Franz and Simcic: Nuclear Reactor Start-Up Simulation

20 volts and the low energy peak at 0 volts. Admittedly,this is a rough calculation, but it is interesting to notethat three such peaks are actually apparent in Fig. 9(silicon diode regulator) at approximately these voltages,whereas in all other tests only two peaks at most wereapparent only with a great deal of uncertainty and atsomewhat lower voltages. Thus, it appears that bothregulating schemes are highly advantageous from aphotometric point of view, but only the silicon dioderegulator is substantially effective in a scintillationcounter. Just why the latter is true the authors are notprepared to say without further investigation, but, they

do feel sufficiently secure in the results to strongly recom-mend either type of regulator for photometric work andthe silicon diode regulator only for scintillation counterwork.

Also, the very fact that the peaks occur at lowervoltages in the unregulated and neon regulated cases thanthey do in the silicon diode regulated case, just as thoughthe voltage applied to the tube had been reduced, raisesanother interesting observation as follows. The silicondiode regulator appears to respond more rapidly to tube-current fluctuations than do other circuits; an attemnpt toverify this fact has been made with no success to date.

Nuclear Reactor Start-Up Simulation*J. P. FRANZt AND N. F. SIMCICt

SummaTy-This paper describes the problem of simulatingnuclear reactor start-up with an electronic analog computer. Thereactor kinetic equations are discussed and the range that theirsolution must cover is defined. Source range simulation and alternatemethods of simulating the intermediate range are presented.The use of reactor period signal in the intermediate range is de-scribed. In the power range, which includes a representation ofthe heating and temperature effects, the inherent advantages of apressurized water reactor are outlined. The portion of the start-upsimulation which applies to nuclear reactors in general is sum-marized. The areas of modification necessary for the simulation ofother than pressurized water reactors is outlined.

EACTOR START-UP simulation is used to studytime variations in neutron flux from source tooperating power level. From source level to the

operating level, a large amount of negative reactivitymust usually be removed by withdrawing control rods.During this rod withdrawal, variables of interest are fluxlevel, reactor period, and reactivity. Fig. 1 shows thetype of reactor under consideration, although the discus-sion could be extended to other types of reactors or plants.For proper evaluation of reactor start-up, it is necessaryto investigate the entire range of operation which maybe as great as 10 decades.The start-up transient is difficult to simulate because

analog computer range with reasonable accuracy islimited to two or, at most, three decades.The kinetic behavior of a reactor is described by the

conventional set of differential equations: (defined inFig. 2)

* Manuscript received by the PGNS, November 8, 1956. Pre-sented at the Third Annual National Meeting of the PGNS, Pitts-burgh, Pa. This paper is based on work performed at Bettis Plantunder U. S. Atomic Energy Commission Contract AT-11-1-GEN-14.

t Westinghouse Electric Corp., Pittsburgh, Pa.

FROMCON D E N SER

Fig. 1-Power plant which was simulated.

dn _KK-fldt 18 n + <AiCi + Sdt -1*

dCioi~~~~idt 1* -xc

Where: n-Neutron density3K Excess reactivityI* --Effective lifetime of neutronfl-Production factor for the ith group of

delayed neutron emitters.Ci-Concentration of the ith group of de-

layed neutron emitters.Xi-Decay constant for the ith group of

delayed neutron emitters.S-Source strength

Fig. 2-Reactor kinetic equations.

111957

IRE TRANSACTIONS ON NUCLEAR SCIENCE

1000

60

HLINEARSIMULATION

LINEAR0OR 100--

.OG SIMULATION

z0-

100

0vNi~~~~~~Fig. 3-Three ranges of start-up simullation.

Fig. 4 Modified reactor kinetic simulator (for linear simulation).

dn 6K 5drl1*dn+ E iCi+ S

dt

dC, f3,d = I* n-XiCi.

Analytical solutions to the reactor kinetic equations,when reactivity is varied in a discontinuous, nonlinearmanner, are tedious and restrictive in that the entirerange of probable periods and flux levels involved cannotbe covered without exorbitant numerical calculations.The analytical solution for a linear reactivity change,however, is used as a check on the solution obtainedusing a start-up simulator.For the purpose of simulation, three ranges are defined:

source, intermediate, and power. These ranges illustratedin Fig. 3 should be distinguished from the ranges thatmay be selected for reactor instrumentation. A standard

z

0

100 ..

0

Fig. 5-Reactor start-up transient linear simulator.

reactor kinetic simulation adapted for start-up studies isshown in Fig. 4. This is defined as a linear simulation.This simulation allows the arbitrary selection of initialconditions, enabling one to cover the entire range on astep-by-step basis. The initial conditions are establishedby n, S, C,, and AR, variables of the reactor kineticequations, which are represented in analog form by outputvoltage E,- source voltage V,, voltage Vi on the capacitorsin the delayed group circuits, and the position of thereactivity potentiometer. Reactivity as a function of timeis usually a known input and is defined by the problembeing studied.The simulator is operated over a range consistent with

the accuracy of dc amplifiers (1 to 100 volts). The tran-sient started at the minimum voltage is run until E,amplifier output, reaches the upper limit, Emax; it is thenstopped and the values of E, V,, Vi, and AK are recorded.This determines the initial conditions for the succeedingrun. It is now necessary to reset the computer by returningthe output voltage to its reference level, E0, and setting

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w

> H

w 10

Li

W c

>10

Li

10'z

° 10'i

I 10Iz

II

T j I

Franz and Simcic: ANuclear Reactor Start-Up Simulation

It

0xY-60

(I)

Fig. 6.

Fig. 7 Reactor kinetic log simulator.

the new initial conditions VI and V' where V' = oV

V' = cTVi where -Emax

Fig. 8-Reactor start-up transient log simulator.

After the setting of the initial conditions, the simulatorn is again operated until the output voltage reaches Emax

Again the E', VI, and V' will be depressed by the scalefactor (a-) and the simulator and initial conditions reset.This process is repeated until the desired range has beencovered. A sample run is shown in Fig. 5.

In the log simulator the logarithm of (n) is simulatedwithout using logarithmic computer components. Thisis accomplished by substituting variables and transformingthe kinetic equations into a form in which (1/n) (dn/dt) isdetermined. Since (dn/dt) (ln n) = (1/n) (dn/dt), thecomputer signal for (1/n) (dn/dt) may be integrated togive (In n). Also, (1/n) (dn/dt) is equal to the reciprocal ofthe reactor period T, which is an important variable in thisrange of the transient. Fig. 6 shows the transformedequations. Fig. 7 shows the analog circuit used in theintermediate range to obtain six decade coverage. Typicalresults are shown in Fig. 8.From either of these two methods we obtain the initial

conditions for the power range where sufficient heat

Log Simulator Reactor Kinetic EquationsDefinition of transformations

1 dnn dtcin

W=Sn

Transformed Equations

u= -1 X-ivi+ w

dt --1*-XiUt-uvialso,

dw- -uwdt

can be used to generate the vrariable, w

131957

IRE TRANSACTIONS ON NUCLEAR SCIENCE

-J

LLJ

LL

w

I

m

0

at

0(U)

w

0

CD

lLU

w

0

Ir-

TIME (SEC)

Fig. 9 Start-up study.

would be produced to -affect the reactivity through thetemperature coefficient of the reactor. The heat producedin the reactor is transferred to the coolant being circulatedby the pumps. Heat is normally removed from the coolantin the boiler. In most start-up situations it is usuallyassumed that the heat will not be taken from the coolant,that is, there is no steam load on the boiler. For a completestart-up study the heat transfer effects are simulatedwith the power plant simulator.Some of the problems studied on a start-up simulator

require consideration of rod reactivity rates and tempera-ture coefficients. Corrective action initiated by a periodsignal is important in the intermediate range. Theseeffects are illustrated in Fig. 9 (above).

It is logical to ask how good a prediction does this typesimulation give. This can best be answered by comparisonwith test data. For example, the period at 1 per centpower level obtained for linear reactivity rates is shownin Fig. 10. Experimental observations for similar con-

ditions are also plotted and show good agreement.The effect of temperature coefficient in limiting power

peaks depends on the period on entering the power range.Simulator results and observed data are shown in Fig. 11.

3Ww

:0

0C

0

w

a:

y

Fig. 10-Sub-power range comparison of simulator and test data.

10 __ ...

8 POWER RANGE TRANSIENT6 PEAK POWER -VS-

PERIOD AT 1% FULL POWER

4 _-___ i

2 _ ___ __ __ _

10 A _.. _ __6 _ _=- _ __ _

4 M*- *SIMULATORX-x -x ACTUAL

2 __

10 ___ 4 ._. ,. _IC_ _-_ _ __

6 __4 -_ -- Xv _

2

2 _4 22 4 6 8 10 2 4 6 8 10 2 4 6 8 10

Fig. 11-P'ower range comparison of simulator and test data.

For other types of reactors the methods can apply withsuitable modifications. In the source and intermediaterange simulation, the nuclear constants which applywould have to be used. The power producing range shouldconsider pressure and temperature reactivity effects ifthey are present. In a boiling water reactor for example,these could both be significant factors.

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