development and verification of batan's standard ...

DEVELOPMENT AND VERIFICATION OF BATAN’S STANDARD DIFFUSION CODE MODULES FOR TREATING TRIANGULAR MESH GEOMETRY1

Liem Peng Hong Center for Multipurpose Reactor, National Atomic Energy Agency ABSTRACT Development and Verification of Batan’s Standard Diffusion Code Modules for Treating Triangular Mesh Geometry. New code modules for treating the triangular mesh geometry have been successfully developed and implemented into the Batan’s standard codes for multi-dimensional multigroup neutron diffusion calculations, Batan2DIFF and -3DIFF. With the new code modules Batan-2DIFF and -3DIFF codes can handle the triangular and hexagonal fuel element/assembly commonly used in the blocktype high temperature gas reactors, fast breeder reactors or advanced water reactors. The solution methods used in the modules follow the ones already implemented in Batan2DIFF and -3DIFF codes, i.e. the standard source iteration method for solving the eigenvalue problem and the finite difference method for treating the spatial variables. The new modules have been verified with other generic diffusion codes and the verification results showed that the new modules produced very accurate results for the effective multiplication, forward and adjoint group neutron flux and power density distributions, neutron balances and other important neutronic parameters.

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Submitted to Atom Indonesia

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Abstrak

Pengembangan dan Verifikasi Module Program Difusi Standar Batan untuk Menangani Geometri Kisi Segitiga. Module-module baru untuk menangani geometri kisi segitiga telah berhasil dikembangkan dan diimplementasikan ke program-program standar Batan untuk perhitungan difusi neutron banyak kelompok banyak dimensi, Batan2DIFF dan –3DIFF. Dengan module-module baru tersebut, program Batan-2DIFF dan – 3DIFF dapat menyelesaikan problem reaktor dengan elemen bakar berbentuk segitiga dan heksagonal seperti yang sering dipakai pada reaktor gas temperatur tinggi tipe blok, reaktor pembiak cepat atau reaktor air maju. Metode penyelesaian yang dipakai pada module-module yang dikembangkan mengikuti metode yang telah diimplementasikan pada program Batan-2DIFF dan –3DIFF, yakni metode iterasi sumber untuk menyelesaikan problem nilai diri dan metode beda hingga untuk menangani peubah ruang. Module-module yang dikembangkan telah diverifikasi dengan program difusi neutron generik lain dan hasil verifikasi menunjukkan bahwa module-module tersebut mampu memberikan hasil yang akurat untuk faktor multiplikasi efektif, distribusi fluks neutron maju dan adjoint serta rapat daya, neraca neutron dan parameter neutronik penting lainnya.

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INTRODUCTION In order to fulfil the Agency’s needs of reactor physics, especially the computational tools for neutronics design and analysis, Batan-1DIFF, -2DIFF and -3DIFF codes have been developed and verified [1,2,3]. Batan-1DIFF, -2DIFF and -3DIFF codes are 1-D, 2-D and 3-D multigroup neutron diffusion codes which solve both the forward and adjoint neutron diffusion problems, external source neutron diffusion problems, and calculate the kinetic parameters calculations and reactivity changes based on the perturbation theory.

Batan-1DIFF diffusion code can handle 1-D planar, cylindrical and spherical reactor geometry, while Batan-2DIFF diffusion code can be used to solve problems in 2-D XY and RZ reactor geometry. However, the only reactor geometry which can be handled by Batan-3DIFF diffusion code in the present version is 3-D XYZ reactor geometry.

For common research reactors, light water reactors (PWRs and BWRs) the 2-D XY or 3-D XYZ reactor geometry is commonly adopted, therefore, the present capability of Batan-2DIFF and -3DIFF codes is adequate. However, at present the Agency is involved also in the assessment of other types of reactors for future applications such as high temperature gas-cooled reactors, fast breeder reactors, and innovative light water reactors [4]. These types of reactor use hexagonal fuel assemblies so that neutronic calculations can not be conducted easily and accurately using the existing XY or XYZ reactor geometry. Therefore, a triangular mesh geometry become an indispensable option for the above mentioned codes.

This paper describes the development, compilation, and verification results of the triangular mesh geometry options for Batan-2DIFF and -3DIFF multigroup diffusion codes. 2DBUM generic diffusion code [4] is used for verification purposes of the present work. The organisation of the paper is as follows. In the next section, the derivation of the mathematical formula used for treating the triangular mesh and hexagonal fuel assembly by combining triangular meshes using the finite difference method are proposed. Then, the implementation of the finite difference formula into Batan-2DIFF and -3DIFF codes is discussed. Verification results for a simple reactor problem is then discussed, and in the last section of the paper, the concluding remark is given.

METHODS

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Since the detailed discussion on the multigroup diffusion theory and its implementation on Batan-2DIFF and -3DIFF codes were already given in Refs. [2,3] they will not be discussed here, except some parts which are related to the present work. The multigroup neutron diffusion problem can be written as (all notations are commonly used in the standard reactor textbooks and they will not be discussed here),

 Dg (r )  g (r )   r , g (r )g (r )  G

g

g ' 1

k eff

  s, g ' g (r )g ' (r ) 

G

  g ' 1

f ,g'

(r )g ' (r )

(1)

g  1,2,  , G

Since we only concern with the spatial variable, r, then all the terms in the right hand side of Eq.(1) can be lumped up into Sg, and if the energy group index g is dropped then one can write,

 D(r )   (r )   r (r ) (r )  S (r )

(2)

Eq.(2) represents the neutron balance between production on the right side and removal on the left side. The removal of neutron is attributed to leakage (1-st term), and removal of neutron from the group (2-nd term). Eq.(2) is solved with finite difference method (FDM) in Batan-2DIFF and -3DIFF codes for XZ, RZ and XYZ mesh geometry [2,3]. In the next subsection the FDM formula for triangular mesh geometry is proposed.

Triangular Mesh Treatment in Finite Difference Method

As shown in Fig. 1, one hexagonal fuel assembly can be composed of six identical triangular meshes, so that if a code can treat the triangular mesh geometry then it can treat the hexagonal fuel assembly geometry as well. Furthermore, in the axial direction (zdirection) the mesh geometry and division are identical with the one in the XYZ geometry, so that it will not be discussed further.

Fig. 1 gives an example of 5 by 4 triangular mesh configuration in the XY plane of a reactor medium. In the figure, one hexagonal fuel assembly is characterised by FTF (flat to flat) distance. Similar to XY, RZ or XYZ mesh geometry, the mesh points where the neutron flux is calculated are located in the center of the triangular mesh while the 4

distance between two adjacent mesh points are identical for all meshes in the medium. It should be noted here that the lower and upper boundary conditions can be set to be reflective or vacuum, while for the left and right boundary conditions, only vacuum boundary condition is recommended (however, the code can still accept the reflective boundary condition).

The finite difference formula for Eq.(2) is derived as follows. Consider one control volume which consist of one triangular mesh. If Eq.(2) is integrated over one control volume then one has

  D(r )   (r )dV   

V

r

(r ) (r ) dV 

V

 S (r )dV

(3)

V

It is assumed here that within one triangular control volume the physical properties (group constants) do not change, i.e. uniform, and the group neutron flux at the mesh center represents the average neutron flux in that control volume.

Then the divergence theorem is applied to the volume integral of the leakage term and if it is assumed that the continuous flux gradient  ( r ) can be approximated by twopoint difference formula then Eq.(3) can be rewritten as















i, j i , j  i 1, j  i, j i , j  i 1, j  i , j i , j  i , j 1  i , j i , j  i , j 1   r ,i , j i , j Vi , j  S i , j Vi , j



(4)

or

 i , j i , j 1  i, j  i 1, j  i , j i , j  i, j i 1, j  i , j i , j 1  Si , j Vi , j with

5

(5)

 i, j 

i , j 

2A x x  Di , j Di 1, j 2A y y  Di , j Di , j 1

 i , j   i, j   i, j  i , j  i , j   r ,i , j V A

1 3

FTF

x  y  V 

(6)

1 4 3

1 FTF 3 FTF 2

Since all triangular control volumes have an identical dimension then A, x , y and V do not depend on the mesh index (i,j). By observation on Fig.1 it can be seen that the coefficients i, j vanish according to the following rule:

If j is odd

If j is even

and

i is odd then

i, j  0

and

i is even

then

and

i is odd then

i, j  0

and

i is even

then

i, j  0 (7)

i, j  0

For example, the area of the normal surface, A, between mesh points (3,2) and (3,3) is zero so there is no neutron current flowing between the two meshes, i.e. . Eq.(7) indicates that the finite difference formula of the Laplacian term for triangular mesh geometry become four-point difference equation, in contrast to five point difference equation for the 2-D XY geometry.

Matrix Equations

Eq.(5) can be rearranged into a matrix form:

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M  s

(8)

where the coefficient matrix M is a symmetric, positive definite, banded matrix, while  and s are group neutron flux and source vectors, respectively. Since the dimensions of matrix M are usually large for practical problems of reactor design, the matrix equation is solved with the successive over-relaxation (SOR) method. The source iteration method is applied to obtain the effective multiplication factor.

VERIFICATION RESULTS The Batan’s standard codes with the newly developed modules for treating the triangular mesh geometry maintain their original capabilities for neutron diffusion calculation, namely, solving the forward and adjoint eigenvalue problems as well as external or fixed source problems. Other important features of the codes cover the strong absorber treatment, kinetic parameter calculations, reactivity calculations based on perturbation theory and several schemes for power peaking factor determination.

The new modules for treating the triangular mesh have been extensively debugged and verified to exclude any possible error in the code. Errors mostly occur during the programming phase. The 2DBUM diffusion code [5] was used to check the new module for 2-D geometry problems, since the code has been already used for Batan2DIFF code validation [2]. However, since there were some technical difficulties on using the 3DBUM diffusion code [6] for verification of the 3-D module, semi analytical problems will be used instead. Some of the verification results are discussed below.

The reactor core-reflector configuration, mesh and material assignment for the 2D sample problem is shown in Figure 2. The reactor consists of central water reflector, core region and outer water reflector. In this case, the flat to flat (FTF) distance is 5 cm. The outer boundary conditions were vacuum boundary condition while the axial buckling is taken to be zero. The cross section set for each material used in the calculation is shown in Table 1. The four group diffusion parameters for core and reflector regions were prepared with WIMSD/4 cell calculation code [5]. The core region is assumed to be composed by MTR-type silicide fuel elements while the reflector region is light water. Under this configuration and composition a high thermal neutron flux will appear in the central water reflector as that region serves as a flux trap. In addition, a high power 7

peaking factor will also appear in the interface between the core and central reflector regions. The reactor power was taken to be 1.0 MWth/cm and one fission produces thermal energy of 200 MeV.

Forward and adjoint criticality calculations were done with Batan-2DIFF and 2DBUM codes. In the calculations the convergence criteria for effective multiplication factor and group neutron flux were 10-5. The calculation results of the two codes are summarised in Table 2.

Firstly, the relative differences for the effective multiplication factors computed by the codes are in the order 10-4 %. Since the relative differences are below the convergence criterion for multiplication factor then practically the effective multiplication factors are identical. Secondly, it can be observed from the table that the relative differences for the group neutron flux (forward and adjoint) at the center of the reactor (center of the inner reflector) are in the order of 10-3 %, i.e. they are in the same order with the convergence criterion, therefore, again the two results are practically the same.

The worst relative differences for the neutron balance over the reactor system in term of neutron production, absorption and leakage, as well as the power peaking factor are found in the order of 10-2 % and 10-3 %, respectively.

The forward and adjoint neutron flux distributions are shown in Figs. 3 and 4, while the power density distribution is shown in Fig. 5. The solid lines represent the calculation results by Batan-2DIFF code while the diamond, block, triangle and cross marks represents the calculation results by 2DBUM code. The two codes produced practically identical distributions of forward, adjoint group neutron flux and power density.

Secondly, the semi-analytical problems are used for the verification of the 3-D module takes the XY core-reflector configuration of the above sample problem (Figure 2). In the first 3-D semi-analytical problem the upper and lower boundary conditions are taken to be reflective so that the results must coinceed with the 2-D ones. This is to check wether the 3-D module FDM scheme in the XY plane, which is identical with the one in the 2-D module, is corrrectly programmed. In the axial direction, 60 meshes of 2 cm length are used. The verification results are summarized in Table 4.

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The relative differences between Batan-3DIFF and 2DBUM calculation results are quite small since in fact for the problem the two codes must produce identical answers. The practically identical forward and adjoint multiplication factors (eigenvalues) also become a strong base for the validity of the module developed. The results on the neutron leakage as the neutrons diffuse also indicated that the FDM scheme for the triangular mesh on the XY plane in the 3-D module is valid both in the inner and boundary meshes.

In the second problem the reactor is assumed to have a finite height of 60 cm, and consequently the axial leakage will no longer zero as in the first problem. The axial leakage in the 2-D diffusion calculation can be accomodated through the axial buckling concept commonly adopted in the reactor physics. In this case, the analytical solution of group neutron fluxes in axial direction follow the cosine distribution (see for example Ref. 7),

 Z ( z)  A cos( BZ z)

(9)

where the axial buckling is derived to be

   B    H  2 

2

2 Z

(10)

The physical height of the reactor is denoted by H, and the extrapolation length  is found from the transport theory

  0.71tr  0.71  3D

(11)

Finally the axial leakage calculated in the 2-D diffusion codes becomes

LZ   DBZ2  Z dV

(12)

Using the above relations the axial buckling values for core and reflector materials used in the problem are shown in Table 4. These values are used in the 2-D diffusion calculation and the calculation results are compared with the ones of Batan-3DIFF as shown in Table 5.

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From Table 5, first, it can be seen that the forward and adjoint muliplication factors calculated by Batan-3DIFF code are consistent, that is, they are very close to each other. Secondly, the approximation in the axial leakage results in slightly large relative differences in the forward and adjoint multiplication factors (around 0.2 %). This is clearly attributed to the leakage terms. Furthermore, among the leakage terms, the axial leakage terms contribute the largest relative differences, i.e. almost one order larger than the other directional leakage terms. This can be explained by to what extent the numerical and analytical flux axial distributions differ to each other. Figure 6 shows the comparison between the analytical flux axial distribution (purely cosine one) and the one calculated by Batan-3DIFF. It can be observed that the numerical flux distribution only approximates the purely cosine analytical flux profile. However, the approximation is considered to be good enough. Although in the figure only the first group of neutron flux in the core is shown, the trend is similar for other groups and for the reflector regions.

Another comment on the second problem is that the reactor height is chosen to be small to enhance the axial leakage so that any discrepancy between the analytical and numerical results will be more pronounced. In the practical reactor problems, the axial buckling for a typical light water power reactor is in the order of 10 -5, that is, two order smaller than the present case. CONCLUDING REMARKS

New modules for treating the triangular mesh geometry have been successfully developed and implemented in Batan’s standard neutron diffusion codes. With the new modules, Batan’s standard neutron diffusion codes can be applied for assessing reactors with hexagonal fuel assemblies such as high temperature gas-cooled reactors, fast breeder reactors and other innovative light water reactors.

The 2-D and 3-D modules were verified by 2DBUM code and semi-analytical problems, respectively. The code verification results showed consistency and good agreement in term of the forward and adjoint effective multiplication factors, local and region-wise group neutron flux, power peaking factor, and neutron balance which proved the validity of the codes.

ACKNOWLEDGMENT

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The author expresses his special gratitude to Ir. Tagor M. Sembiring for providing the macroscopic group constant with WIMSD/4 cell calculation code and for his advise on using the 2DBUM diffusion code, and to Mr. Asnul Sufmawan for preparing some graphs. Keen interest and constant encouragement given by Ir. Zuhair, M.Eng., Ir. T.A. Budiono, Ir. Alfahari Mardi, MSc., Ir. Iman Kuntoro who made very valuable comments on the manuscript, and by all staff members of Reactor Physics Division and Critical Assembly Installation, are highly appreciated.

REFERENCES 1. P.H. LIEM, “Introduction to Diffusion Code Programming”, IAEA Regional Training Course on Calculation and Measurement of Neutron Flux Spectrum for Research Reactors, Lecture Notes, Jakarta (1993) 2. P.H. LIEM, “Development and Verification of Batan’s Standard, Two-Dimensional Multigroup Neutron Diffusion Code (Batan-2DIFF)”, Atom Indonesia, 20, 2 (1994) 3. P.H. LIEM, “Pengembangan Program Komputer Standard Batan Difusi Neutron Banyak Kelompok 3-D (Batan-3DIFF)”, Lokakarya Komputasi dalam Sains dan Teknologi Nuklir V, PPI-Batan, Jakarta (1995) 4. W.W. LITTLE and R.W. HARDIE, “2DB User’s Manual Revision I”, BNWL-831 REV 1 (1969). See also documentation of 2DBUM (2DB Michigan University) included in the tape. 5. J.R. ASKEW, F.J. FAYERS, and P.B. KEMSHELL, “A General Description of the Lattice Code WIMS”, Journ. of the Brit. Nucl. Energy Soc. 5 (4) (1966) 6. 3DBUM documentation on tape. 7. J.R. LAMARSH, “Nuclear Reactor Theory”, Addison Wesley, page 295 (1966)

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Table 1. Macroscopic cross sections for code verification g

 f ,g

1 2 3 4

1.7123E-3* 9.2812E-4 1.2629E-2 1.7972E-1

1 2 3 4

0.0 0.0 0.0 0.0

 f , g

 s, g g 1

 s, g g 2

 a,g

Core (Silicide MTR fuel) 6.1896E-4 7.2244E-2 4.0177E-4 1.0747E-3 3.7764E-4 8.6854E-2 8.4567E-6 7.8325E-4 5.1879E-3 8.0215E-2 0.0 1.4650E-2 7.3945E-2 0.0 0.0 1.0157E-1 Inner and Outer Reflector (light water) 0.0 1.0465E-1 6.5165E-4 5.2009E-4 0.0 1.4625E-1 1.4299E-5 2.1743E-7 0.0 1.4590E-1 0.0 9.7625E-4 0.0 0.0 0.0 1.8944E-2

1  0. 7452496, 2  0. 2545513, 3  2. 006703E  4, 4  0. 0

Dg 2.4904 1.0166 0.8019 0.2781 2.3305 0.7878 0.5633 0.1531

* read as 1.7123 x 10-3

Tabel 2. Verification results of Batan-2DIFF module with 2DBUM code keff (forward) Forward flux at the center of the reactor: 1 (x 1014 cm-1 s-1) 2 (x 1014 cm-1 s-1) 3 (x 1014 cm-1 s-1) 4 (x 1015 cm-1 s-1) Production (x 1016 s-1) Absorption (x 1016 s-1) Leakage (x 1015 s-1) Max. power peaking factor keff (adjoint) Adjoint flux at the center of the reactor: 1 (x 1014 cm-1 s-1) 2 (x 1014 cm-1 s-1) 3 (x 1014 cm-1 s-1) 4 (x 1014 cm-1 s-1)

Batan-2DIFF 0.9787111

2DBUM 0.97870791

Rel. Diff. (%) 3.3 10-4

5.14187 5.15532 5.72584 2.79501 7.59522 7.06015 7.00295 2.1358

5.142296 5.155738 5.726294 2.795224 7.5957 7.0605 7.0044 2.1357

8.3 10-3 8.1 10-3 7.9 10-3 7.7 10-3 6.3 10-3 5.0 10-3 2.1 10-2 4.7 10-3

0.9787111

0.9787038

7.5 10-4

4.12715 4.55327 4.07959 3.16576

4.127413 4.553626 4.079938 3.166070

6.4 10-3 7.8 10-3 8.5 10-3 9.8 10-3

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Tabel 3. Verification results of Batan-3DIFF module for the first semi-analytical problem Batan-3DIFF 0.9787067

2DBUM 0.97870791

Rel. Diff. (%) 1.2 10-4

Production (x 1016 s-1) Absorption (x 1016 s-1) Leakage (x 1015 s-1)

7.5954 7.0603 7.0033

7.5957 7.0605 7.0044

3.9 10-3 3.0 10-3 1.6 10-2

Directional leakage: X-positive (%) X-negative (%) Y-positive (%) Y-negative (%) Z-positive (%) Z-negative (%)

1.9445 1.9450 2.6657 2.6653 0 0

1.9449 1.9444 2.6653 2.6656 n.c. n.c.

2.1 10-2 3.1 10-2 1.5 10-2 1.1 10-2 n.c n.c

Core group flux: 1 (x 1014 cm-1 s-1) 2 (x 1014 cm-1 s-1) 3 (x 1014 cm-1 s-1) 4 (x 1014 cm-1 s-1)

6.1050 6.4896 5.3898 5.6292

6.1049 6.4896 5.3901 5.6295

1.6 10-3 0.0 7.6 10-3 5.3 10-3

Outer reflector group flux: 1 (x 1013 cm-1 s-1) 2 (x 1013 cm-1 s-1) 3 (x 1014 cm-1 s-1) 4 (x 1014 cm-1 s-1)

9.6960 9.6600 1.0578 5.7720

9.6984 9.6641 1.0582 5.7730

2.4 10-2 4.2 10-2 3.8 10-2 1.7 10-2

Inner reflector group flux: 1 (x 1014 cm-1 s-1) 2 (x 1014 cm-1 s-1) 3 (x 1014 cm-1 s-1) 4 (x 1015 cm-1 s-1)

5.7618 5.8908 6.0774 2.2326

5.7615 5.8909 6.0773 2.2325

5.2 10-3 1.7 10-3 1.6 10-3 4.5 10-3

0.9787104

0.9787038

6.7 10-4

keff (forward)

keff (adjoint) n.c. : not computed

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Table 4. Axial bukling calculations results for 2-D diffusion calculation of the second semianalytical problem

Core (MTR fuel)

Reflector (water)

g

D

tr



Bz2

1

2.4904E+00

7.4712E+00

5.3046E+00

1.9796E-03

2

1.0166E+00

3.0498E+00

2.1654E+00

2.3849E-03

3

8.0190E-01

2.4057E+00

1.7080E+00

2.4541E-03

4

2.7810E-01

8.3430E-01

5.9235E-01

2.6364E-03

1

2.3305E+00

6.9915E+00

4.9640E+00

2.0184E-03

2

7.8780E-01

2.3634E+00

1.6780E+00

2.4588E-03

3

5.6330E-01

1.6899E+00

1.1998E+00

2.5348E-03

4

1.5310E-01

4.5930E-01

3.2610E-01

2.6829E-03

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Tabel 5. Verification results of Batan-3DIFF module for the second semi-analytical problem Batan-3DIFF 0.9032518

2DBUM 0.9050440

Rel. Diff. (%) 0.2

Production (x 1016 s-1) Absorption (x 1016 s-1) Leakage (x 1016 s-1)

7.5956 7.0255 1.3838

7.5956 7.0305 1.3620

0.0 7.1 10-2 1.6

Directional leakage: X-positive (%) X-negative (%) Y-positive (%) Y-negative (%) Z-positive (%) Z-negative (%)

1.9086 1.9086 2.6311 2.6311 4.5693 4.5693

1.8985 1.8979 2.6170 2.6174 4.4504 4.4504

0.5 0.6 0.5 0.5 2.7 2.7

Core group flux: 1 (x 1014 cm-1 s-1) 2 (x 1014 cm-1 s-1) 3 (x 1014 cm-1 s-1) 4 (x 1014 cm-1 s-1)

6.3882 6.7128 5.4624 5.6208

6.3213 6.6720 5.4405 5.6232

1.1 0.6 0.4 4.2 10-2

Outer reflector group flux: 1 (x 1013 cm-1 s-1) 2 (x 1013 cm-1 s-1) 3 (x 1014 cm-1 s-1) 4 (x 1014 cm-1 s-1)

9.9360 9.8160 1.0620 5.6634

9.8109 9.7457 1.0580 5.6815

1.3 0.7 0.4 0.3

Inner reflector group flux: 1 (x 1014 cm-1 s-1) 2 (x 1014 cm-1 s-1) 3 (x 1014 cm-1 s-1) 4 (x 1015 cm-1 s-1)

5.9814 6.0534 6.1614 2.2265

5.9102 6.0133 6.1376 2.2300

1.2 0.7 0.4 0.2

0.9032540

0.9050443

0.2

keff (forward)

keff (adjoint) n.c. : not computed

15

x

y

(2,1)

(4,1) (3,1)

(5,1)

(1,2)

(3,2)

(5,2)

FTF

(1,1)

(2,2)

(4,2)

(2,3)

(4,3)

(1,3)

(3,3)

(5,3)

(1,4)

(3,4)

(5,4)

(2,4)

(4,4)

Figure 1. Triangular meshes and their mesh indexing order.

16

2

4

6

8

10

12

14

16

18

20

22

24

26 28

30

32

34

36

X 1

3 2

5

7

9

11

13

15

17

19

21

23

25 27

29

31

33

35

37

1 2

2

3 4 5

1

6 7 8 9

3

10 CL 11 12 13 14 15 16 17 18 19 5 cm 20

Y

1. Core 2. Outer Reflector 3. Inner Reflector

Figure 2. Reactor sample problem for code verification.

17

Radial Group Neutron Flux Distribution 3.00E+15

Group-1

2.50E+15

Group-2

Neutron Flux (/cm^2.s)

Group-3 Group-4

2.00E+15

1.50E+15

1.00E+15

5.00E+14

0.00E+00 1.44

7.22

12.99

18.76

24.54

30.30

36.08

41.86

47.63

53.40

X (cm)

Figure 3. Forward group neutron flux distributions computed by Batan-2DIFF code (solid lines and 2DBUM code (diamond, block, triangle and cross marks).

18

Radial Adjoint Group Neutron Flux Distribution 8.00E+14 Group-1 Group-2

7.00E+14

Group-3 Group-4

Neutron FLux (/cm^2.s)

6.00E+14

5.00E+14

4.00E+14

3.00E+14

2.00E+14

1.00E+14

0.00E+00 1.44

7.22

12.99

18.76

24.54

30.31

36.08

41.86

47.63

53.40

X (cm)

Figure 4. Adjoint group neutron flux distributions computed by Batan-2DIFF code (solid lines and 2DBUM code (diamond, block, triangle and cross marks).

19

Power Density Distribution 2.50E+03

Power Density (W/cc)

2.00E+03

1.50E+03

1.00E+03

5.00E+02

0.00E+00 1.44

7.22

12.99

18.76

24.54

30.30

36.08

41.86

47.63

53.40

X (cm)

Figure 5. Power density distribution computed by Batan-2DIFF code (solid lines and 2DBUM code (diamond mark).

20

1.6 FDM Flux Analytical Flux

1.4

First group neutron flux

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

10

20

30

40

50

60

Axial direction, Z (cm)

Figure 6. Comparison between analytical neutron flux and the one calculated by Batan3DIFF (axial, first group neutron flux in the core).

21