Refrigerators and heat pumps

Moving heat the unnatural way

A refrigerator moves heat from a cold space to a warmer environment. This does not happen spontaneously. The second law says heat naturally flows from hot to cold, so a refrigerator requires work input.

A heat pump uses the same basic idea but focuses on delivering heat to a warm space, such as a house in winter.

Both devices operate in cycles and are essentially heat engines run in reverse.

Energy flow for a refrigerator

Let $Q_C$ be heat removed from the cold reservoir, $Q_H$ be heat delivered to the hot reservoir, and $W$ be work input. Energy conservation gives

$Q_H=Q_C+W$

The desired output of a refrigerator is heat removed from the cold space. Its coefficient of performance is

COP_R=rac{Q_C}{W}

Unlike efficiency, COP can be greater than 1 because the device is moving heat, not converting work directly into heat.

Heat pump coefficient of performance

For a heat pump, the desired output is heat delivered to the warm space:

COP_{HP}=rac{Q_H}{W}

Since $Q_H=Q_C+W$, the heat pump COP is

$COP_{HP}=COP_R+1$

for the same cycle.

This means heat pumps can deliver more thermal energy to a home than the electrical work they consume, because they move heat from outside rather than creating all heat from work.

Carnot refrigerator

The maximum possible refrigerator COP between reservoirs at $T_C$ and $T_H$ is

COP_{R,Carnot}=rac{T_C}{T_H-T_C}

The maximum heat pump COP is

COP_{HP,Carnot}=rac{T_H}{T_H-T_C}

Temperatures must be in kelvins.

As the temperature difference $T_H-T_C$ becomes smaller, the maximum COP increases. It is easier to move heat across a small temperature difference than a large one.

Why freezers need more work

A freezer must maintain a lower cold temperature than a refrigerator. If the room temperature is fixed, then $T_H-T_C$ is larger for a freezer. This lowers the maximum possible COP and usually requires more work per unit heat removed.

Similarly, heat pumps are most efficient in mild weather and less efficient when outdoor temperatures are very low.

Refrigeration cycle

Real refrigerators often use a vapor-compression cycle. A refrigerant evaporates at low pressure inside the cold region, absorbing heat. A compressor does work on the refrigerant, raising its pressure and temperature. The refrigerant condenses outside, rejecting heat. An expansion valve lowers the pressure and temperature, preparing it to evaporate again.

The refrigerant's phase changes allow significant heat transfer.

Second-law view

The refrigerator does not violate the second law because work input causes the heat transfer from cold to hot. Total entropy of the universe still increases for a real refrigerator.

For an ideal reversible refrigerator, total entropy change is zero. Real devices require finite temperature differences, friction, electrical resistance, and fluid flow losses, all of which produce entropy.

The big idea

Refrigerators and heat pumps use work to move heat from colder regions to warmer ones. Their performance is measured by coefficient of performance rather than efficiency. Carnot limits show that performance depends strongly on temperature difference, explaining why small temperature lifts are easier and why real devices fall short of ideal limits.