Working to Heat

When a thermodynamic process such as that described above is run in the opposite direction the result is conversion of work into heat. This process is not as familiar as that described above but nonetheless it is in common use for example in the suburbs of Baltimore where I live. Almost all the houses in Columbia are for better or worse heated by so-called heat pumps wherein an electric compressor performs work on a working substance (Freon) with the goal of heating a house. The efficiency for such a heater, typically called a heat pump, is given by

Where is the heat we want and |W| is the work we must do. Using standard signs for these quantities we have . Again the second law allows us to place limits on how large can be. We start with the observation that the entropy of the working substance cannot change through one cycle of the process. This immediately implies that there must be an intake of heat to balance the entropy which leaves the system in the form of heat. We denote this amount of heat by . Because we have a cyclic process the internal energy of the working substance does not change in a cycle thus we have

Or in terms of unsigned quantities

Clearly the efficiency is greatest if we can avoid doing any work at all but simply transport heat from the cold to the hot reservoir ie.

The second law of thermodynamics clearly forbids this. To prove this we assume that a cyclic process accomplishes this, we will show that it would violate the second law of thermodynamics. The isolated system which we consider consists of the working substance and the two reservoirs, Hot and Cold, which receive and deliver heat respectively. The entropy of the working substance does not change in one cycle of the process. However for the Hot reservoir which receives heat

Where is the highest temperature in the system. In contrary the entropy decreases for the cold reservoir:

Where is the upper bound on the temperature of the cold reservoir(s) which delivers heat. The total change in entropy for the entire isolated system is

is against the second law of thermodynamics and so we have shown that a heat pump has a finite efficiency and that a finite amount of work is required to transfer heat from a cold to a hot reservoir. Another way of stating the second law of thermodynamics is as follows:

No process is possible whose sole result is to transfer heat from one reservoir to a reservoir at a higher temperatures.

One example of a heat pump (not necessarily the most efficient) is to run the Carnot cycle in the reverse sense as compared to the sense used to convert heat into work. The efficiency of this process is

We note that the efficiency of the process diverges as meaning that an infinitely small amount of work is required to induce heat flow between reservoirs at almost the same temperature. It is unfortunate but inevitable that the max efficiency of the heat pump occurs when it is needed least and vise versa.