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Pedagogical implications in the thermal analysis of uniform annular fins: Alternative analytic solutions by series

The dimensionless temperature along annular fins of uniform thickness and constant physical properties is governed by an ordinary differential equation of second order with variable coefficients, called the modified Bessel equation of zero order. This educational article addresses two alternative computational procedures of approximate nature for solving this modified Bessel equation: (a) the Frobenius method, and (b) the power series method coupled with a curve fit. Approximate temperature distributions of good quality have been obtained with available symbolic algebra software such as Maple or Mathematica on a personal computer. The primary specification when designing annular fins of uniform thickness is the amount of heat transfer from a tube to the surrounding fluid. These calculations are traditionally done with the help of a fin efficiency diagram. The fin efficiencies can be computed approximately by integration or by differentiation via the Frobenius and power series methods as well. Students of heat transfer courses can benefit from these two alternative computational procedures that seek to circumvent the use and operations with Bessel functions and still produce approximate analytic results of good caliber. A course on ordinary differential equations is the only mathematical background that students need to have to implement these computational techniques. © 1998 John Wiley & Sons, Inc. Comput Appl Eng Educ 6: 31–40, 1998