Problems in Gas Dynamics

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26 Fundamentals of Gas Dynamics

Exercises

(1) Air enters the diffuser of an aircraft jet engine at a static pressure of20 kPa and static temperature 217 K and a Mach number of 0.9. Theair leaves the diffuser with a velocity of 85 m/s. Assuming isentropicoperation, determine the exit static temperature and pressure.[249 K, 32 kPa]

(2) Air is compressed adiabatically in a compressor from a static pressureof 100 kPa to 2000 kPa. If the static temperature of the air at the inletand exit of the compressor are 300 K and 800 K, determine the powerrequired per unit mass flow rate of air. Also, determine whether thecompression process is isentropic or not.[503 kW, Not isentropic]

(3) Air enters a turbine at a static pressure of 2 MPa, 1400 K. It expandsisentropically in the turbine to a pressure of 500 kPa. Determine thework developed by the turbine per unit mass flow rate of air and thestatic temperature at the exit.[460 kW, 942 K]

(4) Air at 100 kPa, 295 K and moving at 710 m/s is decelerated isentrop-ically to 250 m/s. Determine the final static temperature and staticpressure.[515 K, 702 kPa]

(5) Air enters a combustion chamber at 150 kPa, 300 K and 75 m/s.Heat addition in the combustion chamber amounts to 900 kJ/kg. Airleaves the combustion chamber at 110 kPa and 1128 K. Determine thestagnation temperature, stagnation pressure and velocity at the exit andthe entropy change across the combustion chamber.[1198 K, 115 kPa, 168 m/s, 1420 J/kg.K]

(6) Air at 900 K and negligible velocity enters the nozzle of an aircraft jetengine. If the flow is sonic at the nozzle exit, determine the exit statictemperature and velocity. Assume adiabatic operation.


One Dimensional Flows - Basics 27

[750 K, 549 m/s]

(7) Air expands isentropically in a rocket nozzle from P0 = 3.5 MPa, T0 =2700 K to an ambient pressure of 100 kPa. Determine the exit velocity,Mach number and static temperature.[1860 m/s, 2.97, 978 K]

(8) Consider the capture streamtube of an aircraft engine cruising at Mach0.8 at an altitude of 10 km. The capture mass flow rate is 250 kg/s. Atstation 1, which is in the freestream, the static pressure and temperatureare 26.5 kPa and 223 K respectively. At station 2, which is downstreamof station 1, the cross-sectional area is 3 m2. Further downstream atstation 3, the Mach number is 0.4. Determine (a) the cross-sectionalarea at station 1 (usually called the capture area), (b) the Mach numberat station 2, (c) the static pressure and temperature at stations 2 and 3and (d) the cross-sectional at station 3.[a) 2.5213 m2 b) 0.5635 c) Station2: 32.9 kPa, 236.52 K, Station3:36.496 kPa, 243.74 K d) 3.8258 m2]

(9) The ramjet engine shown in Fig. 2.4 does not have any moving parts.It operates at high supersonic Mach numbers (< 4). The entering airis decelerated in the diffuser to a subsonic speed. Heat is added in thecombustion chamber and the hot gases expand in the nozzle generating

Source: http://www.aerospaceweb.org/question/propulsion/q0175.shtml

Fig. 2.4: Schematic of a ramjet engine

thrust. In an “ideal” ramjet engine, air is the working fluid throughout



and the compression, expansion processes are isentropic. In addition,there is no loss of stagnation pressure due to the heat addition. The airis expanded in the nozzle to the ambient pressure. Show that the Machnumber of the air as it leaves the nozzle is the same as the Mach numberof the air when it enters the diffuser. Sketch the process undergone bythe air on T-s and P-v diagrams.


Normal Shock Waves 43

Exercises

(1) A shock wave advances into stagnant air at a pressure of 100 kPa and300 K. If the static pressure downstream of the wave is tripled, what isthe shock speed and the absolute velocity of the air downstream of theshock?[573 m/s, 302 m/s]

(2) Repeat Problem 1 assuming the fluid to be helium instead of air.[1644.97 m/s, 757.49 m/s.]

(3) Air at 2.5 kPa, 221 K approaches the intake of a ramjet engineoperating at an altitude of 25 km. The Mach number is 3.0. Forthis Mach number a normal shock stands just ahead of the intake.Determine the stagnation pressure, static pressure and temperature ofthe air immediately after the normal shock. Also calculate the % lossin stagnation pressure. Repeat the calculations for Mach number equalto 4. The high loss of stagnation pressure that you see from yourcalculations illustrates why the intake of a ramjet has to be designedcarefully to avoid such normal shocks during operation.[30 kPa, 26 kPa, 592 K, 67%; 53 kPa, 46 kPa, 894 K, 86%]

(4) A blast wave passes through still air at 300 K. The velocity of the airbehind the wave is measured to be 180 m/s in the laboratory frameof reference. Determine the speed of the blast wave in the laboratoryframe of reference and the stagnation temperature behind the wave inthe laboratory as well moving frames of reference.[471.59 m/s, 410 K, 384 K]

(5) A normal shock wave travels into still air at 300 K. If the statictemperature of the air is increased by 50 K as a result of the passageof the shock wave, determine the speed of the wave in the laboratoryframe of reference.[437.46 m/s]

(6) A shock wave generated due to an explosion travels at a speed of 1.5



km/s into still air at 100 kPa and 300 K. Determine the velocity of theair, static and stagnation quantities (with respect to a stationary frameof reference) in the region through which the shock has passed.[1183 m/s, 2.2 MPa, 1370 K, 9.1 MPa, 2067 K]

(7) A bullet travels through air (300 K, 100 kPa) at twice the speed ofsound. Determine the temperature and pressure at the nose of thebullet.Note that although there will be a curved, bow shock ahead of thebullet, in the nose region, normal shock relationships can be used. Alsonote that the nose is a stagnation point![540 K, 565 kPa]

(8) A pitot tube is used to measure the Mach number (M1) of a supersonicflow as shown in the figure. Although a curved shock stands ahead ofthe probe, it is fairly accurate to assume that the fluid in the streamtubecaptured by the probe has passed through a normal shock wave. Itis also reasonable to assume that the probe measures the stagnationpressure downstream of the shock wave (P0,2). If the static pressureupstream of the shock wave (P1) is also measured, then the Machnumber M1 can be evaluated. Derive the relation connecting P0,2/P1

and M1†.

M1

†This is called the Rayleigh pitot formula.


Quasi One Dimensional Flows 67

Exercises

(1) Consider the two tank system in Fig. 6.13. Assume the stagnationtemperature to be 300 K and the throat diameter of the nozzles to be2.54 cm. Sketch the variation of the exit pressure, mass flow rate,exit Mach number and the ambient pressure of nozzles A and B withtime starting from time 0+ until steady state is reached. Althoughthe profiles can be qualitative, key instants should be marked withnumerical values for these quantities. The pressure profiles must beshown together in same figure using the same axes.

(2) Consider again the two tank system in Fig. 6.13. Assume that onlynozzle A is present and that it is a convergent-divergent nozzle of exit-to-throat area ratio 2 with the same throat diameter as before. Sketchthe variation of the exit pressure, mass flow rate, exit Mach number andthe ambient pressure of nozzle A with time starting from time 0+ untilsteady state is reached. Although the profiles can be qualitative, keyinstants should be marked with numerical values for these quantities.The pressure profiles must be shown together in same figure using thesame axes.

(3) A reservoir of volume V initially contains air at pressure Pi andtemperature Ti. A hole of cross-sectional area A develops in thereservoir and the air begins to leak out. Develop an expression for thetime taken for half of the initial mass of air in the reservoir to escape.Assume that, during the process, the pressure in the reservoir is muchhigher than the ambient pressure and also that the temperature remainsconstant.

(4) Consider a CD nozzle with exit and throat areas of 0.5 m2 and 0.25m2 respectively. The inlet reservoir pressure is 100 kPa and the exitstatic pressure is 60 kPa. Determine the exit Mach number.[0.46]

(5) Air at a pressure and temperature of 400 kPa and 300 K contained in alarge vessel is discharged through an isentropic nozzle into a space



at a pressure of 100 kPa. Find the mass flow rate if the nozzle is(a) convergent and (b) convergent-divergent with optimum expansionratio. In both cases, the minimum cross-sectional area of the nozzlemay be taken to be 6.5 cm2.[0.6067 kg/s in both cases]

(6) A student is trying to design an experimental set-up to produce acorrectly expanded supersonic stream at a Mach number of 2 issuinginto ambient at 100 kPa. For this purpose the student wishes to usea CD nozzle with the largest possible exit area. There is a 10 m3

reservoir containing air at 1 MPa and 300 K available in the lab. Thenozzle is connected to the reservoir through a settling chamber. Thesettling chamber is reasonably large and allows the stagnation pressurejust ahead of the nozzle to be fixed at the desired value. Determine thelargest possible exit area for the nozzle that will allow the student to runthe experiment continuously for at least 15 minutes. Neglect frictionallosses in the pipes and assume that the temperature of the air remainsconstant ahead of the nozzle.[9.34× 10−5m2]

(7) What is the stagnation pressure required to run the nozzle described inthe previous question at the desired Mach number?[782.4 kPa]If a supersonic diffuser is now connected to the end of the duct todiffuse the air to ambient pressure (thereby eliminating the normalshock), what is the stagnation pressure required to drive the flow?[100 kPa]


Flow of Steam through Nozzles 91

Exercises

(1) Dry, saturated steam enters a convergent nozzle at a static pressureof 800 kPa and is expanded to the sonic state. If the inlet and throatdiameters are 0.05 m and 0.025 m respectively, determine the velocityat the inlet and exit and the stagnation pressure.[70.56 m/s, 452 m/s, 806 kPa]

(2) Dry saturated steam at 1.1 MPa is expanded in a nozzle to a pressureof 15 kPa. Assuming the expansion process to be isentropic and inequilibrium throughout, determine (a) if the nozzle is convergent orconvergent-divergent, (b) the exit velocity, (c) the dryness fraction atthe exit and (c) the exit to throat area ratio.[1147 m/s, 0.8, 11]

(3) Superheated steam at 600 kPa, 200◦C in a steam chest is expandedthrough a nozzle to a final pressure of 20 kPa. The throat diameteris 10 mm. Assuming the expansion process to be isentropic and inequilibrium throughout, determine (a) the mass flow rate, (b) the exitvelocity, (c) the dryness fraction at the exit and (c) the exit diameter.[0.068 kg/s, 1053 m/s, 0.87, 23.4 mm]

(4) Dry saturated steam at 1.2 MPa is expanded in a nozzle to 20 kPa. Thethroat diameter of the nozzle is 6 mm. If the total mass flow rate is 0.5kg/s, determine how many nozzles are required and the exit diameterof the nozzle. Assume the expansion process to be isentropic and inequilibrium throughout.[10]

(5) Superheated steam at 850 kPa, 200◦C expands in a convergent nozzleto a pressure of 480 kPa. Determine the exit velocity, assuming theexpansion process to be isentropic and in equilibrium throughout.[472 m/s]

(6) Superheated steam at 700 kPa, 250◦C expands in a nozzle to a pressureof 100 kPa. Determine the throat area and the exit area if the mass flow



rate is 0.1 kg/s. Assume the expansion process to be isentropic and inequilibrium throughout.[1.04× 10−4 m2, 1.88× 10−4 m2]

(7) Steam which is initially saturated and dry expands from 1400 kPa to700 kPa. Assuming the expansion to be in equilibrium (n=1.135),determine the final velocity and specific volume. If the expansion is outof equilibrium (n=1.3), determine the final velocity, specific volume,supersaturation ratio and the degree of undercooling.[512 m/s, 0.259 m3/kg; 512 m/s, 0.2593 m3/kg, 1.2, 7◦C]

(8) Superheated steam at 500 kPa, 170◦C is expanded in a nozzle topressure of 170 kPa. Assuming the expansion process to be isentropicand in equilibrium determine the exit velocity. Assuming the flow tobe isentropic and supersaturated, determine the supersaturation ratio,the degree of supercooling and the exit velocity.[622 m/s, 4.92, 43◦C, 613.6 m/s]

(9) For each of the stagnation condition given below, determine thepressure, velocity and degree of supercooling just before the onset ofcondensation shock for a limiting value of supersaturation ratio of 5.Assume the expansion process to be isentropic. (a) 87000 Pa, 96◦C,(b) 70727 Pa, 104◦C and (c)25000 Pa, 85◦C.[45037 Pa, 453 m/s, 35◦C; 29918 Pa, 518.6 m/s, 33◦C; 10127 Pa, 518m/s, 28◦C]

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Problems in Gas Dynamics