1.8.1 Introduction

Sections 1.6 and 1.7 showed how the differential equation describing constant cross-section fins and other extended surfaces is derived. Analytical solutions for these differential equations take the form of an exponential function. In this section, extended surface problems are considered for which the cross-sectional area for conduction and the wetted perimeter for convection are not constant. The resulting differential equation is solved by Bessel functions.

1.8.2 Series Solutions

It is worthwhile asking what the exponential function really is; we take it for granted in terms of its properties (i.e., how it can be integrated and differentiated). With some experience, it is possible to see that it solves a certain type of differential equation. In fact, that is its purpose: the exponential is really a polynomial series that has been defined so that it solves a commonly encountered differential equation. There are other types of differential equations that appear in engineering problems; series solutions to these differential equations have been defined and given formal names like Bessel function and Kelvin function .

The homogeneous differential equation that results from the analysis of a constant cross-sectional area fin is derived in Section 1.6.3

Provided that the solution to Eq. (1-325) is continuous, it can be represented by a series of the form:

By substituting Eq. (1-326) into Eq. (1-325), it is possible to identify the characteristics of the series that solves this class of differential equation. The second derivative of the solution is...