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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 Table 1 - Carnot Cycle Efficiency vs Heat
Addition Temperature
Table 2 - 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 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 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 Table 1 - Brayton Cycle Efficiency vs Compression Ratio
Table 2 - Specific Power Output vs Compression
Ratio
Table 3 - Power Output vs Compression Ratio
Figures
Conclusions The Brayton Cycle efficiency increases with an increase in the compression ratio. 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. 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. References JANAF Thermochemical Data - Tables, 1970 Otto Cycle Introduction 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 Table 1 - Otto Cycle Efficiency vs Compression Ratio
Table 2 - Otto Cycle Power Output
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 Diesel Cycle Introduction 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 Table 1 - Diesel Cycle Efficiency
Table 2 - Diesel Cycle Efficiency
Table 3 - Diesel Cycle Cut Off Ratio
Table 4 - Diesel Cycle Power Output
Figures
   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 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Power Cycle
Analysis
