Learning Goal: To understand that a heat engine run backward is a heat pump that

Question

Learning Goal:

To understand that a heat engine run backward is a heat pump that can be used as a refrigerator.

By now you should be familiar with heat engines--devices, theoretical or actual, designed to convert heat into work. You should understand the following:

Heat engines must be cyclical; that is, they must return to their original state some time after having absorbed some heat and done some work).

Heat engines cannot convert heat into work without generating some waste heat in the process.

The second characteristic is a rigorous result even for a perfect engine and follows from thermodynamics. A perfect heat engine is reversible, another result of the laws of thermodynamics.

If a heat engine is run backward (i.e., with every input and output reversed), it becomes a heat pump (as pictured schematically (Figure 1) ). Work Win must be put into a heat pump, and it then pumps heat from a colder temperature Tc to a hotter temperature Th, that is, against the usual direction of heat flow (which explains why it is called a "heat pump").

The heat coming out the hot side Qh of a heat pump or the heat going in to the cold side Qc of a refrigerator is more than the work put in; in fact it can be many times larger. For this reason, the ratio of the heat to the work in heat pumps and refrigerators is called the coefficient of performance, K. In a refrigerator, this is the ratio of heat removed from the cold side Qc to work put in:

Kfrig=QcWin.

In a heat pump the coefficient of performance is the ratio of heat exiting the hot side Qh to the work put in:

Kpump=QhWin.

Take Qh, and Qc to be the magnitudes of the heat emitted and absorbed respectively.

Part A

What is the relationship of Win to the work W done by the system?

Express Win in terms of W and other quantities given in the introduction.

Part C

A heat pump is used to heat a house in winter; the inside radiators are at Th and the outside heat exchanger is at Tc. If it is a perfect (i.e., Carnot cycle) heat pump, what is Kpump, its coefficient of performance?

Give your answer in terms of Th and Tc

Part E

Assume that you heat your home with a heat pump whose heat exchanger is at Tc=2C, and which maintains the baseboard radiators at Th=47C. If it would cost $1000 to heat the house for one winter with ideal electric heaters (which have a coefficient of performance of 1), how much would it cost if the actual coefficient of performance of the heat pump were 75% of that allowed by thermodynamics?

Express the cost in dollars.

Explanation / Answer

KRefri  = Q C/Win

In a heat pump the coefficient of performance is the ratio of heat exiting the hot side Q h to the work put in:

KPump  = Q h/Win

Take ,Q C and Q h to be the magnitudes of the heat emitted and absorbed respectively.

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