Entropy change in irreversible processes - real refrigerator

Problem Statement:

A refrigerator removes heat from a cold thermal reservoir at 00C and discharges it to the surrounding air at 270C. In order to operate, the refrigerator requires a work input of 2 104 J of work per cycle.

1. Calculate what would be the maximum coefficient of performance of an ideal refrigerator operating between the same two thermal reservoirs.
2. Calculate the heat an ideal refrigerator would remove per cycle from the cold thermal reservoir, its entropy change and the entropy change of the thermal reservoirs as well as that of the universe.
3. Calculate the coefficient of performance of a real refrigerator operating between these two thermal reservoirs, knowing that its coefficient of performance is equal to 75% of the value for a reversible refrigeration cycle. How much heat per cycle does it remove from the cold reservoir? How much heat per cycle does it discard to the hot reservoir?
4. Calculate the entropy change of the working fluid of this refrigerator per cycle, and that of the thermal reservoirs and of the universe.

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Solution:

A refrigerator (or heat pump) is a cyclically operating device which removes heat from a cold thermal reservoir (at temperature T2 in the figure below) and discards it to a hot thermal reservoir (at temperature T1 in the figure) while consuming energy in the form of work input (W<0).

The most efficient refrigerator operating between two thermal reservoirs is the Carnot refrigerator. Its coefficient of performance is (both temperatures in kelvin):

After substituting temperatures T1 and T2 we get:

As you can see, the coefficient of performance of a refrigerator is greater than 1.

We will use the expression for the coefficient of performance of a refrigerator to determine the heat removed (Q2) by the refrigerator from the cold thermal reservoir:

In order to calculate the entropy change of the thermal reservoirs we need to know the amount of heat discarded (Q1) to the hot thermal reservoir. The first law of Thermodynamics must apply so the energy entering the system has to be equal to the energy leaving it (see upper figure), So:

The entropy change of the system (the working fluid of the refrigerator) is zero. Entropy is a state function, so its change depends only on the initial and final states of the system, and in a cycle they are the same.

The entropy change of the thermal reservoirs is given by:

As you can see in the previous expressions, Q1 is positive from the point of view of the hot thermal reservoir (because Q1 is absorbed by the hot reservoir) and Q2 is negative from the point of view of the cold thermal reservoir (as Q2 is removed from it).

In this page you can see in detail how the entropy change of a thermal reservoir is calculated.

The entropy change of the universe is the sum of the entropy change of the system (the refrigerator in this case) and of its surroundings (the thermal reservoirs):

Which is zero as expected because a Carnot refrigerator operates reversibly and the second law of thermodynamics states that for any reversible process, the entropy of the universe remains constant.

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If the coefficient of performance of a real refrigerator operating between the same two thermal reservoirs is lower than the coefficient of performance of a Carnot refrigerator, it means that its operation is irreversible. It will remove less heat from the cold thermal reservoir for the same amount of work input per cycle.

We use the definition of the refrigerator coefficient of performance (COP) to get:

The heat discharged to the hot thermal reservoir (its absolute value) is:

The entropy change per cycle for the working fluid is zero, because entropy is a state function.

The entropy change for the thermal reservoirs is given by:

And the entropy change of the universe is the sum of the entropy changes of the refrigerator and the thermal reservoirs.

In this case the entropy change of the universe is positive because the operation of a real refrigerator is irreversible and whenever an irreversible process occurs in the universe, its entropy increases.

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