**Updated 10/8/2015**

**Extended Surfaces (Fins)**

The normal
textbook treatment of heat transfer from fins involves the solution by
analytical means of an ordinary differential equation describing transport by conduction
along the fin and convection from its surface. Even with a straight,
rectangular fin, there is the quandary about what boundary condition to apply
at the tip (which is usually resolved using an extended length), and the
solution is expressed in terms of hyperbolic functions. In the case of straight
fins of triangular cross-section and annular fins of constant thickness, the
solutions are given in terms of Bessel functions, which the student may or may
not have studied. Also in the latter cases he or she never sees the temperature
distribution along the fin; only an overall parameter, the fin efficiency, is
usually graphed.

**Module Description**

In this module a
finite-volume solution of the governing heat balance equation is used instead.
Terms representing conduction into and out of a short representative segment of
the fin and convection from its surface are included and the necessary
derivatives of temperature in the conduction terms are expressed in terms of
changes over small increments. The resulting triadiagonal
system of equations is solved using publicly available software and the student
has the complete temperature distribution at his or her disposal immediately.
With that plot a good or poor fin design is readily apparent and in the latter
case, may be rectified quickly. Since the algorithm does not require the
evaluation of different functions for each distinct geometry,
but only input of the appropriate areas for axial conduction and for convection
to the fluid, this solution may be easily carried out for virtually any
one-dimensional geometry.

The user selects
the fin type (straight rectangular, cylindrical pin, straight annular or
triangular) and a schematic of that geometry pops up so that he or she can see
how the requisite dimensions are defined. More general two-dimensional, spine
and annular fins having a cross-section that may be described by a simple
polynomial are also handled readily. For all geometries the user may input the
thermal conductivity of the material and the surface convection coefficient for
as many as six cases simultaneously. A scaled cross-sectional profile of each
fin is plotted in the left window, while the resulting 1-D temperature
distribution along the fins is plotted in the right window. Numerical values of
the efficiency, effectiveness, total heat transferred and thermal resistance
are reported in tabular form. This module also provides an input sheet so that
data from a simple accompanying desktop experiment may be plotted along with
the numerical predictions. Photos of a number of extended surface heat transfer
applications may be accessed from the module.