VERA Progression Problem 4

MPACT is the deterministic neutronics code for the Virtual Environment for Reactor Applications (VERA). Several benchmark progression problems are defined to assess the capabilities of VERA. Problem 4 is a 3x3 set of 17x17 PWR assemblies, with control rod insertion into the center assembly. The radial and axial geometry are given in Fig. 4.31a and Fig. 4.31b, respectively.

The green assemblies have 20 pyrex burnable absorber rods and 2.6 wt% enriched UO2. The red assemblies have 2.1 wt% enriched UO2. The assembly is divided into 58 axial planes. In the active core region, the MOC planes are 8.065 cm thick, except for the planes with spacer grids, which are 3.81 cm thick.

The MPACT 51-group cross section library (dated 03/26/2018) is used for this problem. The materials, cross sections, radial, and axial dimensions reflect those of a real PWR. This is a main target problem for MPACT. In this sense, it may be a

(a) VERA P4 radial geometry

(b) VERA P4 axial geometry

Figure 4.31: VERA Problem 4 geometry

better test problem to evaluate the effectiveness of these methods for the practical cases in which we are usually interested.

This is a steady-state problem. The control rods are withdrawn in 10% increments from 0% to 100%. The eigenvalue and power shape are computed for each configura- tion and compared to a Monte Carlo reference solution using KENO-VI [71].

However, in the standard model used in MPACT, the control rod tips are partially inserted into the MOC planes. This leads to a rod cusping error, which is much greater in magnitude than the effects treated by the method in this thesis. There are methods implemented in MPACT to reduce the rod cusping errors, such as a

polynomial decusping or subplane decomposition of the MOC plane [30]. Even with these treatments, the remaining error from the rod cusping is large compared to the error addressed by the improved angular coupling method.

If the axial mesh is refined so that the control rod tips align with the MOC plane boundaries, the rod cusping error will be eliminated. However, this makes the planes near the control rod thinner, and the axial TL between the rodded plane and the unrodded plane becomes high. The combination of a thin plane and high axial TL leads to a negative total 2D source, and in turn the 2D/1D iteration fails to converge. This phenomenon was mentioned in Sec. 3.2.1. We need TL splitting for convergence, but the TL splitting method degrades the accuracy of the 2D/1D solution, counteracting the improved angular coupling. As a result, we have to use the original mesh that introduces rod cusping effects.

The errors are given for three methods:

1. ISOTL: isotropic transverse leakage, isotropic cross sections 2. ISOXS: anisotropic transverse leakage, isotropic cross sections

3. POLXS: anisotropic transverse leakage, (polar) anisotropic cross sections Eigenvalue results are given for 20% increments of rod withdrawal in Table 4.10. The polynomial decusping method is used.

Table 4.10: Eigenvalue error, VERA Problem 4

Rod kef f Error [pcm]

Withdrawal KENO-VI ISOTL ISOXS POLXS

0% (in) 0.97241 -45 -41 -43 20% 0.97936 -57 -50 -55 40% 0.99234 -44 -39 -44 60% 0.99803 -51 -48 -51 80% 1.00058 -52 -50 -53 100% (out) 1.00139 -51 -50 -52

Pin power results are given in Table 4.11. The average uncertainty in the KENO- VI pin powers is 0.3%.

From these results, two things are clear:

1. There is a significant error in the MPACT pin power shape, especially for the cases where the rod tip is near the center of the active core (40% and 60%

Table 4.11: Pin power errors, VERA Problem 4

Rod RMS [%] Max [%]

Withdrawal ISOTL ISOXS POLXS ISOTL ISOXS POLXS

0% (in) 0.66 0.68 0.70 7.44 6.09 6.61 20% 1.57 1.22 1.79 5.62 4.58 5.69 40% 1.45 1.55 2.13 13.31 11.71 13.73 60% 1.49 1.68 2.05 10.00 8.30 9.39 80% 0.53 0.58 0.58 3.73 4.35 4.07 100% (out) 0.50 0.60 0.52 3.80 4.33 3.82

2. The effect of the improved methods on the overall power shape is small relative to the errors in the power shape.

Unlike the previous cases we have studied, the results here do not indicate that any one method is significantly better than the others. The eigenvalue effect of the anisotropic TL and XS is less than 5 pcm for each, and the effects seem to approximately cancel in the eigenvalue.

Based on these results, we may infer that the modeling error caused by the par- tially inserted control rod (i.e., rod cusping effect) is sufficiently large to wash out whatever improvement we might have seen with the new, improved angular coupling method. The effect of the new, improved angular coupling method is smaller here than it was in the other cases studied in this chapter. The differences are larger for the pin power errors than the eigenvalue, but there is no consistent trend in the results.

Even for the 100% withdrawn case, which has no rod cusping effects, the new method does not improve the solution, or make it significantly worse. Thus, it is rea- sonable to say that the errors caused by the isotropic TL and XS approximation are negligible in this case, and other approximations are the source of the salient errors (-51 pcm eigenvalue, 0.50% RMS and 3.80% max pin power error). The multigroup approximation may be the main source of this error. Spatial and angular discretiza- tion error may also be contributing.

The axial power shape for the rodded assembly is given in Fig. 4.32a (40% with- drawn) and Fig. 4.32b (100% withdrawn). There are subtle differences for each method, but the main error in the vicinity of the control rod is present with each method. With the control rod withdrawn, the magnitude of the error is much lower. Thus, the overwhelming majority of the error is caused, either directly or indirectly, by the axial heterogeneity introduced by the partially inserted control rod.

The control rod insertion error in the C5G7 benchmark was effectively reduced by the new method, but in this case it is not. The error in this case is much greater, so we might expect the anisotropic TL and XS to be more important. However, the error does not improve or even change in any significant way when using the higher-fidelity method. Something may be fundamentally different about the error in this case.