**1. NAME AND TITLE**

NESTLE 5.2.1: Code System to Solve the Few-Group Neutron Diffusion Equation Utilizing the Nodal Expansion Method (NEM) for Eigenvalue, Adjoint, and Fixed-Source Steady-State and Transient Problems.

**2. CONTRIBUTORS**

North Carolina State University, Raleigh, North Carolina.

Idaho National Engineering and Environmental Laboratory, Idaho Falls, Idaho.

Los Alamos National Laboratory, Los Alamos, New Mexico.

Oak Ridge National Laboratory, Oak Ridge, Tennessee.

**3. CODING LANGUAGE AND COMPUTER**

Unix Workstations (SUN and IBM RS/6000); Fortran 77 (C00641MNYCP04).

**4. NATURE OF PROBLEM SOLVED**

NESTLE solves the few-group neutron diffusion equation utilizing the NEM. The NESTLE code can solve the eigenvalue (criticality), eigenvalue adjoint, external fixed-source steady-state, and external fixed-source or eigenvalue initiated transient problems. The eigenvalue problem allows criticality searches to be completed, and the external fixed-source steady-state problem can search to achieve a specified power level. Transient problems model delayed neutrons via precursor groups. Several core properties can be input as time dependent.

Two- or four-energy groups can be utilized, with all energy groups being thermal groups (i.e., upscatter exits) if desired. Core geometries modeled include Cartesian and hexagonal. Three-, two-, and one-dimensional models can be utilized with various symmetries. The thermal conditions predicted by the thermal-hydraulic model of the core are used to correct cross sections for temperature and density effects. Cross sections are parameterized by color, control rod state (i.e., in or out), and burnup, allowing fuel depletion to be modeled. Either a macroscopic or microscopic model may be employed.

In April 2004, RSICC updated to NESTLE Version 5.21, which is a modification of version 5.2.0 and includes the following corrections:

1. Corrected bugs in reading of initial number densities

(main.f, file_numden.f)

2. Corrected indexing of alphai array (starter.f)

3. Averaged DL* array for NEM stability for hexagonal core calculation

(geometry.f, nonnemh.f)

4. Corrected bugs in ABD and AB arrays (chain.f and burnnode.f)

**5. 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 *L*_{2} 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.

**6. RESTRICTIONS OR LIMITATIONS**

The nodal option works only for two- and four-energy groups, and depletable isotopes are limited to the principle fissile and fertile isotopes and lumped and transient fission products.

**7. TYPICAL RUNNING TIME**

Test cases ran in ~6 minutes on an IBM RS/6000 Model 270.

**8. COMPUTER HARDWARE REQUIREMENTS**

NESTLE has been tested on SUN and IBM/6000 Unix workstations.

**9. COMPUTER SOFTWARE REQUIREMENTS**

A Fortran 77 compiler is required; the package contains no executables. This release has not been implemented on personal computers. At NCSU, NESTLE was run on a 440MHz Sun Ultra10 under Solaris with SUN's Fortran compiler. At RSICC, it was tested on an IBM RS/6000 Model 270 under AIX 5.1 with xlf Fortran 77 Version 8.1.0.2 and on a Sun UltraSparc 60 under SunOS 5.6 with F77 Version 5.0

**10. REFERENCES**

** a: included in documentation:**

"NESTLE (V5.2.1) Few-Group Neutron Diffusion Equation Solver Utilizing the Nodal Expansion Method for Eigenvalue, Adjoint, Fixed-Source Steady-State and Transient Problems" Electric Power Research Center, North Carolina State University (no report number) (Revised July 2003).

** b: background information:**

Y.A. Chao and Y.A.Shatilla, "Conformal Mapping and Hexagonal Nodal Methods-II: Implementation in the ANC-H code," Nucl.Sci.Eng.,121,210-225 (1995).

M. Knight, P. Hutt, and I.Lewis, "Comparison of PANTHER Nodal Solutions in Hexagonal-z Geometry," Nucl. Sci. Eng., 121, 254-263 (1995).

P. J. Turinsky, R. M. K. Al-Chalabi, P. Engrand, H. N. Sarsour, F. X. Faure, and W. Guo, "NESTLE: A Few-Group Neutron Diffusion Equation Solver Utilizing the Nodal Expansion Method (NEM) for Eigenvalue, Adjoint, and Fixed-Source Steady-State and Transient Problems," EGG-NRE-11406 (June 1994).

**11. CONTENTS OF CODE PACKAGE**

The package is transmitted on a CD in a GNU compressed tar file. The disk contains the referenced document in 10.a plus source code, makefile, and test problem input/output.

**12. DATE OF ABSTRACT**

January 1996, revised December 1996, August 1999, March 2003, April 2004.

**KEYWORDS:** ADJOINT; COMPLEX GEOMETRY; CRITICALITY CALCULATIONS;
DIFFUSION THEORY; WORKSTATION; REACTOR PHYSICS