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# Heat Transfer and Temperature Distribution in a Fin

This Demonstration calculates temperature as a function of distance from the base of a fin of uniform cross-sectional area. The fin is attached to a base, which has the same cross-sectional area and is at a constant temperature of 100 °C. Heat loss from the fin is by natural convection to the surrounding air, which is at 25 °C. The rate of heat transfer is calculated for an adiabatic tip (no heat transfer through the tip surface) and for a tip that has heat transfer by convection. The temperature decreases down the fin due to conduction and to heat being lost by convection. The conduction is assumed to be one-dimensional along the length of the fin. Use sliders to vary the dimensions of the fin.
Use buttons to select the fin cross-section (rectangular or pin fin). Select one of three fin materials (with different thermal conductivities) with buttons. Select "fin" to view a three-dimensional representation of the fin. Select "graph" to view a plot of temperature versus distance from the base; note that the plot range changes for certain conditions.
The heat transfer rates (for adiabatic tip or tip with convection) are shown in the plot. The temperature distribution is shown on the fin surface using a color scale (red is hottest, blue is coolest), and the temperature of the fin tip is shown below the fin along with the temperature legend.

### DETAILS

The axial temperature distributions for a fin of uniform cross section for an adiabatic tip and a tip with convection are:
,
,
,
for a rectangular fin:
,
,
for a pin fin:
,
,
where and are the base and ambient air temperatures, is fin length (m), is position down the fin (m), is the convection heat transfer coefficient, is the thermal conductivity of the material (W/(m K)), is a simplification term, is fin perimeter (m), is the fin cross-sectional area (), and are the width and height of the rectangular fin (m) and is the pin fin diameter (m).
The fin heat transfer rates for each tip condition are:
,
,
,
where is in W, and is a simplification term.
The thermal conductivities for the fin materials are:
,
,
.
Reference
[1] T. L. Bergman, A. S. Lavine, F. P. Incropera and D. P. DeWitt, Introduction to Heat Transfer, 6th ed., Hoboken: John Wiley and Sons, 2011 p. 161.

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