VM99

VM99
Temperature Distribution in a Trapezoidal Fin

Overview

Reference:	P. J. Schneider, Conduction Heat Transfer, 2nd Printing, Addison-Wesley Publishing Co., Inc., Reading, MA, 1957, pg. 164, Article 7-8.
Analysis Type(s):	Thermal Analysis (ANTYPE = 0)
Element Type(s):	2D Thermal Solid Elements (PLANE55)
Input Listing:	vm99.dat

Test Case

A rectangular cooling fin with a trapezoidal cross-section dissipates heat from a wall maintained at a temperature T_w. The surrounding air temperature is T_a and the convection coefficient between the fin and the air is h. Determine the temperature distribution within the fin and the heat dissipation rate q.

Figure 139: Trapezoidal Fin Problem Sketch

Material Properties

Geometric Properties

k = 18 Btu/hr-ft-°F

h = 500 Btu/hr-ft²-°F

w = 0.96 in = 0.08 ft

T_w = 100°F

T_a = 0°F

Analysis Assumptions and Modeling Notes

The finite element model is made the same as the reference's relaxation model for a direct comparison. The solution is based on a fin of unit depth (Z-direction). Only half of the fin is modeled due to symmetry. POST1 is used to extract results from the solution phase.

Results Comparison

	Target[1]	Mechanical APDL	Ratio
T, °F (at Node 4)	27.6	27.8	1.01
T, °F (at Node 5)	32.7	32.8	1.00
T, °F (at Node 6)	9.5	9.5	1.00
T, °F (at Node 7)	10.7	10.7	1.00
q, Btu/hr	3545.[2]	3482.	0.982

Based on a numerical relaxation method.
Solution recalculated.