Heat Pumps and Refrigerators

Heat Pumps and Refrigerators

A heat pump is a device that transfers heat from a cold reservoir to a hot reservoir — the opposite of the natural direction of heat flow — by consuming work. Both refrigerators and heating heat pumps operate on this principle; they differ only in which heat transfer is the desired output.

Refrigerator

A refrigerator's goal is to remove heat $Q_c$ from the cold space (e.g., the inside of a freezer). The Carnot COP for a refrigerator is:

$\text{COP}_{\text{ref}} = \frac{Q_c}{W} = \frac{T_c}{T_h - T_c}$

A higher COP means less work is needed per joule of heat removed. When $T_c$ is close to $T_h$ (small temperature difference), the COP is large.

Heating Heat Pump

A heating heat pump's goal is to deliver heat $Q_h$ to the hot space (e.g., the interior of a building). The Carnot COP is:

$\text{COP}_{\text{hp}} = \frac{Q_h}{W} = \frac{T_h}{T_h - T_c}$

Key Relationship

The two COPs are always related by:

$\text{COP}_{\text{hp}} = \text{COP}_{\text{ref}} + 1$

This follows from energy conservation: $Q_h = Q_c + W$, so $Q_h/W = Q_c/W + 1$.

A heat pump with $\text{COP}_{\text{hp}} = 4$ delivers 4 J of heat for every 1 J of electrical work consumed — far more efficient than a simple resistive heater (which has COP = 1).

Computing Heat Transferred

Given the COP and work input $W$:

$Q = \text{COP} \times W$