This section is a continuation of "Second Law Analysis" in which we attempt to determine theoretical limits of performance of various thermodynamic devices, with the ultimate goal of determining how to best manage our natural resources. In all current thermodynamic texts I find that it is presented in a somewhat confusing and nonstandard manner, thus I have decided to develop the concepts in terms of various examples. We will cover Sections 7.1, 7.2 and 7.3, and not the optional Section 7.4.
We will go over Potter Example 7.1 in
class with modified values specified, since the values chosen
by Potter are not available in the tables (requiring busy work
in interpolation) and 700°C is out of range of our P-h and h-s
diagrams. We restate this example problem as follows:
"An ideal (isentropic) steam turbine is supplied with steam
at 10MPa and 600°C, and exhausts at 600kPa.
Potter Example 7.2
is presented in English units, thus we restate this problem example
in SI units as follows:
"Measurements are made on an adiabatic compressor with supply
air at 100kPa and 27°C, The exhaust air is measured at 500kPa
and 227°C. Can these measurements be correct? [yes]"
Note that we find it more intuitive and simpler to use values
of Specific Heat at the average temperature of the process (taken
from the table of Specific
Heat Capacity of Air) rather than the method presented
by Potter.
Potter Example 7.6
introduces the concept of a second law effectiveness for non work
producing or absorbing devices. We are unfamiliar with this concept
and prefer to deal directly with the concepts of irreversibility
or (in the case of a nozzle) the equivalent concept of a lost
kinetic energy potential. Thus we replace Example 7.6 with
the following, and will go over this in class when we complete
the section on Exergy Analysis
of an Air Nozzle.
"Air enters a nozzle at a steady flow of 300kPa and 87°C
with a velocity of 50m/s, and exits at 95kPa and 300m/s. The heat
loss from the nozzle to the surroundings at 17°C is estimated
to be 4kJ/kg. Determine (a) the exit temperature of the air [40°C] and
(b) the lost kinetic energy potential (or irreversibility) during
this process [58.5kJ/kg]."
We will not cover Examples 7.3, 7.5, 7.7 or 7.8.
The recommended Supplementary Problems for this section - Potter & Somerton Chapter 7: 7.7, 7.10, 7.12, 7.19, 7.21, 7.24. (Note: Problem 7.7 can only be solved if the turbine is adiabatic (not specified). In Problem 7.21 answer given is incorrect. Correct values are: mass flow of air = 0.66kg/s, irreversibility = 1.89kW)
Exergy (also known as Availability) - the maximum Work Potential of a System or Component at a given state in a specified environment. (Note that in Chapter 7 Potter & Somerton define Availability and Exergy as related but different concepts. All other thermodynamic texts that I am familiar with define them both as the same concept and we do the same in what follows.)
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 (77°F).
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 Kremer of the ME department. 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.
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:
entropy:
exergy: we first eliminate q from equations (1) and (2) as follows:
The maximum work will occur for a reversible system in which there is no entropy generated. We define the irreversibility (i) as follows:
Thus when the irreversibility i = 0, the reversible work is:
We now define the Second Law Efficiency hII as follows:
The exergy y (or availability) 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, thus:
Thus the reversible work between two states 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 Dr. Gary Graham (of Ohio University) in 1995. Thus:
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