Application of an Integral Numerical Technique for a Temperature-Dependent Thermal Conductivity Fin with Internal Heat G
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Journal of Engineering Physics and Thermophysics, Vol. 93, No. 6, November, 2020
APPLICATION OF AN INTEGRAL NUMERICAL TECHNIQUE FOR A TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY FIN WITH INTERNAL HEAT GENERATION O. O. Onyejekwe, G. Tamiru, T. Amha, F. Habtamu, Y. Demiss, N. Alemseged, and B. Mengistu
UDC 536.2.001
A numerical study of convective heat transfer in a longitudinal fin with temperature-dependent thermal conductivity and internal heat generation is undertaken. Integral calculations are implemented on each generic element of the discretized problem domain. The resulting systems of nonlinear equations are solved efficiently because of the coefficient matrix sparsity to yield both the dependent variable and its flux. In order to validate the formulation, the effects of the thermogeometric parameter, nonlinearity due to the temperature-dependent thermal conductivity, and of the heat transfer coefficient on the fin temperature distribution are investigated. The results are found to be in agreement with those for similar problems described in the literature and with the physics of the problem. Keywords: convective heat transfer, longitudinal fin, temperature-dependent thermal conductivity, systems of nonlinear equations, discretized problem domain, sparsity of coefficient matrix, integral calculations, generic element, thermogeometric parameter. Introduction. The primary challenge in handling a second-order ordinary differential equation for the fin problem is the presence of nonlinearity arising due to the temperature-dependent thermal conductivity. For considerably long longitudinal fins with uniform cross-sectional circular and rectangular geometries, the temperature variation is assumed to occur only in the axial direction. This consideration explains why most fin heat transfer calculations are one-dimensional and nonlinear, except for the cases when all the thermophysical properties, including the thermal conductivity, are assumed constant [1, 2]. However for most practical applications, the dependence of the heat transfer coefficient on a dependent variable should be expressed in such a way as to accurately represent the energy transfer process according to exponential, linear, or power law. A consideration of these factors with respect to heat transfer in a longitudinal fin constitutes the primary motivation for this study. In addition, we have extended our investigation to include the cases of internal heat generation occurring in current carrying electric arcs, nuclear rods, or any other heat generating components. The studies involving temperature-dependent thermal conductivity and heat transfer coefficient have for a long time provided a fertile ground for researches. Aziz and Benzies [3] applied a regular perturbation technique to study convective heat transfer in a fin with variable thermal conductivity. In a later study, Pakdemirli and Sahin [4] as well as Bokhari, Kara, and Zaman [5] obtained analytic solutions of the fin problem with variable thermal conductivity by the symmetry method. Mebin
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