ORNL-TM-0378.txt

 

3 ' -

 

OAK RIDGE NATIONAL LABORATORY
operated by |

UNION CARBIDE CORPORATION
for the

U.S. ATOMIC ENERGY COMMISSION

‘ORNL- TM— 378

. /-w
COPY NO. - //5

 

DATE - November 5, 1962

TEMPERATURES IN THE MSRE CORE DURING STEADY-STATE
POWER OPERATION

| e o i\ :
" J. R. Engel and P. N. Haubenreich L on I e \‘fi_‘;’h
AR r

ABSTRACT

Over=-gll fuel and graphite temperature distributions were calculated
for a detailed hydraulic and nuclear representation of the MSRE. These
temperature distributions were weighted in various ways to obtain nuclear
and bulk mean temperatures for both materials. At the design power level
of 10 Mw, with the reactor inlet and outlet temperatures at 1175°F and
1225°F, respectively, the nuclear mean fuel temperature is 1213°F. The
bulk average temperature of the fuel in the reactor vessel (excluding

.-the volute) is 1198°F. For the same conditions and with no fuel per-
meation, the graphite nuclear and bulk mean temperatures are 1257°F and
1226°F, respectively. Fuel permeation of 2% of the graphite volume
raises these values to 1264°F and 1231°F, respectively.

The effects of power on the nuclear mean temperatures were combined
with the temperature coefficients of reactivity of the fuel and graphite
to estimate the power coefficient of reactivity of the reactor. If the
reactor outlet temperature is held constant during power changes, the
power coefficient is - 0.018% 2%/Mw. If, on the other hand, the average
of the reactor inlet and outle’ ‘ce@eratures is held constant, the power
coefficient is - 0.04T% é%/m, T

’

 

 

imile Price $§ .4 " ,/// =
MicrofiMpPrice $ / ”

Available fr N

Office echnical ices

De ment of Commerce
ashington 25, D. C.

 

 

NOTICE

This document contains information of a preliminary nature and was prepared
primarily for internal use at the Oak 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 othgrwise given public dis-
semination without the approval of the ORNL patent branch, Legal and Infor-
mationn Control Department,

p"
¥,

 

 

LEGAL NOTICE

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

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

A. Makes any warranty or representation, expressed or implied, with respect to the accuracy,
completeness, or usefulness of the information contained in this report, or that the use of
any information, apparotus, 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 disclossd in this report.

As used in the above, '‘person acting on behalf of the Commission’ includes ony employse or

contracter of the Commission, or employes of such contractor, to the extent that such employae

or contractor of the Commission, or employse of such contracter prepares, disseminates, or
provides access to, any information pursuant to his employment or contract with the Commission,

or his employment with such contracter,

 

 

 

 
 

CONTENTS
Page
ABSTRACT & & o o o o « o o . 1
LIST OF FIGURES & & & v v v 6 v o v s v e v o 5
LIST OF TABLES & « + o s « = 4 v & o o o o o o o o o 6
INTRODUCTION . . » . . 7
DESCRIFTION OF CORE 8

Fuel Channe-ls « a & e < @ . o & . o o & o & ° ¢ ® 4 ® ll

Hydraulic Model . . . . . &« o ¢ ¢ o ¢ o 6 o o o « &« « o 12
Neutronic Model . ¢« « .+ o o & o ¢ o o & o s o o o & o = 15
NEUTRONIC CAICUIATIONS . . & o o o o o & & & o o o o o « » - 18
Flux Distributions . . « ¢ « v o ¢ ¢ o v « 5 « 5 o o » 19
Power Density Distribution . . . . ¢« . « . + « o ¢ . . 22
Nuclear Importance Functions for Temperature . . . . . 22
FUEL TEMPERATURES « .= &+ ¢ 4 o s o 4 v « o o s o o + o 2 o« . 26
Over-all Temperature Distribution . . . . « + + &« .+ + . 30
Nuclear Mean Temperature . . « ¢ ¢« ¢« « &+ o o o = « o 33
Bulk Mean Temperature . « « o+ + o o o « 4+ & = « « « « 36
GRAPHITE TEMPERATURES . & « v & o v v o o o o o o o s o o« 36
ILocal Temperature Differences . . « o « « « « » o « . 36

Over-all Temperature Distribution . . . . . . . . . . . 421

Nuclear Mean Temperature . . . . . + +« + 4 o « o o+ « . 4k
Bulk Mean Temperature . . o« « « + v « o &+ o o+ o « « o« . hk
POWER COEFFICIENT OF REACTIVITY . & & v v v o o v o o o o o 45
DISCUSSION o o « & o o o « o o o o o » o « o o o o o o « » o b7
Temperature Distributions . . . . . . . « . « . « . « . 47
Temperature Control . . + & v ¢ ¢ o o & o « « &+ + & « k7
APPENDIX  + « v o v o o o e e e b e e e e e e e e e e e kO
Nomenclature o « & o+ v & o s o 4 o 4 o« s o o =+« .« bo

Derivation of Equations . . + v v « v v v 4 o o o + o 51
N

LIST OF FIGURES

Cutaway Drawing of MSRE Core and Core Vessel

MSRE Control Rod Arrangement and Typical Fuel
Channels

Nineteen-Regicn Core Model for Equipcise 3A Calcu-
lations. See Table 3 for explanation of letters.

Radial Distribution of Slow Flux and Fuel Fission
Density in the Plane of Maximum Slow Flux

Axial Distribution of Slow Flux at a Position
7 in. from Core Center Line

Radial Distribution of Fluxes and Adjoint Fluxes
in the Plane of Maximum Slow Flux

Axial Distribution of Fluxes and Adjoint Fluxes
at a Position 7 in. from Core Center Line

Axial Distribution of Fuel Fission Density at a
Position 7 in. from Core Center ILine

Relative Nuclear Importance of Fuel Temperature
Changes as a Function of Position on an Axis
Iccated 7 in. from Core Center Line

Relative Nuclear Importance of Graphite Tempera-
ture Changes as a Function of Position on an Axis
Iocated 7 in. from Core Center Line

Relative Nuclear Importance of Fuel and Graphite
Temperature Changes as a Function of Radial Position
in Plane of Maximum Thermal Flux

Channel Outlet Temperatures for MSRE and for a Uniform
Core

Radial Temperature Profiles in MSRE Core Near Midplane

Axial Temperature Profiles in Hottest Channel of MSRE
Core (7 in. from Core Center Line)

10

16

20

21

23

2k

25

27

28

29

3k
he

43
i..-l

LAY

N

LIST OF TABLES

Fuel Channels in the MSRE

Ccre Regions Used to Calculate Temperature
Distributions in the MSRE

Nineteen-Regicn Core Model Used in Eguipoise
Calculations for MSRE

Flow Rates, Powers and Temperatures in Reactor
Regions

Jocal Graphite-Fuel Temperature Differences in
the MSRE

Nuclear Mean and Bulk Mean Texperatures of Graphite

Temperatures and Associated Pcwer Coefficients of
Reactivity

nge

11

1

17

31

Lo
Ly

Lé
INTRODUCTION

This report is concerned with the temperature distribution and zp-
propriately averaged temperatures of the fuel and graphite in the MSRE
reactor vessel during steady operation at power.

The temperature distribution in the reactor is determined by the
heat production and heat transfer. The heat production follows the over-
all shape of the neutron flux, with the fraction generated in the grephite
depending on how much fuel is soaked into the graphite. TFuel channels tend
1o be hottest near the axis c¢f the core because of higher pover densities
there, but fuel velocities are equally important in'determining fuel tem-
peratures, and in the fuel channels near the axis of the MSRE core the
velocity is over three times the average for the core. Variations in
velocity also occur in the outer regions of the core. Graphite tempera-
tures are locally higher than the fuel temperatures by an amount which
varies with the power density and also depends on the factors which gov-
ern the heat transfer into the flowing fuel.

The mass of fuel in the reactor must be kinown for inventory calcu-
lations, and this requires that the mean density of the fuel and the
graphite be known. The bulk mean temperatures must therefore be calcu-
lated.

The temperature and density of the graphite and fuel affect the
neutron leakage and absorption (or reactivity). The reactivity effect
of a local change in temperature depends on where it is in the core,
with the central regions being much more important. Useful quantities
in reactivity analysis are the "nuclear mean" temperatures of the fuel
and graphite, which are the result of weighting the local temperatures
by the local nuclear importance.

The power coefficient of reactivity is a measure of how much the
control rod poisoning must be changed to obtain the desired temperature
control as the power is changed. The power coefficient depends on what
temperature is chosen to be controlled and on the relation of this tem-
perature to the nuclear mean temperatures.

This report describes the MSRE core in terms of the factors which

govern the temperature distribution. It next presents the calculated
temperature distributions and mean temperatures. The power coefficient
of reactivity and its effect on operating plans are then discussed, An
appendix sets forth the derivation of the necessary equations and the

procedures used in the calculations.

DESCRIPTION OF CCRE

Figure 1 1s a cutaway drawing of the MSRE reactor vessel and core.
Circulating fuel flows downward in the annulus between the reactor vessel
and the core can into the lower head, up through the active region of the
core and into the upper head.

The major contribution to reactivity effects in the operating reactor
comes from a central region, designated as the main portion of the core,
where most of the nuclear power is produced. However, the other regions
within the reactor vessel also contribute to these effects and these con-
tributions must be included in the evaluation of the reactivity behavior.
The main portion of the core is comprised, for the most part, of a regular
array of close-packed, 2-in. square stringers with half-channels machined
in each face to provide fuel passages. The regular pattern of fuel chan-
nels is broken near the axis of the core, where control-rod thimbles and
graphite samples are located (see Fig. 2), and near the perimeter where
the graphite stringers are cut to fit the core can. The lower ends of
the vertical stringers are cylindrical and, except for the five stringers
at the core axis, the ends fit into a graphite support structure. This
structure consists of graphite bars laid in two horizontal layers at
right angles to each other, with clearances for fuel flow. The center
five stringers rest directly on the INOR-8 grid which supports the hori-
zontal graphite structure. The main portion of the core includes the
lower graphite support region. Regions at the top and bottom of the core
where the ends of the stringers project into the heads as well as the
neads themselves and the inlet annulus are regarded as peripheral regions.

The fuel velocity in any passage changes with flow area as the fuel
moves from the lower head, through the support structure, the channels,
and the channel outlet region into the upper head. The velocities in most
chainnels are nearly equal, with higher velocities near the axis and near

the perimeter of the main portion of the core. For most passages, the
UNCLASSIFIED
ORNL-LR-DWG 61097

AIR INLET

  
  
 
  
 
 
 
  
  
 
  
  
  
 
 
   
 
  
 

AIR QUTLET
FLEXIBLE CONDUIT TO

CONTROL ROD DRIVE (3)
SMALL GRAPRITE SAMPLE AGCESS PORT

CONTROL ROD COOLING AIR INLETS

 

CONTROL ROD COOLING AIR QUTLETS

COOLING JAGKET AfR INLETS
COOLING JACKET AIR QUTLETS

AGCCESS PORT COOLING JAGKETS
REACTOR ACCESS PORT

FUEL OUTLET

SMALL GRAPHITE SAMPLES

CONTROL ROD THIMBLES {3) HOLD DOWN ROD

LARGE GRAPHITE SAMPLES (5)

CORE CENTERING GRID

FLOW DISTRIBUTOR

GRAPHITE-MODERATOR
STRINGER

REACTOR CORE CAN

REACTOR VESSEL

 

ANTI-SWIRL VANES
MODERATCR
SUPPORT GRID

Fig. 1. Cutaway Drawing of MSRE Core and Core Vessel
10

'l
===l
#

J
.\..

Rt

N
%"\ \i/ _”\ GUIDE BAR

o
. N7, -,
2 Y //J//, ) |

Fig. 2. MSRE Control Rod Arrangements and Typical Fuel Channels

 
Il

greater part of the pressure drop occurs in the tortuocus path through the
horizontal supporting bars. This restriction is absent in the central
passages, resulting in flows through these channels being much higher than
the average.

The variations in fuel-to-graphite ratic and fuel velocity have a
significant effect on the nuclear characteristics and the {temperature
distribution of the core. In the temperature analysls reported here, the
differences in flow area and velocity in the entrance reglons were neg-
lected; i.e., the flow passages were assumed tc extend frcm the top to the
bottom of the main portion of the core without change. Radial variations
were taken inte account by dividing the core intc concentric, cylindrical

regions according to the fuel velocity and the fuel-to-graphite ratio.

Fuel Channels

The fuel channels are of severesl types. The number of each in the
final core design is listed in Table 1. The full-sized channel is the
typical channel shown in Fig. 2. The half-channels occur near the core
perimeter where faces of normal graphite stringers are adjacent to flat-
surfaced stringers. The fractional channels are half-channels extending
to the edge of the core. The large annmulus is the gap between the graphite
and the core can. There are three annuli around the contrcol-rod thimbles.
The graphite specimens, which occupy only the upper half of one lattice
space above a stringer of normal cross section, were treated as part of

a full-length normal stringer.

Table 1. Fuel Channels in the MSRE

 

 

Channel Type Number
Full-size 1120
Half-size 28
Fractional 16
Large annulus 1

Thimble annuli 3

 
12

Hydraulic Model

Hydraulic studies by Kedll on a fifth-scale model of the MSRE core
showed that the axial velocity was a function of radial pesition, pri~
marily because of geometric factors at the core inlet. As a result of
these studies, he divided the actual core chamnels intc several groups
according to the velocity which he predicted would exist. This division
was based on a total of 1064 channels with each of the control-rod-thimble
annuli treated as four separate channels. He did not attempt to define
precisely the radial boundaries of some of the regions in which the chan-
nels would be found.

Since the total number of fuel channels in the core is greater than
the number assumed by Kedl, and since radial position is important in
evaluating nuclear effects, it was necessary tc make a modified division
of the core for the temperature analysis. For this purpose the core was
divided into five concentric annular regions, as described below, based
on the information obtained by Kedl. The fuel velocities assigned by

Kedl to the various regions are used as the nominal velocities.

Region 1

This region consists of the central 6-in. square in the core, with
all fuel channels adjacent to it, plus one-fourth of the area of the
graphite stringers which help form the adjacent channels. The total
crcss~-sectional area of Region 1 is 45.0 in.2 and the equivalent radius
is 3.78 in. The fuel fraction (f) for the region is 0.256. This region
contains the 16 channels assigned to it by Kedl plus the 8 channels which
he classified as marginal. Because of the cylindrical geometry around
the control-rod thimbles, six of the channels which were marginal in
Kedl's model have the same flow velocity as the rest of the region. It
was not considered worthwhile to provide a separate region for the two

remaining channels. The nominal fluid velocity is 2.18 ft/sec.

 

lE. S. Bettis et al., Internal Correspondence.
Region 2

This region covers most of the core and contains only normal, full-
sized fuel channels. All the fuel channels which were not assigned else-
where were assigned to this region. On this basis, Region 2 has 940 fuel
channels, a total cross-sectional area of 1880 in.e, a fuel fraction of
0.224, and equivalent outer radius of 24.76 in. (the inner radius is equal
to the outer radius of Region 1), and a nominal fluid velocity of 0.66

ft/sec.

Region 3
This region contains 108 full-sized fuel channels as assigned to it
>
by Kedl. The total cross-sectional area is 216 in.”; the fuel {raction

is 0.224; the effective outer radius is 26.10 in.; and the nominal fluid

velocity is 1.63 f£t/sec.

Region 4

This region was arbitrarily placed outside Region 3 even though it
contains marginal channels from both sides of the region. All the half-
channels and fractional half-channels were added to the 60 full-sized
channels assigned to the region by Kedl. This gives the equivalent of
78 full-sized channels. All of the remaining graphite cross-sectional
area was also assigned to this region. As a result, the total cross-
sectional area is 245.9 in.g; the fuel fraction is 0.142; the effective

outer radius is 27.58 in., and the nominal fluid velocity is 0.90 ft/sec.

Region 5

The salt annulus between the graphite and the core can was treated
as a separate region. The total area is 29.55 in.a; the fuel fraction
is 1.0; the outer radius is 27.75 in. (the inner radius of the core can),

and the nominal fluid velocity is 0.29 ft/sec.

Bffective Velocities

The nominal fluid velocities and flow areas listed above result in
a total flow rate through the core of 1315 gpm at 1200°F. All the veloci-
ties were reduced proportionately to give a total flow of 1200 gpm. Table
2 lists the effective fluid velocities and Reynolds numbers for the various

regions along with other factors which describe the regions.
Table 2. Core Regions Used to Calculate Temperature Distributions in the MSRE

 

 

 

Number of Total Crosse
Full-sized sectional Area

Region Fuel Channels {;nflfi}

1 12* 45.00

2 o 1880.

3 108 216.0

4 78 245.9

5 o** 29.55

Fuel

Fraction

T T Y R L TR T A " TR B M S e,

0.256
0.224

0.22k

0.142

1.000

Effective

 

 

Effective Flow

Outer Radius Fluid Velccity  Reynolds  Rate
(in:) (£t /sec) Number (gpm)

3.78 1.99 3120 72

24 .76 0.60 o5 791

26.10 1.49 2360 22k
27.58 0.82 1300 89

27.75 0.26 Yol 24

1200

 

*
Plus 3 control-rod-thimble annuli.

%
Annulus between graphite and core shell.

HT
15

Neutronic Model

The neutronic calculations upon which the tempereature distributions
are based were made with Iquipoise 3A,2’3 a8 2-group, 2-dimensional, nulti-
region neutron diffusion program for the IBM-7090 computer. Because of
the limitation to two dimensions and other limitations on the problemn
size, the reactor meodel used for this calculation differed somewhat fromn
tlie hydraulic model.

The entire reactor, including the reactor vessel, was represented in
cylindrical (r,z) geometry. Three basic materials, fuel salt, graphite
and INOR were used in the model. A total of 19 cylindrical regions with
various proportions of the basic materials was used.

Figure 3 is a vertical half-section through the model showing the
relative size and location of the various regions. The region composi-
tions, in terms of volume fractions of the basic materials, are summa-
rized in Table 3.

Regions J, L, M, and N comprise the main portion of the core. This
portion contains 98.7% of the graphite and produces 87% of the total power.
The central region (L) has the same fuel and graphite fractions as the
central region of the hydraulic model. However, the outer boundary of the
region is different because of the control-rod-thimble representation.
bince it was necessary to represent the thimbles as a single hollow cyl-
inder (region K) in this geometry, a can containing the same amount of
INOR as the three-rod thimbles in the reactor and of the same thickness
was used. This established the outer radius of the INOR cylinder at 3 in.
which also is close to the radius of the pitch circle for the three thim-
bles in the MSRE. The central fuel region was allowed to extend only to
the inside radius of the rod thimble. The portion of the core outside
the rod thimble and above the horizontal graphite support region was

homogenized into one composition (regions J and M). The graphite-fuel

 

2T. B. Fowler and M. L. Tobias, EQUIPOISE~3: A Two-Dimensional, Two=-
Group Neutron Diffusion Code for the IBM-7090 Computer, ORNL~3199 (Feb. 7,
1962 ).

3. . Nestor, Jr., FQUIPOISE-3A, ORNL-3199 Addendum (June 6, 1362).
16

Uneclassified
ORNL-LR-DWG. T16k42

 

 
 

Mge 3. Ninetcen-Region Corve Nodel Zor Eguipoise Calculations.
See Table 2 for explanstio: ’
Table 3. 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.% Th.92 76 .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 Th.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 Downcomer

G 3.00 28.00 65.53 66,22 k.6 5.4 0

H 3.00 27-75 6h .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 6L .59 22.5 T7.5 0 Core

K 2.94 3.00 5.50 Th.92 0 0 100 Simulated
thimbles

L 0 2.94 2.00 64 .59 25.6 T4 .4 0 Central
region

M 2.94 27.75 2.00 5 .50 22.5 77.5 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 T4 .92 100 0 0

R 0 2.94 65.53 66 .22 89.9 10.1 0

S 0 2.94 64 .59 65.53 43.8 56 .2 0

 

LT
18

mixture containing the horizontal graphite bars at the core inlet was
treated as a separate region (I).

The main part of the core, as described in the preceding paragraph,
does nct include the lower ends of the graphite stringers, which extend
beyond the horizental graphite bars, cor the pointed tops of the stringers.
Trhe bottom ends were included in a single regicn (O and the mixture at
the top ¢f the core was approximated by 5 regicns E, G, H, R, and 8).

The thickness of the material contained in the upper and lower heads
(regions O, 8, snd P) was adjusted so that thes amcunt of fuel salt was
2guel O The amcunt ccntalned in the reactor heads. As a result of this
a

fugtment the over-all height of the neutronic model of the reactor is

U!

not exactly the same as the physical height. The upper and lower heads

themseives (regions A and C) were flattened cut and represented as metal
disecs cf the nominail thickness of the reactcr material.
iIr the radial directicon ocutside the main part of the core, the core
can (region I), the fuel inlet annulus (regios F) and the reactor vessel
\region B) were included with the actual physical dimensicns of the re-
actor applied to them.
The materials withir each separate regicn were treated as homogeneous

-

mixtures in the calculations. As g resulht, the calculations give only

zhe overeall shape of the flux in the regicas where inhomogeneity exists
beuzuse of the presence of two cr more of the basi: materials.

The first step in the nsutronisc caleulatlons was the establishment
required for criticality. A carrier salt
cortaining TC mel % IiF, %5% BeFo and 5% ZrF4 was assumed. The LiF and
ZriFg voncentrations were keld zonstant and the BeF- concentration was
reduceda as th2 concentraticn of UF4 was increased in the criticality

e
U“ji, 5% U238, 1% U23”,

searck. The uranium was assumed ts censist of 93%

236

and 1% Thi

[/

isotepic ccmposition is bypical of the material ex-

pected to be avgilable for the reactor. The lithium to be used in the
marufaciure of the fuel szlt contairs 73 ppm Llf\° This value was used in

the calsulations. No chemizal impurities were considered in the fuel mix-

ture. Al of the calculations were made for an iscthermal system at 1200°F.
The criticality search was made with MODRIC, a one=-dimensicnal, multi-
group, multiregion neutron diffusior. program. This established the criti-
cal concentration of UF, at 0.15 mol %, leaving 24 .85% BeF,,

program was used for radial and axisl traverses cf the model used for the

The sare

Fquipcise 3A calculation to provide the 2-group cconstants for that program.
The spatial distributions cf the fluxes and fuel power density were cb-
tained directly from the results of the Fguipcise 3A calculation. The

same fluxes were used to evaluate the nuclear impertance functions for

temperature changes.

Flux Distributions

The radial distribution cf the slow neutron flux calculated for the
MSRE near the midplane is shown in Fig. 4. This plane contains the nraxi-
mum value of the flux and is 35 in. above the bottom of the main part of
the core. The distortion of the flux produced by abscrptions in the sinu-
lated control-rod thimble; 3 in. from the axial centerline, is readily
apparent. Because of the magnitude of the distortion this simplified
representation of the control-rod thimbles is probably not adequate for an
accurate description of the slow flux in the immediate vicinity of the
thimbles. However, the over~all distribution is probably reasonably
accurate. The distorticn of the flux at the center of the reactor alsc
precludes the use of a simple analytic expression tc describe the radial
distribution.

Figure 5 shows the calculated aexial distribution of the slow flux
along a line vwhich passes through the maximum value, 7 in. from the verti-
cal centerline. The reference plane for measurements in the axial direc-
tion is the bottom of the horizontal array of graphite bars at the lower
end of the main portion of the ccre. Thus, the outside of the lower heed
is at -10.26 in.; the top of the main portion of the core is at 64.59 ir.;
and the outside of the upper head is at 76.04. The shape of the axisl
distribution in the main portion is closely approximated by a sine curve
extending beyond it at both ends. The equation of the sine approximetion

shown in Fig. 5 is
UNCLASSIFIED

ORNL-LR-DWG 75777

0.9

0.8

0.7

0.6

0.5

FRACTION OF MAX. VALUE

0.4

0.3

0.2

0.1

 

0 5 10 15 20 25 30

RADIUS, in.

Fig. 4. Radial Distribution of Slow Flux and Fuel Fission Density
in the Plane of Maximum Slow Flux
FRACTION OF MAXIMUM VALUE

-10 0 i0 20 30 40 50 60
Z,in

Fig. 5. Axial Distribution of Slow Flux at a Position 7 in.
Core Center Line

70

from

UNCLASSIFIED

ORNL-LR-DWG 75823

 

80

12
22

 

Egzj = Sin[?%[ 7

m

(z + 4.36)} ,

witl the linear dimension given in inches.
The relation of the slow neutrcn flux to the other fluxes (fast,
fast adjoint and slow adjeint) is shown in Figs. 6 and 7. These figures

rresent the absolute values for a reactcr power level cof 10 Mw.

Power Density Distributicn

A function that is of greater interest than the flux distributicn,
from the standpoint of its effect on the reactor temperatures, is the
Gistribution of fissicn power density in the fuel. For the fuel compo-
sition considered here, only 0.87 of the total fissions are induced by
thermal neutrens; the fraction of thermal fissions in the main part of
the core is 0.90. 1In spite of the relatively large fraction of nonthermal
fissions, the over-all distribution of fuel fission density is very simi-
lar to the slow-flux distribution. The radial distribution is shown on
Fig. & for a direct comparison with the thermel flux. The axial fission
density distribution (see Fig. 8) was fitted with the same analytic ex-
Pression used for the axial slow flux. The quality of the fit is about

the same for both functions.

Nuclear Importance Functions for Temperature

The effects urcn reactivity cf local temperature changes in fuel and
graphite have been derived from first-order two-group perturbation theory
and are reported elsewhere.l+ This analysis produced weighting functions
or nuclear importance functions, G(r,z) for the local fuel and graphite
temperatures. The weighted average temperatures of the fuel and graphite
obtained by the use of these functions may be used with the appropriate
temperature ccefficients of reactivity to calculate the reactivity change

associated with any change in temperature distribution.

 

 

uB. E. Prince and J. R. Engel, Temperature and Reactivity Coefficient.
Averaging in the MSRE, ORNL TM=-379 {Octcber 15, 1962).
23

UNCL ASSIFIED
ORNL-LR-DWG 73611

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(x1013) |
\—Lfi

5 |
8 |
- \FAST FLUX
12 o
£
< \\\\\
c
o
5 \\\
2 10
=z
=
o
S
> 8 7 SLOW FLUX
Lid \/‘ — ——
= ,r"— ~~
& T~ \\
o« SLOW ADJOINT \\
E 6 e - —— N
& - I
o S~e
r e
v FAST ADJOINT ~~w
£ 4
>
-
—
L

2

0

0 5 10 15

 

RADIUS (in.)

Fig. 6. Radial Distribution of Fluxes and Adjoint Fluxes in the
Plane of Maximum Slow Flux
24,

UNCLASSIFIED
ORNL-LR-DWG 73612

 

 

 

   
 

 

 

 

 

 

 

 

(x1019) \ |
14 L
l FAST FLUX
|
| .
12 - |
10
. i
SLOW FLUX
8 o | \ | N

FLUX FOR REACTOR POWER OF {0 Mw (neutrons/cmz-sec)

 

  

 

 
 
 
    

 

 

 

 

~ . .
SLOW ADJOINT
T NN

 

 
   

 

 

-

- N -y
- -'h‘

-

-~

-

FAST ADJOIN

~

™

 

    

 

 

 

|
!
|

 

    

 

 

20

30

40

 

Fig. 7.
Position 7

AXIAL POSITION (in.)

80

AxZal Distribution of Fluxes and Adjoint Fluxes at a
in. frcm Core Center ILine
FRACTION OF MAXIMUM VALUE

7 in.

Fig. 8. Axial Distribution of Fuel
from Core Center Line

Fission

Density

at a Position

UNCLASSIFIED

ORNL-LR-DWG 75824

 

Gc
The temperature weighting functions are evaluated in terms of the
* * * *

four flux products: ¢l ¢l’ ¢l ¢2, ¢2 ¢1 and ¢2 ¢2. The coefficients
applied to the varicus terms depend on the material being considered, its
lccal volume fraction and the manner in which temperature affects its
physical and nuclear properties. Since the weighting functions are evalu-
ated separately for fuel and graphite, a particular weighting function
exists cnly in those regions where the material is represented.

Figures 9 and 10 show the axial variation in the weighting functions
fer fuel and graphite at the radial pcsition of the maximum fuel power
censity. For both materials the sxial variaticn in the main part of the

core is very closely approximated by a sine-squared. The functicn:

.. 2] = 1
sin [7§Tfl-(z + 4071)J

is plctted on each ¢f the figures for comparison. The wide variations at
the ends resuit primarily from the drastic changes in veolume fraction
and the apparent discontinuities are the result of dividing the reactor
into discrete regioas for the Equipcise 3A calcuiation.

The radial temperature weighting functicns for both fuel and graphite
near the core midplane are shewn in Fig. 11. Because of the peculiar shape
of the curves; nc attemrt was made tc fit analytic expressions to these
functinns.

FUEL TEMPERATURES

The fuel temperature distribution in the reactor has an cver-all shape
which is determined by the shape c¢f the power density and the fuel flow
pattern. Within the main part cf the core;, where the fuel flows in dis-
crete channels. temperature variations across each individual channel are
superimpczed on this cver-all shape. These lccal effects will be touched
on later (page 37 ) in the section cn graphite-fuel temperature differences.
The present secticn deals with the over-all shape c¢f the fuel temperature
distributicn and average temperatures. These were calculated for a reactor
power level of 1C Mw with inlet and outlet temperatures at 11795 and 1225°F,

respectively. The total fuel flow rate was 1200 gpm.
27

UNCLASSIFIED
ORNL-LR-DWG 73618

 

1.

° | 1 | | ~
EQUIPOISE 3A RESULT : \

08 sin2 ["/79.4 ( 2+ 471 ] // .

) \

 

 

 

 

 

 

0.4 /
_/

O ™ — - —— b
\k&—mwMAiN PORTION OF CORE-——"——————“——*fl

 

 

 

 

 

 

 

 

 

 

 

 

RELATIVE NUCLEAR IMPORTANCE

1 - T ]

 

-0.2

 

 

 

 

 

 

 

 

 

 

 

|
oy }

 

 

 

-0.6 : :
-20 -10 0 10 20 30 40 50 60 70 80

z, AXIAL POSITION (in.)

Fig. 9. Relative Nuclear Importance of Fuel Temperature Changes
as a Function of Posgition on an Axis Located 7 in. from Core Center
TLine
28

UNCLASSIFIED
ORNL-LR-DWG 73619

 

 

 

 

 

 

 

IMPORTANCE

 

 

 

 

 

 

 

 

RELATIVE NUCLEAR

 

 

 

 

 

 

 

 

1.0 | ,
—— EQUIPOISE 3A RESULT
——sin2[T/r0.4 (24 a.71)]
0.8 -
0.6 /
0.4 / \
\
‘ \
0.2 / S \\
\ i
<~ MAIN PORTION OF CORE — 4-4
—0.2 —— : | |
.
—0.4 1 '
-10 0 10 20 30 40 50 60 70 80

z,AXIAL POSITION (in.)

Fig. 10. Relative Nuclear Impcrtance of Graphite Temperature
Changes as & Function of Position on an Axis Located 7 in. from Core
Center Line
RELATIVE NUCLEAR IMPORTANCE

1.0

0.9

0.8

0.7

0.6

0.5

0.4

©
W

0.2

0.1

29

UNCLASSIFIED

ORNL-LR-DWG 73617

 

 

] T

 

 

N\
GRAPHITE\

 

 

 

 

 

 

 

 

 

 

 

 

 

 

N
N\
\
\
\
\
A\
\\
\
5 10 15 20 25 30
RADIUS (in)

Fig. 11.

Relative Nuclear Importance of Fuel and Graphite
Temperature Changes as a Function of Radial Position in Plane of

Maximum Thermal Flux
30

Over-all Temperature Distribution

In describing the over-all fuel temperature distribution it is con-
venient to regard the reactor as corsisting of a main core, in which most
cf the nuclear heat is produced, and a number of peripheral regions which,
tegether, contribute only a small amcunt to the fctal power level. Of the
19 reactcr regicns used for the neutronicicalculations (see Fig. 3), 14
coentain fuel and 10 ¢of these were regarded as peripheral regions. The
four remaining fuel-bearing regions (J, L, M, and N) were combined to
represent the main part of the ccre. On this basis, the main portion ex-
tends radially tc the inside cof the core shell and axially from the bottom
of the herizontal graphite bars to the top of the uniform fuel channels.
This portion accounts for 87% of the total reactor power. The volute on
the reactor inlet was neglected in the temperature calculatjons because

cf its physical separaticn from the rest of the reactor.

Peripheral Regions

 

Since only 13% cof the reactor power is produced in the peripheral
regions, the temperature variations within them are smell and the details
cf the temperature distributions within these regions were not calculated.
The mean temperature rise for each region was calculated from the fraction
of the toctal power produced in the region and the fraction of the tectal
flow rate through it. The inlet temperature to each region was assumed
constant at the mixed mean cutlet of the preceding regicn. An approximate
buliremean temperature, midway between the region inlet and outlet tempera-
tures, was assigned tc each peripkeral region. Table 4 summarizes the

flow rates, powers and fuel temperatures in the varicus reactor regions.

Msin Ceore

The wide variaticns in fuel temperature, both radially and axially,
in the main part c¢f the core necessitate a more detailed description of
the temperature distribution.

The average temperature of the fuel in a channel at any axial position
is equal to the channel inlet temperature plus a rise proportional to the
sum of the heat generated in the fuel and that transferred to it from the
adjacent graphite as the fuel moves from the channel inlet to the specified

pcint. The heat prcoduced in the fuel fcllows very closely the radial and
31

Table 4. Flow Rates, Powers and Temperatures
in Reactor Regions

 

 

o Flow Powerb Tempo Riseb Av. Temp.c

Region (gpm) (kw) (°F) (°F)

D 1157 45k 2.4 1224 .7

E 1157 151 0.8 1223.1

F 1200 251 1.3 1175.6

G 1157 105 0.5 1222.4

H 1157 11k 0.6 1221 .9

J 1157 8287 43 a

L 43 159 22 d

M 1157 192 1.0 a

N 1200 68 3 a

0 1200 69 0.3 1177.1

P 1200 134 0.7 1176.6

Q 43 12 1.7 1200.9

R 43 2 0.3 1199,9

S 43 2 0.2 1199.7

 

aRegions not containing fuel are excluded.

bAt 10 Mw

“with T, = 1175°F; T_ . = 1225°F

out

dActual temperature distribution calculated for this
region. See text.

axial variation of the fission power density. Since the heat production

in the graphite is small, no great error is introduced by assigning the

same spatial distribution to this term. Then, if axial heat transfer in

the graphite ig neglected, the net rate of heat addition to the fuel has

the shape of the fuel power density. The fuel temperature rise is in-

versely proportional to the volumetric heat capacity and velocity. Thus
‘2 (Qf)m

Tf (I‘,Z) = Tf (Z =0) + éf W Pf(r,z) dz , (l)
32

where Qf is an equivalent specific power which includes the heat added to
the fuel from the graphite. The channel inlet temperature, Tf(z =0), is
assumed constant for all channels and its value is greater than the reactor
inlet temperature because of the peripheral regions through which the fuel
passes before it reaches the inlet to the main part of the core. The volu-
metric heat capacity, (pcp)f, is assumed constant and only radial variations
in the fuel velocity, u, are considered. It is further assumed that the

radial and axial variations in the power-density distribution are separable:

P(r,z) = A(r) B{(z) . (2)

Then
Tf(r,z) = 'I‘f(z =0) + r—éajl)l—; fi%%f { B(z) dz . (3)

If the sine approximetion for the axial variation of the power density

(Fig. 8) is substituted for B(z), equation (3) becomes

Tpr,z) = Tp(z =0) + Xxééig-{}os a - COS[T;¢7 (z + 4.36)} }-. (%)

In this expression, X is a collection of constants,

 

 

(Q.)
- 17 fm 5
pf
and
_ e
a = = (0 + 4.36) . (6)

The limits within which (4) is applicable are the lower and upper
boundaries of the main part of the core, namely, O €z < 64.6 in. It is
clear from this that the shape of the axial temperature distribution in
the fuel in any channel is proportional to the central portion of the
curve [1 - cos B). (The axial distribution for the hottest channel in the
MSRE is shown in Fig. 14, where it is used to provide a reference for the

axial temperature distribution in the graphite.)
D
!._j_)

The radial distribution of the fuel temperature at the outlet of the
channels is shown in Fig. 12 for the reference conditions at 10 Mw. This
distribution includes the effects of the distorted power-density distri-
bution (Fig. 4) and the radial variations in fuel velocity. At the refer-
ence conditions the main-core inlet temperature is 1177.3°F and the mixed
mean temperature leaving that region is 1220.8°F. The additional heat re-
quired to raise the reactor outlet temperature to 1225°F is produced in
the peripheral regions above the main part of the core. Also shown on
Fig. 12 for comparison is the cutlet temperature distributicn for an ideal-
ized core with a uniform fuel fraction and uniform fuel velocity. For this
case the radial power distribution was assumed to feollow the Bessel func-
tion, JO(E.h r/R), which vanished at the inside of the core shell. The
distribution was normelized to the same inlet and mixed-mean outlet tem-

peratures that were calculated for the main portion of the MSRE core.

Nuclear Mean Temperature

At zero power the fuel temperature is essentially constant throughout
the reactor and, at power it assumes the distribution discussed in the pre-
ceding section. The nuclear mean temperature of the fuel is defined as
the uniform fuel temperature which would produce the same reactivity change
from the zero-power condition as that produced by the actual fuel tempera-
ture distribution.

The nuclear average temperature of the fuel is obtained by weighting
the local temperatures by the local nuclear importance for a fuel tenpera-
ture change. The importance functicn, G(r,z), includes the effect of fuel
volume fraction as well as all nuclear effects (see p. 22 et seq.). There-
Tore

« J; Tf(r,z) G(r,z) dv
Tf = (7)

f G(r,z) av
v

 

In carrying out the indicated integration for the MSRE, both the numerator
aznd denominator of (7) were split irto 11 terms, one for the main part of
the core and one for each of the fuel-begring peripheral regions. For the

. . . . .
peripheoral reogicns, the fuel tomperaturce in each reogicn was assumcd constant
TEMPERATURE, °F

Core

1290

1275

1250

1225

1200

1175

Fig.

1-20

 

UNCLASSIFIED

ORNL-LR-DWG 75775

RADIUS, in.

Channel Outlet Temperatures for MSRE and for a Uniform
(e
A

at the average temperature (see Table 4). The volume integrals of the
welghting functions were obtained by combining volume integrals of the
flux products, produced by the Equipoise 3A calculation, with coefficients
derived from the perturbation calculations. For the main part of the core,
the temperatures and temperature weighting functions were combined and
integrated analytically (in the axial direction) and numerically (in the
radial direction) to produce the required terms. The net result for the
fuel in the MSRE reactor vessel is
Te = Ty +0.762 Ty = Ty) 6)

A similar analysis was performed for a uniform cylindrical reactor
with uniform flow. In this case the fuel power density was assigned a
sine distribution in the axial direction and a JO Bessel function for
the radial distribution. Both functions were allowed to vanish at the
reactor boundaries and no peripheral regions were considered. A tempera-
ture weighting function equal to the product of the fuel fraction and the
square of the thermal flux (or power density since both have the same shape

in the ideal case) was used. For this case:

T. = T, +0.838 (r . -T ) (9)
The principal reason for the lower nuclear average fuel temperature
in the MSRE is the large volume of fuel in the peripheral regions at the
inlet to the main bart of the core and the high statistical welight assigned
to these regions because of the high fuel fraction. The existence of the
small, high-velocity, low-temperature region around the axis of the core
has little effect on the nuclear average temperature. Actually this
"short~circuit' through the core causes a slight increase in the nuclear
average temperature. There is lower flow and larger temperature rise in
the bulk of the reactor than if the flow were uniform, and this effect

outweighs the lower temperatures in the small region at the center.

 

# See appendix for derivation of equations for the simple case.
36

Bulk Mean Temperature

The bulk mean fuel temperature is obtained by integrating the local,
unweighted temperature over the volume of the reactor. As in the case
of the nuclear mean temperature, the large vclume cf low-temperature fuel
at the inlet to the main part of the MSRE ccre makes the bulk mean tempera-

ture for the MSRE lower than for a simple, uniform core

MSRE: T

Tt 091;6(TGut - Tin) (10)

i

Uniform Core: T, T, * 1/2 (Tout - Tin) (11)

GRAPHITE TEMPERATURES

During steady=-state power operaticn, the mean temperature in any
graphite stringer is higker than the mean temperature cof the fuel in the
adjacent channels. As g result, both the nuclear mean and the bulk mean
temperatures of the graphite are higher than the corresponding mean tem-
peratures in the fuel.

In general, it is convenient tc express the graphite temperature in
terms of the fuel temperature and the difference between the graphite and
fuel temperatures. That is:

Tg = T, + AT , (12)

where AT is a positive number at steady-state.

Nearly all of the graphite in the MSRE (98.7%) is contained in the
regions which are combined to form the main part c¢f the core. Since the
remainder would have only a very smell effect on the system, only the main
part of the core is treated in evalvating grarhite temperatures and dis=-

tributions.

Iocal Temperatiure Differences

In order to evaluate the local graphite-fuel temperature difference,
it is necessary to consider the ccre in terms of a number of unit cells,

each containing graphite and fuel and extending the length of the core.
37

In calculating the local temperature distributions, it is assumed that no
heat is conducted in the axial direction and that the heat generation is
uniform in the radial direction over the unit cells. The temperature
distributions within the unit cells are superimposed on the cover=-all re-
actor temperature distributions. The difference between the mean graphite
and fuel temperatures within an individual cell can, in general; be btroken
down into three parts:

1. the Poppendiek effect, which causes the fuel near the wall
of a channel tc be hotter than the mean for the channel;

2. the temperature drop across the graphite-fuel interface,
resulting from heat flow out of the graphite; and

3. the difference between the mean temperature in a graphite
stringer and the temperature at the interface, which is
necessary to conduct heat produced in the graphite to the
surface.

These parts are treated separately in the fcllowing paragraphs.

Poppendiek Effect

 

As may be seen from the Reynolds nmumbers in Table 2 (page 1), the
fluild flow in most of the core is clearly in the laminar regime and that
in the remainder of the core is in the transiticon range where flow may
be either laminar or turbulent. 1In this analysis, the flow in the ertire
core is assumed to be laminar to provide conservatively pessimistic esti-
nates of the temperature effects.

Fquations are available for directly computing the Poppendiek effect

5,6 (i.e.,

for laminar flow in circular channels or between infinite slabs;
the rise in temperature of fluid near the wall asscciated with interral
heat generation and relatively low fluid flow in the boundary layer).
Since the fuel channels in the MSRE are neither circular nor infinite in
two dimensions, the true Poppendiek effect will be between the results

predicted by these two approaches. The method used here was to assume

 

5H. F. Poppendiek and L. D. Palmer, Forced Convection Heat Transfer
in Pipes with Volume Heat Sources Within the Fluids, ORNL-1395 (Nov. 5,

1953 ).
H. F. Poppendiek and L. D. Palmer, Forced Convection Heat Transfer
Between Parallel Plates and in Annuli with Volume Heat Sources Within the

A L. o ~ T grrea dweo 09 N
Fiulds, ORNL=L7CL May 1i, 1954 ).
38

circular channels with a diameter such that the channel flow ares is equal
to the actual channel area. This slightly overestimates the effect.
For circular channels with laminar flow and heat transfer from

graphite to fuel, the following equation applies:

 

 

 

’ :
' p p 2 [11(1+Pf4;w q_w)-BJ | -

This equation is strictly applicable only if the power density and heat
flux are uniform along the channel and the length of the channel is in-
finite. However, it is applied here to a finite channel in which the
power density and heat flux vary along the length. It is assumed thet
the temperature profile in the fluid instantaneously assumes the shape
corresponding to the parameters at each elevation. This probably does
not introduce much error, because the heat generation varies continuously
and not very drastically along the channel. If it is further assumed
that the heat generation terms in the fuel and in the graphite have the
same over-all spatial distribution, the Poppendiek effect is directly
proportional to the fuel power density. This simple proportionality
results from the fact that, in the absence of axial heat conduction in
the graphite, the rate of heat transfer from graphite to fuel, q.,> is

directly proportional to the graphite specific power.

Temperature Drop in Fluid Film

 

Since the Poppendiek effect in the core is calculated for laminar
Tlow, the temperature drop through the fluid immediately adjacent to the
chennel wall is included in this effect. Therefore, a separate calculation

of film temperature drop is not required.

lemperature Distribution in Graphite Stringers

The difference between the mean temperature in a graphite stringer
and the temperature at the channel wall cannot be calculated analytically
for the geometry in the MSRE. Therefore, two approximations were calcu-
lated as upper and lower limits and the effect in the MSRE was assigned &

value between them.
One approximation to the graphite stringer is a cylinder whose cross-
sectional area is equal to that of the more complex shape. This leads to
2 cylinder with a radius of 0.9935 in. and a surface-to-volume ratio of
2.01 ing"lo If only the fuel-channel surface (the surface through which
21l heat must be transferred from the graphite) of the actual graphite
stringer is considered, the surface-to-volume ratio is 1.8k in.“l. There-
fore, the cylinder approximation underestimates the mean graphite tenpera-
ture. The second approximation is to consider the stringer as a slab,
cooled on two sides, with a half-thickness of 0.8 in. (the distance from
the stringer center line to the edge of the fuel channel). This approxi-
mation has a surface-to-volume ratic of 1.25 ino_l which causes an over-
estimate of the mean graphite temperature. The value assigned to the
difference between the mean graphite temperature and the channel-wall
temperature was obtained by a linear interpolation between the two ap-
proximations on the basis of surface-to-volume ratio.

For the cylindrical geometry with a radially uniform heat source in
the stringer, the difference between the average temperature in the stringer

and the channel wall temperature is given by:

2
\ 1 P rs
T - T = L8 » (lll')
g W 8 k
g
For the slab approximation:
. . P £
T - T = 3 £ (15)

Application of unit cell dimensions for the MSRE and linear interpolation

between the two approximations leads to

y F
T «T = 9.97 x 10 E& (16)
g

Net local Temperature Difference
The two temperature effects described by Fquations (13) and (16) mey
be combined to give the local mean tempersature difference between grephite

and fuel.
Lo

 

2 . -
2 11 (} + b ‘) -8
B -l E& Pele Fe Ty
AT = 9.97 x 10 kg + X, I3 . (17)

 

Since both terms in this equation are directly proportional to the power
density, the local temperature difference may also be expressed in terms
of a maximm value and the local, relative power density.

AT (r, z) = BT Plr, z) | (28)

P
m

Maximum Iocal Values

 

The maximum values of the local graphite-fuel temperasture difference
may be obtained by applying the appropriate specific powers to Equation
(17).* Because of the dependence on specific power, the temperature dif-
ferences are strongly influenced by the degree of fuel permeation of the
graphite which affects primarily the graphite power density. The effects
were eXamined for O, 0.5, and 2.0% of the graphite volume occupied by fuel.
In all cases it was assumed that the fuel in the graphite was uniformly
distributed in the stringers and that the specific powers were uniform in
the transverse direction for individual graphite stringers and fuel chan-
nels. Table 5 shows the maximum graphite-fuel temperature differences

for the three degrees of fuel permeetion, at the 10-Mw reference condition.

Table 5. Iccal Grapkite-Fuel Temperature
Differences in the MSRE

 

 

Graphite Permeation Maximum ILocal
by Fuel Temperature Difference
% of graphite volume ) (°F)
0 62.5
0.5 65.8
2.0 15.5

 

 

*In these calculations, it was assumed that 6% of the reactor power
is produced in the graphite in the absence of fuel permeation. This value
is based on calculations of gamma and neutron heating in the graphite by
C. W. Nestor, (unpublished). The thermal conductivities of fuel and graph-
ite were assumed to be 3.21 and 13 Btu/hr £t°F, respectively.
In the above computations, it was assumed that fuel which soaked into
the graphite was uniformly distributed. This is not the worst possitle
case. Slightly higher temperatures result if the fuel is concentrated
near the perimeter of the stringers. To obtain an estimate of the in-
creased severity, a stringer was examined for the case where 2% of the
graphite volume is occupied by fuel. It was assumed that the salt con-
centration was 15% of the graphite volume in a layer extending intdé the
stringers far enough (0.05 in.) to give an. average.concentration of 2%.
The salt concentration in the central portion of the stringers was assumred
to be zero. The specific power in the graphite was divided into two parts:
the contribution from gamma and neutron heating which was assumed uniform
across the stringer, and the contribution from fission heating which was
confined to the fuel-bearing layer. This resulted in an increase of 2.0°F
in the maximum graphite-~fuel temperature difference at 10 Mw. This in-
crease was neglected because other approximations tend to overestimate

the temperature difference.

Over-all Temperature Distribution

The over-all temperature distribution in the graphite is obtained by
adding the graphite-fuel temperature difference to the fuel temperature.
Figure 13 shows the radial distribution at the midplane of the MSRE for
10-Mw power operation with no fuel soakup in the graphite. The fuel tem-
perature distribution, which is a scaled~down version of the corresponding
curve in Fig. 12 to allow for axial position, is included for reference.
Figure 14 shows the axial temperature distributions at the hottest radial
position for the same conditions as Fig. 13. The fuel temperature in an
adjacent channel is included for reference. The continuously increasing
fuel temperature shifts the maximum in the graphite temperature to a
position considerably above the reactor midplane.

The distributions shown in Figs. 13 and 14 are for the mean tempers-
ture within individual graphite stringers. The local temperature distri-
butions within the stringers are superimposed on these 1n the operating

reactor.
TEMPERATURE (°F)

42

UNCLASSIFIED
ORNL-LR-DWG 73616

 

1290

1280

 

 

GRAPHITE

 

1270

 

 

 

1260

{250

 

 

1240

 

 

 

1230

 

 

1220

1210

  

 

 

 

 

{200

{190

 

 

1180

ll———mfl—-—-—_‘
!
1

 

! |
’ j

 

 

 

Fig. 13.

5 10 15 20
RADIUS (in.)

5 30

Radial Temperature Profiles in MSRE Core Near Midplane
TEMPERATURE (°F)

43

UNCLASSIFIED
ORNL—LR— DWG 73615

TN

/ ~

GRAPHITE
P
1260 -

 

1300

 

 

 

1240 /// 1///

1220 / ///,
FUEL

1200 ’//////

1180 [

 

 

 

 

 

 

 

 

 

 

 

 

1160
0 10 c0 30 40 SO 60 70

DISTANCE FRCM BOTTOM OF CORE (in.)

Fig. 1l4. Axial Temperature Profiles in Hottest Channel of MSRE
Core (7 in. from Core Center Line)
bl

Nuclear Mean Temperature

The nuclear mean graphite temperature is obtained by weighting the
local temperatures by the graphite temperature weighting function. In

general

j' T (r, 2) + AT(r, 2) | G (r, z) av
Tg* = - [ - - ] - (17)
J; Gg(r, z) dv

 

Substitution of the appropriate expressions leads to an expression of

the form

T = T. + b
n

o i T - Tin) + D AT (18)

l( out

vhere bl and b2 are functions of the system.

The nuclear mean graphite temperature was evaluated for three
degrees of fuel permeation of the graphite. Table 6 lists the results

for the reference condition at 10 Mw.

Bulk Mean Temperatures

The bulk mean graphite temperature is obtained in the same way as
the nuclear mean temperature, but without the weighting factor. The
bulk mean graphite temperature for various degrees of graphite permeatiocn

by fuel are listed in Table 6.

Table 6. Nuclear Mean and Bulk Mean Temperature
of Grephite®

 

Graphite Permeation

 

by Fuel Nuclear Mean T Bulk Mean T
% of graphite volume) (°F) (°F)
0 1257 1226
0.5 1258 1227
2.0 1264 1231

 

®At 10 Mw reactor power, T, = 175°F, T_, = 1225°F.
45

POWER COEFFICIENT OF REACTIVITY

The power coefficient of a reactor is a measure of the amount by which
the system reactivity must be adjusted during a power change tc maintain a
preselected contrcl parameter at the contrcl peint. The reactivity adjust-
ment is normally made by changing control rod positicon(s) and the control
parameter is usually a temperature cr a combination of temperatures in
the system. The reactivity effect cof interest is only that due tc the
change in power level; it does not include effects such as fuel burnup
or changing poison level which follcw a power change.

The power coefficient of the MSRE can be evaluated through the use
of the relaticnships between reactor inlet and ocutlet temperatures and
the fuel and graphite nuclear average temperatures develcped above. These
relationships may be expressed as follows for 10-Mw power operaticn with

no fuel permeation:

Out = Tin + 500F # (19)
* m o ~
Tf = .‘l.out - ll 09 F 3 (CO )
* . o
T, = T v 3L6F (21)

Although these equations were develcped for fuel inlet and outlet tem-
veratures of 1175 and 1225°F, respectively, they may be applied cver a
range of temperature levels withcut significant error.

In addition to the temperature relations, it is necessary tc use
temperature coefficients of reactivity which were derived on a basis
compatible with the temperature weighting functions. The fuel and graph-

ite temperature coefficients of reactivity from first-order, two-group

 

- Fy - o -
verturbation theoryT are -h4.b x 1077 °F L and ~7.3 x 10 2 °F l, respectively.
'i'hus
- * - *
Ak = -{4k.4 x 10 5) AT, - (7.3 x 10 5) ATg . (22)
T

B. E. Prince and J. R. Engel, Temperature and Reactivity Ccefficient
Aversging in the MSRE, ORNL TM=-379 (Oct. 15, 194% ;.

 

 
46

The magnitude of the power coefficient in the MSRE is strongly depend-
ent on the choice of the control parameter (temperature). This is best il-
lustrated by evaluating the power coefficient for various choices of the
controlled temperature. Three conditions are considered. The first of
these is a reference condition in which no reactivity change is imposed on
the system and the temperatures are allowed to compensate for the power-
induced reactivity change. The other choices of the control parameter are
constant reactor outlet temperature (To = const ) and constant average of

ut
inlet and outlet temperatures (& T, * T = const). In all three cases

the initial condition was assumed to be zeggtpower at 120C°F and the change
associated with a power increase to 10 Mw was determined. The results are
summarized in Table 7. It is clear that the power ccefficlient for the ref-
erence case must be zero because no reactivity change is permitted. In the
other cases the power coefficient merely reflects the amount by which the
general temperature level of the reactor must be raised from the reference
condition to achieve the desired control.

Fuel permeation of the graphite has only a small effect on the power
coefficient of reactivity. For 2% of the graphite volume occupied by fuel,
the power coefficient based on a constant average of reactor inlet and out-
let temperatures is -0.052% %E/Mw (vs. -0.047 for no fuel permeation}.

Table T. Tem.peraturesa and Associated Power Coefficients
of Reactivity

 

* *

 

T T T T Power
Control ocut ic T g Coefficient
Parameter (°F) (°F) (°F) (°F) (% Ak/k/Mw)
No external control 1184.9 1134%.9 1173.0 1216.5 0
cf reactivity
Tout = const 1200 1150 1188.1 1231.6 - 0.018
Twn Tout
- = const 1225 1175 1213.1 1256.6 - 0.047

 

®At 10 Mw, initially isothermal at 1200°F.
k7

DISCUSSION

Temperature Distributions

The steady-state temperature distributions and average temperatures
rresented in this report are based on a calculational model which is cnly
an approximation of the actual reactor. An attempt has been made tc in-
2lude in the model those factors which have the greatest effect on the
temperatures. However, some simplifications and approximations have been
made which will produce at least miror differences between the predicted
temperatures and those which will exist in the reactor. The treatment of
the main part of the core as a series of concentric cylinders with a
clearly defined, constant fuel velocity in each region is one such simplifi-
cation. This approximation leads tc very abrupt temperature changes at the
radial boundaries of these regions. Steep temperature gradients will un-
doubtedly exist at these points but they will not be as severe as thcse
indicated by the model. The neglect of axial heat transfer in the graphite
also tends to produce an exaggeration of the temperature profiles cover
those to be expected in the reactor. It is obvious that the calculated
axial temperature distribution in the graphite (see Fig. 14) can not exist
without producing some axial heat transfer. The effect of axial heat trans-
fer in the graphite will be to flatten this distribution somewhat and reduce
the graphite nuclear-average temperature.

Some uncertainty is added to the calculated temperature distributicns
by the lack of accurate data on the physical properties of the reactor nate-
rials. The properties of both the fuel salt and the graphite are based on
estimates rather than on actual measurements. A review of the temperature
calculations will be required when detailed physical data become available.

However, no large changes in the temperature pattern are expected.

Temperature Control

It has been shown that the power coefficient of the MSRE depends upon
the choice of the temperature to be controlled. Of the two modes of cor-

trel considered here, control of the reactor outlet temperature appesrs to
48

control-rod motion for a given power change. However, the magnitude of
the power coefficient is only one of several factors to be considered in
selecting a control mode. Another important consideration is the quality
of the control that can be achieved. Recent studies of the MSRE dynamics
with an analog computer8 indicated greater stability with control of the
reactor outlet temperature than when the average of inlet and outlet was

controlled.

 

8

S. J. Ball, Internal correspondence.
+3
I--SICiq

3]

kg

APPENDIX
Nomenclature

specific heat of fuel salt,

volume fraction of fuel,

Total circulation rate of fuel,

Temperature weighting function,

heat produced per fission,

thermal conductivity or neutron multiplication factor,

equivalent half-thickness of a graphite stringer
treated as a slab,

length of the core,

relative specific power,

equivalent specific power (absolute),
rate of heat transfer per unit area,
radius,

equivalent radius cf a fuel channel,

equivalent radius of a graphite stringer treated
as a cylinder,

radius of the core,
tinme,
temperature,

local transverse mean temperature in a single fuel
channel or graphite stringer

fuel nuclear mean temperature,
graphite nuclear mean temperature,
bulk mean temperature of fuel in the reactor vessel

bulk mean temperature of the graphite,
fe

R

@

%a* @ FAM T 2%

Subscrigts
f

g
in

out

Nomenclature - cont'd

velocity,

volume of fuel in the core,
2.405 r
R 2

axial distance from inlet end cf main core,

dimensionless radius,

defined by (6),

fraction of core heat originating in fuel,

defined by (5),

density of fuel salt,

macroscopic fission cross section,

neutron flux,

adjoint flux,

local temperature difference between mean temperature

across a graphite stringer and the mean temperature
in the adjacent fuel,

fuel,

graphite,

fuel entering the reactor

mixed mean of fuel leaving the reactor,
wall, or fuel-graphite interface,

fast neutron group,

thermal neutron group,

maximum value in reactor.
Derivation of Fguations for a Simple Model

Fuel Temperature Distribution

 

Consider a small volume of fuel as it moves up through a channel in

the core. The instantaneous rate of temperature rise at any point is

S ° BT al
S(DCP )
In this expression it is assumed that heat generated in the graphite
adjacent to the channel is transferred directly into the fuel (i.e.,
is not conducted along the graphite;, which accounts for the fraction 6.
In the channel being considered, the fuel moves up with a velocity,
u, which is, in general, a function of both r and z.

e w) w2

The steady-state temperature gradiernt along the channel is then

ar _ arfot g(r, z)

 

= = a3)
dz z/dt e(pCp)f u(r, z) (a3
let us represent the neutron flux by
_ 2.405 ¢ \ . gz
#(r, z) = qm I, (; = ,> sin T : (ak)

Assume further that the channels extend from z = 0 to z = L (ignoring
the entrance and exit regions for this idealized case). The temperature
at z = 0, for all channels, is Tin' The temperature up in the channel

is found by integration
52

T(r, z) = T Jfiz HZ%¢m - <é e ) sin <7~ > dz . (a5)

in T Y e(pc e ulr, Z)

For convenience in writing, let us define

HZ, ¢
@), = L2 (26)

m

= g;&%2;£ ’ (a7)

Then

Qp), L J,(x)
T. +
0 :rr(pcp)f U.(X, z)

 

T{x, z) =

- cos %&1) s (a8)

and the temperature at the outlet of the channel,

2(Qp), L J_(x)

Tin * fl(pcp)f u(x, L) (29)

 

T(x, L) =

Now let us calculate the mixed mean temperature of the fuel issulng
from all of the channels. This requires that we weight the exit tempera-

ture by the flow rate. The total flow rate is, in general;

F = JHR 2 f(r, z) u(r, z) dr . (a10)
o

For the idealized case where f and v are uniform throughout the core:

F = sRfu . (a1l)
The general expression for mixed mean temperature at L is then

R
m - = ji o f(r, L) u(r, L) T(r, L) ar . (al2)
out F 0

Substitution of (a9) for the temperature gives
23

h(Qp) L f R
out Tin + §T5§;7; 5 f(r, L) r Jo(x) dr

H3
§

4(Q.) L 2 .2.405
T'm R
Doy F(pcp)f <2.1+05) _£ f(x, L) x I (x) ax . (al3)

 

0

For the ideal case where f is constant, integration gives

 

l*(Qf)ml'f "RV
T L= T Floc ), ETE,) (2.4) 3, (2.4) (allt)
From (alh) we find that
(T - T, ) F(pC_ ). (2.4)
(Qf)m = out 1n2 P T (315)
4 LfR Iy (2.4)

which may be rewritten as

(F(pc_ ). (T . - T, )
B p’'f ‘Tout in n| |_2.405
Qg = L RAL eJ %LeJlZz.uoflJ ‘ (a16)

The first term in (al6) is simply the average effective value of the

 

fuel specific power (including heat transferred in from the graphite).
The second and third terms are, respectively, the maximum-to-average
ratios for the sine distribution of specific power in the axial direction

and the JO distribution in the radial direction.

Substituting (al6) in (a8) gives

(217)

 

T (x, 2) = T. + - cos —

2.405 © (Toue = Tin) 950 1 nz
AR R® by Jl(2.h05) u(x,z) L .)

This equation will be used in subsequent developments because it leads
to expression of the average temperatures in terms of the reactor in-

let temperature and the fuel temperature rise across the reactor.
Fuel Nuclear Mean Temperature

The nuclear mean temperature is obtained by weighting the local tem-
perature by the nuclear importance, which is also a function of position,
and integrating over the volume of fuel in the core. For the idealized
case the importance varies as the square of the neutron flux. Thus the

nuclear mean temperature of the fuel is given by

R 1,

J‘ jf T (r, z) ¢2 (r, z) 2r £(r, 2) dr dz

© O

Tp = . (a18)
R .L

jfl Jfl ¢ s z) 2w f(r, 2} dr 4z

G C

 

when (aht) is substituted for ¢(r, z), the denominator, which is

0
@ Ve, » becomes

~,
",

R L :
¢2 V, = 2fl¢m2 J ‘é f{r, z) r JO2 (:2;&92;5') sin” £§:> ar dz .
o

fe R L
(219)
For constant f, integration gives
.7;5 V, = Iif:f&fijiffi 3.2 (2.4505) (a20)
fc 2 1 ’ ’

With the substituticn of (alt), (all), (al7) and (a20), and assum-

ing £ and u %o be ccnstant, {al8) becomes

 

 

* (r., -T. ) 2.405 L
T, = T, + cut 31n > Jr X JO3(x) sin” %E *
2.405 Jq (2.405)L ° ©
(i - cos %E‘) dx dy (a21)
Integration with respect to z results in
¥* (Tout = Tin ) . 2 -11-05 3
T, o= T+ J o oxIl) ax . (s22)

2){2.405) Jl3 (2.405) o
25

Graphical integration with respect to x yields

j"g'ho5x JO3(X) ax = 0.564 . (a23)

o
Thus, the final result for the idealized core is
*

Tp o= T, 0+ 0.638 (TOut - Tin) . (a2l )

el Bulk Mean Temperature

 

The bulk mean temperature of the fuel in the core is

R .L
11 Jf Tf(r,z) 2qr f(r,z) dr dz
o "o

 

 

T = R L . (225)
£ jfl Jfi 2qr f(r,z) dr dz
O ©
The denominator is the volume of fuel in the core. Considering only
the idealized case again
- e -
Vfc = g RTLE (az6)
Substituting (all) and (al7) into (225) for constant f and u gives
o = T + 2.405 nt (Tout - Tii’l) ‘]'R fo.J (.2_'&9..5.5.) *
of in 2 Vg, J,(2.505) o Yo o R
c 1
<1 - cos £ Jar az (a27)
which is readily integrated to give
(T T, )
o out - “in
Tp= T, + 5 (a28)

The equations for evaluating the nuclear and bulk mean temperatures
of the fuel have been presented to illustrate the general method which
must be employed. An idealized core was used in this illustration to
reduce the complexity of the mathemetical expressions. This technique
may be applied to any material in any reactor by using the appropriate
temperature distributions and weighting functions. The results obtained
by applying this method to the fuel and graphite in the MSRE are pre-

sented in the body of the revort.

geb
=

o

-

= O O~ O oo

MSRP Director's Office,
Rm. 219, 9204-1
G. M. Adamson

L. G. Alexander
S. E. Beall

. Bender

. Bettis

. Billington

. Blankenship

. Bohlmann

. Bolt

. Borkowski

. Brandon

. Bruce

antor

. Cole
Conlin
Cook
Corbin
Cristy
Crowley
Culler

DeVan

Ditto

Donnelly

Douglas

Dunwoody

Engel

. Epler

. Brgen

. Fraas

. Frye

. Gabbard

Gallaher

Greenstreet
Grimes

. Grindell

Guymon

. Harley

Harrill

. Haubenreich

. Hise

. Hoffman

. Holz

. dJarvis

Jordan

. Kasten

Kedl

. Kelley

FEGUHnygoanEgS =

-

CHEPPHOsEHQO P GE QW

-

*

*
-

-

EWtflfl%"fltflt’d"flfi"fitfl?’flbflDUOQP&EUC-{ZUEIJU)QMC—IQ
HGUnoyYzazsZzuhndoagrwmonYy="RYdDEPQNGH

Distribution

IWEBEG

AR E

2 38 B

62

PO

I':-HL;IPdet:di‘JUbfl"—ibc-li—i"db@}gc—iOUC-amegfi—ctdflbtfl"UEH-BQEUMPO:I:

SRR

\Q
=3

EHEEORREOG Y !

-

-

-

-

. Kirslis

. Krewson
Iane

. Lindauer

. Lundin
Lyon

. MacPherson
. Maienschein
. Mann

. McDonsld

. MeDuffie

. McGlothlan
Miller

. Miller
Moore

. Moyers

. Northup

o

-

mmhoumwwanszbzm

. R. Osborn

Patriarca

. R. Payne

M. Perry

. B. Pike
. E. Prince

L. Redford

. Richardson

. C. Robertson
. W. Savage

. W. Savolainen

E. Savolainen

Scott

. Secoy

. Shaffer

. Skinner

. Slaughter

. Smith
Smith

mzzumm

. Splewak
. A. Swartout
. Taboada

. Tallackson
. Thoma

. Trauger
Ulrich
Weaver
Weaver

. Webster
Weinberg
White
Wilson

-

-

-

< QEHH"unQugi s
98,

99.
100-101.
102-103.
10k,
105-107 .
108.

109-110.
111.
112.

113.
11k,
115-129.

50

Distribution -~ contd

 

C. H. Wodtke

R. Van Winkle

Reactor Division Library
Central Research Library
Document Reference Section
Iaboratory Records

ORNL-RC

External

D. F. Cope, Reactor Division, AEC, ORO

A. W. larson, Reactor Division, AEC, ORO

H. M. Roth, Division of Research and
Development, AEC, ORO

W. L. Smalley, Reactor Division, AEC, ORO

Jd. Wett, AEC, Washington

Division of Technical Information Extension,
AEC