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A TWO DIMENSIONAL, TWO FLUID MODEL FOR SODIUM BOILING IN LMFBR FUEL ASSEMBLIES by Mario Roberto Granziera Mujid S. Kazimi Energy Laboratory Report No. MIT-EL 80-011 May 1980 -1 i REPORTS IN REACTOR THERMAL HYDARULICS RELATED TO THE MIT ENERGY LABORATORY ELECTRIC POWER PROGRAM A. Topical Reports (For availability check Energy Laboratory Headquarters, Room E19-439, MIT, Cambridge, Massachusetts, 02139) A.1 General Applications A.2 PWR Applications A.3 A.4 BWR Applications LMFBR Applications A.1 M. Massoud, "A Condensed Review of Nuclear Reactor ThermalHydraulic Computer Codes for Two-Phase Flow Analysis," MIT Energy Laboratory Report MIT-EL-79-018, February 1979. J.E. Kelly and M.S. Kazimi, "Development and Testing of the Three Dimensional, Two-Fluid Code THERMIT for LWR Core and Subchannel Applications," MIT Energy Laboratory Report MIT-EL-79-046, December 1979. A.2 P. Moreno, C. Chiu, R. Bowring, E. Khan, J. Liu, N. Todreas, "Methods for Steady-State Thermal/Hydraulic Analysis of PWR Cores," MIT Energy Laboratory Report MIT-EL-76-006, Rev. 1, July 1977 (Orig. 3/77). J.E. Kelly, J. Loomis, L. Wolf, "LWR Core Thermal-Hydraulic Analysis--Assessment and Comparison of the Range of Applicability of the Codes COBRA-IIIC/MIT and COBRA IV-l," MIT Energy Laboratory Report MIT-EL-78-026, September 1978. J. Liu, N. Todreas, "Transient Thermal Analysis of PWR's by a Single Pass Procedure Using a Simplified Model Layout," MIT Energy Laboratory Report MIT-EL-77-008, Final, February 1979, (Draft, June 1977). J. Liu, N. Todreas, "The Comparison of Available Data on PWR Assembly Thermal Behavior with Analytic Predictions," MIT Energy Laboratory Report MIT-EL-77-009, Final, February 1979, (Draft, June 1977). A.3 L. Guillebaud, A. Levin, W. Boyd, A. Faya, L. Wolf, "WOSUBA Subchannel Code for Steady-State and Transient ThermalHydraulic Analysis of Boiling Water Reactor Fuel Bundles," Vol. II, Users Manual, MIT-EL-78-024. July 1977. ii L. Wolf, A Faya, A. Levin, W. Boyd, L. Guillebaud, "WOSUBA Subchannel Code for Steady-State and Transient ThermalHydraulic Analysis of Boiling Water Reactor Fuel Pin Bundles," Vol. III, Assessment and Comparison, MIT-EL-78-025, October 1977. L. Wolf, A. Faya, A. Levin, L. Guillebaud, "WOSUB-A Subchannel Code for Steady-State Reactor Fuel Pin Bundles," Vol. I, Model Description, MIT-EL-78-023, September 1978. A. Faya, L. Wolf and N. Todreas, "Development of a Method for BWR Subchannel Analysis," MIT-EL-79-027, November 1979. A. Faya, L. Wolf and N. Todreas, "CANAL User's Manual," MITEL-79-028, November 1979. A.4 W.D. Hinkle, "Water Tests for Determining Post-Voiding Behavior in the LMFBR," MIT Energy Laboratory Report MIT-EL-76-005, June 1976. W.D. Hinkle, Ed., "LMFBR Safety and Sodium Boiling - A State of the Art Reprot," Draft DOE Report, June 1978. M.R. Granziera, P. Griffith, W.D. Hinkle, M.S. Kazimi, A. Levin, M. Manahan, A. Schor, N. Todreas, G. Wilson, "Development of Computer Code for Multi-dimensional Analysis of Sodium Voiding in the LMFBR," Preliminary Draft Report, July 1979. M. Granziera, P. Griffith, W. Hinkle (ed.), M. Kazimi, A. Levin, M. Manahan, A. Schor, N. Todreas, R. Vilim, G. Wilson, "Development of Computer Code Models for Analysis of Subassembly Voiding in the LMFBR," Interim Report of the MIT Sodium Boiling Project Covering Work Through September 30, 1979, MIT-EL-80-005. A. Levin and P. Griffith, "Development of a Model to Predict Flow Oscillations in Low-Flow Sodium Boiling," MIT-EL-80-006, April 1980. M.R. Granziera and M. Kazimi, "A Two Dimensional, Two Fluid Model for Sodium Boiling in MFBR Assemblies," MIT-EL-80-011, May 1980. G. Wilson and M. Kazimi, "Development of Models for the Sodium Version of the Two-Phase Three Dimensional Thermal Hydraulics Code THERMIT," MIT-EL-80-010, May 1980. iii B. Papers B.1 B.2 B.3 B4 B.l General Applications PWR Applications BWR Applications LMFBR Applications J.E. Kelly and M.S. Kazimi, "Development of the Two-Fluid Multi-Dimensional Code THERMIT for LWR Analysis," accepted for presentation 19th National Heat Transfer Conference, Orlando, Florida, August 1980. J.E. Kelly and M.S. Kazimi, "THERMIT, A Three-Dimensional, Two-Fluid Code for LWR Transient Analysis," accepted for presentation at Summer Annual American Nuclear Society Meeting, Las Vegas, Nevada, June 1980. B.2 P. Moreno, J. Kiu, E. Khan, N. Todreas, "Steady State Thermal Analysis of PWR's by a Single Pass Procedure Using a Simplified Method," American Nuclear Society Transactions, Vol. 26 P. Moreno, J. Liu, E. Khan, N. Todreas, "Steady-State Thermal Analysis of PWR's by a Single Pass Procedure Using a Simplified Nodal Layout," Nuclear Engineering and Design, Vol. 47, 1978, pp. 35-48. C. Chiu, P. Moreno, R. Bowring, N. Todreas, "Enthalpy Transfer Between PWR Fuel Assemblies in Analysis by the Lumped Subchannel Model," Nuclear Engineering and Design, Vol. 53, 1979, 165-186. B.3 L. Wolf and A. Faya, "A BWR Subchannel Code with. Drift Flux and Vapor Diffusion Transport,"' American Nuclear Society Transactions, Vol. 28, 1978, p. 553. B.4 W.D. Hinkle, (MIT), P.M Tschamper (GE), M.H. Fontana, ORNL), R.E. Henry (ANL), and A. Padilla, (HEDL), for U.S. Department of Energy, "LMFBR Safety & Sodium Boiling," paper presented at the ENS/ANS International Topical Meeting on Nuclear Reactor Safety, October 16-19, 1978, Brussels, Belgium. M.I Autruffe, G.J. Wilson, B. Stewart and M. Kazimi, "A Proposed Momentum Exchange Coefficient for Two-Phase Modeling of Sodium Boiling," Proc. Int. Meeting Fast Reactor Safety Technology, Vol. 4, 2512-2521, Seattle, Washington, August 1979. M.R. Granziera and M.S. Kazimi, "NATOF-2D: A Two Dimensional Two-Fluid Model for Sodium Flow Transient Analysis," Trans. ANS, 33, 515, November 1979. wt NOTICE This report was prepared as an account of work sponsored by the United States Government and two of its subcontractors. Neither the United States nor the United States Department of Energy, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privately owned rights. .1 lw A TWO DIMENSIONAL, TWO FLUID MODEL FOR SODIUM BOILING IN LMFBR FUEL ASSEMBLIES by Mario Roberto Granziera Mujid S. Kazimi Energy Laboratory and Department of Nuclear Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 Topical Report of the MIT Sodium Boiling Project sponsored U. S. Department by: of Energy, General Electric Co. and Hanford Engineering Development Laboratory Energy Laboratory Report No. MIT-EL 80-011 May 1980 2 ABSTRACT A two dimensional numerical model for the simulation of sodium boiling transient was developed using the two fluid set of conservation equations. A semiimplicit numerical differencing scheme capable of handling the problems associated with the ill-posedness implied by the complex characteristic roots of the two fluid problems was used, which took advantage of the dumping effect of the exchange terms. Of particular interest in the development of the model was the identification of the numerical problems caused by the strong disparity between the axial and radial dimensions of fuel assemblies. A solution to this problem was found which uses the particular geometry of fuel assemblies to accelerate the convergence of the iterative technique used in the model. The most important feature of the model was its ability to simulate severe conditions of sodium boiling, in particular flow reversal, which was shown in the tests performed with the model. Three sodium boiling experiments were simulated with the model, with good agreement between the experimental results and the model predictions. 3 ACKNOWLEDGEMENT Funding for this project was provided by the United States Department of Energy, the General Electric Co., and the Hanford Engineering Development Laboratory. Additional support was also provided to the Commisao Nacional de Energia Nuclear. Mario R. Granziera by This support was deeply appreciated. The authors would also like to thank their co-workers on the MIT Sodium Boiling project for their help and contributions to this work. A very special thanks is due to Bruce Stewart whose intimate knowledge of the concepts of numerical analysis was an invaluable resource. The work described in this report was performed primarily by the principal author, Mario R. Granziera, who has submitted the same report in partial fulfillment for the PhD degree in Nuclear Engineering at MIT. 4 TABLE OF CONTENTS Title Page 1 Abstract 2 Acknowledgement 3 List of Figures 7 List of Tables 10 Nomenclature 11 Chapter 1: 1.1 LMFBR Safety Analysis 14 1.2 Characteristics of Numerical Models for Sodium Boiling 23 1.2.1 Dimensionality 23 1.2.2 Boundary Conditions: Pressure Vs. Inlet Velocity 26 Two Fluid Model 28 1.2.3 Chapter 2: 13 Introduction The Conservation Equations and the Numerical Method 2.1 The Mass, Momentum and Energy Equations Averaged over a Control Volume 31 31 2.1.1 The Mass Equation 31 2.1.2 The Momentum Equation 34 2.1.3 The Energy Equation 46 5 Chapter 3: 2.2 The Finite Difference Equations 54 2.3 The Numerical Scheme 71 2.4 The Pressure Problem 86 2.5 Stability Analysis of the Numerical Method 93 The Constitutive Equations and Functions of State 3.1 Chapter 4: The Sodium Functions of State and Transport Properties 107 107 3.1.1 Saturation Temperature 107 3.1.2 Vapor Density 109 3.1.3 Liquid Density 110 3.1.4 Internal Energies 111 3.1.5 Transport Properties 112 3.2 Mass Exchange Rate 116 3.3 Momentum Exchange 126 3.4 Energy Exchange 131 3.4.1 Fuel Pin Heat Conduction 131 3.4.2 Fuel Pin Material Properties 137 3.4.3 Convective Heat Transfer Coefficient 139 3.4.4 Fuel Assembly Structure Model 145 3.4.5 Interphase Heat Transfer 145 Experimental Tests Simulation 148 148 4.1 The SLSF P3A Experiment 4.2 One Dimensional Analysis of the P3A 4.3 The W 1 Experiment 170 4.4 The GR19 Experiment 199 Experiment 164 6 Chapter 5: Conclusions and Recommendations 207 5.1 Conclusions 207 5.2 Recommendations 208 References 211 Appendix A: NATOF-2D Input Data Manual 216 Appendix B: NATOF-2D Code Structure Description 228 Appendix C: NATOF-2D I/O Examples 237 Appendix D: NATOF-2D Program Listing 256 7 LIST OF FIGURES No. Possible Accident Paths and Lines of Assurance for a Potential CDA 17 Key Events and Potential Accident Paths for Unprotected Loss of Flow Accident 18 Key Events and Potential Accident Paths for Loss of Pipe Integrity Accident 19 Key Events and Potential Accident Paths for Unprotected Transient Overpower Accident 20 Key Events and Potential Accident Paths for Inadequate Natural Circulation Decay Heat 21 Key Events and Potential Accident Paths for Local Subassembly Accident 22 2.1 A Typical Cell Arrangement 59 2.2 Position Evaluation of Variables 60 2.3 Different Positions for the Radial Velocity 67 3.1 Condensation Coefficienct as a Function of Pressure 119 3.2 Bubbly Flow Representation 121 3.3 Low Void Fraction Bubbly Flow Representation 121 3.4 Suppression Factor Vs. Reynolds Number 141 3.5 Reynolds Number Factor 142 1.1 1.2 1.3 1.4 1.5 1.6 8 4.1 Pin Number Location 156 4.2 P3A: Mass Flow Rate Vs. Time 157 4.3 P3A: Experimental Mass Flow Rate 158 4.4 P3A: Temperature Vs. Time, Central Channel 159 4.5 P3A: Temperature Vs. Time; Edge Channel 160 4.6 P3A: Axial Temperature Profile 161 4.7 P A: Radial 162 4.8 Void Fraction Maps for the PBA Experiment 163 4.9 P3A-1D: Temperature Vs. Time 166 4.10 P3A-1D: Temperature Vs. Time 167 4.11 P3A-1D: Axial Temperature Profile 168 4.12 P3A: Comparison Between 169 4.13 Typical Boiling Window Flow Decay for the W Test 177 4.14 WI: Temperature and Mass Flow Rate for Sequence 5 178 4.15 WI: Temperature and Mass Flow Rate for Sequence 6 179 4.16 WI: Temperature and Mass Flow Rate for Sequence 6a 180 4.17 WI: Axial Temperature Profile for Sequence 6a 181 4.18 WI: Temperature and Mass Flow Rate for Sequence 7 182 4.19 WI: Temperature and Mass Flow Rate for Sequence 7a 183 4.20 WI: Axial Temperature Profile for Sequence 7a 184 4.21 WI: Temperature and Mass Flow Rate for Sequence 7a' 185 4.22 Wi: Axial Temperature Profile for Sequence 7a' 186 4.23 WI: Void Maps for Sequence 7a' 187 4.24 WI: Temperature and Mass Flow Rate for Sequence 7b' 188 4.25 WI: Axial Temperature Profile for Sequence 7b' 189 Temperature Profile D and 2D; Mass Flow Rate 9 4.26 Wi: Void Maps for Sequence 7b' 190 4.27 Wi: Temperature and Mass Flow Rate for Sequence 3 191 4.28 Wi: Axial Temperature Profile for Sequence 3 192 4.29 WI: Temperature and Mass Flow Rate for Sequence 4 193 4.30 Wi: Axial Temperature Profile for Sequence 4 194 4.31 WI: Void Maps for Sequence 4 195 4.32 Temperature and Mass Flow Rate for 217-Pin Bundle Under Sequence 7b' Conditions 196 Axial Temperature Profile for 217-Pin Bundle Under Sequence 7b' Conditions 197 4.34 Void Maps for 217-Pin Bundle Under Sequence 7b' Conditions 198 4.35 GR19: Temperature Profile for .311 Kg/sec Mass Flow Rate 202 4.36 GR19: Temperature Profile for .265 Kg/sec Mass Flow Rate 203 4.37 GR19: Temperature Profile for .260 Kg/sec Mass Flow Rate 204 4.38 GR19: Quality Contours for .265 Kg/sec Mass Flow Rate 205 4.39 GR19: Quality Contours for .260 Kg/sec Mass Flow Rate 206 A.1 Cell Arrangement in the r-z Plane 217 A.2 Fuel Pin Numbering 218 A.3 Cell Arrangement for Fuel Pin Heat Conduction 219 B.1 NATOF-2D Subroutine Structure 229 4.33 10 LIST OF TABLES No. 1.1 Sodium Boiling Issues 16 3.1 Units Used in this Work and the Correspondent Usual Ones 108 4.1 SLSF-P3A Test Bundle Data 152 4.2 Assumed Non-Uniform Radial Power Distribution in P3A Test Bundle 154 4.3 Event Sequence Times of the P3A Experiment 155 4.4 W1 Test Bundle Data 173 4.5 Boiling Window Matrix for the W1 Experiment 175 4.6 Design Data for the GR19 Experiment 200 173 11 NOMENCLATURE Void Fraction p Density e Internal Energy s Mass Exchange Rate P Pressure U Velocity f Friction Force g Gravity Acceleration V Volume .~~~~~~~~~~~~~ . A Area M Momentum Exchange Rate Q Heat Exchange Rate D Fuel Pin Diameter Ay Radial Mesh Spacing Az Axial Mesh Spacing At Time Step Size 12 SUPERSCRIPTS N Time Level K Iteration Level INTEGRALS Integral over the Volume Occupied by the Fluid Alone (Fuel Pin and Structure Excluded) dV V Surface Integral AdA Integral Over a Closed Surface ~A d A AVERAGES =V X X V A= I= X(r,z)dV = Volume Averaged Quantity t X(r,z)dA = Surface Averaged Quantity 13 I INTRODUCTION The growing public concern about the nuclear industry places an increasingly large emphasis on the safety aspects of nuclear reactor design. In particular, commercial size liquid metal cooled fast breeder reactors (LMFBR) with its large amount of plutonium fuel, combined with its inherent safety problems, namely the potentially positive void coefficient of reactivity, and the high chemical reactivity of the liquid metal coolant with air and water, must be designed, constructed and operated with large safety margins to assure that the public risk will be acceptably low. In order to accomplish the stringent requirements of safety, designers must have a thorough understanding of the phenomena occurring in all possible reactor situations and the adequate analytical tools to correctly predict the reactor behavior in all possible situations. The objective of this work is to provide an analytical model capable of predicting the transient sodium boiling in LMFBR fuel assemblies under realistic conditions. In order to situate the model proposed in this work in the broad field of sodium boiling a review of LMFBR safety analysis and a general description of the accidents of principal concern will be presented, followed by a review of the present status of analytical models currently available. 14 1.1 LMFBR Safety Analysis The U.S. Fast Breeder Reactor Safety Program approach is to provide four levels of protection, which are aimed at reducing both the probability and consequences of a Core Disruptive Accident (CDA)[1]. These levels of protection are referred to as Lines of Assurance (LOA). Figure 1.1 illustrates the possible accident paths for a potential core disruptive accident. The first line of assurance aims at reducing the probability of occurrence of a serious accident. The emphasis is placed on quality assurance, inservice inspection and monitoring at the level of construction and operation, and at the level of design on providing a multilevel redundant plant protection system, which can quickly respond to faults and place the reactor in a safe shutdown condition without damage to the coretl,2]. The second line of assurance assumes that in spite of the measures taken in the first line, low probability but mechanistically possible events involving failure of the first line systems will occur[1]. The strategy in this line is to provide the reactor design with features which make the system respond inherently to accidents in a way which tends to maintain reactivity control and coolability, containing the damage to a limited number of fuel assemblies. 15 The third and fourth lines of assurance aim at limiting the consequences of a serious accident. It is assumed that the first two lines have failed and two subsequent events form a potential sequence leading to core disruption and release of radioactivity to the environment. The objective of these last lines is to make the consequences of a core disruption accident sufficiently limited by the plant containment capability which combined with the low probability of occurrence of the failure of the first and second lines of assurance makes the risk to the public acceptably small[1,2]. Some of the issues concerning the possible accident paths are still unresolved as are some of the phenomena involved in very low probability events. In order to assess the importance of sodium boiling and two phase flow in the general picture of LMFBR safety analysis we reproduce from a compilation of the state of the art in sodium boiling by Hinkle [2] figures 1.2 through 1.6 illustrating the path of the most serious of the postulated accidents considered in LMFBR safety analysis. In all these accidents, the occurrence of sodium boiling and the stability of two phase flow assume a crucial role in determining the path, speed of events and final consequences. reference In table 1.1, also reproduced from 2], the technical issues which must be resolved related to sodium boiling are presented. H f4 ., 4-i 4-J w4-i a-)44 4-i a) 4. 'IO i UiJrl4'rU4O w -Hc uo .4i a) 3¢> H 'H 4 J4 J 4. 4 4J P ~ 4a) . 4-I c > >H Q) V V,-~ O o r5 .j U :1 u ci > 0<T.CIO 4 0H rj O 0 (n 44 r-l Hct o>U V~a O -H LO O :U > O>1 (J a3) 'ici 'VH Zn4i (ng 0 0, a) - O) H a) FW 4i ,-Q . ¢VH o 4-4o 4 -).HJU C OW44) b-H rW a) CO Uh a) C 0a~ c a) 4-4 a) 'IT :I MU 4d)0 4j CO O ,. 0ooo 3,~ -4a O E, H E~ U)O U U) O O ~ oU)CU u .n-i ,--t C o04P- oo~ Ho C b$o 1 0 p4¢) (J ¢~ I~-- ,' ' a) to U) ,4-4 04,- ,0 0 0U' o 0- o E'-4 .'H 10 CUc H HCl) o O oO 0 0 oCU 0j4-4 $ ca $4 ,4 O a) a) 3 L EO CZ'~:3 tU CUH ) C 1-4 ~4oUa O0j a) H o '4-4 4i 4-i 4-i QU) C 4-4 c- r z 3 C C U 0 n t) *r U)Gc 1 0 r. 4- 0 t-4 UCOO o4.i-44 4 0 a) Ir '0' a) $4 ) - H Ca) 75 C cU C) to 4a i 0 CO C)r C*HCU 4. Q < 0e Cl H '0 C) H?3 0a) I-icH U 4i 2 > . -r- L6 0 3 44 -'4 4 O Sl cI-c a) x U H . C 0Z$4 4-O~- o- a) H 4'e X O V) i 0 4 4J 4 v4 d)0 4 ) 4-4 ' UU) CO O4C) 4-0 O c4 CU4-i 4-i C)HH CU- '-4 0 ' XP4 t$ 4i --:3: 0$ ~ O~'-UH aX4 0C OH C U 4H0 O P U -H | - ' 0 UtO HiU ,C . O4- CO 4 4H a 4- -,e i 4 Q0 O -4 4-4 a) W OC .a) 0 4J U -4 O 0 bi a) 4cU H cU U ) 04-iCV CU : U -4U CO U h4 H4 u 4' ct o U) 9 $4 'H 0 bCo0H 4- 0i 'H O o3 ua) H 4- H H 0 1-0 cd 0) C 0 0 '0 "A H 'c c +4 H 44CU -H 44H o ' CU a$ W H $4 a)4-. a D U OoNU ..I 4,.fOH t0a-iJ 'H '-1O O ca 4,QH 4-4 U) $4 "0 a 0 a) O oH O H 0 r$4 '0 4U$H~ CU 4~~~~i vW 4 r = ' 0 $4 :j CU > 0 i NQ H 'H O) rn CO4-i0 r-{ 0 u 44 H) > 1-- w ) U) CU 0 1 0 44 $4'H 0 = q- r--.n -H J,HUP ·i,-ica )ca4JoCU oa H > V Ha 0e O4 W 4- Hi H a0 '.4 UC u -- U 0 c i 04 CU 0 H -- 1. ::- , 0 0- U) - C 4-3 0 - 0 a) 3-i 4-4 0 u 4 l '' U ~ o3U VO--. rl ~.. ~ ,O O G)HVO U) a) U) On U) 6z H 4-i 0 4 0 U 0 O H) 44tH O 0 U) O~ a) CU4-4 CU4W 4J4 1-44- > O Hw a)' 4- - - - - - - 17 Reactor Operator ; m=1 ]Mt I I I][ll I Fault Occurs I Inherent Response or Fault Detection and Control Action I I I I III J nm I .[ -I I ~ JI I · ,! I[ Ill I I No No Damage Damage . _ PPS Action Required IIII . . A_ K PPS Action No Damage . . LOA1 , PPS Failure, Some Clad/Fuel Melting and Relocation iiiii I Inherent Response Leaves Core and HTS Intact and Coolable .. ~~ ~ ~ ~ _~~~~~~~~ F( i i Limited Core Damage _ LOA2 Core Eeltdown or Di.spersal , , ,,, , , Accident Progression Controlled, Containment Intact Controlled Release to Environment y ff LOA3 [l [. Accident Progression not Cc)ntrolled . Containment Limits Release to Environment -- Limited Release" to Environment I LOA4 I ___ __ Uncontrolled Re-' Containment Fails ase Figure 1.1 Possible Accident Paths and Lines of Assurance For a Potential CDA (From Reference 2) to Environmen I , 1 18 Fault Occurs Leading to Pump/ Flow Coastdown & Core Heatup .I. I I. m II - i. ...- Scram; Reactor Reactivity& Failure to Scram Power Decrease ._ _.... !,. I Boiling Flow Transfer; & Power Two-phase & Heat Reactivity Increase No Damage C , |~~~~~~~2 Flow/Heat Transfer Instability & CHF I i Fuel Pin Failure & Fission Gas/Molten Fuel Release -. . . I 11 Reactivity/Power Increase Due to Fuel Motion lU[ I [ i [, _ Reactivity/Power Decrease Due to Fuel Motion I II I {l MFCI & Subassembly Voiding Some Clad/Fuel Melting but Adequate Cooling Restored ,,. .... Voiding; ,i. I Mechanical Disassembly i Clad Melting & Relocation I I . Bulk Subassembly I_ ,~~~~~ c Limited Core Damage i . -_ ~~~~ Core Geometry Not Coolable; Gradual Meltdown . .i i I FCore Disruption ore Disruption Accident Accident Figure 1.2 Key Events & Potential Accident Paths For Unprotected Loss of Flow Accident 19 Fault Develops with Potential for Loss of Pipe Integrity I -1 II1 Fault Detected Operator Action Fault Undetected Loss of Pipe Integrity I· i i ~ No Damag f_ i 7~~~~~~~ Flow Decay and Heatup I Reactor Scram Reactivity and Power Decrease I i ! i i II I Boiling, Two Phase Flow and Heat Transfer II I Fuel Pin Failure and Fission Gas Release I, i II ,l i Flow/Heat Transfer Instability and CHF II .- , . ~~~~~~I Adequate Cooling No CHF .Limited ore Damage ] -1 II -. l . Bulk Subassembly Voiding; Clad Melting and Relocation 11 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Some Clad/Fuel Melting and Relocation but Limited Adequate Cooling CoreDamage Restored ii · I I 'I' - Core Geometry not Coolable; Gradual Meltdown f I- Figure 1.3 Key Events and Potential Accident Path For Loss of Pipe Integrity Accident (From Reference 2) CDA 3 20 Fault Occurs Leading to Transient Overpower and Core Heatup I il Reactor Scram Power Decay Failure to Scram . .- Fuel Pin Failure and Fission Gas Release Damage I o Damage i -gm - -- - - - I Predominantly High Axial Failure Location Substantial Number of Axial idplane Failures I I I! aft I A__ _ . _ _ . _ II _ _ 'I Fuel Relocation to Form Partial Blockage .r i II II I ____ Reactivity and Power Increase MFCI and Subassembly Voiding Im I II Ill _ ii i I I !! Fuel Relocation to Form Partial Blockage I .. . . . 11 i . 5 Fuel Sweepout No Blockage . 0_ .. .. Reactivity and Power Increase Boiling in Wake _n , Reactivity and Power Decrease Adequate Cooling Without Boiling _ of Blockage .! iI Limited Mechanical Disassembly _ -, Inadequate Cooling Core Damage7 or MFCI: CHF and Blockage Propagation ~~~~~~~~m,1 I .-.-I H C ~-- CDA 7 ii Adequate i -- Cooling No Blockage Propagation .~~~ I I imited Core Bulk Subassembly Voiding and Meltdown ( )anmage c Figure 1.4 Key Events and Potential Accident Paths For Untprotected rantsietrt (Froill Rt'C L' verpower Accident l¢'M'2) CDA ) 21 Reactor Shutdown; Power & Flow Decrease; Cooldown I D Loss of Forced Cooling in Primary Loop Single-Phase Natural Circulation Flow & Heat Transfer I i r II -1 Boiling Natural Circulation Flow & Heat Transfer Adequate Heat Removal with no Coolant Boiling .1_ I _ L I c No Damage II ) Fuel Pin Failure & Fission Gas Release . I . . I Inadequate Heat Removal; Flow/Heat Transfer Instability & CHF dequate Heat Removal .Until Forced Cooling Restored I Bulk Subassembly Voiding; Clad/Fuel Melting & Relocation I - c A. I No or Minor Core Damage A= Core Geometry Not Coolable; Gradual Meltdown ore Disruption\ oAccident Figure 1.5 Key Events and Potential Accident Paths for Inadequate Natural Circulation Decay Heat Removal Accident ) 22 Local Fault Due to Clad Defect, Fission Gas/ Molten Fuel Release &/or Blockage Formation _ .. Local Flow Restriction & Increased Coolant Temperatures in Wake I Fuel Pn Failure & Fission) Gas/Molten Fuel Release; Gradual Blockage Propagation i 0 . _ . Local Boiling in Wake I . _ . I , I iI[ii 1 ,i _ . -Fault Tolerated or Detected; Operator or Control Action Flow/Heat Transfer Instability & CHF _ i i i ~~~~~~~~~~~~~~~~.. i1, ! I _ II I _~~~~~~ No or Minor Core Damage Pin-to-Pin Failure Propagation *C u ,~~~~~~C t' Bulk Subassembly Voiding; Clad/Fuel Melting & Relocation I . . . . .. . I- .. i) [ ,, Molten Steel-Sodium Interaction; Subassembly Wrapper Failure [ I _ Core Geometry Coolable; Adequate Cooling Restored V I III Ill I [[ _ I - Fault Tolerated or Detected; Belated Operator or Control Action , I I III Ill Limited Core Damage Subassembly to Subassembly Propagation C I Core Disruption Accident Figure 1.6 Key Events and Potential Accident Paths For oca1 Subassembly Accident II I ) r , I 23 1.2 Characteristics of Numerical Models for Sodium Boiling In the following paragraphs the most important characterstics of numerical models relevant to LMFBR fuel assembly fluido-dynamic analysis will be discussed, along with a comparison of the capabilities of the models presently available and the one proposed in this work. 1.2.1 Dimensionality . ~~~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ . It-is a well recognized fact that a non-flat radial temperature profile exists with steady-state conditions, as well as at the onset of boiling in loss of flow transients [3]. Calculations made for single phase flow with the COBRA III-C code [4] showed that a temperature difference as high as 450°F may exist between the central and peripheral channels in a typical FFTF fuel assembly (Figure 1.7). Obviously this temperature profile will force boiling to start in the central part of the fuel assembly and progress afterwards in the direction of the periphery. During this process, while part of the fuel assembly is under boiling, and the fuel pins in this region may eventually be suffering some damage, the periphery of the fuel assembly still maintains its coolability. 24 F TEMPERATURE AT EXIT HEATED ZONE 0C0 0 00 _ 00 I I 0 0 0 I 0 _QfI C 000o I I I I I 00 I cn 0 fD (n rt W CD CL $ o M " rt CD D m rt I'd 0 -f CD n 0n' rt rt I Ul * *0 CD F~t 4S 0rttd 0' 0ct 0 0 0 r-f fn (D o ZI (t-j 0o I - z I - u Figure I ~ ~~~~~ .... I 1.7 Radial Temperature Profile for 217 Pin Assembly I]-I I I I I 25 Here is a multidimensional effect the timing of which has a direct effect on the amount of damage resultant from the accident. Also, since this radial incoherence is an inherent design feature of all fuel assemblies, the radial void incoherence is expected to occur in every boiling transient. Although this effect can be well represented by a two dimensional model, covering most of the transients of concern to LMFBR safety analysis, two cases present a non-radial symmetry, thus requiring a full three dimensional model for their representation, namely the transients with a non-unform power profile and an asymmetrical flow blockage. Considering the limited number of cases requiring a three dimensional model compared to those which can be analyzed with an axial-radial representation, and also considering the necessarily larger computational time required by a three dimensional model, it seems clear that a two dimensional model has definitive advantages. As for the present situation in computational modeling, the only existing numerical model which can claim success in applications to sodium boiling is the SAS code, which is a one dimensional code [5]. Other codes such as the HEV-2D [6], COMMIX [7], BACHUS [8], to mention only a few of them, have encountered some difficulties in representing sodium boiling up to the point of flow reversal. Therefore a new code with two dimensional capability seems to be well situated. 26 1.2.2 Boundary Conditions: Pressure Vs. Inlet Velocity The boundary conditions applied to the problem are strongly related to the numerical solutions used in the model. In this way, the marching technique, where the solution of the fluido-dynamic equations is obtained successively in planes along the direction of the main flow, can only operate with inlet velocity boundary condition, whereas the simultaneous pressure matrix inversion can work with both kinds of boundary conditions. The advantages of the marching technique are its numerical stability for arbitrary large time steps combined with its relatively quick and straightforward numerical procedure' Its limitations lie in the assumptions necessary to the validity of the marching method: it requires that the flow be predominantly in one direction and always in that direction, making it impossible to analyze any kind of flow reversal. Also certain transport terms in the transverse momentum equations are ignored, whereas there are some doubts on the validity of these assumptions 9]. The simultaneous pressure matrix inversion method avoids the limitations of the marching technique at the price of using a smaller time step and a more laborious numerical solution. In this method, the solution of the fluido-dynamic equations is performed, at each time step, simultaneously for all mesh cells of the problem. In 27 general, this simultaneous solution can be reduced to a pressure matrix inversion. In this way, upstream propagations can be accounted for, and flow reversal transients can in be in principle analyzed. The method does not impose any limitation on the number of conservation equations, therefore the choice of any model, from homogeneous equilibrium to the full two-fluid model is allowed. The disadvantages of the method are that because of the large number of unknowns involved in the matrix inversion, a fully implicit differencing scheme becomes practically impossible, and a semi-implicit method, with its consequent limitation in time step size, becomes practically the only option. Another problem with this method arises when used in conjunction with a multidimensional model. When the conservation equations are reduced to a pressure problem, the resultant pressure matrix becomes only marginally diagonally dominant, the diagonal dominance being provided only by the compressibility terms, which in some cases may be very small. In these situations the usual techniques of matrix inversion fail to produce a solution in a reasonable computational time, and special procedures must be introduced. Indeed, the ability of the model proposed in this work to produce results in a reasonable aMount of computational time owes much to the special technique devised for this matrix inversion which is presented in section 2.4. 28 1.2.3 Two Fluid Model In the early years of two phase flow modeling, much attention has been given to the homogeneous equilibrium model. This model describes the two phase flow in terms of average quantities, such as the density and velocity. In this way, these quantities are defined to represent an homogeneous mixture of the two phases (or two fluids). There are situations during reactor core transients where the assumptions required for this modeling depart from reality, namely when either phase does not stay close to saturation conditions and more importantly when the phase velocities differ substantially. Attempts to circumvent these limitations were made with the introduction of semi-empirical correlations to describe the unequal phase velocities, the so-called slip correiations, and to allow non-saturation conditions. Because of the semi-empirical nature of these correlations, their accuracy is limited to the range of variables for which they were developed, and their generalization is restricted. A new approach to overcome the limitations of the homogeneous equilibrium model was attempted with the drift flux model. This model stays in between the homogeneous equilibrium and two-fluid models in terms of the number of conservation equations employed. Although some variations on the particular set of equations composing the model exist, in general the drift flux model represents the two phase flow with a set of three mixture conservation equations plus two equations 29 for one of the pairs mass-momentum, mass-energy or energy-momentum for one of the phases. In this model the sophistication in the direction of being closer to first principles is increased over the homogeneous mixture model, and so are the complications and size of the numerical solution technique. Indeed, there are some doubts about the computational time advantage of the drift flux model over the two-fluid model. The two-fluid model represents the fluid flow with two complete, separate sets of consevation equations, treating individually the properties of both phases. Its clear advantage is that no assumption is made on the relationship between the properties of the two phases, and the most general situations can in principle be represented. The model requires constitutive expressions for the interaction between phases, namely the exchange of mass, momentum and energy. Unlike the slip correlation this constitutive expressions do not depend on circumstantial conditions of the particular flow situation, but on the physical principles of the transport phenomena involved. Much work has to be done in the field of the constitutive relations required by the two fluid model, and the work presented here cannot claim to represent accurately the two phase flow phenomena without first solving the problems present in this area. Nonetheless, the model presented here can serve as a valuable tool in developing and testing the much needed constitutive relations for sodium two phase 30 flow. This subject will be readdressed in chapter 5, when the recommendations for future developments related to this model will be presented. A final word has to be said about the controversial issue of the complex characteristic roots of the two fluid model, and its consequences to the stability of its numerical solution. Although this subject will be addressed at length in section 2.5, for the moment it is sufficient to point out the inadequacy of using the techniques of partial differential equations and numerical analysis developed for linear systems in analyzing the thermo-fluido dynamic equations, which are non-linear. Therefore, any conclusion on well-posedness of the two fluid model and the stability of its numerical solution has to be drawn from an anlysis which takes into account all the characteristics of the model, in particular the damping effect of the interphase exchange terms. The discussion of the porous body versus the subchannel approach is deliberately omitted here. It is our belief that this issue does not play any important role in the numerical treatment of reactor fluid flow. Indeed, it is possible to extend the porous body model to the limit of very small mesh cells or lump together subchannels to form larger ones. The two concepts overlap completely and no relevant distinction between them can be made in the numerical aspects of code development. 31 THE CONSERVATION EQUATIONS AND THE NUMERICAL METHOD II. The Mass, Momentum and Energy Equations Averaged over 2.1 a Control Volume. In this chapter the derivation of the differential and difference form of the conservation equations will be given. First all assumptions built into the model will be detailed, providing a Secondly, the clear picture of its limitations and range of validity. precise meaning of each term in the set of equations will be established. As will be seen later, this is particularly important with terms describing the geometry of the interacting cells. For the sake of compactness, and to avoid being monotonous, details will be given for the equations of the vapor phase, mentioning only the final form of the equations of the liquid phase. This will cause no lack of understanding, since the two fluid model is completely symmetric with respect to the liquid and vapor phases. 2.1.1 The Mass Equation The mass equation has the form: t v apv dV+ A+ z+ - A f| v z- apvUdA +fA vz - fA r+ (Se Sc) dV r- pUvrdA vr (2.1) 32 The density as well as its first time derivative are assumed independent of position within the volume occupied by each phase separately and the void fraction is also assumed independent of position within the control volume. It follows that: Pv dV = < Pv a p U > v dA = < Vv l (2.2) dV = < Pv > < a > V f U > A V v (2.2) A (S e - S c ) dV = (< S > - e < S >) V (2.3) c We substitute these equations into our original mass equation, and we get a < a Tt > + A z < a V P v U > vz - AZ+ A 7-- A V v r Ar+ - V A r- <S e > - <S c > (2.5) 33 In the above equation we have introduced the areas A bounding our control volume. A and z The axial cross sectional area A r z poses no problem, since in the particular geometry of fuel assemblies of interest for L4FBR it is a constant throughout the axial length. For the radial cross sectional area however the same is not true. Here we have a highly position dependent area, not only in the macro scale, i.e., from one control volume to another, but also in the particular position with respect to the fuel pin rows chosen for this area. So far this position can be chosen arbitrarely. Later it will be seen that for the averaged radial velocities in the momentum equations to be compatible with those in the mass and energy equations we must impose a precise value for this radial cross sectional area. The choice of this position is postponed until we have developed the momentum equations. Finally it can be easily inferred that the liquid mass equation will undergo the same steps and present a similar form: [(1- -t < pj >) [<(l-a~) +V p iz A+ A Z + z V r+ -O )p 9 Zr Ar A r- < V ( 1 - ) p U< k Zr >< SA r- c >< S > e (2.6) 34 2.1.2 The Momentum Equations Following the same procedure used with the mass equation, the momentum equations in a control volume form are: Axial Directi af at v a PvI U vz Jz a - A AZ+ p.k.^ -r dV +f on ^f ndA p U Ar+ V w dA U v vz vr r- Pv g dV + fMvz dV v A pU a A = - f fvz dA-f A f dA + vzA Az_~ (2.7) v Radial Direction a a at v Pv Uvr dV +f P . r . n dA = A v a Pv Uvz Uvr dA + -f AZ+ -f A w A r dA V fM U a p -f A+ A U2 v dA vr (2.8) dV v To obtain the momentum equations in non-conservative form the mass equation (2.1) multiplied by and < Ur 'r > is subtracted 35 from equations 2.7 and 2.8 respectively: at -< v vz V U ( S >f vz v acc ta PdV + V - ) dV - S c (2.9) a -atJJ p V dA U U VZ vr z- > fA -f AZ c pvUvdA +f-fr A+ At+ f vr a Pv U A r_ dA +f v p M vr fA v s dV - fAr+ -fA w e vZ vr Az+ IA dA n dA = z+ -t vz a Pv vvz U U vr dA vz dV - A > vz = vz I Z(-f > vz dV- v v P at + ( U -< Uv> ~vz f at dJ V (2.11) v >< > -a at V v and a t v Uvr V dV a at J a PdV v = < a >< V v > at < >V yr (2.12) Next consider the convective terms. rr yr ~ 2 f-A a Pv UvzdA Uvz A= z z = a P UU vz A Z r J a Pv vz J dA We define: A 4v z LF. A0va z i -1 (2.13) 4j -1 pUv dA vzdA z (2.14) 37 <c p < a p U > vz A v > U v = 1 A z vr A = z A z A z A dA a P U Pv vz (2.15) a p (2.16) z V U dA vUdA z Assume Uz is position independent in each axial cross vz sectional area A . Also assume that Uz and Ur are axially variable in such a way that: <Uvz vz <vr = 1 ( ~< = vz A ) (2.17) A ) (2.18) + <Uvz > > vz A+ + Uvr >A vr z- or in other words, that these velocities have a linear axial variation. The axial convective terms in both momentum equations become: +-f z+ a p v U2 vz dA - f vz z- = < a > < Uv vz > < UVz >'A Z+ -fA z+ - a p v Uvz vz dA < UVz>A Z- (2.19) 38 cXv A Z+ - A fA - > < UVz > < L vr - >A vr Az A dA = (2.20) A - vr z+ p U z- z+n = < a > < Pv a Z~~~ -1 Using the same procedure to the r-convective terms, we define: A > [ = r r [IA PV ] dvr A vr r [ A ccPv PV v ap vUdrA (2.21) [· (2.22) r aP A 1 [JA U r vr U vz dA a P U vr dA] v r Following the same procedure taken with the terms averaged over Az, we must find an expression relating the properties averaged over the radial area and over the entire control volume. variation of Uvz from A to A+ The linear can be assumed without imposing simplifications beyond the level that has been used up till now. same is also true when it is assumed that U vr radial area A . The is constant over each But due to the particular geometry of fuel assemblies under consideration, it will not be realistic to make a linear variation under consideration, it will not be realistic to make a linear variation 39 of U assumption for any arbitrary radial area, since due to the yr presence of fuel pins, this radial area varies drastically with radial position. = Instead, A 2 ( and A r+ ~ A+ r must be chosen such that: +<Uvr >A ) (2.23) r- Introduce the quantities U*(r) and A*(r): r < Uvr >A(r) yr A (r) r r Ar(r) = U*(r) A* . r r r ' (2.24) where A* . r is the average linarly radial dependent area. r This is equivalent to smearing the fuel pins over the fuel assembly to produce an equivalent homogeneous porous body. The criterion to find A* is r to require the integral of A* . r over one unit cell be equal to the r volume occupied by the fluid: rk+ E2 Vk cell where = A* rk r /3 /=3 r2, r dr 2-r = A*r 2 (2.25) 40 The volume Vcell is: cell A ) (2k+l -- 2'/3 Vk 2) 2 - pin cell ( p Az where Apin includes the transverse area of fuel pin wire wrap and pin other structural materials present r 23 rk=kP From equation (2.25) we get: A A* = Az pin 2 p /3 2 r (2.26) We now can make the more acceptable assumption that A Ar(r) r = U*(r) A* r . r r = constant. <U< vr > <a> 1 It follows: drffr f a Ur dV= ~V dr U r +< =I r V r dA A (r) dr A ~~r r > U vr (r+- = U*(r*) . A* . r* . r vr r V r_) (2.27) 41 and r+. where r* is any value between r A* r* (r r +- s choose r* such that Let (2.28) r ) but from equation 2.25 we have: 2 A* r dr = A* V = | r r 2 r+-r_ r 2 so it follows r r++ (2.29) r* -= 2 Substituting for r* in equation 2.27 we have = (2.30) U*(r*) r but since we assumed U* is a linear function of r this is equivalent r to U*(r ) + U*(r +_ = I 2 vr Ar+ Ar+ A* rv + < A > Ar_ r A* (2.32) r- 42 Finally, the desired criterion for choosing the radial cross sectional area such that the averaging procedures taken with the mass, energy and momentum equations be compatible follows immediately if r+ and r- are chosen to satisfy: A A and r- = A* r+ = A* r- (2.33) then the desired equation 2.23 is obtained + = > Ar 2 with these considerations, after a few algebraic steps, the r convective terms in the momentum equations become: I Ar+ a pv Uvz Uvr dA - > vr v U2 vr (< Uvz > vz Ar+ r+ Uvr dA = vr r- > ) (Ar+ Ar- a Ar- V r-) 2 pv Uvr dA = vvr (2.34) 43 > < < v r > ( - < Ur >A > yr vr A r+A+ - (2.35) vr A82 The remaining terms of the momentum equations are obtained by simple averages: n dA = A P . k. z < >(A Av +A +A r . n^ dA =r P . '~ z+ (sA r - 2 fv A f dA= vz <f vz > A A f Aw w vr d fM dVvz v v f M dV- =< f A A w vz V > f r+ w w w w ( s - S e (2.36) Az- ) - c ( S - S ) dV = < M vz ) dV = <M' > V > V < P A r- ) (2.37) 44 t Where the terms ' include both the momentum exchange between phases due to friction and mass exchange. These terms will be analysed in detail in Chapter 3, when discussing the constitutive equations. Equations 2.9 and 2.10 can now be rewriten as: A < >> >< ~t V A(< A) 2V + (Ar+ +~~ z + U't < VZ vr A vz - A r+ (< > vz > UVZ vz + - < Uz >A) vzA Z-/ + Uvz < A)] r- A - t> < V >) Z- A _w < f V vz > > < Pv >V < r < a > < > Ar+ +Ar + 2V - - r + ~2V / A(< > ( + z > V+ - vr > Ar+- A A ) A r- -< vr Az+ >A - U > vr Az_ + - vr Ar A P~~r U vz A V V < ff vr > - < ~II' vr > (2.39) 45 Similarly for the liquid phase: (1 - < a > ) at P, +A A + r+ 2V I r-)< r < U Z > A +- < > - A+ z - - >A ) ( AZ+ < >A) Z-) A. = 2W < f z 2 > - 1 - < g - < t kz (2.40) > A 1 - - < a > )(< (1- < V kr + Ar+ +A r- A) r- Ar+ > < Qr > (- ) > < Ur >A +)<Ur U ~,> >Ar+ - A r+ >- z A r- > Ar >Ar- ) (2. 41) 46 The Energy Equations 2.1.3 Again we start by writing the energy conservation equation in control volume form: 4# +4 raC t ~tfi V vv+ a U v vr dV + a -fA fA A~~~~~~z+ (e + U v v V U2 V ) dA + UVz ( ev + ) dA = r Ar+ =j ½- U 2 ) Pv (e Qv dV V- Qv v Uv a pv g Uvzd vd v f iW p - P . n.UVdA dA + v A V dV (2.42) V Before proceding with the averaging process, some algebraic manipulations will be made in order to eliminate the kinetic energy terms. Subtract from equation 2.42 equations 2.9 multiplied by Uvz >, and rearranging the result it follows by <multiplied and 2.10 >, and rearranging the result it follows and 2.10 multiplied by vz p at U - a vz vr at v p U vv r 1 dV + V z+ + CX P [U~ ½ Cl p~U [Uv Lv ½ - f -P v U vz 22 _ U vz << vr >> U dA + vrJ vz z- Ov A = 2 v Qv vr U2 - v a dV +f dv -fP P v~~~~ U vz n*.uV -Pn U vr - U2 dA = vr .k<Uvz V~~~~ n . r Uv . f dA- - f VZ VZ - f dA- vr w W J [cPv g Uvz - < Uvz > Pv g dV - V f M + vz vz vr ~~~~~~V M ] vrJ dV (2.43) We will turn our kinetic energy. attention to the terms involving the To avoid the trouble of carrying over the whole expression, we will call: J[ ata V ½ Uv2 -at v Uvz - ata p v Uvr dV = E 48 1E= f i p vt ½ UV.2 ½ _ at Uvrj1 + aa p + at U2 V U VZ VZ < Ur - vr > vr ] dV (2.44) It must be assumed that the spatial variation of U and vz Uvr around their mean values are small or in other words, if U is written as:VZ as: written Uvz VZ (z , r) = + VZ (z , r) then < 2 vz 2 >= 1 V U V vz [< The requirement that 1 > + E(z , r)J f C2 dV =< vz >2 +2f v E2 (z , r) dV V >2 dV be small compared to 2 equation 2.44 becomes: (2.45) 49 1E= V < [a > < Pv> ~~--< ++ [1/2 Ll/2 vz at (< U2 ½ >- > + - < vr _ 2) >' vr J Uvr>1 at vr > + 2 + a fc >2] + < kv >) a PdV (2.46) atv vr the convective terms of the kinetic energy equation, which will be called 2E and 3E are: 2E = f A -f z+ A dAa pv [u U2 v - U2 - r vz UU vr z(2. 47) -f _ 3E = A r+ A dAu ½2 vpUvr - vz U vr - U vrr] r- (2.48) Define: 2> 2< v Az = U2 < vz + > A z < U2 vr f > A z = A Az pv 2 Ir~~ Uu 2 dA /f J ZA z p U dA 50 and equations 2.47 and 2.48 become: 2E = [ ½ < U2 > v [ < U2 v AZ > A, - <Uvz > <Uvz >A - - vr > [½ < U2 v > A r- a p v vz dA - A A aP U vz dA (2.49) A+ r+ vz - Azp A + ><Uvr v > vr - > r Ar+]A r+ r -A U >< > vz A r- | A A a P U dA v vr (2.50) 51 and from equations 2.17, 2.18 and 2.23 23 get: 2E = - 3E = - [u A < z +Aif] A > > ~ vz Ar_ f a pvUvzdA - AZr A+ ~~A_ > A A Az+ A A+ <Uvz >A z+ 2E = - " z- < U2 > etc, and it follows vz a p < u2> v A-f Az+ 3E =- /r < U2 v -f U dA (2.51) a Pv v Uvr yr dA (2.52) v vz r- Combining the terms 1E, 2E and 3E, and recalling equation 2.1, we have: 1E + 2E + 3E = - (< S e > - < S c >) < U2 > v We proceed by noting that some terms in equation 2.43 will vanish uppon the performance of the integrals. These terms are: (2.53) 52 P n V v 4 [ A A v w k [ Pv g U J - vz (2.55) vr dA = 0 - f >fv vr vr VZ < Uvz dA = 0 r < Ur > - P n VZ (2.56) The next step would be to define average properties and Since this procedure obtain the final form of the energy equation. is completely similar to that used for the mass equation, only the resultant energy equations are presented here: -[< t a>< 0vv >< e. a >< e rv 57 1 v Ar+( + - <a p V =v t Qv- > vz AzA - < + a pv ev U vz >A ] A ev Uvr >Ar+ v < > D Dt - r-<a V QV kt p v e v U vr > A r- > where the energy exchange between phases has been grouped under the term Q (2.57) 53 For the liquid phase the nergy equation is: t (1- <a> ) <e>] + (1- a)p e Ar+ +- V - <(1 -)p e U >A] A <(1 - a) p kz e U r>A 2,r A+ - r++ =< Q r- <(1- V>. V > + k +.r a+ < ~Dt Q ) p kV e > 9. U r>A A ( (2.58) 54 2.2 The Finite Difference Equations Having established the properly averaged differential equations, the next step is to approximate the conservation equations by a set of algebraic equations suitable for the numerical solution. Before choosing any particular scheme, it is appropriate to discuss in general terms the various applicable finite difference approaches and identify the kind of problems we expect to solve with our model. For the spacial discretization very little can be said in general: the idea to be followed is that one can find the best spacial differen iation to suit a particular time discretization. There are three broad categories concerning the time level at which the variables are to be evaluated: and semi-implicit (or semi-explicit). fully implicit, fully explicit Associated with each of these categories there is a stability criterion which will relate the time step size with the characteristic roots of the set of equations. A fully implicit method is the one in which all spacial derivatives, as well as all the exchange terms are evaluated at the new time level. With this time discretization it is possible in general to find a spacial arrangement which makes the whole method unconditionally stable, thus enabling the problem to be solved with a time step as large (or as small) as desired. As an exemple, a one dimensional mass equation in this scheme is: n+l n+l ai Pi At n n i Pvi Vi U + ( a v n+l nl Uv)i a Pv Uv Az =Sn+l i 55 Note that the convective terms are evaluated at different locations than the other terms. Since they are evaluated at the new time level, they are unknown, which means the solution at one particular cell is coupled to the solution at its neighbors, thus requiring the numerical solution to be made simultaneously in all locations and all variables. poses a very complex matrix This inversion problem, using relatively large computational times. On the other side is the fully explicit method, in which the spacial derivatives and the exchange terms are evaluated at the old The mass equation would look like: time level. ( Pv) ( n+l n n ( P )i v ( Pv v)i+ ( + At v v)i-- n ~~~~~~= S. Az We can see in this case the terms which are evaluated at locations other than the cell i are in the old time level, and so they are known. The solution at each cell is independent of its neighbors and the numerical solution of the set of equations will be relatively simple. The penalty for that simple solution is that the stability criterion for this category is severe, requiring in general very small time step sizes. Typically it would require that a pressure or temperature perturbation travel no farther than one mesh space in one time step, or in mathematical words it would require: At <-Az c 56 where c is a sonic speed. Typical values are on the order of 10 2 to 10- 1 m for the mesh spacing and 103 m/sec for the sonic speed. Thus, we can expect to be limited to time step sizes of the order of 10 -4 4 or -5 10- 5 seconds with this method. In between these two extremes, the semi implicit methods are those in which some terms are treated implicitly while others explicitly. If liquid convection is treated explicitly, the time step restriction for this class of schemes is the convective limit Az At < A-Z v where v is the phase velocity. The general idea behind this category is to devise a particular balance between implicit and explicit terms which would make the solution of the particular set of equations simple compared to that of the fully implicit method, combined with a less restrictive stability criterion compared to the fully explicit method. Here a very large number of possibilities exist, and a general analysis would prove to be of little value since what might be the best solution for a particular problem may not be a good one for another. Thus, instead of a general study of semi-implicit methods we will just analyse the particular scheme used in our model and show the motivation for its choice. One important consideration for this choice is the time scale of the phenomena to be analysed with the model. A typical loss of flow two-phase transient lasts from onset of boiling to flow reversal for about one second. Therefore a sufficient detailed description of this transient requires that the solution scheme 57 produces information with a time interval of about 10-1 to 10- 2 second. In this kind of transient we can expect to have axial velocities on the order of 10 meter per second. -l order of 10 Thus using a axial mesh spacing on the meter, the convective limit Az/v characteristic of the semi-implicit method will be of the same order of magnitude of the time interval in which we want information, and a method with such time step limitation would fit perfectly our purpose. Much longer simulations can be expected in the case of natural convection decay heat removal. But in this class of phenomena the phase velocities would be much smaller, and again in this case, a time step restriction connected to the phase velocity is of the same order of magnitude of the required information interval. Therefore, with the semi-implicit method we take advantage of a simpler solution of the fluid flow equations, with smaller number of operations performed per time step without increasing the number of time steps required to cover the whole transient. After this brief outline of the general features of numerical methods, we proceed with a detailed description of the particular scheme adopted for the model, explaining how this particularly fits our set of equations and insures the stability of the method. We start by dividing the fuel assembly to be simulated into a two dimensional r - z grid. To allow flexibility of application and a more efficient allocation of time and memory space, this division is made to accept variable mesh spacing in both directions, with the sole restriction that at each radial or axial level the mesh spacing corre- 58 sponding to that direction remains the same for all cells in that level. With this restriction, each cell, except the boundaries cells, will have only one neighbor at each of its four sides. Figure 2.1 shows a typical arrangement of cells. All unknowns of the problem, with the exception of the velocities, are evaluated at the center of the mesh cells, while the velocities are evaluated at the faces of these cells. shows a typical mesh cell where this is illustrated. Figure 2.2 This figure also shows the subscript convention used in the difference equations. In this convention subscripts i and j indicate position in the center of a cell along the axial and radial axis respectively, while subscripts i + and j + indicates position at the faces of the cells corresponding to the z and r directions respectively. Superscripts are used to indicate the time level in which the variables are evaluated. Superscript n indicates evaluation at the old time level, thus corresponding to a known quantity, and n 1 indicates a variable in the new time level, to be determined in this step. The exchange terms, which are in general function of both new and old time variables do not carry any superscript. in Chapter They will be discussed at length 3. With these conventions established, the difference form of the mass and energy equations, which are differentiated about the center of the mesh cells are: 59 z I(.) )I (.- 'I* r,.) II 1% C I I 1 I-oolo K 11-11 .0-00 ,I K ,1I / p,I r Figure 2.1 A Typical Cell Arrangement 60 (i+½,j) Uvz c, Pv (k,ji-½) Uvr UZr P U, z ev' ek V v' P, T , V (k,j+½) T (1, j) Uvr, U r I II I _Jil l I~~~~~~~~ (i-½,j) Uvz9 Figure 2.2 U z Position Evaluation of Variables 61 Vapor Mass; n+l (a n+l n ( tn- Pv j (a v i,~j+ Pv - n n+l Pv uvz . . ._ (A r/V (r )i ~~~~~~~~vr i,j+" n Pn v v p Uvz )i- 1 /VAr a/~nPv un+ )i,j½jvr _ -(A ,i + Az At + (n i+½ ,J . = S e - S c (2.2.1) Vapor Energy: n+l Pv n+l en+l c°t v n Pv n ev1 n i, (nnnu+ lvn Vvn vzn+l + At Az, 1 an pn )i0v en v un+l vzi-,_ Az r n r )i,j- ½ r (n+l _ p n )ili aC ij L At ( r~vr i+ + (Ar/V Qnunl)ij+½ Az i ( A/V n un+l) vr ij- - ] vr i+½ + (nUn+l) ( Pvz )i- ,j -n+ vv . L n v Z' )i+ t, j Pv e un+v )ij+ (A /V i (A/V Pv i+, Qwv W + QQv (2.2.2) 62 Liquid ass: (,_,n+l )P n+l (1_,n)pn] - k f(1- n) n i,j un'] (1_ctn ) Pn k t z 1,i 1, + I At Az I + Atr/V (_,n)n (1 [Ar/Vtun+l' Z) rJ - i,jP = - [Ar/V (1-a )pn un+l] i,j-% S e +S c (2.2.3) Liquid Energy: ,_O.n+l Pn+l en+l _ (,_,n)pn ) 91 k e ,_Cn Pn e n u n+l ) k Z k kz i-0-2, J + At z i + [Ar/V(1- )P enUr] [(_an )Pn un+l] 9g eneg ~z n) - /V~/v~ (_a )Pn en k~ n) eUn+l] r ,_,,n u n+l ) + Zz + p. n pn{(_ | i _cc i-P-2, i n) -Ati -(I l At i j+½ + n+l u ZZ ---- i-'-2 , i + [Ar/V(_n) Uni' Az i - [Ar/V (i-c)_c1 n- u Ar/V U-ar Qwk- Qv j i,5 j- = (2.2.4) 63 Some variables in the above equations are used in a location other than the place where they are primarely defined (see figure 2.2.2). For instance, the void fraction , which is a cell-centered quantity, appears in the convective terms of all four equations located in the cell's faces. So in order to make these equations completely determined we must establish a rule to transport the value of these variables from the center to the faces of the cells. In our model we have used a relationship known as donor-cell differencing. Later on, in section 2.4 we will see that this scheme has an important effect on the stability of the method. To illustrate how this technique works, let a general , pv, Pg, ev, variable X stand for any cell centered quantity such as eZ. The face centered value Xi+ will be given by: > if Xi = if i+1 if (2.2.5) Zi+ <0 Uzi4<i+ In the above rule we have used the axial direction as an example. A similar rule is used to dislocate the variables in the radial direction. The final ambiguity to be removed is in the evaluation of the void fraction. Though it is obvious which of the phase velocities we should use to evaluate the densities and internal energies, this choice is not clear when we refer to the void fraction, which appears in both the liquid and vapor equations. To remove this ambiguity we state that the velocity to be used in the decision of equation 2.2.5 is the one corresponding to the equation in which the variable will appear. Thus for the vapor equations the void fraction 64 will be calculated using the vapor velocity in equation 2.2.5, while for the liquid equation, the liquid velocity will serve in the decision. As a final remark, we note that the donor cell rule is used only to locate quantities at time level n, never at time level n+l, thus the velocity used in the decision is always a known quantity. Further more, the rule of equation 2.2.5 places a cell centered variable x in the cell's face where the velocity used in the decision is defined, so that we will never have ambiguity in this decision. We now turn our attention to the momentum equations. Here a difference with respect to the mass and energy equations should be since the velocities are primarely defined at the faces of the noted mesh cells. Let those faces be the reference points for the differencing of the momentum equations. Then the vapor momentum equations are: (W) )n v _+Un I vz L~~~~~~~~~L ~ ~~ i,1j (A UI + Un ) ( i jAr + an -+ A_ n+l _n+l\ i+lj ii + jAri2,iz vr i, Azi+½ + + v g i+-2,j - M wzv + Mvz i+, j (2.2.6) 65 (Un+1 r vr U - - ) +1 vr i,j+½ + vz ij At ( apv) yv n Az ij (A Uvr) i + Un + z nvr),i,j+ + (A Un r y vr ,3+2 - n+1 + n j42 i, Ar n+l Pij ) (Pij+lArj+½ (2.2.7) = - (Mwrv + Mvr)i,ji+ And the liquid momentum equations: In F/I w I I -r -A I n (Un+1 F JZz I I Qz i+f, " Id i-p-2,j L + + (ArUz)i+,j Un kr i½, j re..~ L- ' ~n I (A z UUZ))Ii+2,j TTn °t z i+ ,j + (lan ) i+½Ij Ar n (_a [,~ .- , At j -_ Az (n+l pn+l Pi+lj - w + AziP- g = - (Mz - Mvz)i, j n Ji, j+ L n+l Ur Ir2zr ) i,j+±½ _ n A.~ ~Qzi,j+½ A L 4-L + (2.2.8) (A Ur)i Az j+ 66 )~ ~ ~ ~~~P. n+l r Ur i j+½ Zr i,j+½ -=- (r r + ) (l- _n+l ij+l) P Ar~~~~~~~Arj+;, - Mvr)i,j+I (2.2.9) Again in the momentum equations some variables are used at a location different from where they were primarily defined. question is how are these quantities evaluated? void fraction a and the densities Pv and pZ The First, consider the Contrary to the mass and energy equations, these quantities do not appear in the momentum equation as difference terms. Thus, they do not influence the stability of the method the way they did in equations 2.2.1 through 2.2.4, and we can use a simple averaging rule such as: Xi AZ + X.AZ. Xi+l i i4+½-1 AZi+ + AZ 3i (2.2.10) 3. i and p. where X stands for the void fraction a and the two densities p A similar rule is used to transfer the variables to the faces j+ in the radial direction, with Ar replacing AZ. We next consider the velocities appearing in our momentum equations. U vz i j+½ First we look at the velocities Uvr i+,j' U i1,j+' U1z Figure ij-'FiurY Utr iyj' 2.3 shows as an example the position of 67 (i+l,j) U Uvr i+l,j-2 U (PRIMARY) (PRIMARY) Uvr i+l,j+½ I U Uvr I i+,j (ij) U U Uvr i,j+½ .. vr i, -I (PRIMARY (PRIMARY) - - Figure 2.3 Different Positions for the Radial Velocity 68 U idcompared with the location where Ur is primarily defined. yr i1, j yr Again these velocities do not appear as difference terms and asimple averaging procedure can be used without compromising the stability of the method. Thus we define: [UVZi+½,j+ vz ij+½ = U = ~z i-,j Uv i+ Uvz i-½1, + U i +l vz i, + (2.2.11) VZ i-½,j+l Uvr yr i+ i+½,j -¼ Uvr ,i-+ Uvr ij- + Uvr i+l,j+ +Uvr ij (2.2.12) and a similar pair of relationships for the liquid phase. Finally those velocities appearing in the difference terms must be evaluated. Here a simple averaging procedure would lead to a differencing scheme unstable. with the donor cell technique. Therefore these velocities are evaluated In this way, the expressions for the difference terms are: -UV i, U vz i+3/2,j if Uz i+j vz i+½,j Azi +1 AU z Vz AZ j < 0 (2.2.13) i~l, i U vz ij - AZ Uvz vz ii-½,ji i if U . vzi+½ > - 69 -U vz Uv i, j+l +1 vz i+½, Ar A U r i+, if 0 Uvr i+<,j 1 vz Ar ifvri+½,j i-2-,j U U i uvzi+,j , vz A 1 +-j1 0 Uvr i+l, j - ar.j_½ U vr i+l,j+½ (2.2.14) -U uvr i,j+2 if U . vz i,j+½ < 0 AZi+½ A U vr AZ (2.2.15) i,j+½ Uvr i,j+ - Uvr i-l, j+½ if AZi-_ U -U uvr i,j+3/2 (A vr i,j+ if .. ifUvz i,j+ -> Uvr i,j+ vr Ar 0 Arj+l U r < 0 (2.2.16) )ij i ,j+½ U yr - U ij+2 Ar. yr i,j-½ ifvr i,j+l- 0 70 where the mesh spacings zi+ r.+½ appearing in the above and expressions are defined as: Azi +Az. Az i+ + = i 3(2.2.17) 2 Ar Ar T + Ar. j+l 2 - (2.2.18) and similar expressions apply to the liquid phase. With those rules the differencing scheme for the fluid flow conservation equations is completed. To complete the set of algebraic equations we need only to specify the relationships for the exchange terms and the equations of state. in Chapter 3. These will be discussed We now turn our attention to the numerical solution of the set of algebraic equations, equations 2.2.1 through 2.2.4 and 2.2.6 through 2.2.9. 71 2.3 The Numerical Scheme In the above difference equations all variables evaluated at the time level n were determine in the previous time level, thus in the present level n+l they are known quantities. The problem is to extract from that set of equations the variables at time level n+l. A quick look at those equations reveal they are non linear, complicate equations, and a numerical iterative technique is practically the only option for their solution. The equations of state represent unique relationships of the densities and internal energies for a given pair of pressure and temperature. We will replace these densities and internal energies by the liquid and vapor temperatures as primary variables, thus reducing the number of unknown to eight, namely the void fraction, the pressure, the vapor and liquid temperatures and the four velocity components. The technique used in the solution of algebraic equation is a multidimensional extension of the Newton iterative solution of algebraic equations. Let us first define a vector whose components are the unknowns of the problem. Then: n+l X = [ , Tv, T, Uvz, Uvr, Uz U ] (2.3.1) and the equations 2.2.1 through 2.2.4 and 2.2.6 through 2.2.9 can be written in abreviated form as: F P (X) = O , p = 1, ... 8 (2.3.2) 72 Now suppose that at a certain iteration k we have come up with k an approximate solution of 2.3.2 X. Since this is not the exact solution, the left hand side of 2.3.2 F (X)is not necessarely equal to zero. p Then, let us make a Taylor expansion of F(X) around the point Xk ( F IX ) )( ( ( kFl -k xk ) =F IXkI+8L(-p Xk Ix - x axk qq=l q q) (2.3.3) q = P If 1, 8 k+l is required to be the solution of equation 2.3.2 it follows: .8 pk+ ( q=l Dx k qX q - xk q = - FP(Xk ), P = 1, ... 8 (2.3.4) With equation 2.3.4 the iterative procedure is defined. that this set of equations is now linear in the unknowns x = x q q If equation 2.3.4 are written explicitly, it follows: [PV AL as 1t DeDP Da F LAt v as DP 6p -T r-k LAtT as TT k+l v Note k - x q 73 as 6T (u-V )i4~, Ai Tz+ 3T 6 + (Ar/Vapv) i j+½ 6Uvr i+½2,j j½ 1i - ½6U vz (tp Azi (Ar/V ap) ( )VZ i-,)+½, + 6U - vr i,j-- - Fk 1 (2.3.5) D pe vk k Lkk6k ev Pv T v + [( ½i-2 Azi V v P T At k k + Ia pk~ekv + a k k a + e v ~ Q P Dk v aP PAt 1 6P+ + .j + 1'vL I,j Ar/V )i, Pii)] i a - T 6TE + v v z i-L- (1 - I 6T )T ), vz1 (x2 + i (v p ~~ ( v ilik )ij½)]6U 6%re= v ,jj-, - Fk (2.3.6) 74 k 1 pAt DS] a~l ) P +P At --V~~~~~~~~~~ 3_S 6T v 6Pij + +[( 1 k)P+-S] T + [(1-a)pij kT aT, [Ar/V (-a)p, - + [Ar/V (1-a))p]ij+ 6Ug z i-½,j [(1-ac)P] - j_% vr i, j-!2 kek P91ek + £P90 _ =- P Pt + 6Ugri,j+½ - (2 3.7) Fk 3 Q kDP, Dek I +ke) + e P ) - P ij aP aP At D At [it)- 6i+ aT LAt v T aTQ T / Q I~~ + 6U z i+2, - At i+-2, ~~1-a 1s ( + (Pt2, e Z1)i±-, ) z Q j ½2,j ( b~. 1 [(-½,i 13 i ( j++ (p e), j4)] r ij+ j+½l i ~ ) ~, 2, +[(Ar/V(l_ kzki-_,j i i-1-21 +(p eQ)i,j)Ugz Y, )i, 75 p - [(Ar/V (1-O))i + (p e )i j 6 ] PI Yi, Ur Z ii ,j-½ j 1] 2' 4 (2.3.8) aM (aPv)i ++ , ,v 6 +AtaU ] vz i,+½,j VZ vz At + ,j Cai-P'2,j AZ. V 6U Qauz tz DU z + - (6 Pi+l,j = 6 Pij ) - F5 aMg At ( 1-i, Uz 8 auz j Az+ aMQ ++ au VZ i+½,j = - 6 vz U+ i-!,j + k ..+ - 6 Pij..) (6 Pi+l,j (2.3.9) F5 .1 1½i-)P+] + i+½,j (2.3.10) F6 13 Az.+ + v (aPV)i,j+i au At + Ar 6U r i,j+ i + Ur k (6 i,j+l - P.i) = ij (2.3.11) F7 j+ At Ar i+½ vr vr i'J' +r(1Nj aM U + aM 68 U au_ U 6iP,j+l- r i,j iau uvr i,j+½ vr 6 Pij ) 13 = - Fk 8 + (2.3.12) 76 Note that the last four of the above equations depend only on pressures and the four velocity components. 2.3.9 and 2.3.10 in a pair and again 23.11 Grouping equations and 2.3.12 we can without difficulty isolate the velocity components in the left hand side: 6 = W vz i,j 6 U z i+½,j = Wz i+,j i+l,j (6 P. 6 - 6Pij) + fz (2.3.14) 6 P) Pij (2.3.15) yr i,j+½ Wr i,j+½ (6 (2.3.13) - vr ij+ i,j+½ (6 Pjl ij+l ij+= 6 Ur yr i,j+½ 6 Ur (6 P vz i+½,j P i,j+l uvz ij) uvr (2.3.16) 6 Pij) + fur where the coefficients W are given by: W vz i+½, j (l-)p - a (1-a) lAz At + a k \ c (1-a) +)aMk + Du aM aM aM X au(i + Az 9, z vz i+-2,j v1 (2.3.17) Aau z At auzvz auvZ i+½2,j and similar expressions for the other component velocities. Now, with equation 2.3.13 through 2.3.16 we can eliminate all velocities in equations 2.3.5 through 2.3.8. Rearranging these equations, they can be written in the matrice form: 77 7 a all a12 a21 a22 a3 a14 a23 a24 6a ST v x a31 a32 a33 a34 a41 42 C43 44 + 6T Z SP i~i w- N _m , i _ b11 b21 j_ 12 b13 b14 SPi-l, j f1 b22 b23 b24 SPi, j-1 f2 b x + b31 b32 b33 b34 b41 b b b (2.3.18) SPi,j+1 f3 i i 41 42 43 44- 6Pi+l,j II f4 The expressions for the coefficients a's and b's are not given here for brevity. We will return to them and show representatives of them when we discuss the diagonal dominance of the pressure problem and the limiting case of only one phase present. If we transform the matrix of the coefficients a in equation 2.3.18 into an upper triangular matrix, this equation becomes: 78 I 1. a2 a13 a14 0 a' a' 1 23 I 24 I - , 1 0 1 a 34 . Ii 6a - ...b. fI 6Pi-lj 6P. 14 1 ft 1 2 - -| 0 ,'I~~~~~~~~~~~~~~~~~ , 1 + AT 6Pij+l ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ *, I 0o o 4144 I 0 .... 1 ~~ I . i,j - - 6Pi+l.i . I f4 f ... (2.3.19) The last line of the above equation is an expression involving only pressures. Since this expression relates the pressure at a cell (i,j) to its four neighbors' pressure, this equation must be solved simultaneously for all mesh cells. The solution of this pressure problem is the subject of section 2.4. It is important to point out that this solution technique reduces the inversion of a matrix with dimensions 8N by 8N, with N being the number of mesh cells, to the inversion of a matrix of dimensions N by N by performing for each mesh cell the inversion of two 2 by 2 matrices and one 4 by 4 matrix. Before the closing of this section, it is appropriate to make three comments. phase flow. The first one concerns the limiting case of single The transformation of equation 2.3.18 into equation 2.3.19 requires that all diagonal elements of matrix of the coefficients a be non-zero. We will explore how those coefficients behave as the void fraction assumes the values S = 0. = 0 and = 1, and the mass exchange rate 79 = 0. First let us consider the case If we look back into equation 2.3.5 we can see that in the first line of equation 2.3.18 all coefficients bq , as well as all alq$ with the exception of all have a factor on them. are zero. If we look into equation 2.2.1 we see that the right hand Therefore, except for all, all those coefficients side of 2.3.5 is also zero, and the first line of 2.3.18 corresponds to the equation: k Pv At Now, consider equation 2.3.6. It is seen that without the presence of vapor this equation is trivial, and all coefficients a 2q and b 2q in second line of 2.3.18 are zero. But in this case, a trivial equation would cause us a problem, since an element of the diagonal of the matrix of coefficients a in 2.3.18 would be zero, thus invalidating the triangularization of this matrix. To avoid this problem, we impose that the interphase heat exchange term be in the form: Qtv = h (Tn+ l) Tnnl with the coefficient h being non-zero even if one of the phases is not present. In this way, equation 2.3.6 reduces to: h~~T -haT h Tv - h k Tt = h (T k - T) which implies the model will force the vapor temperature to be equal to the liquid temperature when we have one of the phases absent. If we. repeat this analysis for the vapor single phase flow, it is easy to see that we will reach the same conclusions. Therefore 80 we can be confident that the matrix of coefficients a in 2.3.18 does not have a diagonal element equal to zero and the triangularization of this matrix is always possible. One last question in this subject concerns the inversion of the submatrices of the velocity components. If we look into equation 2.3.17 we will see that the absence of one phase would lead to a division by zero. We again avoid this problem by imposing the interphase momentum exchange term to be in the form; K K M. n+l N~ = K(U~ Un+l '- v again with the coefficient K being non-zero even if one of the phases is not present. As for the energy equation, this will force the vapor velocity to be equal ot the liquid velocity when one of the phases is not present. The second question we would like to discuss concerns the diagonal dominance of the pressure problem. The solution of this problem requires that the diagonal element of the pressure problem matrix be greater or equal to the sum of the absolute value of the elements in the line corresponding to that diagonal element. In terms of the coefficients of equation 2.3.19 this translate to: b 41 + Ib I+!b42 + I b 3 I+' I+ I I143 b 4~ < 1 An exact proof that this condition is satisfied would require a prohibitive amount of algebraic work. So instead of trying to follow this line, we will present only a partial view, which can bring 81 some understanding to this problem. Then, let us consider the elements of the first two lines of the matrix of the coefficients b in equation 2.3.18. We evaluate these coefficients with the help of equations We get the expressions: 2.3.5, 2.3.6, 2.3.13 and 2.3.15. 1b11 b (A Wv ap =A W bl2 = (V PV b1 3 (V v vr Wvr ) W e)i+ ) j z ~PvVZ b4 = L(z b2 i,j- a b b14 =(tv (Pv e Zi, + P) W a (Pv i,j½ Arev v 23 L rj + P) b 2 [ a (Pve )W l b24 =~ (Pv P w i+½, j From the expression for the value of the coefficients W, equation 2.3.17 we can see that all these coefficients b are negative. 82 Now consider the coefficients of the central pressure, coefficients a14 and a24 in equation 2.3.18. From the same equations we used before we get: aa a a14 A DPv - DS+b++) p (b - DP - - +b 1 +b 1 3 1-(l 12 b13 b14 De a 24 -a+ At v UP -Q e v v P + b (b P 21 22 + b 23 Let us examine in detail each of these coefficients. + b 24 The way equation 2.2.1 was written the mass exchange rate S is positive when we have evaporation. It is easy to see that an increment in pressure will produce a decrease in the rate of evaporation, so the term US/3P is negative. The other term making a1 4 is the vapor compressibility, which is a positive quantity. a14 > I bll Therefore we conclude: + b1 |+ 2 I 3 Consider next the coefficient a24 . + is De /DP, which is a very small quantity. v I b14 The first term in a24 Indeed the equation of state used in our model puts a zero in this derivative. The other term 3pv/3P we have already investigated and seen it is a positive quantity. Finally we have the term /DP. The heat transfers equations used in the model have only one term in the heat exchange rate dependent on the pressure, representing the heat transferred due to evaporation or condensation. In this way D/UP has the same sign as S/UP, which we saw 83 before is a negative quantity. + a 2 4 >1 b2 1 We thus conclude again: b2 2 + l b 23 I +lb 24 I We omit here a similar analysis of the coefficients appearing The general form of them is the same, on in the liquid equations. following a similar reasoning we would reach the same conclusions as we did for the vapor equations. Now, since the pressure problem equation (the last line of equation 2.3.19) was obtained as a linear combination of equations whose coefficient of the central pressure exceeds the sum of the absolute value of the coefficients of the neighboring pressures, this pressure problem equation also has this same property, which shows us the pressure problem matrix is diagonal dominant. Finally there is the question of the boundary conditions. We start with the radial direction. At the fuel assembly centerline there is simply the zero radial flow condition at r = 0. This is accomplished by just putting a zero in the terms Uri ,½ appearing in the divergent differences. At the other radial boundary, corresponding to the fuel assembly hexcan there is also a zero flow boundary condition, which is translated in the model by setting the radial velocities at that boundary equal to zero. Note that since these velocities are identically zero, there is no need to evaluate the momentum equations at these nodes J+½, thus there will be only J-1 radial momentum equations at each level i. Besides the flow conditions at this boundary, 84 there is also a thermal boundary condition, taking into account the heat transferred between the fluid and the structure, represented in the code by the hex can model. Chapter This model will be fully analysed in 3. For the axial direction more complicated conditions appear. To explain this refer to figure 2.4 It can be seen in that figure that two fictitious cells were added to the actual fuel assembly. In these cells the conditions determining a particular problem must be specified. Thus the user of the model needs to specify as a function of time, an outlet pressure in cells i = I + 1 and inlet pressure, vapor and liquid temperatures and the void fraction in cells i momentum equations in cells i ½ and i = I + 0. For the the following conditions are imposed U - ,j = U I+3/2,j= ,j I+,j Finally, to completely determine the particular problem to be studied, the user also needs to specify the fuel pin heat generation rate as a function of time. _ 85 -I I I Ficticious Cell - - U) f-4 4 a~) I I I Ficticious Cell L - Figure 2.4 - The Ficticious Cells - - _ I J 86 2.4 The Pressure Problem So far, we hve collapsed the eight conservation equations, the equation of state and the equations governing the exchange terms into a single equation (i.e. one for each mesh cell), involving the pressure in the cell itself and its neighbours. Because of this coupling between cells, those equations must be solved simultaneously. Since this matrix inversion rests inside an iterative process, which is to be repeated for each time step, it is clear that the overall efficiency of the model is strongly dependent on the way this pressure problem solution is done. The approach to the problem was to take advantage of two particular characteristics of the case at hand. The first one is the fact that most of the elements of the matrix are zeros, the non-zeros being only the elements on five diagonals. with the fact that LFBR The second one has to do fuel assemblies have one of its dimensions, the axial one, much larger than the other. This has a surprisingly strong effect on the time required for the matrix inversion, as explained in the following paragraphs. The large number of zeros in the matrix was used to our advantage by adopting an iterative solution known as block-tri-diagonal, which is an extension of the Gauss-Siedel iterative technique (see Ref. ) Recalling the pressure equation, for each mesh cell we have: A._P , + BP, U· j-1 1] _ + C..P l-AJ 13 1J + D ... P l] -iLJ + EP..., = R.. 13 1J3tI 13 (2.4.1) 87 To perform the kth iteration in the cells at level i, we pass to the right-hand side of the equation the terms containing the pressure at the bottom and tope of cell (i,j). k k k k ij ij-1 + Cij + Eijij+l Note that the term in Pj i- lj k. This is a known quantity k-l Bi j i-lj iji+lj since (2.4.2) takes the value obtained at iteration it was obtained in the previous step of the calculation, when this procedure was applied for the cells at level i-1. With this manipulation, we ended with only three unknowns in the equation, and we now can use the tridiagonal matrix inversion technique (Ref. 42 ) which gives the exact solution of equation 2.4.2 for all values of j, with a very few operations. This procedure is repeated for all values of the subscript i, and the pass over all cells is repeated again until the desired convergence is obtained. The second characteristic which was taken into consideration influences the number of passes required to attain convergence. An iterative solution sets arbitrary initial values for the unknowns, and by recalculating these unknowns with the appropriate set of equations aims to reduce the error contained in the previous value of the unknowns. The smaller the error carried from one pass to the other, the fewer the number of passes necessary to meet the required convergence criterion. 88 In the technique used in the model, the new value obtained for the pressure will have an error because in the right-hand side of equation 2.4.2 the values of the pressure are not the exact solution of the problem, but for each level i, the values of the pressure will have the correct relationship between themselves, since the tri-diagonal technique will give the exact solution for a given right-hand side. If we could make our scheme in such a way as to minimize the influence of the error carried into the right-hand side of equation 2.4.2, we would have the iterations converging quickly. The difference in dimensions for the axial and radial directions provides this way. When a fuel assembly is divided into mesh cells, the radial dimension of these mesh cells will be a few pitches in length, or for usual LMFBR fuel assemblies, this dimension will be of the order of one centimeter. On the other hand, typically a fuel assembly is a few meters in length, and in order to keep the number of cells at a minimum, to shorten the time required for the calculations, we expect the axial dimension of a mesh cell to be of the order of tens of centimenter. In this situation, the pressure at radially neighbouring cells must have a very close value, or in other words, a small increment in the pressure in one cell would be propagated to its radial neighbours almost in full. On the other hand, for the axial direction this propagation of error would not be so strong, since the larger distance between cells would act in the sense of atenuating the propagation. We will try next to express the previous statement in mathematical terms. To avoid the formidable algebraic complication of 89 working with the full set of two fluid equations, we will use a simplified model, keeping only the parts relevant to this analysis. We will consider only the mass and momentum equations for a single phase. We also put all explicit terms, which are not relevant n- to this problem into a generic term R . Then the conservation equations become: U + VU + UVU + 3t (2.4.3) = 1 p VP = -kU (2.4.4) and the equation of state: ap 1 _ (2.4.5) ca C2 with c being the sonic speed. Pi P-(pU) -z)(pUri - i~~~~~~-½j r_pu)ij- Applying the differentiating scheme to these equations we get: n+l pn n+1 n+ n+l , un+l) - +_ 2 cAt = +Ar Az Ar (2.4.6) Un ) Un+l z(U -U) pn+l i-Pj + At 1 ~ Oi+½j+ P +-j p i (n+l _ Un ) Atr At r ij 4a + ~1 P. Pij+±½ Pn+l i+li i3 + k z Az Pn+l i+lj Arr n+l ij + Un+l1. = R zi--J z n+l = Rn rij+- . r (2.4.7) (2.4.8) 9O0 We isolate un+l We isolate U~l and U z (Un+l r i ~z ) (Un+l ) i+_ (r )i+ = 2 1 At Pij+ Az 1 At ij+½ Az - u nsn in equations 2.4.7 and 2.4.8: 1 +k At Z - n+l (P ij 1 n+l l+krAt (Pirjj _n+l R pi+lj ) n + n+l _ _ i~ )+ z n r (2.4.9) (2.4.10) It is possible now to eliminate the velocities in equation 2.4.6 to get an expression involving the pressure alone. If this equation is put in the form of equation 2.4.1 we then have the expression for the coefficients of the pressure problem matrix: At 1 2 (2.4.11a) Aij.. r( Bij At l+k r At 1 2 -(z) A= (2.4.llb) l+k At z - Bij - Dj Cij.. =-Aij ij ii ii ijC At Dij 2 =- (Az E..+ 2 (2.4.11c) 1 (2.4.11d) l+k At z Eij.. At = 'Ar- 1 2 (2.4.11e) l+k At r The first point to be considered in these equations is the coefficient C.. in equation 2.4.llc: j Note that C.. exceeds the sum of 13 91 the absolute values of the other coefficients by the factor 1/c2 . In the numerical analysis language this means that the matrix of the coefficients is diagonal dominant, and it guarantees that the numerical inversion of this matrix will converge. Later on, when discussing the equations of state, we will insist that the equation for the density of both phases reflect some sort of compressibility, or in other words, that the derivative of the density with respect to the pressure be always a real positive number. Looking at equation 2.4.11c it can be seen that this requirement guarantees the diagonal dominance of the pressure problem matrix. We now compare the coefficients Ai.. and Bij..(which are in 13 13 all similar to the pair of coefficients Dij and Eij): As it has been established before, Az is ten or more times larger than Ar, which means Bij will be one hundred or more times smaller than Aij. If we go back to equation 2.4.2 it can be seen that in the proposed scheme the errors contained in the pressure terms in the right-hand side will be multiplied by a coefficient which is very small compared to the coefficients in the left-hand side; therefore, the influence of these errors will be minimized, and the convergence of the scheme will be drastically improved. In the comparison we have just made, the friction terms 1 l+kAt were neglected. This was done first because their influences +aresmall, being the product kt not a large number compared to one. are small, being the product kAt not a large number compared to one. 92 Second, their influence is in the direction to enhance the disparity between the coefficients A.. and B... 11 1J Clearly in all situations of practical interest the axial velocity will be two or three orders of magnitude larger than the radial velocity, which means the axial friction factor k z will be larger than its radial counterpart. Finally, to illustrate this point we ran a case with mesh cells whose dimensions were Az = 30 cm and Ar = 1 cm, with the proposed scheme and with one which did the same procedure but exchanged the z axis by the r axis. In the first case we attained a convergence criterion of 10- 6 in less that 10 iterations. While with the second scheme the same convergence criterion could not be attained in ten thousand iterations. 93 2.5 Stability Analysis of the Numerical Method. This chapter would not be complete without a study on the stability of the numerical method, and in the following paragraphs we will attempt to fulfill this requirement. We want to emphasize at this point that the following analysis is not rigorous in the mathematical sense, nor is it a definitive proof of the two fluid model stability. Because the tools of numerical analysis known to date were developed for systems of linear equations, they cannot be applied to the nonlinear thermohydraulic equations without a few assumptions and simplifications, made to fit into the limitations of our tools. Even with this "local-linear" treatment of the system of equations, sometimes the algebraic complication of the study imposed a few approximations in order that we could have an intelligible conclusion. Nonetheless, this analysis gives a picture, if not rigorous, at least sufficiently clear for the understanding of the stability problems of the two-fluid model. We will be following in this study a line developed by Stewart/ 51 / in which the stabilizing effects of the exchange terms are identified. The first simplification made in this analysis was to reduce the full set of eight equations which make the two-dimensional, twofluid problem to a system of only four equations, by taking the momentum equations in only one direction and neglecting the energy equations. Physically this situation corresponds to a one-dimensional, isothermal flow. 94 As we shall see later, we will be solving in this study, determinants and algebraic equations whose order is equal to the number of equations in our model. It is easy to understand that the algebraic difficulty of working with eighth order determinants and equations would be large enough to make it nearly impossible to visualize any kind of conclusion. We will not be loosing the desired degree of generalization with these simplifications, since the momentum equations are exactly the same for both directions, and the energy equations are differentiated in all similar to the mass equations. Therefore, all the characteristics of the eight-equation model will be represented in this analysis and the simplified system of equations will be, from the numerical point of view, analogous to the full two-fluid model. We then write down the fluid-dynamic equations as: a a atav at ~~ ( = S+-~~~~--aPU=S. + + az (1-)p (-)p apvat Uvaz au (l-a))p [R, t =U ~~~~~~~(2.5.1) (2.5.1) S (253 2) v+1-a-tv z =+k(UUV-a vz ~~~~~~(2.5.3) aup + U 2, 3' + (1-a) aP az k(U -U) V (2.5.4) 95 and the equations of state: apV 1 (2.5.5) 2 cPv (2.5.6) In the canonical form the above equations would appear as: A X + B 3X = f(X) (2.5.7) with X = [~, P, U v , U] (2.5.8) a/c 2 0 0 (1-a)/c 0 0 0 0 apv 0 0 0 0 (1-a) p Pv A = T v V -p (2.5.9) Q pU v v -p U aU /c v aPv v (l-a)U /c 0 0 (1-a)p, B 0 a 0 (1-a)) aU v v 0 0 (1-a)p QU (2.5.10) 96 With this formalism the characteristic roots of the system can be found, which are solutions of the equation: det[B - (2.5.11) A] = 0 The reduction of this characteristic determinant results in the algebraic equation: 2 cap (U ,-AI) + 2 (l-2)p (U -A) - z + (1 a)pv 2 U v -X) 2 2 (U-X) = 0 c% (2.5.12) v Since we are interested only in the qualitative aspect of the roots of this equation, rather than its precise value, we will make some approximations, in order to get a solution of 25.12 which are representative of the true value. We note that for the cases of practical interest the liquid density is much higher than the vapor density. Then it is reasonable to neglect the terms in Pvs and two real roots are obtained, which are approximately: U v (2.5.13) + c v On the other hand, with this model we intend to study only sub-sonic flow, hence both Uv and Uz are much smaller than the sonic velocities. Then, if the terms in 1/c and 1/c2 are neglected, the two v other roots become: 97 2U U + EU iE(U-U) Ug 1+l+2 E - ~ + U vS~~~~~~~ ie((2.5.14) 1+E with 2 (1-a)p (2.5.15) apt It can be seen that whenever the phase velocities are different, the system will have two complex characteristic roots. This means the system of equations is not hyperbolic and consequently not well posed as an initial value problem. Nonetheless, with this conclusion it can only be said that the two-fluid problem failed to meet a sufficient condition, but it cannot be concluded that the problem is necessarily unstable. The previous analysis did not take into consideration the important stabilizing effect of the interphase exchange terms, and as we shall see later on, these terms are responsible for the stability of the two-fluid models. To verify this effect, we will proceed with the Von Neumann analysis of the numerical scheme. The difference equations corresponding to equations 2.5.1 through 2.5.6 are: n+l n+l j vvjAt n n tpvj + n n n+l jPvjUvj+ - n n n+l j-lPvj-lUvjAz = ' S (2.5.16) 98 (1 an+l)p n+l _ Pn (-Y.I I ki + j -j 2 jUz+½ 1 Lt vj-l 2½ = -S Az (2.5.17) .n+l n an Pn IUVJ j vj [ (pn+l + j )] Az + pn+l) = k. Az (Un+l + (1 -n.)'+i (2.5.18) n% (Ujw - Uej 1) Az j-+f At n+l n+l) (Uj-U Uj) in Un+l j - Un vjW A_ vj- vj+½ j+l - (1Z) ( + Un -Uvj At _ n+l Az 1 = k+ n+l - i+A vj+½- n+l ) (2.5.19) j+½ The convective terms in the mass and momentum equations involve donor cell differenciating, so the above equations are written for both U v and U, positive. To apply the Von Neumann method these equations must first be linearized. We thus expand the differences in terms of differences of the four basic variables individually, and treat the coefficient of these differences as constant. For simplicity we will neglect the liquid compressibility, so that we can substitute the difference terms in pressure by terms involving the vapor density alone and treat this variable as a basic one. If we recall the Von Neumann method, the error of any variable at a given time and location is expressed as: 99 n+r n xj+s xj reis 0 where 0 = r/m is the wave number. Applying this formalism to equations 2.5.16 through 2.5.19 it follows: t At -i+ n n At~~~~~~~~~~~~~~~~~~~~~~~~~~~~P(~_l)Ep +a -lc ie)E Uvj+½ aO+n n PVj At j +Az ( Az ~le)pvj ) + 1-e )Uvj+-(l- + vUv -e + Az (1-e )n 0 - (-1) -t- (1-a) P naj+ Az apv[tuvj+½ (1 -ie) (1-e n (2.5.21) - Az (l-ei)j 0 VAt U~ ~ ~~~~~~v-2AUj 3 - Pj U (C-1) n (2.5.20) v 2 aeC nv iO (1-e)cuvj+] + + = k (j+ (e (- U n U.( g (1-0)PZ[2A1)CUZj =k (4Uvj2 - Uj ) =0O with and (o, Uv 2 - EUvj+) (2.5.22) C2 Az1(e + (2.5.23) these equations and putting them into matrix form it follows: pv, )e)n(1-a 2 + Rearranging /AxE e - % UQU j 100 V p V(¢1+ v O v -p~ -pg~-fg O / At. PV(ctl+- v ) (t,l+0v ) O r~zlAt A = 2Az. C2r-i~ 0 -~ P (-l+U K )+ K V V v - vK V I I i I E -Ep K 2Az v mt p (-l+D )+QVK I Where we have abbreviated = Uv At -io uf~1t±-e Ut(l-e 2 sin = ) /2 C C m = CvAt -*2 Az sin /2 K = k At/pg £ 2 (1-c)p = apg In order for the errors in the basic variables not to grow geometrically, the absolute value of the eigenvalues matrix/A must be all less than one. of the amplification To find these eigenvalues we solve the After reducing this determinant we end up with the equation det[A] = 0. algebraic equation 2C2[(<-l+D +2EK)(-1+ m v + l+ ( ) -) v ) + (-l+D 2.)G(-1+U,+2~K) + 2 [ (-i+v+KK)(-+U+) -1+ 2 p/p = 0 (2.5.24) 101 The next step in the analysis would be to find the roots of this characteristic equation and see if their values would be less than one. But the expressions for the exact solution of the quartic equation are so complicated that it would be almost impossible to draw any conclusion from them. Instead we prefer to make some approximations which would give reasonably good values for the roots we are searching, but with the advantage of simple expressions which can give a clear visualization of them. Since we want to emphasize the importance of the interphase exchange terms, we will first evaluate the characteristic roots of 2.5.24 with the momentum exchange coefficient k set to zero, and afterwards compare the results of this analysis with those obtained with a positive real non-zero value of k. With k set to zero, equation 2.5.24 reduces to 2CM2[(-1+v2 )2-1+ + (-+D )2(2+ 2 First consider the high frequency behavior. 0 (2.5.25) As has been said before, the model uses the time step size At equal to the convective limit: At = min (Az/Uv, Az/U) Also notice that the phase velocities are small compared to the vapor sonic velocity. Thus C V At/Az >> 1 and for small m, 102 C2 >> 1. m It follows that equation 2.5.25 will have two roots of magnitude approximately ~ + 1/C , which are smaller than one. The other two roots approximately satisfy: or 1 - U(1 + iUv/U) (2.5.26) = (2.5.26) 1 lic + is: In the complex plane this is represented by a circle of radius U At/Az, touching the point one, tilted by an angle (ce Uv/U ) and back through an angle + arctan . arctan Clearly, with small m, points on this circle will not be outside the unit circle if the limit is satisfied: UYAt < 1 (2.5.27) Az and U At V < 1 (2.5.28) Az We then conclude that even without momentum exchange the high frequency modes will not grow geometrically if the convective limit is observed. Now let us turn to the low frequency modes. C -*+0)and in the limit the roots of 2.5.25 will be: m As m a, 103 =+ 1 - U (2.5.29) and -(2.5.30) =+ 1 - U Then, let us say that for m large but finite the roots of 2.5.25 are: = 1 - U+ (2.5.31) 6 We can evaluate this perturbation 6 by substituting 2.5.31 into 2.5.25, and neglecting the terms of order higher than 6 2 . The resulting quadratic equation will be: 1+2 ]2 - 22(1- v 2 )( v-) - £ 2(1- )2( v)2 = 0 C (2.5.32) m and the roots of this equation are: 2~+ 6 = (1-t 2 2 )(TJ -t)[ - 1+C2 (Dv_ and again using the fact that (U U) V V i as: () -U[ (1-D) [ - (2.5.33) / C << 1 we can write the 2. expression for the characteristic root 2IC2 ( .(53i) 2 l+c' ] (2.5.34) 104 Since - Uand 1 + (U of magnitude for some value of circle. - U)I are of the same order one root ~ may lie outside the unit Therefore, without the momentum exchange term k, the low frequency modes will grow geometrically and the method would be unstable. Nonetheless, with very few spacial mesh cells, i.e., with small m the model may have a well behaved solution even without the momentum exchange term. We now return to equation 2.5.24 to verify the effect of the momentum exchange term. For the high frequency modes, the same considerations are made as in the previous analysis with k equal to zero and it is clear that the roots will be of the same form, only multiplied by a factor which is approximately 1/(1+ K). Since in that analysis we concluded that the characteristic roots were less than one in magnitude, we can extend with confidence this result to the present case and conclude that for small values of m the model will present a well behaved solution. To study the low frequency behavior again consider the limiting case as m Then, for m (-1+U) + - and then introduce a perturbation of order 1/m. equation 2.5.24 becomes: - - ((-+l+U)x [( E-l+Uv+tK) ((-1+UX+K and the roots of this equation are: p/p ~~~~~2 2 )-2 K pv/p] = 0 (2.5.35) 105 = 1-U V (2.5.36a) = 1-UE (2.5.36b) ( ) 2 Uv~)(l-U (2.5.36c) (- ~ )(1-0E -~ VI(l-Ov) (-t (i-J i+K PV/ )(lV-k) 1(2.5.36d) Recall that when the difference equations 2.5.16 - 2.5.19 were formed a donor cell scheme was used, which guarantees the reduced velocities U v and U are always positive, so all four characteristic roots in 2.5.36 are always real positive and strictly less than one. As done before we will investigate the effect of a perturbation 6 in those roots, which stands for a large but finite value of m. It is clear from the expressions of equations 2.5.36 that if we analyze the effect of the perturbation in one of the first two values of , the conclusion obtained in this way will stand for all the other three roots. We then substitute = 1 i - U + 6 into equation 2.5.24 and V keep only the first order terms in 6. This will give a first order equation, and the single root of this equation gives an expression for as: K+C2 i +2C2 m (2.5.37) 106 which is strictly less than unity. To get this result we have assumed: P - UV << P K (2.5.38) This condition establishes a minimum value for the momentum exchange coefficient, in order to avoid exponentially growing modes. Stewart / 51 / showed that the condition in 2.5.38 implies that the wave length mAz will not have a growing mode if it is larger than a certain multiple of the radius of an individual bubble or droplet. To summarize, in this section we have seen that although the two-fluid formulation have at least two complex characteristic roots, this does not imply that a well behaved solution cannot be achieved. With the Von Neumann stability analysis we have shown that the numerical scheme used in our model, with a donor cell differencing will have non-growing high frequency modes for any value of the momentum exchange coefficient, and for the low frequency modes a well behaved solution requires a minimum value for k, expressed in 2.5.38. 107 III. THE CONSTITUTIVE EQUATIONS AND FUNCTIONS OF STATE 3.1 The Sodium Functions of State and Transport Properties. The basic source for the sodium properties is a compilation by Golden and Tokar / 46 /, dated 1966. This source has been used extensively since then in sodium technology with great success. Although a recent compilation by the Argonne National Laboratory / 15 / has come to our knowledge, but not yet published, we decided to stay with that of Golden and Tckar on the basis of its wide use and acceptance. A comparison between the new compilation and the one used by us showed a wider range of validity in terms of temperatures and pressure in favor of the new one, but no significant disagreement between them. A few modifications were made in the original expressions to satisfy program requirements, and all properties were converted to S I units. To help a quick reference to these properties we list them in table 3.1, with the correspondence to usual units. 3.1.1 Saturation Temperature From the several correlations for the saturation temperature listed in / 46 , the one which showed the best agreement in the most important range of temperatures 870 - 1100°C (1600 - 2000°F) is the one from Makansi et al, which is valid in the range 620 - 1150°C. 108 Table 3.1 Units Used in this Work and the Correspondent Usual Ones Property SI Units Equal Temperature OK °C + 273.15 Pressure Pa 14.05 x 10- 5 lbf/in2 Density kg/m3 0.06243 lbm/ft3 Internal Energy J/kg 4.2992 x 10 Viscosity kg/m-sec 0.672 lbm/ft sec Thermal Conductivity W/m OK 0.5778 BTU/hr ft Specific Heat J/kg OK 2.3884 x 10- 4 BTU/lbm °F Surface Tension N/m to 2 -5 -4 BTU/lbm ° F 109 The expression is; T sat (P) = b- a lnP with a = 1.2020 x 104 b = 21.9358 valid for 4.8 x 103 < P < 6.6 x 105 3.1.2 Vapor Density For the vapor density the expression which gives the density at saturation conditions was used and a perfect gas behaviour in the superheated zone was assumed: Tsat(P) Pv ( PIT ) = ( rv + rv 1 P + rv2 P2 ) with rv = 1.605 x 10 2 rv = 2.510 x 10- 6 rv2 = 3.230 x 10 13 valid for 3.4 x 104 < P < 2.3 x 106 T 110 3.1.3 Liquid Density From all correlations we have reviewed for the liquid density none showed a pressure dependence. This can be explained because the compressibility effect for the liquid phase is very small, usually smaller than the accuracy of the expressions themselves. Therefore it is reasonable, if one is interested only in the absolute value of that property, to neglect the liquid compressibility. But as seen in chapter 2, the model requires not only the value of the properties but also their derivatives with respect to pressure and temperature. It is clear from the physical point of view that however small, a liquid compressibility exists (otherwise the sonic speed would be infinity). The estimate of liquid compressibility does not have to be very accurate, since as said before its effect is smaller than the accuracy of the equation of state. Therefore, a simple With this idea in expression will satisfy the program requirements. mind, the approximation was used: /~ ~~=( aL)cntn BP P )constant temperature B~~P constant entropy C 2 where C is the speed of sound. A constant sonic speed was taken, equal to 2,100 m/sec, which corresponds to a temperature of approximatly 900°C, and the expression for the liquid density becomes: 1ill1 p (P,T) = r 2- o~ + rT+r + r 1 T 2 2 + r 3 T +ri 4 P 4 with rQ = 1.0116 x 103 0 rk1 = - 0.2205 r - 1.9224 x 10- 5 2 ri3 = 5.6377 x 10- 9 r = 2.26 x 10- 7 4 which is valid in the range 100 < T < 1370°C 3.1.4 Internal Energies The source of sodium properties gives only the expressions for the enthalpies. Therefore the internal energies were derived as e = h - P/p For the liquid enthalpy the following expression has been used: hQ(T) = h 2- o0 + h with: hQ 0 = - 6.7507 x 104 hl1 = 1.6301 x 103 hQ2 = - 0.41672 h 3 = 1.5427 x 10 in the3valid range valid in the range T + h2 1 T 2 T3 + h 3 112 100 < T < 1500°C The vapor enthalpy is derived from the liquid expression. Again a perfect gas behavior is assumed for the vapor phase, in which the enthalpy of the super heated vapor is equal to that of saturated vapor at the same temperature. h It follows: (T) = h V (T) + hv 0 + hv1 T with hv 0 = 5.089 x 106 hv 1 = -1.043 x 103 valid for < T < 1200°C 600 3.1.5 Transport Properties Following a list of the transport properties used in the model, again from reference / 46 / is presented Liquid Thermal Conductivity K(T) k C 0 + C 11 T + C with CZ = 1.0969 x 10 2 CZ1 = - 6.4494 x 102 C2 2 = 1.1727 x 10- 5 valid for 100 < T < 1370°C 2 T2 113 Vapor Thermal Conductivity (T) = Cv K o v T + Cv + Cv 1 T 2 with =- Cv o 1= Cv Cv 3.2349 x 10- 2 1.5167 x 10 - 5.4376 x 10- 8 2 for 700 < T < 5000 0 C Liquid Viscosity vtl (T) = exp[v2. T + o with vi O = - 5.732 vgi - 508.7 V9 2 = - 0.4925 for 100 < T < 1370°C Vapor Viscosity v (T) = vv + vv 1 T with vv = 1.261 x 10 5 vv1 = 6.085 x 10 9 for 700 < T < 5000°C T + 2 2 in T] 114 Specific Heat Liquid Cpt pQ(T) = C. po + C p11 T T + C p 2 with x 103 = 1.6301 C p o C 9. 1 = - p C p2 0.83344 104 x Q = 4.6281 for 100 < 1500°C < T Heat Specific Vapor = Cpv (T) C v + Cpv 1with 0 p with C v0 = 0.5871 x p o 103 = - 0.83344 Cv -4 x = 4.6281 = 4.6281 x 10 Cp 2 for 600 < T < 1200°C Tension Surface o(T) = t0 + st 1. with t = o 0.18 st1 = -1.0 x 10 4 in the range 100 < T < T + C v 1370°C T 1 T 115 Finally we observed that the vapor Prandtl number showed a very smooth variation with temperature. Thus in order to save computation time a quadratic expression for the Prandtl number was fited Prv (T) = pv 0 + pv p1 (T - pv p 2)2) with pv = 0.7596 pv = 0.810 x 10- 6 pv 2 = 844.4 where the range of validity for this expression is taken as the smallest of the ranges of the properties composing this dimensionless number: 600 < T < 1200°C 116 3.2 Mass Exchange Rate It has been stated in Chapter 2 that the interphase exchange terms play a key role in the stability of the Two-Fluid Model. Of all exchange terms, the mass exchange rate is the most critical one to the code stability. Because of the large difference in densities between the liquid and vapor phases for the usual range of pressures encountered in sodium technology, a small amount of mass transferredbetween phases corresponds to a very large volume change, and consequently large pressure and velocity variations. In particular for this models where the solution of the fluid dynamic equations is reduced to a pressure problem, these large pressure variations must be handled with extreme care. To insure the code stability, a choice is to be made of an adequate model for the mass exchange rate and its most strongly varying terms are to be implicitly treated. In general, the mass exchange rate S will be a function of the void fraction, pressure, temperatures and velocities, evaluated both at the old and new time levels. If the solution technique of chapter 2 is recalled,the derivatives of S with respect to the properties at the new time value are required, therefore the mass exchange rate is to be a continuous, differenciable function in these variables. The mass exchange model used in the code is derived from the principlesof the kinetic theory, in which the net mass flux j crossing an imaginary plane between phases is given by: 117 Pv P (3.2.1) E Tt V2-fR where J = mass flux (mass per unit time per unit area) M = molecular weight R = universal gas constant P and T = absolute pressure and temperature for both phases. For small differences in pressure and temperature, the above expression can be reduced to: = P AP 2,rR i- P M AT (3.2.2) 2T S S the Clayperon equation hfg dT dT (3.2.3) sat T fg is used to eliminate AP in equation 3.2.2 leading to: R (hfS 1 p vfg A (3.2.4) S 2 where hfg = difference in enthalpy between phases fg Vfg = difference in specific volume between phases and where the simplification was made: v = PRT/M 118 For the particular case of sodium, a few more simplifications in First note that pi >> pv, thus: equation 3.2.4 can be made. v = 1 1 _ p fg P v 1 ^ P v second, for the actual values of hfg, P and Vfg if follows hf > 1 2 P Vfg fig Therefore equation 3.2.4 becomes: p2 .V -- (3.2.5) AT fg S The above equation was obtained with the assumptions of ideal conditions embodied in the kinetic theory. Although this model's predictions are in good agreement with experimental data for evaporation, large discrepancies appear when condensation is considered. suggested a correction Silver and Simpson /41/ factor, which modifies equation 3.2.5 for condensation: p2h 2a Jc 2 fg v a - a p AT - (3.2.6) s Figure 3.1 reproduced from reference 4 1shows the value of as a function of pressure. From this figure, it can be seen that for the range of pressures expected to be encountered in LMFBR safety analysis, the value of is relatively small, thus the simplification can be made: 2o 2 -a = a 119 1.0 t:) 4rJ 00 © o '4,4 a) 0 C) U0 . 1 0 001 0 01 0 1 1 . 0 Pressure (bar) Figure 3.1 Condensation Coefficient as a Function of Pressure (From Reference 41) 120 Considering also the small variation of a with the pressure, and the uncertainties involved in obtaining this coefficient, a reasonable approximation is to take a constant value for a. Thus, for the pressure equal to one atmosphere the value of a is: a = 0.005 The next factor to be evaluated in the mass exchange rate is the specific area between phases. Wilson /11/proposed a model which takes into account three flow regimes - bubbly, anular flow and dry out. For the bubbly regime, with void fraction less than 0.6, he assumes the bubbles forming in the middle of each subchannel, packed on top of each other. (Figure 3.2) With this assumption, the expression for the specific area becomes: a <0.6 A = 4 2/3 (P/D) (3.2.7) a<0. where D = fuel pin diameter P/D = pitch to diameter ratio Although this model predicts reasonable values for the specific area at high values of the quality, for small void fractions this model would postulate the existance of unreasonalby small vapor bubbles, thus overestimating the specific area. To correct this we introduced a minimum value for the bubble radius, so that for 121 / / /6 Figure 3.2 / ,/ Bubbly Flow Representation 0 / Q / 0 '..' ra Vapor Fuel Pin Figure 3.3 Low Void Fraction Bubbly Flow Representation 122 small void fractions the model would be pictured as in Figure 3.3. The expression for the specific area becomes: A = 3a V r a < a m (3.2.8) m where a were a m = 3 (rM D 2 Tr i3 (P/P)2 - r/2 was chosen so that the two expressions of equations 3.2.7 and 3.2.8 be continuous at a , and r m is the minimum bubble radius m which was taken in our model equal to 6 x 10 -4 m. For the anular flow, all the liquid is assumed to be flowing in a circular annulus around the fuel rods, and the expression for the specific area becomes: _I A=4 V 2/ D for 0.6 < [2 7T (P/D) (P/D) [23 (P/D) 2 2(a 2 _-iT] 2/i (P/D) 2 (3.2.9) - it < 0.957 Finally in the dryout regime a partial contact of the vapor with the fuel pin walls is assumed, and the expression for the area becomes: 123 A 4 V D ra rr) 2 3 (P/D) - 1 r - a 1 _ - .957 (3.2.10) for a > 0.957 where the dryout transition point were taken from Autruffe /50/ the work by analysing the KFK experiments / 52/. Note that the transition from bubbly to annular flow presents a discontinuity in the specific area, whose magnitude is a function of the pitch to diameter ratio. The transition at a = 0.6 was choosen to minimize this discontinuity for the usual pitch to diameter ratio of 1.25. case a = 0 or Finally note that in the limiting a = 1 the interphase area is obviously zero. would prevent the initiation of boiling or condensation. - ~ This To overcome this difficulty a "seed" void fraction is introduced to account for the initiation of phase transition. In this way a is substituted in equations 3.2.8 and 3.2.10 by a which is defined as: if a > 10- 4 10- 4 if a < 10- 4 if .9999 if .9999 if a < .9999 a > .9999 a m> .9999 124 Now the question of determining which terms are to be evaluated at the new or old time level can be addressed. The specific area must be evaluated at the old time level since the discontinuity in the transition from bubbly to annular flow makes it impossible to obtain the derivative of the mass exchange rate. Both the enthalpy of vaporization hfg and the vapor density does not show a marked dependence on the primary variables pressure and temperatures, therefore they can also be evaluated at the old time level. On the other hand, the temperatures and pressure appearing in the expression of the mass flux have a very important dependence, thus they must be taken at the new time value. Following is a summary of the equations used for the mass exchange rate: S s - S e (3.2.11) c Aa e [P2hfg] [ 2rM - T )(l)]n+ 1 (3.2.12) T S Sc = R A(l-a)ac t e= | 10 (3.2.13) __[__ 1.0 if T 0 if T >T(3.2.14) Z -> v T > T s ac = (3.2.15) 0.0 05 T < T - s 125 A = 3 a r a < for Ir = 8(rm 2 m 3' D (3.2.16) m m vr 2 (P/D) (P /D) (3.2.17) _ 72 Tr/2 - 3 A = 2/i a (P/D) a 2 Tr m (3.2.18) < a < 0.6 , ,_ 2,/3 A = r (P/D) 2V' (P/D) 2 Tr at 1 22V 4\ I D V (2 I (3.2.19) - -) 2/3 (P/D) 0.6 A = c ~ar i < - 11 < 0.957 Tr (P/D) (P/D) r) 2 2 2/I(P/D) a2 2VT (P/D) 2-_ at> 0.957 if if if 10 - 4 a = a .9999 r m -R M = 6 x 10 - 4 361.30 a J/kg < 10 -. 957 4 a > .,9999 OK 1 -5a (3.2.20) 10-4 < .9999 10 ' al << .9999 m O (3.2.21) 126 3.3 Momentum Exchange In this section we identify two kinds of momentum transfer in the fluids dynamic equations. One represents the interaction of the fluid with the fuel pins and fuel assembly structure, and the second one accounts for the momentum exchange between the phases themselves. Fur- thermore, because the fuel assembly geometry presents a very marked difference in the flow path for the axial and radial directions, we will have a different set of correlations for each direction. Starting with the axial direction, a set of correlations developed by Autruffe/ 50 / analyzing the KFK experiments/ 52 / is used. The experiments were a series of steady state, single tube tests for several mass flow rates and qualities. Studying the pressure drop in the unheated zone (thus with no change in quality) the following correlations were proposed. Liquid wall friction: axial direction n F = Zz [0.18 "0.l8 g Re -* 22 PZ ' FR = [ On (3.3.1) U2Z, 2DH [ 18 Re FZz j1 Re 2-H H 2 2 ] (_s)n P*IU ZI(-a n+ )] dry U ']Uz{ za adry (-ery) (3.3.2) with (1-a)> I Rek = 2, 1*n (UZz[ID3 (3.3.3) 127 free volume in tube bank D = 4 x exposed surface area of tubes H adry = 0.957 axial direction Vapor wall friction: F = 0 F = vz v a 0.2 Rev -. 2 ap |U v P adry n |] Un+l vz vz] a > adry (3.3.4) with Re vV = C~Pv[|Uv z | DH (3.3.5) IV BRAT Interphase momentum exchange: M z =z(U z vz - UZz axial direction (3.3.6) n+l with .95 Kz = 4.231 PvIU 2DH Wilson/ 11 zUzI k [(l-o)(1+75(l-ca))] (3.3.7) / introduced another term in the expression for the interphase momentum exchange, taking into account the momentum transport associated with the interphase mass exchange. In this formulation, the equation for the momentum exchange becomes: M z )n+l -U = (K + S)n(U 9,z z Vz where S is the mass exchange rate. (3.3.8) 128 We also introduced in the above set of equations a term to represent a localized pressure drop, thus enabling the model to simulate fuel pin spacers or blockages. The expression, which adds up to the liquid wall friction is: n APL AKtjUj] [= Un + l (3.3.9) where KL is an input parameter. If for the axial direction momentum exchange we could find in the literature a number of sodium experiments, for the radial direction this abundance of data does not exist. But if we look into the dimensionless numbers involved in the momentum exchange models, we note the absence of the Prandtl number. Indeed, this number represents the energy transfer associated with momentum transport, and does not influence the pure momentum transfer we are interested here. Since of all dimensionless numbers involved in transport processes the Prandtl number is the only one which differenciates sodium from the other usually encountered fluids, we can expect to have good results if we use for our sodium momentum exchange a model developed for another fluid. For the wall friction two correlations widely accepted in heat exchanges and boiler technology, were considered. and London/ 48 One is by Kays /and the other by Gunter and Shaw/ 49 Both correlations present approximately the same value for the friction factor, thus we made our choice in favor of the second one because its formulation is more conveniently adapted to our code. correlations adopted are: The 129 Liquid wall friction: f n (Um )n+l zr F radial direction where (3.3.10) r where { 180 Rer fr :Regr < 202.5 (3.3.11) -.145 1.92 ReZr =_ I Ur Rear > 202.5 H Rer and (3.3.12) is the radial velocity at the point of maximum flow constric- Zr tion between rods, and the hydraulic diameter DH is the same as for the axial direction. For the vapor wall friction and interphase momentum exchange we found very little in the literature. Therefore we proposed a formulation for these terms consistent with the one used for the other terms. Vapor wall friction: Fv r =0 a < 'dry (3.3.13) f yr radial direction n U 2D H H I] r n+l n+l a> adry (Uvr) with f vr (180 Re Re vr vr < 202.5 (3.3.14) 1.92 Re ' 1 4 5 vr Re vr > 202.5 130 with Re vr =- pv r nv DH (3.3.15) and here again, Ur is the vapor radial velocity at the point of yr maximum constriction between the fuel pins. Interphase momentum exchange: M r = Kn(r r yr radial direction mr)n+l Zr - (3.3.16) with .95 K r ;I r - Um tr I (I -a) (1+7 5 (I -a)) I 4.31 2DH (3.3.17) To evaluate the velocities at the point of minimum transverse flow area we recall Chapter Two, where the primary radial velocities were defined as being the volume average velocities in the cell. One of the assumptions made in the derivations of that chapter was: Ur(r)Ar(r) = constant Thus the average velocity in the cell is: = Tv V v Ur(r)dV = (r)A,(r)dr V rrk U Ur rArrd r Um Am <Ur> = V (rk+l - rk) or Um r V Ar(r r k+1 k) rk) r (3.3.18) 131 Energy Exchange 3.4 As done for the momentum exchange, here again we divide the energy interactioas into two parts, the energy exchange between phases and the heat exchange between fluid and fuel pins and For the latter, we identify three subdivi- structural materials. sions, the fuel pin heat conduction, the convective heat transfer between the fuel pin walls and the fluid, and finally the fuel assembly structure model. Fuel Pin Heat Conduction 3.4.1 A single rod in each volume (node) is selected to represent the fuel pin heat conduction, which is assumed to be thermally equivalent to any other rod in that cell. Axial heat conduction is neglected, so that the radial heat conduction equation is: C T pCa 1 - 1 pt rr fi 3T ar(rK at ) = q"' (341) For the time being all material properties are assumed to be known quantities and we proceed to analyze the solution of equation 3.4.1. Later in section 3.4.2 these material properties are discussed. The fuel and the clad are now divided into mesh cells, the number of these cells being an input parameter. We only impose that all mesh spacings in the same region, whether fuel or clad, be of the same size, but mesh spacings may be different in different regions. One 132 Fuel temperatures are located at the cell is assumed for the gap. boundaries of mesh cells, represented by the subscript k. Fuel pin properties are evaluated in the center of mesh cells, and are repIf we integrate equation 3.4.1 resented with the subscript k+½. between the center of two adjacent cells we get: ~~~~~~r T [rpC dtt- - a .r+3T. k+ rK~]d (rK a)]dr q,,,r q"'rdr = rk-½ (3.4.2) rk-- Using the approximation: 2 <PC = rk+½ > pk 2 2 r k(C 2pk+½ p ) + + k 2 2 k- (3.4.3) (PC) pk- and rk+ rpC r½ P Tdr at = <pCp> aTk k ~ p k (3.4.4) r k- ½1 Also in equation 3.4.2 we have: r ~~ aT~ ~ ~ ~ 3T rk½ a = rk+½5 r+ [=rrr] dr = [rKS] rk2 rk-½ -T Tk+ -T k k+ l k~k+½ Ark+½1 (rK) (rK) k k-½ k 1 (3.4.5) Ark½ Finally, the right hand side of equation 3.4.2 becomes: 2 I rk- q"'rdr = 2 2 2 qk + + 2 2 k- ½ (3.4.6) 133 and the difference equation corresponding to equation 3.4.2 becomes: n <PCp>n k + <PCP T k) _ -rKn At(Ar)k+½( 2 2 2 rk+- rk 2 ,,,+ rk 2 qk+ + k+l - T _ nn+ k (n+l -h (k )k(T _n+l -1) - n k2 fit (3.4.7) qk- There are four locations where equation 3.4.7 must be modified to accomodate boundary conditions. For the center of the fuel pin, equation 3.4.1 is integrated from r = r 0 to r = rl, and the resulting equation is: n+1 p I . n -T rK T At with n+l n+1 2 1 2 rl½1 2-Y1% (3.4.8) 2 rlp <PC> 1 = 2Cp) 2 (3.4.9) il For the clad outside surface we obtain the difference equation by integrating equation 3.4.1 from r = rN½ to r = rN = outside fuel pin radius, and introducing the clad surface heat flux q". We obtain the equation: n+l T <PC > p N 2 N -T n n+l+ N) + (rK)N (Tn+l (Ar)N-1 N At N-1l + q"rTn rN 2 ~N-½ rN - rN_3 qN-½jfi N2 N3 -qN (3.4.10) 134 2 2 rN - rN_½ 2 NPp)N with <PC >N n (3.4.11) The general expression for the heat flux q" (later in section 3.5 the correlations for the heat flux will be discussed in detail) is: q + hn(Tnl Tn+l) + hn(Tn+l )h -TTn+l) )+h( q=h(T v W V ww -Ti n (Tn+l NB (w TS -TTn+l (3.4.12) where T = TN outside clad temperature w TN T, , TT = liquid, vapor and saturation temperatures vS hE , h v' hNB = heat transfer coefficients Finally for the two equations involving gap properties the term k. k is replaced by hGAP , the gap conductance. Returning to equation 3.4.7 note that a fully implicit differentiating scheme was used in this equation. can be shown to be unconditionally stable. This difference equation In this way we ensure that a time step determined by the fluid equations stability does not cause any stability problem for the heat conduction problem. Equation 3.4.7 couples the temperature at a cell k with its neighbors k+l and k-l, thus the temperature for all cells must be solved simultaneously. We incorporate an efficient technique to save computational time for this solution. This technique, proposed by Reed and Stewart / 21 / is a modification of the matrix inversion. tridiagonal 135 In matrix form, the set of equations 3.4.7 become: _ _ Tn+l Fal 11 a a12 a2 1 a22 o a31 ..... 0 a3 0 Tn+l 2 a34 ... I I 1 0 _ _ r f1 f2 0 x * I : I . · (3.4.13) 0 0 N-1,N aN-1 ,N O° Tn+l NN aNN T fN where the coefficients a's and f's depend only on the fuel geometry, the power density and material properties. All these quantities are evaluated at the old time step, therefore they do not change during the new time step iterations. The usual tridiagonal solution for this equation replaces the matrix of coefficients a's (which we will call by the capital letter A) by a product: A = x with C 1 0 0 ......... C21 1 0. 0 C3 1 1 0 0 ........ 0 0 0 ... = 0 * 0 C 0. .......... C N,N-1 1 136 and 11 0 12 0 b22 . 0 23 0 ... 0 ... 0 b33 b b 00 O 0 0 0 bN-l,N-lbN-lN 0 0 bNN In order for this factorization to be true, we must require: bb11 = a =a11 b -a .12 12 cpp b = a pp b = a 1 pp = a p,p+I -l/bp-lp -c p,p-i 'b p-l,p p,p+l Now define a vector X such that: F = C x X where F is the vector of coefficients f's in equation 3.4.13. This factorization requires: X = fl x p = fp - cpp-l Xp-l In this way, equation 34.13 B x = X becomes: (3,4.14) where B is an upper triangular matrix, and once we have gotten the value of TNN+l all other temperatures are easily obtained by backward substitution. N substitution. ' 137 The important characteristic of all these operations is that they are performed only on explicit terms. Thus this procedure must be carried only once, at the beginning of the new time step. The last line of equation 3.4.14 is the one used to determine the clad outside wall temperature. This is the only equation which involves implicit temperatures in the right hand side. equations 3.4.10 and 3.4.12, If we recall we can write this equation for the wall temperature, isolating the implicit terms: n+l n n n+l n n+l bNNTNN = fN + hTQ + h T + vv n Tn+( NB Ts (3.4.15) Then, after any Newton iteration k we use equation 3.4.15 to calculate the new wall temperature TNN, without the need for calculating all the other temperatures, and only after the Newton iteration has converged we return to equation (3.4.14) to calculate the fuel temperatures. 3.4.2 Fuel Pin Material Properties For the clad heat capacity and thermal conductivity the properties of stainless steel are incorporated in the code. / From reference 53 / the following expressions were selected: (Cp)clad a0 + aT + a22 p clad 1 Kclad b0 + bt K =b +bT (3.4.16) (..7 ~~~~~~~~~~~(3.4.17) 138 with a0 = 4.28- 106 a1 = 3.75.102 a 2 = 7.45.103 b 0 = 16.27 b 1 = 1.204.102 Two axially different zones are implemented in the code to represent the fuel itself. One with the properties of Plutonium-Uranium oxides to represent the active core region, and the second one to simulate the fission gas upper plenum. From reference / 21 / the following expressions were selected to represent the fuel region: %FUEL = + a2T = (a0 + aT (oCpFUEL + a3T ) (1 + 0.0450 (b0 + bT + b2 T ) (1 - (1 - d)-X) with X = 2.74 0pu Pu - 5.8 x 10-4T = fraction of PuO 2 in the mixed oxide fuel e d = fraction of theoretical density a 0 = 1.81 x 10 6 a 1 = 3.72 x 103 -2.51 a2 a3 = 6.59 x 10 4 b 0 = 10.8 b 1 -8.84 x 10 b 2 = 2.25 x 10 6 ). d (3.4.18) (3.4.19) 139 The fission gas plenum is simulated with a zero heat capacity. The gap heat capacity is also assumed to be zero. For its conductance the following expression was incorporated, from reference / 5 /: h hG c Gap (3.4.20) + h r with hr = (T 2 + T2 )(T T)(f r ~f + T)l.70 c (3.4.21) x 108 and dg + 1.32 x 0-4 [dg + 1.32 x 10 Cg h c -4-1 + 0.61 x 10 - 4] 1 + 1.8 x 10 3 (3.4.22) with Cg 15. x 2 dil = where dg = gap thickness di = fraction of helium in gap composition Tf and T fc = outside fuel pellet and inside clad temperature Convective Heat Transfer Coefficient 3.4.3 It has been mentioned in the previous section the expression for the heat transfer between the fluid and the fuel pins as: q hn (Tn+l _ - w n+l + hn (Tn+l _ Tn+l vT n n+l v ) + NB(Tw Tn+l) s This expression is an extension of the correlations proposed by Chen / 23 / for non-metallic coolants. Although this correlation 140 has not been verified by comparison with experimental data, we anticipate good agreement with experiments, based on the great success the assumptions of micro and macro-convective heat transfer mechanism has encountered for non-metallic coolants. Nevertheless, only an extensive experimental program could give a definitive confirmation of this model. The conditions for validity of the correlation are stable, vertical, axial convective flow of saturated liquid, with wetted heat transfer surface. These conditions are in general encountered in convective boiling in annular or mist-annular flow. The model is based on the postulate that there are two mechanisms contributing to the total heat transfer, and these mechanisms interact with each other. The macro-convective mechanism is associated with overall flow heat transfer, and the micro-convective mechanism is associated with bubble growth in the annular liquid film. The expression for the micro-convective heat transfer coefficient is: 75S 5 K~ Cp 0k A ap o~oo12 0.00122 9~. Pt(fAT).24 p~ ~2. ~' .79 .45 P49A 'NB a.5 29~ 2g .I,29 ~ f t~T h2P where Sf the suppression factor defined as: AT Sf =e 99 24 (3.4.23) 141 1. 0 .8 IKX I 1 I I I I I I I I I I. !. I I I .6 .4 .2 n --- U 10 - _- I I I I I I ! . I 5 10 Reynolds Reynolds Number 10 Number0 Figure 3.4 Supression Factor vs. Reynolds Number 6 142 10 2 101 1. 0.1 A\ u. 10 I.U 1 Figure 3.5 The Reynolds Number Factor 10 143 with superheat for bubble growth in annular liquid AT e =effective AT = difference between wall temperature and saturation temperature Ap = difference between pressure at the wall and liquid temperature Figure 3.4 shows the dependence of S on the Reynolds number. From that figure, we extract the correlation for Sf: Sf = 114)1 (1 + .12 ReTp ReTp < 32.5 (1 + .42 ReT78 )- 32.5 < ReTp 0.1 70 (3.4.24) ReTP > 70 and the two phase Reynolds number is defined as: Re-..= F 1' 25 (1-a)P UDH T' (3.4.25) TnZ where F is the Reynolds number factor, shown in Figure 3.5. The analytical expression from this figure is: F = 2.35(.213 + 1- ).736 Xt < 10. t Xtt (3.4.26) F = Xtt 1.0 10. and Xtt is the Martinelli parameter: (X .9 P nV (t Pt nI)( (3.4.27) 144 For the macroscopic heat transfer coefficient, Manahen / 11 / proposed a modified form of the Lyon-Martinelli equation: =h%=F375hs F 375h Z z~~sp h (3.4.28) whre F is the same Reynolds number factor used for the microscopic heat tranfer coefficient and hs transfer coefficient. p is the liquid single phase heat The CHAD correlation was used for this single phase heat transfer coefficient: k hksp Nu DH H with 4.5R N Pe 150 3 (3.4.29) Pe P > 150 RPe with R = -16.15 + 24.96(P/d) - 8.55(P/d) (3.4.30) and Pe is the Perclet number = RePr Finally, for the vapor single phase heat transfer coefficient the Dittus-Boelter correlation is used h v = 0.023 Re' 8pr ' v k v DH (3.4.31) 145 3.4.4 Ftl. Assembly Structure Models For the structural materials in fuel assembly two elements are the wire wrap and the fuel assembly hex can. considered: The wire wrap is modeled by assuming it has the same temperature as the outside clad surface. In this way, a thin layer of stainless steel, corresponding to the wire wrap heat capacity is added to the heat capacity of the last cell of the clad in the fuel pin model. Presently the model considers the fuel assembly hex can as an adiabatic boundary condition, modeling only the effect of its heat capacity in transients, although the model was designed to accomodate changes which would consider a heat sink outside the hex can. The equation used to model the hex can heat capacity is: Tn +l _ Tn ) (PC p) n Tn+l n+l -T ) cc c+ ~~~~~~~~~~~~~t h(Tn Tn+l )+ ',~~~~~~~~~~~~~~~ )+h n(Tn+l~~~ hv c v n+ -Tnl (n+l + .n nB(T ) =0 (3.4.32) where hi, hv, hNB are the heat transfer coefficients discussed in the previous section; Tc, T, Tv , T s are the hex can, liquid, vapor and saturation temperatures. 3.4.5 Interphase Heat Exchange Of all models presented in this section, the interphase heat exchange is the least developed. Whereas other constitutive equations, 146 like those for the momentum exchange or fuel pin heat transfer, are applied to all models of two phase flow, the interphase heat exchange constitutive equation has its only application in the two-fluid model, which has been given attention only in recent years. Thus, because of the lack of experimental data, we had to rely on a purely theoretical basis to produce a correlation for this exchange term. Two mechanisms can be identified in which heat is transferred between phases. One represents the enthalpy transported by the mass exchange between phases, and the other accounts for the convective heat transfer. Then, we propose the following expression for this exchange term: q q Zv s5n+v= n+l e e h v~ - n+l -Sh~ c s n+S + HA(T~ + HAi n+1 3l. -T (3.433) where S S h e c vS = evaporation rate = condensation rate = enthalpy for the saturated vapor his = enthalpy for the saturated liquid H = overall heat transfer coefficient A = interfacial area For the interfacial area, the same model developed for the mass exchange rate is used, and the expressions to evaluate this interfacial area can be found in equations 3.2.7 to 3.2.10. 147 In general, written the overall heat ransfer coefficient 1 can be as: Nu K H = H where Kg liquid thermal conductivity DH = hydraulic diamter Nu = Nusselt number A great deal of uncertainty is embodied in the Nusselt number used in this model, which cannot be resolved without a consistent set of experimental data on the heat exchange between phases. Therefore, we tentatively recommend the value Nu = 100. ,zl 148 CHAPTER 4 Experimental Tests Simulation The models and methods presented in the two previous chapters were assembled into a computer program named NATOF-2D. In order to evaluate the model results, as well as to test the program capabilities, three tests were simulated with NATOF-2D. The first experiment simulated was the SLSF P3A test which was used to evaluate the performance of the constitutive equations and to determine the sensibility of the code to these equations. Next the W-1 experiment was simulated, a test which has been completed recently. Finally a steady-state experiment, the GR19 was analyzed. 4.1 The P3A Experiment The Sodium Loop Safety Facility P3A Experiment was an in-pile test performed in the Engineering Test Reactor in the period July 16, 1977 to September 11, 1977. The experiment was made with a 37-pin bundle simulating an FFTR unprotected loss of flow accident. The test bundle 149 was irradiated for 26 full power says prior to the final experiment. The subassembly power was 1240 KWwith a mass flow rate of 9.2 lbm/sec (4.173 Kg/sec). Coolant boiling was detected at 8.8 seconds into the test. Inlet flow reversal occurred at 10 seconds, followed by inlet flow and temperature oscillations. Non-condensable gas passing through the bundle exit flow meter at 10.8 seconds was indicative of clad failure. Table 4.1 summarizes the design and steady-state operational data for the test. Steady-state measurements made prior to the test indicated the existence of a discrepancy between the actual thermocouple readings and their expected values. A temperature gradient in the radial direction was observed which did not agree with the predicted values. This discrepancy was attributed to a non-uniform radial power distribution in the bundle, due to a non-uniform neutronic flux across the test bundle. Therefore a radial power distribution was assumed in the numerical simulation of the test. Table 4.2 shows the assumed radial power profile [39]. An inlet pressure decay was imposed to simulate the loss of flow transient. The expression used was: Piu(bar) = 1.7187 + 7.4380 exp(-.21t) 150 Table 4.3 shows the timing of events for the NATOF-2D predictions, along with the experimental results and the values obtained with SOBOIL code [10]. Following a series of figures showing the results obtained with NATOF-2D is presented. Figure 4.2 shows the inlet mass flow rate as a function of time. The flow oscillations observed in the test were also predicted by NATOF-2D. Figure 4.3 shows the curve for the mass flow rate obtained from the experimental data. Figures 4.4 and 4.5 show the temperature evolution at the top of the heated zone for the central and the edge channels respectively. Here again the oscillations after the flow reversal encountered in the experiment are also observed. Figures 4.6 and 4.7 show the axial temperature profile at the central channel and the radial temperature profile at the top of the heated zone for different times. In this last figure once can observe an increase in the radial temperature gradient up to the time 9.0 seconds. This is attributed to the effect of the duct wall heat capacity. After 9.0 seconds the boiling in the central channels creates a strong radial flow, with the effect of reducing again the radial temperature gradient. 151 Finally figure 4.8 shows the void fraction maps for the three radial channels. From the numerical method point of view, the most encouraging result was the ability of the model to represent the transient beyond the point of flow reversal without numerical instability, a flow condition which has challenged the sodium two phase flow modeling for years. 4. 152 Table 4.1: SLSF-P3A Test Bundle Data Geometry (British Units) Number of Pins 37 Fuel Pellet OD (m) 4.94 .1945 in x 10- 3 Clad OD (m) 5.842 x 10- 3 .230 in Clad ID (m) 5.080 x 10 - 3 .200 in Wire Wrap OD (m) inner pins 1.422 x 10- 3 .056 in outer pins 7.11 .028 in x 10- 4 Flat to Flat (m) 4.501 x 10- 2 1.772 in Duct Wall Thickness (m) 3.048 10 - 3 .12 in 36.0 Length of Fuel Inlet to Bottom of Fuel (m) 1.857 x 10-1 7.31 in in Top of Fuel to End Cap (m) 5.334 Wire Wrap Lead (m) 3.048 x 10- Fill Gas Helium, 1 atm at 20°C with xenon tag gas Fuel Uranium-Plutonium mixed oxide, Pu 25% of total mass 1 210.0 in 12.0 in 153 Table 4.1 continued Thermo-Hydraulics Inlet Temperature (°C) 422 792°F Outlet Temperature at Steady State (0°C) 658 1216°F Bundle Power (kw) 1240 (Kg/sec) 4.173 9.20 lbm/sec Pressure at Top of Heated Zone (atm) 4.27 62.7 psia Cover Gas Pressure (atm) .957 14.1 psia Net Pump Head (atm) 7.619 Test Bundle Flow 112 psi Numerics of Simulation Number of Axial Mesh Cells Number of Radial Mesh Cells 10 3 154 Table 4.2 Assumed Non-Uniform Radial Power Distribution in P3A Test Bundle Pin Number* Power Factor 1 .90 2 .95 3 1.07 4 1.156 *see figure 4.1 for Pin Number location 155 Table 4.3 Event Sequence Times (Seconds) of the P3A Experiment Boiling Inception Inlet Flow Reversal Experiment NATOF- 2D SOBOIL 8.9 8.9 8.8 10.08 9.9 10.15 156 nel Figure 4.1 Pin Number Location #3 1 57 0 cc _ a:) E ,H or4 00 -- :,X C14 04 0 / C H C) 000 0o0o 0CN 0 ,4. (%) ( aJILw MOUIq SSVN 0 N 0 0N$ l5 5.0 1- 4.0 con tc 3.0 A4 2.0 c 1.0 CD zr 0.0 -1.0 0 5 10 15 20 TIME (sec) Figure 4.3 Experimental Inlet Mass Flow Rate 25 59 O C) r- C)- U C) ,-I <coa) HH C1) C) 0 E 00 q 0 C0 0 co O0 O.) 7qmnlvm2dKHj. ~ H 160 O ,-4 a) C IC ci u a) '.0 U, c a) U, ~z 4. a) CO H Cl, H .T L.l a) -c4 -rI CN Oo 0 Nl (D O 0 C 0C 0 (Do) o aafl. VIldNwal CD C) 0I 161 W o -4 I-E .H 44 v z0 x r4 .~4 :e <C4: O~ H z ee ') -4 (zD) 'lHJVddNA4,L 162 I 1 -'n _ I IUU _ _ t=10.0 900 C- t=9.0 0u A4 t=6.0 _______________________________t=6. 0 t=4.0 700 500 I 0 I -- - 1.5 ----t=0. 2 RADIAL DISTANCE (cm) Figure 4.7 P3A: Radial Temperature Profile 163 0 cc W xJ,J c. o zw H 0 -4 H z 0~ 0~ o U-1 U2 w - I I I N I I 0 I I..z* I '.I I To U C) C4 o JQJ 14 -CA m> co 164 4.2 One Dimensional Analysis of the P3A Experiment In order to determine the importance of the two dimensional characteristic of NATOF-2D, a comparison of the results presented in the previous section was made with a one dimensional analysis of the same test. NATOF-2D was modified to allow a one dimensional representation of the fuel assembly, and the P3A test was reanalyzed under the same conditions. Figure 4.9 shows the inlet mass flow rate as a function of time, figure 4.10 the temperature evolution at the top of the heated zone an and figure 4.11 the axial temperature profile for this one dimensional analysis. Finally figure 4.12 shows the inlet mass flow rate for both one and two dimensional representations in the boiling period. These figures show two interesting results. The onset of boiling occurred at 9.2 seconds for the one dimensional analysis, a delay of 0.3 seconds with respect to the two dimensional case. explained by the fact that in the This can be D case, both the central and the edge channels are represented by a single average temperature which is less than the maximum fluid temperature encountered in the central channel, and it takes longer for the average temperature to reach the saturation conditions. 165 The second result which differed from the two dimensional representation was the time of flow reversal, which occurred at 9.8 seconds, 0.28 seconds before the 2D result. This is explained by the fact that while voiding is taking place in the central channels, the edge channel which is relatively colder maintains a substantial liquid flow for a longer time, thus providing a path for an upwards liquid flow. This effect is lost with the one dimensional representation. 166 0 E -H cO © a; -W I. ,- '.. u U u o zU E-4 I O. $L4 CN 0 0o 0 C) 00 C) Ox 0 0 (%) give CD 0 mold ssvlq 0 a CN I 167 0 cc00 C) E .He a) ru -. 1'~ ~ ~ 'Crz m a) b.4 X H C) - I -tt rr Crt -- I~~~~~. <~~~~~C 0 As- o N' 0 00c00 0 0 C) r 0 O0 o0 (o) aLV-4n cx,) aivliawa 0 0O 01 168 On 41-i z0 w TO N w cI a a) C ¢ 0 z.:4 H EZ; Cl) e4 ¢n I as .,1 ,I.4 C C C C C C C 0% 0 0 -,: 1-4 (90) aufivuK~dNZ1 169 C), 4o ci I a. C 44 ci C:) ,--I C)4 v r~ a 7- C C) -4 '7-4 0 0 0 0 (%) 3,Iv MOOdSSVN 0 t . I ozI v 170 4.2 W1 - SLSF Test The W1 experiment is a test recently conducted under the direction of the Hanford Engineering Development Laboratory. Although the test has been completed, their results are not yet made public. The test was divided in two parts. The first one aimed at determining the fuel pin heat released characteristics during a loss of pipe integrity accident. This part of the test does not involve boiling. The second part of the test was directed to determine stable This part boiling and recovery limits as a function of fuel pin power. is the object of the numerical simulation presented in this section. Table 4.4 shows the relevant design data for the test [36], [38]. A series of flow transients were performed with several values of bundle power and flow decrease. Figure 4.12 is the graph of a typical Boiling Window Test flow transient. Table 4.5 shows the bundle power and percentage of full flow for each of the tests. Following a series of figures present the results of the NATOF-2D simulation of the tests. For each case analyzed a figure shows the evolution in time of the saturation temperature, clad and fluid temperatures for the central channel and the fluid temperature for the edge channel. For sequences 6a, 7a', 7b', 3 and 4 the axial temperature profile for the central channel is also shown. Finally .171 for the cases where substantial voiding occurred, namely sequences 7a', 7b' and 4 a figure showing the void maps for the three channels is also presented. In general the results obtained for the high power tests (14.4 kw/ft) seem to present values which agree with the predictions of the test plan (Table 4.5). On the other hand, for the lower power (and longer) tests, NATOF-2D predicted boiling conditions more severe than the expected in the test plan. One possible explanation for this discrepancy is an overestimating of the gap conductance by NATOF-2D, but of course an analysis of the results will be conclusive only when the test results are made available. As an extension of the test, a 217-pin bundle simulation was performed under the same conditions of test sequence 7b'. The simulation was made with five radial mesh cells and the same geometric and fuel pin design parameters as the ones used for the W results are presented in figures 4.31 through 4.33. test. The Comparing these figures with the correspondent figures for the 19-pin test, figures 4.23 through 4.25, the following conclusions can be drawn: ¶ The onset of boiling occurred at approximately the same time. ¶ The flow reversal occurred earlier and the voiding of the subassembly were much sharper for the 217-pin bundle. 172 These results confirm what was expected, since the onset of boiling occurs in the central channel and is not influenced by the size of the subassembly. The second conclusion was also expected, since in a large fuel assembly, the edge channel which is submitted to a smaller heat flux and also has the hexcan wall as a heat sink, occupies a fraction of the total flow area which is much smaller than the correspondent edge channel for a 19-pin bundle. 173 TABLE 4.4 W1 Test Bundle Data Geometry Number of Pins 19 Fuel Pellet OD (m) 4.94 x 10- 3 .1945 in Clad OD (m) 5.842 x 10- 3 .230 in Clad ID (m) 5.030 x 10- 3 .200 in Wire Wrap OD (m) inner pins 1.422 x 10- .056 in outer pins 7.11 x 10- 4 .028 in Flat to Flat (m) 3.26 x 10 - 2 1.283 in Duct Wall Thickness (m) 1.016 x 10- 4 .040 in Length of Fuel (m) .9144 36.0 in Inlet to Bottom of Fuel (m) .279 11 in Top of Fuel to End of Pins (m) 1.27 50 in .3048 12.0 ix n Wire Wrap Lead (m) 3 Fill Gas Helium-Neon at 68°F Fuel Uranium-Plutonium mixed oxide, Pu 25% of total mass. (10%), 25 psia 174 Table 4.4 continued Thermo-Hydraulics 388 732°F Test Bundle Flow (kg/sec) 1.95 4.29 lbm/sec Cover Gas Pressure (atm) 1.18 17 psia Inlet Pressure (atm) 6.42 91.8 psia Inlet Temperature (C) Numerics of Simulation Number of Axial Mesh Cells Number of Radial NFesh Cells 12 3 175 TABLE 4.5 Boiling Window Matrix for the W Experiment Fuel Bundle Power = 348 kw Peak Pin Power = 7.5 kw/ft Percentage of Full Flow Atz Test Sequence 29 5.0 1 Normal Procedure 24 5.0 2 Fallback Procedure A 24 7.0 2a B 22 4.5 2b Percentage of Full Flow Atz Test Sequence 42 5.0 3 Normal Procedure 35 4.0 4 Fallback Procedure A 35 6.0 4a Fallback Procedure B 33 3.0 4b Approach to Boiling Incipient Boiling Fallback Procedure Fuel Bundle Power = 532 kw Peak Pin Power = 11.1 kw/ft Approach to Boiling Incipient Boiling 176 Fuel Bundle Power = 662 kw Peak Pin Power = 14.4 kw/ft Percentage of Full Flow Atz Test Sequence 53 5.0 5 Normal Procedure 45 3.0 6 Fallback Procedure A 45 5.0 6a Fallback Procedure B 43 2.5 6b Normal Procedure A 42 2.0 7 Normal Procedure B 40 2.0 7" Fallback Procedure A 42 3.0 7a Fallback Procedure B 40 3.0 7a' Fallback Procedure C 40 3.0 7b Fallback Procedure D 38 3.0 7b' Approach to Boiling Incipient Boiling Dryout or Fuel Pin Failure 177 FLOW F1 F2 At, At 2 0.5 sec jAt 70.5 sec 3 TIME Figure 4.13 Typical Boiling Window Flow Decay for the W! Test -a- 178 2.0 U ~C M cc X- 1.0 10 z 0.0 <. 0. 3:: 0 I II I I I I I I I I I I 1000 I o 900 l ;: a i 800 700 600 0 1 2 3 4 5 6 TIME (sec) Figure 4.14: WI: Temperatures and Mass Flow Rate for the Sequence 5 179 1- 2.0 ,£ 1.0 to uV 0.0 - I I I I I I I I I 3 4 I 1000 C-, 0 900 E-4i 800 H 700 600 0 2 1 TIME (sec) Figure 4.15: WI: Temperature and lass Flow Rate for Sequence 6 180 2.0 1.0 0.0 1100 1000 Q 0 -A 900 H PH 800 700 600 TIME (sec) Figure 4.16: WI: Temperatures and Mass Flow Rate for the Sequence 6a 181 1100 i 000 900 800 0 5C, *- S.. 700 m_ O 600 500 400 . 0 .2 .2 .4 .6 .8 1.0 1.2 Distance above bottom of fuel (m) Figure 4.17: W: Axial Temperature Profile for Sequence 6a 182 2 mA -c:C 0 01 ai 4J O z 1000 ° 900 1. 800 0) 700 600 0 1 2 3 TIME (sec) Figure 4.18: W: Temperature and Mass Flow Rate for Sequence 7 183 2.0 1.0 0.0 1200 1100 1000 900 800 700 600 0 1 2 3 4 TIME (sec) Figure 4.19: Wl: TemperatLture and Mass Flow Rate for Sequence 7a 184 1100 1000 900 ' 0 800 600 500 400 0 .2 .4 .6 .8 1.0 Distance above bottom of fuel (m) Figure 4.20: WI: Axial Temperature Profile for Sequence 7a 1.2 185 2.0 C) V ZP c . C 1.0 o 0, 0 -1 1200 lt-. 1100 1000 I0 O c) w 900 AL Ed C) C) 800 700 600 0 1 2 3 4 'I M: (se.'() 'igur 4 .21: WI: Tcmperat.tre and Mass Flow Rate for SctLun'e 7a" 186 1100 1000 900 0 800 ' I- C., 0. E 700 600 500 400 0 .4 .2 .6 1.0 .8 1.2 Distance above bottom of fuel (m) Figure 4.22: Axial Temperature Profile for Sequence 7a' 187 Q cj1 N C) Er, H m ,. _x E-- FQ4 11 co r- I w Q) :I 0C U X /J Ct (n -) H *ro i-- N3 c~~~ ee I I I CN I --I (W 4 :I t --T U O3 H N ( 1 uo panq (ill) DUOZ 0 CN aculs sa DAoqP -- I--, U.,"I 188 2.0 " a) ° c W 1.0 4 ej o 0 c1g 1-- 0.0 uP- i_ x~~~~~~ 1200 1100 1000 u ~O a 0 800 7900 - EH mis 600 800 700 600 0 1 2 3 4 TIME (sec) Figure 4.24: WI: Temperatures and Mass Flow Rate for Sequence 7b 189 1100 I000 900 - 0 . 800 ZI SX 700 e 600 -t 500 400 ft 0 .2 .4 .6 .8 1.0 1.2 Sequence 7b' Distance above bottom of fuel (m) Figure 4.25: WI: Axial Temperature Profile for 190 a)1 H w" 11 .0 a) C) u 0* Q X ea ura)CJ r H i0 Co oa o, "- ~qq .) a) E- C (uW) luo Z pIVeH aAoqV NC Ozu-els(l 191 I- 2.0 ,' tZ J .. 1.0 .: ,. 0.0 I0 V Co p C.) 0 1, 2 3 4 5 6 TIME (sec) Figure 4.27: W: 'Tl'emperature and Mass Flow Rate for Se(Lence 3 192 1100 1000 900 800 - 0 - Cj700 4-, c. 700) 600 500 400 0 .2 ,4 ,6 .8 1.0 1.2 Distance Above Bottom of Fuel (m) Figure 428: Wi1:Axial Temperature Profile for Sequence 3 193 u 2. 0 - 2.0 a\ c: 1.0O \ 0 0.0 w- I I I I I I I I I I - I 1200 1100 1000 0 ri = 900 h 800 700 600 0 1 2 3 4 5 TIME (sec) Figure 4.29: W: Temperature and Mass Flow Rate for Sequence 4 194 IO v - E 0 .2 .4 .6 1.0 .8 1.2 Distance Above Bottom of Fuel (m) Figure 4.30: WI: Axial Temperature Profile for Sequence 4 195 u' O C' 1 W, Xn H o) u N rZ Q~~ MA C) 0 cry ) ~ CZ To .t1 z 0 HF F- A: cn~~~~e X ~ rn -t a) W :5 w .H 44 Ul) C) u: H CI I (tu) uo Z paleat oAoqv auelsil N CIA I 196 , ,,, , ,, ,, u C) W 20.0 -C 0 10.0 cI' Q: 43 O C) C) ,.Pt 0.0 A -- I~ I -I I I I, Il I Il- V~j ,Iv I I, , Ill 0 C) C) Or E5~ 0 1 2 3 4 TIME (sec) Figure 4.32: Temperature and Mass Flow Rate for 217-Pin Bundle Under Sequence 7b' Conditions 197 1100 1000 900 800 .; 700 600 500 400 0 .2 .4 .6 .8 1.0 1..2 Distance Above Bottom of Fuel (m) Figure 4.33: Axial Temperature Profile for 217-Pin Bundle Under Sequentce 7b' Conditions 198 C -Z en gO c C C ¢4~~~~~~4 H ~~C . c-i ° C O~~" .CC s_ I: xQa " g: C: gg: O H * I N N o - -- | o -Zr~ * z U -Z CD CD o_ H * N -~~ ~ ~~ ~ ~ ! I . 0 I I ¢,q ° i (W) auoZ paivall aAoqV aouelsia ! -I -r *l- 199 4.3 The GR19 Experiment As a final test of NATOF-2D the GR19 Experiment was simulated. This is a 19-pin, electrically heated, steady-state test performed on the CFNa loop, France. The test was analyzed with the BACCHUS code [35], and their results are presented here for comparison. Table 4.6 presents the significant design data of the test. Table 4.7 shows the mass flow rate for the different tests performed, along with the measured maximum temperature and the NATOF-2D results. Figures 4.32 through 4.34 show the axial temperature profile for these values of the mass flow rate. Figures 4.35 and 4.36 show the quality contours obtained for the values 0.265 and 0.260 kg/sec of the mass flow rate, along with the results of the BACCHUS code. One interesting feature encountered in this simulation was a stable oscillation of the void fraction for the mass flow rate around the value .320 kg/sec, with the void fraction ranging from 10 to 50%, indicating the presence of a slug flow. 200 TABLE 4.6 Design Data for the R19 Experiment Number of Pins 19 Clad OD (m) 8.65 x10 - 3 Heated Length (m) Downstream Unheated Length (m) Upstream Unheated Length (m) 0.6 0.494 0.12 Wire Wrap OD (m) 1.28x 10- 3 Flat 4.58 x 10- 2 to Flat (m) Inlet Temperature (C) Saturation Temperature at the Top of Heated Zone (C) Power (kw) 400 920 170 (axially uniform) 201 TABLE 4.7 Mass Flow Rate And Temperatures for the GR19 Experiment Tmax(°C) (NATOF-2D) FLOW (kg/sec) Tmax (C) (MEASURED) .606 693 694 .476 766 768 .405 825 827 .350 890 892 .329 918 920 (Boiling) .311 923 921 .293 926 921 .277 926 922 .265 926 925 .260 944 927 202 0 a) -W C; E T4 0) E-m .2 .4 .6 .8 1.0 Distance Above Bottom of Fuel (m) Figure 4.35: GR19: Temperature Profiles For .311 kg/sec Mass Flow Rate 1.2 203 940 920 900 880 C- 0I-, Cj 860 I. E 840 820 800 780 .2 .4 .6 .8 1.0 Distance Above Bottom of Fuel Figure 4.36: GR19: Temperature Profiles For .265 k/sec Mass Flow Rate 1.2 204 9Z 9: 9' 0 = w 8t EW 8~ 8 8 7 .2 .4 .6 .8 1.0 Distance Above Bottom of Fuel (m) Figure 4.37: GR19: Temperature Profiles For .260 kg/sec Mass Flow Rate 1.2 205 NATOF- 2D BACCIHUS .5 I I I I .4 I I I i I .3 I I 0 N .2 *0 c1 il 0C) 0 <r. J-W .1 0 C: 4 Ut ..Hi I Top of Heated Zone I I I 0.1 0.011 I -. 01 -. 0. O. 2 i, '.-II I I v/II I I I 1 2 . 3 1 Channel Number 2 3 Channel Number Figure 4.38 GR19 Quality Contours for 0.265 kg/sec Mass Flow Rate 206 BACCiUS NATOF-2D - I .4 .3 .2 .1 0 - .1 1 2 3 1 2 3 Channel Number Channel Number Figure 4.39 GR19: Quality Contours for 0.260 kg/sec bMass Flow Rate 207 CHAPTER 5 CONCLUSIONS 5.1 AND RECOMIENDATIONS Conclusion A two dimensional computer code for the simulation of sodium boiling transients was developed using the two fluid model of conservation equations. A semi-implicit numerical differencing scheme, capable of handling the problems associated with the ill-posedness implied by the complex characteristic roots of the two fluid model was used, which took advantage of the dumping effect of the exchange terms. The stability of the method was demonstrated theoretically in Section 2.5 and also by the practical results obtained with the model, shown in Chapter 4. The stability of the model imposes an upper limit on the time step size, which is related to the mesh spacing the phase velocity by the expression At <max [Az/ , Ar/r] 2 Of particular interest in the development of the model was the identification of the numerical problem used by the strong disparity between the axial and radial dimensions of fuel assemblies used in the current design of Liquid Metal Fast Breeder Reactors. A solution to this problem was found, which used the particular geometry of fuel assemblies to its advantage, reducing drastically the computation time. Most of the constitutive equations incorporated in the model were 208 obtained through previous work. In general, adequate models were found for most equations, but for a few of them no satisfactory correlations could be produced. These models involve areas of the sodium technology not yet fully understood, and a substantial effort of development must be done in these areas. These models are identified and discussed in the recommendations of this work. The models and methods of this work were incorporated into the computer program called NATOF-2D. experiments ties. With this program three series of were simulated in order to demonstrate the model capabili- The results of this simulation, which were presented in Chapter 4 showed good agreement with the experimental results obtained in the tests. One important capability demonstrated in these simulations was the ability of the model to represent the most severe boiling conditions, including flow reversal. 5.2 Recommendations A word of caution must be said to the eventual users of NATOF-2D. The purpose of this work was to develop a numerical framework capable of solving the set of conservation equations of fluid flow under severe conditions of transient sodium boiling. In this way, most of the effort put into the work was dedicated to developing and organizing the numerical methods and models for solving this set of equations. Of course the system of equations of fluid flow is not closed unless the constitutive relations describing the interaction of the fluids with the structural components and with themselves is provided, and a 209 set of constitutive equations were incorporated into NATOF-2D. Some judgment was exercised in order to select constitutive equations representative of the sodium beahvior, epecially tho;e characterizing the explosive volume change associated with sodim boiling at low pressure. This part of the code development was treated as complimentary to the numerical model construction. Therefore, the constitutive models may not be as realistic as the correct representation of sodium boiling in LMFBR fuel assemblies would require, and the overall results of simulations with NATOF-2D may be improved by the eventual improvement of some of the constitutive models incorporated in the code. Thus, this word of caution. The relatively superficial treatment of the constitutive models is not incidental. Only recently did the interest in LBR safety reach the point where extensive investigation of sodium boiling became justified, and a substantial amount of research is yet to be done. Therefore, the present status of knowledge of the physical phenomena associated with sodium boiling does not lead immediately to significantly accurate models of the constitutive equations involved in sodium boiling. The task of developing these models is not a simple one, requiring a considerable effort in theoretical analysis and experimental work, well beyond the scope of this work. But if NATOF-2D cannot claim to be a complete analytical model for sodium boiling simulation, because of the uncertainties contained in the constitutive models, it is an invaluable tool for the development of these models, where they can be implemented and tested against experimental results. 210 One of the most important benefits which NATOF-2D can provide to the development of sodium boiling is to identify, by the execution of sensitivity analysis, those constitutive models which affect most of the overall results, thus directing the research effort of sodium boiling to the directions which will lead to more fruitful results. From the experience we had with NATOF-2D calculations, by far the most important model affecting the end results of sodium boiling simulation is the one for the interphase mass exchange rate (which unfortunately is the one that showed the widest disagreement between authors). Therefore, we recommend as a first step in the continuation of the work presented here that a substantial effort be made in developing a dependable model for the interphase mass exchange rate. Of the same magnitude in importanceis the two phase heat transfer coefficients. incomplete. Here again the presently available models are few and Thus a theoretical and experimental work in this area is recommended, in order to acquire a thorough understanding of the sodium boiling curve. Another area which could be the object of future investigation is the one related to the interphase heat transfer. Although the direct effect of this exchange term on the overall results is not very marked, the relatively simple model incorporated in NATOF-2D could be replaced by a more refined one. The close relationship between this exchange term and the two previously mentioned would make this model a natural by product of the development of the above-mentioned ones. 211 REFERENCES 1. Griffith, J. 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Rohsenow, W.M., and Sukhatme, S.P., "Condensation," Massachusetts Institute of Technology 33. Brinkmann, K.J. and deVries, J.E., "Survey of Local Boiling Investigations in Sodium at ECN-Petten," Netherlands Energy Research Foundation ECN 34. Garrison, P.W., "Superheat Simulation Requirements for the Next Generation of LMFBR Codes," Oak Ridge National Laboratory, March 1979 35. Basque, G., Grand, D., and Menant, B., "Theoretical Analysis and Experimental Evidence of Three Types of Thermohydraulic Incoherence in Undisturbed Cluster Geometry," Kalsruhe, 1979 36. Baker, A.N., et al, "SLSF W-1 Experiment Test Predictions," GEFR 00047-9(L), December 1977 37. Knight, D.D., "SLSF W-1 LOD1 Experiments Preliminary Evaluation Data," ST-TN-80015, October 1979 38. Henderson, J.M., "Sodium Loop Safety Facility Test Plan HEDL W-1 SLSF Experiment," Hanford Engineering Development Laboratory, September 1978 214 39. Thompson, D.H., et al, "SLSF In-Reactor Experiment P3A," Interim Post Test Report, Argonne National Laboratory, November 1977 40. Kraft, T.E., et al, "Simulations of an Unprotected Loss-of-Flow Accident with a 37-Pin Bundle in the Sodium Loop Safety Facility," Argonne National Laboratory 41. Collier, J.G., Convective Boiling and Condensation, McGraw-Hill, United Kingdom, 1972 42. Clark, M., Jr., and Hansen, K.F., Numerical Methods of Reactor Analysis, Academic Press, New York, 1964 43. Richtmeyer, R.D., and Morton, K.W., Differential Methods for Initial Value Problems, Interscience, New York, 1967 44. Wallis, G.B., One Dimensional Two Phase Flow, McGraw-Hill, New York, 1969 45. Courant, R., and Hilbert, D., Methods of Mathematical Physics, Interscience, New York 1962 46. Golden, G.H., and Tokar, J.V., Thermophysical Properties of Sodium, ANL 7323, August 1967 47. Van Wylen, G.J., and Sonntag, R.E., Fundamentals of Classical Thermodynamics, John Wiley & Sons, New York, 1973 48. Kays, W., and London, A.L., Compact Heat Exchanges, McGraw-Hill, New York, 1964 49. Gunter, A.Y., and Shaw, W.A., A General Correlation of Friction Factors for Various Types of Surfaces in Crossflow, ASME Transactions, 67, pp643-660, 1945 50. Autruffe, M.A., Theoretical Study of Thermohydraulic Phenomena for LMFBR Accident Analysis, MIT thesis, September 1978 51. Stewart, H.B., "Stability of Two Phase Flow Calculations Using Two Fluid Model," Journal of Computational Physics, Vol 33, No 2, November 1979 52. Kaiser, A., and Peppler, W., "Sodium Boiling Experiments in an Annular Test Section under Flow Rundown Conditions," KFK 2389, March 1977 53. E1 Wakil, M.M., "Nuclear Heat Transport," International Textbook Company, 1971 215 54. ., "Thermophysical Properties of Fink, J.K., and Leibowitz, May 1979 Sodium," ANL-CEN-RSD-79-1, 55. Gantmacher, F.R., "The Theory of Matrices," Chelesea Publishing Company, 1977 56. Varga, R.S., "Matrix Iterative Analysis," Prentice Hall, 1962 57. Rohsenow, W.M., and Choi, H., "Heat, Mass and Momentum Transfer," Prentice Hall, 1961 58. Poter, M.C., and Foss, J.F., "Fluid Mechanics," Ronald Press, 1975 59. "CRC Handbook of Chemistry and Physics," 58th Edition, CRC Press, 1978 60. Thompson,D.H.,et al."SLSF In-Reactor Experiment P3A - Interim Posttest Report",ANL/RAS 77-48,November 1977 216 APPENDIX A - NATOF-2D INPUT DATA MANUAL In this section the user supplied information necessary to operate Before showing the description of the input NATOF-2D is presented. cards, it is useful to review the array structure of the code. Figure A.l shows an example of a full assembly and the corresponding cell arrangement in a r-z plane. Quantities appearing in this figure are: NI = number of mesh cells in the axial direction. It includes two fictitious half-cells in the top and bottom of fuel assembly. NJ = number of mesh cells in the radial direction. All dimensioned variables appear in the program with only one index, therefore a single number identifies each cell in full assembly. The cells are numbered from bottom to top and radially from center to hex can. Figure A.2 shows a cross section of the fuel assembly indicating the numbering of the fuel pins. Fuel pin rows are numbered from center to hex can, and the boundary between cells is indicated by the row number where this boundary lies. Figure A.3 shows schematically the cell arrangement for the fuel pin heat conduction. The quantities describing this cell arrangement are: NCF number of mesh cells in fuel. NCLD = number of mesh cells in clad. 217 ----- --F- - I---m I NI I 2xNI I NJxNI-1 2xNI-1 NI-i1 I NJxNI etc. , 3 NI+3 2 NI+2 1 I L_ _ J NI+1 B, i ~~ I .. .... I … Figure Al. Cell Arrangement in the R-Z Plane I ._ 218 Figure A.2 Fuel Pin Numbering 219 NCF Cells 1 Cell NCLD Cells I I I I I I 1 I I I1 Figure A3. Cell Arrangement for Fuel Pin Heat Conduction I 220 A single cell is assumed by the code for the gap between fuel and clad. Following is a presentation of the sequence of cards in the input data. Following the list of variables, in parenthesis, is the corresponding format for these variables. 1. General Description of the Problem 1st CARD: NI, NJ, NCF, NCLD (415) NI = number of mesh cells in axial direction NJ = number of mesh cells in radial direction NCF = number of mesh cells in fuel NCLD = number of mesh cells in clad 2nd CARD: NSET, TSET (I5, E15.4) This card contains information which controls the printed output. the code will print NSET times the flow map, with a time interval TSET. This card can be repeated up to 49 times, so that the time interval between prints can be varied to reflect the desired degree of information at each time. Following these cards, a card containing only zeros in the position corresponding to NSET must be placed, to indicate the end of this subset. 3rd CARD: ITM, IGAUSS, DTMAX, EPS1, EPS2 (2110, 3E15.9) ITM = maximum number of iterations in the Newton iterative solution. 221 IGAUSS = maximum number of iterations in the pressure problem solution. DTAX = maximum value for the time step increment. EPS1 = convergence criterion for the Newton iteration. EPS2 = convergence criterion for the pressure problem. EPS1 and EPS2 are criteria on the absolute value of the sure. 2. res- Their unit is N/m2. Boundary Conditions The next group of cards contains information governing the boundary conditions of the problem as a function of time. The simulation time is divided in up to 50 segments in which different functions can be prescribed for the boundary conditions. For a generic time segment L, the formulas used by the program for the boundary condition are: X = (X1 (L)*DTIME + X2 (L))*exp( OM X(L)*DTIME) + X3 (L) where: DTIME = TIME - TB(L-l) L = Index of current time segment TB(L) = Time at the end of segment L X1 , X2 , X 3, OMX = Input parameters and X stands for: PNB Pressure at the bottom of fuel assembly (N/m2 PNT = Pressure at the top of fuel assembly (N/m ) 222 ALB = Void fraction at the inlet of fuel assembly. TVB = Vapor temperature at inlet (K). TLB = Liquid temperature at inlet (K). HNW = Power density in fuel pins (/m 3) In order to save time, the code has an option to eliminate the exponential part in the formula to calculate the boundary condition. Thus, whenever the logical parameter LP is .TRUE., the boundary conditions are calculated as: X1(L) *DTLME + X2 (L) X (L1, F15.5) 1st CARD: LP, TB 2nd CARD: PNBI, PND2, PNB3, OMP (4E15.9) 3rd CARD: PNT1, PNT2, PNT3, OMT (4E15.9) 4th CARD: ALB1, ALB2, ALB3, OMA (4E15.9) 5th CARD: TVB1, TVB2, TVB3, OMV (4E15.9) 6th CARD: TLB1, TLB2, TLB3, OML (4E15.9) 7th CARD: HNB1, HNB2, HNB3, OMH (4E15.9) This group of seven cards can be repeated for as much as the To indicate the end of this sub- number of segments desired. set, a card containing only a be placed following the data. ' F' in the first position must 223 3. 1st CARD: Geometric Description of the Problem NROW, PITCH, D, E (I5, 3E15.9) NROW = Number of rows of fuel pins in fuel assembly. PITCH = Distance between fuel pin centerlines (m). D = Fuel pin diameter (m). E = Minimum distance between fuel pin surface and hex can wall (m). (see Figure A.2) 2nd CARD: N(J), J 1, 20 (2014) N(J) is the row number where the boundary between cell J and cell J + 1 lies. (see Figure A.2) 3rd CARD: LDATA, DZ(K) (L1, 5E15.9) In this group of cards the axial mesh spacing DZ are written sequentially from 1 to NI, five per card. The logical parameter LDATA must have a .TRUE. value in each card where DZ is written. Following this group of cards, a card containing an 'F' in the first position must be placed to indicate the end of this set of data. 224 4th CARD: LDATA, CN(K) (LI, 5E15.9) The same arrangement of the previous group of cards. CAN = Heat capacity of the hex can per unit area, for each axial mesh cell (J/m2 °K). There must be one value for each axial mesh cell. 5th CARD: LDATA, SHAPE(K) (L1, 5E15.9) The same arrangement as the previous group of cards. SHAPE = Power density shape in fuel assembly. There must be one value of SHAPE for each mesh cell in fuel assembly. 6th CARD: LDATA, SPPD(K) (L1, 5E15.9) The same arrangement as the previous group of cards. SPPD = Spacer pressure drop. There must be one value of SPPD for each mesh cell in fuel assembly. The code will treat the spacer pressure drop as: Ap = SPPD* p 2 2 7th CARD: LDATA, PPP(K) (L1, 5E15.9) The same arrangement as the previous group of cards. PPP = Radial power profile inside fuel pin. There must be one value of PPP for each fuel pin mesh cell, including gap and clad (i.e., there is NCF + 1 + NCLD values). 225 The power density at each fuel pin mesh cell will be the product of the power density specified in the boundary conditions, multiplied by the value of SIHAPEfor the corresponding fuel assembly mesh cell, multiplied by the value of PPP for the corresponding fuel pin mesh cell. 8th CARD: AD, APU, DIL (3E15.9) AD = Fraction of theoretical density of fuel. APU - Fraction of plutonium in full. DIL = Fraction of helium in gap composition. 9th CARD: LPLN-f(I), I = 1, NI (3912) LPLNM is an integer which indicates the axial composition of fuel pin. plenum). LPLN LPLN = 0 indicates gas composition (for upper = 1 indicates mixed oxide U,PuO2 . There must be one value of LPLNM for each axial node. 10th CARD: RADR, THC, THG (3E15.9) RADR - Fuel pin outside radius (m). THC = Clad thickness (). THG = Gap thickness (m). 4. 1st CARD: Initial Conditions LSS, TINIT (Ll, E15.9) LSS is a logical parameter to indicate steady-state or transient problem. 226 LSS = .FAT.SE.idicates transient problem. LSS = .TRUE. indicates steady-state problem. In case LSS is .TRUE., the remaining initial condition input data resume to the next card: 2nd CARD: (4E15.9) PIN, POUT, TIN, TAV 2PIN=at Pressure fuel assembly inlet (N/m ) (N/m ) POUTIN = Pressure at fuel assembly outinlet POUT = Pressure at fuel assembly outlet (N/rn,) TIN = Inlet liquid temperature (K) TAV = An estimate of the average temperature in fuel assembly (K) In case LSS = .FALSE., the next cards follow: (I5, 4E15.9) 2nd CARD: KO, TV, TL, P. ALFA 3rd CARD: KO, UVZ, ULZ, UVR, ULR KO is the cell number. check in the input data. the same mesh cell. (I5, 4E15.9) It appears in both cards to put a Each pair of cards correspond to The group is to be repeated for as many as the number of mesh cells. TV = Vapor temperature (K) TL = Liquid temperature (K) P = Pressure ALFA = Void fraction UVZ = Axial vapor velocity (m/sec) UVR = Radial vapor velocity (m/sec) ULR = Radial liquid velocity (m/sec) 227 4th CARD: LDATA, TR(K) (L1, 5E15.9) The same arrangement as the group of cards for DZ. TR = Fuel pin temperature (K). This array must contain one value for each fuel pin mesh cell. The values of TR are ordered as: TR(1) = Fuel centerline temperature at cell number 1. TR(NCF + 1 + NCLD) = Surface clad temperature at cell number 1. TR(NCF + 1 + NCLD + 1) - Fuel centerline temperature at cell number 2. etc. 5th CARD: LDATA, TCAN(K) (L1, 5E15.9) The same arrangement as the previous group of cards. TCAN = Hex can initial temperature (K). There must be one value of TCAN for each axial node. 228 APPENDIX B NATOF- 2D Programming Information When NATOF-2D was programmed, it was recognized that the field of sodium boiling is presently the subject of a large effort of research, and therefore it can be expected that in the future this research will produce better correlations for the constitutive laws governing the sodium two-phase flow. In order to make changes in the program as easy as possible, NATOF-2D was programmed with its subroutines in a modular structure, particularly the parts of the program dealing with the constitutive laws. In this way, the programmer working on modification of one particular subroutine does not have to worry about the rest of the program, provided the expressions introduced in that subroutine meet the requirements of consistency of the derivatives with respect to new time variables, which were discussed in chapter 2. Following is a description of NATOF-2D subroutines, their functions and structure. The reader is referred to figure B1, which shows the structure of NATOF-2D. 229 ERRMES SAVER 'DISTEP. READI I Figure B. NATOF-2D Subroutine Structure BC I I I EAD2 230 Main: The main program's only function is to allocate memory storage space for the dimensioned arrays and transfer the control of the program to subroutine HEAD. All arrays whose dimensions area function of the number of mesh cells are placed within a single array ORBI. Individual arrays are located by pointers which determine the first element of each array. These pointers are grouped into the integer array M, and the correlation of the pointer to the variable is as following: M(1) = P = New time, pressure, cell centered 1(2) = P0 = Old time, pressure, cell centered M(3) = TV = Vapor temperature, new time, cell centered M(4) = TVO = Vapor temperature, old time, cell centered M(5) = TL = Liquid temperature, new time, cell centered M(6) = TLO = Liquid temperature, old time, cell centered M(7) = ALFAN = Void fraction, new time, cell centered M(8) = ALFAO = Void fraction, old time, cell centered M(9) = ALFAZ = Void fraction, axial face centered M(10) = ALFAR = Void fraction radial face centered M(11) = RHOV = Vapor density, cell centered M(12) = RHOL = Liquid density, cell centered 231 M(13) = RHOVZ = Vapor density, axial face centered M(14) = RHOLZ = Liquid density, axial face centered M(15) = RHOVR = Vapor density, radial face centered M(16) = RHOLR = Liquid density, radial face centered M(17) = EV = Vapor internal energy, cell centered M(18) = EL = Liquid internal energy, cell centered M(19) = EVZ = Vapor internal energy, axial face centered M(20) = ELZ = Liquid internal energy, axial face centered M(21) = EVR = Vapor internal energy, radial face centered M(22) = ELR = Liquid internal energy, radial face centered M(23) = UVZN = Axial vapor velocity, new time, axial face centered M(24) = ULZN = Axial liquid velocity, new time, axial face centered M(25) = UVRN = Radial vapor velocity, new time, radial face centered M(26) = ULRN = Radial liquid velocity, new time, radial face centered M(27) = UVZO = Axial vapor velocity, old time, axial face centered M(28) = ULZO = Axial liquid velocity, old time, axial face centered M(29) = UVRO = Radial vapor velocity, old time, radial face centered M(30) = ULRO = Radial liquid velocity, old time, radial face centered M(31) = UVRZ = Radial vapor velocity, axial face centered M(32) = ULRZ = Radial liquid velocity, axial face centered 232 M(33) = UVZR = Axial vapor velocity, radial face centered M(34) = ULZR = Axial liquid velocity, radial race centered M(35) to M(62) = Implicit terms for the conservation equations M(63) = DH = Axial flow hydraulic diameter M(64) = DHR = Radial flow hydraulic diameter M(65) = DV = Fuel pin specific surface area M(66) = QSI = Maximum-to-average radial velocity coefficient M(67) = TS = Saturation temperature, new time M(68) = TW = Fuel pin wall temperature, new time M(69) = DTW = Increment in heat transfer for unit increment in TW M(70) = HCONV = Vapor heat transfer coefficient M(71) = HCONL = Macroscopic liquid heat transfer coefficient M(72) = HNB = Microscopic liquid heat transfer coefficient = Coefficients for the pressure problem M(73) to M(79) M(80) = TR = Fuel pin temperature M(81) = DTR = Auxiliary array for fuel pin heat conduction M(82) = TWO = Fuel pin wall temperature, old time M(83) to M(89) = Auxiliary arrays M(90) = SPPD = Localized pressure drop coefficienct M(91) = TCAN = Hex can temperature The storage space required by the array ORBI is given in double precision storage word by the formula: [135+ 2(NCF+ NCLD)]NI.NJ 233 HEAD: - Defines the pointers of array ORBI Controls the duration of the run Controls the printouts READ 1: - Reads arrays' dimensions READ 2: - Reads all other information Writes in FILE07 the input data for a restart Calculate parameters which will remain constant throughout the problem SS: - Performs an initial guess for the steady-state problem TMSTEP: - Advances one time step Controls convergence of the Newton iteration Controls time step size. The time step is always kept below the connective limit. If an instability occurs during the run, such as non-convergence of the iterative procedures or a variable outside range of validity, TMSTEP reduces the time step size by a factor of ten and the run is resumed. If the difficulty is removed, the time step will be increased slowly towards the convective limit again. If after three time step reductions the instability still persists, an error message will be printed and the execution terminated. DONOR: - Transfers all centered quantities to face centered positions Calculates explicit terms in momentum equation - WS: - Calculates explicit terms for mass and energy equations 234 ONESTP: - Performs one step of Newton iteration Calculates new values of implicit variables Checks variables against range of validity COEFF: - Calculates momentum exchange coefficients BC: - Calculates boundary conditions as a function of time HTCF: - Calculates heat transfer coefficients STATE: - Calculates sodium thermodynamic properties and its derivatives. The code stability imposes two requirements on the expressions for the sodium functions of state: the expressions for the densities must account for the pressure dependence which corresponds to a real, positive, finite sonic speed. The expressions for the property derivatives with respect to new time variables must be the analytic or numerical derivative of the expressions of the properties (but not approximated expressions). NONEQ: - Calculates the mass and energy exchange rates and its derivatives. The same requirement applied to the derivatives of the properties in STATE also applies here. CONDT: - Calculates the heat transfer between fluid and fuel pin and its derivatives. The requirement concerning the derivatives described above also applies here. HEXCAN: - Calculates the heat transfer between fluid and hexcan walls, and its derivatives. The requirement concerning the derivatives described above also applies here. 235 FPROP: - Finds the fuel pin transport properties FUEL: - Transport properties of fuel GAP: - Transport properties of gap CLAD: - Transport properties of clad FPIN: - Solves first part of heat conduction in fuel pin FTP: - Solves second part of heat conduction in fuel pin THXCN: - Solves the first part of hexcan heat conduction THXCNO: - Solves POWER: - Calculates GAUSIE: - Solves the pressure problem ERRMES: - SAVER: - the second part of hexcan heat conduction the power density as a function of time Prints error messages Saves fluid flow variables at the end of run for eventual restart 236 Functions CONDL - Liquid thermal conductivity as function of temperature CONDV - Vapor thermal conductivity as function of temperature CPL - Liquid specific heat as function of temperature HFG - Enthalpy of vaporization as function of pressure PRL - Liquid Prandtl number as function of temperature PRV - Vapor Prandtl number as function of temperature SAT - Saturation temperature as function of pressure DTSDP - Pressure derivative of saturation temperature as function of pressure SURTEN - Surface tension as function of temperature VISCV - Vapor viscosity as function of temperature VISCL - Liquid viscosity as function of temperature I 23 *7 APPENDIX C NATOF- 2D I/O EXAMPLES Fortran unit numbers for the data files are as follows: 5 is the standard input unit 6 is for the printed output 7 is the dump file to restart After a successful run, the program creates in file 7 an input data set corresponding to an initial value problem starting at the time the last run was finished. This is particularly useful in generating a transient problem input data set, which requires a substantial amount of information for the initial conditions. In this way, a steady-state problem, which requires a relatively small amount of information, produces in file 7 the input data for the transient problem. The user must only change the cards which describe the boundary conditions, to represent the desired transient conditions, and the desired sequence of printouts. Following is an example of the input data set for a steady state problem, a transient problem, and an example of the printed output. These examples were taken from the 217-pin simulation described in section 4. 238 STEADY-STATE INPUT EXAMPLE DATA SET 239 00000~000-0000000000 0000000000000000000 0000000000 00000000000 ++ + +++4+ ++ + + ++ + + + ++ + + + * +++ 000000000000000000 O O 00Q 0 O O O 00000000000000000000 00000000000 0000000000000000000 00000000000 0000000000000000000 00000000000 000000000)0CONO00 00000000000 00000000000000000000 moooooooooot-ot-oov 00000000000 ooooooooooo "0000000000Co0o0m00 W~0m00000o0dt'o0mom0r-00000000000 000000000000 (w~~~ 0000000000000000000 00000000000 000000000000900000 N0 0000000000 0 000000 00000000000 000000. 00000000000000000000 .................. ....... ++++ 0 ++++6++ ooo oo oo oo oo oo 060000 0 OOOOO 00000000000000000000 0 00 000000 000000 O OOOC00 0o0o 0 oOO000000 oooooo 0000000000000000000 00000000NOOCO00'00' 0-000000000000000000 + oooooo~roroo¢ooo-rooje "N + +i 00 +l+0 Om + +l+. O + + ++ C. oooooooooo ooo 00000000000 0000000000000000000 ooooooo 00000000000 00000000000 00000000000 00000000000 ++ * oooooo0000 000+ ++ ooo00o ooooooooooo 000000 -M000000000000000 00000000000 ++-+ ·- ....... I ............... I 000000 00 C0o 0 C;OCo C1~~00 o C C C C~0.0~o~C O O O C C ; ; C 00000O O; ; ; C; O C 0 000000 000000 0000 C0000 0000o 00000000000 0 0000 00000 0 0000000 00000000000 O I000000 O O 0 + .+++ 0 a 0 000000 0C 0000000000000000000 0000000000000000000 00000000000 00000000000 00 000000 0000 000000000000 --000000000 000000000OO400'00 00000000000 00000000000 00000000000 0000000000000000000 000000 *000000 o oo)oO0o00voot-o-0000 00000000000oooooooooo - 0; ° 000000 ° OO o 00"N000000r-0ODW000(-00 oNo wwoooooo0oo0ooo-00 :o oo o cooooo o cooa)oaxo o bo o 00000000000 oo o o o o oo ooo 00000000-00 0o oo0oo 0 000000 0 0+ 00 W(DO00M 00000- 0 0 0 00O000000O0O00 0000t7V00000-0000000 0000000000000000000 .... .. .. 0O 00000000000 0(I020000000 00000000000 .... .. . . 0 ++ .+++oo0o..onoo0oo o00000000000 00a0 0 000000 0 00000w 0 0000000000000000000 0000000000000000000 00000000000 0 ....... o0o0c: o no WOOOOOO)+000OOCOO+O' o o o oooooooooooooooooooooooooo 00000000000 .0 00000W 0000000000000000000 o o 0000o OO 00000000000 * O0000W 00 oo00o 0 0* 0000000000000000000 o 00Zooooo-ooo00 000 ooooo 00000000000 0 0o09LOoow0oo 00000000000 N~ - C-00 J0(DI 0 oo0o ooo0r-0o0 Co 0-000000000 oooo o 000 t0 0I 0 w--OW0000(000000W 00000000000 ooo to*Oww" .oo.oooooo-o oo.ooo ooooooooooo 00 ...... ob0 0 000000000000 000 00000000000 00 00000000 ooooooooooooo0C,¢o 00 +++ ..................... n0 000000000 vcn O+++ o o o ooro o oo .ooooo oZo oooo o c ooo oo oso 00 o+++o oo oo oo oo oo oo oo o00000000000 oooooo000 ooo 0O 00000 00000000000 O 000000 C4. 000000000000000c0000 0000000 00C00c++++ 00000000000++++++ -0 000000 000 00000000000 O OOOOOOOOOOOOOC 000oC 0000000000o 00 0000000 0r--000000000000000C, 00O00000o0Wo00-o000 00000000000 000000-0000 o0000*0ooo ¢00o000oo Co0 o *000000 000000 90 000000000000--WOOOWOWOOWO-00 00000000000 . . . ... . . . .. . .. . .. .. . .. .. . ..... 0 0 0 0 00 u0o o a0C00O 0 sro0o0o0o o o o 0000000000000000000 00000000000 0 Z___ 00 O t 0OOOOa OOOOO DOO(30 o o-~~~~~0 o o o o0 o o0 o o o0~00~000~ oOo o o Oc o oC 0oo 00 ~000000~ ...............................-.......... ..... ID 240 0 +++ + 0 o 0000 0000 ++10 0000 0000 0 0000 0000 0000 00 0000 0000 o o0000 0 0000 0000 0 0000 0000 0 00 0 0-00 0 0000 0000 0000o 0 0000 0000 0 0000 00o o 0 . . 0 + 00 0 a 0 0 00 0 000+0 0 0o00000 0000 0 0 00 000000 +++ + 000000 00000 0 0 0+o 0 000000 o 0ooDo 0 000ooo 00 00 0000000 0000-0 0000000 00000 0 000000 0++0 0 0 . 00 0 +00+ 00 0 + 0+ 0 0 0000 0*0 0 0-000 0000000 0000 0 0000000 00000 0000000 0 00 0 00000 000000 0 000000. (Co000000o oC co ro o C 00000 00000~000 0000 ++-0)+0 000 0 C+ 0000+ 0 0 O I0+ 00000 o..Coo _ 0000 I co o0 00000)-000 00000-C 0000-~-000 00C)?+ C0U 0-Oo 0 o C; C; 00 o~ L ooo - O- O. O- OO OOO - 00 C 0 C; ~ooo O OOfl o- ZD 241 TRANSIENT INPUT DATA SET EXAMPLE 242 oo0 o0oo <ro o o o o 0 r0 0 0 0 0000000000 +++ ++ ++++ 0000000000 00000000000 0000000000 oo00000ooo0000 0000000000 0oooooooo 00000000000 0C')0o0 00000 N 000 0 0 0 000 0 C, a 0 0 000000 000000 ++*+++++ 0 0~ o a) o o o o o0ooooo0o 0000000000 0000000000 00000000000 o, o 000 O 0 00OC) 000000 0 000000 0 . .vr 0 . .. 0. 0. 0 0 0 0 0. 0 C+ +++C)) 0 O +00 +0C] 0000000o 1 0 . . 0+++00 *000000 000000 000000 C) 0000000000 0000000000 0 0000000000 0 0 000 C 00 N 0 000000 000000 00 000 000 000000 0 0 ,0 ++++++ 000000 * 000000 000000 0 000000 0 000000 0 000000 + 000000 000000 00000 o , 000000 00000 oo-oo o o0n 0 o(D co 0 000000 000000 000o0 0 00 000000 · * 00 0 0 NO'-0-- 0+ 0 0 +0 CC 00 tU cJD0-OO W) D 04 CD 0 O OOOC)O WC)000 0000000 0000000 0000 0000 0000 0000 00 +i ++ C) 0 + C) 0 0 co 0CD 0OO OU) 0000 ~0000 0CD oo 0o o0C-- -n 000000 00 o C) - CN0 Iq [, u7 M ) 0 O Iq t n - -r o LO 'T (D LO0 4 0 0o CD D (0 0 0C) C) O 0 tO 00000~ 000000 _ 4 OO 00000 00 oooooooooo 00 o CD)C o o oO) o0 a o on DoDn 0( 0000000000 o0o0LN o+o+ ooooo+o 0 0 0 0 +0o0000000000 o o o oo ooN o C C 0 ++++++ 0 0 0O0 0_ 0 000000 0 000000 0 000000 o0000000 0 . 00000000000 C) O 0000000000 0 00o + + + + + ++ + 000000 . . . .C)a 00a 000000 000000 0 0 Nm O _ IT 000 000 0000 O0o o 0 O 0 0o O0000o0t 00NN000~0000 00 00 00 C)0 000 00000000 0000 0000 ol ++ 00 0 0 oo O 0 o oo O -- O WO O O O ( C 0 0 0 00 0000 00 00 0'C))0000000000 O U')0 .... 00000000000 O OO . O0000000rcrO OO 0 0000000000 I 00 t- tD . . . . . . . . . . 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