Question 19.3: Explain why volume averaging the cross sections in a reactor......
Explain why volume averaging the cross sections in a reactor fuel assembly will not conserve the reaction rates when replacing the fuel assembly with a single homogeneous node. Show how this process should be properly performed to ensure that the reaction rates are conserved.
From Equation 19.6, it can be seen that simply volume averaging the cross sections does not conserve the fission or absorption rates unless the neutron flux is the same in the fuel, the moderator, and the cladding. Thus to conserve the reaction rates, the nuclear cross sections must be both volume weighted and flux weighted to create larger homogenous regions of the core or “nodes” that can then be fed into a nodal method. Many reactor design codes (see Table 19.1) also rely on this fact to ensure that the reaction rates are conserved
\sum\limits_{x}^{cell} = \frac{\sum\limits_{x}^{fuel}\phi ^{fuel}V^{fuel} + \sum\limits_{x}^{clad}\phi ^{clad}V^{clad}+ \sum\limits_{x}^{mod}\phi ^{mod}V^{mod}}{\phi ^{cell}V^{cell}} (19.6)
| TABLE 19.1 | |||
| This Table Provides a Summary of Many Popular Neutron Diffusion Theory Codes in Use Today | |||
| Diffusion Theory Code | Primary Developer | Programming Language or O/S |
Function or Purpose |
| 1-DX | Battelle Northwest Lab, Richland, Washington and Argonne National Laboratory, Argonne, Illinois (USA) |
Fortran IV and Assembler Language; IBM 360/370, UNIVAC 1108 |
1DX is a one-dimensional computer program that can be used to produce collapsed group cross sections and self-shielded cross sections for fast reactor calculations |
| CARMEN | Atomic Energy Commission, Madrid, Spain |
FortranV; UNIVAC 1110 | CARMEN is a computer program for PWR analysis that solves the neutron diffusion equation in 1, 2, or 3 spatial dimensions with space-dependent feedback effects dimensions and two energy groups (fast and thermal) |
| CITATION 2D/3D |
Oak Ridge National Laboratory, Oak Ridge, Tennessee, USA | Fortran – IV; Runs on IBM mainframes, CDC machines, MicroVAXs, and IBM PCs |
CITATION is a very sophisticated and flexible computer program designed to solve the neutron diffusion equation in 1, 2, or 3 spatial dimensions with arbitrary group-togroup scattering cross sections. It can be used for x-y-z, θ-r-z, hexagonal-z, and triagonal-z geometries, and has no limit on the number of energy groups that can be used |
| DIFF3D | Nuclear Engineering DivisionArgonne National Laboratory, Argonne, Illinois |
Fortran 90 and C source code for Linux PCs, MacOSX and SUN |
DIFF3D is a widely used neutron diffusion theory program that uses variational nodal methods and finite difference methods to solve the neutron diffusion equation in one-, two-, and three-dimensional geometries |
| FPZD | Marshall Space Flight Center, NASA, Huntsville, Alabama |
FORTRAN 77 and the IBM PC | FPZD is a code system for multigroup neutron diffusion theory and fuel depletion calculations |
| MOSRA | Department of Nuclear Energy System, Japan Atomic Energy Research Institute, Tokai-mura, Japan |
FORTRAN-77/ FORTRAN-90, UNIX workstations, and LINUX PCs |
MOSRA is a high-speed three-dimensional nodal diffusion code system that can be used for three-dimensional x, y, and z geometries |
| PDQ-7 | Bettis Atomic Power Laboratory, Pittsburgh, Pennsylvania, USA |
FORTRAN-IV and FORTRAN-77; IBM mainframes and IBM PCs |
PQD-7 is a few-group neutron diffusion theory code that can be used to predict the flux shape in LWR fuel assemblies for individual fuel pins. It uses the few group cross sections generated by the GAM computer program, which in turn uses the data provided by the ENDF-B database |
| SIMULATE-2 and SIMULATE 3 |
Studsvik Scandpower AB Hantverkargatan 2A,SE-72212 Västerås, Sweden |
FORTRAN-77, IBM and UNIX |
The SIMULATE coarse-mesh nodal code and the PDQ-7 fine-mesh diffusion theory code have been used for many years to synthesize three-dimensional power distributions and flux shapes in pressurized-water reactors. SIMULATE takes the few group cross sections generated by the PDQ-7 code, and uses them to perform burnup and depletion calculations |