From 4d4cd74651ac40fbfef05178be37ea0af8134061 Mon Sep 17 00:00:00 2001 From: Nathan Glaser Date: Thu, 30 May 2024 20:42:06 -0400 Subject: [PATCH] Fix spelling errors from second round of review comments --- doc/content/getting_started/theory.md | 7 +++---- 1 file changed, 3 insertions(+), 4 deletions(-) diff --git a/doc/content/getting_started/theory.md b/doc/content/getting_started/theory.md index 720a28140..2a869e501 100644 --- a/doc/content/getting_started/theory.md +++ b/doc/content/getting_started/theory.md @@ -17,21 +17,20 @@ the MOOSE finite element framework, enabling highly flexible and scalable reacto ## Multigroup Neutron Diffusion The neutron diffusion equation is an approximation to the Boltzmann transport equation, and is derived by taking the zeroth and first moment with respect to $\hat\Omega$, the direction of neutron travel. -A full derivation of the diffusion equation will not be presented here for conciceness. Importantly, when taking the first moment of the transport equation the angular flux is assumed to be linearly anisotropic. +A full derivation of the diffusion equation will not be presented here for conciseness. Importantly, when taking the first moment of the transport equation the angular flux is assumed to be linearly anisotropic. The approximations reduce the phase space through the elimination of angular dependence, but the resulting equation also has reduced fidelity when compared to the transport equation, particularly in regions where the neutron flux has strong angular dependence. A non-exhaustive list of regions in which the neutron flux strongly depends on angle is near material interfaces between materials with highly dissimilar neutronic properties, within strong absorbers, and within near-void regions. -As a dertiministic method, the neutron diffusion method also requires discretization of the continuous energy dependence into energy groups consisting of non-overlapping, finite energy ranges across the entire energy spectrum. +As a deterministic method, the neutron diffusion method also requires discretization of the continuous energy dependence into energy groups consisting of non-overlapping, finite energy ranges across the entire energy spectrum. This energy discretization creates a system of equations referred to as the multigroup neutron diffusion equations: !equation \frac{1}{v_g} \frac{\partial\phi_g}{\partial t}-\nabla \cdot D_g \nabla \phi_g +\Sigma^R_{g} \phi_g =\sum^G_{g \neq g'} {\Sigma^s_{g'\rightarrow g} \phi_{g'}}+ \chi^p_{g} \sum^G_{g'=1} {\left(1- \beta \right) \nu_{g'} \Sigma^f_{g'}\phi_{g'} }+\chi^d_{g} \sum^I_i {\lambda_i C_i} -where the precursor distributions are governed by: +The delayed neutron precursor distributions are governed by: !equation \frac{\partial C_i}{\partial t}=\sum^G_{g'=1}{\beta_i \nu \Sigma^f_{g'}\phi_{g'}}-\lambda_i C_i-\vec{u} \cdot \nabla C_i - Notably, there are two production terms of neutrons, the prompt fission source and delayed neutron precursor decay source. The first term describes the neutrons immediately born from fission, and the second term describes the neutrons born from the radioactive decay of neutron-emitting radionuclides, commonly called delayed neutron precursors. The multigroup neutron diffusion equations are generally impossible to solve analytically for realistic problems and are therefore typically solved with numerical methods.