Nat msredynamicsanalysis

TH EORETICAL DYNAM ICS ANALYSISOF THE MOLTEN-SALT REACTOREXPERIMENTT.'W. KERLIN, S. J. BALL,Oak Ridge National Laboratory,

Received May 23, 1.970Revised September L4, L970

The dynamic characteristi.cs of the MSRE werecalm,lated for operati,on with 235[I and, 233u fuels.The analysis, i.ncluded calculation of the transi.entresponse for reacti.uity p erturbatiolts, frequencyresponse for reactiuity perturbatioTts, stabitity,and sensiti.uity to parameter uari.ati,ons. Thecalcalations showed that the system dynamic be-lmuior is satisfactory for both fuel loadi,ngs.

I. INTRODUCTION

The dynamic characteristics of the Molten-saltReactor E>cperiment (tusnn) were studied care-fulty prior to the initial 23su fuel loading in lgoband again prior to the 233u fuel loading i" 1968.The first obj ective of these studies was to deter-mine the safety and operabitity of the system. Thesecond obj ective was to establish methods ofanalysis which can be used with confidence inpredicting the dynamic behavior of future, high-performance molten-salt reactors. To satisfy thesecond objective, it was necessary to includetheoretical predictions of quantities amenable toexperimental measurement. The frequency re-sponse results proved most useful for this pur-pose.t

several diff erent types of calculations wereused in these sfudies. In general, they consistedof calculations of transient response, frequencyrespons€, stability, and parameter sensitivities.Four considerations led to the decision to usethis many different types of analysis. These were:

*Presenttanooga,

il8

address: Tennessee Valley Authority, Chat-Tennessee.

and R. C. STEFFY*Oak Ridge, Tennessee 37830

KEYWORDS: reocfors, rsoG-tivity, uronium-233, uronium-235, fronsienfs, di sturbonces,f reguency, voriotions, stobil-ity, fuels, operotion, differen-tiol eguofions, MSRE, reoclork in etics

1. It is hetpful to display system dynamiccharacteristics from different points of view asan aid in understanding the underlying physicatcauses for calculated behavior.

2. Computer costs for the different types ofanalysis were small compared to the expense ofpreparing the mathematical models.

3. The calculations for comparison with exper-iment (frequency response) were essential, butthey did not furnish sufficient information aboutthe system.

4. The experience with a number of methodsprovided insight'on selecting methods which wouldbe most useful in analysis of fufure molten-saltreactors.

The analysis of the system with 233u fuel wasvery similar to the analysis of the "uu-fueledsystem. ,The modeling for the 233U study wasinfluenced :slightly !y_ results from dynamicsexperiments on _ the "urr-fueled system and theanalysis for the "tu-fueled system took advantageof some new methods developed after the com-pletion of the first study.

This paper describes the mathematical modelsused, the computational methods used, ild theresults of the calculations. A companion papertgives results of dynamics experiments and com-parisons with theoretical predictions .

II. DESCRIPTION OF THE MSRE

The MSRE is a graphite-moderat€d, circula-ting-fuel reactor with fluoride salts of uranium,lithium, beryllium, ild zirconium as the fuel.zThe basic flow diagram is shown in Fig. 1. The

NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 19?1

Kerlin et al. THEORETICAL DYNAMICS ANALYSF

dependent reactivity feedback (see Sec. m.C). Theequations ares:

.k) op *.1 ),,0c, *Y (1)i=L '

6cr(t - rr)exp(-tr;rr)Tc

Fig. 1. MSRE basic flow diagram.

molten, fuel-bearing salt enters the core matrixat the bottom and passes up through the core inchannels machined out of 2-in. graphite blocks .

The I MW of heat generated in the fuel and trans-ferred from the graphite raises the fuel temper-ature from ttZO"f at the inlet to LZI0"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 primaryheat exchanger, where it transfers heat to a non-fueled secondary salt before reentering the core.The heated secondary salt travels to an air-cooledra.diator before returning to the primary heat

exchanger.Criticatity was first achieved with 235U fuel

(35 at.Vo 235U) in June of 1965. After 9006 equiva-

Ient full power hours of operation, this uraniumwas removed and the reactor was refueled with233U (91 .5 at.Vo

233U) in October of 1968. BetweenOctober 1968, and shutdown in December 1969'

an additional 4166 equivalent fuII power hourswere achieved with 233U fuel.

Dynamicatly, the two most important charac-teristics of the MSRE are that the core is hetero-geneous and that the fuel circulates. Since thiscombination of important characteristics is llr-commor, a detailed study of system dynamics and

stability was required. The fuel circulation actsto reduce the effective delayed-neutron fractioo,to reduce the rate of fuel temperature change

during a power change, and to introduce delayedfuel-temperature and neutron-production effects.The heterogeneity introduces a delayed feedbackeffect due to graphite temperature changes.

In some of the calculations (determination oflll. SISIET TODELS eigenvalues of the system matrix), it was neces-

sary to eliminate the time delay from the pre-A. Neutronics cursor equation. This was accomplished by

The point kinetics equations for circulating fuel 9t1}latins the tast two terms from Eq' (2) and

reactors were used witlr appropriate tem;;"Jt*"- defining an effective B; as follows:

Fi"ff=e'( )

d\c; pidt =l[ 6n - )t;6c, - y! *

Tc

(2)

where

6n=deviation in neutron population fromsteady state

6c;= deviation in concentration of the i'tJn pre-cursor group from steadY state

Po = reactivity change in going from a circu-tating fuel condition to a stationary fuelcondition

Fr = totat delayed-neutron fraction

I i = importance w e i g h t e d delayed-neutronfraction for the i'th precursor group

A - neutron generation time

0p = change in reactivitY

tri = radioactive decay constant for the i'thprecursor group

rc = fuel residence time in the core

rL = fuel residence time in the external loop.

The term 0p is given bY

op = 6P, *D a;67, ,

where

6P, = reactivity change due to control-rod fiio-tion

ei = temPerahrre coefficient of reactivity forthe i'th section (node) of the core

6T; = temPerahrre change in the i'th section(node) of the core.

l t9

PRI MARYHEAT

EXCHANGER

d\n -

(Oo - '-rurrT=tt/6nRADIATOR w

NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971

Kerlin et al. THEORETICAL DYNAMICS ANALYSIS

Then, the approximate precursor equation is

doc; -W6n-)r;6c;. (g)T-f,-v'e '

This formulation assumes that the fraction of theprecursors which decay in in- core regions isconstant during a transient. Comparison of fre-quency response calculations using this approachand an approach which explicitly treats circulatingprecursor effects showed negLigible differences inthe frequency range of interest.

since the neutron population is proportional tofission power, th€ units on 6n were taken to bemegawatts.

B. Power

An attempt was made to include the effect ofdelayed gamma rays on the total power generationrate. If we assume that the detayed gamma raysare emitted by a single nuclide, then the appro-priate equation is

where

N = €n€fgy stored in gamma-ray emitters (inMW sec)

f = fraction of power which is delayed

n = r€utron population (in tmits of MW)

r-decay constant of gamma-ray emitter(sec- t).

The total power is given by

P-Mr+(l -y)n (b)

For these sfudies the value used for r and ywere 0.0053 and 0.066/sec, respectively.

C. Core Heat Transfer

The core heat transfer wa,s modeled using amultinode approach. The reactor was subdividedinto sections and each section was modeled usingthe representation shown in Fig . 2. This modelwas preferred over a model with a single fuellump coupled to a single graphite lump becauseof difficulties in defining appropriate averagetemperatures and outlet temperatures for a singtefluid tump model.a If the outLet temperafure of asingle fLuid lump model is assumed to be the sameas the average temperatur€, then the steady-stateoutlet temperature is too low. rf the averagetemperafure is taken as a linear average of inlettemperafure and outlet temperahrre, then it ispossible for outlet temperature changes to havethe wrong sign shortly after an inlet temperafure

120

AT TRANSFERFUEL-TO-GRAPH ITE DRIVING FORCE

HEAT TRANSFER

rFr, tN.{- Jrz, oUr

PERFECTLY MIXEDSUBREGI ON S

Fig. 2. Model of reactor core region; nuclear powerproduced in all three subregions.

change. The model using two fluid lumps circum-vents these problems by providing an intermediatetemperature to serve as an average temperatureto use in the solid-to-liquid heat transfer cal-culations. Also, the average temperature in thesecond lump is a better representation of theoutlet temperahrre than the average temperatureof a single [ump.

since ^'7Vo of the heat is generated in the gra-phite by gamma ray and neutron interaction, thegraphite lump equation has an internal heat sourceterm. The equations are:

d\Tfq _ Kf, A 1 -

dt =ffiroP + +lort'(in) -

6r1i

dNn=yn -IN, (4)

*(M)t,(twc1,' [u% - 617 i

doTfz : Kr, 6p + +lor1t_ 671 zldt -(MCb^wr rfzLv

.!9f [or" - 6r1J(MC\ Z

doTG KG 6^ l(M)1r+(hA),rlT=@oP-LWJx [or" - 6T1i ,

where

r = residence time

h = heat transfer coefficient for a lump

A = heat transfer area for a lump

M = maSS

C = sp€cific heat

K - fraction of total power

ft = subscript indicating first fuel lump

f, = subscript indicating second fuel lump

G = subscript indicating graphite.

(6)

(7)

(8)


hil most of the calculations, 9 sections of thetype shown in Fig. 2 were used giving a total of rr, ,N

27 lumps. The arrangement is shown in Fig. 3.The fraction of the total power generated in eachlump was obtained from steady-state calculationsof the power distribution. The local temperaturecoefficients were obtained for each region byimportance weighting the computed overall tem-perature coefficients for fuel and for graphite.

T z, our


rt, our

ra, tN

IIHOLDUP TIME =

Fig. 4. Model of heat exchanger and radiator section.

4t

doT:t = 1 [orr(in)-oz*]dt rn

.Wlorr- o?*l (e)

1-= *[orr,. - 6Tnfd6Tp

dt

d,6Tzt

.W[o?zr-

= *l6rrr(in1 o?,rl

(M),,*@116T, - oZzr]

(10)

6rrJ

o ?rl (11)

dt

( Ip) 1p

Fig. 3. Schematic diagram of 9-region core model.

D. Heat Exchanger and Radiatol

The models for the heat exchanger and theradiator were similar to the core heat transfermodels. The arrangement for a heat exchangersection appears in Fig . 4. The equations for aheat exchanger section are !

d.6Tzz 1

T =;116rr, - orzzl

( h a\^^*ffilorr - o?zr]t- - - ,--

In some of the calculations, it waS assumed thatthe heat capacity of the air in the radiator wasnegligible. (Terms Tzt and Tzz are used for theair side of the radiator.) Ignoring the heat storagein the air leads to the following heat balance:

(WC)rt I Tzz- Tzt(in)] = (hAzt*h,Arr)(77 - Tz) , (14)

where W tsthe mass flow rate of the air.If we assume Tzt = lTr,.(in) * TzzVZ, Eq. (14) be-

comes

( 12)

( 13)

( 15)

121

( rr) our

HEAT TRANSFERDRIVING FORCE

WELL.STIRR E D

TANK (TYPICAL)


ffil2rn 2Tzt(in)l = v, - Trr!


Now, w€ write themental quantities andto obtain:

6Tzt =

equation in termsassuming Tzt(in) is

6Tr

This is then used for 6Tzt in Eq.schematic representation of this typemodel appears in Fig. 5.

E. Piping

Several models were used to represent salttransport in the piping in different stages of thesfudies. The simplest model was a pure timedelay. From some calculations (eigenvalues of thesystems matrix) it was necessary to eliminate thedelay terms. They were represented by Pad6approximationsu in those calculations. In some ofthe more detailed calculations, the heat transferto the pipe walls was included. Since experimentalresultsl obtained after the 235u study indicatedsignificant mixing in headers and piping in thefuel stream, some calculations for the 235u fueledsystem used a model of a mixing chamber at thecore outlet. This model consisted of the followingequation (a first-order lag):

d6T 1 toT =; (07i" - o")

F. Values of lmportant Parameters

some of the important parameters computedfor the 23su and zssu toadings appear in Table r.

G. Overall System Model

The models for the subsystems were combinedto give an overall system model. Several differentoverall system models were used in differentstages of the sfudy. The model shown in Fig. 6was used in the study of the "tu-fueled system.This will be called the reference model. Thismodel resulted in a Af'th-order system matrixwith 4 time delays for heat convection and 6 timedelays for precursor circulation. Major modifi-cations of this model which were used in some

of incre-constant

( 16)1+Zffi(11). The

of radiator

( 17)

HEAT REMOVAL BY AIR STREAM(ASSUMED pROpoRTtoNAL TO CHANGES rru rr)

Fig. 5. Model of radiator for assumed negligible airheat capacity.

TABLE IParameters Used in MSRE Dynamics Shrdies

Parameter 235u 233u

FueI reactivity coefficient (.F-1 )Graphite reactivity coefficient ("F-1 )Neutron generation time (sec)Total effective delayed-neutron fraction (fuel stationary)Total effective delayed-neutron fraction (fuel circulating)Tota1 fuel heat capacity (in core) (MW sec/"f1Heat transfer coefficient from fuel to graphite (MW"F)Fraction of po\il/er generated in the fuelDelayed power fractionCore transit time (sec)

Graphite heat capacity (MW sec/T.)Fuel transit time in orternal primary circuitTotal secondant loop transit time (sec)

-4.94x10-s | -6.18x10-5-8.?oxlo-5 | -3.zgx1o-52,4 xlo-a | +.0 xlo-a0.00666 | O.oo290.00362 | O.00t g

4.Lg0.020.9340.05649.46

3.5816.7321.48

122 NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 19?1

3.77-sec DELAY

MSRE CORE(9 REGIONS)

Z-sec M lX ING POT

FUE LLUMP

FUE LLUMP

KeTlin Et A1. THEORETICAL DYNAMICS ANALYSF

FUEL-SALT HEAT

E XCHA NGE R 8.24-sec DELAY

8.67-sec DELAY

Fig. 6. Schematic representation of the MSRE reference model.

AIRSTR E AM

E

aspects of the shrdy are listed below:

1. The mixing pot was not included in the earlyshrdies for the "utl-fueled systems. It was added

after experimental resultsr indicated significantmixing 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 Pad6 approximation.Effective detayed-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. fV), the heat exchanger and radiator modelswere 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 radiatof, 10 sections(each wittt 3 nodes) were used as with the heat

exchanger.

Calculations showed that results obtained withthe simpler heat exchanger and radiator modelsgave good agreement with results obtained usingthe larger models for these components.

IV. ilIETHODS OF ANALVSIS

A. Transient Response

The tfansient response of the reactor systemwa,s calculated for selected input dishtrbances

RAD IATOR

(usually reactivity steps). The computer code

MATEXP6 (a FORTRAN IV program for the IBM-?090 or IBM-360) was used for these calculations.MATEXP uses the matrix exponential techniqueto solve the general matrix differential equation.For the linear case, th€ general matrix differ-ential equation has the form:

( 18)dx ^-#=Ax +f(t) ,dtwhere

V = the solution vector

t = time

A = system matrix (a constant square matrixwith real coefficients)

Vtl = forcing fturction vector.

The solution of Eq. (18) is

i = exp(AtF(o) + I: explA (t - ?)J lk)dr . (1e)

MATEXP solves this equation using a powerseries for the evaluation of exp @t)z

exp(Ar) = I + tvqt) + i(At)' +. . . . (20)

In MATEXP , f (r) must be a step or representableby a staircase approximation. For the nonlinearcase, ttt€ general matrix differential equation is

dxdt

(21)

123

COOLANTL Ut\,4 P

COOLANTLUMP

COOL A NTLUM P

COOLANTLUMP


= AV + LA6V +70 ,


where

LAG) = / matrix whose elements are changesin the coefficients resulting from ror-Iinear effects.

The procedure used in MATEXP to solve thisequation is to use an approximate forcing fturctionrather than to modify the A matrix continuously.The procedure for proceeding from time-step itotime-step7+1is

i(ti *r) = exp(A tii(t,) + I,'." *1) expl,qu - r )]

'1

x f(r) + aÂli111 )1i(ri)Iar

The time delay vector allows any equation in theset (row in the matrix differential equation) tohave terms containing the value of any state vari-able evaluated at some prior time. Clearly, thisis needed to handle transport delays. The formof this type of term is

r;ixiG ,ri),where

rti = constant coefficient in row i for the term. containing ,c i evaluated at a prior time

,ri = time delay for the effect of )ci in row i.The Laplace transform of the time delay term is

t'lr;i x ,(t , ,i )I = r;i x,(s) er.p(- r;i s) (24)

Thus, the Laplace transform of A in Eq. (23) is

Lldl - L(s, r)r(s) , (25)

where

Z(s,r) = / matrix whose elements are lii =r;i exp(- ti s)

r(s) = Laptace transform ot?1t\.

Equation (23) may be Laplace transformed togive

sr(s) = ê'VG) + z7(s) + g/(s)

This result is analytically integrable and amen-able to computer analysis. This method hasproved to be fast and reliable.

MATEXP uses a similar method for systemswith time delays.

B. Frequency Response

The frequency response for the "uu-fueled sys-tem was calculated with a special-purpose digitalcomputer program, MSFRS (a FORTRAN fV pro-gram for the IBM-7090 or IBM-360), ild alsowith a general purpose program, SFR-f[? (aFORTRAN fV program for the IBM- 7090 or fBM-360). The SFR-III program was used for theanalysis of the "tu-fueled system.

The basic approach in the MSFR program isto program the transfer functions for all the sub-systems and to connect them by the methods ofblock diagram algebra to obtain the overall sys-tem transfer function. This method proved to beefficient (low computing time) and ftexible (sub-system models were changed readily by substi-futing different subroutines for the appropriatepart of the model).

The SFR-III program uses a, state-variableapproach to obtain the frequency response. Thesystem model is expressed in matrix form:

(26)

Initial conditions are zeto because the state vari-ables represent deviations from equilibrium. Thetransfer frmction is obtained from Eq. (26).

(22)

(23)

(27)

(28)

The frequency response is obtained by evaluatingthis equation for s = j @ at selected angular fre-quencies al .

The SFR-III program also furnishes sensitivityto parameter changes. For instanc€, the fractionalchange in G (j u) due to a fractional change incoefficient, a;i , is

aii frUu)TiOq '

This type of sensitivity coefficient is calculatedin SFR-III. The algorithm is obtained simply bydifferentiating Eq. (27).

ag?w) = [s/ - A - LJ-t c( iu)uaii _

L{

where

i - vector of system state variables

A = coefficient matrix (a constant, square ma-trix)

A = vector of time delay terms (see betow)

E = vector of constant coefficient of the forcingfunction

f = the scalar forcing fwrction.

124

dtc ^-7 =Ax +d +gf(t) ,aa

It is noteworthy that the factors on the right sideof the sensitivity equation are evaluated in thenormal frequency response calculation. Thus, thesensitivities are obtained only at the expense of amatrix multiplication of known quantities.


C. Stability Analysis

Three different methods were used for ?il?.-lyzing the linear stability of the system. Thesewere analysis by the Nyquist method, calculationof the eigenvalues of the system matrix, andanalysis by the modified Mikhailov method.

1. The Nyqui,st Metlnd-The stability analysisby the Nyquill method followed standard practice.sThe MSFR code (see Sec. IV) was used to com-pute open loop frequency responses. The Nyquistcriterion requirement for stability is that the netnumber of encirclements of the (- 1 ,i0) point for- oo s crr < oo must be equal to the number of righthaU-plane poles of the open loop transfer function.Thus, it is necessary to lmow the stability char-acteristics of the open loop system prior toanalysis of the closed loop system. In the MSREanalysis, it was assumed that the open loop trans-fer function had no right half -plane poles. Thiswas verified in other analyses.

2. Ei,genaahte Calculati,on-The eigenvalues ofthe A matrix (numerically identical with the polesof the closed loop transfer frurction) must havenegative real parts if the system is stable. Eigen-values were computed using a computer codebased on the QR transform method.e

3. Modified Mi.hhailoa Method-A new methodwas developedlo for stabitity analysis of largestate-variable system models (pure time delaysin the model are allowed). The criterion is thata plot of the function M(iu) for - oo S co -( oo musthave no net encirclements of the origin for M(iu)given by

M(iu;, - det(iwl-A - L) .

fr Ucr: + loo, l)i=i

(2e)

D. Stability Range Analysis

After the analysis of the "uu-fueled MSREusing design parameters indicated that the systemwas stable, a systematic study of the influence ofparameter uncertainties was made. The ma:rimumexpected range on the value on each importantsystem parameter was estimated. Then, an auto-matic optimi zation procedurett was used to findthe combination of parameters in this region ofparameter space which gave the least stable sys-tem. A simplified system model was used for thisshrdy (only one graphite node and two fuel nodesto represent the core). These calculations gavecombinations of system parameters which resultin the least stable configuratiolrs. The parameters


corresponding to this least stable configurationwere then used in a stability analysis using themore detailed model. This analysis indicated thatthe system is stable for any combination of sys-tem parameters within the predicted uncertaintyrange.

V. RESULTS

The methods described in Sec. IV were usedwith the models described in Sec. ffl to analyzethe dynamics of the MSRE. Results for the ttuu-

fueled system are given first, followed by resultsfor the "tu-fueled system. AIso, a comparisoq ofsystem performance with 235U fuel and with 233U

fuel may be obtained by comparison of resultsf rom similar calculations for the two systems.

A. 23sU-Fueled System3

1. Transient Resfonse-The MATEXP code wasused with the state variable model to compute theresponse of the uncontrolled MSRE to a step inputin reactivity. The results for a step input of0.0!Vo 0p for low power operation and for highpower operation appear in Fig . 7. At low powertthe response is a low frequency, lightly dampedreturn to equilibrium. At high power the responseis a higher frequency, more strongly dampedrefurn to equilibrium.

2. Frequency Res4onse-The results of a set offrequency response calculations using the MSFRcode (see Sec. IVB) appear in Fig. 8. The resultsindicate fairty sharp peaks in the amplitude at lowfrequency for low power operation, ffid broaderpeaks at higher frequencies for higher poweroperation. This behavior is consistent with thetransient response results. In general, the fre-quency response plots are rather feahrreless andindicate no dynamics problems for the system.

The results of the frequency response analysisand the transient response analysis indicate thatthe natural period of oscillation of the perhrrbedreactor is a strong fturction of the operatingpower level. This natural period may be obtaineddirectly from the transient response results orfrom the location of the dominant amplitude peakin the frequency response results. The dependenceof nafural period of oscillation on power levelappears in Fig. 9.

2. Stabi,tity-For the "uu-fueled system, the orig-inal stability analysis was based on a Nyquistanalysis and aJI eigenvalue calculation. The Ny-quist plot appears in Fig. 10 for low power oper-ation and for high power operation. The locus is

NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 t25

-

235u

---- 233U

I

Y'omw

-t--tIIt-,t-t-


o.6

-o.?

Fig. 7. MSRE transient response to

2o0 300 400TIME (sec)

a +0.0LVo dp step reactivity input when operating at L and 10

-3Jlrl

td o.4JElrJ

=o(Lz_ o.2

lrlIzIoo

MW.

roo

356zoFJ

droU'ot.|-o5o9(rUJ(L

2

1

o.o5 o.l o.? o.5 to

-. 30CDo3H0o-

l0-2 l0-l 100 l0lFREQUENCY (rod/*c)

/Ve, NUCLEAR POWER (Mw)

Fig. 9. Period of oscillation vs power level for 235u-

fueled MSRE.

complicated near the origin, but it is clear thatno encirclements of (- 1 ,j0) exist. The main eigen-values and their power dependence appear in Fig.11. These were computed using the system matrixcontaining Padd approximations for the transportdelays and precursor equations with effectivedelayed-neutron f ractions rather than expticittreatment of precursor circulation effects. Asbefore, this analysis indicates a low frequen cy,lightly damped behavior at low power and a higher


-90 r-I o-3

Fig. 8. Nuclear power to reactivity frequency responsefor z'uu-fueled MSRE at several power levels.

126

2

rdlo-Iro

6l . rtz

2

rd

5

2

lot

90

60

\-

[] t \L

\I \

[;

\\W

Â N

\:.1 M V \ I

illl-/

ltl+t./ z ERO POWER

||ilil1


IMAGINARY

-1.2 -l.o -o.8 -0.6 -o.4 -O.2 02 0.4 0.6 REAL

UNIT CIRC LE

-r o ooo

-o.?

-o.4

-o.6

-o.8 1rO'2

-t.

at 1 and 10 Mw ("uU fuel).

1.2

1.4

t.6

o.c

o.ooor3_//Mw>/ u

/No

10. Nyquist

u-

No= 1

Fig.

?b(l'v,

UJJo(Ll!ol-E,

tl-

E,

z(9

=

o.o7

o.o6

o.o5

o.o4

o.o3

o.o2

o.ol

o

-o.ol

-o.o2

-o.o3

-o. 04

-o.o5

-o.06

-o.o7

Fig. L1. Major poles for the

VOL. 10 FEBRUARY 1971

\.to Mtv

\ \ \\

\

I__l

-o.

-f

08 -o. c7 -o

L]l--'.106 -Q.

\l-.1.o-_...]I o.rH+o-c5 I -o.

o' REAL

l-;nilI

NEGLTGTBLE powenl I

DEPENDENCE J I

04 -o.o3 -o.o2 -o.

PART OF POLE (sec;lt

lf).=n o')'

&-'lI f__+.--

./

./o

NUCLEAR TECHNOLOGY

'3uu-fueled MSRE.

t2T

KeTlin et al. THEORETICAL DYNAMICS ANALYSIS

frequency, more strongly damped behavior athigher power. Also, stable system operation isindicated at aLl power levels with relative sta-bitity increasing as the power level increases.

B. 2339-Pueled System

1. Transi,ent Response-The MATEXP code wasused with the state variable model to compute theresponse of the uncontrolled, tttu-fueled

MSRE12to a step input in reactivity. The results for astep input of 0.0IVo 6p are shown in Fig . 7 .

Other transients which were calculated for the"tu-fueled system appear in Fig . !2. At thehigher power levels the power rises sharply aftera step increase in reactivity, but the temperatureeffects in the core promptly counterbalance thereactivity input, and the power decreases towardits initial level. Howev€r, before returning to itsinitial level, the power levels out on a transientplateau. ft stays at this level until -l'l sec afterthe reactivity perturbation; then it again beginsto decrease. The power plateau is observedbecause a quasi-steady state exists in the core

\ , T-]---T I'\_ i I I I

l 5MW

\ -

NON Ll NE AR

- - LINEAR

I

f-t t*t

-

\.

4l.r MW\!

-(

-d

Fig.

o{o20304050TIME ( sec )

13. Power response of the "tu-f.reled MSRE to a0.04q0 step reactivity insertion as calculatedwith nonlinear and lineari zed kinetics equa-tions.

-90lO-32 5 l0-2 2 5 lo-l Z 5 lO0 2 5 l0l

FREQUENCY (rod/*c)

Fig. L4. Nuclear power to reactivity frequency responsefor zS3g-lreled MSRE at several power levels.

= {.5

=(ElrJ

=oG r.o

l.2rg r.ol3 0.8te 0.61t

5 o.4l-

I 0.21-

-0.2L9 o.8r: o.el; o.4l-

i '3F< -0.2 L

-- 0.5r

= o4l,\/ 0.311

; 3:?[

I _, ?tt o.a[

= o.sFZ o.zVg o.rFgof< -0.1 L

^. 0.207

3 3:il[

; s:s:h

:" r:F-0.oaot-

rol

5

t03

l-o 5ct .@l o

'22

2

lot90

^30CDo3H0Io-

-30

Fig. L2.

128

f00 zffi 300 400 500 600 7N 800 900 t000TIME (sec)

Calculated power response of the zss g-LeledMSRE to a 0.0270 6k/ & step reactivity insertionat various power levels.

\ t.0 Mw\\

0.5 MW

\

t { \\ \

\il iltl\l,iiii'\

\fN II M w\

I

\ / \I:ER

ll

,(

mllPOWER.

illll

\

-j--

/ 0.1 Mw/


region. The inlet temperature is the same as itwas before the perfurbation, ffid the core nuclearaverage temperature has increased enough tocompensate for the reactivity change. After -L7sec (the transit time of the fuel in the externalloop) the return of higher-temperature salt in-creases the inlet temperature and introducesnegative reactivity through the negative temper-afure coefficient. After sufficient time the reactorreturns to the initial power level, at which timethe net increase in average temperature compen-sates for the step reactivit5r input. This behaviorwas not observed in the "uu-fueled system forthe power levels considered.

A comparison of the step response of the sys-tem for the complete, nonlinear model and for aIineartzed model was made. The results appearin Fig. 13. It is observed that the nonlinear effectsare more important at low power where largerfractional power changes can occur before theinherent temperature feedback calr cancel theinserted reactivity.

2. Freqrcncy Response-The results of a set offrequency response calculations using SFR-IIIappear in Fig. 14. The results are similar to theresults for the "uu-fueled system. In general,the dominant amplitude peaks for the "tu-fueledsystem are lower, broad€r, and at slightly higherfrequencies than for the "uu-fueled system. Thisis mainly due to the greater negative temperaturefeedback in the "tu-fueled system resulting fromthe greater magnitude of the negative fuel temp€r-ature coefficient of reactivity which overrides thedestabilizing effect of the lower delayed-neutronfraction. As with the "uu-fueled system, the fre-quency response results indicate a well-behavedsystem.

The dip in the frequency-response amplitudeat 0.24 rad/ sec is due to the fuel recirculationeffect. The total loop time is 16.73 + 8.46 = 25.19sec (see Table I). The frequency associated withthis is 6.28/25.19 = 0.24 rad/sec. Experimentswith the 235U fueled systemt indicated that the dipwas much smaller than predicted by a modelwhich used pure time delays for fuel transport.Consequently, a first-order lag representation ofa mixing pot was added to the model. Calculationswere made to determine the effect of assigningdifferent fractions of the external loop time to themixing pot holdup time. These results appear inFig. 15 for operation at 8 MW.

The sensitivity of the frequency response toparameter changes was also calculated usingSFR-III. Some results are shown in Fig. 16 foroperation at 8 MW. These clearly show the fre-quency range over which the parameters havean important effect on system dynamics. For


S-sec MIXING PO------.2-ff,rc MIXING PO

--,;;,- ruRF DELA.Y

,/5ro-t25loo

FREQTJENcY (rod/*c)

lo-l@

,51 .

IO1Z

IF

u,|rJof,Fzo:

2000

r 000

s00

200

100

o-tO

o.zl

?r lo3uJ.

$lzozH0G-n

lo-z z rol

-60,frfi ro-t z 5 looFREQUENCY ( rod/nc)

Fig. L5. Frequency-response plot for the z33g-lteled

MSRE operating at 8 MW for various amountsof mixing in the circulating looP.

example, the large changes in the mixing potholdup time and the heat exchanger characteris-tics at -0.24 rad/ sec suggest that fuel saltrecirculation effects are important factors indetermining the amplifude dip at 0.24 rad/sec.

3. Stabitity-For the "tu-fueled system, the sta-bility analysis was based on an eigenvalue cal-culation and a modified Mikhailov analysis. Theeigenvalues appear in Fig . t''7. AII of the eigen-values have negative real parts and the real partof the dominant eigenvalue becomes more negativeas the operating power level increases. Theresults of the modified Mikhailov analysis appearin Fig. 18. (These curves only show the range0 ( crl ( oo . The locus for - oo s crt \< 0 is simplythe mirror image arotutd the real axis.) Noencirclements of the origin were observed for anypower level, indicating stable system operationat all operating power levels.

vl. coNcLusloNs

A detailed analysis of the dynamics and sta-bility indicates that the system is stable andoperable at all power levels. Furthermore, rela-tive stability increases as the operating powerlevel increases. The smaller delayed-neutron

lot

tl\ \ \

I \\ I

\ ttt

\\

\ \

NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 1971 129


\\

t.o

o.8

o.6

o3

o.?

o

-o.2o.4

o.3

o.2

o.l

o

-o.l

-o.?

-o.3o.3

o.2

o.{

o

-olto

o.8

o.6

o4

o.2

o

-o.2o.3

o.2

o.l

o

-o.l

-o.2o.oor o.ol

Fig. 16. Curves showingfueled MSRE.

o.t I

IsREQUENCY (rcdlscc )

o.r2

o.08

o.o4

o

-o.o4

-o.o8

o.o8

o.06

o.o4

o.o2

o

-o.o2

-o.o4

o.2

o.l

o

-o. I

-o.2o.2

o

-o.2

-o.4

-o.6

-o.8

- t.oo

-o.2

-o.4

-o.6

-o.8

-t.oIO o.ool

ElrJFlrJ

=E,

f,

=lrJ(9zI()J

zIF()E,l!

P

3-s^-r!9IFEtrJolF

=o-

=

=lrJ(9z-()JzoF(J

ELl!ooF

E

{o o.or o.t {

FREOUENCY ( rod /scc )

the sensitivity of the amplitude ratio to change various system parameters for 233U-IN

FUEL TEMPERATCOEFFICIENT

l il ill-sEcoNlHEAT E

HEAT ']

h''f-!iJ.il:XCHANGER I

r!II9FFF. II

VI

\t I lill i''i 'l

IIlll

ll

fliV il

\ LI]IJII.,LJJJil

i,?'i"'i'fil' I

llURE

tl

lI

\:

ti

il;

!H

H

Qr

illllllllilllUEL-SALT II-EAT EXCHANGEREAT TRANSFERCEFFICIENT II I

li I

ilI uli

itt iI 1I

illi ilI

I

I

ilI

\ GRAPHITE

ilIil| l

HEAI

illllla PA(

I

)ITY

ll il| lli

I ili

lr 'l,,l,r,,ln' iiTlCHARACTERI

,l'u', i lllll

tlsTrc

ll

itl

ill iiilllI \it t ut ililii I

V \// \

JrtlFUEL }CAPACI

lltl

llllIEATTY

I tltAt /

itl

iii iilltl

\

l,t

.llllll II

rl

I

1r

ilil|IEU'InOfr ;EI NEI

lltir;lIATIOiI TIME ilr

itl, *llj

tiiii

il itl

iitllli \ -r-i

./-t r.uill

rNT L,lll

/ \ ilil|;A',I

l Il\

^lII

r30 NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 19?1

KeTlin et A1. THEORETICAL DYNAMICS ANALYSIS

o.8

t.o

POWER FOR EIGENVALUECALCULATION

.8MTY! 5MW.lMwY o.{ MwO ZERO

SOLID LINES INDICATEPOWER DEPENDENCE OFA PARTICULAR EIGENVALUE

NEGLIGIBLEPOWERDEPENDENCE

Fig.

o.t? o.to o.o8 0.06 0.o4 0.o2REA L AX IS ( sec-I)

L7. Major poles for the z33g-lreled MSRE.

fraction for the "tu-fueled system caused nodynamics or stability problems. This is becausethe stabilizing effect of a more negative fueltemperahrre coefficient of reactivity in the "tLJ-fueled case compensates for the effect of asmaller delayed-neutron fraction.

Numerous analytical methods were used in thestudies. Experience showed that the effort r€-quired to implement the different methods wa,sjustified by the increased understanding of systemcharacteristics made possible by interpretation ofthe various results. It is felt that, in general, a

(nt 0.6

E.

2 o.4(9

- o.2

o o.2 0.4 0.6 0.8 l.o 1.2

REAL AXIS

Mikhailov plot for MSRE oPeratingfuel at zeto power.

Fig, 18a. Modifiedwith 233IJ

Fig, 18c. Modifiedwith 233U

-0.6 -o.4 -o,2

L(l,

3Lx

E.

=(9

t.2

Ix

Ezo o.4q.

=

-o.2-0.6 -o.4 -o.2 0 0.2 04 0.6 0.8 lo

REAL AXIS

Fig. 18b. Modified Mikhailov plot for MSRE operatingwith 233U fuel at 1 kw.

o.8

- o.2o o.z o.4 o.6 o.8 l.o t,?

REAL AXIS

Mikhailov plot for MSRE operatingfuel at L00 kW.

-0.6 -o.4 -o.2

l.?

t.2

r.o

"1

IlI

I

{

t!

Ix

Ez.6 0.4

=o,2

t3l

I

POWER =

I

lkw

./,/

i/- \ArJr^srNGI/ FREQUENCY / ,/ $R\

(( ),

\ \ ./I

\ N .// /

--, \ -/I

POWEF : IOO w

INCREASINGFREQUENCY

lt/,/ \

( / ,/ \ \( ( \

/

\ \ <P

\ \ /a"

(olu =Q \ ./ (r=@


POWER: I MW i


- o.5-o.50 -o.25 0 o.25 0.50 0.75 l.o 1.25

REAL AXIS

Fig. 18d. Modified Mikhailov plot for MSRE operatingwith 233U fuel at 1 Mw.

complete analysis of the inherent dynamic char-acteristics of a new system should include tran-sient response calculations, f requency responsecalculations, stabitity analysis, and sensitivityanalysis.

ACIO{OTLEDGTENT

This research was sponsored by the IJ'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 TestingMethods at the Molten-Salt Reactor Experiment," Nlt'cl.

Technol., 10, 103 (1971).

2. P. N. HAUBENREICH and J. R. ENGEL, "Expef-ience with the Molten-Salt Reactor Experiment r" Nu,cl.

AppL Technol., 8,, 118 (1970).

3. S. J. BALL and T. W. KERLIN, "stabitity Analysisof 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,tt /SA Trans., 3, 38

(January 1964).

5. G. S. STUBBS and C. H. SINGLE, "Transport DelaySimulation Circuits,t' USAEC Report WAPD-T-38 and

Strpplement, Westinghouse Atomic Power Division (1 954).

6. S. J. BALL and R. K. ADAMS, "MATE)G, A Gen-eral Purpose Digital Computer Program for SolvingOrdinary Differential Equations by the Matrix Exponen-tial Method," USAEC Report ORNL-TM-1933, OakRidgeNational Laboratory (Augttst 1967).

Fig. 18e. Modifiedwith zssu

7. T. 'W. KERLIN and J. L. LUCIUS, '3The SFR-3Code-A Fortran Program for Calculating the Fre-quency Response of a Multivariable System and ItsSensitivity to Parameter Changes," USAEC ReportORNL-TM-1575, Oak Ridge National Laboratory (June

1 966).

8. F. H. RAVEN, Automati,c Control Engineering,McGraw-Hill, New York (1961).

9. F. P. IMAD and J. E, Van NESS, "Eigenvalues bythe QR Transform," Share-1578, Share DistributionAgency, IBM Program Distribution, White Plains, NewYork (October 1963).

10. 'W. C. WRIGHT and T. 'W. KERLIN, "An Efficient,Computer-Oriented Method for Stability Analysis ofLarge Multivariable Systems,tt Trans. ASME, J. BasicBng., Series D, 92, 2, 279 (1970).

11. T. 'W. KERLIN, "stability Extrema in NuclearPower Systems with Design Uneertainties,t' Nucl. Sci.'Eng., 27, L20 (1967),

L2. R. C. STEFFY, Jr. and P. J. WOOD, "TheoreticalDynamic Analysis of the MSRE with 283U Fuel," USAECReport ORNL-TM-257L, Oak Ridge National Laboratory(July 1969).

r.5

g l.o><

5 0.sz(9

=oo.5

I><

EO<Lz6

=- o.5

-l.o

-t.5-2 -l ot?3

REAL AXIS

Mikhailov plot for MSRE oPeratingfuel at 8 MW.

,/POWE l:8MW

//t//l/ TNcREAStNG

FREQUENCY

\

o =O ff,Y'='\_ /

\

132 NUCLEAR TECHNOLOGY VOL. 10 FEBRUARY 19?1

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