Blundell Thermal Physics · Problem 13.8

Heat engines and the second law

Physics — Statistical Mechanics · heat-pump-building-steady-state

Problem

A building is maintained at a temperature

T

by means of an ideal heat pump, which uses a river at temperature

T_0

as a source of heat. The heat pump consumes power

W

, and the building loses heat to its surroundings at a rate

\alpha(T - T_0)

, where

\alpha

is a positive constant. Show that

T

is given by

T = T_0 + \frac{W}{2\alpha}\left(1 + \sqrt{1 + 4\alpha T_0/W}\right). \tag{13.44}