Problems
Stirling Cycle
A Stirling cycle with 5 g of air as working fluid is heated by solar radiation and rejects heat to the environment. The highest and lowest temperatures reached in the cycle are 1000 K and 300 K, respectively, and the maximum pressure ratio is 10. Draw the process in a T-s-diagram and in a p-v-diagram, then determine the thermal efficiency and the power produced when the engine runs at 400 rpm. Determine the entropy changes, heat and work per unit mass for all four processes.
Stirling Cycle for Refrigeration
A Stirling engine with helium as working gas is considered for refrigeration purposes. The goal is to withdraw heat at a temperature of TL = 150 K and reject it to the environment at TH = 300 K. Helium is a monatomic gas, which is well described as an ideal gas with constant specific heats.
1. Draw T-s-diagram and p-v-diagram for the cycle.
2. For a volume ratio between largest and smallest volume of 3, compute heat and work per unit mass for all four processes, and the coefficient of performance.
3. The computation of specific heat and work is independent of the pressure.
Discuss the role of the pressure in the performance of the engine, why is a high pressure desirable?
4. Determine the pressure at all corner points when the highest pressure in the engine is 10 bar.
Stirling Cycle for Refrigeration
A Stirling engine with argon as working fluid is used for refrigeration purposes. Heat exchange with the cold and warm surroundings takes place at TH = 27 ◦C and TL = −73 ◦C, respectively. The highest pressure in the cycle is pH = 12 bar and the smallest volume is one third of the largest volume.
1. Plot the Stirling cycle in a T-s-diagram, and in a p-v-diagram, number the corner points.
2. Compute the pressures and specific volumes on all corner points
3. Discuss the regenerator: show that the amounts of heat rejection and heat supply in the two isochoric processes have the same absolute value, but different signs.
4. Compute the coefficient of performance of the cycle.
5. Assume the cylinder of the engine contains an air mass of 40 g, and the engine runs at 300 rpm – what is its refrigeration capacity?
Compression Modes
Air (ideal gas with variable specific heats) is compressed in a compressor from 
Determine the power consumption for the following cases:
1. Isothermal reversible compression.
2. Isentropic compression.
3. Polytropic reversible compression with n = 1.2.
4. Compression in two isentropic stages with intercooling to T1 at pm = √p1p2.
Draw a p-v- and a T-s-diagram which shows the four process curves. Hint:
For computation of isothermal and polytropic case use that w12
Two Stage Compressor with Irreversibilities
A two stage compression system with intercooling is used to increase the pressure of an ideal gas. Specifically, the gas enters the system at p1, T1, and leaves the first compressor (isentropic efficiency ηC1) at pressure p2. It is then isobarically intercooled to T1, and compressed to p4 in the second compressor (isentropic efficiency ηC2). Assume constant specific heats, and determine the pressure p2 that should be chosen to minimize the work requirement of the system.
Gas Turbine Cycle with Regeneration
1. Draw a schematic for a gas turbine system for electricity generation with irreversible single stage compression, two stages of irreversible expansion with reheat, and a regenerator. Enumerate the relevant corner points of the process.
2. Draw the corresponding T-s-diagram.
3. Express the thermal efficiency in terms of enthalpies.
Brayton Cycle with Regeneration
A gas turbine running on the Brayton cycle has an efficiency of 35.9%, at pressure ratio 14.7. The turbine inlet temperature is 1288 ◦C, and the air entering the engine is at 1 bar, 20 ◦C. The engine produces a net power of 174.9 MW and the mass throughput is 1690 t .
1. Determine the isentropic efficiencies of turbine and compressor.
2. Determine the thermal efficiency for this gas turbine for the case that a regenerator with 80% effectiveness is added to the cycle.
Optimal Reheat Pressure
Prove the following statement from the text for a reheat turbine with n-stages:
If the turbines have the same inlet temperature, and the same isentropic efficiency, the maximum work is obtained when they have the same pressure ratio.
Brayton Cycle with Intercooling, Reheat and Regeneration A regenerative gas turbine cycle uses two stage of compression with intercooling, and two stages of expansion with reheating. The pressure ratio for each stage is 3.5, the turbine inlet temperature is 1400 K for both turbines, and between the compressors the air is cooled back to the environmental temperature of 290 K. The isentropic efficiencies of the compressors and turbines are 80% and 85%, respectively, and the regenerator effectiveness is 80%.
Determine:
1. The enthalpies at all principal states.
2. The net work and the back work ratio.
3. The thermal efficiency for the system as described, and for the case that no regenerator is present.
4. The work potential of the turbine exhaust, and of the final exhaust.
As usual: draw schematic and diagrams.
Gas Turbine with Regenerator
A gas turbine with air (non-constant specific heats) as working fluid operates according to the following cycle:
1-2: Adiabatic compression of air at T1 = 300 K, p1 = 1 bar to T2 = 620 K, p2 = 9.74 bar.
2-3: Isobaric heating of the working fluid in the regenerator, the temperature T3 is 40 K below the temperature of the turbine exhaust, T5.
3-4: Further isobaric heating in the combustion chamber to T4 = 1300 K.
4-5: Adiabatic expansion in turbine to pressure p5 = p1, with isentropic efficiency of 92%.
5-6: Isobaric cooling in the regenerator.
1. Draw a schematic, and a T-s-diagram.
2. Make a table with pressures, temperatures and enthalpies at the points 1 to 6.
3. Determine the thermal efficiency and the back-work-ratio of the cycle:
a) when it operates with regenerator
b) when it operates without regenerator
4. Compute the isentropic efficiency of the compressor.
Combined Cycle: Gas Turbine and Steam Power Plant
A combined cycle power plant consists of a gas turbine cycle (thermal efficiency 28%), and a steam power plant (thermal efficiency 46%). The exhaust of the gas turbine is used to provide the heat for generating steam in a heat recovery steam generator. Assume that the HRSG has an efficiency of 92%, and compute the overall efficiency of the system.
Turbojet Engine
A turbojet engine drives an airplane traveling with velocity 290 m at a height where the pressure is 28 kPa, and the temperature is −40 ◦C. The compressor pressure ratio is 11, and the turbine inlet temperature is 1300 K. The mass flow through the engine is 60 kg .
Assume isentropic efficiencies of 82% for compressor and turbine, 95% for the nozzle, and 100% for the diffuser. Determine the velocity of the exhaust gas, the propulsive power, the rate of fuel consumption when the heating value of the fuel is 42000 kJ , the thermal efficiency, and the Froude propulsive efficiency.
Air Engine
An airplane propelled by a standard turbo-jet engine flies at Mach number M = 0.9 in an environment where the pressure is 40 kPa and the temperature is 240 K. The heat added to the air flowing through the engine is q = 550 kJ and the hot air leaves the engine at 650 K. The engine inlet has a diameter of 1 m. Assume that the working fluid is air as ideal gas.
Determine outflow velocity, thrust, propulsive power, thermal efficiency, and Froude efficiency of the engine.
Air Engine
Air at 25 kPa, 225 K enters a turbojet engine in flight at an altitude of 10 000 m, the flight velocity is 290 m . The pressure ratio across the com- pressor is 10. The turbine inlet temperature is 1300 K, and the pressure at the nozzle exit is 25 kPa again. The diffuser and nozzle processes are isentropic, compressor and turbine have isentropic efficiencies of 90% and 95%, respectively, and there is no pressure drop for flow through the combustor.
Consider air as ideal gas with constant specific heats, R = 0.287 kJ , cp = 1.004 kJ .
Neglect kinetic energy except at the diffuser inlet and the nozzle exit.
1. Draw a schematic of the engine, and the corresponding T-s-diagram.
2. Make a table with the values of pressure and temperature at each principal state.
3. Compute the velocity at the nozzle exit.
4. Compute the thrust of the engine and the propulsive power for a mass flow rate of 80 kg .
5. Determine thermal efficiency and Froude efficiency.
Bypass Turbofan Engine
A bypass turbo fan engine has a bypass ratio of 5.5 (the mass flow through the bypass is 5.5 times the mass flow through the gas turbine), and propels an aircraft cruising at 250 m in high altitude where the pressure is 30 kPa and the temperature is 230 K. The mass flow through the gas turbine core is s . Assume variable specific heats.
The flow through the bypass consists of isentropic diffuser, fan, nozzle.
The gas turbine process is as follows:
1-2: Compression in isentropic diffuser.
2-3: Isentropic compressor, pressure ratio p3/p2 = 10.
3-4: Isobaric heating in combustion chamber to 1300 K. 4-5: Turbine TC to drive the compressor.
5-6: Turbine TF to drive the fan.
6-7: Isentropic expansion in nozzle.
1. Make a sketch of the engine, and draw the corresponding T-s-diagram.
2. Determine the power required to drive the fan, when the bypass outflow velocity is 420 m .
Hint: Balance the complete bypass. Pressures at inlet and outlet are equal to environmental pressure. Then, for isentropic operation, the outlet temperature is equal to the inlet temperature (show that!)
3. Determine temperature, pressure, relative pressure, enthalpy, and outflow velocity at all 7 points. Provide a table with the values.
4. Compute the propulsive power of the engine, its thermal efficiency, its Froude efficiency, and the power that could be generated from the exhaust by equilibrating it to the environment.