A solution of the neutron diffusion equation in hemispherical symmetry using the homotopy perturbation method

Abstract

abstract The homotopy perturbation method is used to formulate a new analytic solution of the neutron diffusionequation both for a sphere and a hemisphere of fissile material. Different boundary conditions are inves-tigated; including zero flux on boundary, zero flux on extrapolated boundary, and radiation boundarycondition. The interaction between two hemispheres with opposite flat faces is also presented. Numericalresults are provided for one-speed fast neutrons in 235 U. A comparison with Bessel function based solu-tions demonstrates that the homotopy perturbation method can exactly reproduce the results. The com-putational implementation of the analytic solutions was found to improve the numeric results whencompared to finite element calculations. 2009 Published by Elsevier Ltd. 1. IntroductionThere has been variety of methods used over the last decades tosolve the one-group neutron diffusion equation. Up to our knowl-edge, the nonlinear analytical homotopy perturbation method(HPM) has never been tested as a means to solve reactor physicsproblems. The merit of this contribution is that it provides adescription for the application of HPM to solve time-independentneutron diffusion equations, for different geometries at differentboundary conditions. These boundary conditions are the zero fluxat the boundary ZF, the zero flux at the extrapolated boundaryEBC, and the radiation boundary condition RBC. The numerical re-sults obtained in this work are compared to the outcome of otherBessel function based solutions, both obtained in this work and re-ported by Cassell and Williams (2004), in addition to transport the-ory calculations reported in the same reference. The aim of thiscontribution is to improve, or at least widen the methodologiesused to investigate the flux behavior inside the reactor core.The general theory including the essential HPM derivations, andits application to the sphere and hemisphere are given in Section 2.The RBC boundary condition is also detailed in order to set thegrounds for the calculations. Numerical results are provided forcritical radius calculations, as well as for flux distribution in thecritical hemisphere (Section 3). The accompanying technical issuesfaced during the computations are described together with theircorresponding remedies.2. Theory2.1. The homotopy perturbation method (HPM)The HPM, proposed first by He (1999, 2000) for solving linearand nonlineardifferentialand integral equations, has beenthe sub-ject of extensive analytical and numerical studies. The method,which is a coupling of a homotopy technique and a perturbationtechnique, deforms continuously to a simple problem which is eas-ily solved. This method, which does not require a small parameterin an equation, nor it incorporates unjustified assumptions, has asignificant advantage in that it provides an analytical approximatesolution to a wide range of nonlinear problems in applied sciences,see for example Ganji et al. (2007), Chowdhury and Hashim (2008),Marinca and Hericanu (2008), Cveticanin (2009) and referencestherein. The HPM yields the solution in terms of a rapid convergentseries with easily computable components. For linear equationsthe method gives exact solution, and for nonlinear equations itprovides approximate solution with fairly good accuracy (He,2006). Analysis of the method and different illustrative examples(He, 1999, 2000, 2003, 2006; Odibat, 2007) present HPM as a pro-missing means to solve general differential equations.The HPM is applied to neutron diffusion in a bare sphere andhemisphere, in Sections 2.2 and 2.3, respectively.2.2. Bare sphereWe will consider first the rather simple example of a barespherical reactor of radius a. The spherical symmetry of the systemimplies that the flux is a function of r only. The time-independentdiffusion equation can thus be written as