Refrigerators and Heat Pumps

Heat Flow From a Colder to a Hotter Region

According to the second law of thermodynamics, heat cannot flow from a region of lower temperature to a region of higher temperature on its own. It can, however, be forced to do so by doing work on the working substance. Refrigerators, air conditioners and heat pumps are heat engines working in the backward direction: they convert mechanical work into heat transport.

Usually this is achieved with the aid of a phase change of a fluid called the refrigerant. The refrigerant is forced to evaporate and then condense by successively decreasing and increasing its pressure. It thus pumps energy from a region at lower temperature to a region at higher temperature: it extracts the heat of vaporisation from the cold region and rejects it to the hotter region outside, which cools the cold region further. Figure 4.22 shows this schematically as heat carried by the refrigerant from a cold box on one side to a hot box on the other.

The Refrigerator

Refrigeration is the process of cooling a space or substance and maintaining its temperature below the ambient temperature — in short, artificial cooling. A refrigerator extracts heat from the cold region inside the chamber and delivers it to the surrounding atmosphere. That is why the air behind a working refrigerator feels warm while its interior is cold.

Figure 4.23 (a) shows the mechanism: a compressor, an expansion valve, and a closed tube carrying the refrigerant. The part of the tube called the cooling coil lies in the region to be cooled and is at lower temperature and lower pressure; the other part, exposed to the atmosphere, is at higher temperature and higher pressure. A fluid such as isobutane is used as the refrigerant. Normally both the cold and the hot parts of the coil contain the refrigerant as a mixture of liquid and vapour phases in equilibrium.

The four steps of one complete cycle of refrigeration:

• Step 1: the fluid passes through a nozzle and expands into a low-pressure area. Much as carbon dioxide leaving a fire extinguisher cools, the fluid turns into a gas and cools down. This is essentially an adiabatic expansion.
• Step 2: the cool gas is in thermal contact with the inner compartment of the fridge. It heats up as heat is transferred to it from the contents. This takes place at constant pressure, so it is an isobaric expansion.
• Step 3: the gas is transferred to a compressor, which does most of the work in this process. The gas is compressed adiabatically, which heats it and turns it back to a liquid.
• Step 4: the hot liquid passes through coils on the outside of the fridge and heat is transferred to the atmosphere. This is an isobaric compression.

The compressor is driven by an external energy source and does the work $|W|$ on the working substance in each cycle.

Energy Flow Diagram and Performance

Figure 4.23 (b) is the energy flow diagram: a box at the cold reservoir temperature $T_C$ at the bottom sends a pipeline of width $Q_C$ up into a circle representing the refrigerator; a second pipeline of width $|W|$ enters the circle from the side, representing the work done by the compressor; and a single wider pipeline of width $|Q_H|$ leaves the top into the hot reservoir at $T_H$. The heat rejected at the hot reservoir is always greater than the heat received at the cold reservoir, by exactly the work done.

For a refrigerator the working substance absorbs $Q_C$ and rejects $Q_H$, so $Q_C > 0$, $Q_H < 0$ and $W < 0$. The process is cyclic, so $\Delta U = 0$ and the first law gives

$$Q_H + Q_C = W$$

which, taking magnitudes for the refrigerator, is written

$$|Q_H| = Q_C + |W|$$

This is the same relation as $W = Q_H + Q_C$ derived for the heat engine; it is valid for a heat engine and for a refrigerator alike.

The ratio $Q_C/W$ indicates the performance of a refrigerator and is called the coefficient of performance (CoP), $K$, or quality factor, or $Q$-value of the refrigerator. The larger the ratio, the better the refrigerator: the best performance is when the heat extracted at the cold reservoir is maximum for minimum work in one operating cycle.

$$K = \dfrac{Q_C}{W} = \dfrac{Q_C}{Q_H - Q_C}$$

All the quantities on the right are energies in the same units, so $K$ is a dimensionless number. For a typical household refrigerator, $K \approx 5$.

Air Conditioner

The working of an air conditioner and a refrigerator is exactly similar; they differ only in the volume of the chamber or room cooled. For an air conditioner the evaporator coils are inside the room to be cooled and the condenser is outside; the air cooled by the evaporator coils is circulated by a fan inside the unit. Its performance is defined by $K = Q_C/W$, but in practice one considers the rate of heat removed and the power required.

Defining the heat current, the rate of heat removed, as $H = Q_C/t$, where $t$ is the time in which the heat $Q_C$ is removed, and $P$ as the power required,

$$K = \dfrac{Q_C}{W} = \dfrac{Ht}{Pt} = \dfrac{H}{P}$$

Typical values of $K$ are 2.5 to 3.0 for room air conditioners.

Capacity of a refrigerator is expressed in litres — the volume available inside it. Capacity of an air conditioner is expressed in tonnes, a survival from the days before refrigeration when cooling was done with blocks of ice and a machine's capacity was quoted as the equivalent mass of ice melted in 24 hours.

Heat Pump

A heat pump is a device which works similarly to a refrigerator. It is used to heat a building or a similar large structure by cooling the air outside it — a refrigerator operating inside out. Its evaporator coils are outside and absorb heat from the cold outside air; its condenser coils are inside the building and release the absorbed heat to the air in the rooms, warming the building.

Heat Engine and Refrigerator Compared

Standard Results

$$|Q_H| = Q_C + |W|$$

$$K = \dfrac{Q_C}{W} = \dfrac{Q_C}{Q_H - Q_C}, \qquad K = \dfrac{H}{P} \ \text{for an air conditioner}$$

• $K$ is dimensionless; $K \approx 5$ for a household refrigerator, 2.5 to 3.0 for a room air conditioner.
• Heat flow from a hot object to a cold object is spontaneous, whereas work is always required for the transfer of heat from a colder object to a hotter object.