Abstract
Explicit analytical expressions for the temperature profile, fin efficiency, and heat flux in a longitudinal fin are derived. Here, thermal conductivity and heat transfer coefficient depend on the temperature. The differential transform method (DTM) is employed to construct the analytical (series) solutions. Thermal conductivity is considered to be given by the power law in one case and by the linear function of temperature in the other, whereas heat transfer coefficient is only given by the power law. The analytical solutions constructed by the DTM agree very well with the exact solutions even when both the thermal conductivity and the heat transfer coefficient are given by the power law. The analytical solutions are obtained for the problems which cannot be solved exactly. The effects of some physical parameters such as the thermogeometric fin parameter and thermal conductivity gradient on temperature distribution are illustrated and explained.
Highlights
Fins are surfaces that extend from a hot object to increase the rate of heat transfer to the surrounding fluid
It is well known that exact solutions for ordinary differential equations (ODEs) such as (4) exist only when thermal conductivity and the term containing heat transfer coefficient are connected by differentiation (or if the ODE such as (4) is linearizable) [4]
We have successfully applied the differential transform method (DTM) to highly nonlinear problems arising in heat transfer through longitudinal fins of various profiles
Summary
Fins are surfaces that extend from a hot object (body) to increase the rate of heat transfer to the surrounding fluid. DTM is a computational inexpensive tool for obtaining analytical solution, and it generalizes the Taylor method to problems involving procedures such as fractional derivative (see e.g., [22,23,24]). The dependency of thermal conductivity and heat transfer coefficient on temperature renders such problems highly nonlinear and difficult to solve, exactly. The DTM is employed to determine the analytical solutions to the nonlinear boundary value problem describing heat transfer in longitudinal fins of rectangular, exponential, and convex parabolic profiles. Both thermal conductivity and heat transfer coefficient are temperature dependent.
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