# Heat Transfer in Fins

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This Demonstration calculates the heat transfer rate through a single fin (either a rectangular or a pin fin) mounted on a heat sink at 500 K. Air flows laminarly across the fin in the direction indicted by the arrows in the figure (rotate the fin with a mouse to help visualize the flow pattern). The fin tip is adiabatic. Use sliders to vary the fin length and the air temperature. Select the fin material from the drop-down menu. 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 on the scale. Both the rectangular fin thickness and the pin fin diameter are 1.5 mm. The heat transfer rate through the fin is displayed above the fin, as is the fin effectiveness, which is the ratio of the fin heat transfer rate to the heat transfer rate without the fin.

Contributed by: Rachael L. Baumann (September 2016)

Additional contributions by: John L. Falconer

(University of Colorado Boulder, Department of Chemical and Biological Engineering)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The heat transfer coefficient of air is calculated using Nusselt correlations for laminar flow over a flat plate and a cylinder. Air properties are evaluated at 300 K. The average Nusselt and Reynold's numbers are:

, for a rectangular fin of width (m),

, for a pin fin of thickness/diameter (m),

and is the thermal conductivity of air (W/[m K]), is the velocity of air (m/s), is the kinematic viscosity of air () and is in units of .

For a rectangular fin:

,

for a pin fin:

,

and is the unitless Prandtl number:

,

where is the heat capacity of air (J/[kg K]), and is the dynamic viscosity of air ().

For a fin with a tip that is assumed to be adiabatic:

,

,

where is temperature (K), is the ambient air temperature, is the base temperature, is a simplification term, is the fin length (m), is distance down the fin, is the fin heat transfer rate (W), is the fin perimeter (m), is the fin cross-section area () and is the thermal conductivity of the material (W/[m K]), and .

For a rectangular fin:

,

.

For a pin fin:

,

.

The fin effectiveness is the ratio of the fin heat transfer rate to the heat transfer rate that would exist without the fin:

.

The thermal conductivities (W/[m K]) of the fin materials are:

stainless steel

carbon steel

iron

brass

aluminum

copper

Reference

[1] T. L. Bergman, A. S. Lavine, F. P. Incropera and D. P. DeWitt, *Introduction to Heat Transfer*, 6th ed., Hoboken, NJ: John Wiley & Sons, 2011.

## Permanent Citation