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Thermodynamics of Cycles
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Chapter 6
Thermodynamics of Cycles
An important application of thermodynamics is the analysis of power cycles
through which the energy absorbed as heat can be continuously converted into
mechanical work. A thermodynamic analysis of the heat engine cycles provides
valuable information regarding the design of new cycles or a combined cycle for
improving the existing cycles, or pushing efficiency gas turbine output with
Brayton or combined with topping or bottoming Rankine cycle via heat
exchangers/recuperators respectively. Application of combined cycle driven
power plants, either steam or nuclear, plays a very important role in industry
these days in order to make existence of these power plants more cost effective.
In this chapter we briefly touch upon thermodynamic cycles so readers can have a
general idea both on gas power cycles and vapor power cycles and how heat
exchangers are important components to be considered when these cycles are
combined.
6.1
Introduction
A thermodynamic cycle is a series of processes where the properties of the system
are the same after the cycle as they were prior. Three main properties are tracked
when a system undergoes a set of processes and they are:
• Temperature,
• Pressure, and
• Specific volume
To be considered a cycle, all three properties need to be the same at their initial
state and at the end. One property could remain the same throughout any of the
processes; the cycle is considered isothermal if temperature is constant, isobaric if
pressure is constant, and isochoric or isometric if specific volume is constant. The
© Springer International Publishing Switzerland 2017
B. Zohuri, Compact Heat Exchangers, DOI 10.1007/978-3-319-29835-1_6
bahmanz@aol.com
291
292
6 Thermodynamics of Cycles
most efficient type of cycle is one that has only reversible processes, such as the
Carnot cycle, which is made up of four reversible processes.
As far as the thermodynamic cycle definition in general is concerned, there are
two classes of cycles and they are:
1. Power Cycles and
2. Heat Pump Cycles.
Each of these cycles is defined based on the thermodynamic tasks assigned to
them and briefly can be described as follows:
Power cycles are used when there exists some way of converging some heat
energy input into mechanical work output, while heat pump cycles transfer heat
from low to high temperature stages by using mechanical work as the input source
respectively. Cycles are composed of the quasi-static processes going through the
entire cycle and can operate as power or heat pump cycles by controlling them in
the direction of the cycle process. This direction can be defined as either clockwise
or counter-clockwise which can be indicated using the Pressure-volume (P-V )
diagram or Temperature-entropy (T-s) diagram respectively.
A thermodynamic cycle in respect to net mechanical work by input from heat
energy in a closed loop on the P-V diagram mathematically can be presented as:
I
W ¼ PdV
ðEq: 6:1Þ
Equation (6.1) is the indication of net work that is equal to the area inside the closed
loop of the P-V diagram as depicted in Fig. 6.1, and this is because of the following
argument:
(a) The Riemann sum of work done on the substance due to expansion, minus
(b) The work done to re-compress.
The net work presented by Eq. (6.1) is equal to the balance of heat Q transferred
into the system and mathematically is presented by Eq. (6.2) in the following form:
W ¼ Q ¼ Qin Qout
ðEq: 6:2Þ
Equation (6.2) makes a cyclic process into an isothermal process, even though the
internal energy changes during the course of the cyclic process, and this means the
Fig. 6.1 Depiction of net
work in closed loop of P-V
diagram
P
W = ò PdV
a
b
V
åa PDV – åb PDV
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6.1 Introduction
293
Fig. 6.2 Illustration of Otto
cycle in P-V diagram
cyclic process finishes with the system’s energy in a closed loop on the P-V diagram
with the same amount of energy as when the process began. If the cyclic process
moves clockwise around the loop, then the net work W will be positive, and it
represents a heat engine. If it moves counterclockwise, then the net work W will be
negative, and it represents a heat pump.
In the P-V diagram each point of the cycle process can be presented as shown in
Fig. 6.2 of an Otto Cycle, and the definitions of each process for this cycle are
written as follows:
Otto Cycle Loop and description of each point in the thermodynamic cycles
1 ! 2: Isentropic Expansion: Constant Entropy (s), Decrease in Pressure (P),
Increase in Volume (V ), Decrease in Temperature (T ).
2 ! 3: Isochoric Cooling: Constant Volume (V), Decrease in Pressure (P),
Decrease in Entropy (s), Decrease in Temperature (T ).
3 ! 4: Isentropic Compression: Constant entropy (s), Increase in pressure (P),
Decrease in volume (V ), Increase in temperature (T )
4 ! 1: Isochoric Heating: Constant volume (V ), Increase in pressure (P), Increase
in entropy (s), Increase in temperature (T).
Some of the most important thermodynamics processes that we need to know in
order to deal with any thermodynamic cycle are as follows:
• Adiabatic: No energy transfer as heat (Q) during that part of the cycle would
amount to dQ ¼ 0. This does not exclude energy transfer as work.
• Isothermal: The process is at a constant temperature during that part of the cycle
ðT ¼ constant, dT ¼ 0Þ. This does not exclude energy transfer as heat or work.
• Isobaric: Pressure in that part of the cycle will remain constant.
ðP ¼ constant, dP ¼ 0Þ. This does not exclude energy transfer as heat or work.
• Isochoric: The process is at constant volume ðV ¼ constant, dV ¼ 0Þ. This does
not exclude energy transfer as heat or work.
• Isentropic: The process is one of constant entropy ðs ¼ constant, ds ¼ 0Þ. This
excludes the transfer of heat but not work.
Making certain sequences of assumption as part of modeling a real system using
thermodynamic cycles, which is often necessary to reduce the number of degrees of
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6 Thermodynamics of Cycles
Turbine
4
1
Fresh Air
Exhaust
gasses
st.
co
n
2
q in
3
s=
4
co
ns
st.
con
Work
out
s=
Compressor
3
3
p
2
T
q in
=
P
Combustion
Fuel
t.
1
4
2
q out
1
P-v Diagram
v
st.
on
c
p=
q out
T-s Diagram
s
Fig. 6.3 Presentation an idealized process in P-V and T-s diagrams of a Brayton cycle mapped to
actual processes of a gas turbine engine
freedom associated to the problems at hand, will reduce the problem to a very
manageable form. Such simplified modeling is depicted in Fig. 6.3, which is a
presentation of a real system modeled by an idealized process in P-V and T-s
diagrams of a Brayton cycle mapped to actual processes of a gas turbine engine.
The actual device can be made up of a series of stages, each of which is itself
modeled as an idealized thermodynamic process. Although each stage which acts
on the working fluid is a complex real device, they may be modeled as idealized
processes which approximate their real behavior. If energy is added by means other
than combustion, then a further assumption is that the exhaust gases would be
passed from the exhaust to a heat exchanger that would sink the waste heat to the
environment and the working gas would be reused at the inlet stage.
In summary, an important application of thermodynamics is the analysis of
power cycles through which the energy absorbed as heat can be continuously
converted into mechanical work. A thermodynamic analysis of the heat engine
cycles provides valuable information regarding the design of new cycles or for
improving the existing cycles. In this chapter, various gas power cycles are analyzed under some simplifying assumptions.
Two of the most important areas of application of thermodynamics are power
generation and refrigeration, and they both are usually accomplished by a system
that operates on a thermodynamic cycle.
Thermodynamically, the word “cycle” is used in a procedure or arrangement in
which some material goes through a cyclic process and one form of energy, such as
heat at an elevated temperature from combustion of a fuel, is in part converted to
another form, such as mechanical energy of a shaft, the remainder being rejected to
a lower temperature sink that is also known as a heat cycle.
A thermodynamic cycle is defined as a process in which a working fluid
undergoes a series of state changes and finally returns to its initial state. A cycle
plotted on any diagram of properties forms a closed curve (see Fig. 6.4).
Note that a reversible cycle consists only of reversible processes. The area
enclosed by the curve plotted for a reversible cycle on a P-V diagram represents
the net work of the cycle as we explained in the preceding chapters:
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6.1 Introduction
295
Fig. 6.4 Schematic of a
closed cycle
P 4
1
3
2
V
• The work is done on the system, if the state changes happen in an anticlockwise
manner.
• The work is done by the system, if the state changes happen in a clockwise
manner.
The purpose of a thermodynamic cycle is either to produce power, or to produce
refrigeration/pumping of heat. Therefore, the cycles are broadly classified as
follows:
(a) Heat engine or power cycles.
(b) Refrigeration/heat pump cycles.
A thermodynamic cycle requires, in addition to the supply of incoming energy:
1. A working substance, usually a gas or vapor;
2. A mechanism in which the processes or phases can be carried through sequentially; and
3. A thermodynamic sink to which the residual heat can be rejected.
The cycle itself is a repetitive series of operations.
Any thermodynamic cycle is essentially a closed cycle in which the working
substance undergoes a series of processes and is always brought back to the initial
state.
However, some of the power cycles operate on an open cycle. This means that
the working substance is taken into the unit from the atmosphere at one end and is
discharged into the atmosphere after undergoing a series of processes at the other
end. The following are illustrations of heat engines operating on an open cycle:
• Petrol and diesel engines in which the air and fuel are taken into the engine from
a fuel tank and products of combustion are exhausted into the atmosphere.
• Steam locomotives in which the water is taken in the boiler from a tank and
steam is exhausted into the atmosphere.
The basic processes of the cycle, either in open or closed, are heat addition, heat
rejection, expansion, and compression. These processes are always present in a
cycle even though there may be differences in working substance, the individual
processes, pressure ranges, temperature ranges, mechanisms, and heat transfer
arrangements.
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6 Thermodynamics of Cycles
Many cyclic arrangements, using various combinations of phases but all seeking
to convert heat into work, were proposed by many investigators whose names are
attached to their proposals; for example, the Diesel, Otto, Rankine, Brayton,
Stirling, Ericsson, and Atkinson cycles. Not all proposals are equally efficient in
the conversion of heat into work. However, they may offer other advantages, which
have led to their practical development for various applications. See also Brayton
cycle; Carnot cycle; Diesel cycle; Otto cycle; Stirling engine; Thermodynamic
processes.
Essentially, such devices do not form a cycle. However, they can be analyzed by
adding imaginary processes to bring about the state of the working substance, thus
completing a cycle. Note that the terms closed and open cycles that are used here do
not mean closed system cycle and open system cycle. In fact, the processes both in
closed and open cycles could either be closed or open system processes.
There is a basic pattern of processes common to power-producing cycles. There
is a compression process wherein the working substance undergoes an increase in
pressure and therefore density. There is an addition of thermal energy from a source
such as a fossil fuel, a fissile fuel (a fissile material is one that is capable of
sustaining a chain reaction of nuclear fission), or solar radiation. Work is done by
the system on the surroundings during an expansion process. There is a rejection
process where thermal energy is transferred to the surroundings. The algebraic sum
of the energy additions and abstractions is such that some of the thermal energy is
converted into mechanical work.
Different types of working fluids are employed in power plants. The nature of the
working fluids can be classified into two groups:
(a) Vapors.
(b) Gases.
The power cycles are accordingly classified into two groups:
1. Vapor power cycles in which the working fluid undergoes a phase change during
the cyclic process.
2. Gas power cycles in which the working fluid does not undergo any phase change.
In the thermodynamic analysis of power cycles, our main interest lies in estimating the energy conversion efficiency or the thermal efficiency. The thermal
efficiency of a heat engine is defined as the ratio of the network output W delivered
to the energy absorbed as heat Q and mathematically is presented by symbol η and
can be written as:
η¼
W
Q
and it can be illustrated as Fig. 6.5 below;
In this depiction, we identify the following;
LTER ¼ Low Temperature Energy Reservoir
bahmanz@aol.com
ðEq: 6:3Þ
6.1 Introduction
297
Fig. 6.5 Graphic
illustration of thermal
efficiency
HTER
Q1
Heat
Engine
W
Q2
LTER
HTER ¼ High Temperature Energy Reservoir
Using these definitions and refering to Fig. 6.4, Eq. (6.5) can be written more
precisely as follows;
η¼
W
Q1
ðEq: 6:4Þ
where Q1 is the heat supplied at high temperature.
There is a procedure or arrangement in which one form of energy, such as heat at
an elevated temperature from combustion of a fuel, is in part converted to another
form, such as mechanical energy on a shaft, and the remainder is rejected to a
lower-temperature sink as low-grade heat.
Heat engines, depending on how the heat is supplied to the working fluid, are
categorized in two types:
(a) External combustion.
(b) Internal combustion.
In external combustion engines, such as steam power plants, heat is supplied to
the working fluid from an external source such as a furnace, a geothermal well, a
nuclear reactor, or even the sun [1].
In internal combustion engines, such as automobile engines, this is done by
burning the fuel within the system boundaries [1].
Our study of gas power cycles will involve the study of those heat engines in
which the working fluid remains in the gaseous state throughout the cycle. We often
study the ideal cycle in which internal irreversibility and complexities (the actual
intake of air and fuel, the actual combustion process, and the exhaust of products of
combustion among others) are removed. We will be concerned with how the major
parameters of the cycle affect the performance of heat engines. The performance is
often measured in terms of the cycle efficiency of ηth as the ratio of network Wnet
and energy as heat of Qin. In Fig. 6.6 one can observe an Actual Cycle versus an
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298
6 Thermodynamics of Cycles
P
Fig. 6.6 Illustration of
actual vs. ideal cycle in a
P–υ diagram
Actual cycle
Ideal cycle
v
Fig. 6.7 Basic
thermodynamic cycle
HEAT SOURCE
Working
Substance
Qin
Engine
W
Qout
Pump
HEAT SINK
Ideal Cycle in a P–υ diagram, and using Eq. (6.3) and referring to Fig. 6.7 below
mathematically we can show that;
ηth ¼
W net
Qin
ðEq: 6:5Þ
Several cycles utilize a gas as the working substance, the most common being the
Otto cycle and the diesel cycle used in internal combustion engines. We touch upon
some of these cycles in this chapter such as Otto, Brayton, Carnot, etc., and we will
expand on them among other cycles for further evaluation as well.
6.2
Open Cycle
When internal combustion engine operation is examined, it is seen to differ in the
process of heat supply for a typical heat engine cycle because there is a permanent
change in the working fluid during combustion. Therefore, the fluid does not pass
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6.3 Closed Cycle
299
through a cycle so the internal combustion engine is often referred to as an “open
cycle” device, not a cyclic thermodynamic heat engine.
The term “open cycle”, while meaningless from a thermodynamic perspective,
refers to the fact that energy is supplied to the engine from outside in the form of
petroleum fuel and the unconverted portion of energy remaining in the spent
combustion mixture is exhausted to the environment. “Closing the cycle”, i.e.,
returning the rejected products to the starting point where they can be reused, is
left for nature to accomplish—hence the term “open cycle” comes into play.
An internal combustion engine is therefore a device for releasing mechanical
energy from petroleum fuel using air as the working medium rather than a heat
engine for processing air in a thermodynamic cycle. Heat, as such, is not supplied to
the internal combustion engine, so it cannot be a heat engine in the sense described
in most thermodynamic references.
A simulated heat engine cycle can be constructed to correspond approximately
to the operation of an internal combustion engine by substitution of analogous heat
transfer processes for some of the actual engine processes. The specific mechanism
of such heat transfer is neglected because the simulation is only a theoretical model
of the engine, not an actual device. Such cycles, called air standard cycles, are a
subject of study in thermodynamic cycles and are useful in the elementary study of
internal combustion engines.
6.3
Closed Cycle
Thermodynamic cycles can be categorized yet another way as closed and open
cycles. In closed cycles, the working fluid returns to the initial state at the end of the
cycle and is recirculated. By the same thinking, in open cycles, the working fluid is
renewed at the end of each cycle instead of being recirculated. For example, in
automobile engines, the combustion gases are exhausted and replaced by a fresh
air-fuel mixture at the end of each cycle. The engine operates on a mechanical
cycle, but the working fluid does not go through a complete thermodynamic cycle
[1, 2] (Fig. 6.8).
Fig. 6.8 Illustration of a
thermodynamic closed
cycle
Initial State
Final State
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300
6 Thermodynamics of Cycles
As we said before, any thermodynamic cycle is essentially a closed cycle in
which, the working substance undergoes a series of processes and is always brought
back to the initial state.
6.4
Gas Compressors and Brayton Cycle
The work in a gas compressor is calculated by,
W_ comp ¼ m_ ðhe hi Þ
ðEq: 6:6Þ
If we assume that the gas in the compressor is calorically perfect, then we have,
W_ comp ¼ m_ cp ðT e T i Þ
ðEq: 6:7Þ
In many cases, this is a reasonable approximation. For noble gases, it is very
accurate because they are calorically perfect. For air and similar working fluids, it
is reasonable because the temperature rise is not that great and an average value of
cp is usually adequate. However, the average value of cp should be chosen based on
a temperature between Te and T, not one at 300 K.
If then we assume that a compressor operates isentropically (adiabatic and
reversible), the exit temperature can be related to the pressure rise in the compressor
as shown below in Eq. (6.8);
Te ¼ Ti
γ1
γ
pe
pi
" γ1
#
γ
T
γR
p
e
e
Ti
1 ¼ m_
1
W_ ¼ m_ cp ½T e T i ¼ m_ cp T i
Ti
pi
γ1
ðEq: 6:8Þ
There are basically three types of compressors—reciprocating, centrifugal flow,
and axial flow. In a reciprocating or positive displacement compressor, a piston
slides in a cylinder and valves open and close to admit low-pressure fluid and
exhaust high-pressure fluid. In centrifugal flow and axial flow compressors, the
fluid enters at one end and is compressed by rotating blades and exits at the opposite
end of the compressor. In the centrifugal flow compressor, the flow is in a radially
outward direction and the compression is achieved by forcing the flow against the
outer annulus of the compressor. In an axial flow compressor, a set of rotating
blades move the flow through the compressor, acting as airfoils. They force the flow
through an increasingly narrower channel, thus increasing the density and pressure.
Gasoline and diesel engines are examples of reciprocating compressors, as are
positive displacement pumps. Water pumps are examples of centrifugal flow
compressors, similar to the rotor in a washing machine. Jet engine compressors
are typically axial flow compressors. Reciprocating compressors require no priming
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6.4 Gas Compressors and Brayton Cycle
301
Ideal Brayton Cycle
T - s diagram
p
3
T = Temperature
p = pressure
Turbine
s = entropy
4
Combustor
Nozzle
5
T
p
0
3
8
Compressor
Inlet
2
0
s
Fig. 6.9 Illustration of Brayton cycle. (Courtesy of NASA)
and can reach very high pressures, but only moderate flow rates. Centrifugal flow
and axial flow compressors usually require priming and can reach very high flow
rates, but moderate pressures.
In this section we discuss the Brayton Thermodynamic Cycle which is used in
all gas turbine engines. Figure 6.9 shows a T-s diagram of the Brayton Cycle. Using
the turbine engine station numbering system, we begin with free stream conditions
at station 0. In cruising flight, the inlet slows the air stream as it is brought to the
compressor face at station2. As the flow slows, some of the energy associated with
the aircraft velocity increases the static pressure of the air and the flow is compressed. Ideally, the compression is isentropic and the static temperature is also
increased as shown in the plot. The compressor does work on the gas and increases
the pressure and temperature isentropically to station3, the compressor exit. Since
the compression is ideally isentropic, a vertical line on the T-s diagram describes
the process. In reality, the compression is not isentropic and the compression
process line leans to the right because of the increase in entropy of the flow. The
combustion process in the burner occurs at constant pressure from station3 to
station4. The temperature increase depends on the type of fuel used and the fuelair ratio. The hot exhaust is then passed through the power turbine in which work is
done by the flow from station4 to station5. Because the turbine and compressor are
on the same shaft, the work done on the turbine is exactly equal to the work done by
the compressor and, ideally, the temperature change is the same. The nozzle then
brings the flow isentropically (adiabatic and reversible) back to free stream pressure
from station5 to station8. Externally, the flow conditions return to free stream
conditions, which completes the cycle. The area under the T-s diagram is
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302
6 Thermodynamics of Cycles
proportional to the useful work and thrust generated by the engine. The T-s diagram
for the ideal Brayton Cycle is shown here:
The Brayton cycle analysis is used to predict the thermodynamic performance of
gas turbine engines.
As we know the gas turbine is another mechanical system that produces power,
and it may operate on a cycle when used as an automobile or truck engine, or on a
closed cycle when used in a nuclear power plant [3].
Usage of the Brayton process in a simple gas turbine cycle can be described as an
open cycle operation where air first enters the compressor, and passes through a
constant-pressure combustion chamber, then goes through the turbine, and then
exits as a product of combustion to the atmosphere, as shown in Fig. 6.10a. A
similar situation can be studied when the combustion chamber of a heat exchanger
gets added onto the loop of Fig. 6.10a in order to organize a closed cycle as can be
seen in Fig. 6.10b. Energy from some external source enters the cycle and the
additional heat exchanger that has been added onto the loop transfers heat from the
cycle so that the air can be returned to its initial state, as clearly seen in Fig. 6.10b.
The Brayton cycle is a theoretical cycle for a simple gas turbine. This cycle
consists of two isentropic and two constant pressure processes. Figure 6.11 shows
a
b
Qin
Fuel
2
2
Combustor
3
Combustor
Compressor
Turbine
Wout
Turbine
4
1
Heat exchanger
4
1
Air
3
Compressor
Products of
Combustion
Qout
Open Cycle
Closed Cycle
Fig. 6.10 Illustration of Brayton components for open and closed cycles
Fig. 6.11 Illustration of the Brayton cycle on P-V and T-s diagram
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Wout
6.4 Gas Compressors and Brayton Cycle
303
the Brayton cycle on P-V and T-s coordinates. The cycle is similar to the Diesel
cycle in compression and heat addition. The isentropic expansion of the Diesel
cycle is further extended followed by constant pressure heat rejection.
The following notation gives the thermal efficiency in mathematical format for
the ideal cycle used to model the gas turbine, which utilizes isentropic compression
and expansion in the Brayton process:
ηth ¼
ηth
Heat added Heat rejected Q_ out
¼
Heat added
Q_ in
mCp ðT 3 T 1 Þ mCp ðT 4 T 1 Þ
mCp ðT 3 T 2 Þ
T4 T1
¼1
T3 T2
T 1 ðT 4 =T 1 Þ 1
¼1
T 2 ðT 3 =T 2 Þ 1
ðEq: 6:9aÞ
¼
ðEq: 6:9bÞ
Using the following isentropic process and relations we have;
T2
¼
T1
P2
P1
γ1
γ
T3
and
¼
T4
γ1
P3 γ
P4
ðEq: 6:9cÞ
For ideal gas and observation the P-V diagram of Fig. 6.11a obviously shows that
we can state P2 ¼ P3 and P1 ¼ P4 as a result, using Eq. (6.10c), will induce the
following;
T2 T3
¼
T1 T4
or
T4
T3
¼
T1
T2
ðEq: 6:9dÞ
Then the Thermal efficiency ηth from Eq. (6.9a) can be reduced to the following
form;
ηth ¼ 1 T4
T1
¼1
T3
T2
ðEq: 6:9eÞ
Now if we introduce a term of the pressure ration r p ¼ P2 =P1 the thermal efficiency
from Eq. (6.9e) will take a form of the following;
T4 T4 V2
1
¼
¼
¼ γ1
T 3 T 3 V 1 rp
γ1 ðγ1Þ
1
V2
P1
¼
¼ rp
V1
P2
r γ1
p
bahmanz@aol.com
ðEq: 6:9fÞ
γ1
γ
ðEq: 6:9gÞ
304
6 Thermodynamics of Cycles
ðγ1Þ=γ
T1
P1
ηth ¼ 1 ¼1
T2
P2
ðEq: 6:9hÞ
ηth ¼ 1 r p ðl-γÞ=γ
ðEq: 6:9iÞ
or
Note that the above final expression for thermal efficiency ηth in both forms of
Eqs. (6.9h) and (6.9i) were obtained based on the assumption of using constant
specific heats. For more accurate calculations the gas tables should be utilized.
In an actual gas turbine the compressor and the turbine are not isentropic and
some losses take place. These losses, usually in the neighborhood of 85 %, significantly reduce the efficiency of the gas turbine engine [2].
Considering all the above we can see that the back work ratio is defined for a
Brayton system as Wcomp/Wturb. This is an important feature of the gas turbine that
limits the thermal efficiency that is required for the compressor to have high work
and is measured by this ratio. This can actually be fairly large approaching 1.0. If
the compressor is too inefficient, the Brayton Cycle will not work. Only after
efficient air compressors were developed was the jet engine feasible.
6.5
The Non-ideal Brayton Cycle
The Ideal Air Standard Brayton Cycle assumes isentropic compression and expansion processes. So far this has not been achieved in any real device. The isentropic
efficiency for these processes is defined as
Δhisentropic
Δhactual
Δhactual
Isentropic efficiency ðexpansionÞ ¼
Δhisentropic
Isentropic efficiency ðcompressionÞ ¼
ðEq: 6:10Þ
ðEq: 6:11Þ
Unfortunately the isentropic efficiency of a compressor or turbine will depend on
the pressure ratio for the device. In doing parametric or design studies it is more
useful to define an efficiency that does not depend on the pressure ratio, but only on
the manufacturing tolerances and efficiencies of individual stages. This small stage,
or infinitesimal stage, efficiency is called the polytropic efficiency.
Consider the combined First and Second Law for an infinitesimal process.
dh ¼ vdp þ Tds
ðEq: 6:12Þ
The term Tds represents a heat flow for the process. During a compression, the
inefficiency of the process represents a heat flow into the system. For an expansion
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6.5 The Non-ideal Brayton Cycle
305
the inefficiency represents a heat flow out of the system. So on an infinitesimal basis
we can write,
dh ¼ vdp þ ðTdsÞ ¼ v dp=ec, poly for a compressor
dh ¼ vdp þ ðTdsÞ ¼ et, poly vdp for a turbine
ðEq: 6:13Þ
Then these two equations can be integrated similar to the way the isentropic relation
was integrated. For an isentropic expansion of a calorically perfect ideal gas we
have,
vdp ¼
RT
dp
p
RT
dp
p
dT
R dp γ l dp
¼
¼
T
Cp p
γ p
γl
T2
γ
¼ pp2
1
T1
dh ¼ cp dT ¼
ðEq: 6:14Þ
For a polytropic compression we have,
vdp ¼
RT
ec, poly p
dh ¼ cp dT ¼
dp
RT
dp
ec, poly p
dT
R dp
γ 1 dp
¼
¼
T
C p
ec, poly γ p
p γ1
T2
γ
e
¼ pp2 c, poly
1
T1
ðEq: 6:15Þ
And for a polytropic expansion we have,
et, poly RT
dp
p
et, poly RT
dh ¼ cp dT ¼
dp
p
dT
R dp et, poly ðγ 1Þ dp
¼
¼
T
Cp p
p
γ
et, poly γ1
T2
γ
¼ pp2
1
T1
vdp ¼
ðEq: 6:16Þ
Now for a calorically perfect gas, the isentropic efficiency of a compressor is given
by,
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306
6 Thermodynamics of Cycles
ηc, isen ¼
Cp ðT out, isen T in Þ
¼
Cp ðT out, actual T in Þ
T out, isen
T in 1
T out, actual
1
T in
pout
pin
¼ γ1
γ
pout
pin
1
γ1
γec, poly
ðEq: 6:17Þ
And the isentropic efficiency of a turbine is given by,
ηt, isen ¼
Cp ðT out, actual T in Þ
¼
Cp ðT out, isen T in Þ
T out, actual
1
T in
T out, isen
T in 1
¼
pout
pin
et, polyγ ðγ1Þ
pout
pin
1
γ1
γec, poly
ðEq: 6:18Þ
There are more thermodynamic cycles than what it is described here and explaining
every one of them is beyond the scope of this book, so we encourage all readers to
take a look at the reference by Zohuri and McDaniel [2] and Chap. 14 of that
reference for more details.
6.6
Open Cycle Gas Turbines
A gas turbine is an internal combustion engine that operates with rotary rather than
reciprocating motion. Gas turbines are composed of three main components:
compressor, combustor, and power turbine. In the compressor section, air is
drawn in and compressed up to 30 times ambient pressure and directed to the
combustor section where fuel is introduced, ignited, and burned. Combustors can be
either annular, can-annular, or silo. An annular combustor is a doughnut-shaped,
single, continuous chamber that encircles the turbine in a plane perpendicular to the
air flow. Can-annular combustors are similar to annular combustors, however they
incorporate several can shaped combustion chambers rather than a single combustion chamber. Annular and can-annular combustors are based on aircraft turbine
technology and are typically used for smaller scale applications. A silo combustor
has one or more combustion chambers mounted external to the gas turbine body.
Silo combustors are typically larger than annular or can-annular combustors and are
used for larger scale operations.
The compressor, combustor, and turbine are connected by one or more shafts and
are collectively called the gas generator or gas turbine. Figures 6.12 and 6.13 below
illustrate the typical gas turbine generator configuration and schematic [4].
Hot gases from the combustion section are diluted with additional air from the
compressor section and directed to the power turbine section at temperatures up to
2600 F. Energy from the hot exhaust gases, which expand in the power turbine
section, is recovered in the form of shaft horsepower. More than 50 % of the shaft
horsepower is needed to drive the internal compressor and the balance of recovered
shaft horsepower is available to drive an external load. In the open cycle gas
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6.6 Open Cycle Gas Turbines
Fig. 6.12 Open cycle gas
turbine configuration
307
fuel
combustion
chamber
3
gases
2 air
compressor
turbine
wout
1
air
ambient
4
gases
ambient
Fig. 6.13 Open cycle gas turbine schematic of JR1 engine [4]
turbine, the heat content of the exhaust gases exiting the turbine is discarded as
opposed to using a heat exchanger to preheat the combustion air entering the
combustor (regenerative cycle) or recovered in a heat recovery steam generator to
raise process steam, with or without supplemental firing (cogeneration) or recovered, with or without supplementary firing to raise steam for a steam turbine
(combined cycle or repowering). The open or simple cycle is the most basic
operating cycle of a gas turbine with a thermal efficiency ranging from 15 to
42 %. Open cycle gas turbines are available in a wide range of power outputs
ranging from 300 hp to over 200,000 hp (0.22–149.14 MW).
As alternatives to the use of gas turbine cycles have already been explored (in the
combined cycle section), this section focuses on alternative technologies to the
standard turbine unit itself that render the open cycle turbine more efficient.
Relatively few manufacturers build large machines; among them are Alstom,
General Electric, Mitsubishi Heavy Industries, and Siemens. The high efficiency
gas turbines (H class) and aero-derivative intercooler gas turbines, developed by
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6 Thermodynamics of Cycles
Table 6.1 High efficiency gas turbine models
Manufacture
Alstom
Mitsubishi
General
Electric
General
Electric
Siemens
Siemens
Hitachi
Model
GT24
M501J
7FA
Simple cycle
efficiency
40
41
38.5
Combined cycle
efficiency
58.4
61.5
58.5
Power produced
(simple) (MW)
230.7
327
216
LMS100
44
53.8
103
SGT68000H
SGT62000E
H-25
40
60.75
274
33.9
51.3
112
34.8
50.3
32
these manufactures and considered as possible alternatives to the typical gas turbine
in the open cycle system, are discussed in great detail below.
Five to three hundred and seventy-five megawatt typical sized turbines are sold
by various manufacturers with higher efficiencies for larger models. Smaller sized
turbines are typically used for offshore applications due to lower weight. Gas
turbines are produced in a range of efficiencies, with larger and newer models
being the most efficient. More efficient models, however, cost more due to the
additional advanced components and less per delivered energy; therefore, a complete economic analysis (net present value, discounted payback) should back-up
investment decisions. A major cause of the increased efficiency is a higher operating temperature in the turbines, which is permitted due to the use of advanced
materials and coatings that can handle more heat. Upgraded cooling systems are
essential to handle this heat, and new sealing systems are used to reduce the cooling
air loss. These upgrades, combined with new advanced compressors, result in
expensive but highly efficient gas turbines that may be considered as alternatives
to the traditional turbines. Table 6.1 provides a summary of the high efficiency gas
turbines on the market today.
6.6.1
Aeroderivative Intercooler Gas Turbines
Intercooler systems work to increase efficiency by allowing for higher pressure
ratios in the combustion zone. This is achieved by splitting the compression unit
into two sections: the Low Pressure Compressor (LPC) and the High Pressure
Compressor (HPC). The intake air is first compressed by the LPC, and then sent
to the intercooler where the pressure is held constant but the temperature is
decreased. The air then goes through the HPC and is sent to the combustor. Since
the air in the engine cannot exceed a given temperature due to the material used in
the turbine, there is traditionally a limit on the pressure ratio; since compressing gas
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6.6 Open Cycle Gas Turbines
309
increases its temperature. By cooling the air part way through but not losing any of
the pressure gain, the intercooler allows for a second compression to occur,
allowing air in the combustor to be within the temperature limits but with a much
higher pressure ratio. The higher ratio causes the turbine to generate more power
with the same fuel input, increasing the overall efficiency of the turbine.
An example of new innovations to the aero-derivative gas turbine is the
35–65 MW high pressure turbine (HPT) developed by GE [5]. The LM6000 PG
offers a 25 % simple cycle power increase compared to its predecessor. The
applications of these turbines include oil and gas platforms, university cogeneration
systems, and industrial park combined cycle installations. These turbines are
designed to operate on partial power, withstand voltage swings, and be capable of
faster dispatching.
6.6.2
Operational Issues/Risks
Gas turbines are complex high speed components, with tight dimensional tolerances, operating at very high temperatures. Components are subject to a variety of
potential issues. These include creep, fatigue, erosion, and oxidation with impact
damage an issue if components fail or following maintenance. Creep may eventually lead to failure but is of most concern because of the dimensional changes it
produces in components subject to load and temperature. A major part of maintenance is checking of dimensions and tolerances. Fatigue is of particular concern at
areas of stress concentration such as the turbine blade roots. Therefore, regular
inspection and maintenance is a requirement, particularly for gas turbines operating
in harsh environments such as offshore applications [6]. This would include
electrical and control systems in addition to the gas turbine itself.
6.6.3
Opportunities/Business Case
The general trend in gas turbine advancement has been toward a combination of
higher temperatures and pressures. While such advancements increase the
manufacturing cost of the machine, the higher value in terms of greater power
output and higher efficiency provides net economic benefits. The industrial gas
turbine is a balance between performances and cost which results in the most
economic machine for both the user and manufacturer. Applications in the oil and
gas industry include pipeline natural gas compression stations in the range of
800–1200 psi (5516–8274 kPa) and compression is required as well as oil pipeline
pumping of crude and refined oil. Turbines up to about 50 MW may be either
industrial or modified aeroderivative engines while larger units up to about
330 MW are designed for specific purposes. For electric power applications, such
as large industrial facilities, simple-cycle gas turbines without heat recovery can
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310
6 Thermodynamics of Cycles
provide peaking power in capacity constrained areas, and utilities often place gas
turbines in the 5–40 MW size range at substations to provide incremental capacity
and grid support. A significant number of simple-cycle gas turbine based Combined
Heat and Power (CHP) systems are in operation at a variety of applications
including oil recovery, chemicals, paper production, food processing, and universities. Note that CHP is also known as cogeneration, which is the simultaneous
production of electricity and heat from a single fuel source, such as: natural gas,
biomass, biogas, coal, waste heat, or oil.
CHP is not a single technology, but an integrated energy system that can be
modified depending upon the needs of the energy end user [7].
CHP provides:
• Onsite generation of electrical and/or mechanical power.
• Waste-heat recovery for heating, cooling, dehumidification, or process
applications.
• Seamless system integration for a variety of technologies, thermal applications,
and fuel types into existing building infrastructure.
The two most common CHP system configurations are:
• Gas turbine or engine with heat recovery unit.
• Steam boiler with steam turbine.
The system configuration is shown in Figs. 6.14 and 6.15.
In order to achieve the two above processes for the most common CHP, design
and application of an appropriate heat exchanger, in particular in the form of
Compact Heat Excganher (CHE), is required.
Gas turbine or reciprocating engine CHP systems generate electricity by burning
fuel (natural gas or biogas) to generate electricity and then use a heat recovery unit
to capture heat from the combustion system’s exhaust stream. This heat is
converted into useful thermal energy, usually in the form of steam or hot water.
Steam or Hot Water
Water
Cooling/Heating
Heat Recovery
Unit
Hot Exhaust
Gases
Fuel
Engine
or
Turbine
Electricity
Building
or
Facility
Generator
Grid
Fig. 6.14 Gas turbine or engine with heat recovery unit
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6.6 Open Cycle Gas Turbines
311
Steam or Hot Water
Cooling/Heating
Water
Boiler
Building
or
Facility
Electricity
Steam
Turbine
Generator
Grid
Fuel
Fig. 6.15 Steam boiler with steam turbine
Fig. 6.16 Overall
schematic of CHP with
HRSG. (Courtesy of Energy
Solutions Center)
Cogeneration
Exhaust
Heat Recovery Steam
Generator (HRSG)
Duct burner
Ambient Air
Combustion Turbine
Generator
Steam Turbine
Steam
Gas turbines/engines are ideally suited for large industrial or commercial CHP
applications requiring ample amounts of electricity and heat.
Steam turbines normally generate electricity as a byproduct of heat (steam)
generation, unlike gas turbine and reciprocating engine CHP systems, where heat
is a byproduct of power generation. Steam turbine-based CHP systems are typically
used in industrial processes, where solid fuels (biomass or coal) or waste products
are readily available to fuel the boiler unit.
To function CHP imposes the integration of a power system such as an engine or
turbine and a Heat Recovery Steam Generator (HRSG) usually a boiler, which is
located on or nearby the user’s facility. Figure 6.16 is an overall schematic of such
configuration.
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6 Thermodynamics of Cycles
CHP application can be found throughout literature as well as in Ref. [7] of this
chapter.
6.6.4
Industrial Case Studies for Open Cycle Gas Turbine
The following is a presentation of industrial case studies for open cycle gas
turbines:
1. High Efficiency Gas Turbine
The new line of high efficiency gas turbines has been designated the H class, and
are currently built by few manufacturers. After an extensive validation process,
GE installed their model, the 9H, at Baglan Bay in 2003. This new model
increased efficiency by allowing the firing temperatures to increase 200 F
(93.3 C) higher than previous models, potentially reaching 2600 F
(1426.7 C). The plant has been reliably providing up to 530 MW to the UK
national grid since then, operating at over 60 % efficiency (as part of a combined
cycle system) [8].
Another manufacturer, Siemens, tested their H class model, the SGT5-8000H,
at full load in Ingolstadt, Germany in 2008. The gas turbine unit’s efficiency was
shown to be 40 %, and was part of a combined cycle system reaching a world
record of 60.75 % efficiency [9]. This plant has been providing power to the
German grid since the testing period finished, all at this same efficiency.
Only these H class turbines truly showcase all of the new adjustments that can
be made to increase efficiency, and they have very large footprints and have
specified outputs of 375 MW and higher. However, the technologies behind the
H class turbines (advanced materials, improved cooling, etc.) are available on
smaller systems. These cases were chosen to illustrate that they are all effective
and operational.
2. Aeroderivative Intercooler Gas Turbines
GE has produced the LMS 100, an extremely high efficiency Aeroderivative
engine. Operating at up to 44 % efficiency at full base load, it generates over
100 MW after a 10 min start-up. The Groton Generating Station in South Dakota
was the first plant to begin using the LMS100, and it has been successfully
operational since 2006 [8]. This technology, while currently available from GE,
is the newest and least tested technology identified here. However, due to its
successful initial testing and extremely high efficiency for a simple cycle, it is an
important alternative to consider.
For further information on combined cycle and application of compact heat
exchanger (CHE) driven efficiency of these combines refer to books by Zohuri
[9, 10].
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References
313
References
1. Cengel, Yunus A., and Michael A. Boles. 2011. Thermodynamics: An engineering approach,
7th ed. New York: McGraw Hill.
2. Zohuri, B., and Patrick McDaniel. 2015. Thermodynamics in nuclear power plant systems.
New York: Springer Publisher.
3. Potter, Merle C., and Craig W. Somerton. 2006. Thermodynamics for engineers, McGraw-Hill
Schaum’s outlines series, 2nd ed. New York: McGraw-Hill.
4. http://www.ipieca.org/energyefficiency/solutions/77801
5. Aeroderivative Technology: A more efficient use of gas turbine Technology, Wacke, A,
General Electric, DRAFT – 2010 – Jan-15.
6. Wall, Martin, Lee Richard, and Simon Frost. Offshore gas turbines (and major driven equipment) integrity and inspection guidance notes. Research Report, 430, ESR Technology Ltd for
the Health and Safety Executive 2006.
7. http://www.epa.gov/chp/basic/
8. Reale, Michael J., and James K. Prochaska. 2005. New high efficiency simple cycle gas
turbine—GE’s LMS100. Industrial Application of Gas Turbines Committee, 14 Oct 2005.
Web. 29 Jul 2013.
9. Zohuri, B. 2015. Combined cycle driven efficiency for next generation nuclear power plants:
An innovative design approach. New York: Springer Publishing Company.
10. Zohuri, B. 2015. Application of compact heat exchangers for combined cycle driven efficiency
in next generation nuclear power plants. New York: Springer Publishing Company.
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