A fin of unusual shape protrudes from a surface whose temper

A fin of unusual shape protrudes from a surface whose temperature is 200 degree C. The first section of the fin is rectangular and the second part is triangular as shown in the figure below. The length of the rectangular section is 12.5 cm and the length of the triangular section is 10 cm. The width of the fin is 100 cm and the thickness of the rectangular section is 1 cm. The fin material has a thermal conductivity of 100 W/m-K and the surrounding air is at a temperature of T_inf = 30 degree C and a convective heat transfer coefficient of 30 W/m^2-K. Determine the temperature at the location where the fin changes from rectangular to triangular shape. What is the heat transfer rate from the fin to the fluid?

Solution

a) The temperature of the fin at the the rectangular fin junction is given by,

T_L = cosh m(L - x) / cosh mL (T_b - Tinf) + Tinf

where m = sqrt (hP / kA_c )

m = sqrt (30 x 1.01) / (100 x 0.0100) = 5.504

T_12.5 = [cosh 5.5(0.225 - 0.125) / cosh 5.5 x 0.225] (473 - 303) + 303

T at 12.5cm from the fin base = 408 K or 135 deg C

b) The overall heat transfer for the fin is given by,

Q = Q rec + Q tri

Qrec = sqrt (hPkA_c) (Tb - Tinf) tanh mL

Qrec = 5.504 x 343 x 0.688 = 1298.33 W = 1.3 kW

Qtri = sqrt (hPkA_c) (Tb - Tinf) tanh mL

where P = 0.05 + 0.1 = 0.15 m and Ac = 0.1 x 0.05 = 0.005 m and Tb = 408 K

m = sqrt (30 x 0.15) / (100 x0.005) = 3

Qtri = 1.5 x 105 x 0.29 = 45.88 W = 0.045kW

Q = 1.3 + 0.045 = 1.345 kW