**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
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