Nonconforming Three-Dimensional Model for PWR Control Rod Movements Without Homogenization and Cusping Effect

Based on previous work on the IDT solver, enhanced geometrical capabilities have been developed to handle accurate modeling of pressurized water reactors (PWRs). This work particularly focuses on the tridimensional model of the control rods. We propose a transport-based model to avoid the control rod cusping effect generated by homogenizations. Due to the geometry of a PWR core, the height of a node generally ranges from 10 to 20 cm. On the other hand, the control rods move along the vertical direction within the PWR core with axial steps of 1 to 2 cm. As a result, a control rod cluster may be partially inserted into a set of Cartesian nodes, thus requiring a heterogeneous representation of the cross sections along z.

In traditional three-dimensional (3-D) transport codes, piecewise distributed cross sections are not allowed within a single node. Consequently, despite the heterogeneous nature, the node has to be homogenized. Because of sharp axial heterogeneity, the cross sections cannot be represented by polynomial expansions at the interface between the control rod tip and the water. Currently, the solvers of APOLLO3® do not allow for nonconformal geometries along the z-axis, i.e., all two-dimensional regions have the same number of planes. Lately, high-fidelity transport-based core solvers have dealt with this issue by proposing flux-weighting homogenization techniques. Nevertheless, the simple homogenization triggers a discontinuity in the first derivative of the curve representing the variation of the fundamental eigenvalue as the control rods are inserted into the reactor. This nonphysical phenomenon is known as the control rod cusping effect. Several numerical remedies have been proposed over the years to remove it. For instance, in diffusion, the P_N or SPN method, and Fanning and Palmiotti have proposed a heterogeneous variational nodal method (VNM). Another heterogeneous VNM was developed by Smith et al. in 2003. In this method, each heterogeneous node is split into subelements with uniform cross sections. The flux is then developed by finite spatial trial functions. The nodal functional is constructed by the functional of all subelements in the node.

In this work, discrete ordinates linear short characteristics are applied to 3-D heterogeneous Cartesian cells (HCCs). To ensure accurate representation of the control rod movements, each heterogeneous node, represented by a HCC, is the combination of an inner local tridimensional grid and a set of cylinders representing the fuel rod in its exact shape. The new geometrical model generates local nonconforming geometries. Spatial integrals over the HCC are performed using modular tridimensional ray tracing. The ray-tracing technique is based on the combinatorial geometry composed of cylinders and the local XYZ grid. In this manner, a single HCC can be equipped with several local steps to follow exactly the axial displacement of the control rod.

The IDT provides a conformal 3-D Cartesian mesh with heterogeneous nonconformal 3-D nodes to eliminate such an issue. Each node represents a geometrical pattern named HCC. Neither a homogenization procedure nor a mesh adjustment is needed by taking advantage of the geometrical model provided by the HCC. Numerical results show that the HCC transport model can eliminate the cusping effect and obtain accurate power distribution with relatively higher efficiency and accuracy.