{"id":15150,"date":"2017-08-11T16:21:42","date_gmt":"2017-08-11T16:21:42","guid":{"rendered":"http:\/\/sitepourvtc.com\/?page_id=15150"},"modified":"2022-10-27T10:54:34","modified_gmt":"2022-10-27T10:54:34","slug":"neutron-diffusion-theory","status":"publish","type":"page","link":"https:\/\/sitepourvtc.com\/nuclear-power\/reactor-physics\/neutron-diffusion-theory\/","title":{"rendered":"Neutron Diffusion Theory"},"content":{"rendered":"
Neutron diffusion theory<\/strong> provides a theoretical basis for neutron-physical computing<\/strong> of reactor cores<\/a>. It uses a diffusion equation to determine the spatial flux distributions within power reactors.<\/div><\/div>\n

In the previous section, we dealt with the multiplication system and defined the infinite<\/a> and finite multiplication factors<\/a><\/strong>. This section was about conditions for a stable, self-sustained fission chain reaction <\/strong><\/a>and maintaining such conditions. This problem contains no information about the spatial distribution of neutrons<\/a><\/strong> because it is a point geometry problem. We have characterized the effects of the global distribution of neutrons simply by a non-leakage probability\u00a0(thermal<\/a> or fast<\/a>), which, as stated earlier, increases toward a value of one as the reactor core becomes larger.<\/p>\n

\"Neutron<\/a>
Solution of the diffusion equation in a multiplying system with a control rod insertion. It is assumed that keff is equal to unity in every state.<\/figcaption><\/figure>\n

To design a nuclear reactor <\/a>properly, predicting how the neutrons<\/strong> will be distributed<\/strong> throughout the system is highly important. This is a very difficult problem because the neutrons interact<\/a> differently with different environments (moderator<\/a>, fuel<\/a>, etc.) in a reactor core. Neutrons undergo various interactions when they migrate through the multiplying system. To a first approximation, <\/strong>the overall effect of these interactions is that the neutrons undergo a kind of diffusion<\/strong> in the reactor core, much like the diffusion of one gas in another. This approximation is usually known as the diffusion approximation,<\/strong> based on the neutron diffusion theory<\/strong>. This approximation allows solving such problems using the diffusion equation<\/strong>.<\/p>\n

In this chapter, we will introduce the neutron diffusion theory<\/strong>. We\u00a0will examine the spatial migration of neutrons<\/strong> to understand the relationships between reactor size<\/strong>, shape<\/strong>, and criticality<\/strong><\/a> and determine the spatial flux distributions within power reactors. The diffusion theory provides a theoretical basis for neutron-physical computing<\/strong> of nuclear cores<\/a>. It must be added many neutron-physical codes are based on this theory.<\/p>\n

First, we will analyze the spatial distributions of neutrons, and we will consider a one-group diffusion theory<\/strong> (mono-energetic neutrons<\/strong>) for a uniform non-multiplying medium<\/strong>. That means that the neutron flux<\/a> and cross-sections<\/a> have already been averaged over energy. Such a relatively simple model has the great advantage of illustrating many important features of the spatial distribution of neutrons without the complexity introduced by the treatment of effects associated with the neutron energy spectrum<\/a>.<\/p>\n

See also: Neutron Flux Spectra<\/a>.<\/p>\n

Moreover, mathematical methods used to analyze a one-group diffusion equation<\/strong> are the same as those applied in more sophisticated and accurate methods such as multi-group diffusion theory<\/strong>. Subsequently, the one-group diffusion theory will be applied in simple geometries on a uniform multiplying medium (a homogeneous \u201cnuclear reactor\u201d). Finally, the multi-group diffusion theory will be applied in simple geometries on a non-uniform multiplying medium (a heterogenous \u201cnuclear reactor\u201d).<\/p>\n

\n
<\/span>Neutron Transport Theory<\/div>
Neutron transport theory<\/strong> is concerned with the transport of neutrons<\/a> through various media. As was discussed, neutrons are neutral particles. Therefore they travel in straight lines<\/strong>, deviating from their path only when they collide<\/strong> with a nucleus to be scattered<\/a> into a new direction or absorbed<\/a>.\n

Transport theory is relatively simple in principle,<\/strong> and an exact transport equation governing this phenomenon can easily be derived<\/strong>. This equation is called the Boltzmann transport equation,<\/strong> and the entire study of transport theory focuses on the study of this equation. In general, the neutron balance can be expressed graphically as:<\/p>\n

\"Boltzmann<\/a>
\nBoltzmann Transport Equation<\/h2>\n

The Boltzman transport equation<\/strong> is a balance statement that conserves neutrons<\/strong>. Each term represents a gain or a loss of a neutron, and the balance, in essence, claims that neutrons gained equals neutrons lost. It must be added that it is much easier to derive<\/strong> the Boltzmann transport equation than to solve it<\/strong>. Deterministic methods<\/strong> solve the Boltzmann transport equation in a numerically approximated manner everywhere throughout a modeled system. This task demands enormous computational resources because the problem has many dimensions. But the Boltzmann transport equation can be treated in a rather straightforward way. This simplified version of the Boltzmann transport equation is just the neutron diffusion equation<\/strong><\/a>. Naturally, many assumptions must be fulfilled when using the diffusion equation, but the diffusion equation usually provides a sufficiently accurate approximation to the exact transport equation.<\/p>\n

Nowadays, reactor core analyses<\/strong> and design can be\u00a0performed using nodal two-group diffusion methods. These methods are based on pre-computed<\/strong> assembly homogenized cross-sections<\/strong> and assembly discontinuity factors<\/strong> (pin factors) obtained by single assembly calculation with reflective boundary conditions (infinite lattice). Two methods exist for the calculation of the pre-computed assembly cross-sections and pin factors.<\/p>\n