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Carnot Cycle Introduction This section provides a Carnot Cycle analysis when the working fluid is air. Analysis In the presented Carnot Cycle analysis, only air is considered as the working fluid behaving as a perfect gas -- specific heat has a constant value. Ideal gas state equation is valid -- pv = RT. Air enters a compressor at point 1 and it exits the compressor at point 2. Isentropic compression is considered with no entropy change. Air enters a heat exchanger -- heat addition -- at point 2 and it exits the heat exchanger at point 3. At a constant temperature, heat addition takes place. Air enters a turbine at point 3 and it exits the turbine at point 4. Isentropic expansion is considered with no entropy change. Air enters a heat exchanger -- heat rejection -- at point 4 and it exits the heat exchanger at point 1. At a constant temperature, heat rejection takes place. It should be mentioned that air at point 1 enters the compressor and the cycle is repeated. Figure 1 presents a Carnot Cycle schematic layout.
Figure 1 - Carnot Cycle Schematic Layout Figure 2 presents a Carnot Cycle temperature vs entropy diagram.
Figure 2 - Carnot Cycle Temperature vs Entropy Diagram Figure 3 presents the Carnot Cycle efficiency as a function of the heat addition temperature. It should be noted that the inlet conditions are standard ambient conditions: temperature of 298 [K] and absolute pressure of 1 [atm].
Figure 3 - Carnot Cycle Efficiency vs Heat Addition Temperature Figure 4 presents the Carnot Cycle efficiency as a function of the heat rejection temperature. It should be noted that the turbine inlet temperature is at 800 [K].
Figure 4 - Carnot Cycle Efficiency vs Heat Rejection Temperature One can notice that the Carnot Cycle efficiency increases with an increase in the heat addition temperature when the heat rejection temperature does not change at all. One can notice that the Carnot Cycle efficiency decreases with an increase in the heat rejection temperature when the heat addition temperature does not change at all. Assumptions Working fluid is air. There is no friction. Compression and expansion are isentropic -- there is no entropy change. During heat addition and heat rejection, the air temperature does not change. Ideal gas state equation is valid -- pv = RT. Air behaves as a perfect gas -- specific heat has a constant value. Governing Equations T2/T1 =
(p2/p1)(k-1)/k k = cp/cv efficiency = 1 - T1/T2 Input Data T1 = 298 [K] p1 = 1 [atm] cp = 1.004 [kJ/kg*K] k = cp/cv - for air k = 1.4 [/] Results Carnot Cycle Efficiency vs Heat
Addition Temperature
Carnot Cycle Efficiency vs Heat
Rejection Temperature
Figures
Conclusions The Carnot Cycle efficiency increases with an increase in the heat addition temperature when the heat rejection temperature does not change at all. Furthermore, the Carnot Cycle efficiency decreases with an increase in the heat rejection temperature when the heat addition temperature does not change at all. The Carnot Cycle efficiency is not dependent on the working fluid properties. References
JANAF Thermochemical Data - Tables, 1970 Brayton Cycle -- Power Application Introduction This section provides a Brayton Cycle analysis when the working fluid is air. Analysis In the presented Brayton Cycle analysis, only air is considered as the working fluid behaving as a perfect gas -- specific heat has a constant value. Ideal gas state equation is valid -- pv = RT. A gas turbine is a heat engine that uses a high temperature, high pressure gas as the working fluid. Combustion of a fuel in air is usually used to produce the needed temperatures and pressures in the gas turbine, which is why gas turbines are often referred to as combustion turbines. Expansion of the high temperature, high pressure working fluid takes place in the gas turbine. The gas turbine shaft rotation drives an electric generator and a compressor for the working fluid, air, used in the gas turbine combustor. Many gas turbines also use a heat exchanger called a recouperator to impart turbine exhaust heat into the combustor's air/fuel mixture. Gas turbines produce high quality heat that can be used to generate steam for combined heat and power and combined-cycle applications, significantly enhancing efficiency. Air is compressed, isentropically, along line 1-2 by a compressor and it enters a combustor. At a constant pressure, combustion takes place (fuel is added to the combustor and the air temperature raises) and/or heat gets added to air. High temperature air exits the combustor at point 3. Then air enters a gas turbine where an isentropic expansion occurs, producing power. Air exits the gas turbine at point 4. It should be mentioned that air at point 1 enters the compressor and the cycle is repeated. Figure 1 presents a Brayton Cycle schematic layout .
Figure 1 - Brayton Cycle Schematic Layout Figure 2 presents a Brayton Cycle temperature vs entropy diagram.
Figure 2 - Brayton Cycle Temperature vs Entropy Diagram It should be pointed out that this material deals with the open Brayton Cycle. The way how the T - s diagram is presented, it describes a closed Brayton Cycle -- this would require a heat exchanger after point 4 where the working fluid would be cooled down to point 1 and the cycle repeats. Therefore, the T - s diagram is presented as a closed Brayton Cycle to allow easier understanding and derivation of the Brayton Cycle thermal efficiency -- heat addition and heat rejection. The gas turbine and compressor are connected by shaft so the considerable amount of work done on the gas turbine is used to power the compressor. It can be noticed from the T - s diagram that the work done on the gas turbine is greater than the work necessary to power the compressor -- constant pressure lines in the T - s diagram diverge by going to the right side (entropy wise). Figure 3 presents the Brayton Cycle efficiency as a function of the compression ratio. It should be noted that the inlet conditions are standard ambient conditions: temperature of 298 [K] and absolute pressure of 1 [atm].
Figure 3 - Brayton Cycle Efficiency Here, two general performance trends are considered. First, impact of the gas turbine inlet temperature and compression ratio on the Brayton Cycle specific power output and second, impact of the working fluid mass flow rate on the Brayton Cycle power output. Figure 4 presents the results of the first performance trend, while Figure 5 presents the results of the second trend.
Figure 4 - Brayton Cycle Specific Power Output
Figure 5 - Brayton Cycle Power Output One can notice that the Brayton Cycle efficiency increases with an increase in the compression ratio. One can notice that the Brayton Cycle specific power output increases with an increase in the gas turbine inlet temperature. Furthermore, the increase is greater for the higher compression ratio. One can notice that the Brayton Cycle power output increases with an increase in the working fluid mass flow rate. The increase is greater for the higher compression ratio. Assumptions Compression and expansion processes are reversible and adiabatic --
isentropic. The working fluid has the same
composition throughout the
cycle. Ideal gas state
equation is valid -- pv = RT. Air behaves as
a
perfect gas --specific heat has a
constant value. T2/T1 = (p2/p1)(k-1)/k p2/p1 = (T2/T1)k/(k-1) T3/T4 = (p3/p4)(k-1)/k p3/p4 = (T3/T4)k/(k-1) k = cp/cv w = qh - ql w = cp(T3 - T2) - cp(T4 - T1) W = (cp(T3 - T2) - cp(T4 - T1))m efficiency = 1 - 1/rp(k-1)/k rp = p2/p1 Input Data T1 =
298 [K] T3 = 900, 1,200 and 1,500 [K]
cp = 1.004 [kJ/kg*K] k = cp/cv - for air k = 1.4
[/] Results Brayton Cycle Efficiency vs Compression Ratio
Specific Power Output vs Compression
Ratio
Power Output vs Compression Ratio
Figures
Conclusions References
JANAF Thermochemical Data - Tables, 1970
This section provides a Brayton Cycle analysis when the working fluid is air.
Analysis
In the presented Brayton Cycle analysis, only air is considered as the
working fluid behaving as a perfect gas -- specific
heat has a constant
value.
Ideal gas state equation
is valid -- pv = RT. A gas turbine is a heat engine that uses a high temperature, high pressure
gas as the working fluid. Combustion of a fuel in
air is usually used to
produce the
needed temperatures and
pressures in the gas turbine,
which is why
gas turbines
are often
referred to as combustion turbines. Expansion of
the
high temperature, high pressure working fluid takes
place
in the gas
turbine. The gas turbine shaft
rotation drives an electric generator and a
compressor for
the working fluid,
air, used in the gas turbine combustor.
Many gas
turbines also use a heat exchanger called a
recouperator
to
impart turbine exhaust heat into the
combustor's air/fuel mixture. Gas
turbines produce
high quality heat
that can be used to generate steam for
combined heat and
power and combined-cycle applications,
significantly
enhancing
efficiency.
Air is compressed,
isentropically, along line 1-2 by a
compressor and it
enters a
combustor.
At a constant pressure, combustion takes place (fuel
is added to the combustor and the air
temperature raises) and/or heat gets added to air. High
temperature air
exits the
combustor at point 3. Then air
enters
a gas turbine where an isentropic expansion
occurs,
producing
power. Air exits the
gas turbine at point 4. It should
be mentioned that air
at point 1 enters the compressor and the
cycle is
repeated.
Figure 1 presents a Brayton Cycle schematic layout
.
Figure 1 - Brayton Cycle Schematic
Layout
Figure 2 presents a Brayton Cycle temperature vs entropy diagram.
Figure 2 - Brayton Cycle Temperature vs Entropy
Diagram
In order to keep the scope of thrust analysis simple, air exiting turbine expans to the atmospheric conditions - exit pressure is equal to the ambient pressure (p1 = p4). It should be pointed out that this material deals with the open
Brayton Cycle. The way how the T - s diagram is
presented, it describes a
closed
Brayton Cycle -- this
would require a heat exchanger after
point 4 where
the
working fluid
would be cooled down to point 1 and the cycle repeats.
Therefore, the T - s diagram is presented as a
closed Brayton Cycle to
allow easier
understanding and
derivation of the Brayton Cycle thermal
efficiency -- heat
addition and heat
rejection.
The gas turbine and compressor are connected by shaft so the
considerable amount of work done on the gas turbine is
used to power the
compressor.
Propulsion is provided by the difference of gas turbine expansion minus the compressor power requirements. It can be noticed from the T - s diagram that the work done on the
gas turbine is greater than the work necessary to power
the compressor --
constant pressure
lines in the T - s
diagram diverge by going to the right
side
(entropy wise).
Figure 3 presents the Brayton Cycle efficiency as a function of the compression
ratio. It should be noted that the inlet conditions are standard ambient conditions: temperature of 298 [K] and absolute pressure of 1 [atm].
Figure 3 - Brayton Cycle
Efficiency
Here, two general performance trends are considered. First,
impact of the gas turbine inlet temperature and
compression ratio on the Brayton
Cycle specific propulsion
output and
second, impact of the working fluid mass flow rate on
the
Brayton Cycle propulsion output.
Figure 4 presents the results of the first performance trend,
while Figure 5 presents the results of the second trend.
Figure 4 - Brayton Cycle Specific Propulsion
Output
Figure 5 - Brayton Cycle Propulsion Output
One can notice that the Brayton Cycle efficiency increases with an increase in the compression ratio. One can notice that the Brayton Cycle specific propulsion output
increases with an increase in the gas turbine inlet
temperature.
Furthermore, the
increase is greater
for the higher compression ratio.
One can notice that the Brayton Cycle propulsion output increases with
an increase in the working fluid mass flow rate. The
increase is greater
for the higher
compression ratio.
Assumptions
Compression and expansion processes are reversible and adiabatic --
isentropic. The working fluid has the same
composition throughout the
cycle. Ideal gas state
equation is valid -- pv = RT. Air behaves as
a
perfect gas --specific heat has a
constant value. T2/T1 =
(p2/p1)(k-1)/k p2/p1 =
(T2/T1)k/(k-1) T3/T4 =
(p3/p4)(k-1)/k
p3/p4 =
(T3/T4)k/(k-1)
k = cp/cv w = qh - ql
w = cp(T3 - T2) - cp(T4 - T1)
W = (cp(T3 - T2) - cp(T4 - T1))m efficiency = 1 - 1/rp(k-1)/k
rp =
p2/p1 v2/2 = cp((T3 - T2) - (T4 - T1) ) v/2 = (2cp((T3 - T2) - (T4 - T1)))1/2 Thrust = vm
Input Data T1 =
298 [K] T3 = 900, 1,200 and 1,500 [K]
cp =
1.004 [kJ/kg*K]
k = cp/cv - for air k = 1.4
[/] Results
Brayton Cycle Efficiency vs Compression
Ratio
Compression Ratio Brayton Cycle 5 36.92 10 48.22 15 53.87 20 57.53 25 60.16 Specific Propulsion Output vs Compression
Ratio Specific Propulsion Output Gas Turbine Inlet Temperature Compression Ratio 900 1,200 1,500 5 563 734 872 15 524 774 961 Propulsion Output vs Compression Ratio Propulsion Output Mass Flow Rate Compression Ratio 50 100 150 5 43.61 87.23 130.85 15 48.06 96.11 144.17 Figures
Conclusions
The Brayton Cycle efficiency increases with an increase in the compression ratio. The Brayton Cycle specific propulsion output increases with an increase in the gas
turbine inlet temperature. Furthermore, the increase is
greater for the higher
compression ratio. The Brayton Cycle propulsion output increases with an increase in the
working fluid mass flow rate. The increase is greater for
the higher compression
ratio.
References
JANAF Thermochemical Data - Tables, 1970
This section provides an Otto Cycle analysis when the working fluid is air.
Analysis
In the presented Otto Cycle analysis, only air is
considered as the working fluid behaving as a perfect gas -- specific
heat has a constant value. Ideal gas state equation is valid -- pv =
RT.
Air
enters a cylinder at point 1 when compression starts and it ends at
point 2.
Isentropic compression
is considered with no entropy
change. Heat
addition
starts at point 2 and it ends at
point 3.
At a constant volume, combustion takes place (fuel is added to the cylinder and the air temperature raises) and/or heat gets added to air. Expansion
starts at point 3 and it ends at
point 4.
Isentropic expansion
is considered with no entropy
change. Air heat
rejection starts at point 4
and it ends
at point 1. At a constant
volume, air gets cooled and the working fluid
temperature
goes down. It
should be mentioned that air at point 1 enters
the
compression process again and the cycle is repeated. Figure 1 presents an Otto Cycle pressure vs volume diagram.
Figure 1 - Otto Cycle Pressure vs
Volume Diagram
Figure 2 presents an Otto Cycle temperature vs entropy diagram.
Figure 2 - Otto Cycle
Temperature vs Entropy Diagram
Figure 3 presents the Otto Cycle efficiency as a function of the
compression ratio. It should be noted that the inlet conditions are standard ambient conditions: temperature of 298 [K] and absolute pressure of 1 [atm].
Figure 3 - Otto Cycle Efficiency
Figure 4 presents the Otto Cycle power output as a function of
combustion temperature and compression ratio. It
should be noted that the
number of
revolutions is 60 [1/s]
for a given geometry of a four
cylinder and
four stroke
Otto
engine.
Figure 4 - Otto Cycle Power Output
One can notice that the Otto Cycle efficiency increases with an increase in
the compression ratio. One can notice that the Otto
Cycle power output
increases with an
increase in the
combustion temperature and that the Otto
Cycle
power
output is greater
for higher compression ratio values.
Assumptions
Working fluid is air. There is no friction. Compression and
expansion are isentropic -- there is no entropy change.
During heat
addition and heat
rejection, the air
temperature does change. Ideal
gas
state equation is
valid -- pv =
RT. Air behaves as a perfect gas --
specific heat
has a constant value.
Governing Equations
T2/T1 =
(V1/V2)(k-1) V1/V2 =
(T2/T1)1/(k-1) V4/V3 =
(T3/T4)1/(k-1)
k = cp/cv w = qh - ql
w = cv(T3 - T2) - cv(T4 - T1)
W = (cv(T3 - T2) - cv(T4 - T1))m
efficiency = 1 - 1/compression ratio(k-1)
compression ratio = V1/V2
Input Data
T1 =
298 [K]
p1 =
1 [atm]
T3 = 1,200, 1,500 and 1,800 [K] cp =
1.004 [kJ/kg*K] Results
Otto Cycle Efficiency vs Compression
Ratio
Compression Otto Cycle 2.5 30.69 5 47.47 7.5 55.33 10 60.19 12.5 63.59 Otto Cycle Power Output
Power Output Combustion Temperature Compression Ratio 1,200 1,500 1,800 5 167 246 326 10 151 252 352
Figures
Conclusions
The Otto Cycle efficiency increases with an increase in the compression
ratio. Also, the Otto Cycle power output increases
with an increase in the
combustion
temperature and the
Otto Cycle power output is greater
for higher
compression ratio
values.
References
JANAF Thermochemical Data - Tables, 1970
This section provides a Diesel Cycle analysis when the working fluid is air.
Analysis
In the presented Diesel Cycle analysis, only air is
considered as the working fluid behaving as a perfect gas -- specific
heat has a constant value. Ideal gas state equation is valid -- pv =
RT.
Air
enters a cylinder at point 1 when compression starts and it ends at
point 2.
Isentropic compression
is considered with no entropy
change. Heat
addition
starts at point 2 and it ends at
point 3.
At a constant pressure, combustion takes place (fuel is added to the cylinder and the air temperature raises) and/or heat gets added to air. Expansion
starts at point 3 and it ends at
point 4.
Isentropic expansion
is considered with no entropy
change. Air heat
rejection starts at point 4
and it ends
at point 1. At a constant
volume, air gets cooled and the working fluid
temperature
goes down. It
should be mentioned that air at point 1 enters
the
compression process again and the cycle is repeated. Figure 1 presents a Diesel Cycle pressure vs volume diagram.
Figure 1 - Diesel
Cycle Pressure vs Volume Diagram
Figure 2 presents a Diesel Cycle temperature vs entropy diagram.
Figure 2 - Diesel Cycle
Temperature vs Entropy Diagram
Figure 3 presents the Diesel Cycle efficiency as a function of the
compression ratio and cut off ratio values. It should be noted that the inlet conditions are standard ambient conditions: temperature of 298 [K] and absolute pressure of 1 [atm].
Figure 3 - Diesel Cycle Efficiency
Figure 4 presents the Diesel Cycle efficiency as a function of the
compression ratio and combustion temperature values. Figure 4 - Diesel Cycle Efficiency
Figure 5 presents the Diesel Cycle cut off ratio as a function of the
combustion temperature and compression ratio values. Figure 5 -
Diesel Cycle Cut Off Ratio
Figure 6 presents the Diesel Cycle power output as a function of the
combustion temperature and compression ratio values. It
should be noted that
the number of
revolutions is 60 [1/s]
for a given geometry of a four
cylinder
and four stroke
Diesel
engine.
One can notice that the Diesel Cycle efficiency increases with an increase in
the compression ratio and a decrease in the cut off ratio values. One can notice that the
Diesel Cycle power output
increases
with an increase in
the compression ratio and combustion temperature and that the
Diesel Cycle
power
output is greater
for lower cut off ratio values for given combustion temperature values.
Assumptions
Working fluid is air. There is no friction. Compression and
expansion are isentropic -- there is no entropy change.
During heat
addition and heat
rejection, the air
temperature does change. Ideal
gas
state equation is
valid -- pv =
RT. Air behaves as a perfect gas --
specific heat
has a constant value.
Governing Equations
T2/T1 =
(V1/V2)(k-1) V1/V2 =
(T2/T1)1/(k-1) V4/V3 =
(T3/T4)1/(k-1) k = cp/cv w = qh - ql
w = cp(T3 - T2) - cv(T4 - T1)
W = (cp(T3 - T2) - cv(T4 - T1))m
efficiency = 1 - (cut off ratiok - 1)/(k(compression
ratio(k-1))(cut off ratio - 1))
compression ratio = V1/V2
cut off ratio = V3/V2
Input Data
T1 =
298 [K] p1 =
1 [atm] T3 = 1,500, 1,800 and 2,100
[K]
compression ratio = 7.5, 10,
12.5, 15 and 17.5 [/] cp =
1.004 [kJ/kg*K]
k = cp/cv - for air k = 1.4 [/] Results
Diesel Cycle Efficiency
Diesel Cycle Compression Ratio Cut Off Ratio 7.5 10 12.5 15 17.5 3 41.69 48.03 52.46 55.81 58.45 4 36.57 43.46 48.29 51.93 54.80 Diesel Cycle Efficiency
Diesel Cycle Combustion Temperature Compression 1,500 1,800 2,100 10 53.39 51.12 49.20 15 61.94 60.12 58.49 Diesel Cycle Cut Off Ratio
Cut Off Ratio Combustion Temperature Compression 1,500 1,800 2,100 10 2.00 2.40 2.80 15 1.70 2.05 2.39 Diesel Cycle Power Output
Power Output Combustion Temperature Compression 1,500 1,800 2,100 10 311 417 514 15 297 433 557 Conclusions The Diesel Cycle efficiency increases with an increase in the compression
ratio and a decrease in the cut off ratio values. Also, the Diesel Cycle power output increases
with an increase in
the compression
ratio and combustion temperature and the Diesel
Cycle power output is greater for lower
cut off ratio
values for given combustion temperatrure values. References JANAF Thermochemical Data - Tables, 1970
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Power Cycle
Analysis
