A Division of North American Aviation, Inc.

i Do not removr this sheet

-5011 N AA-SR-M EM0 This document contains -pages 18

This i s copy -of -series -

I J

Security Classification

NAA-SR-MEMO8 are working papers and may be eqanded, modified, or withdrawn at any time,

and are intended for internd u8e only.

This report may not be published without the approval of the Patent Branch, AEC / d h d

LEGAL NOTICE

This mport wos proparod os on ocsount of Government spopsorod work. N.lthor the Unitod Stotos, nor the Commission, nor ony porson octing on boholf of the Commission:

A. Makos m y warmnty or nprorontotion, oxpross or inpliod, with rospct 90 tho occwacy, complotonoss, or usofulners of the informption contained in this roport, or thot tho US. of any informotion, opporotus, mothod, or procost disclosed in this roport may not inking. priwtoly pmod rights; or

B. Assumos ony liabilities with 10s act to tho US. of, or for domagor msulting from tho US. of informotion, opparotus, mothod: or procoss disclosod in this mort.

As usod in tho obovo, "porson octing on boholf of th. Commlssion" includo~ my.-- ploy00 or antroctor of tho Commission to tho oxtont thot such omployoo or contractor proporor, handlos or distributor, or provides occoss to, m y information purswnt to his omploymmt or controct with tho bmmission.

i

1 !

. . . . .

ATOMICS INTERMANONAL A Division of North American Aviation, Inc. .

COPIES~O: 0. R. Anderson, B. E, Aronatein", R. J. Beelep , W e P. Coraoran*, E. W. Ribkinson, K. Poster*, T. KerlinY, 01 Oleen*, E. Spoehel", A. C. W i l l i a m g J. 01 Greenleaf*

SUBJECT:

inary Euolear Analysis of 8 Sodium Cooled Fast Reaotor (SCFR) II

With 00 FU el Elemenfa i n tt Skewed Hexam=% may CONTENTS:

I

I1

1 1 1

I V REFERENCES AND APPENDIQS .................... PACE

STATEEEWI OF PROSLEH ...................... PACE 1

SUPIHARY OF RESULTS AND RECOWENDATIONS ............. PAGE

METHOD USED, DESUIIPTIOti OF EQUIPHEKl, SMPLE CALCUUTIONS ... PAGE -

/

I STATEMENT OF PROBLEM

. To perform s nuolear analyeie on a aodiwn-cooled, enriohed-uranium- carbide fueled fast reaotor core of unique design wi th fue l elements skewed, varying .the l a t t i c e over the length of the core. the ra t io of peak t o average pover deneity, and t o evaluate blanket and re- fl. ea t o r designs.

In particrular, t o deteminc

I f S m E Y OF RESULTS AHD BECOHHEEDATIOHS

I The core, in i t s final form, oonsiats of 199 e ta in leas s teel prooese tabea, 5.00 in. OD x 0.0625 in. wall, In f ixed sksi@d.'positi.onrt between

, f ixed grid-plates and i n a triangular array. thru t o the top of the biological ehield t o f a c i l i t a t e the refuel ing proce- awe. The overal l oore dimensions are l i e t e d i n fable 1.

Exteneiona of these tabes pa88

TABLE I Amwoxlmate Dimeneione end XktsariD t i on of '%he Plnal SCFR Core,

Height of fueled region 7 f t . Thickness of ax ia l blankete 18 in.

fueled region 46 in.* radial blanket 15 .in,* '

Number of fue l elements 66 Xumbsr of oontrol rods 7 Number of blanket elements 126

* Values a r e volums we ghted averages 3c2

._

i

5011 NO. ATE ,February 24, 1960

ATOMICS INTERNATIONAl PAGE 2 OF 18 A Division of North American Aviation, Inc.

i

[I SUKKARY OF RESULTS AND RECOMXElfDATIOHS (Continued)

The l a t t i c e epacing a t the top of the ehield WBB e e t by the shield s t rength requirements a t 31.00 in. and a t the mid-plane of the ac t ive core at 5.018 in. by nuclear requirements. between theee two planes i e 25.00 f t , and the lower grid-plate i e about 7 f t below the mid-plane. grid-plate t o the top of the biological sh ie ld the array of tubes ro t a t e s thru goo, while the l a t t i o e spacing first contracte from 5.625 In. t o 5.018, and them expand8 t o l l o O O In. of tubea and centered on the mid-plane l e a graphite r e f l eo to r

The dietance

In moving up the core from the lower

Around the a r ray

I

i I Direction of Solution Radial xx!Mrl

Median Filesion Energy (Kev) 200 206 I Enrichment( atom percent) 15, 3 1508

; I n i t i a l Convereion Ratio 1.120 0.652 I n i t i a l Convereion Etatio,Including

I n i t i a l Volume Weighted Peak t o Axial Blanket 1.17 -

I f

i

Average Power Denaity Ratio 1 . 566 1 370

Thie oore would contain about 10,000 kg enricrhtsd UC and 31,300 kg depleted (leas than 0.3 atom peroent U-235) UC. The r e f l ec to r

i I contains about 16,200 kg of graphite.

--... . .-. ATOM ICs INTERNATIONAL

A Division of North American Aviation, Inc.

11

XI

SUMMBRY OF €@SUL?pS AND F f E C O ~ D A T I O N S (Continued)

on ooneiderations ot? minimum o r i t i a a l ma88 (It I s greater than 1.6) i t would probably be desirable t o modify the core. t ransfer requirements w i l l not allow a slgnif ioant reduotion i n L without a oorreepondlng increase in D, and einue the radial blanket is ooneiderably more effeot ivs +hem the axial blanket, i t appearol t ha t replaolng thorse blanket elements adJaoent t o f u e l elements with fue l elements and adding a suff lolent number of blanket elements around the outeide t o maintain the s m e blanket thlokness would be advantageous. A rnodifioation of t h i s nature would probably reduue the enrichment necessary, although the t o t a l amount of UC would, of oourse, be inareaaed. The o r l t i o a l ma88 would probably inoreaee also. A reduotion of L/D 6ould also be obtained by inoreasing the blanket thlokness. This would both dearease the e r l t l o a l ma88 and lnoreeree the convereion r a t io . I f e i t h e r o r both of these modificafions were uaed i t would then be desirable t o inorease the dllatanae frola the oore mid-plane t o the top of the biologloal shield BO that the angle whloh the outer prooess tubes make with the v e r t i c a l would not become too large and thure prePrent prOblen8 i n refueling, The oon- oept of a ro ta tab le gr id-p la te eons t l tu tes an effect ive method of eontrol l ing a fast reaotor by changing the l e t t i o e spaolng, method might be used t o allow f o r depletion of the f u e l a8 well as for sorams and normal shutdowns.

Becrauae the value of L/D f o r t h i a core $8 quite unfavorable bailed

S h o e the heat I

1 j

Suoh a

METHOD USED

A. Method

A l l reaotor oalculetions were made using ORU-$1, a multireglon, muliiigrmp one-dimensional neutron diffueion code o r o ~ s seotione, A l l cores were oonsidered t o be ju0f cr i t loa l and olean, t h a t is, without oontrol o r flserion produot poisons present.

After caloulatlon of the aotual geometry of a oorel i t was transformed first i n t o a hyperboloid of revolution, and then i n t o a r ight oirouletr oylinder of about equal height and volume, and the

w i t h 32 group

. materials homogenized in t h i s volume. This model wae used t o obtain the radial aolutlon of the oore. For the axia l solution of the aore, the oylinder was transfdqmed i n t o a reotangular parallelepiped of the same height and volume, and with a equare (Jross seotion I n the plane normal t o the Ulrection of the solution, Msterial denel t ies f o r the axial solut ion were the same a s for the radial solution, and aooount f o r almost a l l of the ma8888 hvolved, slight var ia t ions being due t o the approximations neceseary i n the height baaause of the slight var ie t ion aoross the ends of the crore. f o r the transformations In geometry a re given in Appendix A. Appendix B gives the equations f o r oaloulating the material densi t ies .

The method and equations used

730-V-45 (REV 5 - 5 9 ) ~ ~ ~-~

.-

.-

5011 NO.

704" J. H. Booketoa et . al. QMB 101 General Hotore Corporation.

2. Personal oommunioation w i t h 8 . Burton,

t

r

.- ATOMICS lNTfRNATIONAL A Division of Norfh American Aviation, Inc.

i

i 1

I I '

I I I

I I f

1 i

*

' I V

I11

1

,

PAGE 4 O F '* HETHOD USED ( C O A t h U f i d )

B. Dieoussion of the Calculations

The values e v e n as oonvereion r a t l o refl'ebt1.a oorreotion made t o thie ONU output t o account for thoae absorption8 in V-238 which produoe fialsione. This oorreotion, whioh i a ahown i n de t a i l in Appendix C, allowe the value8 quoted i n Table I t o give a more r e a l i s t i c picrture of' the i n i t i a l produotion of new fuel.

The average power deneity in the core i s oornputed from the power density d is t r ibu t ion BB printed In the Om output. equation uaed for the rad ia l solution2 is:

The

(2s -1XP + P 4- * - I L

2s2

where

P = Average power density

. P = Power density a t point 4 n n = Heeh point number (6 n5 8 )

Po - Power density a t the center

= Power density a t the outer boundary

s = Meah point number a t the outer boundary

For the ax ia l solution, sinae the arose-seaflonal area wae the 8ame at any elevation, t he average power deneity is equal t o the average height of the power deneity d is t r ibu t ion ourve. This i e easily oalouleted by uskng nSimpson'a Rule" f o r the area under 8 curve . REFEREXCES

i

I I

1 . 11. 730-V-45 (REV 5-59) ~ . .

NO, 5011

ATOMICS INTERNATIONAL A Division of North American Aviation, Inc.

DATE February 24, 1960 PAGELOF lQ1

I

APPENDIX

A. A,Method for Trans fomlnR a &ewe d Element Core Into One of S i m l er Oeoms t m

The traneformation of the skewed element core i n t o a r ight c i r cu la r oylinder, or i n t o a rsotangular p&ralleleplped wi th square orose-sectlon,~having app.roxirnstely the 8-8 het.$litt and volume 88 the ac tua l oore can be made using the geometric TeldtiOn8hipS behwen

T h i s trsneformation is done i q two step$. Firart, the ekewed element ooret which has a hexagonal cross-seotioa l e changed i n t o a hyperboloid of revolution, whioh has a oi rcu lar oro~8-areotion. The transformation i 8 such that a t the aame elevation

. the two f igures have the same cross-eectional area. uhanges the hyperboloid in to one o r more r igh t o i rcu lar cylinder8 having approximately the same-total height and volume 88 the hyper- boloid, The geometric relationehipa derived here are baaed, for . eimplioity, upon the central , ver t ica l , element and un element at one of the ve r t io i s s of a regular hezagon which has the r e r f i c l l element a t i t a center (See Figure 1) .

in whieh both elements appear i n t h e i r tms lengths. shown are defined belowt

any two element8.

9he second atep

Figure 2 shove par t of the plan view, and the elevation view The dlmensions

P = is any point on a vertex element

h .I ver t i ca l dlstanoe from the mid-plane of the ac t ive core

fhe E .I ver t i ca l dietanoe frodmid-plane of the ac t ive oore t o

the t o p of the biologi'oal 8 k l d

QA = l i m i t of h for the so t ive core

GR =

Lg =

l i m i t of h for the aria1 blanket

dietanoe along the element from the mid-plane of the ac t ive oore t o the top of the biologioal ehield

I

i I LA = % the ac t ive length of B fue l element minue the axial

blanket

LR - .!! the ac t ive length of a fue l element including the

LR = fi the aotfve length of a blanket element

'- axial blanket

1 = the prodection of' r on the ve r t i ca l plane oontaining the vertex element 2@4j

- 730-V-45 (REV 5-59)

t

,

,

i

1 f i

NO. 5011

ATOMICS INTERNATIONAL A Division of North American Aviation, Inc.

DATE Pebruary 24, 1960 PAGE OF l8

Plan View of the Relative Position8 of the Two Elenente used i n the Derivation of the Equations for the Oeometriu 9renafornatlona

Ver tex Elemen J 1 '

730-V-45 (REV 5-59]

NO. 5011 DATE February 24, 1960 PAGELOF 18

AfOM ICs INTER NATIONAL A Division of North American Aviation, Inc.

The QeoPsetry for a Skewed Element Core, Showing the Center-lines of the Central Element and of a Vertex Element

I

I

' 1

Plane of lower

Top of Biologioal Shiel'b--------

Mid-plane o f act ive core

r H

Grid plate

\ Central Element

1 FIGURE 2

730-V-45 (REV 5-59) k

1 L," __:... ..- ,

I . . . ...

"

I

! 1

NO. 3011 I

c = the horizontal projeotion of t ha t segment of the vertex element between the mid-plane and the top of the biologi-

... . ATOMICS INTERNATIONAL

A Division of North American Aviation, fnc.

1

4 I I i I

AF’PEBDIX (Continued)

r E the horizontal distanue from the ve r t i aa l element t o point P

” I a = the horizontal distancte a t the mld-plane from the

i .

I

v e r t i ca l element t o the vertex element I-

i : b - the horizontal distanoe between the two elements a t the

top of the biological shield

i 1 n - the number of hexagonal ring8, uompleted o r uncompleted, I ,

i at elevation h

i n a o I I I

From the Figure i t can be seen t h a t 1

). DATE February 24, 1960 PAGE 9 OF l.8 ATOM ICs INTERNATIONAL

A Division of North American Aviation, inc.

4 -7

Subatittat& (78) and (sa) i n ( 5 )

I APPEBlDIX (Continued)

t;

1

where

i A e = J 0 0 A (14) ; 1 I 1 i

1

where

Substituting (12) and (13) i n (14)

No =' the number of c e l l e having a l a t t i c e spaoing 8..

A 0 - 4 7 C ( A 2 + & C') 2 N

N" 0

2 2 or c2 = c n

A .I Minimum l a t t i c e epaoing

B - Haxilnqm l a t t i a e ipacing 2 C2 I B2 - d

i

NO. 5011

w

&

ATOM ICs, INT€RNATIONAL -- - A Division of North American Aviation, fnc.

1 1

element

f (19) j

A1 so %* I E* + o*

or subst ibt ing ( 9 8 ) in (19) L~ .I ( H ~ + c 2 2 % n

Subs ti tu t ing ( lga) in (lea)

a q 3 .- .L

DATE February 24, 1960 16 PAGE lo O F

APPEBDIX (Continued) w!

where number 8f elements, and A. is the oroes-seotional area of . t h e oore a t the same elevation.

l e now the number o f oe l le a t elevation h, or twioe the I 1 I

I This:uore, of hexagonal oroes-seation can be traneformed In to t I

a oore with c i roular oross-eeotlon by l e t t i n g ,

I

I (168) : !

f I I where Re is the radiue a t elevation h of a oore of equal volume.

The l imi te on the values of h f o r the vertex elements, as a 1 funotion of n is derived aa followet

From Figure f i t can be g e m t h a t t

NO. 5011

PAGEIIOF 18

i DATE February 24, 1960 ._ ATOMICS INTERNATIONAL . I A Division of North American Aviation, fnc.

APPENDIX (Continued)

This equivalent core (a hyperboloid of revolution) o r seatione of it , may be transformed i n t o a r igh t c i r a u h r cylinder of the same height and voltme. ends have been ne@ected, they oould be taken in to ooneideration by the ~ a m e , o r a imi l a r , method a8 the following:

For bhe cyl indrical model, the volume is;

Although the doming ef fec ts which occur a t the

- nR, 2 A, vM

and h# are the volume, radius and height respedtively 2 e :;:BL%. The volume of the hyperboloid coret VE, i 8 8

o r eubat i ta t ing (17) i n (22)

va .I- eafic (A2+ - hZ C') dh (2%)

Ahoy

4 H= or

0

By assuming that IT,- is a oon8tant and equal t o i t a maximum value we &re i n e f feb t nggleciting the doming a t the ends. N a$ a diecontinuous funotion of h we may take t h i s e f f ec t i n t o agaount .

If we aonsider

However, neglecting t h i s e f f ec t , them

z ( 24)

,

Lett ing % = 2 h x and equating (21) and (24)

730-V-45 (REV 5-59)

ATOMICS INTERNATIONAL A Division of North American Aviation, Inc.

5011 NO. DATE February 24, 1960 PAGE 1 2 O F A

To transform t h i s r ight o i rau lar cylinder i n t o a parallelepiped, with a square oros8-eeetion, which has the eame height and volume, a l l tha t is necessary i s that the cr088 eectional area$ be equal.

That is:

where X i e .the length of the side of the square.

Subst i tut ing (23) in (26)

L I

For the f i n a l oore the follow$ng values were used:

A = 5.018 in. = 12.746 om

B = 11.000 in. .I 27.94 am

H g00.00 in. E 762.0 cm

L I 42.00 in. = 107.78 cm

= 60.00 in. I 152.80 om

II

I 398 for blanketed oore

LR

Bcmar 146 f o r bare aore ( fue l seetion)

The values obtained f o r hmar were:

f o r the fue l eection, haax - 106.0 cm

f o r the blanket seotion hmax - 147.5 om

f o r the ax ia l blanket sect ion hmax .I 151.0 CQL

The value8 of % were:

for the fue l section, 57.87 om

for the r ad ia l blanket, 95.56 OIP

for the ax ia l solution only the value of x f o r the fuel sect ion wae needed. It was 102.57 cm.

* '

I i f !

i ! i 1

I i i i

I

I 1

i

I .I . ___ -,

1 730-V-45 (REV 5-59]

L

NO. 5011

ATOMICS INTERNATIONAL A Division of North American Aviation, fnc.

DATE February 24, 1960 P A G E . ~ O F 18

I ~PPPEEPIX (Continued) .V* B. Caluulstlon of' bversne Dene I t l e e for t h e E Q U ~ ~ Vol ume Hod el

In order t o make the mathematical model described oorreotly the actual oore the model dens i t ies should be suoh t h a t the t o t a l number of atoms in the model i s the same a8 in the actual owe.

Theas follows t

Atome

where

.

expressions for the material densi t ies a re derived as

of I In the model - A t o m of i In the oore

I atom denarity of 1 In the model

atom density of i in actual oore at elevation h

volume of model o r any seotion of it

I

I

I actual volume being deeoribed

- equivalent radius .I 8 funotion of h

c 730-V-45 (REV 5-59)

,

I

ATOMICS INTERNATIONAL A Division of North American Aviation, fnc.

w

P

V' *

NO. 501 1 DATE February 24, 1960 PAGELOF 18

APPEXDIX (Continued)

where

density o f i I n @/ea at elevation h

HA - atcina/gm mole

A, a area of the c e l l at elevation h

A, = gram molecular weight of i

AJ = area of i i n cm i n S ce l l at 2

ala0

elevation h

therefore

30-V-45 (REV 5-59)

5011 NO. L DATE February 24, 1960

ATOMICS INTERNATIONAL PAGE&.OF 18 A Division of North American Aviation, Inc.

ILPPENDIX (Continued)

where Hcah = number o f cells in the eection of the oore being aonsidered

m area of unit cell A.

Substituting (7) in (6) and makine A h I constant over A 4

or

For those materials for which Ai ia conatant in the volume "i.)

deaaribed by ~ h ,

( 9 )

Lf % ~t crose-sectional area of the model

= 7T%2

For those materiale for which Ai is not constant

I A * K (11) 0

"a3 6 L' A

.” w

NO. 5011 1 DATE February 24, 1960

”. ATOM ICs INTERNATIONAL PAGELOF 18 -.: A Division of North American Aviation, Inc.

Lj

5

W

- APPEEDIX (Continued)

where K = the sum.of the areas of the other materials *

Xf IC is a oonstant, independent o f ah,and sinoe

A then

It can be ahown that the integral i n (14) reduces to

, 4 $ ( A 2 + $ c ) - ~ d h = TRG -/aaA thus

A,

A similar method may be used for those cases where K is oonstant, that i s when there i s more than one material for whioh

However, in th is 6aee there was only one 2 t e r i a l for whioh Ai varied with h, so the derivation W ~ E I not extended t o the more general oa8e.

i s a function of h.

730-V-45 (REV 5-59]

Ir

5011 NO. DATE February 26, 1960

ATOM ICs INTERNATIONAL P A G E . 4 O F c

A Division of North American Aviation, Inc.

APPEZDIX (Continued)

For t h i e study, the follawing oonetante were ueedt

Ma t a r i s l Ai *%f5

Sodium ,823 23

Sta in less Steel(Fe) 8.03 55.85

f uc 13.6 249 . 5 (Varied dependin upon enrichment

I 0.6023 x looz4 xA ,

Hinor correotlons were also made t o acoount f o r the materials in the oontrol rod cells,although the oontrol poisons were omitted.

C. The Correction t o the Conversion Ratio t o Aeoount f o r U-238 Fisalona

In the OHU program the oonveraion ratio i r s defined k s the ratio of absorption8 in fert,5.le material t o absorpt ions- in fuel material. A b e t t e r definiti'gri vouXd be the ra t io of non-fission absorptions in f e r t i l e material t o absorySione in fue l material.

a8 f O l l O W 8 , using data from the GNU output and values of+,- and The fieeion absorptions in f e r t i l e material may be oaloulateh

w 4 g or the f e r t i l e material.

For any lethergy group

Ab8 g - &Ee9 (1)

are given in the QHU output then L, Since Aba and g

730-V-45 (REV 5-59)

i !

5011 NO. DATE February 24, 1960

18 ATOM ICs INTERNATIONAL PAGE 18 OF 1

A Division of Norfh American Aviation, Inc.

i \.

t

APPENDIX (Continued)

Fission 8bSOrptiOnt3 &re calculated from

J - where (vc"J values were from the AE3R eross-seetion data book, V8 waa e8 B12-7jl),and ?I is the atom density of fert i le material. Beoause of the method of normalization used by Gm, equation ( 2 ) I s dimen- eionally correct and has the leame un i t s as the abaorptione given i n the GKU output.

mated from the "Six Group Croaa-Sections" (LOB dlamoe,

Thue, the corrsoted oonverelon r a t i o i s

- The value8 used f o r andt-hare given below:

730-V-45 (REV 5-59)