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# Chapter 3 - Power Cycles Notes | EduRev

## : Chapter 3 - Power Cycles Notes | EduRev

``` Page 1

45
C Chapter 3    Power Cycles

Despite recent progress in fuel cell technology which converts the chemical energy of
fuels into electrical energy directly, large scale production of power from fuels involves
the intermediate step of heat generation by combustion. This heat is then converted into
work by a cyclic process in which a working fluid such as water, air or some suitable
chemical absorbs heat at a high temperature and rejects part of this heat at a lower
temperature as shown below. Since the efficiency of converting heat into work is limited
by the second law of thermodynamics, different cycles with different working fluids have
been proposed which strive to approach this theoretical limit as much as possible.

3.1. Fundamentals
System Boundary: Although the heat generated by combustion of a fuel is the most
common heat source for power cycles, other heat sources are also available, such as the
solar radiation collected by concentrating thermal collectors or the heat recovered from
gas turbine exhaust gases. Hence it is important to analyze these cycles independent of
the origin of the heat source in order to evaluate their efficiency in converting heat into
work. For this purpose, the system boundary will be defined such that it is in contact with
the heat source and the heat sink as shown in the figure above. Thus irreversibilities
associated with heat transfer between the working fluid and the heat source/sink are
included but irreversibilities associated with the generation of the heat source itself are
excluded.

FIGURE 3.1: System boundary.
Q
in
Q
out
W
Page 2

45
C Chapter 3    Power Cycles

Despite recent progress in fuel cell technology which converts the chemical energy of
fuels into electrical energy directly, large scale production of power from fuels involves
the intermediate step of heat generation by combustion. This heat is then converted into
work by a cyclic process in which a working fluid such as water, air or some suitable
chemical absorbs heat at a high temperature and rejects part of this heat at a lower
temperature as shown below. Since the efficiency of converting heat into work is limited
by the second law of thermodynamics, different cycles with different working fluids have
been proposed which strive to approach this theoretical limit as much as possible.

3.1. Fundamentals
System Boundary: Although the heat generated by combustion of a fuel is the most
common heat source for power cycles, other heat sources are also available, such as the
solar radiation collected by concentrating thermal collectors or the heat recovered from
gas turbine exhaust gases. Hence it is important to analyze these cycles independent of
the origin of the heat source in order to evaluate their efficiency in converting heat into
work. For this purpose, the system boundary will be defined such that it is in contact with
the heat source and the heat sink as shown in the figure above. Thus irreversibilities
associated with heat transfer between the working fluid and the heat source/sink are
included but irreversibilities associated with the generation of the heat source itself are
excluded.

FIGURE 3.1: System boundary.
Q
in
Q
out
W
46
Carnot, Thermal and Second Law Efficiencies: Thermal efficiency is the work
produced by the cycle divided by the heat input:

Second law efficiency of the cycle can be calculated as:

?
II
= W/ W
ideal
= W/(W + LW)

where lost work or irreversibility can be calculated according to Gouy-Stodola theorem
as the product of the environmental temperature T
o
and the cyclic integral of dQ/T where
T is the temperature of the source(s) or sink(s) exchanging heat with the working fluid.

For each step in the cycle, the entropy balance for a closed or an open system can be
used to calculate entropy production. Then multiplication of this entropy production with
the environmental temperature gives irreversibility for this step.

For a closed system,

I = T
o
[ ( S
final
– S
initial
)
sys
– ? Q
in
/T
source
+  ? Q
out
/T
sink
]

For a steady flow, open system,

I = T
o
[?(ms)
out
–  ?(ms)
in
– ? Q
in
/T
source
+ ? Q
out
/T
sink
]

Now let’s consider a reversible heat engine whose low temperature sink is the
environment. Then the work produced by this heat engine is the same as the ideal work
and the Carnot and the second law efficiencies are:

?
C
= W
rev
/Q
in

?
II
= W
rev
/ W
ideal
= 1
For an actual heat engine operating between the same source and sink as the reversible
one, thermal efficiency is the product of second law and Carnot efficiencies as shown
below:

?
I
= W
actual
/Q
in

?
II
= W
actual
/ W
ideal
= W
actual
/ W
rev
?
I
= (W
actual
/Q
in
)(W
rev
/W
rev
) = (W
actual
/W
rev
)(W
rev
/Q
in
)
?
I
= ?
II
?
C
?
I
= W/Q
in

Page 3

45
C Chapter 3    Power Cycles

Despite recent progress in fuel cell technology which converts the chemical energy of
fuels into electrical energy directly, large scale production of power from fuels involves
the intermediate step of heat generation by combustion. This heat is then converted into
work by a cyclic process in which a working fluid such as water, air or some suitable
chemical absorbs heat at a high temperature and rejects part of this heat at a lower
temperature as shown below. Since the efficiency of converting heat into work is limited
by the second law of thermodynamics, different cycles with different working fluids have
been proposed which strive to approach this theoretical limit as much as possible.

3.1. Fundamentals
System Boundary: Although the heat generated by combustion of a fuel is the most
common heat source for power cycles, other heat sources are also available, such as the
solar radiation collected by concentrating thermal collectors or the heat recovered from
gas turbine exhaust gases. Hence it is important to analyze these cycles independent of
the origin of the heat source in order to evaluate their efficiency in converting heat into
work. For this purpose, the system boundary will be defined such that it is in contact with
the heat source and the heat sink as shown in the figure above. Thus irreversibilities
associated with heat transfer between the working fluid and the heat source/sink are
included but irreversibilities associated with the generation of the heat source itself are
excluded.

FIGURE 3.1: System boundary.
Q
in
Q
out
W
46
Carnot, Thermal and Second Law Efficiencies: Thermal efficiency is the work
produced by the cycle divided by the heat input:

Second law efficiency of the cycle can be calculated as:

?
II
= W/ W
ideal
= W/(W + LW)

where lost work or irreversibility can be calculated according to Gouy-Stodola theorem
as the product of the environmental temperature T
o
and the cyclic integral of dQ/T where
T is the temperature of the source(s) or sink(s) exchanging heat with the working fluid.

For each step in the cycle, the entropy balance for a closed or an open system can be
used to calculate entropy production. Then multiplication of this entropy production with
the environmental temperature gives irreversibility for this step.

For a closed system,

I = T
o
[ ( S
final
– S
initial
)
sys
– ? Q
in
/T
source
+  ? Q
out
/T
sink
]

For a steady flow, open system,

I = T
o
[?(ms)
out
–  ?(ms)
in
– ? Q
in
/T
source
+ ? Q
out
/T
sink
]

Now let’s consider a reversible heat engine whose low temperature sink is the
environment. Then the work produced by this heat engine is the same as the ideal work
and the Carnot and the second law efficiencies are:

?
C
= W
rev
/Q
in

?
II
= W
rev
/ W
ideal
= 1
For an actual heat engine operating between the same source and sink as the reversible
one, thermal efficiency is the product of second law and Carnot efficiencies as shown
below:

?
I
= W
actual
/Q
in

?
II
= W
actual
/ W
ideal
= W
actual
/ W
rev
?
I
= (W
actual
/Q
in
)(W
rev
/W
rev
) = (W
actual
/W
rev
)(W
rev
/Q
in
)
?
I
= ?
II
?
C
?
I
= W/Q
in

47
This relationship is applicable to all heat engines whose low temperature sink is the
environment. Note that if the temperature of the heat source is variable, Carnot efficiency
is not equal to 1 – T
o
/T and should be properly modified as shown in the following
example.
.
Example 3.1: A hot exhaust stream from a process plant is at T
1
and P
o
. Since T
1
is
considerably greater than T
o
, it is considered to construct a heat engine which will be
driven by the heat extracted from this stream as it is cooled down to T
2
. The heat engine
will use the environment as its sink. If this engine were a reversible one, what would be
its thermal efficiency?

Solution:

FIGURE 3.2: Heat engine with a variable temperature heat source.

The  engine is shown above. The  condition for reversibility can be expressed as:

?S + Q
o
/ T
o
= 0 ?  Q
o
= – T
o
?S

W = Q – Q
o

Q = ?H = H
1
– H
2
= ?c
P
dT
?S = S
2
– S
1
= ?c
P
dT/T

Assuming heat capacity is constant, integration yields:

H
1
H
2
Q
Q
o
W
T
o
S
1
S
2
Page 4

45
C Chapter 3    Power Cycles

Despite recent progress in fuel cell technology which converts the chemical energy of
fuels into electrical energy directly, large scale production of power from fuels involves
the intermediate step of heat generation by combustion. This heat is then converted into
work by a cyclic process in which a working fluid such as water, air or some suitable
chemical absorbs heat at a high temperature and rejects part of this heat at a lower
temperature as shown below. Since the efficiency of converting heat into work is limited
by the second law of thermodynamics, different cycles with different working fluids have
been proposed which strive to approach this theoretical limit as much as possible.

3.1. Fundamentals
System Boundary: Although the heat generated by combustion of a fuel is the most
common heat source for power cycles, other heat sources are also available, such as the
solar radiation collected by concentrating thermal collectors or the heat recovered from
gas turbine exhaust gases. Hence it is important to analyze these cycles independent of
the origin of the heat source in order to evaluate their efficiency in converting heat into
work. For this purpose, the system boundary will be defined such that it is in contact with
the heat source and the heat sink as shown in the figure above. Thus irreversibilities
associated with heat transfer between the working fluid and the heat source/sink are
included but irreversibilities associated with the generation of the heat source itself are
excluded.

FIGURE 3.1: System boundary.
Q
in
Q
out
W
46
Carnot, Thermal and Second Law Efficiencies: Thermal efficiency is the work
produced by the cycle divided by the heat input:

Second law efficiency of the cycle can be calculated as:

?
II
= W/ W
ideal
= W/(W + LW)

where lost work or irreversibility can be calculated according to Gouy-Stodola theorem
as the product of the environmental temperature T
o
and the cyclic integral of dQ/T where
T is the temperature of the source(s) or sink(s) exchanging heat with the working fluid.

For each step in the cycle, the entropy balance for a closed or an open system can be
used to calculate entropy production. Then multiplication of this entropy production with
the environmental temperature gives irreversibility for this step.

For a closed system,

I = T
o
[ ( S
final
– S
initial
)
sys
– ? Q
in
/T
source
+  ? Q
out
/T
sink
]

For a steady flow, open system,

I = T
o
[?(ms)
out
–  ?(ms)
in
– ? Q
in
/T
source
+ ? Q
out
/T
sink
]

Now let’s consider a reversible heat engine whose low temperature sink is the
environment. Then the work produced by this heat engine is the same as the ideal work
and the Carnot and the second law efficiencies are:

?
C
= W
rev
/Q
in

?
II
= W
rev
/ W
ideal
= 1
For an actual heat engine operating between the same source and sink as the reversible
one, thermal efficiency is the product of second law and Carnot efficiencies as shown
below:

?
I
= W
actual
/Q
in

?
II
= W
actual
/ W
ideal
= W
actual
/ W
rev
?
I
= (W
actual
/Q
in
)(W
rev
/W
rev
) = (W
actual
/W
rev
)(W
rev
/Q
in
)
?
I
= ?
II
?
C
?
I
= W/Q
in

47
This relationship is applicable to all heat engines whose low temperature sink is the
environment. Note that if the temperature of the heat source is variable, Carnot efficiency
is not equal to 1 – T
o
/T and should be properly modified as shown in the following
example.
.
Example 3.1: A hot exhaust stream from a process plant is at T
1
and P
o
. Since T
1
is
considerably greater than T
o
, it is considered to construct a heat engine which will be
driven by the heat extracted from this stream as it is cooled down to T
2
. The heat engine
will use the environment as its sink. If this engine were a reversible one, what would be
its thermal efficiency?

Solution:

FIGURE 3.2: Heat engine with a variable temperature heat source.

The  engine is shown above. The  condition for reversibility can be expressed as:

?S + Q
o
/ T
o
= 0 ?  Q
o
= – T
o
?S

W = Q – Q
o

Q = ?H = H
1
– H
2
= ?c
P
dT
?S = S
2
– S
1
= ?c
P
dT/T

Assuming heat capacity is constant, integration yields:

H
1
H
2
Q
Q
o
W
T
o
S
1
S
2
48
W
rev
= c
P
(T
1
– T
2
)  – Toc
P
ln(T
1
/T
2
)
?
C
= W
rev
/Q  = ?c
P
(T
1
–T
2
) –T
o
c
P
ln(T
1
/T
2
) ? / c
P
(T
1
–T
2
)
Rearranging and simplifying,

?
C
= ?1 –T
o
ln(T
1
/T
2
) /(T
1
–T
2
) ?
Ideal and Non-ideal Cycles: All ideal cycles are internally reversible. Hence all the
irreversibilities are external and associated with heat flow. Since frictional dissipation is
ignored, pipes connecting different components are assumed to have no heat loss and
processes involving work are assumed to be adiabatic, heat addition and heat rejection
processes are the only sources of irreversibility in ideal cycles. Provided that source and
sink temperatures can be properly defined, second law efficiency of the entire cycle can
be easily calculated and it is not necessary to know the entropy of the working fluid.
A suitable sink temperature is usually the ambient temperature which is constant whereas
the source temperature is usually not constant. Furthermore, assigning a numerical value
to the source temperature is difficult in the case of internal combustion engines where
actual combustion is replaced by external heat addition to simplify the analysis. One
approach is to assign the maximum temperature of the cycle as the source temperature.
Obviously, second law efficiencies calculated in this manner are only approximate and
are useful for a quick comparison of different cycles.
Calculating irreversibilities of heat addition and heat rejection steps separately requires
application of the entropy balance for a closed or steady flow, open system depending on
the characteristics of the cycle. It is customary to employ the steady flow entropy balance
for cycles such as the Rankine and the Brayton cycles whereas cycles such as the Otto
and the Diesel cycles are analyzed as closed systems. In either case, entropy of the
working fluid must be known.
Ideal cycles which are internally reversible do not exist in real life since fluid flow
always involves frictional dissipation. For incompressible fluids flowing in pipes
connecting different pieces of equipment, frictional dissipation is usually ignored by
assuming pressure drops are negligible. However, for compressible flow through
equipment such as compressors and turbines where the ratio between inlet and outlet
pressures is usually greater than 5, frictional dissipation can no longer be ignored, i.e.,
compression and expansion processes can not be assumed isentropic.
Now let us consider non-ideal cycles where compression and expansion processes are not
isentropic. The second law efficiency of the cycle can still be calculated in the same
manner, i.e., by the cyclic integral of dQ/T since the effect of internal irreversibilities are
reflected in the numerical values of heat flows. For example, if the turbine has an
isentropic efficiency of less than 100%, less work than the ideal cycle is produced with
the consequence of rejecting more heat than the ideal cycle.

Page 5

45
C Chapter 3    Power Cycles

Despite recent progress in fuel cell technology which converts the chemical energy of
fuels into electrical energy directly, large scale production of power from fuels involves
the intermediate step of heat generation by combustion. This heat is then converted into
work by a cyclic process in which a working fluid such as water, air or some suitable
chemical absorbs heat at a high temperature and rejects part of this heat at a lower
temperature as shown below. Since the efficiency of converting heat into work is limited
by the second law of thermodynamics, different cycles with different working fluids have
been proposed which strive to approach this theoretical limit as much as possible.

3.1. Fundamentals
System Boundary: Although the heat generated by combustion of a fuel is the most
common heat source for power cycles, other heat sources are also available, such as the
solar radiation collected by concentrating thermal collectors or the heat recovered from
gas turbine exhaust gases. Hence it is important to analyze these cycles independent of
the origin of the heat source in order to evaluate their efficiency in converting heat into
work. For this purpose, the system boundary will be defined such that it is in contact with
the heat source and the heat sink as shown in the figure above. Thus irreversibilities
associated with heat transfer between the working fluid and the heat source/sink are
included but irreversibilities associated with the generation of the heat source itself are
excluded.

FIGURE 3.1: System boundary.
Q
in
Q
out
W
46
Carnot, Thermal and Second Law Efficiencies: Thermal efficiency is the work
produced by the cycle divided by the heat input:

Second law efficiency of the cycle can be calculated as:

?
II
= W/ W
ideal
= W/(W + LW)

where lost work or irreversibility can be calculated according to Gouy-Stodola theorem
as the product of the environmental temperature T
o
and the cyclic integral of dQ/T where
T is the temperature of the source(s) or sink(s) exchanging heat with the working fluid.

For each step in the cycle, the entropy balance for a closed or an open system can be
used to calculate entropy production. Then multiplication of this entropy production with
the environmental temperature gives irreversibility for this step.

For a closed system,

I = T
o
[ ( S
final
– S
initial
)
sys
– ? Q
in
/T
source
+  ? Q
out
/T
sink
]

For a steady flow, open system,

I = T
o
[?(ms)
out
–  ?(ms)
in
– ? Q
in
/T
source
+ ? Q
out
/T
sink
]

Now let’s consider a reversible heat engine whose low temperature sink is the
environment. Then the work produced by this heat engine is the same as the ideal work
and the Carnot and the second law efficiencies are:

?
C
= W
rev
/Q
in

?
II
= W
rev
/ W
ideal
= 1
For an actual heat engine operating between the same source and sink as the reversible
one, thermal efficiency is the product of second law and Carnot efficiencies as shown
below:

?
I
= W
actual
/Q
in

?
II
= W
actual
/ W
ideal
= W
actual
/ W
rev
?
I
= (W
actual
/Q
in
)(W
rev
/W
rev
) = (W
actual
/W
rev
)(W
rev
/Q
in
)
?
I
= ?
II
?
C
?
I
= W/Q
in

47
This relationship is applicable to all heat engines whose low temperature sink is the
environment. Note that if the temperature of the heat source is variable, Carnot efficiency
is not equal to 1 – T
o
/T and should be properly modified as shown in the following
example.
.
Example 3.1: A hot exhaust stream from a process plant is at T
1
and P
o
. Since T
1
is
considerably greater than T
o
, it is considered to construct a heat engine which will be
driven by the heat extracted from this stream as it is cooled down to T
2
. The heat engine
will use the environment as its sink. If this engine were a reversible one, what would be
its thermal efficiency?

Solution:

FIGURE 3.2: Heat engine with a variable temperature heat source.

The  engine is shown above. The  condition for reversibility can be expressed as:

?S + Q
o
/ T
o
= 0 ?  Q
o
= – T
o
?S

W = Q – Q
o

Q = ?H = H
1
– H
2
= ?c
P
dT
?S = S
2
– S
1
= ?c
P
dT/T

Assuming heat capacity is constant, integration yields:

H
1
H
2
Q
Q
o
W
T
o
S
1
S
2
48
W
rev
= c
P
(T
1
– T
2
)  – Toc
P
ln(T
1
/T
2
)
?
C
= W
rev
/Q  = ?c
P
(T
1
–T
2
) –T
o
c
P
ln(T
1
/T
2
) ? / c
P
(T
1
–T
2
)
Rearranging and simplifying,

?
C
= ?1 –T
o
ln(T
1
/T
2
) /(T
1
–T
2
) ?
Ideal and Non-ideal Cycles: All ideal cycles are internally reversible. Hence all the
irreversibilities are external and associated with heat flow. Since frictional dissipation is
ignored, pipes connecting different components are assumed to have no heat loss and
processes involving work are assumed to be adiabatic, heat addition and heat rejection
processes are the only sources of irreversibility in ideal cycles. Provided that source and
sink temperatures can be properly defined, second law efficiency of the entire cycle can
be easily calculated and it is not necessary to know the entropy of the working fluid.
A suitable sink temperature is usually the ambient temperature which is constant whereas
the source temperature is usually not constant. Furthermore, assigning a numerical value
to the source temperature is difficult in the case of internal combustion engines where
actual combustion is replaced by external heat addition to simplify the analysis. One
approach is to assign the maximum temperature of the cycle as the source temperature.
Obviously, second law efficiencies calculated in this manner are only approximate and
are useful for a quick comparison of different cycles.
Calculating irreversibilities of heat addition and heat rejection steps separately requires
application of the entropy balance for a closed or steady flow, open system depending on
the characteristics of the cycle. It is customary to employ the steady flow entropy balance
for cycles such as the Rankine and the Brayton cycles whereas cycles such as the Otto
and the Diesel cycles are analyzed as closed systems. In either case, entropy of the
working fluid must be known.
Ideal cycles which are internally reversible do not exist in real life since fluid flow
always involves frictional dissipation. For incompressible fluids flowing in pipes
connecting different pieces of equipment, frictional dissipation is usually ignored by
assuming pressure drops are negligible. However, for compressible flow through
equipment such as compressors and turbines where the ratio between inlet and outlet
pressures is usually greater than 5, frictional dissipation can no longer be ignored, i.e.,
compression and expansion processes can not be assumed isentropic.
Now let us consider non-ideal cycles where compression and expansion processes are not
isentropic. The second law efficiency of the cycle can still be calculated in the same
manner, i.e., by the cyclic integral of dQ/T since the effect of internal irreversibilities are
reflected in the numerical values of heat flows. For example, if the turbine has an
isentropic efficiency of less than 100%, less work than the ideal cycle is produced with
the consequence of rejecting more heat than the ideal cycle.

49
3.2. Gas Power Cycles
In heat engines, working fluids undergo cyclic processes, extracting heat from a hot
source and rejecting heat to a low temperature sink (usually the environment), producing
some mechanical work in the way (usually shaft work). Gases are the best working
substances because they can easily convert thermal energy into mechanical energy by
compression or expansion, whereas liquids have little compressibility. Gas power cycles
are those where the working substance stays all the time in the gas phase, and vapor
power cycles are those where the gas condenses to liquid in some part of the cycle.
Many gas cycles have been proposed, and several are currently used, to model real heat
engines: the Otto cycle (approximates the actual gasoline engine), the Diesel cycle
(approximates the actual diesel engine), the mixed cycle (a hybrid of the Otto and the
Diesel that is better than both), the Brayton cycle (approximates very well the actual gas
turbine engine), and the Stirling cycle. We will restrict the analysis to the so called ‘air
standard’ model, which assumes air as an ideal gas with temperature-independent
properties is the working fluid, neglecting fuel addition effects (the air to fuel mass ratio
is around 15:1 in Otto engines, 30:1 in Diesel engines, and 60:1 in Brayton engines), and
The Otto Cycle:  In the ideal air standard Otto cycle, air with temperature independent
properties undergoes four main processes as shown in Figure 3.3: (i) isentropic
compression(1?2), (ii) constant volume heat addition(2?3), (iii) isentropic
expansion(3?4) and (iv) constant volume heat rejection(4?1).

FIGURE 3.3: The ideal Otto cycle.

Assuming a thermal reservoir is available at the maximum temperature of the cycle (T
3
)
and the heat sink is at T
1
, it can be shown that the thermal and second law efficiencies are
given by:

Q
in

Q
out

W
in

W
out

V
P
Q
in

Q
out

W
out

W
in

T
S
4
3
2
1
4
3
2
1
```
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