THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF TIE COLLEGE OF ENGINEERING A STABILITY STUDY OF AN ATOMIC POWER PIANT Robert W. Albrecht October, 1958 IP-333

TABLE OF CONTENTS Page LIST OF FIGIURES.........................,.... i.. i I. INTRODUCTION.......................................... 1 A. Description of Problem.............................. 1 B. General Characteristics of Enrico Fermi Reactor Plant *...........................*.................. 1 II. APPROXIMATIONS AND LIMITATIONS......................... 2 III. ILLUSTRATION OF TYPICAL POINT TRANSFER FUNCTION............ 3 A. Symbols to be Used.......................... *. 4.. B. Derivation of Transfer Functions.................. 4 IV. THE REPRESENTATION OF THE SYSTEM........................... V. SOLVING FOR THE OPEN-LOOP FUNCTION......................... 8 VI. CONCLUSIONS OF THIS STUDY................................. 8 BIBLIOGRAPHY. I................. <.... a 16 ii

LIST OF FIGURES Figure Page 1 Block Diagram-,...*.~,..,~.**.........,.. 6 2 Servo Block-Diagrn of Enrico Fermi Plant.........,..., 10 3 Plant Thermal System Block Diagram-,..*.....,.,,.... 11 4 Open-Loop Signal Flow Chart.*,*.........*.,.*....... 12 5 Untouching Loops from Signal Flow Chart.....,.,. 13 6 Forward Gain Paths from Signal Flow Chart,,*........... 14 7 Nyquist Plot of Open-Loop System Function............. 15 iii

I. INTRODUCTION A. Description of Problem This paper presents a relatively simple, straight forward analysis of the stability of a nuclear power plant. The analysis is undertaken for only one operating condition; namely full power with a constant flow rate. This is the type of study which should be carried out early in the design or feasibility stage of power plant development because it yields some approximate information concerning stability, and is valuable for making early decisions as to necessary parameter adjustments before the design has proceeded beyond the point of no return. This type of study would also serve as a rough guide to control engineers who must decide upon the degree of complexity which will be required in the control system. The analysis undertaken here would also serve as a guide in making a more comprehensive computer study as the design proceeds. As an example, the method has been used to study the Enrico Fermi Power Plant. B. General Characteristics of Enrico Fermi Reactor Plant The Enrico Fermi Atomic Power Plant is a fast breeder power reactor which is designed for a power output of 300 megawatts, with 268 megawatts being released in the reactor core and 32 megawatts in the blanket. The coolant is liquid sqdium with a total flow rate of 13,200,000 pounds per hour, resulting in a coolant temperature rise of about 250~F across the core, Heat is removed from the reactor core and blanket by the primary coolant, transferred to the secondary coolant in the intermediate heat exchangers, and then transferred to water and steam in once-through steam -1

-2generators. The overall system is composed of three primary coolant loops having a common point in the reactor vessel and three independent secondary coolant loops having no common hydraulic point. The'primary system sodium flows by gravity from the free surface pool of the reactor upper chamber to the shell qide of the intermediate heat exchanger and then to the pump tank. The sodium is pumped from the pump tank back to the reactor, The secondary coolant system is the intermediate link that transfers the heat from the primary coolant system to the steam generator and consists of the tube side of the intermediate heat exchanger, the steam generator, and the centrifugal sump-type pump, The steam generators are counter-flow, shell and tube, oriethrough type of units with water and steam in the tubes and sodium on the shell side of the tubes. Electricity is then generated in a conventional steam-turbine generator. II, APPROXIMATIONS AND LIMITATIONS This stability analysis uses the concepts of linear servomechanism theory, and is therefore subject to the usual limitations of this technique. Some of the limitations are: l) Because of the linearity assumption, this analysis is only valid for small perturbations in 6K about a nominal operate ing power level, 2) loading between transfer functions Is not considered, 3) since the study does not use a computer, it is necessary to represent distributed components of the system by point transfer functions. Further approximations have been made in this study in order to facilitate the use of a mathematical model of the system which can be handled expediently by analytical methods.

-3Some of the more serious of these approximations are listed below: 1) Reactor core and blanket are represented by point functions. 2) Intermediate heat exchanger and secondary piping represented by point functions. 3) Frequency dependent portion of core upper structure and core lower structure is neglected. 4) Fuel element and coolant reactivity coefficients are both taken as a function of the average temperature of the coolant (for both core and blanket). 5) Core and blanket are represented as independent point functions. 6) Only one equivalent delayed neutron group is considered. -st 7) Coolant transport is represented by the transport term es and the mixing term. J + st. III. ILLUSTRATION OF TYPICAL POINT TRANSFER FUNCTION In order to illustrate the derivation of a typical point transfer function, the reactor core was chosen as a suitable example. In the derivation below, the heat transfer equations are written, Laplace transform notation is employed, some manipulations are carried out, and then the transfer function of the core is displayed as a block diagram.

-4A, Symbols to be used: Mf = mass of fuel Cf = specific heat of fuel Tf = average temperature of fuel element.Qf = kAeat generated in core A = heat transfer area U = heat conductivity Tcc = average temperature of core coolant Mcc = mass of core coolant Ccc = specific heat of core coolant W = weight flow rate through core Tcc. in = inlet temperature of core coolant Tm oLt= outlet temperature of core coolant B. Derivation of transfer function: (ref. 3) 1. Heat transfer equations. a) Rate of heat increase in core = heat content of core - heat transfer from core d Tf MfCf dt = Qf AU (Tf - Tcc) b) Rate of heat increase. of coolant = heat transfer from core + heat increase due to incoming sodium - heat decrease due to outgoing sodium. c) Definition of TCc Tcc in + Tcc out 2

-5rewriting the heat transfer equations; dTf Qf AU- AU - a) Tf + TC dt MfCf MfCf MfCf ~ dTcc AU AU 2W 2W - 1) T -MGcf - T^A cc +,, Tcc in.. Tc dt M c cc M ccc Mc cc Mcc T +~ T cc in +cc-out ) cc 2 now, defining the time constants; AU 1 AU 1 2W 2 MfCf - t1 McCcc t2; McC t, then, substituting these into the heat transfer equations and changing to Laplace transform notation; a) Tf + tl S Tf = Tcc + tl MfCf _2 + _2t2 b) Te (1 + 2 t) + t2 S T Tf + T in c cc.in + Tcc Qut 3 O2 cc now, eliminating T between equations a) and b), we get;,Qf 2t2 Tcc + tl Jf 2t T (1+ -T —) + t S T + t T cc (1 2 t s cc + t in now, substituting c) into the above and separating TCC in and out and arranging; flto to 2 [ t (tl \ \ t to 2 TCc out S + [ ( + l) + S + = Tc in S _ ( tl ) 1] ) St22 MfCf

-6so, solving for Tcc out; 1to 2 0 ti Qf \ S2+ - tS] S + T 2cc in + -~t2 Mf f ~- t]"tO S2 l~ t~~22 l~ I l~ t2 MQFf Tcc out =.... S + 2 t+ 1 + t S + 1 2 -2 factoring the polynomials appearing above, the equation becomes; Icc out i +1 1+S )( S ))TMfcf ] so, the core transfer function can be represented by the block diagram; t t1 ot1,i T1 S) Figure 1 Note that in the above representation, the heat generated in the core is artificially fed into the center of the core between the transfer functions which deal with the transport of coolant through the core. The or's and t's above are functions only of the parameters of the system and the operating condition being considered. Tcc in(s) is a function of the coolant flow throughout the power plant and the value of Qf is a function of the neutron kinetics which in turn is a function of the temperatures in the system and the reactivity coefficients.

7IV. THE REPRESENTATION OF THE SYSTEM The transfer functions of all parts of the system must be derived from basic heat transfer equations and assembled into a block diagram which represents the entire power plant system. Time limitations prevent the discussion of all of these transfer functions, so the system block diagram will merely be presented. Figure 2 shows the system block diagram and Figure 3 shows an enlargement of the block entitled "Plant Thermal System." The block diagram of Figure 2 shows 5K affecting both the core and blanket neutron kinetics. Thus, the outlet temperatures of the reactor (TBC out &Tcc out) are functions of both the inlet temperature and the heat generated in fuel and blanket elements. The reactor coolant is then fed into the plant thermal system and thus back into the reactor inlet. It can be seen that four reactivity coefficients are considered. The core element and coolant reactivity coefficient is taken as a function of the average temperature of the core coolant, the blanket element and coolant reactivity coefficient is taken as a function of the blanket coolant average temperature, and the upper and lower structure reactivity coefficients are taken.as a function of the coolant temperature of the upper and lower structure respectively. The block diagram-of Figure 3 shows the primary piping transfer functions, the intermediate heat exchanger, the secondary sodium piping and the.steam generator. For this study the transfer function of the steam generator was taken as a constant attenuation and a transport delay. In order to investigate stability by use of the N.yquist criterion, the open loop system must be considered. In order to do this, the loop is

-8broken at the reactivity feedback summer, and the open loop system is drawn as an open-loop signal flow diagram (ref. 1). This signal flow diagram is shown in Figure 4. V. SOLVING FOR THE OPEN-LOOP FUNCTION By investigating the forward gain paths and the closed loops of the signal flow diagram the open-loop system function can be solved for by use of a standard form (ref. 4). The closed loops are shown in Figure 5, and the forward gain paths are shown in Figure 6. Using standard techniques, (ref. 4) the open loop function is found by finding the function. Fl(l -E2) + F2 + F3(1 El) + F4(1 - E1) + F5 + F6 KG =........................ 1 E1 E2 where the F's and the E's are shown in Figures 5 and 6. The above open loop function is solved for by drawing a Bode plot for each forward gain path transfer function and each closed loop transfer function and then manipulating these plots to arrive at a Bode plot of the open-loop system function, KG. This plot is then transformed into a Nyquist plot of the open-loop function. The Nyquist plot for this power plant is shown in Figure 7. VI. CONCLUSIONS OF THIS STUDY The Nyquist plot of the open-loop function indicates stability for the operating condition considered. The plot also indicates that at a low value of gain (power level) and the same flow rate, an instability could occur.

I 9 The purpose of this analysis is just to get a feeling for system behavior and to serve as a guide for future computer studies; so in this case a study of this nature wod indicate to a designer that the possibility of instability should be investigated rmore thoroughly at low power levels.

t I1 r-B TC BLANKET INLET PIPING BLANKET a NEUTRON! KINETICS --- - + A _ ---- 1 1 ---- STBCOUT __ ^.+ ^J \- -^ -PLANT - i__ I=__ _^YV THERMAL L_.... --....___..... SYSTEM 8K CORE 8K BLANKET BLOCK DIAG. CORE c -NEUTRON CORE BLOCK DIAG. TccOUT KINETICS rLZ — a - E ——.. I - - C I - Z u + -j-J> J CORE IN ET PIPING w ~ C LOWER STRUCTURE REACTIVITY FDBK. PPER STRUCTURE-REACTIVITY FDBK. BLANKET ELEMENT a COOLANT REACTIVITY FDBK.......... Figure 2. Servo Block-Diagram of Enrico Fermi Plant

PRIMARY SODIUM PIPING SCONDARY SODIUM PIPING FROM I -ST I TO REACTOR (lS,) (-ST) (I+s) (+S TRBINE SYSTEM NTERMEDIATE GEN HEAT -TOR EXCHANGER H DOWNCOMER TO EACTOR I _ WATER FROM Z4 290 SEC. Tjl 5.87 SEC. s 6=.63 SEC. T2 1.83 SEC. T' 1.19 SEC. T 1.16 SEC. T 9r 9.49 SEC. T4 t 0.67 SEC Figure 3. Plant Thermal System Block Diagram.

BLANKET INLET IPING BLANKET THERMAL INLET I BLANKET ELEMENT BLANKET /?LANKET a COOLANT REACTIVTY NEUTRON / THERMAL. COEFFICIENT KINETICS/ EXIT PLANT THERMAL S K ---- (> SYSTEM~ s K, CORE NEUTRON KINETICS \ CORE CORE THERMAL UPPER T S —--- ^TRUCTUR OR~ZE ELEMEIa CORE \ ~ /:R/ a COOLANT REACTIVFTY THERMAL I UPPER STRUCTU COEFFICIENT INLET REACTIVITY ^ ^k COEFFICIE COR LOWER STRUCTURE CORE \.^^CORE EACTIVITY COEFFICIENT LOWE-R VNLET STRUCTURES- PIPING Figure 4. Open-Loop Signal Flow Chart.

-13BLANKET IN / LET~INLET PIPING BLANKET E THERMAL \E INLET BLANKET THERMAL EXIT \PLANT F THERMAL \ b urSrEM PLANT THERMAL. g SYSTEM _ CORE CORE / THERMAL UPPER / / C EXIT _,STRUCJUR / CORE J THERMAL / E INLET CORE CO - RE LOWER ^ INLET STRUCTUR PIPING Figure 5. Untouching Loops from Signal Flow Chart.

V ^ s BLANKET B_ -, S~ "~ ELEMENT a COOLANT BLANKET BLANKET B TREACTIVITY COEFFICIENT NEUTRON, THERMAL KINETICS EXIT F BLANKET ~~~'-"~"~. N LET PIPING BLANKET ELEMENT BLANKET COOLANT REACTIVITY NEUTRON BLANKET COEFFICIENT KINETICS THERMAL 8K EXIT \ \ PLANT THERMAL SK, SYSTEM CORE \ KINETICS THERMAL F3 / COOLANT _ a /,/l' *0^ REACTIVITY COEFFICIENT SK 8K, CORE F4 4 NEUTRON \ CORE CORE ORE UPPER KINETICS \ THERMAL UPPER STRUCTURE _ EXIT STRUCT U REACTIVITY COEFFICIENT 8K s8K COR' PLA NEUTRON CORE CORE ER L KINETICS THERMAL UPPER / NFJ CORE INLET / OE LOWER'5 COREINLETSTRUCTURE REACTIVITY COs~REt~ ^^^ COEFFICIENT sK PLANT THERMAL 8K1 ^ORE CORE CORE NEUTRON THERMAL UPPER RE ELEMENT r LOWER ORE INLET STRUCTURE - PIPING Figure 6. Forward Gain Paths from Signal Flow Chart.

Gs ~~~~~~~~ ~ ~~~~~~Fi gure 7 o iXyqui s tPlot of Open-Loop System Function:R 00 0C

BIBLIOGRAPHY 1. Truxel, J. G., Automatic Feedback.Control System Synthesis, McGraw-Hill Book Company, Inc., New York, 1955. 2. Murphy, G. J,, Basic Automatic Control Theory The Van Nostrand Company, Inc., New York, 1953 3. Schultz, M. A., Control Qf Nuclear Reactors and Power Plants, McGraw-Hill Book Company, Inc. New York, 1955. 4. Kazda, Iouis F., Verbal Communication. 5. Atomic Power Development Associates, Data on Parameters of Enrico Fermi Plant. -16

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