Analysis of the nine-point finite difference approximation for the heat conduction equation in a nuclear fuel element

摘要

The time dependent heat conduction equation in the x-y Cartesian geometry is formulated in terms of a nine-point finite difference relation using a Taylor series expansion technique. The accuracy of the nine-point formulation over the five-point formulation has been tested and evaluated for various reactor fuel-cladding plate configurations using a computer program. The results have been checked against analytical solutions for various model problems;The following cases were considered in the steady-state condition: (a) The thermal conductivity and the heat generation were uniform. (b) The thermal conductivity was constant, the heat generation variable. (c) The thermal conductivity varied linearly with the temperature, the heat generation was uniform. (d) Both thermal conductivity and heat generation vary;In case (a), approximately, for the same accuracy, 85% fewer grid points were needed for the nine-point relation which has a 14% higher convergence rate as compared to the five-point relation. In case (b), on the average, 84% fewer grid points were needed for the nine-point relation which has a 65% higher convergence rate as compared to the five-point relation. In cases (c) and (d), there is significant accuracy (91% higher than the five-point relation) for the nine-point relation when a worse grid was used;The numerical solution of the nine-point formula in the time dependent case was also more accurate and converges faster than the numerical solution of the five-point formula for all comparative tests related to heat conduction problems in a nuclear fuel element.