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Abstract

In the teaching of annular fins of rectangular profile, the main obstacle lies in solving the quasi one-dimensional heat conduction equation, the modified Bessel equation of first kind. In the modified Bessel equation, the variable coefficient $\frac{1}{r}$ multiplying the first order derivative of temperature $\frac{d T}{d r}$ is problematic. The principal objective of the present paper on engineering education is to solve the modified Bessel equation of first kind in approximate analytical form. Specifically, we seek to apply the mean value theorem for integrals to the variable coefficient $\frac{1}{r}$ , viewed as an auxiliary function in the domain of the annular fin extending from the inner radius $r_{1}$ to the outer radius $r_{2}$ . It is demonstrated in a convincing manner that approximate analytical temperature distributions having exponential functions are easy to obtain without resorting to the exact analytical temperature distribution embodying four modified Bessel functions of first kind. Furthermore, the easiness in calculating heat transfer rates in annular fins of rectangular profile for realistic combinations of the two controlling parameters: the normalized radius ratio and the dimensionless thermo-geometrical parameter is verifiable.

Keywords

Annular fin of rectangular profile modified Bessel equation of zero order mean value theorem for integrals transformed ordinary differential equation of second order with constant coefficients approximate analytical temperatures and heat transfer rates

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Approximate analytical treatment of annular fins of rectangular profile for teaching fin heat transfer: Utilization of the mean value theorem for integrals

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