In this Section we combine the First and Second Laws of thermodynamics in an attempt to determine theoretical limits of performance of various thermodynamic components and systems. Thus we introduce the concept of Exergy (aka Availability) - defined as the maximum work potential of a system or component at a given state in a specified environment. The environment is crucial in this definition since once the system or component has reached total thermodynamic equilibrium with its environment, and has used up all of its potential and kinetic energy relative to that environment, it is said to be in the Dead State. The environment is usually specified in terms of pressure and temperature as P0 = 1 atmosphere. T0 = 25°C (298K). In the following we attempt to introduce the concepts in terms of various examples.
This very intuitive first example defines the theoretical maximum available power from a wind generator as that which occurs when the kinetic energy of the air passing through the turbine rotor is reduced to zero. Clearly this is impractical, and in an interesting discussion of wind power on Wikipedia we find that Betz's Law imposes a theoretical limit of 59.3% of this maximum available power when the wind velocity is reduced by 1/3 while passing through the turbine rotor, and in fact the actual energy usage is much less.
An interesting application of wind power generation for home usage is the project of Dr Greg Kremer of the ME department at Ohio University. He has combined wind and solar power with battery backup connected to the electrical grid in his home. Using the conditions defining Dr. Kremer's wind turbine system (rotor diameter 3.53m) we determine the availability of his system as follows:
where:
Notice the dependence on the cube of the wind velocity. The average annual wind velocity in Athens, Ohio is 7mph (3.11m/s) giving a maximum available power of only 174W. However during the winter months (when the solar energy is lower) the velocity reaches 22.5mph (10m/s) giving a maximum available power of 5.79kW! Thus the wind/solar combination system seems like a compatible match, and so far Dr. Kremer has found that his net electrical power usage from the grid is negative! (His system feeds energy into the grid).
Our second example is that of hydroelectric power generation due to potential energy. Unlike wind power as described above, all of the available potential energy can be converted directly into work. Our favorite example is that of the Shoshone Hydro power plant in Glenwood Canyon, Colorado. A delightful description of this power plant is presented in Glenwood Canyon: An I-70 Odyssey by Matthew E Salek. The unique aspect of this plant is that unlike traditional plants which have the dam located at the same location, the Shoshone dam is located two miles upstream, and the water flows through a tunnel in the wall of the canyon to the power plant. At the power plant the water exits the Canyon wall and drops to the hydroelectric turbines to generate power.
where:
The Shoshone plant can provide up to 15MW power, which is enough power for about 15,000 households.
Exergy Analysis of a Control Volume
In our third example we do an exergy analysis of a single-inlet single-outlet steady-flow control volume and define and evaluate the various concepts used. We have ignored kinetic and potential energy terms which simply directly contribute to the exergy as needed. We find it convenient to do the development in terms of specific quantities (by dividing throughout by the mass flow).
Energy (First Law):
Entropy Generation (Second Law):
Exergy Analysis: we first eliminate q from equations (1) and (2) as follows:
Notice in equation (3) that we have defined reversible work (wrev) as that in which no entropy is generated. We thus define a new term Irreversibility (irrev) as follows:
Thus from equation (3), when the irreversibility irrev = 0, the resulting Reversible Work is given by:
We now define the Second Law Efficiency (ηII) for either a work producing or a work absorbing device as follows:
The Exergy ψ (or Availability) of the working fluid at either the inlet or the outlet port is defined as the maximum available work when the state of that working fluid is reduced to the Dead State 0, thus:
Notice that on referring to the Control Volume diagram above, the reversible work equation (5) can be written in terms of the inlet (i) and outlet (e) states as follows:
Thus the Reversible Work of the control volume can also be defined in terms of the difference in exergy between the inlet and exit ports, thus:
In order to get an inuitive understanding of this analysis, consider the following equivalent system in which we use the heat transfer between the system and the surroundings in order to obtain reversible work.
However this reversible work wHE is a function of the temperature T of the control volume, which can vary significantly between the inlet state (i) and the outlet state (e). Thus we will need to sum the work output of an infinite number of elemental reversible heat engines, as shown in the equivalent diagram which follows:
This analysis was first presented to me by the late Gary Graham (of Ohio University) in 1995. Thus:
Note: the following two problems are extensions of Solved Problem 6.11 in which we did an ideal analysis of the T700 helicopter gas turbine. In solving these problems you should derive all equations used starting from the basic energy equation for a flow system, the enthalpy difference (Δh) for an ideal gas, the equation for entropy generated (sgen) and entropy change (Δs) for an ideal gas, and the exergy equations for reversible work (wrev) and Second Law efficiency (ηII). Use values of Specific Heat Capacity (CP) at the average temperature of each process, which are obtained from the table of Specific Heat Capacities of Air.
Problem 7.2 - Recall Solved Problem 6.11 in which we did an ideal analysis of the T700 helicopter gas turbine. In this problem we wish to do a Second Law analysis of the compressor only of this gas turbine. Assume that air enters the compressor at 100kPa and 27°C and exits at 1500kPa and 427°C, with 40 kJ/kg of heat cooling transferred to the surroundings at 25°C. Determine the actual work (from the energy equation) [-451.6 kJ/kg], the reversible work (from an exergy analysis) [-383.4 kJ/kg], and the Second Law Efficiency (ηII) [85%] of this compressor.
Problem 7.3 - Recall again Solved Problem 6.11 in which we did an ideal analysis of the T700 helicopter gas turbine. In this problem we wish to do a Second Law analysis of the turbine which is required to drive the compressor of a gas turbine engine. Assume that the gas leaving the combustion chamber enters the turbine at 1500 kPa and and 927°C and exits at 400 kPa and 627°C, with 50 kJ/kg of heat loss transferred to the surroundings at 25°C. Assuming that the gas is pure air, determine
a) the actual work output (from the energy equation) [292.6 kJ/kg],
b) the entropy generated by this process [0.22 kJ/kg.K],
c) the available reversible work output (from an exergy analysis) [357.5 kJ/kg], and
d) the Second Law efficiency (ηII) of this turbine [82%].
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Engineering Thermodynamics by Israel
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