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, = 1.4 Cp for gas = 1.148 kJ/kg K, R = 0.287 kJ/kg K, = 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. Take calorific value of fuel=43 MJ/kg. [35.1 % & 44.2 %; For Olympus 40.7 % & 55.1 %]

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 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 of the low-pressure compressor. [78 %, 78.8 %; 78 %]

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 temperature range of the expansion of Cp = 1.16 kJ/kg K and = 1.33. [0.957]

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 8.0 Turbine inlet (stagnation) temperature 1200 K Isentropic Efficiencies Intake, D 93 % Compressor, c 87 % Turbine, T 90 % Propelling Nozzle, N 95 % Mechanical transmission efficiency, m 99 % Combustion efficiency, B 98 % Combustion pressure loss, pb 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, D 90 % Compressor, c 90 % Turbine, T 92 % Propelling Nozzle, N 92 % Mechanical transmission efficiency, m 98 % Combustion efficiency, B 98 % Combustion pressure loss, pb 0.14 bar Calorific value of fuel 43 300 kJ/kg Nozzle outlet area 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 afterburner process is 0.07 bar. [0.244 m2, 22 kN]

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 required, the net thrust developed and SFC. [0.0713 m2, 7896 N, 0.126 kg/N-h]

Polytropic efficiencies of compressor and turbine 0.87 Isentropic efficiency of intake 0.95

Isentropic efficiency of Propelling nozzle 0.95 Mechanical efficiency 0.99

Combustion chamber pressure loss 6 % comp.del.pr. Combustion efficiency 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 ambient pressure and temperature are 1.0 bar and 288 K. [71.5 kN, 0.0403 kg/h N]

Overall pressure ratio 25.0

Fan pressure ratio 1.65 By-pass ratio 5.0

Turbine inlet temperature 1550 K Fan, compressor and turbine polytropic efficiency 0.90

Isentropic efficiency of each propelling nozzle 0.95 Mechanical efficiency of each spool 0.99

Combustion pressure loss 1.50 bar Total air mass flow 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 flow remains at 23 kg/s. [0.0763m2, 15.35 kN, 24.9 kN]

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 0.97 and 0.05 bar respectively. [0.0429 kg/ h N; 55.95 kN; 0.128 kg/h N]

Turboprop Engines

18. A turboprop engine is operating under following conditions:

Flight speed at sea-level, standard day 0 Airflow entering the compressor 1.0 kg/s

Compressor pressure ratio (total-to-total) 12 Efficiencies

Diffuser 100 % Compressor 87 %

Turbine to drive compressor 89 % Turbine to drive the propeller 89 %

Nozzle 100 % Turbine inlet temperature (stagnation) 1400 K

Stagnation pressure leaving power turbine 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 100 kg/s

Total pressure 0.04 Mpa Total temperature 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, pr 0.79 Nozzle, n 0.98

Power turbine, pt 0.89

The mechanical efficiency of the gearbox is g = 0.97. Assume the turbine working fluid has = 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 = 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

Find the (a) vehicles mass ratio, (b) mass ratio of the rocket system (c) propellant mass fraction, (d) propellant flow rate, (e) thrust, (f) initial and final thrust-to-weight ratio, (g) acceleration of the vehicle, (h) effective exhaust velocity, (i) total impulse, and (j) the impulse-to-weight ratio.

[0.65, 0.22, 0.778, 54.8 kN, 28, 43, 421 m/s2, 2352 m/s, 164.6 kN-sec, 187]

31. The following measurements were made in a sea level test of a solid rocket motor:

Burn duration 40 sec Initial mass before test 1210 kg Mass of rocket motor after test 215 kg Average thrust 62,250 N Chamber pressure 7.00 Mpa Nozzle exit pressure 0.070 Mpa Nozzle throat diameter 0.0855 m Nozzle exit diameter 0.2703 m

Determine mass flow rate, actual average exhaust velocity, characteristic velocity, effective exhaust velocity and specific impulse at sea level, and effective exhaust velocity and specific impulse at 1000 and 25,000 m altitude. Assume an invariant thrust and mass flow rate and negligible short start and stop transients. (For altitudes of 1 km & 25 km the ambient pressure is 0.0898 and 0.00255 Mpa respectively).

[24.9 kg/s, 2572 m/s, 1613 m/s, 2500 m/s, 255 sec; 2527 m/s, 258 sec; 2727 m/s, 278 sec]

32. The following data are given for a certain rocket unit:

Thrust 8896 kN Propellant consumption 3.867 kg/s Velocity of vehicle 400 m/s Energy content of propellant 6.911 MJ/kg

Determine (a) the effective exhaust velocity, (b) the kinetic jet energy rate per unit flow of propellant, (c) the internal efficiency, (d) the propulsive efficiency, (e) the overall efficiency, (f) the specific impulse and (g) the specific propellant consumption. Assume combustion efficiency to be 98 %. [2300 m/s, 2.645 MJ-sec/kg, 38.3 %, 33.7 %, 13.3 %234.7 sec, 0.00426 sec-1]

Rocket Nozzle Theory

33. An ideal rocket chamber is to operate at sea level using propellants whose combustion

products have a specific heat ratio of 1.3. Find the required chamber pressure and the nozzle area ratio between throat and exit if the nozzle exit Mach number is 2.4. The nozzle inlet Mach number may be considered to be very small. [1.5 Mpa, 2.64]

34. A rocket operates at sea level (p=0.1013 Mpa) with a chamber pressure of 2.068 Mpa and a

chamber temperature of 2222 K and a propellant consumption of 1 kg/s. Assuming a ratio of specific heats to be 1.3 and gas constant to be 345.7 kJ/kg K, show graphically the variation of area, temperature, specific volume and velocity with respect to pressure along the nozzle. Calculate the ideal thrust and ideal specific impulse.

35. The German World War-II A-4 propulsion system had a sea level thrust of 25,400 Kg and a

chamber pressure of 1.5 Mpa. If the exit pressure is 0.084 Mpa and the exit diameter is 740 mm, what is the thrust at 25,000 m, where the ambient pressure is 2.6077 x 103 N/m2.

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36. Derive the equation for the theoretical correction factor for a conical nozzle assuming that all the mass flow originates at the apex of the cone. Hence, calculate the nozzle angle correction factor for a nozzle whose divergent half angle is 130.

37. Design a rocket nozzle to conform to the following conditions:

Chamber pressure 20.4 atm = 2.068 Mpa Atmospheric pressure 1.0 atm Chamber temperature 2861 K Mean molecular mass of gases 21.87 kg/kg-mol Ideal specific impulse 230 sec Specific heat ratio 1.229 Desired thrust 1300 N

Find nozzle throat and exit areas, respective diameters, actual exhaust velocity, and actual specific impulse. [4.66 cm2; 2.43 cm; 15.9 cm2; 4.5 cm; 2074 m/s; 212 sec]

38. The following data are given for an ideal rocket:

Average molecular mass 24 kg/kg-mol Chamber pressure 2.53 Mpa External pressure 0.090 Mpa Chamber temperature 2900 K Throat area 0.00050 m2

Assuming specific heat ratio of 1.3, determine (a) throat velocity, (b) specific volume at throat, (c) propellant flow and specific impulse, (d) thrust, (e) Mach number at throat.

39. For an ideal rocket with a characteristic exhaust velocity of 1200 m/s, a mass flow rate of

73.0 kg/s, a thrust coefficient of 1.50 and a nozzle throat area of 0.0248 m2, compute the effective exhaust velocity, the thrust, the chamber pressure and the specific impulse.

[1800 m/s; 131400 N; 3.530 x 106 N/m2; 183.4 sec]

40. Using a propellant of molecular mass 15 kg/kg-mol and flame temperature 3300 K, determine the rocket nozzle throat and exit areas required for a thrust of 500 kN and an ideal impulse of 300 sec. The ambient pressure is 0.1 Mpa, and specific heat ratio of the propellant is 1.40. (a) How much thrust would this rocket develop if the ambient pressure were changed to 0.03 Mpa? (b) How much thrust would be developed by a rocket designed to expand to 0.03 Mpa if it had the same stagnation conditions, throat area and propellant?

[0.0647 m2, 0.342 m2, 523 kN, 533 N]

Solid Rocket Motors

41. The following requirements are given for a solid propellant rocket motor:

Sea level thrust 8900 N (2000 lbf) Duration 10 sec Chamber pressure 6.8947 Mpa (1000 psia) Operating temperature Ambient (approx. 70 F) Propellant Ammonium nitrate-hydrocarbon

Determine the specific impulse, the throat and exit areas, the flow rate, the total propellant weight, the total impulse, the burning area, and an estimated mass assuming moderately

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efficient design. Properties of this propellant are: = 1.26; T1 = 1755 K; r = 0.254 cm/sec at 6.8947 Mpa; c* = 1219.2 m/sec; b= 1550 kg/m3; molecular mass = 22 kg/kg-mol. Assume a total impulse to weight ratio (It/wG) of 143.

[191 sec, 8.38 cm2, 65.16 cm2,46.6 N/s, 466 N, 485 N, 89000 N-s, 1.187 m2,622.37 N]

42. A solid rocket motor has the following operating characteristics:

Sea level thrust 10,000 N Duration 10 sec Chamber pressure 70.928 bar Specific heat ratio 1.26 Chamber temperature 2500 K Burning rate 6 mm/sec at 70.928 bar Propellant density 1.67 gm/cc Molecular mass 22 kg/kg-mol

Assuming a neutral burning (with a cigarette burning grain) and an adapted nozzle at sea level, find the (a) characteristic velocity, (b) thrust coefficient at optimum expansion, (c) specific impulse at sea level, (d) nozzle throat area, (e) weight of propellant and (f) burning surface area. [1473 m/s, 1.59, 238 sec, 8.867 cm2, 418.4 N, 0.4261 m2]

43. Determine the grain geometry and propellant weight of an internal burning tubular grain

solid propellant unit given the following data:

Specific impulse 240 sec at sea level at 70 ata Burning rate 6 cm/sec at 70 ata Specific gravity 1.65 Ratio of specific heats 1.25 Combustor pressure 70 ata Desired average thrust 6000 kgf Maximum outer diameter 400 mm Ambient pressure 0.25 ata Vehicle pay load 2250 kg Desired duration 6 secs

Determine the nozzle dimensions assuming optimum expansion in the nozzle. 44. The grain in a solid propellant rocket is a hollow cylinder bonded to the casing so that it

burns only on its inner cylindrical surface. Its density is 1650 kg/m3, and its burning rate is characterized by r=13.3 p10.63 mm/s where p1 is in Mpa. At a point in the burning period when p1 = 0.7 Mpa, the grain d/D = 0.4 and L/D = 6, L being the grain length and, d and D being its inner and outer diameters. Determine the rate of change of chamber pressure, assuming the gas temperature stays constant at 2750 K and that the gas specific heat ratio is 1.24 and D = 1 m. [0.0862 Mpa/s]

45. The burning rate of a particular propellant is given by r = c/(T1 - T) p1n in which r is in

mm/s, p1 is in Mpa, and T is in K, and c = 176 T1 = 415 K n = 0.716

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When the propellant initial temperature is 293 K, the chamber pressure is 3 Mpa (steady) for 4 min. If the same propellant grain in the same rocket motor is heated to 318 K, what would be the new steady state pressure level and burning period. Assume the time required to reach steady state is small compared to the burning rate. [1.79 min]

Liquid Rocket Engines

46. A liquid oxygen-liquid hydrogen rocket thrust chamber of 10, 000 lbf thrust operates at a

chamber pressure of 1000 psia, a mixture ratio of 3.40, has exhaust products with a mean molecular mass of 8.9 lbm/lb-mol, a combustion temperature of 43800 F, and a specific heat ratio of 1.26. Find the nozzle area, exit area for optimum operation at an altitude where p3= p2=1.58 psia, the propellant weight and volume flow rates, and the total propellant requirements for 2 minutes of operation. Assume that the actual specific impulse is 97 % of the theoretical value.

47. What will be the volume of air tank required to pressurize the propellant tanks of a 9000 N

thrust rocket thrust chamber using 90 % hydrogen peroxide as a monopropellant at a chamber pressure of 2.00 Mpa for 30 sec in conjunction with a solid catalyst? The air tank pressure is 14 Mpa and the propellant tank pressure is 3.0 Mpa. Allow for 1.20 % residual propellant.

48. Determine the shaft speed and the overall impeller dimensions for a liquid oxygen pump

which delivers 500 lb/sec of propellant at a discharge pressure of 1000 psia and a suction pressure of 14.7 psia. The oxygen tank is pressurized to 35 psia. Neglect the friction in the suction pipe and the section head changes due to acceleration and propellant consumption. The initial tank level is 15 ft above the pump suction inlet.

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