{"id":15629,"date":"2017-10-13T17:15:52","date_gmt":"2017-10-13T17:15:52","guid":{"rendered":"http:\/\/sitepourvtc.com\/?page_id=15629"},"modified":"2022-10-31T13:28:51","modified_gmt":"2022-10-31T13:28:51","slug":"numerical-solution-diffusion-equation","status":"publish","type":"page","link":"https:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/neutron-diffusion-theory\/numerical-solution-diffusion-equation\/","title":{"rendered":"Numerical Solution of Diffusion Equation"},"content":{"rendered":"
The design<\/strong> and safe operation<\/strong> of nuclear reactors<\/a> is based on detailed and accurate knowledge of the spatial<\/strong> and temporal<\/strong> behavior of the core power distribution<\/strong> everywhere within the core. This knowledge is necessary to ensure that:<\/p>\n Nowadays, reactor core analyses and designs are often performed using nodal two-group diffusion methods<\/strong>. These methods are based on pre-computed assembly homogenized cross-sections<\/strong>, diffusion coefficients,<\/strong> and assembly discontinuity factors<\/strong> (pin factors) obtained by single assembly calculation with reflective boundary conditions (infinite lattice). Highly absorbing control elements are represented by effective diffusion theory cross-sections, which reproduce transport theory<\/a> absorption rates. These pre-computed data<\/strong> (discontinuity factors, homogenized cross-sections, etc.) are calculated by neutron transport codes<\/strong> based on a more accurate neutron transport theory<\/strong><\/a>. In short, neutron transport theory is used to make diffusion theory work<\/strong>. The neutron transport equation is the most fundamental and exact description of the distribution of neutrons in space, energy, and direction (of motion). It is the starting point for approximate methods.<\/p>\n Two methods exist for the calculation of the pre-computed assembly cross-sections and pin factors.<\/p>\n The diffusion equation<\/a> can be derived by adding an additional assumption that the angular flux has a linearly anisotropic directional dependence in isotropic sources and scattering problems. \u00a0This allows the removal of the directional variables from the neutron density and simplifies the governing equation and associated numerical methods.<\/p>\n In common practice, these methods can be divided into two classes<\/strong> based on the relative spatial mesh size<\/strong> over which the numerical approximations are valid.<\/p>\n The steady-state core analysis of both new reactor designs and the reload cores of operating reactors involves many\u00a0<\/strong>whole-core calculations to optimize loading patterns and determine to reload safety parameters. It must be noted that:<\/p>\n As a result, at the current speed of computational machinery, it is absolutely impractical to perform all of these calculations by applying fine-mesh transport methods<\/strong> to a model containing detail at the level of individual fuel rods, control elements, and coolant regions in an entire reactor core.<\/p>\n Reference: Scott W. Mosher, A Variational Transport Theory Method for Two-Dimensional Reactor Core Calculations. Georgia Institute of Technology, 2004.<\/p>\n For this reason, nodal methods<\/strong> are currently widely used to predict the neutronic behavior of a reactor core<\/strong>. In general nodal methods<\/strong> are based on a multi-phase approach:<\/p>\n The efficiency of nodal methods is very high and comparable with the finite difference methods. It is very fast and allows us to compute such a huge number of reactor states<\/strong>. It must be added, using this method, one can get only approximate information about the neutron flux<\/a> in a single node (or coarse-mesh area, usually a single fuel assembly of a 20 cm height). This analysis requires many subsequent calculations of the flux and power distributions for the fuel assemblies while there is no need for detailed distribution within the assembly. For obtaining detailed distribution within the assembly, the heterogeneous flux reconstruction<\/strong> must be applied. However, homogenization of fuel assembly properties, required for the nodal method, may cause difficulties when applied to fuel assemblies with many burnable absorber rods due to a very high absorption cross-section <\/a>(especially with Gd <\/a>– burnable absorbers) and very strong heterogeneity within the assembly.<\/p>\n These methods are very efficient and accurate when applied to the current Pressurized Water Reactors (PWRs) <\/a>or Boiling Water Reactors (BWRs)<\/a>.<\/p>\n Core design calculations<\/strong> are a challenging reactor engineering discipline. Such calculations are reactor-specific, and therefore they cannot be transferred from one power plant to another (especially if they have different reactor types). The fuel requirements cannot be based on estimations because the core design has too many variables and restrictions. The nuclear calculations consist of the following aspects:<\/p>\n A mid-term analysis<\/strong> of reloading strategy comprises calculations of the fuel requirements for several reloads in a row<\/strong> using simple nuclear codes based on point kinetics and linear reactivity model<\/strong>. This analysis aims to optimize a number of fresh fuel assemblies, their enrichment, and neutron leakage from the reactor core during several years (Mid-term analysis).<\/p>\n The proposal of a reference loading pattern<\/strong> or transition to a new fuel strategy is based on searching the loading patterns using 3D computational codes. Such outputs are crucial in respect of entire nuclear calculations. They provide detailed knowledge about the behavior of the reactor core during the fuel cycle. The output consists of the proposed fuel loading pattern, which must meet energy \u00a0(cycle length on full power) and all safety requirements such as power distribution<\/strong>, peaking factors<\/strong>, reactivity feedbacks<\/strong>. These calculations can be extended by a cycle optimization, especially meaning searching the low leakage loading patterns with enhanced neutron and fuel economy.<\/p>\n Each change in the project or operation of nuclear power plant requires safety assessment, let alone such a significant change as is the switching of the fuel cycle strategy. The type of the particular safety assessment always depends on the nature of the change. Such calculations are then absolutely crucial for the whole middle part of the fuel cycle. In general, it must be proven that the new fuel strategy meets all safety criteria from the Safety Analysis Report (SAR)<\/strong>. These criteria are divided into three areas:<\/p>\n\n
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Nodal Method in Neutron Diffusion<\/h2>\n
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Core Design and Nuclear Calculations<\/h2>\n
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