Effective one-group coarse-mesh method for calculating three-dimensional power distribution in fast reactors

Annals of Nuclear Eneroy, Vol. 6, pp. 65 to 80. Pergamon Press. 1979. Printed in Great Britain

EFFECTIVE O N E - G R O U P COARSE-MESH M E T H O D FOR CALCULATING THREE-DIMENSIONAL POWER DISTRIBUTION IN FAST REACTORS T. TAKEDA, K. ARAI and Y. KOMANO Department of Nuclear Engineering, Osaka University, Suita, Osaka, Japan (Received 10 June 1978)

Abstract--An effective one-group coarse-mesh method for calculating three-dimensional power distribution in fast reactors has been developed. This method uses direction dependent one-group diffusion coefficients in each region so as to preserve neutron leakage rates before and after energy condensation. Furthermore, to correct coarse meshes of one point per hexagonal assembly used in the three-dimensional diffusion calculation, one-group cross-sectious are modified by applying Askew's method. Based upon this method an effective three-dimeusional diffusion code has been made. Results obtained in test cases on a prototype fast reactor indicate that this method is as accurate as six-group fine-mesh diffusion calculations using six mesh points per assembly in each radial plane and the solution of this method is a factor of 40 faster than the six-group fine-mesh calculations.

I. I N T R O D U C T I O N

cross-sections which preserve not only reaction rates but also leakage rates. To satisfy the leakage preservation, diffusion coefficients are collapsed with radial and axial flux-gradients as weighting functions. Then radial and axial diffusion coefficients are determined in each region. As for coarse-mesh corrections, Finnemann et al. (1975) and Askew et al. (1972) developed new techniques for the solution of multi-dimensional diffusion problems. These techniques, however, require an additional fine-mesh 1-D calculation or a considerable change in the calculating algorithm for the multidimensional finite difference equation. These make it troublesome to incorporate these methods into conventional diffusion codes. Therefore we have developed a modified coarse-mesh method from Askew's method, which can be easily incorporated into conventional diffusion codes such as C I T A T I O N and G A U G E made by Fowler et al. (1971) and Wagner (1968). This coarse-mesh method uses analytically obtained correction factors for the effective one-group cross-sections. Using the above effective one-group cross-sections and the modified coarse-mesh method, we develop an effective one-group coarse-mesh method for calculating 3-D power distributions in fast breeder reactors. The calculational technique for the effective onegroup cross-sections is shown in Section 2, and the modified coarse-mesh method is derived in Section 3. The calculational accuracy of the effective one-

A lot of problems associated with nuclear designs of fast breeder reactors require three-dimensional (3-D) representation o f the cores. Direct numerical solutions of the multi-group fine-mesh diffusion equation in 3-D systems, however, consume a large and often prohibitive amount of computer time. For on-line core surveillance calculations or core burnup scoping studies, it is, therefore, desirable to develop a more rapid calculational method. For this purpose we develop a one-group coarse-mesh diffusion calculation method in a 3-D core model using modified onegroup cross-sections, which has an accuracy comparable with few-group fine-mesh calculations. In conventional energy condensation, diffusion coefficients are collapsed with neutron spectra as a weighting function as shown by Takeda et al. (1973). This collapsing has a drawback in that neutron leakage rates are not preserved before and after condensation over an energy group. Kato (1976) and Takeda (1974) pointed out that the error incurred in fewgroup diffusion calculations is mainly due to an erroneous condensation of diffusion coefficients and showed that a conventional one-group diffusion calculation produces an error of about 5~o in power distributions relative to a multi-group calculation. Because of this large error, a conventional one-group diffusion calculation cannot be used in design calculations. In this paper we improve the one-group diffusion calculation method by using effective one-group A.N.E.

6/2

A

65

66

T. TAKEDA, K. ARM and Y. KOMANO

group coarse-mesh method has been tested on a sodium-cooled prototype breeder reactor by comparing the results with those obtained by six-group finemesh calculations as shown in Section 4.

usually done. Diffusion coefficients in each region are collapsed with a flux-gradient as follows: G

akJ~

g=1 ak Ji

2. CALCULATIONAL M E T H O D OF

where

EFFECTIVE O N E - G R O U P CROSS-SECTION

The purpose of this section is to introduce a calculational technique for effective one-group cross-sections for 3-D diffusion calculations. The cross-sec~ions in fuel assemblies are calculated using multi-group neutron spectra obtained in a 2-D RZ core model, and cross-sections in control rods (or sodium channels) are calculated using those obtained in I-D cylindrical cell models. In the RZ core model, all the control rods are assumed to be withdrawn. The 1-D cylindrical cells consist of a central control rod or a sodium channel (cylindricalized) and six surrounding fuel assemblies (cylindricalized). The central control rod is divided into two regions in order to take account of the heterogeneous effect (lumping effect): an inner absorption region containing B4C, clad and sodium, and an outer structure region containing sodium and guide tube. We first show a calculational method for effective one-group cross-sections in fuel assemblies. RZ multigroup calculations are performed in the 2-D core model shown in Fig. 1, and group dependent neutron fluxes $,~ are calculated in each region i. Absorption and fission cross-sections are collapsed into onegroup by using q~,¢ as a weighting function, as is

akJ~ is an average of g-group flux-gradients in region i, and is calculated by

a4]g = akgi

~,%~.k - 4't~.~ di '

where ~+½.k and ~_½.~ represent neutron fluxes at two boundaries of the region i along a line orienting radial direction (k = r) or axial direction (k = z), and dl is the length between the two boundaries. Thus two kinds of diffusion coefficients (radial and axial) are calculated in each region. Use of equation (1) preserves neutron leakage rates in the radial and axial directions. The introduction of the direction dependent diffusion coefficients improves the calculational accuracy of the one-group theory, and makes it applicable to core design calculations. These diffusion coefficients are prepared, for example, for individual regions shown in Fig. 2. Near interfaces of each zone (inner core, outer core, radial blanket and axial blanket), relatively more cross-section sets are used than around the central region of

Plane number

i

A x i a l shield

2 3

Axial blanket

Corner blanket

4 Radial shield

5

6

Inner core

Outer core

Radial blanket

7 ._. 8 1 , 2 ~ 3 .: 4 .: 5 .: 6 , 7 Core center

8 :. 9

(2)

I0 Ill !12

13

-- R o w n u m b e r

Fig. 1. RZ core model ~ r a prototype ~st breeder reactor.

Effective one-group coarse-mesh method

41

67

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41

41

41

41

41

41

40

,m

•

2

21

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21 21

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21

36

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40

3

20 .o_

20

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31

40

14

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I0

ii

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1

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, i,

i,

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, 5

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1

2

7

8

9

i0

II

12

13

5

~6

3

1

il

,2

4

5

6

Row

number

Fig. 2. Example of identification of cross-section sets for each region.

1

2

3

i

4

i

5

!

Inner

6

I

7

,

8

1

9

I0

II

!

core

12

l

Outer

I

Radial

core

blanket

2.0

I o ~q o =1.5

~ u m ~

D r on

plane

8

....

D z on plane

8

.....

D r on plane

5

=.-..

1.0

........... D z o n

J 1

i 2

t 3

I 4

plane

I 5

5

I 6

7

8

9

I

i0

I

I

II

12

Row

number

Fig. 3. Radial distribution of one-group diffusion coefficients D~, and D~= in planes 5 and 8.

68

T. TAKEDA, K. A R M and Y. KOMANO

the inner core. This is because one-group diffusion coefficients calculated using equation (1) show a larger change near the interfaces. This change is seen in Fig. 3 which shows the radial distribution of onegroup diffusion coefficients D~, and Di, in planes 5 and 8 (see Fig. 2). One-group absorption and production cross-sections for fuel assemblies were obtained from the RZ model. However, in the case where control rods are inserted, the actual neutron spectra are harder than those obtained by the RZ calculation. Thus if we use absorption and production cross-sections obtained from the RZ calculation, the absorption rate in fuel assemblies is overestimated by about 1-2% in the case where control rods are all inserted, and this leads to an overestimation of the control rod worth by about 20-40% in the core considered here. Therefore we must take into account the spectral hardening for the rod-in pattern in order to reasonably calculate the control rod worth. We modify absorption and fission cross-sections of fuel assemblies just adjacent to control rods by multiplying them by the ratio of cross-sections of fuel assemblies in the rod-in and rodout patterns. For the core treated here, the ratios for absorption and fission cross-sections are 0.95 and 0.99, respectively. The effective cross-sections of control rods (or sodium channels) are obtained in the conventional energy collapsing. The use of equation (1) in the 1-D cylindrical cell leads to a negative or unreasonable diffusion coefficient because the neutron current changes sign frequently in the lower energy group as shown by Alcouffe (1971). Therefore we use neutron spectra obtained by the cell calculations when collapsing diffusion coefficients as well as absorption and fission cross-sections. In the calculation of power distribution in fast breeder reactors the effect of heating by gamma rays should be taken into account especially in blanket regions where about half of the power density is caused by gamma heating. We take this effect into account by using heating cross-sections by gamma rays (see Appendix).

3. M O D I F I E D COARSE-MESH M E T H O D BASED ON ASKEW'S PROCEDURE

First, let us briefly review Askew's coarse-mesh method (1972). Figure 4 represents a 3-D hexagonal-z geometry in which points Bi are centers of the assemblies i (i = 0-7) and h, and h= are the mesh intervals along radial and axial directions. Askew added

Fig. 4. Mesh points for calculating neutron balances in three-dimensional geometry.

supplementary mesh points Ai and A~ by dividing the mesh intervals, h, and h,, between the two centers into three equal parts. Neutron fluxes at points B,, Ai and A'i are denoted by ~bBi, ~b,, and ~bA,i, respectively. In the conventional coarse-mesh calculation in which mesh points are B~, the net neutron current through an interface S1 between the assemblies 0 and 1 is given by

""

{

2DoDIS1 \(

:

+

+ " - +.o),

(3)

where D, is the diffusion coefficient of assembly i. Equation (3) yields a relatively large error because of the large mesh interval h,. To reduce this error, Askew used the fluxes at the supplementary points Ai and A~ in calculating the net neutron current. This procedure leads to the following net current through the interface $1:

{

6DoD1S1 "~

Js = - t h , ~

° + ~ - j ) ( t ~ A , - q~A',).

(4)

Effective one-group coarse-mesh method To represent JI in terms of dpBo and dpsx, the neutron balances in regions R0 and Rt (see Fig. 4) are used. After algebraic calculation, it follows that (

2D~,D*,St

J [ = -- \ h,(-~o, + ~ , ) ] (dp* , - dp$o.,),

69

(9) into the form ~ I ,=,

2D* D* Si ']

. dp'o)

(5) ,= 7

where

h,(O* + O~)] (dp*,- dp*o) 1

D~, = Di(1 - r~ 2 "k 2, m 0 2,~ •

~ ",vi J

8 2 = \Keff

(7)

,8,

--

and Y~,~and vEf~ are the absorption and production cross-sections, respectively, in region i, and Kar is the neutron multiplication factor. Use of equation (5) and similar expressions for leakage rates through upper and lower boundaries of assembly 0 leads to the following neutron balance equation in the assembly:

i=l

hr(D~r+ D*,)

,=7 x

1

1 - - "fft 1.202 u rPi

1 - - ~1/..202/ Jur PO/

h,(~ + ~o,)]

dp., --

dp,o

1L 2 0 2 ~juzp i

• • + ZoodpBo * * v = - KenvZY°dpB°V'

(6)

-~ff-2o21 1 -- rmzpo/ 1

+ E~°dpn°V= K ~ vZr°dPn°V

(9)

where D* = Di(l -- y?n~pO,. 2 hZa2~

(10)

The first and second terms represent neutron leakage rates through six side boundaries and two upper and lower boundaries, and V is the volume of each node. Askew used equation (9) to solve for the neutron flux. Equation (9), however, has a different form from the conventional finite difference equation. Therefore, a change in the calculating algorithm for the finite difference equation is necessary to incorporate Askew's method into conventional diffusion codes. Thus, we develop a modified coarse-mesh method which can be easily incorporated into conventional diffusion codes. Assuming that the mesh intervals h, and h, are equal to their average, h, in calculating correction factors 1 - - ~1r 1~ r.p2 o 2i , 1 - - ~ 2u L r 2 p 02 i , 1 - 1 Ln~ 2o2 2 W, and 1 - ~ h,2 #~, we can rewrite equation

(ll)

where dp$, = dp.,/(l - ~ h 2 # 2 )

(12)

D * = D,(1 -

~rh2#2)

(13)

Y ~ = Yo,(1 -

~h2#2)

(14)

vZii(1 - ~h2#2)

(15)

vY,}i =

Equation (11) is very similar to the expression of the conventional coarse-mesh finite difference equation. In this expression, however, D~, X~i, vE)~ and dp~ are substituted for Di, Za~, vXsl and dp~, respectively. Therefore, if we use the modified diffusion coefficient and the modified macroscopic cross-sectious defined by equations (13)-(15) in the conventional coarsemesh finite difference equation, the error due to the coarse-mesh can be reduced. Here it should be noticed that these modified crosssections depend on Kaf (see equation (8)). Therefore, we have to recalculate these cross-sections for each outer iteration in the finite difference equation. Figure 5 shows the calculational flow of the modified coarse-mesh method. Neutron cross-sections and the geometry of the system are given as input data. At first, the usual inner and outer iterations are repeated until the neutron multiplication factor and the neutron flux are converged. Then the modified macroscopic cross-sections Z* and the diffusion coefficient D* are calculated by equations (13)-(15) in terms of Kaf determined by the usual iteration. By using these cross-sections, the solution dp* of the finite difference equation is obtained through inner iteration. The inner and outer iterations are repeated until Kerr and dp* are converged. The routine used to calculate the modified cross-sections takes as much computing time as that of an inner iteration. So the present calculational flow which uses the coarse-mesh correction routine after the usual convergence is achieved, saves computing time compared with the case where the coarse-mesh correction routine is used for every outer iteration.

70

T. TAKEDA,K. ARAI and Y. KOMANO

/(

Input

cross section

/

geometry )

I l Calculation of flux by inner iteration

I

Calculation of modified [ section I

macroscopic cross

I

Calculation of flux ~* by inner iteration

Calculation of Keff J by neutron balance

No

++_

I

Calculation of Keff by neutron balance

~

No

Power d i s t r i b u t i o n /

Fig. 5. Calculational flow of the modified coarse-mesh method.

4. RESULTSOFEFFECTIVEONE-GROUP COARSE-MESH CALCULATION This section describes results of test calculations for the effective one-group coarse-mesh calculation on a 300 MWe prototype fast breeder reactor. The crosssectional view of the core is shown in Fig. 6. The results obtained by the present method were compared with those by reference six-group fine-mesh calculations, six-group coarse-mesh calculations and conventional one-group coarse-mesh calculations. In the conventional one-group coarse-mesh calculations, flux-weighted one-group diffusion coefficients were used. All the calculations were performed for the three control rod patterns (A, B, C); all the control rods are withdrawn for pattern A (Fig. 6), six control rods installed at the 4th row (the row number is enumerated as shown in Fig. 6) are fully inserted for rod pattern B, and twelve control rods installed at the 6th and 7th rows are fully inserted for pattern C.

Table 1 shows the atomic number densities of individual nuclei for each region. At the interfaces between blanket and reflector regions, we use an energy independent extrapolated boundary constant (logarithmic derivative of flux) ~ of 0.08. Table 2 shows values of Keff and the control rod worth obtained by the four methods: the six-group fine-mesh calculations, the effective one-group coarsemesh calculations, the six-group coarse-mesh calculations and the conventional one-group coarse-mesh calculations: Errors in Kdf and the control rod worth for the effective one-group coarse-mesh calculations relative to the reference calculations are within 0.3%AK/K and 5%, respectively. The six-group coarse-mesh calculations overestimate Kerr by about 1-2% and underestimate control rod worth by about 12%. Thus a remarkable improvement is attained by the use of the modified coarse-mesh calculation for the control rod worth. This is due to coarse-mesh

Effective one-group coarse-mesh method I C ' Inner Core

Table 1. Atomic number densities used in 3-D calculations ( x 102* atoms/cm 3)

OC, Outer Cole RB, RodioL I:lonke! $ : Sofety Rod C ' Shim Rod F

'

Fine Red

71

Nuclide

•

nl~Wu ~

239pu 240pU 24]pu z42pu 235 U 238 U

O Na Cr Ni Mo Fe Mn

Inner core (x 10-')

Outer core (x 10 -4 )

10.09 14.21 2,680 3.775 0.6673 0.9398 . . . 0.1125 0.1018 55.42 50.17 134.5 134.5 90.84 90.84 36.45 36.45 22.79 22.79 2.906 2.906 132.8 132.8 4.058 4.058

Radial blanket (x 10-'*)

Axial blanket (x 10-4)

----

----

0.2067 101.8 199.0 75.86 29.61 18.52 2.361 107.9 3.397

0.1517 74.72 146.0 90.84 36.45 22.79 2.906 132.8 4.058

.

mesh calculations produce large errors in Keff and control rod worth, while the effective one-group coarse-mesh calculations produce allowable errors compared with the six-group fine-mesh calculations. Tables 4 - 6 show values of regional power fractions for rod patterns A, B and C, respectively, obtained by the reference calculations, the present calculations, the six-group coarse-mesh calculations and the c o n ventional one-group coarse-mesh calculations. The m a x i m u m error in power fractions in the inner and outer cores for the present calculations relative to the reference calculations is within 1.0% for all the rod patterns, while those for the six-group coarse-mesh calculations and the conventional one-group coarsemesh calculations are a b o u t 2% and 3%, respectively.

Fig. 6. Cross-sectional view of the core used for test calculations. corrections of a b o u t 10°/0 for cross-sections of the control rod used in the modified coarse-mesh method as s h o w n in Table 3. The conventional one-group coarse-mesh calculations overestimate Kaf by a b o u t 2-3% and overestimate the control rod worth by a b o u t 20%. Thus conventional one-group coarse-

Table 2. Comparison of Km and control rod worth obtained by six-group fine-mesh calculation, effective one-group coarse-mesh calculation, six-group coarse-mesh calculation and conventional one-group coarse-mesh calculation Control rod worth (%AK)

Kaf Method Six-group finemesh calculation Effective one-group coarse-mesh calculation Six-group coarsemesh calculation Conventional one-group coarse-mesh calculation

Pattern A (rod-out)

B

C

(rod-in)

(rod-in)

A-B

A-C

1.0409

0.9941

0.9685

4.68

7.23

1.0436 (0.3)t

0.9945 (0.0)

0.9681 (0.0)

4.91 (4.9)

7.55 (4.4)

1.0484 (0.7)

1.0074 (1.3)

0.9847 (1.7)

4.10 ( - 12.3)

6.37 ( - 11.9)

1.0742 (3.2)

1.0195 (2.6)

0.9841 (1.6)

5.48 (17.1)

9.02 (24.8)

t Percent difference from the six-group fine-mesh calculation.

T. TAKEDA, K. ARAI and Y. KOMANO

72

Table 3. Coarse-mesh corrections in each region used in the effective coarse-mesh method Rod-in Regions

Rod-out (B pattern)

(1 - ,~rfl2h2)

(1 - ~f12h2)

0.993 0.981 1.031 1.077 1.001

0.989 0.971 1,047 1,115 1,001

Inner core Outer core Blanket Control rod Sodium channel

~.fflZh2)

(1 -

(1 - ~f12h2)

0.997 0.986 1.032 -1.001

0.995 0.980 1.049 -1.001

h = lattice pitch between assemblies (11.56 cm). The factor (1 - ,z~rl/2h2) is a correction to the diffusion coefficient, and the factor (1 - ~]~2hZ) is a correction to the absorption and production cross-sections.

Table 4. Regional percent power for pattern A

Method Six-group finemesh calculation Effective one-group coarse-mesh calculation Six-group coarsemesh calculation Conventional one-group coarse-mesh calculation

Inner core

Percent power Outer Radial core blanket

54.50

39.53

3.66

2.32

54.50 (0.0)1"

39.65 (0.3)

3.63 ( - 0.8)

2.28 ( - 1.7)

54.12 (-0.7)

40.22 (1.7)

3.48 (-4.9)

2.18 (-6.0)

53.84 ( - 1.2)

40.65 (2.8)

3.50 ( - 4.5)

2.07 ( - 10,6)

Axial blanket

t Percent difference from the six-group fine-mesh calculation.

Table 5. Regional percent power for pattern B

Method Six-group finemesh calculation Effective one-group coarse-mesh calculation Six-group coarsemesh calculation Conventional one-group coarse-mesh calculation

Inner core

Percent power Outer Radial core blanket

49.78

43.84

4.22

2.17

50.08 (0.6)*

43.79 ( - 0.1 )

4.08 ( - 3.3)

2.12 ( - 2.3)

49.90 (0.2)

44.13 (0.7)

3.94 ( - 6.8)

2.04 ( - 5.8)

48.92 ( - 1.7)

45.28 (3.3)

3.96 (-6.2)

1.91 ( - 12.2)

Percent difference from the six-group fine-mesh calculation.

Axial blanket

Effective one-group coarse-mesh method

73

Table 6. Regional percent power for pattern C

Method Six-group finemesh calculation Effective one-group coarse-mesh calculation Six-group coarsemesh calculation Conventional one-group coarse-mesh calculation

Inner core

Percent power Outer Radial core blanket

54.06

39.82

3.94

2.18

54.16 (0.2)t

40.04 (0.6)

3.78 ( - 4.1)

2.12 ( - 2.8)

53.81 ( - 0.5)

40.46 (1.6)

3.69 ( - 6.3)

2.05 ( - 6.0)

53.14 ( - 1.7)

41.10 (3.2)

3.64 ( - 7.7)

1.90 ( - 12.8)

Axial blanket

t Percent difference from the six-group fine-mesh calculation. The maximum error in power fractions in blanket regions for the present calculations is 4.1~o, while those for the six-group coarse-mesh calculations and the conventional one-group coarse-mesh calculations are about 6~o and 12~o, respectively. Thus the errors in regional power fractions for the present calculations are smaller than those for the six-group coarsemesh calculations. Let us compare power distributions obtained by the four calculation methods for rod patterns A, B and C. Figures 7-9 show percent errors in power densities on a central radial plane for the present calculation, the six-group coarse-mesh calculation and the conventional one-group coarse-mesh calculation, respectively, relative to the reference calculation for rod patterns A, B and C. The maximum errors for the present calculation are about 2Vo in the core region and 7~o in the blanket region except corner assemblies. Maximum errors for the six-group coarsemesh calculation in the core and blanket regions are

Table 7. Comparison of computing time for each 3-D calculation for rod pattern A Method

CPU time (sec)t

Six-group fine-mesh calculation Effective one-group coarse-mesh calculation Six-group coarse-mesh calculation Conventional one-group coarse-mesh calculation

5328 142 944 122

t CPU time required in a diffusion calculation on NEAC 2200/700 with the convergence criteria AK/K < 5 x 10 -4, A4V4' _< 5 x 10 -3. A.N.I!. 6/2

I~

about 3~o and 99/0, respectively. Thus, on the plane the present calculation produces calculational accuracy comparable with the six-group coarse-mesh calculation. The conventional one-group coarse-mesh calculation produces a maximum error of about 6~o in the core region and an error of about 10-20Vo in the blanket region. Figures 10-12 show percent errors in power densities on a radial plane through the axial blanket (plane number 4 in Fig. 2) for the three calculation methods for rod patterns A, B and C. The maximum error for the present calculation is about 7~o except for the corner blanket, while the six-group coarsemesh calculation and the conventional one-group coarse-mesh calculation produce maximum errors of about 10~ and 14~o, respectively. Thus in the axial blanket the accuracy of power density obtained by the present calculation is also better than that obtained by the six-group coarse-mesh calculation. Table 7 shows computing times required by the reference calculation, the present calculation, the sixgroup coarse-mesh calculation and the conventional one-group coarse-mesh calculation. The present calculation requires only about one fortieth of the computing time required by the reference calculation. The increase in computing time compared with the conventional one-group coarse-mesh calculation is only 40~o. 5. C O N C L U S I O N S

An effective one-group coarse-mesh method has been developed for calculating three-dimensional power distributions of fast breeder reactors. Numerical results show that this method yields calculational errors of only about 2~o in core regions

74

T. TAKEDA, K. ARAI and Y. KOMANO

A

©

0

Sodium channel

Control rod / / / ~ channel

co

Core center

A: Effective one-group coarse mesh calculation B: Six-group coarse mesh calculation C: Conventional one-group coarse mesh calculation Fig. 7. Percent errors in assembly power densities on a central radial plane obtained by each calculation method for the pattern A.

Effective one-group coarse-mesh method

© 0

75

/ Sodium

A

X

channel

C o n t r o l rod channel

Core center

A:

Effective one-group coarse mesh calculation

B:

Six-group coarse mesh calculation

C: C o n v e n t i o n a l coarse mesh

one-group calculation

Fig. 8. Percent errors in assembly power densities on a central radial plane obtained by each calculation method for the patlern B.

76

T. TAKEDA, K. ARAI and Y. KOMANO

A

©

0

Sodium

channel

Control rod channel

Core center

A:

Effective one-group coarse mesh calculation

B:

Six-group coarse mesh calculation

C: C o n v e n t i o n a l coarse mesh

one-group calculation

Fig. 9. Percent errors in assembly power densities on a central radial plane obtained by each calculation method for the pattern C.

Effective one-group coarse-mesh method

©

S o d i u m channel

Control rod channel

A: Effective o n e - g r o u p coarse m e s h calculation B: Six-group coarse m e s h calculation C: Conventional o n e - g r o u p coarse m e s h calculation

Fig. 10. Percent errors in assembly power densities on a plane 4 obtained by each calculation method for the pattern A.

77

78

T. TAKEDA, K. ARM and Y. KOMANO

©

S o d i u m channel

Control rod channel

A: Effective one-group coarse mesh calculation B: Six-group coarse mesh calculation C: Conventional one-group coarse mesh calculation

Fig. 11. Percent errors in assembly power densities on a plane 4 obtained by each calculation method for the pattern B.

Effective one-group coarse-mesh method

©

Sodium channel

Control rod channel

A: Effective o n e - g r o u p coarse m e s h calculation B: Six-group coarse m e s h calculation C: Conventional o n e - g r o u p coarse mesh c a l c u l a t i o n

Fig. 12. Percent errors in assembly power densities on a plane 4 obtained by each calculation method for the pattern C.

79

80

T. TAKEDA, K. ARAI and Y. KOMANO

and of 7% in blanket regions for power distributions relative to the few-group (six-group) fine-mesh calculation; while six-group coarse-mesh calculations produce errors of about 3% in core regions and of about 9% in blanket regions. Thus the effective one-group coarse-mesh method produces better results than the few-group coarse-mesh calculations. Conventional one-group coarse-mesh calculations produce errors of about 2-3% in Keff and cannot be used as design calculations. The calculation by the effective one-group coarsemesh method is faster than the six-group fine-mesh calculation by a factor of forty, and the increase in computing time relative to the conventional onegroup coarse-mesh calculation is only 40%/0. Thus, the effective one-group coarse-mesh method provides a capability for carrying out rapid and accurate diffusion calculations which can be utilized in on-line core surveillance calculations and core burnup scoping studies. Acknowledgements--Part of this work was done by the first author when he was in Atomic Energy Research Laboratory, Hitachi Ltd. Thanks are due in particular to K. Azekura for his kind help. REFERENCES

Alcouffe R. E. (1971) Nucl. Sci. Engng 43, 173. Askew J. R., Anderson D. W. and Pearson K. G. (1972) CONF-720901, Book 1. F i n n e m a n n H. and Wagner M. R. (1975) International Meeting of Specialists on Methods of Neutron Transport Theory in Reactor Calculations, Bologna, Italy. Fowler T. B., Vondy D. R. and C u n n i n g h a m G. W. (1971)

ORNL-TM-2496. Kato Y., Takeda T. and Takeda S. (1976) Nucl. Sci. Engng 61, 127. Takeda S., Takeda R. and Kobayashi T. (1973) Three dimensional b u r n u p calculation in fast reactors, Inter-

national Symposium on Physics of Fast Reactors, B 11, Tokyo. Takeda T. and Takeda S. (1974) J. nucl. Sci. TechnoL I I , 356. Wagner M. R. (1968) GA-8307, General Atomic. APPENDIX

We take into account the heating by g a m m a rays in a simple manner by assuming that g a m m a rays produced by fission, capture and inelastic scattering yield heat at the point where they are produced (neglecting the transport effect of the g a m m a ray). Using this assumption, the heating cross-section of a nuclide i by g a m m a rays is collapsed into one-group such that G

G'

d?°C ~ {af, Af~," + a~,A~,' + a°i°',A~i}E"~ g=l

o'=l

{T h . i - -

G

g

!

where C is a constant given by C = 1.602 x 1 0 - 1 3 W . s e c / MeV, a}~ = microscopic fission cross-section of nuclide i for neutron with energy group g, a~ = microscopic capture cross-section of nuclide i for neutron with energy group g, a,% = microscopic inelastic cross-section of nuclide i for neutron with energy group g, Ag,~' = number of g a m m a rays with energy group g' produced by fission (n = f), capture (n = c) and inelastic scattering (n = in), E 9' = average energy of g a m m a rays within group g'. In obtaining power distribution we add the power produced by g a m m a rays to that by neutron fissions as follows

P(r) = ~ {ktTfi + ffhilNi(r)dp(r), i-!

where N~ = number density of nuclide i at r, k = energy released by fission less g a m m a ray's energy (about 180 MeV = 2.896 x 10- ~1 W. sec).