Numerical Results I

Numerical results of this section show the batch eigenvalues, the batch averaged eigenvalues, and the standard deviation for the batch averaged eigenvalues. This first numerical test problem [Kap74] is for a single energy group, critical, homogeneous slab, with isotropic scattering, and vacuum boundary conditions on both ends. The half thickness of the slab is 8.3295135616 cm and the cross sections are $\Sigma_t$= 1.0 cm^-1, $\Sigma_s$ = 0.9 cm^-1 and $\bar\nu\Sigma_f$ = 0.11 cm^-1, as shown in figure 2.4. The Monte Carlo cases shown here are for 120 batches, with the initial 40 batches discarded, each batch consisting of 2000 neutrons. Figure 2.5 shows the plot of batch eigenvalue, batch averaged eigenvalue, and the standard deviation of the batch averaged eigenvalue using source iteration. The value of the batch averaged K after 80 active batches is .99988 ${\pm}$0.56E-3. Figure 2.6 shows the same results utilizing the fission matrix approach, where the value of K is .99983 ${\pm}$0.55E-3. It should be noticed that for both cases the estimated eigenvalue is within one standard deviation of the benchmark
(K = 1.0) result. From now on, only the batch averaged eigenvalue and its associated standard deviation will be provided.

 

Figure: Configuration of Homogeneous Slab.

Table 2.1 compares the source iteration and the fission matrix results for different single energy critical homogeneous slab problems [Kap74] with vacuum boundary conditions on both ends. These problems are similar to that shown in figure 2.4 except different cross sections and half thicknesses. All the Monte Carlo results were produced with 40 inactive batches, 80 active batches and 2000 neutrons per batch. We notice that as the half thickness decreases, in terms of mfps, the standard deviation of the eigenvalue increases, even though all the problems are simulated with the same number of batches and neutrons per batch. This is due to the fact that neutrons leak out more easily from smaller systems and hence fewer neutrons contribute to the eigenvalue estimators.

 

Figure: Eigenvalue Calculation Using Source Iteration Algorithm.

 

Figure: Eigenvalue Calculation Using Fission Matrix Algorithm.

 

$${\frac{\Sigma_s+\bar\nu\Sigma_f}{\Sigma_t}}$$ 1|cHalf thickness (mfp) 1|c|Source iteration 1c|Fission matrix

$${\pm} {\pm}$$ 5.6655054562 1.02

$${\pm} {\pm}$$ 3.300263772 1.05

$${\pm} {\pm}$$ 2.113309666 1.10

$${\pm} {\pm}$$ 1.289379285 1.20

$${\pm} {\pm}$$ 0.93772556 1.30

$${\pm} {\pm}$$ 0.73660355 1.40

$${\pm} {\pm}$$ 0.51196298 1.60

$${\pm} {\pm}$$ 0.31102598 2.00

Next in table 2.2 we show the results for single energy group heterogeneous slab test problems with configurations as shown in figure 2.7 and 2.8. The Monte Carlo eigenvalue results (I_i = 40, I_a = 80, 2500 neutrons per batch) are compared to that of the TWODANT code with S₃₂ quadrature sets and inner and outer iteration convergence criteria of 10^-12.

 

Figure: Configuration of Heterogeneous Slab 1.

 

1|cProblem 1|c|TWODANT 1|c|Source iteration 1c|Fission matrix

$${\pm} {\pm}$$ Hetero-slab 1 0.847805

$${\pm} {\pm}$$ Hetero-slab 2 1.00010

$${\pm} {\pm}$$ Hetero-slab 3 0.925597

 

Figure: Configuration of Heterogeneous Slabs 2 and 3.