- Email: [email protected]
Use of the surface harmonics method forcalculation of 2D C5G7 MOX benchmark
Progress in Nuclear EnergN, Vol. 45, No. 2-4, pp. 133-142, 2004 Availableonline at www.sciencedirect.com SClmNCE
~)DIR~Cr.
ELSEVIER www.elseviencom/locate/pnucene
© 2004 Elsevier Ltd. All rights reserved Printed in Great Britain 0149-1970/$ - see front matter
doi: 10.1016/j.pnucene.2004.09.013
USE OF THE S U R F A C E H A R M O N I C S CALCULATION
METHOD FOR
OF 2D C5G7 MOX BENCHMARK
VICTOR F. BOYARINOV
Russian Research Center "Kurchatov Institute", Nuclear Reactor Institute 123182, Moscow, RUSSIA E-mail: [email protected]
ABSTRACT The C5G7 MOX benchmark specifying a sixteen-assembly core with a surrounding water reflector was proposed as a basis to measure current transport code abilities in the treatment of reactor core problems without spatial homogenization. Seven-group cross sections for all materials were used as initial information. Just that fact allows to test an accuracy of solving the neutron transport equation excluding additional errors connected with preparing the group cross sections. In this paper, Surface Harmonics Method (SHM) is applied to calculation of the two-dimensional configuration of this benchmark. Different approximations of SHM were applied, both with and without spatial homogenization. Additionally, this fact allowed evaluating the effect of spatial homogenization of cells. Comparisons were carried out for keel and pin powers both with the reference results and between the results calculated by different SHM approximations.. © 2004 Elsevier Ltd All rights reserved KEYWORDS Transport benchmark, C5G7, MOX, spatial homogenization, Surface Harmonics Method
1. INTRODUCTION The NEA Expert Group on 3-D Radiation Transport Benchmarks proposed the seven-group form of the C5 MOX problem [1] to test the ability of current transport codes to compute reactor core problems without spatial homogenization. Seven-group cross sections for UO2, the three enrichments of MOX, the guide tubes, the fission chamber and the moderator were used as initial information. Just that fact allows to test an accuracy of solving neutron transport equation excluding additional errors connected with preparing the group cross sections. The Expert Group proposed to use the results calculated by Dr. M. Smith (ANL, USA) as reference ones.
V. E Boyarinov
134
In this paper, Surface Harmonics Method [2] is applied to calculate the two-dimensional configuration of this benchmark. SHM is the method for solving the neutron transport equation in a reactor core. A characteristic feature of SHM is that, in general case, it does not use spatial homogenization. In case if the homogeneous cross sections are used as initial information, SHM actually works as a nodal method and solves diffusion equation. In SHM, neutron distribution in a reactor core is presented as a superposition of trial functions. Each trial function is taken as a solution of the neutron transport equation in the internal cell area with certain boundary conditions on the cell surface. Different trial functions differ one from another by the boundary conditions. Since the actual number of the trial functions is limited, at substituting the quest solution in the neutron transport equation, a residual appears. The minimization procedure for this residual gives the finite-difference equations of the SHM. The main limitation of SHM is connected with the restricted number of used trial functions. For computation of trial functions, WlMS-SH [3] system of codes is used. WlMS-SH is the only code containing computational modules for solving the group neutron transport equation in heterogeneous reactor cells with required boundary conditions. Now WIMS-SH calculates only three first spatial trial functions for heterogeneous cells. For the calculation of higher trial functions, the spatial homogenization and diffusion approximation are used and SUHAM-2D [4] code is applied for solving the group diffusion equation. Thus, the most accurate currently available approximation of SHM without spatial homogenization is '3f(het)'. The notation '3f(het)' means that three trial functions are used. In so doing the group neutron transport equation is solved in heterogeneous cells for calculation of these trial functions. Other approximations of SHM are forced to use spatial homogenization and diffusion approximation for calculation of separate trial functions. Actual calculations in the present work were carried out by different approximations of SHM. The most accurate calculation was carried out with use of eight trial functions for each cell, and in so doing the first three trial functions were calculated for heterogeneous cells and the next five - for homogeneous cells (symbolic name '3f(het)+5f(hom)'). Comparisons of calculational results were carried out for kaf value and pin powers both between different approximations of SHM and with the reference solution obtained by Monte-Carlo method.
2. BRIEF DESCRIPTION OF SURFACE HARMONICS METHOD In SHM, the group function of neutron flux distribution in a reactor core is presented as a superposition of group trial matrices I
N
~N (w) = Z Z
¢}", (w)[~.
(1)
i=l n=O
Here ~p}")(w) are the trial matrices describing the neutron field in i-th cell, vectors ~, are the unknown group amplitudes, n - trial matrix number, (N+I) - total number of trial matrices for each cell, w = {7, (2}. Each vector ~2)(w) of each trial matrix qg}n)(w) is taken as a solution of the group neutron transport equation in the internal cell area with certain boundary condition on the cell surface. For calculation of the vector q?}g)(W) (g=l,2,...,G) the following boundary condition is used: inflowing current for group 'g' is distributed on the external boundary as function of coordinates Wk(~) and for each another group this current equals zero for all points of external boundary. Functions W k (~)are presented as follows
135
Su~ace harmonics method
"cos(laj)'~
(2)
Wk(~) = PP (PJ)" .sin(laJ) J ' where Pj = ( - 1 ) J - 1 2 [ ( - 2 ( 2 j - 1 ) ] , - 1 ~ P j
Here Pp(pj) - Legendre polynomials; M - number of lateral sides of a cell; a - length of the lateral side of a cell; j - lateral side number of a cell; ctj - angle between the normal built from the center of a cell to j-th lateral side and X axis. The lateral sides of a cell are numbered against the clock hand; the first lateral side of a cell is the right side (Fig. 1). normal on 2-nd side [
i=2
B
C
i=3
i=1
X
A
D
i=4
Fig. 1. Cell with square external boundary. For a cell with square external boundary, the first eight functions Wk (?s) have the following forms Wo(~.)=l,
t
Wl(r~)=cos(aj),
W3(~. ) = cos(Za~),
W2(K)=sin(aj)
W4(~) = PI(Pi),
W6(~. ) = Pl(pj)sin(ctj),
Ws(~)= P l ( p j ) c o s ( a j ) ,
(4)
WT(~) = P l ( p i ) c o s ( Z a j )
These functions W~ (Fs) are presented on Fig. 2. Inflowing currents corresponding to these first eight functions Wk (i~) are presented on Fig. 3. Since the actual number of the trial matrices is limited, at substituting the quest solution in the group neutron transport equation, a residual appears. The minimization procedure for this residual gives the finite-difference equations of the SHM. Mathematically this procedure for obtaining the finite-difference equations corresponds to the coupling of the angular moments of neutron distributions presented in the form (1) on the adjacent sides of neighboring cells. The number and form of the finite-difference equations depend on the number of used trial matrices. More details about this procedure could be found e.g. in [2, 3]. The finite-difference equations of SHM for the square lattice with eight trial functions have the following form.
136
g E Boyarinov
IB
Ic
IB
C
I
W0 (rs)
A
Wl(rs)
D
A
~
W2 (rs)
I
I
]
: /i",. '.oj io\
i IA
C
' /x. oAj IB\
~-
IA
I
ilA I'B Ic
ilA
ID
A
-x/
W3 (r s )
W4 (r s )
'
olA . /
;S
D
-1
:A
W5 (rs )
~ '
~
C
~
A
W6 (rs )
W7 (rs )
,("
!
/
!
:
Fig. 2. The first eight functions Wk ( ~ ) .
137
Surface harmonics method
r
o
(0) ~ _ _
~
V
q~(1) ~ , ~ I0
(2)
k
0
° I
|
g,..~
~(4)
1 7
,61 -
•
,,,.~,k~.
W
-
-
,
!
'~-!--'
o
-i! - -
i-
--L,
A__ Fig. 3. Infiowing currents corresponding to the first eight functions Wk (F~).
.~(0)
A 0 + ~ - :i:~4,~ + &(o) = 0
~k
~(I)
h ~01"k o ) .,~(1)Xo) -- ~k k + S},) = 0
/~k ~.-(7) .{_~,(2)2(2 ) +~(2) = 0 2"~k ~k k ~k k 3X k{~) + ~~k° ) X °k) +S} 3) = 0 /~0X~ 4) + --k~'(4)X(4)k+ ~k~(4)= 0
_ ,~,~(1) ~
"~l~
k
t
/~, ~7-(6) ,~ ,~{7) ~
~3
~
k = -h6(~bk
(5) =
** ~'I e~( ' k) _?~;X~ ~) + ?~'X(~) 2 k
s~ 6) -- - ~ ' 6 ( x ~ ' ) - x ? )
~ ,,(2) _2(7)X(7) + ~,(7) = 0
~(V) =
k
~k
k
A;~(7)
)
/~ ~-(3) _ ,~(6)X(6) .{..~(6) ----0 2~k --k k ~k k
-- "~2~
, where
S(~) k
3"~k
k
Ap.p(6)
,
Here /~i, ~'i - different types of finite-difference operator, ~'~, 2(~)
(4) + e ~
)
- unknown vectors connected with
the different laws of neutron inflowing. Coefficients ,~(0 ~k ' rSo) ~k are the functionals of trial matrices and the coefficients /)~J) 0=1,2) are concealed in the finite-difference operators.
V. E Boyarinov
138
The finite-difference equations of SHM with four trial functions have the following form
{
A0,I,k -2kqS~ +S~ °~ = 0
'~ i~0) +~°)X~) +g~) = 0 '
,'~0 ~
k
k
(6)
k
The finite-difference equations in the lowest approximation of SHM with three trial functions have the following form Aoq~k -2kq5 k = 0 ,
(7)
In its form, equation (7) differs from the traditional finite-difference equation of the homogenization method by the fact that the group matrix of diffusion coefficients in the finite-difference operator '~o is a full matrix, i.e. it has nonzero non-diagonal elements. In addition, the vector q5k has non-standard sense. Equation (7) is solved by fission source iterations. While solving equations (6) or (5), an additional layer of iterations connected with the higher trial matrices is organized. It should be noted that all SHM characteristics ~(i) /)(j) depend on the unknown value of keff and so k , k another additional iteration layer is organized. Thus, in general case three iteration layers are used.
3. USED CODES WlMS-SH system of codes and SUHAM-2D code were used for computation of this benchmark. WIMSSH is the only code containing computational modules for solving the group neutron transport equation in a heterogeneous reactor cell with required boundary conditions. Now WIMS-SH can calculate only three first spatial trial matrices for heterogeneous cells. For calculation of elements of the zeroth trial matrix q3(°) the RACIA option [3] of WIMS-SH is used. In RACIA code, G3 approximation of Surface Pseudo Sources Method is used for solving the group neutron transport equation. In this method, surface Green function is used in each homogeneous zone of cell. As angular variable concerns, the G3 approximation corresponds to the P3 approximation. For the calculation of the first and the second trial matrices q3°) , q~(2) the newly developed DIC-PN option of WlMS-SH was used. In DIC-PN code, the P2 approximation of Spherical Harmonics method is used. Spatial homogenization was used for the calculation of the higher trial matrices from q3(3) to q~(7). In so doing WIMS-SH code was used for the preparation of homogeneous group cross sections for cells and SUHAM-2D code was used for solving the group diffusion equation in cells with corresponding boundary conditions. In the latter case the cell was divided in 100 square meshes (10xl0). For methodical purposes, the procedure of spatial homogenization and diffusion approximation were also used for the calculation of the first three trial matrices. Let's designate each specific type of calculation by the symbolic name 'nf(het)+mf(hom)'. This means that first n trial matrices (0
Surface harmonics method
139
4. P E R F O R M E D C A L C U L A T I O N S A N D RESULTS. Detailed specification of the benchmark is given in [1] and in the summary paper [5]. The values of keff for the considered 2D benchmark calculated by W I M S - S H and S U H A M - 2 D codes in different approximations of SHM are presented in Table I. The reference benchmark value is 1.186550 with percent error 0.008. Besides, Table I shows the total CPU time for a Pentium4-1500MHz for all calculations.
Table I. Values of keff for considered 2D benchmark Approximation
l%ff
5kerr, %, from reference
5kerr, %, from '3f(het)+5f(hom)'
Total CPU time
3f(hom)
1.18600
-0.046
-0.029
2 rain 1 sec
4f(hom)
1.18641
-0.012
0.006
6 rain 37 sec
8f(hom)
1.18647
-0.007
0.011
60 rain 25 sec
lf(het)+2f(hom)
1.18598
-0.048
-0.030
3 rain 22 sec
lf(het)+3f(hom)
1.18640
-0.013
0.005
7 rain 13 sec
lf(het)+7f(hom)
1.18646
-0.008
0.010
43 min 32 sec
3f(het)
1.18584
-0.060
-0.042
3 rain 10 sec
3f(het)+lf(hom)
1.18628
-0.023
-0.005
8 rain 20 sec
3f(het)+5f(hom)
1.18634
-0.018
--
32 min 51 sec
Reference
1.18655
--
--
--
One can see that the l%ff value of the considered 2D case of the benchmark depends very weakly on the used calculational procedure. Distribution of pin powers was compared with the reference values using the following collective percent error measures: average pin power percent error (AVG), root mean square (RMS) of the power percent error distribution, and mean absolute power percent error (MAE). 1 AVG
N
~-]e,] 777_-,.
=
1
(8)
N
RMS
(9)
N
Zle lpo MAN
~ n=l
Np~ Here N is the number of fuel pins and en is the calculated percent error for the n-th pin power, Pn-
(10)
140
V. F.. Boyarinov
Table II shows these collective percent error measures for all calculations carried out. Table III shows the maximum and minimum pin powers for all calculations, as well as their percent deviation from the reference and m a x i m u m percent error. Table IV shows percentage of fuel pins within the reference confidence interval. In all calculations at solving the finite-difference equations, the following relative tolerances of convergence were used: 10 -6 for the eigenvalue and 10 -5 for group local fluxes. Besides, for the keff value in the additional iteration layer the relative accuracy of 10 .5 was used. It should be noted that Tables I-IV show that all performed calculations are approximately identical, including the calculations ' m f ( h o m ) ' i . e . calculations using spatial homogenization, diffusion approximation and nodal abilities of SHM. Besides, there is no gradual approaching the pin powers calculated by different approximations of S H M to reference results. For obtaining the more exact results it is necessary to use the higher trial matrices for heterogeneous cells and, perhaps, the higher angular approximation at calculation of these trial matrices.
Table II. Collective percent error measures Approximation
AVG
RMS
MRE
3f(hom)
0.97
1.20
0.81
4f(hom)
1.41
1.81
1.15
8f(hom)
1.33
1.73
1.06
lf(het)+2f(hom)
0.90
1.12
0.75
lf(het)+3f(hom)
1.27
1.58
1.06
lf(het)+7f(hom)
1.19
1.49
0.97
3f(het)
0.89
1.11
0.72
3f(het)+lf(hom)
1.25
1.57
1.04
3f(het)+5f(hom)
1.15
1.48
0.93
Table III. Maximum and minimum Pin Powers Approximation
Maximum Percent Pin Power Error
Minimum Percent Pin Power Error
Maximum Percent Error
3f(hom)
2.520
0.88
0.227
-2.00
-3.87
4f(hom)
2.530
1.32
0.229
-0.94
5.42
8f(hom)
2.525
1.10
0.229
-0.96
5.34
1f(het)+2f(hom)
2.517
0.79
0.226
-2.46
-3.93
lf(het)+3f(hom)
2.528
1.21
0.228
-1.36
3.98
lf(het)+7f(hom)
2.523
1.00
0.228
-1.38
3.90
3f(het)
2.514
0.65
0.226
-2.38
-3.79
3f(het)+lf(hom)
2.524
1.07
0.228
-1.31
4.38
3f(het)+5f(hom)
2.519
0.86
0.228
-1.33
4.30
Surface harmonics method
141
Table IV. Percentage of Fuel Pins within the Reference Confidence Interval Approximation
68%
90%
98%
99.8%
3f(hom)
13.1
15.6
16.2
16.2
4f(hom)
13.4
16.0
16.9
17.3
8f(hom)
16.5
16.3
18.5
19.0
lf(het)+2f(hom)
12.9
18.3
20.4
20.6
lf(het)+3f(hom)
12.7
16.6
17.7
17.8
lf(het)+7f(hom)
14.1
17.4
18.0
18.5
3f(het)
13.0
17.0
17.9
18.0
3f(het)+lf(hom)
12.0
15.8
16.7
16.9
3f(het)+5f(hom)
13.5
17.8
19.4
19.4
5. CONCLUSIONS The Surface Harmonics Method was applied to calculate the two-dimensional configuration of the C5G7 MOX benchmark. Different approximations of SHM were used, both with and without spatial homogenization for the calculation of the trial matrices (all of them or only individual ones). Comparisons were carried out for the keffvalue and pin powers. The following conclusions could be drawn from the analysis of the obtained results. • • •
•
• • •
•
In all calculations the eigenvalues are very close one to another, as well as to reference value. In all calculations the maximum of pin powers is located in the points (ny,nx)={(4,5), (5,4)}, and the minimum of pin powers is located in the point (ny,nx)=(33,33). There is no gradual approaching of the pin powers calculated by different approximations of SHM to the reference results. At the same time, there is a gradual approaching of the pin powers calculated by different approximations of SHM to the results calculated by the most accurate SHM approximation, namely '3f(het)+5f(hom)'. The maximum deviation of pin powers in the most accurate approximation of SHM '3f(het)+5f(hom)' from the reference ones reaches 4.3%; in so doing this deviation takes place on the boundary of the MOX assembly with reflector. The maximum deviation of pin powers in the most accurate approximation without spatial homogenization '3f(het)' from the reference pin powers reaches 3.8%. The maximum deviation of pin powers in the best possible approximation using the spatial homogenization '8f(hom)' from the reference pin powers reaches 5.3%. All calculations carried out are approximately identical from the viewpoint of accuracy achieved, including the calculations of 'mf(hom)'-type, i.e. calculations using spatial homogenization together with nodal approach. For obtaining the more exact results it is necessary to use the higher trial matrices for heterogeneous cells and, perhaps, the higher angular approximation at calculation of these trial matrices.
142
V. E Boyarinov
REFERENCES 1.
2. 3. 4.
5.
Lewis E.E., Smith M.A., Tsoulfanidis N., et al "Benchmark specification for Deterministic 2-D/3-D MOX fuel assembly transport calculations without spatial homogenization (C5G7 MOX)", OECD/NEA document, Final specification, NEA/NSC/DOC(2001)4, March 28, 2001. Laletin N.I., "On the Equations of Heterogeneous Reactor", Voprosi Atomnoi Nauki i Tehniki. Ser. Fisika Yadernih Reactorov, 5, 18, 1981, 31-46 (in Russia). Laletin N.I., Sultanov N.V., V.F. Boyarinov, et al, "WIMS-SU complex of codes and SPEKTR code", Proceeding. of 'PHYSOR-90'", v.4, p. PV-148, 1990, Marseilles, France, 23-27 April. Boyarinov V.F., "SUHAM-2.5 Code for Solving 2D Finite-Difference Equations of the Surface Harmonics Method in Square and Triangular Lattices", Nuclear Technology'99, 1999, p. 23, Karlsruhe, Germany, 18-20 May. Smith M.A., Lewis E.E., Byung-Chan Na, "Benchmark on Deterministic 2-D MOX Fuel Assembly Transport Calculations without spatial homogenization", Progress in Nuclear Energy, 2004, to be published in this issue.
~)DIR~Cr.
ELSEVIER www.elseviencom/locate/pnucene
© 2004 Elsevier Ltd. All rights reserved Printed in Great Britain 0149-1970/$ - see front matter
doi: 10.1016/j.pnucene.2004.09.013
USE OF THE S U R F A C E H A R M O N I C S CALCULATION
METHOD FOR
OF 2D C5G7 MOX BENCHMARK
VICTOR F. BOYARINOV
Russian Research Center "Kurchatov Institute", Nuclear Reactor Institute 123182, Moscow, RUSSIA E-mail: [email protected]
ABSTRACT The C5G7 MOX benchmark specifying a sixteen-assembly core with a surrounding water reflector was proposed as a basis to measure current transport code abilities in the treatment of reactor core problems without spatial homogenization. Seven-group cross sections for all materials were used as initial information. Just that fact allows to test an accuracy of solving the neutron transport equation excluding additional errors connected with preparing the group cross sections. In this paper, Surface Harmonics Method (SHM) is applied to calculation of the two-dimensional configuration of this benchmark. Different approximations of SHM were applied, both with and without spatial homogenization. Additionally, this fact allowed evaluating the effect of spatial homogenization of cells. Comparisons were carried out for keel and pin powers both with the reference results and between the results calculated by different SHM approximations.. © 2004 Elsevier Ltd All rights reserved KEYWORDS Transport benchmark, C5G7, MOX, spatial homogenization, Surface Harmonics Method
1. INTRODUCTION The NEA Expert Group on 3-D Radiation Transport Benchmarks proposed the seven-group form of the C5 MOX problem [1] to test the ability of current transport codes to compute reactor core problems without spatial homogenization. Seven-group cross sections for UO2, the three enrichments of MOX, the guide tubes, the fission chamber and the moderator were used as initial information. Just that fact allows to test an accuracy of solving neutron transport equation excluding additional errors connected with preparing the group cross sections. The Expert Group proposed to use the results calculated by Dr. M. Smith (ANL, USA) as reference ones.
V. E Boyarinov
134
In this paper, Surface Harmonics Method [2] is applied to calculate the two-dimensional configuration of this benchmark. SHM is the method for solving the neutron transport equation in a reactor core. A characteristic feature of SHM is that, in general case, it does not use spatial homogenization. In case if the homogeneous cross sections are used as initial information, SHM actually works as a nodal method and solves diffusion equation. In SHM, neutron distribution in a reactor core is presented as a superposition of trial functions. Each trial function is taken as a solution of the neutron transport equation in the internal cell area with certain boundary conditions on the cell surface. Different trial functions differ one from another by the boundary conditions. Since the actual number of the trial functions is limited, at substituting the quest solution in the neutron transport equation, a residual appears. The minimization procedure for this residual gives the finite-difference equations of the SHM. The main limitation of SHM is connected with the restricted number of used trial functions. For computation of trial functions, WlMS-SH [3] system of codes is used. WlMS-SH is the only code containing computational modules for solving the group neutron transport equation in heterogeneous reactor cells with required boundary conditions. Now WIMS-SH calculates only three first spatial trial functions for heterogeneous cells. For the calculation of higher trial functions, the spatial homogenization and diffusion approximation are used and SUHAM-2D [4] code is applied for solving the group diffusion equation. Thus, the most accurate currently available approximation of SHM without spatial homogenization is '3f(het)'. The notation '3f(het)' means that three trial functions are used. In so doing the group neutron transport equation is solved in heterogeneous cells for calculation of these trial functions. Other approximations of SHM are forced to use spatial homogenization and diffusion approximation for calculation of separate trial functions. Actual calculations in the present work were carried out by different approximations of SHM. The most accurate calculation was carried out with use of eight trial functions for each cell, and in so doing the first three trial functions were calculated for heterogeneous cells and the next five - for homogeneous cells (symbolic name '3f(het)+5f(hom)'). Comparisons of calculational results were carried out for kaf value and pin powers both between different approximations of SHM and with the reference solution obtained by Monte-Carlo method.
2. BRIEF DESCRIPTION OF SURFACE HARMONICS METHOD In SHM, the group function of neutron flux distribution in a reactor core is presented as a superposition of group trial matrices I
N
~N (w) = Z Z
¢}", (w)[~.
(1)
i=l n=O
Here ~p}")(w) are the trial matrices describing the neutron field in i-th cell, vectors ~, are the unknown group amplitudes, n - trial matrix number, (N+I) - total number of trial matrices for each cell, w = {7, (2}. Each vector ~2)(w) of each trial matrix qg}n)(w) is taken as a solution of the group neutron transport equation in the internal cell area with certain boundary condition on the cell surface. For calculation of the vector q?}g)(W) (g=l,2,...,G) the following boundary condition is used: inflowing current for group 'g' is distributed on the external boundary as function of coordinates Wk(~) and for each another group this current equals zero for all points of external boundary. Functions W k (~)are presented as follows
135
Su~ace harmonics method
"cos(laj)'~
(2)
Wk(~) = PP (PJ)" .sin(laJ) J ' where Pj = ( - 1 ) J - 1 2 [ ( - 2 ( 2 j - 1 ) ] , - 1 ~ P j
Here Pp(pj) - Legendre polynomials; M - number of lateral sides of a cell; a - length of the lateral side of a cell; j - lateral side number of a cell; ctj - angle between the normal built from the center of a cell to j-th lateral side and X axis. The lateral sides of a cell are numbered against the clock hand; the first lateral side of a cell is the right side (Fig. 1). normal on 2-nd side [
i=2
B
C
i=3
i=1
X
A
D
i=4
Fig. 1. Cell with square external boundary. For a cell with square external boundary, the first eight functions Wk (?s) have the following forms Wo(~.)=l,
t
Wl(r~)=cos(aj),
W3(~. ) = cos(Za~),
W2(K)=sin(aj)
W4(~) = PI(Pi),
W6(~. ) = Pl(pj)sin(ctj),
Ws(~)= P l ( p j ) c o s ( a j ) ,
(4)
WT(~) = P l ( p i ) c o s ( Z a j )
These functions W~ (Fs) are presented on Fig. 2. Inflowing currents corresponding to these first eight functions Wk (i~) are presented on Fig. 3. Since the actual number of the trial matrices is limited, at substituting the quest solution in the group neutron transport equation, a residual appears. The minimization procedure for this residual gives the finite-difference equations of the SHM. Mathematically this procedure for obtaining the finite-difference equations corresponds to the coupling of the angular moments of neutron distributions presented in the form (1) on the adjacent sides of neighboring cells. The number and form of the finite-difference equations depend on the number of used trial matrices. More details about this procedure could be found e.g. in [2, 3]. The finite-difference equations of SHM for the square lattice with eight trial functions have the following form.
136
g E Boyarinov
IB
Ic
IB
C
I
W0 (rs)
A
Wl(rs)
D
A
~
W2 (rs)
I
I
]
: /i",. '.oj io\
i IA
C
' /x. oAj IB\
~-
IA
I
ilA I'B Ic
ilA
ID
A
-x/
W3 (r s )
W4 (r s )
'
olA . /
;S
D
-1
:A
W5 (rs )
~ '
~
C
~
A
W6 (rs )
W7 (rs )
,("
!
/
!
:
Fig. 2. The first eight functions Wk ( ~ ) .
137
Surface harmonics method
r
o
(0) ~ _ _
~
V
q~(1) ~ , ~ I0
(2)
k
0
° I
|
g,..~
~(4)
1 7
,61 -
•
,,,.~,k~.
W
-
-
,
!
'~-!--'
o
-i! - -
i-
--L,
A__ Fig. 3. Infiowing currents corresponding to the first eight functions Wk (F~).
.~(0)
A 0 + ~ - :i:~4,~ + &(o) = 0
~k
~(I)
h ~01"k o ) .,~(1)Xo) -- ~k k + S},) = 0
/~k ~.-(7) .{_~,(2)2(2 ) +~(2) = 0 2"~k ~k k ~k k 3X k{~) + ~~k° ) X °k) +S} 3) = 0 /~0X~ 4) + --k~'(4)X(4)k+ ~k~(4)= 0
_ ,~,~(1) ~
"~l~
k
t
/~, ~7-(6) ,~ ,~{7) ~
~3
~
k = -h6(~bk
(5) =
** ~'I e~( ' k) _?~;X~ ~) + ?~'X(~) 2 k
s~ 6) -- - ~ ' 6 ( x ~ ' ) - x ? )
~ ,,(2) _2(7)X(7) + ~,(7) = 0
~(V) =
k
~k
k
A;~(7)
)
/~ ~-(3) _ ,~(6)X(6) .{..~(6) ----0 2~k --k k ~k k
-- "~2~
, where
S(~) k
3"~k
k
Ap.p(6)
,
Here /~i, ~'i - different types of finite-difference operator, ~'~, 2(~)
(4) + e ~
)
- unknown vectors connected with
the different laws of neutron inflowing. Coefficients ,~(0 ~k ' rSo) ~k are the functionals of trial matrices and the coefficients /)~J) 0=1,2) are concealed in the finite-difference operators.
V. E Boyarinov
138
The finite-difference equations of SHM with four trial functions have the following form
{
A0,I,k -2kqS~ +S~ °~ = 0
'~ i~0) +~°)X~) +g~) = 0 '
,'~0 ~
k
k
(6)
k
The finite-difference equations in the lowest approximation of SHM with three trial functions have the following form Aoq~k -2kq5 k = 0 ,
(7)
In its form, equation (7) differs from the traditional finite-difference equation of the homogenization method by the fact that the group matrix of diffusion coefficients in the finite-difference operator '~o is a full matrix, i.e. it has nonzero non-diagonal elements. In addition, the vector q5k has non-standard sense. Equation (7) is solved by fission source iterations. While solving equations (6) or (5), an additional layer of iterations connected with the higher trial matrices is organized. It should be noted that all SHM characteristics ~(i) /)(j) depend on the unknown value of keff and so k , k another additional iteration layer is organized. Thus, in general case three iteration layers are used.
3. USED CODES WlMS-SH system of codes and SUHAM-2D code were used for computation of this benchmark. WIMSSH is the only code containing computational modules for solving the group neutron transport equation in a heterogeneous reactor cell with required boundary conditions. Now WIMS-SH can calculate only three first spatial trial matrices for heterogeneous cells. For calculation of elements of the zeroth trial matrix q3(°) the RACIA option [3] of WIMS-SH is used. In RACIA code, G3 approximation of Surface Pseudo Sources Method is used for solving the group neutron transport equation. In this method, surface Green function is used in each homogeneous zone of cell. As angular variable concerns, the G3 approximation corresponds to the P3 approximation. For the calculation of the first and the second trial matrices q3°) , q~(2) the newly developed DIC-PN option of WlMS-SH was used. In DIC-PN code, the P2 approximation of Spherical Harmonics method is used. Spatial homogenization was used for the calculation of the higher trial matrices from q3(3) to q~(7). In so doing WIMS-SH code was used for the preparation of homogeneous group cross sections for cells and SUHAM-2D code was used for solving the group diffusion equation in cells with corresponding boundary conditions. In the latter case the cell was divided in 100 square meshes (10xl0). For methodical purposes, the procedure of spatial homogenization and diffusion approximation were also used for the calculation of the first three trial matrices. Let's designate each specific type of calculation by the symbolic name 'nf(het)+mf(hom)'. This means that first n trial matrices (0
Surface harmonics method
139
4. P E R F O R M E D C A L C U L A T I O N S A N D RESULTS. Detailed specification of the benchmark is given in [1] and in the summary paper [5]. The values of keff for the considered 2D benchmark calculated by W I M S - S H and S U H A M - 2 D codes in different approximations of SHM are presented in Table I. The reference benchmark value is 1.186550 with percent error 0.008. Besides, Table I shows the total CPU time for a Pentium4-1500MHz for all calculations.
Table I. Values of keff for considered 2D benchmark Approximation
l%ff
5kerr, %, from reference
5kerr, %, from '3f(het)+5f(hom)'
Total CPU time
3f(hom)
1.18600
-0.046
-0.029
2 rain 1 sec
4f(hom)
1.18641
-0.012
0.006
6 rain 37 sec
8f(hom)
1.18647
-0.007
0.011
60 rain 25 sec
lf(het)+2f(hom)
1.18598
-0.048
-0.030
3 rain 22 sec
lf(het)+3f(hom)
1.18640
-0.013
0.005
7 rain 13 sec
lf(het)+7f(hom)
1.18646
-0.008
0.010
43 min 32 sec
3f(het)
1.18584
-0.060
-0.042
3 rain 10 sec
3f(het)+lf(hom)
1.18628
-0.023
-0.005
8 rain 20 sec
3f(het)+5f(hom)
1.18634
-0.018
--
32 min 51 sec
Reference
1.18655
--
--
--
One can see that the l%ff value of the considered 2D case of the benchmark depends very weakly on the used calculational procedure. Distribution of pin powers was compared with the reference values using the following collective percent error measures: average pin power percent error (AVG), root mean square (RMS) of the power percent error distribution, and mean absolute power percent error (MAE). 1 AVG
N
~-]e,] 777_-,.
=
1
(8)
N
RMS
(9)
N
Zle lpo MAN
~ n=l
Np~ Here N is the number of fuel pins and en is the calculated percent error for the n-th pin power, Pn-
(10)
140
V. F.. Boyarinov
Table II shows these collective percent error measures for all calculations carried out. Table III shows the maximum and minimum pin powers for all calculations, as well as their percent deviation from the reference and m a x i m u m percent error. Table IV shows percentage of fuel pins within the reference confidence interval. In all calculations at solving the finite-difference equations, the following relative tolerances of convergence were used: 10 -6 for the eigenvalue and 10 -5 for group local fluxes. Besides, for the keff value in the additional iteration layer the relative accuracy of 10 .5 was used. It should be noted that Tables I-IV show that all performed calculations are approximately identical, including the calculations ' m f ( h o m ) ' i . e . calculations using spatial homogenization, diffusion approximation and nodal abilities of SHM. Besides, there is no gradual approaching the pin powers calculated by different approximations of S H M to reference results. For obtaining the more exact results it is necessary to use the higher trial matrices for heterogeneous cells and, perhaps, the higher angular approximation at calculation of these trial matrices.
Table II. Collective percent error measures Approximation
AVG
RMS
MRE
3f(hom)
0.97
1.20
0.81
4f(hom)
1.41
1.81
1.15
8f(hom)
1.33
1.73
1.06
lf(het)+2f(hom)
0.90
1.12
0.75
lf(het)+3f(hom)
1.27
1.58
1.06
lf(het)+7f(hom)
1.19
1.49
0.97
3f(het)
0.89
1.11
0.72
3f(het)+lf(hom)
1.25
1.57
1.04
3f(het)+5f(hom)
1.15
1.48
0.93
Table III. Maximum and minimum Pin Powers Approximation
Maximum Percent Pin Power Error
Minimum Percent Pin Power Error
Maximum Percent Error
3f(hom)
2.520
0.88
0.227
-2.00
-3.87
4f(hom)
2.530
1.32
0.229
-0.94
5.42
8f(hom)
2.525
1.10
0.229
-0.96
5.34
1f(het)+2f(hom)
2.517
0.79
0.226
-2.46
-3.93
lf(het)+3f(hom)
2.528
1.21
0.228
-1.36
3.98
lf(het)+7f(hom)
2.523
1.00
0.228
-1.38
3.90
3f(het)
2.514
0.65
0.226
-2.38
-3.79
3f(het)+lf(hom)
2.524
1.07
0.228
-1.31
4.38
3f(het)+5f(hom)
2.519
0.86
0.228
-1.33
4.30
Surface harmonics method
141
Table IV. Percentage of Fuel Pins within the Reference Confidence Interval Approximation
68%
90%
98%
99.8%
3f(hom)
13.1
15.6
16.2
16.2
4f(hom)
13.4
16.0
16.9
17.3
8f(hom)
16.5
16.3
18.5
19.0
lf(het)+2f(hom)
12.9
18.3
20.4
20.6
lf(het)+3f(hom)
12.7
16.6
17.7
17.8
lf(het)+7f(hom)
14.1
17.4
18.0
18.5
3f(het)
13.0
17.0
17.9
18.0
3f(het)+lf(hom)
12.0
15.8
16.7
16.9
3f(het)+5f(hom)
13.5
17.8
19.4
19.4
5. CONCLUSIONS The Surface Harmonics Method was applied to calculate the two-dimensional configuration of the C5G7 MOX benchmark. Different approximations of SHM were used, both with and without spatial homogenization for the calculation of the trial matrices (all of them or only individual ones). Comparisons were carried out for the keffvalue and pin powers. The following conclusions could be drawn from the analysis of the obtained results. • • •
•
• • •
•
In all calculations the eigenvalues are very close one to another, as well as to reference value. In all calculations the maximum of pin powers is located in the points (ny,nx)={(4,5), (5,4)}, and the minimum of pin powers is located in the point (ny,nx)=(33,33). There is no gradual approaching of the pin powers calculated by different approximations of SHM to the reference results. At the same time, there is a gradual approaching of the pin powers calculated by different approximations of SHM to the results calculated by the most accurate SHM approximation, namely '3f(het)+5f(hom)'. The maximum deviation of pin powers in the most accurate approximation of SHM '3f(het)+5f(hom)' from the reference ones reaches 4.3%; in so doing this deviation takes place on the boundary of the MOX assembly with reflector. The maximum deviation of pin powers in the most accurate approximation without spatial homogenization '3f(het)' from the reference pin powers reaches 3.8%. The maximum deviation of pin powers in the best possible approximation using the spatial homogenization '8f(hom)' from the reference pin powers reaches 5.3%. All calculations carried out are approximately identical from the viewpoint of accuracy achieved, including the calculations of 'mf(hom)'-type, i.e. calculations using spatial homogenization together with nodal approach. For obtaining the more exact results it is necessary to use the higher trial matrices for heterogeneous cells and, perhaps, the higher angular approximation at calculation of these trial matrices.
142
V. E Boyarinov
REFERENCES 1.
2. 3. 4.
5.
Lewis E.E., Smith M.A., Tsoulfanidis N., et al "Benchmark specification for Deterministic 2-D/3-D MOX fuel assembly transport calculations without spatial homogenization (C5G7 MOX)", OECD/NEA document, Final specification, NEA/NSC/DOC(2001)4, March 28, 2001. Laletin N.I., "On the Equations of Heterogeneous Reactor", Voprosi Atomnoi Nauki i Tehniki. Ser. Fisika Yadernih Reactorov, 5, 18, 1981, 31-46 (in Russia). Laletin N.I., Sultanov N.V., V.F. Boyarinov, et al, "WIMS-SU complex of codes and SPEKTR code", Proceeding. of 'PHYSOR-90'", v.4, p. PV-148, 1990, Marseilles, France, 23-27 April. Boyarinov V.F., "SUHAM-2.5 Code for Solving 2D Finite-Difference Equations of the Surface Harmonics Method in Square and Triangular Lattices", Nuclear Technology'99, 1999, p. 23, Karlsruhe, Germany, 18-20 May. Smith M.A., Lewis E.E., Byung-Chan Na, "Benchmark on Deterministic 2-D MOX Fuel Assembly Transport Calculations without spatial homogenization", Progress in Nuclear Energy, 2004, to be published in this issue.