Numerical modeling of neutron flux in hexagonal geometry. (English) Zbl 1206.82153

Berka, Tomáš; Brandner, Marek; Hanuš, Milan; Kužel, Roman; Matas, Aleš

Chleboun, J. (ed.) et al., Programs and algorithms of numerical mathematics 14. Proceedings of the seminar, Dolní Maxov, Czech Republic, June 1–6, 2008. Prague: Academy of Sciences of the Czech Republic, Institute of Mathematics (ISBN 978-80-85823-55-4). 15-22 (2008).

In this paper the authors study the two-dimensional two-group neutron diffusion model, which is a generalized eigenvalue problem and is given in the following way
\[ \begin{aligned} &\nabla \cdot {\mathbf j}^1(\mathbf x)+\Sigma_r^1(\mathbf x)\phi^1(\mathbf x) =\frac{1}{k_{\text{eff}}} \big[\nu\Sigma_f^1(\mathbf x)\phi^1(\mathbf x)+\nu\Sigma_f^2(\mathbf x)\phi^2(\mathbf x)\big],\\ &\nabla \cdot {\mathbf j}^2(\mathbf x)+\Sigma_r^2(\mathbf x)\phi^2(\mathbf x)=\Sigma_s^{1\to 2}(\mathbf x)\phi^1(\mathbf x).\end{aligned} \tag{1} \]
Here the unknown quantities are the neutron flux \(\phi^g\) (eigenfunction, \(g=1,2\)) and the reactor critical number \(k_{\text{eff}}\) (inverse of the largest eigenvalue). The superscript \(g\) corresponds to the energy group. The term \({\mathbf j}^g\) is the neutron current and it is a link to the neutron flux \(\phi^g\) through the following constitutive relation,
\[ {\mathbf j}^g(\mathbf x)=-D^g(\mathbf x)\nabla\phi^g(\mathbf x). \tag{2} \]
At the boundary is considered the albedo boundary conditions of the form
\[ \gamma\phi^1(\mathbf x)-{\mathbf j}^1(\mathbf x)\cdot {\mathbf n}(\mathbf x)=0,\qquad \gamma\phi^2(\mathbf x)-{\mathbf j}^2(\mathbf x)\cdot {\mathbf n}(\mathbf x)=0, \tag{3} \]
where \(\gamma\) is a given albedo coefficient.
For numerical solution of the problem (1)–(3) is used the conformal mapping method which is a variant of the CMFD (Coarse Mesh Finite Differences) nodal methods suited particularly for solving the neutron diffusion equation on a hexagonal mesh. The method is built upon the conformal mapping. The authors introduce the integral part of nodal methods, the technique of homogenization. It determines how to transform the general equations with variable coefficients to the equations with node-wise constant coefficients. The authors notes that such a procedure is based on conditions of preserving certain physical quantities.
For the entire collection see [Zbl 1194.65013].

Reviewer: Dazmir Shulaia (Tbilisi)

MSC:

82D75

Nuclear reactor theory; neutron transport