate fitting procedure and graphics capability would be useful not only for finding and displaying the best possibilities for the water parameters, but also for obtaining SIMMER-II input for other materials. (b) A slightly tedious correction (from an algebraic viewpoint) to the liquid's internal energy would be to subtract one-half the internal energy of vaporization rather than one-half the enthalpy of vaporization

59 10*

SESAME

-* 10s 0 EL / \ «w SIMMER- H UJ GC CO CO UJ cc °" 100

10 3.0 4.0 LIQUID INTERNAL ENERGY (MJ/kg)

Fig. 24. Pressure as a function of liquid internal energy taken from the density and temperature describing a fictitious saturation above TCrt.

in Eqs. (A-6) and (A-13). This should significantly improve the agreement in Fig. 15. (c) The formalism to define the gas "constant," 1^, could be made more appealing from a dimensional standpoint. Taking the logarithm of a quantity possessing units is not desirable. (d) The increase in the gas "constant" upon departure from the saturation line needs to depend on pressure as well as temperature. Unfortunately, a little addition to any relaxation equation can add significant complexity to the numerical algorithm. An additional inner iteration to

compute a consistent TSatM, RM, and TG needs to be considered but must be designed carefully to avoid excessive costs.

60 TABLE VI

SESAME RESULTS FOR FIG. 24

THE DSPLX CODE, LASL T-4, 11-27-79, VERSION 2

1 EOS 7150,.07546,600 P E DP/DR DP/DT DE/DR DE/DT 1.079E-02 2.194E+00 1.296E-02 6.478E-05 -1.124E+01 4.861E-03 P T DP/DR DP/DE DT/DR DT/DE 1.172E-02 6.132E+02 1.594E-01 1.436E-02 1.150E+03 1.923E+02 SOUND SPEED = 4.347E-01 KM/SEC 2 EOS 7150,. 3170,647 P E DP/DR DP/DT DE/DR DE/DT 2.558E-02 1.73OE+OO 2.701E-02 2.565E-04 -9.379E-01 4.834E-03 P T DP/DR DP/DE DT/DR DT/DE 2.210E-02 6.472E+O2 7.554E-02 4.561E-02 1.853E+O2 2. 193E+02 SOUr© SPEED = 2.925E-01 KM/SEC 3 EOS 7150,-3438,700 P E DP/DR DP/DT DE/DR DE/DT 4.121E-02 1.947E+00 7.449E-02 2.833E-04 -1.086E-.O0 4.356E-03 P T DP/DR DP/DE DT/DR DT/DE 3.663E-O2 7.028E+02 1.401E-01 6.386E-02 2.6O8E+O2 2.413E+02 SOUND SPEED - 3.999E-O1 KM/SEC 4 EOS 7150,.4593,800 P E DP/DR DP/DT DE/DR DE/DT 9.433E-02 2.187E+00 2.628E-01 4.203E-04 -1.317E+OO 3.319E-03 P T DP/DR DP/DE DT/DR DT/DE 9.178E-02 8.O78E.-O2 4.405E-01 1.454E-01 3.972E+O2 3.265E+02 SOUND SPEED - 7.O97E-O1 KM/SEC 5 EOS 7150,.6218,900 P E DP/DR DP/DT DE/DR DE/DT 2.446E-01 2.252E+00 9.901E-01 7.643E-04 -1.361E+OO 2.443E-03 P T DP/DR DP/DE DT/DR DT/DE 2.396E-01 9.033E+C2 1.338E+OO 3.179E-O1 4.943E+O2 4.040E+02 SOUND SPEED - 1.239E+OO KM/SEC 6 EOS 7150,.8136,1000 P E DP/DR DP/DT DE/DR DE/DT 6.8O4E-O1 2.270E+00 2.737E+00 1.296E-O3 -7.482E-01 2.463E-03 P T DP/DR DP/DE DT/DR DT/DE 6.766E-01 1.001E+03 3.198E+OO 5.354E-01 2.78OE+O2 4.044E+02 SOUND SPEED - 1.935E+OO KM/SEC 7 EOS 7150,1.023,1100 P E DP/DR DP/DT DE/DR DE/DT 1.733E+OO 2.512E+00 6.O8OE+OO 2.018E-O3 4.953E-01 2.822E-03 P T DP/DR DP/DE DT/DR DT/DE 1.734E+OO 1.100E+03 5.700E+00 7.O89E-O1 -1.715E+O2 3.554E+02 SOUND SPEED = 2.622E+00 KM/SEC 8 EOS 7150,1.242,1200 P E DP/DR DP/DT DE/DR DE/DT 3.866E+OO 3.O15E+OO 1.165E+01 2.823E-O3 1.220E+00 3. 139E-03 P T DP/DR DP/DE DT/DR DT/DE 3.844E+OO 1.200E+03 1.071E+0J. 8.378E-O1 -3.761E+O2 3.222E+02 SOUND SPEED = 3.578E+OO KM/SEC

61 (e) In single-phase states, the effective sonic velocity could depend both on liquid energy and the extent of departure from "he saturation line, or density. lecause liquid energy and density are independent variables, a properly considered scheme would be straightforward to implement. (f) We should eliminate the inconsistency in using liquid and saturation temperatures with values above the critical temperature. Besides the EOS modifications to get a proper pressure at higu vapor densities, significant numerical methods changes are desirable to implement such ideas correctly. For example, we should consider immediate transfer of liquid to the vapor field whenever its temperature exceeds the critical temperature. This transfer would require modifications to subroutine EXFL1 to stop the possibility that liquid might convect from a cell simultaneously with this interfield mass transfer. Also, wherever a hypothetical liquid/vapor interface temperature exceeds the critical temperature, only heat transfer and vaporization, not condensation, should be considered. In other words, all present use of situations with To.4. > Tp , would have to be removed. (g) Improvements might be considered to make the gas "constant" more responsive to density changes at high pressures. Hovever, if highly supercritical regimes are truly required for an analysis, continuation of ad hoc modifications may not be prudent. Modilication of both SIMMER-II and the SESAME EOS to allow use of SESAME might be desirable to permit calculations with a more complete representation of thermodynamic states. The problem here is efficiency. At high pressures some degree of implicitness in solving the energy equation is desirable because of the important contribution of tin; work term. This implies the desirability of having updated EOS variables for every cell on every pressure iteration. Such an approach could be expensive, particularly for calculations with many mesh points that run for more than a few milliseconds of real time. From the above suggestions, clearly additional time and effort cculd be expended on the SIMMER-II EOS. Unfortunately, the list of possible and even desirable improvements may be endless. Decisions must be made to focus efforts for improvement on problems that are judged to offer real benefits for analysis in terms of credibility, flexibility, robustness, problem resolution, and physical reasonableness. As an example, one of the interesting situations that arose

62 with the current AEOS is given in Fig. 25. Here the desired convtrged pressure

is above the p* value, meaning that a positive Tgaj. cannot be defined and no solution exists. Fortunately, this situation is physically unreasonable. It arose in SIMMER-II when a single-phase node was rapidly compressed to an <*Q of 10 while retaining the previous time-step vapor density. The SESAME tables predict TQ = 7 100 K and p • 275 GPa for the conditions of Fig. 15. Nature is far too nonlinear to permit success over all possible multicomponent EOS regimes with a few simple analytic formulas that possess densities and internal energies as the independent variables. These difficulties need to be eliminated if they continually frustrate the analyst in using the code.

10*

-104 — •Q-4.60 x 10* J/kg

-10s -

100.0 1000.0 % 0,000.0 TEMPERATURE (K)

Fig. 25. An exampls showing that p > p* is not permitted.

63 8. Conclusions. Part of the deficiency with the 1980 ZIP study was the inadequacy in the SIMMER-II AEOS. A series of corrections to eliminate or reduce these AEOS problems was made for this molten-core/coolant-interactions program as described in this report. The main modification is a variable vapor heat capacity to take the vapor state to the infinitely dilute energy far from the saturation curve. In the domain of interest for this program, five new AEOS input parameters allow pressures to be calculated to within ±20% of steam-table values or SESAME EOS values. Usually the agreement is much better. Further increases in accuracy in some regimes clearly are possible but at a cost of decreased stability. However, we believe the current approach is sufficiently accurate for the reactor meltdown calculations in Chap. VI because other modeling problems lead to greater uncertainty in the calculated results. These prolilems include but are certainly not limited to (a) the initial conditions assumed, (b) the heat-transfer algorithm employed, (c) structural and three-dimensional effects ignored, and (d) the other approximations made in the SIM4ER-II solution algorithm. The correction set formulated contains 527 lines and is given in App. B. Suggested inputs for water are given in App. E. Revisions to provide improved AEOS properties for corium will require additional effort.

D. Modification of SIMMER-II Liquid-Liquid Heat Transfer for Water One large qualitative problem with the SDMiR-II ZIP study was the assumption of a water-lean mixing zone in calculating experimental simulations to avoid immediate thermite overquenching and insufficient pressure development. A second qualitative problem was the somewhat slow risetime in the calculated pressures. SNL experiments suggest an overall water-rich fuel-water-steam mixture. They also give an almost instantaneous pressure risetime in some situations. Both problems appear to originate with the SIM/IER-II liquid-liquid heat- transfer model. The thermal conductivity of water is low, for example, ~0.68 W/(m«K) at about 400 K, leading to a rather low thermal diffusivity, say -1.7 x 10'7 m2/s. In a typical SINMER-II time step, ~10~4 s or shorter, the thermal penetration distance is only a few microns. Instead of bulk liquid water heating, ve should be vaporizing off the surface in a nonequilibrium manner. The large heat capacity present in the liquid water should not become

64 effective until after pressures have developed, causing an increase in the water's surface area. The present liquid-liquid SINMER-II heat-transfer model is described in App. F (reproduced from App. D of the revised SIhA1£R-II manual). As an initial modification, the following ideas have been implemented. There were "IF" statements inserted to turn off liquid-liquid heat transfer to water from another liquid whenever the other liquid is hotter. The calculative flow was diverted to the vaporization/condensation model where heat tranoTvr from the hotter liquid is performed. However, we do not modify single- phase cells or the situations where Tj^m (m * 3) < Tj_g (water). At the beginning of subroutine PHASE, we set up heat-transfer coefficients based on the liquid-liquid formalism modified so that the resistance to heat transfer is only that of the hot droplets (fuel, steel, or zirconium oxide). Consequently, Eq. (F-9) in App. F, defining the overall heat-transfer coefficient, now is approximated by

U k 2 r (38) cLmL3 * Lm - pm

The water temperature is that of the water surface or T

(r 45 pm ^HLmG3 " Lm3 "g" ^— rT~I (r. + rp3 > r lSat,3 '3P3

HLniG3 (TSat,3 (39)

where

^HLmG3 ^s encrgy flux going into vaporization off the water surface,

*p t'ie effective product of the surface area and the heat-transfer coefficient, and

is an input coefficient, the liquid-liquid heat-transfer multiplier.

65 The quantity HIm/~-> is zero whenever m = 3, r or r , are zero, or T, < Ti->. The. first two restrictions are obvious. The last condition is Lm LJ simply a restatement of the idea that contact with a colder liquid should not result in instantaneous condensation. When r 3 < r , Hi^^ will be limited by 15Cm J10, Lm k iL m/rlL pm. basi-d on the available surface area of hot material. Because the liquid-energy equations are treated explicitly, a second limit on Ht^jj *s given by the heat in the hot liquids or

(40)

The problem of vaporization of the available water is discussed below. Section II.E gives the complete formulas to integrate these heat-transfer modifications within the context of a revised vaporization/condensation model. Here we simply review those modifications that appear in the correction set for these heat-transfer revisions, App. G. The mass continuity equation is still approximated by

u/g,M,eff

With the additionst a* 3 and a£ 3 must be redefined by

NCLE _ 2

m=l k=l

and

NCLE 2 a2,3 = -^ («GL3 + HLG3+ I HLmG3 + I C^Sk + HSGk» ' (42) m=l k=l

where T^ is the result of the liquid-liquid heat-transfer model

66 The next modification to the vaporization/condensation treatment is in the adjustment of the vapor-side heat-transfer coefficient based on the new mass flux at the watir surface. This adjustment is described by Eq. (E-ll) of the SIKWER-II manual. The modified adjustment to be evaluated in this iteration is given by (for water, component 3)

tHLG3(TSat,3 " TL3) m=l

" V) =° (43)

c or ne ne a where x = " Dn''lG3L''. ' * S tive of the product of the heat capacity and the mass-transfer rate. The teration is x - x - F /(dF/ds) . The product of the surface area and the new vapor-side heat-transfer coefficient is

The special cases for convergence of this equation set are the same as in the SItoMER-II manual, Ver. 2. The heit-transfer rates simply play a passive role similar to the H^^CTg^ ^L3^ ^erm< The vapor-energy equation 1 so must be modified. First, the steam's mass flux must be incremented by

NCLE

m=l

Second, when all the water vaporizes in one time step the mass flux is known; however, the additional energy transferred from the other liquids or

NCLE

m=l

67 must be given to the vapor. The equations and other details are described in the vaporization/condensation model revisions (Sec. II. E). After convergence of the model, heat raust be taken away from the liquids.

Additions include updating the FORTRAN variable QL by qHLmG3. and the FORTRAN variable SIELN by At^jji^-j/Pj^. Mass transfer is accounted for automatically with the vapor-density terms P/L. and PQI . When all the water vaporizes an extra term,

NCLE " £ iHLmGS • m=l

is added to the FORTRAN variable QG. The final correction in the correction set results from the effect of these heat-transfer modifications on the effective heat of vaporization for the vaporization case. This correction is described in Sec. II.E, but is retained in this correction set to isolate all heat-transfer model changes. This concludes the description of the liquid-liquid heat-transfer model changes. The correction set (App. G), without the last nine lines, was merged with the SIM/1ER-II code, compiled, and loaded. The input for the SNL experiment calculated in the ZIP study was updated and run. The vaporization rate of water was found to be extremely rapid. For a case in which the hot liquid to water heat-transfer multipliers were set to 0.1, the pressure in the node at the bottom center of the interaction zone rose to TO MPa (nearly the critical pressure) in 0.08 ms (see Fig. 26). This rise is much faster than in a conventional (unmodified) comparison case run with all the liquid-liquid multipliers at unity as shown in Fig. 27. In the conventional case, SIMMER-11 achieved 6 MPa by ~1 ms. Unfortunately, the vaporization formalism initially used (that described by the SIMMER-II manual) required water to leave the liquid field at the condensate energy. The energy difference between the condensate energy and the true liquid energy (liquid is subcooled) came from the liquid fie'd. With interface vaporization, the liquid-side heat-transfer coefficient for heat transfer into the bulk water was insufficient to transfer enough energy from the water's surface to make up for the losses. A plot of the water temperature of the node in question is shown in Fig. 28. A decrease of more than 20 K has occurred by 0.08 ms. The temperature from the conventional comparison case is nhown in

68 &

00 1.0 2.0 3.0 4.0 5.0 6.0 TO 8.0 no" Tlme^)

Fig. 26. Pressure in the sample problem with new heat transfer.

8 i i r

0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Time (s) Fig. 27. Pressure in the sample problem with original heat transfer. 69 360

370 -

t 360 -

350 -

340 00 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 *10-5 Time (s)

Fig. 28. Water temperature in the sample problem with new heat transfer.

Fig. 29. Here the water has gained more than 220 K in 1 ms. Although the energy gain in the conventional case is unrealistic, the energy loss in the new formalism also was incorrect. Several options presented themselves to deal with the problem. These options were as follows: CD Ignore the problem. Perhaps the SINMER-II results still could be argued to be qualitatively reasonable. (2) Try to put in some liquid-liquid heat transfer to compensate for the problem. (3) Rewrite a minimal vaporization/condensation model. (4) Because any rewriting involves significant work, develop a more realistic multicomponent vaporization/condensation model.

Because of the limited scope of this program, option (3) appeared to be the best compromise with the goal of attempting to preserve scientific integrity. This* revision is described in Sec. II.E. It assumes that for condensation, vapor is removed from the vapor field at the average vapor energy and appears in the liquid field at the condensate energy. For vaporization, liquid is removed from the appropriate liquid component at the liquid energy and appears in the vapor field at an energy equal to the entha'py of saturated vapor. Consequent- ly, subcooling increases the effective heat of vaporization. A rerun of the 70 600

§ 500 -

300 0.0 1.0 2.0 3.0

Time (s)

Fig. 29. Water temperature in the sample problem with original heat transfer. same sample problem with the revised vaporization/condensation model (and the AEOS changes) produced the results shown in Figs. 30 and 31. The liquid-to- water-surface multipliers were set to 0.1. The pressure rises to a peak value of 60 MPa on a submi 11 isecond time scale (~0.25 ms) and the bulk liquid water only increases in temperature by a moderate amount. The form of this response thus gives promise for calibrating to SNL vapor-explo3ion pressure da'a in a water-rich environment.

E. Vaporization/Condensation Model Changes This section gives the mathematical equations that are solved in the vaporization/condensation model and the reasons for the coding in difficult-to-converge situations. For the purposes of brevity, the equations presented include the liquid-liquid heat-transfer modifications. This is the manner in which the code is expected to be run in the molten-core/coolant-interaction program. How- ever, the heat-transfer changes do constitute a separate correction set /VAPOR/WRBHEAT (see App. G), and ths vaporization/condensation modifications are equally applicable to the old heat-transfer model.

71 I

0.0 2.0 4.0 8.0 8.0 10.0 12.0 14.0 18.0

Time (s)

Fig. 30. Pressure in the sample problem with new revised heat transfer.

Time (s) Fig. 31. Water temperature in the sample problem with new revised heat transfer. 72 The use of the modified liquid-liquid heat-transfer model to cause vaporization directly from a water surface following liquid-liquid contact suggested that water's subcooling energy be included in an effective heat of vaporization. Experience with running the SIMMER-II code has further suggested that in the case of condensation, the vapor superheat should also be included in an effective condensation energy. These two changes have been incorporated in a new correction set /VAPOR/WRBPHAS, listed in App. H. The differential equations treated in the vaporization/condensation model are taken from the basic conservation equations by ignoring convection. These are as follows:

(a) Vapor-Energy Equation

NMAT-1 6Q+1 condensation sn+1 .n+n+1 sn _n AT v G " PG eG " At 2. m=l t ) eCon,M+h5g,M .vaporization

NMAT-1 I m=l k=l

NMAT-1 NCLE I p I m=l m=l

(b) Liquid-Energy Equations

eCol,M • condensation n+l Bn .n c p e _ r K+l Lm Lm ~ Lm Lm x r mrGLm 1 . c Lm

xm , m = 1,2,3,4 ;

NMAT-1 NMAT-1 I (46) m=l m=l

73 (d) Liquid-Continuity Equations

_ __ L - xm) P{^ , m - 1,2,3,4 ; (47)

(e) Structure-Energy Equations

NMAT-1 Sk Sk — Sk Sk ^« m=l

(f) AEOS

NMAT eG = ^ XM tcVap.M + cvGM m=l

(49)

-T*

K+l ~r-t-l nk-4-1 * Sat,^ M pGm

(g) Vapor-Energy Heat-Transfer Coefficients

and

(50)

rK+l ; H*"^ -1 ISfc CpG

where

T !Sat,M " lG HLGm

K+l "GSk I'Sat.M ' lG '

_ u (T""1'1 Tn+1 1 ~ uSGSGkk ^lSatM xSk '

xffi = 1 is the nominal case,

xm = 0 if all of energy component m is vaporizing, and the quimfii are given by the heat-transfer modifications of Sec, II.D, and the remaining terms are defined by the SIMMER-II manual. No generalization of the formalism to include er.ergy transfer at surfaces without mass tiansfer in the vapor-energy equation has yet been accomplished. For example, solid particulate cannot simply heat superheated steam. The available structure surfaces stili contribute to the vapor-energy equation even in the cases where all the appropriate liquid vaporizes, although the Tg^J ^ used for these situations is bated on a temperature yielding balanced energy transfer to a structure's surface. Also, if some of liquid component 3, water, is present at 4he beginning of the time step, qjjimQ3 will always be evaluated for each relevant liquid-energy component. The total phase-transition rate is obtained by summing Eqs. (44), (45), and (48) and equating the separate components. This gives

K+l _ _K+1

r } q + GLm U+l . h*+l e*+l ' «»»Lm eCon,M + h^g,M " eLm

2 NCLE + I ^IGmSk + ilSkGrn^ + 6m3 I k=l k=l

where 6m^ is the Kronecker delta and x is assumed to be 1.

a ae If ^Q^At < -?Lm' ^ *- liquid of component m vaporizes and this equation is presumed not to apply. If an effective heat of vaporiz tion, h^e^M eff" is cie" fined as

75 e?i+1 - efit}, M if condensing ' 1 (52) K+1 +1 K+1 . . eCon.M + h5g!M ' eLm lf vaporizing then Eq. (IV-96) of the SIfcMER-7I manual gives TQ^ with ^^M eff substituted for hyl \»' (The SIfoMER-II manual equations referred to are given in App. I.) The first Newton-Raphson iteration procedure, Eqs. (IV-97)-(IV-102) of the SIMMER-II manual, is now the solution for Eq. (IV-96) of the manual written as

2 NCLNCLEE rGLm " "777 (ilGmLm + ÎLmGm + ^ ÎGmSk + ÎSkGm5 + 6m3 E iHLk h5g!M,eff k-1 k-

(53)

The required modifications are in the evaluation of h^X u «ff 9h5g M eff/'9^Sat M assuming tliat TQ is constant at TQ. TO obtain the derivative term, we must differentiate Eq. (52). Because we are solving the coupled continuity/EOS relationship, CQ is given by Eq. (49) where Xj^ is the updated mass fraction of each component. Because a full evaluation of CQ is only made at the end of the iteration, eS is determined from

NMAT-1 M=l

where

e Vap,M _ c.K + 3TSat,M

u> m Because c^, = CvGM + max {0, ' IiEl2 yun_u aat.w | ^ 1TG " TSat,M! f pM

76 3c vGM = 0 when c.'^u = c 9TSat,M

If cvGM > cvGM ' then we obtain

9eG „« 3eVap,M . = f 'G ? 'Sat.M 3T l3 Sat,M~^ Tsat,M~ ^ lg "ll

(55)

M vGM ^^ • TSat,M>TG 3T Sat,M 'Sat.M " *G

3eVap.M The expressiors for are 3TSat,M

3eVap,M S ' 'Sat.M ^ 3 'Crt.M • 3TSat,M 'Sat.M

and (56)

3eVap.M T* ^ 2T • 'Sat.M > 3'Crt.M • 3TSat,M

The next term in Eq. (52) is CQ^ ^, It is given by Eq. (A-6) and updated eact inner iteration. Its derivatives are

77 Con,M T T = CvLM • TSat,M and (57)

5eCon,M

Obviously, e^. >. as well as its derivatives must be updated each inner iteration. The final new expression is that for e£~j" . Multiplication of Eq. (47) by e£ and substitution in Eq. (45) gives

,K+1 K+l e n K+1 _ gn n -Con.M ' Lm . PgLmeLm " PLmeLm + AtA VGLr m

ess)

Only the JC = 1 case and the vaporizing option need be considered to obtain an eLm estimate for Eq. (52). For this case, we obtain

At ^£»+ A? ^5« . (59)

If the e£m used in this equation is adjusted to be consistent with T. (defined by Eqs. (IV-90-92) of the SIMMER-II manual), then Eq. (59) becomes

L + At

78 (61) 3TSat,,vi 'Lm

where is zero if m = 3 .

This completes the necessary new expressions to evaluate Eq. (52) and its derivatives. A second Newton-Raphson iteration procedure is used to solve for the vapor-side heat-transfer coefficients, Eq. (50). Equation (50) is substituted into the expressions for the mass-transfer rates at a given surface. This gives expressions like Eqs. (IV-104) and (IV-105) of the SIMMER-II manual. The function for the Newton-Raphson iteration is then defined by Eq. (E-ll) of the SIMMER-II manual. This is straightforward to modify. The FORTRAN variable, CPDHFG, which is (c /h^ ^), is simply replaced by (c /h^ ^ eff^1 Because more than one surface can engage in mass transfer, for example, droplets can be vaporizing while structure is condensing the same component, the object is to

maintain h^o ^ eff constant throughout this iteration. Also, for water-droplet surfaces, the quimg? terms contribute to the mass transfer. For such water surfaces, the function for the Newton-Raphson iteration is given by Eq. (43) of Sec. II.D. The solution to the vapor-energy equation is more complex. The starting point is to multiply Eq. (46) by CQ+ and subtract each term of the resulting e- quation from Eq. (44). Using the fact that the converged solution, CQ+ , will

equal eG , this operation yields

NMAT-1 *n K+1 PGeG t eCon,M eG

NMAT-1 (62) m=l k=l

NMAT-1 NCLE I [d - m=l m=l

79 The objective is to determine TQ+ from Eq. (62) with the other variables constant. The value of TQI"^ does not matter with a condensing component. Conse- can be assun + h e 1 from E( quently, b^giM,eff >ed as econ.M %]\A ~ Lm [ J- (52)1, which has no direct dependence on TQ. [We are assuming the saturation temperature to be constant while solving Eq. (62)]. Then, making the approximation, eS+ = e£ + c^+ (TQ - TQ), and putting in the values for the heat-transfer terms, Eq. (62) becomes

NMAT-1 eS-At £ m=l 2 *• LHHSGkSGkuSat,uSMM " !Sk > + "A' " ] k=l

NCLE , . -3 ' lLq -' q=l

condensing

eG " S^CTT1 " TG) vaporizing

NN5AT-1 NMAT-1 2 I xmH^tffi(T^iM-T^)+A? I l m=l m=l k=l

NCLE " x3) I m=l

NMAT-1

x p e e c U 2. U - mJ Lm^ Lm G vG G m=l

To place Eq. (63) in a manageable format, dummy variables are defined as

k=l

80 NCLE I «LqG3«fi{.3 " q=l 2 + I and k=l

0 condensing d = { } l } eCon,M + G ccv G UG G vaporizing

Then, we collect terms on the various powers of TQ , or

NMAT-1 0 coidens ing 2 [I *m { ,+ 1 +1 H m=1 bmcvG /h5g,M,eff vaporizing

NMAT-1 NMAT-1 2 K+1 r sn-' tc+1 G L~ PGcvG SGk m=l m=l k=l

NMAT-1 condensing

Xm vaporizing (64) NMAT-1 NMAT-1 I m=l m=l

NMAT-1

NMAT-1 2 NCLE " At(l - m=l k=l m=l

m=l

NMAT-1 I ( = 0 m=l

81 Eq. (64) has the form of a quadratic AT^ + BTQ + C = 0. As written, A < 0, so that the negative square root of the quadratic formula gives the higher temperature and furnishes the first choice for a correct solution. The special cases adopted in this revision of the vaporization/condensation model are as follows: (a) In the first Newton-Raphson iteration, all previous special cases for treating a near-critical pressure described by App. E of the SIMMER manual have been eliminated. Indeed, with Eq. (IV-100) of the SIMMER-II manual now written by

(65) ^g.M.eff

the FK function can be triple-valued whenever h^- \» »ff is small, not just near the critical pressure. Also, we must avoid oscillations between vaporization on one iteration and condensation on the next. To resolve these problems the solution logic contains the following elements:

(1') h^g M eff is limited to a minimum value of 100 J/kg.

(2') A variable ICOUNT(N) is defined for each component. Normally, ICOUNT(N) = 0. For the normal case, oscillations are limited by using Steffensen's method, Eq. (IV-116) of the SIMMER-II manual, accelerating the convergence to an average value.

(3') An abnormal case is defined by ICOUNT(N) = 1. This occurs if TQ £ TCrt M and TLm > TCrt,M •o r if h5g,M,eff « 1 00° J/k8 or dF*/dPL > "<>. 1 • Under these conditions maximum and minimum permissible densities are updated each iteration based on whether FK is negative or positive in the previous FK calculations. A bisection procedure is used if the Newton-Raphson iteration tries to extrapolate outside the range where a solution is known to exist. The bisection algorithm is also used every 20 iterations if ICOUNT(N) = 1 to stop the rare pathological case of oscillations just

82 within the lower and upper bounds. Finally if TQ, T^m, and Tga^. ^ are

above Tr. i v, and the value of Tea*. u implies vaporization, the minimum AC vapor density is initialized so that the converged value of T

(4) Oscillations between vaporizing and condensing from iteration to iteration are further limited by demanding that the first term in brackets in Eq. (65) be positive while the second term be negative for condensation. For vaporization, the first term must be negative, and the second term positive. Once a selection is made, a switch in the manner of calculating

h^. ^ eff requires satisfying of both opposing conditions.

(5') Although the special tightening of the convergence criterion with Tga^. near L i has been eliminated, a new criterion has been added. Now, the relative difference between successive density iterates must be within 10 in addition to the standard criteria. See the discussion following Eq. (IV-102) of the SIMMER-II manual in App. I. This allows the necessary extra iterations when macroscopic densities are small. (b) In the second Nev/ton-Raphson iteration, the special cases described following Eq. (E-12) of the SI^ER-II manual remain unchanged with one exception', any species that completely vaporizes in a time step is assumed not to contribute to FiXl in Eq. (50) at a structure surface. The

reason for this is consistency with the treatment of Tga£ ^ in the vapor-energy equation for such a case. (c) The special cases in the vapor-energy equation associated with the discussion of Eq. (E-15-18) of the SIKMER-II manual also remain the same, with one change. Because the structure is now presumed to transfer energy even for the case of all the liquid vaporizing, a special value for Tg^ ^ is used in calculating structure/vapor heat transfer. Conse- quently, when all the liquid is vaporizing, the value of Tcl'J »« used in

the termHSi(T§;J>M is

83 2 2 n n V IIK+1 TK+1 . V H T I HSGk 'G + i HSk 'Sk TAC+1 k=l k=l TAAI TSat,M = 2 " C } I < k=l

This definition means that the energy flows balance, so no mass transfer occurs at the structure's surface. For other uses when all the liquid of a given component vaporizes, T<^ ^ is defined so that Eq. (51) is satisfied, which is higher than the true saturation temperature. The remaining change is In the update of the liquid energies following the vaporization/condensation to prepare for the melting-freezing calculations. Equation (58) is used for this purpose. There are no problems for the vaporizing case. The q{jLmG3 *erm *s evaluated and added separately. The case where x is zero is unchanged. The condensing case, with x a 1, is evaluated implicitly from

At qji t r GLm An+1 eCon ,m p L Lm e? (67) -Lm i+l h eLm eLm 1 3Lm

where

e"Lm is the temporary updated liquid energy,

e"Lm includes liquid-energy updates from previous energy transfers, and

*Lm is tlie ^i(lui^ energy at T^.

This concludes the description of vaporization/condensation model changes. Their improvement over the version in the SIMMER-II manual is demonstrated in Sec. II.D. Difficulties arise where the vapor-energy equation and vapor-continuity equations dre closely coupled; in other words, where TQ is a strong

84 function of T

F. Miscellaneous Corrections for the Molten-Core/Coolant-Interaction Program A separate correction set was generated to deal with problems not treated in the previous AEOS, heat-transfer, and vaporization/condensation discussions. This correction set is given in App. J. A summary of the corrections is as follows: (a) A crude fit to the steam tables for the liquid water's thermal conductivity was made. The programming assumes that liquid water is described by energy component 3. Because the thermal conductivity of liquid water is so much lower than that of the fuel, it tends to be one of the key parameters controlling water temperature and water expansion. (b) Corrections were inserted allowing the maximum water-droplet size to be a multiple of the input maximum. This allows both fuel and water to have fixed but different droplet sizes in the same cell. Such input was used to fit the SNL MD-19 teat data. (c) The IMFLUD subroutine was found to ignore boundary cell pressures after convergence of the pressure iteration. This omission can lead to spurious results in the Weber number calculation for the liquid's droplet size. A temporary correction to reflect pressures on the left, top, and right-hand boundaries was inserted to permit better calculation of the shallow-pool experiments. (d) A time-step control allowing the vapor-energy work term to affect the time-step control on internal energy changes in vapor, DTL (11), was inserted. This control is

*n+l n+1 Atn+1 = f] (Min («S+1, 0.1) • G ^G ) , (68) IWI

where

85 Aj <«LVL> w A

and the notation is that of the SIMMER-II manual. This time-step change was made necessary because of the oscillations that otherwise developed in the anal sis of the shallow-pool experiments and because the enormous pressures of an in-vessel steam explosion can readily lead to spurious vapor-energy values unless short time steps are t »ken. A better procedure, although beyond the scope of this study, is to include the work term implicitly within the pressure iteration. (e) Because densities at different times are used in its construction, the quantity,

'Lm

which determines the minimum vapor pressure for a component in a single-phase cell, -^as found to go spuriously negative in some situations. The inverse of this quantity cannot control the pressure in these situations, and a 10 bound was inserted.

86 III. LOWER HEAD MODEL FOR SIMMER-II

A. Introduction The lower head model developed for the steam-explosion study is a Dtodifica- tion of the plug model developed to track material from a reactor vessel to the containment building during an LMFBR core disruptive accident (CDA). That plug model is described by Bell, A description also is planned for revision 2 of ths SIMMER-II manu.-il. The modifications made to this plug model to create the present lower head model are (a) transforming the boundary treated from the upper to the lower interface, (b) making the hmindary of the movir.r structure nonuniform as a function of radius, (c) inserting a single-degree-of-freedom head-failure model based on a correlation to results calculated with the ADINA code, (d) generalizing a radial continuative inflow/outflow boundary condition for material release in the vessel's piping and in the cavity below the vessel, and (e) inserting new output statements for fores over the nonuniform upper head and for plotting other transient details.

B. Input A description of the model is given through an explanation of the input for a particular problem. The problem examined is the SIMMER-II structural setup for lower head failure shown in Fig. 32, with the input description in Table VIL From axial nodes 13 to 66 the structural setup was taken from the Zion/Indian-Point study. The features represented are the vessel's interior, the core support forging, the radial shield and cere barrel, the vessel wall, and the inlet and outlet piping. The extra input to represent the failure and motion of the lower head replaces the PLUG DATA, currently described by SIMMER-II input cards 85-88. In this problem, the movable lower structure is defined by setting ISO = 10, JSO = 12, and JPLUG(I) = 0,0,0,0,1,1,2,3,4,5. The lower head thus defined is modeled by Newton's second law with a constant mass, given as PLUGM, or 11 000 kg, on card 88. To monitor the forces on the upper head, an outline of upper head structure is also input. For this case, ISOT = 11, JSOT = 67, and JPLUGT(I) = 0,0,0,0,-1,-1,-2,-3,-4,-5,-6. In this model, the movable "structure" must be input mainly as "no-flow" volume in a special region parameter set. The "no-flow" volume must exceed the infinite drag limit, ALDRG. This means a small amount of cladding must be present, exceeding RSCLAD, which should be set to a small number. Because the expanding fluid from the vessel will bs highly two-phase (mostly vapor), only a

87 LOWER HEAD FAILURE PROBLEM ISOT H JSOT

OUTLET PIPING

SOLID STRUCTURE

12.54 m INLET PIPING DOWNCOMER

CORE SUPPORT FORGING

JSO

MOVABLE STRUCTURE

OUTLET TO KEYWAY

Fig. 32. Mesh and structure geometry for the quasi-mechanistic reactor steam- explosion problem. (Note: Nodes are expanded in the radial direction.)

88 TABLE VII

LOWER HEAD MODELING INPUT

Card No. 85 (FORMAT: 416) ISO, JSO, ISOT, JSOT

Columns Variable Description

1-6 ISO Right-hand boundary of the lower head. 7-12 JSO Top boundary of the lower head. 13-18 ISOT Right-hand boundary of the upper head. 19-24 JSOT Bottom boundary of the upper head.

Card No. 86 (FORMAT: 1016) (JPLUG(I), 1=1, ISO)

Columns Variable Description

1-60 JPLUG Radially dependent addition to JSO defining the top boundary of the lower head.

Card No. 87 (FORMAT: 1116) (JPLUGT(I), 1=1, ISOT)

Columns Variable Description

1-66 JPLUGT Radially dependent addition to JSOT defining the lower boundary of the upper head.

Card No, 88 (FORMAT: 1216) (JCONT(J), J=l, JBAR)

1-72 JCONT JCONT = 0 Rigid radial boundary condition at this axial location. JCONT = 1 Continuative inflow/outflow boundary condition at this axial location. 1-12 PLUGM Mass of lower head, kg.

89 minimal liquid-volume fraction is desired in the initial movable "structure" region. A saturated steam environment was considered reasonable for this class of problems. The JCONT(J) variable allows outflow (inflow) at specified nodes along the radial boundary. This is more completely explained in the section on boundary conditions. In the current problem, the three axial locations where JCONT(J) equals one and consequently allows motion are 1, 18 and 66.

C. Head Failure Before the lower head can move, it must fail. Finite element calculations (see App. M) indicate that failure can be expected at the radius where the outermost vessel penetrations are located. A circumferential split would be expected to proceed around the head at this radius and leave the inner portion of the head as a free body. A simple, single-degree-of-freedom spring-mass model was correlated to these finite-element results. This model accepts the average pressure loading over the the lower head, and furnishes the time and downward head velocity when the head disengages from the vessel. A summary description of this model is given in App. K. Other information useful for understanding the programming used in calling the failure model is as follows: (a) The pressure differential is computed assuming 0.1 MPa exists outside the vessel. (b) The average pressure used by the model has units of psi, displacements are in inches, and velocities in inches/s. (c) The failure flag is NFAIL. When NFAIL = 1, failure is indicated. The initial velocity of the moving head is the final velocity computed by the failure model, after conversion to meters/s.

D. Lower Head Motion As indicated in the input description, lower head motion following failure is computed by Newton's second law. The acceleration is computed from the mass of the head and the integrated forces acting on it. SIMMER-II computes the pressure at the upper surface of the lower head. This is the pressure in the axial node where the surface actually exists; or, to avoid over-expansion problems, one node above the surface level if the head occupies more than 80% of the local cell volume. An opposing pressure of 0.1 MPa is assumed to act on the

90 lower surface of the head. Gravity is included. The initial velocity comes from the head-failure model. The variable ZPLUG, defined at head failure as zero, is used to define the lower head displacement. Each radial node is given a bias, ZBIAS(I), relative to ZPLUG, from the input array JPLUG(I) and the axial mesh spacing. This locates the structure location, ZPLUG + ZBIAS(I), between axial interfaces given by ZTOP(I) and ZBOT(I). When the quantity ZPI.UG + ZBIAS(I) is determined to pass an interface, a flag (JPLUGS(I)) is set to begin a new node. Upward as well as downward motion of the lower head is permitted, although only downward motion is anticipated. Besides charging the interface indices ZTOP(I) and ZBOT(I), downward motion requires the fluid velocity at a former ZBOT(I) interface to be set equal to the lower head velocity when the head passes that interface. The array JPLUG(I) is redefined in the transient to represent the actual axial node occupied by the lower head interface. To avoid problems with zero indices, JPLUG(I) is not allowed to be decreased below 1. The lower head structure simply disappears when it reaches the bottom of the mesh. Figure 33 gives a pictorial view of this arrangement. As can be inferred from the above discussion, the lower head dynamics are coupled explicitly to the SIMMER-II fluid dynamics by having the no-flow volume time depenient. The SIMMER-II fluid dynamics then follows a natural course in which the fluid maintains contact to continue the acceleration. Numerical problems have been anticipated when the head has a high velocity and passes into a new cell. A time-step control is applied to help stabilize this problem. Using the quantity FDZ(I) defined in Fig. 33, a time-step cutback, prevents the fluid-volume fraction change caused by the moving head from being greater than 10% in one time step. Consequently, when the fluid-volume fraction is small, the time step will tend to be small. Further changes to the SIMMER-II pressure iteration (the solution of the momentum equations) are required to limit unphysical pressure reductions at the head/fluid interface. In the formulation of the momentum equations, the area over which the pressure gradient acts is the minimum area for the two half cells that form the momentum cell. This is shown in Fig. 34. SIMMER-II would use Aj for this area. As a result, the force driving fluid into cell JPLUG(I) is continually too small. The physical situations should be as shown on the left side

of Fig. 34 in which A2 equals Aj. The changes made in SIMMER-II provide for the maximum area from either cell JPLUG(I) or cell JPLUG(I) + 1 to be used.

ni Z TOP® FLUID

P LOWER HEAD BOUNDARY (ZPLUG + ZBIASQ))

STRUCTURE (NO-FLOW VOLUME) ZBOT0)

1-1 I-r 1

FDZ/JN = ZPLUG + ZBIASffl - ZBOTfl) ZTOPQ - ZBOT(I)

= LOWER HEAD VOLUME FRACTION

Fig. 33. Geometric indices for a radial node in the SIMMER-II moving lower head modelt

92 CELL I BOUNDARIES

\ r i r

HEAD , INTERFACE N

ENLARGING CHANNEL

JPLUG(I)

Fig. 34. SltoMER-II treatment of the fluid dynamics at the lower head interface.

Finally, with reference to Fig. 32, sideways venting to the lower cavity-keyway volume starts when the lower head is displaced by one node. The total uass of materials moved downward into the cavity volume can be obtained by analyzing of the results with the SINWER-II postprocessor.

E. Boundary Conditions and Edits The below-reactor cavity-keyway volume is highly three-dimensional and cannot be represented by SIMER-II. Flow relief exists, so a rigid wall boundary condition is incorrect. However, as a consequence of flow restrictions into the containment, a constant (low) pressure boundary condition will allow too much relief to take place. The option selected was the use of a continuative cutflow/inflow boundary condition. This condition/option was applied in three places on the radial boundary as shown in Fig. 32. Thes > places are at

93 the end of the inlet piping, the end of the outlet piping, and the outlet to the keyway. The manner in which this radial boundary condition was defined and applied is as follows. An input variable, JCONT(J), is read for each axial node. A value of zero means a rigid wall. A value of one means continuative outflow/inflow. When JCONT(J) = 1, the pressure in the radial boundary cell, p, ., is fixed over each time step to the beginning-of-time-step value in the cell to the left, or pu_i ;• In other respects the solution can proceed as if p. • were fixed by an input table. This device allows acceleration or decelera- tion of outflow/inflow momentum depending on the quantity (p^tj j " Pb 0' ^ this difference is unchanged, the outflow/inflow momentum will only change as a consequence of unbalanced momentum convection, friction, or other terms in the momentum equation. Consequently, once a high internal pressure has built up within the cavity (or piping), acceleration of fluid out of the radial boundary will cease. Five edits associated with the lower head failure model deserve mention. First, there is the edit of the input data. For the example problem in Fig. 32, this edit is given in Table VIII. The numbers correspond to those described in the input section. Second, before lower head failure, one extra line is printed each time step. An example is shown in Table IX. This line mainly contains data from the head-failure taodel, as XPLUG is displacement, VPLUG is the head velocity, PAVERG is the average pressure acting over the failure region, FPLUG is the force (in newtons) on the failure region, and FHEAD is the force (in newtons) on the upper head. According to the current criterion, lower head failure occurs when the displacement, XPLUG, reaches 5.0 in. The third edit is that occurring after failure. An example is given in Table X. This edit con- sists of two lines. The first line gives JPLUG(I), the axial nodes where the head interface is located. The second line gives additional information on head motion. In this case, the signs indicate the direction, and SI units are employed. The quantities are FPLUG, the net force (excluding gravity) acting on the lower head; APLUG, the lower head's acceleration; VPLUG, the lower head's velocity; ZPLUG, the lower head's displacement (initialized to zero at head failure); and FHEAD, the force on the upper head. The fourth edit is to a binary file, TAPE3, before lower head failure. The output line is NFAIL, T, PXPLOT, XPLUG, VPLUG, PAVERG, FHEAD, and FPLUG. The quantity NFAIL is the failure flag, which is zero for this case. The quantity T

94 TABLE VIII

PLUG DATA FOR BOTTOM/TOP HEAD CALCULATION

MASS OF LOWER HEAD FOR PLUG CALCULATION (PLUGM)= 1.1OOOE +04 RIGHT-HAND BOUNDARY OF THE PLUG (ISO)=» 10 TOP BOUNDARY OF THE PLUG (JSO) = 12 RIGHT-HAND BOUNDARY OF THE UPPER HEAD (ISOT)= 11 BOTTOM BOUNDARY OF THE UPPER HEAD (JSOT)= 67

AXIAL BIAS NODES AS A FUNCTION OF RADIUS 0000112345

AXIAL BIAS NODES FOR THE UPPER HEAD 0 0 0 0-1-1-2-3 -4 -5 -6

RADIAL CONTINUATIVE OUTFLOW ( INFLOW) BOUNDARY NODES

I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 1 0 0 0 0 0 0 0 0 0 0 0000001000000000000000000000

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 000000000000000000000000001 en

TABLE IX

SAMPLE OUTPUT EDIT BEFORE LOWER HEAD FAILURE

T= 7.O1259E-O1 CYCLE= 30 DT= 4.99523E-05 NUMINO 109 3 CELL= 0 ITR0= 0 ISTEP= 3 CT= 5.54O73E+OO DTL( 1)= 1 ,OOOOOE+20 DTL( 2)= 7.9 UO7E-O4 DTL( 3)= 4 .99523E-05 1= 7 J=26 DTL( 4)- 1.44739E-04 1= 5 J=23 DTL( 5)= 1 .O3536E-U DTL( 6)= 1.00000E-03 DTL( 7)= 6 .90241E-05 1= 4 J=27 DTL( 8)= 6.13778E-04 1= 8 J-24 DTL( 9)= l .OOOdOE+20 DTL(10) = 1.OOOOOE+20 DTL(11)= 1 .20962E-04 I- 5 J=23 DTL(12)= 5.O828OE-O3 1=11 J=23 PLUG DATA: XPLUO 4 .35992E-01 IN. VPLUO 1.O4296E+O3 IN/S PAVERO 1.01982E+04 PS1 FPLUO 5.7487OE+O8 FHEAD= 1.O5574E+O6 T= 7.O13O8E O1 CYCLB= 31 DT= 4.8913OE-O5 NL'MINO 110 ITP= 3 CELL= 0 ITRO= 0 ISTEP= 3 CT- 5.3 5297E+00 UTL( 1)= 1.00000E+20 DTL( 2)= 7.41155E-04 DTL( 3)= 4 .89130E-05 1= 7 J=27 DTL( 4)= 1.45508E-04 I- 3 J-21 DTL( 5)= 9.99045E-05 DTL( 6)= 1.00000E-03 DTL{ 7)= 6• 66030E-05 1= 5 J=24 DTL( 8)= 4.08316E-04 I- 9 J-25 DTL( 9)= 1 .OOOOOE+20 DTL(10) = 1.00000E+20 DTL(ll)= 3..84647E-04 £=• 9 J=25 DTL(12)= 6.61452E-03 1= 1 J=21 PLUG DATA: XPLUG= 4 .90653E-01 IN. VPLUO 1.14557E+03 IN/S PAVERO 1.03493E+O4 PSI FPLUO 5.83383E+08 FHEAD= 1.05581E+06 T= 7.01354E 01 CYCLE= 32 DT= 4.59729E-05 NUMINC= 110 ITP= 3 CELL= 0 ITRO 0 ISTEP= 3 CT= 5.38411E+00 DTL( 1)= 1 .OOOOOE+20 DTL( 2)= 6.92242E-04 DTL( 3)= 4.• 59729E-05 1= 7 J»27 DTL( 4)= 1.45776E-04 1= 3 J=21 DTL( 5)= 9 78261E-05 DTL( 6)= 1.00000E-03 DTL( 7)= t.52174E-0. 5 1= 1 J=22 DTL( 8)= 5.04903E-04 1= 9 J=25 DTL( 9)= 1 .OOOOOE+20 DTL(10) = 1.000OOE+20 DTL(11)= 3.9 72O4E-O4 1= 5 J=22 DTL(12)= 9.O4537E-O3 1= 7 J=20 PLUG DATA: XPLUO 5 .49196E-01 IN. VPLUO 1.24819E+03 IN/S PAVERO 1.06422E+04 PSI FPLUO 5.99898E+08 FHEAD= 1.05586E+06 T= 7.O1398E O1 CYCLE- 33 DT= 4.40846E-05 NUHINC= 112 ITP= 3 CELL= 0 ITRO 0 ISTEP= 3 CT" 5.39615E+O0 DTL( l)= 1 .OOOOOE+20 DTL( 2)= 6.46269E-04 DTL( 3)= 4•. 40846E-05 1= 7 J=27 DTL( 4)= 1.45981E-O4 1= 5 J=22 DTL( 5)= 9.19459E-0. 5 DTL( 6)= 1.00000E-03 DTL( 7) = 6.12973E-0. 5 1= 5 J=23 DTL( 8)= 4.33985E-04 1= 7 J=24 DTL( 9)= 1.G0000E+20 DTL(10)= 1.00000E+20 DTL(11)= 4.09790E-0. 4 1= 7 J=26 DTL(12)= 1.10611E-02 1= 2 J=21 PLUG DATA: XPLUG= 6.08866E-0. 1 IN. VPLUG- 1.34769E+03 IN/3 PAVERO 1.10244E+04 PSI FPLUO 6.21440E+08 FHEAD= 1.05590E+06 T= 7.O1441E O1 CYCLE= 34 DT= 4.27969E-05 NUMINO 112 ITP= 3 CELL= 0 ITRO 0 ISTEP= 3 CT= 5.29815E+00 DTL( 1)= 1.00000E+2. 0 DTL( 2)= 6.O2184E-04 DTL( 3)= 4.2 7969E-05 1= 7 J=27 DTL( 4)= 2.20904E-04 1= 8 J=18 DTL( 5)= 8.81692E-O. 5 DTL( 6)= l.OOOOOE-03 DTL( 7)= 5..87794E-05 I» 6 J=23 DTL( 8)= 5.33728E-04 1= 7 J=24 DTL( 9)= 1.00000E+2. 0 DTL(10)= 1.00000E+20 DTL(11) = 4..09971E-04 1= 7 J-26 DTL(12)= 1.12226E-02 1= 2 J=21 PLUG DATA: XPLUG= 6.7O467E-O. 1 IN. VPLUO 1.44698E+03 IN/S PAVERO 1.15262E+04 PSI FPLUO 6.49726E+O8 FHEAB= 1.05592E+06 T= 7.01483E- 01 CYCLE= 35 DT- 4.1914r!E-05 NUMINO 112 ITP= 3 CELL- 0 ITRO 0 ISTEP= 3 CT> 5.45580E+00 DTU 1)= 1.OOOOOE+2. 0 DTL( 2)= 5.59387E-04 DTL( 3)= 4..19142E-05 1= 7 J-27 DTL( 4)= 1.4 7607E-04 1= 5 J=22 DTL( 5)= 8.5 5939E-O5 DTL( 6)= 1.00000E-03 DTL( 7)= 5..70626E-05 I- 6 J=23 DTL( 8)= 4.94355E-04 1= 9 J=18 DTL( 9)= 1.OOOOOE+2. 0 DTL(10)= 1.OOOOOE+20 DTL(ll)- 3..55843E-04 I- 9 J=25 DTL(12)= 3.94931E-03 1= 5 J-23 PLUG DATA: XPLUO 7.34561E-0. 1 IN. VPLUO 1.5483OE+O3 IN/S PAVERO 1.21850E+04 PSI FPLUO 6.86863E+08 FHEAD= 1.05593E+06 4.13572E-05 NUMINO 112 ITP= T= 7.01524E- 01 CYCLE= 36 DT- 3 CELL= 0 ITR0= 0 ISTEP- 3 CT- 5.37331E+OO DTL( 2)= 5.17473E-O4 DTL( 3)= 4. DTU 1)= 1.OOOOOE+2. 0 .13572E-05 I» 7 J-27 DTL( 4)- 9.16051E-05 1= 1 J=21 DTU 6)= 1.00000E-03 DTL( 7)= 5. DTU 5)= 8. 38283E-05 58855E-05 I- 8 J-18 DTL( 8)= 6.47546E-05 1= 1 J=21 DTL(10)= 1.OOOOOE+20 DTL(11)= 4. DTU 9)- 1. OOOOOE+20 13887E-04 1= 9 J=19 DTL(12)= 4.86724E-O3 1= 4 J=16 IN. VPLUO 1.65325E+03 IN/S PLUG DATA: XPLUO 8.O1657E-O1 PAVERO 1.28992E+04 PSI FPLUO 7.27123E+08 FHEAD= 1.05592E+06 TABLE X

SAMPLE OUTPUT EDIT AFTER LOWER HEAD FAILURE T= 7.03641E-01 CYCLE= 87 DT= 2.54010E-05 NUMINO 121 ITP=> 3 CELL= 0 ITRO= 0 ISTEP= 7 CT= 2.3 5751E+OO DTL( 1)= 1.OOOOOE+20 DTL( 2)= 3.84229E-04 DTL( 3)=> 7 .40338E-05 1= 7 J=27 DTL( 4)= 8.28694E-05 1= 5 J=13 DTL( 5)= 3.81015E-05 DTL( 6)= 1.00000E-03 DTL( 7)= 2 .54010E-05 1° 3 J=14 DTL( 8)- 3.42714E-O5 1= 9 J»23 DTL( 9)= 1.OOOOOE+20 DTL(1O)= 1.OOOOOE+20 DTL(11) = 1 .02821E-04 1= 3 J=14 DTL(12)* 3.15950E-04 I- 5 J=12 PLUG DATA: JPLUGI= 12 12 12 12 12 12 13 14 15 17 FPLUG= -.14261E+09 APLUG= --12974E+05 VPLUG= -.17963E+03 ZPLUG= -.16358E+00 FHEAD= JO43OE+O7 T= 7.O3671E-O1 CYCLE= 88 DT= 2.971OOE-O5 NUMINC= 122 ITP= 3 CELL= 0 ITRO= 0 ISVKP= 8 CT» 5.71277E+OO DTU 1)= 1.OOOOOE+20 DTL( 2)= 3.58328E-04 DTL( 3)- 3 .0 2444E-05 1= 7 J=27 DTL( 4)= 5.A2986E-05 1= 5 J=28 DTL( 5)= 5.08020E-05 DTL( 6)= 1.00000E-03 DTL( 7)= 3,.38680E-05 1= 5 J=30 DTL( 8)= 2^71OOE-O5 1= 3 J=14 DTL( 9)= 1.OOOOOE+20 DTLUO)= 1.OOOOOE+20 DTL(11)= 7 .31387E-05 I« 5 J=28 DTL(12)= 2.fcl806E-03 1= 6 J=27 PLUG DATA: JPLUGI= 12 12 12 12 12 12 13 14 15 17 FPLUG= -.17688E+09 APLUG= -.16090E+05 VPLUG= -.18004E+03 ZPLUG= -.16815E+00 FHEAD= .104.7E+07 T= 7.03701E-01 CYCLE= 89 DT= 3.02895E-05 NUMINC= 122 ITP= 3 CELL= 0 ITR0= 0 ISTEP- 3 CT= 5.56995E+00 DTL( 1)= 1.OOOOOE+20 DTL( 2)= 3.29118E-04 DTL( 3)= 3,.02895E-05 1= 8 J=27 DTL( 4)= 1.04/UE-04 1= 5 J=28 DTL( 5)= 5.94200E-05 DTL( 6)= 1.00000E-03 DTL( 7)= 3..96133E-05 1= 5 J=29 DTL( 8)= 4.52857E-05 1= 3 J=14 DTL( 9)= 1.OOOOOE+20 DTL(10)= 1.OOOOOE+20 DTL(11)= 1..48345E-04 1=10 J=17 DTL(12)= 1.18448E-03 1= 7 J=13 PLUG DATA: JPLUGI= 12 12 12 12 12 12 13 14 15 17 FPLUG= -.24538E+09 APLUG= -.22317E+05 VPLUG= -.18070E+03 ZPLUG= -.17351E+00 FHEAD= .10425E+07 T= 7.O3731E-O1 CYCLE= 90 D1>= 3.03O61E-05 NUMIMO 123 ITP= 3 CELL= 0 ITR0= 0 ISTEP= 3 CT= 5.56712E+OO DTL( 1)= 1.OOOOOE+20 DTL( 2)= 2.98823E-04 DTL( 3)= 3,• 03061E-05 1= 3 J=27 DTL( 4)= 1.05773E-04 1= 5 J=28 DTL( 5)= 6.05789E-05 DTL( 6)= 1.00000E-03 DTL( 7)= 4. 03859E-05 1= 5 J=28 DTL( 8)= 7.62297E-05 1= 9 J=15 DTL( 9)= 1.00000E+20 DTL(10)= 1.00000E+20 DTL(11)= 2, 13984E-04 1= 9 J=31 DTL(12)= 2.44220E-0J 1= 9 J=15 PLUG DATA: JPLUGI= 12 12 12 12 12 12 13 14 15 17 FPLUG= -.42679E+09 APLUG= -.38809E+05 VPLUG= -.18188E+03 ZPLUG= -.17900E+00 FHEAD= .10422E+07 T= 7.03751E-01 CYCLE= 91 DT= 1.93693E-05 NUMINC= 122 ITP= 3 CELL= 0 ITRO= 0 ISTEP= 8 CT= 5.48010E+00 DTL( 1)= 1.OOOOOE+20 DTL( 2)= 2.68522E-04 DTL( 3)= 3.03411E-05 1= 8 J=27 DTL( 4)= 1.26733E-04 I=iO J=27 DTL( 5)= 6.06122E-05 DTL( 6)= 1.00000E-03 DTL( 7)= 4.04081E-05 1= 5 J=29 DTL( 8)= 1.93693E-05 1= 5 J=12 DTL( 9)= 1.00000E+20 DTL(10)= 1.00000E+20 DTL(11)= 1.58564E-04 1= 9 J=16 DTL(12)= 1.23808E-03 1= 5 J=12 PLUG DATA: JPLUGI= 12 12 12 12 12 12 13 14 15 17 FPLUG= -.5 5908E+09 APLUG= -.50835E+05 VPLUG= -.18342E+03 ZPLUG= -.18454E+00 FHEAD= .10420E+07 T= 7.O3765E-O1 CYCLE= 92 DT= 1.45636E-05 NUMINC= 122 ITP= 3 C2LL= 0 ITR0= 0 ISTEP= 8 CT= 5.5 9 308E+00 DTL( 1)= 1.00000E+20 DTL( 2)= 2.49153E-04 DTL( 3)= 3.O3772E-O5 1= 8 J=27 DTL( 4)= 9.21114E-05 1= 2 J=21 DTL( 5)= 3.87386E-05 DTL( 6)= 1.00000E-03 DTL( 7)= 2. 58258E-05 1=10 J=18 DTL( 8) = 1.4 5636E-O5 1= 5 J-12 DTL( 9)= 1.OOOOOE+20 DTL(10)= 1.00000E+20 DTL(11)= 5. 13908E-05 1= 2 .T=21 DTL(12) = 1.71904E-03 1= 3 J=14 PLUG DATA: JPLUGI= 12 12 12 12 1.2 12 13 14 15 16 FPLUG= -.27910E+09 APLUG= -.25383E+05 VPLUG= -.18391E+03 ZPLUG= -.18809E+0C FHEAD= .1O419E+O7 is the time in seconds. The quantity PXPLOT is an array of 13 pressures for future plotting purposes. These pressures have been set by data statements to coirespond to specific nodes in the problem of Fig. 32. These nodes are listed in Table XI. The remaining variables are as defined under the prefailure print edit. The fifth edit is to a binary file, TAPE3, after lower head failure. The output line here is T, PLPLOT, FHEAD. FPLUG, and ZPLUG. These variables have all been previously defined.

F. Correction Set Table XII contains a copy of the correction set that embodies the modifications described. When the program library is updated using this correction set, the PLUG option mus.t be defined to activate the modifications. If the PLUG option is not defined, the standard SI&MER-II code will be obtained. Variables added to common blocks are defined in App. L. Other variables used are C. R. Bell's original correction set, and are defined in the new SIMMER-II manual.

G, Sa pie Problem Fluids were added to the configuration described in the structural diagram in Fig. 32. The resulting problem was run. Output figures and a discussion were first given by this author in App. C of the Steam-Explosion Review Group (SERG) report. The SERG report is a compilation of opinions written by various steam-explosion "experts." Contributions and correspondence reproduced in the SERG report and referenced here will be attributed to the salient expert

TABLE XI DEFINITION OF THE PXPLOT ARRAY

PXPLOT (1) ) represents (15,1) Cavity bottom (2) M (1.13) Inlet plenum bottom (3) M (9,17) 45° along bottom (4) t? (15,18) Inlet pipe end (5) M (1,22) Above core support (6) If (13,25) 90° along bottom (7) 11 (13,38) Side of vessel at 4.4 M (8) II (11,47) Shroud boundary (9) || (13,47) Downcomer end (10) || (11,60) Vessel flange (ID || (6,65) Head curvature (12) II (1,66) Top of vessel (13) II (15,66) ouLlet pipe end (i,j) are external dimensions from Fig. 32.

98 TABLE XII

CORRECTION SET FOR THE SIMMER-II LOWER HEAD MODEL

1 *IDENT BAA4 2 */ 3 */ CHANGE THE COMMON BLOCKS FOR THIS PROBLEM 4 */ 5 *D E1M9.4.5 6 *,PLUGM,ZPLUG,APLUG,FPLUG,VPLUG,FDZ(ll),ZBOT(ll),ZTOP(ll) 7 *,ZBIAS(11),FHEAD,XPLUG,PAVERG 8 *D E1M9.7.8 9 *,JSO,ISO,JSOT,ISOT,JPLUG(ll),JPLUGS(ll),JPLUGT(12),JCONT(70) 10 *,NFAIL 11 V 12 */ CHANGE THE INPUT FOR THE PLUG PROBLEM 13 */ 14 *D E1M9.19 15 *IF DEF.PLUG.ll 16 *D E1M9.24,25 17 READ (NINP.831) ISO,JSO,ISOT,JSOT 18 ISOP=ISO+1 19 ISOTP = ISOT + 1 20 READ CNINP.831) (JPLUG(I),I=2,ISOP) 21 READ (NINP.831) (JPLUGT(I),1=2,ISOTP) 22 READ (NINP.831) (JCONT(J),J=2,JP1) 23 */ 24 V CHANGE THE OUTPUT EDIT 25 */ 26 *D E1M9.32,34 27 WRITE (KT.7803) ISO,JSO, ISOT,JSOT 28 ISOP = ISO + 1 29 WRITE (KT.7804) (JPLUG(I),I=2,ISOP) 30 ISOTP = ISOT + 1 31 WRITE (KT.78O5) (JPLUGT(I),1=2,ISOTP) 32 WRITE (KT.78O6) (J,J=1,39),(JCONT(J),J=2,40) 33 WRITE (KT.7807) (J,J=40,JBAR) 34 WRITE (KT.78O7) (JCONT(J),J=41,JP1) 35 WRITE (KT.78O8) 36 7808 FORMAT (1H0.1X//) 37 7802 FORMAT (//19X.41HPLUG DATA FOR BOTTOM/TOP HEAD CALCULATION//) 38 *D E1M9. 35,43 39 7803 FORMAT(3X,31HRIGHT HAND BOUNDARY OF THE PLUG,39X,6H(ISO)=, 112/ 40 *3X,24HTOP BOUNDARY OF THE PLUG,46X,6H(JS0)=I12/ 41 *3X,37HRIGHT HAND BOUNDARY OF THE UPPER HEAD,32X,7H(ISOT)=,112/ 42 *3X,33HBOTTOM BOUNDARY OF THE UPPER HEAD,36X,7H(JSOT)=,112//) 43 *D E1M9.44,55 44 7804 FORMAT (3X.40HAXIAL BIAS NODES AS A FUNCTION OF RADIUS,3X, 1013//) 45 7805 FORMAT (3X,35HAXIAL BIAS NODES FOR THE UPPER HEAD,8X, 1113///) 46 7806 FORMAT(12X,52H RADIAL CONTINUATIVE OUTFLOW(INFLOW) BOUNDARY NODES 47 1//3X.39I3/3X.39I3//) 48 7807 FORMAT (3X.40I3)

99 TABLE XII (CONT.)

49 */ 50 */ INITIALIZATION MODIFICATIONS 51 */ 52 *D E1M9.61.62 53 ISO - ISO + 1 54 JSO - JSO + 1 55 ISOT - ISOT + 1 56 JSOT - JSOT + 1 57 NFAIL = 0 58 *D E1M9.63.73 59 *D PC13.7 60 *D E1M9.80 61 XPLUG - ZERO 62 *D PC13.8 63 *I SETI.324 64 *1F DEF.PLUG t5 */ 66 V INITIALIZE 1MESH DEPENDENT PLUG VARIABLES 67 */ 68 DO 2055 1=2,ISO 69 JPLUGI - JPLUG(I) 70 ZBIAS(I) - ZERO 71 IF (JFLUGI .EQ. 0) GO TO 2045 72 DO 2050 N=l,JPLUGI 73 2050 ZBIAS(I) - ZBIAS(I) + DZ(JSO+N) 74 2045 CONTINUE 75 ZTOP(I) - ZBIAS(I) 76 ZBOT(I) - ZTOP(I) - DZ(JSO+JPLUGI) 77 JPLUG(I) - JSO + JPLUGI 78 FDZ(I) - ONE 79 JPLUGSCI) - 0 80 2055 CONTINUE 81 DO 2056 I=2,IS0T 82 JPLUGT(I) - JSOT + JPLUGT(I) 83 2056 CONTINUE 84 *ENDIF 85 */ 86 */ XCHANJ MODS 87 */ 88 •/ MODIFICATIONS TO PLUG MOTION 89 •/ 90 *D E1M9.214,215 91 IF (I .GT. ISO) GO TO 2002 92 IF (J .NE. JPLUG(I)) GO TO 2002 93 *D PC13.33,34 94 IF (FDZ(I) .GT. CP8) PPLUG = P(IJP) - EP5 95 IF (FIJL(IJ) .GE. AKINFI) PPLUG - P(IJP) - EP5 96 *D PC13.39 97 IF (NFAIL .EQ. 1) GO TO 3001 98 PAVEOD-PAVERG 99 PAVERG . FPLUG/(PIE*RIP(IS0)*RIP(ISO))

100 TABLE XII (CONT.)

100 PAVERG = PAVERG*1.45074E-04 101 IF (NCYC .EQ. 0) GO TO 3002 102 CALL SMQDCXPLUG.VPLUG,PAVERG.PAVEOD.OTKACH.NFAIL) 103 IF (NFAIL .EQ, 0) GO TO 2000 104 VPLUG - -VPLUG/39. 371 105 3001 CONTINUE 106 *B E1M9.226 107 FPLUG - -FPLUG 108 *D E1M9.233,245 109 C DO EACH RADIAL MESH SEPARATELY 110 C 111 *IF DEF.DBL.l 112 VPLUGI - ONE/DABS(VPLUG^ 113 *IF -DEF.DBL.l 114 VPLUGI - ONE/ABSCVPLUG) 115 DO 2010 I-2,ISO 116 IF (ZPLUG+ZBIAS(I) .LT. ZBOT(I)) GO TO 2003 117 IF (ZPLUG^ZBIAS(I) .LE. ZTOP(I)) GO TO 2001 118 ZBOT(I) - ZTOP(I) 119 JPLUGI « JPLUG(I) + 1 120 JPLUG(I) = JPLUGI 121 ZTOP(I) -> ZBOTCI) + DZ(JPLUGI) 122 JPLUGS(I) « 1 123 GO TO 2001 124 2003 JPLUGI - JPLUG(I) - 1 125 JPLUGI » MAXO(JPLUGI ,1) 126 JPLUG(I) - JPLUGI 127 ZTOP(I) -• ZBOTfn 128 ZBOT(I) =• ZBOTCI) - DZ(JPLUGI) 129 JPLUGS(I)i - 2 130 2001 CONTINUE 131 •D E1M9.2S0 132 JP'UGI - JPLUG(I) 133 FDZ(I) - ZERO 134 IF (JPLUGI .GT. 1) 135 *FDZ(I) - (ZPLUG+ZBIASCI)•ZBQTCI))*RDZ(JPLUGI) 136 *IF DEF.DBL,! 137 FDZ(I) « DMAXKFDZ(I).ZERO) 138 *IF -DEF.DBL.l 139 FDZ(I) - AMAX1(FDZ(I),ZERO) 140 *D PC13.41.45 141 »D E1M9.252 142 DTNN - (ONE-FDZ(I)«ALNOFL(IREG))*EM1*VPLUGI 143 *IF DEF.DBL.l 144 DTN - DMINl(DTN.DTNN) 145 *IF -DEF.DBL.l 146 DTN - AMIN1 (DTN.DTNN) 147 2010 CONTINUE 148 */ 149 */ EXFLUD MODIFICATIONS IN DEFINING A NEW PLUG INTERFACE NODE 150 V

101 TABLE XII (CONT.)

151 *D E1M9.124,125 152 IF (JPLUG(I) .EQ. 1 .AND. J .EQ. 2) ANOFL(IJ) = ZERO 153 IF (I .GT. ISO) GO TO 1OOO 154 IF (J .NE. JPLUG(I)) GO TO 1000 155 *D PC13.12 156 *D E1M9.127 157 *D PC13.13 158 *D E1M9.129.13O 159 ANOFL(IJM) = ONE 160 ANOFL(IJ) = FDZ(I) 161 ANOFL(IJP) = ZERO 162 IF (JPLUGS(I) .EQ. 0) GO TO 1000 163 IF (JPLUGS(I) .EQ. 2) GO TO 1001 164 *D PC13.14 165 *D E1M9.134 166 JPLUGS(I) = 0 167 «D E1M9.138 168 ANOFL(IJ)=ONE+VPLUG*DT*RDZ(J) 169 JPLUGS(I) = 0 170 */ 171 */ IMFLUD MODIFICATIONS 172 */ 173 *D E1M9.146,147 174 IF (I .GT. ISO) GO TO 1002 175 IF (J-l .EQ. JPLUG(I)) 176 *D E1M9.154,155 177 IF (I .GT. ISO) GO TO 1001 178 IF (J .EQ. JPLUG(I)) 179 *D E1M9.159.16O 180 IF (I .GT. ISO) GO TO 1000 181 IF (J .EQ. JPLUG(I)) 182 *D E1M9.163 183 «D PC13.27,29 184 *B IMFL.1559 185 *IF -DEF.PLUG.l 186 *I IMFL.1559 187 *IF DEF.PLUG.l 188 IF (J .NE. JP1) GO TO 3039 189 »D E1M9.169,177 190 *IF DEF.PLUG '.91 C 192 C CALCULATE FORCE ON THE HEAD 193 C 194 3039 CONTINUE 195 AREA = RAREA(I)*DT 196 IF (I .GT. ISOT) GO TO 3040 197 IF (J+l .NE. JPLUGT(I)) GO TO 3040 198 'ENDIF 199 *D E1K9.x78 200 *D E1M9.180,205

102 TABLE XII (CONT.)

201 *I E1M9.207 202 *IF -DEF.PL'JG.l 203 *I IMFL.1603 204 *IF DEF.PLUG.l 205 PTPTMP = PTPTMP/(RIP(IS0T)**2*PIE) 206 V 207 */ HYDRO MODIFICATIONS 208 */ 209 *D E1M9.91 210 *IF DEF.PLUG.20 211 *D E1M9.95,99 212 IF (NFAIL .EQ. 1) GO TO 4176 213 WRITE (N0FS.4174) XPLUG.VPLUG.PAVERG.FPLUG.FHEAD 214 4174 FORMAT (12H PLUG DATA:,9H XPLUG=,1PE12.S.4H IN., 215 1 9H VPLUG=,1PE12.5,5H IN/S.10H PAVERG=,1PE12.5,4H PSI, 216 2 9H FPLUG=,1PE12.5,,9H FHEAD=,1PE12.5) 217 GO TO 4178 218 4176 CONTINUE 219 DO 4172 1=2,ISO 220 JPLUGS(I)=JPLUG(I)-1 221 4172 CONTINUE 222 WRITE (N0FS,4171)(JPLUGS(I),I=2,IS0),FPLUG,APLUG,VPLUG,ZPLUG,FHEAD 223 4171 F0RMAT(12H PLUG DATA:,10H JPLUGI=,1013/ 224 *D E1M9.101,105 225 »,E12.5,9H FHEAD=,E12.5) 226 4178 CONTINUE 227 DOO 4173 1=2,IS1=2 O 228 JPLUGS(I) - 0 229 4173 CONTINUE 230 C 231 */ 232 */ CHANGE CONTINUOUS INFLOW/OUTFLOW BOUNDARY CONDITIONS 233 */ 234 *B IMFL.70 235 *IF -DEF.PLUG.l 236 *I IMFL.70 237 *IF DEF.PLUG.l 238 IF (I.EQ.IP1 ,AND. JCONT(J).EQ.O) GO TO 2015 239 *B IMFL.499 240 *IF -DEF.PLUG.l 241 *I IMFL.499 242 *IF DEF.PLUG.l 243 IF (I.EQ.IP1 ,AND. JCONT(J).EQ.O) GO TO 2230 244 *B B1C0.26 245 *IF -DEF.PLUG.l 246 *I B1C0.26 247 *IF DEF.PLUG.l 248 IF (I.EQ.IP1 .AND. JCONT(J) .EQ. 0) 249 *B IMFL.864 250 *IF -DEF.PLUG.l 251 «I IMFL.864 252 *IF DEF.PLUG.l

103 TABLE XII (CONT. )

253 IF (I.EQ.IP1 .AND. JCONT(J).EQ.O) ROLBRT(N+NQ)=ROLBRT(N) 254 «B IMFL. 870 255 *IF -DEF.PLUG.l 256 *I IMFL. 870 257 *IF DEF.PLUG.l 258 IF (I.EQ.IP1 .AND. JCONT(J).EQ.O) ROGBRT(N+NQ)=ROGBRT(N) 259 *B IMFL.908 260 *IF -DEF.PLUG.l 261 *I IMFL. 908 262 *IF DEF.PLUG.l 263 IF (I.EQ.IP1 .AND. JCX)NT(J).EQ.O) ROLBRT(IJ1P+NQ)=ROLBRT(IJP) 264 «B IMFL. 913 265 *IF -DEF.PLUG.l 266 *I IMFL.913 267 *IF DEF.PLUG.l 268 IF (I.EQ.IP1 • AND. JO0NT(J).EQ,,0) ROGBRT(IJP+NQ)=R0GBRT(IJP) 269 *B B1C0.28 270 *IF -DEF.PLUG.2 271 *I IMFL. 922 272 *IF DEF.PLUG.2 273 IF (JCONT(J) EQ. 0) P(IPJ)=P(IJ) 274 IF (JCONT(J) ,EQ. 0) GO TO 2550 275 *B B8D4.2 276 *IF -DEF.PLUG.l 277 *I B8D4.2 278 *IF DEF.PLUG.l 279 IF (I.EQ.IP1 ,AND. JCONT(J) .EQ. 0) P(IPJ)«P(IJ) 280 *B IMFL. 1523 281 *IF -DEF.PLUG.l 282 *I IMFL. 1523 283 *IF DEF.PLUG.l 284 IF (JCONT(J) . EQ. 0) GO TO 3006 285 *B IMFL.1503 286 »IF -DEF.PLUG.l 287 *I IMFL. 1503 288 *IF DEF.PLUG.l 289 IF ((IVIS.EQ.O) .OR. (JCONT(J) . EQ. 1)) GO TO 3015 290 *B IMFL. 1512 291 *IF -DEF.PLUG.l 292 *I IMFL. 1512 293 *IF DEF.PLUG.l 294 IF ((IVIS.EQ.O) .OR. (JCONT(J). EQ. 1)) GO TO 3018 295 *B EXFL. 1395 296 *IF -DEF.PLUG.l 297 «I EXFL. 1395 298 *IF DEF.PLUG.l 299 IF (JCONT(J).EQ. 1) GO TO 1504 300 *B RINP.566 301 *IF -DEF.PLUG.l

104 TABLE XII (CONT.)

302 *I RINP.566 303 *IF DEF.PLUG.l 304 IF (JCONT(J) .EQ. 1) GO TO 342 305 *B SETI.364 306 *IF -DEF.PLUG.l 307 *I SETI.364 308 *IF DEF.PLUG.l 309 IF (JCONT(J) -EQ. 1) RCONT(J)=ONE 310 */ 311 */ ADD SPECIAL VARIABLES FOR PLOTTING 312 V 313 *I XCHA.26 314 *IF DEF.PLUG.2 315 DIMENSION PXPL0T(13) 316 DIMENSION IXPL0T(14),JXPL0T(14) 317 *B XCHA.31 318 *IF DEF.PLUG.2 319 DATA IXPLOT/15,1,9,15,1,13,13,11,13,11,6,1,15,0/ 320 DATA JXPLOT/1,13,17,18,22,25,38,47,47,60,65,66,66,0/ 321 *I XCHA. 36 322 *IF DEF.PLUG.l 323 ICPLUG = 1 324 *I B989.172 325 *IF DEF.PLUG 326 IF (1-1 .NE. IXPLOTCICPLUG) .OR. J-l.NE.JXPLOT(ICPLUG)) 327 1G0 TO 4170 328 PXPLOT(ICPLUG)=P(IJ) 329 ICPLUG=ICPLUG+1 330 4170 CONTINUE 331 *ENDIF 332 *I 0V00.7 333 *IF DEF.PLUG.l 334 *.TAPE3 335 *I E1M9.253 336 IF (NFAIL .EQ. 0) WRITE (3) 337 1 NFAIL,T,PXPLOT,XPLUG,VPLUG,PAVERG,FHEAD,FPLUG 338 IF (NFAIL .EQ. 1) WRITE (3) T.PXPLOT.FHEAD.FPLUG.ZPLUG 339 3002 CONTINUE 340 »D E1M9.26 341 READ (NINP,811) PLUGM 342 *I XCHA.632 343 *IF DEF.PLUG 344 */ 345 */ PUT IN SUBROUTINE TO CALCULATE FAILURE 346 */ 347 SUBROUTINE SM0D(X,V,FTN,FTO,DT,NFAIL) 348 DATA SKO.AMO.PSKO.AMOP /33694. ,0. 01456,187.,0.00996/ 349 C THIS SUBROUTINE ASSUMES NO UNLOADING AND ZERO INITIAL CONDITIONS 350 CNF2=SKO/AM0 351 XOLD=X 352 VOLD=V

105 TABLE XII (CONT.)

353 IF(XOLD.GT.0.2) SKO=PSKO 354 IF(XO!^ GT.0.2) AMO=AMOP 355 IFCXOLD.GT.0.2) CNF2-SKO/AMO 356 IF(ABS(X0LD).GT.5.0) GO TO 2005 357 ANEW=-CNF2*XOLD+FTO/AMO 358 AOLD=ANEW 359 1001 AIP1=ANEW 360 VNEW=VOLD+(AOLD+AIP1) *DT*0. 5 361 XNEW=XOLD+VOLD*DT+O. 25*A0LD*DT**2+0, 25*AIP1*DT**2 362 ANEW=-CNF2*XNEW+FTN/AMO 363 IF(ABS(ANEW-AIP1).GT.1.E-7*ABS(AIP1)) GO TO 1001 364 C UPDATE VARIABLES 365 V=VNEW 366 X=XNEW 367 RETURN 368 2005 NFAIL=1 369 V=VOLD 370 X=XOLD 371 RETURN 372 END 373 *ENDIF 374 *IDENT BCX4 375 */ 376 */ ADDITIONAL PRINT STATEMENT 377 */ 378 *B BAA4.15 379 WRITE (KT.7809) PLUGM 380 7809 FORMAT (3X.39HMASS OF LOWER HEAD FOR PLUG CALCULATION,29X, 381 1 8H(PLUGM)=,1PE12.4) when possible. Additional details regarding the model are in Apps. K through N. Further application of the model is described ir Chap. VI. The particular problem presented in the SERG report is in Sec. VLB. 3.

106 IV. CORRELATION TO SNL STEAM-EXPLOSION EXPERIMENTS

Thrse initial observations can be made regarding the SNL tests. First, many experiments were conducted. There are more than 75 FITS tests at intermediate scale. Second, there is a significant stochastic component present. Later tests have not tended to reproduce earlier results. Third, detailed time-dependent data on material volume fractions and size distributions are beyond the capability of the instrumentation employed (cameras and pressure transducers). Because of the lack of precision and reproducibility in the experimental results as well as the lack of models known to represent the correct phenomenology, the ability to correlate to experimental data is limited. For the purposes of this study, test MD-19 was selected as representative. Some characteristic results of other tests are discussed in App. 0. Test MD-19 produced well-characterized pressuie data and has previously been analyzed by Mitchell using the SNL wavecode CSQ, which calculates mass, momentum, internal energy, and kinetic energy fluxes between Eulerian cells. Calculations simulating test MD-19 also were performed by this author using a three-velocity field algorithm, and by Oh using a parametric nonequilibrium model. The present limited comparison with SIMMER-II results is judged to be useful for obtaining initial estimates of model parameters, and for gaining an appreciation of the type and magnitude of discrepancies between SIMMER-II calculations and experimental data. The overall geometric setup for this test is shown in Fig. 35. This was a test in which 5.11 kg of iron-alumina thermite were dropped into 224 kg of water contained in a Lucite tank, shown in Fig. 36. The configuration at the time of th« explosion (the initial SIMMER-II configuration) was taken from Mitchell et A 1 al. They report a mixture volume of 0.032 m , a mixture diameter of 0.42 m, and a mixture melt density of 159 kg/m at the time of the explosion. Assuming that (1) equal volumes of steam and water exist in the exploding mixture as has been suggested by Oh and Corradini, and that (2) a two-phase (50% steam/50% water) chimney exists above the interaction zone, the assumed coarse-mixed configuration for this test is shown in Fig. 37. The water temperature was taken as 299 K, the thermite temperature was taken as 3 000 K, and constant-pressure boundary conditions of 83 kPa were used at the top and right-hand boundaries. The thermite drop size was taken as 0.3 mm in diameter corresponding to the peak in the debris distribution from tests FITS3A and FITS5A. Also, this debris size is consistent with that used in the previous ZIP study.*

107 AIR CYLINDER DELIVERY)

TEST STAND

Fig. 35. EXO-FITS apparatus.

108 CAMERA REF. (STADIA)

MELT VELOCITY SENSORS

TRANSPARENT WATER CHAMBER TEMP T.C. WATER LEVEL ( SWELL MEAS.^ (OPTICAL) WATER PHASE PRESSURE TRANSDUCER

EXTERNAL^ TRIGGER RIGID SUPPORT (SE-1DET FOR WATER CHAMBER (in some tests) or quartz pressure guag© CHAMBER .CHAMBER BASE (PLEXIGLASS)

12.7mm

WATER CHAMBER SUPPORT

DETONATOR

DETONATOR MOUNTING (NOT USED IN ALL TESTS)

Fig. 36. Typical instrumented water chamber.

109 50% WATER

100% 50% STEAM 100% WATER WATER o 2

4 T 48% WATER E 48% STEAM CO 2_ 4% THERMITE O 'l ll

|-— 0.21 m —H

• 0.34 m -

1 2 3 4 5 6

RADIAL

Fig. 37. Initial configuration for the SIMMER-II representation of test MD-19.

110 The main data used for comparison were the water chamber's base pressure reproduced in Fig- 38. The location of the transducer is directly below the explosion.* The two parameters varied in the calculations were the water-droplet radius and the heat-transfer coefficient multiplier governing thermite droplet to water surface heat transfer. The water-droplet size controls the surface area of the water. For each droplet size the goal was to reach an approximate peak pressure of 17.5 MPa in the base node (1,1). Once this was obtained, the calculations were examined for the desired width of the pressure pulse at 8 MPa of approximately 0.35 ms, The results of these parametric variations are shown in Table XIII. These calculations suggest that a best fit might be achievable

*M. L. Corradini, University of Wisconsin (June 4, 1984).

too

CO < 100 - CO

0.00200 0.00400 TIME (s) Fig. 38. Base pressure of the water chamber in experiment MD-19. Ill TABLE XIII INITIAL CALCULATIONS FOR FITTING TEST MD-19

Water-Droplet Heat-Transfer Peak Pulse Width (ms) Diameter (am) Multiplier Pressure (MPa) at 8 MPa

300 1 24.3 0.1 8.1 0.5 19.5 1.2

150 1 28.7 0.5 13.0 0.7 16.9 1.0

75 1.0 10.6 5.0 27.1 2.5 23.9 1.8 21.4 1.7 17.4 0.47

20 0.5 4.4 10.0 19.6 0.10

for a water-droplet diameter between 0.02 and 0.075 mm. Unfortunately, an equilibration effect influences the results at these low water-droplet sizes. Figure 39 gives the pressure trace for the 0.075-mm case. The high-pressure tail resulting from bulk water heating docs not appear in the experimental results. This tail becomes worse for the 0.02-mm case because temperature equilibration is then much faster than any expansion eflict. The SIMMER-II input for this case is given in App. P. A partial improvement on the above results is to specify a nonuniform fuel distribution. An experimental distribution is not available. Consequently, a calculation must be used. The computation selected was that done in a previous study, demonstrating the feasibility of a threerfield calculation of the coarse premising phase of a steam explosion. Additionally, some SIMMER-II premixing calculations were performed. These calculations should be helpful in extrapo- l_.ing to the reactor configuration and larger scale experiments. They are reported in App. Q and compared with the thres-field and experimental results. The thermite distribution from the three-field calculation as well as some experimental results are shown in Fig. 40. The node with the peak fuel density is considerably more fuel rich (27% thermite, 36% water, and 37% vapor) than the smeared distribution of Fig. 37.

112 20.0-i PRESSURE AT THE BASE

16.0-

(0 Q. 12.0-

D 10 8.0- a.

4.0-

0.0 H o.o 1.0 2.0 3.0 4.0 5.0 TIME (ms)

Fig. 39. Best fit for a uniform iteration zone (water-droplet size 75 i/M't heat-transfer multiplier 1.7).

For the explosion calculations a 0.3-mm fuel-droplet diameter and a 0.075-mm water-droplet diameter were used. Peak pressures calculated as a function of the heat-transfer multiplier are shown in Fig. 41. A multiplier of 0.2 appeared to be adequate. A plot of pressure vs time in the base node for this case is shown in Fig. 42. The width of the pressure pulse at 8 MPa is approximately 0.39 ms, which is reasonable. Data from the water phase pressure gauges is shown in Fig. 43. The calculation at a similar location gives an initial pressure rise of about the same magnitude (Fig. 44). Time delays are irrelevant, becauss no attempt was made to calculate propagation of a detonation wave.

The reported conversion ratio estimate (1.3-2.0%) is not directly comparable to the SlkSAER-II calculations (0.5% for the nonuniform case). First, SIMMER-II used a constant-pressure boundary condition at the sides of the tank. Fluid could not be accelerated after exiting the problem boundaries. Second, one reported method of obtaining the experimental conversion ratio was to integrate the base-pressuTe transducer curve. The experimental value (from Fig. 38)

113 ORIGMAL WATER LEVEL

L, EXPLOSION WAVE POSITIONS GAGE P3 LOCATION I^WATER CHAMBER MELT-WATER MIXTURE OUVUNE GAGEP2 0.61 H LOCATION CHAMBER BASE

Original Water Level

16 fl MD-18 Thermi te Vo I time Fraction

Fig. 40. Initial thermite distribution for the three-field calculation compared with the reported experimental results.

114 0.2 0.6 1.0 1.4 1.8 HEAT-TRANSFER MULTIPLIER Fig. 41. Sensitivity of the peak pressure to the heat-transfer multiplier for a 0.3-mm fuel droplet and a 0.075-mm water droplet.

1.0 2.0 TIME (MS) Fig. 42. Best fit starting from a nonuniform interaction zone with a 0.3-mm fuel droplet and a 0.075-mm water droplet. 115 09 S SD

.00200 .00400 .00200 .00400 TIDE M SECONDS TME I* SECONDS (a) GAGP P2 (b) GA4SE P3

Fig. 43. Water phase pressure for experiment MD-19. is at most 8 000 (N«s)/m2. The calculated value (from Fig. 42) is about 8 700 (N«s)/m2. If the experimental value is multiplied by the entire base area (0.36 m ), which is unrealistic because the measurement represents only one point, the axial impulse is less than 3 000 N's. The value reported by the experimentalists doing this exercise is 10 000 N*s. Because the kinetic energy goes as the square of the impulse, the reported kinetic energy based on this calculation appears suspicious. The SIMMER-11 input for the case shown in Figs. 42 and 44 is given in App. T. This fitting exercise suggests that reasonable base-case parameters for use in a reactor study should be a fuel-droplet diameter of 0.3 mm, a water-droplet diameter of 0.075 mm, and a heat-transfer multiplier of 0.2. The case with the uniformly lean fuel distribution was judged to be less useful as compared with

116 TIME (MS)

Fig. 44. Pressure at location (r,z) - (0.27 m, 0.30 m) measured from the bottom of the Lucite container. inputting the computationally (three-field) determined fuel density distribution. The results of using these fitting parameters on a second nonuniform coarse premixture, that calculated by SIMdER-II (two-field), are in App. T. If a vapor explosion is possible only with such water-rich or fuel-lean environments, any in-vessel steam explosion cannot develop sufficient energy for a containment challenge to occur because of the quenching capacity of the abundant water and simply the lack of sufficient volume within the vessel to allow a significant fraction of the fuel to participate. If a uniform, fuel-rich environment is assumed, even a multiplier of 0.2 may exaggerate the rate of heat transfer with the new heat-transfer asaumptions. A calculation of test 43 of Buxton's SNL series used for calibration in the old ZIP study with a 50:25:25 (fuel :water:vapor) premixed region and the present fitted input produces a peak pressure of 12.6 bars at the wall, about twice the desired value for that test. Two final observations should be made. First, in reality only some fraction of the fuel interacts efficiently to produce steam. This fraction will fragment to sizes smaller than 0.3 mm. Because no method of calculating this fraction is known, all premixed fuel is used to transfer energy to the water in

117 our calculations. Some parametric variations are desirable to allow for the implied uncertainties in the reactor calculations. Second, the previous ZIP study suggested that the reactor case should be more highly constrained than the heretofore experimental simulations. For reference, the effects of increasing constraint in this simulation of test MD-19 are shown in App. S.

118 V. ANALYSIS OF LOS ALAMOS EXPERIMENTAL DATA FOR SHALLOW POOLS

A. Introduction A brief SIMMER-II study examined the shallow-pool experiments performed at Los Alamos. The objective of the SIMMER-II calculations of the experiments was to compare calculated and actual slug breakup dynamics, so that judgments could be made on whether the slug breakup calculated in reactor accident situations is exaggerated or underestimated. This chapter is organized following the time sequence used in performing the calculations. First, we made a correlation to a reference case. Second, some hypothetical cases were run with nonuaiform interfaces. Different breakup modes were obtained, and the question of SIMMER-II scaleup to the reactor was examined. Third, an experiment with a different pressure ratio was calculated. Fourth, the experimental apparatus was modified to present an initially nonuniform interface. A calculation of this modified configuration was performed with SIMMER-II. The final section treats conclusions. A schematic of the test apparatus is shown in Fig. 45. The depth of the pool and the height of the free space above the pool are scaled using the actual dimensions of the reactor vessel and the total amount of fuel available. As indicated in the schematic, a very thin (1/4-mil) diaphragm supports the pool in a 102-mm-i.d. Plexiglas tube. The pool is 50 mm deep and there are 185 mm of free space above the pool. The bottom of the tube is separated from the nitrogen driver gas by a 4-mil Mylar diaphragm. The experiment is started by cutting the lower diaphragm; the pressure rises in the bottom tube and ruptures the thin diaphragm that supports the water. The increased pressure then induces the motion of the water that is of interest. Three pieces of data are collected during the test: the pressure history just under the top diaphragm (PI), the pressure history at the endplate (P4), and a high-speed movie.

B. Experimental Correlation In the case used for correlation to SIMMER-II, the water was to be accelerated by an expected 70-psig (81.3-psia) nitrogen gas pressure. The space above the water was at ~0.82 psia, with the partial pressures of water and nitrogen being equal. This gives a pressure ratio across the water slug of 100:1 and a resulting acceleration scaled to that expected for a core pool following a steam explosion.

119 1/4-mil water Mylar diaphragm

Mylar diaphragm

diaphragm nitrogen cutter

/"// /////////Ar////S///////SL

Fig. 45. Experimental setup for the shallow-pool tests.

120 The desire to examine water slug breakup in various initial configurations under rapid acceleration dictates the use of as many radial nodes across the water as possible. To limit the expense of running these simulations, only the area above the PI pressure transducer was modeled, and the experimental PI data were input as a pressure vs time boundaiy condition. The initial SIMMER-II setup chosen ccntains 10 radial by 47 axial nodes and is shown in Fig. 46. The actual output from the PI transducer is shown in Fig. 47. To avoid the calculation of mostly irrelevant but time-consuming high-velocity gas flow, time zero for the purposes of SIMMER-II was defined as 10.9 ms on Fig. 47. The resulting boundary condition for SIMMER-II is shown in Fig. 48. This curve was simulated by 23 input points. By linearly interpolating between these 23 points the SIMMER-II approximated time integral of Fig. 48 from 0 to 8 ms is 4.155 kPa*s. The actual time integral of Fig. 48 is 4.157 kPa«s. Consequently, the average inlet pressure was 75.4 psia over these 8 ms. The same definition of time zero was used to graph the results of the P4 pressure transducer, shown in Fig. 49. To fit this response, close coupling of the liquid and vapor fields in SIMMER-II was required to hold the slug together as long as possible. A constant 100-^m droplet diameter was chosen to achieve this objective. This is as small as we can reasonably expect the droplet diame- ters to be in the molten-core/cooiant-interactions problem. (We recognize that this drcp diameter is approximately an order of magnitude smaller than the Taylor wavelength of the fastest growing instability, which is

2ir(3a/ji(Pc - *>)) "• 900 nm. Implications of such required close liquid-vapor coupling are discussed in subsequent sections.) A second adjusted parameter was

aQ, the vapor-volume fraction at which the two-phase to single-phase transition occurs. This was lowered to 0.025 to try to maintain cells in a two phase condition as long as possible during liquid impact with the upper endplate. The calculated pressures at the center of the upper endplate, node (1,47), are shown in Fig. 50. The peak magnitude and impact duration compare favorably to the P4 data. The experimental impact occurs slightly earlier than that of the SIMMER-II calculation. The water probably begins to move before the SIMMER-II time zero. This type of delay is also consistent with previously reported SIMMER-II calculations of Stanford Research Inst:tute (SRI) experiments.1 To further check the SIMMER-II results, we constructed a simple slug model. In this model the water was driven upward by the pressures shown in Fig. 48, reduced by the 0.82 psi above the water. Impact occurred at 5.99 ms at a

121 SHALLOW-POOL SETUP

185 30 LOW P GAS

13 S555

50 WATER

HIGH P GAS 1 2 10

Fig. 46. SIW4ER-II mesh using equal area radial nodes and a a pressure vs time bottom boundary condition.

122 0.7-1

0.6-

0.5-

o.-; - I

02-

0.1-

0.0 0.0 5.0 1O.0 15.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0 Time (frig)

Fig. 47. Experimental pressure recorded at location PI.

u • PRESSURE FOR SIMMER INPUT SP7014 07-

0.6 : f ^

(0 / Q_ 0 5 -

Q; 0.4-

a: a.

0.2 :

01:

00 : 0.0 1.0 2 0 3 0 4.0 5.0 6.0 7.0 8.0 TIME (ms)

Fig. 48. PI recorded pressures over the time range of interest for SIMriER-II calculations.

123 20.0 -i PRESSURE RESPONSE FOR TEST SP7014

16.0-

CO CU 12.0- w en en 8.0-

4.0-

0.0 0.0 1.0 20 3.0 4.0 5.0 I 6.0 7.0 8.0 TIME (ms)

Fig. 49. Experimental P4 pressure trace.

20 0 PRESSURE RESPONSE FOR TEST SP7014

1G0

120

•X.

4 0

00 —I—i 1—|—i—|—i—i—|—| p—| 1—| 1—i—j—|—|—,—|—|—|—[ill—i—I—i—|—i—|—|—|—|—|—| 1 0 0 10 2 0 3 0 4 0 5 0 6 0 7.0 8.0 TIME (ms) Fig. 50. Calculated pressure for comparison at the P4 transducer.

124 velocity of 58.69 m/s. The peak SIMMER-II impact pressure was at 5.87 ms. The slug model should calculate a later time because the effect of any decrease of the PI pressure is felt instantly. Also, in the slug model, all the water impacts simultaneously. The SIMMER-II peak occurs at more of an averaged time. Another calculation of interest is the time integral of the P4 pressure trace. The integral under the pressure trace of Fig. 49 is 4.34 kPa«s. The pressure time integral averaged over the top boundary in the SIMMER-II calculation is 4.42 kPa's. Again, this is favorable agreement. (Both numbers are larger than the PI integral. At the end of the calculation the water is moving downward. ) Finally, we must examine slug breakup. Both the experiment and calculation show significant water dispersal before impact. The water's hammer pressure, pcv, from the simple slug model is ~88 MPa. Less than 16 MPa are measured and calculated. However, this SIMMER-II calculation does not model the proper physical mechanism for slug breakup, that is, the classic Taylor instability obtained when a light fluid accelerates a dense fluid, and the SIMMER-II result is therefore artificial. In this small-droplet SIMMER-II calculation, the slug disperses as a consequence of numerical diffusion, caused by implicit first-order donor-cell mass convection, and from numerical instability, caused by the explicit evaluation of the work term coupled with the small mesh and

augmented by the reduced ctQ. Liquid-volume-fraction contour plots obtained from SIMMER-II are shown in Fig. 51. Both axial spreading, or diffusion, and radial instabilities in the small outer meshes are shown. To correct the instability problem, an additional time-step control based on the work term was inserted. This control is described in the coding modifications write-up in Sec. II.F. The resulting liquid-volume fraction contour plots are shown in Fig. 52. The numerical instabilities have indeed been eliminated except for a minor residual problem following impact. A revised pressure trace is given in Fig. 53. Although the peak value is no longer as high with the more stable calculation, the comparison is still reasonable, given the experimental uncertainties. The pressure-time integral is now 4.32 kPa«s. A listing of the SIMMER-II input for the problem is given in Table XIV.

125 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME OOOMJL TIME 2 OOOMS

MINr 9 58E-06 MAXr 1 OOE+OO CI= 1 OOE-Ol M1N: 9 58E-06 MAX: 1 GOE+OO CI= I OOE-01

(a) (b)

VOLUME FRACTION OF LIQUID VOLUME FRACTiON OF Li QUID TIME 1 OOOMS TIME 5 OOOMS.

MIN= 9 58E-06 MAX= 1 OOE+OO C|: 1. OOE-O! : 9 58E-06 MAX: 1 OOE+00 CI= 1 OOE-01

126 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 5 500MS TIME 6 OOOMS_

= 9 58E-06 MAX= 1 OOE+OO CI- 1 OOE-O1 MIN= 9 S8E-06 MAX- 1 OOF+OO OOE-O! (e) (f)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 7 OOOMS TIME 8 OOOMS

MIN= 9 58E-06 MAX= 1 OOE+OO OOE-O1 MINr 9 58E-06 MAXr 1 OOE+OO CI= 00E-01 (5) (h) Fig. 51. (cont.)

127 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TI ME OOOMS TIME 2 OOOM.S_

MIN= 9 58E-06 MAX= 9 89E-01 CI= 9 89E-02 MIN= 9 58E-O6 MAX= 9 89E-01 CI= 9 89E-02 (a) (b)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 4 000M^_ TIME 5 OOOMS

MIN= 9 58E-06 MAX= 9 89E-01 CI= 9 89E-02 MIN= 9 58E-06 MAX= 9 89E-01 CI= 9 89E-02 (c) (d)

Fig. 52. Contour plots with time-step control revisions.

128 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 5 500MS_ TIME 6 OOOMS

MIN= 9 58E-06 MAX- 9 89E-01 9 89E-02 MIN: 9 S8E-06 MAX= 9 89E-01 CI= 9 89E-02 (f)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 7 OOOMS TIME 8

MINr 9 58E-06 MAXz 9 89E-01 CI= 9 89E-02 58E-06 MAX= 4 89E-01 CI= 9 89E-02 (g) (h) Fig. 52. (cont.)

129 11.0 T PRESSURE RESPONSE TEST SP7014

8.0

(0 7.0 a. 60

CO 50 CO LJ OS 40 Q_ 3.0

20 1.0 -,

00 T~r~ ""1 o.o 10 20 3.0 4.0 5.0 6.0 7.0 8.0 TIME (ms)

Fig. 53. Revised calculation for comparison with the P4 transducer.

C. Hypothetical Cases with Nonuniform Interfaces Three additional calculations were performed with distorted lower interfaces to assess water slug breakup under differing initial conditions. The goal was to develop recommendations for a revised experimental configuration. Any steam explosion occurring as a consequence of core melt will not possess the flat interfaces of Fig. 52. First, a concave surface was modeled, where the center was raised 1 cm relative to the side. The split involved one-half the area, and the total mass of water was conserved. The liquid-volume-fraction contour plots for this case are shown in Fig. 54. The instability rapidly grew, and much of the liquid was de- posited along the wall. Because no friction was modeled in this problem, the liquid continued unimpeded to the top of the tube. The impact pressure at the center of the endplate was significantly reduced. The impact dynamics was more representative of a two-phase spray. The pressure plot for the center of the endplate is given in Fig. 55. The peak pressure occurred earlier, because the smaller amount of water in the center was

130 TABLE XIV SIMMER-II INPUT FOR THE REFERENCE CALCULATION

0 -105Q07WRB AIR/WATER SLUG BREAKUP BASE CASE 10 0 0 0 32 200 3 AIR/WATER SLUG BREAKUP BASE CASE 4 .0080 10.0 5 10 47 6 FLUID DYNAMICS INTEGER INPUT 7 3 47 -23 0 0 8 0 9 93 1 1 1 0 1 0 1 1 1 1 10 1 1 1 2 1 3 1 4 1 5 1 6 11 1 7 1 8 1 9 1 10 1 11 1 12 12 1 13 1 14 1 15 1 16 1 17 1 18 13 1 19 1 20 1 21 1 22 1 23 1 24 14 1 25 1 26 1 27 1 28 1 29 1 30 15 1 31 1 32 1 33 1 34 1 35 1 36 16 1 37 1 38 39 1 40 1 41 1 42 17 1 43 1 44 1 45 1 46 1 47 2 5 18 3 5 4 5 5 5 6 5 7 5 8 5 19 9 5 10 5 2 6 2 13 3 13 4 13 20 5 13 6 13 7 13 8 13 9 13 10 13 21 2 23 3 23 4 23 5 23 6 23 7 23 22 8 23 9 23 10 23 2 40 3 40 4 40 23 5 40 6 40 7 40 8 40 9 40 10 40 24 2 47 3 47 4 47 5 47 6 47 7 47 25 8 47 9 47 10 47 26 10 500 50 20 4 0 -1 00 27 PROBLEM DIMENSIONS AND OPERATIONAL CONTROLS 28 0.016128 1 0.0066803 2 0.005126 3 29 0.0043214 4 0.0038072 5 0.0034420 6 30 0.0031652 7 0.0029461 8 0.0027671 9 31 0.0026172 10 32 0.005 12 0.0052857 47 33 0.5 0.0 -9.8 34 .000001 ,0000010 .000001 ,000001 35 1.0E-12 1.0E-12 l.OE-12 1-0E-5 0.1 36 .025 .99 0.5 10.0 0.1 10C.0 37 EDIT CONTROLS AND POSTPROCESSOR CONTROLS 38 1.5 1.0 2.0 0.0 4.0 39 0.0010 0.0005 40 41 0.005 2.0 42 43 0.00025 0.00001 0.00005 44 45 0.0055 0.0065 0.0080 46 47 .0020 .0010 48 49 0.004 2.0 50 51 0.0

131 TABLE XIV (CONT.)

52 0.0 53 0.0 54 0.0 55 0.0 56 0.0 57 0.0 58 \HEM POINT PARAMETERS 59 60 TIME STEP CONTROLS 61 0.0 1.00000E-06 l.OOOOOE-09 i0.100 62 1.0E-4 0.25 10.0 1.0 1.0 1.0 63 64 STRUCTURE AND FAILURE PARAMETERS 65 0.0 0.0 0.0 0.0 0.0 0.0 66 0.0 67 1.0 1.0 1.0 1.0 68 1.0E10 1.0E10 1.0E10 1.0E10 1.0E10 1.0E10 69 1.0E10 1.0E10 1.0E10 70 0.0 71 HATER PROPERTIES AND EQUATION OF STATE 72 913.02 1926.0 273.13 3. 337E+5 2.22 73 997.61 4217.1 0.0757 6.0E-1 1.789E-3 74 2.26756E+10 4.70579E+03 0.0 3.22689E+06 647.286 0.390597 75 1402.0 1.329 3.737 2.93390E+6 18.0 32.0 76 316.957 95.77 316.957 200. 77 1346. 0.3302 5.OE+4 50. 0.5 78 0.OOOOOE+00 0. OOOOOE+00 0.OOOOOE+00 79 0. OOOOOE+00 1.OOOOOE+00 0. OOOOOE+00 80 1.00000E-30 1.OOOOOE+00 14.98O0E+02 81 STEEL PROPERTIES AND EQUATION OF STATE 82 8001.0 712.00 1700.0 2.472OOE+O5 25.0 83 6100.0 75C.O 1.6 20.0 5. 36OOOE-O3 84 1.338OOE+11 4.33700E+04 0.0 8.170OOE+O6 10000.0 0. 360 85 492.0 1.26 1.64000E-00 0.0 56.0 7700. 86 87 88 0.OOOOOE+00 0. OOOOOE+00 0.OOOOOE+00 89 0.OOOOOE+00 1.OOOOOE+00 0.OOOOOE+00 90 SODIUM PROPERTIES AND EQUATION OF STATE 91 0.0 92 705.0 1300.0 !J99.9 50.0 1.50000E-04 93 3.76OOOE+O9 1.21300E+04 10.0 4.816OOE+O6 2503.0 0,341 94 1460. 1.650 3.567OOE-OO 0.0 23.0 1375. 95 96 97 0.OOOOOE+00 0.OOOOOE+00 0.OOOOOE+00 98 0.OOOOOE+00 1.OOOOOE+00 0.OOOOOE+00 99 CONTROL MATERIAL PROPERTIES AND EQUATION OF STATE 100 2520.0 1893.0 2623.0 3.OOOOOE+O5 83.74 101 2520.0 1890.0 1.0 80.0 1.00000E-03 102 4.286OOE+14 8.36800E+04 0.0 5.OOOOOE+06 7107.0 0.350 103 500.0 1.50 1.46000E-00 0.0 55.3 5472. 104

132 TABLE XIV (CONT. )

105 106 INJECTOR GASPROPERTIES ,AND EQUATION OF STATE 107 0.0 108 1.0 109 1.00000E+12 4.00000E+03 0.0 0.0 1.0 0,3 110 741.0 1.4 3.798E-00 3.0E06 28.0 85.0 111 112 113 COMPONENT PROPERTIES 114 913.02 913.02 913.02 913.,02 8001.0 8001.0 115 2520.0 0.0 0.0 116 997.61 997.61 6100.0 705,,0 2520.0 913.02 117 913.02 997.61 0.0 118 14.98OOE+O2 14.9800E+O2 14.98OOE+O2 14.!J8OOE+O2 14.'98OOE+O2 14.9800E+02 119 14.9800E+O2 14.98OOE+O2 14.98OOE+O2 120 HEAT TRANSFER CORRELATION DATA 121 1.0 1.0 1.0 1.0 1.0 1.0 122 1.0 1.0 1.0 1.0 1.0 1.0 123 1.0 1.0 1.0 124 2.3OOOOE-O2 8.00000E-01 4.00000E-01 0.0 125 2.SOOOOE-02 B.OOOOOE-01 8.00000E-01 5.0 126 2.5OOOOE-O2 8.00000E-01 8.00000E-01 5.0 127 2.3OOOOE-O2 8.00000E-01 4.00000E-01 0.0 128 0.000 0.8 0.4 0.0 129 3.7E-1 0.6 0.33 0.0 130 DRAG CORRELATION DATA 131 laOOlEOO 12.0 1.3E-5 l.OE-6 1.0 132 4.0 1.0 0.5 1.0 .95 133 0.046 -0.20 l.E-5 0.046 -0.20 l.E-5 134 4.1736E+O5 4.2500E+05 4.7963E+05 5.4044E+05 6.4119E+05 6.8182E+05 135 6.2885E+O5 5.4886E+O5 4.9178E+O5 4.438OE+O5 4.1O61E+O5 4.O61OE+O5 136 4.0336E+O5 4.3O68E+O5 4.3176E+05 4.532OE+O5 4.6710E+05 6.0428E+05 137 6.359OE+O5 6.2866E+O5 6.3923E+O5 6.14O7E+O5 5.4837E+O5 138 0. 2.0000E-04 6.0000E-04 1.OOOOE-03 1.4000E-03 1.8OOOE-O3 13$ 2.2000E-03 2.6OOOE-O3 3.OOOOE-03 3.4000E-03 3.8OOOE-O3 4.2000E-03 140 4.6OOOE-O3 5.OOOOE-03 5.4000E-03 5.8OOOE-O3 6.2000E-03 6.6000E-03 141 7.OOOOE-03 7.40O0E-03 7.8OOOE-O3 8.2000E-03 8.600OE-O3 142 REGION 1 EVERYTHING (NO STRUCTURE IN THIS PROBLEM) 143 7.0 0.0 0.0 0.0 0.0 0.001 144 0.0 0.0 1.0 0.0 0.0 l.E-5 145 0.10200 0.10200 0.10200 l.E-5 l.E-5 l.E-5 146 195.00 0.64 .00 2.3E-5 l.OE-17 .00005 147 0.5E-4 1.E19 148 VAPOR AT© LIQUID VELOCITIES ON THE BOTTOM BOUNDARY 149 0.0 0.0 0.0 0.0 0.0 0.0 150 0.0 0.0 0.0 0.0 151 0.0 0.0 0.0 0,0 0.0 0.0 152 CO 0.0 0.0 0.0 153 INLET (VARIABLE PRESSURE) 154 1 2 1 10 1 1 0 0 1 155 0.0 0.0 0.0 0.0 0.0 0.0 156 0.0 0.0 0.0 157 O.G 0.0 0.0 0.0 0.0 133 TABLE XIV (CONT.)

158 0.0 0.0 0.0 0.0 0.0 0.0 159 0.0 .01 160 0.0 0.0 0.0 0.0 0.0 296. 161 0.0 0.0 0.0 0.0 0.0 4.75709 162 296.0 163 0.0 0.0 0.0 0.0 .0001 .0001 164 WATER 165 3 12 1 10 1 0 166 0.0 0.0 0.0 0.0 0.0 0.0 167 0.0 0.0 0.0 168 0.0 0.0 0.0 0.0 0.0 169 998.0 0.0 0.0 0.0 0.0 0.0 170 0.0 0.0 171 296.0 0.0 0.0 0.0 0.0 0.0 172 0.0413989 0.0 0.0 0.0 0.0 0.0 173 296.0 174 0.0 0.0 0.0 0.0 .0001 .0001 175 UPPER GAS COLUMN 176 13 47 1 10 1 0 177 0.0 0.0 0.0 0.0 0.0 0.0 178 0.0 0.0 0.0 179 0.0 0.0 0.0 0.0 0.0 180 0.0 .01 0.0 0.0 0.0 0.0 181 0.0 0.0 182 296.0 0.0 0.0 0.0 0.0 0.0 183 0.0206994 0.0 0.0 0.0 0.0 0.0322125 184 296.0 185 0.0 0.0 0.0 0.0 .0001 .0001 given a higher acceleration. The pressure was sustained as a consequence of water moving inward from the sides of the tube. A second parametric modeled a convex surface, with the water level depressed by 1 cm in the center of the tube. Again, one-half of the area was involved. The liquid-volume-fraction contour plots are given in Fig. 56. Here also an instability grew. For this case, much of the liquid was calculated to stay in the center of the tube. Because of the concentration of the liquid toward the center, the peak pressure at this location was relatively high, as shown in Fig. 57. The third parametric was to place some structure v- plate) below the depressed center liquid in the convex case. This condition was felt to lead to a more achievable experimental situation. The liquid-volume fractions for this case are shown in Fig. 58. With the plate, a volume of liquid perhaps somewhat unrealistically stayed in the centerline node. A turbulence model might be desirable to better calculate this case. The predicted P4 pressure response is

134 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME OOOMJL TIME 2 OOOM_J

MIN- 2 78E-06 MAXr 9 79E-G1 CI= 9 79E-02 MIN = 2 78F-0& MAX:: 9 7?E-01 = 9 7SE-0? (a) (b)

VOLUME FRACTION CF LiQUlD VOLUME FRACTION OF LIQIHJ TINE 4 000MS TIME 5 OOOMS

MIN= 2 78E-06 MAX= 9 79E-O1 CI: 9 79E-02 MIN; 2 78E-06 MAXr 9 79E-01 CI= 9 79E-02

Fig. 54. Liquid-volumc-fraction contour plots. Concave initial conditions. 135 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIOUID TIME 5 500MS_ TIME 6 OOOMS

2 78E-06 MAXr 9 79E-01 CI= 9 79E-02 = 2 78E-06 MAX= 9 ""ii£-01 CI= 9 79E-02 (e) (f)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TiME 7 OOOMS TIME 7 500MS

MIN; 2 7SE-06 MAX = 9 79E-01 CI= 9 79E-02 MINr 2 78E-06 MAXr 9 79E-01 CI= 9 79E-02 (g) (h)

Fig. 54. (cont.)

136 5.0-1 PRESSURE CONCAVE INITIAL CONDITIONS

4.0-

a. s 3.0- u cc

CO S3 2.0 4 OS

1.0-

0.0 1 i ' • • ' i • • i M r 1 i ' 0.0 1.0 2.0 3.0 40 5.0 6.0 7.0 8.0 TIME (ms)

Fig. 55. Pressure trace at the center of the upper endplate for the concave case. shown in Fig. 59. A reduction from the case without the plate is observed as expected. These three auxiliary calculations indicate that the pressure distribution on the top plate is a function of the initial shape of the lower water interface. Also meaningful for comparison with the reference case is an integral quantity, the force on the head. This is defined by SIKMER-II as

dA (69) Head Area

The results for the four cases discussed so far are given in Figs. 60-63. Initially, the force is negative because the upper chamber is subatmospheric. Further, as a consequence of the conservation of momentum, the integral F(t) will be approximately the same for all cases if the integral is extended far enough in time, for example, 8 ms. This impulse will not compare exactly because of differing degrees of reflection following impact. The cases do significantly differ however. Without the initially flat interface, the impact

137 VOLUME FRACTION OF LIQUID TIME OOOMS_

MIN: 5 77E-06 MAX= 9 83E-01 CI= 9 83E-0Z

(a)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 2 OOOMS TIME 4 OOOMS

MIN= 5 77E-06 MAXr 9 83E-01 CIr 9 83E-02 MIN= 5 77E-06 MAX= 9 83E-01 CI- 9 83E-02

(b) (c)

Fig. 56. Liquid-volume fraction contour plots, Convex initial condi t ions. 138 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID VOLUME FRACTION Oi' '.. lOL'lG TIME S OOOMJ TIME 5 5OOMS T I ME b OOOMS

MIN; 5 77E-06 MAXz 9 83E-01 CI= 9 83E-02 MIN= 5 77E-06 MAX: 9 83E-0I C\- 9 83E-02 MIN: 5 77E-06 MAX- 9 83E-01 CI= 9

(d) (e) (f)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 7 OOOMi. TIME 8 OOOMS^

MINr 5 77E-06 MAX= 9 83E-01 CI= 9 83E-O2 MIN= 5 77E-06 MAX: 9 83E-01 CI= 9 83E-02

(g) (h)

Fig. 56. (cont.)

139 11.0 PRESSURE IN THE CONVEX CASE 100

60 tx. •J-J 50 C/j 40 CL 30

2.0

00 00 10 20 3.0 4.0 5.0 6.0 7.0 8.0 TIME (ms)

Fig. 57. Pressure trace at the center of the endplate for the convex case. incoherence means that the peak force is reduced by a factor of 2 or more, and the previously narrow peaks are spread out in time. Consequently, these calculations with an initially nonuniform interface imply a signif ict-at increase in the two-phase character of any "slug" impact. In examining the effects of a nonuniform interface, the influence of scaling is of some importance. The scaling was checked by increasing both the dimensions and the driving pressure by a factor of 40 for the convex case. This increases the tube radius to 2 m or to reactor scale while maintaining the same acceleration. The impact pressures at the center of the tube are shown in Fig. 64. Peak pressure has increased by a factor of 40 over that in Fig. 57, as expected. Of some concern is the configuration development during the expansion. The liquid-volume fractions for this case are shown in Fig. 65. After dividing out the expected /40 in the time scale, we see that the upscaled case is considerably more coherent. In terms of the calculation, the biggest difference compared with the small tube results is in the slip between liquid and vapor. A simplified SIMdER-II momentum equation for either the liquid or vapor field is

140 VOLUME FRACTION OF LIQUID TIME OOOMS_

MIN= 3 81E-06 MAX = 9 76E-01 CI= 9 76E-02

(a)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 2 OOOMS TIME •« OOOMS

MINr 3 81E-06 MAX= 9 76E-01 CI- 9 76E-02 MINr 3 81E-06 MAXr 9 76E-01 C\- 9 76E-02

(b) (c) Fig. 58. Liquid-volume fractions for the plate obstruction case.

141 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME S OOOMS TIME 5 SOOM

MIN= 3 81E-06 MAX: 9 76E-01 CI= 9 76E-02 MIN= ? 81E-06 MAX: 9 76E-01 = 9 76E-02 (d) (e)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME & OOOMS TIME 6 SOOMS

r 3 81E-06 MAXr 9 76E-01 Ci= 9 76E-02 MIN= 3 8iE-06 MAX= 9 76E-O1 CI= 9 76E-Q2 (g) Fig. 58. (cont.)

142 6.01 PLATE OBSTRUCTION CASE

5.0-

(0 4.0- a.

3.0-

a. 2.0-

1.0-

0.0 0.0 10 2 0 3.0 4.0 5.0 6.0 7 0 8.0 TIME (ms)

Fig. 59. Pressure trace at the centei' of the endplate with a permanent obstruction.

8.0 n

4.0 x o 3-0

2-0

i.o v \A

0.0-

-10 • i i i i i • ' •-[-' ' ' ' i T"1"1 ' ' I 0.0 l.C 2.0 3 0 4.0 5.0 6 0 7.0 8.0 TIME (ms)

Fig. 60. Reference case pressure integral,

143 5.0-1

4.0-

3.0-

2.0- o

1.0-

0.0-

-10 I I I I • I I ' I ' ' ' ' I ' ' ' ' I ' ' • ' I • ' ' ' I • ' ' ' I ' ' • ' I 0.0 10 2.0 3.0 4.0 5.0 6 0 7.0 8.0

TIME (ms)

Fig. 61. Concave case pressure integral.

r FORCE ON HEAD

3.0-

2.0-

O o 10-

0.0-

-1.0 00 10 2.0 30 40 5.0 6.0 7.0 8.0 TIME (ms)

Fig. 62. Convex case pressure integral.

144 200-i FORCE ON HEAD

16.0-

12.0-

u 8.0- o

4.0

0.0-

-4.0 • • • i • i • • i i • • ' i ' ' i • i • ' ' ' i ' ' ' ' i ' ' • ' i • • • • i 0.0 10 2.0 3.0 4.0 5.0 6.0 7.0 8.0 TIME (ms) Fig. 63. Plate obstruction case pressure integral.

500.0-1 UPSCALE OF THE CONVEX CASE

400.0 -

£ 300.0 -

S 200.0 - a.

100.0-

0.0 000 001 002 0D3 0.04 0.05 TIME (s) Fig. 64. Center of tube pressure trace for the scaled-up case.

145 VULUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME OOOMS_ TIME 12 65OMS

MINr 5 G3C-O6 MAXz 1 OOE+00 Cl OOE-O1 MIN= 5 53E-06 MAX: 1 OOE+00 C|: 1 00E-01 (a) (b)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 25 3OOMS TIME 31 620MS

MIN= 5 53E-06 MAX: 1 OOE+OO CI: 1 OOE-01 MINr 5 53E-O6 MAX= 1 OOE+00 r 1 00E-01 (c) (d)

Fig. 65. Liquid-volume fractions for the scaled-up case.

146 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 34 790M.S, TIME 31 630MJL

MSN: 5 53E-06 MAXr 1 OOE+OO Cl: 1 00E-01 MIN= 5 53E-06 MAX= 1 OOE+OO CI= 1 OOE-O1

(e)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 3/ 940MS TIME 41 26OMS TIME 50 O0OMS_

HINz 5 53E-06 MAXr 1 OOE^OO Cl; 1 OOE-01 MIN- 5 53E-Ofc MAXz I OOL'-Jc'ci: I OOE-01 MIN: S S3E-C>6 MAX; 1 OOE'OO CI:

147 f -J v_ , (70) 3t ^ * r r r where P is the macroscopic density, v is the velocity, a is the volume fraction, p is the pressure, f is a function of densities, volume fractions, and the drag coefficient, r is an effective droplet size, and v is the relative interfield velocity.

Because the velocity scales as /40, all terms scale (rexai \ unchanged) except for interfield drag. If the effective droplet siae is assumed to be unchanged, because the overall pressure gradient remains the same, the force-per-unit volume from interfield drag will increase by a factor of 40. Less liquid-vapor slip is apparent in the results shown in Fig. 65. Further, a larger droplet size than required to achieve comparable results with those from the small scale would not be justified because any tendency toward increasing interfield velocity differences should decrease the droplet size producing less slip. In terms of a more theoretical approach, Tennant has performed a dimensional analysis to obtain (for a flat interface) an equation of the form

i = K U/d)a (Re)b (Re/We)c (Fr)e , (71)

where s is the breakup distance, •i is the initial length, of the pool, K depends on the experimental procedure, d ;s the diameter of the pool, Re is the Reynolds number, We is the Weber number,

148 Fr is the Froude number, and a, b, c, and e are constants.

Over the range of parameters investigated experimentally by Tennant, he concluded that "a" should be unity, and b, c, and e appeared to be zero. These results suggest that breakup should occur over the same scaled distance. Three additional considerations deserve attention. First, the upscale in the SIHMER-II calculation was perfarmed by simply increasing the mesh dimensions. This approach increases the effective numerical diffusion in the radial direction. Slug breakup would probably occur faster if it were possible to use the same size nodes at both scales. Second, a factor producing less of a liquid slug impact in the calculation (as well as decreased slip) was the increased nitrogen vapor density used to accelerate the water. For proper scaling, the nitrogen's internal energy should have been increased. Because such an energy increase would have unrealistically produced steam, the density was changed instead. This change meant an increased proportion of the kinetic energy went into a acceleration of the nitrogen, whose density was now an appreciable fraction of that of liquid water. Third, a model by Corradini et al. gives the a rate of entrairnwnt from Taylor instabilities as dV/dt = 4.6 A (a7 ) , where 7^ is the critical Taylor instability wavelength defined as 1//3 times the wavelength of the fastest growing instability given in Sec. V.B, V is the volume en- trained, and A is the area of the pool or slug. For a one-dimensional case, V = A z. This model predicts complete entrainnient of the 50-mm experimental slug at 4.5 ms, which is approximately correct. It also predicts only a weak dependence, (a) , from acceleration on the rate of entrainment, but it does suggest less entrainment at large scale. In other words, at constant acceleration, increasing the dimensions by a factor of 40 only scales the axial entrainment distance by /40 based on the Corradini model. The lesson from all these scaling considerations is that a combination experimental/analytical program is desirable if truly precise statements are to be made on scaling effects in a unique physical system. For the current calculation, Fig. 66 gives the force on the head for the upscaled case. The magnitude is somewhat greater than might be expected based on a scaling factor of 40 or 64 000, probably as a consequence of the more coherent impact.

149 2 350. FORCE ON HEAD

30 0-

25 OH

20.0- o cr o 15.0-1

100-

50-

0.0- • • . • 0 00 0 01 0 02 0.03 004 005 TIME (s)

Fig. 66. Upscaled case pressure integral.

D. Experimental Comparison with a Pressure Ratio of 50^1 The next activity was a calculation of an experiment run with a 50:1 pressure ratio using the SIMdER-II code as previously correlated to the 100:1 pressure-ratio data. Here times should scale as /2. The PI pressure, starting at 11 ms into a typical transient, is shown in Fig. 67. For SIKMER-II input, time zero was defined as 1.3 ms into Fig. 67 (12.3 ms into the transient). The resulting inlet pressure, PI, which can be compared with Fig. 48, is shown in Fig. 68. The previously used time-step control was insufficient to remove instabilities generated by the work term, so this problem was simply run with a maximum time-step size of 10 us. The resulting impact pressure at the center of the endplate is shown in Fig. 69. The timing was about right. The maximum pressure occurred at 8.27 ms. The simple slug model obtained impact at 8.44 ms with a velocity of 43.05 m/s. This was reasonable given that the driving pressures were not quite reduced by a factor of 2. The magnitude of the calculated pressure response was as expected. A reduction of a factor of /2 to 2 from Figs. 50 and 53 to Fig. 69 was anticipated and observed. Because the magnitude of the peak pressure was somewhat a consequence of instabilities,

150 0.4-1

03-

a a. :s u K 02- 3

0.1-

015 0.0 5.0 10.0 15.0 200 25.0 30.0 35.0 TIME (nu)

Fig. 67. Experimental pressures recorded at location PI for the 50=1 pressure ratio test.

0.4-1 PRESSURE RESPONSE FOR TEST SP3052

0.2 m

0.1-

0.0 r~rT, , 00 20 40 60 BO 10 0 12 0 TIME (ms)

Fig. 68. PI record over the time range of interest for SIMtfER-II calculations in the 50:1 pressure ratio test.

151 accuracy was limited. Of concern was the reduction seen in the experimental P4 response, showu in Fig. 70. While the timing of the peak was reasonable, a much more two-phase response was observed. The SIKMER-II impact characteristics were more like those at the 100:1 pressure ratio, as could be inferred from the head-force plot, Fig. 71. Further study on the difference between the 100:1 and 50:1 pressure ratio cases lead to the following: (a) The numerical diffusion of the SIMMER-II code does scale as expected. Fig. 72 shows the liquid-volume-fraction plots at scaled times. At least the first four plots are very similar to those in Fig. 52. Although the pressure was modified by changing the gas density rather than its internal energy, all terms in the momentum equation, Eq. (70), do scale by approximately the same factor. (b) The experimental breakup occurs at about the same distance in the 50:1 and 100:1 tests. If anything, the slug in the 100:1 test disperses faster with a finer spray existing behind it than in the 50:1 test just before impact. Essentially Tennant's conclusion on the exponents in Eq. (71) appears valid in comparing these two cases. Although the

100 CALCULATED P4 PRESSURE SP3052 90

8.0

60 (MPa )

5.0 JR E

ESS l 40 a: CL-

00 0 0 2 0 4 0 6.0 8.0 100 120 TIME (ms)

Fig. 69. Results at location P4 for the 50:1 pressure ratio test. 152 EXPERIMENTAL P4 PRESSURE TEST SP3052

30-

(0 a.

2.0- a:

1.0-

00 "I T T T IHTI ['ITU I oo 2.0 4.0 6.0 8.0 10.0 12.0 TIME (ms) Fig. 70. Experimental results in the 50=1 pressure ratio test.

o 7.0-

6.0-

5.0-"

4 0 -]

±1 3.0- o u. 2.0-

1.0-

0.0-

-1.0 0.0 2.0 4.0 6.0 8.0 10.0 120

TIME (ms) Fig. 71. Pressure integral in the 50:1 pressure ratio case.

153 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 000MS TIME 2 830MS_

MIN= 9 58u-06 MAX= 9 82E-01 CI= 9 82E-02 MIN: 9 53E-06 MAX: 9 82E-01 CI= 9 82E-02

(a) (b)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 5 660MS TIME 7 070MS

MIN= 9 58E-06 MAX= 9 82E-01 CI= 9 8?E-02 MIN= 9 58E-06 MAXr 9 82E-01 CIz 9 82E-02 (c) (d)

Fig. 72. Contour plots for the 50:1 pressure ratio case.

154 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 7 780MS_ TIME 8 4 90MJL

MINr 9 58E-0& MAX; 9 82E-01 CI= 9 82E-02 MINr 9 5SE-06 MAX= 9 CI = 9 82E-Q2 (e) (f)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 9 900MS TIME 11 300MS

MIN= 9 58E-06 MAX= 9 82E-01 CI= 9 82E-02 M1Nr q 58E-06 MAX= 9 82E-01 CI= 9 82E-02 (g) (h) Fig. 72. (cont.)

155 Corradini model predicts the correct direction of increasing entrainment with increasing acceleration, quantification usir.j an acceleration raised to the one-fourth power was not possible utilizing the results from these two tests. (c) It does not appear possible to match both the 50:1 and the 100:I experimental P4 pressure transducer responses with consistent SIMMER-II input. A mechanistically calculated droplet size would give a trend in the correct direction for the 50:1 pressure ratio. However, results of an exploratory calculation suggest that a two-phase response would be observed in all cases if the SIMMER-II calculations are done with a droplet size chosen internally, based on the Weber criterion. This pressure trace would not agree with that of Fig. 49 for the 100:1 test.

E. Experimental Comparison with a Nonuniform Initial Interface Insertion of a plate as an obstruction is experimentally feasible.. The results shown in Fig. 58 suggest it may also provide an interesting comparison with a SIMMER-II calculation. Experimentally, a 2.5-in.-diameter plate was inserted, which lowered the center of the thin upper diaphragm by 1.3 in. With zero defined as the interface between nodes 2 and 3 in Fig. 46, the SIMMER-II representation of the initial experimental configuration is shown in Fig. 73. The total number of axial nodes was retained at 47 by increasing the mesh spacing above the water. Data from four presumably identical tests were examined. The Pi pressure traces with a zero time of 10.9 ms are shown in Figs. 74-77. These curves have relatively similar shapes. Figs. 78-81 give the P4 pressure response curves, plotted on similar scales. Based on the previous hypothetical calculations, the data of test 19, shown in Fig. 81, were selected as representative. For the purpose of defining a SIMMER-11 input pressure, a zero time of 12.3 ms (or 1.4 ms on Fig. 77) was selected. The calculated pressure at the center of tae toppl&te is shown in Fig. 82. The timing of impact compared with experiment is acceptable, noting that Fig. 81 needs to be shifted to the left by 1.4 ms because of the difference in time zero. Hoover, the peak pressure going up to 3.6 MPa is missing in the calculated pressure response. The integrated force over the area of the head is shown in Fig. 83. Increased smoothing over previous cases is apparent. The calculated pressure-time integral over the top boundary is 4.01 kPaJs. This is within 90% of the value calculated in the reference case (see Fig. 60). The

156 LOW PRESSURE

PLATE

12 3 4 5 RADIUS (CM)

Fig. 73. SIM/lER-il representation of the experimental configuration with a plate.

157 pressure-time integrals for the four experiments ^ s 4.25 kPa*s for test 16, 7.26 kpa-s for test 17, 4.22 kPa«s for test 18, and 4.49 kPa«s for test 19. These integrals are for 8 ms with zero time defined as the moment the PI transducer reaches 0.04 MPa. In view of their uncharacteristic quality, the data from test 17 are probably spurious. Otherwise, the calculated time integral is simply a little low. The SI^ER-II calculated liquid-volume-fraction plots are shown in Fig. 84. In comparison with the movie films, breakthrough of the slug occurs more rapidly. Also, the films are not axisymmetric, and the spray of water left behind the disintegrating slug tends to obscure furt her details. However, having water concentrated along the axis may be exaggerated by SIMMER-II.

F. Conclusions (1) Reasonable agreement was obtained to the referenced shallow-pool experiment (pressure ratio of 100"l and a flat lower interface for the slug). 1R As in previous simulations of this type of experiment, this agreement was obtained as a consequence of the numerical approximations, rather than from actual modeling of physical instabilities1.

0.8-, PI PRESSURE TRACE FOR TEST SP7016 0.7-

0.6-

05- (MPa )

0.4- UH E - :ss i a: 0.3- ; 02-

01-

Fig, 74. PI pressiic in test .16 with a stationary plate. 158 u cc d a. SURE ( MPa) c a. a: Fig. 75PIpressureintest17withastationaryplate 0.8 n 07- 06- Fig. 76,P I pressureintest1 8 withastationaryplate . 0.4- 05- 03- 0.2- 0.0 0.1- 0.0 1.2.3.456 0.0 1 PI PRESSURETRACFORTESTSP7017 J PI PRESSURETRACFORTESTSP7018 2 034.56 07 TIME (ms) TIME (ms) -' I• 7.0 80 8.0 159 PI PRESSURE TRACE FOR TEST SP7019

a. D en a8: a.

0.0 1.0 2.0 3.0 4.0 50 TIMS (ms)

Fig. 77. PI pressure in test 19 with a stationary plate.

PRESSURE RESPONSE FOR " EST SP7016 9 0

8.0-J

70 ~

6.0^ -I a:

72 a: 4 0- a. 30-j 4

1.0 J

00 • r-r j T- r-^fT-1—I—I—f , . . . | . , . , | , , , I | I T • I |-r • I I | 0.0 10 2 0 3 0 4 0 5 0 6 0 7 0 8.0 TIME (ms)

Hg. 78. P4 pressure in test 16 with a stationary plate.

160 lO.O-i PRESSURE RESPONSE FOR TEST SP7017

ffl •L 1

00 4.0 5.0

TIME (ms) Fig. 79. P4 pressure in test 17 with a stationary plate.

100- PRESSURE RESPONSE FOR TEST SP7018 90:

(0 Q.

CO 4.0 0u1 Q.

20 -.

1.0-

0.0-^ ••i •-*••••• i •> i i iii i"' i i i | i i i i | i 0.0 10 20 30 4.0 5.0 6.0 7.0 8.0 TIME 'ms) Fig. 80. P4 pressure in test 18 with a stationary plate.

161 lOOn j PRESSURE RESPONSE FOR TEST SP7019 90 H

BOH -I

70-

6.0-1 U a: '/: a. "I 30-

20-

0 0 i r -—I1- v 0 0 10 2 0 3 0 4 0 5 0

TIME (ms)

Fig. 81. P4 pressure in test 19 with a stationary plate.

4.0 n CALCULATION P4 PRESSURE TEST SP7019 3.5-

30-

2.5 (MP a

cc 2.0

1.5 PRESS L

1.0

0.5-j

00 00 1.0 2.0 3.0 4.0 5.0 6^0 7.0 8.0 TIME (ms)

Fig. 82. Calculated P4 pressures for test ' *,

162 o 12.0-q * 11.0 T 10.0-1 9.0-i 80- 7.0 -. 6.0 J u 5.0-i o 4.0 J 30^ 2 0 - 10- 0.0-i

-1.0 ^H 11 I ' 0 0 1.0 20 3.0 4.0 5.0 6.0 7.0 8.0 TIME (ms)

Fig. 83. Calculated pressure integral, test 19.

(2) The constant, K, in the formula used by Tennant in Eq. (71) is small for these shallow-pool tests. The correlation in Ref. 16 would give an "s" of 6.5 cm, with a "•(" of 5 cm and a "d" of 10 cm. In the current ex-

periments, br r does not occur until the approximate impact distance, or 18 cm. Also, the same SIMMER-II assumptions cannot be used to match both the tests of Ref. 18, which also obtain more rapid breakup, and the current experiments. (3) The SIMMER-Ii results apparently scale "reasonably" in going from a 100:1 pressure ratio tc a 50:1 pressure ratio. Howevei, the experimental pressure signature changes significantly. Current opinion is that the difference in the P4 pressure transducer response is not the consequence of a different mode of impact, but rather a characteristic of the apparatus-fluid-pressure-tran sducer coupling. The apparatus is not rigid and its acceleration would influence the results in a sufficiently energetic case. Also, individual water-droplet impacts could affect the readings obtained from a 0.5-in.-diameter transducer. Consequently, the pressure peak measured in tb 100^1 test may be high. The small droplet size and tight liquid-vapor coupling required to match this peak would 163 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME OOOM$_ TIME 2 OOOMS

MIN; 2 92E-06 MAXr 9 79E-01 CI: 9 79E-02 = 2 92E-06 MAX: 9 79E-01 CI= 9 79E-( (a) (b)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 3 OOOMS TIME 3 500MS

MIN= 2 92E-06 MAXr 9 79E-01 CI= 9 79E-02 MINr 2 92E-0fe MAXr 9 79E-01 CI: 9 79E-02 (c) (d)

Fig. 84. Slk/MER-II liquid-volume-fraction plots for the comparison with the experimental plate case. 164 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 4 DOOMS TIME 5 OOOMS.

MIN: 2 92E-06 MAXr 9 79E-01 Clz 9 79E-02 MIN: 2 92E-06 MAXz 9 79E-O1 Clz 9 79E-02 (e) (f)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 6 OOOMS TIME 8 OOOMS

MIN; 2 92E-06 MAX: 9 79E-01 CIr 9 79E-02 MIN= 2 92E-06 MAX: 9 79E-01 CI= 9 79E-0Z

(g) ;h) Fig. 84 (cont.)

165 not be required if the calibration could have been made to data without such problems. This also agrees with the idea of identifying the droplet radius as one-half the Taylor wavelength of the fastest growing instability. (4) Even with excessive liquid-vapor momentum coupling, 5IMMER-II slug breakup occurs too quickly when compared with the prestnt experimental data starting with a nonuaiform interface. With a small-scale experiment having part of the flow area obstructed by a plate, bieakup is a consequence of the lack of turbulent mixing effects in the SIMMER-II formalism. An idealized example of thi3 problem is shown in Fig. 85. Here an area expansion occurs in a multinode structure. SIMMER-II is content to have the entering fluid maintain the same velocity, trans- ferring no momentum to the stagnant fluid along the area blocked by the wall. (5) A nonunifcrm interface clearly mitigates the peak impact pressures in this experimental configuration and spreads out forces over time. More work must be performed in examining extension-to-reactor scale. With reactor scale nodes and similar fluid accelerations, SIMMER-II indicates a significant reduction in the rate of slug breakup. Although this reduction does have some theoretical support, small nodes may be required to obtain the proper development of the large-scale instabilities. (6) With the small radial nodes used in this problem, an instability developed resulting from the current algorithm for treating the work term in the vapor-energy equation. An ad hoc tim*-step control was sufficient to suppress these instabilities in one case. In changing the driving pressure while maintaining geometry, this time-step control scaled as v"1, similar to the SIMMER-II DTL(3) and DTL(4) time-step controls. This as inadequate and more study is required. As a consequence of the difficulty involved in revising the SIMMER-II algorithm to treat the work term implicitly, a formulation for a new time-step control based on the node size and EOS properties is indicated. (7) Presenting the detailed results of all tests would be too voluminous for this report. However, some degree of random behavior was observed. When +his behavior is coupled to the numerical approximations that must be made in any multiphase numerical fluid-dynamics algorithm, exact

166 -\ ~ r - i - 1 r ~ i ~ - r ~ i i i i i i i i i STRUCTURE i i i i i i i /V =

_ i _ j _ L _ 1 _ J _ L _ 1 _ j _ L _ i i 1 1 i i t -\ - r - i - -i " r ~ i ~ r 1 I I I i

1 1 t 1 1 1 1 1 I I I I I I I I I I 1 u -J _ l_ _l_ J _ L _ 1 J _ t_ _l_ J _ L. _l_ J _ L _l_ J _ L _ 1 1 1 1 1 1 I I I I I I I I I I I

~ i - -\ ~ r -1- - r - i - -i r

i i i 1 1 i i i i i i i 1 1 i i i i f f _ 1 _ J _ L _ 1 _ J _ i _ j _ u _

* 1 1 1 1 i i i i

Fig. 85. SIMMER-II results for an idealized expansion in a multinode environment. agreement or correlation to this type of experiment may be wishful thinking. Overall, current results suggest that a one-dimensional slug expansion calculated by SINMER-II with tight liquid/vapor momentum coupling will exaggerate the magnitude of eventual fluid impact pressures. Two-dimensional slug breakup will smear the impact over time. At small scale SIMMER-II will exaggerate such smearing from lack of turbulent mixing. At large scale with large nodes, SIMMER-II may underestimate the growth of large-scale instabilities if strong liquid/vapor momentum coupling is present. The proper conclusion to draw if weak liquid/vapor momentum coupling is present, or large droplet sizes are used, is less clear. As in other aspects of the steam-explosion problem, either larger scale tests are required or models/numerical techniques must be developed to represent more of the dominant physics if the eventual claims of high confidence are to be justified.

168 VI. ANALYSIS OF STEAM EXPLOSIONS WITH SIMMER-11 This chapter presents the details of the SIMMER-II calculations for the reactor case. For each of the five cases summarized in Chap. I, a discussion is given to put these calculations in context. In the first two cases, several additional SIMMER-II calculations were performed. The results of these additional runs are discussed to show the significance of the single calculation quoted for each case in Chap. I. A table of all results is presented in a summary section.

A. Case 1 - 20% Premixed The first case examined was based on the 20% premixed configuration used in the 1980 SIMMER-II ZIP study.1 Cases 1 and 2 were conceived in the ZIP study as appropriate for assessing whether steam explosions represented a serious threat to containment integrity. Mixing ~20% of the corium with water was identified as a minimal requirement in the 1975 Reactor Safety Study1197 for an interaction leading to containment failure. The initial configuration for the present case 1 is shown in Fig. 86. The structural setup is that described in Chap. III. The cylindrical mixing zene is shown more clearly in Fig. 87, reproduced from the ZIP study and here representing axial nodes 13 to 66. This is a volume fraction plot with the radial direction to the right and the axial direction upward. A small vapor buffer layer was assumed to exist beiween the core pool and the material below because of the initially large temperature differences, that is, the water could not have been in contact with the pool of molten-core materials. The mixing region was 1.85 m in diameter and the molten-cors pool was about 2-m deep. The initial distribution of liquid water is shown in Fig. 88. The lower head was assumed to be filled except for the vapor buffer region. The volume of water displaced by the core material and the film-boiling vapor layers in the mixing region was transferred to the downcomer. Tie core's material temperature was assumed to be 3 100 K, which is similar to that assumed in the 1975 Reactor Safety Study. The water was assumed to be saturated at 10^ Pa. The mixture region was assumed to be composed of 50% fuel, 25% water, and 25% steam by volume and to be fragmented to 300-jjm-diameter globules. Three separate investigations were carried out starting with this initial configuration. First, sensitivity to the water-droplet size was examined, because of the difficulty in justifying the water flow regime assumed in SIMMER-II. Second, a calculation without lower head failure was run to compare

169 OUTLET PIPING

SOLID STRUCTURE

12.54 m INLET PIPING DOWNCOMER MOLTEN CORE WATER

PREMIXED REGION CORE SUPPORT FORGING

MOVEABLE STRUCTURE

•-OUTLET TO KEYWAY

Fig. 86. Configuration model for case 1,

170 Holten Core Pool

Prenixed Renion

Vapor Buffer

Core Support Forging

Fig. 87. Initial core material distribution.

171 Downconer

Lower Plenum

Premixed Reoion

Fig. 88. Initial water distribution.

172 results from the new SIMMEK-II models with the ZIP calculational results. Third, the configuration was modified to deteimine how much dissipation might be anticipated if effects omitted in the ZIP study are included.

1. Investigation of Water Surface Area. In Chap. IV where the SNL experiments are evaluated, there appears to be a model sensitivity to the surface area of water (actually water-droplet size in the model). The quenching effect of heat transfer into the bulk water was important in the experiments for limiting the duration of the pressure pulse. The kinetic energy produced from previous reactor scale SIMMER-II calculations (the ZIP study) possessed an insensitivity to heat transfer. However, part of this result could be related to the hsat-transfer-model/EOS combination employed. The standard fuel-droplet size (3GO ^m) was used in two SItoMER-II cases that were run only through the early part of the interaction. One used the standard (ZIP study) water-droplet size of 300 #m. The other case used a water-droplet size itduced by an order of magnitude. This reduction applied both within the interaction zone and in the adjacent water. Although the initial vaporization rate within the interaction zone increased by a factor of 2.7, the peak pressure in the two cases was within a factor of 1.1 after 2 ms (125 MPa vs 115 MPa). By this time the water in the interaction zone was all vaporized in one case and nearly so in the other. Con- sequently, in this reactor corium/water configuration, the water-droplet size is not a sensitive parameter.

2. Comparison with the ZIP Study Using the New Models. To provide a basis for comparison, a complete recalculation of the ZIP case was made with the new models and with only minor adjustments in input. Explicitly, the changes were as follows: (a) The water EOS was updated and the new heat-transfer and vaporization/condensation modifications were included. (b) All droplet sizes were set to 100-nm diameter based on the investigation with the shallow-pool experiments (Chap. V). (c) The freezing point of corium was lowered by 10 K (to 3 090 K). Unless fuel vapor exists and unless fuel is more than 1 degree above the melting temperature, heat transfer from fuel to vapor does not occur (in SINMER-II). This change guarantees such a heat-transfer path.

173 The liquid-volume-fraction plots for this SIMMER-II calculation are shown in Fig. 89. Each contour line represents a 10% change in liquid-volume fraction. Because the lower head was assumed not to fail, axial nodes 1 to 12 were eliminated from the calculation. The characteristics of this expansion are similar to those the ZIP study with the water proceeding up the downcomer and down the inlet pipe, while the corium makes a two-phase impact concentrated toward the center of the head. The major qualitative difference in the expansion characteristics is the increase in the vaporization of water in the vessel. The ZIP case had a water-rich corium/water liquid mixture along the sides of the expanding corium. Its absence here is mainly the consequence of the water's surface vaporization. Also, the ZIP case assumed larger droplets, up to 1 mm in radius, above the premixed region. Pressures produced at various locations within the vessel for this calculation are shown in Figs. 90-94. The total force on the upper head is shown in Fig. 95. (As in the ZIP study these curves have a resolution of 1 ms after the first millisecond.) At the inlet plenum bottom, the spurious single-phase peak pressure of the ZIP study is eliminated; however, the pressure remains above 100 MPa for about the same time, -15 ms. The upper head impact pressures are higher toward the center of the vessel. The increased water vaporization and the smaller droplet size during the expansion more than compensate for the excessive steam pressures present in case 2 of the ZIP study. A stronger loading bias toward the apex of the upper vessel head was produced in this calculation with the peak pressure in the center being 550 MPa (vs 375 MPa in the ZIP study) while the integrated peak force is 2.8 GN (vs ~2.6 GN in the ZIP study). The total fluid kinetic energy is plotted in Fig. 96. As expected from the higher impact pressures, the peak value of 3 820 MJ is higher than the 3 400 MJ of the ZIP study. Of interest is the fact that only 889b of this peak kinetic energy (3 350 MJ) is associated with upward moving fluid, as shown in Fig. 97. (For the present purpose, upward moving fluid is defined by that fluid existing in axial nodes 25 to 66 and radial nodes 1 to 12 in Fig. 86.) The remaining kinetic energy, shown in Fig. 98, is mainly associated with water in the downcomer and inlet pipe. If detailed analysis of loads on the head structure is desired, the difference in timing between loads on the bolts produced by water in the downcomer and loads delivered directly on the vessel head by corium impact requires consideration.

174 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIOU1D VOLUME FRAC'ION Of LlQj'.D TIME 5 300MS T1ME 1 TIME OOOM£

MINz 7 95E-05 MAX; 1 00E<00 Cl: 1 OOE-01 MlN= 7 SSE-O'. MAXz 1 OOE'OO CIz 1 OOE-01 MINz 7 95E-05 MAXr 1 00E-00 Clz 1 OOE-01

(a) (b) (c)

VOLUME FRACTION Of LiPl'I VOLUME FRACTION OF LIQUID VOLUME FRACTION OF |_ I Qu I D TIME IS OOOJSS TIME 20 000MS TIME !i JGOMJ

•»«: I OOt'OO Cl: I OOE-01 MIN: 7 95E-05 MA«; 1 00E-00 Cl: I OOE-ul MINr 7 q?,E MAxr | COE»OO Clz I JOE-OI

(d) (e) (f)

Fig. 89. Liquid-volume-fraction plots for comparison with the ZIP study.

175 VOLUME FRACTION OF L10U1D VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIOUlO TIME 30 OOOMS_ TIME 35 OOOMS_ TIME 40 OOOMS_

MIN; 7 95E-0S MAX: 1 OOE-00 Cl: 1 OOE-OI - 7 9SE-05 MAX= 1 OOE'OO Cl: 1 OOE-01 MIN= 7 9SE-05 MAX= 1 OOE'OO C|: I OOE-01

(g) (h) (i)

VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME aS QOOHS_ TIME 50 OOOMS_

MIN= 7 95E-05 MAXr 1 00E"00 Cl: 1 OOE-01 MIN: 7 95E-05 MAX; 1 OOE'OO Cl= I OOE-01

(j) (k)

Fig. 89. (cont.)

176 5000 PRESSURE AT INLET PLENUM BO""1""1'

400.0 -

300.0 -

Sj 2000 a.

100.0

0.00 0.01 0.02 0.03 0.04 0.05 TIME (s)

Fig. 90. Pressure at the inlet plenum bottom for comparison with the ZIP study.

200 0 -. PRESSURE IN DOWNCOMER

160.0 -

s. 120.0 -

i 80.0-

400-

00 0.00 0.01 002 0.03 TIME (s) Fig. 91. Pressure at the downcomer top for comparison with the ZIP study.

177 ruu.u . PRESSURE AT TOP OE VESSEL 600 Or (I 500 o -:

300.0 - PRESSUR E 200.0 - / 100 0 -

-T-r-n-r-pr o.o00- 0 001 0.02 J0.03 0.04 0.05 TIME (s)

Fig. 92. Pressure at the top of vessel for comparison with the ZIP study.

600.0 -i PRESSURE AT HEAD CURVATURE

~~0.0 T" T" "I ! 0.00 0.01 0.02 0.03 0.04 0.05 TIME (s) Fig. 93. Pressure on the head at 30° to the vertical for comparison with the ZIP study.~~

~~178 uuu PRESSURE AT VESSEL FLi\NGE~~

~~o.o- -Till r i i I I I j i i I 1 1 1 iT^T i' • 1 ' ' ' 1 1 0.00 0.01 0.02 0.03 0.04 0.05~~

~~TIME (s)~~

~~Fig. 94. Pressure on the head at 70° to the vertical for comparison with the ZIP study.~~

~~FORCE ON HEAD~~

~~00 ooo 001 002 003 0.05 TIME (s) Fig. 95. Force on the head for comparison with the ZIP study.~~

~~179 TOTAL FLUID KINETIC ENERGY FOR NEW MODEL RERUN~~

~~4 0E.09~~

~~MINIMUM VALUE= 0 MAXIMUM VALUE= 3.82094E+09 Fig. 96. Total fluid kinetic energy (KE) for comparison with the ZIP study.~~

~~UPWARD FLUID KINETIC ENERGY FOR NEW MODEL RERUN~~

~~» 0E»09, r-~~

~~MINIMUM VALUE= 0. MAXIMUM VALUE= 3.35139E+O9 Fig. 97. Upward fluid kinetic energy for comparison with the ZIP study.~~

~~180 MISCELLANEOUS FLUID KINETIC ENERGY FOR NEW MODEL RERUN~~

~~TIME (S)~~

~~MINIMUM VALUE: MAXIMUM VALUE' 8.06170E+08 Fig. 98. Kinetic energy of materials not in the core and above core regions for comparison with the ZIP study.~~

3. Case 1 Update. The comparison with the ZIP study shows that the new heat-transfer and new EOS modifications give overall results that are comparable. Within the limitations of the SIKMER-II formalism, reduction in head loadings can be achieved only by consideration of lower head failure and other input changes. The changes made for the updated case are as follows: (a) The lower head failure model was inserted. Peak inlet plenum pressures exceeded the static failure pressures by more than an order of magnitude in the reference case 1. Therefore, lower head failure should occur. (b) The droplet sizes were changed to those of the ZIP study. Calculations showed that these sizes overpredict the SNL experiment used for calibration in the ZIP study when the new heat-transfer formalism is used. Consequently, they should still be conservative. (c) Steel particles were inserted in axial nodes 46 through 60 to represent the structure above the reactor core. Although residual strength of such a structure cannot be accounted for in SIMMER-II, inclusion of 35 000 kg of steel will introduce inelastic collisions and change the character of the expansion.

181 (d) The solid corium heat capacity was changed to 618.5 J/(kg*K), the UO2 value just below the melting temperature, because excessive quenching of solid corium does not occur with the water-lean premixture in this problem. The U(U melting temperature was somewhat arbitrarily set at 3 000 K. Some discussion of the results of this case is given in Chap. I. The averaged pressure differential over the lower head is shown in Fig. 99. The static failure pressure quoted in the ZIP study, 44 MPa (6 400 psi), was exceeded relatively early (0.7 ms). Calculated vessel displacement is given in Fig. 100. The failure value, 5 in., was exceeded at 3.5 ms. The liquid-volume fractions are given in Fig. 101 for different times. Following failure, the main mass moving downward was the failed part of the lower vessel head (11 000 kg for the failure radius at 1.6 m). The fluid that moved downward was a two-phase spray. Compared with the results of the reference case 1 (see Fig. 89), water was vaporized more slowly from the sides of the upward moving corium, less water was forced up the downcomcr, and the steel particles made the impact of the upper head more coherent radially and in time.

~~2 250^ PRESSURE DIFFERENTIAL~~

~~00 0 0 0 5 10 1.5 2.0 2.5 3.0 3.5 TIME (ms)~~

~~Fig. 99. Pressure differential across the lower head in case 1 (update).~~

~~182 6.0 -n DISPLACEMEN7 OF HEAD~~

~~c 4.0 T E-~~

~~3.0- U < Q. 20-~~

~~1.0-~~

~~1.5 2.0 2.5 3.0 3.5 TIME (ms)~~

~~Fig. 100. Lower head displacement before failure in case 1 (update).~~

However, a lowei tail of liquid developed from the existence of the relief path in the downward direction, increasing impact incoherence. Pressures (forces) produced at various in-vessel locations to compare with Figs. 90-95 are shown in Figs. 102-107. (These plots have the resolution of every SIKMER-II time step.) The lower plenum pressure is drastically reduced following lower head failure. The magnitude of the peak upper head loading was more biased toward the center, although the relative loading at 70° to the vertical was increased at the time the peak center loads were obtained. The kinetic energy produced is divided into two parts, the lower head's kinetic energy and the fluid's kinetic energy. The lower head's kinetic energy reached a maximum of 640 MJ at 24 ms. At this point the last radial node of the head reached the bottom of the calculative mesh, and the moving head no longer appeared in the calculation. The fluid's total kinetic energy is shown in Fig. 108. However, because kinetic energies appear in various parts of the problem and then are dissipated, this total is not a good indication of the total mechanical energy produced. To obtain a reasonable total, the kinetic energy was divided into the four parts shown in Table XV. Plots for these four regions are in Figs. 109-112. Summing these maximums gives 1 130 MJ for the

~~183 VOLUME FRACTION OF LIQUID iJWE FRUCT ION OF LlOUlC VOLUME FRACTiON OF LIQUID TIME 10 DOOMS •IE ODOMS TIME S DOOMS~~

~~M-. "•' 78E-0S MAX; 1 OCE-00 Cl: 1 OOE-OI MIN- 7 78E-05 MAX; 1 OOE + 00 Cl: 1 OOE-01 MIN: 7 78E-°S MAX= '• OOE-00 Cl= 1 ODE-01~~

~~(a) (b) (c)~~

~~VOLUMt FRACTION CF LIQUID VOLUME FRACTION OF LIQUID TIME 15 OOOMS TIME 20 OOOMS~~

~~MIN= 7 78E-05 MAX= 1 OOE+00 OOE-01 MIN= 7 78E-05 MAX= 1 OOE+00 CI= 1 OOE-01 (d) (e)~~

~~Fig. 101. Liquid-volume-fraction plots for case 1 (update).~~

~~184 VOLUME FRACTION OF LIQUID VOLUME FRACTION Of LIQUID TIME 25 OOOMS TIME 30 OOOMS~~

~~MINU 7 78E-05 MAX; 1 OOE+00 Cl= 1 OOE-Oi - 7 782-05 MAX- 1 OOE+00 C] OOE-01~~

~~; (g)~~

~~VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 35 OOOMS^ TIME 40 OOOMS~~

~~i I MINI:: 7 78E-05 MAX: . 00E*00 Clz ] OOE-01 MIN= 7 78F-05 MAXr I QOE^QQ CI = I OOE-01 (h) (i)~~

~~Fig. 101. (cont.)~~

~~185 VOLUME FRACTION OF LIQUID1 VOLUME FRACTION OF LIQUID TIME 50 OOOMS TIME 45 OOOMSr~~

~~MIN= 7 78E-05 MAX: 1 OOE+00 CI= 1 OOE-01 MINr 7 78E-05 MAX = 1 OOE+00 CI= 1 OOE-01 (3) (10~~

~~VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 60 OOOMS TIME 55 OOOMSr~~

~~MIN= 7 78E-0S MAX= 1 OOE+00 CI= 1 OOE-01 MIN= 7 78E-05 MAX= 1 OOE+00 CI= 1 OOE-01 (l) ("0~~

~~Fig. 101. (cont.)~~

~~186 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 65 OOOMS TIME 70 OOOMS.~~

~~MIN= 7 78E-05 MAX; 1 OOE+00 CI: 1 OOE-01 MINr 7 78E-05 MAX; 1 OOE+00 U: 1 OOE-01~~

~~(n) (o)~~

~~VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 75 OOOMS TIME 80 00OMS_~~

~~MIN= 7 78E-05 MAX= 1 OOE+00 CI = 1 OOE-01MIN; 7 78E-05 X- ! OOE+00 CIr ! OOE-0! (P) (q)~~

~~Fig. 101. (cont.)~~

~~187 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 85 OQQMS TIME 90 OOOMS~~

~~MIN: 7 78E-Q5 MAX: 1 OOE+00 C] Q0E_Q1 MIN= 7 78E-05 MAX = 1 OOE+00 CI= 1 OOE-01~~

~~(r) (s)~~

~~VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 100 OOOMS TIME 95 OOOMS~~

~~MIN= 7 7SE-05 MAX= 1 OQE+00 CI= 1 OOE-01 MIN: 7 78E-05 MAX= 1 OOE+00 CI= 1 OOE-01~~

~~(t) (u)~~

~~Fig. 101. (cont.)~~

~~188 c PRESSURE IN PLENUM 1 T 200.0-~~

~~<0 Qu 150.0 -; w OS 100.0- ae, ou~~

~~50.0-^~~

~~o.o0.0- 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 TIME (s)~~

~~Fig. 102. Pressure at the inlet plenum bottom for case 1 (update).~~

~~60.0 -i END OF DOWNCOMER~~

~~50.0 -i~~

~~30.0 -. I a.pa a- 20.0 -j~~

~~IO.O-;~~

~~0.0 • • ' I ' ' ' ' I ' ' ' • I ' • ' ' I ' ' ' ' I ' • T^r-j-r-r-t • | • • • • | • ^ • • | • • • • | 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 TIME (s)~~

~~Fig. 103. Pressure at the top of downcomer for case 1 (update).~~

~~189 250.0 -i TOP OF THE VESSEL~~

~~200 0 -~~

~~iO a. 150.0 - w D 100 0 -~~

~~50.0-~~

~~0.0 —i i i i i i i i i i i i i i i i i i i i r i i i i i i i i i i i i i i i i i i i i i i i i i i i i i 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10~~

~~TIME (s)~~

~~Fig. 104. Pressure at the top of the vessel for case 1 (update).~~

~~175.0 -q PRESSURE AT THE HEAD CURVATURE 150.0J~~

~~0.0 0.00 001 002 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 TIME (s)~~

~~Fig. 105. Pressure on the head at 30° to the vertical for case 1 (update).~~

~~190 700- PRESSURE AT VESSEL FLANGE~~

~~60.0^~~

~~500- (0 a. 5 40.0 u a: 30.0 Of a. 20.0~~

~~10.0 -.~~

~~0.0 " i | i i I I | iTTTi i i i i | i i i • | i i • • | • • i i | i i i i-| i i i i i 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 010 TIME (s)~~

~~Fig. 106. Pressure on the head at 70° to the vertical for case 1 (update).~~

~~r 'j FORCE ON HEAD~~

~~10.0^ 9.0-; 8XH~~

~~g 6.(H a: ° 5.0-3 4.0-^~~

~~2.0 H~~

~~0.0 | ' ' " • J I < I Tj-r-l-T-r [TTT I I I I 1 T-| f TTT-pTT I J J T- 1"'T T~p-1~rTr"] 0.00 0.01 0.02 0.03 0.04 0.05 006 0.07 0.08 0.09 010 TIME (s)~~

~~Fig. 107. Force on the head for case 1 (update).~~

~~191 TOTAL FLUID KINETIC ENERGY FOR CASE 1 i. ot «~~

~~e.octoa -~~

~~2 0E*08 -~~

~~,,,,,,,., I ,., I ... 0.00 04 06 TIME (S)~~

~~MINIMUM VALUE= 0. MAXIMUM VALUE= 8.26685E+08~~

~~Fig. 108. Total fluid (SIMMER-II liquid and vapor velocity fields) kinetic energy for case 1 (update).~~

~~TABLE XV FLUIP KINETIC ENERGY PARTITION~~

Label Peak Upward i - 1 to 12 647 MJ j - 25 to 66 Downward i - 1 to 15 373 MJ j - 1 to 16, and i - 1 to 12 j - 16 to 24 Downcomcr and Inlet Pipe i - 13 to 15 104 MJ j - 18 to 47 Outlet Pipe i - 13 to 15 5 MJ j - 49 to 66

~~192 FLUID KINETIC ENERGY FOR CASE 1~~

~~9 • QC*O«i—, 1 , 1 , 1 1 [—i 1 1 1 1 1 1 p~~

~~0.00 .02 .04 06 TIME (S)~~

~~MINIMUM VALUE= 0. MAXIMUM VALUE= 6.47069E+08 Fig. 109. Upward fluid kinetic enerev for case 1 (update). ow»iH..*tMJ iwNtllC ENERGY FOR CASE 1 4.0£»08,—i—,—r-~~

~~.04 06 TIME (S)~~

~~MINIMUM VALUE= 0. MAXIMUM VALUE= 3.73363E+08 Fig. 110. Downward fluid kinetic energy for case 1 (update).~~

~~193 OOWNCOMER AND INLET PIPE KINETIC ENERGY FOR CASE 1~~

~~1. 5E*06i—T 1 1-~~

~~04 06 TI ME (S )~~

~~MINIMUM VALUE= 0. MAXIMUM VALUE= 1.04419E+08~~

~~Fig. 111. Downcomer and inlet pipe kinetic energy for case 1 (update). OUTLET PIPE KINETIC ENERGY FOR CASE 1~~

~~6 0E»O6~~

~~TI ME (S )~~

~~MINIMUM VALUE= 0. MAXIMUM VALUE= 4.92049E+06 Fig. 112. Outlet pipe kinetic energy for case 1 (update).~~

194 fluid's kinetic energy or 1 770 MJ for the total kinetic energy. Consequently, for case 1 (update) approximately 57% of the total kinetic energy was directed downward and 37% of the total kinetic energy was contained in the upward moving two-phase "slug." Finally, actual coherency of impact can be judged by the peak force squared divided by the peak in the upwardly directed kinetic energy, F*VKE . The force represents the time rate of change in impulse, and the kinetic energy is related to the square of the impulse. Consequently, F /KE is a measure of the maximum fractional change in impulse over the period of fluid impact. The reference case 1 for comparing with the ZIP study gives 2.3 GN/m. With the modifications, case 1 (update) gives 1.6 GN/m. The fluid tail shown in Fig. 101 does result in a less coherent impact.

B. Case 2 - Conservative Premixing, Explosion, and Expansion No transparent argument exists to quantify the degree of conservatism to associate with the postulated conditions in case 1. Case 2 is an attempt to produce a plausibly conservative case, starting with an unmixed configuration so that the effects of fluidization may be consistently included. The initial configuration used for this case is shown in Fig. 113, although axial nodes 1 to 13 used for modeling lower head motion and venting were eliminated from the premixing calculation. The corium temperature was 3 100 K, the corium's heat capacity was 0.54 J/(gm»K), the corium's heat of fusion was 276 J/gm, and the corium's melting temperature was assumed to be 2 700 K. This may not be the upper limit for the corium temperature, which might be as high as the saturation temperature of steel at some elevated pressure; however, this temperature is higher than those calculated as "best estimates" for a molten pool in MELPROG. • The amount of corium was assumed to be 131 760 kg, in other words, the entire core. Although crusts must exist so that radiation losses do not lower the temperature, and consequently the entire core cannot be molten at once, an unknown amount of steel could have melted in from tue upper core structure. Also, it is probably the inertia of the movable material, not the actual amount of molten corium, that is important. The 18 000 kg of water assumed to be present in the lower plenum was assumed to be saturated at an assumed vessel

~~Information provided by W. J. Camp, Sandia National Laboratories, November 27, 1984.~~

~~195 ••• M • •1MB~~

~~•M « am 60~~

~~tarn m mm m OUTLET PIPING •1 H •i m~~

~~•1 LV •1 • .SOLID STRUCTURE~~

~~50 •i • INLET PIPING -• 12.54 m i • i - DOWNCOMER~~

~~CORIUM BLOCKAGE VAPOR WATER CORE SUPPORT FORGING~~

~~5.20 MOVABLE STRUCTURE~~

~~•••Ml •** OUTLET TO KEYWAY 1 5 10 15 h—1.613—^ I p 2.537 H~~

~~Fig. 113. Initial configuration for the SDMER-II premising calculation in case 2.~~

196 pressure of 1 atm. Elevated pressures could well be present, which would allow less steam production and more mixing during a pour of corium into the lower plenum, but with 1 atm, questions regarding the lack of a trigger are reduced. A partial blockage was assumed leading to a pour diameter of 1.8S m. This is not as large as allowed by SNL is a recent study but considerably larger than judged to be present by most experts in a recent review of steam explosions. Unmodified SIMMER-II heat-transfer assumptions and models were used with a droplet size of 20 mm in diameter for the corium and the water. These heat-transfer assumptions were shown to underestimate steam production in a SIMMER-II calculation of a SNL test that was run with 15-mm-diameter droplets see (App. Q). Decreasing heat transfer further should assure a low steam production rate here, also thereby maximizing the extent of mixing. The mssh size in the Fig. 113 mixing region tends to be greater than 10 cm. This may be considered a SIMMER-II mixing length scale for this problem. Theofanous has argued that 10 cm is an upper limit over which premixing can occur. Contour plots of fuel and coolant (water plus steam) densities over the first second of fuel (coriusn)/water contact with these heat-transfer assumptions are shown in Figs. 114-119. The corium is seen to push the water away and up the downcomer. In contrast to the experiment in which all the melt mixed with water, in this calculation mixing only occurred around the edges of the downward pouring corium. Not until 1 s did enough vapor pressure develop to cause counterflow through the corium. In addition to the limited heat transfer and the downcomer escape path, this time delay is a consequence uf the inertial constraint posed by the corium pool. This inertial effect is important. It is not considered in common quasi-steady-state fluidization arguments. The first preliminary SIKMER-II explosion and postexplosion calculation was started after 1 s of mixing.

1. Explosion After 1 s of Mixing (Scoping Calculation). At 1 s an explosion was assumed to occur. The SIMMER-II calculation was rezoned so that the new SIMMER-II heat-transfer and vaporization/condensation models could be used. The droplet sizes of 300-^m diameter for coriucj and 75-jtm diameter for water, obtained from the correlation to SNL experiments, were used everywhere. The steel blockage restricting core motion to the 1.85-m diameter value was changed to adiabatic steel particulate at 300-ym diameter. No lower head failure was allowed for this scoping calculation, and this part of the mesh was ignored.

~~197 TI ME .000 MS STEAM EXPLOSION EVENT SEQUENCE FUELDENSI TV COOLANT DENSITY~~

~~MINI MUM 1.00E+00 5.00E-02 MA X I MUM 7.41E+03 1 07E+03 CONTOUR INTERVAL 7.41E+02 1.07E+02 PEG ION-RADIAL { 1.15) AXIAL ( 1.54)~~

~~Fig. 114. Initial conditions, SIMAER-II conservative premixing calculation (case 2). TIME 300.000 MS STEAM EXPLOSION EVENT SEQUENCE FUEL DENSITY COOLANT DENSITY~~

MINIMUM~ 1 .00E+00 5.00E-02 MA/I MUM 7.41E+03 1.07E+03 CONTOUR INTERVAL 7.41E+02 1.07E+02 PEOtON-RADIAL ( 1.15) AXIAL ( 1.54) Fig. 115. Water-fuel contact, SIMMER-II conservative premixing calculation (case 2).

~~198 STEAM EXPLOSION EVENT SEQUENCE FUEL DENSITY COOLANT DENSITY~~

MINI MUM 1 .OOE+OO 5.00E-02 MAX I MUM 7.41E+03 1 07E+03 CONTOUR INTrRVAL V . 4 1E+O2 1 .O7E+O2 PEGION-RADIAL ( 1.15) AXIAL ( 1.54) Fig. 116. Fuel contact with the support forging, SINMER-II conservative premixing calculation (case 2) (time = 500 ms).

~~tvPLU'jIUN EVENT SEQUENCE FUEL DENSITY COOLANT DENSITY~~

~~1 .OOE + OO 7.4 1E + 03 CONTOUR INTERVAL 7.41E+02 PECION-RADIAL ( 1.15) AXIAL ( 1 54~~

~~Fig. 117. Initial contact of fuel with the lower head, SIMMER-II conservative premixing calculation (case 2) (time = 700 ms).~~

~~199 T I ME 900.000 MS STEAM EXPLOSION EVENT SEQUENCE FUEL DENSITY COOLANT DENSITY~~

~~l.OOE+00 5.00E-02 7.41E+03 1.07E+03 CONTOUR INTERVAL 7.41E+02 1.07E+02 PEGION-RADIAL ( 1.15) AXIAL ( 1.54)~~

~~Fig. 118. Beginning of fuel pool breakup, SIKMER-II conservative premising calculation (case 2). I I ME 1000.000 MS~~

~~STF.AM EXPLOSION EVFNT SEQUENCE~~

~~FUEL DENSITY COOL AN I DENSITY~~

V 1 111 Ml JM 1 . 001+ 00 5 .OOE-02 lft./ l yiJM 7. 4 IF • 01 1 07E+03 ' Vi'O'J'-' 1NTEPVAL 7 .4 11+02 1 .O7E+O2 r ~E 0' O> 1- 'ADlAP L < 1 l j ) AX 1 AL ( i. e>4) Fig. 119. Two-phase fuel pool formed, SIMMER-II conservative premizing calculation (case 2).

200 Contour plots for the expansion fuel density and coolant density are given in Figs. 120-126. The expansion is similar to the shallow-pool calculation simulating a hypothetical plate obstruction (see Chap. V). Corium that poured into the lower plenum but did not mix with the water had only limited motion. Fluid was primarily accelerated up the center as well as up the sides of the vessel and then coalesced at the top. Pressures and forces for comparison with those of other cases are given in Figs. 127-132. (With the exception of the first millisecond, these plots have 1-ms resolution.) The pressures at the inlet plenum again would be sufficient to result in lower head failure. The peak impact pressures are reduced compared with case 1. However, although the impact is diffuse, the pressures at various spatial locations on the upper head are much more uniform, leading to a peak force of 0.94 GN. The fluid's kinetic energy plots are in Figs. 133-135. Total and upward kinetic energy have the same meaning as in Table XV. The miscellaneous category includes everything else but the upward kinetic energy. Except for the early peak coming from the acceleration of materials in the lower plenum, it represents the kinetic energy of water in the downcomer and inlet pipe. For T I ME 1020.000 MS STEAM EXPLOSION EVENT SEOUENCE FUEL DENSITY COOLANT DENSITY

~~1.00E+00 5 00E-02 7.41E+03 1 07E+03 CONTOUR INTERVAL 7.4IE+02 1.O7E+O2 PEG I ON-RADIAL ( 1.15) AXIAL ( 1 54) Fig. 120. Initial expansion instability, fluidized explosion conditions.~~

~~201 TI ME 10 10.000 MS STEAM EXPLOSION EVENT SEQUENCE FUEL DENSITY COOLANT DENSITY~~

MINI MUM I.OOE+00 5.00E-02 MAXIMUM 7.4 1E+03 1.07E+03 CONTOUR INTERVAL 7.41E+O2 I.07E+02 PEG I ON-RADIAL ( 1.15) AXIAL ( 1.54) Fig. 121. Instability development, fluidizcd explosion conditions. TIME 1030.000 MS STEAM EXPLOSION EVENT SEQUENCE FUEL DENS ITY COOLANT DENSITY

~~1.00E+00 5.OOE-O2 7.41E+03 1.O7E+O3 CONTOUR INTERVAL 7.41E+02 1.O7E+O? PEGION-RADIAL ( 1.15) AXIAL ( 1,54) Fig. 122. Venting beginning under fluidized explosion conditions.~~

~~202 TI ME 1040.000 MS STEAM EXPLOSION EVENT SEQUENCE FUEL DENSITY COOLANT DENSITY~~

l.OOE+00 5.00E-02 MAX I MUM 7.41E+03 1.07E+03 CONTOUR INTERVAL 7.41E+02 1.07E+02 REGION-RADIAL ( 1.15) AXIAL ( 1.54) Fig. 123. Configuration just before impact, fluidized explosion conditions. TIME 1050.000 MS STEAM EXPLOSION EVENT SEQUENCE FUEL DENSITY COOLANT DENS IIY

~~1.00E+00 5 UOt -0J~~

~~7 41F. + 03 1 O7ttU.5 CONTOUR INTERVAL 7.41E+02 1 O/L+02 REG I ON-RADIAL ( 1.15) AXIAL ( I .54) Fig. 124. Material impact beginning, fluidized explosion conditions.~~

~~203 TIME 1060.000 MS STEAM EXPLOSION EVENT SEQUENCE FUEL DENSITY COOLANT DENSITY~~

MINI MUM 1.00E+00 5.00E-02 MAXIMUM 7.41E+03 1.07E+03 CONTOUR INTERVAL 7.41E+O2 1.07E+02 REGION-RADIAL ( 1.15) AXIAL ( 1.54) Fig. 125. Material impact complete and rebound beginning, fluidized explosion conditions. TI ME 1 100.000 MS STEAM EXPLOSION EVENT SEQUENCE FUEL DENSITY COOLANT DENSITY

~~MINI MUM 1 .OOE-t-00 5 OOE-02 MAX I MUM 7.41E+03 1.07E+03 CONTOUR INTERVAL 7.41E+O2 1 07E+02 REGION-RADIAL ( 1.15) AXIAL ( 1,54) Fig. 126. Final configuration, fluidized explosion conditions.~~

~~204 150.0 T PRESSURE AT INLET PLENUM BOTTOM~~

~~0.0 • ''' '* i' ' ' ' i ' ' ' ' i ' '' ' i' ' '' i ' ' ' ' I ' ' ' ' I ' ''' i ' ' ' * i ' ' * ' i 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10~~

~~TIME (s)~~

~~Fig. 127. Inlet plenum pressure, fluidized explosion conditions.~~

~~180.0-1 PRESSURE IN DOWNCOMER~~

~~0.0 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 TIME (s) Fig. 128. Pressure at the top of downcomcr, fluidized explosion conditions.~~

~~205 140.0 - PRESSURE AT TOP OF VESSEL~~

~~120.0 -~~

~~100.0 - (0 80.0 :~~

~~g? 6o.o H u a. 40.0-~~

~~20.0~~

~~0.0 . . I | M I I | iT | | | | 1.00 1.01 102 1.03 1.04 1.05 1.06 1.07 1.0B 1.09 1.10 TIME (s)~~

~~Fig. 129. Pressure at the top of vessel, fluidized explosion conditions.~~

~~120.0 -i PRESSURE AT HEAD CURVATURE~~

~~0.0 • • '• i • • ' • i ' * ' * i * * * * i * * * * i ' ' * ' i ' ' ' ' i * * * * i * * ' * i ' ' ' * i 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 TIME (s)~~

~~Fig. 130. Head pressure at 30° to the vertical, fluidized explosion conditions.~~

206 Fig. 131Headpressureat70°othvertical,fluidizeexplosionconditions Fig. 132Forceonth e head,fluidizedexplosionconditions. Q_ o w o {ESSURE (MPa "2 u. Ol 100.0 - 80.0-J 90.0-1 70.0^ 60.0^ 40.0 T 50.0-^ 30.0-^ 20.0-1 10.0 -j 10.0-I 0.0 - 9.0 -. 7.0-| 8.0^ 4.0 -. 6.0 -i 5.0 -. 3.0 -j 0.0 1inI•[, i, 1.00 1234567891.1 1.00 12345678 1.091.10 PRESSURE ATVESSELFLANG FORCE ONHEAD TIME (s) TIME (s) 207 TOTAL FLUID KINETIC ENERGY FLUIDIZED EXPLOSION

~~J OE*09i~~

~~5 at•OB -~~

~~100 1 02 104 1.06 108 1.10~~

~~MINIMUM VALUE- 3.71798E+06 MAXIMUM VALUE= 1.72872E+09 Fig. 133. Total fluid kinetic energy, fluidized exp.osion conditions.~~

~~UPWARD FLUID KINETIC ENERGY FLUIDIZED EXPLOSION~~

~~1 I • ' I I • I • I i i i I I i 00 1 02 I 04 I 06 1 08 I 10~~

~~MINIMUM VALUE= 1.71403E+06 MAXIMUM VALUE= 1.49003E+09 Fig. 134. Upward fluid kinetic energy, fluidized explosion conditions.~~

~~208 MISCELLANEOUS FLUID KINETIC ENERGY FLUIDIZED EXPLOSION~~

~~3 0E«0»r—i 1—r~~

~~2.0E40B~~

~~I . , , ' 04 1.08 I.OB TIME (S)~~

MINIMUM VALUE= 2.Q0395E+Q6 MAXIMUM VALUE= 2.73141E+08 Fig. 135. Miscellaneous fluid kinetic energy, fluidized explosion conditions. this calculation, the coherence parameter for fluid impact, F*/KE , is 0.59 GN/m, reflecting the diffuse nature of the impacting spray on the upper head.

2. Explosion After 0.7 s of Mixing (Second Scoping Calculation). By 1 s into the premixing calculation, the fuel began to fluidize. Fig. 136 shows the corium mass that moved to an elevation below that of the bottom of the initial corium pool as a func tion of the premixing time. A value of 40 metric tons was reached at about 0.7 s. Beyond this time, counterflow occurred and little additional corium was added to the lower plenum. As shown in Fig. 117, a time of 0.7 s also is meaningful, because that is when fuel initially contacted the bottom of the lower plenum and could be associated with triggering the explosion. The assumptions uade in this calculation are the same as those in the scoping calculation made after 1 s of mixing. The one exception is that the water-droplet diameter was set to the same value as the fuel diameter (300 #m) through an oversight. Again, the most important assumption was the exclusion of lower head failure.

~~209 60.0~~

~~0.2 0.4 0.6 0.8 1.0 TIME(s) Fig. 136. Corium below the bottom of the initial corium pool, SIKMER-II conservative premixing calculation (case 2).~~

Liquid-volume-fraction plots are given in Fig. 137. As in the previous calculation, corium that poured into the lower plenum but was not mixed with water had only limited motion. However, unlike the explosion at 1 s, a single tongue of liquid was accelerated up to the top of the head. Plots for comparing pressures and forces are given in Figs. 138-143, with the same resolution as before, 1 ms. In general, the driving pressures are reduced, because of the reduced extent of mixing. The impacting fluid produced higher pressures, but a narrower pulse width. Fluid motion was less diffuse and the peak pressures were more concentrated toward the center of the vessel head. Kinetic energy plots are given in Figs. 144-146. The total kinetic energy was reduced, although energy partition remained about the same. For this second scoping calculation F /KE =1.8 GN/m, comparable to the coherence obtained in case 1.

3. Case 2 - Final. The case in which premixing is calculated, although including many conservative aspects, resulted in reduced steam-explosion energetics compared with case 1. The evaluation of this case was completed by making the following modifications to the scoping calculation that started at 0.7 s.

~~210 T I ME 70Q.Q00 MS TI ME 7 10.000 MS STEAM EXPLOSION EVENT SEQUENCE STEAM EXPLOSION EVENT SEQUENCC »I1H LCMICa MEAD FAILURE WITHOUT HF,AD WITH ifJWER HC»D FAILURE WIlHOUT MEAD FAILUBE~~

~~(a) (b)~~

~~MINIUUU I OOE-01 MAXIMUM l Qu[»UO I OOEtOO CQWOUR inUKVAL CONTOUR INTERVAL 9 99E-02 9.99E-02~~

~~TIME 720 000 MS TI ME 7JO.000 MS STEAU CAPLOSLUN EVENT SEQUENCE F.IEAM EXPLOSION EVENT 'jECIUENCE WITH LOIUL.' HEAD TAILUBE HIIHOUI HEAD FAILURE WITH LOWER HEAD FAILURE WITHOUT MEAD FAILUHt~~

~~(c) (d)~~

~~UI NI MUM' I OOE-Oi UAKI MUM I OUL.OO I OOEtOO CONTOUR INTERVAl CONlLIUf) INTERVAl 9 iK-ai 9 99E-O2~~

~~Fig. 137. Liquid volume fractions, explosion at 0.7 s.~~

~~211 T IME 740 000 MS TIME 750.000 MS STEAM EXPLOSION EVEN' SEQUENCE STEAM EXPLOSION EVENT SEQUENCE WITH LOWER HEAD FAILURE WITHOUD7~T HEAD FAILUR-E WITHOUP T HEAD FAIL~~

~~(e) (f)~~

~~I OOE-03 I.OOE»00 CONTOUR INTERVAL 9 99E-02~~

~~TIME 760.000 MS TIME 770.000 MS STEAM EXPLOSION EVENT SEQUENCE STEAM EXPLOSION EVENT SEQUENCE URE WITHOUT HEAI WITH LOWER HEAD FAILURE WITHOUT~~

~~(g) (h)~~

~~1 OOE-03 1 OOE-03 I DOE•00 I OOE*00 I 0OE*0O l.OOEtOO CONTOUR INTERVAL CONTOUR INTERVAL 9 99E-0? 9 99E-02 9 99E-02 9 99E-02~~

~~Fig. 137. (cont.)~~

~~212 TI ME 780 000 MS TIME 7qo 000 MS STEAM E»Pin5l0N FVLNI '.EavJENCE MIMA C~~

~~I 00E-03 1 OOE-OJ~~

~~I O0E»00 onEOo I OOE*00 C0M0UR INTERVAL 9 99E-02 9.99E-0?~~

~~(i) (j)~~

~~T I ME 800 . i_'00 >-S TIME 5U0.00O MS STEAM EXPLOSION EVENT SEQUENCE STEAM EXPLOSION EVENT SIOUENCE *l TM LOWER HEAD FAILURE Wl TH MTH LOWER HEAP FAILURE~~

~~I 00E«00 CONTOUR INTERVAL 9 99E-02~~

~~(k) (1)~~

~~Fig. 137. (cont.)~~

~~213 1800-1 PRESSURE AT INLET PLENUM BOTTOM~~

~~150.0-~~

~~120.0 - a a. :£ u K 90.0-~~

~~a. 60.0-~~

~~30.0-~~

~~0.0 i i [ i • i i i i i i i i ' i i i i • i i i i • i i • i ' i i • i i i i i i i i i i i i i i i i i i i i i 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81~~

~~TIME (s)~~

~~Fig. 138. Inlet plenum pressure, explosion at 0.7 s.~~

~~PRESSURE IN DOWNCOMER~~

~~0.0 ~| 1111)111111-111 j—T~T~I I | I TT-T | 1 I IT f T T'l-f-J 0.70 0.71 0.72 0.73 074 0.75 0.76 0.77 0.7B 0.79 0.80 0.81~~

~~TIME (s) Fig. 139. Pressure at the top of the downcomer, explosion at 0.7 s.~~

~~214 300.0-1 PRESSURE AT TOP OF VESSEL~~

~~250.0-~~

~~200.0 - <£ ns 150.0 3~~

~~Q. 100.0 -~~

~~50.0-~~

~~0.0 —i i i i [ i i i i i i i i i i n i i ) i i i ' i i ' ' i i i i i i i i i i i i i i i i i i i i i i i i i i i 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 TIME (s)~~

~~Fig. 140. Pressure at the top of vessel, explosion at 0.7 s.~~

~~250.0-, PRESSURE AT HEAD CURVATURE~~

0.0 • • 1111 • 11' i' 1111111111 -i 1111111111) 11111111111111111 ii | 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 TIME (s) Fig. 141. Head pressure at 30° to the vertical, explosion at 0.7 s.

~~215 120.0-1 PRESSURE AT VESSEL FLANGE~~

~~100.0-~~

~~80.0-~~

~~u 60.0- i a.OS 40.0-~~

~~20.0-~~

~~Fig. 142. Head pressure at 70° to the vertical, explosion at 0.7 s~~

~~18.0-1 FORCE ON HEAD~~

0.0 -| ' 11' i •' i' i'''' I*1 lT11 ' • • ' i'''' I'' ' ' I' ' ' ' I ' ''' I' ''' I''' ' I 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 TIME (s) Fig. 143. Force on the head, explosion at 0.7 s.

~~216 TOTAL FLUID KINETIC ENERGY FOR CASE 2 NO FAILURE~~

~~-| i i i |-~~

~~MINIMUM VALUE= 6.40460E+05 MAXIMUM VALUE= 1 . 523J51E+09 Fig. 144. Total fluid kinetic energy, explosion at 0.7 s.~~

~~UPWARD FLUID KINETIC ENERGY FOR CASE 2 NO FAILURE~~

~~?a ao~~

~~MINIMUM VALUE= 1 .82975E+05 MAXIMUM VALUE= 1 . 33477E+OQ Fig. 145. Upward fluid kinetic energy, explosion at 0.7 s.~~

~~217 MISCELLANEOUS FLUID KINETIC ENERGY FOR CASE 2 NO FAILU~~

~~1 0t«08i~~

~~.74 76 TIME (S)~~

~~MINIMUM VALUE= 4 57484E+05 MAXIMUM VALUE= 2.82198E+08 Fig. 146. Miscellaneous fluid kinetic energy, explosion at 0.7 s.~~

(a) The lower head failure model was inserted. Unfortunately, the dynamic amplification factor of 2 was omitted in the stiffness coefficient of Eq. (M-7) in App. M so failure may be slightly early. Rerun of a corrected version of case 2 was not performed, because of the bounding character of case 4. (b) The oversight in the water-droplet size was corrected. The water size was uniformly set to 75-ym diameter. (c) Steel particles were inserted to represent upper core structure as in case 1. The density of particles was adjusted so that the total of 35 000 kg included the steel particles formed by breakup of the core-supporting blockage. Lower head failure for this case occurred at 2.7 ms. Figure 147 gives the liquid-volume fraction during the expansion process. More liquid progressed downward in this case compared with case 1. The pressures and forces are given in Figs. 143-153. (These plots have the resolution of every SIMMER-II time step.) The peak pressure at the apex of the vessel divided by the peak pressure at 70° from the centerline is about 2.4, compared with about 3.7 in case 1. The lower head's maximum kinetic energy is 613 MJ. The four components

~~218 VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIOUID VOLUME FRACTION OF LIQUID TIME 7QQ OQOMS_ TIME 705 DOOMS TIME ^^0 OOQMS_~~

~~MIN= Z 07E-0S MAX; I OOE'OO Cl: 1 00E-Q1 uINr 2 07E-0S MAX: 1 OOE'OO Cl: I 00E-0I MIN: Z 07E-05 MAX: 1 OOE'00 Cl= I 00E-01~~

~~(a) (b) (c)~~

~~VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME ^S OOOMS TIME 750 OOOMS~~

~~MIN= Z 07E-0S MAX= 1 OOE+00 CIr 1 OOE-OI MIN= 2 07E-05 MAX= 1 OOE+00 CI= 1 OOE-01~~

~~(d) (e)~~

~~Fig. 147. Liquid-volume fractions during the expansion process, case 2 (final).~~

~~219 VOLUC FRACTION OF LIOUIO VOLUME FRACTION OP LIQUID VOLUME FRACTION OF LIQUID T|MC 760 OOOWS TIME 770 OOOMS TIME 780 OOOMS,.~~

~~MINI 2 07E-0S «*»: I OOE'OO Cl: I QOE-01 MIN- j Q7E-0S M* : I OOE'OO CI= 1 OOE-01 HIN= Z 07E-0S M*Xr I 00E*00 Cl= I OOE-OI~~

~~(f) (g) (h)~~

~~VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 795 OOOMS. TIME 810 OOOMS~~

~~MINr Z 07E-05 MAXr 1 OOE+00 CI= 1 OOE-01 MIN= 2 07E-05 MAX= 1 OOE+00 CI= 1 OOE-01~~

~~(i) (j)~~

~~Fig. 147. (cont.)~~

~~220 250.0 - PRESSURE IN PLENUM~~

~~200.0 -~~

~~CO ex 150.0 - w OS m m Cd 100.0 - 05~~

~~50.0-~~

~~0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 TIME (s)~~

~~ro Fig. 148. Inlet plenum pressure, case 2 (final). 60.0-3 END OP DOWNCOMER~~

~~0.0 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 TIME (s) Fig. 149. Pressure at the top of the downcomer, case 2 (final).~~

~~:uu.u - PRESSURE AT TOP OF THE VESSEL~~

~~160.0 -j \ a. :s 120.0- u~~

~~CO u 80.0 : a; a.~~

~~40.0 : )~~

~~0.0 0 70 071 072 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 TIME (s)~~

~~Fig. 150. Pressure at the top of vessel, case 2 (final).~~

~~222 200.0 -i PRESSURE AT HEAD CURVATURE~~

~~200.0-n PRESSURE AT VESSEL FLANGE~~

~~0.0 0.70 0.71 0.72 0.73 0.H 0.75 0.76 0.77 0.78 0.79 0.80 0.81 TIME (s)~~

~~Fig. 152. Head pressure at 70° to the vertical, case 2 (final).~~

223 to.o- FORCE ON HEAD 9.0 -I 8.0 -I 7.0-1 6.0 -i u 5,0 i o 4.0- 3.0 -I 2.0- 10- 0.0 0.70 0.71 0.72 0.73 0.74 0.75 0.78 0.77 0.78 0.79 0.80 0.81 TIME (s) Fig. 153. Force on the head, case 2 (final). of the fluid's kinetic energy are given in Figs. 154-156. The sum of their psak magnitudes gives 1 052 MJ for the fluid's peak kinetic energy, or 1 665 MJ fcr the kinetic energy produced by the explosion. Consequently, for case 2, 62% of the kinetic energy was directed downward while 31% of the kinetic energy was contained in the upward moving two-phase "slug." Figure 157 shows how much of the fluid's total kinetic energy is realized at any one time. The coherence parameter for this case, F2/KE , is 1.3 GN/m. Reduction from case 1 might be anticipated because more fuel moved downward. Finally, we should emphasize that the explosion at 0.7 s followed a premixing calculation in which steam production was minimized. A complementary calculation, in which a comparison with the SNL experiments suggests steam production to be exaggerated, is given in App. U. At high steam generation rates, the calculated extent of premixing of corium and water in a reactor situation was very limited. With such configurations, large steam explosions do not seem possible.

~~224 UPWARD FLUID KINETIC ENERGY FOR CASE 2~~

~~TIME (S)~~

~~MINIMUM VALUE- 6.33803E+05 MAXIMUM VALUE' 5.I5547E+08 Fig. 154. Upward fluid kinetic energy, case 2 (final).~~

~~DOWNCOMER FLUID KINETIC ENERGY FOR CASE 2~~

~~MINIMUM VALUE= 1 79022E+05 MAXIMUM VALUED 1.04989E+08 Fig. 155. Downcomer fluid kinetic energy, case 2 (final).~~

~~225 MISCELLANEOUS FLUID KINETIC ENERGY FOR CASE 2 REGION SETS 1 JO 3~~

~~TIME (S)~~

MINIMUM VALUED 7.25921E+01 MAXIMUM VALUE = 4.J984IE+O8 Fig. 156. Downward and outlet pipe fluid kinetic energy, case 2 (final). (Curve 1 is the downward fluid kinetic energy; curve 2 is the fluid kinetic energy in the downcomer; and curve 3 is the fluid kinetic energy in the outlet pipe.)

~~TOTAL FLUID KINETIC ENERGY FOR CASE 2 i.oc*o»r~~

~~MINIMUM VALUE= 1.09154E+O6 MAXIMLMA VALUE= 8 . 72598E+08 Fig. 157. Total fluid kinetic energy, case 2 (final).~~

226 C. Case 3 - 75% Premised The explosion assumed to occur at 1 s in case 2 suggests that, if the corium on top of any premixture is fluidized, upwardly directed kinetic energy will be less effective in causing significant head loadings. Case 3 takes this assumption to the extreme case of total fluidization. A report by SNL6 that addressed the uncertainties in steam-explosion phenomenology proposed that the upper limit on corium-watcr premixing was 75% of the core, or 94 OOO kg. In case 3 all this corium was premixed with water. The configuration assumed is shown in Fig. 158. The mass of water in the premixture was 20 000 kg. An extra 8 000 kg of water was assumed to exist unmixed in the downcomer. The remaining solid corium (25%) was assumed to consist of unmovable crust and was ignored. The corium's temperature was assumed to be 2 800 K. Its heat capacity was 500 J/(kg«K), with no heat of fusion so that its energy above the water temperature of 4C0 K was 1.2 MJ/kg, the assumed best-estimate energy in the SNL study. To limit the height of the premixture, the initial steam-volume fraction was assumed to be 0.19 in the corium/water mixture. This led to a premixture height equal to that shown in Fig. 113, and used for the initial conditions in case 2. The corium was assumed to exist as 100-jtm-diameter solid particles and the water was assumed to exist as 100-jim- diameter droplets. The new models for SIMMER-II heat transfer and water vaporization were employed. These assumptions cause all the premixed water to vaporize within the first millisecond, producing 194 MPa steam at 922 K with a steam-volume fraction of 0.69. Because the SIMMER-II models do not have heat transfer between solid particles and steam, the particle-steam expansion is adiabatic (except in the downcomer) following the initial steam production. The lower head was assumed not to fail in this calculation. Density profiles of corium for the expansion are shown in Figs. 159-160. Initially a spray developed, then material decelerated and collected near the head, and finally a smeared shock front delineating upward moving corium from downward moving material developed. Water movement in the downcomer limited the rate of corium penetration in that direction. Pressures and forces are shown in Figs. 161-166. Although loading on the head was fairly uniform as in the previous fluidized case, the peak force on the head stayed below 1 GN. Lower head failure also should occur. The fluid's kinetic energy plots are given in Figs. 167-169, Although the upward fluid's peak the kinetic energy (1 834 MJ) significantly exceeded that of case 1 (647 MJ) and case 2 (516 MJ), the

~~227 60 = = =T | P OUTLET PIPING~~

~~'SOLID STRUCTURE 50---~~

~~12.54 : • •, "•• -> 1 INLET PIPING -. : :••: v-- n •. |]~~

~~.... "p s| 40 DOWNCOMER PREMIXED REGION WATER 30S~~

~~CORE SUPPORT FORGING~~

~~MOVABLE 5.20 STRUCTURE~~

~~OUTLET TO KEYWAY 1 5 10 —1.613—*| 2.537~~

~~Fig. 158. Configuration model, case 3.~~

~~228 TOTAL DENSITY OF FUEL TOTAL DENSITY OF FUEL TIME 000MS TIME 5 OOQMS~~

~~MINr 0 MAX= 2 18E+03 Ci = 2 18E+02 MIN= 0 MAXr 2 18E+O3 CI= 2 18E+02~~

~~1OIAL DENSITY OF FUEL TOTAL DENSITY OF FUEL TIME 1Q DOOMS TIME 15 OOOMS~~

~~MIN= 0 MAX: 2 18E*03 Cl: 2 I8E+02 MINr 0 MAXr 2 18E+03 CI: 2 18E+02~~

~~Fig. 159. Initial expansion of fuel in case 3.~~

~~229 TOTAL DENSITY OF FUEL TOTAL DENSITY OP FUEL TIME 25 DOOMS TIME 30 000MS~~

~~= 0 MAXr 2 18E+03 Cl = 2 18E»02 M1N- 0 MAX; 2 18E*03 CI- 2 18E+02~~

~~TOTAL DENSITY OF FUEL TOTAL DENSITY OF FUEL TIME 35 000MS TIME 50 000MS~~

~~MINI: 0 2 18E + 03 CIr 2 18E + 0? MINr 0 MAX. : 1BE+03 CI= 2 18E+02~~

~~Fig. 160. Head loading and reflection ic case 3.~~

~~230 250.0-1 PRESSURE AT INLET PLENUM BOTTOM~~

~~00 "'' I 0.00 0.01 TIME (s)~~

~~Fig. 161. Pressure at the inlet plenum bottom for case 3.~~

~~180.0 T PRESSURE IN DOWNCOMER~~

~~0.0 | i i i i i i i i i | i i I I i i I i I i i i r i i i i 0.00 0.01 TIME (s) Fig. 162. Pressure of the top of the downcomer for case 3.~~

~~231 lOO.O-n PRESSURE AT THE TOP OF THE VESSEL~~

~~0.0 000 0.01 0.02 0.03 0.04 0.05 TIME (s) Fig. 163. Pressure at the top of the vessel for case 3.~~

~~100.0 -i PRESSURE AT HEAD CURVATURE~~

~~0.0 0.00 0.01 0.02 0.03 0.04 0.05 TIME (s)~~

~~Fig. 164. Pressure on the head at 30" to the vertical for case 3.~~

~~232 100.0-1 PRESSURE AT VESSEL FLANGE~~

~~0.0 r-r-r-rt 000 001 0.02 0.03 0.04 0.05 TIME (s) Fig. 165. Pressure on the head at 70° to the vertical for case 3.~~

~~FORCE ON HEAD~~

~~rr-|rrrrr 0.00 0.01 0.02 0.04 0.05 TIME (s)~~

~~Fig. 166. Force on the head for case 3.~~

~~233 TOTAL FLUID KINETIC ENERGY FOR CASE 3~~

~~0E+09i- 1 1 1 1 1~~

~~MINIMUM VALUE= 0. MAXIMUM VALUE= 2.45919E+09 Fig. 167. Total fluid kinetic energy, case 3.~~

~~UPWARD FLUID KINETIC ENERGY FOR CASE 3~~

~~2 OC.OS~~

~~MINIMUM VALUF= 0 MAXIMUM VALUE= 1 83355E+O9 Fig. 168. Upward fluid kinetic energy, case 3.~~

~~234 MISCELLANEOUS FLUID KINETIC ENERGY FOR CASE 5 REGION SETS I TO 2 a octaa~~

~~TIME (S)~~

MINIMUM VALUE= 0. MAXIMUM VALUE= 6.26136E+08 Fig. 169. Miscellaneous fluid kinetic energy, case 3. (Curve 1 is the fluid kinetic energy in the downcomer and remaining parts of the vessel, and curve 2 is the fluid kinetic energy in the outlet pipe.) coherency factor, F2/KE , is 0.33 GN/m. An increase in pre-explosion fluidization apparently does reduce effectiveness of energy transmittal into head loadings.

D. Case 4 - Simulation of an Incoherent Explosion One of the questions raised in the ZIP study was whether a large missile could be generated from a steam explosion by in-vessel acceleration of liquid at driving pressures below those required to fail the lower head. Case 4 attempts to obtain a conservative limit on upper head loads that might be anticipated if pressures are not sufficient to cause lower head failure. Figure 136 from case 2 provides the starting point to formulate initial conditions. The 40 000 kg of corium that fell below the bottom of the initial corium pool was assumed to be homogenized with the remaining water (10 000 kg) in the lower plenum at 0.7 s. A slow "explosion" was then calculated in which droplet sizes were increased an order of magnitude over SNL experimentally correlated values. This reduced the heat transfer by two orders of magnitude. The objective was to simulate an incoherent, multiple explosion environment that could be representative of the reactor meltdown situation. As described in

235 App. D of the SERG report,12 coupling length considerations prohibit such homogenization from producing a highly efficient explosion when the premixing calculation showed most of the bulk corium separated from the bulk water. Bow- ever, the chaos introduced by incoherent explosions might lead to progressive mixing. Although the coherence introduced by the assumed homogenization is exaggerated, the large fuel constraint on the system would favor progressive mixing before the water separated beyond recovery. Figure 170 shows the initial configuration. The premixed region was assumed to occupy the entire space below the remaining single-phase corium pooL The steel particles that represent the broken-up core blockage were smeared across the radius of the core to make the expansion more one-dimensional, and consequently conservative, based on the SIMdER-11 analysis of the shallow-pool experiments. Premixed corium and water were included in these steel particles to generate pressure. Steel particles were included above the corium pool to represent upper internal structure. The -8 000 kg of water in the downcomer permitted the water mass to be conserved. The averaged pressure differential across that part of the lower head used in the head-failure calculation is shown in Fig. 171. The loading time corresponds to roughly quasi-static loading in terms of the model, meaning the failure threshold is below but close to 6 740 psi (the failure pressure from the model if the driving pressure rises with infinite slowness). Although the applicability of the failure model in this situation could probably be improved, the no-failure prediction is reasonable. Liquid-volume-fraction plots are shown in Fig. 172. Because the lower head does not fail, only axial nodes 13-66 are plotted. The expansion was more one-dimensional than case 2. Also, a single-phase liquid region developed at the corium-pool/premixed-region interface. In this region, water must undergo bulk heating (surface vaporization cannot occur in the current single-phase model). When the "water" vapor pressure became sufficient to disperse this zone, starting at "50 ms, an extra boost was given to the upward moving corium. The effect of the buildup of this pressure can also be seen in Fig. 171. The liquid-volume-fraction plots exhibit considerable impact coherence at the upper head. Figures 173-178 give the calculated pressure and force transients for comparison with other cases. The resolution is that of every SIMdER-II time step. The peak pressure at the bottom of the lower plenum was slightly higher than the averaged value used to calculate lower plenum failure, showing the effect of 236 OUTLET PIPING STEEL PARTICLES >SOLID STRUCTURE 50E

~~INLET PIPING 12.54~~

~~DOWNCOMER S3' ^ s s CORIUM 5 2 ' 5 S WATER 30 I PARTICLES & PREFIXED i MIXTURE REGION CORE SUPPORT FORGING - 13~~

~~MOVABLE 5.20 1 STRUCTURE~~

~~OUXLET TO KEYWAY 10 15~~

~~«-1.613-*! 2.537~~

~~Fig. 170. Configuration model, case 4.~~

~~237 6500.0 -a PRESSURE DIFFERENTIAL 60000^ 5500.0 7 5000.0 ^ 45000 -. a. 4000.0 -. m 3500.0 -:~~

~~T- 3000.0 -. •A 2500.0 -j a. 2000.0 •=~~

~~1500.0T 1000.0 - 500.0 -i~~

0.0 1TTTT I I I I I t I I I I I • 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 TIME (s) Fig. 171. Driving pressure for lower head failure, case 4. relief up the downcomer. Downcomer impact pressures were relatively low. The peak pressure at the apex of the vessel divided by the peak pressure at 70° from the centerline is about 2.7, greater than cane 2. The kinetic energy plots are given in Figs. 179-181. The upwardly directed kinetic energy reached a maximum of 765 MJ, leading to a coherence parameter, F /KE_, of 3.0 GN/m. Case 4 is further considered in the discussion on the probability of containment failure as the upper bound on plausible head loadings.

E. Case 5 - Upper Bound If limits to mixing are ignored, and large premised regions undergo energy equilibration before expansion (as predicted by detonation theory), much more energetic events can be calculated. In case 5 such excessively pessimistic assumptions are made. The expected result, a containment challenge, is obtained. Details of case S are in App. V. Case 5 is simply a reminder that physical limits or arguments on the assumed initial conditions must be made if

~~238 VOLUME FRACTION OF LlflUlO VOLUME FRACTION Of LIQUID VOLUME FRICTION OF LIQUID VOLUME FRACTION OF LlOUID TIHE 5 OOOMi. IIME 10 OOOMJ TIHE IS 000~~

~~07E-0S M*X= 9 95E-01 Cl= 9 95E-02 MIN= 2 07E-05 |: 9 9SF-02 MIN= 2 07E-0S MAXz 9 9SE-01 C|: 9 9SZ-02 MINr 2 07E-05 M*x= 9 «E-OJ Cl: 9 95R-DZ~~

~~VOLUME FRACTION Of LIQUID VOLUME FRACTION OF LIOLlC VOLUME FPiC'iO'. jF LIS'JID VOLUME FRACTION OF LIQUID TIME JO OOu£i TIME 25 O00MS_ TIME 35 DOOMS. TIME 30 no:":~~

~~MIN= ; O^E-OS KH- 9 9SE-01 CI= 9 9SE-02 MINr 2 07E-05 MAX; 9 9SE-0I CI: 9 9SE-02 95E-01 Ci= 9 9SE-02 «1N= 2 07E-05 Mi~~

~~Fig. 172. Liquid-volume-fraction plots for post-explosion/expansion, case 4. t\3 CO 10 o o~~

~~i'i~~

~~240 VOLUME F«»CI!ON Of LIQUID VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 80 OOOMS II ME 3S O00MS_ TIME 90 000N£_ TIME 95 OOOMS_~~

~~V/ MIN; ^ 07E-05 MAx= 9 95E-0I Cl= 1 95E-02 MIN= Z 07E-06 MAX: 9 9SE-0I Clr 9 95E-0Z H1N= 2 O'E-OS MAX= 9 9SE-01 Cl: 9 95E-0Z «1N= T 07E-05 MAX: 9 95E-0I CIs 9 9SE-0Z~~

~~VOLUME FRACTION OF LIQUID VOLUMF FRACTION OF LIQUID VOLUME FRACTION OF LIOUIO VOLUME FRACTION OF LIQUID TIME 100 000 S. TIME 110 100MS_ TIME 105 OOOHS TIME 115 000MJ_~~

~~MIN: H1N= 2 07E-0S MAX= 9 95E-01 C[= 9 9SE-02 H1N: 2 O7E-O6 MAX= 9 9SE-01 Cl= 9 9SE-0Z * 07E-0S MX: 9 KE-01 CI: 9 HE-02 HIN= Z 07E-0S MAXr 9 95E-0I CI: 9 95E-DZ~~

~~Fig. 172. (cont.) -P. 60.0-1 PRESSURE IN PLENUM~~

~~0.0 I • • • • i • • • • i • • • • i • •'' i' •' • i " " i " " 1" " I " " I " " I " •' I " 000 0.01 0.02 003 0.04 0.05 0.06 0.07 008 0.09 010 Oil TIME (s)~~

~~Fig. 173. Inlet plenum pressure, case 4.~~

~~60.0-1 END OF DOWNCOMER~~

~~50.0-~~

~~I 40.0- u a: 30.0-~~

~~a: a. 20.0-~~

~~10.0-~~

~~0.0 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 TIME (s)~~

~~Fig. 174. Pressure at the top of the downcomer, case 4.~~

~~242 350.0 -i TOP OF THE VESSEL 300 0 -~~

~~250 0 - a. 200.0- a; 150.0 - wa: Q. 1000-~~

~~50.0-~~

1 1T 1 0.0 • • I • • • • i •' • • i' •' • i • ' T • • • • i • • • • i " • • i • • • • r • • • i • • • • i " 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 O.OB 0.09 0.10 0.11 TIME (s) Fig. 175. Pressure at the top of vessel, case 4.

~~250.0-1 PRESSURE AT THE HEAD CURVATURE~~

~~0.0 »[IIIP|IIIIIII iTrrrrrrm t i i i n i rm i T T I ? rlTTTT 0.00 0.01 002 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 TIMK (s)~~

~~Fig. 176. Head pressure at 30° to the vertical, case 4.~~

~~243 PRESSURF AT VESSEL FLANGE~~

~~r r n i I i i i • I • » n I •"" • I " " I " " I " • • I " " I " " I ' 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 010 Oil~~

~~TIME (s)~~

~~Fig. 177. Head pressure at 70° to the vertical, case 4.~~

~~"b 16.0- FORCE ON HEAD 14.0 -~~

~~0.0 1" I " " I " " I " " I' • " I' • " I' " ' I''' • I' " ' I'' " I " '' I'' 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 TIME (s) Fig. 178. Force on the head, case 4.~~

~~244 TOTAL FLUID KINETIC ENERGY FOR CASE 4~~

~~1 0C«O» —r—T—'•—i—•—' •—r-i • > * i • • • i • • •~~

~~/ y / i i . . . i 01 04 0 oa io TIME (S)~~

~~MINIMUM VALUE- 0. MAXIMUM VALUE- 8 94921E+O8 Fig. 179. Total fluid kinetic energy, case 4.~~

~~UPWARD FLUID KINETIC ENERGY FOR CASE 4~~

~~04 oa TIME (S)~~

~~MINIMUM VALUE* 0 MAXIMUM VALUE= 7 65293E+OB Fig. 180. Upward fluid kinetic energy, case 4.~~

~~245 MISCELLANEOUS FLUID KINETIC ENERGY FOR CASE 4~~

~~2 OC«~~

MINIMUM VALUE= 0 MAXIMUM VALUE= I.53253E+O8 Fig. 181. Miscellaneous fluid kinetic energy, case 4. the treatment of post-steam-explosion/expansions is to be consistent with maintaining containment integrity.

F. Summary of SIMMER-II Steam-Exploaion Cases To provide a reference point for the discussion on containment failure in the next chapter, the results of all SIMMER-11 cases that were run until head impact are presented in Table XVI. In attempting .0 draw conclusions from these cases, we must recall that the general philosophy used to set up cases was to pick configurations that could be argued to be plausible upper bounds, or, that on the basis of previous study, were suggested as useful for an upper bound discussion. In other words, the types of assumptions made for these cases are those causing a failure threshold to be approached. Determining the uncertainty in these calculations is difficult. An attempt to judge the extent of the conservatisms present resulted in the use of probability arguments. This discussion and its conclusions are presented in Chap. VII. Before starting our probability discussion, one result from Table XVI might be noted. In case 5 a combination of overly pessimistic assumptions made results approach thermodynamics limits. Although the other SIMMER-II cases also

246 TABLE XVI SUKMARY OF STEAM-EXPLOSION CASES KE Total Impacting Peak KE Head Force Case Description (MJ) CMJ) (GN) 1 Comparison with ZIP study. 3 820 3 350 2.8 Small, 100-jim drop sizes. No lower head failure. 1 (Update) Includes lower head failure. 1 770 647 1.04 Includes steel particles. 300-ym ZIP droplet sizes. 2 Calculated conservative 1 729 1 490 0.94 premising. Explosion after 1 s of premising. No lower head failure. 2a Same as case 2 except 1 S23 1 335 1.55 explosion is after 0.7 s of premising. 2 (Final) Add lower head failure 1 665 516 0.81 is after 0.7 s of premising 3 Total fluidization with 2 459 1 834 0.78 94 000 kg of corium. No lower head failure. No corium/steam heat transfer. 4 Plenum water homogenized 895 765 1.52 with 40 000 kg of corium. Slow heat transfer. Lower head failure threshold not exceeded. 5 Upper bound pessimistic 18 048 7 253 12.4 assumptions made to approach thermodynamic limits.

247 are believed to be conservative, perhaps excessively so, they reduce head loading by an order of magnitude compared with case 5. Consequently, we believe future mechanistic calculations of even a "conservative" large-scale steam explosion and the subsequent expansion of the materials will suggest, as does the limited experimental database, that maximum yields, based on complete mixing and thermodynamic arguments, are physically unreasonable in the reactor meltdown environment. Finally, these cases fere set up and run in the fall and winter of 1984-85. Subsequent development in describing core melt accidents and expected reactor head thermal conditions would lead to some changes in the geometries assumed and the amount of materials present. The probability discussion in Chap. VII also could be modified with differing results for some branch points. However, because of the tradeoffs and uncertainties, no clearly defined overall changes can be recommended in the level of concern a reader should infer regarding either steam-explosion consequences during core melt accidents or in the proposed research activities in Chap. VIII.

248 VII. PROBABILITY OF CONTAINMENT FAILURE The original intent of this study was to address the upper bound energetics from a steam explosion occurring during a reactor meltdown. Chapter VI allows the conclusion that current technology requires the addition of a degree of subjectivity, depending on the analyst, to obtain a satisfactory answer on this issue. Knowledge of the physics and chemistry of severe (core melt) accidents is limited, and analysis tends to a "bootstrap" process. Regarding the possibility of containment failure from a steam explosion, Theofanous has stated in App. F of the SERG report that the word "unlikely means nothing." To provide something other than additional phraseology that could be misinterpreted, this chapter presents a formalism to quantify the word "unlikely." We then offer judgments using this formalism, on what the conditional probability of containment failure might be, give n core melt. The method used to address the conditional probability of containment failure by a missile from a steam explosion is an expansion of the approach formulated by the Clinch River Breeder Reactor study22 done for the USNRC on core-disruptive accidents. An initial presentation and use of this method was prepared for the SERC and is documented in App. C of NUREG-1116 (see Ref. 12). This chapter is a revision of that appendix because it includes a discussion of all results obtained in the Los Alamos molten-core/coolant-interaction program. A generic progression diagram was devised, and each branch was assigned a probability as defined in Table XVII. The algorithm to assign the probabilities was as follows: (a) Assume that an integral, accident-analysis computer program exists with models that describe the correct meltdown phenomena to the precision expected from our current uncertainties.

TABLE XVII DEFINITION OF PROBABILITY SPLIT LEVELS 1/10 Behavior within known trends but obtainable only at the edge-of- spectrum parameter values. 1/100 Behavior cannot be positively excluded but outside the spectrum of reason. 1/1000 Physically unreasonable behavior violating well-known reality and its occurrence can be argued against positively. 249 (b) Do gedanken (or thought) computations on this program varying the input parameters. (c) Based on the results of these computations, assign probability ranges from the definitions in Table XVII. A preliminary diagram of an accident's progression, focused on the alpha mode of containment failure, is given in Fig. 182. This diagram assumes a ZION-type PWR. The effects of differing accident sequences, for example, high pressure vs low pressure, are discussed where appropriate. The branch points are annotated on the diagram for reference. The reasons for the indicated choices are as follows: 1. A loss-of-coolant accident (or LOCA) leading to core melt was postulated. A station blackout or pipe break are possible accident initiators. Although the details of the accident's progression are important in assessing the likelihood of an eventual steam explosion, they were beyond the scope of this investigation except as discussed in branch 2. In this study we did not .Include the possibility that a given accident initiator might not result in core melt, or that operator intervention might stop the accident's progression. 2. A rubble bed leading to pool formation seems likely. This was the 19 expected behavior in the 1975 Reactor Safety Study. No prototypic experimental evidence apparently exists that justifies use of alternative assumptions. Eventual pool formation is consistent with the behavior observed in the Three Mile Island accident, and the results reported for Power Burst Facility tests SFD-ST,24 SFD 1-1,25 SFD 1-3,* and the damaged fuel test DF-1. From papers by Bisanz et al. and Schmidt and Bisanz, the German best-estimate core meltdown code, KESS-2 with MELSIM-3, apparently slumps fuel rods in each zone once a slumping temperature is reached, assumes blockage formation in disrupted core regions, and then allows coherent downward motion into the lower plenum once failure of the core-supporting structure occurs. IDGOR apparently assumes intact geometry and the consequences of conduction-limited freezing to perform calculations with the MAAP code indicating incoherent fuel meltout. We cannot rule out this possibility. It may

~~•Information on severe fuel damage test 1-3 provided by R. J. Henninger, Los Alamos National Laboratory Group N-9 (November 1984).~~

~~250 LOFA~~

~~Vio CORE MELT~~

~~(V10-I) CORIUM POOL~~

~~OTHER APPROPRIATE PHENOMENOLOGY -MOLTEN CORIUM -WATER INCOHERENT. -CORE BARREL UMITED, OR NO -STRUCTURE EXPLOSIONS (I/10-1/100)~~

BIG STEAM EXPLOSION LOWER HEAD (I/10-1/100) EFFECTIVELY 1/100 SIGNIFICANT UPPER 0-V1O) NO LARGE VENTS HEAD LOADING ENERGETIC AND/OR MISSILE EFFECTIVE (1/10-1) SLUG BREAKUP OCCURS LARGE ENERGETIC (Vioo-1) MISSILE m MISSILE STOPPED

~~0-/1O) CONTAINMENT FAILS~~

~~CONTAINMENT INTEGRITY ASSURED~~

~~Fig. 182. Diagram of an accident's progression focused on the alpha mode of direct containment failure by a steam explosion.~~

251 be reachable with edge-of-spectrum assumptions. However, the judgment here is that IDCOR's assumption, that as fuel rods melt they drop as In blobs into the bottom of the reactor vessel, is more of a hypothesis than a verified model based on scientific evidence of corium's behavior. Pool formation is also consistent with LMFBR integral loss-of-flow (LOF) tests with much shorter fuel pins at much higher power levels, although steel does not wet uranium dioxide, whereas molten zirconium not only wets the fuel but also dissolves it. Access to studies done with ANCHAR CANL/NSAC),31 CORMLT (SAI/EPRI),32 and MELPROG (SNL/NRC)33 should clari- fy the core meltdown picture in the future. At present, our gedanken program needs quite improbable assumptions, at least edge-of-spectrum, not to form a pool. 3. Several other characteristics in the meltdown sequence must be present besides pool formation for a dangerous steam explosion to be likely. In view of the uncertainties, these prerequisites can only bo discussed qualitatively. They include the following: (a) Molten corium is in the pool. At commonly quoted coriuii* temperatures much of the uranium dioxide may still be solid. For example, the sample PWR-large-dry-coniainment problem analyzed by the containment loads working group possessed a pool temperature of 2 533 K. One reason speculated for weak or nonexplosive behavior of some fuel/coolant interaction (FCI) tests with corium stimulants has been the presence of solid material. Theoretically a steam explosion involves rapid fragmentation in a liquid-liquid system, viot a liquid-solid environment. A significant quantity of corium above the liquidus temperature appears essential. (b) Sufficient water is in the lower plenum. A small, preliminary explosion must not blow the water away from the mixing region if this necessary condition is to be met. Some water must vaporize from downward heat transfer, but IDCOR's argument of extensive downward thermal radiation that boils water away seems implausible if a solid crust is supporting a molten pool. (c) Core barrel or molten corium openings to the downcomer must not lead to incoherent coriurn/water contact. Water would be expelled and any subsequent explosion would be small. Core barrel meItthrough is a possibility with the large potential for thermal radiation heat

252 transfer from molten corium. Also, radial heat transfer from convection within the pool is apparently larger than downward heat transfer in LMFBK molten pool problems, and some of the same phenomenology should occur in thermal reactor corium pools. One consequence could be collapse of the core as shown in Fig. 183 from the SNL ZIP study, although this ignores the core's secondary support system. A second, more likely, possibility is shown in

~~FROM SNL Z!P STUDY~~

~~STEAM~~

~~FRACTURED CORE BARREL~~

~~WATER LEVEL~~

~~FRACTURED CORE BARREL~~

~~VIGOROUS BOILING BOTTOM SUPPORT PLATE RESTING ON BOTTOM HEAD~~

~~Fig. 183. Visualization of the stats resulting from a failure of the core barrel prior to penetration of a coherent molten mass through the below-core structure.""~~

253 Fig* 184 from from a more recent SNL study by Berman ct al. Esca- lation to a big steam explosion becomes far more difficult in these situations. (d) The failure of the structure to hold up the pool must be sufficiently coherent to permit the initiation of large-scale liquid-liquid contact. The explosion from a small pour is inconsequential to the

~~NUREG/CR - 3369~~

~~CRUST, SINTERED RUBBLE~~

~~FRACTURED FUEL, ZrO2 INTACT FUEL RODS 7~~

~~WATER LEVEL~~

~~MELT FLOW INTO WATER~~

Fig. 184,. Melt flpw into the l^wer plenum by sideways penetration of the core barrel 6 Z54 alpha mode of containment failure unless it can cause more coherent contact. The Los Alamos molten-core/coolant-interaction program did not have a task to provide quantitative calculations on these phenomena. It is our understanding that MELPROG results are tending toward a largs "semi-molten" pool with temperatures below the melting temperature of uranium dioxide but with a rather coherent failure of the lower grid plate.36 In our opinion, the WASH-1400 conclusion that up to 80% of the core may be molten before a massive pour occurs is possible given the information available. Therefore, our gedanken calculation would satisfy the required conditions for edge-of-spectrum parameters, perhaps using best-estimate parameters in some accident sequence. A 1 to 1/10 probability range was consequently selected for this branch, although the value of unity may be a result of ignorance of progress in other research programs. 4. A discussion of branch 4 requires defining a big steam explosion. A big steam explosion is defined as an explosion possessing the potential of sustained supercritical pressures that drive the expansion until head impact. The reasons for this definition arc as follows: The critical pressure of water is 22 MPa. The potential expansion volume in the vessel is approximately 80 tn . As stated in Chap. 2 of the SERG report, containment failure limits lie in the range of 1-3 GJ of energy released in a steam explosion. The median value, 2 GJ, can be obtained by expansion with a constant 25-MPa pressure differential. In practice, because of slug breakup and reduction of pressure as a consequence of expansion, higher initial pressures are required to obtain 2 GJ in a 80-m expansion. Using the words "potential of" in describing the sustained supercritical pressures is a consequence of the pressure relief available through failure of the lower head. This pressure relief is a mitigation feature considered separately in branch S. The occurrence of lower head failure also is judged to satisfy the requirements for a big steam explosion. Provided the initiation of a coherent pour or molten corium/water contact can be obtained, +he standard arguments against a large steam explosion come from fluidization phenomena. If fluidization furnishes the only arguments that can be made to limit

~~255 coarse mixing, and therefore big steam explosions, then a 1/10 probability would be appropriate.~~

Illustration of a calculation that uses cdge-of-spectrum parameters but achieves a big steam explosion is given in case 2 of Chap. VI. Starting the premixing calculation with an entirely molten core at 3 100 K is clearly edge-of-spectrum; minimizing steam production during premixing by the heat-transfer models employed is also edge-of-spectrum. However, the resulting explosion using SNL correlated parameters generates sufficient pressure to satisfy the lower head's failure criterion in both the 0.7 and 10 s explosions. Also, when lower head failure was ignored, the explosion at 1.0 s generated sufficient pressure to maintain the bottom of the lower plenum at supercritical pressures until head impact.

k big steam explosion should still be obtainable even if some of the edge-of-spectrum characteristics of the case 2 premixing calculation are relaxed. The inertia of a large mass of coiium is not accounted for in standard fluidization arguments. Nevertheless, the calculation with alternative high heat transfer shows the potential for corium dispersal by steam generation without extensive mixing. Additional help tc decide whether extensive, mixing is possible starting with a large molten pool could be obtained by constructing a more consistent calculational capability with three-velocity fields. From past experience, edge-of-spectrum parameters would be one expected requirement to obtain a big steam explosion from such a computational tool. Our gedanken results should be consistent with these expectations.

An alternative scenario can be considered. Several SNL tests have shown a tendency toward early detonation. Although little is known about how to model triggering phenomena in the context of a integral computer program, if confirmation of this early triggering tendency cou'd be obtained from larger scale tests, achievement of a 1/100 probability would be possible. The presence of structures in the lower plenum should act as further potential triggering sites. Indeed, our gedanken program suggestr that a likely outcome is a series of incoherent explosions. Although the possibility cannot be ruled out, that a little explosion may be just the correct magnitude to be the precursor of a big

~~256 explosion, modeling such a sequence with consistent assumptions in a reactor configuration seems difficult. A big steam explosion could well lie outside the spectrum of reason.~~

We observe that the more probable accident sequences (TMLB' and the small-break LOCA) may be speculated to lead to a somewhat elevated ambient pressure at the time of a steam explosion. Also, water in the lower plenum will likely be saturated. Both the existence of high pressure and the presence of saturated water increase film-boiling stability and decrease the tendency toward early triggering. If the resulting film-boiling stability means that a steam explosion cannot be triggered, the problem is eliminated. In the Three Mile Island accident, the high ambient pressure apparently did supress the development of a steam explosion. However, because the "olume of steam is reduced at high pressure, higher concentrations of fuel and water can exist at lower vapor-volume fractions. If a steam explosion is triggered, it could be more efficient. Also, increased mixing could simply result from the time delay furnished by trigger suppression. In any case, triggering requirements stiil are not clearly understood, and maintaining a probability range of 1/10 to 1/100 seems reasonable at this time.

Finally, s. correlation exists between branch 3 and branch 4. The larger any molten pool becomes and the more purely liquid characteristics it exhibits, the greater the likelihood of a big steam explosion. For an upper limit discussion, an edge-of-spectrum character should probably be associated with the branch 3-4 combination. Multiplying independent numbers may excessively lower any probability estimate. 5. A discussion of branch 5 requires defining significant head loading as an averaged force over the head of greater than ~1-GN. Some minimum impulse is also required, but this impulse is small given the duration of the expected two-phase loadings. The value of 1 GN comes from the following: (a) The SNL uncertainty study suggests that the force required to excesd the boit-failure tension is 1 170 MN, corresponding to a static-failure pressure of 80 MPa.

257 (b) The Los Alamos ZIP study1 using less conservative assumptions found a hydrostatic pressure capability of 100 MPa before bolt failure would occur. (c) In a computer simulation, the Los Alamos ZIP study also obtained substantial plastic deformation of the upper head at a uniform pressure load of 70 MPa. (d) Because the thermal conditions of the upper head are so uncertain and depend on the accident sequence, only a rough number is possible. A 1-GN failure should be possible in some cases, but without the excess momentum present to form an energetic large missile. If the temperatures are significantly higher than assumed in either Ref. 1 or 6, further considerations are necessary as discussed in branches 6 and 7.

With this definition, the SIMMER-II cases support the results of the gedanken calculations shown in Fig. 182. The case run for comparison with the ZIP study is outside the spectrum of reason for not allowing lower head failure and venting. The postulated expansion conditions in case 1 are edge of spectrum in limiting local compliance for the steam explosion, and in assuming the explosion could occur throughout such a water-lean system. The explosion/expansion starting from fluidized pool conditions (case 2) exhibited effective slug breakup. Both it and the 0.7-8 explosion scoping case would have been mitigated by inclusion of lower head venting. Case 2 "final," although using an edge-of-spectrum premixing calculation, did use best-estimate postexplosion assumptions; however, the peak force was 0.81 GJ. Case 3 used premixing assumptions that are beyond the spectrum of reason, but the possible influence of lower head venting can be assumed as limited, and the expansion assumptions ate not extreme. In case 3 effective slug breakup limited the peak force to 0.78 GN. Case 4 is the plausible bounding case in this study. The postulated premixing is edge of spectrum, and the expansion can be claimed to be edge-of-spectrum. The problem was deliberately made as one-dimensional as possible to avoid spurious slug breakup. The late expansion of supercritical "water" provided as much maintenance of high driving pressures as can probably be achieved without artificially increasing surface area as a function of time. The upper head loading went to 1.52 GN, but this might be expected for a edge of-spectrum case.

258 Case 5 uses premising assumptions that are beyond the spectrum of reason, as in case 3. Although both effective lower head venting and effective slug breakup are calculated, loadings are severe. This is a consequence of the thermal equilibration at constant volume throughout the premixed region and of pessimistic EOS assumptions used to expand the premixture. Thes? expansion conditions are thought to be beyond the spectrum of reason. The combination of beyond-reason premixing and expansion assumptions gives this case the flavor of physical impossibility.

Thus, the conclusion for branch 5 is a 1/10 to 1/100 probability range. This choice cannot be made with high confidence because of the difficulty in defining what should be edge-of-spectrum SIMMER-II expansion assumptions, the uncertain thermal conditions of the upper head, and the correlation to branch 4, which deals with how much the definition of a big steam explosion has been exceeded. 6. Because a quantitative analysis of the path of an energetic missile through the containment was beyond the sc ope of the current study, a large, energetic missile was defined as a missile in which the vessel head is given an initial velocity of greater than 25 m/s. This is the velocity under which a missile could reach the top of the containment with only gravity as a limitation. The effects of impeding structures for a large missile are- considered in branch .7* The initial kinetic energy associated with 25 m/s is not large. For a 65 OOO-kg head with associated structure, 1/2 mv = 20.3 MJ.

Assuming that the two-phase force on the head is greater than 1 GJ, so that failure occurs before dissipation of the impacting momentum, the principal concern is the mode of failure, not satisfaction of some energy requirement. The character of head loading in case 4, where edge-of-spectrum assumptions produced a peak load of 1.52 GN, was to first produce a peak load in the area of the bolts (80 MPa at 75 ms). Then, material was concentrated at the apex of the vessel producing 310 MPa at 85 ms. The peak force followed soon after wbsn liquid and solid particulate collected around the surface of the head. However, the secondary peak at the bolt area was only 115 MPa at 88 ms. This loading bias of approximately 2.7 favors heavy shrapnel being produced

259 rather than the intact head becoming a missile. Of course, a detailed quantitative analysis ia desirable. At some large loading level, the spatial distribution of the loading is of little importance.

Additional uncertainty is introduced from the thermal conditions, fluid impact in the downcomer, and variations in the shape of the expanding two-phase "slug." At higher temperatures, a force of less than 1 GN could produce a 25-m/s missile. The area of the downcomer is such that an approximate 400-MPa pressure at the top of the downcomer is required to produce a 1-GN force on the bolts. Except for case 5, calculated pressures in all cases were more than a factor of 2 below 400 MPa. In case 4 the peak downcomer pressure was ~50 MPa. A remaining concern was whether the timing of the downcomer loading would correspond to that of the fluid loading on the upper head. For the SIKMER-II runs performed so far, this was not the situation when premixing was either calculated or assumed to mechanistically force water partially up the downcomer before the explosion.* The downcomer pressure pulse occurred much earlier than the direct fluid loading of the upper head. Finally, although the apex loading bias seems to be a rather general characteristic resultir? from a center pour of corium into the lower plenum, a significant variation in loading patterns was observed in the SIM4ER-II cases. Dogmatic statements regarding the beneficial effects of such & bias are unwarranted at this time.

From the above discussion, our gedankan calculation is seen to be indeterminate at this time about the path at branch 6 that requires sdge-of-spectrum parameters. A 1 - 1/10 estimate was consequently chosen for both paths. 7. The assumption was that once the vessel head or the vessel's head bolts failed, all or part of it (different pieces of the failed head) would move vertically and strike the missile's shield assembly. (The term

*Other concerns resulting from differing postulated initial conditiona could not be considered within the scope of this study. For example, the downcomer could be completely full of water at the time of the explosion. A pressure pulse transmitted directly through the liquid could strain the bolts enough to reduce the required failure force to be associated with later fluid impact on the upper head. Such a case requires a more detailed structural analysis.

260 vertically is used rather loosely here because, as discussed in the SNL uncertainty study, differences in bolt failure times would cause some asymmetries). The equipment on top of the head (control rod drive mechanisms, cooling system, and so on) would contact the missile shield first and absorb some energy as it crushed. The head itself would contact the two, shield-supporting 36-in.-wide flange I beams before it contacted the concrete missile shield sections. Again, deformation of these structural members and the head at the contact area would absorb some energy in plastic deformation. Connection of the concrete missile shield slabs (three 6- by 18- by 3-ft-thick slabs with a 1-in.-thick steel plate on the bottom) to the I beams is with relatively small bolts that ensure the slabs do not move during seismic events. These bolts would rapidly fail and allow the two outside slabs and the I beams to slip off the upward moving head. The middle slab could stay in place until the head/slab combination contacted the polar crane structure. The missile shield structure probably varies considerably from reactor to reactor but almost all PVR and boiling water reactor (BWR) designs include a polar crane that will be positioned directly above the center of the reactor vessel. The mass of the polar crane is typically two to three times the mass of the head of the reactor vessel. This structure may deform considerably but could bring the velocity of the head down to levels that will preclude significant damage to containment. Because of the potential stoppage of even a large, energetic missile by the polar crane, k 1 to 1/10 probability was adopted for the containment-failure path. A realistic estimate requires knowing by how tuch the initial head velocity exceeds 25 m/s. The SNL uncertainty study suggests that containment failure will occur if the head becomes a missile with a vel>city greater than 50 m/s. Because of potential containment failure from those slabs of the missile shield that do not proceed straight up, this assumption is reasonable. An indication of the energy requirements in such a case can be made as follows. If a 50 000-kg "slug" has an inelastic collision with a 65 OOO-kg head, the resulting unit will have a velocity of 50 m/s if the kinetic energy is 144 MJ. The 50 OOO-kg slug must have had an energy of 330 MJ (after all losses from upper internal structure and from the head-failure mechanics) before the inelastic collision. With case 4 producing 760 MJ in upvardly directed

261 energy, absorption of energy in deformation and failure by materials of unknown temperature are important. In any case, for high enough velocities actual stoppage may prove to be outside the spectrum of reason, so a 1/100 to 1 probability was selected for this alternative.

In conclusion, we suggest that with current uncertainties, the conditional probability of direct containment failure from a steam explosion can range from essentially 0 to 0.01. The higher number is obtained by assuming that only achievement of (a) a big steam explosion and (b) significant upper head loadings justify edge-of-spectrum parameters in the gedanken code. The unlikelihood of a big steam explosion is based primarily on coarse mixing difficulties, while SINf1ER-II cases indicate edge-of-spectrum parameters are required in an explosion/expansion sequence to generate the 1-GN loads that are regarded as significant. High confidence is difficult because of the lack of knowledge concerning meltdown sequences, the lack of data on large-scale steam explosions, the inability (limited scope of the study) to perform extensive SIKMER-II parametrics, the difficulty in quantifying the distortion that may be caused by SINMER-II model assumptions and input parameters, and the absence of a quantitative analysis of the actual calculated loads, or even the temperatures of structures to which these loads should be applied. Because of these difficulties and the existence of some correlations among the branches of our diagram of an accident's progression, a 0.1 limit might be proposed as a more confident upper bound. However, failure of containment by an in-vessel steam explosion is almost by definition an edge-of-spectrum phenomenon. Some events of our accident-progression diagram may be more improbable than we assumed. For example, catastrophic failure of the boundary of a large, hot, molten pool of corium may be unreasonable. If a best-estimate probability is desired, the current 0.01 bounding estimate on the containment failure path might be expected to be reduced at one or two branches by an additional order of magnitude. Care is required here, because the combination of three edge-of-spectrum calculations does not necessarily imply physically unreasonable behavior. Again, we must not permit our probabilities to become too low simply through branch proliferation. However, a best-estimate probability is judged to be in

262 •a A the range of 10" to 10, meaning that containment failure from a in- vessel steam explosion i physically unreasonable. VIII, RESEARCH PRIORITIES ON STEAM EXPLOSIONS The fundamental problem in discussing the consequences of steam explosions is the issue of uncertainty. The question that needs to be addressed is how much residual uncertainty is acceptable. To approach this question, we introduce here a simplified concept defining levels of knowledge. Five levels that seem apparent to this observer follow. Level 1 - The problems are not discussed seriously. This seems to have been the position of the nuclear industry on class 9 accidents before the "safety-related occurrence" at Three Mile Island. Defense in-depth was used as a philosophy but never really had a quantified or unique meaning. Even today, those attending plenary sessions at severe accident conferences are likely to hear that reactors are inherently safe, catastrophic accidents cannot happen, and that we must stop this self-defeating investigation of the science fiction involved in meltdown behavior. Level 2 - Expert opinion dominates the discussion of severe accidents. At level 2 simple quasi-steady-state models are introduced and experimental data are uncritically presented to support the conclusions drawn. Accident scenarios are constructed using engineering judgment; or simple models are connected together in a computer program, as*for example in the MAAP code. This is the level at which the Clinch River Breeder Reactor (CRBR) project presented its transition-phase analysis. It is also the level at which the SERG operated. The recent SNL study argued that an approach to steam explosions at this level still possesses too much uncertainty. The current Los Alamos study suggests that the all-important quasi-steady-state level 2 fluidization arguments limiting mixing in large-scale systems are flawed because of not considering inertia. Level 3 - Detailed transient calculations are performed on pieces of the problems. These calculations are supported where possible with data. The data are more quantitative than at level 2 and it can be used more directly to check theoretical speculations on what the phenomenology should be. Also, at level 3 there is more of an appreciation that direct extrapolation of experimental results to reactor accident situations may be misleading. Some aspects of the NRC assessment of CRBR core-disruptive accident energetics were performed at level 3.

265 Level 4 - Actual detailed, integrated code systems exist to calculate complete accident sequences based on accepted phenomenology. These code systems presumably would be accurate enough to precalculate confirmatory in-pile experiments, or relevant out-of-pile experiments. This appears to be the level that the NRC was at one time targeting in programs to obtain a predictive capability for large-scale, pipe-break accidents in light water reactors (LWRs). Level 5 - A fast-running code (or codes) exists that is suitable to determine all the sensitivities required in a risk assessment. This calcviational system would simulate the dominant phenomena involved in a severe accident so the required source term distributions as a function of **me, space, and input conditions would be readily obtainable. Uncertainty would be given a precise mathematical definition. Now it should be apparent to even a casual observer that dismissal of concerns regarding uncertainties about steam explosions on a level 1 basis is not a viable long-term solution. The SERG suggested in its report that further research was required beyond the level 2 arguments and experiments that existed for its review. This Los Alamos study, and any future SIMMER-II study on stears explosions carried out with only currently available knowledge, is flawed by the necessary application of significant engineering judgment to the meaning of the SIMdER-II calculations. Clearly, a level 5 approach is desirable for a future study. Assuming level S calculations exi>st, accurate statistical results CCK Id be obtained as a function of accident initiators, reactor characteristics, and other initial and boundary conditions. Unfortunately, while there have been claims of approaches that will yield level 5 results for the steam-explosion problem, all of those approaches known to this author appear to contain overly optimistic assumptions. As Berman points out in App. E of Ref. 12, "The history of steam-explosion analyses is replete with examples of results of 'current thinking* which have later been shown to be false." Without actually knowing the dominant phenomena, how can we accurately include them in a computer calculation? Parametric exercises are no substitute for knowledge. Because the manner of leaping to level 5 is unknown, we propose in this chapter that level 3 research activities receive future emphasis. Because a partial level 3 approach was judged to be satisfactory on rlass 9 accidents in the CRBR licensing proceedings, activities of this type should be appropriate for reducing residual uncertainties sufficiently in the steam-explosion area.

266 The first level 3 suggestion is to improve modeling the meltdown process. Table XVIII lists accident features that could be used as a basis for SIMMER-II steam-explosion parametrics. Many of these features relate to how the initial phases of an accident occur. For example, when and how does a high-pressure accident sequence become a low-pressure accident sequence from failure of the primary system? What does the steam circulation mean, while the system is at high pressure, to the temperatures of the upper core structure, the vessel head, and the head's holddown bolts? Further, reliable mechanistic calculations of the meltdown processes that are currently under NRC investigation could be useful for other aspects of source term characterization. For example, the amount of unoxidized Zr is important for determining of the extent of direct containment heating following vessel failure. Estimates of unoxidized Zr at vessel failure are currently made with the MARCH code. The results of the MARCH code have been questioned in application to crucial high-pressure scenarios leading in the direction of core meltdown. The second level 3 suggestion is for large-scale steam-explosion experiments. Some SIMMER-II scoping calculations on proposed SNL large-scale tests, the SEALS test series, are reported in App. C of the SERG report (Ref. 12). To save space, the results of these calculations are not reproduced here. The conclusions from these SIMMER-II calculations are as follows: (a) SEALS and the full-seals reactor case both have significantly augmented fluidization potentials relative to existing SNL tests (FITS), if coarse fragmentation becomes extensive. Fuel may be levitated by the stean produced before an explosion occurs. (b) As scale is increased beyond the FITS level, more fuel has limited access to water. Explosive interactions that can occur face increased constraint. (c) There is a water-quenching effect in the FITS tests. This quenching effect is reduced in the SEALS simulations, and is further reduced in full-scale calculations. Simulation of the quenching effect in calculations of FITS tests depends on correlation to data. SEALS could significantly influence the choice of parameters. (d) The fluid "slug" resulting from a steam explosion is calculated to differ between the reactor case and the experimental simulations. A water slug is predicted far FITS and SEALS experiments. A fuel slug is evident in the full-scale reactor case.

267 TABLE XVIII FEATURES THAT GOULD BE USED IN SIMMER-II PARAMETRICS OF STEAM EXPLOSION/EXPANSIONS (1) Differing core, vessel, and containment designs -PWR -BWR -Dimensional and geometric details (2) Differing meltdown sequences -Accident initiator -Extent of pool before the pour (upper crusts, side crusts, and steel melted in from the upper internal structures) -Effect of Cd, In, and Ag control rod meltdown -Melt composition and property uncertainties -Melt temperature -Ambient pressure at the time of an interaction -Amount of water (3) Differing pouring modes intc the lower plenum -Center -Sides (through the core barrel) -Preliminary explosion causing massive collapse (4) Triggering time and location (effects of structure) (5) Propagation velocity of explosion (6) Effectiveness of explosion -Surface area of fuel and water as a function of time and location -Multiple explosion possibilities -Fragmentation and heat-transfer characteristics (7) Mode of lower head failure and consequences -Time and location -Venting and expansion characteristics (8) Size distribution of fragments during expansion (9) State of upper core internals (10) Temperature and general condition of upper head of the reactor vessel (11) Modeling of the containment if the upper head is projected to become a missile.

268 The critical heat flux theory of Henry and Fauske states that only about 100 kg (order of magnitude) of melt could mix with water without violating their fluidization criterion. Theofanous has proposed that a linear dimension of 10 cm is the upper limit over which coarse premixing can occur that will support a steam explosion. Large-scale tests with 2 000 kg of melt would allow these qualitative arguments to be examined, as well as significantly increase the level of confidence in SIMMER-II extrapolations to reactor scale. Two differences from the previous SIMMER-II write-up on the SEALS proposal should be noted. First, support for SEALS was previously tentative. It now appears to be a desirable experimental activity. Second, the previous write-up in Ref. 12 proposed that transient steam-explosion pressures of several hundred megapascals are required to obtain containment failure directly from an in-vessel steam explosion. This is still the case if lower head failure and effective venting occur. However, case 4 of Chap. VI achieved significant upper head loads through sustained supercritical pressures that were below the ~50 MPa level required for lower head failure. Further investigation of the consequences of case 4 calculated loads is recommended. The previous recommendations to the SERG are repeated here. The confinement in the SEALS tests should be increased, and a serious modeling program maintained. Figure 185 reproduces our suggestion on an axial constraint. Although 2 000 kg of thermite does not simulate the inertia that is important in the coarse premixing phase of a react or-scale corium pour, both SNL and Los Alamos experiments as well as SIMMER-II calculations indicate confinement does increase the energetics involved in a steam explosion. An initial modeling program using three velocity fields and applied to ana- lyze SNL experimental data was reported at Lyon. Improvements achieved by this program are shown in Table XIX. This program was cancelled in 1981. A renewed three-field code-development effort was started in FY 1986. Its limited scope, duration, and resources may be insufficient to reduce the steam-explosion uncertainties as desired. If the SEALS tests are performed, some further improvement and use of the three-field capability should be considered. Other acceptable modeling piograms are an expanded MIMAS (MELPROG) effort20 or writing a two-dimensional TEXAS code as has been proposed by Young. The third level 3 suggestion is to continue some form of smaller scale testing. Although a continuation of exploratory testing where thermite is simply poured into water to see what will happen would not be very helpful (as

~~269 ACTUATOR CONFINEMENT BODY SAND~~

~~UNATTACHED~~

~~CRUCIBLE GUIDE SKIRT 203 m~~

~~RETRACTABLE CRUCIBLE~~

~~VENTS~~

~~MOUNTING BRACKET STEEL CRUCIBLE SHELL~~

~~STEAM/WATER RELEASE CABLES RELIEF PATH BOTTOM LID THERMITE LEVEL (SEGMENTED) RIGID INTERACTION VESSEL MELT LEVEL LID ARMS WATER REFRACTORY LINER REPLACEABLE LOWER VESSEL RING SECTION~~

~~MgO REFRACTORY LINER IN FINAL 25 cm OF LOWER STEEL BASE VESSEL RING AND BASE PLATE PLATE CONCRETE BAS PAD~~

~~Fig. 185. Proposed SEALS experimental configuration.~~

~~270 TABLE XIX~~

~~PRESENTLY INVESTIGATED IMPROVEMENTS COMPARING THE INITIALLY PROGRAMMED THREE-FIELD FLUID-DYNAMICS ALGORITHM AND SINMER-II~~

SIMMER-II New Algorithm 1. Two velocity fields Three velocity fields 2. One flow regime Three flow regimes (Dispersed flow) -Bubbly flow -Churn-turbulent flow -Dispersed flow 3* Pressure iteration performed Pressure iteration accommodates with a constant pressure EOS nonlinearities resulting from gradient for each cell. cell compositional changes and the two-phase/single-phase transition.

4. EOS and the pressure iteration ctQ concept expanded in the EOS. cannot accommodate gas/vapor Implemented consistently in the at the two-phase/single-phase pressure iteration to remove problems transition. with material conservation. 5. Energy equation explicitly Both mass and energy equations finite differenced with beginning- consistently finite-differenced of-time-step velocities. Mass with a semi-implicit algorithm. equations implicitly differenced using end-of-time-step velocities. 6. Full donor-cell differencing A step donor-cell option added with the associated smearing reducing numerical diffusion. the only usable option. 7. Only a nonequilibrium vapor- An optional (faster) quasi-equilibrium ization/condensation model. heat-transfer-limited vaporization/ condensation algorithm. 8. Exchange rates can force time- Algorithm written to accept arbitrary step size reductions. exchange rates.

271 is explained in App. 0), well-planned tests could be very useful to confirm or disprove theories on steam explosions, or to provide ideas for modeling. The coupling length scale proposed by Theofanous and Corradini in the SERG report seems amenable to experimental investigation. Entering solid spheres into water and allowing film boiling to take place in a two-dimensional geometry could be used to check flow regimes and momentum coupling. More information is needed on the time-dependent breakup of thermite and surface area formation during coarse premixing. Besides a "business as usual" continuation of FITS tests, continuation of some other previous activities may not be cost effective. Some examples are the following. Setting up additional committees of experts to merely manipulate incomplete and insufficient data can be expected to produce simply more word engineering, net credible issue resolution. Performing further extensive parametric studies with SIMdER-II or any other code in which the conclusions reached depend so critically on the judgment of the investigators will not be convinc- ing. Level 2 quasi-steady-state arguments, which claim to eliminate this issue, apparently have not gained wide endorsement for the complex area of steam explosions. Their uncritical endorsement will lead to loss of credibility as other nations continue to investigate this area and perhaps disprove simple models. Finally, we suggest that the response of the vessel head to case 4 loads be evaluated in detail. If the head-failure modes do not appear to suggest a containment threat at various "reasonable" vessel head temperatures, the proposed 0.01 upper bound would have considerably increased confidence. Branch 6 in Chap. VII would be less indeterminate. Although it is obvious that much additional work could be done in the areas analyzed in this study, we cannot guarantee that performing this work will allow closure on the steam-explosion issue. Additional knowledge of meltdown sequences, of what novel characteristics steam explosions might exhibit at large scale, and of how these characteristics are to be included in a future analysis seems to be a prerequisite for truly reducing the outstanding uncertainties.

~~272 APPENDIX A THE SIMMER-II MANUAL AEOS TREATMENT~~

The material presented in this appendix has beer, reprinted from the SIMMER-U manual, Rev. 2. Although the nomenclature has been retained, the equation numbers have been made consistent with the current report. The equations of motion presented in the SIMMER-II manual are not complete without the EOS relating the microscopic densities and internal energies of the components in the three fields to pressure and temperatures. Furthermore, EOS information is required for the phase-transition models to predict the mass- transfer rates. Two types of EOS are available in SIMMER-II. First, the AEOS of SIMMER-1 has been extended to include control material and fission gas as well as fuel, steel, and sodium. Second, a tabular EOS is included as an option to provide realistic material properties that can be revised as new supporting experimental data become available. Currently, the tabular approach is nonfunctional. The AEOS used in SIMMER-II contains simplified models that are not sufficiently general to apply to all thermodynamic states. Several of the simplifications arise from assumptions made in the development of the fluid-dynamics methods. Other simplifications are caused by the analytic nature of the EOS. This nature assists in understanding the thermodynamic phenomena occurring during an accident calculation. More complex EOS do not have this advantage.

I. MATERIAL TEMPERATURES The thermodynamic relation between the temperature and the energy of a material is obtained by integrating the specific heat of the material temperature from zero K to the material temperature. For the solid state, the specific heats are assumed constant. Thus,

~~eSm " cvSM TSm • m " X MCSB • (A"1)~~

~~where~~

~~cvSM is *ne constant-volume specific heat for the solid phase of material M,~~

273 M • M(m) is the EOS material number for structure-field energy component m, and Tc_ is the temperature of structure-field energy component m. At the melting temperature, Eq. (A-l) yields the definition of the solidus energy,

~~eSol,M* cvSMTMelt,M • M " * where~~

cSol M ia *nc s°lidus energy for material M, ^Melt M *s *^eme lting temperature of material M, and NMAT -5 is the number of materials. During the melting process, the material temperature is assumed to remain constant as the energy equivalent to the material heat of fusion is absorbed. Com- pletion of the melting process defines the liquidus energy,

~~eLiq,M " cvSM TMelt,M + nf,M • M " X NMAT ' (A"3) where~~

eLiq M *s *ne 1iquidus energy of material M and hf it is the heat of fusion of material M. The transition from the liquid to the vapor state is similar to the solid-liquid transition. However, the transition temperature is the saturation temperature for the material and the saturation temperature is dependent on the local pressure. In SIMMER-II, this vapor pressure-temperature relation is

TM*

~~PM e TSat'M~~

~~274 where~~

pvM is the vapor pressure for material M, pjyj and T|y| are input parameters for material M, and ^Sat M is *^c s&turation temperature for material M. Equation (A-4) is easily inverted to yield the saturation temperature,

~~T* TSat,M " — ' (A"5)~~

~~PvM~~

The energy for each liquid material at the saturation temperature is obtained using a constant liquid-heat capacity until the saturation temperature reaches two-thirds of the critical temperature. For higher temperatures, the average enthalpy between the liquid and vapor phases of a material, neglecting pressure-volume effects, is approximated by extrapolating linearly to the assumed critical point energy. Thus, the condensate energy is obtained by subtracting one-half the heat of vaporization from the average internal energy extrapolated to the critical energy. The condensate energy is defined by

~~eCon,M " eLiq,M + cvLM (TSat,M "~~

~~TSat,M * 3 TCrt,M •~~

~~cCon,M " cLiq,M + cvLM (TSat,M " 3 TCrt,M) + aL,M " 2 h-~~

~~TSat,M > 3 TCrt,M~~

~~M = 1 NMAT ,~~

~~275 where~~

~~cLia M *8 *^e liquidus energy of material M, h^_ w is the enthalpy of vaporization of material M,~~

~~cvLM *8 *^e con8tan*"v°lume specific heat for the liquid phase for material M,~~

~~Tc-i u is the saturation temperature for material M,~~

~~M *8 ***« critical temperature for material M,~~

~~M is *^e mclt^ng temperature for material M,~~

~~aL M *8 defined to obtain continuity in eCon M a* ^Sat M • 2/3 TCrt,M« and~~

~~c tu is defined so that the input critical point energy, er_f M, is obtained.~~

~~The saturated vapor energy then is defined for each material by adding the heat of vaporization to the condensate energy and subtracting the work involved in the expansion process, or~~

~~eVap,M - eCon,M + h^g,M " PAvM • M -~~

~~Unlike the solidus and liquidus energies, the condensate energy and saturated~~

~~vapor energy vary with pressure because Tga^ ^ depends on the pressure. In addition, the heat of vaporization is a function of the saturation temperature,~~

~~T t~~

~~Crt ,M~~

~~where~~

~~h^ u and {M are input parameters for material M.~~

~~276 Thus, aL M and cyLM are easily evaluated as~~

~~CA-9)~~

~~- 3 *CrtlM lilL , M - 1 NMAT. (A-10) TCrt,M~~

~~The pressure-volume work term in Eq. (A-7) is obtained by differentiating the vapor pressure-temperature relation, Eq. (A-4), and then using the Clapeyron equation,~~

~~AT Sat,M~~

~~Consequently, we obtain~~

~~The equations for the internal energies of the liquid components are evaluated in a fashion similar to those equations involved in obtaining the condensate energies. They are~~

~~eLm " eLiq,M + cvLM (TLm " TLm < 3 TCrt,M •~~

~~1 1 + 2 Lm " eLiq,M + cvLM (TLm " 3 aL,M " 2 h- 3 TCrt,M~~

~~m • 1, ..., 4 ,~~

~~277 where~~

~~W1 • TH>~~

~~TLm * TCrt,M , and~~

~~Tj^ is the temperature of liquid-field energy component m.~~

Only four liquid-field energy components are indicated for Eq. (A-13) because only the first four are in the liquid state. The solid-fuel and solid-steel energies are determined from Eq, (A-l). Addition of energy to the vapor increases its energy above saturation according to

eG,M " eVap,M + cvGM • M " *

~~where~~

cvGM ia *^c constant-volume specific heat of vapor phase of material M and TQ is the vapor temperature for the vapor mixture. For fission gas (M - 5), the equation for the vapor energy is revised to neglect pressure dependence on the internal energy. The assumed relationship is

~~eG,5 " eCrt,5 + cvG5(TG " ^rt.S^ •~~

~~This form is obtained by always evaluating eG 5 from~~

~~eG,5 " eLiq,5 + cvL5 (TSat,5 - 3 TCrt,5) + aL,5 + 2~~

~~- pAv5 + CvG5 (TG - T3at5) , and~~

~~278 piesetting the parameters in this equation with the constants~~

~~cvL5 " cvG5 •~~

~~aL,5 " «Crt,5 " 3 QvG5 TCrt,5 " cLiq,5 • h-~~

Equation (A-14) is written for each material. In general, SIMMER-11 computes a mixture of vapors having a single energy and temperature. To obtain a relationship between the vapor temperature and vapor energy, we define

~~eG,M " eG,M " cvGMTG • M - 1 NMAT . (A-17)~~

~~Then we assume that the mixture's total is the sum of the energies of the mixture's components, or NMAT CG " I *m eG,M • (A-18) m-1 where~~

~~PGm xm is the mass fraction of vapor material m in the vapor mixture. ?G~~

~~Because the same EOS is used for fertile-fuel vapor and fissile-fuel vapor, their mass fractions have been combined to form Xj. Substitution of Eq. (A-17) in Eq. (A-18) results in~~

~~NMAT NMAT eG ' TG I( II xxm cCvGMVGM^) + + EI xm eG,M • m-1 m»l~~

~~279 Equation (A-l) is inverted easily for the structure and solid particle temperatureSi~~

~~, m - 1 NCSE, and (A-20) :vSM~~

~~eLm m cvSM~~

~~Below 2/3 T^rt M, Eq. (A-13) can be inverted for as~~

Above 2/3 TCrt M, Eq. (A-13) is solved for T^ by a Newton-Raphson procedure as is explained in Sec. VI. Equation (A-19) contains the vapor temperature both explicitly and implicitly (through the pressure). Again, Sec. VI details the algorithm for solving for the vapor temperature.

II. TWO-PHASE FLOW PRESSURES For two-phase flow, the local pressure is obtained from the vapor EOS. Be- cause the vaporization and condensation models described in App. I are not equilibrium models, the pressure is for the vapor alone, rather than for the vapor in equilibrium with its liquid. Thus, the vapor pressure equation, Eq. (A-4), is not evaluated for the vapor pressure using the liquid temperature, and instead, each vapor material is assumed to obey a modified polytropic gas relationship with pressure given as

~~P(3m " NMAT~~

~~280 where~~

pGm *s *^c Partial pressure of vapor material m, pGm is *^e microscopic density of vapor material m, and 8w is a variable gas "constant" for vapor material M. The most important parameter in this equation is R^. The algorithm chosen to evaluate RLJ is based on the observation that in the analysis of LMFBR hypothetical core-disruptive accident (HCDA) situations, high vapor densities aid significant deviations from ideal behavior are generally calculated near saturated conditions. For example, it appears very difficult to obtain highly superheated sodium vapor except at low density and pressure. However, in some cases SIMMER-II has been employed with vapor temperatures well above the critical temperature. For example, in the expand-on phase of hypothesized steam explosions, large amounts of fuel can exist at 3 000 K, but the critical temperature of water is 647.3 K. Therefore, variations in R^ are assumed to be based on saturation conditions with a correction for vapor temperatures above the critical temperature.

For TQ below TQTI ^, R^ is defined by a function f«, where f« is a simple empir- ical fit based on the product of the vapor density and pressure. Also, each component is assumed to be fit independently using saturated vapor conditions. For TQ above TQ_^ ^, the Rw is assumed to approach the infinitely dilute value monotonically. The formulation chosen may be summarized by

~~fM (A-24)~~

~~where~~

~~T T gM G < Crt,M~~

~~" TCrt,M TG>TCrt,M~~

~~fM < ROGCUT~~

~~aMl0 aM30 R0GCUT < pGmTG~~

~~281 M - aMll + aM21 (pGoTG)1/2 + aM31pGmTG •~~

~~fM " ^rt.M • pGmTG > (pGmVcrt~~

~~with~~

~~RMi - <7M~~

~~PCrt.M trt.M (pCrt,MTCrt,M)~~

~~pCrt,MTCrt,M •~~

~~(PQQ^TQ)0 = the product of the vapor density and tempeTature at~~

~~TSat,M " °*95 TCrt,M • i~~

~~aM10' &M20« aM30' aMll« aM21« and aM31 are fittin8 constants,~~

~~I™ • —^— «= the ratio of vapor specific heats at infinite dilution for material M,~~

~~M = ^^c critical pressure for material M,~~

~~pCrt M = *^e critical density for material M, and~~

~~ROGCUT and aG M - SIMMER input variables.~~

~~The six fitting constants are determined along the saturation curve with from the following conditions.~~

~~(1) For continuity, K^ » RJ^J at PQ^Q * ROGCUT.~~

~~(2) At infinite dilution, Rw should be constant; therefore,~~

~~282 d(R») iL-- = 0 at PrJTr. - ROGCUT .~~

~~C3> At TSat,M " °'95 TCrt,M- *M * RMo where RMo is defined using the vapor-pressure relationship and (PQBJQ^Q'~~

(4) Values for Rw must be continuous at C^Gm^G^o*

~~(5) For continuity, 1^ - RcrttM at (PQJQ)^.~~

~~(6) When a component's partial pressure equals the critical pressure, infinite compressibility is assumed; therefore,~~

~~Again, the one exception to this fitting procedure is the fission-gas component, where Rj is assumed to be R^, or~~

~~R5 - (75 - 1)CvG 5 . (A-26)~~

~~The total pressure now can be expressed as the sum of the partial pressures~~

~~NMAT NMAT P = I VQm-l ^GBMO ' CA"27) m=l m=l~~

~~The gas constant, pressure, and vapor temperature are obtained from Eqs. (A-19) and (A-27) by a Newton-Raphson procedure outlined in Sec. VI.~~

283 HI. SINGLE-PHASE FLOW STATES For single-phase flow, the local pressure is obtained from the EOS for the liquid components. This pressure should be continuous at the transition between single-phase and two-phase flow. For single material systems, such continuity is obtained by requiring equilibrium conditions between the liquid and the vapor components. Multimaterial systems are more complicated and require additional assumptions to obtain continuity. In SIMMER-II, the materials are assumed to be immiscible; and thus, at equilibrium, the vapor pressure for the material mixture is the sum of the vapor pressures for the individual materials. As derived in Sec. VII, the resulting pressure for single-phase flow is given by

~~r NCLE L - os) k=l a, k"m L~~

~~284 and where~~

cti is the liquid-field-volume fraction at the assumed transition point between single- and two-phase flow, &^ is the square of the effective sonic velocity for the liquid- field mixture, p> is the sum of the liquid-field vapor pressures, PvM(m) *s **** effective vapor pressure for liquid-field energy component m, jjjj) is the vapor pressure [Eq. (A-4)] for liquid-field energy component m evaluated at the component temperature, and

cLm *s *^c liquid sonic velocity for component m. With the decrease in liquid fuel density as a function of temperature, single-phase liquid pressures can sometimes develop to limit the energy production in a fast reactor disassembly calculation with SIMdER-II. These pressures may be unrealistically high if a constant sonic velocity of fuel is employed. Consequently, the actual decrease in the sonic velocity of fuel is programmed via the algorithm

~~c^j - max~~

~~(A-34)~~

~~CL1 = C » TL1 * TCrt,l •~~

~~where A, B, and C are input constants, Tj^ is the temperature of liquid fuel,~~

T^r£ * is the critical temperature of liquid fuel, and c^j is the sonic velocity of liquid fuel. A continuous thermodynamic state at the transition from single- to two-phase flow imposes further conditions on the vapor state at the transition. These conditions are relatively trivial for single-component systems, that is,

~~285 the liquid and vapor components have the same pressure and temperature. For the AEOS the single-component conditions are expressed as~~

~~TrG. - T,L , and (A-35)~~

~~P R T PG - G G - PL -~~

For multicomponent systems, the situation is somewhat different. In SIMMER-II, the state of each vapor material is assumed to be independent of other vapor materials; and each vapor material is assumed to be in saturated equilibrium with its liquid counterpart. This implies that the vapor's internal energy should be defined by

~~NMAT NMAT _t (pGmeVap,MHI P^) • (A-37) m=l~~

~~For the AEOS, Eq. (A-37), Eq. (A-19), and the definition of CQ » are used to eliminate ey ^ and give TQ as~~

~~NMAT NMAT _j TG " E pGmcvGMTSat,M( X pGmcvGM) • (A~38)~~

In Eq. (A-38), P^ and To-* u remain arbitrary. The condition of saturated equilibrium requires that the saturation temperatures equal the respective liquid temperatures. The vapor density is chosen by assuming

~~PvM(m) " PGm = pGm % TG~~

When TQ is below TQT^ M> % is only a function of the product P^ TQ, and R^ can be calculated iteratively from Eq. (A-39) for known values of Py^^)- If an estimated value of TQ is above TQ^ M> *ne iteration for R^ involves gj^ from

Eq. (A-25). In the present version of SIMMER-II, gM is estimated using a TQ calculated from Eq. (A-19) using the vapor densities and internal energy from 286 the previous cycle. After R^ is obtained, Eq. (A-39) is solved for P^, which is substituted into Eq. (A-38) to obtain TQ from known quantities as

~~NMAT-1 - y [PvMfmi cvGM~~

~~V rPvMfmi cvGMi R 1~~

This summation does not include the fission-gas component, m = NMAT. In single-phase cells, the fission-gas's vapor density is assumed to be zero. Fur- ther, this result does not guarantee thermal equilibrium conditions at the single-phase/two-phase transition. The vapor-mixture temperature obviously cannot equal all the liquid temperatures simultaneously; hence, temperature differences will exist to drive phase transitions. For a given EOS and a single vapor temperature, no other recourse seems available; hopefully, the temperature differences will not produce large phase-transition rates. In the previous discussions, the vapor pressure, p^, was evaluated at the liquid temperature, T^. To account for superheat, which delays the single-phase/two-phase transition until the liquid temperature reaches the normal boiling temperature plus the superheat temperature, the saturation temperature and vapor pressure actually are evaluated at the liquid temperature minus the superheat-temperature increment. When the switch from single to two phase occurs, the pressure is continuous. However, because the saturation temperatures are lower than the liquid temperatures from the superheat-temperature increment, rapid vaporization and pressurization of the two-phase region can occur.

IV. LIQUID MICROSCOPIC DENSITIES The use of constant microscopic densities does not allow the calculation of many single-phase expansion effects, which are important in determining the neu- tronic behavior of reactor transients. However, calculation of variable microscopic densities for all components would involve extensive code modifications to be efficient. As a compromise, temperature-dependent, liquid, microscopic densities are calculated for the first three liquid energy components, or fuel, steel, and sodium. The formulas used are

~~287 2 3 2 pLm * alm + a2m TLm + a3m TLm + a4m TLm« TMelt,M < TLm < 3 TCrt,M~~

~~pLm ' pCrt,M t1 + blm *n?* + b3m $ • f TCrt,M < TLm < TCrt,M~~

~~pLm - pCrt,M • TLm > TCrt,M • where~~

~~" X " TLm/TCrt.M pCrt M ™ *be cri*ical density, and alm« a2m- a3m' a4m' blm- b2m' and b3m are inPut parameters.~~

~~The component temperature for Eq. (A-41) are obtained from the internal energies by inversion of Eq. (A-13) as described by Sec. VI.~~

V. AEOS INPUT The vapor state is determined in one of four ways, depending on the value of the input variable IINP. In the first method (IINP - -1), the liquid temperatures are specified and one vapor-field component is selected through the input variable IOOMPO to be in equilibrium with its liquid-field counterpart. Because the two fuel vapors, fertile and fissile, are assumed to have the same EOS, the variable ICOMPO refers to the first four liquid-field energy components'- fuel, steel, sodium, and control, in that order. Thus, the vapor-field temperature is set equal to the temperature of liquid material ICOMPO and the cell pressure, vapor density, and vapor energy result from the EOS. Finally, the total component mass is conserved by subtracting the vapor mass generated from the liquid field. For the AEOS, the calculation for IINP » -1 is done as follows. First, the vapor temperature and saturation temperature for component m - ICOMPO are determined from

~~TG " TSat,M - TLm~~

~~288 where~~

~~M • M(m) (m * ICOMPO) is the material number for vapor-field material component m. This allows the cell pressure to be determined from~~

~~-T* P " PM e*P (T ) " (A-43) TSat,M~~

~~The vapor's internal energy is now evaluated from~~

~~eG " cvSM TMelt,M + hf ,M + cvLM (TSat,M " TMelt,lf)~~

~~2 if TSat,M < 3 TCrt,M • or~~

~~eG " cvSMTMelt,M + hf ,M + cvLM (TSat,M " 3 TCrt,M)~~

~~aL,M + 2 h^g.~~

~~TSat,M>|TCrt,M • CA-44)~~

~~The initial microscopic density for vapor-field material component ICOMPO is obtained from~~

~~FCpGmV " P "~~

~~This is solved using a Newton-Raphson iteration where the iteration scheme is~~

~~289 The derivative has the form~~

~~dF (A-47)~~

~~where~~

~~p T dB Gm G M (A-48)~~

~~is evaluated from~~

~~"GmTG < R0GClJT •~~

~~aM20 + \ aM30 R0GCUT « pGmTG <~~

~~- \ aM21 pGmTG , and~~

~~(A-49)~~

In the current version of SIhWER-II the option has not been programmed to input the vapor temperature above a component's critical temperature with &Q ^ from Eq. (A-2S) smaller than a large number, for example, 10 . To determine the new component densities, we first conserve volume,

~~•g (A-50)~~

~~and then conserve mass,~~

~~pGm 4m Lm (A-51)~~

~~290 where~~

«Q is the initial (cycle zero) vapor-volume fraction, a^n is the initial volume fraction for liquid-field component m, ag is the structure-field's total volume fraction, a£ is the input liquid-field's volume fraction, which necessarily accounts for the remaining liquid components,

~~p£m is the input macroscopic density for liquid component M, Pj^ is the microscopic density for liquid component M, and~~

~~a0 is the fraction of the flow volume fraction below which single-phase calculations are performed. These equations can be solved for CCQ as~~

~~«G-|O. -I'a-Oj • tA-52) um o J~~

~~Hence, the initial macroscopic density of vapor for component m is~~

~~pGm- and the initial liquid macroscopic density becomes~~

If the vapor- and liquid-field component specified by IGOMPO is fuel, then the densities must be determined for the fertile- and fissile-fuel vapor. The mass fractions of the fertile fuel and fissile fuel in the vapor field are assumed to be the same as the mass fractions in the liquid fuel. These mass fractions are

~~291 and hi~~

~~(A-55) hi where~~

is *^c mass frac*ion of fertile fuel in the liquid fuel, x-t2Ll *s ***« ma8S ^'action of fissile fuel in the liquid fuel, Pjl and ?^2 arc *he macroscopic densities of the liquid fertile-fuel and the liquid fissile-fuel density components, respectively, and ?j_l is the macroscopic density of the liquid fuel energy component. Then, the initial fuel's vapor densities for fuel are calculated as

*gm " x-ftnLl?Gi • a - 1, 2 . (A-56)

For the second method (IINP - 0), the vapor-field pressure is input and this pressure again is assigned to a single vapor-field component through the input variable ICOMPO. The vapor-field energy component, then, is calculated to have a vapor temperature equal to the saturation temperature for the input pressure. In addition, the temperature of the liquid-field energy component also is set to the same saturation temperature, thus satisfying Eq. (A-42). For the AEOS, the major difference between this option and the IINP <• -1 path is that we now solve for the inverse of Eq. (A-43) or

~~T* TSat,M-—V- CA-57)~~

~~292 where~~

~~p1 is the input vapor-field pressure.~~

For the third method (IINP » 1), the vapor state is specified by the vapor temperature and the microscopic densities input for the vapor-field components. By use of the EOS, both the initial vapor energy and the pressure then are obtained. For the AEOS, the desired pressure is assumed to be obtained by using the gas constant at infinite dilution or

~~PGm - pGm <7M " « cvGMYG • where~~

~~pGm *s *^e ^nPu* microscopic vapor density and~~

~~NMAT P - I PC, • (A-59) m-1~~

~~The actual microscopic vapor density is then determined from~~

~~using a Newtcn-Raphson algorithm as was done to solve Eq. (A-45). The partial pressures of the vapor-field components determine the corresponding saturation temperatures~~

~~T* TSat,M" V • CA-61)~~

~~from which the vapor's internal energy can be evaluated by~~

~~293 NMAT e C + c G-I *m < G,M vGM V ' CA-62)~~

Because the input density has been changed, the legitimate question to ask is whether mass conservation should be attempted. In this situation, setting ICOMPO to one will force such an attempt for two-phase cells. The equations that must be solved are

~~m m~~

~~G pGm + *Lm 'iV*1 " «o> " «G pGm + «Lm "Lm^1 " *o> ' (A"64)~~

where CEQ is the input vapor-volume fraction before mass adjustments, the a}m arc the input liquid-field component volume fractions, and we consider the four energy components that can vaporize. Formally, Eqs. (A-63) and (A-64) can be solved to obtain

~~«G U " 1 — 1 Bpl ^ - . (A-65) 4 pPn. (1 - a (1 "Z Gm c Lm~~

Unfortunately, a negative a^ may result if insufficient liquid density is available, for example, in a cell that contains sodium vapor but no liquid sodium. Therefore, Eq. (A-64) is valid for those components where

~~«G ^Gm " "L> < «Lmp~~

~~294 This still allows errors, because~~

~~L «o> > «GpGm " «G is not guaranteed. However, an iteration is avoided, and this restriction is believed adequate for the IINP - 1 option. Application of Eq. (A-6S) therefore defines c&, and then~~

*Gm " «GpGm • m " * 5 , and

*u - max (0, P^ - PQJ , m - 1 4 . (A-67)

For those cells initially having single-phase flow, the initial pressure is determined in the above manner, but the single-phase state is calculated using the single-phase EOS. For a cell to have single-phase flow, the user must have specified sufficient structure and liquid for the vapor-field volume fraction CCQ to be less than the single-phase/two-phase transition volume fraction, atg • a (1 - a

~~PvM(m) - PM e ^ SuP'M' . m-1 4 , (A-68)~~

~~where~~

~~^nn M is the superheat of material M.~~

~~The total vapor pressure at the transition is formed from these liquid pressures~~

~~4 PL " E PvM ' (A-69) m-1~~

~~295 Equations (A-28) and (A-32) are written in terms of density components and are then solved for the liquid-field volume fraction that yields the desired cell pressure~~

~~NCLR «L " «Lo •i where~~

~~a£ is the initial liquid-field volume fraction,~~

• 1 " - OQQ is the liquid-field volume fraction at the transition between single- and two-phase flow, ?• is the total macroscopic density, input for the liquid-field im inPut volume fraction of liquid-field density component m

c^m is the square of the sonic velocity for liquid-field density component m, and p° is the initial cell pressure determined from Eq. (A-S9). The adjusted liquid-field volume fraction is assumed to apply uniformly to all liquid-field components so that

~~-1 NCLR~~

~~where~~

~~jm is the initial macroscopic density of liquid-field density component m and~~

~~_fm is the input macroscopic density of liquid-field density component m.~~

~~296 The adjusted liquid-field volume fraction also leads to a new initial vapor-fieId volume fraction,~~

~~c£ - 1 - as - «£ . (A-72)~~

The initial vapor state for the single-phase cells is computed in the manner discussed in Sec. III. The fourth method (IINP • 2) is similar to IINP = l; however, it permits conservation of mass to be maintained with significantly less effort in two-phase cells. In this case, the gas constant, R^, is computed directly from PQ^Q- The pressure thus is computed from

without an iteration. Because the microscopic vapor density is unchanged in two-phase cells, no modification of the liquid density is performed. Otherwise, this option proceeds identically to the IINP = 1 approach.

VI. DETERMINATION OF COMPONENT TEMPERATURES Before the exchange functions can be evaluated, an EOS call is required to evaluate the component temperatures. The simple cases for the AEOS are given in Eqs. (A-20)-(A-22). The more complex cases involve solving Eq. (A-19) for the

~~vapor temperature and solving Eq. (A-13) for T^ above 2/3 TQT^ ^. Equa- tion (A-19) can be rearranged as~~

~~NMAT eG " I xmeG,M 1 £ xmcvGM m=l~~

~~297 The Ncwton-Raphson algorithm is~~

~~CA-75)~~

~~where~~

~~NMAT d • _ y G.M 2 -1 . (A-76) dTG NMAT xmmccvGM~~

~~The explicit expressions for efi M are~~

~~eG,M " eLiq,M + cvLM~~

~~" PAvM " cvCMTSat,M ' TSat,M < 3 TCrt,M •~~

~~eG,M = eLiq,M + cvLM (TSat,M " 3 TCrt,M) + aL,M~~

~~1 2 + 2 h^g.M ' pAvM " cvGMTSat,M • TSat,M > 3 TCrt,M ' (A-77)~~

~~Using Eq. (A-12) for the work term expression pAv ,~~

~~298 deG,M dTSat,M , TM TSat,M vovGMw f f G G XM Tc t M Ts t~~

~~for ,M * 3 TCrt,M • and~~

~~de dT G,M Sat,M S ( ) dXr dTr T l TM Crt,M " Sat,M~~

~~for TSat,M > 3 TCrt,M (A-78)~~

~~where~~

~~dTSat,M "TSat,M (1 + IV.M ) and (A-79)~~

~~TG dRM~~

~~Due to the form adopted for is the same as D^ given in Eq. (A-49) if TG < TCrt,M" If TG > TCrt,M'~~

~~~ TCrt,M + (A-80)~~

~~where g^j, K^^, f^, and aQ ^ are defined in Eqs. (A-24) and (A-2S), and D^ is given by Eq. (A-49).~~

~~299 The two difficulties with this iteration are that convergence is slow or a com onen s initially, and if Tgat ^ a ^Crt M ^ ^ P t » then~~

~~dF dTG~~

The first problem is avoided by storing the value of the vapor temperature obtained on the previous call to the EOS as an additional cell variable and by using this value as an initial estimate for TQ. The second problem is avoided by defining a minimum value of -cvgu + 10 J kg K to (de/j U/^TQ). This insures that (dF/dTg) remains sufficiently far from zero to avoid numerical difficulties. 2 In the case, when > - , we define a similar function, F, by M

~~- eLiqM) - \ - 0~~

~~The derivative is evaluated as~~

~~dF u dTLLm 2(T Crt.~~

~~and the initial estimate of for the iteration is~~

~~T^-min [TCrtM - 2K, max (TSatM, |TCrtM)] (A-83)~~

~~Because the iteration is done only if e» < e^. * w, the iteration is only appli- 1 cable when Tjjjj < T^rt ^. If any TJ^J > TCrt M (because of a poor initial estimate, perhaps), the algorithm becomes~~

~~300 CA-84)~~

~~until the Newton-Raphson iteration can obtain a Tj^ below~~

VII. DETAILS IN THE EVALUATION OF SINGLE-PHASE LIQUID STATES In SIVMER-II the transition from a single- to a two-phase flow calculation occurs when the vapor-volume fraction exceeds a specified percentage of the available flow-volume fraction. The available flow volume fraction is given by 1 - otg, where et§ is the structure-field's total volume fraction. The fraction of this that determines the transition between single- and two-phase flow is denoted by aQ, that is,

~~OQ > aQ (l - a~~

ctQ < aQ (l - as) indicates a mesh cell having single-phase flow. (A-86) For mesh cells with two-phase flow, the cell pressure is determined from the vapor EOS, for example, the modified perfect-gas EOS if the AEOS is used.

~~If the vapor-volume fraction cutoff, aQ, is very small, the pressure calculation from the V;por EOS can be very sensitive to changes in the vapor-volume fraction~~

~~as the transition between single- and two-phase flow is approached. Thus aQ equal to zero is never used and most often a value of 0.05 is input to the SHMER-II code.~~

A. Single-Component Systems At the single-phase/two-phase transition it is necessary to have a continuous theimodynamic state. Otherwise, a mesh cell could oscillate between single- and two-phase flow. Because the pressure determines the fluid-dynamics state, pressure continuity must be maintained. In addition, thermal continuity is necessary so that large temperature gradients do not induce high phase-transition rates. Continuity in pressure and temperature implies that the liquid and vapor are in equilibrium at the transition from two phase to single phase. That is, for a single-component system,

~~PL " Pv = PG * PGRMTG • and (A-87)~~

~~301 TL - TG , (A-88) where Pi is the single-phase liquid pressure,~~

pv is the vapor pressure, PQ is the vapor state pressure determined from the EOS, PQ is the microscopic density of the vapor-field Bu is the ratio of the specific heats in the vapor-field TQ is the vapor-field temperature, and Tj^ is the liquid-field temperature.

~~In SIMMER-II the vapor pressure pv is given by the two-parameter approximation~~

~~, (A-89)~~

~~where p* and T* are input parameters determined from a fit to appropriate two-~~

phase data and Tgaj. is the saturation temperature for the pressure py. To develop the single-phase EOS, a single-component system is considered first and the treatment then is extended to a multicomponent system. Equa- tion (A-87) indicates the appropriate liquid pressure at the single-phase/two- phase transition, that is,

~~PL " Pv (V when "G " "Go " "o U " *s) ' (A-90)~~

To develop the EOS further, liquid must be compressed to add more liquid to the mesh cell, thus increasing the pressure in the cell. This path is illustrated in Fig. A-l. In this pressure-volume diagram, point A indicates the pressure-density state at the single-phase/two-phase transition. Point B is the state achieved through compression of the liquid. That is

~~302 PL~~

~~Pv(TL)~~

~~V, V LO V~~

~~Fig. A-l. Pressure-volume relation for the single-phase EOS.~~

~~PL - Sv (A-91) or~~

~~3pL 6p (A-92) PL " Pv~~

The microscopic density is used instead of the specific volume because SIKMER-II 3PL uses densities. The derivative -— is the square of the liquid-field's sonic velocity, c^, and is assumed constant in SINMER-II except for case of liquid fuel where it depends on the liquid temperature. Thus Eq. (A-92) becomes

~~PL " Pv CTL) + c£ (PLeff - Pj (A-93)~~

~~where P^ t^f is the effective microscopic density of the liquid field and P^ is the microscopic density of the the liquid field at the single-phase/two-phase~~

303 transition. Because the normal microscopic density used in SIMMER-II determines the single-phase/two-phase transition point, P^ *s the normal microscopic density. Because the normal microscopic density generally is desired to be associated with a liquid's density reported in the literature, for example, as in steam tables, we define the liquid volume fraction, «L, by

~~(A-94)~~

~~where fij is the macroscopic density of the liquid field. Consequently, at the single-phase/two-phase transition P^ becomes P^Q, which is given by~~

ho - and the macroscopic density of the liquid field is the same value as if it occupied all the volume available for fluid in a cell. This definition means s ven a or e mass er un that P^ eff * £i by 'i/(* " S^' *^ P ** °^ available fluid volume. Consequently, we may write

~~PL - Pv~~

~~Using the definition of the volume fraction at the single-phase/two-phase transition, aLo, we obtain~~

*Lo " 1 ~ aS " «Go

~~- 1 - as - a0 (1 - as) (A-97)~~

~~- (1 - «o) (1 - as) . which gives [by using of Eq. (A-94)]~~

~~304 (A-98)~~

~~(l - «c] - OR (l ^ *»]>, . (A-W~~

~~In addition, the SIM4ER-II fluid-dynamics method needs the change in pressure with respect to macroscopic density. This change is obtained from Eq. (A-96) as~~

~~(A-10O)~~

B. Multicomponent System For a multicomponent system the components are assumed to be immiscible. Hence the mesh cell pressure at the transition is the sum of the liquid-field component pressures as given by the component vapor pressure curves. Additional mass added to a mesh cell again causes compression of the liquid, increasing the pressure. For a given liquid-field component m, the pressure in that component is

~~PLn - I Pvk + cLm Spm • U where PjjQ is the pressure in liquid-field component m, c^ is the square of the sonic velocity for liquid-field component m, and~~

~~&Pm is the change in microscopic density from the state at the single- phase/two-phase transition to the current liquid state.~~

305 To obtain the single-phase pressure, all liquid-field component pressures are assumed to be equal and the sum of the macroscopic densities is assumed to be the total macroscopic density for the liquid field. First

~~h'l Pvk CA"102) k~~

~~Is defined. Then, by assuming that the actual volume occupied by each component is proportional to its volume fraction, the multicomponent equivalent to Eq. (A-96) is~~

~~P " PLm ' PL + cLm t~~

~~where the m subscript has been added to indicate component dependence.~~

~~Equation (A-103) is multiplied by a^o/c^ and summed over m to yield~~

~~2 p A 104 (P - PL) I^T„* " Z (JJ. ~ we^* • «Lm Lm) • ( " > m CiuMu m O~~

~~The multi component version of Eq. (A-94) is~~

~~KLm~~

~~306 Substitution of Eq. (A-1O5) into Eq. (A-104) gives~~

~~CA 106) Y Lm S m " L -5-~~

~~Use of the definition for «G allows Eq, (A-106) to be written as~~

~~p . $ + _i_ [_£_L ILS] fi , (A-107) L l a L v «Lm • S~~

~~i - I -^H~ • (A-10S)~~

~~m «L ^Lm~~

In the current version of SIM4ER-II, the pressure iteration is performed assuming all single-phase cells retain constant composition. By assuming aim/ai is a constant, Eq. (A-104) can be differentiated to obtain

~~. (A-109) 3PL * " «S~~

~~307 This is evaluated in the code by using the definition of a^, Eq. (A-105), and~~

~~«Lo, Eq. (A-97). to yield~~

~~(A-110)~~

~~m PLm cLm~~

Under some conditions SIMMER-II will compute that a small amount of material at a high temperature is present in a mesh cell. Hence the sum of the liquid pressures, pL in Eq. (A-102), can be due primarily to a small amount of liquid. To offset this anomaly, we submit the following arguments. The high pressure of a small amount of material will compress the other material in a cell. If this pressurization were considered to come from additional single-phase mass added to the mesh cell (that is, not from the vapor pressure of the small amount of material), then the corresponding density increase could be found from

~~where~~

~~B L «L Lm m*k~~

If the macroscopic density P^k °f *^c small amount of material is less than 6?^ in Eq. (A-lll), then the small amount of material must vaporize to occupy the volume, decreasing its contribution to the total vapor pressure. Consequently, for the AEOS the pressure used in Eq. (A-102) for component k is the minimum of

~~3Q8 Pvk » Pme"Tm/Tu the relationship for the vapor pressure,~~

~~pvk (TLk) - *Lk~~

~~for replacement by compressed liquid, or~~

~~Pvk - N*Gk + hi) «M(k) for complete vaporization, where~~

~~?Gk + ?Lk is the total macroscopic density of component k,~~

s as IWk) * *ke E constant for component k, TQ is the vapor temperature, and M is the material corresponding to liquid energy component k. Finally, we note that SI&MER-II FORTRAN treats the liquid's sonic velocity in the form of a density component rathet than an energy component. The original intent of this division was to allow different sonic velocities for fertile and fissile fuel. However, because the most recent version of SIIMER-II forces one temperature-dependent function for the fuel's sonic velocity, the density subdivision has become irrelevant, and Sec. VII has not been complicated by these considerations. Equation (A-70) illustrates how Eq. (A-106) could look when written in terms of density components and solved for an a^ based on an assumed pressure, p°.

~~309 APPENDIX B CFS NODE /VAPOR/WRBAEOS (EOS CORRECTION SET)~~

The following is a listing of the actual modifications to the SIMMER-11 program library required to implement the equations of Sec. II.C. This correction set is stored on the Los Alamos Common File System (CFS).

•IDENT b714 •/ */ CORRECTION SET FOR THE AEOS */ •/ ACVG IS THE FIRST COEFFICIENT ON THE VAPOR HEAT CAPACITY */ BCVG IS THE TEMPERATURE DEPENDENT COEFFICIENT ON THE VAPOR */ HEAT CAPACITY */ CORC IS THE CONSTANT CORRECTION TERM FOR THE VAPOR ENERGY IN */ IN THE DILUTE DENSITY LIMIT */ BPARA IS THE RELAXATION CONSTANT FOR THE HEAT CAPACITY */ CPARA IS THE MULTIPLIER TO ADJUST THE HIGH DENSITY GAS CONSTANT */ ALCVG2 IS LOG(CVGGM1*(ONE+CPARA)) */ DENARG IS AN INTERPOLATION VARIABLE USED IN COMPUTING RMAT */ *I EOS.10 *, ACVG(5) ,BCVG(5) ,CORC(5) ,BPARA(5) ,CPARA(5), ALCVG2(5) ,DENARG(5) *,CVGR(5),TSTARM *I B989.29 READ (NINP.811) ACVG(M),BCVG(M),CORC(M),BPARA(M),CPARA(M) IF (ACVG(M) .EQ. ZERO) ACVG(M)=CVG(M) IF (BPARA(M) .EQ. ZERO) BPARA(M)=EP10 *D AEOS.21 *D AEOS.28 •/ */ CHANGES TO MODIFY THE ITERATION FOR THE VAPOR TEMPERATURE */ *I B4K3.304 TGMIN - EM10 TGMAX - TSTARM *I AEOS.43 CVG(M)-ACVG(M)+TG*BCVG(M) *D BAK9.9 *I AEOS. 41 SUMDCG - ZERO SUMDE - ZERO *D B4K3.309 C DRDRG- (ROGPM/RMAT)* (DERIVATIVE OF RMAT WITH RESPECT TO ROGPM) *D B473.58,61 *IF -DEF.DBL.l RMAT(M) - EXP(ALRCRT(M)*GFUNC+ALCVG2(M)*(0NE-GFUNC)) *IF DEF.DBL.l RMAT(M) - DEXP(ALRCRT(M)*GFUNC+ALCVG2(M)*(0NE-GFUNC)) *D B473.63 DRDTG(M) = GFUNC*TG*(ALCVG2(M)-ALRCRT(M))*RGFUNC(M) 311 *D B473.70.75 OOEFC - ONE + ((ARG-T9SR0G(M))»DENARG(M))*CPARA(M) •IF -DEF.DBL.2 COEFM - ALOG(OOEFC) RMAT(M) - EXP(FFUNC*GFUNC+(ALCVGM(M)-»COEFM)»(ONE-GFUNC)) *IF DEF,DBL,2 COEFM - DLOG(COEFC) RMAT(M) - DEXP(FFUNC*GFUNO(ALCVGM(M)-KX)EFM)*(ONE-GFUNC)) DRDRG(M)-GFUNC»( ARGDUM*HALF+AEOS31(M)*ARG) 1 +((ONE-GFUNC)*CPARA(M)*DENARG(M)*ARG)/O0EFC DRDTG(M)=DRDRG(M)+GFUNC*TG*(ALCVGM(M)+€OEFM-FFUNC)*RGFUNC(M) •I B473.92 ESENT - SENT(M)*(ONE-TSAT(M)»RTSTAR(M)) *D AEOS.86 *IF -DEF.DBL.2 CVGR(M)- CVG(M) - AMIN1(ZERO,((CVL(M)-CVG(M))*TSAT(M) + ESENT 1 - HSTAR(M) + CORC(M))/(ABS(TG-TSAT(M))+BPARA(M))) *IF DEF.DBL.2 CVGR(M)- CVG(M) - DMIN1(ZERO,((CVL(M)-CVG(M))*TSAT(M) + ESENT 1 - HSTAR(M) + CORC(M))/(DABS(TG-TSAT(M))+BPARA(M))) SUMDE - SUMDE -I- CVGR(M)*Y(M) ESTAR - SIELS(M) - CVGR(M)*TSAT(M) + ESENT •D AEOS. 91.93 CVGR(M)- CVG(M) DTSDTG(M) - ZERO ESENT - ZERO SIELS(M) - SIELIQ(M)+AEOSLM(M)-tCVLP(M)*(TSAT(M)-TWO3RD*TCRIT(M)) GO TO 447 445 CVSAT - CVLP(M)*(TSAT(M)-TWO3RD*TCRIT(M))+AEOSLM(M) SIELS(M) - SIELIQ(M) - EALF*SENT(M) + CVSAT •IF -DEF.DBL.3 CVGR(M)- CVG(M) - AMIN1(ZERO,(CVSAT-CVG(M)*TSAT(M)+CVL(M)*TMLT(M) 1 + SENT(M)»(HALF-TSAT(M)*RTSTAR(M))-HSTAR(M)4C0RC(M)) 2 /(ABS(TG-TSAT(M))+BPARA(M))) *IF DEF.DBL.3 CVGR(M)- CVG(M)-DMIN1(ZERO, (CVSAT-CVG(M)*TSAT(M)4CVL(M)*TMLT(M) 1 + SENT(M)*(HALF-TSAT(M)*RTSTAR(M))-HSTAR(M)4CORC(M)) 2 /(DABS(TG-TSAT(M))+BPARA(M))) 447 CONTINUE SUMDE - SUMDE + CVGR(M)*Y(M) ESTAR - SIELS(M)-CVGR(M)*TSAT(M)+ESENT *I AEOS.99 CVSATF - CVLPP - SENT(M)*FACTOR*RTSTAR(M) DCVGRT - BCVG(M) IF (CVGR(M).LE. CVG(M)+EM10) GO TO 455 IF (TG .GE. TSAT(M)) 1DCVGRT - -((CVGR(M)-CVG(M)+DTSDTG(M)*(CVSATF-CVGR(M))) 2 /(TG-TSAT(M)+BPARA(M) ) )+BCVG(M) * (ONE+TSAT(M) 3 /(TG-TSAT(M)+BPARA(M))) IF (TG .LT. TSAT(M)) 1DCVGRT - ((CVGR(M)-CVG(M)-DTSDTG(M)*(CVSATF4CVGR(M)-TW0*CVG(M))) 2 /(TSAT(M) -TGfBPARA(M) ) )+BCVG(M) * (ONE+TSAT(M) 3 /(TSAT(M)-TG+BPARA(M))) 455 CONTINUE

312 *D AEOS. 101,106 SUMDCG « SUMDCG + Y(M)*DCVGRT SUMNU - SUMNU + Y(M)*(DTSDTG(M)*(CVSATF-CVGR(M))-DCVGRT*TSAT(M)) *D AEOS. 108 RSUMDE « ONE/SUMDE FCOEF - (SIEGL - SUMI)*RSUMDE FW FCOEF - TG SUMNU - SUMNU + FCOEF*SUMDCG *I AEOS.109 •IF -DEF.DBL.l DFDTG - AMINK-EM6, DFDTG) •IF DEF.DBL.l DFDTG « DMINK-EM6.DFDTG) •I AEOS.110 *IF -DEF.DBL.2 IF (F .LT. ZERO) TGMAX - AMIN1(TG,TGMAX) IF (F .GT. ZERO) TGMIN - AMAXl(TG.TGMIN) •IF DEF.DBL.2 IF (F .LT. ZERO) TGMAX <- DMINl(TG,TGMAX) IF (F .GT. ZERO) TGMIN « DMAX1(TG,TGMIN) IF (TGN.LT. TGMIN .OR. TON ,GT. TGMAX) 1TGN - (TGMAX + TGMIN) »HALF */ */ CHANGES IN THE ENTRY TO COMPUTE MATERIAL DERIVATIVES •/ •D B473.100,103 •IF -DEF.DBL.l RMAT(NZ) - EXP(ALRCRT(NZ)*GFUNC+ALCVG2(NZ)*(0NE-GFUNC)) •IF DEF.DBL.l RMAT(NZ) - DEXP(ALRCRT(N2)*GFUNC+ALCVG2(NZ)*(0NE-GFUNC)) •D B473.110.114 COEFC - ONE + ((ARG-T95R0G(NZ))«DENARG(NZ))*CPARA(NZ) •IF -DEF.DBL.2 COEFM - ALOG(COEFC) RMAT(NZ) - EXP(FFUNC*GFUNC+(ALCVGM(NZ)4C0EFM)*(0NE-GFUNC)) •IF DEF.DBL.2 COEFM « DLOG(COEFC) RMAT(NZ) » DEXP(FFUNC*GFUNC+(ALCVGM(NZ)-fCOEFM)*(ONE-GFUNC)) DRDRG(NZ)-GFUNC*(ARGDUM*HALF+AE0S31(NZ) »ARG) 1 +( (ONE-GFUNC) *CPARA(NZ) *DENARG(NZ)*ARG) /COEFC */ */ CHANGES IN THE ENTRY TO DEFINE THE VAPOR ENERGY •/ DURING VAPORIZATION/CONDENSATION */ *D B473.133.136 •IF -DEF.DBL.l RMAT(M) - EXP(ALRCRT(M)*GFUNC+ALCVG2(M)*(ONE-GFUNC)) •IF DEF.DBL.l RMAT(M) - DEXP(ALRCRT(M)*GFUNC+ALCVG2(M)*(0NE-GFUNC)) *D B473.143.147 COEFC - ONE + ((ARG-T95R0G(M))»DENARG(M))*CPARA(M) •IF -DEF.DBL.2 COEFM - ALOG(COEFC) RMAT(M) - EXP(FFUNC*GFUNC+(ALCVGM(M)+COEFM)*(ONE-GFUNC))

313 *IF DEF.DBL.2 COEFM = DLOG(COEFC) RMAT(M) = DEXP(FFUNC*GFUNC+(ALCVGM(M)+COEFM)*(ONE-GFUNC)) DRDRG(M)=GFUNC*(ARGDUM»HALF+AEOS31(M)*ARG) 1 +((ONE-GFUNC)*CPARA(M)*DENARG(M)*ARG)/COEFC *I AEOS.288 CVGR(M) = ZERO CVG(M) - ACVG(M) + BCVG(M)*TG *I AEOS.334 *IF -DEF.DBL.3 CVGR(M)= CVG(M) - AMIN1 (ZERO, ((CVL(M) -CVG(M))«TSAT(M) -HSTAR(M) 1 + SENT(M)*(ONE-TSAT(M)*RTSTAR(M))4CORC(M)) 2 /(ABS(TG-TSAT(M))+BPARA(M))) •IF DEF.DBL.3 CVGR(M)= CVG(M) - DMIM(ZERO,((CVL(M)-CVG(M))*TSAT(M)-HSTAR(M) X + SENT(M)*(ONE-TSAT(M)«RTSTAR(M))+CORC(M)) 2 /(DABS(TG-TSAT(M))+BPARA(M))) •I AEOS.338 •IF -DEF,DBL,3 CVGR(M)- CVG(M) - AMINKZERO, (SIELS(M) - SIELIQ(M) + CORC(M) 1 + SEMT(M)*(ONE-TSAT(M)*RTSTAR(M))+CVL(M)*TMLT(M) 2 - HSTAR(M)-CVG(M)*TSAT(M))/(ABS(TG-TSAT(M))+BPARA(M))) •IF DEF.DBL.3 CVGR(M)= CVG(M) - DMIN1(ZERO,(SIELS(M) •• SIELIQ(M) + CORC(M) 1 + SE:Tr(M)»(ONE-TSAT(M)*RTSTAR(M))+CVL(M)*TMLT(M) 2 - HSTAR(M)-CVG(M)*TSAT(M))/(DABS(TG-TSAT(M))+BPARA(M))) •D AEOS.341 SUMI - SUMI + (SIEGS(M)-(CVGR(M)+SENT(M)*RTSTAR(M))*TSAT(M))*X(M) •D AEOS.347,348 CVGBAR - X(M)»CVGR(M)+ CVGBAR CPGBAR - X(M)*(CVGR(M)+RMAT(M)) + CPGBAR •/ */ INPUT,PRIMING,AND ADJUSTMENTS •/ •I B989.52 WRITE (NOUT.682) (BPARA(M),M-1,4) WRITE (NOUT.683) (CPARA(M),M=1,4) WRITE (NOUT.684) (ACVG(M),M=1,5) WRITE (NOUT.685) (BCVG(M),M-1,5) WRITE (NOUT.686) (C0RC(M),M=l,4) •I B989.63 682 FORMAT (36X.6H BPARA,4(3X,1PE12.5)) 683 FORMAT (36X.6H CPARA,4(3X,1PE12.5)) 684 FORMAT (36X.6H ACVG,5(3X,1PE12.5)) 685 FORMAT (36X.6H BCVG,5(3X,1PE12.5)) 686 FORMAT (36X.6H CX)RC,4(3X,1PE12.5)) •I B989.72 WRITE (NFILM.682) (BPARA(M),M=1,4) WRITE (NFILM.683) (CPARA(M),M=1,4) WRITE (NFILM.684) (ACVG(M),M=1,5) WRITE (NFILM.685) (BCVG(M),M=1,5) WRITE (NFILM.686) (C0RC(M),M=l,4) *D B473.2 •IF -DEF.DBL.3 ALCVG2CM) - ALOG(CVGGM1(M)*(ONE-K:PARA(M))) 314 •D B473.5 *IF DEF.DBL.3 ALCVG2CM) - DLOG(CVGGM1(M)*(ONE+CPARA(M))) *D B4K3.56,58 C C DO THE NECESSARY ITERATIONS ALLOWING FOR TG GREATER THAN TCRIT C *I EINP.198 IF (TG .LT. TCRIT(M)) GO TO 874 GFUNC m APARA(M)/(TG-TCRIT(M)+APARA(M)) •IF -DEF.DBL.l RMAT(M) - EXP(ALRCRT(M)*GFUNC+ALCYG2(M)*(ONE-GFUNC)) •IF DEF.DBL.l RMAT(M) » DEXP(ALRCRT(M)*GFUNC+ALCVG2(M)»(ONE-GFUNC)) 874 CONTINUE •I EINP.2O9 IF (TG .GT. TCRIT(M)) GO TO 882 •I EINP.22O 882 RGFUNC(M) = ONE/(TG-TCRIT(M)+APARA(M)) GFUNC - APARA(M)*RGFUNC(M) •IF -DEF.DBL.2 FFUNC .- AEOS1O(M)+AEOS2O(M)*ALOG(ZIP)+AEOS3O(M)*SQRT(ZIP) RNEW - EXP(FFUNC*GFUNC + ALCVGM(M)*(ONE-GFUNC)) •IF DEF.DBL.2 FFUNC - AEOS10(M)+AEOS2O(M)*DLOG(ZIP)+AEOS30(M)*DSQRT(ZIP) RNEW - DEXP(FFUNC*GFUNC + ALCVGM(M)*(ONE-GFUNC)) ZIPNEW - ZIP + (PNI-RNEW*ZIJP)/ •IF -DEF.DBL.l 1(RNEW*(ONE+GFUNC*(AEOS2O(M)+HALF*AEOS3O(M)*SQRT(ZIP) •IF DEF.DBL.l l(RNEW*(0NE4GFUNC*(AE0S20(M)+HALF»AE0S30(M)*DSQRT(ZIP) 2 •-)))' •IF -DEF.DBL.l IF (ABS((ZIPNEW-ZIP)/ZIPNEW) .GT. EM1O) GO TO 880 •IF DEF.DBL.l IF (DABS((ZIPNEW-ZIP)/ZIPNEW) .GT. EM10) GO TO 880 GO TO 890 •I EINP.224 IF (TG .GT. TCRIT(M)) GO TO 887 •I EINP.235 887 RGFUNC(M) - ONE/(TG-TCRIT(M)+APARA(M)) GFUNC - APARA(M)*RGFUNC(M) •IF -DEF.DBL.4 COEFC - ONE + AMAX1(ZERO,((ZIP-T95ROG(M))*DENARG(M))*CPARA(M)) COEFM - ALOG(COEFC) FFUNC - AEOS11(M)+AEOS21(M)*SQRT(ZIP)+AEOS31(M)*ZIP RNEW » EXP(FFUNC*GFUNC+(ALCVGM(M)4C0EFM)*(0NE-GFUNC)) •IF DEF.DBL.4 COEFC - ONE + DMAX1(ZERO,((ZIP-T95ROG(M))*DENARG(M))*CPARA(M)) . COEFM - DLOG(COEFC) FFUNC - AEOSU(M)+AEOS21(M)»DSQRT(ZIP)+AEOS31(M)*ZIP RNEW - DEXP(FFUNC*GFUNC+(ALCVGM(M)4C0EFM)*(0NE-GFUNC)) ZIPDOM =

315 *IF -DEF,DBL,1 l(ENEW*(0NE4GFUNC*(AE0S21(M)*HALF»SQRTCZIP)+AE0S31(M)*ZIP *1F DEF DBL 1 1(RNEW*(ONE+GFUNC*(AE0S21(M) *HALF*DSQRT(ZIP)+AE0S31(M)*ZIP 2 ))) IF (COEFC .GT. ONE) 1ZIPDOM - ZIPDOM + ((ONE-GFUNC)*CPARA(M)*DENARG(M)*ZIP)/COEFC ZIPNEW - ZIP + (PNI - RNEW*ZIP)/ZIPDOM *IF -DEF,DBL,1 IF (ABS((ZIPNEW-ZIP)/ZIPNEW) .GT. EM1O) GO TO 885 *IF DEF,DBL,1 IF (DABS((ZIPNEW-ZIP)/ZIPNEW) .GT. EM1O) GO TO 885 IF (ZIPNEW .LT. T95R0GCM)) GO TO 880 GO TO 890 *I EINP.18 DIMENSION ESTA(4),RGFUNC(5),DRDTG(5) *I EINP.536 ESTA(M) - SIELS(M) + SENT(M)*(ONE-TSAT(M)«RTSTAR(M)) 1 - HSTAR(M) - SIELIQ(M) + CVL(M) *TMLT(M) + CORC(M) 5900 CONTINUE C C ITERATE TO CALCULATE A CONSISTENT VAPOR TEMPERATURE. C NITNO = 0 TGN - TG 420 CONTINUE NITNO - NITNO + 1 IF (NITNO .GT. 50) GO TO 789 SUMNU - ZERO SUMDE - ZERO SUMDR1 - ZERO SUMDR2 - ZERO SUMDR3 - ZERO SUMDR4 - ZERO TG - TGN DO 460 M * 1.NCLEM2 ARG - PARPL(M)/RMAT(M) CVG(M) - ACVG(M) + BCVG(M)*TG IF (ARG .LT. R0GCUT*EM3) GO TO 435 IF (TG .GT. TCRIT(M)) RGFUNC(M)-ONE/(TG-TCRIT(M)+APARA(M)) IF (ARG .LT. T95ROG(M)) GO TO 430 IF (ARG .LE. TCRIT(M)*ROGCRT(M)) GO TO 425 IF (TG .GT. TCRIT(M)) GO TO 423 RMAT(M)-RCRIT(M) DRDTG(M) - ZERO GO TO 440 423 GFUNC = APARA(M)*RGFUNC(M) *IF -DEF,DBL,1 RMAT(M) - EXP(ALRCRT(M)*GFUNC + ALCVG2(M)*(0NE-GFUNC)) *IF DEF,DBL,1 RMAT(M) = DEXP(ALRCRT(M)*GFUNC + ALCVG2(M)*(0NE-GFUNC)) DRDTG(M) - GFUNC*TG*(ALCVG2(M)-ALRCRT(M))*RGFUNC(M) GO TO 440

316 »IF DEF,DBL,3 425 ARGDUM - AE0S21(M)*DSQRT(ARG) IF(TG.GT.TCRIT(M)) GO TO 428 RMAT(M) =DEXP(AEOS11(M)+ARGDUM+AEOS31(M)*ARG) »IF -DEF.DBI.,3 425 ARGDUM = AEOS2l(M)*SQRT(ARG) IF (TG .GT. TCRIT(M)) GO TO 428 RMAT(M) = EXP(AEOS11(M)+ARGDUM+AEOS31(M)«ARG) DRDTG(M)=ARGDUM*HALF + AEOS3l(M)*ARG GO TO 440 428 GFUNC - AFARA(M)*RGFUNC(M) FFUNC = AEOSll(M) + ARGDUM + AE0S31(M)*ARG *IF -DEF.DBL.2 COEFM - ALOG(COEFC) RMAT(M) = EXP(FFUNC*GFUNC+(ALCVGM(M)+CX)EFM)*(ONE-GFUNC)) »IF DEF.DBL.2 COEFM - DLOG(COEFC) RMAT(M) - DEXP(FFUNC»GFUNC+(ALCVGM(M)+COEFM)*(ONE-GFUNC)) DRDRG(M)=GFUNC*(ARGDUM*HALF+AEOS31(M)*ARG) 1 +((ONE-GFUNC)*CPARA(M)'DENARG(M)*ARG)/COEFC DRDTG(M)=DRDRG(M)-K5FUNC»TG* ( ALCVGM(M)4C0EFM-FFUNC) *RGFUNC(M) GO TO 440 •IF DEF.DBL.3 430 ARGDUM = AE0S30(M)«DSQRT(ARG) IF(TG.GT.TCRIT(M)) GO TO 432 RMAT(M) =£>EXP(AE0S10(M)+AE0S20(M)*DLOG(ARG)+AP.GDUM) •IF -DEF.DBL.3 430 ARGDUM - AE0S30(M)*SQRT(ARG) IF (TG .GT. TCRIT(M)) GO TO 432 RMAT(M) = EXP(AEOS10(M)+AEOS20(M)*ALOG(ARG)+ARGDUM) DRDTG(M) - AE0S20(M) + HALF*-\RGDUM GO TO 440 432 GFUNC = APARA(M)*RGFIM:(M) •IF -DEF.DBL.2 FFU1C = AE0S10(M)+AE0S20(M)*AL0G(ARG) + ARGDUM RMAT(M) = EXP(FFUNC*GFUNC + ALCVGM(M)*(ONE-GFUNC)) SIF DEF.DBL.2 FFUIC = AEOS10(M)+AEOS2O(M)»DL0G(ARG) + ARGDUM RMAT(M) = DEXP(FFUNC*GFUNC + ALCVGM(M)*(ONE-GFUNC)) DRDRG(M) = GFUNC'(AE0S20(M)+HALF*ARGDUM) DRDTG(M) = DRDRG(M) + GFUNC*TG*(ALCVGM(M)-FFUNC)*RGFUNC(M) GO TO 440 435 RMAT(M)=CVGGM1(M) DRDTG(M) = ZERO GO TO 460 440 CONTINUE DRDTG(M) = DRDTG(M)/TG *IF -DEF.DBL.2 CVGR(M) « CVG(M) - AMIN1(ZERO,(ESTA(M)-CVG(M)*TSAT(M)) 1 /(ABS(TG-TSAT(M))+BPARA(M))) SIF DEF.DBL.2 CVGR(M) = CVG(M) - DMINl(ZERO,(ESTA(M)-CVG(M)*TSAT(M)) 1 /(DABS(TG-TSAT(M))+BPARA(M))) FACTOR = ARG FACT02 = FACTOR*TSAT(M) 317 DCVGRT = BCVG(M) IF (CVGR(M) .LE. CVG(M)+EM10) GO TO 455 IF (TG .GE. T3AT(M)) 1DCVGRT = DCVGRT+(CVG(M)-CVGR(M)+BCVG(M)*TSAT(M)) 2 /(TG-TSAT(M)+BPARA(M)) IF (TG .LT. TSAT(M)) 1DCTGRT=DCVGRT+(CVGR(M)-CVG(M)+BCVG(M)*TSAT(M)) 2 /(TSAT(M)-TG+BPARA(M)) 455 CONTINUE SUMNU = SUMNU + FACTO2*CVGR(M) SUMDE = SUMDE + FACTOR*CVGR(M) SUMDR1 = SUMDR1 + FACTO2*DCVGRT SUMDR2 = SUMDR2 + FACTO2*CVGR(M)*DRDTG(M) SUMDR3 = SUMDR3 + FACTOR*DCVGRT SUMDR4 = SUMDR4 + FACTOR*CVGR(M)*DRDTG(M) 460 CONTINUE RSUMDE = ONE/SUMDE F = SUMNU*RSUMDE - TG DFDTG = (SUMDR1-SUMDR2-(FVTG)*(SUMDR3-SUMDR4))*RSUMDE - ONE TGN = TG - F/DFDTG *IF -DEF.DBL.l IF (ABS((TGN-TG)/TG) .GT. CAEOS) GO TO 420 *IF DEF.DBL.l IF (DABS((TGN-TG)/TG) .GT. CAEOS) GO TO 420 GO TO 790 *D B4K3.71,74 *D EINP.537,573 *D EINP.578,582 *D B4K3.75,77 *D EINP.583 TG = TGN *D EINP.587,588 M=IEOSLE(N) ROGTN(N) = PARPL(M)/RMAT(M) IF (PARPL(N) .EQ. ZERO) GO TO 5950 */ */ THIS PART IS TO CORRECT THE INPUT INITIALIZATION WHEN */ ONE COMPONENT EXISTS AND TG EQUALS TSAT */ *D B4K3.43 C HAS NOW BEEN CONSISTENTLY PROGRAMMED *D EINP.93 *D B3O3.26 GFUNC = APARA(M)/(TG-TCRIT(M)+APARA(M)) *IF -DEF,DBL,1 RNEW = EX?(ALRCRT(M)*GFUNC+ALCVG2(M)*(0NE-GFUNC)) *IF DEF.DBL.l RNEW = DEXP(ALRCRT(M)*GFUNC+ALCVG2(M)*(0NE-GFUNC)) ZIPNEW = PNI/RNEW */ */ FINALLY WE MUST CORRECT THE TRANSIENT •/ THIS IS THE SINGLE PHASE CALCULATION IN EXFLUD */ *D EXFL.847

318 31 EXFL. 860 CVG(M) = ACVG(M) + BCVG(M)*TG ESTAR = SIELS(M) + SENT(M)«(ONE-TSAT(M)*RTSTAR(M))-CVG(M)*TSAT(M) •IF -DEF.DBL.2 CVGR(M)- CVG(M) - AMINl(ZER0,(-HSTAR(M)-SIELIQ(M)+CVL(M)*TMLT(M) 1 + CORC(M) + ESTAR)/(TG - TSAT(M) + BPARA(M))) •IF DEF.DBL.2 CVGR(M)= CVG(M) - DMIN1(ZERO,(-HSTAR(M)-SIELIQ(M)+CVL(M)*TMLT(M) 1 + CORC(M) + ESTAR)/(TG - TSAT(M) + BPARA(M))) ZIP = (PARPL(N)*CVGR(M))/RMAT(M) *D 8473.13,16 *IF -DEF.DBL.l RMAT(M) = EXP(ALRCRT(M)*GFUNC+ALCVG2(M)*(0NE-GFUNC)) *IF DEF.DBL.l RMiVT(M) = DEXP(ALRCRT(M)«GFUNC+ALCVG2(M)*(0NE-GFUNC)) *D B473.33 2 ))) *I B473.38 *IF -DEF.DBL.2 COEFC =• ONE + AMAX1(ZERO,((ZIP-T95ROG(M))*DENARG(M))*CPARA(M)) COEFM - ALOG(COEFC) •IF DEF.DBL.2 COEFC =. ONE + DMAX1(ZERO,((ZIP-T95ROG(M))*DENARG(M))*CPARA(M)) COEFM = DLOG(COEFC) *D B473.41 RNEW = EXP(FFUNC*GFUNC + (ALCVGM(M)+COEFM)*(0NE-GFUNC)) *D B473.44 RNEW = DEXP(FFUNC*GFUNC + (ALCVGM(M)+COEFM)*(ONE-GFUNC)) *D B473.45 ZIPDOM = *D B473.50 2 ))) IF (COEFC .GT. ONE) 1ZIPD0M = ZIPDOM + ((ONE-GFUNC)*CPARA(M)*DENARG(M)*ZIP)/COEFC ZIPNEW = ZIP + (PARPL(N) - RNEW*ZIP)/ZIPDOM IPATH = 2 *I EXFL. 990 IPATH = 1 •I EXFL. 992 . IF (IPATH .EQ. 2 .AND. PARPL(N)/RNEW . LT. T95R0G(M)) GO TO 780 *D EXFL. 1001,1007 *D EXFL.1010,1012 CyG(M) = ACVG(M) + BCVG(M)*TG ESTAR = SIELS(M) + SENT(M)*(ONE-TSAT(M)*RTSTAR(M))-CVG(M)*TSAT(M) •IF -DEF.DBL.2 CVGR(M)= CVG(M) - AMINKZERO, (-HSTAR(M)-SIELIQ(M)+CVL(M)*TMLT(M) 1 + CORC(M) + ESTAR)/(TG - TSAT(M) + BPARA(M))) SIF DEF,DBL,2 C¥GR(M)= CVG(M) - DMINl(ZERO,(-HSTAR(M)-SIELIQ(M)-fCVL(M)*TMLT(M) 1 + CORC(M) + ESTAR)/(TG - TSAT(M) + BPARA(M))) ZIP = (PARPL(N)*CVGR(M))/RNEW SIEGTE = SIEGTE + TSAT(N)*ZIP ROGTEM = ROGTEM + ZIP

319 2410 CONTINUE TG = SIEGTE/ROGTEM TEMGAS(IJ) = TG SIEGTE - ZERO DO 2415 N = 1.NCLEM2 IF (PARPL(N) .EQ. ZERO) GO TO 2415 M = IEOSLE(N) *I P5EO.3 DENARG(M) = ONE/(ROGCRT(M)*TCRIT(M)-T95ROG(M)) */ V DO NOT WANT TO DO THE VAPOR ITERATION IN SINGLE PHASE CELLS V *B AEOS.14 DIMENSION ALPHG(1),ALPHS(1) EQUIVALENCE (ALPHG(1),AASC(1)),(ALPHS(1),AASC(2)) *I AEOS.107 IF (ALPHG(IJ) ,LT, (ONE-ALPHS(IJ))SALPHO) GO TO 480 *I EXFL.1146 ALPHG(IJ) - AGSAVE ALPHS(IJ) = ASSAVE *I EXFL. 1144 AGSAVE = ALPHG(IJ) ASSAVE = ALPHS(IJ) ALPHS(IJ)= ALPHSN ALPHG(IJ) = ALPH */ */ MISCELLANEOUS INITIALIZATIONS */ *I EXFL. 888 TEMGAS(IJ) = TG *I AEOS.31 CVGR(M) = ZERO *I B2B0.15 TGMAX=EP10 TGMIN=EM10 *I B473.91 SUMDE = SUMDE + CVG(M)»Y(M) * IDENT BCH4 */ */ MAKE ADDITIONAL CORRECTIONS */ */ *D AEOS.34 NITNO = -6 *I SETI.135 *IF DEF,AEOS,1 TSTARM = TSTAR(l) 31 SETI. 156 ?IF DEF.DBL.l TSTARM = DMIN1(TSTARM,TSTAR(M)) *IF -DEF.DBL.l TSTARM = AMIN1(TSTARM,TSTAR(M))

~~320 APPENDIX C REVISED SIMMER-II INPUT DESCRIPTION FOR THE AEOS~~

Cards 42-47 specify the material properties and EOS data for each of the materials: fuel, M = l; steel, M = 2; sodium, M = 3; control, M = 4; and fission gas, M = 5. A complete set of these cards must be input for each material in the above order. Additional cards are required for fuel, steel, and sodium as indicated.

~~Card No. 42 (FORMAT: 18A4) (IDCARD(I), I = 1, 18)~~

~~Columns Var iable Descr ipt ion~~

~~1-72 IDCARD(I) Message identifying the material whose properties and~~

~~EOS data are to be input.~~

Card No. 43 (FORMAT: 5E12.5) ROSE(M), CVS(M), TMLT(M), HFUS(M), THCONS(M) ^olumns Var iable Descript ion 1-12 ROSE(M) The solid microscopic density of material M (kg/m ). 13-24 CVS(M) The solid constant-volume specific heat of material M (J/kg • K).

~~25-36 TMLT(M) The melting temperature of material M (K).~~

~~37-48 HFUS(M) The heat of fusion of material M (J/kg).~~

~~49-60 THCONS(M) The thermal conductivity of material M in the solid phase (W/m • K).~~

~~Card No. 44 (FORMAT: 5E12.5) ROLE(M), CyL(M), SIG(M), THCONL(M), XMUL(M)~~

~~Columns Variable Description~~

~~1-12 ROLE(M) The liquid microscopic density of material M (kg/m ).~~

~~13-24 CVL(M) The liquid-phase constant-volume specific heat of material M (J/kg • K).~~

~~25-36 SIG(M) The surface tension of material M (kg/s ).~~

~~37-48 THCONL(M) The thermal conductivity of material M in the liquid phase (W/m • K).~~

~~49-60 XMUL(M) The viscosity of material M in the liquid phase (Pa • s).~~

~~321 Card No. 45 [FORMAT(FORMAT: 6E12.5) PSTAR(M), TSTAR(M), TSUP(M), HSTAR(M), TCRIT(M), ZETA(M)~~

~~Columns Variable Descr i pt ion~~

~~1-12 PSTAR(M) The p* parameter in the vapor pressure-temperature relation for material M (Pa).~~

~~13-24 TSTAS(M) The T* parameter in the vapor pressure-temperature relation for material M (K).~~

~~25-36 TSUP(M) The superheat of material M (K).~~

~~37-48 HSTAR(M) The h* parameter in the heat-of-vaporization equation for material M (J/kg).~~

~~49-60 TCRIT(M) The critical temperature of material M (K).~~

~~61-72 ZETA(M) The exponent parameter in the heat-of-vaporization equation for material M.~~

~~Card No. 46 (FORMAT: 6E12.5) CVG(M), GAM(M), ATOM(M), ENCRIT(M), WTMOL(M), EPSK(M)~~

~~Columns Variable Description~~

~~1-12 CVG(M) The vapor-phase constant-volume specific heat of material M (J/kg • K). If ACVG(M) is defined (Card 47a), CVG(M) is only used to compute the infinitely dilute gas constant.~~

13-24 GAM(M) The ratio of the constant-pressure specific b"it to constant-volume specific heat for material M in the vapor phase. o 25-36 ATOM(M) The molecular diameter of vapor material M (A). (Suggested values: fuel, 4.4; steel, 1.64; sodium, 3.567; control, 1.46; fission gas, 4.047; nitrogen, 3.798.)

~~37-48 ENCRIT(M) The internal energy of material M at the critical point (J/kg). Default: ENCRIT(M) = max [HSTAR(M), TMLT(M) * CVS(M) + HFUS(M) + CVL(M) * (TCRIT(M) - TMLT(M)). ]~~

~~49-60 VTMOL(M) The molecular weight of material M.~~

~~61-72 EPSK(M) The molecular force constant (K) of material M in the vapor phase. (Suggested values: fuel, 6468; steel, 7700; sodium, 1375; control, 5472; fission gas, 231.)~~

~~322 Card No. 47 (FORMAT: 4E12.5) ROGCRT(M), ROGP95(M), ROLCRT(M), APARA(M)~~

~~Columns ¥ar iable Descri pt ion~~

~~1-12 ROGCRT(M) The critical density (kg/m3)of material M. [Default: 0.375*ROLE(M).]~~

~~13-24 ROGP95(M) The saturated vapor density (kg/m ) of material M when TSAT(M) = 0.95*TCRIT(M). [Default: 0.25*R0GCRT(M).]~~

25-36 ROLCRT(M) The liquid critical density (kg/m3) of material M. If a constant liquid density is desired for consistency with previous versions of SIMMER, ROLCR'lXM) should be set to RHOAl(M) from Card 48. [Default: ROGCRT(M).]

~~37-48 APARA(M) The parameter used to increase the gas constant above the critical temperature. This is O.Q >« in Eq. (A-25). [Default: 1010.]~~

~~Card No. 47a (FORMAT: 5el2.5) ACVG(M), BCVG(M)- CORC(M), BPARA(M), CPARA(M)~~

~~Columns Variable Descr ipt ion~~

~~1-12 ACVG(M) Coefficient a^ in the Eq. c ~M = a^ + K»(TG - 273.16). [Default: CVG(M).]~~

~~13-24 BCVG(M) Coefficient bj^ in the c^^ equation.~~

~~25-36 CORC(M) Correction term for the vapor energy (J/kg) in the~~

~~dilute vapor density limit.~~

~~37-48 BPARA(M) Relaxation constant (K) for the vapor heat capacity.~~

~~49-60 CPARA(M) Multiplier to adjust the gas constant at high vapor~~

~~densit ies.~~

~~Card No. 48 (FORMAT: 4E12.5) RHOAl(M), RHOA2(M), RHOA3(M), RHOA4(M)~~

[Note: This card only read for materials 1, 2, and 3, that is, fuel, steel, and sodium, j Columns Var iable Descr ipt ion 1-12 RHOAl(M) First coefficient for the density vs temperature equation at low temperature. Variable ajm in Eq. (A-41). [Default: ROL(l) for M = 1. ROL(M + 1) for M > 1. See Card No. 53 for the definition of ROL. ]

~~13-24 RHOA2(M) Second coefficient for the density vs temperature equation a low temperature. Variable a-. in Eq. (A-41). m~~

~~323 Columns Var iable Description~~

~~24-36 RHOA3(M) Third coefficient for the density vs temperature equation at low temperature. Variabale a^ in Eq. (A-41).~~

~~37-4S RH0A4(M) Fourth coefficient for the density vs temperature equation at low temperature. variable a, in Eq. (A-41),~~

~~Card No. 49 (FORMAT: 3E12.5) RHOBl(M), RHOB2(M), RHOB3(M)~~

~~[Note: This card only read for materials 1, 2, and 3, that is, fuel, steel, and sodium.]~~

~~Columns Variable Description~~

~~1-12 RHOBl(M) First coefficient for the density vs temperature equation at high temperatures. Variable b* in Eq. A-41).~~

~~13-24 RHOB2(M) Exponent for the density vs temperature equation at high temperatures. Variable b2_ in Eq. (A-41).~~

~~25-36 RH0B3(M) Last coefficient for the density vs temperature equation at high temperatures. Variable b^ in Eq. (A-41).~~

~~Card No. 50 (FORMAT: 3E12.5) ASOUND, BSOUND, CSOUND~~

~~[Note: This card only read for material 1, fuel.]~~

~~Columns Variable Descr ipt ion~~

~~1-12 ASOUND Coefficient to compute velocity of sound as a function of temperature. Variable A in Eq. (A-34). [Default: 2212.339 m/s.]~~

~~13-24 BSOUND Exponent for the velocity of sound equation. Variable B in Eq. (A-34). [Default: 0.3539176.]~~

~~25-36 CSOUND Minimum fuel velocity of sound (m/s). Variable C in~~

~~Eq. (A-34). [Default: 400 m/s.]~~

~~Card No. 51 (FORMAT: 18A4) (IDCARD(I), I = 1, 18)~~

~~Columns Var iable Descr i pt ion~~

~~1-72 IBCARD(I) Message identifying component properties to be input next.~~

~~324 Card No. 52 (FORMAT: 6E12.5) (ROS(N), N = 1, NCSR)~~

~~Columns Variable Descript ion~~

~~1-72 ROS(N) The microscopic density of the solid phase of structure-field density component N (kg/mJ). NCSR 9.~~

N = 1. fabricated fertile fuel, N - 2, fabriated fi ^ile fuel, N = 3, frozen fertile fuel, N - 4, frozen fissile fuel, N = 5, cladding, N = 6, can wall, N = 7, control, N = 8, intragranular fission gas, and N = 9, intergranular fission gas.

~~Card No. 53 (FORMAT: 6E12.5) (ROL(N), N = 1, NCLR)~~

~~Columns Variable Descript ion~~

~~1-72 ROL(N) The tnicroscopic density of iiquid-field density component N (kg/m°). NCLR = 8.~~

~~N = 1, liquid fertile fuel, N - 2, liquid fissile fuel, N = 3, liquid steel, N = 4, sodium, N = 5, control, N = 6, solid fertile, N = 7, solid fissile fuel, and N = 8, solid steel.~~

For liquid fuel, steel, and liquid sodium the ROLs are to be used as default uensities as indicated in Card No. 48. Also, for liquid fuel and steel they furnish the microscopic densities while melting structure or particles. For example, sec Eqs. (1II-5) and (III-6) of the SIMMER-II manual.

~~Card No. 54 (FORMAT: 6E12.5) (SVEL(N), N = 1, NCLR)~~

~~Columns Variable Descript ion~~

~~1-72 SVEL(N) The sonic velocity of liquid-field density component N (m/s). NCLR = 8. APPENDIX D~~

~~AEOS SIMULATION PROGRAM~~

This is a standalone FORTRAN program to evaluate water properties for the SIMMER-II AEOS so that comparisons with other data may be performed and properties of the solution algorithm evaluated. This program was used to produce the numbers for Sec. II. C. It is stored on the Los Alamos CFS as node /VAPOR/AEOS3.

1 SCHAT «ME,AFVST,BFNS,LIST,QO7,S,S,,C 2 CALL CHANGE(5H+FVST) 3 PROGRAM NAME(TAPE58,TAPE22) 4 DIMENSION PV(2) 5 6 THIS IS A PROGRAM TO SIMULATE THE SIf/MER AEOS FOR WATER(H2O). 7 THE NOTATION REMAINS AS IN SI&/MER-II. 8 9 ZERO = 0. 10 TG = 373. 11 CORC = 5.0E+4 12 ROGCUT = l.E-10 13 EM3 = l.E-3 14 ACVG = 1346. 15 BCVG = 0.3302 16 ONE = 1. 17 EM2 = l.E-2 18 EP2 = 100. 19 EM7 = l.E-7 20 NOUT = 22 21 CLIQ = 1498. 22 NIN = 58 23 CVG = 2181.6 24 T95ROG = 5.889196E+4 25 TCRIT = 647.296 26 RTCRIT = ONE/TCRIT 27 AE0S11 = 6.38828 28 AE0S21 = -3.12508E-3 29 AE0S31 = -1.42444E-6 30 AEOS10 = 6.13434 31 AE0S20 = 3.83349E-6 32 AE0S30 = -2.42451E-3 33 ROGCRT = 316.957 34 ROLCRT = 316.957 35 RCRIT = 107.8264 36 HALF = .5 37 TW03RD = 2./3, 38 SIEIIQ = 9.04304E+5 39 TMLT = 273.16

327 40 EP4 = l.E+4 41 TV/0 = 2. 42 EP1 = 10. 43 HFUS=3.3?4E+5 44 CVS=2090. 45 RCVS = 4.78469E-4 46 S1ES0L = 5.70904E+5 47 SIELCR = 1.572169E+6 48 RCVL = 2.3713E-4 49 EICRIT - 2.9339E+6 50 EM10 = l.E-10 51 EP10-1.E+10 52 CP9 =• .9 53 EMI » .1 54 EMS = l.E-8 55 CVGGM1 = 461,408 56 PSTAR = 3.17771E+10 57 RPSTAR = ONE/PSTAR 58 TSTAR = 4.7O579E+O3 59 RTSTAR = ONE/TSTAR 60 PCRIT = PSTAR*EXP(-TSTAR*RTCRIT) 61 CP95=0.95 62 HSTAR = 3.0898O5E+O6 63 ZETA = 0.3660361 64 AEOSLM = 1.70124E+6 65 CVL = 4217.1 66 CVLP = 1521.82 67 ROLE = 1050. 68 RH0A1 = 838.607921 69 RH0A2 - 1.3483123 70 RH0A3 = -.00274914042 71 RH0R1 = 2.85605286 72 RH0B2 » .381221451 73 RH0B3 = -.0748002905 74 EPSOK = 32. 75 WTMOL = 18. 76 RWTMOL = 5.555555556E-2 77 GAM = 1.2115 78 RATMSQ - 7.160672E-2 79 CP9249 = 0.92495 80 C207M3 = 2.O7368E-3 81 CP7192 - 0.719288 82 C1P151 = 1.15049 83 C546M2 = 5.46452E-2 84 C831M2 = 8.314E-2 85 C267M6 = 2.671E-6 86 C1P32 = 1.32 87 C2P5 = 2.5 88 ALRCRT - ALOG(RCRIT) 89 ALCVGM = ALOG(CVGGMl) 90 DENARG^ONE/(ROGCRTSTCRIT-T9:5ROG) 91 CAEOS = l.E-10 92 Y= 1. 93 C 94 C N IS THE NUMBER OF VARIABLES TO INPUT

328 95 C IOPTN = 1 WILL OBTAIN PROPERTIES FROM THE VAPOR DENSITY AND 96 C THE VAPOR AND LIQUID INTERNAL ENERGIES 97 C IOPTN = 2 OBTAINS SATURATION PROPERTIES FROM TSAT 98 C IOPTN = 3 OBTAINS SATURATION PROPERTIES FROM THE PRESSURE, PNI 99 C IOPTN = 4 OBTAINS THE LIQUID MICROSCOPIC DENSITY FROM TL 100 C IOPTN = 5 CONTAINS THE SIMMER CALCULATION OF THE VAPOR THERMAL 101 C CONDUCTIVITY AND VAPOR VISCOSITY 102 READ (NIN.23) N, IOPTN, APARA.BPARA.CPARA 103 ALCVG2=AL0G(CVGGMl*(0NE4CPARA)) 104 GO TO (6,7,7,9,10),IOPTN 105 6 CONTINUE 106 DO 24 NZQ=1,N 107 IF (NZQ .EQ. 15) CALL Q8QPAU 108 1 READ (NIN.3) ROGPM.SIEGL.SIELL.ROLPM 109 2 FORMAT (1P8E1O.3) 110 3 FORMAT (1P4E12.5) 111 IF (SIEGL .EQ. ZERO) GO TO 510 112 NITNO = 0 113 TGMIN=EM10 114 TGMAX = EP10 115 TGN = TG 116 420 CONTINUE 117 NITNO = NITNO + 1 118 IF (NITNO .GE. 15) GO TO 470 119 SUMI = ZERO 120 SUMNU = ZERO 121 TG = TGN 122 CVG = ACVG + BCVG*(TG-273.16) 123 ARG = ROGPM*TG 124 IF (ARG .LT. ROGCUT*EM3) GO TO 435 125 IF (TG .GT. TCRIT) RGFUNC=ONE/(TG-TCRIT+APARA) 126 IF (ARG .LT. T95ROG) GO TO 430 127 IF (ARG .LE. TCRIT*ROGCRT) GO TO 425 128 IF (TG .GT. TCRIT) GO TO 423 129 RMAT=RCRIT 130 DRDTG = ZERO 131 DRDRG = ZERO 132 GO TO 440 133 423 GFUNC = APARA*RGFUNC 134 RMAT = EXP(ALRCRT*GFUNC + ALCVG2»(ONE-GFUNC)) 135 DRDRG = ZERO 136 DRDTG = GFUNC*TG*(ALCVG2-ALRCRT)*RGFUNC 137 GO TO 440 138 425 ARGDUM = AE0S21*SQRT(ARG) 139 IF (TG .GT. TCRIT) GO TO 428 140 RMAT = EXP(AEOS11+ARGDUM+AEOS31*ARG) 141 DRDTG=ARGDUM*HALF + AE0S31*ARG 142 DRDRG = DRDTG 143 GO TO 440 144 428 GFUNC = APARA*RGFUNC 145 FFUNC = AF0SH + ARGDUM + AE0SJ1*ARG 146 COEFC = ONE + ((ARG-T95ROG)*DENARG)*CPARA 147 COEFM = ALOG(COEFC) 148 RMAT = EXP(FFUNC*GFUNC + (ALCVGM+COEFM)*(ONE-GFUNC)) 149 DRDRG = GFUNC* (ARGDUM*HALF+AE0S31*ARG)

329 150 1 +((ONE-GFUNC)«CPARA*DENARG*ARG)/COEFC 151 DRDTG = DRDRG + GFUNC*TG*(ALCVGM+COEFM-FF17X:)*RGFUNC 152 GO TO 440 153 430 ARGDUM = AE0S30*SQRT(ARG) 154 IF (TG .GT. TCRIT) GO TO 432 155 RMAT = EXP(AE0S10+AE0S20*AL0G(ARG)+ARGDUM) 156 DRDTG = AE0S20 + HALF*ARGDUM 157 DRDRG = DRDTG 158 GO TO 440 159 432 GFUNC = APARA*RGFUNC 160 FFUNC = AE0S10+AE0S20*AL0G(ARG) + ARGDUM 161 RMAT = EXP(FFUNC*GFUNC + ALCVGM*(ONE-GFUNC)) 162 DRDRG = GFUNC*(AEOS20+HALF*ARGDUM) 163 DRDTG = DRDRG + GFUNC*TG*(ALCVGM-FFUNC)*RGFUNC 164 GO TO 440 165 435 RMAT=CVGGM1 166 DRDTG = ZERO 167 DRDRG = ZERO 168 GO TO 460 169 440 CONTINUE 170 DENOM = ALOG(ARG»RMAT*RPSTAR) 171 TSAT=-TSTAR/DENOM 172 SENT=ZERO 173 IF (TSAT .LT. TCRIT) 174 1 SENT = HSTAR*(ONE-TSAT*RTCRIT)**ZETA 175 DTSDTG = -(TSAT*(DRDTG-K)NE))/(DENOM*TG) 176 IF (TSAT .GT. TWO3RD*TCRIT) GO TO 445 177 SIELS = SIELIQ+ AMAX1(TSAT-TMLT,ZERO)*CVL 178 CVGR = CVG -AMIN1(ZERO,(CVL*TSAT + SENT*(ONE-TSAT*RTSTAR) 179 2 -CVG*TSAT - HSTAR+CORC)/(ABS(TG - TSAT) + BPARA)) 180 RSUMDE = ONE/CVGR 181 ESTAR=SIELS-CVGR*TSAT+SENT*(ONE-TSAT*RTSTAR) 182 FACTOR = ZETA*(TSTAR-TSAT)/(TCRIT-TSAT)+ONE 183 CVLPP = CVL 184 GO TO 450 185 443 TSAT - EP4 186 445 SIELS-SIELIQfAEOSLM-HALF*SENT 187 1 + CVIP*(TSAT-TWO3RD*TCRIT) 188 CVGR=CVG-AMIN1(ZERO,(CVLP*(TSAT-TWO3RD*TCRIT)+AEOSLM -CVG*TSAT 189 2 + SENT*(HALF-TSAT*RTSTAR) + CVL*TMLT - HSTAR+CORC) 190 3 / (ABS(TG-TSAT)+BPARA)) 191 RSUMDE = ONE/CVGR 192 ESTAR = SIELS-CVGR*TSAT+SENT*(ONE-TSAT*RTSTAR) 193 FACTOR = ZERO 194 IF (TSAT .LT. TCRIT) 195 1FACT0R = ZETA*HALF*(TSTAR-TWO*TSAT)/(TCRIT-TSAT)+ 196 1 ONE 197 CVLPP = CVLP 198 450 CONTINUE 199 DCVGRT = BCVG 200 IF (CVGR .LE. CVG+EM10) GO TO 455 201 IF (TG .GE. TSAT) 202 1DCVGRT = -((CVGR-CTG+DTSDTGS(CVLPP-CVGR-SENT*FACTOR*RTSTAR)) 203 1 /(1G-TSAT+BPARA)) + BCVG*(ONE+TSAT/(TG-TSAT+BPARA)) 204 IF (TG ,LT. TSAT)

330 205 1K:VGRT=((CVGR-CVG-DTSDTG*(CVLPP+CVGR-TWO*CVG-SENT*FACTOR*RTSTAR)) 206 1 /(TSAT-TG+BPARA)) +BCVG*(ONE+TSAT/(TSAT-TG+BPARA)) 207 455 CONTINUE 208 SUMI = SUMI + ESTAR*Y 209 SUMNU = SUMNU + Y*DTSDTG*(CVLPP-CVGR-SENT*FACTOR*RTSTAR) 210 1 -Y*DCVGRT*TSAT+((SIEGL-ESTAR*Y)*DCVGRT)*RSUMDE 211 460 CONTINUE 212 F = (SIEGL-SUMI)*RSUMDE-TG 213 DFDTG = -SUMNU*RSUMDE-ONE 214 DFDTG=AMIN1(-EM2,DFDTG) 215 TGN = TG - F/DFDTG 216 IF (F .LT. ZERO) TGMAX=AMIN1(TG,TGMAX) 217 IF (F .GT. ZERO) TGMIN=AMAX1(TG,TGMIN) 218 IF (TGN .LT.TGMIN .OR. TGN. GT.TGMAX) 219 1TGN=(TGMAX+TGMIN)*HALF 220 IF (ABS((TGN-TG)/TG) .GT. CAEOS) GO TO 420 221 GO TO 480 222 470 CALL Q8QPAU 223 480 CONTINUE 224 TG = TGN 225 PV(1)=ZERO 226 PV(2)=ZER0 227 DTSDRG = ZERO 228 DHSDTS = -EM3 229 PV(2) = PV(2) + AMAX1(DRDRG+ONE,EM2)*Y*RMAT 230 PV(1)=PV(1)+ROGPM*RMAT 231 IF (RMAT .EQ. CVGGM1) GO TO 195 232 DENOM = ALOG(TG*ROGPM*RMAT*RPSTAR) 233 DTSDRG = -(TSAT*(DRDRG+ONE))/(DENOM*ROGPM) 234 IF (TSAT . LT. TCRIT) 235 1DHSDTS = -(ZETA*SENT)/(TCRIT-TSAT) 236 GO TO 200 237 195 IF (ROGPM .EQ. ZERO) GO TO 200 238 RTSAT = ONE/ALOG(ROGPM*RMAT*TG*RPSTAR) 239 TSAT=-TSTAR*RTSAT 240 DTSDRG=-(TSAT*RTSAT)/ROGPM 241 IF (TSAT .GE. TCRIT) GO TO 200 242 SENT = HSTAR*(ONE-TSAT*RTCRIT)**ZETA 243 DHSDTS=-ZETA*SENT/(TCRIT-TSAT) 244 200 CONTINUE 245 PV(2)=PV(2)*TG 246 PV(1)=PV(1)*TG 247 IF (ROLPM .EQ. ZERO) ROLPM = ROGPM 248 TL=SIELL*RCVS 249 ZIP=SIELL 250 RHOLNA = ROLE 25! IF(ZIP.LE. SIESOL) GO TO 60 252 ZIP=ZIP-SIELIQ 253 IF(ZIP.GT.ZERO) GO TO 20 254 TL=TMLT 255 GO TO 28 256 20 CONTINUE 257 IF(SIELL.GT.SIELCR) GO TO 31 258 TL=TMLT+ZIP*RCVL 255 28 CONTINUE

331 260 RHOLNA = RHOA1 + TL*(RHOA2+TL*RHOA3) 261 GO TO 60 262 31 CONTINUE 263 IF(SIELL.GE.ENCRIT) GO TO 50 264 NITNOO 265 TLT=0.8333333*TCRIT 266 ZIP=ZIP-AEOSLM 267 35 CONTINUE 268 DELI=HALF*HSTAR*(ONE-TLT*RTCRIT)**ZETA 269 DFDTLT=-((ZETASDELI)/(TCRIT-TLT)) 270 F=ZIP-CVLP*(TLT-TVO3RD*TCRIT)+DELI 271 TL=TLT-F/(DFDTLT-CVLP) 272 IF(ABS((TL-TLT)/TL).LT.EM10) GO TO 40 273 IF (TL .GE. TCRIT) TL = CP9*TCRIT + EM1*TLT 274 IF(NITNO.EQ. 50) GO TO 45 275 TLT = TL 276 NITNO=NITNO+1 277 GO TO 35 278 40 CONTINUE 279 IF(TL.GT.TCRIT-EM8) TL=TCRIT-EVS8 280 RHOLNA=ROLCRT*(ONE+RHOB1S(ONE-TL*RTCRIT) 281 1**RHOB2+RHOB3*(ONE-TL»RTCRIT)**2) 282 GO TO 60 283 45 CONTINUE 284 CALL Q8QPAU 285 GO TO 40 286 50 CONTINUE 287 TL=(SIELL-ENCRIT)/CVLP+TCRIT 288 51 CONTINUE 289 RHOLNA = ROLCRT 290 60 CONTINUE 291 PLIQ = PV(1) 292 ALPHL = 0. 95"ROLPM/RHOLNA 293 IF (ALPHL . LT. 0.95) GO TO 70 294 PLIQ = PLIQ + (CLIQ*«2)*(ALPHL-0.95)*ROLPM/ALPHL 295 70 CONTINUE 296 WRITE (N0UT.2) PV(1),PV(2).SENT,TG.TSAT.TL,RHOLNA,PLIQ 297 C 298 C PV(1) IS THE VAPOR PRESSURE 299 C PV(2) IS THE DERIVATIVE OF THE PRESSURE WITH RESPECT TO DENSITY 300 C SENT IS THE HEAT OF VAPORIZATION 301 C TG IS THE VAPOR TEMPERATURE 302 C TSAT IS THE SATURATION TEMPERATURE 303 C TL IS THE LIQUID TEMPERATURE 304 C RHOLNA IS THE LIQUID MICROSCOPIC DENSITY 305 C [ORE VARIABLES MAY BE PRINTED IF DESIRED 306 C 307 24 CONTINUE 308 23 FORMAT (2I5.3F1O.2) 309 STOP 310 510 CONTINUE 311 READ (NIN.3) TG 312 SUMI=ZERO 313 CVGBAR=ZERO 314 CPGBAR=ZERO

332 315 TSAT=ZERO 316 DTSDRG=ZERO 317 DHSDTS=-EM3 318 X=l. 319 CVG = ACVG + BCVG*(TG-273.16) 320 IF (ROGPM .EQ. ZERO) GO TO 550 321 ARG=ROGPM*TG 322 IF (ARG . LT. ROGCUT) GO TO 535 323 IF (TG .GT. TCRIT) RGFUNC=ONE/(TG-TCRIT+APARA) 324 IF (ARG .LT. T95ROG) GO TO 530 325 IF (ARG . LE. TCRIT*ROGCRT) GO TO 525 326 IF (TG .GT. TCRIT) GO TO 523 327 RMAT = RCRIT 328 DRDRG = ZERO 329 GO TO 540 330 523 GFUNC = APARA'RGFUNC 331 RMAT - EXP(ALRCRT«GFUNC + ALCVG2«(ONE-GFUNC)) 332 DRDRG=ZERO 333 GO TO 540 334 525 ARGDUM = AE0S21*SQRT(ARG) 335 IF (TG .GT. TCRIT) GO TO 528 336 RMAT = EXP(AEOS11+ARGDUM+AEOS31*ARG) 337 DRDRG=ARGDUM*HALF + AE0S31*ARG 338 GO TO 540 339 528 GFUNC = APARA*RGFUNC 340 FFUNC = AE0S11 + ARGDUM + AE0S31*ARG 341 COEFC - ONE + ((ARG-T95ROG)*DENARG)*CPARA 342 COEFM = ALOG(COEFC) 343 RMAT - EXP(FFUNC*GFUNC + (ALCVGM+C0EFM)*(ONE-GFUNC)) 344 DRDRG - GFUNC'(ARGDUM*HALF+AE0S31*ARG) 345 1 +((ONE-GFUNC)*CPARA*DENARG*ARG)/COEFC 346 GO TO 540 347 530 ARGDUM = AEOS3OSSQRT(ARG) 348 IF (TG .GT. TCRIT) GO TO 532 349 RMAT = EXP(AEOS10+AEOS20*ALOG(ARG)+ARGDUM) 350 DRDRG = AE0S20 + HALF*ARGDUM 351 GO TO 540 352 532 GFUNC = APARA*RGFUNC 353 FFUNC = AE0S10+AE0S20*AL0G(ARG) + ARGDUM 354 RMAT - EXP(FFUN€*GFUNC + ALCVGM*( ONE -GFUNC)) 355 DRDRG = GFUNC*(AE0S20+HALF*ARGDUM) 356 GO TO 540 357 535 RMAT = CVGGM1 358 DRDRG = ZERO 359 540 ARGP= ARG*RMAT*RPSTAR 360 IF (ARGP.LE.ZERO) GO TO 545 361 DENOM = ALOG(ARGP) 362 TSAT=-TSTAR/DENOM 363 DTSDRG = -(TSAT*(DRDRG+€NE))/(DENOM*ROGPM) 364 SENT=ZERO 365 IF (TSAT.GE. TCRIT) GO TO 541 366 SENT = HSTAR*(ONE-TSAT*RTCRIT)**ZETA 367 DHSDTS = -(ZETA*SENT)/(TCRIT-TSAT) 368 541 CONTINUE 369 IF (TSAT .GT. TW03RD*TCRIT) GO TO 542

333 370 SIELS = SIELIQ+AMAX1(TSAT-TMLT.ZERO)*CVL 371 CVGR=CVG-AMIN1(ZERO,(CVL*TSAT + SENT*(ONE-TSAT*RTSTAR) 372 1 -CVG'TSAT - HSTAR+CORC)/(ABS(TG-TSAT)+ BPARA)) 373 GO TO 544 374 542 SIELS = SIELIQ+CVLP*(TSAT-TW03RD*TCRIT) 375 1 + AEOSLM - HALF*SENT 376 CVGR=CVG-AMIN1(ZERO,(CVLP*(TSAT-TWO3RD*TCRIT) + AEOSLM -CVG*TSAT 377 1 + SENT*(HALF-TSAT*RTSTAR) + CVL'TMLT - HSTAR+CORC) 378 2 /(ABS(TG-TSAT)H-BPARA)) 379 544 SIEGS = SIELS + SENT 380 IF (ARG .LT. ROGCUT) GO TO 545 381 SUMI = (SIEGS-(CVGR+SENT*RTSTAR)*TSAT)*X 382 SIEGS = SIEGS + AMAX1(EP2-SENT,ZERO) 383 545 CONTINUE 384 CVGBAR=X*CVGR+CVGBAR 385 CPGBAR = X*(CVGR+RMAT)+CPGBAR 386 550 CONTINUE 387 SIEGL=CVGBAR*TG+SUMI 388 WRITE (N0UT.2) ROGPM,TG,RMAT,DRDRG,SIEGL 389 TGN = TG 390 GO TO 480 391 7 CONTINUE 392 EX) 895 NZQ=1,N 393 IF (IOPTN .EQ. 3) GO TO 8 394 READ (NIN.3) TSAT 395 PNI = PSTAR*EXP(-TSTAR/TSAT) 396 GO TO 860 397 8 CONTINUE 398 READ (NIN.3) PNI 399 860 CONTINUE 400 RNEW = CVGGM1 401 ZIPNEW = PNI/RNEW 402 NITNO = 0 403 IF (ZIPNEW .LT. ROGCUT) GO TO 890 404 IF (PNI .LT. PCRIT) GO TO 875 405 GFUNC = APARA/(TSAT-TCRIT+APARA) 406 RNEW = EXP(ALRCRT*GFUNC + ALCVG2*(0NE-GFUNC)) 407 ZIPNEW = PNI/RNEW 408 GO TO 890 409 875 CONTINUE 410 P95CRT = PSTAR*EXP(-TSTAR/(CP95*TCRIT)) 411 IF (PNI .GT. P95CRT) GO TO 885 412 880 NITNO - NITNO + 1 413 IF (NITNO .GE. 50) GO TO 889 414 ZIP = ZIPNEW 415 RNEW=EXP(AEOS10+AEOS20*ALOG(ZIP)+AEOS30*SQRT(ZIP)) 416 ZIPNEW=ZIP+(PNI-RNEW*ZIP)/ 417 1 (RNEW*(ONE+AEOS2O+HALF*AEOS3O*SQRT(ZIP))) 418 IF (ABS((ZIPNEW-ZIP)/ZIPNEW) .GT. EM10) GO TO 880 419 GO TO 890 420 885 CONTINUE 421 NITNO = NITNO + 1 422 IF (NITNO . GE. 50) GO TO 889 423 ZIP = ZIPNEW 424 RNEW = EXP(AEOS11+AEOS21*SQRT(ZIP)+AEOS31*ZIP)

334 425 ZIPNEW=ZIP+(PNI-RNEW*ZIP)/ 426 1 (RNEW*(ONE+AEOS21*HALF*SQRT(ZIP)+AEOS31*ZIP)) 427 IF (ABS((ZIPNEW-ZIP)/ZIPNEW) .GT. EM10) GO TO 885 428 GO TO 890 429 889 CALL Q8QPAU 430 890 CONTINUE 431 TSAT = -TSTAR/(ALOG(PNI*RPSTAR)) 432 ROGPM = PNI/(RNEW*TSAT) 433 RHOLNA = ROLE 434 IF (TSAT .LT. TMLT) GO TO 710 435 IF (TSAT . LT. TWO3RD*TCRIT) GO TO 703 436 IF (TSAT .GT. TCRIT-l.E-8) GO TO 704 437 RHOLNA=ROGCRT*(ONE+RHOB1*(ONE-TSAT*RTCRIT) 438 1**RHOB2+RHOB3*(ONE-TSAT*RTCRIT)**2) 439 GO TO 710 440 703 CONTINUE 441 RHOLNA = RH0A1 + TSAT*(RHOA2+TSAT*RHOA3) 442 GO TO 710 443 704 CONTINUE 444 RHOLNA=PSTAR*EXP(-TSTAR/TSAT)/(RCRIT*TSAT) 445 710 CONTINUE 446 SENT = ZERO 447 IF (TSAT . LT. TCRIT) SENT=HSTAR*(ONE-TSAT*RTCRIT)**ZETA 448 IF (TSAT .GT. TOO3RD*TCRIT) GO TO 745 449 SIELS = (TSAT-TMLT)*CVL + SIELIQ 450 GO TO 750 451 745 SIELS = AEOSLM - HALF*SENT + CVLP*(TSAT-TW03RD*TCRXT) 452 1 + SIELIQ 453 750 SIEGS = SIELS + SENT*(ONE-TSAT*RTSTAR) 454 WRITE (N0UT.2) ROGPM,TSAT,RNEW.PNI.RHOLNA,SIELS,SIEGS 455 895 CONTINUE 456 STOP 457 9 CONTINUE 458 DO 910 NZQ=1,N 459 READ (NIN.3) TL 460 RHOLNA = ROLE 461 IF (TL .LT. TMLT) GO TO 910 462 IF (TL .LT. TW03RD*TCRIT) GO TO 903 463 IF (TL .GT. TCRIT-l.E-8) GO TO 904 464 RHOLNA=ROGCRT*(0NE+RH0B1*(ONE-TL*RTCRIT) 465 l**RHOB2-s-RHOB3*(ONE-TL*RTCRIT)**2) 466 GO TO 910 467 903 CONTINUE 468 RHOLNA = RH0A1 + TL*(RHOA2+TL*RHOA3) 469 GO TO 910 470 904 CONTINUE 471 RHOLNA=PSTAR*EXP(-TSTAR/TL)/(RCRIT*TL) 472 910 WRITE (N0UT.2) TL,RHOLNA 473 STOP 474 10 CONTINUE 475 DO 915 NZQ=1,N 476 READ (NIN.3) TGC 477 TKOEPS=TGC/EPSOK 478 ROMUK=RATMSQ/(CP9249-^-,O7M3*TKOEPS+CP7l92*(TKOE«'S**(-ClP151)) 479 * -C546M2*SQRT(TK0EPS)) 480 THCONG=C831M2!>SQRT(RWTMOLSTGC)*ROMUK 481 XMUG=€267M6*SQRT(Vm0L8TGC)*R0MUK 482 THCONG=THCONG+C1P32*(CVG*GAM-C2P5* 483 • CVG*(GAM-ONE))«XMUG 484 WRITE (N0UT.2) TGC,THCONG,XMUG 485 915 CONTINUE 486 STOP 487 END APPENDIX E

A summary of the input suggested by the evaluation in Sec. II.C is given in Table E-I. The numbers correspond to the format and definitions of App. C. The water property data has been substituted for the aodium data, M •= 3.

~~TABLE E-I WATER AEOS INPUT~~

WATER PROPERTIES AND EOS 1000. 2090.00 73. 16 3. 334E+5 .68 1001.78 4217. 1 0.0727 .68 l.OE-4 3.17771E+1O 4. 7O579E+O3 0.0 3.O898O5E+6 647.296 0.3660361 2181.6 1.2115 3.737 2.9333OE+6 18. 32. 316.957 95.77 316.957 200. 1346. 0.3302 5.E+O4 50. .5 838.607921 1.34831230 -.0027491404 2.85605286 .381221451 -.0748002905 APPENDIX F

~~STANDARD SIlvMER-II LIQUID-LIQUID HEAT-TRANSFER MODEL~~

The liquid-liquid heat-transfer model is based on a droplet-droplet collision concept similar to that assumed for coalescence. The total collision r-ite per unit volume is derived in the following discussion along with the heat- transfer area, overall heat-transfer coefficient, and contact time per collision. The collision rate between liquid droplets of components m and k is formulated from the physical representation shown in Fig. F-l. Each droplet is assumed to have an average random velocity given by V and V^ that differ's from the liquid-field velocity. Differing accelerations resulting from both droplet-size distributions and density variations might be expected to produce a spectrum of liquid velocities. As a result each droplet sweeps a volume of space per unit time given by the product of the random velocity and the projected area of the droplet. The volume swept per unit time and volume by all droplets or particles of component m is given by

~~V0Lm = nm "Jm Vm • CF-1)~~

~~where n is the number density of droplet component m. The number of collisions~~

with droplets or particles of component k in VOLm is simply VOL times the number density of type k. The collision rate per unit volume of component m with component k caused by the movement of component m is given by

~~"pm ^m nm n~~

~~= "pk \ nm nk~~

~~339 O~~

~~Fig. F-l. Physical representation of liquid-liquid interaction. The total collision rate is approximated by 0.5 times the sum of the rates given by Eqs. (F-2) and (F-3),~~

~~pk \) °'5~~

~~The factor of 0.5 accounts for counting each collision twice. The number density of droplets of each component is related to the respective volume fractions, a. and radii by~~

~~m i .fin~~

~~(F-5)~~

~~nk = 3aLk/4trrpk~~

~~Substituting Eq. (F-5) in Eq. (F-4) gives the collision rate in terms of the variables computed by SIMMER-II in each mesh cell. Thus,~~

~~a a 9 Lm Lk , _2 $ + ,2 l3 (rpm Vm rpm rpk~~

The collision rate does not behave well in the limit of the radii approaching zero. In the following, this situation is considered below in terms of physical limitations on the heat-transfer rates. We consider the area of contact as the area for heat transfer that may be expected for each collision. Fig. F-2 shows the assumed collision configuration

for droplets of components m and k. In general, the contact area, Acj M, may be highly variable due to its dependence on surface tension, velocity of impact, relative densities of the two components, and angle of impact. Though the mechanics of the impact may b; complicated by the above factors, a reasonable assumption is that the contact area is the order of the cross-sectional area of the smaller droplet. Therefore v/e define the contact area to be

~~342 AcLmLk " Cl "Jia rpm < rpk • and~~

~~(F-7)~~

~~AcLmLk = Cl ffrpk rpm * rpk '~~

The value of C, may range from 0.1 for glancing collision or low impact velocities to rjk/rjm if rpm < rpk or rjm/rjk if rpffl ? rpk- This upper limit corresponds to the situation in which the smaller droplet deforms and /ers an area equal to the cross-sectional area of the larger droplet. The heat-transfer rate per unit area during the collision is assumed to be quasi-static. This is consistent with other heat-transfer modeling in SIMMER-II and approximates the process reasonably well for small droplets. In this particular case of spherical droplet interaction, it can be argutd that because most of the heat capacity in a sphere is near the outer surface, the effective conduction path length is generally very small (about 0.2 *,,)• For droplets with radii of about 10 m, the thermal response time can be approximated by

~~(F-l 2 (0.2 rp)~~

where a^ is the thermal diffusivity of the liquid. Fueli steel, and sodium droplets (10"J m radius) respond thermally ia time intervals of 0.070, 0.008, and 0.0008 s, respectively. The quasi-static apTroximation is therefore not good for fuel and steel droplets of 10 m radius but would be adequate for those with 10 m radii. For short contact times and these relatively large droplets, the average heat-transfer rate during a collision will be substantially underestimated by this model. The overall heat-transfer coefficient is based on conduction between two slabs shown in Fig. F-3 and assumed to have thicknesses of 0.2r and 0.2r v.. These thicknesses represent the distance from the outvi surface of a flattened or deformed sphere to the centroid surface (one-half of the mass is inside this surface). The overall heat -1ransfer coefficient is approximated by

~~343 CONTACT CONTACT BEGINS ENDS~~

Fig. F-3. Assumed contact .ntervai. T kLm kLk The contact time per collision also is uncertain due to the nature of the collision; that is, the velocity of collision, the angle of collision, surface tension, and thermal interaction at the contact surface (film boiling, and so on). We conveniently estimate the contact time on the basis of a mutual residence time, considered to be proportional to the time that the two interacting droplets or particles are in possible contact. Figure F-4 illustrates the situation in which droplet k is moving relative to droplet m with an average velocity V. The contact could begin when droplet k is at position A and end when it is at position B. The distance droplet k travels is about 2r L + 2r . The contact time, t-.iir. is given by

The constant C2 represents the fraction of the residence time that would charac- terize th. average collision. The average velocity is divided by two because in general droplet m will be moving, and as a result, the relative velocity will be reduced. The uncertainty in the average contact time is related to the uncertainty in the heat transfer per collision becausie of the thermal time-constant considerations discussed previously. To some extent the uncertainties cancel. The error in the heat-transfer rate during a collision, qcLmLjc. will be inversely proportional to the contact time due to the transient temperature-gradient effect. Because the total heat transferred per collision is the product of ''cLmLk anc* ^cLmLk' *^e magnitude of the error in the total heat transferred may

be considerably less than the individual errors in qcLmLk and t ^ LJ.. The overall heat-transfer rate per unit volume is given by the product of the heat transferred per collision and the. number of collisions per unit volume and time,

~~= UcLmLk AcLmLk ^LmLk (TLk ' TLnJ ZLmLk ' (F-ll)~~

~~345 ASSUMED TEMP. PROFILE~~

~~TLM TRANSIENT TEMP. PROFILE o LK~~

~~LIQUID, L I QU IDk~~

~~0.2r~~

~~Fig. F-4. Heat-transfer treatment during contact. Substituting Eqs. (F-4), (F-7), (F-9), and (F-10) in (F-ll) gives the heat- t ransfer rate,~~

~~45 1 aLm a~~

~~(F-12)~~

~~whe re rc = rpkk if rpkk < rrpm rc = rpm if rpk > rptr C = Cj C2 .~~

In Eq. (D-12) Vm and V^ have been approximated by V. Thus, the random velocities of various types of droplets 01 particles are characterized by a single average value V that cancels in Eq. (D-12). The (1 - a

~~k a a ,r45 Lk Lm 1HL..U - C 7 — ^r^~~

The heat-transfer rate approaches infinity as r approaches zero, resulting from the product of the number density of component m (Eq. D-5) and the contact area per collision, irr . Physically, the average instantaneous contact area exceeds the total surface area of all droplets of component k. This effect gives a heat-transfer rate greater than that which could occur with zero thermal resistance in the liquid component m. The limiting heat-transfer rate for this latter case is given by

~~347 kLk~~

~~or if r u approaches zero,~~

~~rpm~~

~~348 APPENDIX G CORRECTION SET FOR MODIFIED LIQUID-LIQUID HEAT TRANSFER~~

1 MDENT B5E4 2 */ 3 */ CORRECTION SET TO FORCE VAPORIZATION FOR LIQUID/LIQUID 4 •/ HEAT TRANSFER 5 */ 6 •/ 7 •/ THE HEAT TRANSFER IN TSHTR IS MODIFIED FIRST 8 */ 9 *I TSHT.333 10 IF (ALPHG(IJ) .LT. ALPHO*(ONE-ALPHS(IJ)) .OR. NCVC .EQ. 0) 11 1 GO TO 7704 12 IF (N .EQ. 3 .AND. TL(M) .GT. TL(N)) OHTC=ZERO 13 IF (M .EQ. 3 .AND. TL(N) .GT. TL(M)) OHTC=ZERO 14 7704 CONTINUE 15 SD TSHT.423 16 DO 627 N=l,2 17 • I TSHT.443 18 IF (M .EQ. 3 .AND. TL(N) .GT. TL(M) .AND. NCVC .GT. 0 19 1 .AND. ALPHG(IJ) .GE. ALPH0*(ONE-ALPHS(IJ))) OHTC = ZERO 20 */ 21 */ INITIALIZATION OF VARIABLES FOR VAPORIZATION/CONDENSATION 22 V 23 *B PHAS.28 24 DIMENSION HL3(6) 25 *I PHAS.46 26 HL3T = ZERO 27 HL3TN = ZERO 28 DO 1058 N=1,NCLE 29 HL3(N)=ZER0 30 IF (N .EQ. 3) GO TO 1058 31 IF (TL(N) .LT. TL(3)) GO TO 1058 32 IF (RP(N) .LE. ZERO) GO TO 1058 33 IF (RP(3) .LE. ZERO) GO TO 1058 34 MM=IEOSLE(N) 35 THCON = THCONL(MM) 36 CVLL=CVL(MM) 37 IF (N .GT. NCLEM2) THCON=THCONS(1^) 38 IF (N .GT. NCLEM2) CVLL=CVS(MM) 39 0HTCL2 = ROLP(N)*CVLL*RDTT 40 HL1 = THCON*HX1(N) 41 A12 = HL1*HALPS*VOLFX(N)*VOLFX(3) 42 iH(RP(N)**2+RP(3)**2)»(RP(N)+RP(3))*RP(N)**2)/(RP(N)**3*RP(3)**3) 43 OHTCL = C15P*HL1*VOLFX(N)/RP(N) 44 IF (RP(3) .LT. RP(N)) 45 1A12 = A12*(RP(3)**2/RP(N)«*2) 46 1057 CONTINUE 47 RLMULT = RLL(N,3) 48 IF (N .GT. 3) RLMULT = RLL(3.N) 49 *IF -DEF.DBL.l 50 HL3(N) = AMIN1(AMIN1(OHTCL,A12)*RLMULT,OHTCL2) 349 51 *IF DEF.DBL.l 52 HL3(N) = DMIN1(DMIN1(OHTCL,A12)*RLMULT,OHTCL2) 53 HL3T = HL3(N)+HL3T 54 HL3TN = HL3TN + HL3(N)*TN(NCSE+N) 55 1058 CONTINUE 56 HL3T = HL3T*DTT 57 HL3TN = HL3TN*DTT 58 */ 59 */ FIRST, WE MODIFY THE CONTINUITY EQUATION 60 */ 61 *I PEAS.982 62 IF (N .NE. 3) GO T) 7704 63 AIRHO(N) = AIRHO(N) + HL3TN 64 A2RH0(N) = A2RHO(N) - HL3T 65 7704 CONTINUE 66 */ 67 */ NEXT, WE MODIFY THE VAPOR SIDE HEAT TRANSFER COEFFICIENT 68 */ ITERATION 69 */ 70 *I PHAS.1322 71 FOSUM = ZERO 72 IF (M .NE. 3) GO TO 34 73 DO 35 M=1,NCLE 74 35 FOSUM = FOSUM + HL3(M)*(TTSAT(N)-TN(NCSE+M)) 75 ARGG= ARGG- CPDHFG(N)*FOSUM 76 34 CONTINUE 77 *I PHAS.1332 78 FO = FO + CPDHFG(N)*FOSUM 79 */ 80 */ THIRD, WE PUT THE MASS TRANSFER INTO THE VAPOR ENERGY EQN. 81 */ 82 *I PHAS.886 83 FOSUM = ZERO 84 IF (N .NE. 3) GO TO 36 85 DO 37 M=1,NCLE 86 FOSUM = FOSUM + HL3(M)*(TSAT(N)-TN(NCSE+M)) 87 37 CONTINUE 88 36 CONTINUE 89 *I PHAS.888 90 AMM = AMM + FOSUM 91 *I PHAS.905 92 SUMC = SUMC - FOSUM*DTT 93 */ 94 »/ AFTER CONVERGENCE WE TAKE THE HEAT AWAY FROM THE LIQUIDS 95 */ 96 *I PHAS.1528 97 FOSUM = ZERO 98 DO 38 N=1,NCLE 99 IF (HL3(N) .EQ. ZERO) GO TO 38 100 QLCIJ+N-1) = QLCIJ+N-1) + HL3(N)*(TTSAT(3)-TN(NCSE+N)) 101 SIELN(N)=SIELN(N)+(HL3(N)*(TTSAT(3)-TN(NCSE+N)))/ROLPP(N) 102 F0SUM=F0SUM+HL3(N)*(TTSAT(3)-TN(NCSE+N)) 103 38 CONTINUE 104 *I PHAS.1604 105 IF (N .EQ. 3) 350 106 1QG(IJ)-QG( IJ)-FOSU 107 V 108 V MODIFY THE HEAT OF VAPORIZATION RESULTING FROM ENERGY LOSS 109 */ 110 *I B754.67 111 IF (N .NE. 3) 112 1SENT(N) - 5Cr,T(N) - DTT*HL3(N)*(TSAT(3)-TN(NCSE+N))*RROLP(N) 113 *I B754.109 114 IF (N .NE. 3) 115 ISENT(N) SENT(N) - DTT*HL3(N)*(TSAT(3)-TN(NCSE+N))*RROLP(N)

~~351 APPENDIX H~~

~~VAPORIZATION/CONDENSATION CORRECTION SET~~

1 *IDENT B754 2 V 3 */ THIS CORRECTION SET MODIFIES THE PHASE TRANSITION MODEL 4 */ 5 *D PHAS.983 6 *D B1FO.25 7 *D PHAS.984 8 *D B4K3.277,280 9 *D PHAS.985,1048 10 *I PHAS.962 11 IVPP = IVPALL(N) 12 ICOUNT(N) = 0 13 *D P8M0.15 14 C 15 C IDIR(N) = +1 IMPLIES CONDENSATION 16 C IDIR(N) = -1 IMPLIES VAPORIZATION 17 C 18 IF (IVPP .EQ. 1) GO TO 443 19 SETDR - AIRHO(N) + A2RH0(N)*TSAT(N) 20 IDIR(N) « +1 21 IF (SETDR .GT. ZERO) IDIR(N) - -1 22 443 CONTINUE 23 ROGPPP(N) = ROGPP(N) 24 IF (NITNO .EQ. 1 ) GO TO 7705 25 IF (TG.GT.TCRIT(N) .AND. TLIQ(N).GT.TCRIT(N)) ICOUNT(N) = 1 26 IF (ICOUNT(N) .EQ. 0) UO TO 7705 27 XLROG(N) - EM10 28 IF (IDIR(N) .EQ. -1 .AND. TSAT(N) .GT. TCRIT(N)) 29 IXLROG(N) - (PCRIT(N)*ALPHG(IJ))/(RMAT(N)*TG) 30 XHROG(N) = EP10 31 *D PHAS.1051,1053 32 »D B7S0.1,3 33 *D PHAS.1060,1071 34 SIEGLN = SIEGL 35 DO 447 NN=1,NCVC 36 N- ICGC(NN) 37 SIEGLN « SIEGLN + A3RH0(N)»(TSAT(N)-TTSATP(N)) 38 447 CONTINUE 39 *D PHA-S. 1076,1077 40 *D B7G9.56 41 *D PHAS.1096,1112 42 *D P240.26 43 *D PHAS.1114 44 *D P240.27 45 *D PHAS.1116,1120 46 *D P240.28 47 *I PHAS.1140 48 DCODTS = CVL(N) 49 IF (TSAT(N).GT.TW03RD*TCRIT(N)) DCODTS - CVLP(N) - HALF*DHSDTS(N) 50 SETDR=A1RHO(N)+A2RHO(N)*TSAT(N) 353 51 IF (SETDR.GT.ZERO .AND. ROGPP(N). GT. ROGP(N)) IDIR(N)=-1 52 IF (SETDR. LT.ZERO .AND. ROGPP(N). LT=ROGP(N)) IDIR(N)=+1 53 IF (IDIR(N) .EQ. -*1) GO TO 635 54 SENT(N) = SENT(N) + SIELS(N) - SIELNP(N) 55 1 - DTT*KD(N)sRROLP(N)d(TSAT(N)-TLIQ(N)) 56 DHSDTS(N) = DHSDTS(N) + DCODTS - DTT»HD(N)*RROLP(N) 57 GO TO 640 58 635 CONTINUE 59 SENT(N) = SIEGLN - SIELS(K) 60 DHSDTS(N) = -DCODTS + A3RH0(N) 61 640 CONTINUE 62 aI PHAS.1144 63 IF (SENT(N) . EQ. EP2) DHSDTS(N) = -EM3 64 *D PHAS.1146 65 FMED - SETDR*RSENT(N) 66 *D PHAS.1152,1156 67 *D AEOS.275 68 IF (TSAT(NZ) .GE. TCRIT(NZ)) GO TO 342 69 *I AEOS.277 70 IF (TSAT(NZ) ,GT. TW03RD*TCRIT(.NZ)) GO TO 342 71 *IF -DEF.DBL.l 72 SIELS(NZ)=SIELIQ(NZ)+AMAX1(TSAT(NZ)-TMLT(N2),ZE8O)*CVL(NZ) 73 »IF DEF.DBL.l 74 SIELS(NZ)=SIELIQ(NZ)+DMAX1(TSAT(NZ)-TMLT(NZ),ZERO)*CVL(NZ) 75 GO TO 390 76 342 SIELS(NZ) = SIELIQ(NZ)+CVJ.P(NZ)*CTSAT(NZ)"TW03RD*TCRIT(NZ)) 77 1 + AEOSLM(NZ)-HALF*SENT(NZ) 78 *D PHAS.1229,1242 79 *D PHAS.1227 80 *D PHAS.1598 81 «D PHAS.872 82 GO TO 3365 83 *D PHAS.943 84 *I PHAS.1160 35 ROC-P2(N) - ROGPPP(N) 86 SI PHAS.1162 87 IF (IKHOIT .EQ. 0) GO TO 77032 88 IF (ICOUNTCN) . EQ. 0) GO TO 77031 89 *IF -DEF.DBL.2 90 77024 IF (F . LT. ZERO) XHROG(N) *= AMIN1(R»3GPPP(N),XHROG(N)) 91 IF (F .GT. ZERO) XLROG(N) = AMAX1(ROGPPP(N),XLROG(N)) 92 "IF DEF.DBL.2 93 77024 IF (F .LT. ZERO) XHROG(N) = DMIN1(ROGPPP(N),XHRCG(N)) 94 IF (F .GT. ZERO) XLROG(N) = DMAX1(ROGPPP(N),XLRQG(N)) 95 IF (ROGPP(N) .LE. XLROG(N) .OR. MOD(IRHOIT,20) . EQ. 5 .OR. 96 1 ROGPP(N) .GE. XHROG(N)) ROGPP(N) = (XHROG(N) +XI.ROG(N))*HALF 97 IF (IPHOTT .EQ. 50) PAUSE 98 GO TO 77032 99 77031 CONTINUE 100 IF (IRH0T2 . F.Q. ZERO) GO TO 77032 101 DR0GP2 = ROGPPP(N) - ROGP?.(N) 102 DROGP =. ROGPP(N) - ROGPPP(N) 103 IF (DROGP2SDROGP .GE. ZERO) GO TO 77032 104 DENOM = ROGPP(N) - TWO*ROGPPP(N) + R0GP2(N) 105 aIF -DEF.DBL.2 354 106 IF (ABS(DENOM).GT. EM6) ROGPP(N) = R0GPP(N)~(DROGP*DROGP)/DEN0M 107 FMAX = AMAX1(FMAX,ABS(DROGP)*RTH) 108 'IF DEF.DBL, '. 109 IF (DABS(DEN0M).GT.EM6) ROGPP(N) = ROGPP(N)-(DROGP*DROGP)/DENOM 110 FMAX = DMAX1(FMAX,DABS(DROGP)*RTH) 111 77032 CONTINUE 112 * I PHAS.1222 113 SENTP(N) = SENT(N) 114 IF (IDIR(N) .EQ. +1) GO TO 645 115 SENT(N) - SENT(N) + SIELS(NJ - SIELNP(N) 116 1 - DTT*HD(N)*RROLP(N)*(TSAT(N)-TLIQ(N)) 117 GO TO 650 118 645 SENT(N) = SIEGL - SIELS(N) 119 650 CONTINUE 120 *D PHAS. 890 121 DMM = ZERO 122 IF (IDIR(N) .EQ. -1) DMM=SIEGS(N)-SIEGL+CVGBAR*TG 123 *D B151.3 124 IF (IDIR(N) .EQ. +1) GO TO 8564 125 *D PHAS.902 126 SUMB = SUMB - ((AM4fBMM*TSAT(N))*CVGBAR+BIiai«DIiM)*RSENT(N) 127 8564 CONTINUE 128 SUMB = SUMB + BMM 129 *D B7G9.49 130 8565 SUMB = SUMB + BMM 131 IF (IDIR(N) .EQ. -1) SUMB = SUMB - (ROGPP(N)-ROGP(N))*CVGBAR 132 * I PHAS.906 133 IF (NCSC .EQ. 0) GO TO 8575 134 BMMP = ZERO 135 DO 8573 M = l.NCSC 136 BMMP = BMMP + HVS(M.N) 137 8573 CONTINUE 138 SUMB = SUMB - DTT*BMMP 139 SUMC = SUMC + DTT*BMMP*TSAT(N) 140 8575 CONTINUE 141 *D PHAS.27 142 DIMENSION IDIR(4),SIELNP(4),RR0LP(4),A3RH0(4),ROGP2(4), 143 1XLROG(4),XHROG(4),SENTP(4) 144 *I PHAS.764 145 SIELNP(N) = SIELCU+N-1) + CVL(N)!!(TLIQ(N)-TL(N)) 146 RROLP(N) = ONE/ROLP(N) 147 SENTP(N) = SENT(N) 148 »D PHAS.1122,1136 149 *D PHAS.972,973 150 *D PHAS.964,965 151 *D PHAS.975 152 *D PHAS.816 153 *D PHAS.1191 154 * I PHAS.976 155 SENT(N) = SENTP(N) 156 A3RH0(N) = CVL(N) + DHSDTS(N)*(ONE-TTSAT(N)*RTSTAR(N)) 157 IF (TSAT(N) .GT. TWO3RD*TCSIT(N)) 158 1A3RHO(N) = CVLP(N) + DHSDTS(N)*(HALF-TTSAT(N)*RTSTAP(N)) 159 A3RKO(N) = A3RH0(N) -SENT(N)SRTSTAR(N) -CVGR(N) 160 IF (CVGR(N) .LE- CVG(N) + L'^O) GO TO 445

355 161 IF (TG .GE. TTSAT(N)) 162 1A3RHO(N) = (A3RH0(N)*BPARA(N))/(TG-TTSAT(N)+BPARA(N)) 163 IF (TG .LT. TTSAT(N)) A3RH0(N) = A3RH0(N) 164 l+((-A3RH0(N)+TW0*(CVG(N)-CVGR(N)))*(TG-TTSAT(N))) 165 2/(TTSAT(N)-TG+BPARA(N)) I6o 445 A3RH0(N) = (A3RH0(N)«R0GPP(N)«RTH)/R0G 167 *D PHAS.1577,1579 168 IF (IDIR(N) .EQ. 1) GO TO 7735 169 QG(IJ) - QG(IJ) + QGC(N) - GAMCE(IJNL)*SIEGS(N) 170 QL(IJNL)=QL(IJNL)-K)LC(N)+GAIC:E(IJNL)*(SIELNP(N)+Drr*RROLP(N) 171 1*QLC(N)) 172 SIELN(N) - SIELN(N) + QLC(N)*DTT»RSOLP(N) 173 GO TO 7740 174 7735 QG(IJ) = QG(IJ) + QGC(N) - GAMCE(IJNL)*SIEGL 175 QL(IJNL) = QL(IJNL) + QLC(N) + GAMCE(IJNL)*SIELS(N) 176 SIELN(N) - SIELN(N) + (SIELNP(N)+ 177 1 C(QLC(N)+GAICE(IJNL)*SIELS(N))«DTTaRROLP(N))) 17S 2 /(ONE-K}A!CE(IJNL)*DTT*RRQLP(N))-SIELNP(N) 179 7740 CONTINUE 180 *D PHAS. 788 181 *D B7G9.33,36 182 *I PHAS.960 183 IRH0T2 - 0 184 •! PHAS.1205 185 »IF DEF.IBM.l 186 IRH0T2 - MODCIRHOT2.2) 187 *IF -DEF.IBM.l 188 IRH0T2 = IRHOIT.AND. 1 189 *D PHAS.1196,1198 190 *I PHAS.1263 191 IF (NITNO .GE. 50) PAUSE 192 aI PHAS.1535 193 IF (IVPALL(N) .EQ. 0) GO TO 670 194 SUM1 = ZERO 195 SUM2 - ZERO 196 DO 665 M-l.NCSC 197 IF (ISTRHT(M.N) .EQ. 1) GO TO 670 198 SUM1 = SUM1 + HS(M,N)*TST(M) + HVS(M,N)*TGN 199 SUM2 = SUM2 + HS(M,N) + HVS(M.N) 200 665 CONTINUE 201 TTSAT(N) = SUM1/SUM2 202 670 CONTINUE 203 SD PHAS.1416 204 IF (ISTRHT(M.N) .EQ. 1 .OR. IVPALL(N) . EQ. 1) GO TO 34055 205 */ 206 */ AVOID TROUBLE BY CORRECTING PREVIOUS IDENT 207 V 208 'IDENT B814 209 *I B754.89 210 IF (SENT(N) .GT. EP3 .AND. DFDRG .LT. -EMI) GO TO 454 211 XHROG(N) = EP10 212 XLROG(N) = EM20 213 ICOUNT(N) - 1 214 GO TO 77024 215 454 CONTINUE

356 216 *I B754.86 217 IF (XHRCG(N) . LT. EM10) ICOUNT(N) = 0 218 *I B754. 16 219 AMMP = ZERO 220 RMMH = ZERO 221 *I B754.18 222 AMMP = AMMP + HS(M,N)*TST(M) 223 EtMl = BMMH + HS(M,N) 224 *D B754.20.21 225 IF (BMMP+BMMH .LT. ONE) GO TO 8575 226 SUMB = SUMB - (DTT*BMMP*BKMO/(BMMP+BMMH) 227 SUMC = SUMC + (DTT*BMMP*AMMP)/ (BMMP+BMMH) 228 »D B754.113 229 *D B754.87 230 *D PHAS.1150 231 *IF -DEF,DBL,1 232 DFDRG = AMIN1(DFDRG,-EM2) 233 "IF DEF.DBL.l 234 DFDRG = DMINl(DFDRG,-EM2) 235 *D PHAS.1185,1186 236 IF (IRHOIT .GT. 0 .AND. M0D(IRHOIT,20) .NE. 5) GO TO 77039 237 IPHOIT = IRHOIT + 1 238 SD PHAS.1199,1204 239 *IF DEF.DBL.3 240 IF ((DABS(TTSAT(N)-TSAT(N)).LT.EMS .OR. FMAX. LT. FTEST) 241 1 .AND. DABS(ROGPPPOO-ROGPP(N))/ROGPP(N) . LT. EM6) 242 2 GO TO 77082 243 «IF -DEF.DBL.3 244 IF ((ABS(TTSAT(N)-TSAT(N)).LT.EM8 .OR. FMAX. LT. FTEST) 245 1 .AND. ABS(RGGPPP(N)-ROGPP(N))/RQGPP(N) . i-i. EM6) 246 2 GO TO 77082 247 *D B754.76 248 *D PHAS.1161 249 *I PHAS.1147 7.50 R0GP2(N) = ROGPPP(N) 251 ROGPPP(N) = ROGPP(N) 252 *IDENT BCI4 253 */ 254 */ MAKE ADDITIONAL CORRECTIONS 255 •/ 256 «I B754.2 257 DIMENSION TSATH(4) 258 «I PHAS.969 259 IF (IVPP .EQ. 0) GO TO 424 260 TSAT(N) = TSATH(N) 261 TTSAT(N) = TSATH(N) 262 424 CONTINUE 263 *I PHAS.1228 264 TSATH(N) = TSAT(N) 265 »I PHAS.1314 266 IF (IVPALL(N) .EQ. 1) GO TO 34001 267 *I PHAS.1319 268 TSAT(N) = TTSAT(N) 269 TSATH(N) - TTSAT(N)

~~357 APPENDIX I~~

~~SIMMER-11 MANUAL TREATMENT FOR THE VAPORIZATION/CONDENSATION MODEL~~

~~The material in this appendix is taken directly from the SIMMER-II manual. It has~~

7. Methods for the Simple Vaporization-Condensation Model. The method used in SIMMER-II for determining the vaporization and condensation rates differs considerably from that used in SIMMER-I. To obtain a phase-trans it ion rate that automatically accounts for the change in saturation propeities from phase transition, the vapor components are treated implicitly. Only the liquid and structure components are treated explicitly. This also differs from the SIMMER-I technique in that the solution is obtained iteratively rather than in a single step. Compared with the SIMMER-I method, the iterative solution method provides more consistent answers and intermediate results that are easier to interpret when debugging. However, the iterative technique is more complicated to describe and the solution requires more computer time. The equations are presented in terms of the iteration values of the vapor properties. These va.'ues approach the end-of-time-step values as the iteration converges. The heat flows for phase transition at a liquid-field energy- componrnt interface are

~~Sat,M~~

~~for the heat flow to the vapor field fiom the interface and~~

~~IL = H (TSa1,M~~

~~for the heat flow to the 1 ^t-.: id-field energy component fiom the interface. Here~~

359 K is the iteration index for heat flow, and the heat-transfer coefficients and sin-face areas have been combined into the single variable H. The solution method assumes that the liquid possesses sufficient thermal inertia such that liquid temperatures are not affected significantly by vaporization or 'jondensat ion in any one time step. This is not a good approximation as a component's liquid volume fraction approaches zero. Examination of impiic:tly formulated liquid component energy equations reveals that even for infinite liquid-side heat-transfer coefficients, the energy-transter rate to the liquid should not be larger than

~~P c Lm vLM fTn+l Tn \ At~~

Because other processes can change the liquid temperature, experience has indicated that some fraction of this maximum xate is desirable. Therefore, the maximum liquid-side heat-transfer coefficient is restricted to

The vapor-field temperature in Eq. (IV-86) is evaluated at che advanced time. The. liquid-field energy-component temperature also could be evaluated implicitly, but this would complicate the solution procedure further. To account partially for the thermal response of the liquid-field energy component, the temperature used in Eq. (IV-87) depends on the change in the liquid-field energy-component temperature during the time step from other heat-transfer modes. This change is predicted by the implicit heat-transfer calcination. The basis for the choice "f temperature 13 that toe liquid-side hedt-transfer coefficient is usually significantly larger -,Lian the vapor-side heat-transfer coeffic: it. Thus, the outcome of the comparison of the liquid-field energy-competent temperature with the saturation temperature determines whether valorization or condensation is likely to occur. Also considered in the choice is that the component temperature p/jbably will increase during c. ^denaation and decre?se during vaporization. These considerations result in the following 360 selections for the liquid-field energy-component temperature Lo be used in Eq. CIV-87).

~~xn+1 : c xn -, Tn+' -, Tn !Lm • 1X 'Sat.M > 'Lm > 'L (IV-9O)~~

~~n n n+ w >• > !T -v T 'SatSat.M > !LLm =" 'Lra~~

~~X — Xn i F r> i . i 11 ^ 'Lm " 'Sat.M ' lt 'Lin !Sat,M > Xn+1 ^ Tn ^ 01 JLm < 'Sat.M <~~

In general, Tj^J and Tj^ differ little, and the above choice has little effect compared with using the advanced-time temperature. However, \*/hen the thermal response time of the liquid-fiei nergy component is small, this approach provides stability by anticir the change ir liquid-field component temperatures resulting from vapc tion or condensation. The heat flows from the siructure-field energy component interface are evaluated in a similar manlier'

~~(IV-93)~~

~~for heat flow to the vapor field from ihe interface and~~

~~"-ISkCm - HsGk (T&!.M- ^ ) (IV-94)~~

~~for the heat flow to the structure-field ener component from the interface.~~

361 Again, the products of the heat-transfer coefficient and the surface area are combined. The component temperature used ir. Eq. (IV-94) is the advanced-time temperature because the frozen material resistances are added to the resistance for the underlying component when the thermal response of the frozen material is rapid. To obtain the expected change in the saturation conditions during the time step, the vapor-field material component continuity equations and the ,apor mixture energy equation are used in a truncated f-:m that includes only terms from phase transitions. After differencing in time, the truncated continuity equations are

~~rGLm • m = 1, . . . , NMAT , (IV 95)~~

~~whe re~~

~~PQ is the macroscopic density of vapor-field material comronent m after /c+1 iterations,~~

At is the estimated time step, and K+ 1 FQJ is the total mass-transfer rate from vapor-field material component m after K+1 iterations aud the fission gas (component 5) is neglected. The total phase-trunsition rate is given by

~~= + ^ U" -°~~

~~where h^p"^ is the heat of vaporization for material M after K+1 iterations.~~

Equations (IV-95) and (IV-9o) are the first equations solved in the phase- transition model because a good estimate of the muss-transfer rate is required to solve the vapor eneigy equation. To evaluate the heat flows in Eq. (IV-96), T Harid ^GSk are aPProximated by TQ, HQI , and HQ^^. Then v/e note a strong dependence of vapor field density's on saturation conditions. Conse- quently, a simple evaluation of saturation properties from the equation of state and their insertion into the equations generally overestimates the degree of mass transfer. An iteration if. required to couple the nonlinear relationship

~~7 (-. ? 1 between PQ* and Tg+J M in the equation of state with Eq. (IV-95). The iteration proceeds by writing the differenced, truncated continuity equation for each vapor-field component as~~

~~n al.m + a2,m 'JSat.M (IV-97) Gm Gm 1~~

~~whc re~~

~~m " A? (HGLm TG + HLGm TLnJ + " I ^Sk TG + HSGk k=l~~

~~and~~

~~2 a2,m - "** [HGLm+ HLGm + I («GSk + HSGk)] • (IV"99) k=l~~

To find the saturation temperature, heat of vaporization, and vapor-field component density that are consistent for satisfying Eq. (IV-97) and the equation-of-state relation between the saturation temperature and the density, the Newton-Raphson technique is used to determine the vapor-field material component density. The function used involves a rearrangement of Eq. (T.V-97) as

~~where k is the inner iteration index. The iteration equation then is~~

~~dPGm~~

~~363 whe :e~~

~~3T a + a dF* 1 , Sat.M,Jc r * t l.m 2,m a T m~~

~~(IV-102)~~

where the partial derivatives on the right-hand-side of Eq. (IV-102), the saturation temperature, and the heat of vaporization are obtained from the equation of state after each inner iteration. During the iterate i\, the vapor-field material component density is restricted to be between zero and the total liquid and vapor mass of the material in the mesh cell. Tf the component density is computed to be less than zero in an iteration, the component density is set to 0.001 of the value for the previous iteration. The convergence precision achieved in this iteration is crucial for overall convergence of the model. The criteria are (1) A minimum of two inner iterations must be performed because consistency between PQ and Tc.i « is not guaranteed when the fii3t ^Gm calculation is performed. (2) Either the saturation temperature must be converged tj within 10" K, or the change in each material component APr* , evaluated by Eq. (IV-101), must be le3S than 6,

~~where~~

~~7 5 = 60 max (lO~ , PQm) ^£ , except that~~

~~4 6=5* ID' 60PGm, if ?CrtiM - 0.0625 K < TSatfM < TCrt>M , and~~

5Q = EPHASE, an input parameter In some cases, for example, for Tr i u near T^ i u, the derivatives change so rapidly that convergence will not occur unless the initial estimated conditions are sufficiently close to the converged solution. Methods for these cases are discussed in App. E. Also, a ouasi-equiJibrium approximation for very large heat -1ransfer coefficients has been eliminated from this version of SIMMER-II because of difficulties with energy conservation.

364 The second set of equations solved corrects for the vapor-side heat- transfer coefficient based on t _e mass transfer at tne various surfaces modeled £q. (1V-96). As shown in Sec, III,D.6.d, the conected heat-transfer coefficient for condensation on liquid-field energy component m is

~~(IV-103) /H pG GLiJ -1~~

~~where~~

~~is the corrected product of the vapor-side heat-transfer coefficient and interfacial area per unit volume,~~

~~r'lGmLm is the condensation mass-transfer rate at the surface,~~

~~c Q is the average constant-pressure specific heat of the vapor mixture, ?nd~~

~~is the product of the vapor-side heat-transfer coefficient and the interfacial area per unit volume in the absence of phase transi tions.~~

If vaporization rather than condensation is occurring, then FTQ , in Eq. (IV-103) is replaced by 'ÎLmGm' ôr *ê structure-field surfaces, the mass-transfer rate in Eq (IV-103) is the total mass-transfer rate at the surface for all vapor-field components. Again, a high degree of coupling exists between the heat-transfer coefficient and the phase-transition rates, and instabilities can arise if Eq. (IV-103) is included either directly in the energy equation iteration or as part of an outer iteration. To provide stability, v/e use the definition of the phase-transition rate,

~~1 ( f sc-S-l HH T IGmLm K+l i LGm I'Sat.M~~

~~-tt+1 c IGmLm pG [Ta * IGmLm ''pG '"GLin^~~

365 which gives he additional coupling between the heat-transfer coefficient and the phase-transition rate. Equation (IV-104) is written for the condensation rate on liquid surfaces. The equation for vaporization is

~~K+1 _ 1 III f-rK+1 _ -f ILmGm ^Tf~ ! LGm LlSat,M " Lm h~~

~~r~K+\ rK+ ' ILmGm pG ^71-K+ ' ILmGm cpG~~

~~__ NMAT G t. GLm l Con.M -fg»M; m=i~~

~~^MAT 2 " ' ""' " "^ * , (IV-1O6) m=l k=l~~

~~vhe re This yields~~

~~NMAT-1 PG LeG + cvG ^TG = PGeG m=l~~

~~2 -K+l fT c T ^ ^IGmS ' vG 1 G k=l~~

~~^MAT-l G " cvG m=l~~

~~where 1 is the nominal case and~~

~~xm = 0 when all of P"m vaporizes. To place Eq. (1V-1O9J into a manageabl'- format, three dummy variables (a , b , and d ) are defined as~~

~~2 [H I LGffi k=l~~

~~2 bm= Z . and (IV-111) k=l~~

~~dmm - eCon,M + vG~~

The quantities am, b , and d are regarded as constants during the solution of the vapor energy equation. They use the undated saturation quantities from the first Newtor-Raphson iteration and updated vapor-side beat transfer coefficients from the second iteration. Then it is possible to show that

~~367 NMAT is the total macroscopic density of the v^por-field m=l components after evaluating Eq. (IV-95);~~

~~is the specific internal energy of the vapor mixture for eG which the equation is solved, and~~

~~e K+1 's *;^e sPecific internal energy of liquid-field material Con.M component m at the saturation temperature determined in evaluating Eq. (IV-95).~~

Only the energy-transf ~.x terms from mass transfer and the associated heat transfer are included in B"!. (IV-106). The phase transition is assumed to occur at the liquid-field and structure-field energy component interfaces, which are assumed to be at the saturation temperature. Therefore, the mess transferred f'.om the vapor field is assumed to leave the field at saturated conditions. Multiplication of Eq. (IV-95) by 6QT". summation over m, and subtraction from F.q. (IV-106) yields

~~NMAT gn -K+l „ *n n AT v rK+I / _#C+1 . hK+l . PG eG ~ PG eG " At i rGLm lcConM + h#gM e m=l~~

~~NMAT~~

~~m=l k=l~~

Equation (IV-107) now is modified as follows. (1) Special energy-transport terms are not written for the fission gas component. Except for a special case of direct structure-to-vapor heat transport described in App. E, fission gas merely adds to the vapcr density and heat capacity and modifies the vapor thermophysical properties (and consequently influences the vapor-side heat-transfer coeffie ients). (2) The case when a liquid component can vaporize totally is handled bv inserting terms that place the liquid energy directly into the vapor field. (3) We wish to solve Eq. (IV-107) for TQ+1 and eg+1. We approximate by

~~5vG! (T«^.+ 1 " Tr.) • (IV-108)~~

~~36h NMAT-1 b 5K+I r r- ra vG 1 L ^ m=l~~

~~in l m m=l~~

~~NMAT-1 v I cvG Lm I1 " xmjJ lG m-1~~

~~1"PG leG " fG " cvGT G~~

~~NM4T-1 + x~~

~~m=l~~

~~= 0 . (iV-113) This is a quadratic cf the form~~

~~AT? + BTr + C = 0 (IV-114)~~

that is solved using the quadratic formula, assuming TQ is the only unknown. The quantity A < 0 by definition, so the negative square root gives the higher temperature and, in most cases, furnishes the correct solution. The exceptions and the procedure followed if R - 4AC is less than ^ero are Jiscussed in App. E. The one majcr special case requiiing immediate discussion is wh^re we have a large vapor-side heat-transfer coefficient. In this ?-ase, tns vapor continuity and vapor energy equations are coupled strongly. Experience has indicated that the best apprcach to this problem is to evaluate At Fy^ directly

~~3:9 by using the previous expression for PQ^ - PQ£^ . This reduces Eq. (IV-113) to a linear equation in Tjj+1. The criterion for this case is that~~

~~K+1 bmv cG "xm~~

for any component m. Three special situations should be mentioned regarding the overall iteration. First, if all the liquid of a given component vaporizes, the corresponding saturation temperature returned from the model is increased such that FQJJJJ still can be calculated correctly from temperature differences. (This artificial saturation temperature appears on the SIKMF.R-II full print.) Second, convergence of the model can occur either with the variables oscillating about an apparent converged value from iteration to iteration or with the variables changing monotonically in a given algebraic direction. Convergence is 39 accelerated in the oscillatory case by Steffensen's method, where for a given variable, y:

~~y"+1 = y" - (y ' *) , . (IV-116) yK _ Kl K2~~

an< a Third, if complete convergence does not occur, Tga^ ^ is set to -a? m/&2 m * message is printed. This error exit eliminates vaporization/condensation for such a cell for the time step in question. Generally, this warning message indicates a problem; the input and/or coding should be investigated if excessive warning messages are given. In summary, the vaporization-condensation model is solved by successively performing three major computational operations. There is an iteration scheme to obtain phase-transition rates consistent with the saturation temperatures. A second set of iterations adjusts these rates for consistency with the vapor-side heat-transfer coefficients. The third computation takes these saturation temperatures and heat-transfer coefficients and updates the vapor temperature using the vapor energy equations. These operations are repeated until

370 convergence is obtained. The steps in the algorithm can be related to the equations as follows. (1) Estimate the liquid-field energy-component temperatures from the beginning-of-time-step temperatures, the advanced-tin;? temperatures, or the saturation temperatures using Eqs. (IV-90) through (IV-92). (2) Perform the inner iteration between the saturation temperature and the vapor-field material components continuity equations using Eq. (IV-101) and the equation of state. (3) If the outer iteration is converged, evaluate the converged heat fluxes and phase-transition rates and exit from the phase-transition calculation. (4) Perform the inner iteration between the corrected vapor-side heat- transfer coefficients and the phase-transition-rate definition using Eqs. (IV-104) and (IV-105) and the associated equations for structure surfaces if required. (5) Solve the vapor-field energy equation for the new vapor-field temperature using Eq. (IV-113). (6) Return to Step 2. Step (5) is performed at least once. The convergence criteria required in Step (3) are that the mass-transfer fractional change between successive outer iterations associated with HQJJJJ must be converged to within an input parameter (EVAPOR) and that the vapor-side heat-transfer coefficients in Step (4) must converge within 0.1% between successive outer iterations. The vaporization-condensation model has many special cases that arise when a small amount of a liquid-field energy component is present or when the liquid- field energy-component temperatures approach or exceed the critical temperature for the material. These special cases and their treatment are detailed in App. E. APPENDIX J MISCELLANEOUS CORRECTIONS

1 *IDENT B774 2 */ 3 */ CHANGE THE LIQUID THERMAL CONDUCTIVITY 4 */ 5 *I XCHA. 113 6 IF (TL(3) .LT. 448.) GO TO 4100 7 IF (TLC3) .GT. TCRIT(3)) GO TO 4150 8 THC0NL(3)-0. 71473-1.1781E-4*EXP(0.012693*TL(3)) 9 GO TO 4160 10 4100 THC0NL(3) - .680 11 GO TO 4160 12 4150 THC0NLC3) - .275 13 4160 CONTINUE 14 */ 15 */ PUT IN THE DROPLET RADIUS RATIO 16 */ 17 *I B7G9.1 18 *,RPRAT 19 *I SETU.174 20 *,RPRAT 21 IF (RPRAT . EQ. ZERO) RPRAT - ONE 22 *I TSHT.212 23 RPRATN = ONE 24 IF (N .EQ. 3) RPRATN - RPRAT 25 *D B811.13 26 RP(N) - AMIN1(RP1,RP2,RP3,RP4,RP5,RPMAXCIREG)*RPRATN) 27 *D B811.17 28 RP(N) = DMIN1(RP1.RP2,RP3,RP4,RP5, RPMAXC IREG) *RPRATN) 29 *D DMPS.17 30 *,OMP,ROGCUT,ALPH0JALDRG,ALCSCO,R0SFAL,RPRAT 31 *I B7G9.12 32 *,RPRAT 33 *D B7G9.15 34 29HCEVAP0R)-, 1PE12.5/31H THE WATER-FUEL DROPLET RATIO, 39X, 35 39HCRPRAT )=,1PE12.5) 36 *IDENT B8D4 37 */ 38 •/ TEMPORARY CORRECTION FOR THE BOUNDARY CONDITIONS 39 */ 40 *I XCHA. 50 41 IF CI -EQ. 2) PCIMJ) - PCIJ) 42 IF CI .EQ. IP1) PCIPJ) - PCU) 43 IF CJ .EQ. JP1) PCIJP) - PCU) 44 *IDENT B8I4 45 */ 46 */ PUT IN ANOTHER TIME STEP CONTROL ON THE VAPOR ENERGY 47 */ 48 *I IMFL.1290 49 C 50 C PUT IN ANOTHER VAPOR ENERGY TIME STEP CONTROL 373 51 C 52 *IF -DEF.DBL.l 53 PTEA = ABS(PTE) 54 *IF DEF.DBL.l 55 PTEA - DABS(PTE) 56 *I IMFL.1292 57 QGWORK(IJ) = PTEA 58 *I IMFL.1445 59 IF (ALPHG(IJ) .LT. ALPHO*(ONE-ALPHS(IJ))) GO TO 2654 60 PTEA - QGWORK(IJ) 61 IF (PTEA .LT. EP3) GO TO 2654 62 *IF -DEF.DBL.l 63 DTRIAL - (AMIN1(ALPHG(IJ),EM1)*RHS(K))/PTEA 64 *IF DEF.DBL.l 65 DTRIAL = (DMIN1(ALPHG(IJ).EM1)*RHS(K))/PTEA 66 IF (DTRIAL .GT. DTL11) GO TO 2654 67 DTL11 - DTRIAL 68 IVI(ll) = IMS 69 JVJ(ll) = JMS 70 2654 CONTINUE 71 *I IMFL.39 72 DTL11 = EP20 73 *I HYDR.470 74 IW11 = IVI(ll) 75 JW11 - JVJ(ll) 76 *I HYDR.529 77 1X11 - IVI(11) 78 JX11 - JVJ(ll) 79 *I HYDR.535 80 IVI(ll) = IW11 81 JVJ(ll) = JW11 82 *D HYDR. 112 83 DTL(ll) = DTL11 84 *D HYDR. 300 85 *IF -DEF.DBL.l 86 DTL11 = AMIN1(DTL11,(DTMPM*DTN)/DTMPG) 87 *IF DEF.DBL.l 88 DTL11 = DMIN1(DTL11,(DTMPM*DTN)/DTMPG) 89 *I HYDR. 547 90 IVI(ll) = 1X11 91 JVJ(ll) = JX11 92 *D FAM.243,244 93 IW11 - 1-1 94 JW11 = J-l 95 *I HYDR.302 96 IVI(ll) = IW11 97 JVJ(ll) - JW11 98 *D TIME. 20 99 *,DTALPH,DTL11 100 *D TIME. 35 101 *.LNITIM.IW11,JW11 102 *I SETI.135 103 DTL11 = EP20 104 *IDENT B8J4 105 */ 374 106 */ CORRECT AN ERROR IN EXFLUD 107 */ 108 *D EXFL.939 109 *IF -DEF,DBL,1 110 ATEMP = AMAXKACOMP - RLCMP,EM20) 111 *IF DEF DBL 1 112 ATEMP'= DMAXKACOMP - RLCMP,EM20) APPENDIX K

~~SUMMARY DESCRIPTION OF THE HEAD-FAILURE MODEL SUBROUTINE~~

The single degree of freedom (SDOF) head-failure model integrated into the SIMMER-II code in this study is based on a series of calculations using the nonlinear finite element (FE) computer program ADINA. An axisymmetric FE representation of the reactor vessel was developed that included the lower head, the cylindrical core barrel, the stiffened pipe loop and support region, and the reactor head. This FE model was subjected to five different, transient, uniform-interior pressure-time histories that ranged from very large impulse spikes to a near-quasi-static loading that would produce failure. In addition, a sixth time history that approximated the spatial distribution of the pulse was used. Also a triaxial stress failure criterion was implemented into ADINA based on the work cited in Ref. 40. All ADINA failure modes began in the lower head near the discontinuity radius for the lower head penetrations. Examination of the stress states in the "failed" elements indicated that a circumferential split of this lower head around this radius would be the failure mode, thus allowing the lower cap to move away from the vessel and vent the pressure. The equation of motion for the simple, spherical-head SDOF model is given by

~~mx" + Kx - P(t) , (K-l)~~

where m is the equivalent mass per unit area, K is the stiffness per unit area, and P is the pressure. The equivalent mass was established by calculating the frequencies and mode shapes from the ADINA model using first, elastic properties and second, partial plastic properties. The stiffaess for the bottom head is taken to be

~~2? • CK-2) R2~~

377 where E is either the elastic (29 x 10° psi) or the plastic strain hardening (16*2 x 104 psi) Young's modulus, T is the thickness (5.375 in.), and R is the average radius (96.2 in.). A failure "displacement" for this equation of motion is calculated based on the equivalent uniaxial failure criteria calculated from the ADINA triaxial failure model and the meridional and hoop strain expression. The equation of motion can then be integrated numerically to approximate the ADINA result.

~~378 APPENDIX L DEFINITIONS OF VARIABLES IN THE PLUGW CORRECTION SET~~

~~PLUGM Lower head mass to be moved, input in kg.~~

~~ZPLUG Distance (meters) from reference that the lower head has moved.~~

~~APLUG Acceleration (meters/s ) of the lower head.~~

~~FPLUG Force (newtons) on the lower head as a consequence of unbalanced pressures. Stored as a positive number before head failure, and as a negative number thereafter.~~

~~VPLUG Velocity of the lower head. VPLUG is a positive number in inchcs/s before head failure and a negative number in m/s after head failure.~~

~~FDZ(I) The fraction of the current axial node occupied by the lower head at the head's interface with the fluid, as a function of the radial subscript I.~~

~~ZBOT(I) The axial location (in meters) of the bottom of the interface node between the lower head and the fluid, as a function of the radial subscript I.~~

~~ZTOP(I) The axial location (in meters) of the top of the interface node between the lower head and the fluid, as a function of the radial subscript I.~~

~~ZBIAS(I) The axial location (in meters) of the lower head at radial location I relative to ZPLUG.~~

~~FHEAD Force (newtons) integrated over the upper head. Presently FHEAD is computed and output for informative and plotting purposes only.~~

~~379 XPLUG Displacement (in inches) calculated by the failure subroutine SMOD.~~

~~PAVERG Average pressure (in psi) at the end of the time step used for input into the failure subroutine SMOD.~~

~~JSO Input integer for the top boundary of the lower head.~~

~~ISO Input integer for the right-hand boundary of the lower head.~~

~~JSOT Input integer for the lower boundary of the upper head.~~

~~ISOT Input for the right-hand boundary of the upper head.~~

~~JPLUG(I) Radially dependent integer added to JSO defining the top boundary of the lower head (input).~~

~~JPLUGS(I) Temporary variable to transfer information. Main use is in EXFLD1, determining whether the lower head has crossed an axial mesh cell boundary at a given radial location. The choices are~~

~~JPLUGS(I) » 0 No boundary crossed JPLUGS(I) - 1 Upper boundary crossed JPLUGS(I) « 2 Lower boundary crossed~~

~~JPLUGT(I) Radially dependent integer added to JSOT defining the bottom boundary of the upper head (input).~~

~~JCONT(J) Axial index for the radially continuative inflow/outflow boundary condition. If~~

~~JCONT(J) - 0 Radial boundary is rigid JCONT(J) - 1 Radial boundary has continuative inflow/outflow~~

~~380 APPENDIX M LOWER HEAD DYNAMIC ANALYSIS~~

~~I. INTRODUCTION AND SUMMARY OF RESULTS OF ADINA STUDY~~

An ADINA model was developed and was used to run parameter studies and to develop a reasonable SDOF model that could be used (with some judgment and care) in the SIMMER-II code. Thus, a coupled analysis could be made of the pressure relief caused by lower head failure and consequent venting. The pressure transients were selected to cover a spectrum of steam-explosion intensities and characteristics. Six cases were calculated. In cases 1-5 the pressure was applied uniformly as a time history over the entire interior surface of the axisymmetric finite element model of the vessel. Pressure histories for these cases are given in Table M-I.

~~TABLE M-I PRESSURE TRANSIENTS USED IN LOWER HEAD DYNAMIC ANALYSIS Time Pressures (MPa) (ms) Case 1 Case 2 Case 3 Case 4 Case~~

0 0.1 0.1 0.1 0.1 0.1 1 6 6 100 0.1 0.1 2 1000 1000 200 100 50 3 150 150 200 100 50 4 90 90 200 100 50 5 150 100 200 100 50 6 170 100 200 100 50 7 160 100 200 100 50 8 150 100 200 100 50 9 150 100 200 100 50 10 150 100 200 100 50 12 120 100 200 100 50 14 110 100 200 100 50 16 100 80 200 100 50 18 90 60 200 100 50 20 80 50 200 100 50 25 60 40 100 50 25 30 46 35 100 50 25 35 42 35 100 50 25 40 40 35 100 50 25 50 35 30 100 50 25 60 35 25 ICO 50 25

381 Case 6 used the results from the SIMMER-11 scoping calculation that was exploded after 0.7 s of mixing (see Sec. VLB.2). In case 6, the space-time effects were approximated by applying four different pressure-time histories over four of the meridional sections of the vessel as shown schematically on the geometry in Fig. M-l. Table M-II summarizes the major results of the study with ADINA, and of a simple specific-impulse calculation. The first column gives the time at which enough specific impulse has been delivered to a spherical segment to exceed a fracture strain (for this hand calculation, set at 0.08%). The reason for the double entry for cases 1 and 2 is that the first ms of loading for these cases was deemed negligibly small in contributing to the specific impulse, and the hand computation took the ramp pulse to begin at 1 ms. The second column gives the time at which ADINA computes that the failure criterion has been exceeded. In all cases, examination of the stress state at the initiation point of failure indicates & split of the vessel along the circumferential direction (that is, across a meridian) with a portion of the lower head coming off intact and being accelerated downward. The third column

~~j-, LOWER HEHD DYNPMIC FfllLURE INVEST. GRID T- 0.001800 STEP- 6 DSF- 5.000~~

~~-100.0 0.0 100.0 200.0 300.0 400.0 500.0 600.0 wi-it nmt i com R~~

Fig. M-l. Case 6- Regimes used to approximate the spatial equation of the Table M-I SIMMER transient with four separate pressure-time histories. 382 TABLE M-II SUMMARY OF LOWER HEAD FAILURE STUDY FOR A GENERIC PWR VESSEL ADINA Approx. Time (ms) Time at velocity of Impulsei = I min which failure lower head to fail initiates (ms) (in./s)

Case 1 0.5 (1.5) 1.8 11 000 Case 2 0.5 (1.5) 1.8 11 000 Case 3 1.43 2.1 5 250 Case 4 2.03 3.0 3 460 Ca3e 5 3.06 9.6 2 250 Case 6 __ 2.1 3 060 gives the approximate initial velocity that this lower head cap would have as a rigid body. The following sections discuss the analyses and assumptions that were used in arriving at the Table M-II results and a simple SDOF model that can approximate the result.

~~II. THE FAILURE MODEL AND CRITERIA~~

The failure model is based on the recommendations of Ju and Butler40 and this approach has significant experimental verification. From the state of stress at each integration point, a triaxiality factor (TF) is computed. The TF is defined as a dilational stress (tfjj) (sum on repeated indices) divided by the equivalent stress

~~(M-l)~~

~~with Sjj being the deviatoric stress and a-x the principal stress. For a TF greater than or equal to unity, a damage factor f(TF) is computed as~~

~~383 sinh [V ^ Cl - n)] f (TF) = T—2 . (M-2) sinh [y| (1 - n) TF]~~

In this study, we took n = 0.2. The triaxial fracture strain is then computed as if » 0.7 ifu f(TF), where ifu is the uniaxial true failure strain, in this case assumed to be 20%. When the maximum principal strain at this point exceeds the triaxial fracture strain, then a ductile material failure is declared. In the case for TF less than unity, failure is declared if the effective stress exceeds the uniaxial ultimate stress. All of these criteria were programmed into the ADINA code and evaluated at each time step and each integration point. Generally the triaxiality factors at failure ranged from 1.8 to 2 and the fracture strains were about 5%. Study of Fig. IS in Ref. 40 reveals that the criteria were programmed into ADINA correctly.

~~III. BASIS FOR SIMPLE MODEL TO ESTIMATE REQUIRED FAILURE IMPULSE (COLUMN 1 OF TABLE M-II).~~

~~i - mv , (M-3)~~

~~where m is the mass per unit area and v is the velocity of the spherical segment. Therefore,~~

384 i^ 2 ^ Kinetic Energy 2m A Unit Area where M is the total moving mass. If we assume that enough impulse is delivered to just fail the sphere at zero velocity, then ail kinetic energy is absorbed by the strain energy. Thus

~~where Uf is the strain energy per unit volume at failure and V is the volume. For an expanding sphere a.. - OQQ (« a) and ef 6f ef~~

which is two times the area under the uniaxial stress-strain curve to the biaxial cut-cff failure strain. For biaxial conditions and AS33B Steel, s 3 ef 0.08, and Uf is easily computed to be 10 574 (in.-lb/in. ). Thus for a 1 in. segment of an expanding sphere, the failure impulse is

~~(2 1/2 ^ail " *UfT) . (M-6)~~

can e where T is the thickness of the sphere, and a value for ifan ^ computed as 14.97 (lb-s/in. ). But the impulse per unit area is the area under the pressure vs time diagram. Thus, by solving for the time from each p-t transient at which the impulse per area given by Eq. (M-6) has been delivered, the results in column 1 of Table M-II were determined.

~~IV. A SDOF MODEL THAT APPROXIMATES FAILURE TIMES~~

A study of the ADINA results reveals that for these severe transients, failure always initiated in the hemispherical portion of the lower head in the region marked on Fig. M-2. Study of the actual vessel indicates that there is a

~~385 §-,LOWER HERO DTNRMIC TfllLURE INVEST. o •»- ni GRID T- 0.001800 STEP- 6 o DSr- 5.000 a-~~

§•

~~General reoion for failure Initiation in ADINA studies.~~

~~-100.0 0.0 100.0 200.0 300.0 400.0 500.0 600.0 R~~

Fig. M-2. Failure initiation region from ADINA parameter studies. discontinuity in the lower head geometry because of the lower head's penetrations at about R - 64 in. When the dynamic stress state is perturbed by this geometry change, it is likely, then, that the actual vessel failure will be around this discontinuity. This failure mode was suggested for the SINMER-II code, that is, a splitting of the lower head along this coordinate with venting of the vessel occuring as this cap i, *«d downward. However, a model is needed for the SIM4ER-II code that gives a reasonable prediction of failure time and initial velocity of the cap. The equation of motion for an expanding sphere under uniform internal pressure P(t) is

~~2FT (M-7) 2 2±R UD - P(t) 2 K~~

386 where P is the mass density, T is the thickness, R is the average radius, Up is the radial displacement, E is the modulus of elasticity, and t is time, and a dot indicates differentiation with respect to time. It is on this principle that we based the SDOF model suggested for SXMMER-II. If we let m • PT be the mass per unit area and K = (2ET/R ) be the stiffness per unit area, then the motion of a spherical segment can be represented, with Ug = x, by

~~mk" + Kx = P(t) . (M-8)~~

For an elastic/plastic model, E i3 either the elastic modulus (20 x 10° psi) or plastic strain hardening modulus (16.2 x 10 psi). Because the model is to represent the motion of the lower head as it relates to (and is affected by) the entire vessel, (that is, it should approximate the ADINA result), Eq. (M-8) was treated as follows. The stiffness per unit area, K, was found as above. The mass per unit area, m, was taken to be the modal mass as derived from

~~K -~~

where the frequency used corresponds to the frequency for the mode shape of the "expanding vessel" as in Fig. M-3. This frequency turns out to correspond to the second mode. Thus, the ADINA model was used with the same boundary conditions as used in the transient runs to determine the first four mode shapes and frequencies. The second mode, which corresponds to the "expanding" vessel mode, gave u = 1521/s, which gives an equivalent mass for Eq. (M-8) of

~~m - 0.01456 -^^- in~~

~~387 J-.LOHER HEflO DYNAMIC FRILURE INVEST. 8 GHID T- 0.001800 STEP- 6 DSF- 5.000 a- .~~

~~o a Initial „ a- vessel shape~~

~~8 Displaced •*. shape~~

~~-100.0 0.0 100.0 200.0 330.0 400.0 500.0 600.0 R~~

~~Fig. M-3. "Expanding vessel" mode.~~

Note that this mass is nearly twice the actual mass per unit area that a 5.375-in. -thick head would have. An elastic limit (yield) displacement based on 2 0.2% strain and o xt^ • 60 400 lb/in. gives xyieid - 0.2 in. for Eq. (M-8). A failure displacement value of S.O in. was set based on a triaxial failure strain of 5.3%, Eq. (M-8) was integrated numerically (the Newmark beta method), and the failure times were compared to ADINA predictions of Table M-II. These comparisons are shown in column 2 of Table M-III under "elastic" mass. The failure time comparisons on this case are not considered good, but initial velocities are worse. In an effort to improve this model, further examination of the ADINA results revealed the SDOF model to be quite accurate up to first yielding of the vessel. Furthermore, large portions of the vessel's lower and top heads yield first with the thicker cylindrical portions remaining elastic. Based on this observation, a new eigenvalue problem was run, for which the plastic material modulus was used in portions of the mesh approximately corresponding to the plastic region that appears during the time from first yield until failure is predicted in the ADINA model. From this run, a new frequency corresponding to the expansion mode was identified and an third mass

388 was derived for the SDOF model based on this value (137 rad/s). The SDOF model was rerun for all cases using the value of the mass parameter in the plastic regime (that is, displacement beyond 0.2 in.). The results are shown in Table M-III under the third column. Significant improvements in time to failure and initial velocities were noted. As a sidenote, one might be tempted to use the actual mass/unit area of the spherical head in the SDOF model. This comparsion is also shown in Table M-III, and though failure times are probably not too bad for the larger impulse cases, initial velocity predictions are very poor. On the whole, SDOF Model 2 with a set of elastic and plastic stiffness and mass parameters was judged to give the best approximation of the ADINA results. With respect to case 5, we noted that it is approximately the static, vessel-design pressure case, applied dynamically. Since the dynamic load factor increases the response by nearly two for this shape of pulse, the fact that the vessel fails is not suprising. But it also should be expected that different portions of the vessel will go plastic at significantly different times (that is, spatial response-time effects are important), resulting in a deviation from the SDOF model. This conclusion is reflected in Table M-III. In fact, the primary reason that the model doesn't give as good a result for case 5 is that significant spatial effects do begin to appear. Case 6 also shows this trend. The results from SIMMER-11 using this model must then be interpreted with some caution. If significant spatial variation in pressure magnitudes for "long" periods of time are expected, then the predictions will not be very good. For steam-explosion work, pressure magnitude differences disappear rapidly because the pulse propagation speed is high. It should be noted that the failure times are slightly conservative, with the exception of case 5. Another restriction on the SDOF model, as suggested for SXMMER-II, is that there can be no unloading of the vessel (that is, incremental displacements must be always positive) since the vessel would unload elastically. Because the steam explosions under study have invariably resulted in vessel failure, this restriction is not deemed to be significant, but SIMMER-11 results in any parameter study should be examined with this restriction in mind.

~~389 GO o~~

~~TABLE M-III~~

~~COMPARISON OF SDOF MODEL AND ADINA RESULTS~~

~~ADINA RESULT ELASTIC MODAL MASS PLASTIC MODAL MASS ACTUAL MASS~~

~~T ( : T (ms) Velocity T (ms) Velocity T (ms) Velocity f..i - (in. ^ail (in./s) Tfail (in./s) Tfail Cin./s)~~

Case 1 1.8 11 000 2.5 8 841 2.3 10 920 1.9 16 360 Case 2 1.8 11 000 2.5 8 841 2.3 10 920 1.9 16 360 Case 3 2.1 5 250 3.3 4 364 3.0 5 275 2.1 7 583 Case 4 3.0 3 460 4.3 3.048 3.9 3 721 2.8 5 835 Case 5 9.6 2 250 6.1 2 140 5.4 2 582 3.9 4 107 Case 6 2.1 3 060 3.4 5 317 3.0 4 503 2.0 6 664 V. CONCLUSION

The ADINA studies are believed to give a fairly accurate picture of both failure times and mode. The automation of the failure model from Ref. 40 is largely responsible for allowing a confident prediction of the strain and mode of vessel failure. Strain rates, by the way, were estimated to be about 44/s (case 1), which means the material properties used are still accurate (that is, strain-rate effects are negligible). The SDOF model suggested for use with SIKMER-II can be used to study the venting effects parametrically with at least as much confidence in the result as there is in the model of the physics of the steam explosion. APPENDIX N CONSIDERATIONS REGARDING THE RESULTS FROM THE SDOF LOVER HEAD FAILURE MODEL

~~The equation for the SDOF model, Eq. (M-8), is~~

~~x + «2 x m P(t)/m . (N-l)~~

~~Three cases are considered. First, if P(t) is assumed to be a constant, and x acd x are zero at t - 0, the solution to this equation is~~

~~£ (l - sin (ut + ir/2)) , (N-2)~~

~~where~~

~~K - m u2.~~

The pressure that can be applied is related to the time the vessel goes plastic, in other words, when x = 0.2 in. With the constants given, further displacement to failure in the plastic regime is almost assured. Here the minimum P to cause yielding is at t = ir/ca, or

~~Pmin = Kx/2 ' CN-3)~~

~~Second, if the loading builds up gradually so that x is small, the load to produce failure is~~

~~Pfail - K* • (N-4)~~

~~393 Third, if loading has the form of P(t) = P sin(ut), and the same initial conditions are applied as with the constant load situation, the solution is~~

~~r? (sin(ut) - ot cos(ut)) . (N-5)~~

~~The maximum forcing function is obtained when t = n72u. Therefore, the maximum pressure that could be exerted before yielding occurs is~~

~~Pmax - 2 Kx . (N-6)~~

~~The value for K obtained from App. M is 33 694 psi/in. When x = 0.2 in., we~~

~~obtain 3 369.4 psi for Pmin by Eq. (N-3), 6 738.8 psi for Pfail by Eq. (N-4),~~

and 13 47?. pai for Pmax by Eq. (N-6). These solutions indicate the range over wiiuii the applied pressure differential might be varied and produce failure. The static load to cause lower head failure in the ZIP study was estimated at 44 MPa or 6 400 psi.

~~394 APPENDIX 0 A LIMITED REVIEW OF SNL STEAM-EXPLOSION EXPERIMENTS~~

This appendix reflects experiments done at SNL and reported at the time of this review, September 1984. Two types of experiments had been conducted at SNL by that date. These were small-scale experiments involving O.OS to IS g of melt looking at the behavior of single drops and "intermediate-scale" experiments involving 0.6 to 20 kg. The small-scale experiments examined such phenomena as steam-explosion triggerability, droplet fragmentation, conversion ratio, debris size distributions and characteristics, and hydrogen generation rates. Some qualitative correlations are available; see Fig. 0-1.* Also, the careful scientific nature of these experiments has led to some detailed models of the fragmentation process. Bankoff provides a review of these modeling efforts, and presents an additional multiple-bubble fragmentation model based on the experiments in Ref. 43. Inclusion of the level of detail represented by these experiments in SIMMER-II is beyond the scope of our program. However, the experimental information obtained doe3 not preclude a significant steam explosion in a reactor meltdown environment. Intermediate-scale experiments are divided into two types; partially instrumented tests and fully instrumented tests (FITS). The partially instrumented tests were performed by Buxton and Benedick. The first 48 tests involved 1-27 kg of iron-alumina thermite poured into an open tank of water. The primary conclusion drawn was that an energetic explosion can occur under these circumstances. Other interesting results were that saturated water appeared to have no effect and that the highest reported efficiency was with a cover plate providing a confinement effect. The next 11 tests involved 13.6 or 19.4 kg of iron-alumina thermite or corium-A+R thermite, which contained 53

w/o U02, 17 w/o ZrO2 and 30 w/o stainless steel. The melt was poured into a larger tank of water buried in the ground. The corium was found to produce either no explosions or mild explosions compared with the iron-alumina mixture. This was speculated to be the consequence of either noncondensable gas generation, or, more satisfactorily, the consequence of oxide solidification.

~~•Figure provided by M. Berman, Sandia National Laboratories (August 16, 1984).~~

~~395 5.0~~

~~4.0 —~~

~~0.2 0.4 0.6 0.8 1.0 1.2 AMBIENT PRESSURE (MPa)~~

~~Fig. 0-1. Effect of pressure on explosion yield.~~

Numerous FITS tests have been run. An almost complete list is reproduced in Table 0-1.* The first smaller tests, involving 1-5 kg of iron-alumina thermite in the melt drop (MD) and FITSA series, are the most interesting from the viewpoint of analysis. The MD tests were performed in the EXO-FITS apparatus shown in Fig. 35 of the main body of this report. The FITSA series was performed in a containment chamber shown in Fig. 0-2. Both series were run and reported by Mitchell. Although some random behavior is present, a common

~~•Table provided by M. Berman, Sandia National Laboratories (August 16, 1984).~~

~~396 > T1 -rl Tl t-l s w pi >~~

~~o MFLT MASS FUEL/CDOLANT MASS RATIO~~

~~•* • O • o AMBIENT PRESSURE FUEL COMPOSITION -4 m o •• •< — so MELT ENTRY CHAMBER GEOMETRY en 6 •< ! WATER TEMPERATURE 1 tn si en 3D o o tn m OTHER (LID. HOLD TIME. ETC.) CONTACT MODE~~

~~FUEL TEMPERATURE rn Iff en o o COARSE MIXING o GAS PHASE PRESSURE o o WATER PHASE PRESSURE HYDROGEN GENERATION STEAM GENERATION o CONVERSION RATIO CO DEBRIS CHARACTERISTICS Melt Transport~~

~~Graphite Crucible r & Charge " Igniter & Melt Observ.~~

~~Closure Valve & Actuator " Melt Chamber 32 cm Chamber Static_ Hi-Cam 6000 fpa Pressure & Temp.~~

~~Removeable Head~~

~~Access Walk~~

~~154 cm Chamber Chamber Pressurization Pressure Port~~

~~Main Chamber 2.4 MPa W.P. 5.6 M3 Volume Hi-Cam 6000 fps Instrument 276 cm Feed-Thru Ports Hi-Cam 6000 fps~~

~~Chamber Pressure~~

~~Fig. 0-2. FITS containment chamber.~~

398 result was triggering a steam explosion at the base of the water container followed by a propagating wave, similar to a detonation wave, at a velocity of 250 to 560 m/s through the melt-coolant mixture. This is similar to the idea proposed by Buxton that spontaneous initiation of an explosion probably involves wall or similar other solid surface contact by the melt. Not much information was found on the FITSG-series tests. Evidently "G" stands for steam generation. FITS-1G involved 21.4 kg and FITS-2G 13.5 kg of Aft iron-alumina thermite poured into saturated water. The water masses were relatively small, 44 kg in FITS-1G and 88 kg in FITS-2G. No explosions occurred. It was reported that the saturated water pool and fog created by fuel-water contact made visual observation impossible. The fact that there were no explosions with saturated water possibly contradicts the results of the earlier Buxton tests; however, these apparently were experiments performed at special request of NRC in such a manner that nonexplosive interactions could be studied. The MDC-Series and were similar to the MD-Series, but involved corium rat'er than iron-alumina thermite. Four of the experiments had explosions, each of them triggered at the base of the tank. The conversion ratio in the tests that exploded was reported to be similar to that of iron-alumina. An idea was advanced that a minimum melt volume (0.5 liter or 1 pint) delivered in a coherent fashion is required to suppress a noncondensable gas effect and form an explosive mixture. No further discussion on the previous difficulties of exploding corium in the Buxton tests was given. Although these earlier tests did use more than 1.5 liters of corium, at that time the melts were poured in thin streams into the water. One goal of the early FITS tests was to deliver the melt in a fairly coherent lump. The FITSB tests confuse the picture obtained from earlier results, although this may be because of the phenomenological complexity, not because of inherent test deficiencies. These tests used a nominal 18.7 kg of iron-alumina thermite with varying amounts of water. Initiation sites (explosion triggers) were observed at random locations in the melt-water mixture, as opposed to the more common base triggering phenomena in earlier tests. Some of the experiments indicated a wave propagating away from the trigger sites. However, the wave front could not be distinguished clearly in these experiments, and, in some cases, could only be observed as a change in light intensity transmitted to the cameras. The experiments with the largest amounts of water gave double explo-

399 si cms, with an increase in explosive yield. Two experiments were done in saturated water, resulting in no explosions. Another level of confusion was added by the FITSC4^ series. The primary purpose of these experiments was to compare the corium A+R and iron-alumina thermites in terms of explosiveness and resulting debris formation. A nitrogen atmosphere was used so that hydrogen generation could be determined. The two corium experiments produced weaker explosions than in the MDC tests. The speculation was that "...with corium in air, the atmospheric oxygen reacts with the absorbed hydrogen (~200 ppm) in the starting constituents; this liberates heat and enhances the thermite reaction so that the temperature of the final in melt stays above the liquidus point until after the melt is released." The CM-series consist of more recent tests (FY83-FY84).50 The purpose of these tests was to distinguish between existing models for coarse mixing. Twelve tests were run involving 4-18.5 kg of iron-alumina. Both saturated and subcooled water were used. The unexpected results were water-surface events of sufficient violence to expel melt from the water, as well as prevent some melt from entering the water, which occurred in every test. Why these events occurred in these tests and not in earlier ones is cot clear. One speculative hypothesis is that some change in the melt temperature, water temperature, or melt composition (metal vs oxide), preparation, or delivery may have been responsible. The OM tests involved an oxide melt using the reaction,

3Fe + KC104-> FejC^ + KC1, with the KC1 product driven off as a vapor. An initial test using this reaction, MDF-1, performed earlier, had delivered -800 g of material to the water with no explosion. In the new tests, about 20 kg of thermite was loaded with about one-half this amount delivered to the water. Violent and unambiguous explosions occurred in all four OM tests run, although the experimental details in each test were unique. It was concluded that rapid oxidation of the metallic component in the CM series was strongly influencing the triggering and/or propagation of the explosive and nonexplosive interactions that were observed. Some final confusion is provided by the RC and ACM tests. The two rigid confinement (RC) tests involved a (61-cm-o.d. , schedule-60 steel pipe 55.9 cm i.d.) instead of a Lucite box for the water container. The melt was

~~20 kg of Fe3O4. Test RC-1 did not explode. Test RC-2 produced an explosion qualitatively more energetic than in the previous CM/OM tests. This was~~

400 unexpected. The suggestion was made that confinement was allowing more of the melt to participate in heat transfer. The two alternate contact mode (ACM) tests involved pouring water on top of iron-alumina. In the first test a violent explosion occurred. In the second test, no explosion occurred. Because water should exist in film boiling under these conditions, the violent explosion was not anticipated. No speculative explanation is given by the experimenters. The actual final tests run (by the summer of 1984) have been the FITSD* series, which are to investigate characteristics of an FCI that potentially could lead to indirect contahjnent failure, such as steam generation, hydrogen generation, and direct containment heating. How these questions are to be addressed by pouring 20 kg of iron-alumina thermite into water is unclear. In any case, of the four tests run by August 1984, the only results possibly meaningful to the Los Alamos molten-core/coolant interaction program were that no spontaneous explosions at high ambient pressure or low water subcooling were observed. Apparently, all the more recent experiments involved inexpensive, rapid- turn-around, scoping tests. Their objectives were to reduce a very large number of independent variables to a much smaller and more managable set. Unfortunately, the tests produced new effects, and, in some cases, unusual results that were not readily explainable. Because of this situation, test MD-19 cannot be considered a typical or a conservative test. However, we must start somewhere, and test MD-19 resulted in a well-characterized explosion. Its selection in Chap. IV for correlation with SIMMER-II seems justified in order to obtain some indication on how to model this phenomenon. Any explanation of all the test data is probably not possible without considering many experimental details that are not germane to the main issues of steam explosions in reactors. Some test characteristics which are more universal may be of additional use in the case of reactor meltdown. These include 1. some tendency for explosions to trigger on walls, other solid, or liquid surfaces, 2. reduction of yield when solids may be present in the melt, and 3. a tendency for energetic steam explosions not to develop in saturated water.

~~•Information provided by M. Berman, Sandia National Laboratories (August 16, 1984).~~

~~401 We hope that if future tests are performed, they can be done so as to raise fewer questions and provide more answers.~~

~~402 APPENDIX P SIMJER-II INPUT FOR THE ANALYSIS OF THE MD-19 EXPERIMENT USING A UNIFORM MIXING ZONE~~

1 0 -105Q07WRB ANALYSIS OF EXPERIMENT MD-19 TRY 1 2 20000 32 11 100 0 3 SIMMER-2 STUDY OF THE SANDIA VAPOR EXPLOSION TESTS 4 ANALYSIS OF EXPERIMENT MD-19 TRY 1 5 0.0045 1.0 6 6 13 7 FLUID DYNAMICS INTEGER INPUT 8 0 -2 0 -2 9 10 30 1 1 1 0 1 0 1 1 1 1 11 1 1 2 1 3 1 4 1 5 1 6 1 12 1 2 1 3 1 4 1 5 1 6 1 7 13 1 8 1 9 1 10 1 II 1 12 1 13 14 4 4 5 4 2 5 3 5 4 5 6 5 15 2 9 3 9 4 9 5 9 6 9 5 5 16 4 10 500 50 20 5 0 -1 2 6 6 1 17 PROBLEM DIMENSIONS AND OPERATIONAL CONTROLS 18 0.0525 4 0.06426 6 19 0.05774 4 0.05878 13 20 0.5 0.0 -9.8 1.0E-4 0.25 21 .0001 l.OE-6 1.0E-4 l.OE-3 22 1.0E-12 l.OE-12 l.OE-12 1.0E-5 0.1 23 0.05 0.90 0.3 100. 0.1 100. 24 EDIT CONTROLS AND POSTPROCESSOR CONTROLS 25 1.5 1.0 3.0 0.0 2.5 26 0.0002 0.0005 0.0010 27 0.0 28 0.0010 0.0035 0.0045 29 0.0 30 .00001 .00005 0.0001 31 0.0 32 0.0005 0.001 0.0045 33 0.0 34 0.010 0.02 35 0.0 36 4.0 8.0 37 0.0 38 0.0 39 0.0 40 0.0 41 0.0 42 0.0 43 0.0 44 0.0 45 46 VIEW FACTORS 47 48 TIME STEP CONTROLS 49 0.0 l.OE-6 l.OE-9 0.249 50 .000100 0.5 10.0 1.0 1.0 1.0 1.0 0.96 0.02 0.02 0.0 0.0

403 51 STRUCTURE AND FAILURE PARAMETERS 52 0.0 53 0.0 54 1.0 1.0 1.0 1.0 55 l.E+10 l.E+10 l.E+10 l.E+10 l.E+10 l.E+10 56 1.E+10 l.E+10 l.E+10 57 58 FUEL PROPERTIES AND EQUATION OF STATE 59 1000. 2090.00 273.16 3.334E+5 l.OE-12 60 1001.78 4217.1 0.0727 l.OE-12 1.0E-4 61 3.17771E+10 4.7O579E+03 0.0 3.22689E+06 647.286 0.390597 62 1402. 1.329 3.737 2.93390E+6 18. 32. 63 316,957 95.770 316.957 64 65 66 67 2212.339 0.3539176 400.00 68 IRON PROPERTIES AND EQUATION OF STATE 69 7365.0 639.0 1700.0 2.60000E+05 25.0 70 6100.0 750.0 1.6 30.0 5.36000E-03 71 1.338O0E+11 4.33700E+04 0.0 8.17OOOE+O6 10000.0 0.360 72 492.0 1.26 1.64 0.000000000 56.0 7700. 73 74 75 76 77 WATER PROPERTIES AND EQUATION OF STATE 78 1000. 2090.00 273.16 3.334E+5 .68 79 .68 1.0E-4 80 1001.78 4217.1 0.0727 3.17771E+10 4.70579E+03 0.0 3.089805E+6 647.296 0.3660361 81 3. 737 2.93390E+6 18. 32. 82 2181.6 1.2115 316.957 95.77 316.957 200. 83 5. E+04 50. 84 1346. 0.3302 .5 85 838.607921 1.34831230-. 0027491404 86 2.85605286 . 381221451-. 0748002905 87 IRON-ALUMINUM THERMITE PROPERTIES AND EQUATION OF STATE 88 4000. 1060. 200.0 2.76000E+05 22. 89 3800.0 1060. 1.00 22. 4.300O0E-03 90 1.44000E+11 5.17080E+04 0.0 2.62000E+06 8400.0 0.597 91 492.0 1.26 4.4 0.000000000 75.0 6468. 92 93 94 HYDROGEN GAS PROPERTIES AND THE EQUATION OF STATE 95 0.0 0.0 0.0 1.0E+4 0.0 96 1. 97 1.0E+12 4.OE+03 0.0 0.0 1.0 0.3 98 10164. 1.405 2.915 0.0 2.016 38. 99 100 101 COMPONENT PROPERTIES 102 9890.0 9890.0 9890.0 9890.0 7365.0 7365.0 103 4000.0 0.0 104 1000.0 1000.0 6100.0 1000.0 3800.0 9890.0 105 9890.0 7365.0 0.0 2000. 2000. 2000. 1498. 2000. 2000. 404 106 2.OO000E+O3 2,,OOOOOE+03 2.OOOOOE+03 107 HEAT TRANSFER CORRELATION DATA 108 1.0 1.0 1.0 1.0 1.0 1. 109 1.700 1.700 1.700 1.70 1.700 1.700 110 1.0 1.0 1.0 111 0.023 0.8 0.4 0.0 112 0.023 0.8 0.4 5.0 113 0.023 0.8 0.4 0.0 114 0.023 0.8 0.4 0.0 115 0.023 0.8 0.4 0.0 116 0.6 0.5 0.33 0.0 117 DRAG CORRELATION DATA 118 1.0 12.0 2.0E-4 9.2E-7 1.0 119 4.7 1.0 0.5 .001 1.0 l.E+20 120 0.046 -0.2 0.001 0.046 -0.2 0.001 121 PRESSURE BOUNDARY CONDITIONS 122 0.83E+5 0.83E+5 123 0.0 5.0 124 0.83E+5 0.83E+5 125 0.0 5.0 126 PARAMETER REGION 1 PREMIX REGION 127 7. 0.0 l.OE+5 0.0 0.0 0.1 128 0.0 34. 34. 0.0 0.125 0.75 129 0.3 0.3 0.2 1000. 1.0E+4 1000. 130 1950. 0.64 l.OE+5 2. 3E-5 l.E-17 .00015 131 1.5E-4 132 PARAMETER REGI0N2 CHIMNEY 133 7. 0.0 l.OE+5 0.0 0.0 0.1 134 0.0 34. 34. 0.0 0.125 0.75 135 0.3 0.3 0.2 1000. 1.0E+4 1000. 136 1950. 0.64 l.OE+5 2.3E-5 l.E-17 .00015 137 1.5E-4 138 PARAMETER REGION 3 WATER 139 7. 0.0 l.OE+5 0.0 0.0 0.1 140 0.0 34. 34. 0.0 0.125 0.75 141 0.3 0.3 0.2 1000. l.OE+4 1000. 142 1950. 0.64 l.OE+5 2. 3E-5 l.E-17 .00015 143 1.5E-4 144 LOWER BOUNDARY INITIAL VELOCITIES 145 0.0 0.0 0.0 0.0 0.0 0.0 146 0.0 0.0 0.0 0.0 0.0 0.0 147 MESH SET 1 COMPACTED CORE 148 1 4 1 4 1 1 0 0 1 149 150 151 152 0.0 0.0 0.0 478.5 159.0 153 154 0.0 0.0 299. 3000. 155 0.0 0.0 0.0 0.4914125 156 366.0551 157 0.0 0.0 0.0 0.0 .0010 .001 158 MESH SET 2 CHIMNEY 159 5 13 1 4 1 1 0 0 2 160 405 161 162 163 0.0 0.0 0.0 498.4 164 165 0.0 0.0 299.0 166 0.0 0.0 0.0 0.4914125 167 366.0551 168 0.0 0.0 0.0 00..0 0.001 0.001 169 MESH SET 3 WATER 170 1 13 5 6 1 0 0 3 171 0.0 0.0 0.0 0.0 0.0 0.0 172 173 0.0 0.0 0.0 0.0 0.0 0.0 174 0.0 0.0 0.0 999. 175 176 0.0 0.0 299.0 177 0.0 0.0 0.0 0.4914125 178 366.0551 179 0.0 0.0 0.0 0.0 0.001 0.001

406 APPENDIX Q COMPARISON OF THREE-FIELD AND SIMMER-II CALCULATIONS OF COARSE PREMIXING A quantitative limit to the amount of melt that can coherently participate in a steam explosion is one of the more important areas of disagreement among contemporary investigators. Where quantitative arguments have been developed to give limits to premixing, they have utilized a number of simplifying assumptions, notably those of a steady state and a one-dimensional flow pattern. This appendix discusses some numerical calculations of the coarse premixing process in the context of experiment MD-19. Although these numerical simulations are both transient and two-dimensional, no claim is made that the correct forms of the energy and momentum coupling terms are present in the conservation equations. To some extent these mechanistic formulas are still speculative, for example, for the rate of fragmentation of the melt. We believe that considerable progress could be made in modeling these relationships; however, this type of investigation is beyond the scope of the current program. What is offered here are some calculations which may help to suggest bounds when extrapolated to a larger scale. The first calculation was an exploratory, three-field calculation with separate velocities for vapor, water, and thermite, and was first reported at Lyon, France. The initial volume fractions are shown in Fig. Q-l. The melt is elongated and smeared initially. The elongation appears to be real from the films. To what extent the density is reduced by remaining gas is not known. The degree of smearing adopted here was intended to obtain a more representative calculation once the melt contacted the water by avoiding excessive heat transfer. The initial downward velocity of the melt was 5.9 m/s. Figures Q-2 through Q-6 show the downward progression of the thermite. The motion is quite two-dimensional, with water tending to move into and outward around the thermite at the leading edge and radially inward into the steam chimney at the top. This calculation used a radiation heat-transfer formalism with a reduced emissivity to achieve the desired rate of penetration. The thermite was prefragmented to a 15-mm diameter. The second calculation used the standard SIMMER-II code without any model changes. The SIMMER-II input is given in Table Q-I. Initial conditions are the

~~o Fig. Q-2. Three-field calculation following impact with water. WATER~~

~~VAPOR~~

~~100 S~~

~~THERMITE~~

~~Fig. Q-3. Three-field calculation showing initisl total removal of water before the advancing melt. WATER~~

~~VAPOR~~

~~150 S~~

~~THERMITE~~

~~Fig. Q-4. Three-field calculation after the melt density is reduced sufficiently to stop total water vaporization (removal) at the melt front. WATER~~

~~VAPOR~~

~~200 S~~

~~THERMITE~~

~~Fig. Q-5. Three-field calculation when the container bottom is being approached. WATER~~

~~VAPOR~~

~~224 S~~

~~THERMITE~~

~~Fig. Q-6. Three-field calculation at the time of a presumed steam explosion. co TABLE Q-I~~

~~SIMMER-11 INPUT MD-19 PREMIXING CALCULATION~~

1 0 -105Q071 ANALYSIS» OF 1PRE-M1X]ING JLN MD-19 2 2 0 0 0 0 32 1 1 100 0 3 SIMMER-2 STUDY OF PREMIXING IN EXPERIMENT MD-19 4 ANALYSIS OF EXPERIMENT MD-19 5 <3.225 1.0 6 12 28 7 FLUID DYNAMICS INTEGER INPUT 8 7 28 0 -2 5 0 3 0 0 0 0 0 9 10 78 1 1 1 0 1 0 1 1 1 1 11 1 1 2 1 3 1 4 1 5 1 6 1 12 7 1 8 1 9 1 10 1 11 1 12 1 13 1 2 1 3 1 4 1 5 1 6 1 7 14 1 8 1 9 1 10 1 11 1 12 1 13 15 1 14 1 15 1 16 1 17 1 18 1 19 16 1 20 1 21 1 22 1 23 1 24 1 25 17 1 26 1 27 1 28 2 2 3 2 2 3 18 2 17 3 17 2 18 3 18 19 3 19 19 2 20 3 20 2 21 3 21 2 22 3 22 20 2 23 3 23 2 24 3 24 2 25 3 25 *» 21 2 26 3 26 2 27 3 27 2 28 j 28 22 3 3 4 4 5 5 6 6 7 7 8 8 23 9 9 10 10 11 11 12 12 4 2 2 4 24 4 10 500 50 20 5 0 -1 2 6 6 1 25 PROBLEM DIMENSIONS AND OPERATIONAL CONTROLS 26 0.0282 12 27 0.0305 21 0.075 28 28 0.5 0.0 -9.8 l.OE-4 29 .0001 l.OE-6 1.OE-4 1.0E-3 30 1.OE-12 1.OE-12 1.OE-12 l.OE-5 0.1 31 0.05 0.90 0.3 100. 0.1 100. 32 EDIT CONTROLS AND POSTPROCESSOR CONTROLS 33 0.0 1.0 2.0 3.0 4.0 34 0.0225 I0.0225 35 0.0 36 0.225 0.50 37 0.0 38 0.003 0.003 39 0.0 40 0.225 0.5 41 0.0 42 0.010 0.02 43 0.0 44 4.0 8.0 45 0.0 46 0.0 47 0.0 48 0.0 49 0.0 50 0.0 414 TABLE Q-I (cont.)

51 0.0 52 0.0 53 VIEW FACTORS 54 55 TIME STEP CONTROLS 56 0.0 l.OE-5 l.OE-9 0.249 57 .001000 0.5 10.0 10. 10. 10. 58 1.0 0.96 0.02 0.02 0,0 0.0 59 STRUCTURE AND FAILURE PARAMETERS 60 0.0 61 -0.0 62 1.0 1.0 1.0 1.0 63 l.E+10 l.E+10 l.E+10 l.E+10 l.E+10 l.E+10 64 l.E+10 l.E+10 l.E+10 65 0 66 FUEL PROPERTIES AND EQUATION OF STATE 67 1000. 2090.00 273.16 3.334E+5 l.OE-12 68 1001.78 4217.1 0.0727 l.OE-12 1.0E-4 69 3.17771E+1O 4.70579E+03 0.0 3.22689E+06 647.286 0.390597 70 1402. 1.329 3.737 2.9339OE+6 18. 32. 71 316.957 95.770 316.957 72 73 74 75 IRON PROPERTIES AND EQUATION OF STATE 76 7365.0 639.0 1700.0 2.60000E+05 25.0 77 6100.0 750.0 1.6 30.0 5.36000E-03 78 1.338OOE+11 4.33700E+04 0.0 8.17000E+06 10000.0 0. 360 79 492.0 1.26 1.64 0.000000000 56.0 7700. 80 81 82 83 WATER PROPERTIES AND EQUATION OF STATE 84 1000. 2090.00 273.16 3.334E+5 .68 85 1001.78 4217.1 0.0727 .68 1.0E-4 86 3.17771E+10 4. 7O579E+O3 0.0 3.0898O5E+6 647.296 0.3660361 87 2181.6 1.2115 3.737 2.93390E+6 18. 31. 88 316.957 95.77 316.957 89 838.607921 1.34831230-. 0027491404 90 2.85605286 .381221451-. 0748002905 91 IRON-ALUMINUM THERMITE PROPERTIES AND EQUATION ()F STATE 92 4000. 1060. 200. 2.76000E+05 22. 93 3800.0 1060. 1.00 22. 4.30000E-03 94 1.44000E+11 5.17080E+04 0.0 2.62OOOE+O6 8400.0 0. 597 95 492.0 1.26 4.4 0.000000000 75.0 6468. 96 97 AIR PROPERTIES AND THE EQUATION OF STATJ E 98 0.0 0.0 0.0 0. OE+4 0.0 99 1. 100 1.0E+12 4. OE+03 0.0 5.OE+06 4.0 0.3 101 716.67 1.4 3.798 4.E+6 28.967 85. 102

~~415 TABLE Q-I (cont.)~~

103 COMPONENT PROPERTIES 104 9890.0 9890.0 9890.0 9890.0 7365.0 7365.0 105 4000.0 0.0 106 1000.0 1000.0 6100.0 1000.0 3800.0 9890.0 107 9890.0 7365.0 0.0 108 2000. 2000. 2000. 1498. 2000. 2000. 109 2.00000E+03 2.00000E+03 2.OOOOOE+O3 110 HEAT TRANSFER CORRELATION DATA 111 1.0 1.0 1.0 1.0 1.OE-10 1.0 112 1.0 1.0 1.OE-10 1.0 1.0 1.OOE-10 113 1.0 1.OE-10 1.OE-10 114 0.023 0.8 0.4 0.0 115 0.023 0.8 0.4 5.0 116 0.023 0.8 0.4 0.0 117 0.023 0.8 0.4 0.0 118 0.023 0.8 0.4 0.0 119 0.6 0.5 0.33 0.0 120 DRAG CORRELATION DATA 121 1.0 12.0 2. OE-4 9.2E-7 1.0 122 4.7 1.0 0.5 .001 1.0 l.E+20 123 0.046 -0.2 0.001 0.046 -0.2 0.001 124 PRESSURE BOUNDARY CONDITIONS 125 0.83E+5 0.83E+5 126 0.0 5.0 127 PARAMETER REGION 1I AIR 128 7. 0.0 l.OE+5 0.0 0.0 0.1 129 0.0 34. 34. 0.0 0.125 0.75 130 0.3 0.3 0.2 1000. 1.0E+4 1000. 131 1950. 0.64 l.OE+5 2. 3E-5 l.E-17 .00750 132 7.5E-3 133 PARAMETER REGI0N2 THERMITE 134 7. 0.0 l.OE+5 0.0 0.0 0.1 135 0.0 34. 34. 0.0 0.125 0.75 136 0.3 0.3 0.2 1000. 1.0E+4 1000. 137 1950. 0.64 l.OE+5 2. 3E-5 l.E-17 .00750 138 7.5E-3 139 PARAMETER REGION .3 WATER 140 7. 0.0 l.OE+5 0.0 0.0 0.1 141 0.0 34. 34. 0.0 0.125 0.75 142 0.3 0.3 0.2 1000. 1.0E+4 1000. 143 1950. 0.64 l.OE+5 2.3E-5 l.E-17 .00750 144 7.5E-3 145 LOWER BOUNDARY INITIAL VELOCITIES 146 0.0 0.0 0.0 0.0 0.0 0.0 147 0.0 0.0 0.0 0.0 0.0 0.0 148 149 150 MESH SET 1 WATER 151 1 20 1 12 1 1 0 0 3 152 153 154

~~416 TABLE Q-I (cont.)~~

155 0.0 0.0 0.0 996.00 156 0,0 0.0 157 0.0 0.0 299.00 158 0.0 0.0 0.0 0.601619 159 299. 160 0.0 0.0 0.0 0.0 0.0075 0.0075 161 MESH SET 2 AIR BETWEEN THERMITE AND WATER 162 21 21 1 4 1 1 0 0 163 164 165 166 167 0.0 .1 168 0.0 0.0 0.0 0.0 0.0 299.0 169 0.0 0.0 0.0 0.0 0.0 0.0965112 170 3000. 171 -5.90 -5.90 0.0 0.0 0.0075 0.0075 172 MESH SET 3 THERMITE RING 1 173 22 27 1 1 1 0 0 2 174 175 176 177 0.0 0.0 0.0 0.0 673.3743 178 179 0.0 0.0 0.0 3000. 180 0.0 0.0 0.0 0.0 0.0 0.0965112 181 3000. 182 -5.90 -5.90 0.0 0.0 0.0075 0.0075 I 183 MESH SET 4 THERMITE RING 2 AND 3\ 184 22 27 2 3 1 1 0 0 2 185 186 187 188 0.0 0.0 0.0 0.0 336.6871 189 190 0.0 0.0 0.0 3000. 191 0.0 0.0 0.0 0.0 0.0 0..0965112 192 3000. 193 -5.90 -5.90 0.0 0.0 0.0075 0.0075 194 MESH SET 5 THERMITE RING 4 195 22 27 4 4 1 1 0 0 2 196 197 198 199 0.0 0.0 0.0 0.0 168.3436 200 201 0.0 0.0 0.0 3000. 202 0.0 0.0 0.0 0.0 0.0 0.0965112 203 3000. 204 -5.90 -5.90 0.0 0.0 0.0075 0.0075 TTC 205 MESH SET 6 GAS OUTSIDE OF THERMITlibE 206 21 28 5 12 1 1 0 0 1

~~417 TABLE Q-I (cont.)~~

207 208 209 210 211 0.0 .1 212 0.0 0.0 0.0 0.0 0.0 299. 213 0.0 0.0 0.0 0.0 0.0 .96834 214 299. 215 0.0 0.0 0.0 0.0 0.0075 0.0075 216 MESH SET 7 GAS ABOVE THERMITE 217 28 28 14 1 1 0 0 1 218 219 220 221 222 0.0 .1 223 0.0 0.0 0.0 0.0 0.0 299. 224 0.0 0.0 0.0 0.0 0.0 0.0965112 225 3000. 226 -5.90 -5.90 0.0 0.0 0.0075 0.0075 same as in the three-field calculation. The downward melt progression is shown in Figs. Q-7 to Q-12. There is a significant similarity to the three-field results, although the vapor volume fraction is obviously low toward the bottom of the premixture at the end of the calculation. The agreement in the calculations may be a consequence of the heat-transfer assumptions. If the fuel's surface area controlled the heat transfer, a radiation heat flux per unit volume can be expressed by

~~3a R If (Q-D~~

~~where a^f is the fuel-volume fraction, r is the fuel drop radius, a is 5.67 x 10"8 W/m2K4, Tf is the fuel surface temperature,~~

~~TgUf is a water temperature, water surface temperature, or steam~~

~~Fig. Q-7. Initial volume fractions for the standard SINWER-II calculation. 10 ro o WATER~~

~~VAPOR~~

~~050 S~~

~~THERMITE~~

~~Fig. Q-8. Melt-water contact with standard SIMflER-II modeling. WATER~~

~~VAPOR~~

~~100 S~~

~~THERMITE~~

~~Fig. Q-9. Melt penetration of water with standard SIIMER-II modeling. ro no WATER~~

~~VAPOR~~

~~150 S~~

~~THERMITE~~

~~Fig. Q-10. Start of low vapor production with standard SIKWER-II modeling. WATER~~

~~VAPOR~~

~~ZOO S~~

~~THERMITE~~

~~Fig. Q-ll. Results with standard SIKMER-II modeling as the container bottom is being approached. CO WATER~~

~~VAPOR~~

~~224 S~~

~~THERMITE~~

~~Fig. Q-12. Results at time of the steam explosion with standard SlbMER-II modeling. temperature, and e is some sort of emissivity view-factor product.~~

~~If Tf ~3 000 K and Tgur ~400 K, we can factor a (Tf - TSur) quantity and obtain~~

~~3a Lf J R , (_J£) (1.766 x 10 ) [Tf - TSur] c . (Q-2)~~

~~On the other hand, the heat-transfer rate per unit volume with the standard SIMMER-II liquid-liquid formalism, with equal radii, is~~

~~~ [T f/^ ] (Tf " V ' (Q"3) lkf + V~~

~~where~~

~~aw is the water-volume fraction, kf is the thermite's thermal conductivity, 22 W/(m-K),~~

~~kw is the water's thermal conductivity, 0.68 W/(m-K), and~~

~~Tw is the water temperature.~~

~~If we assume Tw - TSur, Eq. (Q-2) and (Q-3) will be equal if~~

~~(|) (.66) — - 1.766 x 103 e . (Q-4) 2 r~~

If r - 7.S mm, e - 0. 374aw. Consequently, although the SIMMER-11 formalism does not possess the proper functional form for film-boiling heat transfer driven by radiation, the water's low thermal conductivity does lead to heat-transfer rates similar to those with radiation using a reduced emissivity. The third calculation used the new SIMMER-II modeling of this study. Specifically, the heat transfer was revised to calculate both the energy transfer to the water surface following contact with a fuel droplet as well as

425 the energy transfer into the bulk water. The difference between these two energy-transfer rates was used to produce steam on the surface of the water droplet. Effectively, the steam production rate is increased by a factor of approximately 20 over that implied by Eq. (Q-3). The results of this calculation are shown in Figs. Q-13 to Q-18. Here the vapor production is so strong that the melt does not reach the bottom of the container during the course of the computation. Overall, the three-field calculation probably furnishes the best agreement to experiment as is shown by Fig. 40 in the main text. However, its further development and use is beyond the scope of this program. The standard SIMMER-11 calculation clearly does not produce enough vapor, while the revised heat- transfer model produces excessive vapor. This suggests that limits can be obtained on the extent of premixing at larger scale, if triggering is ignored. Finally, these calculations also suggest a direction for the most cost-effective numerical modeling formalism for coarse premixing, if future develop- ments in this area are judged to possess technical merit. The following facts are available. The two-field SIMMER-II calculations, even with water and melt moving with the same local velocity, probably could be made to resemble closely the three-field calculations with adjustments of model parameters. The experimental information is crude for detailed modeling purposes. Improved models for energy and momentum coupling terms will not be simple. A three-field calculation will be significantly more expensive than a two-field calculation. With this knowledge, a new code with two velocity fields, one for the melt and one for the water-steam-hydrogen, may represent the appropriate approach for this problem, although the ability of steam to carry away liquid water will produce some bias in the results.

~~Fig. Q-13. Initial conditions with revised SIMMER-II heat transfer. 00 a a: o •d §~~

~~Fig. Q-16. Beginning of melt breakup with revised SIKMER-II modeling. WATER~~

~~VAPOR~~

~~200 S~~

~~THERMITE~~

~~Fig. Q-17. Melt breakup with revised SIMJER-II modeling. O~~

~~SIMMER-II INPUT FOR THE ANALYSIS OF THE MD-19 EXPERIMENT USING A NONUNIFORM MIXING ZONE~~

1 0 -105Q07WRB ANALYSIS OF EXPERIMENT MD-19 TRY2 2 2 0 0 0 0 32 1 1 100 0 3 SIMMER-2 STUDY OF THE SANDIA VAPOR EXPLOSION TESTS 4 ANALYSIS OF EXPERIMENT MD-19 TRY 2 5 .004500 1.0 6 12 28 7 FLUID DYNAMICS INTEGER INPUT 8 1 0 0 -2 5 0 3 0 0 0 -2 0 Q y 54 1 1 1 0 1 0 1 1 1 1 10 11 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 10 1 11 1 12 1 13 1 2 1 3 1 4 1 5 1 6 1 7 14 1 8 1 9 1 10 1 11 1 12 1 13 15 1 14 1 15 1 16 1 17 1 18 1 19 16 1 20 1 21 1 22 1 23 1 24 1 25 17 1 26 1 27 1 28 2 2 3 2 2 3 18 3 3 4 4 5 5 6 6 7 7 8 8 19 9 9 10 10 11 11 12 12 4 2 2 4 20 4 10 500 50 20 5 0 -1 2 6 6 1 21 PROBLEM DIMENSIONS AND OPERATIONAL CONTROLS 22 0.0282 12 23 0.0305 21 0.075 28 24 0.5 0.0 -9.8 1.OE-4 0.25 25 .0001 l.OE-6 l.OE-4 1,OE-3 1.OE-8 1.OE-4 26 1.OE-12 1.OE-12 1,OE-12 1.OE-5 0.1 27 0.05 0.90 0.3 100. 0.1 100. 28 EDIT CONTROLS AND POSTPROCESSOR CONTROLS 29 0.0 0.5 1.0 1.5 2.0 30 0.0002 0.0005 0.0010 31 0.0 32 0.0010 0.0035 0.0045 33 0.0 34 .00001 .00005 0.0001 35 0.0 36 0.0005 0.001 0.0045 37 0.0 38 0.010 0.02 39 0.0 40 4.0 8.0 41 0.0 42 0.0 43 0.0 44 0.0 45 0.0 46 0.0 47 0.0 48 0.0 49 VIEW FACTORS

433 50 51 TIME STEP CONTROLS 52 0.0 l.OE-6 l.OE-9 0.249 53 .000100 0.5 10.0 10. 10. 10. 54 1.0 0.96 0.02 0.02 0.0 0.0 55 STRUCTURE AND FAILURE PARAMETERS 56 0.0 57 0.0 58 1.0 1.0 1.0 1.0 59 l.E+10 l.E+10 l.E+10 l.E+10 l.E+10 l.E+10 60 l.E+10 l.E+10 l.E+10 61 0 62 FUEL PROPERTIES AND EQUATION OF STATE 63 1000. 2090.00 273.16 3. 334E+5 l.OE-12 64 1001.78 4217.1 0.0727 l.OE-12 1.0E-4 65 3.17771E+1O 4.7O579E+O3 0.0 3.22689E+06 647.286 0.390597 66 1402. 1.329 3.737 2.93390E+6 18. 32. 67 316.957 95.770 316.957 68 69 70 71 72 IRON PROPERTIES AND EQUATION OF STATE 73 7365.0 639.0 1700.0 2.60000E+05 25.0 74 6100.0 750.0 1.6 30.0 5.36OOOE-O3 75 1.33800E+11 4.33700E+04 0.0 8.17000E+06 10000.0 0.360 76 492.0 1.26 1.64 0.000000000 56.0 7700. 77 78 79 80 81 WATER PROPERTIES AND EQUATION OF STATE 82 1000. 2090.00 273.16 3.334E+5 .68 83 1001.78 4217.1 0.0727 .68 1.0E-4 84 3.17771E+10 4.7O579E+O3 0.0 3.0898O5E+6 647.296 0.3660361 85 2181.6 1.2115 3.737 2. 9339OE+6 18. 32. 86 316.957 95.77 316.957 200. 87 1346. 0. 3302 S.E+04 50. .5 88 838.607921 1.34831230--.0027491404 89 2.85605286 .381221451--.0748002905 90 IRON-ALUMINUM THERMITE 1PROPERTIES AND EQUATION ()F STATE 91 4000. 1060. 200. 2.76OOOE+O5 22. 92 3800.0 1060. 1.00 22. 4.3OOOOE-O3 93 1.44000E+11 5.17O8OE+O4 0.0 2.62OOOE+O6 8400.0 0.597 94 492.0 1.26 4.4 0.000000000 75.0 6468. 95 96 97 HYDROGEN GAS PROPERTIES AND THE EQUATION OF STATE 98 0.0 0.0 0.0 1.0E+4 0.0 99 1. 100 1.0E+12 4.OE+03 0.0 0.0 1.0 0.2 101 10164. 1.405 2.915 0.0 2.016 38. 102 103 104 COMPONENT PIJOPERTIES

434 105 9890.0 9890.0 9890.0 9890.0 7365.0 7365.0 106 4000.0 0.0 107 1000.0 1000.0 6100.0 1000.0 3800.0 9890.0 108 9890.0 7365.0 0.0 109 2000. 2000. 2000. 1498. 2000. 2000. 110 2.00000E+03 2.OOOOOE+O3 ,00O0OE+O3 111 HEAT TRANSFER CORRELATION DATA 112 1.0 1.0 1.0 1.0 l.OE-10 1.0 113 0.2 0.2 1.0E-10 0.2 0.2 l.OOE-10 114 1.0 1.0E-10 1.0E-10 115 0.023 0.8 0.4 0.0 116 0.023 0.8 0.4 5.0 117 0.023 0.8 0.4 0.0 118 0.023 0.8 0.4 0.0 119 0.023 0.8 0.4 0.0 120 0.6 0.5 0.33 0.0 121 DRAG CORRELATION DATA 122 1.0 12.0 2.0E-4 9.2E-7 1. 0 123 4.7 1.0 0.5 .001 1. 0 l.E+20 124 0.046 -0.2 0.001 0.046 -0. 2 0.001 125 PRESSURE BOUNDARY CONDITIONS 126 0.83E+5 0.83E+5 127 0.0 5.0 128 0.83E+5 0.83E+5 129 0.0 5.0 130 PARAMETER REGION PREMIX REGION 131 7. 0.0 l.OE+5 0.0 0.0 0.1 132 0.0 34. 34. 0.0 0.125 0.75 133 0.3 0.3 0.2 1000. 1.0E+4 1000. 134 1950, 0.64 l.OE+5 2.3E-5 l.E-17 .00015 135 1.5E-4 136 PARAMETER REGI0N2 CHIMNEY 137 7. 0.0 l.OE+5 0.0 0.0 0.1 138 0.0 34. 34. 0.0 0.125 0.75 139 0.3 0.3 0.2 1000. 1.0E+4 1000. 140 1950. 0.64 l.OE+5 2.3E-5 l.E-17 .00015 141 1.5E-4 142 PARAMETER REGION 3 WATER 143 7. 0.0 l.OE+5 0.0 0.0 0.1 144 0.0 34. 34. 0.0 0.125 0.75 145 0.3 0.3 0.2 1000. l.OE+4 1000. 146 1950. 0.64 l.OE+5 2.3E-5 l.E-17 .00015 147 1.5E-4 148 LOWER BOUNDARY INITIAL VELOCITIES 149 0.0 0.0 0.0 0.0 0.0 0.0 150 0.0 0.0 0.0 0.0 0.0 0.0 151 152 153 MESH SET 1 COMPACTED CORE 154 1 28 1 12 0 155 91 82 56 57 156 157 158 159 0.0 0.0 0.0 478.5 159.0

436 215 5.99972E+02 6.89423E+02 7.216O8E+O2 6.51515E+O2 4. 91102E+02 3.47718E+02 216 3.00001E+03 2. 99999E+03 2.99901E+03 2. 19105E+03 4. 95648E+02 3.81789E+02 217 4. 23O78E+O2 4.96239E+02 5.1278OE+O2 4. 65402E+02 3.79407E+02 3.19310E+02 218 3.00000E+O3 2.99997E+O3 2.99521E+O3 2. 79039E+03 7. 34766E+02 3.55102E+02 219 3. 58394E+O2 3.84519E+O2 3.9O889E+O2 3.66773E+02 3.28857E+02 3.O6873E+02 220 3.00O0OE+03 3.00O00E+03 2.99811E+03 2. 52781E+03 5.17162E+02 3.44625E+02 221 3. 32196E+02 3.35426E+02 3.34773E+02 3.23434E+02 3.09135E+02 3.O21O8E+O2 222 WATER VAPOFI DENSITY 223 2.40272E-01 3.777O3E-O1 5.74169E-01 5.98670E-O1 5.99592E-01 5.99613E-01 224 5.99613E-01 5.99613E-01 5.99613E-O1 5.99613E-O1 5.99613E-01 5.99613E-01 225 2.18O68E-O1 2.36160E-01 4.13358E-01 5.82537E-O1 5.99O53E-O1 5.99602E-01

~~233 2.O9155E-O1 2.17432E-01 2. 33OO1E-O1 3.65030E-01 t/ i 46930E-01 5.95463E-01~~

234 5.99421E-01 5.99607E-01 5.99613E-01 t/ i 99613E-01 5.99S13E-01 5.99613E-01 235 2.08665E-01 2.11732E-01 2.26352E-01 3.17654E-O1 5.01549E-01 5.88966E-01 236 5.99OO7E-O1 5.99590E-01 5.99613E-01 5.99613E-01 5.99613E-01 5.99613E-01 237 2.14718E-01 2.02194E-01 2.24848E-01 2. 46657E-01 4.,00924E-01 5.57856E-01

238 5.96833E-01 5.99498E-01 S.99610E-01 5.99613E-O1 t/ i 99613E-O1 5.99613E-01 239 2.13801E-01 1.97181E-01 2.18886E-O1 2,,39615E-01 2.,8143OE-O1 4. 5O3O9E-O1 240 5.80301E-01 5.98781E-01 5.99588E-O1 5.99613E-01 5.,99613E-01 5.99613E-01 241 1.95723E-01 1.8871OE-O1 2.O23O5E-O1 2..68972E-O1 2,,29555E-01 3.29422E-01 242 5.11763E-01 5.94144E-01 5.99420E-01 5,,99608E-01 5,.99613E-O1 5.99613E-01 243 1.78835E-O1 1.29110E-01 1.68981E-O1 3,,05922E-01 2,.57683E-01 2. 49741E-01 244 4.05261E-01 5.74630E-01 5.98536E-01 5,.99581E-01 5,.99613E-01 5.99613E-01 245 1.61335E-01 1.22213E-01 1.40085E-01 1,.95237E-01 2,. 91973E-012. 25518E-01

~~246 3.23182E-O1 5.28852E-01 5.95885E-O1 5,.99473E-01 t/ i .99610E-01 5.99613E-01 247 1.23550E-01 9.11358E-02 1.24047E-01 1.44792E-01 3.09611E-01 2. 45527E-01~~

248 2.72465E-01 4.71612E-01 5.91395E-01 t/ i .99231E-01 5.99603E-01 5.99613E-01 249 9.95553E-02 8.32430E-02 1.02468E-01 1.30948E-01 2.O3579E-O1 2. 71323E-01 250 2.49953E-O1 4.3O83OE-O1 5.8671OE-O1 5. 98923E-O1 5.99593E-01 5.,99613E-01 251 9.38739E-O2 8.00299E-02 8.40494E-02 1.18237E-O1 2.OOO63E-O1 2.,88193E-01 252 2.49801E-01 4.27445E-01 5.84855E-O1 5.98728E-01 5,99585E-01 5.,99613E-01 253 9.41018E-02 7.94488E-02 7.47786E-02 1. 04354E-01 1.74681E-01 3..27447E-01 254 2.78571E-O1 4.67500E-01 5. 87263E-01 5•988O2E-O1 5.99584E-01 5..99613E-01 255 8.19867E-02 7.71182E-02 6.74837E-02 8.95897E-02 1.66173E-O1 3,,92056E-01

~~256 3.49303E-01 5.30919E-01 5.91166E-01 t/ < .99024E-01 5. 99590E-01 5,.99613E-01 257 6.23735E-02 6.59313E-02 6.32O97E-O2 7.42942E-02 1.64481E-01 4,.77926E-01~~

258 4.40291E-01 5.66365E-01 5.94408E-01 (A .99287E-O1 5. 99617E-01 5,.99623E-01 259 5.99585E-O2 6.12O62E-O2 6.09326E-02 6.838O1E-O2 1. 59291E-01 2,,29948E-01 260 5.34892E-01 5.7OO62E-01 5.97739E-01 6.OO63OE-O1 6. OO836E-O1 6,.00411E-01 261 5.99591E-02 6.00864E-02 6.01113E-02 6.53140E-02 1.47726E-01 1,.53032E-01

262 2.66266E-01 3.92921E-01 5.35716E-01 t/ i .99O75E-O1 6.10065E-01 6.16763E-01 263 5.99594E-02 5.99599E-02 5.99711E-02 6.14745E-02 9. 42369E-02 1,.22220E-01 264 1.5O674E-O1 2.O3358E-O1 2.65832E-01 3.861O9E-O1 4.94965E-01 6,.13198E-01 265 5.99598E-02 5.99601E-02 5.99603E-02 6.00689E-02 6. 75759E-02 7,.22617E-02 266 9.85947E-02 1.47346E-01 1.84579E-01 2.81867E-O1 4. 44509E-01 5,.37070E-01 267 5.996O3E-O2 5.99603E-02 5.99604E-02 6.06799E-02 8.45944E-02 1.OO379E-01 268 1.3O213E-O1 1.48604E-01 1.61401E-01 1.6798OE-O1 1.89O28E-O1 3.23817E-01 269 5.996O8E-O2 5.99609E-02 5.99614E-02 6.49579E-02 1. 9O368E-O1 2.83554E-01

438 325 2.37249E-04 2. 85148E-04 2. 75972E-04 7.O2797E-O5 4.54440E-05 1.15975E-O2 326 1.32340E-15 4.08184E-13 6. 39237E-13 7.81798E-O9 2.49131E-O6 1.51668E-03 327 3.O8224E-03 1.56O69E-03 2.82898E-04 6.6O584E-O5 3.89653E-03 2.86797E-O2 328 4.29744E-15 1.72068E-12 8.41355E-12 4.64551E-07 3.O1928E-O4 3.92561E-03 329 9.58391E-O3 1.06539E-02 1.19253E-02 2.06183E-02 3.62991E-O2 5.56474E-02 330 1.24988E-14 5.92268E-12 1.83521E-10 1.51317E-05 3.76551E-04 3.29841E-03 331 1.O31O3E-O2 1.77110E-02 2.66204E-02 3.71233E-O2 4.67954E-02 5.9O387E-O2 332 2.41098E-14 1.32861E-11 2.65668E-09 1.93384E-O7 2.66823E-05 8.22482E-04 333 5.54136E-03 1.41593E-02 2. 38271E-02 3.08024E-02 3.35236E-O2 3.42671E-02 334 3.16737E-16 1.49041E-12 1.62318E-09 8.22020E-O7 3.01110E-05 4.04746E-04 335 2.20810E-03 5.93761E-03 1.01272E-02 1.16783E-O2 1.12693E-02 1.O87OOE-O2 336 LIQUID CONTROL DENSITY 337 3.53384E+02 6.75696E+01 2.41430E+00 3.97925E-O2 6.02120E-O4 5.178O1E-O6 338 2.90027E-08 1.O891OE-1O 2.72493E-13 4.36494E-16 4.O578OE-19 1.66678E-22 339 9.5O778E+O2 2.98244E+02 3.00924E+01 6.81745E-O1 7.31O98E-O3 4.14416E-05 340 4.05668E-07 2. 37634E-09 8.46989E-12 1.82863E-14 2.22574E-17 1.18262E-20 341 1.02336E+03 4.24416E+02 7.20073E+01 3.49224E+00 5.27263E-02 4.98404E-04 342 5.04134E-06 3.5O781E-08 1.66802E-10 5.08319E-13 8.79099E-16 6.48344E-19 343 9.46388E+02 4.65097E+02 1.03451E+02 8.47642E+OO 1.77835E-O1 2.92526E-03 344 3.08945E-05 3.29854E-07 2.04717E-09 7.63378E-12 1.61453E-14 1.49773E-17 345 7.S9580E+02 4.67774E+02 1.3O176E+O2 1.47916E+O1 5.27055E-01 1.19123E-02 346 1.15130E-04 1.58279E-06 1.40849E-08 7.44598E-11 2.11464E-13 2.48568E-16 347 6.32534E+02 4.4335SE+02 2. 27706E+02 2.8O878E+O1 2.00373E+O0 5.8OO86E-O2 348 8.07250E-04 1.00048E-0S 8.92583E-08 S.41002E-10 1.94340E-12 3.O1618E-15 349 5.43102E+02 4.19673E+02 4.02206E+02 8.67871E+O1 1.O2396E+O1 5.90282E-01 350 1.05869E-02 6.48471E-05 6.39111E-07 4.87794E-O9 2.15797E-11 4.16394E-14 351 3.94517E+02 3.86468E+02 4.09539E+02 2.92759E+O2 5.29688E+O1 6.47146E+00 352 2.10603E-01 1.89728E-03 1.17186E-05 7.O7SO9E-O8 2.46449E-10 3.10409E-13 353 1.983O6E+O2 3.O26O7E+O2 3.63945E+02 3.6O175E+O2 1.96476E+O2 3.78982E+O1 354 2.90191E+00 3.55337E-02 1.98636E-04 7.61647E-O7 2.63890E-09 8.64968E-12 355 2.81211E+00 9. 86067E+01 2.43758E+02 3.16600E+02 2.97925E+O2 1.15735E+O2 356 1.64908E+01 3.40710E-01 2.92792E-03 1.81553E-O5 7.7845OE-O8 8.85933E-11 357 3.71190E-04 2.53062E-01 3. 78753E+O1 2.05132E+02 2.6OO87E+O2 2.35549E+02 358 4.98774E+01 2.29638E+00 2.03616E-02 7.38531E-O5 5.79333E-O7 2.30002E-09 359 4.31042E-04 8.97415E-02 1.04426E-01 2.13129E+01 1.63272E+O2 2.68737E+02 360 1.O3187E+O2 7. 62832E+00 1.89038E-02 8.41559E-O4 2.8O667E-O6 1.58332E-O8 361 1.51658E+00 2.O7830E+00 1.74221E-02 3.2795OE-O1 7.13358E+O1 2.22341E+02 362 1.60660E+O2 1.62578E+01 2.51327E-03 2.49264E-03 1.28428E-0S 1.O1499E-O7 363 2.31610E+00 2.36267E+00 1.65590E-01 1.00552E-01 7.18374E+OO 1.27010E+02 364 1.657O1E+O2 2.06674E+01 4. 76567E-03 4. 72357E-03 4.63002E-06 2.76699E-07 365 2.25477E+00 2.32891E+00 1.21966E+00 5.23243E-O1 6.14120E+00 3.63394E+01 366 1.25639E+02 1.66854E+01 6.94994E-03 6.89892E-O3 5.68858E-O7 5.67940E-07 367 1.51424E+00 2.01644E+00 1.78019E+00 1.10106E+00 3.8O377E+OO 1.70314E+00 368 5.24546E+01 6.64211E+00 6.44285E-03 6.41999E-03 8.16455E-07 8.16268E-07 369 8.30326E-01 1.23448E+00 1.81532E+OO 1.4O926E+OO 2.32436E+OO 1.41228E+00 370 4.89456E-01 3.47195E-01 3.83932E-03 3.76236E-O3 7.59497E-07 7.40445E-07 371 9.40690E-02 4.54619E-01 1.28875E+00 1.22369E+O0 1.24110E+00 1.85O53E+OO 372 4.83879E-01 3.44997E-01 1.71620E-02 4.46786E-04 1.3O918E-O5 4.82O96E-O7 373 3.O3332E-13 9.90619E-03 6.8O338E-O1 1.O2997E+OO 2.O5O51E-O1 1.40168E+00 374 1.86923E+OO 2.73126E-01 3.92499E-03 7. 38776E-O5 1.07109E-05 2.O6885E-O7 375 2.93792E-13 1.64204E-04 1.8227OE-O1 4.37988E-O1 2.O1692E-O1 3.66970E-01 376 1.78515E+OO 4.14699E-01 6.41385E-03 4.02985E-04 4.27668E-06 8.41920E-08 377 1.77678E-13 4.97436E-07 3.91171E-03 1.923O2E-O1 2.37384E-O1 2.47690E-01 378 8.55383E-01 1.04961E+00 1.99951E-01 5.85477E-O3 3.08753E-05 2.11124E-06 379 1.45650E-13 6.08828E-06 9.24386E-06 2.09404E-03 1.03392E-01 1.06536E-01

439 380 7.13576E-01 9.76218E-O1 5.75951E-O1 5.40512E-02 5.04346E-04 2.63053E-04 381 2.51404E-13 6.71099E-11 7.00046E-11 1.58473E-09 7.52150E-04 3.52108E-02 382 6.85370E-02 8.58232E-02 8.44766E-O2 8.48574E-O2 8.88946E-02 5.42624E-02 383 1.O195OE-12 3.42177E-10 5.03818E-10 1.20210E-07 2.15861E-O2 1.46991E-01 384 2.40797E-01 3.19276E-01 2.49681E-01 1.42505E-01 9. 75386E-O2 3.51297E-02 385 3.16836E-12 1.28360E-09 5.96725E-09 7.12325E-O6 4.97137E-O2 2.37958E-01 386 2.92003E-01 2.68447E-01 1.623O5E-O1 1.2679OE-O1 6.44773E-02 1.41685E-02 387 8.84I3OE-12 4. 3O13OE-O9 1.27O18E-O7 2.35475E-04 S.33646E-03 5.32348E-02 388 1.O3855E-O1 1.2OO83E-O1 794645E-O2 6.428O4E-O2 2.O3235E-O2 4.42988E-03 389 1.65Oi3E-ll 9.06828E-09 1.81195E-06 1.25454E-04 5.28684M-O3 4.59084E-02 390 3.61204E-02 3. 87212E-02 3.2786SE-O2 2,11223E-O2 4.88595E-O3 1.O8O23E-O3 391 4.67762E-14 2.26375E-10 6.67744E-O7 3.65463E-04 1.1O6O5E-O2 4.21063E-02 392 1.33969E-02 1.11702E-02 1.10504E-02 4.63393E-O3 9.74431E-04 2.11308E-04

~~440 APPENDIX S EFFECTS OF CHANGING THE RADIAL AND AXIAL CONSTRAINTS IN THE SIMULATION OF MD-19~~

The RC-2 experiment was performed with a steel pipe rather than a Lucite box being the water chamber. In this experiment the test apparatus was destroy- ed. Exit velocities of particles were estimated to be as high as ~1 km/s. The inference was drawn that a substantially higher energy conversion ratio was obtained in experiment RC-2 compared with tests without radial confinement. The postulated reactor meltdown steam explosions simulated in the ZIP study1 were strongly confined by the presence of the ~100 tons of fluid in the corium pool. The inertial effects tended to make the results somewhat insensitive to the energy transfer rates. Because of these effects, additional calculations were performed to investigate the consequences of increasing the boundary contraints in the Slkl/IER-II representation of experiment MD-19. The first calculation assumed a rigid radial boundary, with all other parameters fixed at the values used in producing Fig. 42. The resulting base

~~2.0 3.0 TIME (MS)~~

~~Fig. S-l. Results starting from a nonuniform interaction zone with a rigid-wall radial constraint.~~

441 pressure is shown in Fig. S-l. The high pressure duration as a consequence of the wall reflection was substantial. Increased axial momentum was generated. Increased heat transfer occurred in the nodes along the axial centerline. However, overall the rate of energy transfer was reduced (the thermite tended to vaporize water locally rather than to follow the radially expanding water), and slightly less kinetic energy was produced (there was an increased tendency to vent up the axial chimney rather than accelerate water radially). Although the radial boundaries must be extended and the calculation run longer to accurately quantify the kinetic energy, there is clearly no dramatic increase in yield with the current model. A more mechanistic treatment of fragmentation is required to generate this effect. Currently, fragmentation is modeled as independent of the local water- volume fraction, reflected pressure waves, the ability of a detonation wave to propagate, or any transient conditions. Also, the insensitivity calculated in the ZIP study is not obtainable when simulating an experiment with a radial constraint. Heat-transfer rates are still very important, with significant quenching of peak pressures calculated to occur in the water-rich environment of the steam explosion independent of the radial boundary conditions. The second calculation assumed a movable axial constraint in addition to the radial constraint. The SIM4ER-II slug model was used to place a movable structure (a slug) above the water, as shown in the initial vapor volume fraction plot of Fig. iS-2. The slug was given a mass of SO kg. This is about 10 times the mass of the thermite, or a mass ratio of premixed fuel to fuel slug similar to the reactor meltdown case. Also, with a mass of 50 kg, the slug was expected to move no more than two axial nodes during the calculation, which was desirable to avoid changing the number of axial nodes or their length. This calculation reproduced the pressure trace shown in Fig. S-l. The axial constraint had no effect until late in the calculation because of the compliance introduced by the vapor chimney. A plot, of pressure in the node below the slug shows this quite clearly (see Fig. S-3). Impact does not occur until 3.5 ms, when the pressure has decreased at the base transducer to only a few bars. In conclusion, current modeling does not predict significant changes in the kinetic energy produced by varying the degree of constraint in simulations of the water-rich steam-explosion experiment, MD-19.

~~442 VOLUME FRACTION OF LIQUID TIME OOQMS~~

~~MIN= 1 Z9E-05 MAX= 9 50E-01 C I = 9 50E-02~~

~~Fig. S-2. Initial conditions for the case with axial constraint.~~

~~2.0 4.0 6.0 TIME (MS)~~

~~Fig. S-3. Pressures below the slug in the axially/radially constrained case. APPENDIX T EXPLOSION CALCUUTION STARTING WITH A STANDARD SIM/IER-II PREMIXED CONFIGURATION~~

Case 2 of the SIMMER-II calculated reactor atsam explosions (see Chap. VI, Sec. B) used initial conditions that were determined from a SIMMER-II coarse premising run. The standard SIMMER-II code was used for this run as in App. Q. Because this reactor steam-explosion calculation is claimed to be edge-of- spectrum as a consequence of the SIMMER-II premixing determination, we thought an additional MD-19 simulation would be desirable as a calibration. The additional calculation explodes the configuration of Fig. Q-12 with the reference parameters (a fuel diameter of 300 jim, a water diameter of 75 jtm, and a heat-transfer multiplier of 0.2). The pressure pulse corresponding to Fig. 42 is given in Fig. T-l. A higher pressure and more kinetic energy is produced. The peak pressure is now 19% higher, and this case has 5.7% more kinetic energy than the case with the three-field assumptions of Fig. Q-6. However, the width of the pressure pulse at 8 ms is reasonable.

~~30.0~~

*• 10.0 -

~~1.0 2.0 TIME (MS)~~

~~Fig. T-l. Explosion results starting with standard SIMMER-II calculated premixing.~~

445 Evidently, the low amount of steam production and consequent low vapor volume fraction during premising led to less immediate expansion and made local heat transfer more effective in the initial stages of the explosion. This calculation suggests that the lack of the explicit calculation of separate velocities for thermite and water does not necessarily result in a degraded steam explosion. There is still the question of scale that has to be addressed, but the large nodes used in the reactor problem might again exaggerate water/fuel mixing.

~~446 APPENDIX U~~

~~SIMAER-II PREMIXING WITH HIGH STEAM PRODUCTION RATES~~

This appendix documents the results of a second corium-water premixing case performed with SIKMER-II for a postulated reactor meltdown situation. Initial conditions are the same as described in case 2 of Chap. VI. In case 2, steam production was limited as a consequence of the water's low thermal conductivity. Liquid-to-liquid heat transfer could only progress as dictated by the heat conduction into a 20-mm-diameter water droplet. In the present case, heat transfer to the water surface is given by conduction from the corium and the corium-water contact features. The heat that cannot be conducted into the water is used to produce steam. In Chap. IV it was found that this computational algorithm, assuming steam production on contact, tends to exaggerate the rate of steam production compared with SNL test MD-19. Alternatively, the case 2 assumptions seem to underestimate the dispersive effects of steam production on a thermite melt in the film-boiling regime. In this sense, these two calculations tend to bound the extent of fuel mixing under such assumed initial low-pressure conditions. Figures U-l to U-9 give density contours of the corium and coolant (water plus steam) in this second premixing case. Initially, the results are similar to those of case 2. Water tends to be pushed away from the falling corium. However, by 500 ms the results of the increased steam production are evident, and an instability starts to grow in the corium pool. By 700 ms, instead of the corium beginning to reach the bottom of the vessel, a two-phase steam-corium mixture begins to expand toward the top of the vessel. Finally, at 1 s, fuel is spread around the sides of the vessel, while liquid water is present only at the vessel's bottom, in the downcomer, and especially in the inlet piping. In conclusion, for this case of high steam production rates, coarse premixing of corium-water is very limited. This should preclude the possibility of a containment threatening steam explosion with such model assumptions.

~~447 T I ME .000 MS~~

~~STEAM EXPLOSION EVENT SEQUENCE HIGH HEAT TRANSFER NORMAL HEAT TRANSFER FUEL COOLANT FUEL COOLANT~~

~~MINI MUM 1 OOE-t-00 5 OOE-02 1 00E+00 b .OOE-02 MAXIMUM 7.25E+03 9 62E+02 7 25E+O3 9 .62E+02 CONTOUR INTERVAL 7 25E+02 9 6IE + 0' 7 25E*02 9 62E+01 REGION-RADIAL ( 1 .15) AXIAL ( 1.54)~~

~~Fig. U-l. Initial conditions, high steam production premixing case (time 0.0 ms).~~

~~T I ME 100.000 MS STEAM EXPLOSION EVENT SEOUENCE HIGH HEAT TRANSFER NORM"! HEAT TRANSFER FUEL COOLANT r • . COOLANT~~

~~MINI MUM I 00E+00 5 OOE-02 1 00E+00 5 OOE-02 MAXIMUM 7 25E+03 9.62E+O2 7.25E+03 9.62E+02 CONTOUR INTERVAL 7 25E+02 9 61E + 01 7 25E+02 9.62E+01 REGiON-RADlAL ( 1.15) AXIAL ( 1.54)~~

~~Fig. U-2. Initial flow, high steam production premixing case (time « 100 ms).~~

~~448 T IME 200.000 MS~~

~~STEAM EXPLOSION EVENT SEQUENCE HIGH HEAT TRANSFER NORMAL HEAT TRANSFER FUEL COOLANT FUEL COOLANT~~

~~MINIMUM 1 OOE+00 5 OOE-02 1 OOE+00 5 OOE-02 MAXIMUM 7 25E+03 9 62E+02 7.25E+O3 9.62E+02 CONTOUR INTERVAL 7.25E+O2 9 61E-I-01 7.25E+02 9.62E+0I REG I ON-RADIAL ( 1 . 15) A.~~

~~Fig. U-3. Initial mixing, high steam production premising case (time - 200 ms).~~

~~T IME 300.000 MS STEAM EXPLOSION EVENT SEQUENCE HIGH HEAT TRANSFER NORMAL HEAT TRANSFER FUEL COOLANT FUEL COOLANT~~

MINI MUM 1 OOE+00 5.OOE-02 1 OOE+00 5 OOE-02 MAX I MUM 7 25E+03 9 62E+02 7 25E+03 9 62E+02 CONTOUR INTERVAL 7 25E+02 9 61E + 01 7 25Et-O2 9 62E+01 REGION-RADIAL ( 1.15) AX IAL ! .54) Fig. U-4. Premizture formed above support forging, high steam production premizing case (time - 300 ms).

~~449 T I ME 400.000 MS~~

~~bTEAM EXPLOSION EVENT SEQUENCE HIGH HEAT TRANSFER NORMAL HEAT TRANSFER FUEL COOLANT FUEL COOLANT~~

~~MINI MUM 1.0OE+00 5 OOE-02 1.00E+00 5.OOE-02 MAX I MUM 7.25E+0J 9.62E+02 7.25E+03 9.62E+02 CONTOUR INTERVAL 7 25E+02 9 61E+0I 7.25E+02 9.62E+01 REGION-RADIAL ( 1.15) AXIAL ( 1.54)~~

~~Fig. MS. Counterflow beginning, high steam production premixing case (time 400 ms).~~

~~TIME 500.000 MS STEAM EXPLOSION EVENT SEQUENCE HIGH HEAT TRANSFER NORMAL HEAT TRANSFER FUEL COOLANT FUEL COOLANT~~

MINIMUM 1 OOE+00 b OOE-02 1 OOE+00 5.OOE-02 MAX I MUM 7.25E+03 9 62E+02 7 25E+03 9.62E+02 CONTOUR INTERVAL 7 25E+02 9 61E+01 7 25E+02 9 62E+01 REGION-RADIAL ( 1.15) AXIAL ( 1.54) Fig. U-6. Pool breakthrough close, high steam production premixing case (time 50(0 ms).

~~450 TIME 600.000 MS STEAM EXPLOSION EVENT SEQUENCE HIGH HEAT TRANSFER NORMAL HEAT TRANSFER FUEL COOLANT FUEL COOLANT~~

~~MINI MUM t OOE+00 5 OOE-02 1.OOE+00 5.OOE-02 MAX I MUM 7.25E+03 9 62E+O2 7.25E+03 9.62E+02 CONTOUR INTERVAL 7 25E+02 9.6IE+0I 7 25E+02 9 62E+01 REG 1 ON-RADIAL ( 1,15) AXIAL ( 1,54)~~

~~Fig. U-7. Corium pool fluidization beginning, high steam production premixing case (time - 600 ms).~~

~~TIME 800.000 MS STEAM EXPLOSION EVENT SEQUENCE HIGH HEAT TRANSFER NORMAL HEAT TRANSFER FUEL COOLANT FUEL COOLANT~~

MINI MUM 1.OOE+00 5 OOE-02 1.OOE+00 5.OOE-02 MAX I MUM 7.25E+O3 9.62E+0Z 7 25E+03 9.62E+02 CONTOUR INTERVAL 7.25E+02 9 61E + 01 7 25E+02 9.62E+01 REGION-RADIAL ( 1,15) AX IAL 1,54) Fig. U-8. Fuel spray reaching the top of the vessel, high steam production premixing case (time - 700 ms).

~~451 TI ME 1000.000 MS STEAM EXPLOSION EVENT SEQUENCE HIGH HEAT TRANSFER NORMAL HEAT TRANSFER FUEL COOLANT FUEL COOLANT~~

~~MINI MUM 1 OOE+00 5.OOE-O2 1.00E+00 5.00E-02 MAXIMUM 7.25E+O3 9 62E+02 7.25E+03 9.62E+02 CONTOUR INTERVAL 7.25E+02 9 61E+01 7 25E+02 9.62E+01 REGION-RADIAL ( 1,15) AXIAL ( 1.54)~~

~~Fig. U-9. Final conditions, high steam production premixing case (time 1000 ms).~~

~~452 APPENDIX V DETAILS ON THE UPPER BOUND SIMMER-II STEAM-EXPLOSION CALCULATION~~

One objective of the Los Alamos steam-explosion program was to examine the upper bound issue. The purpose of this examination was to determine if the mitigative features associated with the postexplosion expansion would be sufficient to eliminate concern regarding containment failure. This appendix describes a calculation illustrating that limits must be placed on initial conditions in a SIMMER-II calculated postexplosion expansion, if the calculated results are not to be suggestive of containment failure. The geometric setup is shown in Fig. V-l. This problem has the lower head failure model added to case 3, along with additional modifications. The amount of premixed corium was 94 000 kg, the value SNL chose to represent 75% of the core and the upper bound on the amount of corium that can mix with water. The corium's smear density was 2 177.9 kg/m , in regions of no structure. With an assumed corium/water mass ratio of 4.644, an initial corium temperature of 2 800 K, a corium heat capacity of 500 J/(kg«K), a value of 7 000 kg/m3 for fully dense corium, and an initial water temperature of 400 K, the SESAME tables were used to equilibrate corium and water (steam) temperatures at 1 482.5 K. At a steam smear density of 470 kg/m or a microscopic (real) density of 680.8 kg/m3, the SESAME tables gave a pressure of 906 MPa for the equilibrated temperature. The remaining 25% of the core, plus 10% (3 500 kg) of the assumed steel structure was assumed to exist as a fully dense debris layer (at 2 000 K) above the premixed material. Remaining steel, 31 500 kg, v/as assumed to be particles in the regions where upper core structure exists. This starting pressure is beyond the domain over which the water EOS was fit for SIMMER-II. For this problem different AEOS constants were obtained with the following procedure. (a) To minimize the initial vapor energy in the SIMMER-Ii AEOS, the

~~objective was to have the saturation energy and consequently Tga^ 3 as low as possible. This is the initial mixture temperature, where TQ is~~

1 482 K. When Tga^ 3 differs from TQ, the vapor energy to obtain a given pressure is higher. Therefore, the p*, T* relationship was fit at two points. These points are the critical point where p = 22 MPa and T = 647 K, and the starting point for this problem, or p » 906 MPa and

~~453 U- ISOT—4 LOWER HEAD FAILURE PROBLEM JSOT~~

~~OUTLET PIPING~~

~~STEEL PARTICLES SOLID STRUCTURE~~

~~INLET PIPING DEBRIS 12-54 & STEEL DOWNCOMER WATER PREMIXED REGION CORE SUPPORT FORGING~~

~~JSO~~

~~MOVABLE 5.20 STRUCTURE~~

~~OUTLET TO KEYWAY~~

~~iso~~

~~Fig. V-l. Geometric setup for bounding case illustration.~~

454 T - 1 482 K. This fit resulted in T* - 4.2695 x 103 K and p* - 1.6154 x 1O10 Pa. (b) The gas parameter, R^, needs to be large as suggested by the high pressure domain of the Rougen-Watson-Ragatz chart. To accomplish this, the variable &., the high pressure augmentation parameter on R^, was increased by a factor of 2 to un

ity. The variable am was reduced by an order of magnitude to 20 K. (c) The heat capacity's relaxation parameter was assumed to remain at 50 K. No particular advantage appeared obtainable by changing this relaxation curve. An isentropic expansion was computed from the SESAME tables by expanding the high pressure steam through 80 ra , the approximate in-vessel expansion volume. This is shown in Fig. V-2. Also shown in Fig. V-2 are pressures from the revised SIMMER-II AEOS, assuming the internal energies and densities from the values along the SESAME curve. More expansion work, will be obtained from the revised SIMMER-II AEOS because of its flatter relationship with pressure in this domain. (The standard fit corresponds better at low pressures and significantly underestimates high pressures. A true match to the details of the expansion curve requires more sophistication.) Although the integrity of the sides of the reactor vessel are clearly questionable with pressure loadings approaching 10 kbars, this case was run with the standard SIMMER-II radial boundary conditions. The lower head failure parameters were those shown in Table XII of Chap. III. Lower head failure occurred at 0.825 ms. The pressure at the bottom of the vessel dropped rather quickly in response to this failure as shown in Fig. V-3. However, the premixed zone is long and much of the expansion is directed upward. High impact pressures were obtained. The pressure at the vessel's flange, node (11,60) where the head curvature begins, is given in Fig. V-4. It reached 640 MPa at 13.0 ms. A focusing toward the center of the vessel's head then occurred. The pressure at node (6,65) in the middle of the head's curvature region is shown in Fig. V-5, and the pressure at the center of the head is given in Fig. V-6. These pressures go well beyond that required for vessel failure. The integrated force on the head is shown in Fig. V-7. It peaked above 12 GN. Liquid-volume fractions are shown in Fig. V-8. Despite the downward progression of a low-density spray, head loadings were high along the entire surface at 15 ms. Fluid kinetic energies for the four regions defined in Table XV

~~Q. O 0.6 SIMMER-H UJ 0.5 CO CO o.4 £L~~

~~0.3 SESAME 0.2~~

~~0.1~~

~~1 I I I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 DENSITY (g/cm3)~~

Fig. V-2. Comparison of an isentrope from the SESAME EOS with an adjusted SIMMER-II AEOS using the same input densities and internal energies. of case 1 are given in Figs. V-9 to V-12. Summing the peak values gives IS 165 MJ. The maximum kinetic energy given to the lower head was 2 883 MJ. The explosion energy can consequently be estimated as 18 048 MJ. Assuming the available work potential is 1.2 MJ/kg of corium or 1.128 x 10 MJ, the energy conversion is 16%, apparently close to the thermodynamic limit for an in-vessel expansion. However, it is difficult to relate this energy conversion to those

~~456 1000.0 -= PRESSURE IN PLENUM 900.0 A 1 800.0 i~~

~~700.0 -j 10 a. 600.0 -j~~

~~5000 A UR E 400.0 A JES S a. 300.0 4~~

~~200.0-~~

~~100.0-~~

~~0.0- • • • • i • > > > i i i • « i • • ' • i • • • • i"1""1 • • i • • • 'I 0000 0.002 0.004 0.006 0.O0B 0010 0.012 0.014 0.016 TIME (s)~~

~~Fig. V-3. Pressure at the bottom of the inlet plenum for case 5.~~

~~10000- PRESSURE AT VESSEL FLANGE 900.0 i~~

~~800.0 -j~~

~~700.0 -I~~

~~Q. 600.0 \~~

~~OL 500.0 A cn u 400.0 ij a. a. 300.0 A~~

~~200.0 A~~

~~100.0 -I~~

~~0.0 0.000 0.002 0.004 0.006 0.008 0010 0.012 0.014 0016 TIME (s)~~

~~Fig. V-4. Pressure at node (11,60), where the head curvature begins for case S.~~

457 Fig. V-5Pressureinnod(6,65),thmiddlofheadcurvaturregiofo r 458 u K case 5. Ed Q. (0 tESSl (MPa 2000.0 - 1800.0 i 1400.01 1600.0-i 1200.0 -i 1000.0 -i 7000 - 6000.0 -; 2000.0 3000.0 '-. 4000.0- 5000.0 \ 1000.0- 400.0 -1 600 i 8000 -j 200.0 i ! 0 0.0 Fig. V-6Pressurea t thecenterofheadfocas5. 0.000 24680.01 0000 0.00246 00080.010246 T I |•1<' • 'Irq i PRESSURE ATTHHEADCURVATUR TOP OFTHEVESSEL TIME (s) TIME (s) °b 16.0 FORCE ON HEAD 14 0

~~0.0 • I I I • I I I I I I I • I I • I • I I • I • I • ' ' ' f • ' ' ' I ' ' ' • I 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 TIME (s)~~

Fig. V-7. Integrated force on the head for case 5. quoted elsewhere because of the pessimistic EOS parameters and the extra volume available in the downward direction (more than an extra 100 m exist as can be determined from Fig. V-l). The impact coherence parameter, F /KE , is 20.5 GN/ci. Impact here compres- ses the upward moving liqui' into a single-phase domain different from the other cases. Clearly containment integrity may be questioned for this upper bound case.

~~459 O VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIOUIO TIME OOOMS. VOLUME FRACTION OF LIQUID TIME 8 OOOMS TIME « OOOMS~~

MIN: I 29E-OS MAX: I 0OE*O0 Cl= I 0OE-0I MIN: I 29E-O5 MAX: I OOE*OO Cl= I OOE-OI M1N: I 21E-OS WAX: I OOE'OO C|: I OOE-OI VOLUME FRACTION OF LIQUID VOLUME FRACTION OF LIQUID TIME 12 OOOMS, TIME IS OOOMS

~~i i~~

~~MIN- I 29C-0S MAX: I OO£*OO C|: I OOE-OI MIN: I 29E-05 MAX: I 00E*O0 Cl= I OOE-OI~~

~~Fig. V-8. Liquid-volume fractions for case 5. UPWARD FLUID KINETIC ENERGY FOR CASE 5~~

~~> OC»O9t—I 1 1 1 [-~~

~~005 .010 TIME (S)~~

~~MINIMUM VALUE- 0. MAXIMUM VALUE* 7.253*1E+09~~

~~Fig. V-9. Upward fluid kinetic energy for case 5.~~

~~DOWNWARD FLUID KINETIC ENERGY FOR CASE 5~~

~~« 0C»09, r-~~

~~MINIMUM VALUE= 0. MAXIUUM VALUE= 5 86273E+09~~

~~Fig. V-10. Downward fluid kinetic energy for case 5.~~

~~461 OOWNCOMER AND INLET PIPE FLUID KINETIC ENERGY FOR CA~~

~~2 ot.os, 1 r-~~

~~MINIMUM VALUE= 0 MAXIMUM VALUED I.9551OE+O9~~

~~Fig. V-ll. Downcomer and inlet pipe fluid kinetic energy for case 5.~~

~~OUTLEI PIPE FLUID KINLIIC ENERGY I Ok CASE b~~

~~S OE«O>~~

~~6 DE*O7 J - • OE*O' - / :~~

~~2 0£*O? -~~

~~/, . i . , . , D 00!) 0 10 TIME (S)~~

~~MINIMUM VALUE= 0 MAXIMUM VALUE^ 9 423I6E*O7~~

~~Fig. V-12. Outlet pipe fluid kinetic energy for case 5.~~

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~~466~~

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~~467~~