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 P_{0} = 1 atmosphere. T_{0} = 25°C (77°F). 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 (w_{rev}) 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** (η_{II})
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 w_{HE} 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 6.10** 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 (s_{gen}) and
entropy change (Δs) for an ideal gas, and the exergy equations
for reversible work (w_{rev}) and Second Law efficiency
(η_{II}). Use values of Specific
Heat Capacity (C_{P}) 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 6.10** 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 (η_{II})
[85%]
of this compressor.

**Problem 7.3 - **Recall again **Solved
Problem 6.10** 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 (η
_{II}) of this turbine [82%].

______________________________________________________________________________________

Engineering Thermodynamics by Israel Urieli is licensed under a
Creative Commons Attribution-Noncommercial-Share
Alike 3.0 United States License