The normal textbook treatment of heat transfer from fins involves the solution by analytical means of a single ordinary differential equation. That equation describes transport by conduction along the fin and convection from its surface. Even with a straight, rectangular fin, there is a quandary about what thermal boundary condition to apply at the tip. Most practitioners resolve that question using an “extended length.” In the case of straight fins of triangular cross-section and annular fins of constant thickness, the textbooks give solutions in terms of Bessel functions, which the student may not have studied. Also, in the latter cases users never see the temperature distribution along the fin. Graphical representation at best shows an overall (and confusing) parameter, the fin efficiency. The major application of extended surface heat transfer is in arrays of fins (heat sinks.)
Module Description
Algorithm
In this module we use a finite-volume solution of the governing heat balance equation instead. The calculation includes terms representing conduction into and out of a short representative segment of the fin and convection from its surface. We express the necessary derivatives of temperature in the conduction terms in terms of changes over small increments. The program solves the resulting tridiagonal system of equations using publicly available software. Unlike conventional methods of extended surface analysis, the student has the complete temperature distribution at their disposal immediately. With that plot he or she can spot a good or poor fin design immediately and, in the latter case, rectify it quickly.
This algorithm does not require the evaluation of different functions for each distinct geometry. The only inputs are the appropriate areas for axial conduction and for convection to the fluid. That being the case, we can implement the calculation easily for virtually any one-dimensional geometry. A separate spreadsheet that shows clearly the procedures for using the finite-volume method on the fin equation (as implemented in HTTextnd) may be downloaded here.
Inputs/Outputs
The user selects the fin type (straight rectangular, cylindrical pin, straight annular or triangular). A schematic of that geometry pops up so that he or she can see how the requisite dimensions are defined. The algorithm also handles more general two-dimensional, spine and annular fins having a cross-section that may be described by a simple polynomial. 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. The output includes a scaled cross-sectional profile of each fin in the left window. Meanwhile, the program provides a plot of the resulting 1-D temperature distribution in the right window. Below the graphs, the program returns numerical values of the fin efficiency (η), effectiveness (ε), total heat transferred, and thermal resistance. A link on the module leads to photos of a number of extended surface heat transfer applications.
Virtual Laboratory
Jorge Navalho has developed a virtual laboratory based on the HTTextnd module. You can find that writeup here.