VALIDATION RFSULTS FOR TW SDX CELL -.-- EMOGENIZAT~~~~ co= FoR PIN GEOmTRY -

R. Da t4cKnight

Argonne National Iabomtory

ZACRP-/%-(+-f--3

u1040001

,

NUCLEAR SCIENCE AND ENGINEERING: 75, 1 i l-12s (1980)

Techniccsl Note

been in prqxss by \Vade,z Gelbard et al.,’ Stxey et al.,’ Wade ad Gelbard~5 and G?lbxd et aL6 This effort ha compaxd SDX results against ?.K*-2 (Ref. 7) and Vl!vI (continuous cnerSy Monte Carlo) result? and lhx system- atically progressed from the study of intinite homogeneow media to the study of heteroSenwus 1xt:ices (first with zero lakage, then with noniwo la&e). The cults reported pwiously have investigated the standard zero power reactor (Zl’R) plate-type unit cell. Tlx present study repwents a compi;mentary validation effort for the ZPR pm caland& type unit cell. The widcd pin calandria unit ccl1 used in the gas-cooled fast reactor p~ogun was selected for this validation stJdy,9 Section II oi this pzpcr provides a brief description oi the pin calandria unit cell with d&&Is of the three-dinwnsioxal and one-dbwnsional modeling procedures. Section III dis- cusses cz!cuiations for a single infinite pin. These calcuiarions (SDX vwsus VI?vl) test the accurxy of SDX in treating the one-dimensional cyiindr!cal model, The results of the SDX calculations of [he sin@ pin model are compared in Sec. IV with the results of the Vlh% calcokations of the threwiimen. s?x~al pin caiandria unit cell. Sections V and Vl compare SDX!Benoist vcrsu VI&l caladations for low and high- buckling syswns. The% results indicate that the inclusion of Benoist anisotropic difcusion coefficients underpredicts both

010400~02

112 TECHNICAL NOTE

the magnitude and asymmetry in the leakage effects. An alternate one-dimensional unit cell modelin&! procedure is defined in Sec. VII, and results obtained using Gelbard diffusion coefficients” for both high- and lowbuckliw systems are presented. The VIM &<onte Carlo results for foils in the pin calandria model are discussed in Sec. VIII: Section IX provides a brief summwy of results and c~ndu-

sions. A discussion and comparison of Monte Carlo analyses for the single pin model using various cell boundaries and boundary conditions are given in the Appendix.

“E. M. GELBARD and R. LELL, ,Vuc/. Sci. &, 63, 8(1977).

Il. “NIT CELL MODEL PROCEDURES

The unit cell loadin selected for this validation study of pin geometry ccxisted of a 5.08- X 5.0% X 30.4&m voided .caiandria ioadcd with a 4 X 4 array of 0.957.cm (diam) X :5.24ar nliwd-oxide rods il5% PuO#J02). Figures i and 2 show thz dimensions of the calandtia unit cell. The atom densities by region are given in Table I.

This three-dimensional calandria unit cell was modeled in full detail for thz reference VIM Monte Carlo calculations. 1: may be noted that the fuel region retains the physical composition and dimemions (radius) of the fuel pins, while

MATRIX TU8E

\

l STAINLESS-STEEL TUBE-

0.52 OUTER RADIUS, 0.49 INNER RADIUS

PIN JACKET- VOID 0.4.8 OUTER RADIUS,

0.45 INNER RADIUS

e-PELLET-O.43 l?ADiUS

, ’

TECHNICAL NOTE 113

TABLE I Atom Densities for 1he S- X S- X 30.cm Voided Cala”drka Unit Cell

1 ~‘%I ~.003073l45 0.00000l =%I+ 0.000418l63 q”* 0.000089SS6 2%” 0.000045923 0.000001 XqJ 0.020237668 0.00000~ '60 0.04757i992 O.OOl9Sl653

h” 0.0579ls300 Nickel o.oo7osl7ls Chromium 0.016278032

, Nzlng3nese 0.001246223

0.003327831 O.OQli27334 O.OOl728g30 0.002567583 0.002448l89. 0.0627486SO 0.032571045 0.036802066 0.064953794 0.061933406 ’ 0.008SSX89S O.O0?4427S3 0.004947984 O.OOSS49304 O.OOSlSl7S6 1 O.OlSl77720 O.O09<3SS37 0.0106OS964 O.OlSS46423 0.017684004 O.OOll90827 , O.WO6180.27 O.OOW9417?. O.OOlZl2OlS 0.001 IS5659

the fuel cladding! calandria tubes, and saps are smeared into a si”$? a”nu!ar clzxidins region for each pi”.

A one-dimensional cylindrical model of a single pin unit cell was required for the SDX cross-section rqneration. This o”e.di”~ensional model is show” in Fig,. 3. The atom density

code ATD3 ws used to obtai” the “umber de”sities by region. This incorporaws the following modeli”g procedure. Two cylindrical rcl+x~s are mod&d representins a single fuel pin and the surrwnding diluent material. The radius of the inner regioti is the physical radius of the (tmcPad) fue! pin and is comprised oi the physical (i.e., unstrerched) “umber densities of the fuel constituents. The surrounding a”nulx region has a volume equivalent to ox-sixteenth the total x”fue! vo!w”e within the matrix tube, ca!andria, ad fuel cladding, i.e.. one-sixteenth the mass of the nonfue! rexions in the unit cell. The ztcx” densities by qiun for t!x one-dimensional modei of the 5.08. X 5.0% X 30.48~cm voided cala”dria are give” in Table Il.

.4lthough the ultimate goal was to ca”lwre the o”e. dimcnsicrul SDX rewlts a@nst the three-dimensio”al VIM Monte Carlo resulu, as an i”wmediate step in the vaiidatio” of the S’JX calculations of the we-dimasional cylindrial unit &I, this en&$ pin model (as show” i” Fig. 3) was also us?d for VIM Mogte Cxlo calculztioxs. Si:< ws of VIM calc&tio”s were pzrforr”Ed fo: various cell boundaries a”d boundary cond;tiux. These included

I. two cylindrical rqions with white boundary conditions

I.>,,., = 0.4290 an

Q,L = 0.7792 ml

2. two cyliud:ical retions with reflective boundary condi- tions

3. sin& pi” in square cell with reflective boundary conditions

qatN = 0.4290 cm

widt~hce,, = I .38 I I cm

4. single pi” in sqawz ccl1 with periodic bou”duy condi- tions

rp,N = 0.4290 ‘3”

widthcell = I.381 I cm

. . ,

,‘~ , .

114 TECHNICAL NOTE

TABLE 11

Alom Densities for the Single Pin Model of the Voided Pi? Czlandria Unit Cell

Region 1 Region 2 (Pin) ~Dil~ent) Homogeneous

O.liO4003 (-5)b 0.5l348l4 (-61 0.2936136 (-2) OS47705 (-3) 0.3912260 (-3) 0.117s9l4 C-3) 0.4791544 (-4) 0.1443876 (-4) 0.6589506 (-5) O.I%3567l (-5)

.a 0.3764785 (-4) 0.1134474 (-4) 0.4387609 (-4) 0.1322155 (-4) 0.1933542 (-.I) OSS26504 C-1) 0.45451 I I (-1) 0.6691683 (-3) 0.1416369 C-1)

0.1751377 C-1) 0.1223620 (-I)

0.22?6084 (-2) O.lS902l2 l-2) 0.49Y5939 C-2) 0.3490471 C-2) 0.3456332 (-3) 03i4807 (-3)

5. homogeneovs so,uare ceil with periodic boundary con- ditions

6. single pin with cladding in voided square cell with periodic boundary conditions

rp,N = 0.4290 ml

rcLAD = 0.5 I7906 cm

wid&tt = I .3S I 1 cm.

Cases I. 3, and 4 were shown to be equivalent,~’ Case I represents the same cell geometry arxl bounday conditions as used in the SDX cziculations, l~ovxver, since ase 4 was show! to be equivalent, iI was taken 2s the reference calculation Sor comparison with SDX, The periodic boundary conditions upon the square-cell pwnit the direct ca+lation, by the VIM code, of I>: the mean square distance from neutron birth by fission to death by fission. The quivalwx of these two cell representations is also sigificzdnt since SDX requires the cylindrical boundary in the sin@ pin calculation, whereas the physical arrangxnent of the pin calandria is a square lattice (with pitch = I.2395 cm). Case 6 includes an ovtcr region of ww void (instad of the homogenised low density stainless-steel diluent r&on) anJ alI structurdl materiak hzs been collapsed into an amula qion (with thickness equal to one-Mf the difference between the outside dianeter (o.d.) of the calandtia tube and t!x o,d, of t!x fuel rods1 surrounding the central fuel pin. The rexon for this model becomesevident later.

-.-

TECHNICAL NOTE

TABLE Ill

Resul!s of VIM Czlcuiations for the Single Infinite Pin Models

:

Single Pin Hete~ogeneot!s-Sme~red~

I.28947 k 0.0024l (90 000 histories)

I.08029 + O.OOSO4 0.021229 * O.OOOl31 O.OlS724 t 0.00079 0.28061 t 0.00191 02006S f 0.00081 I.40147 i 0.00653

2243.2 5 13.0 747.7 k 4.3 741.0 k 7.7 745.0 k 7.0 75'1.2 * 7.8

743.0 * 5.2

0.9937 ? 0.0090

I.0127 i 0.0119 I.0191 + 0.0127

l- Single Pin

Hom.ogt3leol~i~

I.28505 i 0.00244 (90 000 histories)

I.07980 t 0.00475 0.02lOS9 5 O.OOOll6 0.15916 ? 0.00075 0.28236 k 0.00171 0.19941~ t 0.00079’ I.39967 + 0.00614

2197.5 t 13.1 732.5 * 4.4 739.0 t 7.9 734.0 k 7.5

724.S i 1.4

736.5 t 5.4

I .0055 i 0.0095~

0.989 1 ? 0.0117~

0.9s37 3z 0.0124~

aUncertainties regreswt one standard deviation. ‘Fiere, f~ fission, c 5 capture; 2S G *W; 49 s ‘39P~; 28 E *“U, etc. cThese ratios are unity within the statistics.

sir?le infinite pi!> models xe swnmarized in Table V, This t ’

*

also shows the comparison with VIM calculations for the mode!. It should be noted that the unceTtaintic3 listed

inc ude only t!x uncertainties from the VlM statistics. The eiwwalu.? (k-1 for tlx l~omogewo~~s cm is 0.00354 Sk low relative to the VIM calculation, which has a one-sigma uncertainty .of 0.00244 Sk. The eigenvalue for the hetwo- gtmeotwsnmmd case is 0.00284 8k low relative to the Vl!vl calculation. which has 8 one-sigma uncertainty of 0.09241 Sk. The heterogeneity effect, 8+,*,, is to.00542, which is cam wrabie to the Vlh! result. All the calculated reaction rate ratios are in good agreement (within I<;;) with the correspond- ing VI% results. The values for p produced by the SDX! Benoist method are also summxized in Table V. ‘These v&ties correspond to low (near zero) buckling and .wece obtained iq the relation

E!genGalues were calada ted fog three bt!ckling values (B2 = 0.0, 2.0 X 10-j cm-, and 4.0 X !O-’ cm-‘) and the slop? 6k/SB' was obtained k,y parabo%z interpolatiox As snowd, the uncertainties listed in Table V for the conlparison of SDX and VIM inc!ude only the one-sigma VIM statistics; however,

Single Pin Heterogell~otls-Voids

--

i28Sl9 t 0.00267 (30000 histori~s~

I ,OS! I7 t O.CO645 02l200~ 0.000~19 O.lS764i O.CO~li 0.28144 2 0.00245 0.200l0 t O.OOG95 I.40225 i 0.1X820

2472.4 k 34. I 824. I ? 11.4 770.1 2 10.7 so7.3 i :7.2 895.0 2 27.4

788.T 2 IO.1

0.9570 * O.OlttO

I.0860 k 0.0365

I.1348 2 3.0277

01040006

116 TECHNICAL NOTE

0.000488487 0.000i7l028 0.000039538 0.4685485 I6 I .07 I030732 0.36547SO24 0.103073492 0.023771536 0.02946481 S O.OOY636185 0.0l413s3s2

0.025133Ol6 O.t’O84SSS29 O.OOl604l52 0.00037332s 0.0~0124747 0.000174479 o.oo34l9xss 0.00:045397 0.003lSSO57

0.007846412 O.Ol4S42623 0.0060s04s6 0.001795956 0.427255284 0.143808l68 0.051946686 0.375308598

--- 0.002925735 0.026553343 --- --- 0.026SS3343

--- --- 0.01 I l8208i --- --- O.Ol4lO7523

0.003003335

l.000000 + 0.00017 1.2X8561024+ 0.0041 0.442908352 i 0.0014 0.557091648 t 0.0014

0.0000433l8 0.102375297

O.GO9349725 O.Ol3646SlS 0.001561762 0.0002024x7

0.01 I149173 0.0:34s4213 0.00289252l

0.438399240 + 0.0012 0.561600760 k 0.0012

Single Pin Heterogeneous-Void

mp" O.OOOOS83lO =gPll 0362OG8566 O.lO2l67l28 T’u 0.023539434 0.0304Sl572 0.009?94285 O.Ol3.54Sl49 t -‘PlI O.OlOS72797 0.026419025 0.008S793S4 0.001693443 TQ 0.000232731 O.OOOl989E8 I~.000067807 0.0~30 I64925 -‘Am 0.00428896S 0.0030286D8 ’ 0.00093258i 0.003356384 y! 0.007214737 0.013608985 0.005546728 0.00l668008 238” 0.43365422S 0.146393336 0.053! !9326 0.38OS34899 ‘60 0.00299230l --- 0.00299230:

IIOZl 0.026598563 --- 0.026S98563 Nickel 0.010306482 --- . 0.010306482 Clxomium 0.013598013 --- O.Ol3598Ol3 Manganese 0.002593327 --- 0.002S93327

Totals I .oooooo * o.ooaz7 I .28199059 I+ 0.0063 0.44Q?23368 t 0.0021 OS59276932 % 0.0021 ---

~I’!. COW.4RlSO~ OF SDX AND VW CALCULATIONS FOR A VOIDED Pls CALANDRIA

B

B

, ’

TECHNICAL NOTE 117

TABLE v Results of the SDX/Benoist Calculations for the .%I& Infinite Pin Model

l- SDX

I.28693

I.0824 1.0020 i 0.0047 0.02109 0.9935 k 0.0061 O.iS675 0.996Y k 0.0050 0.28280 I.0078 k 0.0069 0.19945 0.9940 * 0.0940 1.40331 I.001 3 k 0.0047

2198.06

732.67Sd

lS4.383 143.941

1.0140

l.OlS4

I .0296 2242.3

1.02Ol

0.9953

SDX/VIMb

0.9980 k 0.0019

0.9799 t 0.0057 0.9799 t 0.0057

O.Y963 5 O.OlO2 I .oc13 2 0.00~0

0.99so k 0.0124

1.02lS t 0.0092

1.0167 * 0.0120

0.9996 t C.0058

!.OOl6 kO.0091

expliktly, The surrounding cladding represents the camins for

l- Single Pin

SDX

l.28l5l

I.0789 0.02iO~ O.lS842 0.28181 0. I9966 I .39942

- .---

1 SDX/VIMb

0.9¶72 * 0.0019

0.9992 e 0.0044 0.9977 k 0.0055 0.99s4 k 0.0047 0.9981 ? 0.0060 I .oo 13 i 0.0040 0.9998 * 0.0044

0.9985 k O.OOGO 0.99X5 t 0.0060

OlO40008

II8 ?‘ECliNlCAL NOTE

TABLE VI

Comparison of Results for the VIM Calculation of the Pir! Caiandria Uait Cell and the SDX/Benoist Calctilations of the Si@ Pin Model

- Three-Dim&sional One-Dimcnsiomd

Pin Cakmdria Single Pin VW Hetero~elleous-Smeared SDX SDX/ViI@

- Eigenvaiue

k.a l.3OlS4 k 0.00318 1.29935 0.9981 k 0.0024 (SO O@O histories)

Reaction rate wios’ 2Yl~9r I.07857 k 0.01973 I.081 17 I.0024 i 0.0182 =w~Y 0.021342 * 0.000517 0.021 163 0.9916 2 0.0242 =%Pf 0.1563s t 0.00348 0.15669 1.0022 i 0.0226 %Pf 0.27984 k 0.00773 .0.28199 I ,0077 i 0.0276

:$:I; 0.2Ol20 1.39678 t k 0.00291 0.02536 1 0.19997 i.4OlSt3 0.9939 I.0034 t + 0.014s 0.0181

Diffusion chxacteristics (in tmits oi cm’)

i2 764.2 + 10.9 c 763.6 k 9.9 z 800.3 * 9s 755.779 0.9444 !i 0.01 I9

iz =+(z+q) 763.9 i 7.4 YJ4s.314 0.9757 i 0.0097

7 2328.1 .t 17.5 2202.ld 0.9459 t 0.0075

-TECHNICAL NOTE ll9

TABLE Vii

Comparison of SDX/Benoist Versus VIN Calculations of kcff for High Buckling Systems

Buckling’ z’& cm-*

B; , cm.‘2 I---- B!, crnes

Eigenvaiues, &ff

,

VIM pin calandria mod&

SDi one-dimexional diffusion theov with isotropic diffusion coefficients

SDX/Benoist on?-dimensional diffusion theory with Benoist anisotropic diffusion coefficients

0.0

0.0

0.0

1.29935

1.29935

: 1

0.0 3.3 X lOma 2.33 x 10-4

0.0 3,s x so-+ 2.33 x 10-4

7.0 x 10-d 0.0 2.33 x 10-q

1.0!023 kO.00416 l.O?Ol2?0.00429 ~.01544~0.0045.

I .02574 1 i.oxi4 I .02574

1 ,.021*5 :.01915 1 l.o*231

bQuoted uncertainties are one-sigma intervals.

ca!cl!lated p appears to result from the modeling procedure, whereby t!x stainlas-steel cladding materials are homogenised tiuoughout the dih&c:nt region that surrounds the pin.

r-

“,. COMPARISON OF SLWBWOIST AND Vlhf CALCULATIONS FOR HlGt.1 BUCKLING SYSTEMS

The eigenvalues calculate+ for an array of voided pin ulandGa unit ceils as a function of the buckling using the VIM site :apc are summxized in Table VII. To obtain SDX/Benoiu c?,lct!lated eigenraiues at near critic21 buckling vz!ues, broad grwp cross sxtiox were generated by collapsing over the ZxF,XO,Xi~k (buckled) slxctm (& = 7.83 I87 X lO-a) using SEF ID. Bidirectional (r-z) diffusion coefficient modifiers obtained from the Ben&t code are given ln Table VIII. Apprc~priaw combinations of these diffusion coefficicn? modifiers were used for the various systems listed in Table Vii. A buckling vahx of 7.0 X I Om4 cm-* w&s s&x&d to lxoduce a k<rr near I.0 for the system. Three cases were

2. i?: =P= 7.0 X iO~4cm~*,B~= 0.0.

3. B; = 2Bj, B2 z 7.0 X IO-’ cm-’

The SDX/Benoist c:alculated eigenvalws given in Table VII are consistent witk! the results of the calculated F values con:puted at low bucklings (3s discussed in Sec. V). Note that the use of isotropic diifusion coefficienls (i.e., total ncgleci of a streaming effect) consistcntiy underpredicts the leakage with eig~nvilues too hi& relative to VIM from I.03 !o 1.55% 6k. In all cases, the estimate of the icakage obtained by inclusion of the Ben&t :inix:ropic diffusion coefficients is too small (and therEfore k*fr is too iarge relative to VIM). This b& is lugcst (0.9% Sk) for case 1 in which all the diffusion is assumed to be in the : direction (i.e., parallel to the axis of the pins). l=or case 2, with diffusion only in t!x .yy plane, the the kefc bias (SDX r&tiw to VIM) is +0.2% ?& which is xGthin Ihe statisrics of the Monte Carlo calculation (I0 = 0.00429 Sk). For use 3, t!x SDX ~ulue of k<:g is 0.6% sk high relative to VIM.

All of these welts are consiswnt with the obsawtion

TABLE VIII

Benolst Diffusion Coefficient Modifiers with Critically.Buckied* SDX Cross Sections

I 1.00395 2 1.00347 3 1.00397 4 1.00383 5 1.00704

I.00777 1.0068 1 1.00777 1.00748 !.01374

6 l.01653 1.03233 7 l.01221 1.02384 8 1.32<89 l.OSOl6 9 1.01973 1.03804

IO 1.02185 1.04209

11 12 13 I4 15

l.Oli86 1.01853 I.01176

I.03592 1.0351 I 1.02198

1.023x 1.04489 I.01328 I.02478

!6 17 I8 !9 20

I.00169 1.0045l 1.00566 1.01007 1.00953

1.00307 1.00824 I.01034 I.01854 1.01749

21’ 1.00896 22 l.OU238 23 i.01153 24 I.01646 25 1.02537

1.01645 1.00432 1.020?4 I.02978 1.04s53

26 1.02390 %7 I.00691 28 I.02281 29 1.40793

1.04291 1.01264 1.04093 1.63506

pxdlel (axial) Jir&ons, reqcctively, and Eg is the homogc- neous diffusion coefficient for the cell.

01040010

120 TECHNICAL NOTE

that the one-dimensional modeling procedure, whereby sur- rounding structural material is smeared homogeneously throu&out the void channels. is inadequate and results in an underprediction of the streaming effects.

W. CALCL!LATtOxS FOR T”E VOlDED PIN C.4LASDRlA. USING GELBARD DlFFL%lOS COEFFICIENTS

The conclusions, which have been presented to this point, are generally consis:ent wirh the results of the plate studies by Wade et al.’ For the voided unit cell the modelins procedure used in SDX adequately computes A& and reaction rates How- ever, diffusion chamcteristics (and eigenvalue for a high buck- ling system) are not well predicted. InchIsion of Benoist anisotropic diffusion coefticients, althowh an improvement on homogeneous diffusion theory, still underpredicts (both the magnitude of and the asymmetry in) the leakage effects. A model& procedure that retains a true void region was used in the Gelbard method for anisotropic diffusion coefficient%‘6 The structural material associated with b sin~?le gin (namely,

calandria unit cell) was homogxized into zin annuh~r wion (with thickness equal to one-half the difference between the ad. of th$ calandria tube and the o.d. of the fuel rods) sur- roundins the caltrzl fuel pin. This is the hetero&x?eous-void prescription of Sec. lli. It should he stressed that the SDX model used to generate the cell-averaged cross sections is still tke h~tero~eneo~ls-smeared model. The heterogeneous-void model is used only to gnerate the Gelbard anisotropic diffu- sion coefticients.

The use of Gelbard diffusion coefficients was tested both for systems with high or low buckling. The following buckling combinzdons were considered for the high buckling system:

I. B; = 0.0, B;= B1 = 7.0 X lO-“~m-~.

2.B~=Bz=7.0X~~4cm~2,B~=0.0.

3. Bf+ = 2B&Bz = 7.0 X 10~4cm~z.

TABLE IX

Gelbard Diff,lsion Coefficient Multipliers

0.9843a 0.71 0.9836 0.74 0.9856 0.86 1.0002 0.80 I .0252 0.86

D!/P * a(%) --

0.993s 0.74 0.992s 0.92 0.9982 0.98 I .0048 0.96 I ,032l 0.99

1.0616 I.93 i.0457 I.00 I.0944 1.22 I.0995 I.18 1.1140 I.26

i.1415 I.50 I.!347 I.45 I.1449 IS6 I.!395 1.33 I.1734 I.58

0.9822 03 I 0.9873 0.57 0.9934 0.66 i .0050 0.61 I.0310 O.G4

I.0493 0.66 !.04l2 0.67 1.0738 0.76 1.0738 0.72 I .0798 0.76

1 4.9329’

: 4.6123 3.9928 4 3.8! I7 5 3.2809

6 2.7309 1.0313 0.87

.; 2.920s 1.030s. 0.90 I.9318 I.0574 0.96

9 I.9555 I.0610 0.85 IO I .8660 1.0596 0.96

11 I.6109 1.0737 0.97 12 I.5005 I.0676 I .OO I3 I .2820 .I.0883 1.1 1 I4 I.5572 1.0701 0.92 IS 1.2489 l.lOSS 1.15

I6 0.833S7 1.1432 1.33 I7 0.92157 I.1471 1.48 I8 0.94946 1.1248 1.2G 19 1.0612 i.lllG I.21 20 1.0129 1.1064 I.%6

21 I.0150 I.1377 I.36 22 0.84018 I.1590 I.41 23 0.90688 I.1436 1.40 24 0.86743 I.1400 I.53 25 0.79493 I.1597 I.39

26 0.8043 I I.1682 l.GZ I .2234. 2.05 27 0.97330 1.1349 I.21 1.2334 I.93 28 9.79504 I .2048 1.90 1.3S82 2.58 29 0.24G3S I .8612 4.62 2.5244 6.27 --- -

‘@, D{, and IJ! represent the homogeneous! the radial, and the axial Gelbxd.diffusion coeifici.xts, respectiveiy.

I.2781 2.29 1.28% 2.29 1.23bO ?.Ol i.2!21 1.67 I.2095 I.76

I.1001 0.84 1.0894 0.83 1.1 I22 0.93 I.0971 0.82 1.14.77 I.05

I.1895 1.22 1 .I 764 I.36 I.1700 I.24 l.iSli 1.13 i.lS46 I.14

12177 !.83 I.2858 2.35 I.2727 2.33 I.2697 2.26 I.2938 2.28

I.1679 I.13 I.2108 I.45 1.183s 1.39 I.1885 I.48 I.2140 1.1.9

1.2164 1.33 I.1614 I.04 1.274! I.69 2.0779 4.08

Bz = 7 X IO+ cr0.B; = 2B:

TECHNICAL NOTE

The corresponding G&xrd diffusion cwfticient multipliers l.E~=B2=I!.0X10~4cm~2~B~=0.0.

I21

are listed in Tabie IX. These values were obtained from 300 000 neutron histories and the listed uncertainties (one- sigma intcrvais) are generally <I% above group IS. h’ormalized leakage utes are given in Table X for wrious buckling combinations. It is evident that -96.4% of :he total leakage occws in the top I5 groups (i,?.. above Y keV) and -99.8% in lhe top 20 groups (i.e., above 0.?5 keV), Therefore. the poorer statistics associated xvith the diffusion coefficiwts in lower energy groups are much less important. Eigenvalucs obtained using these diffusiott coefftcients are compared in Table Xl with the results of the VIM calculation for the voided pin caiandria unit cell, As noted previously, the use of Benoist diffusion coefficients consistently (and significantly) underpredicts the Aklk due to lcakagc. The use of the heterogeneous-void one.dimensionai model& and the Gelbard diffusion coefficients greatly improves the agreement with the full three-dimensional Uonte Carlo calcuiarion, In each case, the AL/k obtained with Gelbard diffusion coefficients is s!!g/zl/J, small but well within VIM statistics.

The following buckiing combinations were considewd for t@’ buckling system

2. B; = 0.0, E= = Bj = 2.0 X 10-j a”-’

Gelbard diffusiort-coef~cient mu!tipliers were recomputed for each of these systems using the SDX cross secticms that were gcnerxted with zero b<xklil;g. The Gelbard diffusion c@ efficiepts were obtaiwd irom ZOO 000 ncutron histories and the one-sigma uncertainties are generally <I.5 to :.O% above group :5. Eigenvalues obtained using these diffusion co- efliciwts and the Benoist diifusion coefficients (whic!t were also obtairxd with the /3’ = 0.0 SDX cross sections and the heterogeneous-secured one-dimensional model) are compared in Table XII with the rewlts of the VIM cz!cuPdtions for the voided pin calandria unit cell. .4s noted b&ore (with the high buckling systems), the Sk due to !eakage is underpredicted with the Ben&t methods. In this case, however, the effect is

TABLE X

Normalised Leakage Rates (by Group) for the Voided Pin Caundria Unit Cell

Group

I 0.00047 2 0.00624 3 0.02 I80 4 0.05485 5 0.06758

0.00046 0.00610 0.02 139 0.05342 0.06594

6 0.07731 0.07703 7 0.17062 0.16749 8 0.08902 0.08904 9 0.1126s 0.1 I261

i0 O.lO74Y 0.10878

0.08002 0.06497 0.039GO

O.OXl6Y 0.066 I5 0.03985

0.04588 0.~4661 0.02680 0.02702

16 0.00954 O.OlOl4 i7 0.00909 0.00968 I8 0.00637 0.00663 I9 0.005 14 0.00529 20 0.00244 O.SO252 21 22 23 24 25

0.00129 0.00042

0.00l30 0.00044

26 27 28 29 !

0.0002G o.oocol 0.00000

0.000G0 0.00000 0.00000 0.00000

0.00027 0.0000l 0.00000

0.00000 0.00000 0.00000 0.00000

I .ooooo

0.000~2 0.0057c 0.02026 0.04886 0.05879

0.0004l 0.00559

0.0 I888 O.Ol953 0.04721 0.04883

0.06709 0. I5443 0.08343 0.11-237 0.1 I653

0.0583G

9.06480 0.!46?3 Osl83 15 O.! 1784 0.11584

0.05765

O.O6%S 0.18146 O.Of33l9 0.1 I097 O.llSl3

0.!?8511 0.08596 0.08.559 0.07~55 0.07i8Y 0.07138 0.0443 I 0.0463s 0.04570 0.05436 0.055i4 0.05517 ‘0.03: 7Y 0.0334? 0.03217

0.01 lY3 0.0l151 0.00838 0.00698 0.00342

0.00174 o.t?Joos7 o.lJoo35 0.0000l 0.00000

0.00000 0.00000 0.00000 0.00000

0.01233 0.0 1362 0.00872 0.00743 0.003S2

O.OOlY5 0.0006 I 0.00036 0.00002 0.00000

0.00000 0.00000 0.00000 0.00000

I .0oooa I .ooooo

B; = 2B;, B= = I .o x 10-s cm-*

O.Oli79 0.01284 0.00873 0.00743 0.003s7

0.00 I87 0.000.58 0.00036 0.00002 0.00000

0.00000 0.00000 0.00000 0.00000

I .ooooo

small and. althougl~ the bias (rebdtive to ViM) is larger than the Monte Carlo uncertainties, the overall impact on calculated eisenvalue is not significant. Furthermore, the ratio of the axial-tomdid compone!~t~ of the Sk due m ieakage gives a tnmsurc of the czlcu~ated axial-to-radial asymmetry in dif- fusion, The Benoist diffusion coefficietxs indicate an axi&t”- radix1 asymmetry of 1.013, as compared ‘to I.047 (51.7%) from the VIM calcu!ation. The results given in Table Xl1 fa the SDX/Gelbard methods consistently owpredict tlx ?G due to leakage, The bizs frelatiw to :he V!M calculation) is comparable in nugitude to the bias obtained with the Benoist methods with similar insigificant overall impact on the calculated ei&!envalue. The Gelbard diffusion coefficients indicate an axial-to.radial x~ntnetry of 1,074, which is 2.6% high relative to VIM, The xe oi either Ben&t or C&xrd diffusion coefficients appears to provide re~onable resul:s (and therefore considw~b!e improvement over conventional diffusion theory) for systems with low buck!ing,

VIII. VIM WYTE CARLO RESULTS FOR FOIL.5 tN %ltZ VOIDED PIN CALANDlS, &lODl:L

Small edit regions simulating fission foils were inclruded in the pin calandria model. These foils were positioned in two axial locations of the calandria, namely, the axial midp!ane of the cxlandria and the axial midp!ane of each fuel pin. Each “i the foils had a thickness of O.OlOl6 cm (0.004 in.) and con- tai”e<, >w,+,, mu, a”d 23s U at a density of lO’a atom/cm’. The foils ioct~ted zt the axiai midplane of the calandria were sandwiched between the chtddin- : ends of the two 15.24.cm (‘&in.>long fuel pfns loaded in each alandria tube. i-his position is physically convenient. and therefore a reaiistic 10mtiofl for : fission foil. One foil was p&cd in tllis position in each of the alandria pins l thro@ 8 as numbered in

TECHNICAL NOTE

TABLE XII

Comparison of Eigemalues for a Low Ruckling System

123

“IM ~three-*ime”sio”~n s”x;Be”oisr (““e.diJne”ca”al~ sDx,Gelbml ,me-dime”simlal, Pin Cahndria Single Pin Single Pin

P = 2.0 x ,0-s ml-~ l?a,ic fMltive to ,‘I%, btio KelXive to \‘I!4

:ase 1: B; = 2.0 x IF5 c”l-.B! = 0.0

&k’ k -0.00759 O.OOOIOb -O.O0?4l * 0,976, 2 0.0132 -0.00780 1.0277 * 0.0,;:

hi I.29196 kO.OO3iX I.28946 cl.998i *o.oEs ,.23x95 0.9Y77 A 0.003

23se 2: Bi = 0.0, B: = 2.0 X IO-' a"-'

& k -0.00795 t 0.00009 0.0075, 0.9449 ,3 f 0.01 -0.00838 I.0541 : 0.01 i3

kr ,.29,49 5 0.00318 I.28932 0.9983'~ 0.0025 l.2SSl9 O.Y974 2 0.0025

LO4CS t 0.0128

0.9976 2 0.0025

l.03XS z 0.0129

0.9Y77 * 0.002s

l.o2lo*o.ol35

0.9973 io.0025

010400~14

124 TECHNICAL NOTE

I. In 2n asymproric, lwmo~twfxx~s fast reactor composi- tion, the SDX code produces cross sections and diffusion coefticients that correctly predict reaction rates, leakage rates, llux spxtra, and eigenvalue (see the last two columns of Table V)~ This conclusion generally supports the results’* of previous studies.2.3

2. For a single pin fast reactor unit cell consisting of a fuel pin surrounded by a low density diluent, and in lhe asymptotic cast:

a. The SDX code produces cell-averged cross sections that correctly predict the reaction rates. However, the leakage rates are underpredicted.

b. By using Ben&t anisotropic diffusion coefficients for this unit cell, the leakage rates at low buckling are also correctly predicted (see the first two columns of Table V).

3. The voided l6epin calandria unit cell used in ZPR studies cannot be adequately modeled by the single pin/two region one-dimensional model mentioned in item 2. Allhou& reaction rates are adequately predicted, leak:j:e rates using the Benoist diffusion coefficienrs are underprcdic%d (see Tables Vi and VII).

4. Tlw following procedure has been shown, for the voided pin caiandria ZPR unit cell, to produce cell-averaged cross sections and (Wbard) anisotropic diffusion coefficients, which correctly predict ceil-rwerased reaction rates, llux spx- tra, and ci$envalues and leakage rates ar both low and high buckling in [he asymptotic situation:

a. For the purpose of :I:? SDX alculation-to produce ce~~.averaged cross section.-the pin cahmdria is mod- cled as a single fuel pin of physical radius and composition surrounded by an anxuix diloent wgion, which hx a volume equiwlent to one-s&teenth the tool nonfuel volume in the p,in cahmdria unit cell and the composition of whicl: comprises one-sixtewtl! of the nonfucl pin material in the pin ca!andria. (>!“t only are c&avera@d cross sections ~en&ted; but fuel pin cross sections are as well to be used in step b below.)

b. For the purposes of .~eneratin$ the Gelbard anis”. tropic diffusion> coefficients, the single pin ceil of step a is modified to collapse 511 diluent into a cladding of thickness equal to the difference between the outer radius of the caiandria tube and the <wter radius of the fuel rod. The remainder of the diluent retion in the cell of swp a becomes a true void.

c. The cross sections and the Gelbard diffusion cw efficicnts are generated for the buckling vector expected to be encountered in the subsequent dif- fusion theory calculation since both the cross sections and diffl.lsi”n coefficients are buckling dependent.‘,*

5, The heterogeneity effect in the pin cell is small for eigenvahw but nontrivial for leakage, from Table III:

a. cLPi” =,, - L~omo~~“eo”$~ =am4

6. The (peak pin)/( 16-pin weraze) ratio of reaction rates in the asymptotic, voided pin calandria is near unity for 49j; l*f: z8~; and 5. Foils that sample the reaction rates appear to show more structure, but the Monte Carlo statistics are tog poor i” draw.firm conciuGons.

APPENDIX

MOliTE CARLO CALCULATIOM FOR A O~E.DlMEMiONAL CYLIXDRICAL UNIT CELL

The VIM Monte Carlo calculations were performed for a single infi!!ite pin model for direct comparison with SDX results. Fox distinct VIM calculations were made with various cell boundaries (cylindrical versus square) znd boundary conditions (white: reliective, and periodic). The models for these VIM calculations and the results are summarised in Table p\.l. Cases 1 and 2 have the same geometric representa- tion (i.e., two cylindrical regions) as is used in SDX. Case ! ais” has the appropriate bomldary condirion on the outer ceil boundary (i.e., white or isotropic). Cases 3 aud 4 Irive a square cell boundary. The sn,uare cell was defined to !a) eliminate the deleterious effecls of uGng reflective boundary conditions with a cylindric?.t ceil (case 2), and (b) permit the use of the VIM site tape utility code to generate ? using periodic bo!lnd&ry conditions (case 4). The results shown in Table A.1. represent 30 000 histories for each oi the VIM calculations with the exception of cae 4 for wh!ch 90 000 histories were obtained.‘9 The eignvalues (k-) and reaction rates are in exc?lle”t egrww*t Kxcept for case 2, It is also 1noted that 1:. s>+xtr~.m~ is much Iharder (as indicated by the s&uitican a higher werag~ fission energy) for GW ?.. These comparisons co?l:rm that the us! uf reflective boundary conditions with the cylindrical cell is not adequate, Cases I, 3, and 4, however, may be considered equivalent, -

. . . .

hkxm Carlo Calculations for a Single lnfinitc Pin Unit Cell -

case I cisc 2 case 3 czse 4

Xl boundary Cylindrical Cylindrical square square

Q.,N = 0.4290 cm rp,x = 0.4290 cm i-p,N = 0.4290 ml ,p,x = 0.4290 cm q),L= 0.7792 ml Q,L = 0.7792 cm widtl~<~,,= I.381 I cm widlbLYi, = I.381 I cm

I t3oundary rmditim White Rellxtive Retlective Periodic

Eigenvalue L I .26245 2 0.00454 1.27606 ? 0.00395 I.28924 * 0.00381 1.2605S 2 0.00224

Avenge fission mergy, MeV 0.6934 t 0.01 I6 0.7S86 k 0.0134 0.6788 zk 0.0100 Reaction rate ratioa , 0.6772 + 0.0054

TECHNICAL NOTE 125

TABLE .A.1

yyf I .09084 1.09040 I .08889. I .os907 2Fpf 0.020205 0.022388 0.019983 0.020084 x+,“,ej 0.159s2 O.lS599 0.15935 0.15932

q::; -pf 0.29361 1.41237 0.19490

0.2SS67 1.41201 0.2OOS6

0.29 I .409x9 105 0.19~568

0.28970 1.4lS88 0.19S73

-

l