ML20002E401
| ML20002E401 | |
| Person / Time | |
|---|---|
| Site: | Clinch River |
| Issue date: | 09/30/1980 |
| From: | Cazzoli E, Imtiaz Madni BROOKHAVEN NATIONAL LABORATORY |
| To: | NRC OFFICE OF NUCLEAR REGULATORY RESEARCH (RES) |
| References | |
| CON-FIN-A-3015 BNL-NUREG-51280, NUREG-CR-1738, NUDOCS 8101270827 | |
| Download: ML20002E401 (105) | |
Text
{{#Wiki_filter:NUREG/CR-1738 BNL-NUREG-51280 l AN, R-7 l AN ADVANCED THERM 0 HYDRAULIC SIMULATION CODE FOR P0OL-TYPE LMFBRs (SSC-P CODED 1.K. MADNI AND E.G. CAzzou Manuscript Completed - August 1980 Date Published - September 1980 CODE DEVELOPMENT AND VERIFICATION GROUP DEPARTMENT OF NUCLEAR ENERGY BROOKHAVEN NATIONAL LABORATORY UPTON, NEW YORK 11973 Prepared for the UNITED STATES NUCLEAR REGULATORY COMMISSION OFFICE OF NUCLEAR REGULATORY RESEARCH , WASHINGTON, D.C. 20555 UNDER CONTRACT NO. DE-AC02-76CH00016 i NRC FIN N0. A-3015 j i
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f FOREWORD As a part of the Super System Code (SSC) development project for simulat-ing thermohydraulic transients in LMFBRs, the SSC-P code has been developed at Brookhaven National Laboratory. This code is intended to simulate system response to a malfunction anywhere in the heat transport system of a pool-type LMFBR design. This topical report describes the modeling and coding efforts for the SSC-P code. A users' manual is under preparation and will be issued as a separate report. This work, covered'under budget activity No. 60-19-20-01-1, was performed for the Office of the Assistant Director for Advanced Reactor Safety Research, Division of Reactor Safety Research, United States Nuclear Regulatory Com-mission.
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TABLE OF CONTENTS. FOREWORD l .iii LIST 0F. FIGURES- - viii ; LIST OF-TABLES ix NOMENCLATURE -x fACKNOWLEDGMENT xiv
- 1. ABSTRACT xiv.
2'. ; INTRODUCTION 1
- 3. ~ STEADY-STATE MODELS 9 3.1. GLOBAL THERMAL BALANCE 10 -
3.2 ENERGY BALANCE IN_P0OLS 15 3.2.1 Hot Pool Balance 17 3.2.2 Cold Pool Balance 12 1 22 3.2.3 Barrier. Heat Transfer 25 3.2.3.1- Areas 3.2.3.2 - Film Coefficients 27 3.3 PRIMARY HYDRAULICS 30 3.3.1 Intermediate Heat Exchanger 30 3.3.2 Cold Leg Pipin9 34 3.3.3 Liquid Levels in Pools 34 3.3.4 External Bypass 35 3.3.5 Pressure Drops in Pools 36 3.3.6 Pumps 36 I
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- 4. . TRANSIENT MODELS 38 l 4.1 PRIMARY COOLANT DYNAMICS 38 4.1.1 Flow Equations 39 1
4.1.1.1 Intact System 39 4.1.1.2 Damaged System 40
'4.1.2 Liquid Levels in Pools 41 4.1.3 External Bypass 43 4.1.4 Reactor Inlet Pressure 43 4.1.4.1 Intact System 44 4.1.4.2 Damaged System 46 4.2 INTERMEDIATE SYSTEM 46 4.2.1 Spherical Pump Tank Model 47 4.2.1.1 Solution of Cubic Equation 49 4.3 ENERGY BALANCE IN P0OLS 50 4.3.1 Hot Pool Stratification 50 4.3.1.1 Heat Transfer Areas 52 4.3.1.2 Volumes of Zones 54 4.3.1.3 Energy Equations 54 4.4 INTERMEDIATE HEAT EXCHANGER 58 4.4.1 Pressure Losses 59 4.4.2 Heat Transfer 62 4.5 SOLUTION PROCEDURE 65 4.6 SDME CONSIDERATIONS FOR COLD POOL CONCEPT 72
- 5. INPUT DATA PREPARATION 75 5.1 CORE DATA 75 5.1.1 Flow Fractions 75 5.1.2 Power Fractions 76 4
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5.1.3 ' Fuel Hot Channel ~77 5.1.4 Axial Power Profile in Fuel 78 5.2 PRIMARY SYSTEM 80 5.2.1 Pump Parameters !80J 5.2.2 Data on Structures 81
- 6. CODE DESCRIPTION 84 6.1 CODE DEVELOPMENT APPROACH 84 6.2 FLOW CHARTS AND SUBROUTINE DESCRIPTIONS 86 REFERENCES 90 4
1. I t i f e 'l J I i
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LIST OF FIGURES Figure Title Page 1 Primary Systs schematics a) Pool-type LMFBR b). Loop-type LMFBR
. 3 2 Configuration options in pool-type LMFBR primary system 4 ,
3 Schematic diagram of' Phenix heat transport circuit 8 4 Logic for SSC-P overall steady-state solution 11 5 Plant schematic for global balance 11 6 Iterative logic for global themal balance 14 7 Primary system configuration for tank enenjy balance 16 8 Driver logic for steady-state energy balance in the sodium pools (hot pool design) 18 9 Logic for steady-state thermal balance in hot pool 20 10 Solution prmedure for thermal balance in cold pool 23 11 Thermal barrier configuration 23 12 Logic to calculate overall barrier heat transfer coefficients 26 13 Configuration for calculation of barrier heat transfer area 26 14 Hydraulic profile of primary tar- and components 31 15 Steady-state logic for primary hydraulic calculations 32 : 16 Phenix expansion tank and secondary pump 4g 17 Model configuration for spherical expansion tank 48 18 Configuration for calculation of heat transfer areas fro the two zones to the barrier 53 } 19 Profile of intemediate heat exchanger 60 20 Driver for transient themal-hydraulic calculations in primary system 66 , i 21 Calculational logic for primary hydraulics 67 ) l.
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Figure Title Page 22 Calculational logic for. energy balance within
. primary tank 68 ,
23 Three main' driver programs of SSC-P 70 24 MAIN 9S flow diagram 70 25 MAIN 9T flow diagram 71 26- DRIVIT flow diagram 71 ., LIST OF TABLES i l-Table Title Page I Steady-state power and flow fractions 78 i l 1 1 +
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NOMENCLATURE Symbol Description Unit-2 A Cross-sectional area or heat transfer area m C Specific heat capacity J/kg-K D Diameter m D h Hydraulic diameter m e' Specific enthalpy J/kg f Fraction, factor -- Fp Number of pumps lumped into a flow path -- F X Number of IHXs lumped into a flow path --- ! 2 g Gravitational acceleration N/m Gr Grashoff number -- H Pump head m h Film heat transfer coefficient W/m -K I Moment of inertia kg-m K Loss coefficient -- k Thermal conductivity W/m-K L Length of pipe section m L f. Length m M, m Mass kg N Number of nodes in a pipe run or IHX section -- N p Number of plates in bart er -- 1 N Specific speed of pump -- s ny Number of intermediate loops -- np Number of pumps in primary system -- n t Number of active tubes in IHX -- n Number of IHXs -- X Nu Nusselt number -- P Pitch m P Pressure N/m P Power W 2 P ext Pressure external to the break N/m l 1 l l X -- l
NOMENCLATURE (cont) Symbol Description Unit Pr Prandtl number -- q Heat transfer rate W r, Discharge radius of jet m R Gas constant 2 2 m /s ,g g R T Composite barrier resistance K/W Ra Rayleigh number (RePr) -- Re Reynolds. number (WD/Au) -- T Temperature K t Time s U Overall heat transfer coefficient W/m -K - 3 V Volume m 3 Vg Volume of liquid in spherical pump tank m
; W Mass flow rate kg/s Wg Intermediate loop flow rate kg/s W
Ptot Total flow from all primary pumps kg/s W Xtot Total flow from all IHXs kg/s x Axial coordinate m Z Vertical coordinate measured from reactor inlet as reference m z Vertical distance m Jet penetration distance m z) , zg Liquid level in spherical pump tank m l a Normalized pump speed (0/0 ) ~~ 9 R , S Bypass fraction -- A Difference or loss -- 6 Thickness .. O Angle made with horizontal degrees L e g Angle associated with liquid level in spherical tank degrees u Dynamic viscosity 2 N-s/m v Kinematic viscosity m /s p Coolant density kg/m p H Density of hot pool sodium kg/m 3 pC Density of cold pool sodium kg/m T Torque N-m n Pump speed rad /s
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NOMENCLATURE (cont) Subscript Description A,a Sodium in zone A a,amb Ambient acc - Acceleration B,b Sodium in zone B b At break location barr Barrier value BP External bypass flow C Core flow c Contraction cg Cold pool to cover gas CP,c Cold pool value CW Coolant to wall CV Check valve e Expansion f Fuel. f Friction f,g Friction, Gravity, etc. g Cover gas value HP,h Hot pool value j I Intermediate i Value at nodal point i in Inlet m1,m2,m3 Netal in hot pool j m4 Metal in cold pool ] Na Sodium o Outlet P Pump value p Primary Pin Pump inlet ! Po Pump outlet j psh Primary to shell wall pt Primary to tube wall R Rated value
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l NOMENCLATURE (cont) Subscript Description Rin Reactor inlet Ro Reactor outlet S Steam , s Secondary
's Steel sh Shell st Secondary to tube wall t Tube <
w Wall X IHX value x Co-ordinate in flow direction Xin IHX inlet Xo IHX outlet IX- Primary side of IHX l '2X Secondary side of IHX , l I i l a 4 i i I a i i
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- 1. ABSTRACT ,
Models for components and processes that are needed for simulation of thermohydraulic transients in a pool-type liquid metal fast breeder reactor (LMFBR) plant are described in this report. A computer code, SSC-P, has been dev. oped as a part of the Super System Code (SSC) development project. A users' manual is being prepared as a separate document. ACKNOWLEDGMENT A note of appreciation and thanks is due to A. K. Agrawal, S. F. Carter and M. Khatib-Rahbar for their assistance in the initial phase of this work. Thanks are also due to R. J. Cerbone and J. G. Guppy for reviewing the report and to P. Siemen for her skillful typing of the manuscript. This work was performed under the auspices of the U.S. Nuclear Regulatory Commission. l l l
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- 2. INTRODUCTION BasM w current liquid metal-cooled fast breeder reactor (LMFBR) tech-nology, a Nuercial LMFBR power plant could be either of the loop-type or
; pool-type design. Primary interest in the J.S. has centered on the loop con-cept, while in Europe and Asia interest has been divided. The French and British have selected the pool concept for their large plants, and the Gemans and Japanese have selected the loop concept, while the Russians are trying one of each.- Judging from the successful experiences with pool systems, e.g.,
EBR-II ii. the 'J.S., Phenix in France, and PFR in the United Kingdom, the pool design stands as a viable option to the loop design. The Snherent difference between the pool and loop design options lies in the primary heat transport system. In the fomer, the entire radioactive primary system including pumps, intermediate heat exchangers (IHXs), reactor and associated piping is enclosed in a large tank filled with sodium. This is in contrast to the latter arrangement, in which all the primary components are connected via piping to form loops attached externally to the reactor vessel. Figure 1 illustrates this difference. The intermediate and tertiary systems of both cptions are generically identical. Some of the pool-type primary sys-tem characteristics and advantages stemming from its integral and enclosed de-sign are delineated be'.ow: (i) All components are submerged in the coolant. This would render the consequences of a pipe rupture accident less severe because of a con-stant back pressure exerted against the coolant flow out of the break. (ii) Piping lengths are fairly short. i l l
s (iii) There.is a large inventory of sodium (iv) Pumps as a rule are located in the cold leg of the heat transport ci rcuit. (v) Due to the large volume of the cold pool, the reactor inlet plenum is not important. (vi) The check valve is usually included in the removable pump unit. (vii) There is only one cover gas space. This eliminates the need for separate cover gas systems over liquid levels in pumps and IHX. (viii) Due to the large thermal inertia of the pool, emergency cooling of the pool is totally decoupled from the emergency cooling requirements of the core which simplifies the design. However, this implies that this very large thermal inertia may keep the pool cool, even though the core sodium is boiling away. Therefore, natural convection is just as important as in loop-type primary systems. There are soma disadvantages stemming from problems of accessibility for inspection and maintenance, thermal expansion effects, and plant lifetime insulation requirements. There are currently two key alternative primary system concepts for pool designs based on the choice of reactor outlet plenum; these are namely, the hot pool concept and the cold pool concept which are illustrated in Fig. 2. Only important components have been shown. In both concepts, the pump draws sodium from an open pool and supplies core flow. In the hot pool concept, there is no pipe connection between the reactor , l outlet and the IliX inlet. The coolant exiting the core enters an open pool; during steady state operation, this open pool temperature is at approximately the reactor mixed mean outlet temperature. The sodium in this hot pool is 1 l
MOT 0g COVER GAS 11 !
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PUMP OUTLET PLENUM (OPEN OR CLOSED) s II II h il i l h CORE ll ll g PRIMARY TANK
- INLET PLENUM COLD POOL (a) Pool-type LMFBR COV R GAS MOTOR Il
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FROM' F SOOluy TO IHX LEVEL ' CTOR
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OUTLET PLENUM [s
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CORE k ') , INLET PLENUM ( ) GUARD VESSEL (b) Loop-type LMFBR Figure 1. Primary System schematics i
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( 3 csa f_00g -t k- JJJ n- ))) OLO PO LD POO (a) Hot Pool Concept (b) Cold Pool Concept Figure 2. Conf 3f uration options in pool-type LMFBR primary system i t i i l 1 i t l l
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separated from the cooler sodium in the pump suction region of the tank by an insulating barrier. In-general, the liquid levels in the hot and cold pools, are different; this level difference accounts for the hydraulic losses and gravitational heads o,ccurring during flow through the IHX. These levels will vary with changes in the total coolant flow through the IHXs and flow through ths core. The cold pool concept utilizes an enclosed reactor outlet plenum, similar to that in the loop design except for the absence of cover gas. Hot sodium leaves the outlet plenum and flows to the IHX via a short length of insulated piping. The bulk sodium of the pool is at the reactor inlet temperature and only one liquid level exists. In either concept, the pump outlet is con-nected via piping to the reactor inlet. The IHX outlet to the pump inlet is in the cold pool environment. In both concepts, an elevated IHX arrangement is used to promote natural convection in the absence of forced cooling. As mentioned earlier, the insulated barrier (hot pool concept) and the hou leg piping (cold pool concept) have to operate in a sodium environment for the life of the plant. The hot pool concept has been implemented in operating prototype plants, namely, Phenix (France), PFR (United Kingdom), it will also be implemented in the Superphenix (France) and CFR (United Kingdom) commercial plants. In con-trast, the cold pool concept has only been implemented in the experimental, albeit successful EBR-II plant (U.S.A.). Furthermore, all U.S. EPRI-sponsored pool-type prototype large breeder reactor (PLBR) design studies have been based on the hot pool concept. Restricted analytical models and associated computer codes such as MELANI,1 2 3 NATDEM0 , and DEM0-P00l have been developed by other organizations to simu-
i i
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late the overall resnonse for specific plants. The first code, MELANI, was designed for the Prototype Fast Reactor (PFR). NATDEMO models the EBR-II
. plant, and is' a combination of NATCON" describing the primary system and DEM0,-
l an ada'ptation of the original DEMO ?, ode' to model the secondary and tertiary l S systems. DEM0-P0OL is an adaptation of the DEM0 code - to model the con-ceptual pool-type PLBR design study by the General Electric Company. More re- , cently, an extension of the fast-running CURL code 6 to model pool-type plants 7 has been reported . Since an LMFBR plant using the pool concept is a viable option for the U.S., a general system code is required by NRC for the safety analysis of pool-type LMFBRs. A program was therefore initiated at BN18 to develop an advanced thermohydraulic system transient code T0r pool-type LMFBR plants, capable of predicting plant -response under various off-normal and accident
- conditions. This code is designed to analyze the system response to a mal-function anywhere in the heat transport system of the plant. In all events, computations are terminated when loss of core integrity is indicated (i.e.,
when clad temperature reaches its melting point). Another key feature of the code is the capability of performing steady-state or preaccident initialization. The initialized conditions are calculated from user-specified design parameters (9d operating conditions. A restart capability is provided so that a . series of transient analyses can be made from a single steady-state compu.. Mon; a restart option during transient ? analyses is also available. Although emphasis of the code development was on transient analyses re-quired for safety,' this code can be utilized for other purposes such as (a) design scoping analyses and (b) specification of various components.
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The models developed for the SSC-P code are reported. The essential com-ponents and their arrangement in a pool system, such as the Phenix reactor plant, are schematically shown in Fig. 3. Under nonnal operation,11guld sod-ium flows in the primary and intermediate (secondary) systems as indicated oy the direction of the arrows. Although only one circuit is explicitly shown in the figure, a plant does have more than one circuit. There can be more than one pump feedinr sodium into the core, and more than one IHX transferring heat to the secondary system. In general, the number of pumps and IHXs are not re-quired to be the same and are specified as input parameters in the SSC-P code. The number of secondary loeps in the plant is determined by the number of IHXs serving each loop. A comprehensive description of the models is given in Sections 3 and 4; the steady-state plant characterization, prior to the initiation of transients is described in Chapter 3 and their transient counterparts are discussed in Section 4. The analytical effort thus far has been guided largely by the Phenix design; the Phenix input data decx has evolved from an extensive search of the available literature. In Section 5 a discussion is presented on the ways in which some of the datt, which were not directly available in the literature, were obtained. Section 6 presents a summary of the development approach used for SSC-P. Since many portions of SSC-P utilize the same methods and models as its parent code SSC-L,9 the primary emphasis in the development and its description in this report has been to focus on the differences between the two codes. In particular, it is our purpose to highlight the new models and modifications < required for pool type LMFBR simulation. i l l l l
T COVER GAS
. Nm BUFFER TAN K PUMP, ,
STEAM EXP TANK I GENERATOR g
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} TURBINE COVER GAS l (Ps 0.104 MPa)
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Cy / y (uy I . PRivARY IN TER MEDI AT E TERTI ARY Figure 3. Schematic diagram of Phenix heat transport circuit l l 1 l l 1 1
- 3. STEADY-STATE MODELS As shown in Fig. 3, the primary heat transport system of a pool-type LMFBR plant is one in which consists of all of the essential components are located within.the primary tank. This includes reactor, picmps, check valves, primary side of the intermediate heat exchangers, sodium pools, cover gas blanket, and piping. Excluded is the secondary side of the IHX, which is considered part of the secondary system. The centrifugal pump is a variable speed pump. It provides the driving head to offset the pressure losses opposing coolant flow.
There is no separate pump tank with its own cover gas volume. The absolute pressure is the primary system is determined by the specification of the cover gas blanket pressure. Typically, the cover gas pressure is very close to atmospheric pressure. The intermediate heat transport system is similar to that_ used in the loop-type designs. It includes the secondary side of the IHX, the sodium side of the steam generator, the buffer , surge) tank to accommodate changes in sod-ium volume, a pump with its own tank and cover gas volume, and the piping con-necting all these components. The steam generator constitutes the essential part of the tertiary or water / steam circuit. The modeling for the inter-mediate and tertiary circuits has essentially been carried over from the SSC-L code. In the initial part of the transient calculation, a stable and unique ste;dy-state or pretransient solution for the entire plant must be obtained, start?ng from user-specified design parameters and operating conditions. As a result, the continuity, energy, and momentum conservation equations in time-independent form are reduced to a set of nonlinear algebraic equations. These 1
equations are solved in two steps. First, the global parameters are obtained. More detailed characterization is achieved by using the global conditions obtained in the first step, as boundary conditions. Figure 4 is a schematic illustration of the overall SSC-P steady-state solution procedure. Details of the global thermal balance, primary system thermal-hydraulics, and individual component models, are discussed in the following subsections. 3.1 GLOBAL THERMAL BALANCE Calculations required for global thermal balance are not significantly dif-ferent from those used in SSC-L calculations. The main differences arise from the absence of direct piping connections between the IHXs and the reactor in the hot pool concept. In addition, the logic is slightly altered to allow for more than one IHX per intermediate loop in the pool system. For clarity of presentation, the entire global balance as implemented in SSC-P will be de-scribed here, instead of ,just the differences from SSC-L. A schematic diagram of the heat transport circuit for thermal balance is shown in Fig. 5. The reactor is missing in the figure, the reasons for which will become clear later. Gross energy balance may be represented by the fol-lowing five independent equations: , PX*WlX e(TXin) - e(TXo) (3-1) lM (3-2) PX = UXXA aTX P=WI e(TIo) - e(Tlin) (3-3) , _ _ l P=UgA30 3 AT3G lM (3-4) P=W3 eso - eFW (3-5) l I l
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$$4 P STEADY STATE GLOBAL THERMAL BALANCE i,
'I Wg , eg ,, P STEAM GEN, CONVERGED T,,,,W,, Tg ,,wt f I STEAM
,, A g GEN A 3, , - ss ENERGY B AL ANCE IN POOLS u
2 l } CORE C AL Cul ATIONS 'F W ' F* 7 z 7r .. u Figure 5. Plant schematic for PRlWARY H Y OR A ULICS global balance 1 SECONDARY THERMAL HYDR AULICS RETURN Figure 4. Logic for SSC-P overall steady-state solution where ATLM denotes the log-mean temperature' difference. The. sodium en-thalpies are assumed to be functions of temperature only, while the water-side enthalpies are taken to te functions of both temperature and pressure. In Eqs.(3-2)and(3-4),UX and Usg are the overall heat transfer coef-ficients at the IHX and steam generator, respectively. In actual steady-state computations, these equations are replaced by a series of nodal heat balances, which constitute the detailed thermal models for the components. In solving Eqs. (3-1) through (3-5), it is assume?. that the heat transfer aress AX and ASG are known from the design. Also, the steam generator computational module presently assumes that eFW
- fl(WS ) (3-6) and W3 = f 2(Psg) (3-7)
The relationships (3-6) and (3-7) are user-supplied input. Thus, the nine key plant variables to N determined are P, WlX, TXin, T Xos WI , TI o, Tlini eSo, and Psg. Five ci these car, ae determined using Eqs. (3-1) through (3-5). The remainir.a four must ca specified as input. Since some confusion and uncertainty can arise on classifying an operating condition as known or unknown, the user is allowed some flexibility in the selection of plant variables which are input and those which are to be calculated. However, to keep the number of options within bounds, several constraints are placed on the choices. Since the reactor side and steam generator side contain the more important parameters from a plant design and operational point of view, the intermediate loop is left unspecified (i.e., to I l l n
be calculated). Thus, Tro, Tlin and WI may not be specified as input. Furthermore, since the total power (or power per loop P) is the prime variable, it is always assumed to be specified. The pressure at the module (e.g., evaporator, superheater) endpoints (PSG) is also assumed to be specified. With P and PSG assumed to be specified on input, only two more variables out of the remaining four (i.e., W ,X TXin, TXo. eSo) remain to be specified. This yields the following six combinations: _0ption Parameters specified I WlX. eSo 2 TXo eSo 3 TXin, eso 4 TXin WlX 5 TX o, TXin 6
.TXo. WlX The user can select any of these options. The iterative schemes and calculation procedure for the various options are shown in Figure 6.
Some of the changes in this flowchart as compared to the SSC-L flowchart are
- 1) WlX replaces loop flow Wp as the global variable.
ii) Once a converged intermediate loop power is obtained from the steam generator iteration loop, the power transferred per 'HX is calculated from PX = P/nX. This power is then the convergent point for all i options, for obtaining the IHX primary side parameters. This is re-quired due to the presence or likelihood of more than one IHX per intermediate loop in the pool system. iii) TRin, TRo can no longer 'oe directly determined from TXo, TXine as part of the plant thermal balance because there is no direct piping l t i
k l lGLOgAL THEnts& L B AL A NC E l lsuESS S . T, .. [ ir SOLVE SG T/M SALANCE IS Poe 4 INCREMENT
;
ONVERGED y INCREMENT ? f ra TES NO PTION y8 *
- i.2 04 s 'e Jr p
l P, a P/n, j
. S . .. NO 1 CONVERGE -
C A LC.fs. ,was .O R T,. ? (OPfl0N 4.S.046 2 R E 3 P.). E 0. 8 3 - 6 ) l P, a P/n, l y,, t VES GUESS CALC.Tn;. SOLVE 8HE IS Ps OPTt0N E gg,gg.gg W THERMAL BAL. ONVER SOLVE IME 3, Y,g THERMAL S AL. TO C ALC. P TO CALC. P s TES NO NO (RETURN ) INCREMENT N0 f a se C 15 Pm CONVERGED
? ,r YES S GutSS CALC.w as* SOLVE IMX iS P, OPfl0N y .J Le % THE R M AL SAL. C ONVE
' TO C ALC. P, ?
I ( RETURN ) YES (RETURNJ NO NO INCREMENT _ 7,. l' SOLVE IHz 15 P, GUESS u CALC.W is . CON RM THERMALBAL. Ts. EO. ( 3 - a ) TO C ALC. Ps VES (RETURN) Figure 6. Iterative logic for global thermal balance l l i 14 -
connection between the reactor and IHX. Hence, TX o, TXin replace , TRin, TRo as the global variables. Note that the user can input approximate values for Tlin, and WI , and they will be used as initial guesses. Equations to calculate flow through the core, external bypass and pump, knowing WlX and the external bypass fraction, have also been introduced as follows: WC = nXW lX (3-8) WBP
- fBPWC (3-9) and Wp = (WC+WBP)/DP (3-10)
These equations reflect the need to define individual component flow rates in the primary system due to ihe absence of well-defined loops. WBP is the bypass flow external to the core barrel. The Phenix design employs this con-cept, whereby a small precentage of core flow is allowed to leak below the core support structure and up around the cold pool to cool the main tank wall; more details will ba given later. 3.2 ENERGY BALANCE IN P0OLS The components of the primary bstem required for energy balance are shown in Fig. 7. The main tank is covered by a roof generally welded to it. The coolant exiting the core mixes with the sodium in the hot pool. Cooled sodium from the IHXs and external bypass flow (if any) mix with the sodium in the cold pool. The hot and cold pools are physically separated by a thermal bar-l r
UPPER INTERNAL STRUCTURE (m i) ROOF (m 3) sxxsssss, ssss sd Assxx .ss s s ssxx v
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l , T,.. [ COVER GAS V M;
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! i y t [i J- t - J y l i l
s A l 7 T,, IHX
- \ B ARRIER (m 2)
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wp,,, CORE v - W,,, l ( T x l T ai, 7/////////// MW eP LOWER STRUCTURE (m4) Figure 7. Primary system configuration for tank energy balance 1 rier. However, their energy balances are coupled via heat transfer through this barrier. There is also heat transfer between the pool sodium and other structures and with the cover gas. All these processes have to be included in the energy balance for the sodium pools. The overall solution is determined in (Fig. 8). As shown in Fig. 7, the instrument tree, control rod drive mechanism, control assembly, and other structures immersed in the hot pool are lumped together as a single mass ml. The thermal barrier is denoted mass m2. The roof, together with the portion of tank wall exposed to cover gas, is denoted mass m3. In the cold pool, the core .upport structure and the tank wall in contact with the cold pool sodium are all lumped together as mass m4. , 3.2.1 Hot Pool Balance Energy balance in the hot pool is written as A NC 'Ro - "X 1X*Xin - U hghg(THP - Tg ) - U hm2A hm2(THP - Tm2) (3-11)
-U hm1A hm1(THP - T,1) = 0 '
- Here, eXin = e(TXin) (3-12) eRo = e(TRo) (3-13)
Also, during steady state THP = TXin (3-14) In the above equations, TXin W lX and WC (=nXW lX) are known from global thermal balance. Other energy equations are i i l l
.- - .m -
ENERGY BALANCE IN POOLS ir
- GUE SS T e , ( T,, )
;'
HOT POOL BALANCE 1 ir COLD POOL BALANCE i IS
'T c , NO- ; CONVERGED i
< ? YES I ir T,;, s T,, Figure 8. Driver logic for steady-state energy balance in the sodium pools (hot pool design)
;
4 l
Cover gas U hgA hg(THP - Tg ) + U cgA cg(TCP - Tg ) - U gm3A deck (Tg -Tm3)
+ Ug ,1 A
g,7(T,3 -T)=0g (3-15) Internal structure _ U hmiAhml(T HP .Tml} - UgmlAgm1(Tml - Tg ) = 0 (3-16) Barrier Uhm2A hm2(THP - Tm2) - Ucm2A cm2(Tm2 - TCP) = 0 (3-17) Deck Ug m3 A deck (Tg-T m3) - V am3 Adeck (Tm3 - Tamb) = 0 (3-18) A complete specification of the hot pool requires determination of the seven variables eRo, Tml, Tm2, Tm3, THP, TCP, andg T . If TCP is known, six can be obtained using Eqs. (3-11), (3-14) -(3-18). In the above equations the overall heat transfer coefficients U hg , Ucg , Ug m3, Uh ml , Ugml , and Uam3 are all user-input constants. However, the coefficients for the barrier heat transfer are calculated by the code. Details on this, and area calculations, are provided in Section 3.2.4. The solution procedure to determine the hot pool parameters, begins with guessing TCP. The procedure is shown in the flow chart of Fig. 9. Note that if heat losses through the deck are neglected, Equation (3-18) can be dropped, and the term describing the heat transfer with the deck material can be excluded from the cover gas energy equation, Eq. (3-15). Currently we assume that both the roof and tank are perfectly insulated from the outside environment. This assumption is reasonable for Phenix where the roof is covered by insulation followed by a nitrogen-filled gap between it and the l i l L
HOT POOL BALANCE u T,,a T ,,, GUESST.: = 0.5 ( T. , + Tc , ) if CALCULATE BARRIER U a , Us.3 CO M P UT E T. , , E q ( 3 -17 ) 15 Tot NO CONVERGE 0
?
YES GUESS T, 0.5(T,,+T c,) COVER GAS
+ COMPUTE T. , , E e. ( 3 -18 )
STRUCTURES u COMPUTE T, , E q. ( 3 - I S) o S To NO CONvlRGEO
?
YES MOT POOL COMPUTE e , , E q . ( 3 - 11 ) 1 T,, a T ( e , ) RETURN Figure 9. Logic for steady-state thermal balance in hot pool 4._ shield deck; the main. sodium tank wall is surrounded by an argon gas layer between it'and the leak tank. For long-term transients, however, one should account for the impact of radiative heat losses to the water circuit surrounding the' leak tank in Phenix. 3.2.2 Cold Pool Balance Energy balance in the cold pool includes heat transfer to the barrier and cover gas. It also includes energy additions and subtractions due to IHX and external bypass flows entering and pump flow leaving the pool. Allowances are made in the formulation to include the possibility that some part of the IHX flow can directly stream into the pump suction without any mixing with the cold pool sodium. This is done through a user-specified fraction Sx. Under steady-state conditions, there is no net mass accumulation in the
- cold pool. Therefore, mass balance yields WBP + nXW X " DPWp (3-19)
Energy balance in the cold pool can now be written as (1 - S )nX X W XeXo + WBPeBP + Ucm2 Acm2(Tm2 -TCP) (3-20)
+U cgcg(Tg-TCP)
A - (WBP + (I - S )nX X W X)eCP = 0-Here, e (3-21) xo = e(Txo) and esp = e(TPin + ATp) (3-22) l l l I g , --
TX o is known from global thermal balance. Note that under steady-state - conditions, the cold pool sodium is assumed to be in thermal equilibrium with the lower structural metal m4. The enthalpy of sodium entering the pump suc-tion is obtained from PWPePin " S X nXWXeXo + (W BP + (1-6 X)nX WX)eCP or W (3-23) ePin = npWpe CP + 8XXX D W (eXo -ECD)/DP P Using the initial guess of TCP, Eq. (3-20) is solved to get eCP-Thereafter, Eq. (3-20) is iterated on until a converged value of eCP is-obtained, based on Tm2 and Tg determined from the hot pool thermal bal-ance. The solution procedure is illustrated in Fig. 10. 3.2.3 Barrier Heat Transfer The overall heat transfer coefficients (U hm2, Ucm2) for the barrier are evaluated assuming a multi-plate configuration with stagnant sodium or any other medium in between plates (see Fig.11).
I l i i l i COLD POOL BALANCE ir GUESS e,,ae(TcP+AT,) ir C ALCUL ATEg o ,, E q. ( 3-20) ir COMPUTE e,,, , Eq. ( 3-2 3) i 2 3 4 T
+ vs q ' %
T,;, s T ( e,;, ) HOT POOL (
+
dj 8 x - T O T CP COLD POOL
,g COMPUTE e,, , E q. ( 3-22) s
\
$]
Figure 11. Thermal barrier configura 'on IS e c, NO CONVERGED
?
YES i Tg ,sT(e c,) [ ir T.e : Te, RETURN Figure 10. Solution procedure for thermal balance in cold pool r [ i
_; ___ __ _ ___- -__ _
-The model is developed, assuming the':following:
(i) Sodium (or other medium) between plates is stagnant' i.e., there is
.-only conduction heat transfer.
.(ii)- The plates.have the same thickness and are equidistant from each other.
(iii) The overall heat transfer coefficients for the composite structure can be calculated using the flat plate assumption. (iv) ..There is no axial temperature variatic . in the barrier, i.e., radial , conduction only. (v) -The properties are based on the-average barrier temperature.
. Let Np ,be the number of plates,'Tm2 the average temperature in the
~ barrier, and q the power transferred, in watts, through the barrier.
From.the concept of thermal resistances,-we can write: IN 6 fNp -- 1. 6 Na 1 J lh_ + l _P_)i_kS +1 2 k (3-24) U hm2 g (2 j 3 ( Na / and IN ) 6 fN -1 6
" + ' * (3-25)
U 2 k cm2 o \ / s ( / Na The power transfer q can be expressed in terms of the surface heat transfer coefficients hj, ho as q = hjAhm2(THP - Tj) (3-26) and q=hAo cm2(To - TCP) (3-27)
In terms of the-' composite barrier resistance, excluding the surface (film) l
- coefficients hj,.ho :
q = (Tj - To)/RT (3-28) Lwhere 6
, -Ry=A -N p[6 + (Np - 1) k (3-29) 4 barr _ s Na ,
-'There are five unknowns Uhm2, U cm2, Tj, To , q (provided hj, hocan be determined as functions of the temperatures Tj,o T , respectively). All five con be determined using Eqs. (3-24)-through (3-28). The solution procedure is illustrated in Fig. 12.
3.2.3.1 Areas' The barrier is' allowed to be composed of two or more parts, (see Fig.13) 2 one inclined at an angle e to the horizontal, and the others v,trtical. e can
- vary, depending on the barrier design. For Phenix, e = 19 .
f +
l T I 1 l B ARRIER ME AT TRANSFER .
'GUE S S T; a O.S ( T , + T,, )
ir C ALCULATE h; _a h ( T;) 4. C ALCULATE q , Eq.( 3-26)
- ]-
Aa 2 C ALCUL AT E 7,, E q. ( 3- 2 7)
"L
; 0, /, :
sr CALCUL ATE q Eq.( 3 -28) 0 u +D,4 p r
' r, COMPUTE T; , E q. ( 5-26) -/ w
, s 'y l 1
ARE i' 0 "O Figure 13. Configuration for calculation CONVERGED 7 of barrier heat transfer area - YES ir CALCUL ATE h; , h, 1r CALCULATE U. ,U,,, l ?
.( RE URN I
' Figure 12. Logic to calculate overall barrier heat. transfer coefficients
- e. ,
!- e I-l' l l
, , , , , . - ~.r . , _ . , . - . , , . . . . . . . . . - - ., , . . . .
The surface area of the inclined-redan is Af 'nci = f (D2 *2 - Dy3 1)
=
-01)
~
' 4cose (D2 where Di , D2 are user-input dimensions.
' Avert = wDi zj + xD 24Z2 (3-31)
Then, Ah m2 = Acm2 = iA ncl + Avert
- Abarr (3-32)
An option is' provided in t;ie code which al' lows the user to either (i) input Ai ncl, Avert, Ab arr or (ii) input Di , 0 2, e , 21, and Az2, in which case the code calculates the areas. 3.2.3.2 Film Coefficients Equation (3-26) can be rewritten as q=h vert Avert (THP - Tj) + h incl Aincl (THP -Tj) (3-33) Combining Eqs. (3-28), (3-29), we get hi = (bvert Avert + h ncl i inc1)/A A m2 h (3-34) i-Equation (3-34) also holds for ho . l l ( I i
, ._ - - - - . - - + _
w h vert, h nci i are obtained from established correlations .for Nusselt number.
-hv ert For laminar boundary layer flow over a vertical flat plate immersed in a IU body of 1iquid,'Eckert derived
-0*"
Gr Pr2 Nu, =.0.508 0.952 + Pr,
.For Pr+ 0, this becomes-Nu x . = 0. 514 Ra,0. 2 5 pp o.2s for Ra 1 108 (3-36)
But L h= h,dx (3-37) o and L
=
* ; Gr =gsxfAT (3-38)
Nu x k x v Substituting Eq. (3-38) into (3-27)-and integrating yields E=hNu x (3-39) or E = 0.685 Ra o.25pec.25 (3- N Ra 5 108 II For turbulent flow, Eckert derived: _ _ 2/s Pr7/6 Gr x Gr > 108 "x *
- 2 Ra > 108 (3-41)
_1 + 0.494 Pr / c
;
- For : Pr + 0 Nu x = '0.0295 Pr .467Gr x 0*" '(3-42) r Evaluating average Nu as before gives Nu = 0.833 Nu x (3-43)
O r. I Nu = 0.025 Pr *
- Gr
- Gr'>:10 '(3-44)
Ra > 10
.Here,- for either the hot pool side or the cold pool side, we have Grj = g(pj - pHP)Lvert /Pivi (3-45) and Gro=g(pCP-Po)bvert /Po Vo .(3-46)
Also, Ra = Gr Pr (3-47) and v = p/p (3-48) hincl The same equations derived above, are used for hi ncl, except that Lvert in the formulation for Grashof number is Lv ert " l nci i sin e (3-49) A study was made to evaluate the sensitivity of the barrier overall heat transfer coefficients to temperature, e.g., U m2 h to Tm2 and Tgp; an option has been added to the code that allows the user to specify U m2 h and l
Ucm2, altogether bypassing the. barrier heat transfer calculations. These coefficients would then remain constant during transient computations. Another option being considered is to allow the user to specify a higher sodium thermal conductivity between plates to represent an active barrier, i.e., where sodium between plates is moving causing convective heat transfer. 3.3 PRIMARY HYDRAULICS The steady-state hydraulic calculations in the primary system involve the determination of pressure losses (or gains) in the different components, pump operating head and speed, liquid levels in the primary tank, and the absolute pressures at important locations in the circuit. The flow rates have already been detennined through global thermal balance. Figure 14 shows the hydraulic profile of the primary tank and components. The overall steady-state logic for the primary hydraulic calculations is presented in Fig.15. The logic. includes both hot and cold pool design concepts. Note that for the cold pool design, both thermal and hydraulic balances are perfonned simultaneously bypassing the energy balance in the pools. Since the commercial pool LMFBR will most likely be of the hot pool concept, the first version of SSC-P has been developed to simulate the hot pool concept. 3.3.1 Intermediate Heat Exchanger During steady-state conditions, the volume-averaged momentum equation re-duces to Pressure drop, i.e., Pin - Po = pressure losses. l
I 9 jk.---= ~ Q h , , V
/ ZE -
(jl a l PX IHX n ZHP d
" d /2 P Ro s
d 7 ZCP Ppo" CORE
]
v d l Z Xin p I" P Wp "X IX 0 WC /
'l I PR Wgp Figure 14. Hydraulic profile of prinary tank and components I
l
- . - . .-.
l PRIMARY HYDRAULICS l L 6 P'S IN HOT AND COLO POOLS l 1r J*I br THI IHX IHX C ALCULATION S l e NO PIPE CALCULATIONS HIS COLD r LEG PIPE f NO ir TH: NO POOL I IS THIS YES UPPER PLEN. CALCULATIONS ygg UPPER (THERMAL 8 HYDRAULIC) PLENUM
?
l lJ so J+6 l NO IS YES HOT LEG PIPE CALCULATIONS THIS HOT LEG PIPE (THERMAL & HYDRAULIC)
?
"O, IS THIS COLD YES COLD POOL CALCULATIONS POOL ( THERMAL & HYDRAULIC) f NO 1
iS S YES THISlHX NO THIS YES OR : LAST rl, CHECK VALVE OP l 807 POOL g L EVEL CALCUL AflUNS ADJUST 8,C. *S h l l COMPUTE PUMP HE AD 1 l CALCULATE PRESSURES l 4 l BYPASS CALCULATIONS l 4 ( RETURN ) i Figure 15. Steady-state logic for primary hydraulic calculations l l. l l
For,the IHX, the' sum of losses in either the primary or secondary side, fis expressed as (APf ,g)X = (acceleration loss) + (frictional loss) + (gravity loss (gain)) + (inlet loss) + (exit loss) + contraction, expansion losses + other losses.
' Note that a negative value obtained for gravity loss indicates a gain. Other losses are expressed as K N AP 1X 1X IX other " pA (3-50)
IX for the primary side, and similarly for the secondary side. In most cases, the IHX pressure drop corresponding to full flow conditions is known a priori, through flow model testing performed by the manufacturer. However, this pressure drop may correspond only to losses with the unit placed in a horizontal position, for which effect t of gravity are not included. For this reason, in the option where K1 x is either specified or calculated by the code, the following equation is used: AP = AP + AP (3-51) 77 acc + ^ f + ^ in + AP c,e other Earlier, in the SSC-L formulation, the right hand side of Equation (3-51) included gravity. The user should exercise caution when specifying the value of APi x if known, to ensure that it does not already include the gravity term. The code adds on the gravity term to give the final value of APi x for the hydraulic calculations. Further details on the fonnulation have been reported earlier, and will not be presented here. I i j I
c3.3.2' Cold leg Piping The pressure loss calculation during steady-state. consists of combining the losses due' to friction ~'and gravity, and losses' due.to bends, fittings, etc.
~ ~
The coefficient representing the latter loss is always user-specified. Note s that-the length and elevation changes assigned to the. piping include the elevation changes, .if any, occurring across the pump (azp) and check valve
' (AzCV), while those components are treated as point volumes.
3.3.3 Liquid Levels in Pools From Fig.14 we can write, from static balance: PXin = Pg + pHg(ZHP - ZXin) (3-52) PXo = Pg + pC9(ZCP - ZXo) (3-53) Subtracting Eq. (3-53) from Eq. (3-52) and rearranging yields the sodium level in the cold pool as
- hP Z * +P C Xo P CP P
H( HP - Xin) - g C The level in the hot pool, ZHP, is assumed known, along with IHX elevations ZXin and ZX o. Note that here AP1X is the value provided by the IHX hydraulic calculations and includes the gain in pressure due to gravity as the primary coolant moves downward in the unit. }- .The volume of cover gas is obtained simply as i ; Vg = (Z tank - ZHP)A hg + (Z tank - ZCP)Acg. (3-55) l i 1 l
where Ahg, Acg are the areas of hot pool and cold pool in contact with the cover gas, respectively. For a. cold-pool concept, only one liquid level exists, and the volume of cover gas would simply be equal to (Z ankt - ZCP)Acg* The mass of cover gas blanket is obtained, assuming the perftet cas law, t
'as m g = P gg V /R Tgg (3-56) 3.3.4 External Bypass The code models external bypass flow, a small fraction of the total pump flow that may be allowed to leak down from the core inlet plenum region and up atween the outer baffle and the tank wall (see Fig.14).
The calculations are set up to allow bypass flow to be evaluated dynamic-ally during a transient. With steady-state core flow and external oypass fraction known, the loss coefficient is calcu3ated to account for all losses occurring in the bypass region. The formulation is described below. Momentum balance reduces to: K BP BP (3-57) AP BP
*PC 9I BPo ~ BPin) + p C
Since the bypass flow originates at the reactor inlet, ZBPin is zero. Re-arranging Equation (3-57) yields 2 KBP = (APBP - PC 9 2BPo; C/WBP (3-58) Here, APgo = PBPin - PBPo (3-59) and PBPin = PRin' (3-60) PBPo = Pg + pCg (ZCP - 2BPol (3-61) The loss coefficient KBP in Equation-(3-58) is held constant.to enable . computation of WBP during the transient. 3.3.5' Pressure Drops in the Pools
.The. pressure drop experienced by the coolant in the hot pool from the core outlet to the IHX inlet is simply APHP " PHg(ZXin - ZR o) (3-62)
.In the cold pool, the pressure drop between tN IHX outlet and pump inlet is formulated as 2
APCP " PC9(ZPin - ZXo) + KWP /PCA (3-63) The additional term in Eq. (3-62) accounts for any losses occurring as the fluid is forced to turn. In Phenix, for example, there is an annulus sur-rounding the pump inlet and extending downwards in the pool. Its apparent purpose is to minimize the impact of cold pool stratification (during IHX undercooling events) on the inlet temperature to the core. In PFR, the coolant emerging from the IHX is distributed around the heat exchanger shell and sweeps the tank surface before flowing to the pump suction. In this way, the risk of hot-sodium stratification is reduced. 3.3.6 Pumps The sodium pumps used in pool-type LMFBRs as in the loop type are also vertically mounted, variable speed, centrifugal units. In the pump model developed for SSC-L,2the 1 impeller behavior is characterized by homologous-head an'd torque relations encompassing all regions of operation. The homologous
- l. characteristics were derived from independent model test results with a centrifugal pump of specific speed (Ns ) equal to 35.(SI units), and are applicable to LMFBR pumps in general.
! The model was shown to give very good agreement with measured data for FFTF pumps (Ns = 27.2), and with vendor calculations for the CRBR pump l (Ns = 42.8). It is not anticipated therefore, that the characteristic coefficients built into the code will need to be changed for future applications, unless a pump is encountered with sN drastically higher or lower than 35. The primary and secondary pumps in Phenix have a specific j speed of approximately 36 (SI units). Details on the procedure for obtaining pump head and speed can be found in Ref. 13. l l l l l
- 4. TRANSIENT MODELS 4.1 PRIMARY COOLANT DYNAMICS The dynamic response of the primary coolant in a pool-type LMFBR, particularly the hot pool concept, can be quite different from response in the loop-type LMFBR. This difference arises primarily from the lack of direct piping connections between components in the hot and cold pools (see Fig.14).
Even though there are free surfaces present in the reactor vessel and pump tank of loop-type designs, the direct piping connections permit the use of basically a single flow equation to characterize the coolant dynamics in the primary loop, except in a transient initiated by a pipe' rupture or similar asymmetric initiator. In the pool-type designs under discussion where both hot and cold pools are free surfaces there is direct mixing of the coolant with these open pools prior to entering the next component, two different fiows would have to be modeled to characterize the coolant dynamics of the-primary system. During steady-state the two flow rates are related by a simple algebraic equation. During a transient, however, the flow in the up-leg from the pump would respond to the pump head and losses in that circuit including losses in the core; the IHX Ilow we id respond to the level difference between the two pools, as well as losses and gravity gains in the unit. The gravity gain could be significant for low-flow conditions, l particularly if the IHX gets overcooled due to a mismatch of primary and I secondary flows. j i In addition to the above considerations, the number of heat exchangers, i nX, is a design choice and can be expected to be different from.the number of pumps, np. The number nx is determined by pressure drop and thermal
! rating per unit requirements. Also, the number of primary pumps is
-independent of the number of secondary loops in both hot pool and cold pool concepts; there is no pipe connection in the cold pool. The Phenix plant has three pumps and six IHXs operating in parallel. For symmetric transients, such as a loss-of-electric power (L0EP) event, all parallel components can be
[ expected to behave identically, and only one flow equation needs to be modeled
. for each set of parallel components. However, in cases of asymmetric events, I
it is necessary to distinguish between the components that are directly affected by the postulated accident from those that are not. Examples of such events are a pipe rupture in a pump discharge line to the reactor, a single pump malfunction or a malfunction in an intermediate circuit causing the affected IHXs to behave differently from the others. 4.1.1 Flow Equations We introduce the concept of flow paths (Npath), which defines the number of pump or IHX flow equations to be solved. In each flow path, we can lump a number of IHXs (Fx(k), k = 1, Npath) and a number of pumps (F p (k), k = 1, Npath) that are expected to have the same reaction to the transient. 4.1.1.1 Intact system For an intact system, volume-averaged equations can be written for k = 1 to Npath, as follows: i Pump flow l i dWdt (k) E (k) = P pg (k) - PRin - AP f,g(k) (4-1) P P l l 1 l )
1 IHX flow q (k) = P Xin A
- Xo - f,9 Note that for symmetric transients, there is only one flow path, hence only one flow equation for each component. In the cam of a. single pump malfunction, two paths would have to be modeled. The first path would have the damaged pump and an IHX, and the second path would include the remaining pumps and IHXs.
In the above equations, Ppo, the pomp exit pressure, is obtained from Ppo = PPin + Pin gH (4-3) where H is the pump head, obtained from the pump characteristics . The IHX terminal pressures are obtained from static balance as PXin = Pg+ pHg (ZHP - 2Xin) (4-4) Po=Pg X + pC 9(ZCP - ZXo) (4-5) The reactor inlet pressure, PRin, is obtained from a complicated algebraic equation (see Section 4.1.4 for derivation). 4.1.1.2 Damaged System In case of pipe rupture (for the hot pool concept, this can only happen in the pump discharge line to the reactor), Eq. (4-1) gets modified, for the broken path, to
=P pg -P bin - A f,9 d uob uob An additional. equation is needed to describe the flow downstream of the break:
P ,p _p l 9 Rin " f,g dob doo Some discussion of the formulation for a generic cold pool design is presented f I in a later section. The inlet and outlet pressures at the break location, Pbin and P b o, respectively, are calculated by the break model. Details of the model can be found in an earlier report ". The pressure external to the break, which is needed to compui.a these pressures, is obtained from a static balance as Pext = Pg + pcg(ZCP - Zb ) (4-8) This is also the back pressure opposing the flow out of the break. Eq. (4-8) yields a much larger value for Pext when compared to loop-type designs (where it is generally equal to atmospheric pressure, unless the break occurs within a guard vessel, in which case it would remain atmospheric until the vessel fills to the break elevation). This is a contributing factor to the less serious nature of pipe break accidents in pool-type LMFBR designs. Another contributing f actor is the generally low hydraulic resistances through the core and IHX, allowing higher core flow from the intact flow paths, relative to the loop design. 4.1.2 Liqu ; Levels in Pools During steady-state, the level difference between hot and cold pools sup-ports the net losses occurring in the intermediate heat exchanger, thereby J maintaining flow through it. A high level difference would be necessary to drive flow through a high pressure drop unit. Hence, this dictates the 7 r i
requirement of low pressure drop units for these designs. During transient conditions, any reduction in IHX losses alone will tend to increase flow through it. However, this will also reduce the level difference driving the , flow. This competition between two opposing forces determines the dynamic state of levels and flow through the IHX. During flow coastdown transients, ( the levels approach each other, implying a net increase of cold pool mass at d the expense of the hot pool. When the levels restabilize under low flow ; conditions, the level difference once aosf9 maintains the IHX losses. While it is not generally expected that the liquid levels will cross each other, it has been seen analytically that if the IHX is overcooled due to very high intermediate flow, the resulting gravity nead in the unit, available at much higher densities, is sufficient to overcome all frictional and other losses. The levels then eventually cross each other while the flow is maintained positive. The total flow through all the IHXs and all the pumps can be detennined from the summations
" path FX (k),Wy (k) (4-9)
W Xtot
=
{ k=1 and path (4-10) Ptot ={ k=1 Fp (k) Wp (k) Mass balance at the cold pool gives A d~ cg dt (PC CP} " WXtot - NPtot=+ NBP + Nb Note that W b , the break flow, is zero for an intact system. Eq. (4-11) as-sumes that all the level changes likely to occur during a transient are confined to a constant cross-sectional area. l
(_ Mass balance in the hot pool gives d A ! hg at IPH HP} " CN - Xtot (4-12)
- 'When equations (4-11) and (4-12) are solved simultaneously with the flow
, . equations, the results yield the liquid levels ZCP ZHP during the
- transient.
4.1.3' External Bypass The external bypass flow is dynamically described by the flaw equation l dW ' 8P BN
- AP -K BD 'p (4-13) j dt XL=PRin - PBPo g l
l !. Currently, the fannulation has two restrictions: (i) the flow is not allowed to reverse (ii) no thermal interactions are represented The former restriction can be removed, while the latter may not be significant to the s'ystem behavior.
As before, i l PBPo = Pg + pCg(ZCP - ZBPo) (4-14) and. AP g = pRin92BPo (4-15) KBP is the loss coefficient determined during plant initialization l calculations. PRin is obtained from Eq. (4-23). l' 4.1.4 Reactor Inlet Pressure Due to the tight hydraulic coupling of the reactor inlet with the rest of l
;
l l
the primary system, the solution of the system flow equations, as well as coolant dynamics in the core, requires the reactor inlet pressure, PRin, to. be known at all times. In _the following sections, the equation used to calculate PRin is de-rived for both an intact and a damaged system. f
. 4.1.4.1 Intact System <
Mass conservation at the reactor inlet yields Fp(k)Wp(k) - WBP WC" k 4 Differentiating both sides gives-d
=
p Fp (k)' d (k) - (4-17) The core flow can be expressed in terms of channel flows as N ! ch W C" W. (4-18) j=1 J where Nch represents the number of channels simulated in the core. Dif-i ferentiating both sides, we get 1 (4-19) J Furthermore, for each channel j we can write, from momentum balance i
=P Rin -P Ro - APf ,g (4-20) j 3 dt where j Po=Pg R + pH g(ZHP - ZR o). (4-21) '(
i^ Combining Eqs. (4-17), (4-19) and (4-20) gives-d BP,{PRin - pro
-( APf,0) j pg
[ kF (k) p (k) - j g j LSubstituting Eqs. (4-1) and (4-13) into the left hand side of Eq. (4-22) and simplifying yields the reactor inlet pressure as P. = (A + B + C)/(D + E + F) (4-23) where pro + i A= b [ Lj j (4-24) j
\ A/J ;
PPo(k).- f.9( } B= [ kF (k) p p l [ (k) P (4-25) .; _ [PBPo + AP +K BPN BP / PC C= L (4-26) 1 BP A / ,
~
0= (4-27) [ (~D)3 J ! E= { k P(k) (4-28). AP7,(k) l F= ~ (4-29) , BP
l , l l
-4.1.4.2 Damaged System In case of a pipe rupture in one of the pump discharge lines, mass con- l servation at the reactor inlet has-to account for one of the flows which is 1
the break to the reactor. Thus,
" path
- 0)
W C
=
{ k=Pr Fp (k)Wp (k) + Wdob ~ BP This alters the formulation of Eq. (4-23) so that path I Po (k) - P aP,g(k) P I'9 B= [ k=2 Fp (k) P 1, , bo dob (4,333' _ _ _ k(k) k Npath-I Fp (k) E.= {Pf,g(k) k P dob The rest of Eq. (4-23) remains the same. 4.2 INTERMEDIATE SYSTEH The thermo-hydraulic modeling of the intermediate circuit is essentially unchanged from that in SSC-L, as reported earlier9. The presence or likelihood of more than one intermediate heat exchanger operating in parallel for each intermediate loop has necessitated the inclu-sion of branching. Currently, the fonnulation does not allow for dissimilar branches. The flow through each IHX is then Wg n; N
- 2X ny (4-33) ,
Similar branching was also built in earlier to allow for a a. coecified S _ _ _.
l i number of superheaters and evaporators. Another addition is the fonnulation of a model to represent spherical pump tanks. This has'been necessitated by the presence of such a pump tank design
~
in the Phenix secondary circuit. 4.2.1 Spherical Pump Tank Model h- . Fig.16 shows' the Phenix secondary pump including the tank. The inlet pressure to the pump, and hence, the base pressure in the intermediate
' circuit, depends on the level in the tank. This requires the level to be known-at all times.
We can derive an expression relating the volume of sodium in the tank, Vg, and its operating level, zg, as follows (see Fig.17): dV = w( hsin )2 dz. (4-34) The incremental height dz can be written in terms of as dz=d(f-fcos ). (4-35) Combining Eqs. (4-34), (4-35) and integrating yields y =
"f sin 3ede. (4-36) o 2
=
V g 24 cos3e g - 3coseg+2 (4-37) l i Equation (4-37) relates Vg to the angle eg , whereas the operating level is related to eg by D zg = y (1 - cos g) (4-38) e l PONY MOTOR _--m_ - ll - ll MOTOR gh' TI di
**..e.
' 5" '
PUMP (EXPANSION)
]{ TANK
. -. =, :.w .. -.
'.**:: *.~.;'
s M u i3 *
\ -
IMPELLOR s N. p \ S0oluu OUTLET K $00 lum INLET Figure 16. Phenix expansion tank and secondary pump COVER LAS Os l
}cos6 6 D/2
,L ,
y s.n 9 A d 1I I / / f f fIlli dr Figure 17. Model configt. ration for spherical expansion tank i l l l l
-If zg, the operating level is known during nominal, steady-state conditions, the angle og can be evaluated from Eq. (4-38), and subsequently the volume of sodium obtained from Eq. (4-37).
During a transient, the level tracking is achieved as described below. 4- The sodium volume in the tank is governed by the mass balance equation h(pV)=.Wn'N g g o (4-39) The ' integrated parameter here is pV, as opposed togpz for the verti-cal cylindrical tanks. To:obtain zg, we need first to solve Eq.- (4-37) for e.g Eq. (4-37) can be rearranged in the form cos3 eg - 3cose g + (2 24 y),0 g (4-40) wD This is a cubic equation ine g, and has to be solved for its real roots. i 4.2.1.1 Solution of Cubic Equation Eq. (4.*0) is in the normal form x3 + ax + b = 0 where x = cos o g a= -3 4 b=2- V wD3 A To apply the test for roots, we evaluate the expression 12V g b2 a3 4 +H*- nD3 I Since Vg > 0, this means that b2 a3 y+p<0 Eq.(4-40)thereforehasthreerealandunequalrootsgivenb/5 cose,k=2cos(h+120*k) 1 k = 0, 1, 2 (4-41) where 4is to be computed from cos = 2(1 - *) nD3 and where the upper sign is to be used if b is positive and the lower sign if b is negative. Withe g obtained as described above, Eq. (4-38) can be used to determine the transient level. The inlet pressure to the pump impeller is related to zg by u PPin Pg + pin 921 4.3 ENERGY BALANCE IN P0OLS 4.3.1 Hot Pool Stratification Stable stratification occurs when hot, hence lighter fluid, forms a layer on top of cold fluid. This can occur in the hot pool region if the entering coolant is colder than the pool coolant and it experiences a large decrease in momentum. Since the hot pon1 forms a link in the primary flow circuit, it is necessary to predict the pool coolant temperature distribution with sufficient accuracy to determine its contribution to the net buoyancy head. It is also needed for the computation of the inlet temperature conditions for the components in the circuit. During a normal reactor scram, the heat generation is reduced almost s instantaneously while the coolant flow rate follows the pump coastdown. l l l
i
This mismatch between power and flow results in a situation where the core flow entering the hot pool is at a lower temperature than the temperature of the bulk pool sodium. This temperature difference leads to stratification when the decaying coolant momentum is insufficient to overcome the negative buoyancy force. Instead of penetrating upwards, the cool, dense sodium is then deflected downward and outward in a stratified pattern. Currently, the stratification of core flow in the hot pool is represented 'by a two-zone model, based on the model for mixing in the upper plenum of loop-type LMFBRs in SSC-L l6 Accordingly, the hot pool is divided into two perfectly mixed zones, determined by the maximum penetration distance of the core flow. This distance, zj, is related to the initial Froude number, Fro, of the average core exit flow. The temperature of each zone is com-puted from energy balance considerations. The temperature of the upper por-tion, T A, will be relatively unchanged; in the lower region, however, TB will be changed and be somewhere between RT o and TA due to active mixing with the core exit flow as well as heat transfer with the upper zone. TA is mainly affected by interfacial heat transfer. The location of the interface will determine whether the upper region has any influence on the IHX inlet temperature. Full penetration is assumed for flow with positive buoyancy. Further work on this aspect of the modeling is planned, and it is expected that the current representation of mixing will be improved with a two-dimen-sional, more rigorous treatment. The 1-D approach could then be retained as an option. Compared to the upper plenum representation, the hot pool analysis is more complicated because the pool cross-sectional area is not uniform. The volumes
of the lower and upper zones, as well as their areas for heat transfer with the thermal barrier, have to be evaluated during the transient. 4.3.1.1 Heat Transfer Areas The heat transfer area between the lower mixing zone 8 and the barrier (see Fig. 18)'is derived as follows: zj < zi Abm2 = wDi zj if or
" -2 2 AM2 = 3D zy y + 4c se (Dt -D) if z y < z) < z_2 (4~44) where D=
2(zy-z) y
+0 1 (4-45) tane 3 2 2 AM2 = nD z1 y + 4cose (D 2-D)+302 if zj > zg (4-46) 1 (z) - z2) where z
(D2-D) 1 tane (4-47) 2 * *1 + 2 Here, zi, Di , D2 and the angle of the redan, e , are user-input quantities. For a horizontal redan (0 = 0*), z2 " 21* For zone A, the area is evaluated from : Aam2 = Ah m2 - A bm2 (4-48) i
.For the case of full penetration, A m2 b = A m2 h and Aam2 = 0.
\
4 f 4 DIAMETER D 2 a s l ZONE A l
/4 D
- - - - - - _ 4 ZONE B l 2
/4 Z l
s Dy 21 m j TOPOF CORE Figure 18. Configuration for calculation for heat transfer areas from the two zones to the barrier I
i-i 4.3.1.2 Volumes of Zones l
\
- - -The volume of sodium in zone B is given by 2
" {4'49)
VB" *j 1# *j * *1 or 2 3 3 VB"
*1*b.tane-(D -D)i if Il ' *j * *2 (4-50) or 2 2 z 1+htane(D2-Of)+wD,(*j~*2) ># (4-51)
VB" 4 if *j 2 The volume of zone A is then obtained from VA=VHP - VB (4-52) For the case of full penetration, i.e. zj = ZHP - ZRoe VB*VHP and VA = 0. ; 4.3.1.3 Enerqy Equations i The governing equations which determine the various temperatures in the hot pool are given below: Lower mixing zone B: de PgVB d "Nc ('Ro ~ 'B) - (UA)bm2 (TB-Tm2)
- (UA)hm1 (1 - f)(TB - T,1) - (hA)ba (TB-T) A Note that this equation includes mass conservation.
Upper mixing zone A: de o VA d "0 1 Nc *B ~ 'A) + (hA)ba (TB - T A) - (UA)am2 (TA-Tm2)
- (UA)gy f(TA-Tml) - (UA)hg (TA-T) g (4-54)
Upper internal structure.(metal ml):
'dT -
.(MC)ml . =(UA)hm1 f TA + (1-f) Tg-TM -(UA) 1 (T,1 - Tg ) (4-55)
~
l . i Barrier (metal m2): dT- "A T +A T (MC)m2 d = (UA)hm2 A m2 - (UA)cm2 (Tm2 - TCP)(4-56) _ 2 Based on sensitivity analysis, Uh m2 is not very sensitive to changes in sodi-um temperature, and so this equation is derived assuming U m2 b = Uam2 - Roof (metal m3): dT (MC)m3 d = (UA) gm3(T g -Tm3) (4-57) In this equation, the heat transfer from the roof to the outside ambient has ! been neglected. I 4 Cover gas: dT (MC) gas =(UA)hg(TA - Tg) - (UA)cg (T g -TCP}
+ (UA)g,g (T,1 - Tg) - (UA)gm3 ( g
~
( m3) The auxiliary equations required by the above governing equations are 2 Ab a = wD /4 (4-59) f = 1 - zj(t)/(ZHP(t) - zRo) (4-60) Liquid sodium densities pA' PB, etc. are obtained from the constitutive relationship p = p(T), where the sodium temperature, T, is related to its enthalpy by T = T(e). L
-..w-- .- - - . -
7-
The heat transfer areas Ahml, A gml, Acg, Ahg and Ag m3 are user-input quantities. Abm2, Aam2 are calculated as described earlier. The control index si represents the location of the jet penetration relative to the heat exchanger inlet. Thus, 81 = 0 j i l
> for zj > (ZXin -ZRo) l I
andeXin"eB/
-(4-61) 81 " 1 '
l
\' for zj < (ZXin -ZRo)
I and eXin " eA - zj is defined as l7 zj = (1.0484 Fr8 785)ro+zh c where Fro is the discharge Froude number, defined as 2 ' I C P B pp = (4-63) o p (wr g2 p9 ) gp9(pp9 . pB)_ For full penetration (f r, 0.0), the equation for zone B is replaced by:
- de P
B d *Nc ('Ro - *B) - IUA)bm2 (TB-Im2)
- (UA)gy (TB - T,1) - (UA)hg (T -T)
NN B g and de A , de B (4-65) dt dt
For no penetration (f = 1.0), the equation for zone A is replaced by
.de o A "c I'Ro 'A) -
d A)am2 (TA-Tm2)
-(UA)gg(TA ~
ml) ~ ( A)h'g (TA
~
g
.and deg. de g dt " dt (4-67)
'4.2.2 Cold Pool Currently, we assume perfect mixing of the IHX flow with the cold pool sodium, but allow for a user-specified fraction BX of the: flow to go directly to the adjacent pumps. Based on this, the energy equation is derived j as t
de ( ] '(pV)CP dt P = (1 - SX) NXtot(*Xo ~ 'CP}
- BP ("BP ~ 'CP
+(UA)cm2(Tm2 - TCP} * ( A)cg (Tgas -T CP) - (UA)cm4 (TCP - T,4)
'Also, due to the possibility of shortcircuiting, the enthalpy of fluid en-tering the pumps is not necessarily equal to the enthalpy of the cold pool sodium. Rather, it is given by the expression:
(4-69) , Pin - Ptot 'CP WPtoti* OX Xtot ('Xo - 'CP For 8X = 0, this reduces to ePin
- eCP (4-70) t Lower structures (metal m4):
dT (4-71)
.(MC)m4 dt (UA)cm4 (TCP ~ m4)
, , - ,- er-. - - - . - - , --e. , c ..-m4 w-m -. .- , , - ,
l l The mass heat capacities of all structures (m1 to m4), as well as the heat j capacity of cover gas, mass of sodium in hot and cold pools, areas for heat ] transfer, are needed for transient calculations. j l The total volume of sodium in the hot pool is calculated during steady j state from: 2
-Z go - z2) WW v gp - z 1+htane(D2-o!)+$(z gp and during transients from y new ,y old , 4 g (7 pnew -Z HP The volume of cold pool sodium during steady-state is obtained from m
tot ( )HP CP " p (4-74) C During transients, it is obtained from V
- CP
+A cg (zCP - CP }
CP In the energy equations presented above, axial heat conduction through the
; walls is neglected. For example, in the Tm3 equation, the effect of including axial conduction in the tank wall on Tm3 is less than 1K. Its effect on system temperatures was even less.
4.4 INTERMEDIATE HEAT EXCHANGER The intermediate heat exchanger in pool-type LMFBRs is identical in func-tion, and very similar in design, to that in loop-type designs. The only dif-ference arises from the different configuration in the primary system, where
the IHX draws coolant from an open pool (in the hot pool concept) and dis-charges to another open pool- The liquid levels in the hot and cold pools re-flect the hydraulic flow resist ance through the IHXs. The liquid-level , difference in Phenix, for example, is 66 cm. under normal operation. In PFR, L
- the difference is somewhat higher. In both cases, however, the differential level-is low. This of course, requires the IHX to have a low pressure drop on the primary side. The main concern with a low-pressure-drop heat exchanger is its effect on flow distribution. Poor flow distribution can adversely affect operational reliability by causing temperature distributions and resultant thermal stresses that could exceed design allowances.
In most IHX designs, the primary coolant flows in the shell side, while the secondary coolant flows in the tubes. However, for pool-type designs, particularly the hot pool concept, where the pressure losses in the primary side are limited as discussed above, it may be aavantageous to. send primary flow through the tubes to ensure good flow distribution. PFR is an example of this design choice. The problem of flow distribution is then transferred to the secondary side of the unit. Since high pressure drops can be more readily accommodated by the secondary system, flow baffles such as the disc-and-donut type could be considered to improve the flow distribution. The IHX model, therefore, has to be extended to allow primary flow in the tubes. 4.4.1 Pressure losses Figure 19 shows the IHX flow profile. Primary coolant flows downward in the active heat transfer region and exits at the bottom of the unit. The secondary coolant flows down the central downcomer into the bottom header
1 1 I i _ ,- : ;- i r SECONDARY _- ~ .~ fN
~ ~
.-:B ?..=S.
- m-j yEZ21- c c .
SECONDARY l OUT ' H l_ u i h I,._P
- SHIELD DECK
_> ~.
< '.-,-....A f .
> '. . ,j 6 '
. , .j
- s ':
- ;
G (:.e;;.,. Syf *'f ::y'i
.i * .'...~'.. ' ' .~:
e ~
~.
e: , , e';r
. .: , ; o n . . >
2 _ E I L '?.7,- e,..~
^
' I% ' ,:; ..
gD:0_.Is C l e-
'. ' 1 -'f r*:0 * ];
j.'e'.0h... :_ = fk f A 'k. 9 $,8 ' ' %I , u
?.' : Q 72-Q-,l ~
h ? k ~ ^= ~ '", ROOF
'"l:--L 'l % '
"{ ; y ] .[. j /
i_: = _ . - - f 1 _=.__r.. SODIUM LEVEL l II I T , III k- ' s
'f PRIMARY IN w
DOW NCOME R REGION
'I l
ll l l ! If l jR hh PRIM AR Y OUT Figure 19. Profile of intermediate heat exchanger i I I l i
(inlet plemum region) where it turns upward to present a'counterflow arrangement in the heat transfer region. In all cases, the primary flow is downward, and this helps .to simplify the formulation. l t The pressure losses can be expressed as described in Section 3.3.1. For i the primary side, this gives L 2 W1x l l 1 g1x g1x f a
- f,g IX A2p PN P Y DA2 i 9 1
pp 9 W1x W1x ~
+ inlet loss + exit loss + Klx -pA 2 p
And, for the secondary side, L W2X W2X 1 1 1 SX f I^ f,g 2X* As ' P 'P N g 1 DA2 ss o o 2X
+ inlet loss + exit loss + K 2X (4-77) 2X pA 2 s
In Eqs. (4-76), (4-77), the primary flow area Ap, hydraulic diameter Dp, secondary flow area A s , and hydraulic Diameter sD are defined for shell-side primary flow as follows:
=n t wD ~ 2E [P ,1 A
p T_w (D2j _ 4A p
=
D "twD2 (4-78) 2
=U A
3 th D =D 3 3 l l l I _ . _ . m a ,.__ . - _ _ , .
where D t , D2 are the inner and outer diameter of the tubes respectively. To extend the model for primary flow through the tubes, these parameters need to be re-specified as follows: 3D,2 A P
= "t 4 D =D 1 I4-79) p 2
w D, A = "t 4 2E . [ P__\ _1-s _w (D2j . 4 D =D 1 s The pitch-to-diameter ratio, , is taken to be the average value in case the pitch is not uniform throughout the tube bundle. Contraction and ex-pansion losses occurring at the entrance and exit regions of the tube bundle can be lumped into the ' uncertainty absorber' loss coefficient. 4.4.2 Heat Transfer The model is essentially unchanged from SSC-L. The energy equations are written using nodal heat balance with the ' donor cell' differencing approach. The slight modifications are extensions to allow tube-side primary flow. The. equations in the active heat transfer region are: Primary coolant
-T t pV p he p
=W h "Pj ~ "P
-U ptA pt T pj. i (4-80)
-kU pshApsh T -T sh ' '
j5 g
Secondary coolant
+U stAst -T s pV s 's g *Wd 's g ~ 's j T t j j I - (1 - k)U sshAssh T ~
sh g
' "I'"
s,91 Tube wall N
- A T -T t -Ust^st T ~
s tt j tg pt t pj t j iH1 g g_g i = 1, N - 1 Shell wall C sh sh, T sh g
= U A psh sh T -T + (1 - k) U sshAssh s
-T sh q { sh, yg1 9 (4-83)
In the above equations, k = 1 for shell-side primary flow, and o for tube-side primary flow. V,V p are s the control volumes between i and i + 1 on prim-ary and secondary sides, respectively. Upt U st, U psh, U ssh denote the overall heat transfer coefficients and Apt, Ast, Apsh and Assh are the heat transfer areas per length Ax. The heat-transfer areas are defined as A pt = kwD2 ntax = (1-k)wDi ntax (4-84) Ast = kwDi ntax = (1-k)nD2ntax (4-85) h Assh, Ap sh " Ax (4-86) i l
where nt = number of active heat transfer tubes, Ash is the shell heat transfer area, and L is the length of the active _ heat transfer regien. The overall heat transfer coefficients are defined, based on the re-sistance concept, by: U "h film,p +rwall 'p
- h ~
}
pt- foul ,p
* +r -
V st h film,s wall's + h foul ,s I I (4-89) U psn
=h fj) ,,,
1 1
, (4-90)
U - h ssh Olm s o where the film coefficients are calculated in terms of Nu by Nu k
=
pt p h fj) ,, 0 (4-91) P and Nu k st s h dl s" D s (4-92) Dp , Ds have been defined in the previous subsection. The flusselt numbers, Nup t and Nus t, are obtained from established correlations. The wall resistance terms are obtained by dividing the tube wall thickness equally between nrimary and secondary sides, since Tt is defined at the mid-point of wall thickness. If 2D 2
+
r * (4-93) A k t i'
and in [Di+D) 2 p ,D1 \ 2D i / (4 94) B 2 k t Then, r wal'1,p = krA + (1-k)r8 (4-95) and r wall ,s = krB + (1-'k)rA (4-96) where k = 1 for shell-side primary flow. Equations (4-80) to (4-83), along with the plena equations, are integrated by a fully implicit simple-layer scheme. The heat flux terms in Eqs. (4-80), (4-81) are allowed to be determined explicity, thereby uncoupling them. These f equations are then solved in a marching fashion without resorting to matrix inversions. The energy equations for the piping are written and solved in the same manner. For details of the piping model, the interested reader is referred to Ref. 18. 4.5 SOLUTION PROCEDURE During a transient the system hydraulic equations and equations for energy balance within the primary tank are solved together by a fifth-order predictor-corrector scheme. The solution procedure is shown in Figs. 20-22 by means of simple flow charts. Only the main calculations which are involved during each timestep are indicated. Details of the algorithm are not shown as they can be found elsewhere 19 The thermal equations for the IHX and pipings (both primary and intermediatr 3 . ems), are solved in a marching fashion in the direction of
( ENTER FLOWIT PRIMARY H' R AULIC S l l E NERGY BAL NCE IN POOLS l sti DEnivATIVEFOR INTEGRATOR ( RE RN Figure 20. Driver for transient thermal-hydraulic calculations in primary system r
Pe want **D#avia Cl f I Coupo? . , , , , . . . . , el vf 5 I futet a gegas *C.54.44 305
?
40
.g . E s.14 - a4 6 L
[ Cowruf f Arg,, t AP,.,1,. . t or,,,I,,,, 1 CC upwit u , e.,,, f. , A C0wPuff P ..P,... P, . P,, il f=E *E a 'ES gefaa , . g,g,.g3 7
% h -
.0 C o m puT E P,.. . P,,, .,
I s t OP,,,1,. f e.14 - s $ D I is t
"g. g a .* I P,.. . t e. t e -2 51 r,/ L 0
s P,,,. Es.I4 ISI L 5 Cow *uit P,
-s
, f e. ( 4 I SI i
7 ,f4.I4-21 il
*ES TnE#f a I 4]
B#fa5 g et . E e. 4 4 el 1
,Ee.t4-65
- 7. Ee. t e Fi L
G e< P v.,, e:P 2,,i C0mpWf t
- 1
( aftv== ) Figure 21. Calculational logic for primary hydraulics ( ENERGY 8AL ANCE IN POOLS ) i l s$ THis go .l HOT POOL DESIGN RETURN ) VES. a 3 . Eg. ( 4-62) y,,"'" Ee. ( 4 - 73) r IS YES CNLY ZONE A A.. a
- Anet 23 $ 2" PRESENT Va *V nP
?
NO
'S VES ONLY 2ONE 8 Aset
- Anet i "' "' - PRESENT Ve ' VnP NO 3r COMPUTE A .,
A,,, s A..,- A 3 if _ COMPdTE A net v, s v,, - v, g i/ ir A., s ( O' ve ,"'", E o. ( 4-75) y oto a2 "'" P MP ~ MP I 2c , s Z e ,"' " i l I i ! Figure 22. Calculational logic for energy balance within primary tank l i I l l- ..._ . _ _ , - ~ _ _ _ _ . . . , . , _ _ . _ . . _ . . .,
A COMPUTE
"' 'C' **
i di di dt IS NO ZONE A dTo de s COMPUTE PRESENT di 7. Ee. (4 -64) YES
if dTg dea des COMPUTE di di di f
IS ZONE 8 NO de, PRESENT
}
di
?
-l YES u u de* , Ee. ( 4- 53) d's , de, dt di de if de a
, Ee. ( 4-S4 )
di
\r _
j u dT,,
' ** ~ '
df Nf dT at di 4 '. Nr ( RETURN ) Figure 22 (cont'd) . Calculational logic for energy balance within primary tank l l ____j
GENERAL FLOWCHARTS l MAIN 9R l SAME AS $$C-L. ONLY A00's l POOLou l l S$c-P l ; MuN,5 l l tooPiu l l l MAIN 97 l Figure 23. Three main driver programs of SSC-P. d STGN3S l l PSAL95 l H IMxiS l l PINTIS l H HTPLIS h
! DRIVIS l H -uSRRIU l H CDPLIS b-H LPLN65 l l MAIN 95 l COOL 65 } l OPTN65 HH CORE 65 l l FUELSS l H OPAS65]
H HYDRIS l
-{ PIPE IS l
-4 UPLNIS l l POOLIS l
-j P:PHIS l H CDPLIS l
-i E N D 1 $ l
-{ CVALIS l
-l LEVLIS l
-{ PUMPIS l
-l PRESIS l
-l SPASIS l -{ PIPE 25 l H RITEIS l -4 SPHT2S [
-j EVAP2S l l LOOP 2S l l PUMP 2S l d RES 25 l
-] PRES 2S l l PRNT9S l H TANK 2S l
-{ E N D 2 5 l M RITE 2S l l Figure 24. MAIN 95 flow diagram l
t
i XIIT l d RSETIT l i INIT9T l l WAINST I d DEFNIT l l DRIV 9T H H FLOWIT l H DRIVIT }- H FLOW 2Tl d STON3T l H SavE9T I d INTF9T l d FUEL 5T l , d C00L6T l d LOOPIT l d LOOP 2T l Figure 25. MAIN 9T flow diagram
! INTGli l l EQiviT l
'dQiV2T l d DEFNIT j l PIPWIT l l DRIVIT l ---{ PDFGIT l l HYDRIT l l STORIT l -{ RESIT l l CVAI.lT l H PUMPIT l l FLOWIT l -{ PLOSIT l
--{ VESLIT l l
d PRESIT l I d FUNCIT l d ZONEIT l i d POOLIT H f d HTPLIT l l H EQiVIT l l FLOW 2T H H RES2T l Figure 26. DRIVIT flow diagram
flow. Since the IHX and pipings are highly nodalized, the number of thermal equations to be solved is too large to be handled by the predictor-corrector algorithm. The combined solution is made possible by judiciously taking advantage of the properties of the flowing medium. Thus, in the liquid sodium circuits, the time-dependent energy and momentum equations can be decoupled since the effect of pressure on subcooled liquid sodium properties is considered negligible. The energy equations have only a weak influence on the momentum equations through the sodium properties. This allows the hydraul e equations (along with the energy balance equations in the primary tank) to be solved first, using coolant properties from the IHX and pipe as boundary conditions, evaluated at the previous tinestep. The component energy equations can then be readily solved, using the most recent flow value. More details on the numerical technique employed in SSC can be found in Ref. 9. 4.6 SOME CONSIDERATIONS FOR COLD POOL CONCEPT In a generic cold pool concept (see Fig. 2), there is also a hot leg pipe connecting the reactor outlet plenum with the IHX inlet. Since a substantial temperature gradient, equal to the difference between reactor outlet and inlet temperatures, exists between the hot sodium in the pipe and cold pool sodium, the pipe wall must be insulated. To protect the insulation from corrosion in a sodium environment, it could be imbedded within the wall. Pool-type systems have an enormous inventory of sodium within the primary tank. The heat capacity of the sodium inventory is available to absorb heat following a postulated loss of IHX cooling. In this respect, the cold pool concept has a larger margin, since its bulk sodium is at reactor inlet temperature. Further, for the cold pool concept, any leakage from a reactor outlet pipe to the bulk sodium of the pool would only heat up the pool. Since the pool has such enomous thermal capacity, safety would not be compromised. The presence of a hot-leg pipe, and the absence of a free liquid level in the outlet plenum, will somewhat modify the formulation -for primary system coolant dynamics. This is briefly summarized below. Some considerations for the energy balance calculations have been briefly mentioned with reference to the flow chart of Fig.15. Since the reactor outlet plenum is pressurized with no free surface, the flow rate in the IHX will be related to the pump flow rate by means of a simple algebraic relation. As a result, ouring a transient', the inventory of
;adium in the cold pool remains constant, and its volume, hence level can only change due to thermal expansion effects (i.e., coolant density changes with temperature, hence time).
For an intact system, the reactor inlet pressure is obtained in much the same way as for the hot pool concept, by applying mass conservation at the reactor inlet, and then relating the derivatives of flow through the pump and reactor. It would also be given by Eq. (4-23). The difference is that, due to the absence of a free liquid level, P oR in Eq. (4-23) is not known and has to be determined. Assuming that PR o is at the same elevation as the reactor outlet pipe, an expression can be derived for pro in terms of PRin and flow losses, by applying mass conservaton at the reactor outlet and then relating the flow deriviatives through the core and IHX. The simultaneous I solutions of. the two equations would then yield PRin and PR o. The rest of 1 I l l l r , f i
.the-fonnulation for' coolant dynamics remains unchanged f rom the ' hot pool concept.'
To simulate the transient following a primary pipe rupture accident event, the fonnulation should also allow for a break in the hot leg pipe.
- 5. INPUT DATA PREPARATION The first intended application of the SSC-P code is to analyze, in detail, any one or more of the independent EPRI sponsored design studies of a 1000 MWe pool-type LMFBR. Perhaps the first design selected for this purpose will be the conceptual study perfonned by the GE-Bechtel team2 . However, the code de-velopment effort to date has been guided primarily by the Phenix design and, as such, it would be helpful if this could lead to some trend comparisons.
Towards this end, the Phenix input data deck has evolved through an extensive search of the available literature. The numbers have been uncovered in bits and pieces from a wide spectrum of references, with cross-checking to minim-ize the chance of large discrepancies. Where direct numbers were unavailable, they were generally estimated frca related information, or else assumed. In case of unknown dimensions, they were estimated from a diagram believed to be drawn to scale 20, A sample input data deck for Phenix will be presented in the Users' Manual for SSC-P, to be issued subsequently to this report. However, it was felt it would be useful to devote a small Section to discuss the ways in which some of the data, not available in the literature, were obtained. 5.1 CORL DATA 5.1.1 Flow Fractions ! The average flow per fuel assembly is 22.7 kg/s 21 . Thus, the flow through 103 fuel assemblies is 2338 kg/s. With the total core flow at rated power i being 2760 kg/s, the flow fraction in the fuel channel is calculated to be l l l l I
0.845.. This leaves 0.155 for the blanket channel. Here, the flow through the control and shielding assemblies is assumed to be lumped with the blanket channel. 5.1.2 Power Fractions 2 From the natural circulation results for CRBRP using SSC-L,3 it is seen that, at t=400 seconds following scram, the ratio normalized power in blanket = 1.4839 (5-1) normalized power in fuel Under full power conditions (t=0 sec), this ratio is 1.0. Also, at t=400 sec., the power generation is due exclusively to decay power. Therefore, we can write P E - bd _fd =.1.4839 P P bo fg or P P bo ,g3 bd (5-3) P P fg fd where Pfd, Pbd are the decay power in fuel and blanket assemblies, re-spectively, and Pfo, Pb o are the respective powers corresponding to full-power conditions. We assume that Eq. (5-3) also applies to the Phenix reactor. During re-fuc'ing operations in Phenix 22 ,17 fuel assemblies give out 40 KW of decay heat, and 11 blanket assemblies give out 7 KW of decay heat (assuming power due to control assemblies is lumped in with the blankets). Using this information and taking Pfd, Pbd as power per respective assembly, we de-rive the ratio of full power conditions per assembly as
= 0.182 (5-4) fo
There are 90 radial blanket assemblies and 103 fuel assemblies in the Phenix core, producing nearly 563 MW of power. We can write this as 90Pbo + 103Pfo = 563 HW (5-5) Combining Eqs. (5-4), (5-5) gives P o f= 4.728 MW and P ob = 0.8445 MW. Thus, the power fractions in fuel and radial blanket regions are calculated to be 0.865 and 0.135, respectively. 5.1.3 Fuel Hot Channel The radial peaking factor for Phenix is 1.3 24 . Using this, we obtain the power generated in the fuel hot channel as 6.1464 MW. Assuming the hot chan-nel to represent only one assembly, the power fraction in the hot channel is calculated to be 0.01092. This leaves 0.85408 for the remaining 102 fuel as-semblies. From the analytical results of Freslon, et. al.25 , we obtain'the steady-state temperature rise in the nominal fuel hot channel (ATfhc) as 235.24K. Energy balance for the coolant in the channel can be written as WfhcC ATfhe = Pfhe (~} where C is the average heat capacity of the coolant in the hot channel. With Pfhe and ATfhe known, Eq. (5-6) can be used to determine the flow Wfhce and hence the flow fraction kIfhc. Note that since C is the integrated aver-age over the channel, and hence not readily obtain41e by hand calculations, the desired flow fraction was obtained after a few computer trials. Table I presents the power and flow fractions in the fuel het channel (Ch.1), average fuel channel (Ch. 2), and blanket channel (Ch. 3). The coolant temperature l l t l rise values in the last column were obtained from a steady-state run using i SSC-P. i i Table I Steady-state Power and Flow Fractions Channel Number Power Flow Coolant No. of Fraction Fraction Temp. Rise, K Assemblies 1 1 0.01092 0.007365 235.32 2 102 0.85408 0.837635 161.43 3 90 0.135 0.155 137.76 5.1.4 Axial Power Profile in Fuel 21 Heat flux on fuel element surface = 143W/cm (av.)
= 208W/cm (max.)
This can be used to evaluated the ratio of maximum to average power fraction in the fuel element as ax
= 1.4545 (5-7) q A parabolic profile of local power fraction q' can be constructed as q'=3=A + Bz2+ C (5-8) q i
where' A, B, C are constants and z is the axial distance along the fuel ele-ment of length L. l
\-
Boundary conditions on Eq. '(5-8) are (1) q'=0- at z = 0
..(ii) q' = 0 at z = L-(iii) 'q' = 1.4545 at z = L/2
- Applying (1), (ii), (iii) to Eq. (5-8), with L' = 0.85m, the profile is ob -
tained as 2 q' = -8.053z + 6.845z (5-9)
. The active fuel region within.the fuel pinLis divided into equidistant nodes. Therefore, we can construct a table of FSPAX (i.e. q') corresponding to midpoints in each nodal volume as shown below.
N F5 PAX 1 0.2764 2 0.7418 3 1.0909 4 1.3236 5 1.4400 6 1.4400 7 1.3236 8 1.0909 9 0.7418 10 0.2764 These can be read-in for the 10 axial nodes of the fuel region. Axial blanket, etc., do not have significant contributions, and have been left as before. i 5.2 PRIMARY SYSTEM
' 5.2.1 Pump Parameters The rated motor torque was extimated from information on.the rated speed NR (rpm) and power consumption P(W) of the motor at that speed. The formula is 60P (5-10) r mR = 2nN R (N-m)
From this, the rated torque, T R. (the hydraulic torque at rated speed) is obtained from T R= T mR T f,R (5-11) T whereT f,R is assumed to be a known, small fraction of R. The pump inertia, Ip , has been evaluated with the aid of a few trial-and-error runs to achieve the time required to reach half-speed, T 1/2, of. 7.5 sec. as re-ported in the lit arature 25 . With each run, the next guess is obtained using the approximated relation I To I ',' (5-12) p [h About two or three guesses were sufficient. It is also known that the pump reaches approximately 10% speed in 60 26 seconds . This information was used to revise the coefficients in the frictional ' torque correlation "
' fr * 'R (co + c a) (5-13)
The first guess was obtained assuming a linear coastdown between 20 and 60 seconds, and using a test run to obtain the values of a at 20 and 40 seconds. Then,-the pump speed equation _can be written as
~
2nl (n60 - n20) - Cg +C3 a + 0.44652a 2 T6 40 ~R 2
+ 0.5065;2(--) + 0.59643a2 (g) 4 (5-14) 2 2
-0.64055a({}3 i -+0.11531a(g)'_.0.025531a(g) 2 i
With a evaluated at t=40 sec as the linear average between a60 and a20 Eq.(5-14)simplifiesto co + 0.045c1 = 0.0056631. (5-15) Further, from rated conditions, we have that co + c1 = O.035 (5-16) Eqs.(5-15),.(5-16),solvedsimultaneously, yield 4 co = 0.001, c1 = 0.034 for 0.017 < a i 1.0 (5-17)
- To assure continuity, co , ci are redefined for the other 2 regions as co = 0.117, c1 = -9.458 for 0.005 < a 10.0117 (5-18) co = 0.035, c1 = 12.942 for 01 a 10.005 5.2.2 Data on Structures The mass heat capacities of structures are needed for transient energy bal-ance calculations. ~ Since the thermal barrier (metal m2) is a combination of l structure and sodium (or some other medium), it was decided to input (MC)m2 rather than Mm2 alone. The same was also done for metals ml, m3 and m4.
9 Specific heat of structure is given by C(T) = 380.962 + 0.535104T - 6.01413 x 10' T (5-19)
+ 3.02469 x 10' T w - - . - 7
and for sodium, by
- 1, 2 C(T) = 1630.22 - 0.83354T + 4.62838 x 10 T (5-20)
The heat capacities are then evaluated at expected operating temperatures for al, m2, ,m3, m4. Metal m2, Mstr = Ahm2 Np 6p pstr (5-21) M pNa Na =Ahm2 (N p - 1) 6 Na The density of the metal is assumed to be 7750 kg/m3, Then (MC)m2 = (MC)str + (MC)Na (5-22) Due to lack of available data on upper internal structures, (MC)ml is assumed to be the same as the value for CRBRP. - Note that the part of IHX and pump wells in hot pool sodium can, on user choice, be lumped together with metal ml. Metal m3 The mass of metal m3 is Mm3 = Mroof + mass of tank wall. exposed to cover gas. The area of contact between cover. gas and m3 is Agm3 = A hg + Acg + wDt ank(Ztank - ZCP) (5-24) Metal m4 Mm4 was obtained as the sum of the mass of tank wall exposed to cold pool temperature (Mwall) and the mass of core support structure. Further, it was assumed that 70% of the core support platfonn is solid steel, and that the mass of the support cone is negligible in comparison. The heat transfer coefficients between sodium and the structures were as-sumed constant. l l l l l l I il
Lin computing heat transfer area,' the area of the support platform was- as-sumed negligible. The area of support cone is-Acone " 4cose-(D, -Dj) (5-25)
~
where' e is the angle the' cone makes with the horizontal. Finally,.
Acm4 = Acone + Awall ( -26) i 4 f 4 4 4 0 i { l-k-
.l 6.- CODE DESCRIPTION 6.1 CODE DEVELOPMENT APPROACH ,
SSC was conceived as a' series of advanced computer codes to analyze'the transient behavior for a variety of LMFBR plant designs. In order to easily accommodate all the models and integration methods needed for such a diversified analysis, and to allow for maximum flexibility in the system to be analyzed, the parent code, SSC-L, was designed to be a highly modular and variably dimensioned code. SSC-L can only be applied to study loop-type LMFBRs. However, by exploiting the highly modular structure and flexibility of SSC-l, another version of SSC, to analyze pool-type LMFBRs, could be re-adily developed. A complete and new code was therefore not necessary for SSC-P since only the modules and submodules necessary to describe the characteristics specific to the pool design needed to be changed or added. The remainder of the code was obtained from the existing and tested SSC-L library. Code development using this approach, however, was not without its pit-falls since SSC-L was undergoing continual upgrading and modifications. The most serious question was how to continue the P-development around a given L-cycle and yet keep pace with the latest cycle generated. While there was no unique and simple solution, the method chosen would have to reflect the , minimum effort required to achieve the desired objective. The method adopted was to consolidate all the pool modules, common blocks and data blocks into a
- separate program library. In this context, a ' pool module' is any routine used in the pool version that differs in any fashion from the corresponding routine in i
the loop version. In this way, the routines common to both versions,l but unaffected by the pool version are kept exclusively in the SSC-L program library, and stay current with its development without any extra effort. Two-program libraries describe the current state of the SSC-P code. They are the current ' official' cycle of SSC-L and the most current cycle of SSC-P. The program library 'SSC-P' consists mainly of the following submodules:
-- Primary heat transport system input, output, restart.
-- Steady-state and transient core plena
-- Steady-state and transient pool hydraulics, and energy balance in the primary tank.
In addition, several other subroutines are also included which were affected due, in some instances, to a different nomenclature (e.g., flow rates, pressure losses, levels, etc. in the primary system), and sometimes to a dif-ferent model employed. In creating the SSC-P library, all programming features of SSC-L were preserved, namely:
-- adherence to ANSI - standards
-- naming convention that uniquely identifies modules and variables in the code
-- highly modular structure
-- variable dimensioning scheme The code development approach described above, while strictly following the philosophy of SSC-L, allows for easy maintenance of pool-related sub-routines, without concern and duplication of effort in the upgrading of the re-maining parts of the code. In addition, less than one third of the space that i
t
would be required by a single, completely independent SSC-P library is needed to store the code in either.a permanent file, tape or disk, and in case of any change made to the program, the compilation time is considerably less within the facilities of a computer such as the CDC-7600. 6.2 FLOWCHARTS AND SUBROUTINE DESCRIPTIONS Flow chart's for the main drivers, along with their modules and submodules, are shown in Figs. 23-26. New and modified subroutines are presented in the logical order of their calls within the global SSC code. All submodules that 27 are exclusive to the SSC-L program library , have been omitted. A short de-scription of the subroutines in SSC-P is given below. Many of the drivers, modules and submodules modified from SSC-L and which form a part of the SSC-P
' library have been omitted because their basic function, hence definition, has not changed from SSC-L. These definitions can be found in the documentation on SSC-L. '
LOOPlU This subroutine sums up the elevation differences of both primary and intermediate systems, based on user-input geometric profile data, and checks for convergence against a user-specified criterion. This ensures a consistent profile specification before the code goes into detailed system computation. P00LlU This subroutine displays, on request, all the pool common blocks along with their contents, during transient. PSAV90 this subroutine writes all pool common blocks on a restart file, during either steady-state or transient. . I
i 5 - PRST9U This subiodtine reads all pool common blocks from a restart file. DRIVIS'
- lThis subroutine 1is.the driver-for.the solution of energy balance in the pools during steady-state. -Starting from'an initial' guess for cold pool tem-perature (TCP), it iterates.between HTPLlS and CDPLlS until a converged
- value of TCP is obtained.
HPTLIS-This subroutine performs energy balance in the hot pool and iterates to calculate the steady-state temperatures in the hot center pool, barrier,;cov-er gas and structures'for a given value of cold pool temperature.
- CDPLlS This subroutine iterates to converge on a new value for cold pool tem-perature, based on barrier and cover gas temperatures determined from HTPLlS.
1 It also determines primary pump inlet and outlet enthalpies. UBRRlU This is a subroutine, currently called by HTPLIS, that computes overall heat transfer coefficients (U m2, h Ucm2) for the thermal barrier. i HBRRid
-This. function computes the film heat transfer coefficient from barrier to either hot or cold pool, as part of the computations for overall heat transfer coefficient in VBRRIV.
BPASIS I (- This subroutine, called from PBAL95, computes steady-state pool external
' bypass flow rate and inlet and outlet pressures.
I 1 __ _
RITEls
)
This subroutine prints primary pool system steady-state information. ! 9 P00LIS j i This subroutine drives the steady-state hydraulic calculations for the 1 primary system. In addition to interpreting logical variables to arrange the calling sequence to appropriate component submodules, it also evaluates pres-sure drops in the regions of hot and cold pool lying between the reactor and IHXs. Both hot pool and cold pool design options are covered. In case of the cold pool option, both thermal and hydraulic calculations would be included. LEVLIS For the hot pool concept, this subroutine computes the steady-state level in the cold pool, with hot pool level assumed user input. It also computes the volume and mass of the cover gas blanket above the free surfaces. RES2S This subroutine computes the coolant level in a spherical pump given the volume of coolant in it, and sets the inlet pressure to the intermediate sodi-um pump. The mass of cover gas above the coolant level is also calculated. TANK 2S This subroutine computes the level of coolant in the buffer (surge) tank, given the pressure of cover gas from input and the pressure at the tank loca-tion from PRES 2S. It also calculates the mass of cover gas in the tank. P00 LIT This subroutine is the driver for transient thermal balance in the pools. ZONEIT This subroutine computes the penetration of core flow, the volumes of the two zones in the hot pool heat transfer area for each zone and the barrier heat transfer coefficients. It'also updates the sodium volume in the hot and cold pools during the transient. HTPLlT This subroutine computes the time derivatives of enthalpy'(or temperature) in the hot pool, cold pool, cover gas, barrier, and all other structures. i .I i 1 4 I, 4 J 4 l t l l .
, , ,, m ,, - -. --- -
..~v ,- .~ . . . . - - - . . _ . - ---- - . -..
REFERENCES
- 1. M. E. Durham, " Optimization of Reactor Design for Natural Circulation De-cay Heat Removal in a Pool-Type LMFBR", Optimization of Sodium - Cooled Fast Reactors, British Nucl . Energy Soc. , London (1977) .
- 2. D. Mohr and E. E. Feldman, "A Dynamic Simulation of the EBR-II Plant Dur- )
ing Natural Convection with the NATDEM0 Code", Proceedings of the Specialists Meeting on Decay Heat Removal and Natural Convection in 1 FBR's, Upton, NY, February,1980.(to be published).
- 3. " Pool-Type LMFBR Plant,1000 Ne Phase A Design", EPRI NP-646, Vol 1, 1 Part II, Electric Power Resea'ch Institute (1978). j
- 4. D. Mohr, "NATCON, A Dynamic Code for Evaluating the Thermal-Hydraulic Be-havior of the EBR-II Primary System, Including the Effect of Natural Con-vection", Reactor Develoment Progress Report, ANL-RDP-74, p.1.11 ( August 1978).
- 5. "LMFBR Demonstration Plant Simulation Model, DEM0", Westinghouse Advanced Reactors Division, WARD-D-0005, Rev. 4 (1976) .
- 6. M. Khatib-Rahbar, " System Modeling for Transient Analysis of Loop-Type Liquid-Metal-Cooled Fast Breeder Reactors", Cornell University Nuclear Reactor Laboratory, CURL-53 (1978) .
- 7. J. R. Raber, " System Modeling for a Pool-Type Liquid Metal-Cooled Fast Breeder Reactor", Cornell University Nuclear Reactor Laboratory, CURL-56 (1980).
- 8. I. K. Madni and A. K. Agrawal, "A Work Plan for the Development of SSC-P", Brookhaven National Laboratory, BNL-NUREG-24225 (1978).
- 9. A. K. Agrawal, et al ., "An Advanced Themohydraulic Simulation Code for Transients in LMFBRs (SSC-L Code)", Brookhaven National Laboratory, BNL-NUREG-50773 (1978).
- 10. E. R. G. Eckert and R. Drake, Jr., " Analysis of Heat and Mass Transfer",
McGraw-Hill, Inc. (1972).
- 11. B. Gebhart, " Heat Transfer", McGraw-Hill, Inc., 2nd. Edition (1971).
- 12. I. K. Madni and E. G. Cazzoli, "A Single-Phase Ptap Model for Analysis of LMFBR Heat Transport Systems", Brookhaven National Laboratory, BNL-NUREG-50859 (1978).
- 13. I. K. Madni, E. G. Cazzoli and A. K. Agrawal, "A Single-Phase Sodium Pump Model for LMFBR Themal-Hydraulic Analysis", Proceedings of the Inter-national Meeting on Fast Reactor Safety Technology, Seattle, Washington, August 19-23, 1979.
l l l 1
- 14. V. Quan and A. K. Agrawal, "A Pipe-Break Model for LMFBR Safety An-alysis", Brookhaven National Laboratory BNL-NUREG-50688 (1977).
- 15. R. S.-Burington, Handbook of Mathematical Tables and Formulas, 4th Edi-tion, McGraw-Hill, 1965.
- 16. J. W. Yang, "An Analysis of Transient Themal Response in the Outlet Pen-um of an LMFBR", Brookhaven National Laboratory, BNL-NUREG-50521 (1976).
- 17. J. W. Yang, " Penetration of Core Flow in Upper Plenum of an LMFBR",
Trans, Am. Nucl. Soc. 23, 414 (1976).
- 18. I. K. Madni, " Transient Analysis of Coolant Flow and Heat Transfer in LMFBR Piping Systems", Brookhaven National Laboratory, BNL-NUREG-51179 (1980).
- 19. F. T. Krogl, " Variable Order Integrators for the Numerical Solution of Ordinary Differential Equations", det Propulsion Laboratory Report, May 1, 1969.
- 20. A. Anorosi, et al ., "An Overview of Pool-Type LMFBRs, General Characteristics", Argonne National Laboratory, ANL-76-71 (1976).
- 21. " PHENIX" in Directory of Nuclear Reactors, Vol . IX, Power Reactors, IAEA (1971).
- 22. A. K. Agrawal, et al ., " Dynamic Simulation of LMFBR Plant Under Natural Circulation", ASME Paper No. 79-HT-6,1979.
- 23. Phenix, Prototype Fast-Neutron Nuclear Power Station, Bulletin d'Informa-tion Scientifiques et Techniques No.138 (June 1969); AEC-tr-7130 (Febur-a ry - 1970) .
- 24. W. B. Wolfe, Monthly Report No.11, Atomics International, Nov.1973.
- 25. H. Freslon, et al., " Analysis of the Dynamic Behavior of the Phenix and Super Phenix Reactors during certain Accident Sequences", Proc. Int. Mtg.
on Fast Reactor Safety Technology, Vol . III, Seattle, Wash., Aug.19-23, 19/9.
- 26. E. R. Appleby, " Compilation of Data and Descriptions for U.S. and Foreign Liquid Metal Fast Breeder Reactors", Hanford Engineering Development Laboratory, HEDL-TME-75-12 ( Aug.1975).
- 27. A. K. Agrawal, et al ., " Users' Manual for the SSC-L Code", Brookhaven National Laboratory BNL-NUREG-50914 (1978) .
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