Analysis of a Nonlinear Model of Heat Transfer Through a Rectangular Fin: Exact Solutions and Their Multiplicity
-
Mohammad Danish
,
Shashi Kumar
Abstract
Exact analytical solutions for the temperature profile and the efficiency of a nonlinear rectangular fin model have been obtained in the forms of well-known algebraic/non-algebraic functions. In the considered nonlinear fin model, the thermal conductivity and the heat transfer coefficient have been assumed to vary as distinct power-law functions of temperature thereby yielding a nonlinear BVP in a 2nd order ODE (ordinary differential equation). These exact solutions have been obtained by employing the derivative substitution method which not only include the solutions of previously studied simplified cases of the same problem but also the solutions of a similar problem of reaction-diffusion process occurring in a porous catalyst slab. These exact solutions have been successfully validated against their numerical counterparts. Besides, effects of various parameters on the obtained solutions have been studied, and the conditions for their existence, uniqueness/multiplicity and stability/instability are analyzed and discussed in detail.
Nomenclature
heat transfer area, m2
fin thickness, m
convective heat transfer coefficient, J/s. m2.K
thermal conductivity of the system, J/s. m.K
fin length, m
dimensionless parameter
dimensionless fin parameter
fin perimeter, m
temperature, K
distance from the fin tip along the fin length, m
dimensionless function of
- Greek letters
dimensionless parameter
fin efficiency
dimensionless temperature
dimensionless temperature at fin tip
dimensionless distance from the fin tip
- Subscript
- a
ambient
- b
fin base
- min
minimum
- max
maximum
Acknowledgement
One of us (MD) is highly indebted to the parent institution, i. e. AMU, Aligarh, for granting the study leave for pursuing PhD at IITR, Roorkee, during which this study was carried out.
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