Multiple K Calculation Using CSFM

We have shown in the previous sections of this chapter that multiple $\Delta$Ks can be calculated from a single Monte Carlo simulation with good accuracy using the CSFM approach. In this section we will utilize this to calculate multiple Ks and show the applicability of this method for design calculations. Our purpose is to apply the CSFM method to evaluate eigenvalues of multiple systems that closely resemble each other, i.e., all the different systems are slightly perturbed versions of a single system. This single system can be referred to as the unperturbed system. We assume that the eigenvalue of the unperturbed system is determined with arbitrary accuracy from either a S_N or a Monte Carlo calculation or perhaps an analytical solution if the unperturbed system is simple enough. The CSFM technique can then be applied to compute multiple $\Delta$Ks relative to the unperturbed system's eigenvalue. Using the known K of the unperturbed system and the multiple $\Delta$Ks computed by the CSFM method, we can evaluate the absolute Ks of the multiple perturbed systems.

The test problem is a 5cm X 5cm square region with one group cross sections and vacuum boundaries. The square region is divided into 25 square cells of dimension 1cm X 1cm. The 16 cells along the boundaries represent fuel cells, and the center cell represents poison. Rest of the 8 cells represent moderator. The objective of this design problem is to observe the effect of varying the scattering ratio of the poison material on the eigenvalue of the system. The geometric configuration and cross sections for the problem are shown in figure 4.1.

 

Figure: Dimension and Cross Sections for K Calculation Problem.

According to a fine mesh (S₁₆) TWODANT calculation, with inner and outer iteration convergence criteria of 10^-12, the K for the unperturbed system is 1.03006. Next, the CSFM method is used to compute the three $\Delta$Ks corresponding to the three different scattering ratios of the poison material at the center cell of the system. Using these three $\Delta$Ks and the known K of the unperturbed system, we evaluate the three Ks. These results are shown in table 4.9. The Monte Carlo runs utilize 100 inactive batches, 260 active batches, and 4000 neutrons per batch. We have also calculated the three eigenvalues corresponding to the three different scattering ratios of the poison material using fine mesh S₁₆ TWODANT simulations. These TWODANT results are also shown in table 4.9 along with relative errors ${\left({\frac{TWODANT(K) - Monte Carlo(K)}{TWODANT(K)}}\right)}$. The Monte Carlo eigenvalues agree within one tenth of a percent with the fine mesh TWODANT eigenvalues. This example shows application of the CSFM approach for design problems.

 

1|c|Cases 1|c|TWODANT K 1|c| Monte Carlo K 1|c|error (%) Case 0: poison

$$\frac{0.1}{4.0}$$ 1.03006 1.03006 Case 1: poison

$$\frac{0.7}{4.0} {\pm}$$ 1.031031 0.003

$$\;$$ Case 2: poison

$$\frac{1.3}{4.0} {\pm}$$ 1.032291 0.006

$$\;$$ Case 3: poison

$$\frac{1.9}{4.0} {\pm}$$ 1.034017 0.01