can someone help me You are asked to design a fin to remove

can someone help me....

You are asked to design a fin to remove heat from a CPU. The fin will be made out of 2024-T6 aluminum, and must extend exactly 8 cm from the CPU. The fin can either be a circular or a rectangular cross-section, but due to weight and cost restrictions, its total volume is restricted to 10 cm^3. The fin base (i.e., CPU) temperature is T_b = 320K, and it is exposed to airflow at T_infinity = 280K and an approach velocity of V = 5 m/s. Calculate the average heat transfer coefficient for a cylindrical fin, and for a rectangular fin with a thickness t = 2 mm. Treat the rectangular fin as a flat plate, What is the rate of heat transfer dissipated through the fin by each shape (i.e., fin heat transfer rate, q_f, in W)? Assume that the tip of the fin is exposed to the same convection coefficient as in part a). Is the result surprising? Comment on your answer. (Partial answer: q_f for cylinder=7.0 W)

Solution

a) The average heat transfer coefficient U is given by,

U = 1 / [x / k + 1 / h] where k = 151 W/m2K and h = 10.45 - v + 10 v^1/2 W/m2K

For v = 5m/s, h = 27.81 W/m2K

Since the volume of the rec plate equals the cylinder we get radius of the cylinder as,

4/3 x pi x r^3 = 10

r = 1.33 cm

Simlarly for plate,

l x w x t = 10

8 x w x 0.2 = 10, w = 6.25 cm

Heat transfer for the cylinder:

Uc = 1 / [(0.013 x 2) / 151 + 1 / 27.81] = 1 / 41.66 = 27.77 W/m2K

Heat transfer for the fin:

Uplate = 1 / [(0.002) / 151 + 1 / 27.81] = 27.799 W/m2K

b) Heat trafer rate for fins:

Cylinder:

Q = sqrt (h x k x P x Ac) [Tb - Tinf] tanh (mL)

m = sqrt (hP / kA) = sqrt (2.27/0.08) = 5.32

Q = sqrt ( 27.81 x 151 x 0.0816 x 0.00053) [320-280] tanh (5.32 x 0.08)

Q, for cylinder = 6.84 .W

Rec plate:

Q = sqrt (h x k x P x Ac) [Tb - Tinf] tanh (mL)

m = sqrt (hP / kA) = sqrt (1.79/0.0188) = 9.97

Q = sqrt ( 27.81x 151 x 0.129 x 0.000125) [320-280] tanh (9.97x 0.08)

Q, for plate = 6.89 W

c) The heat transfer of the plate is equivalent to that of the heat transfer of the cylinder. The efficiency of the cylindrical and rec plate fins are equivalent in performance.