# Heat pump efficiency problem

In the year 2050, very efficient heating systems involving heat pumps are commonly used. When people leave their heated office rooms, heat pumps will pump energy from the vacated rooms to the occupied rooms. Instead of conventional heat pumps where on needs to provide work to pump heat from low temperature to a high temperature, this system uses only the temperature difference between the outside environment, the empty rooms, and the occupied rooms. The question is then what fraction of the energy that would be lost from the vacated office rooms to the outside environment can in theory be made available to the occupied rooms. Here we assume a constant heat capacity as a function of temperature and an infinite heat capacity of the occupied rooms and the environment.

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#### Solution Preview

Let's denote the outside temperature as T1, the temperature in the occupied rooms as T3, and the temperature in the unoccupied rooms from which the heat is pumped to the occupied rooms is denoted as T2. Initially T2 will be equal to T3. The heat capacity of these rooms is denoted as C.

Consider a general infinitesimal process where an amount of heat dq1 is added to the environment, an ...

#### Solution Summary

A detailed solution from first principles is given.