This section is mainly concerned with an attempt
to develop an intuitive understanding of the exergy equations
developed in the previous section, by considering reversible equivalent
circuits of some common adiabatic components. We repeat equations
(3), (4) and (5) developed in **Chapter
7a** on the exergy analysis of a control volume.

Note that the reversible work w_{rev}
will either be the maximum available output work for work producing
devices, or the minimum possible input work (negative value) for
work absorbing devices.

We now consider the Energy and Entropy Generation equations for adiabatic components (q = 0):

Applying all the above analysis to evaluating
the Second Law Efficiencies (η_{II}) of adiabatic
work absorbing and work producing components we obtain:

We now apply the above Second Law analysis
to an adiabatic refrigeration compressor. We wish to determine
the minimum work w_{C rev} required to drive the compressor
between the inlet state (1) and the exit state (2). Note that
the isentropic compression that we evaluated in **Chapter
6** will not provide the answer, since state (2s) is not
the same as the actual state (2).

The above equations are in fact correct however
we have difficulty in understanding their significance. In examining
the adiabatic compressor above we cannot understand why the environment
(dead space) temperature T_{0} features so prominently
in the equations, when in fact there seems to be no obvious interaction
between the adiabatic compressor and the environment. Note that
as far as the adiabatic compressor is concerned we will assume
that the surroundings temperature T_{0} is equal to the
exit temperature.

In an attempt to find some intuitive meaning to these equations we consider a reversible system having the same inlet and exit states as our actual compressor. This comprises a three component system consisting of an inlet heat exchanger, a reversible heat engine and an isentropic compressor as shown below:

A typical *h-s* diagram for this system
is shown below, in which we have used typical inlet conditions
of 140kPa, -10°C and exit conditions of 700kPa, 60°C.
The reversible heat engine will provide extra work to drive the
compressor, absorbing its heat from the environment temperature
T_{0} while rejecting heat to the heat exchanger. The
exit state (2) from the heat exchanger has been chosen such that
the compression process (2) - (3) will be isentropic.

We now derive the exergy equations for the
three component system above, and consider first the heat engine.
Since the temperature T of the heat exchanger varies from the
inlet temperature T_{1} to the outlet temperature T_{2},
we use the differential energy equation form for the reversible
heat engine.

Since T_{0} is constant, this equation
can be integrated from the inlet state (1) to the outlet state
(2), leading to:

This familiar final form was to be expected. The net minimum work required to drive the compressor is thus:determined as follows:

Notice that this result is identical to that shown above for the actual adiabatic compressor, since we added the heat exchanger, thus state (3) is in fact equivalent to the original state (2).

**Problem 7.4 - **Recall **Solved
Problem 6.5** in which we evaluated the Adiabatic Efficiency
(η_{C}) of a refrigeration compressor, and determined
it to be 92%. Redo this problem and determine the Second Law Efficiency
(η_{II}) of this compressor. [92%]

We now apply the above Second Law Analysis
to an adiabatic steam turbine. We wish to determine the maximum
available turbine work output w_{T rev} between the inlet
state (1) and the exit state (2). We will then be able to determine
the second law eficiency by comparing the actual work output to
the reversible (maximum available) work output as follows:

Once again, in an attempt to find some intuitive
meaning to these equations we consider a reversible system having
the same inlet and exit states as the actual turbine, comprising
an isentropic turbine, a heat pump pumping heat from the surroundings
to the heat exchanger in the exit stream. Note that as far as
the adiabatic turbine is concerned we will assume that the surroundings
temperature T_{0} is equal to the exit temperature.

The *h-s* diagram for this system is shown
below, in which we have chosen as an example a steam turbine having
inlet conditions 6MPa, 600°C, and outlet conditions 50kPa,
100°C.

Notice from the h-s diagram that the heat exchanger temperature varies from the saturation temperature at 50 kPa (81°C) to 100°C at state (3). In order to accommodate that change we develop the differential form of the heat pump as follows:

Since T_{0} is constant, this differential
equation can be integrated from state (2) to state (3).

We can then subtract the work provided to the heat pump from the output work of the turbine leading to the final form of the maximum available work, as follows:

Notice that this result is identical to that shown in the box above for the actual adiabatic turbine, since state (3) is in fact equivalent to the original state (2).

**Problem 7.5 -**
Recall **Solved
Problem 6.1** in which we evaluated the Adiabatic Efficiencies
(η_{T}) of both the High Pressure and the Low Pressure
steam turbines of the supercritical steam power plantand found
them to be 77% and 90% respectively. Redo this problem and determine
the respective Second Law Efficiencies (η_{II}) of
both turbines. [77%, 90%]

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Engineering Thermodynamics by Israel Urieli is licensed under a
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