Chapter 19, Problem 67P

(III) A cylindrical pipe has inner radius R₁ and outer radius R₂. The interior of the pipe carries hot water at temperature T₁. The temperature outside is T₂ (< T₁). (a) Show that the rate of heat loss for a length L of pipe is

$\frac{dQ}{dt}=\frac{2\pi k(T_1-T_2)L}{\ln (R_2/R_1)},$

where k is the thermal conductivity of the pipe. (b) Suppose the pipe is steel with R₁ = 3.3 cm, R₂ = 4.0 cm, and T₂ = 18°C. If the pipe holds still water at T₁ = 71°C, what will be the initial rate of change of its temperature? (c) Suppose water at 71°C enters the pipe and moves at a speed of 8.0 cm/s. What will be its temperature drop per centimeter of travel?