Chapter 19, Problem 62P

A Carnot engine extracts heat from a block of mass m and specific heat c initially at temperature T_h0 but without a heat source to maintain that temperature. The engine rejects heat to a reservoir at constant temperature T_c. The engine is operated so its mechanical power output is proportional to the temperature difference T_h − T_c:

$P=P_0\frac{T_{\text{h}}-T_{\text{c}}}{T_{\text{h0}}-T_{\text{c}}}$

where T_h is the instantaneous temperature of the hot block and P₀, is the initial power. (a) Find an expression for T_h as a function of time, and (b) determine how long it takes for the engine’s power output to reach zero.