4. METHOD OF SOLUTION
The few-group neutron diffusion equation is spatially discretized utilizing the Nodal Expansion Method (NEM). For Cartesian geometry, quartic polynomial expansion for the transverse integrated fluxes are employed. Transverse leakage terms are represented by a quadratic polynomial. For hexagonal geometry, a conformal mapping based hexagonal nodal method is employed. The transverse integrated flux expansion consists of trigonometric, hyperbolic trigonometric, and polynomial functions. The transverse leakage term in the axial direction is presented by a quadratic polynomial for both Cartesian and hexagonal geometries. The quadratic polynomial is also utilized for radial directed leakage in Cartesian geometry but is expressed in terms of the mapping scale function and the physical currents for hexagonal geometry. Discontinuity Factors (DFs) are utilized to correct for homogenization errors.
Transient problems utilize a user-specified number of delayed-neutron precursor groups. Time-dependent inputs include coolant inlet temperature and flow, soluble poison concentration, and control bank positions. Time discretization is done in a fully implicit manner utilizing a first-order difference operator for the diffusion equation. The precursor equations are analytically solved, assuming the fission rate behaves linearly over a time step.
Independent of problem type, an outer-inner iterative strategy is employed to solve the resulting matrix system. Outer iterations can employ Chebyshev acceleration, Weilandt Shift acceleration with flux extrapolation, and the Fixed Source Scaling Technique to accelerate convergence. Inner iterations employ either color line or point successive over relaxation iteration schemes, dependent upon problem geometry. Values of the energy group-dependent optimum relaxation parameter and the number of inner iterations per outer iteration to achieve a specified L2 relative error reduction are determined a priori.
The nonlinear iterative strategy associated with the nodal method is utilized. This has advantages in regard to reducing FLOP count and memory size requirements versus the more conventional linear iterative strategy utilized in the surface response formulation. In addition, by electing to not update the coupling coefficients in the nonlinear iterative strategy, the Finite Difference Method (FDM) representation, utilizing the box scheme, of the few-group neutron diffusion equation results. The implication is that NESTLE can be utilized to solve either the nodal or FDM representation of the few-group neutron diffusion equation.
Thermal-hydraulic feedback is modeled employing a homogeneous equilibrium mixture (HEM) model, which allows two-phase flow to be treated. However, only the continuity and energy equations for the coolant are solved, implying a constant pressure treatment. The slip is assumed to be one in the HEM model. A lumped parameter model is employed to determine the fuel temperature. Decay heat groups are used to model decay heat. All cross sections are expressed in terms of a Taylor's series expansion in coolant density, coolant temperature, effective fuel temperature, and soluble poison number density.