Serway & Jewett — The First Law of Thermodynamics: Problem 45.

(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 \(2\pi rL\). (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 \(4.00 \times 10^{-5} \text{ cal/s} \cdot \text{cm} \cdot \text{°C}\). 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?