Chapter 19, Problem 45CP

(a) The inside of a hollow cylinder is maintained at a temperature T_a, and the outside is at a lower temperature, T_b (Fig. P19.45). The wall of the cylinder has a thermal conductivity k. Ignoring end effects, show that the rate of energy conduction from the inner surface to the outer surface in the radial direction is

$\frac{dQ}{dt}=2\pi Lk\left[\frac{T_a-T_b}{\ln (b/a)}\right]$

Suggestions: The temperature gradient is dT/dr. A radial energy current passes through a concentric cylinder of area 2πrL. (b) The passenger section of a jet airliner is in the shape of a cylindrical tube with a length of 35.0 m and an inner radius of 2.50 m. Its walls are lined with an insulating material 6.00 cm in thickness and having a thermal conductivity of 4.00 × 10⁻⁵ cal/s · cm · °C. A heater must maintain the interior temperature at 25.0°C while the outside temperature is −35.0°C. What power must be supplied to the heater?

Figure P19.45