3% . :‘\'4__,\; - " QASTER OAK RIDGE NATIONAL LABORATORY operated by UNION CARBIDE CORPORATION NUCLEAR DIVISION for the U.S. ATOMIC ENERGY COMMISSION ORNL- TM- 3102 MSBR CONTROL STUDIES: ANALOG SIMULATION PROGRAM W. H. Sides, Jr. NOTICE This document contains information of a preliminary nature and was prepared primarily for internal use at the Qak Ridge National Laboratory. 1t is subject to revision or correction and therefore does not represent a final report. UISTRIBUTION OF THIS DOCUMENT IS UNLIMITER; This report was prepared as an account of work sponsored by the United States Government. Neither the United States nor the United States Atomic Energy Commission, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completaeness or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privately owned rights, ORNL-TM-3102 Contract No. W-7405-eng-26 INSTRUMENTATION AND CONTROLS DIVISION MSBR CONTROL STUDIES: ANALOG SIMULATION PROGRAM W. H. Sides, Jr. This report was prepared as an account of work sponsored by the United States Government, Neither the United States nor the United Staies Atomic Energy Commission, nor any of their employees, nor any of their contractors, subcontractors, or their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, com- pieteness or usefulness of any iaformation, apparatus, produci{ or process disclosed, or represents that its use waould not infringe privately owned rights. MAY 1971 OAK RIDGE NATIONAL LABORATORY Oak Ridge, Tennessee operated by UNION CARBIDE CORPORATION for the U. 8. ATOMIC ENERGY COMMISSION SISTRIBUTION OF THIS DOCUMENT IS UNIJMIT}EP iii CONTENTS CIErOdUCION . . e e e e 1 . Description of the Plant and Control System . ......... ... .. . . i, 1 . Simulation of the SysStem .. .. ... e e 7 B MoOdEl .o e e e 7 3.2 System Equations and Analog Computer Program ............ ... ... .. ... L. 9 320 Reactor Core ..o i e e 9 3.2.2 Primary Heat Exchanger ... ....... ... ... ... .. . ... e 15 3.2.3 Steam Generator ..o v ittt c e ettt et e e e e e 19 324 Control SYstem . ..ottt e e e 25 MSBR CONTROL STUDIES: ANALOG SIMULATION PROGRAM W. H. Sides, Jr. ABSTRACT This report describes the mathematical model and analog computer program which were used in the preliminary study of the dynamics and control of the 1000-MW(e) single-fluid Molten-Salt Breeder Reactor. The results and conclusions of the study were reported earlier in Control Studies of a 1000-Mwfe) MSBR, ORNL-TM-2927 (May 18, 1970), by W. H. Sides, Ir. 1. INTRODUCTION A preliminary investigation was made of the dynamics and possible control schemes for the proposed 1000-MW(e) single-fluid Molten-Salt Breeder Reactor (MSBR). For this purpose an analog computer simulation of the plant was devised. In this report the system, simulation model, and analog computer program are described. The specific transients investigated using this simulation, the results, and the conclusions were presented in another report.’ For the purposes of the analysis, the MSBR plant consisted of a graphite-moderated circulating-fuel (primary salt) reactor, a shell-and-tube heat exchanger for transferring the generated heat to a coolant (secondary salt), a shell-and-tube supercritical steam generator, and a possible control system. The analog simulation of the plant consisted of a lumped-parameter heat transfer model for the core, primary heat exchanger, and steam generator; a two-delayed-neutron-group model of the circulating-fuel nuclear kinetics with temperature reactivity feedbacks; and the external control system, The simulation was carried out on the ORNL Reactor Controls Department analog computer. So that the model would have the maximum dynamic range, the system differential equations were not linearized, and as a result the requisite quantity of equipment required that the model be severely limited spatially to minimize the number of equations. In addition, the pressure in the water side of the steam generator, as well as in the rest of the plant, and the physical properties of the salts and water were taken to be time invariant. The temperature of the feedwater to the steam generators was also held constant. 2. DESCRIPTION OF THE PLANT AND CONTROL SYSTEM The proposed 1000-MW(e) MSBR steam-electric generating plant consisted of a 2250-MW(t) graphite-moderated molten-salt reactor, 4 shell-and-tube primary heat exchangers, and 16 shell-and-tube supercritical steam generators (Fig. 1). The reactor core contained two zones: a central zone, a cylinder ~14.4 ft in diameter and ~13 ft high with a primary-salt volume fraction of 0.132; and an outer zone, an annular region ~1.25 ft thick and the same height as the central zone. The salt volume fraction in this region was 0.37. The primary salt, bearing 2?3U and 232 Th, flowed upward through the graphite core in a single pass and then to the tube side of one of four vertical single-pass primary heat exchangers, each ~19 ft long, 5 ft in diameter, and constructed of Hastelloy N. The salt flow rate at design point was 9.48 X 107 Ib/hr. The design-point temperature of the salt entering the core was 1050°F and that at the core outlet was 1300°F. The liquidus temperature of this salt was approximately 930°F, The heat generated in the primary salt in the core was transferred from the tube side of the primary heat exchangers to a countercurrent secondary salt passing through the shell side. This salt flowed in a closed secondary loop to one of four horizontal supercritical steam generators. The four secondary loops, 1w, H. Sides, Jr., Control Studies of a 1000-Mw(fe) MSBR, ORNL-TM-2927 (May 18, 1970). PRIMARY SALT PUMP (4) n"" REACTOR VESSEL=a| = | EVEL CONTROL® 9.48 X107 Ib/hr at 1300°F PRIMARY GRAPHITE HEAT EXCHANGER MODERATOR—" (4) 9.48 X 10" Ib/hr at 1050°F Fig. 1. Flow Diagram of MSBR Plant, ORNL-DVWG 70-8883 SECONDARY SALT PUMP (4) 1X 10" Ib/hr STEAM ai 1000°F — i — . i ety . —— . i o o l96X10° 1o/hr . 5X10° b/ hr 712 X10" 1b/br at 1150°F 6.16 X 10" Ib/hr at 1150°F REHEATER 850°F STEAM GENERATOR 19 @) REHEAT STEAM at 650°F The quantities shown are totals for the entire plant. ORNL DWG. 71-3380 + PDEMAND ROD T (1000°F) DRIVE Y Mset SERVO & P, /D PRIMARY ., STEAM HEAT GENERATOR j EXCHANGER — Bec REACTOR % °F - 3 Tei l ri < I 700°F Fig. 2. Simulation Model of Plant and Control System. one for each primary heat exchanger, were independent of each other, with each loop supplying heat to four steam generators. Thus there was a total of 16 steam generators in the plant. The design-point flow rate of secondary salt in each loop was 1.78 X 107 Ib/hr. At the design point the secondary-salt cold-leg temperature was 850°F, and the hot-leg temperature was 1150°F. The liquidus temperature of this salt was ~725°F. The shell-and-tube supercritical steam generators were countercurrent single-pass U-tube exchangers ~73 ft long and ~18 in. in diameter and were constructed of Hastelloy N. Feedwater entered the steam generators at the design point at 700°F and a pressure of about 3750 psi. The outlet steam conditions at the design point were 1000°F and 3600 psi. Each steam generator produced steam at the design point at a rate of 6.30 X 10° Ib/hr. Reference 2 gives a complete description of an earlier, but quite similar, version of the steam generator and primary heat exchanger. The load control system used in this study maintained the temperature of the steam delivered to the turbines at a design value of 1000°F during all steady-state conditions and within a narrow band around this value during plant transients. The rudimentary control system used in this simulation is shown in Fig. 2. It consisted of a reactor outlet temperature controller similar to that used successfully in the MSRE® and a steam temperature controller. Steam temperature control was accomplished by varying the secondary-salt flow rate. This method was chosen because of the relatively tight coupling which existed between steam temperature and secondary-salt flow rate. The measured steam temperature was compared with its set point of 1000°F, and any error caused the secondary-salt flow rate to change at a rate proportional to the error if the error was 2°F or less. If the error was greater than 2°F, the rate of change of the secondary-salt flow rate was limited to its rate of change for a 2°F error, which was approximately 11%/min. 2General Engineering Division Design Analysis Section, Design Study of a Heat Exchange System for One MSBR Concept, ORNL-TM-1545 (September 1967). 31 R, Tallackson, MSRE Design and Operations Report, Part IIA: Nuclear and Process Instrumentation, ORNL-TM-729 (February 1968). To control the reactor outlet temperature, an external plant-load demand signal was used to obtain a reactor outlet temperature set point. The outlet temperature set point was a linear function of load demand, varying between 1125 and 1300°F for loads above 50% and between 1000 and 1125°F for loads below 50%. The measured value of the reactor inlet temperature was subtracted from the outlet temperature set point, and, since the primary-salt flow rate was constant, a reactor (heat) power set point was generated by multiplying this AT by a proportionality constant. The reactor power set point was a function of inlet temperature during a transient and thus a function of dynamic load. The measured value of reactor power (from neutron flux) was compared with the reactor power set point, and any error was fed to the control rod servo for appropriate reactivity adjustment. Under normal conditions, the control rod servo added or removed reactivity at a rate proportional to the reactor power error if the error was 1% or less. If the error was greater than 1%, the addition or removal rate was limited to the rate fora 1% error, which was about 0.01%/sec. The maximum magnitude of reactivity that the simulation allowed was £1%. The physical constants used in this simulation are summarized in Table 1. The various system volumes, masses, flow rates, etc., calculated from the constants are listed in Table 2. Table 1. Physical Constants Properties of Materials Cp p k [Btulb ™! (°F)7!] (b/ft3) {Btu hr ™! (°F)~! ft 1) Primary salt 0.324 207.8 at 1175°F Secondary salt 0.360 117 at 1000°F Steam 726°F 6.08 227 750°F 6.59 114 850°F 1.67 6.78 1000°F 1.11 5.03 Hastelloy N 1600°F 0.115 548 9.39 1175°F 0.129 il1.6 Graphite 042 115 Reactor Core Central Zone Outer Zone Diameter, ft 144 16.9 Height, ft 13 13 Salt volume fraction 0.132 0.37 Fuel 233y Graphite-to-salt heat transfer coefficient, 1065 Btuhr7! ft72 (°F)~! Temperature coefficients of reactivity, (°F)~! Primary salt -1.333 X 1075 Graphite +1.056 X 1073 Thermal-neutron lifetime, sec 3.6 X107 Delayed neutron constants, g = 0.00264 i B; A; (sec™!) 1 0.00102 0.02446 2 0.00162 0.2245 Heat Exchangers Primary Heat Steam Generator Exchanger Length, ft 18.7 Triangular tube pitch, in. 0.75 0.875 Tube OD, in. 0.375 0.50 Wall thickness, in. 0.035 0.077 Heat transfer coefficients, Btu hr ™! ft =2 (°F)~1 Steam Qutlet Feedwater Inlet Tube-side fluid film 3500 3590 6400 Tube-wall conductance 3963 1224 Shell-side fluid film 2130 1316 Table 2. Plant Parameters (Design Point) Heat flux, Btu/hr Primary-salt flow rate, Ib/hr Steady-state reactivity, pg Externatl loop transit time of primary salt, sec Heat generation, MW(t) Salt volume fraction Active core volume, ft3 Primary-salt volume, ft3 Graphite volume, ft3 Primary salt mass, Ib Graphite mass, b Number of graphite clements Heat transfer area, ft? Average primary-salt velocity, fps Core transit time of primary salt, sec Reactor Core 7.68 X 107 [2250 MW(t)] 9.48 x 107 0.00140 6.048 Zone I 1830 0.132 2117 279 1838 58,074 212,213 1466 30,077 ~4.80 2.71 Primary Heat Exchanger Total for each of four exchanges, tube region only Secondary-salt flow rate, ib/hr Number of tubes Heat transfer area, ft? Overall heat transfer coefficient, Btu hr™! ft =2 (°F)~! Tube metal volume, ft3 Tube metal mass, 1b Volume, ft3 Mass, ib Velocity, fps Transit time, sec Primary Salt (Tube Side) Steam Generator 1.78 x 107 6020 11,050 993 30 16,020 Zone 11 420 0.37 800 296 504 61,428 58,124 553 14,206 ~1.04 12.5 Secondary Salt (Shell Side) 57 11,870 10.4 1.80 Total for each of 16 steam generators, tube region only Steam flow rate, 1b/hr Number of tubes Heat transfer area, ft2 Tube metal volume, ft3 Tube metal mass, lb Volume, ft3 Mass, 1b Transit time, sec Average velocity, fps Steam (Tube Side) 7.38 X 103 434 4102 22 12,203 295 34,428 2.68 6.97 Secondary Salt (Shell Side) 20 235 1.15 ~62.8 102 11,873 9.62 7.50 3. SIMULATION OF THE SYSTEM 3.1 Model A spatially lumped parameter model used for the heat transfer system (Fig. 3) consisted of the reactor core, one primary heat exchanger, one steam generator, the nuclear kinetics, and a control system as shown in Fig. 2. All 4 primary heat exchangers were combined into 1 and all 16 steam generators into 1. In the core the primary salt in the central zone was divided axially into four equal lumps, and the graphite was divided into two. The outer zone was divided equally into two primary-salt lumps and one graphite lump. Since the primary-salt density varied only slightly with temperature, the four central-zone lumps were of equal mass, as were the two outer-zone lumps. The two central-zone graphite lumps were of equal mass as well. The mass flow rate of the primary salt in the two zones of the core was determined by the heat generation rate in each zone, so that the temperature rise of the primary salt in the two zones was equal. Thus, 81.4% of the flow passed through the central zone and 18.6% through the outer zone. A two-delayed-neutron-group approximation of the circulating-fuel nuclear kinetics equations* was used in the model. This allowed the delayed-neutron precursor concentration term C/(t — 7, ) (see Sect. 3.2.1) to be simulated directly with two of four available transport lag devices. The delayed-neutron fraction for 232U was 0.00264, and the prompt-neutron generation time was 0.36 msec. The coefficient of reactivity for the primary salt was —1.33 X 107° 8k/k per °F, which was divided equally among the six primary-salt lumps of the core model. The temperature coefficient for the graphite was +1.06 X 107° §k/k per °F, which was divided equally among the three graphite lumps. The model was designed to accommodate a variable flow rate of the primary salt as well as the secondary salt and steam. The required variations of film heat transfer coefficients with the various salt and steam flow rates were included.” The film coefficient for secondary salt on the shell side of the primary heat exchanger and steam generator was proportional to the 0.6 power of the flow rate. The film coefficient for steam on the tube side of the steam generators was assumed to be proportional to the 0.8 power of the flow rate. The variation of the film coefficient in the reactor core and on the tube (primary salt) side of the primary heat exchangers decreased with flow, as shown in Fig. 4. The heat conductance across the tube wall in both exchangers was assumed to be constant. The primary and secondary salts in the primary heat exchanger were divided axially into four equal lumps, with the tube wall represented by two lumps. As did the primary-salt density, the secondary-salt density varied only slightly with temperature, and thus the masses of the salt lumps were assumed to be equal and constant. A variable transport delay was included in the hot and cold legs of the secondary-salt loop to simulate the transport of secondary salt between the primary heat exchanger and the steam generator. The secondary salt in the steam generator was axially divided into four lumps of equal mass, as in the primary heat exchanger. The steam on the tube side was likewise divided into four equal lumps spatially, but of unequal mass. Under design conditions the supercritical steam density varied from 34 Ib/ft* at the feedwater inlet to 5 1b/ft® at the steam outlet.® The density of the steam in the lump nearest the feedwater entrance was taken as the average density in the quarter of the steam generator represented by that lump, or 22.7 Ib/ft?. The densities of the remaining three steam lumps were determined in a similar manner. The #J. MacPhee, “The Kinetics of Circulating Fuel Reactors,” Nucl. Sci. Eng. 4, 588-97 (1958). >Private communication from H. A. McLain, ORNL. SPrivate communication from T. W. Pickel, ORNL. ¢ OF CORE T Tro S— Tss A Bl | ---------- — - e e = —— -——)f————_— e, ——e— r— ey e | e . sttt ZONE T T ZONE TT i T | [ i PRE SALT| | ' ! PRITSALT SECTSALT ‘ i SECTSALT - Toa | | | PN se | ss N | . \ ! \\ | i 1073 |\ | GRAPHITE | PRISALT| | | 4 TUBE r | N ] T '\ | Tps i i Tt M | l | ™\ |PRi sALT | : | {PRI SALT Nsec sauf | | |sec saLt | N Tes i | ! Tpe Ts 3 | I Tsg ! ! ' 996 GRAPHITE ! ! I (i N I - | ' i PRI SALT | i I PRI SALT sscsmi l SEC SALT 1 Tpa [ \ : ' Tpe N Ts» , ! Tst N I PHIT ‘ | | “«| TUBE | | N . [GRAPHITE [ PRI SALT] , \ | ! ! Tg2 \\ i Tos ! | Ttz M ! | | N PRESAT] | | | [PRISALT e saT || | [SEC sALT I 4 TPT ' A ' { Tri T l i Tss 1 I ! ! 1 \ 850 I_ _________ — _-J [ —_ — I | Yy “ I_T._.l Y it} - 2 | T . W REACTOR CORE PRIMARY HEAT EXCHANGER STEAM Fig. 3. Lumped-Parameter Model of MSBR Plant. ORNL DWG, 71-3378 7 /// L 1.0 : > [N / o S 0.8 - / 2 // L o Vi < £ // =y / w = 0,6 7 S 2 CORE / = S e / /" M-PRIMARY oo / HEAT EXCHANGER w T 04 / L. I . vy // = O =" / '—- '—.. =L [T xr | . o o '— L o 0 0.2 0.4 0.6 0.8 1.0 RATIC OF REDUCED FLOW RATE TO DESIGN FLOW RATE Fig. 4. Variation of Film Heat Transfer Coefficient with Primary-Salt Flow Rate in the Reactor Core and Primary Heat Exchanger. axial temperature distribution in the steam was nonlinear also and was determined from the average design-point temperatures in each half of the steam generator. The specific heat of each lump was calculated from the enthalpy and temperature distributions. In the model, these resulting design-point values of density and specific heat were assumed to remain constant during part-load steady-state conditions and during all transients. 3.2 System Equations and Analog Computer Program The spatially lumped parameter model used to describe the plant heat transfer system is shown in Fig. 3. 3.2.1 Reactor Core. In the graphite the heat transfer equation was: T Myi Cpg dt =hfg Agl(Tp — Tgi)+kgiPr > (1) where Ty; = temperature of graphite lump i, T, = temperature of primary salt lump to which heat is being transferred, Mgy =M, =mass of graphite lump in core zone I, M, 3 = mass of graphite lump in core zone I, Cpg = specific heat of graphite, 10 hfg = graphite-to-primary-salt heat transfer coefficient, Agy = Agy = heat transfer area of lump in zone I, Ag3 = heat transfer area of lump in zone II, kg; = fraction of fission heat generated in graphite lump 7, P, = reactor heat generation rate. In the primary salt: dT,; Mpi Cpf'?;fl =fF1 Cp]‘{Tpi-—l — Tpt) + hfg Api(Tg — Tpl) + kpiPr s (2) where Tpl- = temperature of primary-salt lump i, Mpy =My =Mpy3 = My, = mass of primary-salt lump in core zone I, Mp5 =Mp6 = mass of primary-salt lump in core zone II, Cp F= specific heat of primary salt, F, =total primary-salt mass flow rate, f = fraction of primary-salt mass flow rate in zone I, Apy =Apy =Ap3 = Ay = heat transfer area of lump in zone I, Aps =A,6 = heat transfer area of lump in zone II, kp ; = fraction of fission heat generated in primary-salt lump /. The reactor outlet temperature T, , was given by: TFO:pr4+(1_f)Tp6‘ (3) The heat transfer coefficient hg, varied with the primary-salt mass flow rate, as follows: hg = hrgo Mg s where %z, is the design-point value of the graphite-to-salt heat transfer coefficient. A diode function generator was used to simulate the function s, (F', ), which is shown in Fig. 4 labeled “‘core.” The fraction of the total primary-salt mass flow rate that passed through zone I (and the fraction of the total heat generated in zone I) was given in Sect. 3.1 as 0.814. The fraction of heat generated in the primary salt as compared with the graphite in both zones was 91.9% in the salt and 8.1% in the graphite. Thus in zone II, kg3 =0.186 X 0.081 =0.0151, 4) and kys =k, =0.186 X 0.919 X 0.5 =0.0857 . (5) 11 In zone |, kg1 = kg2 =0.814 X 0.081 X 0.5=0.0329. (6) The axial distribution of the reactor heat generation rate in the primary salt in core zone I was assumed to be a chopped cosine distribution with an extrapolation distance of 8 ft. Then, in the primary salt in zone I, 0.3887 /ro.sx kp1 =kps =0.814 X 0919(-[0.276« sin x d’x/[o.”(m smxdx) X 0.5 (7) =0.1749, 0.5 0.5 kyy =kp3 =0.814 X 0919(-[0.38817 s1nxdx/[‘0'2761r smxdx) X 0.5 (8) =0.1992. The constant coefficients in the equations were calculated using Tables 1 and 2. Thus in Eq. (1), hrgo g1 _ 1065 Btu X 15,039 ft* X 1b-°F X hr ©) M1 Cpg hrft?-°F X 1.06 X 10° 1b X 0.42 Btu X 3600 sec =0.0998 sec ! , where 715, = design-point primary-salt-to-graphite heat transfer coefficient, and kg1 P _0.0329 X °FX 7.68 X 10° BtuX hr (10) My Cpy 0.4452X 10° Btu X hr X 3600 sec =1.577°F/sec, where P,y = design-point power level [2250 MW(t)]. The coefficients in the remaining graphite equations (i = 2, 3) were similarly calculated. For the primary-salt heat transfer equation (2) the constant coefficients are fFio_ 0814X9.48X 107 Ib X hr 11 M, 0.25X 58,074 1b X hr X 3600 sec () =1.476sec™" | where ', , = design point value of primary-salt mass flow rate, hfgo Ap1 ~ 8.008 X 10° BtuX Ib-"F X hr 12) M,y Cpy 1451916 X hr-"F X 0324 Btu X 3600 sec =0.4729 sec”! | 12 and kp1 Pro _0.1749 X °F X 7.68 X 10° Btu X hr Myy Cpr 4704 Btu X hr X 3600 sec =79.32°F/sec . The coefficients for the remaining primary-salt equations (f = The unscaled heat transfer equations for the core were thus dT, L 272l _ 59998 hfg(Fl) (Tp3 — Tg1)+ 1.577—L dt PrO T2 = 0.0998 i (F ) (Toy - Ton)+ 157721 ar OO ) gy~ Tga) + 1. By’ T,y P —E2= 01722 by (F) (Tys — Ty3) + 1. 3”0P— : dT F dlzl = 1_476F_1 (T,; ~ Tp1) + 04729 hyo(F, ) (T, — 10 dT, Pl 476 ( ar dT, F d?3 = 1-476F_1 (Tpa — Tp3) + 04729 hp (F ) (T — 10 dT, F p4 _ 1 _ b4 1.476_FT0 (T3 — Tpa) + 04729 hpy(F) ) (T dT 71;5 = 0. 1599 ( — Tys) + 02112 g, (F)) (T3 dT p6 — —ps 0. 1599 ( Tpe) + 02112 hey (F ) (T3 T,q =0.814T,, +0.186T,¢ P Tp2) + 04729 hyy(F)) (Tgy = Tpy)+90.24 57, 2—6) were similarly calculated. P T, )+7932 L ro ro P Tp3)+ 90.24__" | Pro PrO P 7;5)+1836_1., Pr() Pr Tps)+18.36 5, ro (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) A two-delayed-neutron-group approximation of the circulating-fuel point-kinetics equations was used for the nuclear behavior of the core. The equations were: P, p—8 _____P+ + —F = A C NGy, (24) 13 —AgT %%=§11P,-—7\,-C,-—-T-I;Ci+e CILC,.(tMrL), (25) p=p0+Eapl~ATpl-+Eagl-ATgi+pc , (26) 1 I where P, = reactor heat generation rate, C; = modified delayed-neutron precursor concentration, B; = delayed-neutron fraction, A; = delayed-neutron precursor decay constant, ! = neutron generation time, T, = transit time of primary salt through core, 7, = transit time of primary salt through external loop, p = reactivity, po = steady-state design point reactivity (associated with flowing fuel), ap,; = temperature coefficient of reactivity of salt lump pi, a,; = temperature coefficient of reactivity of graphite lump g, AT),; = variation from design-point temperature of salt lump pi, AT,; = variation from design-point temperature of graphite lump £, p. = control rod (or other externally introduced) reactivity. Variation of the primary-salt flow rate through the core varies the value of the transit times 7, and 7, inversely proportional to flow rate. Provision was made in the simulation for these variations to enable study of transients in the primary-salt flow rate, The variation of the transit times with flow rate was given by T, = Teo (Fillc)—l 27 and T, =TLo (%)—l , (28) where T.o = transit time of primary salt through the core at design-point flow rate, Ty o = transit time of primary salt through the external loop at design-point flow rate, F|[F,, = relative primary-salt flow rate. The value of 7, was as given in Table 2, and the transit time of the primary salt through the active core was taken to be 3.57 sec (ref. 5). 14 The negative temperature coefficient of reactivity for the primary salt was divided equally among the six primary-salt lumps in the core model. Similarly, the positive coefficient of the graphite was divided equally among the three graphite lumps. The nuclear constants for the two-delayed-neutron-group approximation for 2?3U were as given in Table 1. The steady-state design-point reactivity p, was given by? po=6- 5 M @9) SNt (/r) (1 —eMTLO) which yields po = 0.001403 . (30) After calculation of the various coefficients, the unscaled kinetics equations for the relative reactor power (neutron density) and relative delayed-neutron precursor concentrations are d(P,/P P P c ¢ d Py ro Cio Ca0 d(C,/C P F (€,/Cy0) _ 0.0630 -7 — 0.0245 S gagor ‘1 dt PFO ClO 10 ¥10 c (r 6.048 ) F 0.1479 1\" +0.2801 [_1 exp (_ )J Fi[Fyo (32) FlO FI/Fl() CIO p F, C d(Cz/C20)= 04326 —" — 0.2245 Ez_ 028011 =2 dr P, Cro 10 C20 6.048 C, [t - —=—— F 1.358 2( Fl/Flo) +0.2801 | =L exp (- ) ’ (33) [Fxo ( F1/F10 ] C20 6 3 p =0.001403 — L 0.2222 AT, + L 03520 AT, +p, . (34) i=1 i=1 The functions in brackets containing the exponentials in Eqs. (32) and (33) were generated by means of diode function generators. Equations (14) through (23) and (31) through (34) formed the complete 'set for the thermal and neutronic behavior of the reactor core in this model. The system variables in these equations and in those for the primary heat exchanger, steam generator, and control system given in the three following sections were scaled for the analog computer as follows. 15 The allowable computer voltage range from 0 to 100 V corresponded to a temperature range from 0 to 2000°F for all temperatures in the simulation. Therefore the computer variable representing temperature, V., corresponded to system temperature, 7, as T VT:Z). (35) For all flow rates the computer voltage range of 0 to 100 V corresponded to a relative flow-rate range of O to 1.25. Thus, the design-point flow rate (1.0) corresponded to a computer voltage ¥V of 80 V, or F Vp=80— . FUF, (36) The relative power level, P/P,, was similarly scaled. The computer variable representing power was P = 80— . V=805 (37) The computer variables, ¥, representing the relative heat transfer coefficients, 4(F), were generated using diode function generators with the relative flow rate ¥ as the input. The computer variables were directly proportional to the relative heat transfer coefficients such that the design-point value of A(F) corresponded to a computer voltage V;, of 80 V. The scaling for the relative delayed-neutron precursor concentration, C;/C,, in the nuclear kinetics equations was similar to that for flow rate and power level. The computer variable ¥ ; was Vi =80 C& . (38) I Reactivity was scaled such that the range of the computer variable ¥, of 0 to 100 V corresponded to reactivity of 0 to 1.0%, or V,=10°p. (39) The resulting analog computer program for the heat transfer and nuclear kinetics of the core is shown in Fig. 5. 3.2.2 Primary Heat Exchanger. The heat transfer model of the primary heat exchanger is shown in Fig. 3. For the primary-salt stream: dT,; Mp; Cpr —d% =Fy Cop (Tpiy =Tyt hpAp Ty — Ty} - 40) For the tube walls: dT,; Mti Cpt”d_tn =thp (Tpi m th)+ hsp Ap (Tsi - Tti) . (41) F, (80.00v* DESIGN POINT) Fig. 5. Analog Computer Program for the Heat Transfer and Nuclear Kinetics Equations for the Core. ORML WG, 77-3374 91 17 For the secondary-salt stream: aT; M Cps 7;‘1' =F, Cps (Tsi~l — Tt hsp Ap (T;; — Tsj) ; (42) where T,; = temperature of secondary-salt lump 7 , T,; = temperature of tube wall lump i, M7 =M,g =M, =M,; = mass of primary-salt lump, M,; =M,, = mass of tube wall lump, My =My =M.3 =My, = mass of secondary-salt lump, C,¢ = specific heat of tube wall (Hastelloy N), C p F, = secondary-salt mass flow rate, ¢ = specific heat of secondary salt, h¢ = primary-salt-to-tube-wall heat transfer coefficient, hg p= tube-wall-to-secondary-salt heat transfer coefficient, Ap, = heat transfer area of primary heat exchanger lump. Heat transfer coefficients in the above equations were calculated as follows. The term hyin Egs. (40) and (41) included the film resistance inside the tube and one-half of the tube wall resistance. The other half of the tube wall resistance and the outside-film resistance were included in the term hsp in Egs. (41) and (42). When the flow rate of the primary salt was changed, the film coefficient varied with flow, as shown in Fig. 4, while the tube wall conductance was maintained constant. Similarly, the film-coefficient part of the term hg,, varied as the 0.6 power of the secondary-salt flow rate, and the tube-wall-conductance part was constant. The calculation was performed on the analog computer by means of a diode function generator as hp=hey he (F)) (43) and hsp = hspo hsp (F,), (44) where Az, is the design-point primary-salt-to-tube-wall heat transfer coefficient and hspo is the design-point tube-wall-to-secondary-salt heat transfer coefficient. For example, from Table 1: 1 05 \™* hey = [——+ —= 45 fo (3500 3963) (42) =2428 Btuhr ! ft7 (°F) 7! . A diode function generator performed the following calculation: 1 0.5 }"’ 46) = —-—+_ iFL) [1316G(F1) 3963 18 where G(F, ) is the curve for the primary heat exchanger given in Fig. 4. The coefficients for Eqgs. (40) through (42) were calculated using Tables 1 and 2. For example, in Eq. (40), Fio 9.48 X 107 Ib X hr 47 M,; 025X 47,480 1b X hr X 3600 sec =2218sec™!, and heo Ay _ 2428 Btu X 11,050 ft*> X 1b-°F X hr 48) Mpq Cpf hr-ft*-°F X 11,870 1b X 0.324 Btu X 3600 sec =1.938sec”! . The unscaled heat transfer equations for the primary heat exchanger were dT F d!;7 = 2218F_1_ (TrO - Tp7) + 1.938 hf(Fl) (Tl‘l - Tp7) 3 (49) 10 dT, F _ dI;S = 2_213F_1 (T — Tpg) + 1.938 he (Fy)(T;) — Tpq) (50 10 dT F —2=2218 L (Tyg ~ Tpo) + 1938 hp (Fy) (Tyy — Tpo) 1) 10 dT,; F 10 dTy 7 7 =3.608 hf' (Fl) (Tp'] — Ttl) + 2495 hSp (F2) (TS3 — Tfl) , (53) dT,, 54 L2 =3.608 hy (F)) (Tyg — Trp )+ 2495 hyy (F3) (Tyy —Tpa), (4) dT, F dst‘l = 0'5737F_2 (T, — Ty)+ 04159 hsp(F?.) (T — Tgy) s (55) 20 dT, F d;2 = 0'5737F_2 (T51 — T52) + 04159 hey (Fy) (Tyy — Tg1) s (56) 20 dT, F df = 0.5737# (Tgy — Ty3) + 04159 hy, (F) (T — Ty3) (57) 20 dT F _djf‘. =0.5737 ?i (T3 — Tgq) + 04159 hyp, (F) (T4 ~ Ti3) - (58) 20 19 The scaling of these equations for the analog computer is given in Sect. 3.2.1. The analog computer program for these equations is shown in Fig. 6. 3.2.3 Steam Generator. The heat transfer model of the steam generator is shown in Fig. 3. For the secondary-salt stream: dT; Mg; Cps dSI =F, Cps (Tsz'—l sz) + hssA (Tn sz) (59) For the tube walls: dTy; ti Cpt—5 = tgg At Ty — Teg) + gy A (T — Ti) - (60) For the steam: dT,,; _ . My priT_F3 Cowi (Typi -1 = Tyi) + Ry Ag (T — Ty 5 (61) where T,,; = temperature of steam (water) lump i, M =M =M.7 = Mg = mass of secondary-salt lump, M,,; = mass of steam lump i, Cpwi = specific heat of steam lump 4, F4 = steam mass flow rate, h,, = secondary-salt-to-tube-wall transfer coefficient, h,,1 = tube-wall-to-steam heat transfer coefficient for the water inlet half of the steam generator, h,,3 = tube-wall-to-steam heat transfer coefficient for the steam outlet half of the steam generator, A = heat transfer area of the secondary-salt and steam lumps, A4, = heat transfer area of the tube lumps. The heat transfer coefficients in these equations were calculated in the same manner as were those in the primary heat exchanger (Sect. 3.2.2); namely, half of the tube wall resistance was included with the secondary-salt-to-tube-wall coefficient and half with the tube-wall-to-steam coefficient. The tube wall resistance was maintained constant under all flow and temperature conditions. The film coefficient on the secondary-salt (shell) side varied as the 0.6 power of the secondary-salt flow rate, and that on the tube (steam) side varied as the 0.8 power of the steam flow rate. The simulation of these coefficients was performed with a diode function generator as hss = hssO hss (F 2) s (62) hywt =hy1o M1 F3) (63) hw3 = hw30 hw3 (F3) ’ (64) “Teo & IC. ORNL BWG. 71-3373 05750 05199 Tl4 D Fig 7 o717 he T F2 Te3~Taa Ty-Ter 05883 3.8 “T X0 1C. 05053 “Tae 05199 Tez o7l “Toa he LS. T2 T hsp | Fa - Ts1~Taz Tia~Toe 0.5207 318 “Tiz2 4510 Fy -Ts DFG DFG hy(Fy) hsp{Fal } F, TIME | 7, F, (80.0Ov= DESIGN POINT} @ DELAY Tz Fig. 6. Analog Computer Program for the Heat Transfer Equations for the Primary Heat Exchanger. 0T 21 where the “zero” subscript refers to the design-point value. For example, Ao, was calculated as follows, using the information in Table 1: hooo L, 053 - 6 ss0 7 V1316 1224 (65) =855.9 Btu hr ! ft™2 (°F)™! . A diode function generator performed the following calculations: 1 05 ]! F,)= + _ hs (F3) [1316 (F,/F, )05 1224} (66) The other heat transfer coefficients were similarly calculated. The coefficients of Eqgs. (59) and (60) were calculated using Tables 1 and 2. For example, in Eq. (59): F, 711X 10°IbX hr ) MsS 47,492 1b X hr X 3600 sec =0.4159 sec”! . hoo A, 8559 BtuX 16,408 ft> X 1b-°F X hr ) M5 C, 47,4921b X hr-ft>-"F X 0.36 Btu X 3600 sec =0.2282sec”! . The masses and specific heats of the steam lumps in Eq. (61) were calculated as follows. The temperature and pressure distributions of the steam in the steam generator at design point were obtained from ref. 6. At the midplane in the steam generator the temperature and pressure were, respectively, 750°F and 3711 psi. From the steam tables, the enthalpy here was 1087 Btu/lb. At design point, the enthalpy of the water was raised from its inlet value of 770.8 Btu/lb to the steam outlet value of 1421 Btu/lb, an increase of 650.2 Btu/lb. The enthalpy rise from 770.8 to 1087 Btu/lb in the water inlet half of the steam generator was equally divided between lumps w1 and w2, and that from 1087 to 1421 Btu/lb equally divided between lumps w3 and w4. Thus, 48.6% of the heat was transferred in the water inlet half and 51.4% in the steam outlet half of the steam generator. The average temperature of the steam in the water inlet half was (from ref. 6) ~726°F. The average temperature in the steam outlet half was about 850°F. The design-point temperatures of lumps wl, w2, w3, and w4 were, therefore, 726, 750, 850, and 1000°F respectively. The total change in temperature of the secondary salt in the steam generator at design point was 300°F. Since the specific heat of this salt was constant, the 48.6% of the heat transferred in the steam outlet half of the steam generator represented a change in salt temperature of 145.9°F, or a temperature at the midplane of the exchanger of 995.9°F, which compared well with that given by ref. 6 (997°F). This was the design-point temperature for salt lump s6. The temperatures of lumps s5 and s7 were the average in the two halves of the exchanger, or 1073 and 923°F respectively. The overall temperature difference for heat transfer between salt and steam in the water inlet end of the exchanger (between lumps s7 and w1) was 197°F. In the steam end the difference (between lumpss5 and w3) was 223°F. The values of the average heat transfer coefficients for the steam generator were also 22 obtained from ref. 6. The average secondary-salt (shell side) film heat transfer coefficient and average tube wall conductance over the entire exchanger were 1316 and 1224 Btu hr ™" ft ™ (°F)™' respectively. The average film heat transfer coefficient on the steam (tube) side over the water inlet half of the steam generator was taken to be 6400 and that for the steam outlet half was 3590 Btu hr ™! ft ™2 (°F)™'. The average overall heat transfer coefficients in the water inlet half and the steam outlet half were about 577 and 539 Btu hr ™! ft™ (°F) 7! respectively. The heat transfer area in each half was, then, 2051 ft2. The area of each lump was 2051 X 16 A =" " = (69) = 16,400 ft* . The enthalpy and temperature of each steam lump were used to calculate the effective specific heat. For example: ] Cpw1 =7 (1087 — 770.8)/(726 — 700) (70) = 6.08 Btulb™ (°F)7! . The value of each C is given in Table 1. The value for the density of the steam for each of the four wl steam lumps was obfained from ref. 6 as the average density of the steam in each quarter of the steam generator. The steam generator was about 72 ft long. The number of 0.5-in.-OD tubes was calculated from the heat transfer area to be 434 tubes in each steam generator. From this information and the tube wall thickness and tube pitch given in Table 1, the mass and volume of the salt, metal, and steam were calculated for the tube region in each steam generator. These values are given in Table 2. The coefficients of Eq. (61) were then calculated. For example, the total flow area in the 434 tubes of each steam generator was 0.284 ft. The mass of steam lump w1 was M, =16 X 0.284 ft* X 74—2ft X 22.7 Ib/ft? (71) = 1860 1b. Similarly, M., =9341b, (72) M,,4 =555 Ib, (73) M4 =4121b. (74) The design-point flow rate of the steam was 1.181 X 107 Ib/hr. Then Fy, 1.181X 107 Ib X hr 79) M, 18601b X hr X 3600 sec =1.764 sec! . 23 The term h,, ;o was calculated in a manner similar to that for &, [see Eq. (65)]. Its value was 1771 Btu hr~! ft72 (°F)™'. Then hwiods _ 1771 BtuX 16,400 ft*> X 1b-"F X hr 76) M1 Cpyy 1860 1b X hr-ft*- °F X 6.08 Btu X 3600 sec =0.7134sec”!. The unscaled analog computer equations for the steam generator were: deS F2 —2 = 04159 2 (T, — Ty5)+ 02282 hy, (F))(Ty3 — Tys), (77) 20 dT F d;f» =0.4159 =2 (Tys — Tg) + 02282 1y (F,) (Tyy = Tis) (78) 20 dT, F 51 = 0.4159 22 (Ty — Tyg) + 02282 hyg () (Ty4 — Ti) (79) 20 dT, F __d;S =0.4159 }-;-,-?— (Tg7 ~ Tgg) + 02282 hy (F,) (Tyq — Tq7) (80) 20 dT,, —22 206950 g (F,) (Tys — Ty3)+ 1107 kg (F3)(T,,3 - Ts3), (81) dT 44 4= 0.6950 hes (F,) (Tyy — Tpa) + 1.570 hyyy (F3) (Tyyy — Tra) (82) dT F ____d‘;’l = 1.764_F_i (700 — T, )+ 0.7134 hy (F3) (Ty4 — T,,1), (83) 30 dT F T\;f% =3.513 F_3_ (Ty1 — Tyyp) ¥ 131200 F3) (Trg — Tp1) (84) 30 dT F 03 2591222 (T, = Ty3) + 7161 3 (3) (Ty3 = Tya) (85) 30 dT F 2 =7.969 =2 (Tyy3 — Typa) + 1448 hyy3 (F3) (Ty3 = Tyy3) (86) 30 Transport delays were used in the hot and cold legs of the secondary-salt loop to account for the transit time of the secondary salt between the primary heat exchanger and the steam generator. Thus, from Fig. 3, Toq =Tyt +71), (87) Tog =To(t+72). (88) Fig. 7. Analog Computer Program for the Heat Transfer Equations for the Steam Generator. QRNL DWG. -T, TIME | 7, G 7 50.00 18.10 2852 2.96| 0' —_— Tss hee <. Tia~Ty3 Nw3 F3 " T3 Tus TiaTas 46,80 1.384 ~Tia 08688 “Tus 05199 \7 = 3 DFG s ha(Fa) Fs 0.2852 1. F Ty hes 3 T Taz 2032 1.963 “Tia 0.8688 05199 F3 DFG DFG F LMY Ry (Fa) F Fyl Fp T Fig. 8 71-337% {8000y =DESIGN POINT) ¥C 25 The values used for 7, and 7, were 14.5 and 11.9 sec, respectively, at the design-point flow rate of the secondary salt and were inversely proportional to that flow rate at off-design conditions. No transport delay was included for the flow of primary salt between the reactor core and the primary heat exchanger. The scaling of the equations for the analog computer is given in Sect. 3.2.1. The analog computer program for the steam generator is shown in Fig. 7. 3.2.4 Control System. Steam Temperature Controller. The steam temperature was controlled by varying the secondary-salt flow rate and thus the heat input to the steam generator. An error in steam temperature caused the flow rate to change at a rate proportional to the error, that is, dF —7 = —a(T,,4 — 1000°F) (89) where F, is the secondary-salt flow rate, T, 4 is the outlet steam temperature, and « is the controller gain. The controller gain a used in the simulation was approximately 5.5%/min change in flow rate for each 1°F error in steam temperature for errors of 2°F or less. For errors greater than 2°F, the rate of change of flow rate was limited to 11%/min. No optimization was carried out to obtain these values, but they produced reasonably good system response. Reactor Qutlet Temperature Controller. The reactor outlet temperature set point was determined by the plant load demand (Fig. 2). The set-point equations were: nget =350P3omang T 950 (0.5 A reactor-power-level error was obtained by subtracting the set-point value from the measured value (from neutron flux), that is, €e=P, - P, . (93) 26 This power error € was the input signal to a control rod servo described by the second-order transfer function: T = ) s2 +2tws + w? els)’ (94) where G is the controller gain, w is the bandwidth, { is the damping factor, and O(s) is the Laplace transform of the servo output, dp,/dt. In this simulation the bandwidth was 5 Hz and the damping factor was 0.5. These values are typical of the kind and size of servo which may be used in this control-rod-drive service. The gain of the controller & was such that, for [e] = 1% of full power, the control reactivity addition or withdrawal rate was about 0.01%j/sec; that is, 7—0.01%/36(3 s (95) where p,. is the control reactivity. For power level errors greater than 1% of full power, the reactivity addition or withdrawal rate was limited to 0.01%fsec. Integration of Eq. (95) yields the value for the control reactivity in the kinetics equations of Sect. 3.2.1. The steam flow rate F; was obtained directly from the power demand, Pye,.04- A S-sec first-order lag was included to obtain some simulation of the response of the steam valve: dF, a 1 3= dr _;Pdemand (7) _;FB ? (96) where g is a proportionality constant and 7 is the time constant (5 sec). The scaling of the equations for the analog computer is given in Sect. 3.2.1. The analog computer program for the control system part of the siraisiation is shown in Fig. 8. ORNL DWG. 71-3377 { E ) Fig 6 @ Fig. 7 STEAM TEMPERATURE CONTROLLER F, (80.00v = DESIGN POINT } COMPARATOR 0.2188 6400 : -p, LOAD oot DEMAND =Q, ~ ) \ P —Ref =) 4 2 | 0.4750 \ O Tro,,, Tri=T LOAD 383 o o - T O 31,42 DEMAND, & n ! | —Ref. -0 l J 0 +001% /SEC. | LIMIT 31,42 1 ’ : 1 9 CONTROL ROD DRIVE SERVO LOAD -Ref, . 0.4000 REACTOR OUTLET TEMPERATURE CONTROLLER Fig. 8. Analog Computer Program for the Control System. LT 29 INTERNAL DISTRIBUTION 1. J. L. Anderson 35. 2. C. F. Baes 36. 3. H. F. Bauman 37. 4. S. E. Beall 38. 5. C. E. Bettis 39. 6. E. S. Bettis 40. 7. E. G. Bohlmann 41-60. 8. C.J. Borkowski 61. 9. G. E. Boyd 62. 10. R. B. Briggs 63. 11. O. W. Burke 64. 12. F. H. Clark 65—66. 13. C. W.Collins 67. 14—15. D. F. Cope 68. 16. F. L. Culler 69-70. 17. S.J. Ditto 71. 18. W.P. Eatherly 72. 19. J. R. Engel 73. 20. D. E. Ferguson 74. 21. L. M. Ferris 75. 22. W.K. Furlong 76. 23. W. R. Grimes 77. 24. A. G. Grindell 78. 25. P. N. Haubenreich 79. 26. R. E. Helms 80. 27. P. G. Herndon 81. 28. E. C. Hise 82. 29. P. R. Kasten 83. 30. T. W. Kerlin 84-85. 31. R. B. Korsmeyer 86. 32. M. 1. Lundin 87-89. 33. R. E. MacPherson 90. 34. H. E. McCoy EXTERNAL DISTRIBUTION ORNL-TM-3102 = cLain cNeese Wherter gZZZ oore icholson akes arker erry ickel C Robertson M. W. Rosenthal G. S. Sadowski Dunlap Scott W. H. Sides, Jr. M. J. Skinner O. L. Smith D. A. Sundberg R. Tallackson ggwor*r'u’fimrp Qo Z AP ICOHIT-C T "U'-U"U E. B. Trauger M. Watson M. Weinberg R. Weir M. E. Whatley J. C. White — A. S. Meyer L. V. Wilson Gale Young Central Research Library Document Reference Section Laboratory Records Department Laboratory Records, ORNL R. C. J. R. D. G. A. J. 91. C. B. Deering, Black and Veatch Engineers, 1500 Meadowlake Parkway, Kansas City, Missouri 64114 92. Ronald Feit, Instrumentation and Control Branch, Division of Reactor Development and Technology, U. S. Atomic Energy Commission, Washington, D.C. 20545 93. George McCright, Black and Veatch Engineers, 1500 Meadowlake Parkway, Kansas City, Missouri 64114 94-95. T. W. MclIntosh, Division of Reactor Development and Technology, U. S. Atomic Energy Commission, Washington, D.C. 20545 96. M. Shaw, Division of Reactor Development and Technology, U. S. Atomic Energy Commission, Washington D.C. 20545 97. 1. A. Swartout, Union Carbide Corporation, New York, N.Y. 98. Laboratory and University Division, AEC, ORO 99—-100. Division of Technical Information Extension 3 10017