A long, hollow cylinder is constructed from a material whose thermal conductivity is a function of temperature according to k = 0.15 + 0.0018 T, where T is in C and k is in W/(m K). The inner and outer radii of the cylinder are 12.5 and 25 cm, respectively. Under steady-state conditions, the temperature at the interior surface of the cylinder is 427°C and the temperature at the exterior surface is 93°C. (a) Calculate the rate of heat transfer per meter length, taking into account the variation in thermal conductivity with temperature. (b) If the heat transfer coefficient on the exterior surface of the cylinder is 17 W/(m2 K), calculate the temperature of the air on the outside of the cylinder.

Question

GIVEN

• A long hollow cylinder

• Thermal conductivity (k) = 0.15+ 0.0018 T [T in °C, k in W/(m K)]

• Inner radius (ri) = 12.5 cm

• Outer radius (ro) = 25 cm

• Interior surface temperature (Twi) = 427°C

• Exterior surface temperature (Two) = 93°C

• Exterior heat transfer coefficient ( oh ) = 17 W/(m2 K)

• Steady-state conditions

FIND

(a) The rate of heat transfer per meter length (q/L) (b) The temperature of the air on the outside (T?) ASSUMPTIONS

• Steady state heat transfer

• Conduction occurs in the radial direction only

SKETCH

Answer 1

(a) The rate of radial conduction is given





Integrating this from the inside radius to the outside radius



(b) The conduction though the hollow cylinder must equal the convection from the outer surface in steady state



Solving for the air temperature