OAK RIDGE NATIONAL LABORATORY
operated by

UNION CARBIDE CORPORATION @
for the |

U.S. ATOMIC ENERGY COMMISSION

ORNL- TM- 379
copYno. - [/ G

DATE - October 15, 1962

 

TEMPERATURE AND REACTIVITY COEFFICIENT AVERAGING IN THE MSRE

B. E. Prince and J. R. Engel

MAST

ABSTRACT

Use is made of the concept of 'nuclear average temperature"” to re-
late the spatial temperature profiles in fuel and graphite attained during
‘ high power operation of the MSRE to the neutron multiplication constant.
Ty Based on two-group perturbation theory, temperature weighting functions
. for fuel and graphite are derived, from which the muclear aversge tempera-

tures may be calculated. Similarly, importance-averaged temperature co-
efficients of reactivity are defined. The values of the coefficients cal-
culated for the MSRE were -4.b x 10"2/°F for the fuel and -7.3 x 10=D for
the graphite. These values refer to & reactor fueled with salt which does
not contain thorium. They were sbout 5% larger than the values obtained
from a one-region, homogeneous reactor model, thus reflecting the varia-
tion in the fuel volume fraction throughout the reactor and the effect of
the control rod thimbles on the flux profiles.

r.
NOTICE
¢ This document contains information of a preliminary nature and was prepared

primarily for internal use at the Qak Ridge National Laboratory. It is subject
to revision or correction and therefore does not represent a final report. The
information is not to be abstracted, reprinted or otherwise given public dis-
semination without the approval of the ORNL patent branch, Legal and infor-
mation Control Depariment.
 

 

 

LEGAL NOTICE

This report wos prepared as an account of Government sponsored work, Neither the United States,

ner the Commission, nor any person acting on behalf of the Commission:

A. Makes any warranty or representotion, expressed or implied, with respect to the accuraey,
completeness, or usefulness of the information contained in this report, or that the use of
any information, apparatus, method, or process disclosed in this report may net infringe
privately owned rights; or

B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of
any information, apparatus, method, or process disclosed in this report,

As used in the above, *‘parson acting on behalf of the Commission’ includes any employee ot

contractor of the Commission, or employee of such contractor, to the extent that such employee

or contractor of the Commission, or employee of such contractor prepares, disseminotes, or
provides access to, any information pursuant to his employment or contract with the Commission,

or his employment with such contracter,

 

 
INTRODUCTION

Prediction of the temperature kinetic behavior and the control rod
requirements in operating the MSRE at full power requires knowledge of
the reactivity effect of nonuniform temperature differences in fuel and
graphite throughout the core. Since detailed studies have recently been
made of the core physics characteristics at isothermal (1200°F) conditions,l
perturbation theory provides a convenient approach to this problem. Here,
the perturbation is the change in fael and graphite temperature profiles
from the iscthermal values. In the following section the ahalytical
method is presented. Specific calculations for the MSRE are discussed
in the final section of this report.

ANALYSIS

The mathematical problem considered in this section is that of
describing the temperature reactivity feedback associated with changes
in reactor power by use of average fuel and graphite temperatures rathe:
than the complete temperature distributions. The proper averages are
derived by considering spatially uniform temperatures which give the same
reactivity effect as the actual profiles. The reactivity is given by the

following first-order perturbation formula:3

¥*
8k k [&, oMB]
e -~ e
o= o T ()
Te ® , 1]
Equation 1 relates a small change in the effective multiplication con-
stant k.e to a perturbation 6M in the coefficient matrix of the two-group
equations. The formulation of the two-group equations which defines the

terms in (1) is:

 

 

vZ v
2 £l £2 _ ,
- Dl V ¢l + 2&1 ¢l = ke ¢l = ke ¢2 = 0 (2‘3')

D,V Bk T By - Ty Ay = O (20)
or, writing (2) in matrix form:

M= M- 2= O (3)
e
where:
2
(-nlv +zm+zal) 0
A = .
- Iy =D,V + Bp
v Zfl v Zfe
F =
0 0
A
d =
7
OF
6M = O0A - X

*
In equation 1, & is the adjoint flux vector,

3 = (8 &)
and the bracketed terms represent the scalar products;
oM, oM, A
[Q*’ o] = l;eactor(fil* ¢2*) av

oMy My, A

fReactor (¢1* oM, B+ ¢1* oM, Bp ¢2* oMy B

+ ¢2* M, #,) av (&)
*
A similar expression holds for [® , PP} with the elements of the F matrix

replacing OM.

Temperature Aversging

Starting with the critical, isothermal resactor (ke =1, T = TO),
consider the effects on the neutron multiplication constant of changing
the fuel and graphite temperatures to Tf(r,z) and. Té(r,z). These effects
may be treated as separate perturbations as long as 6k/k produced by each
change is small. Consider first the fuel. As the temperature shifts firom
v, to Tf(r,z), with the graphite temperature held constant, the reactivity

fo
chenge is:

*
ok _ [®, oM(T, ~T,) ] (s)
pf = Xk - 5

- *
Tpo = Tf(r,z) [, Fb]

 

If the nuclear coefficients comprising the matrix M do not vary rapidly
with temperature, OM can be adequately approximated by the first term

in an expsnsion about T, , i.e.,

6M(Tfo - Tf) - m(Tfo) (T:E‘ - Tfo) (6)

Tn equation (6), m is the coefficient matrix:

oMy M, ,
M, T,
m(T,,) = (7)

ale 8M22

 

T f f fo
Thus:
*
ok ['1) ’ m(TfO ) (Tf - TfO ) (I’]
o - - . (8)
Now, consider a second situation in which the fuel temperature is changed

*
from Tfo to Tf uniformly over the core. The reactivity change is:

& oM(T,_ - Tf*) 5]

 

 

ok = -

£\t - Tp &, 8]
R m(ry) (T - 7)) @] (o)
= - o 9

*
we may define the fuel nuclear average temperature Tf as the uwniform
temperature which gives rise to the same reactivity change as the actual

temperature profile in the core, i.=.;
¢ ) (r.” Yo = |& ) (T, (r,z) Yo | (o)
3, m(Tfo (Tf - T, <I>J = |®, m(TfO e (r,2) - Tfo) | )

*
Sirce Tf is independent of position 1t may be factored from the scalar
product in the left hand side of (1d):

2t - @, m*(TfO) T, (r,z) @] o)
&, m (Tg,) 2]

 

In an analogous fashior, the naclear average temperature for the
graphite is defined by:
%
* (@ y 1 (Tgo) Tg(r,z) ®]

T = _ (12)
g [, m(T,,) @]

 

with

OM, M, ,

g Tg
m(?go) =
BMQJ_ aM22
T, Ty

3
or
I
R 3
0
Temperature Coefficients of Reactivity

Importance~-averaged temperature coefficients may be derived which
are consistent with the definitions of the nuclear average temperatures.
Again consider the fuel region. Let the initial reactivity perturbation
correspond to Tf assuming a profile Tfl(r,z) about the initial value Tfo:

@, (T, ) (Tg (r,2) - Ty ) 8]
= - »* (lh‘)
3, F]

 

Py
1f a second temperature change is now made (Tf a-Tfe),

[é*) m(TfO) (Tfe(r‘?z) = Tfo) @]
@, F]

 

(15)

Ps

subtracting, and using the definition (11) of the fuel nuclear average

temperature:
« .
[ mrg) (1p602) - 14 602) g o
@ , F&]

&

VP,

 

*
= - e (T, - Ty)

*

@, M)
This leads to the relation defining the fuel temperature coefficient of
reactivity:

&, n(T,,) @]

L0 = (172)

*
8T, &, 7]

and a similar definition mey be made of the graphite temperature coeffi-

cient:

2 (17b)
It may be seen from the preceding analysis that the problem of ob=- —
taining nuclear average temperstures is reducible to the calculation of
the weighting function contained in the scalar product:

@, m] = fReactor W (F, my B+ B my B+ By myy )+ B myy 6
= j~ av G(r,z) (18)
Reactor
G(r,z) = & uwd (19)

In the two-group formulation, the explicit form of the m matrix is:

a d
—a-T_(-Dlv2+le+za.l-vzfl)T=To 'Efl?’("zfa)T=To \
m o= (20) .
d d 2
- & e & DY Il
o 0/ &

where the derivatives are taken with respect to the fuel or graphite

temperatures in order to obtain Gf

venient for numericel evaluation to rewrite the derivatives in loga-

or Gg’ respectively. It is con-

rithmic form; e.g.,

 

a >
a2
aT - Zé2 5(252)
where
W 1 4z, ) d (4n 29.2)
‘a2 Eée aT aT

Thus, carrying out the matrix multiplication implicit in (19):

6(r2) = 80y { (- 2,77 )} (Fast 1eakage)

+ plZy) { Zr1 ¢1* A= Ty ¢2* ¢1} (Slowing down)
+ B(Z) { . 8" ¢l} (Resonance Abs.)

+ a(vzfl) {- v 2, ¢l* ¢1} (Resonance fission)

+ B(viy,) {- vV I, ¢l* 952}’ (Thermal fission)

+ B(5,) { z, ¢2* ¢2} (Thermal Abs. )

+ p(D,) {¢2* (-D?_v2 ¢2)} (Thermal Leakage) (21)

To evaluate the leakage terms in (21), a further simplification is ob-
tained by using the criticality relations for the unperturbed fluxes:

- D, 7" o= ~Zp Pt In g (22)
-Dlvzpjl: —ZRl¢l—2al¢l+vZfl¢l+vZf2¢2 (23)

Incerting the above relations into (21) end regrouping tems results in:
a(em) = { (8 () - 80,0 )3y + (B (5 - 80)) 3,
- (Bvzg) - 30) ) vig 4" 4
r{(80y) - B05)) voes} 4" 4,
{ () - 8(5y)) 7y } 8" 4

A (i) - 50p) .0 17 4, o)
10

FEquation (24) represents the form of the weighting functions used in
numericel calculations for the MSRE. The evaluation of the coefficients
B for fuel and graphite is discussed in the following section.

APPLICATION TO THE MSRE: RESULTS

Utilizing a calcuwlational model in which the reactor composition was
assumed uniform, Nestor obtained values for the fuel and graphite tempera-
ture coefficients.u The purpose of the present study was to account for
the spatial variations in temperature and composition in a more exact
fashion. In this connection, two-group, 19-region calculations of fluxes
and adjoint fluxes have recently been made for the MSRE,l using the
Equipoise-3A 253 program. These studies refer to fuel salt which con-
tains no thorium. The geometric model representing the reactor core
configuration is indicdted in Fig. 1. Average compositions of each
region in this figure are given in Table 1.

The resulting flux distributions, which are the basic data required
for calculation of the temperature weighting funetions, are given in
Figs. 2 and 3. These figures represent axial and radial traverses, along
lines which intersect in the regior J of Fig. 1. The intersection point
occurs close to the point R =7 in., Z = 35 in. of the grid of Fig. 1,
and corresponds to the position of maximum thermal flux in the reactor.

To compute the tempersture weighting functions (Eq. 24), the log-

arithmic derivatives

o
1
M|
|8
+3
0
3

x = D, vZ, I, I

For each group must be numerically evaluated. Here, certain simplifying
approximetions may be made. It was assumed that the diffusion and slow-
ing down parameters D and Zh vary with temperature only through the fuel
and graphite densities and not through the microscopic cross sections.

Thus, using:

2 o= Zf o+ Zg + EIn
ORNL-LR-Dwg 74858

11

Unclassified

 

»

19-Region Core Modei for Equipoise Calcuiation

Fig. |
Table 2.

Nineteen-Region Core Model Used in EQUIPOISE Calculations for MSRE

 

 

 

 

 

radius Z Composition
(in.) (in.) (Volume percent) Region
Region inner outer bottom top fuel graphite INOR Represented
A 0 29.56 Th.92 T76.04 0 0 100 Vessel top
B 29.00 29.56 - 9,14 Th .92 0 0 100 Vessel sides
c 0 29.56 -10.26 -9.14 0 0 100 Vessel bottom
D 3.00 29,00 67 .47 T4 .92 100 0 0 Upper head
E 3.00 28.00 66 .22 67 .47 93.7 3.5 2.8
F 28.00 29.00 0 67 .47 100 0 0 Downconmer
G 2.00 28,00 65.53 66.22 al,6 5.4 0
H 3.00 27.75 64 .59 65.53 63.3 36.5 0.2
I 27.75 28.00 0 65.53 0 0 100 Core can
J 3.00 27.75 5.50 64 .59 22.5 7.5 0 Core
K 2.9 3.00 5.50 .92 0 0 100 Simulated
thimbles
L 0 2.94 2.00 64 .59 25.6 Th.b 0 Central
region
M 2.94 27.75 2.00 5.50 22.5 T7.5 0 Core
N 0 27.75 0 2.00 23.7 76.3 0 Horizontal
stringers
0 0 29.00 -1.41 0 66.9 15.3 17.8
P 0 29.00 -9.14 -1 .41 90.8 0 9.2 Bottom head
Q 0 2.94 66.22 Th.92 100 0 0
R 0 2.94 65.53 66.22 89.9 10.1 0
S 0 2.94 64 .59 65453 %3.8 56.2 0

 

.

gl
ORNL-LR-Dwg 74859
Unclossified

13

.

 
ORNL-LR-Dwg 74860

fassifi

i

 

 
vhere £, g, and In refer to fuel salt, graphite, and Inor;

dp

B(D) = 5 3%

-
Nt

L
D

£
Z%r

z:'tr

 

~
-

’ g
5 P (:tr‘>'+ P Ztr :) *

Blp,) +

-+ (3.7

(Groups 1, 2)

g
z%r

ztr

 

B(pg)

In the above épproximation, the effect of temperature on the Inor density

has also been neglected.
efficients of fuel and graphite were:

Fuel temperature: Bp. ) = 2 8
S o 4
S £
ip
l {
Blog) = o= o
g T
dp
- 1 s
Graphite temperature: B(p ) = — —
S p_ 4T
5 8
de
1
Bloy) = == 7=
g &

.The remaining coefficients were

1

B(vZ,, )

ey

B(z,, )

B(p,

B(p,

The numerical values used for the density co-

—
—

-1.26 x 10“”/°F

I

- L4,0x 10-6/°F

calculated as follows:

)+ B(voy )

)+ plo,)

For the MSRE fuel, the temperature coefficients of the fuel resonaace
cross sections, B(vo&l) and B(Gél)’ are of the order of 10"5/°F, a factor
16

of ten smaller than the fuel density coefficient. In addition, resonance
Tissions contribute only sbout 12% of the total fissions in the reactor.
Thus 6(Vaf1) and B(Uél) were neglected in the present calculations.

For the thermal fission and absorption terms:

B(vZ,,) = Blo ) + B(voy,)

2>

Z
~ a2 25
B(Z,,) 2:2 [B(ps) + B(o, )]

 

+ 2 [B(ps)+ B(cra;)}

v 2 [B(pg)+ a(o-azg)}

t oy B(G.nln)

o

In the above expression, the thermal cross section of the salt was sepa-
rated into 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 sectlon.

Evaluation of the temperature derivatives of the thermal cross sec=
tions gives rise to the question of the relationship between the neutron
temperature Tn and the fuel and graphite temperatures, Tf and Té. As
seen from Table 1, except for the oubter regions of the core, graphite
comprises about 75 to 77 per cent of the core volume. TFollowing Nestor's
calculations, it was assumed that within these regions the neutron tem-
perature was equal to the graphite temperature. These regions comprise
the major part of the core volume (Table 1). For those external regions
with fuel volume fractions greater than 50% (an arbitrarily chosen di-
viding point), the neutron temperatuvre was assumed equal to the bulk

temperature; i.e.,
17

T ~ v, T +v T
g g

 

- n ff
where v is the volume fraction. The cross sections were given by
To b
a2 = c.a.El(To)<'f'_>
n
with b = 0.5 for y232 and b = 0.%50 for the remaining salt constituents;
on this basis the result for B is:
Blo ) = = Va2 _ ( 1 %o ar_
a2 O'a‘2 aT O'a‘2 dTn It
_ . b Ty
Tn aT
Thus, for the fuel temperature:
. do
8o ) = -——-Gl ET—?‘—Q-= 0 Vo< 0.50
8 a2 f
v_b
= - X v, > 0.50
T f
0
T = 1200°F
o
¥or the graphite temperature:
ao
1 a2 b <
Blo.,)) = =— === -+ v. < 0.50
a2 0.0 dTé To f
g ®
= - To 'V'f > 0.50

Based upon the preceding approximations, the results of calculations
of the fuel and graphite temperature weighting functions are plotted in
Figs. 4 through 6. Figure L4 is a radial plot, and Figs. 5 and 6 are axial
plots of the fuel and graphite functions, respectively. In the latter
ORNL-LR-Dwg 74861

 

nclassified

 
 
19

ORNL ~LR-Dwg 74862

Unclassified

 
 

pG!“SSDIDUQ
£967Z BMQ-Y1-TNIO
figures, the "active core" is the vertical section over which the uniform
reactor approximation, sin2 g-(z + zo) may be used to represent the weight
function.

Temperature coefficients of reactivity consistent with these weighting
functions were calculated from Eq. 17. These values are listed in Table 2,
along with the coefficients obtained from the uniform reactor model as used
for the calculations reported in reference (4). In the latter case, a uni-
form fuel volume fraction of 0.225 was used; i.e., that of the largest
region in the reactor. Also, the effective radius and height of the re-
actor were chosen as closely as possible to correspond to the points where
the Equipoise thermal fluxes extrapoclate to gero from the active core.

Table 2. MSRE Temperature Coefficients of Reactivity

 

Fuel Gra%hite
10=2/°F 10™2/°F

T ———rrvd - r——

Perturbation theory -l 45 -7.27

Homogeneous reactor model -4 .13 -6.92

 
22

APPLICATION TO THE MSRE -~ DISCUSSION OF RESULTS

The relative importance of temperature changes on the reactivity
varies from region to region in the reactor due to two effects. One is
the change in the nuclear importance, as measured by the adjoint fluxes.
The other is the variation in the local infinite multiplicetion constant
as the fuel~-graphite-Inor 8 composition varies. The latter effect leads
to the discontinuities in the temperature weighting function. For ex-
ample, region O of Fig. 1 contains a relatively large volume fraction of
Inor 8 (see Table 1). This results in net subecriticality of this region
in the absence of the net inleakage of neutrons from the surrounding
regions. Thus the reactivity effect of a temperature increment in this
region has the opposite sign from that of the surrocundings.

It should be understood that the validity of the perturbation caicu-
lations in representing the local temperature-reactivity effects depends
upon how accurately the original flux distributions and average region
compositions represent the nuclear behavior of the reactor. Also, it
was necessary to assume a specific relation between the local neutron
temperature and the local fuel and graphite temperatures. More exact
calculations would account for a continuous change in thermal spectrum
as the average fuel volume fraction changes. This would have the effect
of "rounding off" the discontinuities in the temperature weighting func-
tions., These effects, however should be relstively minor and the re-
sults presented herein should be a reasonably good approximation.

Nuclear aversge temperabures hsve been calculated from computed
MSRE temperature distributions, usirg the weighting functions given in
5

this report and are reported elsewhere.
23

NOMENCLATURE

Static multiplication constant

Coefficient matrix of absorption plus leaskage terms

in diffusion equations

Coefficient matrix of neutron production terms in
diffusion equations

Matrix of nuclear coefficients in diffusion equations
for unperturbed reactor

Temperature derivative of M matrix

Temperature weighting functions of position:

Subscripts £ = fuel salt, g = graphite
Diffusion coefficient, group Jj = 1,2

Temperature: Subscripts f = fuel salt, g = graphite,
o = initial, n = effective thermal neutron temperature

Nuclear average temperature
Reactor volume element

Volume fraction

Neutron flux, groups j =1, 2
Neutron flux vector

Adjoint flux, groups § = 1, 2
Adjoint flux vector
Reactivity

Density of fuel 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
gchb

2k

REFERENCES

MSRP Prog. Rep. August 31, 1962, (ORNL report to be issued).

T. B. Fowler and M. L. Tobias, Equipoise-3: A Two-Dimensional,
Two=-Group, Neutron Diffusion Code for the IBM-T090 Computer,
0RNI.I"3199’ Feb- 7, 1%2-

C. W. Nestor, Jr., Equipoise 3A, ORNL-3199 Addendum, June 6,
1962.

MSRP Prog. Rep. August 31, 1961, ORNL-3215, p. 83.
J. R. Engel and P. N. Haubenreich, Temperatures in the MSRE

Core During Steady State Power Operation, ORNL-TM-37C (in
preparation).
l-2¢

- * -

- - -

O o= O\ F i

10.
11l.
12.
13.
14,

16.

25.
26.
27«
28,
29,
30.
31.
32.
33.
34,
35
36.
37.
38.
39.
40.
41.
Lo,
b3,
4,
s,
L6,
47,
L8,
L9,

?wamwsz%wamtflflg

Internal Distribution

MSRP Director's Office.
Rm. 219, Bldg. 9204-1

G.
L
S.
M.
c.
E.
D.
N
E.
S.

-

*

M. Adamson

G. Alexander
E. Beall
Bender

E. Bettis

S. Bettis

S. Billington
F. Blankenship
G. Bohlmann
E. Bolt

J. Borkowski
A. Brandon

. R. Bruce

W. Burke
Cantor

E. Cole
A. Conlin

. Ho Cook
. T. Corbin

A. Cristy
L. Crowley

. L. Culler

H. DeVan

G. Donnelly
Douglas
. Dunwoody
. Engel
Epler
Ergen
Fergason
. Fraas

. Frye
Gabbard
Gallaher
Greenstreet
. Grimes
Grindell
Guymon
Harley
Harrill
Haubenreich
. Hise
Hoffman
Holz
Howell
Jarvis

. Jordan

o

-

oM g X g Do

G

mhEgEQ=E0nimm

25

50
51
52.
53.
55.
6.
57+
53.
59.
60.
61.
62.
63.
64 .
65.
66.
67.
66.
69.
T70.
T1.
T2.
73
Th.
5.
T6.
17
78.
79.
80.
81,
82.
83'
8L,
85.
86.
87.
88.

90.
9.
92.
93.
gl .,
95.
96.

8.

P. R.
R, J.
M. T.
M. dJ.
S. 5.
J. W.
J. A.
W. J.
R. B.
M. I.
R. N.
H. G.
F. C.
E. R.
W. B.
H. F.
C. K.
A. J.
E. C.
R. L.
J. C.
T. E.
W. R.

Kasten
Kedl
Kelley
Kelly
Kirslis
Krewson
Lane
Leonard
Lindauer
Lundin
Lyon
MacPherson
Mainensciaein
Mann
McDonald
McDuffie
McGlothlan
Miller
Miller
Moore
Moyers
Northup
Osborn

P. Patriarca

H. R.
A, M.
W. B.
B. E.
Jd. L.

Payne
Perry
Pike
Prince
Redford

M. Richardson

R. C.
T. X.
H. W.
A. W.
J. E.

Robertson
Roche
Savage
Savolainen
Savolainen

D. Scot

C. H.
J. H.
M. Jd.
G. M.
A. N.
P. G.

Secoy
Shaffer
Skinner
Slaughter
Smith
Smith

1. Spiewak

J. A.

Swartout

A. Taboada

J. R.
R. E.
D. B.
W. C.

Tallackson
Thomea
Trauger
Ulrich
26

Internal Distribution -~ cont'd

99.

100.

10L.
102.
103.
104,
105.
106-107.
108-109.
110.
111-113.
11k,

B.
c.
B.
A.
Je
L.
c.

S.
F.
HV.
M.
c.
V.
H.

—_——

Wegver
Wegver
Webster
Weinberg
White
Wilson
Wodtke

Reactor Division Library
Central Research Library
Document Reference Section
Leboratory Records

ORNL-RC

External

115-116. D. F. Cope, Reactor Division, AEC, ORO
117. H. M. Roth, Division of Research and Development,

AEC, ORO

118. F. P. Self, Reactor Division, AEC, ORO
119-133. Division of Technical Information Extension,

AEC, ORO

13%. J. Wett, AEC, Washington