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

##### Date

2014-01-09

##### Authors

Mhlongo, Mfanafikile Don

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##### Abstract

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.

##### Description

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.