Symmetry reductions and approximate solutions for heat transfer in slabs and extended surfaces
Date
2023-06
Authors
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Publisher
University of the Witwatersrand, Johannesburg
Abstract
In this study we analyse heat transfer models prescribed by reaction-diffusion equations. The focus and interest throughout the work is on models for heat transfer in solid slabs (hot bodies) and extended surface. Different phenomena of interest are heat transfer in slabs and through fins of different shapes and profiles. Furthermore, thermal conductivity and heat transfer coefficients are temperature dependent. As a result, the energy balance equations that are produced are nonlinear. Using the theory of Lie symmetry analysis of differential equations, we endeavor to construct exact solutions for these nonlinear models. We will employ a number of symmetry techniques such as the classical Lie point symmetry methods, the nonclassical symmetry, nonlocal and nonclassical potential symmetry approach to construct the group-invariant solutions. In order to identify the forms of the heat source term that appear in the considered equation for which the principal Lie algebra (PLA) is extended by one element, we first perform preliminary group classification of the transient state problem. Also, we consider the direct group classification method. Invariant solutions are constructed after some reductions have been performed. One-dimensional Differential Transform Method (1D DTM) will be used when it is impossible to determine an exact solution. The 1D DTM has been benchmarked using some exact solutions. To solve the transient/unsteady problem, we use the two-dimensional Differential Transform Method (2D DTM). Effects of parameters appearing in the equations on the temperature distribution will be studied.
Description
A thesis submitted in fulfilment of the degree of Doctor of Philosophy, to the Faculty of Science, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, 2023.
Keywords
Heat transfer, Lie point symmetry, Differential transform methods, Approximate analytical solutions, UCTD
Citation
Nkwanazana, Daniel Mpho. (2023). Symmetry reductions and approximate solutions for heat transfer in slabs and extended surfaces. [PhD thesis, University of the Witwatersrand, Johannesburg]. https://hdl.handle.net/10539/42992