- Email: [email protected]
Twodant solutions for the 2-D C5G7 MOX benchmark
Progress in Nuclear Energy, Vol. 45, No. 2-4, pp. 201-213, 2004 Availableonline at vca~w.sciencedirect.com SCIENCE
ELSEVIER
f"--d} D IR E C T "
© 2004 Published by Elsevier Ltd Printed in Great Britain 0149-1970/$ - see front matter
doi:10.1016/j.pnucene.2004.09.010
www.elsevier.com/locate/pnueene
TWODANT SOLUTIONS FOR THE 2-D C5G7 MOX BENCHMARK
HONG-CHUL KIM*, CHI YOUNG HAN, and JONG KYUNG KIM
Department of Nuclear Engineering, Hanyang University 17 Haengdang, Seongdong, Seoul 133-791, Korea *
.
E-mad: [email protected]
ABSTRACT The C5G7 MOX benchmark was proposed to test the ability of commercial transport codes to treat reactor core problems without spatial homogenization. The benchmark requires solutions in the form of normalized pin powers as well as the eigenvalue. In the work, the two-dimensional benchmark calculation using the TWODANT code within the DANTSYS code package has been performed with proper spatial and angular approximations. The TWODANT code solves the multigroup discrete ordinates form of the Boltzmann transport equation in twodimensional geometry. The calculation results show a good agreement in comparison with the reference solution obtained from a seven-group MCNP calculation. In addition, sensitivity studies on mesh and angular refinements have been performed to produce a higher quality solution. In the results, it is found that in the TWODANT calculation the spatial approximation in a staircase form of the circular fuel pin with relatively high quadrature order of SN is a viable method for solving the2-D C5G7 benchmark. © 2004 Published by Elsevier Ltd KEYWORDS C5G7 Benchmark, TWODANT, Discrete Ordinates, Mesh Refinement, Angular Refinement
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1. INTRODUCTION Systems loaded with plutonium in the form of mixed-oxide (MOX) fuel show somewhat different neutronic characteristics compared with those using conventional uranium fuels. Even though many countries have sufficient experiences in using MOX fuel within certain technical boundaries, it is essential to accurately predict the characteristics of MOX-fuelled systems and to further validate both the nuclear data and the computational methods used if the extension of the current constraints is envisaged. In 2001, the C5G7 MOX benchmark for deterministic 2-D and 3-D MOX fuel assembly transport calculations (Lewis et al, 2001) was proposed by the OECD/NEA in order to test the ability of current available transport codes to treat reactor core problems without spatial homogenization. It is based on a seven-group form of the C5 MOX fuel assembly problem specified in the benchmark calculations in 1996 (Cathalau et al, 1996). The domain is made up of four fuel assemblies and reflector, and proper vacuum and reflective boundary conditions. The benchmark requires solutions in the form of normalized pin powers as well as the eigenvalue. The reference solutions for both 2-D and 3-D problems have been obtained from a seven-group MCNP calculation. The benchmark results were reported in 2003 (OECD/NEA, 2003). In this work, the DANTSYS code package (LANL, 1997) was applied to analyze the 2-D and 3-D C5G7 MOX benchmark problems. The DANTSYS code package is a modular computer program package designed to solve the time-independent, multigroup discrete ordinates form of the Boltzmann transport equation in several different geometries. In this paper, the benchmark results are discussed in detail for 2-D problem only. The TWODANT code within the DANTSYS code package, which solves the twodimensional transport equation in x-y, r-z, and r-theta geometries, was employed to perform the calculation for 2-D problem. To analyze the benchmark problem with the TWODANT code, proper spatial and angular approximations were made. The results were compared with the reference solution for the 2-D problem. In addition, several calculations were performed to investigate the effects of the different spatial and angular approximations on the accuracy. The results from these sensitivity studies were analyzed and discussed.
2. TWODANT MODELAND SOLUTION The benchmark geometry consists of two MOX and two U02 fuel assemblies and reflector as shown in Figure 1. Since TWODANT provides only regular mesh features, the circular fuel pin must be approximated by an X-Y Cartesian grid in order to describe the 17x17 lattice of square pin cells. The simplest square shape was generated by preserving the fuel area as shown in Figure 2. A geometrical model composed of a total of 160x160 meshes, corresponding to a 4x4 grid in each pin cell and 24 mesh cells in each direction of the water reflector, was developed for TWODANT calculation. This model is revised for mesh refinement compared with the model of 86x86 meshes submitted to the benchmark (OECD/NEA, 2003). The angular approximation of $8 quadrature order was applied. The convergence criterion of eigenvalue as well as pointwise flux and fission source was set to be 1.0E-5. All calculations have been performed on a HP C3700 model workstation. The accuracy of the TWODANT calculation is summarized by tabulating the pin power results along with the eigenvalue and by comparing with the 2-D reference solution from the seven-group MCNP calculation, as presented in Tables 1 and 2. In the tables, the percent error information for the reference MCNP values represents the statistical error bounds associated with the 98% confidence interval of the Monte Carlo solution. To assess the overall pin power distribution, the following collective errors were measured: average (AVG), root mean square (RMS), and mean absolute (MAE) errors.
203
TWODANT solutions
Reflective B . ~ MOX
~5
~J MOX
UO~
~ I
Reflector [ ] UO2Fuel [ ] 4.3%MOX Fuel
Vacuum B. C.
[] []
7.0%MOXFuel 8.7%MOXFuel
• •
Guide Tube Fission Chamber
Fig. 1 Core Configuration for the 2-D C5G7 Benchmark Problem
Fig. 2
Pin Cell Represented by a 4 x 4 Cartesian Grid
Table 1 TWODANT Calculation Results for Eigenvalue and Pin Power Error Eigenvalue [%]
Maximum Error Value [%]
Reference MCNP
1.18655 +_0.008
2.498
TWODANT
1.18543
2.524
-0.09
1) Pin cell index in each direction as shown in Figure 1
Normalized Pin Power Minimum Error Minimum Maximum Value [%] Position 1) Position 1)
+_0.16 (4, 5) or (5, 1.05
(4, 5) or (5,
0.232 0.232
+_0.58 (33, 33) 0.03
(33, 33)
H.-C. Kim et el
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Table 2
TWODANT Calculation Errors for Pin Power Distribution Normalized Pin Power Error Reference MCNP
TWODANT
Overall
Inner U02
MOX
Outer UO2
Overall
AVG Error [%] 1)
0.32
0.64
0.79
0.29
0.62
RMS Error [%] 2)
0.34
0.70
0.88
0.37
0.73
MAE Error [%] 3)
0.27
0.66
0.70
0.29
0.63
1) A VG = ~Nle,, I/N' where N is the number of fuel pins and e,, is the calculated error for the nth pin power. _
2
N
3) MAE= ~ l e l" P,,/N" Pov~, wherep, is the nth pin power and Pavg iS the averagepin power.
The resulting eigenvalue is 1.18543 and gives a good agreement of -0.09% error in comparison with the reference value of 1.18655. The calculated pin powers were normalized to an average pin power. The maximum pin power with 1.05% error is beyond _+0.16% error bounds of the reference value for this pin, but the minimum pin power with 0.03% error is well within _+0.58% error bounds. In addition, this calculation provides the same positions of the maximum and minimum pin powers as the prediction of reference. The overall pin power distribution agrees well with the reference solution, showing AVG, RMS, and MAE errors of less than 1%. The maximum error of pin power is -1.73% at position (32, 5) or (5, 32). This calculation took a computational time of about 1 minute on the computing machine. An investigation into the pin power error distribution in Figure 3 shows that this calculation gives higher values of the normalized pin powers with 0.70% RMS error in the inner UO2 assembly than the reference values. The results again indicate an overall overestimation of the pin powers in the inner UO2 assembly. This trend is lessened when moving toward the neighboring MOX assembly and on the contrary, it gives underestimated pin powers in the MOX assembly with 0.88% RMS error. The pin power distribution of the outer UOz assembly adjacent to the reflector have slightly negative discrepancies and gives more negative values for the UO2 fuel pins near the MOX assembly.
205
TWODANT solutions
Pin Cell Index in X Direction 17
34
Error [%] 1.5
g
0.5
.9
-g
-0.5
r~
-1.5
Fig. 3
Pin Power Error Distribution of TWODANT Calculation
3. SENSITIVITY STUDIES ON MESH AND ANGULAR REFINEMENTS The choice of finer mesh grid and higher angular quadrature order was motivated to produce a higher quality solution. Therefore, sensitivity studies on mesh and angular refinements were performed to improve the TWODANT calculation results reasonably with a better approximation. 3.1
Mesh refinement
Compared with the 4x4 grid as previously approximated in Figure 2, two different forms of grid structure with the same fuel area were characterized on the pin cell domain in order to approximate the circular fuel pin. In the first grid structure, the square fuel pin was constructed instead of the circular fuel pin in the same way as the 4x4 grid, but the pin cell domain was covered by some grid sets of finer refinement such as 6x6, 8x8, 12x12, 16x16, and 20x20 meshes. For convenience, this grid structure was named "square fuel pin model" after the shape of the fuel pin represented by the spatial approximation. Then, the 4x4 grid is the coarsest grid of the square fuel pin model. Some examples of the pin cell grids refined by the square fuel pin model are shown in Figure 4.
(a) 6x6 Fig. 4
(b) 12x12
(c) 20x20
Examples of Pin Cell Grids Refined by the Square Fuel Pin Model
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In the other grid structure, a staircase form of fuel pin was generated rather than creating the square form to represent the circular fuel pin. This grid structure was named "staircase fuel pin model" from the same point of view as the square model. Procedures for the grid sets of the staircase fuel pin, corresponding to the same mesh sizes (numbers) as the square model, were devised as described below: (1) Set an initial point on the circular fuel pin borderline, at which the distance between the fuel pin borderline and the square pin cell line is minimized (Figure 5-(a)). (2) On the basis of the initial point, determine an octant of the circular fuel pin and divide it into some equal parts in fan shape. Obtain an intersection point between the tangent line on the initial point and the second partition line from the initial point (Figure 5-(b)). (3) Consider variable points such that they are placed on the other partition lines and can be used as vertexes for the staircase model (Figure 5-(c)). (4) Correspond the equal fans in the octant to the cells made by the partition lines and the points in steps above. Compute definite positions of the variable points such that the area of each cell is equal to that of the corresponding fan, but the sum of areas in case of the first and second cells (Figure 5-(d)). (5) Complete pin cell grids refined by the staircase fuel pin model after symmetrizing the staircase approximation for the octant.
(a)
(b) Fig. 5
(c)
(d)
Procedures for Mesh Refinement by the Staircase Fuel Pin Model
The procedures are applicable to the pin cell grids such as 8x8, 12x12, 16×16, and 20x20, and the 6x6 pin cell grid can be obtained simply. Some examples of the pin cell grids refined by the staircase fuel pin model are shown in Figure 6. If a 4×4 grid coarser than the 6x6 grid as shown in Figure 6-(a) is considered for the staircase fuel pin model, the 4x4 grid in Figure 2 can be also regarded fairly as the coarsest grid of the staircase model as well as the square model.
(a) 6×6 Fig. 6
(b) 12×12
(c) 20×20
Examples of Pin Cell Grids Refined by the Staircase Fuel Pin Model
TWODANT solutions
207
The calculations for sensitivity study on mesh refinement were also performed with the same quadrature order of $8 as the previous calculation with the 4x4 grid. The behavior of the benchmark eigenvalue was firstly investigated with respect to the spatial approximation. As shown in Figure 7, the mesh refinements by the two fuel pin models provide less accurate results than by the first 4x4 grid. It was, particularly, found that the staircase model underestimated the eigenvalue by about 100 pcm rather than the square model. The sensitivity of the eigenvalue to mesh size per pin cell is not high in the square model, especially, to mesh size over the 6x6 grid in the staircase model. However, the curves of two models appear to converge to a wrong value with mesh refinement. In brief, the results indicate that the low spatial approximation gives good results more or less by accident.
1.1880 --*--
1.1875
Reference
--m-- Square Model 1.1870
--- •--- S t a i r c a s e M o d e l
1.18651.1860 • 1.1855,
1.1850
I
1
~
1
1
III
1.1845 "O-.
_ ...... • ..............
1.1840
• ................
• .......
1.1835 1.1830
I
I
I
I
I
I
4x4
6x6
8x8
12x 12
16x 16
20x20
M e s h S i z e pe r Pin C e l l
Fig. 7
Eigenvalue Behavior as a Function of Mesh Size for the Two Spatial Models with $8 Quadrature Order
In Figure 8, the accuracy behavior of the pin power results is plotted as a function of mesh size per pin cell with the three quantities: AVG, RMS, and MAE errors. The square model does not improve the accuracy of the pin power noticeably though the grid resolution is increased. Moreover, the pin power results do not converge toward the reference solution. The results from the staircase model rev,sal the rapid convergence of the pin power with the mesh refinement of the 6x6 grid, but slower convergence with increasing mesh size. Hence, the 6x6 grid per pin cell is sufficiently refined so that very little benefit is obtained from improving either of the two fuel pin models. In Figure 9, further investigation into the pin power error distribution with respect to each fuel assembly shows that for each fuel pin model, the RMS value of the pin power errors in the MOX assembly is still larger than the ones in the inner or outer UO2 assemblies. The difference in pin power error between the square and staircase models is large in the MOX and inner UO2 assemblies compared with the outer UO2 assembly.
H.-C. Kim et el
208 1,0-
0.90.80.7-
• ". ,'7~'----~.~m ~
m
m---~__~m__
m
0.6.
0.5-
"N
0.2 0.1
........• ............. '0 . . . . . . . . . . .
- ~ : : : : : : : : :. . . . . . . . . . . . . . . .
0.4 0.3
.
• ........... ""--O . . . . . . . . . . . . . .
"::','.::~:-':--.---.--~:-_0?.~?.,_.:::::::~
--t~-- Square Model - A V G m-- S q u a r e M o d e l - R M S Square Model - M R E ---o - Sta ir c a se M o d e l - A V G ~--e--- Staircase M o d e l - R M S ---e--- Sta ir c a se M o d e l - M R E
0.0
4x4
6x6
8x8
12x12
16x16
20x20
Mesh Size per Pin Cell Fig. 8
Pin Power Error Behavior as a Function of Mesh Size for the Two Spatial Models with $8 Quadrature Order
].0" --n-0.90.8 ~,
M O X - Square Model
---o--- M O X - Sta ir c a s e M o d e l
! "~, ~
i
e
i
•
i---
--i
0.7 "',,,
0.6
• . . . . . . . . . . . . . . . . . .
• . . . . . . . . . . . . . . . .•
0.5 " ' " O . . . . . - . . . . . . . . . . O - - . . . . . . . ----..
' 0 . . . . . . . . . . . . . . . . 0 . . . . . . . . . . . . . . rO
~- 0.4 0.3 e~
- - t 3 - - I nne r U • 2 - S q u a r e M o d e l 0.2 0.1
---o--- I nne r U • z - Staircas e M o d e l --~--
Outer U • 2 - Square Model
---e--- O u t e r U • 2 - Sta ir c a se Mode l 0.0 4x4
6x6
8x8
12x 12
16Xl 6
120X20
Mesh Size per Pin Cell
Fig. 9
Pin Power RMS Error Behavior as a Function of Mesh Size in Each Fuel Assembly for the Two Spatial Models with $8 Quadrature Order
209
TWODANT solutions
3.2
Angular refinement
Based on the above results, the sensitivity study on angular refinement was performed with the mesh size of 6x6 grid per pin cell. The angular refinement was applied to all of the square and staircase models with increasing the quadrature order from $4 to $t6 for the traditional built-in quadrature sets provided in the TWODANT by default. Figure 10 shows that all of the TWODANT solutions for the benchmark eigenvalue converge toward the reference MCNP solution as the quadrature order for the angular refinement is increased. Convergence with respect to the quadrature order is, however, not very fast and even the results indicate that substantially more angular refinement is necessary to come within the error bars of the Monte Carlo solution. The difference in eigenvalue between both spatial models is not reduced by even higher order angular approximation and, therefore, the staircase model still underestimates the eigenvalue by about 100 pcm rather than the square model.
1.1880
- e - - Reference --•-- Square Model ---•--- Staircase Model
1.1875, 1.1870, 1.1865 1.1860 ~ 1.1855
f i •
i~
. ~ 1.1850 .... • .................... 1.1845
......... •
• ............................. • .........................
1.1840 1.1835 1.1830 NI=4
Nu=8
N~12
N~16
Sr~Order Fig. 10
Eigenvalue Behavior as a Function of SN Order for the Two Spatial Models with 6×6 Grid per Pin Cell
The RMS error of the pin power distribution is plotted in Figure 11 against increasing quadrature order. The result shows that all of the square and staircase models reduce the RMS error of the pin power distribution with the angular refinement. Especially, it was found that the RMS error of the staircase model was converging to the reference value more rapidly than that of the square model, and ultimately approached nearly 0.34% which is the RMS value of the statistical errors with the 98% confidence interval in the reference pin power distribution. Therefore, provided that the loss in accuracy for the eigenvalue is acceptable, the staircase model with the 6x6 grid per pin cell is very beneficial to the 2-D C5G7 benchmark problem. The downward trend according to the quadrature order in the pin power RMS error shows a similar tendency to reduce the pin power error with respect to each fuel assembly as shown in Figure 12. However, the error in the MOX assemblies is still larger than inside the UO2 assembly, although the error has
210
H . - C . K i m et e l
certainly been substantially reduced. It also seems that all the pin power RMS errors converge to a value different from the reference solution, even though they are calculated with higher quadrature orders.
1.0
0.9-
--m-I ~ •---
•
Square Model 1 Staircase Model
0.80.70.60.5 o
0.4
-'-O .......
•
.....................
0.3
..= 0.2 0.1
0.0 Nl=4
Nu=8
N=Jl2
N~16
S N Order
Fig. 11
Pin Power Error Behaviors as a Function of SN Order for the Two Spatial Models with 6x6 Grid per Pin Cell
1.0 0.9
l - ~ O..,., ~
I - - i - - M O X - SquareModel [ __.o__. MOX _ Staircase Model k
0.8 0.7 0.6 0.5 o
' - o ......
' []~'"--. 0.4- ~ - - . _
"--.
0.3
"'~
"0 .....
- .....................
................. . ... . ..... .
~ . .... ..... . ... . . . . . . . . . . . . . . . ~
•
-. . . . . . . . @
SquareModel -- o--- Inner UO z - Staircase Model --[]-- OuterUO 2 - SquareModel ~-e--- Outer UO z - Staircase Model --m-- Inner UO 2 -
.=. 0.2
0.1 0.0
NU=4
Nl=8
N=ll2
N216
S N Order
Fig. 12
Pin Power RMS Error Behavior as a Function of SN Order in Each Fuel Assembly for the Two Spatial Models with 6x6 Grid per Pin Cell
211
TWODANTso~tions
3.3
Optimal TWODANT solution
From the sensitivity studies on mesh and angular refinements, the benchmark result, obtained from the TWODANT calculation using the staircase fuel pin model with the 6x6 grid per pin cell and $16 quadrature order, was selected for an optimal TWODANT solution and is summarized in Table 3. The reference solution and the three other calculation results were also tabulated for comparison.
Table 3
TWODANT Calculation Results for Eigenvalue and Pin Power Using the Staircase Model Normalized Pin Power
Eigenvalue
Error [%]
1.18655
±0.008
2.498
±0.16
0.232
±0.58
0.34
6x6 Grid, $8 Order
1.18431
-0.19
2.512
0.57
0.232
0.04
0.50
3
6x6 Grid, $16 Order 2)
1.18492
-0.14
2.508
0.41
0.232
0.05
0.37
10
20x20 Grid, Ss Order
1.18417
-0.20
2.510
0.47
0.232
0.02
0.47
204
20x20 Grid, $16 Order
1.18520
-0.11
2.505
0.29
0.231
-0.07
0.31
396
Reference MCNP
Maximum Error Value [%]
Minimum Error Value [%]
RMS [%]
CPU Time [minute(s)]
TWODANT 1)
1) TWODANT results calculated with the staircase fuel pin model 2) The optimal solution
The TWODANT calculation for the optimal solution measured a computational time of about 10 minutes. The results from the staircase model with two 20x20 grids in the table indicates that the computational cost would increase dramatically in order to resolve the remaining inaccuracies in eigenvalue and pin power distribution by further refining the mesh grid. Therefore, the major advantage of the less fine mesh approximation of 6x6 grid is the computational time reduction; the CPU time for the 6x6 grid calculation in the staircase model with S~6 approximation is over 20 times faster than those for the 20x20 grid calculations. The optimal solution gives an eigenvalue of 1.18492 with a good agreement of-0.14% error in comparison with the reference value, even though the absolute value of the error is a little higher than the absolute value of -0.09% presented previously in Table 1. In addition, the eigenvalue error of the optimal solution makes a difference of only 0.03% point from the error in the model with the 20x20 grid and $16 order. The overall pin power distribution agrees well with the reference solution, showing that the 0.37% RMS error of the optimal solution approached nearly the 0.34% RMS value of the reference pin power. However, the maximum pin power with 0.41% error is still beyond the error bounds of the reference value for this pin. The maximum error of pin power is -1.09%. Figure 13 shows the pin power error distribution for the optimal solution. As expected, the pin power error distribution gives a smooth gradient compared with the distribution shown in Figure 3.
H.-C. Kim et el
212
Pin Cell Index in X Direction 17
34
Enor[%] .....
1.5
ii~iiiii!ilili! ii~iiiiiiiiiiiii
!iiiii!i!il °O ~
0.5
"1:3
-0.5
-1.5
Fig. 13
Pin Power Error Distribution from the TWODANT Calculationfor the Staircase Model with 6x6 Grid per Pin Cell and $16 Quadrature Order
4. SUMMARY AND CONCLUSIONS A TWODANT calculation with proper spatial (4x4 grid per pin cell) and angular ($8) approximations has been performed for the 2-D C5G7 benchmark problem. The resulting solution has given a good agreement by showing the eigenvalue of -0.09% error and the overall pin power distribution of less than 1% error in comparison with the reference MCNP solution. In addition, the sensitivity studies on mesh and angular refinements have presented an optimal TWODANT solution with higher quality. The results from the sensitivity study on mesh refinement showed that the eigenvalue of the C5G7 benchmark from the TWODANT calculations by using the square and staircase fuel pin models on the circular fuel pin would not converge to the reference value though the mesh size was increased. Besides, it was found that since the pin power error was roughly independent of the mesh size, the gains from using finer mesh refinements were not as significant as the effect of improving the angular approximation by higher quadrature order. However, it was found that the staircase model could improve the accuracy of the pin power more than the square model. On the other hand, the results from the sensitivity study on angular refinement showed that the accuracy of the TWODANT calculations could be improved by the angular refinement of higher quadrature order. All of the resulting solutions for the benchmark problem exhibited convergence characteristics going toward the reference solution as the quadrature order for the angular refinement was increased. Especially, it was found that the staircase model was very beneficial to the calculation of the pin power distribution, if the loss in accuracy for the eigenvalue was acceptable. In the result, the 6x6-$16 solution presents a good trade-off between accuracy and CPU usage, although the resulting eigenvalue error was a little lower than that of the first solution.
TWODANT solutions
213
Consequently, the above results have demonstrated that in the TWODANT calculation the spatial approximation by the staircase model with higher order of Sr~ is a viable method for solving the 2-D C5G7 benchmark.
ACKNOWLEDGEMENTS The authors wish to acknowledge the financial support from the Innovative Technology Center for Radiation Safety (iTRS)
REFERENCES Cathalau S., Lefebvre J. C., and West J. P. (1996), Proposal for a Second Stage of the Benchmark on Power Distributions within Assemblies, NEA/NSC/DOC(1996)2, Nuclear Energy Agency. LANL (1997), DANTSYS 3.0: One-, Two-, and Three-Dimensional, Multigroup, Discrete Ordinates Transport Code System, RSICC Code Package CCC-547, Los Alamos National Laboratory. Lewis E. E., Smith M. A., Tsoulfanidis N., Palmiotti G., Taiwo T. A., and Blomquist R. N. (2001), Benchmark Specification for Deterministic 2-D/3-D MOX Fuel Assembly Transport Calculations without Spatial Homogenization (C5G7 MOX), NEA/NSC/DOC(2001)4, Nuclear Energy Agency. OECE/NEA (2000), Benchmark on the VENUS-2 MOX Core Measurements, NEA/NSC/DOC(2000)7 Nuclear Energy Agency. OECD/NEA (2003), Benchmark on the Deterministic Transport Calculations Homogenisation, NEA/NSC/DOC(2003)16, Nuclear Energy Agency.
Without Spatial
ELSEVIER
f"--d} D IR E C T "
© 2004 Published by Elsevier Ltd Printed in Great Britain 0149-1970/$ - see front matter
doi:10.1016/j.pnucene.2004.09.010
www.elsevier.com/locate/pnueene
TWODANT SOLUTIONS FOR THE 2-D C5G7 MOX BENCHMARK
HONG-CHUL KIM*, CHI YOUNG HAN, and JONG KYUNG KIM
Department of Nuclear Engineering, Hanyang University 17 Haengdang, Seongdong, Seoul 133-791, Korea *
.
E-mad: [email protected]
ABSTRACT The C5G7 MOX benchmark was proposed to test the ability of commercial transport codes to treat reactor core problems without spatial homogenization. The benchmark requires solutions in the form of normalized pin powers as well as the eigenvalue. In the work, the two-dimensional benchmark calculation using the TWODANT code within the DANTSYS code package has been performed with proper spatial and angular approximations. The TWODANT code solves the multigroup discrete ordinates form of the Boltzmann transport equation in twodimensional geometry. The calculation results show a good agreement in comparison with the reference solution obtained from a seven-group MCNP calculation. In addition, sensitivity studies on mesh and angular refinements have been performed to produce a higher quality solution. In the results, it is found that in the TWODANT calculation the spatial approximation in a staircase form of the circular fuel pin with relatively high quadrature order of SN is a viable method for solving the2-D C5G7 benchmark. © 2004 Published by Elsevier Ltd KEYWORDS C5G7 Benchmark, TWODANT, Discrete Ordinates, Mesh Refinement, Angular Refinement
201
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H.-C. Kim et el
1. INTRODUCTION Systems loaded with plutonium in the form of mixed-oxide (MOX) fuel show somewhat different neutronic characteristics compared with those using conventional uranium fuels. Even though many countries have sufficient experiences in using MOX fuel within certain technical boundaries, it is essential to accurately predict the characteristics of MOX-fuelled systems and to further validate both the nuclear data and the computational methods used if the extension of the current constraints is envisaged. In 2001, the C5G7 MOX benchmark for deterministic 2-D and 3-D MOX fuel assembly transport calculations (Lewis et al, 2001) was proposed by the OECD/NEA in order to test the ability of current available transport codes to treat reactor core problems without spatial homogenization. It is based on a seven-group form of the C5 MOX fuel assembly problem specified in the benchmark calculations in 1996 (Cathalau et al, 1996). The domain is made up of four fuel assemblies and reflector, and proper vacuum and reflective boundary conditions. The benchmark requires solutions in the form of normalized pin powers as well as the eigenvalue. The reference solutions for both 2-D and 3-D problems have been obtained from a seven-group MCNP calculation. The benchmark results were reported in 2003 (OECD/NEA, 2003). In this work, the DANTSYS code package (LANL, 1997) was applied to analyze the 2-D and 3-D C5G7 MOX benchmark problems. The DANTSYS code package is a modular computer program package designed to solve the time-independent, multigroup discrete ordinates form of the Boltzmann transport equation in several different geometries. In this paper, the benchmark results are discussed in detail for 2-D problem only. The TWODANT code within the DANTSYS code package, which solves the twodimensional transport equation in x-y, r-z, and r-theta geometries, was employed to perform the calculation for 2-D problem. To analyze the benchmark problem with the TWODANT code, proper spatial and angular approximations were made. The results were compared with the reference solution for the 2-D problem. In addition, several calculations were performed to investigate the effects of the different spatial and angular approximations on the accuracy. The results from these sensitivity studies were analyzed and discussed.
2. TWODANT MODELAND SOLUTION The benchmark geometry consists of two MOX and two U02 fuel assemblies and reflector as shown in Figure 1. Since TWODANT provides only regular mesh features, the circular fuel pin must be approximated by an X-Y Cartesian grid in order to describe the 17x17 lattice of square pin cells. The simplest square shape was generated by preserving the fuel area as shown in Figure 2. A geometrical model composed of a total of 160x160 meshes, corresponding to a 4x4 grid in each pin cell and 24 mesh cells in each direction of the water reflector, was developed for TWODANT calculation. This model is revised for mesh refinement compared with the model of 86x86 meshes submitted to the benchmark (OECD/NEA, 2003). The angular approximation of $8 quadrature order was applied. The convergence criterion of eigenvalue as well as pointwise flux and fission source was set to be 1.0E-5. All calculations have been performed on a HP C3700 model workstation. The accuracy of the TWODANT calculation is summarized by tabulating the pin power results along with the eigenvalue and by comparing with the 2-D reference solution from the seven-group MCNP calculation, as presented in Tables 1 and 2. In the tables, the percent error information for the reference MCNP values represents the statistical error bounds associated with the 98% confidence interval of the Monte Carlo solution. To assess the overall pin power distribution, the following collective errors were measured: average (AVG), root mean square (RMS), and mean absolute (MAE) errors.
203
TWODANT solutions
Reflective B . ~ MOX
~5
~J MOX
UO~
~ I
Reflector [ ] UO2Fuel [ ] 4.3%MOX Fuel
Vacuum B. C.
[] []
7.0%MOXFuel 8.7%MOXFuel
• •
Guide Tube Fission Chamber
Fig. 1 Core Configuration for the 2-D C5G7 Benchmark Problem
Fig. 2
Pin Cell Represented by a 4 x 4 Cartesian Grid
Table 1 TWODANT Calculation Results for Eigenvalue and Pin Power Error Eigenvalue [%]
Maximum Error Value [%]
Reference MCNP
1.18655 +_0.008
2.498
TWODANT
1.18543
2.524
-0.09
1) Pin cell index in each direction as shown in Figure 1
Normalized Pin Power Minimum Error Minimum Maximum Value [%] Position 1) Position 1)
+_0.16 (4, 5) or (5, 1.05
(4, 5) or (5,
0.232 0.232
+_0.58 (33, 33) 0.03
(33, 33)
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Table 2
TWODANT Calculation Errors for Pin Power Distribution Normalized Pin Power Error Reference MCNP
TWODANT
Overall
Inner U02
MOX
Outer UO2
Overall
AVG Error [%] 1)
0.32
0.64
0.79
0.29
0.62
RMS Error [%] 2)
0.34
0.70
0.88
0.37
0.73
MAE Error [%] 3)
0.27
0.66
0.70
0.29
0.63
1) A VG = ~Nle,, I/N' where N is the number of fuel pins and e,, is the calculated error for the nth pin power. _
2
N
3) MAE= ~ l e l" P,,/N" Pov~, wherep, is the nth pin power and Pavg iS the averagepin power.
The resulting eigenvalue is 1.18543 and gives a good agreement of -0.09% error in comparison with the reference value of 1.18655. The calculated pin powers were normalized to an average pin power. The maximum pin power with 1.05% error is beyond _+0.16% error bounds of the reference value for this pin, but the minimum pin power with 0.03% error is well within _+0.58% error bounds. In addition, this calculation provides the same positions of the maximum and minimum pin powers as the prediction of reference. The overall pin power distribution agrees well with the reference solution, showing AVG, RMS, and MAE errors of less than 1%. The maximum error of pin power is -1.73% at position (32, 5) or (5, 32). This calculation took a computational time of about 1 minute on the computing machine. An investigation into the pin power error distribution in Figure 3 shows that this calculation gives higher values of the normalized pin powers with 0.70% RMS error in the inner UO2 assembly than the reference values. The results again indicate an overall overestimation of the pin powers in the inner UO2 assembly. This trend is lessened when moving toward the neighboring MOX assembly and on the contrary, it gives underestimated pin powers in the MOX assembly with 0.88% RMS error. The pin power distribution of the outer UOz assembly adjacent to the reflector have slightly negative discrepancies and gives more negative values for the UO2 fuel pins near the MOX assembly.
205
TWODANT solutions
Pin Cell Index in X Direction 17
34
Error [%] 1.5
g
0.5
.9
-g
-0.5
r~
-1.5
Fig. 3
Pin Power Error Distribution of TWODANT Calculation
3. SENSITIVITY STUDIES ON MESH AND ANGULAR REFINEMENTS The choice of finer mesh grid and higher angular quadrature order was motivated to produce a higher quality solution. Therefore, sensitivity studies on mesh and angular refinements were performed to improve the TWODANT calculation results reasonably with a better approximation. 3.1
Mesh refinement
Compared with the 4x4 grid as previously approximated in Figure 2, two different forms of grid structure with the same fuel area were characterized on the pin cell domain in order to approximate the circular fuel pin. In the first grid structure, the square fuel pin was constructed instead of the circular fuel pin in the same way as the 4x4 grid, but the pin cell domain was covered by some grid sets of finer refinement such as 6x6, 8x8, 12x12, 16x16, and 20x20 meshes. For convenience, this grid structure was named "square fuel pin model" after the shape of the fuel pin represented by the spatial approximation. Then, the 4x4 grid is the coarsest grid of the square fuel pin model. Some examples of the pin cell grids refined by the square fuel pin model are shown in Figure 4.
(a) 6x6 Fig. 4
(b) 12x12
(c) 20x20
Examples of Pin Cell Grids Refined by the Square Fuel Pin Model
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In the other grid structure, a staircase form of fuel pin was generated rather than creating the square form to represent the circular fuel pin. This grid structure was named "staircase fuel pin model" from the same point of view as the square model. Procedures for the grid sets of the staircase fuel pin, corresponding to the same mesh sizes (numbers) as the square model, were devised as described below: (1) Set an initial point on the circular fuel pin borderline, at which the distance between the fuel pin borderline and the square pin cell line is minimized (Figure 5-(a)). (2) On the basis of the initial point, determine an octant of the circular fuel pin and divide it into some equal parts in fan shape. Obtain an intersection point between the tangent line on the initial point and the second partition line from the initial point (Figure 5-(b)). (3) Consider variable points such that they are placed on the other partition lines and can be used as vertexes for the staircase model (Figure 5-(c)). (4) Correspond the equal fans in the octant to the cells made by the partition lines and the points in steps above. Compute definite positions of the variable points such that the area of each cell is equal to that of the corresponding fan, but the sum of areas in case of the first and second cells (Figure 5-(d)). (5) Complete pin cell grids refined by the staircase fuel pin model after symmetrizing the staircase approximation for the octant.
(a)
(b) Fig. 5
(c)
(d)
Procedures for Mesh Refinement by the Staircase Fuel Pin Model
The procedures are applicable to the pin cell grids such as 8x8, 12x12, 16×16, and 20x20, and the 6x6 pin cell grid can be obtained simply. Some examples of the pin cell grids refined by the staircase fuel pin model are shown in Figure 6. If a 4×4 grid coarser than the 6x6 grid as shown in Figure 6-(a) is considered for the staircase fuel pin model, the 4x4 grid in Figure 2 can be also regarded fairly as the coarsest grid of the staircase model as well as the square model.
(a) 6×6 Fig. 6
(b) 12×12
(c) 20×20
Examples of Pin Cell Grids Refined by the Staircase Fuel Pin Model
TWODANT solutions
207
The calculations for sensitivity study on mesh refinement were also performed with the same quadrature order of $8 as the previous calculation with the 4x4 grid. The behavior of the benchmark eigenvalue was firstly investigated with respect to the spatial approximation. As shown in Figure 7, the mesh refinements by the two fuel pin models provide less accurate results than by the first 4x4 grid. It was, particularly, found that the staircase model underestimated the eigenvalue by about 100 pcm rather than the square model. The sensitivity of the eigenvalue to mesh size per pin cell is not high in the square model, especially, to mesh size over the 6x6 grid in the staircase model. However, the curves of two models appear to converge to a wrong value with mesh refinement. In brief, the results indicate that the low spatial approximation gives good results more or less by accident.
1.1880 --*--
1.1875
Reference
--m-- Square Model 1.1870
--- •--- S t a i r c a s e M o d e l
1.18651.1860 • 1.1855,
1.1850
I
1
~
1
1
III
1.1845 "O-.
_ ...... • ..............
1.1840
• ................
• .......
1.1835 1.1830
I
I
I
I
I
I
4x4
6x6
8x8
12x 12
16x 16
20x20
M e s h S i z e pe r Pin C e l l
Fig. 7
Eigenvalue Behavior as a Function of Mesh Size for the Two Spatial Models with $8 Quadrature Order
In Figure 8, the accuracy behavior of the pin power results is plotted as a function of mesh size per pin cell with the three quantities: AVG, RMS, and MAE errors. The square model does not improve the accuracy of the pin power noticeably though the grid resolution is increased. Moreover, the pin power results do not converge toward the reference solution. The results from the staircase model rev,sal the rapid convergence of the pin power with the mesh refinement of the 6x6 grid, but slower convergence with increasing mesh size. Hence, the 6x6 grid per pin cell is sufficiently refined so that very little benefit is obtained from improving either of the two fuel pin models. In Figure 9, further investigation into the pin power error distribution with respect to each fuel assembly shows that for each fuel pin model, the RMS value of the pin power errors in the MOX assembly is still larger than the ones in the inner or outer UO2 assemblies. The difference in pin power error between the square and staircase models is large in the MOX and inner UO2 assemblies compared with the outer UO2 assembly.
H.-C. Kim et el
208 1,0-
0.90.80.7-
• ". ,'7~'----~.~m ~
m
m---~__~m__
m
0.6.
0.5-
"N
0.2 0.1
........• ............. '0 . . . . . . . . . . .
- ~ : : : : : : : : :. . . . . . . . . . . . . . . .
0.4 0.3
.
• ........... ""--O . . . . . . . . . . . . . .
"::','.::~:-':--.---.--~:-_0?.~?.,_.:::::::~
--t~-- Square Model - A V G m-- S q u a r e M o d e l - R M S Square Model - M R E ---o - Sta ir c a se M o d e l - A V G ~--e--- Staircase M o d e l - R M S ---e--- Sta ir c a se M o d e l - M R E
0.0
4x4
6x6
8x8
12x12
16x16
20x20
Mesh Size per Pin Cell Fig. 8
Pin Power Error Behavior as a Function of Mesh Size for the Two Spatial Models with $8 Quadrature Order
].0" --n-0.90.8 ~,
M O X - Square Model
---o--- M O X - Sta ir c a s e M o d e l
! "~, ~
i
e
i
•
i---
--i
0.7 "',,,
0.6
• . . . . . . . . . . . . . . . . . .
• . . . . . . . . . . . . . . . .•
0.5 " ' " O . . . . . - . . . . . . . . . . O - - . . . . . . . ----..
' 0 . . . . . . . . . . . . . . . . 0 . . . . . . . . . . . . . . rO
~- 0.4 0.3 e~
- - t 3 - - I nne r U • 2 - S q u a r e M o d e l 0.2 0.1
---o--- I nne r U • z - Staircas e M o d e l --~--
Outer U • 2 - Square Model
---e--- O u t e r U • 2 - Sta ir c a se Mode l 0.0 4x4
6x6
8x8
12x 12
16Xl 6
120X20
Mesh Size per Pin Cell
Fig. 9
Pin Power RMS Error Behavior as a Function of Mesh Size in Each Fuel Assembly for the Two Spatial Models with $8 Quadrature Order
209
TWODANT solutions
3.2
Angular refinement
Based on the above results, the sensitivity study on angular refinement was performed with the mesh size of 6x6 grid per pin cell. The angular refinement was applied to all of the square and staircase models with increasing the quadrature order from $4 to $t6 for the traditional built-in quadrature sets provided in the TWODANT by default. Figure 10 shows that all of the TWODANT solutions for the benchmark eigenvalue converge toward the reference MCNP solution as the quadrature order for the angular refinement is increased. Convergence with respect to the quadrature order is, however, not very fast and even the results indicate that substantially more angular refinement is necessary to come within the error bars of the Monte Carlo solution. The difference in eigenvalue between both spatial models is not reduced by even higher order angular approximation and, therefore, the staircase model still underestimates the eigenvalue by about 100 pcm rather than the square model.
1.1880
- e - - Reference --•-- Square Model ---•--- Staircase Model
1.1875, 1.1870, 1.1865 1.1860 ~ 1.1855
f i •
i~
. ~ 1.1850 .... • .................... 1.1845
......... •
• ............................. • .........................
1.1840 1.1835 1.1830 NI=4
Nu=8
N~12
N~16
Sr~Order Fig. 10
Eigenvalue Behavior as a Function of SN Order for the Two Spatial Models with 6×6 Grid per Pin Cell
The RMS error of the pin power distribution is plotted in Figure 11 against increasing quadrature order. The result shows that all of the square and staircase models reduce the RMS error of the pin power distribution with the angular refinement. Especially, it was found that the RMS error of the staircase model was converging to the reference value more rapidly than that of the square model, and ultimately approached nearly 0.34% which is the RMS value of the statistical errors with the 98% confidence interval in the reference pin power distribution. Therefore, provided that the loss in accuracy for the eigenvalue is acceptable, the staircase model with the 6x6 grid per pin cell is very beneficial to the 2-D C5G7 benchmark problem. The downward trend according to the quadrature order in the pin power RMS error shows a similar tendency to reduce the pin power error with respect to each fuel assembly as shown in Figure 12. However, the error in the MOX assemblies is still larger than inside the UO2 assembly, although the error has
210
H . - C . K i m et e l
certainly been substantially reduced. It also seems that all the pin power RMS errors converge to a value different from the reference solution, even though they are calculated with higher quadrature orders.
1.0
0.9-
--m-I ~ •---
•
Square Model 1 Staircase Model
0.80.70.60.5 o
0.4
-'-O .......
•
.....................
0.3
..= 0.2 0.1
0.0 Nl=4
Nu=8
N=Jl2
N~16
S N Order
Fig. 11
Pin Power Error Behaviors as a Function of SN Order for the Two Spatial Models with 6x6 Grid per Pin Cell
1.0 0.9
l - ~ O..,., ~
I - - i - - M O X - SquareModel [ __.o__. MOX _ Staircase Model k
0.8 0.7 0.6 0.5 o
' - o ......
' []~'"--. 0.4- ~ - - . _
"--.
0.3
"'~
"0 .....
- .....................
................. . ... . ..... .
~ . .... ..... . ... . . . . . . . . . . . . . . . ~
•
-. . . . . . . . @
SquareModel -- o--- Inner UO z - Staircase Model --[]-- OuterUO 2 - SquareModel ~-e--- Outer UO z - Staircase Model --m-- Inner UO 2 -
.=. 0.2
0.1 0.0
NU=4
Nl=8
N=ll2
N216
S N Order
Fig. 12
Pin Power RMS Error Behavior as a Function of SN Order in Each Fuel Assembly for the Two Spatial Models with 6x6 Grid per Pin Cell
211
TWODANTso~tions
3.3
Optimal TWODANT solution
From the sensitivity studies on mesh and angular refinements, the benchmark result, obtained from the TWODANT calculation using the staircase fuel pin model with the 6x6 grid per pin cell and $16 quadrature order, was selected for an optimal TWODANT solution and is summarized in Table 3. The reference solution and the three other calculation results were also tabulated for comparison.
Table 3
TWODANT Calculation Results for Eigenvalue and Pin Power Using the Staircase Model Normalized Pin Power
Eigenvalue
Error [%]
1.18655
±0.008
2.498
±0.16
0.232
±0.58
0.34
6x6 Grid, $8 Order
1.18431
-0.19
2.512
0.57
0.232
0.04
0.50
3
6x6 Grid, $16 Order 2)
1.18492
-0.14
2.508
0.41
0.232
0.05
0.37
10
20x20 Grid, Ss Order
1.18417
-0.20
2.510
0.47
0.232
0.02
0.47
204
20x20 Grid, $16 Order
1.18520
-0.11
2.505
0.29
0.231
-0.07
0.31
396
Reference MCNP
Maximum Error Value [%]
Minimum Error Value [%]
RMS [%]
CPU Time [minute(s)]
TWODANT 1)
1) TWODANT results calculated with the staircase fuel pin model 2) The optimal solution
The TWODANT calculation for the optimal solution measured a computational time of about 10 minutes. The results from the staircase model with two 20x20 grids in the table indicates that the computational cost would increase dramatically in order to resolve the remaining inaccuracies in eigenvalue and pin power distribution by further refining the mesh grid. Therefore, the major advantage of the less fine mesh approximation of 6x6 grid is the computational time reduction; the CPU time for the 6x6 grid calculation in the staircase model with S~6 approximation is over 20 times faster than those for the 20x20 grid calculations. The optimal solution gives an eigenvalue of 1.18492 with a good agreement of-0.14% error in comparison with the reference value, even though the absolute value of the error is a little higher than the absolute value of -0.09% presented previously in Table 1. In addition, the eigenvalue error of the optimal solution makes a difference of only 0.03% point from the error in the model with the 20x20 grid and $16 order. The overall pin power distribution agrees well with the reference solution, showing that the 0.37% RMS error of the optimal solution approached nearly the 0.34% RMS value of the reference pin power. However, the maximum pin power with 0.41% error is still beyond the error bounds of the reference value for this pin. The maximum error of pin power is -1.09%. Figure 13 shows the pin power error distribution for the optimal solution. As expected, the pin power error distribution gives a smooth gradient compared with the distribution shown in Figure 3.
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Pin Cell Index in X Direction 17
34
Enor[%] .....
1.5
ii~iiiii!ilili! ii~iiiiiiiiiiiii
!iiiii!i!il °O ~
0.5
"1:3
-0.5
-1.5
Fig. 13
Pin Power Error Distribution from the TWODANT Calculationfor the Staircase Model with 6x6 Grid per Pin Cell and $16 Quadrature Order
4. SUMMARY AND CONCLUSIONS A TWODANT calculation with proper spatial (4x4 grid per pin cell) and angular ($8) approximations has been performed for the 2-D C5G7 benchmark problem. The resulting solution has given a good agreement by showing the eigenvalue of -0.09% error and the overall pin power distribution of less than 1% error in comparison with the reference MCNP solution. In addition, the sensitivity studies on mesh and angular refinements have presented an optimal TWODANT solution with higher quality. The results from the sensitivity study on mesh refinement showed that the eigenvalue of the C5G7 benchmark from the TWODANT calculations by using the square and staircase fuel pin models on the circular fuel pin would not converge to the reference value though the mesh size was increased. Besides, it was found that since the pin power error was roughly independent of the mesh size, the gains from using finer mesh refinements were not as significant as the effect of improving the angular approximation by higher quadrature order. However, it was found that the staircase model could improve the accuracy of the pin power more than the square model. On the other hand, the results from the sensitivity study on angular refinement showed that the accuracy of the TWODANT calculations could be improved by the angular refinement of higher quadrature order. All of the resulting solutions for the benchmark problem exhibited convergence characteristics going toward the reference solution as the quadrature order for the angular refinement was increased. Especially, it was found that the staircase model was very beneficial to the calculation of the pin power distribution, if the loss in accuracy for the eigenvalue was acceptable. In the result, the 6x6-$16 solution presents a good trade-off between accuracy and CPU usage, although the resulting eigenvalue error was a little lower than that of the first solution.
TWODANT solutions
213
Consequently, the above results have demonstrated that in the TWODANT calculation the spatial approximation by the staircase model with higher order of Sr~ is a viable method for solving the 2-D C5G7 benchmark.
ACKNOWLEDGEMENTS The authors wish to acknowledge the financial support from the Innovative Technology Center for Radiation Safety (iTRS)
REFERENCES Cathalau S., Lefebvre J. C., and West J. P. (1996), Proposal for a Second Stage of the Benchmark on Power Distributions within Assemblies, NEA/NSC/DOC(1996)2, Nuclear Energy Agency. LANL (1997), DANTSYS 3.0: One-, Two-, and Three-Dimensional, Multigroup, Discrete Ordinates Transport Code System, RSICC Code Package CCC-547, Los Alamos National Laboratory. Lewis E. E., Smith M. A., Tsoulfanidis N., Palmiotti G., Taiwo T. A., and Blomquist R. N. (2001), Benchmark Specification for Deterministic 2-D/3-D MOX Fuel Assembly Transport Calculations without Spatial Homogenization (C5G7 MOX), NEA/NSC/DOC(2001)4, Nuclear Energy Agency. OECE/NEA (2000), Benchmark on the VENUS-2 MOX Core Measurements, NEA/NSC/DOC(2000)7 Nuclear Energy Agency. OECD/NEA (2003), Benchmark on the Deterministic Transport Calculations Homogenisation, NEA/NSC/DOC(2003)16, Nuclear Energy Agency.
Without Spatial