12.10 Refrigerators and heat pumps

A refrigerator is the reverse of a heat engine. Here the working substance extracts heat \(Q_2\) from the cold reservoir at temperature \(T_2\), some external work \(W\) is done on it and heat \(Q_1\) is released to the hot reservoir at temperature \(T_1\) (Fig. 12.10).

In a refrigerator the working substance (usually, in gaseous form) goes through the following steps:

• Sudden expansion of the gas from high to low pressure which cools it and converts it into a vapour-liquid mixture.
• Absorption by the cold fluid of heat from the region to be cooled converting it into vapour.
• Heating up of the vapour due to external work done on the system, and
• Release of heat by the vapour to the surroundings, bringing it to the initial state and completing the cycle.

The coefficient of performance \((\alpha)\) of a refrigerator is given by
\(
\alpha=\frac{Q_2}{W} \dots(12.21)
\)
where \(Q_2\) is the heat extracted from the cold reservoir and \(W\) is the work done on the system-the refrigerant. \(\alpha\) for heat pump is defined as \(Q_1 / W\) ) Note that while \(\eta\) by definition can never exceed \(1, \alpha\) can be greater than 1 . By energy conservation, the heat released to the hot reservoir is
\(Q_1=W+Q_2\)

\(
\text { i.e., } \quad \alpha=\frac{Q_2}{Q_1-Q_2} \dots(12.22)
\)

In a heat engine, heat cannot be fully converted to work; likewise, a refrigerator cannot work without some external work done on the system, i.e., the coefficient of performance in Eq. (12.21) cannot be infinite.