Both ends of a 0.6-cm copper U-shaped rod, are rigidly affixed to a vertical wall as shown in the accompanying sketch. The temperature of the wall is maintained at 93°C. The developed length of the rod is 0.6 m, and it is exposed to air at 38°C. The combined radiation and convection heat transfer coefficient for this system is 34 W/(m2 K). (a) Calculate the temperature of the midpoint of the rod. (b) What will the rate of heat transfer from the rod be?

Question

GIVEN



FIND

(a) Temperature of the midpoint (TLf)

(b) Rate of heat transfer from the rod (M)

ASSUMPTIONS

The system is in steady state

Variation in the thermal conductivity of copper is negligible

The U-shaped rod can be approximated by a straight rod of equal length

Uniform temperature across any section of the rod

SKETCH

Answer 1

By symmetry, the conduction through the rod at the center must be zero. Therefore, the rod can be thought of as two pin fins with insulated ends as shown in the sketch above.

(a) The temperature distribution for a fin of uniform cross section with an adiabatic tip is



Evaluating the temperature of the tip of the pin fin



The length of the fin is half of the wire length (Lf = 0.3 m)



The temperature at the tip of the fin is the temperature at the midpoint of the curved rod (49.2°C).

(b) The heat transfer from the fin is



The rate of heat transfer from the curved rod is approximately twice the heat transfer of the pin fin