Question

A thin electrical heater is wrapped around the outer surface of a long cylindrical tube whose inner surface is maintained at a temperature of 10∘C. The tube wall has inner and outer radii of 30 and 80 mm, respectively, and a thermal conductivity of 5 W/m⋅K. The outer surface of the heater is exposed to a fluid with T[infinity]​=−15∘C and a convection coefficient of h=125 W/m2⋅K. Determine the heater power per unit length of tube required to maintain the heater at T0​=20∘C.

Answer 1

To determine the heater power per unit length of tube required to maintain the heater at T0=20∘C, we need to calculate the heat transfer through the wall of the cylindrical tube and the heat transfer to the fluid through convection.

Step-by-step explanation:

To determine the heater power per unit length of tube required to maintain the heater at T0=20∘C, we need to calculate the heat transfer through the wall of the cylindrical tube and the heat transfer to the fluid through convection.

The rate of heat conduction through the wall can be calculated using the formula:

Q = (k * A * (T0 - T_infinity)) / (ln(r2/r1))

Where Q is the heat transfer rate, k is the thermal conductivity of the wall, A is the surface area of the wall, T0 is the temperature of the heater, T_infinity is the temperature of the fluid, and r1 and r2 are the inner and outer radii of the tube wall, respectively.

The rate of heat transfer through convection can be calculated using the formula:

Q = h * A * (T0 - T_infinity)

Where Q is the heat transfer rate, h is the convection coefficient, A is the surface area of the heater, T0 is the temperature of the heater, and T_infinity is the temperature of the fluid.

The heater power per unit length of tube required to maintain the heater at T0=20∘C is the sum of the heat transfer rates through the wall and through convection divided by the length of the tube.