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 (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 (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 6.11** 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 (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 6.11** 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
(η_{II}) [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 (η

_{II}) of this turbine [82%].

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Engineering Thermodynamics by Israel
Urieli is licensed under a Creative
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