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Multigroup Neutron Diffusion Equation

Let us consider a 1D reactor core which is divided into various energy groups and different regions having constant material properties. The discretization of the reactor core into various groups is shown in Figure 7.1.

Using the previous procedure, the shape functions are multiplied with Equation 7.6, and we will have

Integrating Equation 7.11 over the domain, we get

Further simplification of Equation 7.13 gives a system of algebraic equations. In matrix form, these algebraic equations look like

Here,

[K?] is the stiffness matrix for the coupled neutron diffusion equation {ф} is the fuzzy neutron flux vector

In steady case [Q], the matrix is zero.

Bibliography

Aydin, M. and Atalay, M. A. 2007. Inverse neutron diffusion problems in reactor design. Journal of Nuclear Science and Technology 44:1142-1148.

De Oliveira, C. R. E. 1986. An arbitrary geometry finite element method for multigroup neutron transport with anisotropic scattering. Progress in Nuclear Energy 18:227-236.

Fletcher, J. K. 1981. A solution of the multigroup transport equation using a weighted residual technique. Annals of Nuclear Energy 8:647-656.

Glasstone, S. and Sesonke, A. 2004. Nuclear Reactor Engineering, 4th edn., Vol. 1. CBS Publishers and Distributors Private Limited, New Delhi, India.

Militao, D. S., Filho, H. A. and Barros, R. C. 2012. A numerical method for monoenergetic slab- geometry fixed-source adjoint transport problems in the discrete ordinates formulation with no spatial truncation error. International Journal of Nuclear Energy Science and Technology 7:151-165.

Riyait, N. S. and Ackroyd, R. T. 1987. The finite element method for multigroup neutron transport: Anisotropic scattering in 1-D slab geometry. Annals of Nuclear Energy 14:113-133.

Sjenitzer, B. L. and Hoogenboom, J. E. 2013. Dynamic Monte Carlo method for nuclear reactor kinetics calculations. Nuclear Science and Engineering 175:94-107.

Wood, J. 1986. Multigroup anisotroping scattering in the finite element method. Progress in Nuclear Energy 18:91-100.

Wood, J. and De Oliveira, C. 1984. A multigroup finite element solution of the neutron transport equation—I: X-Y geometry. Annals of Nuclear Energy 11:229-243.

Ziver, A. K. and Goddard, A. J. H. 1981. A finite element method for multigroup diffusion-transport problems in two dimensions. Annals of Nuclear Energy 8:689-698.

 
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