THEORETICAL DYNAMICS ANALYSIS OF THE MOLTEN-SALT REACTOR EXPERIMENT T. W. KERLIN, S. J. BALL, and R. C. STEFFY* Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 Received May 23, 1970 Revised September 14, 1970 The dynamic characteristics of the MSRE were calculated for opervation with U and U fuels. The analysis included calculation of the transient response for reactivity perturbations, frequency rvesponse for reactivity perturbations, stability, and sensitivity to parvameter variations. The calculations showed that the system dynamic be- havior is satisfactory for both fuel loadings. . INTRODUCTION The dynamic characteristics of the Molten-Salt Reactor Experiment (MSRE) were studied care- fully prior to the initial U fuel loading in 1965 and again prior to the ***U fuel loading in 1968, The first objective of these studies was to deter- mine the safety and operability of the system. The second objective was to establish methods of analysis which can be used with confidence in predicting the dynamic behavior of future, high- performance molten-salt reactors. To satisfy the second objective, it was necessary to include theoretical predictions of quantities amenable to experimental measurement. The frequency re- sponse results proved most useful for this pur- pose. Several different types of calculations were used in these studies. In general, they consisted of calculations of transient response, frequency response, stability, and parameter sensitivities. Four considerations led to the decision to use this many different types of analysis. These were: *Present address: Tennessee Valley Authority, Chat- tanooga, Tennessee. 118 REACTORS KEYWORDS: reactors, reac- tivity, vranium-233, uranium- 235, transients, disturbances, frequency, variations, stabil- ity, fuels, operation, differen- tial equations, MSRE, reactor kinetics ' 1. It is helpful to display system dynamic characteristics from different points of view as an aid in understanding the underlying physical causes for calculated behavior. 2. Computer costs for the different types of analysis were small compared to the expense of preparing the mathematical models. 3. The calculations for comparison with exper- iment (frequency response) were essential, but they did not furnish sufficient information about the system. 4. The experience with a number of methods provided insight on selecting methods which would be most useful in analysis of future molten-salt reactors. » The analysis of the system with 233U fuel was very similar to the analysis of the **°U-fueled system. The modeling for the **°U study was influenced slightly by results from dynamics experiments on_the *%U-fueled system and the analysis for the ***U-fueled system took advantage of some new methods developed after the com- pletion of the first study. This paper describes the mathematical models used, the computational methods used, and the results of the calculations. A companion paper’ gives results of dynamics experiments and com- parisons with theoretical predictions. Il. DESCRIPTION OF THE MSRE The MSRE is a graphite-moderated, circula- ting-fuel reactor with fluoride salts of ura.niumé lithium, beryllium, and zirconium as the fuel. The basic flow diagram is shown in Fig. 1. The NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 PUMP ‘ PUMP ' N\ i ( " | PRIMARY HEAT EXCHANGER RADIATOR 'CORE’ ‘ | l Fig. 1. MSRE basic flow diagram, molten, fuel-bearing salt enters the core matrix at the bottom and passes up through the core in channels machined out of 2-in. graphite blocks. The 8 MW of heat generated in the fuel and trans- ferred from the graph1te raises the fuel temper- ature from 1170°F at the inlet to 1210°F at the outlet. When the system operates at low power, the flow rate is the same as at 8 MW, and the temperature rise through the core decreases. The high-temperature fuel salt travels to the primary heat exchanger, where it transfers heat to a non- fueled secondary salt before reentering the core. The heated secondary salt travels to an air-cooled radiator before returning to the prlmary heat exchanger. Criticality was first achieved with *°U fuel (35 at.% 2°U) in June of 1965. After 9006 equiva- lent full power hours of operation, this uranium was removed and the reactor was refueled with 23317 (91.5 at.% 2**U) in October of 1968. Between October 1968, and shutdown in December 1969, an additional 4166 equivalent full power hours were achieved with ***U fuel. Dynamically, the two most important charac- teristics of the MSRE are that the core is hetero- geneous and that the fuel circulates. Since this combination of important characteristics is un- common, a detailed study of system dynamics and stability was required. The fuel circulation acts to reduce the effective delayed-neutron fraction, to reduce the rate of fuel temperature change during a power change, and to introduce delayed fuel-temperature and neutron-production effects. The heterogeneity introduces a delayed feedback effect due to graphite temperature changes. . SYSTEM MODELS A. Neutronics The point kinetics equations for circulating fuel reactors were used with appropriate temperature- Kerlin et al. THEORETICAL DYNAMICS ANALYSIS dependent reactivity feedback (see Sec. III.C). The equations are®: dt A A doc; Bi C be; eyt - ) exp(-Niy) a7 —Aén-xiéci-TC + T , (2) where 6n= deviation in neutron population from steady state = deviation in concentration of the 7’th pre- cursor group from steady state po = reactivity change in going from a circu- lating fuel condition to a stationary fuel condition Br = total delayed-neutron fraction B; = importance weighted delayed-neutron fraction for the 7’th precursor group A = neutron generation time 6p = change in reactivity A,; = radioactive decay constant for the i’th precursor group 7. = fuel residence time in the core 7. = fuel residence time in the external loop. L The term 0p is given by 6p = Op, + 24 a,0T; , where 6p, = reactivity change due to control-rod mo- tion a; = temperature coefflc1ent of reactivity for the #’th section (node) of the core 6T, = temperature change in the ¢’th section (node) of the core. In some of the calculations (determination of eigenvalues of the system matrix), it was neces- sary to eliminate the time delay from the pre- cursor equation. This was accomplished by eliminating the last two terms from Eq. (2) and defining an effective 3; as follows: ~ B .. =B X delayed neutrons emitted in core at steady state ieff — total delayed neutrons emitted in the system at steady state | ° NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 119 Meaacdl AR it el Had s A1 ot aenniss W v s s A ) = it sttt eI Uik A e B S gk e bl e o dos o ap it e A it A A AR, e o o, ettt TS (T 15— ST e Kerlin et al. THEORETICAL DYNAMICS ANALYSIS Then, the approximate precursor equation is ddC; _ B; g dt A This formulation assumes that the fraction of the precursors which decay in in-core regions is constant during a transient. Comparison of fre- quency response calculations using this approach and an approach which explicitly treats circulating precursor effects showed negligible differences in the frequency range of interest. Since the neutron population is proportional to fission power, the units on on were taken to be megawatts. on - \;0c; . (3) B. Power An attempt was made to include the effect of delayed gamma rays on the total power generation rate. If we assume that the delayed gamma rays are emitted by a single nuclide, then the appro- priate equation is dN Ez'yn"w, (4) where N = energy stored in gamma-ray emitters (in MW sec) v = fraction of power which is delayed n = neutron population (in units of MW) A =decay constant of gamma-ray emitter (sec™?). The total power is given by P=MN+(1-ym . (5) For these studies the value used for A and v were 0.0053 and 0.066/sec, respectively. C. Core Heat Transfer The core heat transfer was modeled using a multinode approach. The reactor was subdivided into sections and each section was modeled using the representation shown in Fig. 2. This model was preferred over a model with a single fuel lump coupled to a single graphite lump because of difficulties in defining appropriate average temperatures and outlet temperatures for a single fluid lump model.* If the outlet temperature of a single fluid lump model is assumed to be the same as the average temperature, then the steady-state outlet temperature is too low. If the average temperature is taken as a linear average of inlet temperature and outlet temperature, then it is possible for outlet temperature changes to have the wrong sign shortly after an inlet temperature 120 GRAPHITE 7 HEAT TRANSFER FUEL-TO-GRAPHITE ) DRIVING FORCE HEAT TRANSFER ! i i TF1, IN == FUEL_LUMP 1 ', FUEL LUMP 2 ——Tr2, out ] TFy = Tr2 F1 QPERFECTLY MIXED SUBREGIONS Fig. 2. Model of reactor core region; nuclear power produced in all three subregions. change. The model using two fluid lumps circum- vents these problems by providing an intermediate temperature to serve as an average temperature to use in the solid-to-liquid heat transfer cal- culations. Also, the average temperature in the second lump is a better representation of the outlet temperature than the average temperature of a single lump. Since ~7% of the heat is generated in the gra- phite by gamma ray and neutron interaction, the graphite lump equation has an internal heat source term. The equations are: AT _ Kp 1 . T (MC)/-l oP +?f—1 [6Tl1(1n) - 5Tf1] (hA),1 +(7—C)f1 6T, - 67}.] (6) doT;2 Kja 1 = (Mé)fz o + (67,1 - 67} (RA)y2 +'("A/75;,—; [6T¢ - 6T}.] (7) doT; K, (hA) ;1 + (RA);2] dt ~ MC); = (MC), X [GTG - 5Tf1] ’ (8) where T = residence time h = heat transfer coefficient for a lump A = heat transfer area for a lump M = mass C = specific heat K = fraction of total power 1 = subscript indicating first fuel lump f2 = subscript indicating second fuel lump G = subscript indicating graphite. NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 In most of the calculations, 9 sections of the type shown in Fig. 2 were used giving a total of 27 lumps. The arrangement is shown in Fig. 3. The fraction of the total power generated in each lump was obtained from steady-state calculations of the power distribution. The local temperature coefficients were obtained for each region by importance weighting the computed overall tem- perature coefficients for fuel and for graphite. (7Te)out J H B = A J T FLOW (rF)|N r-—l-.— - e Fig. 3. Schematic diagram of 9-region core model. D. Heat Exchanger and Radiator The models for the heat exchanger and the radiator were similar to the core heat transfer models. The arrangement for a heat exchanger section appears in Fig. 4. The equations for a heat exchanger section are: | NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 Kerlin et al. ¥THEORETICAL DYNAMICS ANALYSIS f Tvoin==] T _1!.. T2 —= /1 OUT | | ‘ HEAT TRANSFER f_r a— TUBE ’ HEAT TRANSFER I DRIVING FORCE | T T ‘ T T2y 1N 2 OUT —p— | 22 " — 21 ——— ‘f 2 WELL-STIRRED /‘ TANK (TYPICAL) (HOLDUP TIME = —’é;—) Fig. 4. Model of heat exchanger and radiator section. dé;;ll - ?1; (67 (in) - 6744 +((7\-/Ih‘-45))-’-1£1- (6T - 6T 1] (9) Qd_%’;-lfi = —771—1; [67T11 - 6T 2] ) ((]”“/IAC);;"-Z (6T - 0Tn] (10) L (h%\?cfl;ThAm) (672 - 074 +£h_A_(21M%)_hT£%.22 [57;21 - 65T;] (11) dfi;;m _ '%171 [6 T2 (in) - 67.] + (%))2211 (67 - 6Ta1) (12) dfi;;zz - 712—2 [6T2; - 6T2) ((Mh“é))z:z (67 - 6T=a] . (13) In some of the calculations, it was assumed that the heat capacity of the air in the radiator was negligible. (Terms T2, and T2: are used for the air side of the radiator.) Ignoring the heat storage in the air leads to the following heat balance: (WC)a1[T22 - T21 (in)] = (RA21 +hA2) Ty - T21) | (14) where W is the mass flow rate of the air. If we assume T21 = [Ty (in) + T22)/2, Eq. (14) be- comes | ' (T-A%S%Z—Z)[ZTzl - 2iT'21 (in)] = [Ty - T2i) . | (15) 121 ol e - g < el Y Y " PR . A G Kerlin et al. @THEORETICAL DYNAMICS ANALYSIS Now, we write the equation in terms of incre- mental quantities and assuming T (in) is constant to obtain: 5T (WC)21 hAzl + hAzz This is then wused for 672 in Eq. (11). The schematic representation of this type of radiator model appears in Fig. 5. 0T2; = (16) 1 +2 11 12 HEAT REMOVAL BY AIR STREAM (ASSUMED PROPORTIONAL TO CHANGES IN T1) Fig. 5. Model of radiator for assumed negligible air heat capacity, E. Piping Several models were used to represent salt transport in the piping in different stages of the studies. The simplest model was a pure time delay. From some calculations (eigenvalues of the systems matrix) it was necessary to eliminate the delay terms. They were represented by Padé approximations® in those calculations. In some of the more detailed calculations, the heat transfer to the pipe walls was included. Since experimental results’ obtained after the ?*°U study indicated significant mixing in headers and piping in the fuel stream, some calculations for the **°U fueled system used a model of a mixing chamber at the core outlet. This model consisted of the following equation (a first-order lag): 46T 1 S == (6T, - oT) . (17) F. Values of Importa.nt Parameters Some of the important parameters computed for the ***U and ***U loadings appear in Table I. G. Overall System Model The models for the subsystems were combined to give an overall system model. Several different overall system models were used in different stages of the study. The model shown in Fig. 6 was used in the study of the **U-fueled system. This will be called the reference model. This model resulted in a 44’th-order system matrix with 4 time delays for heat convection and 6 time delays for precursor circulation. Major modifi- cations of this model which were used in some TABLE 1 Parameters Used in MSRE Dynamics Studies Parameter 2%y 233y Fuel reactivity coefficient (°F ) -4.84 x 107° -6.13 X 107> Graphite reactivity coefficient (°F_1) -3.70 x 107> -3.23 x 107° Neutron generation time (sec) 2.4 x10~* 4.0 x107* Total effective delayed-neutron fraction (fuel stationary) 0.00666 0.0029 Total effective delayed-neutron fraction (fuel circulating) 0.00362 0.0019 Total fuel heat capacity (in core) (MW sec/°F) 4.19 Heat transfer coefficient from fuel to graphite (MW/ °F) 0.02 Fraction of power generated in the fuel 0.934 Delayed power fraction 0.0564 Core transit time (sec) 8.46 Graphite heat capacity (MW sec/°F) 3.58 Fuel transit time in external primary circuit 16.73 Total secondary loop transit time (sec) 21.48 122 NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 3.7 7-sec DELAY 2-sec MIXING POT Kerlin et al. THEORETICAL DYNAMICS ANALYSIS 4 71-sec DELAY FUEL-SALT HEAT ' \ 8.67-sec DELAY i AIR 1 4 7 9 STREAM FUEL COOLANT COOLANT A LUMP LUMP LUMP 3 6 MSRE GORE ! (9 REGIONS) 8 -| METAL [} --- . - - - AMETALE --1 o 5 \ Y FUEL COOLANT COOLANT LUMP LUMP | LUMP \ x ; RADIATOR EXCHANGER 8.24-sec DELAY Fig. 6. Schematic representation of the MSRE reference model. aspects of the study are listed below: 1. The mixin§ pot was not included in the early studies for the ***U-fueled systems. It was added after experimental results' indicated significant mixing of the fuel salt. ' 2. For computing the eigenvalues of the sys- tem matrix, each pure time delay for fluid trans- port was replaced by a Padé approximation. Effective delayed-neutron fractions were deter- mined and Eq. (3) was used instead of Eq. (2). 3. In the models used in the MSFR code (see Sec. IV), the heat exchanger and radiator models were expanded. Instead of a single 5-node repre- sentation for the heat exchanger, 10 sections (each with 5 nodes) were used. Instead of a single 3- node representation for the radiator, 10 sections (each with 3 nodes) were used as with the heat exchanger. Calculations showed that results obtained with the simpler heat exchanger and radiator models gave good agreement with results obtained using the larger models for these components. IV. METHODS OF ANALYSIS A. Transient Response The transient response of the reactor system was calculated for selected input disturbances NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 (usually reactivity steps). The computer code MATEXP® (a FORTRAN IV program for the IBM- 7090 or IBM-360) was used for these calculations. MATEXP uses the matrix exponential technique to solve the general matrix differential equation. For the linear case, the general matrix differ- ential equatien has the form: = = Ax + f2) (18) where x = the solution vector t = time | A = system matrix (a constant square matrix with real coefficients) At) = forcing function vector. The solution of Eq. (18) is x = exp(Af)x (0) + f; explA(t - 7)] Alr)dT . (19) MATEXP solves this equation using a power series for the evaluation of exp(At): exp(Af) =T + (Af) + 3(A8)* +. .. . (20) In MATEXP, f(7) must be a step or representable by a staircase approximation. For the nonlinear case, the general matrix differential equation is g;; Ax + AA(X)x +F() (21) 123 e e e - Kerlin et al. @THEORETICAL DYNAMICS ANALYSIS where AA(x) = a matrix whose elements are changes in the coefficients resulting from non- linear effects. The procedure used in MATEXP .to solve this equation is to use an approximate forcing function rather than to modify the A matrix continuously. The procedure for proceeding from time-step j to time-step j + 1 is x(tj 1) = exp(at)e(t)) + [ 9 expla(t - 7)] 7 X {f(7) + AALx(7;)]x(7;)}dT . (22) This result is analytically integrable and amen- able to computer analysis. This method has proved to be fast and reliable. MATEXP uses a similar method for systems with time delays. B. Frequency Response The frequency response for the ***U-fueled sys- tem was calculated with a special-purpose digital computer program, MSFR® (a FORTRAN IV pro- gram for the IBM-7090 or IBM-360), and also with a general purpose program, SFR-III' (a FORTRAN IV program for the IBM-7090 or IBM- 360). The SFR-III program was used for the analysis of the ***U-fueled system. The basic approach in the MSFR program is to program the transfer functions for all the sub- systems and to connect them by the methods of block diagram algebra to obtain the overall sys- tem transfer function. This method proved to be efficient (low computing time) and flexible (sub- system models were changed readily by substi- tuting different subroutines for the appropriate part of the model). ‘ The SFR-III program uses a state-variable approach to obtain the frequency response. The system model is expressed in matrix form: =Ax +d +gf(t) , (23) S& where x = vector of system state variables A = coefficient matrix (a constant, square ma- trix) d = vector of time delay terms (see below) g = vector of constant coefficient of the forcing function f = the scalar fdrcing function. 124 The time delay vector allows any equation in the set (row in the matrix differential equation) to have terms containing the value of any state vari- able evaluated at some prior time. Clearly, this is needed to handle transport delays. The form of this type of term is 7% &= Tii) where i = constant coefficient in row ¢ for the term containing x; evaluated at a prior time T = time delay for the effect of x; in row . The Laplace transform of the time delay term is L{ri]. 7 (¢ - 'ri].-)} = V% (s) exp(-7; s) . (24) Thus, the Laplace transform of d in Eq. (23) is L{d} = L(s, 7)%(s) , (25) where L(s,7) =a matrix whose elements are [;; = 7, exp(=T;; s) x(s) = Laplace transform of x(2). Equation (23) may be Laplace transformed to give sx(s) = Ax(s) + Lx(s) + gf(s) . (26) Initial conditions are zero because the state vari- ables represent deviations from equilibrium. The transfer function is obtained from Eq. (26). G(s) = x(s)=[sI-A-L]'Z. (27) 1 f(s) The frequency response is obtained by evaluating this equation for s =jw at selected angular fre- quencies w. The SFR-III program also furnishes sensitivity to parameter changes. For instance, the fractional change in G(jw) due to a fractional change in coefficient, a;;, is % 3G(jw) This type of sensitivity coefficient is calculated in SFR-III. The algorithm is obtained simply by differentiating Eq. (27). GUw) _ [sI -A - L] ' G(jw) . (28) aa,-,- It is noteworthy that the factors on the right side of the sensitivity equation are evaluated in the normal frequency response calculation. Thus, the sensitivities are obtained only at the expense of a matrix multiplication of known quantities. NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 C. Stability Analysis Three different methods were used for ana- lyzing the linear stability of the system. These were analysis by the Nyquist method, calculation of the eigenvalues of the system matrix, and analysis by the modified Mikhailov method. 1. The Nyquist Method—The stability analysis by the Nyquist method followed standard practice.’ The MSFR code (see Sec. IV) was used to com- pute open loop frequency responses. The Nyquist criterion requirement for stability is that the net number of encirclements of the (-1,j0) point for - o < @ < © must be equal to the number of right half-plane poles of the open loop transfer function. Thus, it is necessary to know the stability char- acteristics of the open loop system prior to analysis of the closed loop system. In the MSRE analysis, it was assumed that the open loop trans- fer function had no right half-plane poles. This was verified in other analyses. 2. Eigenvalue Calculation—The eigenvalues of the A matrix (numerically identical with the poles of the closed loop transfer function) must have negative real parts if the system is stable. Eigen- values were computed using a computer code based on the QR transform method.’ 3. Modified Mikhailov Method—A new method was developed'® for stability analysis of large state-variable system models (pure time delays in the model are allowed). The criterion is that a plot of the function M(jw) for - ©* < w < must have no net encirclements of the origin for M(jw) given by det (jwI-A - L) M(jw) = (29) n . I (jw + la;; ) 1=t D. Stability Range Analysis After the analysis of the **°U-fueled MSRE using design parameters indicated that the system was stable, a systematic study of the influence of parameter uncertainties was made. The maximum expected range on the value on each important system parameter was estimated. Then, an auto- matic optimization procedure'’ was used to find the combination of parameters in this region of parameter space which gave the least stable sys- tem. A simplified system model was used for this study (only one graphite node and two fuel nodes to represent the core). These calculations gave combinations of system parameters which result in the least stable configurations. The parameters NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 Kerlin et al. THEORETICAL DYNAMICS ANALYSIS corresponding to this least stable configuration were then used in a stability analysis using the more detailed model. This analysis indicated that the system is stable for any combination of sys- tem parameters within the predicted uncertainty range. V. RESULTS The methods described in Sec. IV were used with the models described in Sec. III to analyze the dynamics of the MSRE. Results for the **U- fueled system are given first, followed by results for the ***U-fueled system. Also, a comparison of system performance with 235 fuel and with **°U fuel may be obtained by comparison of results from similar calculations for the two systems. A. 25U-Fueled System3 1. Transient Response—The MATEXP code was used with the state variable model to compute the response of the uncontrolled MSRE to a step input in reactivity. The results for a step input of 0.01% dp for low power operation and for high power operation appear in Fig. 7. At low power, the response is a low frequency, lightly damped return to equilibrium. At high power the response is a higher frequency, more strongly damped return to equilibrium. 2. Frequency Response—The results of a set of frequency response calculations using the MSFR code (see Sec. IVB) appear in Fig. 8. The results indicate fairly sharp peaks in the amplitude at low frequency for low power operation, and broader peaks at higher frequencies for higher power operation. This behavior is consistent with the transient response results. In general, the fre- quency response plots are rather featureless and indicate no dynamics problems for the system. The results of the frequency response analysis and the transient response analysis indicate that the natural period of oscillation of the perturbed reactor is a strong function of the operating power level. This natural period may be obtained directly from the transient response results or from the location of the dominant amplitude peak in the frequency response results. The dependence of natural period of oscillation on power level appears in Fig. 9. 2. Stability—TFor the ***U-fueled system, the orig- inal stability analysis was based on a Nyquist analysis and an eigenvalue calculation. The Ny- quist plot appears in Fig. 10 for low power oper- ation and for high power operation. The locus is 125 ey Kerlin et al. THEORETICAL DYNAMICS ANALYSIS 0.8 235, —_——— 233 0.6 0.4 10 MW 0.2 v I MW CHANGE IN POWER LEVEL (MW) / I7 -0.2 0 — ———'-'-————‘—:::—-_fi—+-d e O 100 200 300 400 - TIME (sec) Fig. 7. MSRE transient response to a +0.019 6p step reactivity input when operating at 1 and 10 MW, PHASE (deg) w o o ! w o 90 1073 Fig, 8. 126 ZERO POWER 1072 107! 'FREQUENCY (rad/sec) Nuclear power to reactivity frequency response for 23°U-fueled MSRE at several power levels, 100 50 20 PERIOD OF OSCILLATION (min) 005 01 02 0.5 1 2 5 0 20 No, NUCLEAR POWER (MW) Fig. 9. Period of oscillation vs power level for 235U- fueled MSRE. complicated near the origin, but it is clear that no encirclements of (-1,j0) exist. The main eigen- values and their power dependence appear in Fig. 11. These were computed using the system matrix containing Padé approximations for the transport delays and precursor equations with effective delayed-neutron fractions rather than explicit treatment of precursor circulation effects. As before, this analysis indicates a low frequency, lightly damped behavior at low power and a higher NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 IMAGINARY PART OF POLE (sec™") Kerlin et al. THEORETICAL DYNAMICS ANALYSIS IMAGINARY 16 14 12 40 -08 -06 -04 -02 |0 02 04 06 REAL ——t ! ' : : : : : : —t— "‘—0.2 UNIT CIRCLE wp-00145\ Ng= | MW wp = 0.078 P \/ 14 No = 10 MW Lq 6 IMAGINARY ~10 000 b ReaL - o.oooz\/ 0.0008 - w= 0.0003—mg Ng = 1 MW /,/ 1-10 000 Ng = 10 MW Fig. 10. Nyquist diagrams for complete model at 1 and 10 MW (?°°U fuel). 0.07 0.06 0.05 0.04 0.03 0.02 0.01 -0.07 -0.06 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 0. 0.1 04 DEPENDENCE ¥ } 0. ® { 10 MW ‘'~ ol O 5 NEGLIGIBLE POWER %005 | -004 -003 -002 -001 Fig. 11. Major poles for the 235-fueled MSRE. NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 127 Kerlin et al. THEORETICAL DYNAMICS ANALYSIS frequency, more strongly damped behavior at higher power. Also, stable system operation is indicated at all power levels with relative sta- bility increasing as the power level increases. B. 233U-Fueled System 1. Transient Response—The MATEXP code was used with the state variable model to compute the response of the uncontrolled, ***U-fueled MSRE" to a step input in reactivity. The results for a step input of 0.01% 6p are shown in Fig. 7. Other transients which were calculated for the U-fueled system appear in Fig. 12. At the higher power levels the power rises sharply after a step increase in reactivity, but the temperature effects in the core promptly counterbalance the reactivity input, and the power decreases toward its initial level. However, before returning to its initial level, the power levels out on a transient plateau. It stays at this level until ~17 sec after the reactivity perturbation; then it again begins to decrease. The power plateau is observed because a quasi-steady state exists in the core 233 A POWER (MW) 0 100 200 300 400 500 TIME (sec) 700 800 900 1000 Fig, 12, Calculated power response of the 235 y-fueled MSRE to a 0.02% 6k/ k step reactivity insertion at various power levels. 128 2.5 2.0 \\ — NON LINE AR - — |_INEAR 215 s \\ z-e MW o W 5 N\ a5 Ry qQ \ \\ 0.5 N = 0 0 410 20 30 40 50 TIME (sec) Fig. 13. Power response of the 233U-fueled MSRE to a 0.04% step reactivity insertion as calculated with nonlinear and linearized Kkinetics equa- tions. &n No w o Y 2> 2 PHASE (deg) o N N\ N PPN _60 OW v ZERO PC T 1 W o ) | X X < - m ~ & T l -90 ] 103 2 5 1072 2 5 107 2 5 100 2 5 10 - FREQUENCY (rad/sec) Fig. 14. Nuclear power to reactivity frequency response 233U for -fueled MSRE at several power levels. NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 region. The inlet temperature is the same as it was before the perturbation, and the core nuclear average temperature has increased enough to compensate for the reactivity change. After ~17 sec (the transit time of the fuel in the external loop) the return of higher-temperature salt in- creases the inlet temperature and introduces negative reactivity through the negative temper- ature coefficient. After sufficient time the reactor returns to the initial power level, at which time the net increase in average temperature compen- sates for the step reactivity input. This behavior was not observed in the ***U-fueled system for the power levels considered. A comparison of the step response of the sys- tem for the complete, nonlinear model and for a linearized model was made. The results appear in Fig. 13. It is observed that the nonlinear effects are more important at low power where larger fractional power changes can occur before the inherent temperature feedback can cancel the inserted reactivity. 2. Frequency Response—The results of a set of frequency response calculations using SFR-III appear in Fig. 14. The results are similar to the results for the **°U-fueled system. In general, the dominant amplitude peaks for the *3yu-tueled system are lower, broader, and at slightly higher frequencies than for the **U-fueled system. This is mainly due to the greater negative temperature feedback in the ***U-fueled system resulting from the greater magnitude of the negative fuel temper- ature coefficient of reactivity which overrides the destabilizing effect of the lower delayed-neutron fraction. As with the **°U-fueled system, the fre- quency response results indicate a well-behaved system. The dip in the frequency-response amplitude at 0.24 rad/sec is due to the fuel recirculation effect. The total loop time is 16.73 + 8.46 = 25.19 sec (see Table I). The frequency associated with this is 6.28/25.19 = 0.24 rad/sec. Experiments with the **°U fueled system® indicated that the dip was much smaller than predicted by a model which used pure time delays for fuel transport. Consequently, a first-order lag representation of a mixing pot was added to the model. Calculations were made to determine the effect of assigning different fractions of the external loop time to the mixing pot holdup time. These results appear in Fig. 15 for operation at 8 MW. The sensitivity of the frequency response to parameter changes was also calculated using SFR-III. Some results are shown in Fig. 16 for operation at 8 MW. These clearly show the fre- quency range over which the parameters have an important effect on system dynamics. For NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 Kerlin et al. ¥THEORETICAL DYNAMICS ANALYSIS Q w n w ------- 2-sec MIXING Y Z 1000 ————— SN— 500 MAGNITUDE RATIO 102 2 5 107 2 5 100 2 5 10 FREQUENCY (rad/sec) 5 /) N S PHASE ANGLE (deg) o (f‘ B o, 7 L N N 3 8 N 10-2 2 5 107" 2 5 100 2 5 10 FREQUENCY (rad/sec) -60 Fig. 15. Frequency-response plot for the 2?*U-fueled MSRE operating at 8 MW for various amounts of mixing in the circulating loop. example, the large changes in the mixing pot holdup time and the heat exchanger characteris- tics at ~0.24 rad/sec suggest that fuel salt recirculation effects are important factors in determining the amplitude dip at 0.24 rad/sec. 3. Stability—For the **U-fueled system, the sta- bility analysis was based on an eigenvalue cal- culation and a modified Mikhailov analysis. The eigenvalues appear in Fig. 17. All of the eigen- values have negative real parts and the real part of the dominant eigenvalue becomes more negative as the operating power level increases. The results of the modified Mikhailov analysis appear in Fig. 18. (These curves only show the range 0 0 4-4—>—>-A . »o “7 (1'et / E < — 5 / NEGLIGIBLE g e POWER = o’ DEPENDENCE -0.4 -0.2 /‘,’ ° &AV-§ " -0.3 044 OA2 040 0.08 006 004 002 0 REAL AXIS (sec”!) Fig. 17. Major poles for the ?**U-fueled MSRE. fraction for the 2**U-fueled system caused no dynamics or stability problems. This is because the stabilizing effect of a more negative fuel temperature coefficient of reactivity in the **U- fueled case compensates for the effect of a smaller delayed-neutron fraction. Numerous analytical methods were used in the studies. Experience showed that the effort re- quired to implement the different methods was justified by the increased understanding of system characteristics made possible by interpretation of the various results. It is felt that, in general, a NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 IMAGINARY AXIS Kerlin et al. THEORETICAL DYNAMICS ANALYSIS 1.0 - | ] INCREASING | FREQUENCY | ZERO POWER 0.8 ] m | < 0.6 | - I < x | 4 204 - 5 , T . ! 2 | < ' = | = 0.2 ; e . 0 l [ L ' o | . w=0 | " “ ] | -0.2 1 1 . -06 -04 -0.2 0 02 04 06 08 10 12 REAL AXIS Fig. 18a. Modified Mikhailov plot for MSRE operating with 233U fuel at zero power, '.2 l I POWER =1 kW 1.0 — 08 /| \\ ACREASING ? 06 FREQUENCY IMAGINARY AXIS U NN w=0 REAL AXIS Fig. 18b. Modified Mikhailov plot for MSRE operating with 233U fuel at 1 kW. {.2 I POWER = {00 kw 1.0 =] - 'Péé‘&?é&%f/ v _\\\\ | - (/ /)573 ol | * \ ozl N \ 7 o _ \\ / ’ w|=0 \ / w=® _0.2 (a) -0.6 -04 -0.2 0 02 04 06 08 10 1.2 REAL AXIS Fig. 18c. Modified Mikhailov plot for MSRE operating with 233U fuel at 100 kW, 131 wo ol e i ot 3 oW e (. gling v ( T o il il o - B b i Ot g st b kARG Sl 23, -y P - s 4 Kerlin et al. THEORETICAL DYNAMICS ANALYSIS 1.5 ! POWER = { MW | INCREASING . A// O \ & 8D | IMAGINARY AXIS /. 0 TN w=0 w =0 \u_._// -0.5 -0.50 -0.25 0] 0.25 0.50 0.75 1.0 1.25 REAL AXIS Fig. 18d. Modified Mikhailov plot for MSRE operating with 233U fuel at 1 MW, complete analysis of the inherent dynamic char- acteristics of a new system should include tran- sient response calculations, frequency response calculations, stability analysis, and sensitivity analysis. ACKNOWLEDGMENT This research was sponsored by the U.S. Atomic Energy Commission under contract with the Union Carbide Corporation. REFERENCES 1. T. W. KERLIN, S. J. BALL, R. C. STEFFY, and M. R. BUCKNER, ‘‘Experiences with Dynamic Testing Methods at the Molten-Salt Reactor Experiment,’’ Nucl. Technol., 10,103 (1971). | 2. P. N. HAUBENREICH and J. R. ENGEL, ‘‘Exper- jence with the Molten-Salt Reactor Experiment,”’ Nucl. Appl. Technol., 8, 118 (1970). 3. S. J. BALL and T. W, KERLIN, ‘‘Stability Analysis of the Molten-Salt Reactor Experiment,’”’ USAEC Re- port ORNL-TM-1070, Oak Ridge National Laboratory (December 1965). 4. S. J. BALL, ‘“Approximate Models for Distributed- Parameter Heat-Transfer System,’”’ ISA Trans., 3, 38 (January 1964). 5. G. S. STUBBS and C, H. SINGLE, ‘‘Transport Delay Simulation Circuits,”’ USAEC Report WAPD-T-38 and Supplement, Westinghouse Atomic Power Division (1954). 6. S. J. BALL and R. K. ADAMS, “MATEXP, A Gen- eral Purpose Digital Computer Program for Solving Ordinary Differential Equations by the Matrix Exponen- tial Method,”’ USAEC Report ORNL-TM-1933, Oak Ridge National Laboratory (August 1967). 132 ' 5 v ’\ % NCREASING FREQUENCY _ POWER =8 MW 1.0 0.5 © % [\ vme E 0 UJ:O\ U < : / O < = -0.5 \ o \\/ -2 —1 0 { 2 3 REAL AXIS Fig. 18e. Modified Mikhailov plot for MSRE operating with 2%%U fuel at 8 MW. 7. T. W. KERLIN and J. L. LUCIUS, ‘The SFR-3 Code—A Fortran Program for Calculating the Fre- quency Response of a Multivariable System and Its Sensitivity to Parameter Changes,’”” USAEC Report ORNL-TM-1575, Oak Ridge National Laboratory (June 1966). 8. F. H. RAVEN, Automatlic Control Engineering, McGraw-Hill, New York (1961). 9. F. P. IMAD and J. E. Van NESS, ‘‘Eigenvalues by the QR Transform,’”’ Share-1578, Share Distribution Agency, IBM Program Distribution, White Plains, New York (October 1963). 10. W. C. WRIGHT and T. W. KERLIN, ‘‘An Efficient, Computer-Oriented Method for Stability Analysis of - Large Multivariable Systems,”’” Trans. ASME, J. Basic Eng., Series D, 92, 2, 279 (1970). 11. T. W. KERLIN, ‘‘Stability Extrema in Nuclear Power Systems with Design Uncertainties,”” Nucl. Sci. Eng., 27, 120 (1967). 12. R. C. STEFFY, Jr. and P, J. WOOD, ‘‘Theoretical Dynamic Analysis of the MSRE with 2**U Fuel,”” USAEC Report ORNL-TM-2571, Oak Ridge National Laboratory (July 1969). NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971