Temperature/Heat Analysis of Annular Fins of Hyperbolic Profile Relying on the Simple Theory for Straight Fins of Uniform Profile

This technical brief addresses an elementary analytic procedure for solving approximately the quasi-1D heat conduction equation (a generalized Airy equation) governing the annular fin of hyperbolic profile. The importance of this fin configuration stems from the fact that its geometrical shape and heat transfer performance are reminiscent of the annular fin of convex parabolic profile, the so-called optimal annular fin. To avoid the disturbing variable coefficient in the quasi-1D heat conduction equation, usage of the mean value theorem for integration is made. Thereafter, invoking a coordinate transformation, the product is a differential equation, which is equivalent to the quasi-1D heat conduction equation for the simple straight fin of uniform profile. The nearly exact analytic temperature distribution is conveniently written in terms of the two controlling parameters: the normalized radii ratio $c$ and the dimensionless thermogeometric parameter $M2$⁠, also called the enlarged Biot number. For engineering analysis and design, the estimates of temperatures and heat transfer rates for annular fins of hyperbolic profile owing realistic combinations of $c$ and $M2$ give evidence of good quality.