**Solved Problem 5.1 - Reversible Home Air
Conditioner and Hot Water Heater**

Recall **Problem 4.12**
in which we evaluated the following proposed heat pump system
designed for summertime hot water heating and space cooling.

Under the conditions shown in the diagram we determined that the
Coefficient of Performance of the hot water heater (COP_{HW})
was 4.41, and that of the space cooling air conditioner (COP_{AC})
was 3.41. Assuming that this system is required to maintain the
home air at 20°C and the hot water at 50°C, we wish to
determine the maximum possible theoretical Coefficients of Performance
that could be obtained under these conditions.

- a) Draw a diagram representing the heat pump
system showing the flow of energy and source and sink temperatures
and determine the following:

- b) the maximum possible Coefficient of Performance
of a hot water heater (COP
_{HW}) that could be obtained by a reversible heat pump,

- c) the maximum possible Coefficient of Performance
of a space cooling air conditioner (COP
_{AC}) that could be obtained by a reversible heat pump.

- d) Comparing the actual Coefficients of Performance shown above to those of the reversible heat pump determine if the actual heat pump shown above is feasible. State the reasons for your conclusion.

Derive all equations used starting from the
basic definition of COP_{HW} and COP_{AC} and
the Carnot relations for the ratio of heats of a heat pump.

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**Solution:** For
a), b), and c) we need to reduce the system complexity shown above
to an energy flow diagram showing only the basic requirements
- cool the air to 20°C and heat the water to 50°C. Given
the temperature if the heat source (20°C) and the heat sink
(50°C) we can evaluate the respective reversible Coefficients
of Performance.

For question d) we use the accumulated knowledge that the conditions for thermal and mechanical reversibility are so difficult to realize that no practical heat pump or heat engine can attain more than 50% to 60% of the equivalent reversible (Carnot) performance. Thus comparing the actual to the reversible Coefficients of Performance we obtain:

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Engineering Thermodynamics by Israel Urieli is licensed under a
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