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# Jet & Rocket Propulsion AE 4451

GPA 3.79

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This 0 page Class Notes was uploaded by Demond Hoppe on Monday November 2, 2015. The Class Notes belongs to AE 4451 at Georgia Institute of Technology - Main Campus taught by Staff in Fall. Since its upload, it has received 25 views. For similar materials see /class/234312/ae-4451-georgia-institute-of-technology-main-campus in Aerospace Engineering at Georgia Institute of Technology - Main Campus.

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Date Created: 11/02/15

Example Problems dealing with jet propulsion These are intended to introduce some of the concepts needed for the course and to let the student see some of the numbers involved and their variations The calculations involve some exponentials The student is also asked to do repetitive calculations in order to plot functions this is suited to spreadsheet applications Example 1 Calculate the density of air at a pressure of 1 atmosphere and a temperature of 25 deg C Solution Pressure 1 atmosphere 101300 Newtons per square meter Temperature 25 deg Celsius 27315 25 29815 deg Kelvin The thermal equation of state relates the pressure P absolute temperature T and density p of a gas P p RT where R RuMW Ru being the Universal Gas Constant 83143 in SI units and MW the molecular weight of the gas Air is composed of 79 Nitrogen and 21 Oxygen The molecular weight of Nitrogen N2 is 28 and that of Oxygen 02 is 32 Thus the mean molecular weight is MW 079 28 02132 2884 Thus the gas constant for air R 83143 2884 28829 mK39ls392 p 10130029815 28829 11785 kgm3 It is useful to remember that in SI units atmospheric pressure at sea level is approximately 100000 temperature is 300 and density is 12 Example 2 The ideal ef ciency of a jet engine can be found by representing the processes in the engine as a quotBrayton cyclequot The cycle ef ciency is related to the ratio of the highest pressure in the engine PB to the lowest pressure P A Thus if an engine cycle takes air at 03 atmospheres and the highest pressure in the engine is 10 atmospheres the ideal cycle ef ciency is n 1 PBPA1 YY where g is the ratio of the speci c heats at constant pressure and volume 14 for diatomic gases such as air n 1 10 03 02857 06328 01 6328 This is a measure of how efficiently the heat put into the engine can possibly be converted to work if there are no losses Actual ef ciency of course will be lower This shows why it is desirable to have as high a quotpressure ratioquot as possible in the engine and why compressors are needed Modern engines have compressor pressure ratios as high as 40 Problem for Spreadsheet Practice Plot the variation of ideal cycle efficiency as a function of pressure ratio for ratios ranging from 1 to 40 Example 3 Find the temperature at the compressor eXit if the compressor inlet temperature is 100 deg C and the compressor pressure ratio is 32 Solution The change in temperature of air in the compressor is related to the change in pressure by the quotisentropic relationquot TB TA PBPAY 1Y TA 27315100 37315K PBPA 32 Therefore TB 1004 K Example 4 Plot the stagnation temperature at the nose of an aircraft as a function of flight Mach number from Mach 0 to 25 when the atmospheric temperature is 35 deg C Sample solution When highspeed air is slowed down to stagnant conditions its temperature will rise according to the relation T0 T 1 05 y lM2 where M is the Mach number and T0 is the quotstagnation temperaturequot the temperature reached when the Mach number is reduced to zero 7 is the ratio of specific heats at constant pressure and volume and is equal to 14 for air at moderate temperatures Atmospheric temperature T 27315 35 30815 K At Mach 25 stagnation temperature To 30815 1 02625 69334 K Problem 1 A jet engine cycle is designed to have the following features ambient pressure 1 atmosphere ambient temperature 300K highest temperature 2000K exhaust temperature 500K Calculate the Brayton cycle efficiency and the highest pressure in the engine Problem 2 At a standard altitude of 15000 In an ideal ramjet engine is ying at maximum thrust at Mach 25 The highest temperature in the engine is 2500K The fuel is methane CH4 Find 1 the thrust per unit mass ow rate of air 2 the fueltoair ratio 3 the specific fuel consumption 4 the actual thrust if the nozzle throat diameter is 03m 5 the Brayton cycle efficiency 6 the propulsive efficiency 7 the thermal efficiency 8 the exhaust temperature Problem 3 The compressor of a turbojet engine has a pressure ratio of 28 The upstream stagnation temperature and pressure are 300K and 100000Nm2 respectively The stagnation temperature at the compressor exit is 850K Find the stagnation pressure at the exit and the compressor efficiency 35 Problem 4 A turbojet engine is ying at Mach 18 so that there is 1 oblique shock and a normal shock at the inlet There are some small losses in the diffuser after the shock The compressor and turbine exchange work with the uid with no losses There is some stagnation pressure loss in the combustor The afterbumer is OFF but causes some loss in stagnation pressure The exhaust nozzle is underexpanded Sketch the variation of a Stagnation pressure b Stagnation temperature c static pressure as a function of distance through the engine going from ahead of the normal shock to downstream of the nozzle exit Mark on the sketch the various stations and state what they are eg compressor burner etc Pay careful attention to the slopes and relative heights of the lines that you draw I will Problem 5 Answer in complete sentences and show formulae where appropriate a How do you nd the limiting ight Mach number of an ideal ramjet engine at a given altitude b How do you calculate the limiting mass ow rate of a turbojet engine at a given operating condition at a speci ed altitude and Mach number given the total thrust at takeoff Problem 6 The processes occurring in an engine can be represented as follows The working uid air which is initially at a pressure of 105 Nm2 and a temperature of 300K is compressed isentropically so that its temperature increases by a factor of 3 Heat is then added at constant pressure until the temperature reaches 1200K Work is then extracted from the uid in an isentropic process until the pressure returns to 105 Nm2 a Calculate the cycle efficiency b Calculate the temperature at the end of the work extraction Problem 7 A turbojet engine is ying at Mach 15 so that there is a normal shock at the inlet The heat addition in the burner can be assumed to occur with no pressure losses There are no losses in the diffuser after the shock The compressor and turbine exchange work with the uid with no losses The exhaust nozzle is isentropic Sketch the variation of a Stagnation pressure b Stagnation temperature as a function of distance through the engine going from ahead of the normal shock to downstream of the nozzle exit Mark on the sketch the various stations and state what they are eg compressor burner etc Problem 8 A turbojet engine is designed to y at Mach 05 at an altitude where the temperature is 270K and the pressure is 4 x 104 Nm2 The stagnation pressure of the exhaust is 12 x 104 Nm2 and the stagnation temperature is 800K Find the propulsive efficiency Assume that the fuelair ratio is very small compared to 1 Assume that the nozzle is fully expanded The value of the gas constant for air may be taken as 287 in SI units and the ratio of specific heats may be taken as 14 Problem 9 A jet engine takes in air at 300K and 1 atmosphere pressure and compresses it to 11 atmospheres before adding heat at constant pressure The highest temperature reached is l500K Find the Brayton cycle efficiency If the nozzle exit pressure is the same as the ambient pressure what is the exhaust temperature

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