Question

A Carnot heat pump extracts heat from the environment at 250 K and supplies 6 kW of heat to a room maintained at temperature $T_H$. The heat pump requires a power input of 1 kW. The temperature of the room $T_H$ is _________ K (round off to nearest integer).

Hint:

For a Carnot heat pump, $COP = \frac{T_H}{T_H - T_C}$ always gives a simple linear equation for $T_H$.

Answer 1

Correct Answer: 300

For a heat pump, the coefficient of performance is: \[ COP_{HP} = \frac{Q_H}{W} \] Given: \[ Q_H = 6\ \text{kW}, \qquad W = 1\ \text{kW} \] \[ COP_{HP} = \frac{6}{1} = 6 \] For a Carnot heat pump: \[ COP_{HP} = \frac{T_H}{T_H - T_C} \] where: \[ T_C = 250\ \text{K} \] Set equal: \[ 6 = \frac{T_H}{T_H - 250} \] Cross-multiply: \[ 6(T_H - 250) = T_H \] \[ 6T_H - 1500 = T_H \] \[ 5T_H = 1500 \] \[ T_H = 300\ \text{K} \] \[ \boxed{300\ \text{K}} \]