Temperature coefficients of reactivity of reactors


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Temperature coefficients of reactivity of reactors
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Physical Description:
24 p. : ill. ; 27 cm.
Chernick, J
Brookhaven National Laboratory
U.S. Atomic Energy Commission
United States Atomic Energy Commission, Technical Information Service
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Oak Ridge Tenn
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Nuclear reactors -- Reactivity   ( lcsh )
federal government publication   ( marcgt )
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by Jack Chernick.
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Cover title.
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Originally published 1953.
General Note:
"January 9, 1953."
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"Subject Category: Physics."
General Note:
"Brookhaven National Laboratory, Upton, New York."
General Note:
"Work performed under Contract No. AT-30-2-Gen-16."--P. 2 of cover.
General Note:
"Date Declassified: December 2, 1955."--P. 2 of cover.

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University of Florida
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Subject Category: PHYSICS



Jack Chernick

January 9, 1953

Brookhaven National Laboratory
Upton, New York

Technical Information Service, Oak Ridge, Tennessee

Work performed under Contract No. AT-30-2-Gen-16.

Date Declassified: December 2, 1955.

This report has been reproduced directly from the best
available copy.

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for declassification of classified material of the same or
similar content and title by the same authors.

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disclosed in this report.


By Jack Chernick


The subject of temperature effects on the reactivity of a nus-

lear reactor is of cmnaiderable practical importance since the stability

of a reactor under operating condition depends upon its temperature coef-

ficients. The actual operating temperature coefficient of a reactor in-

Wolves its particular structure and cooling system too much to lead it-

self to a general discussion. Instead we shall confine our remarks to

the so-called unrform teamerature coefficient, i.e., the reactivity change

per degree rise in temperature of all the reactor constituents in some

limited and defined temperature range.

Early Theoretical Resulto

The physical effects contributing to the uniform temperature

coefficient of a reactor are nov generally understood. It is custom-

ary to separate the coefficient into two parts, the nuclear temperature

coefficient which is determined ly the change in nuclear cross-sections

with temperature, and the density temperature coefficient which is due

to thermal expansion of the reactor.

The density temperature coefficient was discussed by Soodak

(M-2966) who pointed out that the neutron leakage from a reactor varied

inversely with the 4/3 power of the density. He thus derived the formula

(1) ( ~~ T d

for the density temperature coefficient in term of the amltiplication

factor k., and the coefficient of volume expansion 4. of the system.

For graphite moderated, normal uraniiu reactors the density tam-

perature coefficient is of the order 10 -'C and can be neglected In com-

parison with the nuclear temperature coefficient which is of the order
10o/oC. On the other hand, enriched fluid fuel reactors have density

temperature coefficients which range from -10 4,C to -10-l3C.

The neutron leakage from a reactor is also affected by the change

in neutron cross-sections with temperature since the migration area of

the reactor is then altered. For a 1/v absorber, the diffusion area of

the reactor increases with the square root of the absolute temperature

while the age to theraal energy is slightly, decreased. hdis leads to the

formula (Glasstone: Elemonts of Nuclear Reactor Theory, p.341),

(2) ( ( I
T k. To 2 3 s o

for the contribution of the neutron leakage to the nuclear temperature


coefficient of the reactor. In equation (2), B2 is the geometrical

buckling of the reactor, 1 the diffusion area evaluated at the neu-

tron temperature To. For a graphite-oraniu reactor, To 385fK at

room 4mperature. The increase in neutron leakage is due mainly to

the increase in diffusion area rather than in the neutron age and,

near critical, the reacetivity coefficient runs about -3 x ID'/DC.

The nuclear temperature coefficient of a reactor depeads on

changes in k, as well as on changes in neutron leakage. In the early

literature two such effects were recognisedt

1. Leveliwn or flattening of the iUtracell flux dis-

tribution. This effect occurs with increase in tem-

perature since the diffusion of neutrons through the

moderator is eased, thus enhancing the probability of

absorption in the fuel. Since the thermal utilisation

is increased, one now obtains a positive contribution

to the reactor temperature coefficient. The leveling

effect is in general due to increase in the absorption

mean free path with temperature. However, in water-wod-

erated reactors, the effect is caused by the large change

in the scattering mean free path in the thermal region.

The thermal utilization may also vary with tampers-

ture in the presence of aon-I/v absorbers. For graphite,

natural uranium reactors, the departure from the 1/v lai

is aall and is generally neglected.

2. Doppler broadening of ;he resonance bends in uranmur.
The resonance absorption of neutrons in U2" increases

with temperature due to the broadening of the resonance

bands. We have here the first contribution to the uni-

form temperature coefficient which depends primarily an

the temperature of the fuel rather than that of the mod-

erator. The effect is important to the stability of

graphite-uranium reactors since the fuel temperatures

are the first to respond to any sudden power rise.

The West Stas Reactor.

The first measurement of the uniform temperature coefficient

of a reactor is mentioned by Fermi in an early (January 15, 1943) pro-

gress report (CP-416). By the simple experiment of opening the windows,

the temperature of the West Stands reactor was brought down from 22~OC

to 14C. The reactor was then brought back to room temperature, the en-

tire operation consuming about three weeks. From the shift in the criti-

cal position of a control rod, and after correcting for barometric effects,

a uniform temperature coefficient of -3.8 x 10/DC was obtained.

Since this result was approximately that expected from increased

neutron leakage alone, Fermi concluded at the time that the Doppler broad-

ening effect must be large enough to cancel the leveling effect, i.e.,

about -6 x 10c5/C.

In a February 6, 1943 progress report (CP-455), however, we find

that the fuel temperature coefficient was measured and led to a value of

only -1 x 10-5/C. The experiment consisted of raising the temperature

of insulated metal lumps in the central portion of the reactor about

300C and measuring the change in reactivity. This result meant that

they had to look elsewhere for a large temperature effect and Fermi

suggested izedietely that it might be found in the variation of 'I with

neutron energy.

Following this experiment a theoretical discussion of temperature

effects in graphite-uranium reactors vas given in a short paper by Mor-
rison (CP-478). He considered both lumped cubic lattices and rod lat-

tices. On the basis of theoretical results of Teller and Metropolis

(CP-387) on chemical binding effects in graphite, he estimated the mean

temperature of these lattices in the neighborhood of 4000K. This value

is in good agreement with later experimental determinations of the neu-

tron temperatures. Early estimates of the best value to use had ranged

from 3000K to 4500K.

Morriaon's estimates of the temperature coefficients of reactivity

of those lattices is given in the following tables

latteie T3pe T
1l 8" cubic lattice, 6 lb lumps 4000K
#2 12" cubic lattice, 48 lb lumps 4100
#3 n8" rod lattice, 1.6 cm radius 412
Temperature Coefficient of Reactivity
T-+4 le akaUe e velig Doppler Broadening Eta Effect ,_UI
fl -4.5xl0-~ /C 10.axlO-5/C -1.3 x 10- /oc -xl0-/oc -4x0-5/C
#2 -2.9 15.3 -5 6
#3 -2.7 4.6 -9.7 -910X-5/OC

The first table gives Harrison's estimate of the neutron temperature

of these lattices. His estimate of the 1-coefficient of reactivity

was based on the experimental value of the uniform temperature coef-

ficient of the '"est Stands reactor (lattice 1l). The large value of

the *)-coefficient is at variance with earlier estimates and Is a re-

flection of the large value assumed by Morrison for the leveling ef-

fect of this lattice.

Morrison also considered for the first time the hardening of

the neutron spectrum in the fuel elents at least insofar as it af-

fected the calculation of the -coefficient. The hardening of the

neutron spectrum is caused by the preferential absorption of low en-

ergy neutrons as they penetrate into the fuel. The opposite effect

of hardening on the thermal utilization was not considered by Morri-

son. Be came to the conclusion, sh b by the above table for lattice

v2, that a reactor with large metal elements would be unstable with

regard to temperature.

The Branmdom es-1arshall lkerimt

An attempt to determine the dependence of on temperature

was carried out by Dragdon, Hughes and Marshall early in 1944. By

using folls of normal and depleted uranium located at the center of

a spherical cavitr in a the-mal oolu, they measured the ratio of

thermal fission in 8 to radiative captures in S. In order to

obtain the sae ratio at a higher neutron temperature, they them sow-

rounded the foil with a spherical shell of silver of 108 mil thickneas.


The silver acted as a filter for low energy neutrons and Bragdon,

Hughes and Marshall es timated the effective neutron temperature

change at 1600C. They obtained a value of

1.0211 t 0.0046

4fr the ratio of capture to fission at the higher neutron tempera-

ture compared to the same ratio at the neutron temperature of the

thermal column. The change in -) with temperature was then esti-

mated as

Sd = -5.2 x 10-5 .

At the time, the authors assumed that the radiative capture cross-

section of U235 was zero. However, re-analysis of their work by

Friedman and Kaplan at Brookhaven (BNL-152) on the basis of a fin-

ite but temperature independent value of X= (25)/'f(25) results

only in a slight reduction in the value of the "-temperature coef-


Attempts have since been made by Wigner (DPW-2071) and by

Sampson and Hrvwits in August 1951 (KAPL-511) to determine the tem-

perature dependence of -C in the thermal region by the use of approxi-

mate formulas for the temperature dependence of r,(28) and 0-(25).

Wigner obtained a value of -22 x 105/OC on the basis of the
c T
Bragdon-Hughes-Marshall experiment which he believed to be unlikely

Sampson and Hurvitz concluded that the best value indicated by the

Bragdon experiment was

"~ -10 a 5 x 10-5/o


Sampson and thrvits also attempted to obtain A fro neutron spee-
d T
tromter data which they thought indicated a positiw value of about

10 x 10-5/0C. They then obtained a value of A a- -29 z 107C5/ which
d T
is in complete diaagrpemnt with the results of the early reactor e -

per ants uhich w have already daecussed. However, even for -# 0,
the cross-eotion formulas assumed of these authors led to a value of

-1 x 10o-5/C for .- Cross-seetion data available to ua at took-
) T
haven through D.J.Bugh s and his group lead to more reasonable values

of 2U However, it appears that at present we must still turn to
a T
reactor temperature coefficient experinant for our beat asti.atei of

the variation of ) with temperature.

Theory of the Donler Broadeninr gEfect

In January 1942, Vigner (C4) pointed out that resonamee ab-

sorption in uranium lattices could, in goad approxiation, be repre-

sented by a volume absorption and a surface abaorptioe term. atew a-

peiaents 1by Cruts (C-116), Hughee (cP-35oo), Haelhawise (AXL.4323) @

Risser ani others ((FML-958) have borne out this division of the resoe-

ance absorption and the method is universally used in resonamee escape

factor calculations. In addition, Wipger gave the formulas for the

shape of the Doppler broadened resonance lines. Thus if the level atrue-

ture were know, the Doppler temperature coefficient could be readily

computed. Since the actual resonance structure was largely unknown,

Viper and later Dancoff (CP-1589) worsad with various simple models

of the resonanee struetare to obtain the temperature eoeffielmt of

the volume absorption term.


According to Wigner's method the effective resonance integral

is given by

A + B surfle

where A and B are respectively the volume and surface absorption inte-

grals. Wigner obtained a value of 3 x 10"'/'C for the value of I dT.
A dTo
Dancoff, in refining the method on the basis of experimental data on

the low lying resonances estimated that the temperature coefficient

of volume absorption lay between lo2 and 1.7 x 10-4/C. For graphite-

uranium reactors, Kaplan and Friedman (BIL-152) showed that this esti-

mate leads to values of

,1 2 -1.4 to -2.0 x I10/ C

for the effect of the Dopplor broadening on the resonance escape factor.

Thore also exists some U29 activation measurements by Creuts

(0-110) and Mitchell (CP-597) at elevated temperatures which yielded

values of 1 dA of 1.7 x lO-/0C and 1.1 x IO4/MC respecLively. The
A dT
spread of the experimental resulLts almost coincides with Daneoff's

theoretical results.

BExoriments on the X-10 Reactor

During the start-up !P- early operation of the Oak Ridge graphite-

uranium reactor, an extensive study of the reactivity coefficients was

undertaken by L.B.Borst (obnP-o60 and Ch. 13 of the National '-clear

Energy Series IV-5).

Several attempts vere made to obtain the barometric coefficient

of the reactor. Of these, the most accurate value was determined in a

long run at constant power during which a change of atmosph-ric pres-

sure of 5.8 = Rg occurred within a period of 6 hours. From these em-

poriments a value of -0.4 0.05 inh/m Hg was established for the

barometric coefficient. In subsequent msnon experiments, it was found

that values ranging from -0.3 to -0.5 inb/a Hg vere required in order

to correct satisfactorily for slow reactivity changes. In this con-

nection, it should be pointed out that, while the barometer will re-

spond to humidity as well as partial nitrogen pressure, the poisoning

effect on a reactor is almost wholly due to the nitrogen content of the

air. Thus barometric pressure alone can not be an exact indication of

the nitrogen poisoning effect.

An attempt was made to obtain the uniform temperature coefficient

of the 1-10 reactor by placing steam radiators in the inlet air chamber

in order to increase the equilibrium temperature of the reactor. A

coefficient of -0.75 inhour/OC was obtained which was quite low compared

to data obtained in later experiments. Similarly, an effort was made to

estimate the metal temperature coefficient of the reactor from data on a

heated slug. Fermi had suggested following the transient temperature

changes in a metal slug as a method of power calibration cnd from the

associated control rod movements, Borst fourd a uniform metal temperature

coefficient of -1.2 inhoura/0C which appeared rather high. Finally,

Borst relied on various metal and graphite temperature transients in

order to obtain more accurate values of the reactivity coefficients. On

the basis of extensive operating experience, he concluded that the uni-


form tempe-ature coefficient of reactivity of the X-IC reactor (in

vacuo) vas -5.9 x IO10AC, of which only

-1.0 x 10"5/O

vcs attributed to the metal teaerature coefficient.

It. is of interest to compare Borst's resutts vith Fermi's

e-timate of Fabruary 1944 (CP-13U9) of the uniform temperature ooef-

ficiant of the Clinton raaclor. He split the various effects up as

leveling: 6.5 x lC05/OC

leakage: -3.1 x 10"5/C

Doppler Broadening: -2 x IC-5/OC

Eta-Effect: -5.5 x 10-5/C

kff: -4.1 x 10-5/oc

The estian',e of the --Tffect was based on the experimental results

of Bra-dcn, Hughes and Marshall. In early design calculations for

the BKL reactor, K.plen and F-rled.n (BNL-152) obtained considerably

smaller values fcr the leveling effect by using elementary diffusion

thecry. These were re.; ectively 3.4 x 10-5/oC for the Oak Ridge re-

actor and 2.6 x 12-5/C fcr the DBL reactor. On this basis, the Oak

Ridge results are consis-ent idth those of the Bra3don-Hughes-'iarshall

ex7 ardent.

Prookhaven R.actor Exro E-ricnta

During the Etart-ur cf the ML reactor a considerable program

of experiments of use in recctor evaluation were carried out. The


organisation of these experiments under the direction of L.B.Borst

was so efficient thet all the low power work, including mch of the

loading of the reactor to criticality, was carried out in about two

weeks. We shall discuss here only those experiments which are per-

tinent to our present topic.

The Baroetric Coefficient of the BN-L Reactor

The first opportunity to obtain the barometric coefficient

of the BEL reactor occurred. on the night of August 23, 1950, while

the reactor was temporarily idle at a slightly suberiticel loading.

The effective multiplication factor of the reactor, measured by: its

free period was 0.9997 at the time. The experiment arose from a sug-

gestion of the theoretical group who noted that at such a loading the

steady state neutron level of the reactor, being inversely proportional

to l-keff, could be expected to vary by Pbout 10% during the night be-

cause of changes in barometer alone. The flux level of the free re-

actor was therefore followed throughout the night by means of neutron

counters. The curves were compared with accurate barometric pressure

end partial nitrogen pressure data supplied by the Meteorology group.

Graphite and metal temperatures at various points in the reactor were

read frequently but no detectable changes in the thermocouple rend ings

were recorded, possibly because of the insensitivity of these instru-

ments to small temperature changes. A total barometric change of 1.5

m Hg occurred during the night.

The correlation of the neutron counting rate with barometric

pressure was fairly good, yielding a barometric coefficient of 0.4 0.1


inhours/m Eg. The correlation with -:artial nitrogen pressure was not

nearly as good, thus indicating that there was little actual mixing of

the outside air with that inside the reactor. If more time were avail-

able, it vwuld have been qc-iite interesting to pursue this study for

longeT reriods of, time end for loadings closer to criticality in order

tc learn more about the ter-orol behavior of a free reactor which is

just sufficently suberitcrl to be stable a&rinst diurnal barometric

ai thhernal fluctuations.

Efforts to determine the barometric coefficient of the BNL re-

etcr ;', control rod chians necessary to maintain constant power during

siz- *er.thr changes led to no great ii-rroverent in the precision of

the bcrcetric coefficient. It was therefore decided to use the reactor

fors to simnlete the barometric effect. By sealing off the inlet air

ducts :nd -"errting one or more fens, a uniform pressure dror ranging

fron 15 to 53 mm g was maintained over the reactor. These pressure

changes were much greater than could be provided by the most severe

weather conditions and the entire experiment was completed in about 3

hours. The results are ahovn in the following table.

Barometer Coefficient of ENL Reactor

Critical Barometric
Position Pressure Drop Change in change in Coefficient
of #9 Rod (=m BE) Pressure (am HN) Reactivitz (Ih) (Ih/mHa)

425.2 ea 0
/45.5 15.4 15.4 5.6 0.36
47C.8 38.3 22.9 7.0 0.31
490.7 53.2 14.9 5.5 0.37
419.8 C 53.2 19.7 0.37


The mean value obtained was 0.35 0.014 inh/m g*. Extensive control

rod calibrations were carried out, both previous to sad during the vs-

periment. It was estimated that errors in the sensitivity of the #9

control rod were about 4%. A possibly more serious source of error of

about .03 nl-h/m g is indicated by the fact that the final critical

position of the control rod differed somewhat from its initial position.

A change of 1C in the over all reactor temperatures, which were not reed

during the xperimant, could account for the discrepancy.

l-ifoas Tamarature C oeffiset of the ~L Ractor

The uniform temperature coefficient of the BEL reactor was meas-

ured on two different occasions at essentially sero power. The method

used was that of drawing in the cold night air into the reactor by use

of the fans and following the reactivity changes with a calibrated con-

trol rod. Graphite, metal and air temperatures and barometric pressures

were recorded as a function of time, the former by means of thermocouples

placed in representative parts of the reactor. We used several thermo-

couples since we were cognizant of the difficulties experienced at Oak

Ridge in similar experiments of this type.

As expected, the major changes in the metal temperatures occurred

during the early part of these runs, while the reverse was true of the

graphite temperatures. Temperature changes during a given interval of

time wre found to vary appreciably at different points in the reactor.

The uniform temperature coefficient of the reactor uncorrected

to vaum was found to be -1.26 0.09 inhours/C. Of this coefficient,


the contribution of the metal coefficient was -0.78 0.16 inhours/OC

while that of the graphite coefficient was -0.48 0.14 inhours/C.

After correcting the latter for barometer we find a uniform tempera-

ture reactivity coefficient of -5.7 x 10-5/oc of which -2.0 x 10-5/C

is attributable to the metal temperature coefficient. On the basis of

operating experience, we would tend to increase the graphite coefficient

somewhat. In any case there is good agreement with theory and in par-

ticular with the result of the Bragdon-Hughea-Marshall experiment.

The Metal TeMDerature Cbefficient

Our first attempt to determine the metal temperature coefficient

oat he BIL reactor involved the use of insulated metal cartridges heated

to about 450C and then placed in three of the central channels of the

reactor. With the hot cartridges in place, the control rods were ad-

justed to produce a slightly falling reactor period at a negligible

power level. As the cartridge temperature decayed, the reactivity of

the reactor increased until a positive period was observed. The control

rod was them adjusted to bring the reactor slightly below critical. This

procedure as repeated several times during the experiment which ran for

about 4 hours. The technique used was suggested by Borst and proved to

be a very sensitive one for measuring the exceedingly small changes of

reactivity which were involved. Under the condition of the experiment

a reactivity coefficient of -0.0093 0.0C04 inhours/'C was observed.

The use of statistical weight factors, however, yields a metal tempera-

ture reactivity coefficient of -1.4 x 105/0C, which is about 30% lower


than the values ob nined in other experiments.

Tep--oretu're F3ash Am'eriments

The most accurate value of the metal temperature coefficient

of the BNL reactor was obtained in temperature flneh experiments during

uhich the reactor was permitted to rum away and brought under control

by the increase in metal temperatures alone. The method used was sim-le

in conception. The reactor cooled to ambient teaperatTre, was started

up without fans from a noZ igible power level with an initial period

'Aich ranged from 36 sec to 4 minutes. There was little change in

metal temper-tures until the integrated power output of the reactor

hid risen to the 10 kw min level. Thus the initial reactivity of the

reactor could be determined from the galvqnometer period. At the time

of mxiamn gelvenoncter reading, the reactivity would be zero and the

corresponding metal temperature at the position of maximnu flux eould

be used to obtain a metal temperature coefficient for the reactor.

Of considerable interest in these er-eriments was the fact that

the rover output and reactor temperature oscillated through several dis-

tinct maxima and minima before damping out. The grarhite thermocourles

registered no increase in temperature until one or more such oscillaticns

had occurred. This was expected beec-use of the nmch larger heat capacity

and 'might of the moderetor relative to the fuel and the fact that the

maximum power output reached was only of the order of a W. For the low

temaorr-ture flashes, the tedium of waiting for the reactor to reach sig-

nificant o'. r levels was reduced by bringing the reactor up rapidly to

the XE region and then setting the controls to the proper position. Since


the uniform temperature coefficient of a reactor depends on the dis-

tribution of temperature as well as flux, readings of available metal

thermocouples other than the one at maximum flux were occasionally re-

corded. It was found that during the run, the temperature rise at

points in the same rlane parallel to the reactor air gap was closely

proportional to the theoretical flux distribution.

In the first temperature flash which took place on October 10i

1950, the metal thermocounles were attached not to uranium but to the

midpoint of the upper aluminum fins. Thus it was possible

for an appreciable temperature gradient to exist between the uranium

surface and the fins. In the second set of experiments (Novenber 10,

1950), thermocouples were placed on the fuel surface as well. It was

found during the run that the difference in temperature between the

uranium end adjacent Lluminum fins was actually negligible and the pre-

vious value obtained for the metal temperature coefficient was confirmed.

Because of the absence of xenon and graphite temperature effects, the

temperature flesh experiment appears to be the best possible method of

determining the metal temperature coefficient of a grphite-urmniun

reactor. Despite the fact that under the different initial control

settings the maximum metal temperatures ran from 600C to 2000C, the

metal temperature coefficients varied very little wdth temperEture.

An average vali.e of -0.49 inhour/OC rise in maximum metal temperature

was obtained whichh corresronds to a uniform reactivity coefficient of


-2.0 x la/oC. Uider strictly adiabatic conditions, the theoretical

value of the mazimm overshoot in temperature in a reactor temperature

flash experiment is twice the equilibrium value and this value was nearly

attained when the reactor was started up with an initial period of 36


Temieratur Coeffiielant, of Natural Uranium Water Latticea

Before leaving the subject of temperature coefficients of re-

actors we should like to quote some of the results obtained some years

ago at Oak Ridge in water, normal uranium, exponential experiments.

The uniform temperature coefficients of these assemblies were found

to be negative for tight lattices and positive at large water to metal

ratios. The cross-over from negative to positive values of reactivity

oeourred at a water-metal volume ratio of 1.6. Experimental values

at other geometrical arrangements are given in the following table:

Water Metal Ratio Reactivitr Coefficient

1.2 -1 x 104/oC
2.1 +1 x I C-4/c
3.1 +2 xIO-4C

The experimental results were based on measured buckling of the lat-

tices at about 25C and 75C00. Changes in the migration area with den-

sity were calculated. The experiamnts show the tendency of leveling

effects in water le -tioes to produce positive temperature coefficients.

Thoory of To1,erature Dependent Reactor Kinetics

We should now like.to consider briefly some reactor problems

which involve temperature effects. To begin with, reactor transients


under ordinary and emergency conditions are of great importance both

in the design and sefe onerption of a reactor. When temperature ef-

fects are considered, the equations governing the system are non-lin-

ear and generally must be solved by numerical methods. All of the AEC
laboratories have done considerable work on these problem as they af-

fect their porticuler reactor types.

For small deviations from steady state conditions, perturbation

methods such as those outlined by Nordheim (MDDC-35) are useful. For

large deviations from equilibrium, more exact methods are required and

mzny of the problems have been solved only with the aid of high speed

electronic computing machines.

The damped, almost periodic nature of the temperature and flux

curves obtained in the BNL temperature flash experiments aroused our in-

terest in the non-lineer asnects of the problem.

If the delayed neutrons are assumed to act only to increase the

effective neutron generation time as is the case if the reactor periods

are not too short, then the transient equations take the simple form

dn/dt Akmexn

ka (kex) -ST

dT/dt = fn -AT

where 1/ is the aver-ge neutron generation time, ,e is the temperature

coefficient of the reactor and X is the relaxation constant for heat

removal. In the ease that (kge)c, the excess k introduced by a control

rod is constant, the temperature equation mBy be reduced to the dimension-

less non-linear form


W + (1-0)6 6e -e(0 + @2)

In the single parameter e 4 (k)g)/A .

Solutions of this equation, obtained by the method of isoclines,

are ubown in Slide No. 1-116-3 (Unclassified)

The first elide shows a phase diagram (rate of change of tem-

perature vs temperature) when the reactor control setting is above the

cold critical setting. If the reactor has a positive temperature coef-

ficient, the solution curves all diverge with time, regardless of initial


There are two possible equity ibrium positions for a reactor with

a negative temperature ooefficient. One occurs at zero flux and sero,

i.e., ambient temperature and is unstable in the same sense that a in-

verted pendulum is in unstable equilibrium. The second equilibrium point

is a stable one. The equilibrium point is a spiral if the reactor cool-

ing rate is sufficiently low. On a time basis, the solutions conveaging

to the stable equilibrium point are then of damped, almost periodic type.

At higher cooling rates, critical damping occurs and the equilibrium point,

in the language of non-linear mechanics, becomes a node. The limiting

spiral connecting both equilibrium points corresponds to the ease of the

reactor temperature flash experiments which we have previously discussed.

Slide No. 1-115-3 (Unclassified)

Thie slide aso what happens when one attempts to shut down a

reactor. The control setting is below the cold critical position and the


solutions for a reactor with a negative teoper.ture coefficient are

very uninLeresting from the mathematical viewpoint. Regardless of

initial conditions, everything converges to the stable equilibrium

point. A reactor with a positive temperature coefficient, however,

has fto possible equilibrium positions one which is an unstable sad-

dle point. One of the two curves which cross the saddle point is

very important since it separates the phase plane into stable and un-

stnble regions. Only in the stable region do the solution curves con-

verge to the stable equilibrium point.

In BNL-173, we have investigated the nature of the solutions

of non-linoar equations of the present type. Of practical interest

is the fact thnt the special solutions which separate the stability

regions of a reactor are of relatively sisnie form.

Statics of a Poactor vith V&riable Local ul.tinlleatiqn

In a reactor o-erating at power, the fuel and moderator tem-

persturos ns wo-1 as poison concentrations vary with position. The

local mult!plic.tion factor is therefore variable and the flux dis-

tribution ass-cirtod with the clean reactor is distorted.

Rei ctors mna be capable of several quasi-steady states which

are fr rly well separated with time. Thus during the start-up of the

BML reactor after a long shutdown, the fuel temperatures omne into

equilibrium in a few minutes, the moderator temperatures then come in-

to equilibrium and finally the xenon production becomes important level-

ing off after about a day's operation. The variation in the local multi-


plication factor is different for each of these approziaate equili-

brium positions.

In BNL-126, the statics of a reactor with variable local

multiplication have been considered on the basis of one and two group

methods for certain cases and the results were compared with nertur-

bation theory approximaticns. Thus the solution of the one-group

equation for a critical slab reactor with Sk local proportional to

flux was shoun to be given by an elliptic integral of the first kind.

The xenon problem with burn-up was solved exactly for various one-

dimensional cases. Equilibrium temperature conditons for a raeator

in which the inpor tnt variation of temperature occurs only along a

cooling channel v-s reduced to a Integro-differential equation which

can be solved by numerical methods.

In general, it ws. found that except in very extreme cases,

the usual statistical weight formulas gave very good agreement with

exact solutions.

The statistical weight formulas for the effect of xenon at

flux levels were burn-up is important "rre considered in Canadian re-

ports by Goldstein and CGugnhein (FT-2:a) and PRennie (CRT-272). Ir

ENL-126, we have checked their formulas over a wide range of flux levels

by an alternateo methodd rnd obtained good agreement.



Slide No. 1-116-3


k \\\\ \ \\i z I/II

Slide No. 1-115-3



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