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ORNL-TM-

378

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COPY NO.

DATE

TEMEgERATURES

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/ Lf-----

November 5, 1962

IN THE MSRE CORE DURING STEADY-STATE POWER OPERATION

ABSTRACT

k

Over-all f u e l and graphite temperature d i s t r i b u t i o n s were calculated f o r a d e t a i l e d hydraulic and nuclear representation of t h e E R E , These temperature d i s t r i b u t i o n s were weighted i n various ways t o obtain nuclear and bulk mean temperatures for both materials. A t t h e design power l e v e l of 10 Mw, with t h e r e a c t o r i n l e t and o u t l e t temperatures at ll75'F and The 1225OF, respectively, the nuclear mean &el temperature i s l2l3'F. bulk average temperature of t h e f u e l i n t h e reactor v e s s e l (excluding For t h e same conditions and with no fuel per.-the volute) i s 119'F. meation, t h e graphite nuclear and bulk mean temperatures are 1257'F and 1226'F, respectively, h e 1 permeation of 2$ of t h e graphite volume r a i s e s these values t o 1264'F and 1231째F9 respectively.

The e f f e c t s of power on t h e nuclear mean temperatures were combined with th.e temperature c o e f f i c i e n t s of r e a c t i v i t y of t h e f u e l and graphite t o estimate t h e power c o e f f i c i e n t of r e a c t i v i t y of the r e a c t o r . If t h e r e a c t o r o u t l e t temperature i s held constant during power changes, t h e power c o e f f i c i e n t is o.o18$ '+Mw. If, on t h e other hand, t h e average i s held constant, t h e power o f t h e r e a c t o r i n l e t and outlet-eratures 0.&7$ '--\ coefficient is

-

-

s i m i l e Price

v

$

-

h ment ashington

of Commerce

25, D. C.

,, NQTICE

This

document contains information of a preliminary nature and was prepared primarily for internal use at the Oak Ridge National Laboratory. I t i s subject to revision or correction and therefore does not represent a final report. The information i s not to be abstracted, reprinted or o t b r w i s e given public dissemination without the approval of the ORNL patent branch, L e g a l and Information Control Deportment.

4


T h i s report was prepared as an account of Government sponaored work.

Neither the United States,

nor the Commission, nor any person acting on behalf of the Commission: A.

Mokes any warranty or representotion, completeness, any

expressed or implied, w i t h respect to the accuracy,

or usefulness of the information contained i n this report, or that the uae of

information,

apparatus,

method, or process disclosed i n this report may not infringe

privotely owned rights; or

6. 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 informotion, apparatus, w t h o d , or pracssr disclosed i n t h i s report.

As

used i n the above, "person octing on beholf 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, ony information pursuant t o his employment or contract w i t h the Commission, or h i s employment with such Contractor.

r

.


CONTENTS

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Page

ABSTRACT...^.....^................

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LISTOFFIGURES

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LISTOFTABUS

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I)

INTRODUCTION.

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Fuelchannels

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HydraulicModel..

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Neutronic Model

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(I

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Flux D i s t r i b u t i o n s

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APPENDIX

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... ., ,. ..,,. . Derivation o f Equations . . . . . , ... . .. Nomenclature

e

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a

V

12

22

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e

11

19

....... ... Nuclear Mean Temperature . . . . . . . , .. BulkMean Temperature . . ... ..... GRAPHITE TENPERATURES . , . .. . ... Local Temperature Differences .. . . . Over-all Temperature D i s t r i b u t i o n . . .... Nuclear Mean Temperature . . . . , ..... .. Bulk Mean Temperature . . ...,... . . POWER COEFFICIENT OF REACTIVITY . . . . . . . . . . . DISCUSSION . ... ..... ... Temperature D i s t r i b u t i o n s . , ........... Temperature Control . . . . . . . ... .. . Over-all Temperature D i s t r i b u t i o n

8

15 18

e

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5 6 7

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it

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Nuclear Importance Functions f o r Temperature FLIELTEMPEMTURES

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Power Density D i s t r i b u t i o n

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NEUTRONIC CALCUUTIONS

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.

DESCRXFTZON O F CORE

a

1

22 26

30 33 36 36 36

41 44 44 45 47 47 47 49 49 51


5 LIST OF FIGURES

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Page

I

1. Cutaway Drawing of MSRE Core and Core Vessel

9

MSRE Control Rod Arrangement and Typical Fuel Channels

10

3. Nineteen-Region Core Model f o r Equipoise 3A Calcu1ati.ons. See Table 3 f o r explanation of l e t t e r s .

16

2.

4.

Radial D i s t r i b u t i o n o f Slow Flux and Fuel F i s s i o n Density i n t h e Plane of Maximum Slow Flux

5 . Axial D i s t r i b u t i o n of Slow Flux a t a Position 7 i n . *om Core Center Line

20

21

6 . Radial D i s t r i b u t i o n o f Fluxes and Adjoint Fluxes i n t h e Plane of Maximum Slow Flux

23

7. Axial D i s t r i b u t i o n of Fluxes and Adjoint Fluxes a t a P a s i t i o n 7 i n , from Core Center Line

24

8 , Axial D i s t r i b u t i o n o f Fuel Fission Density a t a P o s i t i o n 7 i n . from Core Center Line

25

Relative Nuclear Importance of Fuel Temperature Changes as a Function of P o s i t i o n on an Axis U c a t e d 7 i n , &om Core Center Line

27

9.

10. Relative Nuclear Importance of Graphite Temperat u r e Changes as a Function of P o s i t i o n on an Axis Located 7 i n . from Core Center Line

28

11,. Relative Nuelear Importance o f Fuel and Graphite Temperat.ure Changes as a Function of Radial P o s i t i o n i n Plane of Maximum Thermal Flux

29

Channel O u t l e t Temperatures for MSRE and f o r a Uniform Core

34

13. Radial Temperature P r o f i l e s i n MSRE Core Near Midplane

42

12.

1,4, Axial. Temperature P r o f i l e s i n Hottest Channel of MSRE Core

(7 i n . from Core Center Line)

43


LIST OF’ TABLES

1. Fuel Channels i n t h e MSRE 2.

3.

Ccre Regions Used t.o Calculate Temperature D i s t r i b u t i o n s i n t h e i4SPl.E

14

Nineteen-Regicn Core Model Used i n Equipoise Calculations for MSRE

17

k . Flaw Rat,es, Powers and Tempera,tures i n Reactor Regions

31

5.

Local Graphite-Fuel Temgerature Differences i n t h e MSRE

40

6.

Nuclear Mean and Bulk Mean Terrgeratures o f Graphite

44

7.

Temperatures and Associated Pcwer Coefficients o f Reactivity

46

V


7 INTRODUCTION

W I

This r e p o r t i s concerned with t h e temperature d i s t r i b u t i o n and app r o p r i a t e l y averaged temperatures of t h e f u e l and graphite i n t h e MSRE r e a c t o r v e s s e l during steady operation a%power, Tke temperature d i s t r i b u t i o n i n t h e r e a c t o r i s determined by t h e

heat production and heat t r a n s f e r ,

The heat production follows t h e over-

all shape of t h e neutron flux, with t h e fYaction generated i n t h e gre.phi.te depending on how much f i e 1 i s soaked i n t o t h e g r a p h i t e ,

Fuel channels tend

io be h o t t e s t near t h e axis of t h e core because of higher power d e n s i t i e s t h e r e , but f u e l v e l o c i t i e s are equally important i n determining f u e l temperatures, and i n t h e f u e l channels near t h e axis of t h e MSRE core t h e v e l o c i t y i s over three times t h e average f o r t h e core. v e l a c i t y a l s o occur i n t h e outer re6;ions of t h e core.

Variations i n Graphite tempera-

t u r e s a r e l o c a l l y higher than t h e f u e l temperatures by an amount which v a r i e s with t h e power density and a l s o depends on t h e f a c t o r s which gov-

ern t h e heat t r a n s f e r i n t o t h e flowing f'uel. The mass ol" f a e l

iii

tiie r e a c t o r must Be kilown f o r inventory calcu-

l a t i o n s , and t h i s requires t h a t t h e mean density of t h e fuel and t h e graphite be known.

The bulk mean temperatures must t h e r e f o r e be calcu-

lated Tkae temperature and density of t h e graphite and f u e l a f f e c t t h e

nellctrcn leakage and absorption ( o r r e a c t i v i t y )

The r e a c t i v i t y e f f e c t

u t a l o c a l change i n temperature depends on where it i s i n the core,

with t h e cetitral regions being much more importante Useful q u a n t i t i e s i n r e a c t i v i t y a n a l y s i s are t h e "nuclear mean" temperatures of t h e f i e 1 and. graphite, which are t h e r e s u l t of weighting t h e l o c a l temperatures

by t h e l o c a l nuclear importance. The power c o e f f i c i e n t of r e a c t i v i t y i s a measure of how much t h e

eon*rol rod poisoning must be changed t o obtain t h e d e s i r e d temperature c o n t r o l as t h e power i s changed.

The power c o e f f i c i e n t depends on what

temperature i s chosen t o be controll-ed and on t h e r e l a t i o n of t h i s t e m perature t o t h e nuclear mean temperatures. This r e p o r t describes t h e MSRE core i n terms of t h e f a c t o r s which W

govern f h e temperature d i s t r i b u t i o n

It next presezts tiie calculated


8

The power c o e f f i c i e n t

temperature d i s t r i b u t i o n s and mean temperatures.

An

of r e a c t i v i t y and i t s e f f e c t on operating plans a r e then discussed, appendix sets f o r t h t h e derivation of t h e necessary equations and t h e procedures used i n t h e calculations.

DESCRIPTION OF CORE Figure 1 i s a cutaway drawing of the MSRE r e a c t o r vessel and core. Circulating f i e 1 flows downward i n t h e annulus between t h e r e a c t o r v e s s e l and the core can i n t o the lower head, up through t h e a c t i v e region of t h e core and i n t o the upper head. The major contribution t o r e a c t i v i t y e f f e c t s i n the operating r e a c t o r

cames from a c e n t r a l region, designated a s t h e main portion of t h e core, where most of the nuclear power i s produced. However, t h e other regions --wiihin t h e reactor v e s s e l a l s o contribute t o these e f f e c t s and these cont r i b u t i o n s must be included i n the evaluation of t h e r e a c t i v i t y behavior, The main portion of t h e core i s comprised, f o r t h e most p a r t , of a regular a r r a y of close-packed, 2-in. square stringers with half-channels machined i n each face t o provide me1 passages.

The regular p a t t e r n of f h e l chan-

nels i s broken near t h e a x i s of t h e core, where control-rod t h h b l e s and graphite samples are located (see Fig. 2), and near t h e perimeter where t h e graphite s t r i n g e r s a r e cut t o f i t t h e core can.

The lower ends of

t h e v e r t i c a l s t r i n g e r s are c y l i n d r i c a l and, except f o r t h e f i v e s t r i n g e r s a t the core axis, t h e ends f i t i n t o a graphite support s t r u c t u r e .

This

s t r u c t u r e consists of graphite bars l a i d i n two horizontal l a y e r s a t r i g h t angles t o each other, with clearances for f u e l flow. f i v e s t r i n g e r s rest d i r e c t l y on the zon.tal graphite s t r u c t u r e .

The center

INOR-8g r i d which supports t h e h o r i -

The main portion of t h e core includes t h e

lower graphite support region.

Regions a t t h e t o p and bottom of t h e core

where t h e ends of t h e s t r i n g e r s project i n t o t h e heads as well as the heads themselves and the i n l e t annulus are regarded as peripheral regions. The f u e l velocity i n any passage changes w i t h flow area as t h e f u e l

m v e s frm t h e lower head, through the support s t r u c t u r e , t h e channels, and the channel o u t l e t region i n t o t h e upper head.

The v e l o c i t i e s i n most

ciimnels a r e nearly equal, with higher v e l o c i t i e s near t h e a x i s and near t h e perimeter of the main portion of t h e core.

For most passages, the


9 UNCLASSIFIED ORNL-LR-DWG 61097

/

REACTOR CORE CAN

Fig. 1. Cutaway Drawing of IGRE Core and Core Vessel


10

REACTOR $

UNCLASSIFIED ORNL-LR- DWG 60845A

CONTROL ROD -GUIDE TUBE -GUIDE BAR TYPICAL FUEL CHANNEL7 REACTOR

0.200-in. R’

0.400 in. -

4 GRAPHITE IRRADIATION SAMPLES ( 718-1.. D l A ) ”

Fig. 2. MSRE Control Rod Arrangements and Typical f i e 1 Channels

5


g r e a t e r p a r t of t h e pressure drop occurs i n t h e tortuous path through t h e h o r i z o n t a l supporting b a r s .

This r e s t r i c t i o n i s absent i n t h e c e n t r a l

passages, r e s u l t i n g i n flows through t h e s e channels being much higher than

Vie average. The v a r i a t i o n s i n fuel-to-graphite r a t i o and f i e 1 v e l o c i t y have a s i g n i f i c a n t effect, on t h e nuclear cktaracterFs+,fcs and t h e temperature d i s t r i b u t i o n of t h e core.

In t h e temy;era+,ure analysis reported here, t h e

d i f f e r e n c e s i n flow a r e a and v e l o c i t y i n t h e entrance regions were negl e c t e d ; i . e . , t h e flow passages were ass-med t o extend from %he t o p b c - t t o m of t h e main p o r t i o n of t h e core wittowb change

.

t.0

the

Radial v a r i a t i o n s

were taken i n t o account by dividing t h e core into concentric, c y l i n d r i c a l regions according t o t h e f u e l v e l o c i t y and t h e fUel-%o-graph%$e r a t i o . Fuel Channels The fuel channels a r e of s e v e r e l t y p e s .

The number of each i n t h e

f i n a l core design i s l i s t e d i n Table 1. The f u l l - s i z e d channel i s t h e t m i c a l channel shown i n F i g , 2.

The half-channels occur near t h e core

perimeter where faces 02 normal g r a p h i t e si;ringers a r e acigiacent t o f l a t surfaced s t r i n g e r s .

The f r a c t i o n a l channels a r e half-channels extending

t o t h e edge of t h e core. and t h e core can.

The l a r g e annulus I s t h e gap between t h e g r a p h i t e

There a r e t h r e e annuli around t h e contro2-rod thimbles.

Tine g r a p h i t e specimens, which occupy only t h e upper h a l f of one l a t t i c e sgace above a s t r i n g e r of normal cross section,, were t r e a t e d as p a r t of

a Pall-length noma1 stringer. Table 1. Fuel Channels i n t h e MSRE Channel Type

Full-s i z e Half-size Frac+,i o n a l &age annulus Thimble annuli

Number 1120

28

16 1

3


12 Hydraulic Model Hydraulic s t u d i e s by K e d l1 on a f i f t h - s c a l e model of t h e MSRE core showed t h a t t h e a x i a l v e l o c i t y w a s a f'unction cf r a d i a l p c s i t i o n , p r i marily because of geometric f a c t o r s a t t h e core i n l e t .

A s a r e s u l t of these s t u d i e s , he divided t h e a c t u a l core channels i n t o s e v e r a l groups

according t o t h e v e l o c i t y which he predicted would e x i s t .

This d i v i s i o n was based on a t o t a l of 1064 channels with each of t h e control-rod-thimble annuli t r e a t e d as four separate channels.

He d i d not attempt t o define p r e c i s e l y t h e r a d i a l boundaries of some of t h e regions i n which t h e chann e l s would be found. Since t h e t o t a l number o f f u e l channels i n t h e core i s g r e a t e r than t h e number assumed by K e d l , and s i n c e r a d i a l p o s i t i o n i s important i n evaluating nuclear e f f e c t s , it was necessary t o make a modified d i v i s i o n of t h e core f o r t h e temperature a n a l y s i s .

For t h i s purpose t h e core was

divided i n t o f i v e concentric annular regions, as described below, based on t h e information obtained by K e a .

The fuel v e l o c i t i e s assigned by Kedl t o t h e various regions are used as t h e nominal v e l o c i t i e s . Region 1 This region c o n s i s t s of t h e c e n t r a l 6-in. square i n t h e core, with

a l l f u e l channels adjacent t o it, plus one-fourth of t h e area of t h e g r a p h i t e s t r i n g e r s which help form t h e adjacent channels.

The t o t a l

crcss-sectional area of Region 1 i s 45.0 i n e 2 and t h e e q u i v d e r k radius i s 3.78 i n .

The f u e l fract.ion ( f ) for t h e region i s 0.256.

This region contains t h e 16 channels assigned t o it by Kedl p l u s t h e 8 channels which he c l a s s i f i e d as marginal.

Because of t h e c y l i n d r i c a l geometry around

t h e control-rod thimbles, s i x of t h e channels which were marginal i n Kedl's model have t h e same flow v e l o c i t y as t h e r e s t of t h e region.

It was not considered worthwhile t o provide a separate region f o r t h e two remaining channels.

'E.

The nominal f l u i d v e l o c i t y i s 2.18 f t / s e c

S. Bettis e t a1

I n t e r n a l Correspondence.


W

Region 2

This region covers most of t h e core and contains only normal, fill-

All t h e f u e l channels which were not assigned elsewhere were assigned t o t h i s region. On t h i s b a s i s , Region 2 has 940 f u e l channels, a t o t a l c r o s s - s e c t i o n a l area of 1880 i n . 2 , a f u e l f r a c t i o n of s i z e d f b e l channels.

0.224, and equivalent outer radius of 24.76 i n . ( t h e inner radius i s equal t o t h e outer radius of Region l), and a nominal f l u i d v e l o c i t y of 0.66 Tt/sec. Region 3 This region contains 108 f u l l - s i z e d f u e l chmnels a s assigned t o it

by Kedl.

The t o t a l c r o s s - s e c t i o n a l area is 216 i n . 2 ; the fie]. 2racl;Lon

i s 0.224; t h e e f f e c t i v e outer radius i s 26.10

in-j

and t h e nominal f l u i d

v e l o c i t y i s 1.63 f t / s e c . Region

4

This region w a s a r b i t r a r i l y placed outside Region 3 even though it contains marginal channels from both s i d e s of t h e region. channels and f r a c t i o n a l half-channels were added t o t h e channels assigned t o t h e region by Kedl.

A l l the h a l f -

60 f i l l - s i z e d

This gives t h e equivalent of

78 f i l l - s i z e d channels. A l l of t h e remaining graphite c r o s s - s e c t i o n a l a r e a w a s a l s o assigned t o t h i s region.

A s a r e s u l t , t h e t o t a l cross-

s e c t i o n a l a r e a i s 247.9 in.2; t h e f u e l f r a c t i o n i s 0.142; t h e e f f e c t i v e outer radius i s 27.58 i n . , and t h e nominal f l u i d v e l o c i t y i s 0.90 f t / s e c . Negion 5 The s a l t annulus between t h e g r a p h i t e and t h e core can was t r e a t e d as a separate region.

The t o t a l a r e a i s

29.55 i n . 2 ; t h e fuel f r a c t i o n

i s 1.0; t h e outer radius i s 27.75 i n , ( t h e inner radius of t h e core can),

and t h e nominal f l u i d v e l o c i t y i s 0.29 f’t/sec. Effective Velocities The nominal f l u i d v e l o c i t i e s and flow areas l i s t e d above result i n a t o t a l flow r a t e through t h e core of 1315 gpm a t 1200’F.

All t h e v e l o c i -

t i e s were reduced proportionately t o give a t o t a l flow of 1200 g p m .

Table

2 l i s t s the e f f e c t i v e f l u i d v e l o c i t i e s and Reynolds numbers f o r t h e various

regions along with other f a c t o r s which describe t h e regions.


Table 2 .

Region

Number of All1-s i2ed Fuel Channels Y

12"

1

Core Regions Used t o Calculate Temperature Distributions in the MSRE

--

~-

-

Effective Fluid Velocity

ToSa.1 Cross sect,iona,!Area - - (im,z) --

Fuel Fraction =--

E-SfectAve Outer Radius ( I n ._)-

45 -00

0 -256

3.78

1.99

3150

72

--

. ? -

--- --

(fi/sec

Flow Reynolds Number

Rate (gpm)

2

940

1880

0.224

24.76

0.60

945

791

3

108

216 .o

0 -224

26 .io

1.49

2360

224

4

78

245.9

0 .l42

27.58

0.82

1300

89

29 55

1 .om

27 *75

0.26

421

24

5

0

1200

*

Plus 3 control-rod-thimble annuli.

-x+

Annulus between graphite and core shell.


Neutronic Model The neutronic c a l c u l a t i o n s upon which t h e temperature d i s t r i b u t i o n s

are based were made with Equipoise 3A, 2,3 a 2-group, 2-dimensional, multiregion neutron d i f f u s i o n program f o r t h e IEM-7O9O computer. Because of t h e l i m i t a t i o n t o two dimensions and other l i m i t a t i o n s on t h e problern s i z e , t h e r e a c t o r model used f o r t h i s c a l c u l a t i o n d i f f e r e d somewhat f r o n the hydraulic model

-

The e n t i r e reactor, including t h e r e a c t o r vessel, w a s represented i n c y l i n d r i c a l (r, z ) geometry.

Three basic materials, f u e l s a l t , graphite

and INOR were used i n t h e model.

A t o t a l of 19 c y l i n d r i c a l regions v i t h

various proportions of t h e basic materials w a s used. Figure

3 i s a v e r t i c a l h a l f - s e c t i o n through t h e model showing t h e

r e l a t i v e s i z e and l o c a t i o n of t h e various regions.

The region composi-

t i o n s , i n terms of volume f r a c t i o n s of t h e basic materials, are summar i z e d i n Table

3.

Regions J, L, M, and N comprise t h e main portion of t h e core.

This

portion contains 98.7% of t h e graphite and produces 87% of t h e t o t a l pover The c e n t r a l region (L) has t h e same f u e l and graphite f r a c t i o n s as t h e

c e n t r a l region o f t h e hydraulic model.

However, t h e outer boundary of -the

region i s d i f f e r e n t because of t h e control-rod-thimble representation. Since it was necessary t o represent t h e thimbles as a s i n g l e hollow c y l inder (region K ) i n t h i s geometry, a can containing t h e same amount o f INOR as t h e three-rod thimbles i n t h e r e a c t o r and of t h e same thickness

w a s used.

This e s t a b l i s h e d t h e outer r a d i u s o f t h e INOR cylinder a t

3 in.

which a l s o is c l o s e t o t h e radius of t h e p i t c h c i r c l e f o r t h e t h r e e thimbles i n t h e MSRE.

The c e n t r a l f u e l region was allowed t o extend only t o

t h e i n s i d e r a d i u s o f t h e rod thimble.

The p o r t i o n o f t h e core outside

t h e rod thimble and above t h e h o r i z o n t a l g r a p h i t e support region w a s homogenized i n t o one composition (regions J and M ) .

The graphite-fuel

3C. W . Nestor, Jr., EQUIPOISE-3A, ORNL-3199 Addendum (June 6, 1962).


16 Unc lassi f i e d OR??-LR-DWG. 71642


Table 3.

Region A

B c

radius (in. 1 inner outer 0

29 .oo 0

D E F

3 .oo 3 -00 28 .oo

G

3 -00 3 .oo

H I J K

Nineteen-Region Core Model Used in EQUIPOISE Calculations for E R E

27 -75 3 .oo 2.94

L

0

M N

0

2.94

29.s 290% 29-56 29 -00 28 .oo 29 .oo 28 -00 27 -75 28 -00 27 -75 3 -00

r7 -75 27 *75 29 -00 29 .oo 2.94 2 -94 2 -94

Z

Composition

~& (Volume percent ) top

74 * 92

0 0 0 100

0 0 0 0

100

93 e 7

3 -5

2.8

100

0

0 0 0 -2

Downcomer

-

0

Q

100

22.5

77 -5

0

5.P

76 -04 74.92 - 9.14 74.92 67 -47 67 -47 66.22 65.53 65.53 64 59 74.92

0

0

100

2 .oo

64 59

25.6

74 -4

0

2 .OQ 0

2 .oo

5-50

22.5 23 -7

77 - 5 76.3

0 0

Core can Core Simulated thimbles Central region Core Horizonta1 stringers

0 -1A1

66.9 90 08

15-3

17.8 9.2

74 -92 66.22 65.53

100

- 9.14

-io. 26 67 -47 66 -22 0

65 - 53

-

64 59 0

5-50

- 1.41

- 9.14

66 -22 65 * 53 64 59

fuel

94.6 63 93

89.9 43 -8

graphite

5.4 36.5

0 0 10 -1 56 * 2

INOR

Region Represented

bottom

100 100 0

0 0 0

Vessel top Vessel sides Vessel bottom Upper head

Bottom head


18 mixture containing the horizontal graphite bars at the core inlet was treated as a separate region (N). The main part of the core, as described in the preceding paragraph, Y does noC, incbde th.2 lower ends cf the grapk I*.Ice s5r.mgers t which extend beyond the horizcn",ai graphite bars, or She -TS~ cG,sted tops of the stringers. 'The bottom ends were included in a sin&e region (3; and the mixture at tit-~ ,.A* tc,y: c.f c,he -core was approx%mated by 5 regl,r_::rs (Eg C9 H, R, and S). The thickness of the material ccntained in the upper and lower heads (~.~-gi,yf~S0, &> and T) was ad3usted so that the amu~ct oi fue; salt was "q&L -6.3%he &qccl~~r~t. ecntained in the reactor heads. A; a result of this ad&&me.nf the over-all height of the neutronir- model of the reactor is not: exx%ly the same as the physical height. The upper and lower heads themselves (regions A and C) were flattened out and represented as metal discs of the nominal thic-kness of the reactcr mterial. ou-ixide the main Far-5 cf ,fLe core? the core Xc 5be radial direci.lm cat (region I), 5Le f~e1. inlet annulus (regicx F) and the reactor vessel (region S) were included tith the aztuai physfcal. dimensions of the react.0-s applied to them" The materiais within each separate regfcn Tqere treated as homogeneous mixtures in the calculations o As a result. 9 +,Le caLculations give only Yre over.,all shape I:f i;ke 3~. in the regicrs &:;:re inhomogeneity exists

TT1Lefirst step in the neu?zcni~ ealcv.:Lat;fons was ,?he establishment of i,k.e fuel-aa1-L ccm~csi-tlcn required for criticality. A carrier salt p- c.r,-'i -Ii Y I ba3,,ing 70 mcl i;' LiF, 23% 3eF2 and $I ZrF4 was assumed. !I'he LiF and erLf;rae,icri,s 'Krere keld zanst.ant and the BeF2 concentration %rF4 ::o:.~;: was reduce& aE 5t.z zon,cenfrat:cn c-F"UF4 was increased in the criticality sear& a Tke uxmium :$as aszz-aed to consist cf 93s II 235, $I u 238 I ) 1% u 234 ) 1sotcpic zcqosi%ion is 5sz.5a.i GE the material expected to be asailable for the reactor e The lithiu~m to be used in the manufacture of the fuel salt contains 73 p-am Ii-.6 This value was used in the cakulations. No chemiaal impur5ties were considered in the fuel mixix.re e Al;. of ,thc calculat5on.s vere made for an is&Jie~rmal system at 1200'F. and

15

,236

0

“b� LA18


V

The c r i t i c a l i t y search was made with MODRIC, a one-dimensional, multigroup, multiregion neutron diffusior. program,

4 a t O a 1 5 mol $,

c a l concentration of UF

T h i s established t h e c r i t i -

leaving 24.87% BeF2.

'The sarre

program w a s used for radial. and axiel t r a v e r s e s cf t h e mGdel used for t h e Equipoise 3A calcula$ion t o provide t h e 2-group constants f e r t h a t program. Fne s p a t i a l d i s t r i b u t i o n s of t h e fluxes and f u e l pcwer d e n s i t y were cbt a i n e 6 d i r e c t l y from t h e r e s u l t s of t h e Fquipcise

3A c a l c u l a t i o n . The

same fliuxes were used t o evaluate t k e nuclear importance f'unctions f o r

temperature changes. Flux D i s t r i b u t i o n s The r a d i a l d i s t r i b u t i o n of t h e slow neukron f l u x calcu2ar;ed f o r t h e IURE near t h e midplane i s show- i n Fig.

4.

This plane contains t h e xaxi-

m u value of t h e f l u x and i s 35 i n . above t h e bottom o f t h e main p a r t of' the core,

The d i s t o r t i o n of t h e flL-x produced by abscrptions i n t h e sinu-

i a t e d control-rod thimble, 3 i n . from t h e axial c e n t e r l i n e , i s r e a d i l y tipnzrent .

Because of t h e magnitude of t h e d i s t o r t i o n t h . i s s i m p l i f i e d

representation of t h e conmoi-rod thimbles i s probably noz adequate for an a c c u r s t e d e s c r i p t i o n of t h e slow f b x i n t h e immediate v i c i n i t y of t h e thimbles.

However, t h e o v e r - a l l d i s t r i b u t i o n i s probably reasonably

accurate.

The d i s t o r t i o n of t h e flux a t t h e cefiter 6f t h e r e a c t o r a l s o

precludes t h e use o f a simple analyt.ic expression t c describe t h e r a d i a l distribution. Figure 5 shows t h e zalculated axial d i s t r i b u t i o n of t h e slow f l u :.,long a l i n e which passes through t h e maximum value, 7 i n . from t h e v e r t i cal centerline.

The reference plane f o r measurements i n t h e a x i a l d i r e c -

t i c n i s t h e bottom o f t h e h o r i z o n t a l a r r a y o f graphite bars a t t h e lower end of t h e main p o r t i o n of t h e core.

Thus, t h e outside of t h e lower hee.d

i s at -10.26 in.; t h e t o p of t h e main portion of t h e core i s a t

and t h e outside of t h e upper head i s a t 76.04.

64.29 ir-.;

The shape of t h e a x i a l

d i s t r i b u t i o n i n t h e main portion i s c l o s e l y approximated by a s i n e curve extending beyond it a t both ends. shown i n Fig.

> is

The equation of" t h e s i n e approxime.tion


lIC

3 n l V A 'XVW d0 N0113WtJd

m

0

N

m

0

-

0

0

0

k


21

0 (c

E

0,


22

with t h e i i n e a r dimension given i n inches. The r e l a t i o n of t h e slow neutrcn flux t o t h e other fluxes (fast,

fast a d j o i n t and slow adjcint! i s shown i n Figs. 6 and 7.

These f i g u r e s

z r s s e n t t h e absolute v a h e s f o r a r e a c t c r power l e v e l of 10 Mw. Power Density D i s t r i b u t i o n A function t h a t i s ef g r e a t e r i n t e r e s t than the flux d i s t r i b u t i o n ,

from t h e standpoint of i t s effec5 on t h e r e a c t o r temperatures, i s t h e d i s t r i b u t i o n of f i s s i o n power density i n t h e fuel.

For t h e fuel compo-

s i t i o n considered here, only 0.87 of t h e t o t a l f i s s i o n s are induced by thermal neutrcns; t h e f r a c t i o n o f thermal f i s s i o n s i n t h e main p a r t of th.e core i s 0 . 9 .

I n s p i t e of t h e r e l a t i v e l y l a r g e f r a c t i o n o f nonthermal

f i s s i o n s , t h e o v e r - a l l d i s t r i b u t i o n of f u e l f i s s i o n d e n s i t y i s very s i m i lar t o t h e slow-flux d i s t r i b u t i o n .

The radial d i s t r i b u t i o n i s shown on

Fig. 4 for a d i r e c t comparison w i t h t h e t h e m a l f l u x .

The axial f i s s i o n

d e n s i t y d i s t r i b u t i o n (see Fig. 8 ) w a s f i t t e d with t h e same a n a l y t i c expression used f o r t h e a x i a l slow f l u x .

The q u a l i t y of t h e f i t i s about

t h e same f o r both functions. Nuclear Importance Functions f o r Temperature The e f f e c t s uscn r e a c t i v i t . y

G f

locai temperature changes i n fuel and

g r a p h i t e have been derived fram f i r s t - o r d e r two-group p e r t u r b a t i o n theory and are reported elsewhere .4 This a n a l y s i s produced weighting functions

o r zuc1ea.r importance f'mctions, G ( r , z ) f o r t h e l o c a l fie1 and g r a p h i t e temperatures,

The weighted average temperatures o f t h e fuel and g r a p h i t e

obtained 3y t h e use of t h e s e functions may be used with t h e appropriate temperature c o e f f i c i e n t s of r e a c t p r i t y t o c a l c u l a t e t h e r e a c t i v i t y change associated with any change i n temperature d i s t r i b u t i o n .

4,B. E. Prince and J . R . Engel, Temperature and R e a c t i v i t y C o e f f i c i e n t A.veraging i n t h e MSRE, ORNL TM-379 (October 15, 13623.

J


23

UNCLASSIFIED ORNL-LR-DWG 7 3 6 4 1

14

V

W

v)

l u

E

12

\ 0 u)

C

z

+ 3

W

c

v

to

3

H

0 LL

0

cc

8

W

3 0

a cc 0

+

------i----_

6

0

a W cc K 0

LL

FAST ADJOINT--* 4

X 2 1 LL

2

0 0

5

to

(5

20

25

RADIUS (in.)

Fig. 6. Radial Distribution of Fluxes and Adjoint Fluxes in the Plane of M a x i m u m Slow Flux

30


24

UNCLASSIFIED ORNL-LR-DWG 7 3 6 f 2

14

12

10

8

6

4

2

0 -10

0

10

20

30 40 A X I A L POSITION (in.)

50

60

70

Fig. 7. .&la1 DLstrCbutlon of Fluxes and Adjoint Fluxes a t a Posltion 7 i n . f r m Core Center Line

80


Y.

4

2 m

0

N

0

0

0

I

0 N 0

I

ni

p 0 b m

Q

0

FRACTION OF MAXIMUM VALUE

-0

z

C

E n

u)

r D

0


The temperature weighting flrnctions a r e evaluated i n t e r a s o f t h e

four flux products:

*

*

$$, $l

*

*

$d2, Pl2 (ifl

and (if2 (if2. The c o e f f i c i e n t s applied t o t h e various terms depend on t h e m a t e r i a l being considered, i t s l o c a l volume f r a c t i o n and the manner i n which temperature a f f e c t s i t s

physical and nuclear p r o p e r t i e s .

Since t h e weighting functions a r e evalu-

a t e d separately for fuel and graphite, a p a r t i c u l a r weighting function e x i s t s only i n those regions where t h e m a t e r i a l i s represented. Figures 9 and 3.0 show t h e a x i a l v a r i a t i o n i n t h e weighting f’unctions

a‘cr f u e l and graphi+,e a t t h e r a d i a l p o s i t i o n o f t h e maximum f u e l power

Fcr bctk materials t h e a x i a l v a r i a t i o n 9, t h e main p a r t of t h e

censity,

core i s very c l o s e l y approximated by a sine-squared, r

1

Ls g c t t e d on each cf t h e f i g u r e s f o r comparison,

t h e endE r e s u l t p i m a r i l y f!mm

The function:

The wide v a r i a t i o n s a t

t h e d r a s t i c changes i n v a h a e f r a c t i o n

and the a-gFarent; d i a c c n t i n u i t i e s a r e t h e result. cf dividing t h e r e a c t o r

a

i n t o d i s c r e t e regions f e r t h e Equipcise 3A c a l e v l a t i e n The r a d i a l temperature weighting f’unctfcns for both f i e 1 and g r a p h i t e near t h e cere midplane a r e shcwn i n Fig. 11. Because o f t h e p e c u l i a r shape

sf t h e cLII’ves, no aatemFt was made t.c f i t ana1yt.f- expressions t o t h e s e €unctions 0

The f u e l temperature d i s t r i b u t i o n i n t h e r e a c t o r has an o v e r - a l l shape whicb i s Cietemined by t h e shape cf t h e power d e n s i t y and t h e f u e l flow gzttern.

Within t h e main F a r t cf t h e ccre, where t,he f u e l flows i n dis-

w e f e charnelsr temperature v a r i a t i o n s across each i n d i v i d u a l channel a r e

sti;eriinps& cln

02

t h i s c v e r - a l l shape.

‘These l o c a l e f f e c t s w i l l be touched

Later (page 37 ) i n t h e s e c t i o n c n graphite-fuel temperature d i f f e r e n c e s

The present s e c t i c n deals with t h e o v e r - a l l shape of t h e f u e l temperature

d i s t r i b u t i c n acd average temperatures.

These were c a l c u l a t e d for a r e a c t o r

power l e v e l of 10 Mw wi%h i n l e t and o u t l e t temperamres a t resgeetively.

1175 and

1225’F,

Tk.e t o t a l P ~ e lflow r a t e w a s 1200 g y m J


27

UNCLASSIFIED ORNL-LR-DWG 73618

1 .O

0.8

W 0

z

EQUIPOISE 3 A R E S U L T s i n 2 p / 7 9 . 4 ( z + 4.71

I]

~

0.6

a I-

[L

0

a I -

0.4

K

a W

0.2 3

z W

L

+ a

0

-I W K

-0.2

-0.4

-0.6 -20

-10

0

40

20 I ,AXIAL

30

40

50

60

70

80

POSITION ( i n . )

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


2s

UNCLASSIFIED ORNL-LR-DWG 73619

1.o

0.8 W 0

2

0.6

t-

LL

0

a I

-

0.4

LL

a W -I

0

= z

0.2

W

3 a

t1

0

W

LL

- 0.2

-0.4 - 10

0

10

20

30

z , AXIAL

50

40 POSITION

60

70

( in.)

F i g . 10. Relative N x l e a r Importance o f Graphite Temperature Changes as z F a c t i o n of P o s i t i o n on an Axis Located 7 i n . from Core Center Line

80


29 UNCLASSIFIED ORNL-LR-DWG 7 3 6 47

V

1 .O

0.9

0.8

0.7 W 0

z 0.6

g

2 -

5

0.5

W -I 0

3 Z W

0.4

2

5-I

'$

0.3

0.2

0.1

0 0

5

40

15

20

25

RADIUS (in.)

Fig. 11. Relative Nuclear Importance of Fuel and Graphite Temperature Changes as a Function of Radial Position i n Plane o f Maximum Thermal Flux

30


Over-all Temperature Distribution

I n describing t h e over-all f u e l temperature d i s t r i b u t i o n it is convenient t o regard the reactor as c o r s i s t i n g of a main core, i n which most c ? t h e nuclear hea% i s produced, and a number o f peripheral regions which,

tcgether, contribute only a small mount to the tctal. power l e v e l o O f the

19 r e a c t c r regicns used f o r t h e neutronic 'calculations (see Fig. 3 ) > 14 er;n%%inf b e l a d 10 of these were regarded as peripheral regions. remaining fuel-bearing regions

fc)u

represent the m i n part of' the core.

(J9 L9 M9

The

and N ) were combined t o

On t h i s basis, t h e main portion ex-

teads r a d i a l l y tc t h e i c s i d e of t h e core s h e l l and axitally from t h e bottom of t h e h c r i z o n t a l graphite bars t o t h e top of t h e uniform f u e l channels.

This portion accounts f o r 87% of t h e t o t a l r e a c t o r power.

The volute on

t h e r e a c t o r i n l e t was neglected i n t h e temperature calculatJons because cf i t s physical separation from t h e r e s t o f t h e r e a c t o r .

Feripheral Regions Since only l3$ c f t h e r e a c t o r power i s produced i n t h e peripheral regions, t h e t e q e r z t u m v a r i e t i o n s within them m e

s~llelle,n,d

the d e t a i l s

cf t h e temperat.ure d i s t r i b u t i o n s within these regions were not calculated. The mean temperature r i s e f o r each region was calculated from t h e f r a c t i o n

of t h e t c t a l power Froduced i n the region and t h e f r a c t i o n of t h e t c t a l

f i c w r a t e through i t .

The i n l e t temperature t o each region was assumed

ccdstan? at, t h e mixed mean c u t l e t of t h e preceding regicn.

-.?-nea.n tenperatuse,

h,.7 1

An approximate

midway between the region i n l e t and o u t l e t tempera-

t-tires: wa.s assigned t c each peripheral region.

Table

4 summarizes

the

f L c w r a t e s , powers and f u e l temperatures i n t h e various r e a c t o r regions.

The wPde v a r i a t i c n s i n f u e l temperature, both r a d i a l l y and a x i a l l y , i n tlhe main p a r t of t h e core n e c e s s i t a t e a more d e t a i l e d description of t h e kemperature d i s t r i b u t i o n . The average temperature of t h e f'uel i n a channel a t any a x i a l position i s equal t o t h e channel i n l e t temperature plus a rise proportional t o t h e mm a f the heat generated i n t h e fuel and t h a t t.ransferred t o it from t h e

d 2 a c e n t graphite as t h e f'uel moves from t h e channel inlet t o the specified pcint.

%r l=,-zt ~ r a & ~ c ci nd t h e fie1 fcllows very closely the raciial a d


Table

w

a Region

4.

Flow Rates, Powers and Temperatures i n Reactor Regions

Flow

( a mB

Temp. Riseb

Power (kwl

("F)

Av. Temp. C (OF)

D

1224.7

E

1223 1

F

1175.6

1200

G

1222 4

H

1221 9

5

d

c

L

43

d

M TJ

1157

d

1200

d

0

1200

P

1200

1177-1 1176.6

Q

43 43 43

R S

1200.9

1199 9 1199.7

a

Regions not containing f u e l are excluded.

bAt 10 14w C

With Tin

= 1173째F, Tout

= 1223째E'

'Actual temperature d i s t r i b u t i o n calculated for t h i s region. See t e x t . a x i a l v a r i a t i o n of t h e f i s s i o n power d e n s i t y .

Since t h e heat production

i n t h e graphite i s small, no g r e a t error i s introduced by assigning t h e

same s p a t i a l d i s t r i b u t i o n t o t h i s term.

Then, i f axial heat t r a n s f e r i n

t h e g r a p h i t e i s neglected, t h e n e t rate o f heat a d d i t i o n t o t h e f b e l has t h e shape of t h e f u e l power d e n s i t y .

The f u e l temperature rise i s i n -

versely proportional t o t h e volumetric heat capacity and v e l o c i t y .

Thus


32

where Q is an equivalent specific power which includes the heat added to f 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 volumetric heat capacity, (pC ) is assumed constant and only radial variations P fJ in the fie1 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)

Then

If the sine approximation for the axial variation of the power density (Fig. 8) is substituted for B ( z ) , equation (3) becomes A r) Tf(r,z) = Tf(z = 0) + &$5{cos

vr

In this expression,

X

a

- cos

(z + 4.36)]

) . (4)

is a collection of constants,

a = - ‘ (0 + 4.36) 77 -7

.

The limits within which (4)is applicable are the lower and upper bcundaries of the main part of the core, namely, 0 G z < 64.6 in. It is clear from this that the shape of the axial temperature distribution in the fuel in m y channel is proportional to the central portion of the

curve [1

-

cos p].

(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 temTerature distribution in the graphite )


The r a d i a l d i s t r i b u t i o n of t h e f u e l temperature a t t h e o u t l e t of t h e channels i s shown i n Fig. 12 f o r the reference conditions a t 10 Ptw.

This

d i s t r i b u t i o n includes t h e e f f e c t s o f t h e d i s t o r t e d power-density d i s t r i bution (Fig. k ) and t h e r a d i a l v a r i a t i o n s i n f u e i v e l o c i t y .

A t the refer-

ence conditions t h e main-core i n l e t temperature i s 1177,3"Fand t h e xixed mean temperature leaving t h a t regior: i s 2220.8*F0 The a d d i t i o n a l heat required t o raise t h e r e a c t o r o u t l e t temperature t o 1223'F

i s produced i n

Vne p e r i p h e r a l regions above t h e main p a r t o f t h e core.

Also shown on

Fig. 1 2 f o r comparison i s t h e o u t l e t temperature d i s t r i b u t i o n f o r an i d e a l ized core with a uniform f u e l f r a c t i o n and uniform f i e 1 v e l o c i t y .

For t h i s

case t h e radial power d i s t r i b u t i o n vas assumed t o follow t h e Bessel f'unct i o n , J0(2.4 r / R ) 9 which vanished a t t h e i n s i d e o f t h e core s h e l l .

The

d i s t r i b u t i o n was normalized t o t h e same i n l e t and mixed-mean o u t l e t t e m p e r a t u r e s t h a t were calculated for t h e main portion of t h e l4SRE c o r e . Nuclear MeaE Temperature A t zero power t h e f u e l temperature is e s s e n t i a l l y constant throughout

t h e r e a c t o r and, a t power it assumes t h e d i s t r i b u t i o n discussed i n t h e preceding s e c t i o n .

The nuclear mean temperature of t h e f u e l i s defined as

t h e uniform f u e l temperature which vould produce t h e same r e a c t i v i t y chmge from t h e zero-power condition as t h a t produced by t h e a c t u a l f u e l temperature distribution. The nuclear average temperature of t h e f u e l i s obtained by weighting t h e l o c a l temperatures by t h e l o c a l nuclear importance f o r a f u e l texpera-

ture change.

The importance function, G ( r , z ) ,

includes t h e e f f e c t of f u e l

volume f r a c t i o n as w e l l as a l l n u c l e w e f f e c t s (see p . 22 e t s e q . ) .

There-

fore

J'

G(r,z) dv

V

I n carrying o u t t h e indicated i n t e g r a t i o n f o r t h e MSRE, both t h e numerator m d denominator o f (7

were s p l i t ir.to 11 terms, one f o r t h e main p a r t of

t h e core and one for each of the fuel-bearing p e r i p h e r a l regions. p c r f p h e r d rci;ims, tk.2 3 ~ ~C,eqcra;t.ac 2, ic cach rcgio::

For t h e

vas zss-mcd constant


34

UNCLASSIFIED ORNL-LR-DWG

75775

1290

I275

I250 LL 0

W

a 3

t-

a

LL W

a E W c

I225

I200

I175 0

5

10

15

20

25

30

RADIUS, i n .

Fig. 12. Channel 0;rtlet T a p e r a t - x e s for 1;LSRF: 2nd for a Uniform Core


V

a t t h e average temperature ( s e e Table

4).

The volume i n t e g r a l s of t h e

weighting functions were obtained by combining volume i n t e g r a l s of t h e

f l u x products, produced by t h e Equipoise 3A calculation, with c o e f f i c i e n t s derived from t h e perturbation c a l c u l a t i o n s .

For t h e main p a r t o f t h e core,

t h e temperatures and temperature weighting f’unctions were combined and i n t e g r a t e d a n a l y t i c a l l y ( i n t h e axial d i r e c t i o n ) and numerically ( i n t h e

radial d i r e c t i o n ) t o produce t h e required terms.

The n e t r e s u l t f o r t h e

i%el i n t h e MSRE r e a c t o r v e s s e l i s Tf

*-

Tin + 0.762 (Tout

-

Tin)

(8)

A similar a n a l y s i s was performed f o r a uniform c y l i n d r i c a l r e a c t o r

with uniform flow.’

I n t h i s case t h e f u e l power density w a s assigned a

s i n e d i s t r i b u t i o n i n t h e a x i a l d i r e c t i o n and a Jo Bessel function f o r the radial distribution.

Both functions were allowed t o vanish a t t h e

r e a c t o r boundaries and no p e r i p h e r a l regions were considered.

A tempera-

t u r e weighting f’unction equal t o t h e product of t h e f u e l f r a c t i o n and t h e square of t h e thermal f l u x (or power d e n s i t y since both have t h e s m e shape i n t h e i d e a l c a s e ) w a s used. Tf

*-

Tin

For t h i s case: -t-

0.838 (Tout

- Tin)

( 91

The p r i n c i p a l reason f o r t h e lower nuclear average f u e l temperature i n t h e MSRE i s t h e l a r g e volume of f u e l i n t h e p e r i p h e r a l regions a t t h e

i n l e t t o t h e main p a r t of the core and t h e high s t a t i s t i c a l weight assigned

t o t h e s e regions because of t h e high f u e l f r a c t i o n .

The existence of t h e

small, high-velocity, low-temperature region around t h e axis o f t h e core has l i t t l e e f f e c t on t h e nuclear average temperature.

Actually t h i s

“ s h o r t - c i r c u i t ” through t h e core causes a s l i g h t increase i n t h e nuclear average temperature.

There is lower flow and l a r g e r temperature r i s e i n

t h e bulk of t h e r e a c t o r than i f t h e flow were uniform, and t h i s e f f e c t outweighs t h e lower temperatures i n t h e small region a t t h e c e n t e r .

f See appendix f o r d e r i v a t i o n of equations f o r t h e simple case.


Bulk Mean Temperature The bulk mean f u e l temperature i s obtained by i n t e g r a t i n g t h e l o c a l ,

As i n t h e case

unweighted temperature over t h e volume of the r e a c t c r .

rf t h e nuclear mean temperature, the l a r g e volunre cf low-temperacure f u e l

a t the i n l e t t o t h e main p a r t of the IGRE mre makes t h e bulk mean temperat a r e f o r the MSRE lower than f o r a simple, unlfcmn core

Uniform Core:

-

Tf =

T

in

+

1/2 (Tout

- Tin$

GRAPHITE TEMPERATURES

During steady-state power opera.tion, the mean teqera-kure i n any graphite s t r i n g e r i s higker than the mean temperatare off t h e f u e l i n the adjacent channels.

As a r e s u l t , both t h e nuclear mean and the bulk mean

temperatures of t h e graphite a r e higher than the Corresponding mean t e m peratures i n the f u e l . I n general, it i s convenient tc: express t h e graphite temperature i n terms o f t h e f u e l temperature and t k e difference between t,he graphite and fuel temperatures.

That is: T

g

=

TIf

+

AT

where AT i s a p o s i t i v e number at steady-state Nearly a l l o f t h e graphite i n t h e E R E

.

(98.7%)i s

contained i n the

regions which a r e combined t o form t h e main p a r t of t h e core.

Since t h e

remainder would have only a very s m U effect, on the system, only t h e main p a r t of the core i s t r e a t e d i n evalv.ating graphite temperatures and d i s tributions. b c a l Temperatue Differences

I n order t o evaluate t h e l o c a l graphite-,fuel temperature difference, it is necessary t o consider t h e core i n terms of a. number o f u n i t c e l l s , each coataiping graphite and f k l and ex-kending tke l e r g t h of t h e core.


U

I n calculating t h e l o c a l temperature d i s t r i b u t i o n s , it i s assumed t h a t no heat i s conducted i n t h e axial d i r e c t i o n and t h a t t h e heat generation i s

L

uniform i n t h e r a d i a l d i r e c t i o n over t h e u n i t c e l l s .

The temperature

d i s t r i b u t i o n s within t h e unit c e l l s are superimposed on t h e wfer-abl rea c t o r temperature distribu'cions.

The difference between t h e mean graphite

and f u e l temperatures within an individual c e l l can, i n general, be kroken down i n t o t h r e e p a r t s :

1. t h e Poppendiek e f f e c t , which causes t h e f'uel near t h e w a l l of a channel t o be h o t t e r than the mean fa? t h e channel; 2.

t h e temperature drop across t h e g r a p h i t e - f i e 1 i n t e r f a c e , r e s u l t i n g from heat flow 0 L . t of t h e pap'nfte; and

3. t h e difference between t h e mean temperature i n a graphite s t r i n g e r and t h e temperature at t h e i n t e r f a c e , which i s necessary t o conduct heat produced i n t h e graphite t o t h e surface.

These p a r t s are C,reated separately i n the following paragraphso Pogpendiek E f f e e t As m y be seen f r s m t h e Ee-flolc?s numbers i n T&le

2 (page

14))

the

f l u i d flow i n most of t h e core i s c l e a r l y i n t h e laminar regime and t h a t I n t h e remainder of t h e core i s i n t,he t r a n s i t i o n range where flow may be e i t h e r laminar

G r

turbulent.

I n t h i s analysis, t h e flow i n t h e eztire

core i s assumed t o be l a n i n a r t o provide conservatively pessimistic e s t i -

mates of t h e temperature e f f e c t s . Equations are available f o r d i r e c t l y computing the Poppendiek e f f e c t f o r laminar flow i n c i r c u l a r channels o r between i n f i n i t e slabej5j6 (i.e

e

t h e r i s e i n temperature of f l u i d near t h e w a l l associated with i n t e r c a l heat generation and r e l a t i v e l y low f l u i d flow i n t h e boundary l a y e r ) . Since t h e f u e l channels i n t h e MSRE are n e i t h e r c i r c u l a r nor i n f i n i t e irtwo dimensions, t h e t r u e Poppendiek e f f e c t w i l l be between the r e s u l t s pi-edicted by these two approaches.

The method used here w a s t o assume

'H. F . Poppendiek and L. D . Palmer, Forced Convection Heat Transfer i n Pipes with Volume Heat Sources Within t h e Fluids, ORNL-1393 (Nov. 3,

1953 3

5H. F. Poppendiek and L. D . Palmer, Forced Convection Heat Transfer -

&tween P a r a l l e l P l a t e s and i n Annuli with Volume Heat Sources Withic t k e n n ~ ~ W i 1 L r L - I . 1 V I (Way 11, ;9% ) s

W

T l,.-2 7 ~CILLIOb)

I -

3oni

0

,


38 c i r c u l a r channels with a diameter such t h a t t h e channel flow area i s equal t o t h e a c t u a l channel area.

This s l i g h t l y overestimates t h e e f f e c t .

For c i r c u l a r channels with laminar flow and heat t r a n s f e r from graphite t o f u e l , t h e following equation applies:

This equation i s s t r i c t l y applicable only if t h e power density and heat

flux a r e uniform along the channel and t h e length of t h e channel i s i n f i n i t e . However, it i s applied here t o a f i n i t e channel i n which t h e power density and heat flux vary along t h e length. It i s assumed t h a t t h e temperature p r o f i l e i n t h e f l u i d instantaneously assumes t h e shape corresponding t o t h e parameters at each elevation.

This probably does

not introduce much e r r o r , because t h e heat generation v a r i e s continuously snd not very d r a s t i c a l l y along t h e channel.

If it i s f u r t h e r assumed

t h a t t h e heat generation terms i n the f u e l and i n t h e graphite have t h e same o v e r - a l l s p a t i a l d i s t r i b u t i o n , the Poppendiek e f f e c t i s d i r e c t l y proportional t o t h e f u e l power density.

This simple proportionality

r e s u l t s from the f a c t t h a t , i n the absence of axial heat conduction i n t h e graphite, t h e r a t e of heat t r a n s f e r fâ&#x20AC;&#x2122;rom graphite t o f u e l ,

Q,

is

d i r e c t l y proportional t o t h e graphite s p e c i f i c power. Tezperature Drop i n Fluid Film Since t h e Poppendiek e f f e c t i n t h e core i s calculated f o r laminar flow, t h e temperature drop through t h e f l u i d immediately adjacent t o t h e chennel w a l l i s included i n t h i s e f f e c t .

Therefore, a separate calculation

of film temperature drop i s not reqaired. Temperature Distribution i n Graphite Stringers The difference between t h e mean temperature i n a graphite s t r i n g e r and t h e temperature a t t h e channel w a l l cannot be calculated a n a l y t i c a l l y for t h e geometry i n the MSRE.

Therefore, two approximations were calcu-

l a t e d as upper and lower l i m i t s and the e f f e c t i n t h e MSRE was assigned a value between them.


One approximation t o t h e graphite s t r i n g e r i s a cylinder whose c r o s s -

s e c t i o n a l a r e a i s equal t o t h a t o f t h e more complex shape. B

This leads t o

cylinder with a radius of 0.9935 i n , and a surface-to-volume r a t i o of

2.0i i n .

-1.

If only t h e fuel-channel surface (the surface through which

a l l heat must be t r a n s f e r r e d from t h e g r a p h i t e ) of t h e a c t u a l graphite s t r i n g e r i s considered, t h e surface-to-volume r a t i o i s 1.84 i n .

.

-I

There-

f~re,t h e cylinder approximation und.erestimates the mean graphite texperat u r e . The second approximation i s t o consider t h e s t r i n g e r as a slab, cooled on two s i d e s , with a half-thickness of 0.8 i n . ( t h e distance fkâ&#x20AC;&#x2122;or. t h e s t r i n g e r center l i n e t o t h e edge of t h e f u e l channel). n a t i o n has a surface-to-volume r a t i o of 1.25 i n . estimate of t h e mean g r a p h i t e temperature.

-I

This approxi-

which causes an over-

The value assigned t o t h e

difference between t h e mean graphite temperature and t h e channel-wall temperature was obtained by a l i n e a r i n t e r p o l a t i o n between t h e two approximations on t h e basis of surface-to-volume r a t i o .

For t h e c y l i n d r i c a l geometry w i t h a r a d i a l l y uniform heat source i n t h e s t r i n g e r , t h e difference between t h e average temperature i n the s t r i n g e r and t h e channel w a l l temperature i s given by:

T g

1

P

r

2

u s k

-Tw-

g

For t h e slab approximation: 1

g

1 P

R2

5 -gi;-* g

(15 1

Application of u n i t c e l l dimensions f o r t h e MSRE and l i n e a r i n t e r p o l a t i o n between t h e two approximations leads t o P

i;& .

(16 1

g Ret b c a l Temperature Differ enc e The two temperature e f f e c t s described by Equations (13) and

(16) mzy

be combined t o give t h e l o c a l mean t,emperature d i f f e r e n c e between graphite and fixel.


40

L

-J

Since both terms i n t h i s equation a r e d i r e c t l y proportional t o t h e power density, t h e l o c a l temperature difference may also be expressed i n terms

cf n maxhm value and the l o c a l , r e l a t i v e power density.

Maximum Local Values

The maximum values of t h e l o c a l graphite-fuel temperature difference niay be obtained by applying t h e appropriate s p e c i f i c powers t o Equation

(17).

*

Because of t h e dependence or* s p e c i f i c power, t h e temperature d i f -

ferences a r e strongly influenced by t h e degree of f u e l permeation of the graphite which a f f e c t s primarily t h e graphite power density.

The e f f e c t s

were examined f o r 0, 0.5, and 2.05 of t h e graphite volume occupied by f u e l . I n a l l cases it w a s assumed t h a t the f u e l i n t h e graphite w a s u n i f o d y d i s t r i b u t e d i n t h e s t r i n g e r s and t h e t the s p e c i f i c powers were uniform i n t h e transverse d i r e c t i o n for individual graphite s t r i n g e r s and f u e l chan-

-n-els. Table 5 shows the maximum graphite-fuel temperature differences

for the three degrees of f u e l permeation, at the 10-Mw reference condition. Table 5

Lccal Graphite -Fuel Temperature Differences i n t h e MSRE

Graphite Permeation by Fuel 4 of graphite volume)

*

Maximum Local Temperature Difference

(OF)

0

62.5

0.5

65.8

I n these calculations, it was assumed t h a t 6% of t h e r e a c t o r power i s sroduced i n t h e graphite i n t h e absence of f u e l permeation. This value i s based on calculations of gamma arid neutron hca%iag i n t h e graphite by C . W. Nestor, (unpublished). The t h e m 1 conductivities of f u e l and graph%-bewere assumed t r 3 be 3.21 and 13 Btu/hr fâ&#x20AC;&#x2122;tâ&#x20AC;&#x2122;F, respectively.


I n t h e above coriiputations, it was assumed t h a t f u e l which soaked i n t o

W

t n e graphite w a s uniformly d i s t r i b u t e d . case.

This i s not t h e worst possible

S l i g h t l y higher temperatures r e s u l t i f t h e f u e l i s concentrated

near t h e perimeter of t h e s t r i n g e r s .

To obtain an estimate of t h e i n -

creased s e v e r i t y , a stringer w a s examined f o r t h e case where 2% of t h e g r a p h i t e volume i s occupied by f u e l .

It was assumed t h a t t h e s a l t con-

c e n t r a t i o n w a s 15% of t h e graphite volume i n a l a y e r extending i n t o ' t h e s t r i n g e r s fa? enougkr (0.02 ih. ) - t o give an aveSage concentrhtion of 2%. The salt concentration i n t h e centre1 portion of t h e s t r i n g e r s was asswred

t o be zero.

The. s p e c i f i c power i n t h e graphite w a s divided i n t o two p a r t s :

t h e Contribution from g a m a and neutron heating which was assumed uniform across t h e s t r i n g e r , and t h e contribution from f i s s i o n heating which w a s confined t o t h e fuel-bearing layer.

This r e s u l t e d i n an increase of 2,0째F

i n che maximum graphite-fuel temperature difference a t 10 M w .

This i n -

crease was neglected because other approximations tend t o overestimate t h e temperature d i f f e r e n c e . Over - a l l Temperature D i s t r i b u t i o n The o v e r - a l l temperature d i s t r i b u t i o n i n t h e graphite i s obtained by adding t h e g r a p h i t e - f u e l temperature d i f f e r e n c e t o t h e f u e l temperature. Figure 13 shows t h e radial d i s t r i b u t i o n at t h e midplane of t h e )ERE f o r 10-Mw power operation with no f i e 1 soakup i n t h e g r a p h i t e .

The fuel. tem-

p e r a t u r e d i s t r i b u t i o n , which i s a scaled-down version o f t h e corresponding carve i n Fig. 1 2 t o allow f o r a x i a l position, i s included f o r reference. Figure 1 4 shows t h e a x i a l temperature d i s t r i b u t i o n s a t t h e h o t t e s t radial p o s i t i o n f o r t h e same conditions as Fig. 1 3 .

The f u e l temperature i n an

adjacent channel i s included f o r reference.

The continuously increasing

r a e l temperature s h i f t s t h e maximum i n t h e graphite temperature t o a p o s i t i o n considerably above t h e r e a c t o r midplane. The d i s t r i b u t i o n s shown i n Figs. 13 and ture within individual g r a p h i t e s t r i n g e r s .

14 a r e f o r t h e mean tempera-

The l o c a l temperature d i s t r i -

butions within t h e s t r i n g e r s a r e superimposed on these i n t h e operating

reactor.


42 U N C L A S S l F l ED O R N L -LR- DWG 73 6 46

1290

I I

I

I

i280

1270

1260

I250

i240

1230

1220

1210

0

5

to

15 RADIUS (in.)

20

25

Fig. L3. Radial Tempersture P r o f i l e s in E R E Core Near Midplane

30

r,


43

UNCLASSIFIED ORNL-LR- DWG 73615

1300

1280

1260

-

1

GR/

I

LL 0

v

1240 3

t

2W

a

?

1220

1200

1180

10

20

30 40 50 DISTANCE FROM BOTTOM OF CORE (in.)

60

Fig. 14. Axial Temperature Profiles in Hottest Channel of MSRE Core (7 in. from Core Center Line)

70


44

Nuclear Mean Temperature The nuclear mean graphite temperature is obtained by weighting the local temperatures by the graphite temperature weighting function. In general

T

9

g

( [Tf(r, =

z>

J' V

+ AT(r,

z)

G (r, z ) dV Q

Substitution of the appropriate expressions leads to an expression of the form

T

*

Q = Tin + bl(Tout

- Tin) + b2ATm

where b1 and b2 a r e fbnctions of the system. The nuclear mean graphite temperature was evaluated for three degrees of fie1 permeation of the graphite. Table f o r the reference condition at 10 Mrr.

6 lists the results

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 Ic.arious degrees of graphite permeation by 2uel are listed in Table 6. Table 6. Nuclear Mean znd Bulk Mean Temperature of Graphite" ~

Graphite Permeation by Fuel '$ of graphite volume)

Nuclear Mean T

~~~

Bulk Mean T

(OF)

1226 1227

0

0.5 2.o

(OF)

1264

a At 10 Mw reactor power, Tin = 117>*F, To=% = 1223*F.

1231 V


45 POWE3 COEFFICIENT OF REACTIVITY The power c o e f f i c i e n t o f a r e a c t o r i s a measure of t h e amount by which t h e system r e a c t i v i t y must be adjusted during a power change t e maintain a The r e a c t i v i t y a d j u s t -

preselected control parameter a t t h e c o n t r o l p o i n t .

ment i s ncrmally made by changing c o n t r o l rod p c s i t i o n ( s ) and t h e control parameter i s u s u a l l y a temperature o r a combination t h e system.

c\f

temperatures i n

The r e a c t i v i t y e f f e c t o f i n t e r e s t i s only t h a t due t c t h e

change i n power l e v e l ; it does not include e f f e c t s such as f u e l burnup o r changing poison l e v e l which follow a power change,

The power c o e f f i c i e n t of t h e MSRE ean be evaluated through t h e use o f t h e r e l a t i o n s h i p s between reactor i n l e t and o u t l e t temperatures and

cne f i e 1 and graphite nuclear average temperatures develcped above.

These

r e l a t i o n s h i p s may be expressed as f o l l o w s f o r 1 0 - N ~power operatisn with no f u e l permeation:

(so)

Although these equations were develcped f o r f u e l i n l e t and o u t l e t t e m peratures of 1175 and 1225"F, respectively, they may be applied over a raage o f temperature l e v e l s WithGut s i g n i f i c a n t e r r o r

e

I n a d d i t i o n t o t h e temperature r e l a t i o n s , it i s necessary to use temperature c o e f f i c i e n t s of r e a c t i v i t y which were derived on a b a s i s eoypatible with t h e temperature weighting functions.

The f u e l and graph-

ite temperature c o e f f i c i e n t s of r e a c t i v i t y from f i r s t - o r d e r , t y o - g r o q

p r t u r b a t i o n theory7 are -4.4 x lo-''

and -7.3 x 10-5 QF-l respectively.

OF-'

YfiUS

Ak =

-(4.4 x

1 0 " ' )

ATf *

-

(7.3 x

ATg 9

e

(22)

7

I

E. E. Prince and J. R . Engel, Temperature and R e a c t i v i t y Coefficient _. Av=.raging i n %he MSRE, ORNL TM-379 (Oct. 1 3 j l.g62)e


46 The magnitude of t h e power c o e f f i c i e n t i n t h e MRE i s s t r o n g l y depende n t on t h e choice of t h e c o n t r o l parameter (temperature).

*

This i s b e s t il-

l u s t r a t e d by evaluating t h e power c o e f f i c i e n t for various choices of t h e con-holled temperature.

Three conditions a r e considered.

The first of

t h e s e i s a reference condition i n which no r e a c t i v i t y change i s imposed on t h e system and t h e temperatures are allowed t o compensate f o r t h e powerinduced r e a c t i v i t y change.

The other choices of t h e c o n t r o l parameter are

= c o n s t ) and constant average of

constant r e a c t o r o u t l e t temperature (Tout

(3 T~~ + 3

T = c o n s t ) . I n a l l t h r e e cases out t h e i n i t i a l condition w a s assumed t o be zero power a t 1200'F and t h e change i n l e t and o u t l e t temperatures

associated with a power increase t o 10 Mw was determined. summarized i n Table

7.

The r e s u l t s are

It i s c l e a r t h a t t h e power c o e f f i c i e n t f o r t h e r e f -

erence case must be zero because no r e a c t i v i t y change i s permitted.

In the

other cases t h e power c o e f f i c i e n t merely r e f l e c t s t h e amount by which t h e general temperature l e v e l of t h e r e a c t o r must be r a i s e d from t h e reference condition t o achieve t h e d e s i r e d cofitrol. Fuel permeation of t h e graphite has only a small e f f e c t on t h e power coefficient of reactiviiy.

For 246 of t h e g r a g h i t e voluiie occupied by' fuel,

t h e power c o e f f i c i e n t based on a constant average of r e a c t o r i n l e t ar,d outl e t temperatures i s -0.052% Ak T/MW (z.-0.047 f o r no f u e l permeation).

Table

7 . Temperatures a and Associated Power Coefficients o f Reactivity rn

I

Control Parameter

out

TJs external control c f reactivity 'i! = const out

Tin + Tout 2

= const

rn I

ir,

m

*

m A

*

(OF)

("p')

(OF)

Q (OF)

1184.9

1134.9

1173.O

1216.3

1200

llY

1188.1

1231

1225

117=,

1213.1

1256.6

If

a A t 10 Mw, i n i t i a l l y isothermal a t 1200'F.

6

Power Coefficient ($h Ik f k / r n )

0

- 0.018 -

0.047

.


47

DISCUSSION

Temperature Distributions The steady-state temperature d i s t r i b u t i o n s and average temperstures p-esented i n t h i s r e p o r t are based cn a calciilationalmodel which i s cnly an approximation of t h e a c t u a l r e a c t o r .

An attempt has been made t c i n -

clude i n t h e model those f a c t o r s which have t h e g r e a t e s t e f f e c t on t h e temperatures.

However, some simplif'ications and approximations haTre been

made vhich w i l l produce a t least miror differences between t h e predicted

temperatures and those which w i l l e x i s t in t h e r e a c t o r .

%e treatment of

t h e main p a r t of t h e core as a s e r i e s of concentric cylinders with

EL

c l e a r l y defined, constant f u e l v e l o c i t y i n each region i s one such s i m p l i f i cation.

This approximation leads t c very abrupt temperature changes a-t t h e

r a d i a l boundaries of these regions.

Steep temperature gradients w i l l un-

cioubtedly e x i s t a t these points but they w i l l not be as severe as thcse indicated by t h e model.

The neglect, of axial heat t r a n s f e r i n t h e graphite

a l s o teciis t o p o i i i x e z-n_exeggeratian of t h e teqeratw-e p - o f i l e s over those t o be expected i n the r e a c t o r ,

It i s obvious t h a t t h e calculated

a l a 1 temperature d i s t r i b u t i o n i n the graphite (see Fig. 1 4 ) can not e x i s t

~ 6 . t n o u tproducing some a x i a l heat t r a n s f e r .

The e f f e c t of a x i a l heat t r a n s -

fer i n t h e graphite vi11 be t o f l a t t e n t h i s d i s t r i b u t i o n somewhat and reduce

t h e graphite nuclear-average temperzhure. Some uncertainty i s added t o t t e calculated temperature d i s t r i b u t k n s by t h e l a c k of accurate data on t h e physical properties of t h e r e a c t o r x a t e -

rials.

'The properties o f both t h e f u e l salt and t h e graphite are based can

estimates r a t h e r than on a c t u a l measurements.

A review of t h e tempera%.cre

calculations w i l l be required when d.etailed physical d a t a become Etvailakle However, no l a r g e changes i n t h e temperature p a t t e r n are expected. Temperature Control

It has been shown t h a t t h e power c o e f f i c i e n t of t h e E R E depends upon t h e choice of t h e temperature t o be controlled.

O f t h e two modes of cor--

trol considered here, control of t h e r e a c t o r o u t l e t temperature appeess t,o 5~ k re ",ecirzble 5 , n c a ~ ~t h~ ee snsller ,n=:pr coez"f'icient reqLires

.


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 8 with an analog computer indicated greater stability with control of the reactor outlet temperature than when the average of inlet and outlet was controlled.

8 S. J. B a l l , Internal correspondence.


49

Nomenclature s p e c i f i c heat of f'u.el salt,

C P f

volume f r a c t i o n of f u e l ,

F

Total c i r c u l a t i o n r a t e of f u e l ,

G

Temperature weighting f'unction,

H

heat produced per f i s s i o n ,

k

thermal conductivity

a

equivalent half-thickness of a graphite s t r i n g e r t r e a t e d as a slab,

L

length of the core,

P

r e l a t i v e s p e c i f i c power,

Q

equivalent s p e c i f i c power (absolute

9

rate of heat t r a n s f e r per u n i t area,

r

radius,

rW

equivalent radius of a f u e l channel,

r

equivalent radius of a graphite s t r i n g e r t r e a t e d a s a cylinder,

S

neutron multiplication f a c t o r ,

>,

R

radius o f t h e core,

t

time,

T

temperature,

Ts

l o c a l transverse mean temperature i n a single f u e l channel o r graphite s t r i n g e r

.x

Tf

T

g

*

f u e l nuclear mean temperature, graphite nuclear mean temperature, bulk mean temperatwe of f i e 1 i n t h e reactor v e s s e l bulk mean temperatvre of t h e graphite,


-

Nomenclature U

vf c

cont 'd

velocity, volume of f u e l i n t h e core,

2,405 r

X

dimensionless radius,

z

a x i a l distance from i n l e t end of main core,

0:

defined by ( 6 ) )

8

f r a c t i o n of core heat o r i g i n a t i n g i n f u e l ,

x

defined by

P

d e n s i t y of f u e l salt,

5

macroscopic fissiol?, cross section,

$

neutron flux,

$*

a d j o i n t flux,

AT

l o c a l temperature difference between mean temperature across a graphite stringer and t h e mean temperature i n t h e adjacent f'uel,

R

9

(5)j

Subscripts f

fuel,

g

graphite,

in

f u e l entering t h e r e a c t o r

out

mixed mean of f u e l leaving t h e r e a c t o r ,

W

w a l l , o r fuel-graphite i n t e r f a c e ,

1

fast neutron group,

2

thermal neutron group,

m

maximum value i n r e a c t o r .


Derivation o f Equations f o r a Simple Model Fuel Temperature D i s t r i b u t i o n Consider a small volume o f f i e 1 as it moves up through a channel i n t h e core.

The instantaneous rate of' temperature rise a t any point i s

In t h i s expression it i s ass-med t h a t heat generated i n t h e graphite adjacent t o t h e channel i s t r a n s f e r r e d d i r e c t l y i n t o t h e f'uel ( i . e . , i s not conducted along t h e g r a p h i t e ) , which accounts f o r t h e f r a c t i o n 8.

I n t h e channel being considered, t h e fuel moves up with a velocity,

u, which i s , i n general, a function of both r and z .

The s t e a d y - s t a t e temperature gradierLt along t h e channel i s then

Let us represent t h e neutron f l u x by

) sin Assume f u r t h e r t h a t t h e channels extend from z =

o

xz

t o z = L (ignoring

tile entrance and e x i t regions f o r t h i s idealized c a s e ) .

a t z = 0, f o r a l l channels, i s Tin. i s found by i n t e g r a t i o n

U

,

The temperature

The temperature up i n t h e channel


Y

T(r, z ) =

Tin

+

sin

(5) . dz

0.5)

For convenience i n writing, l e t us define

x

=

2.405 r R

men

sad t,he temperature at t h e o u t l e t o f t h e channel,

Now l e t us c a l c u l a t e t h e mixed mean temperature of t h e fuel i s s u i n g

f r o n a l l of t h e channels. t.ure by t h e flow r a t e .

This requires t h a t we weight t h e e x i t tempera-

The t o t a l flow rate is, i n general,

For t h e idealized case where f and F

L

=

are uniform thrcughout t h e cere: 2

&fu.

The general expression f o r mixed mean temperature a t L i s then m

1

out

--- 1 F

s" o

2-

f(r, L ) u ( r , L ) T ( r , I,) d r

Substit>utionof (ag) for t h e temperature gives

e


53

4(Qf ImL Tout =

in

+

P f

J Rf(r,

L) r Jo(x) dr

O

Tor the ideal case where f is constant, integration gives

Tout -. Tin

+

4(Qf)mlf( j'

F(~c)

2.4

(2.4)J1 (2.4)

Pf

From (a14) we find that

which may be rewritten as

The first term in (a16) is simply the average effective value of the f u e l specific power (including heat transferred in from the graphite).

The second and third terms are, respectively, the maximum-to-average ratios f o r the sine distribution of specific power in the axial direction ar,d the J distribution in the radial direction, 0

Substituting (82.6)in (a8) gives

'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 Y

The nuclear mean temperature i s obtained by weighting t h e l o c a l temperature by t h e nuclear importance, which i s a l s o a f'unction of position, and i n t e g r a t i n g over t h e volume of fuel i n t h e core. For t h e idealized case t h e importance v a r i e s as the scuare of t h e neu+,ron flux.

Thus t h e

nuclear mean temperature of t h e fuel i s given by R

J'0

*

0

When (a4) i s s u b s t i t u t e d f o r #(r,

#' Vfc , becomes 2

Vfc

23-@ 2 m

=

f ( r , z ) tir dz

J L T (r, z ) #2 (r, z ) 2-

Jn R o

j "f(r, o

21,

z > I' Jo2

t h e denominator, which i s

( 2.405

r

sln2 ( 5 ) d r

dz

For constant f, i n t e g r a t i o n gives

With t h e s u b s t - i t u t i c n of (ab), (all), (a17) and (azo), and assumiilg f and u 50 be ccnstant, ( a 8 ) becomes

( 1

- cos F

) ax dy

0

I n t e g r a t i o n with respect t o z r e s u l t s i n

* Tf

-.

-

Tin

+

-

(Tout Tin) {2)(2.403) J? (2.405)

12'4:'J o

0

3 ( x ) dx

(a221


Graphical i n t e g r a t i o n with respect t o x y i e l d s V

,

2.405

18

J

Jo3(x:) dx = 0.564

0

.

Thus, t h e f i n a l r e s u l t f o r t h e i d e a l i z e d core i s 3c.

Tf = Ti n

+

0.â&#x201A;Ź138 (Tout

- Tin)

e

F x l B u l k Mean TemDerature

The bulk mean temperature of t k e f u e l i n t h e core i s

-

-

.IR.fâ&#x20AC;? 250: f ( r , z )

d r dz

The denominator i s t h e volume of fie1 i n t h e core.

Considering only

t h e i d e a l i z e d case again =

vfc

fi

(a26 )

R2Lf

S u b s t i t u t i n g (all an& ( d ~ i n) t o (823) f o r c o n s t a t f and u gives

-Tf

=

Tin

+

2.405 nf (Tout - Tin) J1(2.405) (1

-

cos E L ) d r dz

,

which i s r e a d i l y i n t e g r a t e d t o give

-Tf

= Tin

+

cTout

- Tin)

2

The equations f o r evaluating t h e nuclear and bulk mean temperatures

of t h e f u e l have been presented t o i l l u s t r a t e t h e general method which must be employed.

An i d e a l i z e d core was used i n t h i s i l l u s t r a t i o n t o

reduce t h e complexity o f t h e mathemehical expressions.

This techniqv.e

may be applied t o any m a t e r i a l i n any r e a c t o r by using t h e appropriate temperature d i s t r i b u t i o n s and weighting functions.

The r e s u l t s obtained

by applying t h i s method t o t h e f u e l and g r a p h i t e i n t h e MSRE a r e pre-, sented i n t h e body of t h e r e p o r t . gcb


Distribution V

1. MSRP Director ' 8 Office, RL 219, 9204-1 G . M. Adamson 3. L e G . Alexander 4. S o E. Beall 5 . M. Bender 6* E . S . B e t t i s 7 . D . S . Billington 8 , F. F. Blankenship 9. E . G . Bohlmann 1 0 . S . E . Bolt 11 C . J . Borkowski 12 C . A . Brandon F. R . Bruce l 3' 14. S . Cantor 15 T. E. Cole 16. J . A . Conlin 17 W. H. Cook 18 L, T. Corbin 19 G . A . C r i s t y 20 J L. Crowley 21. F. L. Culler 22. J . H. DeVan 23 * S . J . Ditto 24. R . G. Donnelly D . A . Douglas 25 26. N . E . Dunwoody J. R . Engel 27 28. E. P. Epler 29 * W. K . Ergen 30 ' A , P o Baas 31. J . H. B y e 32. C . H. Gabbard 33 R . B. Gallaher 34. B. L. Greenstreet 35 W. R. Grimes 36 A . G. Grindell 37 * R . H. Guymon 38 P. H. Harley 39 C . S. H a r r i l l 40. P. N . Haubenreich 41 E. C . Hise 42. H. W . Hoff'man 2.

0

0

9

a

.

0

9

43 P. P. Holz 44. J . P. J a r v i s 45. W . H. Jordan

46* 47* 48. e

P. R . Kasten R . J. Ked1 M. T. Kelley

49.

S. S. Kirslis

9. J .

W . Krewson

51. J . A . Lane 52. 53. 5h* 55.

R . B. Lindauer M , I . Lundin R o N, Lycn H. G. MacPherson 26. F, C Maiensehein 57. E . R . Mann 9.W . B. McDonald 59. H. F . McDuffie 60. C . K. McGlothlan 61. A . J . Miller 62. E . C. Miller 63. R. L. Mosre 64. J . C . Moyers 65. T. E . Northup 66. W . R . Osborn 67. P. Patriarca 68. H. R . Payne 69. A . M. Perry 70. W . B. Pike 71. B. E. Prince 72. J . L. Redford 73. M. Richardson 74. R . C . Robertson 7 5 . H. W. Savage 76. A . W . Savolainen 77. J . E . Savolainen 78. D. Scott 79. C . H. Secoy 80. 3. H. Shaffer 81. M. J . Skinner 82. G . M. Slaughter 83. A . N . Smith 84. P. G . Smith 85. I. Spiewak 86. J. A . Swartout 87. A. Taboada 88. J . R . Tallackson 89. R . E . Thoma 9.D. B. Trauger 91. W. C. Ulrich 92. B. S, Weaver 93. C . F, Weaver 94. B. H. Webster 95. A . M. Weinberg

.

96. J, C. White 97. L. V . Wilson


Distribution

- contd

9.C. H. Wodtke 99. R. V a n Winkle 100-101. Reactor Division Library 102-103 Central Research Library 104. Document Reference Section 105-107. Laboratory Records 108. ORNL-RC e

109-110. D. F. Cope, Reactor Division, AEC, OR0 111. A . W. Larson, Reactor Division, AEC, OR0 112. R . M. Roth, Division of Research and Development, AEC, OR0 113. W. L. Smalley, Reactor Division, AEC, OR0 114. J. Wett, AEC, Washington 115-129. Division of Technical Information Extension, AEC

>


ORNL-TM-0378