# Sheet 8:Chapter C It is less than the thermal efficiency of a Carnot cycle.

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1 Thermo 1 (MEP 261) Thermodynamics An Engineering Approach Yunus A. Cengel & Michael A. Boles 7 th Edition, McGraw-Hill Companies, ISBN , 2008 Sheet 8:Chapter 9 9 2C How does the thermal efficiency of an ideal cycle, in general, compare to that of a Carnot cycle operating between the same temperature limits? 9-2C It is less than the thermal efficiency of a Carnot cycle. Solution 9 3C What does the area enclosed by the cycle represent on a P-v diagram? How about on a T-s diagram? 9-3C It represents the net work on both diagrams. Solution 9 4C What is the difference between air-standard assumptions and the cold-air-standard assumptions? Solution 9-4C The cold air standard assumptions involve the additional assumption that air can be treated as an ideal gas with constant specific heats at room temperature 9 5C How are the combustion and exhaust processes modeled under the air-standard assumptions? 9-5C Under the air standard assumptions, the combustion process is modeled as a heat addition process, and the exhaust process as a heat rejection process. Solution

2 9 6C What are the air-standard assumptions? 9-6C The air standard assumptions are: (1) the working fluid is air which behaves as an ideal gas, (2) all the processes are internally reversible, (3) the combustion process is replaced by the heat addition process, and (4) the exhaust process is replaced by the heat rejection process which returns the working fluid to its original state. Solution 9 7C What is the difference between the clearance volume and the displacement volume of reciprocating engines? Solution 9-7C The clearance volume is the minimum volume formed in the cylinder whereas the displacement volume is the volume displaced by the piston as the piston moves between the top dead center and the bottom dead center. 9 8C Define the compression ratio for reciprocating engines. 9-8C It is the ratio of the maximum to minimum volumes in the cylinder. Solution 9 9C How is the mean effective pressure for reciprocating engines defined? 9-9C The MEP is the fictitious pressure which, if acted on the piston during the entire power stroke, would produce the same amount of net work as that produced during the actual cycle. Solution 9 12C What is the difference between spark-ignition and compressionignition engines? 9-12C Solution The SI and CI engines differ from each other in the way combustion is initiated; by a spark in SI engines, and by compressing the air above the self-ignition temperature of the fuel in CI engines. 9 13C Define the following terms related to reciprocating engines: stroke, bore, top dead center, and clearance volume. Solution 9-13C Stroke is the distance between the TDC and the BDC, bore is the diameter of the cylinder, TDC is the position of the piston when it forms the smallest volume in the cylinder, and clearance volume is the minimum volume formed in the cylinder.

3 9 14 An air-standard cycle with variable specific heats is executed in a closed system and is composed of the following four processes: 1-2 Isentropic compression from 100 kpa and 27 C to 800 kpa 2-3 v = constant heat addition to 1800 K 3-4 Isentropic expansion to 100 kpa 4-1 P = constant heat rejection to initial state (a) Show the cycle on P-v and T-s diagrams. (b) Calculate the net work output per unit mass. (c) Determine the thermal efficiency The four processes of an air-standard cycle are described. The cycle is to be shown on P-v and T-s diagrams, and the net work output and the thermal efficiency are to be determined. Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with variable specific heats. Properties The properties of air are given in Table A-17. Analysis (b) The properties of air at various states are Solution

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5 9-16 The four processes of an air-standard cycle are described. The cycle is to be shown on P-v and T-s diagrams, and the maximum temperature in the cycle and the thermal efficiency are to be determined. Assumptions 1 The air-standard assumptions are applicable. 2 Kinetic and potential energy changes are negligible. 3 Air is an ideal gas with constant specific heats. Properties The properties of air at room temperature are c p = kj/kg.k, c v = kj/kg K, and k = 1.4 (Table A-2). Analysis (b) From the ideal gas isentropic relations and energy balance, Solution

6 Discussion The assumption of constant specific heats at room temperature is not realistic in this case as the temperature changes involved are too large.

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10 9 24 Repeat Problem 9 23 using helium as the working fluid.

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12 9 26C What four processes make up the ideal Otto cycle? 9-24C SolutionThe four processes that make up the Otto cycle are (1) isentropic compression, (2) v = constant heat addition, (3) isentropic expansion, and (4) v = constant heat rejection.

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16 9 37 The compression ratio of an air-standard Otto cycle is 9.5. Prior to the isentropic compression process, the air is at 100 kpa, 35 C, and 600 cm 3. The temperature at the end of the isentropic expansion process is 800 K. Using specific heat values at room temperature, determine (a) the highest temperature and pressure in the cycle; (b) the amount of heat transferred in, in kj; (c) the thermal efficiency; and (d) the mean effective pressure. Answers: (a) 1969 K, 6072 kpa, (b) 0.59 kj, (c) 59.4 percent, (d) 652 kpa Solution

17 9 38 Repeat Problem 9 37, but replace the isentropic expansion process by a polytropic expansion process with the polytropic exponent n = 1.35.

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19 9 42C How does a diesel engine differ from a gasoline engine? Solution A diesel engine differs from the gasoline engine in the way combustion is initiated. In diesel engines combustion is initiated by compressing the air above the self-ignition temperature of the fuel whereas it is initiated by a spark plug in a gasoline engine. 9 43C How does the ideal Diesel cycle differ from the ideal Otto cycle? Solution The Diesel cycle differs from the Otto cycle in the heat addition process only; it takes place at constant volume in the Otto cycle, but at constant pressure in the Diesel cycle An air-standard Diesel cycle has a compression ratio of 16 and a cutoff ratio of 2. At the beginning of the compression process, air is at 95 kpa and 27 C. Accounting for the variation of specific heats with temperature, determine (a) the temperature after the heat-addition process, (b) the thermal efficiency, and (c) the mean effective pressure. Answers: (a) K, (b) 56.3 percent, (c) kpa

20 9 48 Repeat Problem 9 47 using constant specific heats at room temperature.

21 9 51 An ideal diesel engine has a compression ratio of 20 and uses air as the working fluid. The state of air at the beginning of the compression process is 95 kpa and 20 C. If the maximum temperature in the cycle is not to exceed 2200 K, determine (a) the thermal efficiency and (b) the mean effective pressure. Assume constant specific heats for air at room temperature. Answers: (a) 63.5 percent, (b) 933 kpa

22 9 52 Repeat Problem 9 51, but replace the isentropic expansion process by polytropic expansion process with the polytropic exponent n = 1.35.

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24 9 56 The compression ratio of an ideal dual cycle is 14. Air is at 100 kpa and 300 K at the beginning of the compression process and at 2200 K at the end of the heat-addition process. Heat transfer to air takes place partly at constant volume and partly at constant pressure, and it amounts to kj/kg. Assuming variable specific heats for air, determine (a) the fraction of heat transferred at constant volume and (b) the thermal efficiency of the cycle. Solution

25 9 58 Repeat Problem 9 56 using constant specific heats at room temperature. Is the constant specific heat assumption reasonable in this case? Solution 9 60C Consider the ideal Otto, Stirling, and Carnot cycles operating between the same temperature limits. How would you compare the thermal efficiencies of these three cycles? 9-58C Solution The efficiencies of the Carnot and the Stirling cycles would be the same, the efficiency of the Otto cycle would be less. 9 61C Consider the ideal Diesel, Ericsson, and Carnot cycles operating between the same temperature limits. How would you compare the thermal efficiencies of these three cycles? 9-59C Solution The efficiencies of the Carnot and the Ericsson cycles would be the same, the efficiency of the Diesel cycle would be less Consider an ideal Ericsson cycle with air as the working fluid executed in a steady-flow system. Air is at 27 C and 120 kpa at the beginning of the isothermal compression process, during which 150 kj/kg of heat is rejected. Heat transfer to air occurs at 1200 K. Determine (a) the maximum pressure in the cycle, (b) the net work output per unit mass of air, and (c) the thermal efficiency of the cycle. Answers:

26 (a) 685 kpa, (b) 450 kj/kg, (c) 75 percent Solution 9 66 An ideal Stirling engine using helium as the working fluid operates between temperature limits of 300 and 2000 K and pressure limits of 150 kpa and 3 MPa. Assuming the mass of the helium used in the cycle is 0.12 kg, determine (a) the thermal efficiency of the cycle, (b) the amount of heat transfer in the regenerator, and (c) the work output per cycle.

27 9 73 A simple Brayton cycle using air as the working fluid has a pressure ratio of 8. The minimum and maximum temperatures in the cycle are 310 and 1160 K. Assuming an isentropic efficiency of 75 percent for the compressor and 82 percent for the turbine, determine (a) the air temperature at the turbine exit, (b) the net work output, and (c) the thermal efficiency

28 9 75 Repeat Problem 9 73 using constant specific heats at room temperature.

29 9 76 Air is used as the working fluid in a simple ideal Brayton cycle that has a pressure ratio of 12, a compressor inlet temperature of 300 K, and a turbine inlet temperature of 1000 K. Determine the required mass flow rate of air for a net power output of 70 MW, assuming both the compressor and the turbine have an isentropic efficiency of (a) 100 percent and (b) 85 percent. Assume constant specific heats at room temperature. Answers: (a) 352 kg/s, (b) 1037 kg/s

30 9 77 A stationary gas-turbine power plant operates on a simple ideal Brayton cycle with air as the working fluid. The air enters the compressor at 95 kpa and 290 K and the turbine at 760 kpa and 1100 K. Heat is transferred to air at a rate of 35,000 kj/s. Determine the power delivered by this plant (a) assuming constant specific heats at room temperature and (b) accounting for the variation of specific heats with temperature.

31 9 78 Air enters the compressor of a gas-turbine engine at 300 K and 100 kpa, where it is compressed to 700 kpa and 580 K. Heat is transferred to air in the amount of 950 kj/kg before it enters the turbine. For a turbine efficiency of 86 percent, determine (a) the fraction of the turbine work output used to drive the compressor and (b) the thermal efficiency. Assume variable specific heats for air.

32 9 79 Repeat Problem 9 78 using constant specific heats at room temperature.

33 9 82 A gas-turbine power plant operates on the simple Brayton cycle with air as the working fluid and delivers 32 MW of power. The minimum and maximum temperatures in the cycle are 310 and 900 K, and the pressure of air at the compressor exit is 8 times the value at the compressor inlet. Assuming an isentropic efficiency of 80 percent for the compressor and 86 percent for the turbine, determine the mass flow rate of air through the cycle. Account for the variation of specific heats with temperature.

34 9 83 Repeat Problem 9 82 using constant specific heats at room temperature. Solution

35 9 85C How does regeneration affect the efficiency of a Brayton cycle, and how does it accomplish it? Solution Regeneration increases the thermal efficiency of a Brayton cycle by capturing some of the waste heat from the exhaust gases and preheating the air before it enters the combustion chamber. 9 87C Define the effectiveness of a regenerator used in gas-turbine cycles. Solution 9 93 An ideal Brayton cycle with regeneration has a pressure ratio of 10. Air enters the compressor at 300 K and the turbine at 1200 K. If the effectiveness of the regenerator is 100 percent, determine the net work output and the thermal efficiency of the cycle. Account for the variation of specific heats with temperature.

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