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ORNL- TM- 379 COPY NO. DATE
-
-
// 7
October 15, 1962
T?B@EXUWAMD REACTIVITY COEF'FICIEMT AVERAGING IN THE E R E B. E. Prince and J. R . Engel
ABSTRACT
Use i s made of the concept of "nuclear average temperature" t o rel a t e the s p a t i a l temperature profiles i n f u e l and graphite attained during high parer operation of the WRE t o the neutron multiplication constant. Based on two-group perturbation theory, temperature weighting m c t i o n s for f i e l and graphite are derived, from which the rmclear average temperatures may be calculated. Similarly, importance-averaged temperature coe f f i c i e n t s of r e a c t i v i t y are defined. The values of the coefficients calfor the f u e l and -7.3 x 10-5 for culated f o r the MSRE were -4.4 x lO+/'F the graphite. These values r e f e r t o a reactor f'ueled with salt which does not contain thorium. "hey were about 5% larger than the values obtained from a one-region, homogeneous reactor model, thus reflecting the variat i o n i n the f i e l volume fraction throughout the reactor and the effect of the control rod thimbles on the flux profiles.
NOTICE This document contains information of a preliminary nature and was prepared primarily for internal use at the Oak Ridge National Laboratory. It i s subject to revision or correction and therefore d a r s not represent a final report. The information i s not to be abstracted, reprinted or otherwise given public dissemination without the approval of the ORNL patent branch, Legal and Infarmation Control Department.
I.
LEGAL NOTICE T h i s report was prepared as an account of Government sponsored work.
Neither the United States,
nor the Commission, nor any perroo acting on behalf of the Commission:
A.
Makes
any warranty or representation,
completeness, any
expressed or implied, w i t h respect t o the accuracy,
or usefulness of the information contained i n this report, or that the use of
information,
apporotus, method, or process disctosed i n this report may not infringe
privately owned rights; or
B,
Assumes any l i a b i l i t i e s with respect t o the use of, or for damages resulting from the use of any information, apparatus, method, or process disclosed i n this report.
A s used i n the above, "perron acting on behalf of the Commission" includes any employee or contractor of the Commission. or employee of such contractor, t o the extent that such employee or
contractor
of
the Commission,
or employee of such contractor prepares, disseminates, or
provides access to, any information pursuant to his employment or contract with the Commission, or h i s employment with such contractor.
.
3
INTRODUCTION
Prediction of the temperature kinetic behavior and the control rod requirements in operating the =RE at AiLl power requires knowledge of the reactivity effect of nonuniform temperature differences in fie1 'and graphite throughout the core. Since detailed studies have recently been 1
W e of the core physics characteristics at isothermal (1200OF) conditions, perturbation theory provides a convenient approach to this problem. He::e, the perturbation is the change in fael and graphite temperature profile:; from the isothermal values. In the following section the analytical nethod is presented. Specific CalciLations for the MSRE are discussed in the final section of this report. ANALYSIS
.
' v
The mathematical problem considered in this section is that of describing the temperature reactivity feedback associated with changes ir, reactor power by use of average fie1 and graphite temperatures ra-the:: than the compiete temperature distributions. The proper averages a r e derived by considering spatidly uniform temperatures which give the sane reactivity effect as the actuaJ,profiles. The reactivity is given by the fallowing first-order perturbation formula:3
Equation 1 relates a s m a l l change in the effective multiplication constant ke to a perturbation 6M in the coefficient matrix of the two-grou-? equations, The formulation of the two-group equations which defines the terms in (1) is:
I 4
4
-
or, writing (2) i n matrix form: &lj-
-m= ke
o
where :
A = [ ,
- D2
-332.
* =
(4
*
In equation 1, @ i s the adjoint flux vector,
m d the bracketed terms represent the scalar products;
.
I
5
m
A similar expression holds for [@
it-
, Pb]
w i t h the elements of the F matrix
replacing 6M. Temperature Averaging S t a r t i n g with the c r i t i c a l , isothermal reactor (ke = 1, T = To),
consider the e f f e c t s on the neutron multiplication constant of changing the f u e l and graphite temperatures t o Tf(r,z) and T (r,z). â‚Ź5
These e f f e c t s
may be treated as separate perturbakions as long as 6k/k produced by each change i s s m a l l . I '
If0
Consider first t h e f i e l .
As the temperature s h i f t s from
t o Tf(r,e), with the graphite temperature held constant, the reactivity
chemge is:
If the nuclear coefficients comprising the matrix M do not vary rapidly
with temperature, 6M can be adequately approximated by the first tenn i n an expansion about,,T* "
i.e.,
1x1 equation ( 6 ) , m i s the coefficient matrix:
1 T aM22
Thus :
Tf = Tfo
6
Now, consider a second s i t u a t i o n i n which t h e PJel temperature i s changed
from Tfo t o Tf
*
.rr.
uniformly over the -.ore. The r e a c t i v i t y change i s :
6k
*-
I
E-
[a*,
Tfo --+ Tf
[&*I
m3
ml
*
Ve may define t h e f u e l nuclear average temperatwe T as t h e uniform f temperature which gives rise t o t h e same r e a c t i v i t y change as the aetua:l
temperature profile i n t h e corep i.c.;
*
i s independent of p o s l t i m it may be factored from the s c a l a r Since T f product in the l e f t hand side sf (10):
(:11)
In an analogous fashioll, t h e n x l e a r average temperature f o r the graphite i s defined by:
vith
b
7 t
'clr
Temperature Coefficients of Reactivity Importance-averaged temperature coefficients may be derived which m e consistent with the definitions of the nuclear average temperatures.
Again consider t h e f u e l region.
L e t the i n i t i d r e a c t i v i t y perturbakion
correspond t o T assuming a p r o f i l e Tfl(r,z) about the i n i t i a l value Tfo: f
If a second temperature change i s n m made (Tf
-P
Te),
subtracting, and using the d e f i n i t i m (11)of the fuel nuclear average temperature :
"=
E
-
c.", m(Tfo)
(T.$2(r,z)
E@*,
- Tfl(r,z) )d
(16)
=I
This leads t o t h e r e l a t i o n d e f h i n g the f i e 1 temperature coefficient of
reactivity:
and a similar definition may be made of the graphite temperature coePfi,cient :
8 s
It may be seen from the preceding analysis that the problem of obtaining nuclear average temperatures is reducible to the calculation of the weighting function contained in the sca;lar product:
= $
-*
dV G(r,z)
Reactor
*
(19 1
G(r,z) = i[, m CP
In the two-gruup formulation, the e.xplicit form of the m matrix is: \
>.&ere the derivatives a r e taken with respect to the fie1 or graphite temperatures in order to obtain G Dr G respectively. It is conf g' irenient for numerical evaluation to rewrite the derivatives in logarithmic form; e.g.)
- =a2 - dT
Xa2
Bc=a2)
%-here
dtiplication iqlicit in (19):
ad}
(Fast leakage ) (Slowing do-m) 1 ,
9
r
.
(Resonance Abs )
(Resonance f i s s i o n )
(me-
fission)
r
(Thermal Abs.)
( T h e m Leakage)
To evaluate the leakage terms i n
(a), a f'wther
simplification i s
(a)
013-
tained by using the c r i t i c a l i t y relations f o r the unperturbed fluxes:
Inserting the above relations i n t o (U) and regrouping terms r e s u l t s in:
8
#
10 m
Equation (24) represents the form of the weighting functions used i n ilumerical calculations f o r the MSRE.
The evaluation of t h e coefficients
f3 f o r fuel and graphite i s discussed i n the fallowing section.
APPLICATION TO THE MSIiE:
RESULTS
Utilizing a calculational model i n which the reactor composition was
assumed uniform, Nestor obtained values f o r the f u e l and graphite tempera-
ture coefficients.4
The purpose of‘ the present study w a s t o account f o r
t h e s p a t i a l variations i n temperature and composition i n a more exact
In t h i s connection, two-group, 19-region calculations of fluxes and adjoint fluxes have recently been made f o r t h e MSRE,l using the Equipoise-3 2,3 program. These studies refer t o fie1 s a l t which conk i n s no thorium. The geometric m c d e l representing t h e reactor core configuration i s indicated i n Fig. 1. Average compositions of each region i n t h i s figure are given i n Table 1. %e resulting flux distributions, which are t h e basic data required for calcul.e,%io~~ of t h e %eqere%ureweightip4 fi;n,c+,ims, ere given i n Figs. 2 and 3. These figures represent axial and radial traverses, along l i n e s which intersect i n the regior, J of Fig. 1. The intersection point occurs close t o the point R = 7 in., 2 = 35 i n . of the g r i d of Fig. 1, ctnd corresponds t o the position of maximum thermal flux i n the reactor. To compute the temperat-ae weighting f’unctions (Eq, 24), the logarithmic derivatives fashion.
T = Tf, T g
For each group must be numerically evaluated. Here, certain simplifying ap$roximat ions may be made. It w a s assumed t h a t the diffusion and slowing down parameters D and vary with temperature only through the fuel
5
and graphite densities and not through t h e microscopic cross sections. mus, using:
c=
2 4. 9
f
.XIn
*
1,
Fig. I
ORNL-LR-Dwg 74858
19-Region Core Model for Equipoise Calculation
Table 2.
Nineteen-Region Core Model Used i n EQUIPOISE Calculations f o r MSRE -
radius
(in. m-1
-
A B
inner
-7-
0
29.00 0
C D E
3 .oo
F
28.00
G
3 .MI 3-00
H
I J K L
M
N 0
P Q
3.00
27 975 3 .oo
2 094
0 2 -94 0 0 0 0
R
0
S
0
1 outer
-
bottom
29.56 74 992 29.56 9-14 29.56 -10.26 29 00 67 047 28.00 66.22 29.00 0 28 .@@ 65 * 53 27 *75 64.59 28.00 0 27 e75 5-50 3 .oo 5.50
-
2 .a4
27 975 27 075 29.00 29.00 2-94 2 -94 2 -94
top
76 .o4 74 9 9 2 -9 -14 74 -92 67 -47 67 -47 66 -22 65 53 65 53 64.59 74 -92 9
f'uel
_ I _
0 0 0
Composit ion (volume percent ) araehite
INOR
93 07
3 -5
2.8
0
0 0 0.2
Downcomer
100
Core can Core Simulated thimbles
100
9.6 63 03 0
22.5 0
5.4 36.5 0
77 05 0
100 0
0
100
74.4
0
2.00 0
5.50 2.00
22.5 23 07
77.5 76.3
0 0
66.9 90.8
15.3
17.8 9.2
9
74 992 66.22 65 53
Vessel t o p Vessel sides Ves sel bottom Upper head
100
25.6
0
Region Represented
100 100
64.59
-1A1
~-~
0 0 0 0
2.00
-1e41 -9 -14 66.22 65 53 64 59
~
100
89.9 43 .a
0 0 10.1
56.2
0 0 0
Central region Core Horizonta,l stringers
Bottom head
lk
OR NL-LR -Dwg 74860
8
:&ere f, g, and In r e f e r t o fuel salt, graphite, and Inor; m
1 dCtr - -
D @(D) = 1- d -z D dT
'tr
dT
f Y
=tr
(Groups 1, 2 )
-
'tr i n the above approximation, the effect of temperature on the Inor density
has a l s o been neglected. The n m r i c a l values used for the density (:oe r t i c i e n t s of fuel and graphite were:
f i e 1 temperature:
p(ps) =
1 (IPS -
@(ps) =:
@(PJ
~o-~/oF
4Tf
Ps
Graphite temperature:
- 1.26
=
1 = Ps (ITg
0
Pg
- 4.0
'IPS
-s
= 1 %
=
x
lom6/',
The remaining coefficients were calculated as follows:
@ ( v c , ) = B(Ps) +
B(vcfl)
For the E R E f i e l , t h e temperature coefficients of the f'uel resonajace cross sections, @ ( v g f l ) and B(c&),
are of the orrfer of 1 0-5 /OF, a factor
16
.
I n addition, resonance fissions contribute only about 12$ of the t o t a l fissions i n the reactor. ~ h u s@(vua) and p(ual) were neaecteii i n the present CalcUlations. For the thermal fission and absorption terms: of ten smaller than the fuel density coefficient.
"
In
1
V
i n the abwe expression, the thermal cross section o f t h e salt was separated i n t o components; one was U235 and the other was the remaining salt constituents (labeled s ) . This was done because of the non-l/v behavior
of the U235 cross section. Evaluation of the t e q e r a t u r e d.erivatives of the thermal cross sec-
%ions gives r i s e t o the question of the relationship between the neutron temperature Tn and t h e fuel and $raphiSe temperatures, Tf and T see;? frm
Q
.
As
Table 1, except f o r the cr;.ter regions of the core, graphite
comprises about 75 t o 77 per cent of the core volume. Tollowing Nestor's cdlculations, it was assumed t h a t within these regions the neutron temperature was equal t o the graphite temperature.
These regions comprise
the major part of the core volume (Table 1). For those external regions
with fuel volutne fractions greater tkian 50% (an a s b i t r a r i l y chosen d i viding point), the neutron temperatwe was assumed equal t o the bulk temperature; i .e.,
T + v
Tn'vff
g
T g
where v is the volume fraction. The cross sections were given by
with b 2' 0.9 for U235 and b = O.:iO
for the remaining salt constituents;
on this basis the result for f3 is:
) 3
f 3 ( ' a 2
de 1 a2 = U dT
1
a2
3
-
dT
b - dTn Tn dT
-
Thus, for the fuel temperature: de
g(ca2)
1 a2 = -
=
0
ua2 dTf
I
v b f
-
-
T
0
To = 1200'F For the graphite temperature:
f3(ua2)
=
1 Q
e = -
=t
g
- b-
vf
v b
L
0.50
TO
Vf
> 0.50
TO
Based upon the preceding approximations, the results of calculations of the fie1 and graphite temperature weighting functions a r e platted in F i g s , 4 through 6 , Figure 4 is a radisl plot, and Figs. 5 and 6 are axial plots of the f'uel and graphite f'unctions, respectively. In the latter
Unclassified
ORNL-LR-Dwg 74862
20
0 RN L -LR -Dwg 74863 Unclassified
t
\.
-
figures, the "active core" i s the vertical: section over which the uniform
rr
reactor appraximation, sin2 H (z f m c t ion
+
zO) may be used t o represent the weight
Temperature coefficients of r e a c t i v i t y consistent w i t h these weighting f'unctions were calculated from Eq. 17. Tlzese values are l i s t e d i n Table 2, along w i t h the coefficients obtained fsom the uniform reactor model as used for the calculatiozx reported i n reference
(4)* In
form fuel volume fraction of 0,225 was used; i.e.,
the l a t t e r case, a uni-
that of the l a r g e s t
region i n the reactor.
Also, the effective radius and height of the reactor were chosen as closely as possible t o correspond t o the points where the Equipoise thermal fluxes eartsapolate t o zero from the active core. Table 2, MsRF: -ratwe
Coefficients of Reactivity Fuel 10"5/OF
Perturbation theory Homogeneous reactor mociei
-4 *45 -4 .ij
Gra h i t e 10' /OF
3
-7 027 -6.92
22
APPIJCATION TO
THE MSRE
- DISCUSSION OF RESULTS
The r e l a t i v e importance of temperature changes on the r e a c t i v i t y varies from region t o region i n the realctor due t o two e f f e c t s . One i s the change i n the nuclear importance, as measured by the adjoint fluxes. The other i s the variation i n the l o c a l i n f i n i t e nultiplication constant as the fuel-graphite-Inor 8 camposition varies.
The l a t t e r e f f e c t leads
t o the discontinuities i n the tenperatwe weighting function.
For ex-
ample, region 0 of Fig. 1 contains a r e l a t i v e l y large volume fraction of Inor 8 (see Table 1). This results i n net s u b c r i t i c a l i t y of t h i s region i n the absence of the net, inleakage of neutrons fromthe surraunding regions.
Thus the r e a c t i v i t y effect of a temperature increment i n t h i s
region has the opposite sign frm t h a t of the surroundings. It should be understood that the v a l i d i t y of the perturbation caicul a t i o n s i n representing the l o c a l temperatare-reactivity e f f e c t s depends upon haw accurately the original f l u x distributions and average region c a p o s i t i o n s represent the nuclear behavior of the reactor.
Also, it
was necessary t o assume a specific r e l a t i o n between the l o c a l neutron temperature and t h e l o c a l fuel and graph-%etemperatures,
More exact
calculations w o u l d acccrmt f o r a coc.%inuausc h a q e i n thermal spectrum as the average fuel volume fraction charges.
This would have the effect
of "rounding o f f " the d i s c o n t i m i t i e s i n the temperature weighting fluac-
%ions. These effects, ho-ever s f i ~ a d3e r e l a t i v e l y minor and the resd%s
presented herein should be a reasombiy good approximation.
Nuclear average temperatures h8;sre been calculated fmsm computed MSRE temperature distributions, usirg $be weigkting functions given i n t h i s report and are reported elsewhere. 5
23
NOMENCLATURE rrc
ke
S t a t i c multiplication cmstant
A
Coefficient matrix of absorption plus leakage terms i n diff’usion equations
F
Coefficient matrix of neutron production terms i n diffusion equations
M
Matrix of nuclear coefficients i n diffusion equations f o r unperturbed reactor
Temperature derivative of M matrix Temperature weighting flrnctioas of position: Subscripts f = f i e 1 s a l t , g = graphite
D
j T
T*
Diffusion coefficient,, group j = 1 , 2 Temperature: Subscri.pts f = fuel salt, g = graphite, o = i n i t i a l , n = effective thermal neutron temperature Nuclear average temperature
dV V
Volume f’raction Neutron f l u x , groups j = 1, 2 Neutron f l u x vector
’lsj*
Adjoiat flux, groups j = 1, 2 Ad j o i n t flux vector
P PS
Reactivity Density of f i e 1 salt Density of reactor graphite Macroscopic cross section, groups j “1, 2; Subscripts a = absorption, f = fission, R = removal Temperature derivative of in x Number of neutrons per fission
24
REFERENCES
1. MSRP b o g . Rep. Augu s t 31, 1962, (ORNL report t o be issued).
3. C. W. Nestor, Jr., Equipoise 3, ORNL-3199 Addendum, June 6, 1962.
4.
NRP Frog. Rep. Augu s t 31, i
s QRNL-3215,p. 83.
5. J. R. Engel and P. N. Haubenreich, Temperatures i n the
Core During Steady State Power Operation, ORNL-TM-378 ( i n preparation).
W
Internal Distribution 1-2. MSFP Director's Office. Rm. 2l9, B l d g . 9204-1 3. G. M. Admson
4. L, G. Alexander 5. S . E. Beall 6. M. Bender 7. C. E. Bettis 8. E. S. Bettis
. .
90 D S Billington 10. F. F. Blankenship 11 E. G. Bohlmann 1 2 . S. E. Bolt 1 3 C. J. Borkowski 14. C. A . Brandon 1 5 F. R. Bruce 9
16. 17
0. S. 3.8. T. 19 J. 20. W. 21. L. 22. G. 23. J. 9
24.
F.
25
J.
26 27
28.
R. D. N.
29 30 31 32
J. E. W.
.
33
D A. J. C. R. B. W. A. R. P. C. P. E. H. P. L.
W. Burke
Cantor E. Cole A . Conlin H. Cook T. Corbin A. Cristy L. Crowley L. Culler H. D e V a n G. Donnelly A. Douglas E. Dunwoody R. Engel P. Gpler K. Ergen E:. Fergison 2 . Fraas H. *ye H. Gabbard B. Gallaher L. Greenstreet R. Grimes
34 35 36 37 38 G. 39 40. H. 41. H. 42. S. N. 43 44. C. W. 4% P. 46. N. 47 48. J. P. 49 W. H. 9
0
Grindell Guymon Harley Harrill Haubenreich Hise Hoffman
50. P, R. Kasten 31.
%.
9.R.
B. Lindauer
59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81.
I. Lundin N. Lyon G. MacPherson
82.
83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93.
94.
Jarvis Jordan
98.
Howell
S. S. Kirslis
55. J. $I. Krewson 56. J . A . Lane 57. W. J. Leonard
95. 96. 97.
Holz
R . J. K e d l
52. M. T. Kelley 53. M. J. Kelly
M. R. H. F. E. W. H. C. A. E. R. J. T. W.
P, H. A.
W. B.
.
J M.
R. T. H. A.
J.
C. Mainensciein? R. Mann B. McDonald F. PIckffie K, M c G l O t h l ~ J. Miller C. Miller L. Moore C. Mqers E. Northup R . OsSorn Patriarca R . Payne M. Ferry B. Pike E. -Prince L. Redford Richardson C. Robertson K. Roche W. Savage W. Savolainen E. Savolainen
D. S c o t t . C. H. Secoy J . H. Shaffer M. J. Skinner G. M. Slaughter A. N. Smith P. G. Smith I. Spiewak J. A. Swartout A . Taboada J. R . Tallackson R. E. Thoma D. B. Trauger W. C. U lr ic h
26
Internal Distribution
- cont'd
B. S. Weaver C F. Weaver B, H. Webster A . M. Weinberg J. C. White L. V. Wilson C. H. Wodtke Reactor Division Library lo8-log, Central Research Library 110. Document Reference Section 1u-u3. Laborat ory Records 114. ORNL-RC
99
100. 101 * 102. 103. 104. 105. 106-107
.
External
115-116, D. F. Cope, Reactor Division, AEC, OR0 117. H. M. Roth, Division of Research and Development, AEC, OR0 3.18. F. P. Self, Reactor Division, AEC, OR0 119-133. Division of Technical Information &tension, AEC, OR0 134. J. Vett, AEC, Washington 135. W. L. SmaUey, AEC-OR0