Numerical Results

We have used the approach described in previous sections of this chapter to solve multiple eigenvalue perturbation problems from a single Monte Carlo simulation. Monte Carlo $\Delta$Ks are compared to that of the discrete ordinate code TWODANT [O'De82]. To generate N $\Delta$Ks, due to N perturbations, the TWODANT code was run (N+1) times, whereas the Monte Carlo code needed to be run only once. For some of the test problems, we also compare the wall clock time required to compute one $\Delta$K versus N $\Delta$Ks using the Monte Carlo approach. For comparison of wall clock timings we made an effort to choose dedicated machines and hence are able to show timing results only for the test problems that were simulated on dedicated machines. This comparison shows how much reduction in computational effort is achieved from the multiple Monte Carlo perturbation approach. Cross sections of the reference systems for all the test problems are chosen as the average of that of the unperturbed and all perturbed systems, i.e., according to equation (4.2). Then the reference systems $\delta$-scatter was chosen according to equation (4.5) or (4.6).
$\delta$-scatters for the unperturbed and all perturbed systems are then calculated from equation (4.7) and (4.8). Value of $\delta^{ref}$ is given for each case with the perturbation result. The following different test problems, similar to chapter 3 test problems, are studied using the multiple Monte Carlo perturbation approach:

• Test problem 1: homogeneous slab, one energy group
• Test problem 2: heterogeneous slab, one energy group
• Test problem 3: homogeneous X-Y geometry, one energy group
• Test problem 4: heterogeneous X-Y geometry, one energy group
• Test problem 5: homogeneous slab, two energy group
• Test problem 6: homogeneous X-Y geometry two energy group (Millstone assembly cross sections)

Test problem 1: homogeneous slab, one energy group

This case is a one energy group homogeneous slab problem with vacuum boundary conditions on both ends. Dimension and unperturbed cross sections for the slab are shown in table 4.1. Cross section perturbations were done over the entire slab. Calculated results of two $\Delta$Ks from a single Monte Carlo simulation are also shown in table 4.1, along with the corresponding TWODANT results. The Monte Carlo runs utilize 40 inactive batches, 100 active batches, and 2000 neutrons per batch.

 

1|cHomogeneous slab, 1c1 group, 1cT=16 cm 1c|

1|cUnperturbed 1ccross sections: 1c 1c|

$$\Sigma_t \nu\Sigma_f \Sigma_s$$ 1c|

1|cPerturbed 1|c|TWODANT 1|c|Two correlated 1c|error

$$\Delta \Delta$$ 1|ccross sections 1c|(%)

$${\Delta\nu\Sigma}_f {\pm}$$ 0.00903 .04

$${\Delta\nu\Sigma}_f \mp$$ -0.009028 .06

$$\delta^{ref}$$

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ -0.469124 1.7

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ -0.000888 2.5

$$\delta^{ref}$$

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ -0.008807 0.8

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ -0.000888 0.8

$$\delta^{ref}$$

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ -0.000888 0.9

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ 0.000889 1.1

$$\delta^{ref}$$

Test problem 2: heterogeneous slab, one energy group

This is the same problem as test problem 1 of chapter 3. Calculated results of two $\Delta$Ks from a single Monte Carlo simulation are shown in table 4.2 along with the corresponding TWODANT $\Delta$Ks. The Monte Carlo results utilize 50 inactive batches, 100 active batches, and 6000 neutrons per batch.

 

1|cHeterogeneous slab, 1c 1 group; 1c 1c|

1|cT=6.0cm 1c 1c 1c|

$$\Sigma_t {\Sigma_{a}} \nu\Sigma_f {\le} {\le} {\le} {\le}$$ 1c|

$$\Sigma_t {\Sigma_{a}} \nu\Sigma_f$$ 1c all other X 1c 1c|

1|c|Perturbation in 1c|TWODANT 1c|Correlated 1c|error

$${\Delta}K {\Delta}K$$ 1|c| middle 0.4cm 1c|(%)

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ -0.019339 1.6

$${\Delta\Sigma}_s \;$$

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ -0.007613 1.3

$${\Delta\Sigma}_s \; \delta^{ref}$$

Test problem 3: homogeneous X-Y geometry, one energy group

This is test problem 2a of chapter 3. Calculated results of three $\Delta$Ks from a single Monte Carlo simulation are shown in table 4.3 along with the corresponding TWODANT results. The first three Monte Carlo perturbations cases utilize 30 inactive batches, 70 active batches, and 2000 neutrons per batch, while the last case utilize 40 inactive batches, 160 active batches, and 2000 neutrons per batch.

 

1|cHomogeneous X-Y geometry, 1c 1 group 1c 1c|

1|c|Perturbed 1c|TWODANT 1c|Correlated 1c|error

$${\Delta}K {\Delta}K$$ 1|c| cross sections 1c|(%)

$${\Delta\nu\Sigma}_f \mp$$ -0.016475 .11

$${\Delta\nu\Sigma}_f {\pm}$$ 0.0065902 .11

$${\Delta\nu\Sigma}_f {\pm}$$ 0.090615 .11

$$\delta^{ref}$$

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ -0.0036524 1.8

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ -0.007276 1.8

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ -0.0007328 1.8

$$\delta^{ref}$$

$$\Delta\Sigma_a {\Delta\Sigma}_s {\pm}$$ 0.00059756 .12

$$\Delta\Sigma_a {\Delta\Sigma}_s {\pm}$$ 0.0007471 .11

$$\Delta\Sigma_a {\Delta\Sigma}_s \mp$$ -0.0007459 .11

$$\delta^{ref}$$

$$\Delta\Sigma_t {\Delta\Sigma}_s {\pm}$$ 0.026820 .07

$$\Delta\Sigma_a {\Delta\nu\Sigma}_f$$

$$\Delta\Sigma_t {\Delta\Sigma}_s {\pm}$$ 0.005958 2.3

$$\Delta\Sigma_a {\Delta\nu\Sigma}_f$$

$$\Delta\Sigma_t {\Delta\Sigma}_s {\pm}$$ 0.0095791 .01

$$\Delta\Sigma_a {\Delta\nu\Sigma}_f$$

$$\delta^{ref}$$

Test problem 4: heterogeneous X-Y geometry, one energy group

This is test problem 2b of chapter 3. Calculated results of three $\Delta$Ks from a single Monte Carlo simulation are shown in table 4.4 along with the TWODANT $\Delta$Ks. The Monte Carlo cases have 40 inactive batches, 160 active batches, and 2000 neutrons per batch.

 

1|cHeterogeneous X-Y geometry, 1c 1 group 1c 1c|

1|c|Perturbed 1c|TWODANT 1c|Correlated 1c|error

$${\Delta}K {\Delta}K$$ 1|c| cross sections 1c|(%)

$${\Delta\nu\Sigma}_f \mp$$ -0.009909 2.5

$${\Delta\nu\Sigma}_f {\pm}$$ 0.0039554 1.2

$${\Delta\nu\Sigma}_f {\pm}$$ 0.056820 2.0

$$\delta^{ref}$$

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ -0.011035 2.5

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ -0.0044901 3.0

$$\Delta\Sigma_t \Delta\Sigma_a \mp$$ -0.008872 2.6

$$\delta^{ref}$$

$$\Delta\Sigma_t {\Delta\Sigma}_s {\pm}$$ 0.012602 2.5

$$\Delta\Sigma_a {\Delta\nu\Sigma}_f$$

$$\Delta\Sigma_t {\Delta\Sigma}_s {\pm}$$ 0.005765 0.8

$$\Delta\Sigma_a {\Delta\nu\Sigma}_f$$

$$\Delta\Sigma_t {\Delta\Sigma}_s {\pm}$$ 0.001373 .54

$$\Delta\Sigma_a {\Delta\nu\Sigma}_f$$

$$\delta^{ref}$$

Test problem 5: homogeneous slab, two energy group

This case is for two energy group homogeneous slab problem with vacuum boundary conditions on both ends. The slab is 10 cm thick and the unperturbed cross sections for the slab are shown in table 4.5. Cross section perturbations were done over the entire slab. Calculated results of two $\Delta$Ks from a single Monte Carlo simulation are also shown in table 4.5 along with the corresponding TWODANT results. The Monte Carlo results have 30 inactive batches, 100 active batches, and 2000 neutrons per batch.

 

1|cHomogeneous slab, 1c2 group 1c 1c|

$${\Sigma}_{t1} {\nu\Sigma}_{f1} {\Sigma}_{s11} {\Sigma}_{s12}$$

$${\Sigma}_{t2} {\nu\Sigma}_{f2} {\Sigma}_{s21} {\Sigma}_{s22}$$

1|c|Perturbed 1|c|TWODANT 1|c| Correlated 1|c|error

$${\Delta}K {\Delta}K$$ 1|c| cross-sections 1|c|(%)

$${\Delta\nu\Sigma}_{f1} {\pm}$$ 0.1602 .08

$${\Delta\nu\Sigma}_{f2} {\pm}$$ 0.3203 .05

$$\delta^{ref1} \delta^{ref2}$$

$${\Delta\nu\Sigma}_{f1} {\Delta\Sigma_{a1}=.001} {\pm}$$ 0.1427 .04

$${\Delta\nu\Sigma}_{f2} {\pm}$$ 0.3203 .3

$$\delta^{ref1} \delta^{ref2}$$

$${\Delta\nu\Sigma}_{f1} {\Delta\Sigma_{a1}=.001} {\pm}$$ 0.0235 1.3

$${\Delta\nu\Sigma}_{f2} {\pm}$$ 0.3203 .06

$$\delta^{ref1} \delta^{ref2}$$

Test problem 6: homogeneous X-Y geometry (Millstone assemblies' cross sections), two energy group

This is the same as test problem 3b of chapter 3. The unperturbed cross section is for 2.9 w/o, 0 bp Millstone [NRC85] reactor assembly and the perturbed cross sections are for 2.9 w/o, 20 bp and 2.9 w/o, 24 bp assemblies. The cross sections are given in table 4.6 and the $\Delta$K results are given in table 4.7. The Monte Carlo runs utilize 40 inactive batches, 100 active batches, and 3000 neutrons per batch.

 

1|c 1c 1c 1c|

$${\Sigma}_{t1} {\nu\Sigma}_{f1} {\Sigma}_{s11} {\Sigma}_{s12}$$

$${\Sigma}_{t2} {\nu\Sigma}_{f2} {\Sigma}_{s21} {\Sigma}_{s22}$$

1|c 1c 1c 1c|

1|c 1c 1c 1c|

$${\Sigma}_{t1} {\nu\Sigma}_{f1} {\Sigma}_{s11} {\Sigma}_{s12}$$

$${\Sigma}_{t2} {\nu\Sigma}_{f2} {\Sigma}_{s21} {\Sigma}_{s22}$$

1|c 1c 1c 1c|

1|c 1c 1c 1c|

$${\Sigma}_{t1} {\nu\Sigma}_{f1} {\Sigma}_{s11} {\Sigma}_{s12}$$

$${\Sigma}_{t2} {\nu\Sigma}_{f2} {\Sigma}_{s21} {\Sigma}_{s22}$$

1|c 1c 1c 1c|

 

1|cHomogeneous 1c X-Y geometry, 1c|2 group

1|c|TWODANT 1|c| Correlated 1|c|error

$${\Delta}K {\Delta}K$$ 1|c|(%)

$$\mp$$ -0.165136 3.3

$$\;$$

$$\mp$$ -0.194535 3.5

$$\delta^{ref1} \delta^{ref2}$$