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STATE RESEARCH CENTEROF RUSSIA

Institute of Theoretical and Experiment~L.Physics

FERMILAl:S

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fl,PR 8 1997 47 -96

'.,1BRAR'~

V. I. LYASHUK

SELF-SHIELDING FACTORS OF 24<1>u FOR THE HEAVY WATER FUEL CELL

OF THE BLANKET OF· ELECTRONUCLEAR

FACILITY

M 0 S C o· w 1996

UDX 621.039 M-16

SELl-SHIELDING PACTOllS OJ' 240P\1 )'OR' THE HEAVY WATD J'lJEL CELL 0' THE BLANXET OJ! RLECTltONUCLEAll PACILITY: l'reprint ITEP '7-96/

"/.V.Lyashuk - M., 1996 - 28p.

The method for calculation of Pu-240 self-shielding factors for the

heavy water fuel cell of the blanket of electronuclear facility for long-living

radioactive waste transmutation are presented and these results of computing

are analyzed. Variants of simplification for the cell geometry are considered.

The correctness of considered approaches is confIrmed by results of the

calculation. The modeling of neutrons transport is carried out under the

reactor MCV-code (Monte Carlo method).

Fig. - 16, ret. - 6 name.

The self-shielding of resonances can considerably (or strongly in

number of cases) reduce a rate of corresponding reactions in a resonance

interval of energy. Self-shielding factors [ I ] are the measure of change for a

resonance interaction rate at a resonance self-shielding. The given work is

concerned with determination of capture self-shielding factors fOf the Pu-240

isotope and is performed within development of the· project for transmutation

of radioactive nuclides. The absence of these data for dilution cross sections

Uo > 104 b and presence of a heterogeneous medium (where a transmutation

occurs) have caused necessity of these calculations.

The isotope Pu-240 has the large resonance capture integral I"y =

8475.0 b, fission fesonance integral is small - I"f= 9.369 b [ 2,3 ]. The main

contribution to the value of IllY gives the first resonance with the average

energy E, = 1.056 eV [4], Le. at the bound of 23-rd and 24-th neutron groups

of the 28-neutron-group structure BNAB [ 1,5 ].

The given method of calculation for resonance self-shielding factors is

based on that at the small concentrations C/ , C2 , C) (where C3 > C2 > Cj )

of a researched resonance nuclide the rates of the resonance reaction

Ric, Ric, Ric ( where k - is the neutron group) is directly proportional to the 123

concentration of this nuclide R ic • . Ric .. Ric -- C3'. C 2'. C 1,3 2 I

2

Rk Rki.e C1 / R" C2 = 1 and C1 / R" C3 = 1. 2 1 3 1

With increase of concentration of the nuclide ( let us at the

concentration C4 ) the effect of cross sections self-shielding begins

display and the linear dependence is disturbed: Rk : R" < C4 : C1 ,

4 I

The value

f k = R" C / Rk c·ill I

is the required self-shielding factor for this cross section in the group k at the

concentration C i of the cosidered resonance nuclide.

Thus for Pu-240 the dependences of capture resonance self-shielding

factors on concentration Pu-240 were determined for different neutron groups.

These dependences are obtained for infmite lattice fonned from hexahedral

cells in a real and simplified geometries ( ring-shaped geometry and

homogenizated one).

The cell in the real geometry is presented in fig. 1. The zones are

separated by zirconium tubes. A heavy water solution of isotopes Pu-239 and

Pu-240 (zone 1 and 11) is the fuel. In a functioning facility it is provided

circulation of the fuel between the central tube (zone I) and thin fuel tubes

(zone 11). In the zones 3 and 5 the heavy water is pumped under the pressure

of 100 atmospheres and is a heat-carrier. The heavy water in the intercell

space ( zone 9) is a moderator functionally.

Modeling perfonned for the cell in the real geometry is compared

below with results for the ring-shaped (marked as 'rings' geometry on figures

below) and homogenizated geometry. At modification of the real geometry

into ring-shaped ( fig. 2 ) the three layers of thin fuel tubes in the real

geometry are transformed into the three cylindrical fuel layers with

3

conservation of the radius for the middle of the layer. I.e., the three values of

radiuses r = 5.0,6.3 and 7.6 em (see Fig. 1) at transition into the ring-shaped

geometry were not changed. The thickness of three fuel layers and them

zirconium jackets ( fig. 2 ) are designed from the requirement: nuclear

concentration must remain invariant. I.e., the isotope concentrations in the

zones 3, 10 and II of the ring-shaped geometry are equal to the concentrations

for the zones 3, 10 and 11 of the real geometry.

The further simplification of the design-basis geometry consists in

homogenization for the zones 1, 2, 3, 10 and II of the real geometry. The

geometry of the homogenizated cell is presented in fig. 3.

The modeling of the neutron field is executed under the reactor code

MCV verificated in calculations about 300 experimental assemblies [6]. The

height of cells is assumed equal 4 m. In calculations it was set mirror reflection

on bounds of cells and absorption at end faces. In the task speCification for

MCV the temperature of cells is determined equal 300o K.

At the first stage of work the Pu-240 isotope was excluded from the

compozition of the fuel in the real geometry and the concentration of Pu-239

was brought up to the value 6.564 E-6 ( here and further the nuclear

concentration is given in units - 1024 nuclei/em3 ) that provided ktIJ = 1.

Further the concentration of Pu-239 remained constant in the all

calculations and the concentration of Pu-240 varied only. The modeling

was performed at the next relative concentration:

X = ( concentration of Pu-240 ) I ( concentration of Pu-239) = 0.00025, 0.0025,

0.025,0.05,0.1,0.2,0.3,0.5 specified on the all figures. The first point­

x =0.00025 is used for normalization of the reaction rates and, therefore, it is

absent in the figures. Nuclear concentration of the zones for the real,

ring-shaped and homogenizated geometries are given in the tables I and 2 of

the appendix where for the value ux" appropriate concentrations of Pu-240

4

are specified.

The self-shielding factors Ie of Pu-240 in the zone II (thin fuel tubes)

are presented in fig. 4 for the groups 21+26.The self-shielding is displayed

maximally in the group 23, in a smaller degree -in the group 24 and it is

expressed very slightly in the group 22. These groups 22+24 are the groups in

which the powerful resonance (with E, = 1.056 eV) presents. In the group 21

the action of this resonance is not displayed, Le. the value Ie is practically

equal to unity. In the intervals of 2S-th and 26-th energy groups this

resonance is also absent.However, the strong neutrons absorption in the 23-rd

and 24.th groups results in significant decrease of neutrons number scattered

into 2S-th and 26-th groups. It causes decrease of absorption rate in the 2S-th

and 26-th groups. In groups where resonances are not present (group 21, 25,

26 ) it is more properly to tell about calculation of a functional Ie, instead of

a self-shielding factor. Therefore, the some figures for these groups are

denoted just as the functional Ie. Self-shielding factors (with statistical errors) for the groups 21+26 in the

real, ring-shaped and homogenizated geometry are presented in fig. 5-10. For

clearness the error bars for the ring-shaped geometry are inclined left and in

homogenizated geometry - right to the vertical. Results for the real and

ring-shaped geometry are close and the error bars are overlaped. Values of

factors in the homogenizated geometry, as a rule, more than in the real

one.The maximum distinctions are exist in the 23-rd group where at X =0.2,

0.3 and 0.5 the error intervals in the real and the homogenizated geometries

are not overlaped. Such distinctions confmn the expected physical effect:

decreasing of the self-shielding at homogenization of Pu-240.

Intervals of errors and fited curves ( by exponential curves y ==

A exp(Bx) ) of Pu-240 self-shielding factors are presented in fig. 11 for the

groups 17+21.The 21-st group values does not practically differ from unity and

5

it confirms the correctness of the fit. The maximum self-shielding is displayed

in the 19-th group. The smaller self-shielding ( with respect to decrease) - is

displayed in the groups 18, 17, 20. Such behaviour of the self-shielding factors

corresponds roughly to the group cross sections of absorption [5]: 67.65

b ( group 19), 42.29 b ( group18), 27.000 b (group 17), 30.71 b ( group 20),

0.78 b ( group 21). The given figure is presented for demonstration of the

tendencies in behaviour of Ie factors ( but not the value of Ie) in these

groups. In order to obtain the values of Ie factors the statistics must be

increased considerably.

The considered above self-shielding factors are presented for the zone

of the thin fuel tubes (zone 11). It is necessary to compare these self-shielding

values of Pu-240 with the values of the self-shielding in a more voluminous

central fuel tubes (zone I) where greater effect of the self-shielding is expected.

In a tube of a bigger diameter the large part of volume ( in comparison with

thin fuel tubes) is shielded from resonances. Really, the calculations (see

fig. 12+16) give considerably smaller Ie -factors for the central tube - zone I.

In the group 22 ( where the self-shielding is small ) and group 26 ( where the

effect from neutron absorption in the first resonance is considerably reduced )

theIe -values in the zones 1 and 11 approach.

The value of self-shielding factors are obtained at dilution cross

sections 0'0 are well large than in the Ref [ 5 ]. The dilution cross sections at X

= 1.0 in the real, ring-shaped and homogenizated geometry for groups 22+24

(Pu-239 has not resonances in these groups) are presented in table I. In

order to obtain dilution cross sections for the next relative concentration

of Pu-240 x = 0.0025, 0.025, 0.05, 0.1, 0.2, 0.3, 0.5 it is necessary the values

in the table to divide into appropriate values of "x".

6

Table 1

Neutron Group Cross section of

Dilution (bam) for Real, Ring-shaped

and Homogcnizated Geometrv (zone 1)

Cross section of Dilution (bam) for

Homogenizated Geometry (zone 3)

22 43252 86331

23 43268 86347

24 43384 86464

The results presented in figures are obtained at statistics given in the

table 2 where the concentrations of Pu-240 are specified as x~va1ue (Le. as in

the figures)

Table 2

concentration of Number of Histories for the Cell Geometries Pu-240

Real Ring-shaped Homogenizated

x=O.OO25 963000 720000 -x =0.025 1656000 744000 -x;; 0.05 1998000 876000 720000

x= 0.1 1953000 720000 720000

x=0.2 1350000 732000 720000

x=0.3 2241000 720000 720000

x:: 0.5 1512000 900000 720000

The values kfSJ (accordingto Brissenden evaluation - k8rlll ) obtained for

these geometries are presented in table 3 where errors are specified in

brackets ( i.e., for example the value 1.0031(5) means 1.003I±O.OOO5). The

closeness of values kBrla in the ring-shaped and homogenizated geometry to

the values in the real geometry confirms correctness of simplifications for this

7

lattice geometry.

Table 3

concentration of

Pu-240

Comparison of kfIJ - values for the Cell in the Real,

Ring-shaped and Homogenizated Geometry

Real Ring-shaped Homogenizated

x:: 0.0025

x.:: 0.025

x =0.05

x =0.1

x ==0.2

x=0.3

x. = 0.5

l.OO31( 5)

0.99851(36)

0.99230(33)

0.98126(33)

0.96084(42)

0.94273(29)

0.90739(37)

1.0036( 6)

0.99719(63)

0.99162(56)

0.97913(65)

0.96007(62)

0.94228(61)

0.90781(56)

--

0.99191(61)

0.98159(57)

0.96013(63)

0.94146(59)

0.90680(62)

The author is sincerely grateful to N.I.A1exeyev, L.P.Abagyan,

V.V.Veretenov, EA.GolDin, M.I.Gurevich and M.S.Yudkevich for the help at

work with the code MCU.

The author expresses thanks to P.P.Blagovolin for fruitful discussion of

considered questions.

8

1\

\ \

\ \

\

\

\

\ \

\

\

\

\

\

\

\

\

\

\

\

II 4.15 Jj II4.3 ltiJ .. I

.15~.3 7.6

8.25 8.45

9.7 10.5 --- I

-\10.9

I11.1 ---t

Fig. I. Geometry ofthe cell (dimensions in em). I • liquid fuel; 2 - central tube for liquid fuel; 3 - moderator (D20), zone ofheat rdeasing; 4 - tube separated flows ofheat transpon heavy water; 5 - cooled volume of beat transpon heavy water (zone of heat removal); 6 -load-bearing tube; 7 - heat-insulated gap: 8· calandretube; 9 - cold moderator (020); 10 - fud tube; II -liquid fuel.

9

1\ 1 \

9~ \ 1 \

1

10~31:===;===~

10 3---~ 10--=---~~~

10-----~

10 3

3

\

\

\

\

\

\

\

\

\

\

\

\

\ \

\ , \

Fig.2. Ring-shaped geometry oftbe cell (dimensions in em). I -liquid fuel; 2 - central tube for liquid fuel; 3 - moderator (02°). zone of heat releasing; 4 - tube separated flows of heat transport heavy water; 5· cooled volume of heat transport heavy water (zone of heat removal); 6 -load-bearing tube; 7 - heat-insulated gap; 8 - calandre tube; 9 - cold moderator (020); 10 - (Zr+Nb)-layer; II -liquid fuel.

10

1\ 1 \

9~ \ 1 \

\

\

\

\

\

\

\

\

1

1

1

1

8.25 .45

.7 10.5

1 . 11.1

\

\

\

\

\

\

\

\

\

Fig.3. Geometry oftbe cell (dimensions in em). I -liquid fuel; 2 - central tube for liquid fuel; 3 - rone of homogenization moderator (fuel + D

2 0 + zirconium-niobium

tubes); 4 - tube separated flows ofheat transport heavy water; 5 - cooled volume ofheat transport beavy water (zone of beat removal); 6 - load-bearing tube; 7 - beat-insulated gap; 8 - calandre tube; 9 - cold moderator (D20);

::. 1.00 D.

~ '­~ !! i ~ 0.90

~ "5 Q) c -0 ~

~ J!! 0.80

g» =a 1):c II)

~

~ 0.70

----------;l'I'

A

Self-shielding factor fc for Pu-240 for the zone 11 of the cell in the

~ ~REAL GEOMETRY.

Relative concentrations of Pu-240 <:hoaen for calculations are mar1ced al "X-values" under the curve for the group 23.

I I I ! , I , , , ! I I0.00 ' I , , I ! ,I I , • I •

0.01 0.10 (concentration of Pu-240) I (concentration of Pu-239)

Fig.4 Self-shielding factor ofneutron capture for Pu-240 depending on the relative concentration ofPu-240 for different neutron energy group. Results are given for the zone 11 of the cell in the real geometry.

12

23-rd Neutron Group

Self-shielding factor fc for Pu-240

for the zone 11 ofthe cell in the REAL GEOMETRY ( 0 );

for the zone 11 ofthe cell in the RINGS GEOMETRY (- • - ); for the zone 3 ofthe cell in the HOMOGENIZATED

GEOMETRY ( --.g.--- );

Error bars for the REAL GEOMETRY are vertical. Error bars for the KINO GEOMETRY are inclined left to the vertical. Error bars for the HOMOGENIZATIID GEOMETRY are inclined right to the vertical.

Relative concentrations ofPu-240 cholJCll for calculations J are lIHIIked 88 ·X-valuesn under the curves. .

0.70 '--_-'--------'-------L--'--'............-'-'-_--' .....L.-~.......................L__ ___'___'~__'_..........

0.00 0.01 0.10 (concentration of Pu-240) I (concentration of Pu·239)

Fig.5 Comparison ofself-shielding factors ofneutron capture for Pu-240 for three types of the cell geometries (Real, Rings. Homogenization) depending on the relative concentration ofPu-240 for the 23-rd neutron energy group.

13

24-th Neutron Group

1.00

Self-shielding factor fc for Pu-240

for the zone 11 ofthe cell in the REAL GEOMETRY ( 0 );

for the zone 11 of the cell in the RINGS GEOMETRY (- • - ); for the zone 3 ofthe .cell in the HOMOGENIZATED

GEOMETRY ( ---E!---- );

Error barB for the REAL GEOMETRY .-e vertical. Error bars for the RING GEOMETRY .-e inclined left to the vertical. Error bars for tile HOMOOENIZATBD GEOMETRY In

inclined risht to the vertical.

Relative concentndions ofPu-240 cholell for calculations are l1Ullked as "X-values" under the curves.

0.88 L.-_......L...-~--J.-....L...I--L..L....L.L.._---" ...L--l--1-...............I.-_............---J.---'-..................

0.00 0.01 0.10 (concentration of PU-240) I (concentration of Pu-239)

Fig.6 Comparison ofself-shielding factors ofneutron capture for Pu-240 for three types of the cell geometries (Real, Rings, Homogenimtion) depending on the relative concentration ofPu-240 for the 24-th neutron energy group.

14

:::s Q. o ~ ... J2 t!.a 1.00 ~

~ c e:::s -CD c '0 ... -8 J! Q C =a 0.99 :iz::. (I)

-.!­16 tn

Self-shielding factor fc for Pu-240

for the zone 11 ofthe cell in the Real Geometry (---4~­for the zone 11 of the cell in the Ring Geometry ( ­for the zone 3 ofthe cell in the

Homogenizated Geometry (--....f3---); Error bars for the Reel Geometry ere vertical. Error bars for the Ring GeometIy are inclined left to the vertical. Error bars for the Homogcnizated Geometry arc inclined right to the vertical. Relative conceotrstiODS ofPu-240 chosen for caleuJationa arc marked as "X_values" undec the curves.

0.98 L..-_...L---l---'--...L...L...L...L..~_........I-.--J.--'---'--.L.....L..L..L.L_--J_...L-...I..-J.---L..J

0.00 0.01 0.10 (concentration of Pu-240) I (concentration of Pu-239)

Fig.? Comparison of self-shielding factors ofneutron capturefor Pu-240 for three types of the cell geometries (Real, Rings, Homogenization) depending on ~e relative concentration ofPu-240 for the 21-th neutron energy group.

15

2S-'th Neutron Grogp

1.00

-0.99

:::J 0..

~ a- 0.98 S Functional fc for Pu-240 ....u for the zone 11 of the cell in the 1i REAL GEOMETRY ( 0 );r:: 0 for the zone 11 of the cell in the0.9713 c RINGS GEOMETRY (- • - );.2 for the zone 3 ofthe cell in the

HOMOGENIZATED GEOMETRY ( --8---- );0.96

Error bars for the REAL GEOMETRY are v.ncal. Error bars for the RING GEOMETRY are inclined left to th vertical. Error bars for the HOMOGENIZATED GEOMETRY are

0.95 inclined right to the vertical. . j I

Relll1i................. ofPu-240.booonll><_.... J are marked 88 "X-values" under the curves. X=O.5

0.94 0.00 0.01 0.10

(concentration of Pu-240) I (concentration of Pu-239) Fig.8 Comparison of functional f for Pu-240 for three types of the cell c

geometries (Real, Rings, Homogenization) depending on the relative concentration ofPu-240 for the 25-th neutron energy group.

1-...--_....L...-.......J....---I--L-I--L.-I.....L..l..._----..I_~..L-I__'_L....L..L.........._ __L____l._I.._1_..L...J

16

1.00

0.99

0.98

0.97

j

g. 0.96

~

S a­

__0 0.95

"iii c: 0 0.9413 c: .a

0.93

0.92

0.91

0.90

0.89 '--_-L---'--"--.......................I..I..-_..........----''--''--'-~........__.L__'___'___~

0.00 0.01 0.10 (concentration of Pu-240) I (concentration of Pu-239)

Fig.9 Comparison offunctional f for Pu-240 for three types oftbe celle geometries (Real. RiDgB, Homogenization) depending on the relative concentration ofPu-240 for the 26-th neutron energy group.

17

1.00 ~

~ .e l ­

...0

c for the zone 11 ofthe cell in the REAL GEOMETRY ( 0 ); for the zone 11 ofthe cell in the

Functional for factor f for Pu-240:

0.99 RINGS GEOMElRY (- • - ); for the zone 3 of the cell in the HOMOGENIZATED

GEOMElRY (---s---- );

Error bars for the REAL GBOMJrrIlY are vertical. Boor baus for the RING GEOMETRY are inclined left to the vertical. Error bars for 1bc HOMOOENIZAT£D GEOMIITRY are indinecl right to the vertical. Kelative COIM:ClIIInlIions ofPu-240 diosc:D for calc:ulaticm 8ftl muted • "X-values" UDder the cum:s.

0.98 I--_....&.-----'----...--'-...........~L..__ ........_'_____L___'___'_.........................L....__~____L__'___.._

0.00 0.01 0.10 (concentration of Pu-240) I (concentration of Pu-239)

Fig. IO.Comparison of functional values for self-shielding factor of neutron capture for Pu-240 for three types oftbe cell geometries (Real, Rings, Homogenization) depending on tbe relative concentration ofPu-240 for the 21-st neutron group.

::s B­~ ... .g ! .. .a 1.00 D.

8 c fit for the 17-th groupe -X

o~ ~ ! ~ q¢ i ¢ ....'0 §~ ~o :>< ~¢ ~ 0 00 ... i 0 ><0 ]' ~-8 0.95 >< >< i as 0 >< ~ SeJf-shielding factor fc for Pu-240 for the zone 11 ofthe cell in the Real Geometry. Fiting (exponential ~ method) were made by middles oferror intervals for every relative concentration ofPu-240 CI) Interval oferrors for the: 17-thneutrongroup- 0 19..funeu1rongroup- X :c 18-th neutron group - 0 2O-th neutron group - ¢ ep 21 -st neutron group - •i Relative concentrations ofPu-240 chosen for calculations are marked as "X-values".

0.90 L-__........_a.-................~........L..L..__--a._--'-.-.I.-.Ii-.L..............a.-__..L-_...................L-........

0.00 0.01 0.10 (concentration of Pu-240) I (concentration of Pu-239)

Fig.lt Comparison of self-shielding facton ofneutron capture for Pu-240 for the REAL cell geometry depending on thereIative concentration ofPu-240 for the 17, 18, 19,20 and 21-st neutronen.ergy group.

::1 1.00 n­o ~

2 ~

I) 0.90 ... :I a ~

NEUTRON GROUP 23. a0.80

Comparison of Self-shielding factor fc for Pu-240~ CI) c for the zone 1(central fuel tube) and zone 11 (thin ,...'0 fuel tubes) ofthe cell in the REAL GEOMETRY. co~ 0.70

J! Relative concentrations ofPlJ..240 chosen for calculations 0) are marked as "X-values" under the curves.c

Error bars for the Zone 1curve are incline left to the vertical=s 0.60 at X9>.OO25, 0.025, 0.05..~

.r:.

I '" I I ! ! , 1 !!:t~0.50 "'"'' ". '" 0.00 0.01 0.10

(concentration of Pu-240) I (concentration of Pu-239) Fig.12 Self-shielding factor ofneutron capture for Pu-240 depending on the relative concentration ofPu-240

for 23-rd neutron energy group. Results are given for the zones 1and 11 of the cell in the real geometry.

:;:, a. o ~

1.00

~

S ! ~ ~ 0.95

c: e ~ c: '0 ~ o

1:5 0.90 .! CJ)c: i; .m i

to..:> o

0.01 0.10 (concentration of PU-240) I (concentration of Pu-239)

Fig.13 Self-shielding factor ofneutron capture for Pu-240 depending on the relative conca1tration of Pu-240 for 24-th neutron energy group. Results are given for the·zones I and 11 ofthe cell in the real geometry.

~ I , I I! ,CD I t I , I " I0) 0.85 c !! I I , I I I I

0.00

l'

21

1 \ \

I§ g 0Vl

N CO'l .... I i 0 ?

1

\ ' \ \

\ ' \ ~

/ ,\ , \

, \1 , ,\ - '\,

'on on -

\

, >< ~ ~ ~ ~ ~ ><

I

NEUTRON GROUP 22 Comparison ofSelf-shielding factor fc for Pu-240 for the zone 1 (central fuel tube) ofthe cell ( 0 );

in the Real Geometry 0.98 for the zone 11 (thin fuel tubes )ofthe cell in (- .. - );

the Real Geometry Error bars for the Zone 8 are vertical. Error bars for the Zone 1 are inclined left to 1he vertical. Relative concentnIt:iooa ofPu-240 chosen fur calculations are marked as ·X-vaIues" under-the curves.

0.00 0.01 0.10 (concentration of Pu-240) I (concentration of PU-239)

Fig.14 Self-shielding factors ofneutron capture for Pu-240 depending on the relative concentration of Pu-240 for the 22-nd neutron group. Results are given for the zones 1 and 11 of the cell in the Real Geometry.

1.00

0.99

~ 0.98

~ ... NEUTRON GROUP 25. .e 0.97 Comparison offunctional f for Pu-240 ,••.0 c

for the zone 1 (cen1ral fuel tube) and zone 11 (thin1i ~ ~6 0.96 fuel tubes) of the cell in the REAL GEOMETRY.

~ ::J Relative concentrations ofPu-240 chosen for calculations

\Ioiio 0.95 are marked as "X-values" under the curves. Error bars for the Zone 1 curve are incline left to the vertical at X-o.OO2S, 0.025, 0.05. 0.1. 0.2 and 0.3.

0.94

0,,93 C I I I , I I It' I! I I I , I I I I I , I '-,

0.00 0.01 0.10 (concentration of Pu-240) I (concentration of Pu-239)

Fig.lS Comparison offunetional f for Pu-240 depending on the relative concentration of Pu-240c for 25-th neutron energy group. Results are given for the zones I and 11 of the cell in the real geometry.

", "

1.00

0.99

0.98

:J 0.97 Q.

~ 0.96 '­

.e 0.95

....0

16 0.94

~ 0.93 c .2 0.92

0.91

0.90

11'\

8 ~

NEUTRON GROUP 26. Comparison offunctional f for Pu-240c for the zone 1(central fuel tube) and zone 11 (thin fuel tubes) ofthe cell in the REAL GEOMETRY.

Relative concentrations ofPu-240 chosen for calculations are marked as "X-values" under the curves. Error bars for the Zone 1 curve are incline left to the vertical at X=(lOO25, 0.025.0.05.0.1,0.2 and 0.3.

0.89 r= I !, , , , , , " I I I I I I ,

0.00 0.01 0.10

t-.) ~

J I , I I ='

(concentration of Pu-240) I (concentration of Pu-239) Fig.16 Comparison offunctional f for Pu-240 depending on the relative concentration ofPu-240 for c

26-th neutron energy group. Results are given for the zones I and 11 of the cell in the real geometry.

24

APJ>eIldix

Table 1

Naclear CompositioD Or tile CeO ZoBes ror tie Rnl .Bd RiIIr;-sllaped Geometry

NUCLEAR NUCLEAR ZONE ISOTOPE CONCENTRA· ZONE ISOTOPE CONCENTRA­

TION TION I H 2.120B-4 1 0 1,(JOB-7

D S.311E-2 0 2.6645B-2

Pu-239 6.564&-6 Pu-240: see below:

X=O.OOO2S 1.641&9 X=O.OO2S 1.6418-8 X=O.025 1.641E-7 X=O.os 3.282B-7 X=O.I 6.564B-7 X=O.2 1.313:8-6 X=O.3x..o,S

1.969E-6 3.282B-6

2 Zr 4.2S0£-2 8 'b 4.250£-2 Nb 4.2S0E-4 Nb 4.250B-4

3 H 2.12OB-4 9 H 2.6S6E-4 D 5.300'8-2 D 6.640B-2 0 2.700E-2 0 3.333B-2

4 Zr 4.25OB-2 10 Zr 4.2S0B-2 Nb 4.250B-4 Nb 4.2508-4

5 H 2.120E4 11 H 2.120:&4 0 5.312&2 D S.311B-2 0 2.667£-2 0 2.666E-2

Pu-239 6.564E-6 Pu-240: see below:

X=O.OOO2S 1.641B-9 X=O.OO25 1.641£-8 X=O.02S J.641B-7 x=o.OS 3.282E-7 X=O.I 6.564B-7 X=O.2 1.313&6 X=O.3 1.969~

X-o.5 3.2828-6 6 Zr 4.250£-2

Nb 4.2.50:&4

25

Table 2

Naelear Composition of the CeU Zones for the BomogeDDted Geometry

NUCLEAR NUCLEAR ZONE ISOTOPE CONCENTRA­ ZONE ISOTOPE CONCENTRA­

TION nON I H 2.120&4 6 'h 4.250E-2

0 5.311:&.2 Nb 4.2S0E-4 0 2.666E·2

Pu-239 6.S64E-6 Pu-240: see below:

X=O.OO2S 1.641:&.8 X=O.05 3.2828-7 X=O.1 6.564:&.7 X=O.2 1.313E-6 X=O.3 1.969&-6 X-O.5 3.282B-6

2 Zr 4.2S0E-2 7 0 1.00B-7 Nb 4.250£..4

3 H 1.918:&.4 8 Zr 4.250£..2 D 4.800£-2 Nb 4.2S0E-4 0 2.427E-2 'h 4.0SlE-3 Nb 4.051B-5

Pu-239 2.979&-6 Pu-240: see below:

X=O.OO2S 7.448£-9 X=O.OS 1.490:&.7 X=O.l 2.979E-7 X=O.2 5.959£..7 X=O.3 8.938£-7 X-O.S t.490E-6

4 'b: 4.2S0B-2 9 H 2.656£-4 Nb 4.250&4 D 6.640£-2

0 3.333E-2 5 H 2.t20E-4

D 5.312£..2 0 2.667:&.2

26

REFERENCES

I.Abagyan L.P., Bazazyants N.O., Bondarenko I.I.,and Nicolaev M.N.

Group Constants for Nuclear Reactor Calculations, Consultants Bureau, New

York (1964).

2. Galanin A.D. Introduction to the theory of thermal reactors.

Moscow.: Energoatomizdat, 1984 ( in Russian ) (rWUlHHH A.J];. BBeAeHHe B

TeOpmo SI,llepHblX peaKTOpoB Ha TennOBbIX HCihpoHax. M.:

3HeproaTOMll3AaT, 1984). M.: 3HeproaTOMH3AaT, 1984.

3. Lyashuk V. I. Calculations of Neutron Field Functionals in the

Heavy Water Fuel Cell of Electronuclear Facility. Moscow, Preprint ITEP,

1996, NQ39.

4. Kon'shin V.A. Nuclear Constants of Fissile Nuclei. Moscow.:

Energoatomizdat, 1984 ( in Russian ) KOHLIUHH B.A. JlnepHo-4>H3H'1eCKHe

ICOHCTaHThI neIDIIQUXclIlI,lI;ep. M.: 3HeproaTOMH3.n;aT, 1984.

~. Abagyan L.P., Bazazyants N.O., Nikolaev M.N., Tsibulya A.M. ,

Group Constants for Nuclear Reactor and Shielding Calculations. Moscow.:

Energoatomizdat, 1981 (in Russian) ( A6arm JI.n., Da3a3l1HU H.O.,

HHKOJIaeB M.H., UH6yIDI A.M. rpynoBble KOHCTaHTbl .zvu: pacqeTa

peaKTOpOB H 3anutThI. M.: 3HeproaTOMH3.n;aT, 1981.

6. Abagyan L., Alexeyev N. t Bryzgalov V. t Glushkov A., Gomin E., ~

Gurevich M., Kalugin M., Maiorov L., Marin S., Yudkevich M. MCV

MONTE CARLO CODE FOR NUCLEAR REACTOR CALCULATIONS.

VERIFICATION. Moscow, Preprint of Russian Research Centre

"Kurchatov Institute" Institute of Nuclear Reactors, 1994, IAE-5751/5.