KAHIBARO
Table of Contents
Cooling by Doing Work
A refrigerator is a device that removes heat from a colder region and delivers it to a warmer region. This is the opposite of the natural direction of heat flow, so a refrigerator must use external work to make this happen.
The basic idea is simple. Heat $Q_C$ is taken from the cold interior, work $W$ is supplied by an electric motor or compressor, and a larger amount of heat $Q_H$ is released to the warmer surroundings. Energy conservation gives
$$
Q_H = Q_C + W
$$
This means the heat dumped into the room is equal to the heat removed from the inside plus the work done on the system.
A refrigerator does not create cold. It removes heat from a low temperature region.
For a refrigerator,
$$
Q_H = Q_C + W
$$
with $W > 0$ supplied from outside.
How a Refrigerator Fits into Thermodynamics
A refrigerator is a heat engine run in reverse. Instead of taking heat from a hot reservoir and turning part of it into work, it uses work to transfer heat from a cold reservoir to a hot one.
The cold reservoir is the refrigerated space, and the hot reservoir is usually the surrounding room. The working substance, often called the refrigerant, goes through a cycle so that it can repeatedly absorb heat from the cold side and release heat to the hot side.
This process follows the second law of thermodynamics. Heat does not spontaneously flow from cold to hot, so work input is necessary.
Performance of a Refrigerator
For refrigerators, we do not usually talk about efficiency in the same way as for heat engines. Instead, we use the coefficient of performance, abbreviated as COP.
For a refrigerator, the coefficient of performance is
$$
\text{COP}_R = \frac{Q_C}{W}
$$
This tells us how much heat is removed from the cold space for each unit of work supplied.
A larger COP means better performance. If a refrigerator removes a lot of heat using little work, it is performing well.
Using $W = Q_H - Q_C$, we can also write
$$
\text{COP}_R = \frac{Q_C}{Q_H - Q_C}
$$
For a refrigerator, the coefficient of performance is
$$
\text{COP}_R = \frac{Q_C}{W}
$$
This is not the same as efficiency of a heat engine.
Refrigerator and Heat Pump
A refrigerator and a heat pump are physically very similar devices. The difference is what we consider the useful output.
In a refrigerator, the useful effect is removing heat from the cold region, so the important quantity is $Q_C$.
In a heat pump, the useful effect is delivering heat to the warm region, so the coefficient of performance is
$$
\text{COP}_{HP} = \frac{Q_H}{W}
$$
Since $Q_H = Q_C + W$, the two are related by
$$
\text{COP}_{HP} = \text{COP}_R + 1
$$
| Device | Useful effect | Coefficient of performance |
|---|---|---|
| Refrigerator | Remove heat from cold space | $\text{COP}_R = Q_C/W$ |
| Heat pump | Deliver heat to warm space | $\text{COP}_{HP} = Q_H/W$ |
Ideal Refrigerator
The best possible refrigerator operating between two temperatures is the Carnot refrigerator. It is an ideal model, not a practical machine, but it gives the maximum possible coefficient of performance.
If the cold reservoir has temperature $T_C$ and the hot reservoir has temperature $T_H$, both in kelvin, then for a Carnot refrigerator
$$
\text{COP}_{R,\text{Carnot}} = \frac{T_C}{T_H - T_C}
$$
This formula shows two important things. First, refrigeration becomes harder when the temperature difference $T_H - T_C$ is large. Second, if the cold and hot temperatures are close together, the COP can be high.
For an ideal Carnot refrigerator,
$$
\text{COP}_{R,\text{Carnot}} = \frac{T_C}{T_H - T_C}
$$
Temperatures must be in kelvin.
Main Parts of a Vapor Compression Refrigerator
Most everyday refrigerators use a vapor compression cycle. The refrigerant circulates through four main parts.
The compressor does work on the refrigerant and raises its pressure and temperature. The condenser allows the refrigerant to release heat to the room and condense into liquid. The expansion valve suddenly lowers the pressure, causing the refrigerant to cool. The evaporator is inside the cold region, where the refrigerant absorbs heat and evaporates.
The cycle then repeats.
Physical Picture of the Cycle
In the evaporator, the refrigerant is cold and at low pressure. Because it is colder than the inside of the refrigerator, heat flows into it. This absorbed heat is $Q_C$.
The compressor then compresses the vapor. Compression raises both pressure and temperature. Now the refrigerant is hotter than the room.
In the condenser, heat flows from the refrigerant to the room. This released heat is $Q_H$. As it loses heat, the refrigerant condenses.
The expansion valve lowers the pressure again. The refrigerant cools sharply and returns to the evaporator, ready to absorb more heat.
Why Refrigerators Feel Warm at the Back
The back or bottom of a refrigerator often feels warm because that is where the condenser releases heat. Since
$$
Q_H = Q_C + W
$$
the heat given to the room is greater than the heat removed from the cold interior. The extra amount comes from the electrical work supplied to the compressor.
Example Calculation
Suppose a refrigerator removes $Q_C = 600\ \text{J}$ of heat from its inside while consuming $W = 200\ \text{J}$ of electrical work.
Its coefficient of performance is
$$
\text{COP}_R = \frac{Q_C}{W} = \frac{600}{200} = 3
$$
The heat delivered to the room is
$$
Q_H = Q_C + W = 600 + 200 = 800\ \text{J}
$$
So the refrigerator removes $600\ \text{J}$ from the cold space and releases $800\ \text{J}$ to the room.
Practical Meaning of COP
A COP greater than 1 is normal for refrigerators. This does not violate energy conservation, because the refrigerator is not converting work entirely into heat removal. It is using work to move heat from one place to another.
For example, if $\text{COP}_R = 4$, then each $1\ \text{J}$ of work removes $4\ \text{J}$ of heat from the cold region.
This is why refrigerators and heat pumps can be very effective devices.
A refrigerator can have $\text{COP} > 1$.
This is possible because COP measures heat transferred per unit work, not conversion of work into useful work.
Temperature Difference Matters
The greater the temperature difference between the inside and the room, the harder the refrigerator must work. If the room is very hot, or if the desired inside temperature is very low, the COP becomes smaller.
This is consistent with the Carnot result,
$$
\text{COP}_{R,\text{Carnot}} = \frac{T_C}{T_H - T_C}
$$
As $T_H - T_C$ increases, the denominator gets larger, so the COP decreases.
Summary Relations
The main thermodynamic relations for refrigerators are gathered here.
| Quantity | Formula | Meaning |
|---|---|---|
| Energy balance | $Q_H = Q_C + W$ | Heat rejected equals heat absorbed plus work input |
| Refrigerator COP | $\text{COP}_R = Q_C/W$ | Cooling obtained per unit work |
| Heat pump COP | $\text{COP}_{HP} = Q_H/W$ | Heating obtained per unit work |
| Relation between COPs | $\text{COP}_{HP} = \text{COP}_R + 1$ | Same device, different useful output |
| Carnot refrigerator COP | $\dfrac{T_C}{T_H - T_C}$ | Maximum possible COP |
Final Idea
A refrigerator is a thermodynamic cycle that uses work to transfer heat from a colder place to a warmer one. Its performance is measured by coefficient of performance, not ordinary efficiency. Real refrigerators are less effective than the ideal Carnot refrigerator, but the same energy balance always applies.
Key refrigerator formulas:
$$
Q_H = Q_C + W
$$
$$
\text{COP}_R = \frac{Q_C}{W}
$$
$$
\text{COP}_{R,\text{Carnot}} = \frac{T_C}{T_H - T_C}
$$