We repeat equations (7) and (8) developed in Part 1 on the exergy of a control volume. Recall that exergy y is defined as the maximum available work when the working fluid at the exit port of the control volume is at the dead state 0
The reversible work between two states can thus be defined in terms of the difference in exergy between the inlet and exit ports:
Note that wrev will be either a maximum available output work for work producing devices, or a minimum possible input work (negative value) for work absorbing devices.
We now apply exergy analysis to an adiabatic
refrigeration compressor. We wish to obtain the minimum work wC
required to drive the compressor between the inlet state (1) and
the exit state (2). Note that the isentropic compression will
not provide the answer, since state (2s) is not the same as the
actual state (2).
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Applying the equations (7) and (8) above to
a work absorbing device we find that the minimum work required
to drive the compressor is: However, since the compressor is adiabatic,
we have from the energy equation: Thus the Second Law Efficiency and Irreversibility
are given by: |
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 T0 features so prominently in the equations, when in fact there seems to be no obvious interaction between the adiabatic compressor and the environment. 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 T0 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 T1 to the outlet temperature T2, we use the differential energy equation form for the reversible heat engine.
Since T0 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:
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).
The throttle of a refrigeration system is that
magical device that causes a restriction to flow of the liquid
refrigerant into the saturated mixture phase, and in so doing
enables a temperature drop of around 50°C. At first sight
one wonders at the relevance of evaluating the lost work potential
of this device, in particular since it performs its function without
any change of enthalpy, which we normally associate with work
production.
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Once again we apply the exergy equations (7)
and (8) above with the following results: However, since the throttle is adiabatic,
we have from the energy equation: Thus the Second Law Efficiency and Irreversibility
are given by: |
Again we attempt to find some intuitive meaning to these equations by considering a reversible system having the same inlet and exit states as our actual throttle. This comprises a three component system consisting of an inlet isentropic turbine, a reversible heat engine and a heat exchanger as shown below:
Since we are dealing with a system in the liquid and low quality saturated mixture regions we cannot use the h-s diagram. We consider the P-h diagram below, in which we show a typical example of saturated liquid refrigerant being throttled from 700 kPa to 140 kPa. Notice in particular how the constant entropy lines follow the constant temperature lines in the subcooled (compressed) liquid region, and then deviate to lower enthalpy in the mixture region. We see that the isentropic turbine follows the constant entropy line from state (1) to state (2), the difference in enthalpy allowing work to be produced. Additional work is obtained from the heat engine operating between the surroundings temperature T0 and the low refrigerant temperature, with the heat rejection to the heat exchanger returning the enthalpy of the refrigerant back to its original value at state (3).
The work produced by the isentropic turbine is given by the energy
equation:
The additional work provided by the reversible heat engine:
Since temperature T is constant, we can substitute
Notice that this result is identical to that shown in the box above for the actual adiabatic throttle, since state (3) is in fact equivalent to the original state (2).
Our final example is that of an adiabatic steam turbine. The exergy analysis allows us to determine the maximum available turbine work output between the inlet state (1) and the exit state (2)
 |
Applying the exergy equations (7) and
(8) above to the turbine we obtain: However, since the turbine is adiabatic, we have from the
energy equation: Thus the Second Law Efficiency and Irreversibility are given
by: |
We now consider the three component 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, as follows:
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 T0 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).
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