PROBLEMS ON JET PROPULSION •

Unless otherwise stated, the following data may be used: Cp for air = 1.005 kJ/kg K, R = 0.287 kJ/kg K, γ = 1.4 Cp for gas = 1.148 kJ/kg K, R = 0.287 kJ/kg K, γ = 1.33

Aircraft Performance Criteria 1.

An advanced fighter engine operating at Mach 0.8 and 10 km altitude (ambient temperature of 223.15 K) has the following performance data, and uses a fuel with calorific value 42800 kJ/kg: Thrust = 50 kN, air mass flow = 45 kg/s, and fuel mass flow = 2.65 kg/s. Find the specific thrust, TSFC, exit velocity, thermal, propulsive and overall efficiencies (assume exit pressure equal to ambient pressure). [1111 m/s, 0.053 kg/s/kN, 239.6 m/s, 1275.6 m/s, 33.04 %, 31.97 %, 10.56 %]

2. Find the propulsive efficiency for the following two engines at cruise (a) an RB 211 at 30000 ft (ambient pressure and temperature of 28.52 kPa and 226.4 K), flight Mach number 0.85, approximate jet velocity 390 m/s. [79.4 %] (b) an Olympus 593 (in Concorde) at 51000 ft (ambient pressure and temperature of 11 kPa and 216.7 K) flight Mach number 2.0, approximate jet velocity 1009 m/s. [73.8 %] 3.

If the SFC at cruise for a version of the RB 211 is about 0.60 kg/h/kg and for the Olympus 593 is about 1.19 kg/h/kg, find the overall efficiency and the thermal efficiency in each case. [35.1 % & 44.2 %; For Olympus 40.7 % & 55.1 %] Take calorific value of fuel=43 MJ/kg.

4. A turbofan engine on a test stand in the laboratory operates continuously at a thrust level of 60,000 lb with a thrust specific fuel consumption of 0.5 h-1. The fuel reservoir feeding the engine holds 1000 gallon of jet fuel (1 gallon = 6.7 lb). If the reservoir is full at the beginning of the test, how long can the engine run before the fuel reservoir is empty? [0.223 h] 5. The Allison T56 turboprop engine is rated at 4910 equivalent shaft horsepower at zero velocity at sea level. Consider an airplane with this engine flying at 500 ft/s at sea level. The jet thrust is 250 lb, and the propeller efficiency is 0.9. Calculate the equivalent shaft horsepower at this flight condition. [5163 hp]

Thermodynamic Relations 6.

An axial flow air compressor is designed to provide an overall total-to-total pressure ratio of 8:1. At inlet and outlet the stagnation temperatures are 300 K and 586.4 K respectively. Estimate the overall total-to-total efficiency and the polytropic efficiency for the compressor. Assume that γ for air is 1.4. [0.85, 0.886]

7. A compressor has an isentropic efficiency of 85% at a pressure ratio of 4.0. Calculate the corresponding polytropic efficiency, and thence plot the variation of isentropic efficiency over a range of pressure ratio from 2.0 to 10.0. [0.876; 0.863 at 2 bar and 0.828 at 10 bar] 8. A low-pressure air compressor develops a pressure of 0.147 bar. If the initial and the final states of air are p1=1.02 bar, T1=300 K, and T2 =315 K, estimate the isentropic and infinitesimal stage efficiencies. A second compressor changes the state of air from initial states of p1=1.02 bar, T1=300 K to p2=2.5 bar with an efficiency of 75 %. Find the infinitesimal efficiency of this compressor. Explain the large deviation in the efficiency of this compressor from that [78 %, 78.8 %; 78 %] of the low-pressure compressor. 9. An aircraft is flying at a cruise speed of 250 m/s at an altitude of 5000 m where the ambient pressure is 54.05 kPa and ambient temperature is 255.7 K. The ambient air is first decelerated

in a diffuser before it enters the compressor. Assuming both the diffuser and the compressor to be isentropic, find (a) total pressure at the compressor inlet and (b) the compressor work per unit mass if the total pressure ratio of the compressor is 8. [80.77 kPa, 233.9 kJ/kg]

10. Gas enters the nozzles of a turbine stage at a stagnation pressure and temperature of 4.0 bar and 1200 K and leaves with a velocity of 572 m/s and at a static pressure of 2.36 bar. Find the nozzle efficiency assuming the gas has the average properties over the [0.957] temperature range of the expansion of Cp = 1.16 kJ/kg K and γ = 1.33.

Turbojet Engines 11. Determine the specific thrust and SFC for a simple turbojet engine having the following component performance at the design point at which the cruising speed and altitude are M=0.8 and 10000 m (with ambient temperature & pressure of 223.3 K and 0.2650 bar). Compressor pressure ratio Turbine inlet (stagnation) temperature Isentropic Efficiencies Intake, ηD Compressor, ηc Turbine, ηT Propelling Nozzle, ηN Mechanical transmission efficiency, ηm Combustion efficiency, ηB Combustion pressure loss, ∆pb

8.0 1200 K 93 % 87 % 90 % 95 % 99 % 98 % 4 % Comp. Del. Pr. [589.7 Ns/kg, 0.121 kg/h-N]

12. A turbojet aircraft is flying at 800 km/h at 10 700 m where the pressure and temperature of the atmosphere are 0.24 bar and –500 C respectively. The compressor pressure ratio is 10:1 and the maximum cycle temperature is 8200 C. Assuming a convergent nozzle, find the thrust developed and the specific fuel consumption, using the following data: Isentropic Efficiencies Intake, ηD Compressor, ηc Turbine, ηT Propelling Nozzle, ηN Mechanical transmission efficiency, ηm Combustion efficiency, ηB Combustion pressure loss, ∆pb Calorific value of fuel Nozzle outlet area

90 % 90 % 92 % 92 % 98 % 98 % 0.14 bar 43 300 kJ/kg 0.08 m2 [6453 N, 0.0291 kg/KN s]

Turbojet with Afterburner 13. A turbojet aircraft is traveling at 925 km/h in atmospheric conditions of 0.45 bar and –260C. The compressor pressure ratio is 8, the air mass flow rate is 45 kg/s, and the maximum allowable cycle temperature is 8000C. The compressor, turbine and jet pipe stagnation isentropic efficiencies are 0.85, 0.89, and 0.9 respectively, the mechanical efficiency of the drive is 0.98, and the combustion efficiency is 0.99. Assuming a convergent propelling nozzle, a loss of stagnation pressure in the combustion chamber of 0.2 bar, and a fuel with calorific value of 43300 kJ/kg, calculate: (i) the required nozzle exit area, (ii) the net thrust developed, (iii) the air-fuel ratio and (iv) the specific fuel consumption. [0216 m2, 19.94 kN, 70.87, 0.0319 kg/kN s]

When an afterburner is used to obtain an increase in thrust, calculate the nozzle exit area now required to pass the same mass flow rate and the new net thrust assuming that

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stagnation temperature after the afterburner is 7000C and the pressure loss in the [0.244 m2, 22 kN] afterburner process is 0.07 bar. 14. A simple turbojet engine is operating with a compressor pressure ratio of 8.0, turbine inlet

temperature of 1200 K and a mass flow of 15 kg/s, when the aircraft is flying at 260 m/s at an altitude of 7000 m (with ambient temperature and pressure of 242.65 K and 41.06 kPa). Assuming the following component efficiencies, calculate the propelling nozzle area [0.0713 m2, 7896 N, 0.126 kg/N-h] required, the net thrust developed and SFC.

Polytropic efficiencies of compressor and turbine Isentropic efficiency of intake Isentropic efficiency of Propelling nozzle Mechanical efficiency Combustion chamber pressure loss Combustion efficiency

0.87 0.95 0.95 0.99 6 % comp.del.pr. 0.97

The gases in the jet pipe of the above engine are reheated to 2000 K, and the combustion pressure loss incurred is 3 % of the pressure at the outlet of the turbine. Find the % increase in nozzle area required if the net flow is to be unchanged, and also the % increase in net thrust. [48.3 %, 64.5 %]

Turbofan Engines 15. The following data apply to a twin-spool turbofan engine (non-mixed type) with the fan driven by the LP turbine and the compressor by the HP turbine. Separate cold and hot nozzles are used. Determine the thrust and SFC under sea-level static conditions where the [71.5 kN, 0.0403 kg/h N] ambient pressure and temperature are 1.0 bar and 288 K. Overall pressure ratio Fan pressure ratio By-pass ratio Turbine inlet temperature Fan, compressor and turbine polytropic efficiency Isentropic efficiency of each propelling nozzle Mechanical efficiency of each spool Combustion pressure loss Total air mass flow

25.0 1.65 5.0 1550 K 0.90 0.95 0.99 1.50 bar 215 kg/s

16. Under take-off conditions when the ambient pressure and temperature are 1.01 bar and 288 K, the stagnation pressure and temperature in the jet pipe of a turbojet engine are 2.4 bar and 1000 K, and the mass flow is 23 kg/s. Assuming that the expansion in the converging propelling nozzle is isentropic, calculate the exit area required and the thrust produced. For a new version of the engine, the thrust is to be increased by the addition of an aft fan, which provides a separate cold exhaust stream. The fan has a by-pass ratio of 2.0 and pressure ratio of 1.75, isentropic efficiencies of the fan and the fan-turbine sections being 0.88 and 0.90 respectively. Calculate the take-off thrust assuming that the expansion in the cold nozzle is also isentropic, and that the hot nozzle area is adjusted so that the hot mass [0.0763m2, 15.35 kN, 24.9 kN] flow remains at 23 kg/s.

Turbofan Engines with Duct Heater

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17. Extending the problem 16 on turbofan engine with additional information that the combustion efficiency is 0.99, determine the SFC. Also, calculate the thrust and SFC when a combustion chamber is incorporated in the bypass duct and the cold stream is heated to 1000 K. The combustion efficiency and pressure loss for this process may be assumed to be [0.0429 kg/ h N; 55.95 kN; 0.128 kg/h N] 0.97 and 0.05 bar respectively.

Turboprop Engines 18. A turboprop engine is operating under following conditions: Flight speed at sea-level, standard day Airflow entering the compressor Compressor pressure ratio (total-to-total) Efficiencies Diffuser Compressor Turbine to drive compressor Turbine to drive the propeller Nozzle Turbine inlet temperature (stagnation) Stagnation pressure leaving power turbine

0 1.0 kg/s 12 100 % 87 % 89 % 89 % 100 % 1400 K 1.724 bar

Calculate (a) the power delivered by the engine to the propeller (b) the thrust developed by the engine (c) the equivalent shaft power (d) the equivalent specific fuel consumption 19. Thought is being given to developing a new turboprop engine with an eight bladed propeller specially designed for flight at M=0.7 at an altitude of 12 Km. An existing turbojet engine has a gas generator design that (with the addition of a free turbine, gear reducer, propeller and new propulsion nozzle) would be used for the engine. At the above altitude and flight Mach number the gas generator exit conditions are Mass flow Total pressure Total temperature

100 kg/s 0.04 Mpa 1200 K

If these same conditions were to apply at entrance to the free-power turbine of the turboprop, determine the best combination of the propeller thrust and nozzle thrust for the turboprop engine given the following expected adiabatic efficiencies: Propeller, ηpr Nozzle, ηn Power turbine, ηpt

0.79 0.98 0.89

The mechanical efficiency of the gearbox is ηg = 0.97. Assume the turbine working fluid has γ = 1.33 and molecular weight of 30. [59 kN (propeller)]

Ramjet Engines 20. Compare the specific fuel consumption of a turbojet and a ramjet that are being

considered for flight at M = 1.5 and 50,000 ft altitude (with ambient pressure and temperature of 11.6 kPa and 205 K respectively). The turbojet pressure ratio is 12 and the maximum allowable temperature is 1400 K. For the ramjet the maximum temperature is 2500 K. For simplicity, ignore aerodynamic losses in both engines. Conventional hydrocarbon fuels are used with heating value of 45,000 kJ/kg. Assume γ = 1.4 and Cp = 1.0 kJ/kg K. [0.0558 kg/kN.s (ramjet), 0.0252 kg/kN.s (turbojet)]

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21. A ramjet engine is to propel an aircraft at Mach 3 at high altitude where the ambient pressure is 8.5 kPa and the ambient temperature is 220 K. The turbine inlet temperature is 2540 K. If all the components of the engine are ideal-that is, frictionless-determine (a) the thermal efficiency, (b) the propulsive efficiency and (c) the overall efficiency. Let the specific heat ratio be 1.4 and make the approximations appropriate to f