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1 Engineering Software P.O. Box 2134, Kensington, MD Phone: (301) Web Site: Carnot Cycle Analysis Carnot Cycle Analysis by Engineering Software Course Category: Engineers Course Level: Intermediate Credit: 1 Hour -- i --

2 Course Description The Carnot Cycle is an ideal simple cycle consisting of isentropic compression and expansion and isothermal heat addition and heat rejection. The Carnot Cycle thermal efficiency determines the highest thermal efficiency that a heat engine can achieve. In this one hour course, the closed, simple Carnot Cycle used for stationary power generation is considered. The Carnot Cycle thermal efficiency is presented only for the air as the working fluid. The thermal efficiency derivation is presented with a simple mathematical approach. The Carnot Cycle is presented in a T - s diagram and its major performance trends are plotted in a few figures as a function of heat addition and heat rejection temperature values. In this course, the student gets familiar with the Carnot Cycle, its components, T - s diagram, operation and major performance trends. This course includes a multiple choice quiz at the end. -- ii --

3 Performance Objectives At the conclusion of this course, the student will: Understand basic energy conversion engineering assumptions and equations Know basic components of the Carnot Cycle and its T - s diagram Be familiar with the Carnot Cycle operation Understand general Carnot Cycle performance trends -- iii --

4 Introduction The Carnot Cycle is an ideal simple cycle consisting of isentropic compression and expansion and isothermal heat addition and heat rejection. The Carnot Cycle thermal efficiency determines the highest thermal efficiency that a heat engine can achieve. Therefore, for a heat engine, the maximum thermal efficiency is determined by the Carnot Cycle efficiency and it is not dependent on the physical properties of the working fluid. The Carnot Cycle efficiency can be very helpful in when studying the impact of ambient temperature, on the installed power generation capacity and available electric power generation -- what happens with the amount of available electric power generation capacity when the ambient temperature changes one way or the other. -- iv --

5 Table of Contents Carnot Cycle... 2 Analysis... 2 Assumptions... 7 Governing Equations... 8 Input Data... 8 Results... 9 Conclusions

6 Carnot Cycle 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 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

7 Figure 1 - Carnot Cycle Schematic Layout Figure 2 presents a Carnot Cycle temperature vs entropy diagram

8 Figure 2 - Carnot Cycle Temperature vs Entropy Diagram The thermal cycle efficiency can be given as a function of specific external work (specific net power output) and heat added to the working fluid as follows: η = w/qh = (wt - wc)/qh = (qh - ql)/qh = 1 - ql/qh = 1 - T1Δs/T2Δs = 1 - T1/T2 = 1 - TR/TA where η - thermal efficiency [/] w - specific external work (specific net power output) [kj/kg] wt - expansion specific power output [kj/kg] wc - compression specific power input [kj/kg] qh - heat added to the working fluid [kj/kg]

9 ql - heat rejected from the working fluid [kj/kg] Carnot Cycle Analysis Δs - entropy change during heat addition and heat rejection [kj/kg*k] TA - temperature during heat addition [K] TR - temperature during heat rejection [K] For isentropic compression and expansion: T2/T1 = (p2/p1) (ϰ-1)/ϰ = T3/T4 = (p3/p4) (ϰ-1)/ϰ and ϰ = cp/cv - for air ϰ = 1.4 [/] p1, p2, p3, p4 - pressure values at points 1, 2, 3 and 4 [atm] T1, T2, T3, T4 - temperature values at points 1, 2, 3 and 4 [K] Again, it follows that η = 1 - T1/T2 = 1 - TR/TA The Carnot Cycle efficiency is not dependent on the working fluid properties. 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]

10 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]

11 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

12 Governing Equations T2/T1 = (p2/p1) (ϰ-1)/ϰ T3/T4 = (p3/p4) (ϰ-1)/ϰ ϰ = cp/cv cp - cv = R pv = RT qh = T2Δs = TAΔs ql = T1Δs = TRΔs η = 1 - T1/T2 = 1 - TR/TA Input Data T1 = 298 [K] p1 = 1 [atm] R = [kj/kg*k] cp = [kj/kg*k] ϰ = 1.4 [/]

13 Results Carnot Cycle Efficiency vs Heat Addition Temperature Heat Rejection Temperature = 298 [K] Heat Addition Temperature [K] Carnot Cycle Efficiency [%] , Carnot Cycle Efficiency vs Heat Rejection Temperature Heat Addition Temperature = 800 [K] Heat Rejection Temperature [K] Carnot Cycle Efficiency [%]

14 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. Please use the material you just read to answer the quiz questions at the end of this course. When you get a chance, please visit the following URL: The above URL provides lots of free online and downloadable e-material and e-solutions on energy conversion

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