Three pin-shaped fins are attached to the aluminum wall to
dissipate heat to the surrounding air. Aluminum wall is maintained at 650
degrees Fahrenheit, while surrounding air is at 75 degrees Fahrenheit. Each fin has 1/8-inch diameter and is 1 inch
long. Thermal conductivity of the aluminum is 120 BTU per hour per foot per
degree Fahrenheit. Determine the
percentage increase in the heat removal, if the coefficient of heat transfer is
28 BTU per hour per square foot per degree Fahrenheit.
Solution:
When additional metals pieces are attached to the ordinary
heat-transfer surfaces such pipes or tubes, they extend the surface available
for heat transfer. While extended
surface increases the total transmission of heat, its influence as surface is
treated differently from simple conduction and convection. Pieces, which are employed to extend the
heat-transfer surfaces, are known as fins.
For a plane wall, with out any fins, the heat dissipation is
given by h times (Ts minus Ta). Plugging in the values of these terms, we obtain heat removal of
16,100 BTU per hour per square foot.
For longitudinal fins, the case in hand, a fin parameter, M,
needs to be calculated that is give by square root of h over k rH,
where rH is the hydraulic radius and for a circular fin, this
relationship becomes square root of 4 h over k d. Its value is found to be 9.47 per foot.
If we consider one foot the aluminum wall, then, number of
pins is 3 times 144 or 432. Area of
cross section of the fin is 8.52 times 10-5 square feet. Heat transfer through finned surface is k
times M times tanh (ML) times N AC times
(Ts minus Ta). This value is found to be 15,825.2 BTU per hour per square foot.
We can see that finned are is N times AC
or 0.0368 square foot. It implies that
bare-area, AB is only 0.9632 square foot. So the contribution from the bare-area is 0.9632 times 16,100 BTU
per hour or 15,507.5 BTUs per hour.
Total heat removal is found to be combination of these two
terms, and is equal to 31,333 BTU per hour per square foot. So, there is 94.61 percent increase in the
heat dissipation rate when using fins.