Applications of lie symmetry techniques to models describing heat conduction in extended surfaces

Mhlongo, Mfanafikile Don
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In this thesis we consider the construction of exact solutions for models describing heat transfer through extended surfaces (fins). The interest in the solutions of the heat transfer in extended surfaces is never ending. Perhaps this is because of the vast application of these surfaces in engineering and industrial processes. Throughout this thesis, we assume that both thermal conductivity and heat transfer are temperature dependent. As such the resulting energy balance equations are nonlinear. We attempt to construct exact solutions for these nonlinear models using the theory of Lie symmetry analysis of differential equations. Firstly, we perform preliminary group classification of the steady state problem to determine forms of the arbitrary functions appearing in the considered equation for which the principal Lie algebra is extended by one element. Some reductions are performed and invariant solutions that satisfy the Dirichlet boundary condition at one end and the Neumann boundary condition at the other, are constructed. Secondly, we consider the transient state heat transfer in longitudinal rectangular fins. Here the imposed boundary conditions are the step change in the base temperature and the step change in base heat flow. We employ the local and nonlocal symmetry techniques to analyze the problem at hand. In one case the reduced equation transforms to the tractable Ermakov-Pinney equation. Nonlocal symmetries are admitted when some arbitrary constants appearing in the governing equations are specified. The exact steady state solutions which satisfy the prescribed boundary conditions are constructed. Since the obtained exact solutions for the transient state satisfy only the zero initial temperature and adiabatic boundary condition at the fin tip, we sought numerical solutions. Lastly, we considered the one dimensional steady state heat transfer in fins of different profiles. Some transformation linearizes the problem when the thermal conductivity is a differential consequence of the heat transfer coefficient, and exact solutions are determined. Classical Lie point symmetry methods are employed for the problem which is not linearizable. Some reductions are performed and invariant solutions are constructed. The effects of the thermo-geometric fin parameter and the power law exponent on temperature distribution are studied in all these problems. Furthermore, the fin efficiency and heat flux are analyzed.
A research thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfillment of the requirement for the degree of Doctor of Philosophy. August 7, 2013.