TUTORIAL 6 (15/03/2016) Thermodynamics for Aerospace Engineers (AS1300) Second Law of Thermodynamics (Carnot engine) and Entropy

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1 TUTORIAL 6 (15/03/2016) Thermodynamics for Aerospace Engineers (AS1300) Second Law of Thermodynamics (Carnot engine) and Entropy 1. A Carnot engine operates between two reservoirs at temperatures T H K and T C K. The work output of the engine is 0.6 times the heat rejected. Given the difference in temperature between the source and sink is 200 C, calculate a) The source temperature b) The sink temperature c) Thermal efficiency 2. A quantity of air undergoes a thermodynamic cycle consisting of three processes. Process 1-2: constant volume heating from P 1 = 0.1 MPa, T 1 = 15 C, V 1 = 0.02 m 3 to P 2 = 0.42 MPa. Process 2-3: Constant pressure cooling. Process 3-1: Isothermal heating to initial state. Employing the ideal gas model with C p = 1kJ/kgK, evaluate the change of entropy for each process. Sketch the cycle on P-v and T-s coordinates. 3. Air initially occupying 1 m 3 at 1.5 bar, 20 C undergoes an internally reversible compression for which PV n = constant to a final state where the pressure is 6 bar and the temperature is 120 C. Determine a) The value of n b) The work and the heat transfer and c) The change in entropy, Take c v = kj/kgk 4. 1 kg of air at a pressure of 10 bar and 100 C undergoes a reversible polytropic process which may be represented as pv 1.1 = constant, p is in Pa and v is in m 3 /kg. The final pressure is 2 bar. a) Evaluate the final specific volume, the final temperature and the change in entropy b) Evaluate the work done and the hat transfer c) Repeat a) and b) assuming the process is adiabatic reversible between the same states. 5. A certain gas has c v = 1.2 kj/kgk. When it is expanded isentropically from a specific volume of m 3 /kg and a temperature of 540 K to a specific volume of m 3 /kg, its temperature falls by 170 K. When it is expanded in an adiabatic process with friction, from the same initial state to the same final specific volume, its temperature falls only by 30 K. Find the change in entropy of 1 kg of gas in the adiabatic process. 6. Air enters an insulated compressor at 1.05 bars, 23 C with a mass flowrate of 1.8 kg/s and exits at 2.9 bars. KE and PE changes are negligible. Determine the minimum power input required and the corresponding exit temperature. If the isentropic efficiency is 80%, determine the power input and the exit temperature. 7. Two kg of water at 80 C is mixed adiabatically with 3 kg of water at 30 C in a constant pressure process of 1 atmosphere. Find the increase in entropy of the total mass of water due to the mixing process (take c p of water = kj/kgk) 8. Calculate the entropy change of the universe as a result of the following processes: a) A coper block of 600 g mass & with c p 250 J/kgK at 100 C is placed in a lake at 8 C b) The same block, at 8 C, is dropped from a height of 100 m into the lake. c) Two such blocks at 100 C and 0 C are joined together. 9. Following the case studies in the class obtain an expression for maximum work obtainable from a reversible heat engine operating between a finite source at initial temperature T i and a sink which is a constant temperature thermal reservoir at T 0

2 TUTORIAL 6, SOLUTIONS 1. A Carnot engine operates between two reservoirs at temperatures T H K and T C K. The work output of the engine is 0.6 times the heat rejected. Given the difference in temperature between the source and sink is 200 C, calculate a) The source temperature b) The sink temperature c) Thermal efficiency 2. A quantity of air undergoes a thermodynamic cycle consisting of three processes. Process 1-2: constant volume heating from P 1 = 0.1 MPa, T 1 = 15 C, V 1 = 0.02 m 3 to P 2 = 0.42 MPa. Process 2-3: Constant pressure cooling. Process 3-1: Isothermal heating to initial state. Employing the ideal gas model with C p = 1kJ/kgK, evaluate the change of entropy for each process. Sketch the cycle on P-v and T-s coordinates.

3 3. Air initially occupying 1 m 3 at 1.5 bar, 20 C undergoes an internally reversible compression for which PV n = constant to a final state where the pressure is 6 bar and the temperature is 120 C. Determine a) The value of n b) The work and the heat transfer and c) The change in entropy, Take c v = kj/kgk 4. 1 kg of air at a pressure of 10 bar and 100 C undergoes a reversible polytropic process which may be represented as pv 1.1 = constant, p is in Pa and v is in m 3 /kg. The final pressure is 2 bar. d) Evaluate the final specific volume, the final temperature and the change in entropy e) Evaluate the work done and the hat transfer f) Repeat a) and b) assuming the process is adiabatic reversible between the same states. 5. A certain gas has c v = 1.2 kj/kgk. When it is expanded isentropically from a specific volume of m 3 /kg and a temperature of 540 K to a specific volume of m 3 /kg, its temperature falls by 170 K. When it is expanded in an adiabatic process with friction, from the same initial state to the same final specific volume, its temperature falls only by 30 K. Find the change in entropy of 1 kg of gas in the adiabatic process.

4 6. Air enters an insulated compressor at 1.05 bars, 23 C with a mass flowrate of 1.8 kg/s and exits at 2.9 bars. KE and PE changes are negligible. Determine the minimum power input required and the corresponding exit temperature. If the isentropic efficiency is 80%, determine the power input and the exit temperature. 7. Two kg of water at 80 C is mixed adiabatically with 3 kg of water at 30 C in a constant pressure process of 1 atmosphere. Find the increase in entropy of the total mass of water due to the mixing process (take c p of water = kj/kgk) 8. Calculate the entropy change of the universe as a result of the following processes: a) A coper block of 600 g mass & with c p 250 J/kgK at 100 C is placed in a lake at 8 C b) The same block, at 8 C, is dropped from a height of 100 m into the lake. c) Two such blocks at 100 C and 0 C are joined together.

5 9. Following the case studies in the class obtain an expression for maximum work obtainable from a reversible heat engine operating between a finite source at initial temperature T i and a sink which is a constant temperature thermal reservoir at T 0