Circular cooling fins of diameter D = 1 mm and length L = 25.4 mm, made of copper (k = 400...

Circular cooling fins of diameter D = 1 mm and length L = 25.4 mm, made of copper (k = 400 W/m · K), are used to enhance heat transfer from a surface that is maintained at temperature Ts1 = 132°C. Each rod has one end attached to this surface (x = 0), while the opposite end (x = L) is joined to a second surface, which is maintained at Ts2 = 0°C. The air flowing between the surfaces and the rods is also at T∞ = 0°C, and the convection coefficient is k = 100 W/m2 · K. For fin with prescribed tip temperature, the temperature distribution and the rate of heat transfer are given by

(a) Express the function θ(x) = T(x) − T∞ along a fin, and calculate the temperature at x = L/2.

(b) Determine the rate of heat transferred from the hot surface through each fin and the fin effectiveness. Is the use of fins justified? Why?

(c) What is the total rate of heat transfer from a 10-cm by 10-cm section of the wall, which has 625 uniformly and distributed fins? Assume the same convection coefficient for the fin and for the unfinned wall surface.