Compressible pipe fl ow with heat addition, Sec. 9.8, assumes constant momentum (p 1 V2

An ideal gas fl ows adiabatically through a duct. At section 1, p1 5 140 kPa, T1 5 2608C, and V1 5 75 m/s. Farther downstream, p2 5 30 kPa and T2 5 2078C. Calculate V2 in m/s and s2 2 s1 in J/(kg ? K) if the gas is (a) air, k 5 1.4, and (b) argon, k 5 1.67.

Solve Prob. P9.1 if the gas is steam. Use two approaches: (a) an ideal gas from Table A.4 and (b) real gas data from the steam tables [15]

If 8 kg of oxygen in a closed tank at 2008C and 300 kPa is heated until the pressure rises to 400 kPa, calculate (a) the new temperature, (b) the total heat transfer, and (c) the change in entropy

Consider steady adiabatic airfl ow in a duct. At section B, the pressure is 600 kPa and the temperature is 1778C. At section D, the density is 1.13 kg/m3 and the temperature is 1568C. (a) Find the entropy change, if any. (b) Which way is the air fl owing?

Steam enters a nozzle at 3778C, 1.6 MPa, and a steady speed of 200 m/s and accelerates isentropically until it exits at saturation conditions. Estimate the exit velocity and temperature.

Methane, approximated as a perfect gas, is compressed adiabatically from 101 kPa and 208C to 300 kPa. Estimate (a) the fi nal temperature, and (b) the fi nal density.

Air fl ows through a variable-area duct. At section 1, A1 5 20 cm2 , p1 5 300 kPa, 1 5 1.75 kg/m3 , and V1 5 122.5 m/s. At section 2, the area is exactly the same, but the density is much lower: 2 5 0.266 kg/m3 and T2 5 281 K. There is no transfer of work or heat. Assume one-dimensional steady fl ow. (a) How can you reconcile these differences? (b) Find the mass fl ow at section 2. Calculate (c) V2, (d ) p2, and (e) s2 2 s1. [Hint: This problem requires the continuity equation.]

Atmospheric air at 208C enters and fi lls an insulated tank that is initially evacuated. Using a control volume analysis from Eq. (3.67), compute the tank air temperature when it is full.

Liquid hydrogen and oxygen are burned in a combustion chamber and fed through a rocket nozzle that exhausts at Vexit 5 1600 m/s to an ambient pressure of 54 kPa. The nozzle exit diameter is 45 cm, and the jet exit density is 0.15 kg/m3 . If the exhaust gas has a molecular weight of 18, estimate (a) the exit gas temperature, (b) the mass fl ow, and (c) the thrust developed by the rocket.

A certain aircraft fl ies at 609 mi/h at standard sea level. (a) What is its Mach number? (b) If it fl ies at the same Mach number at 34,000 ft altitude, how much slower (or faster) is it fl ying, in mi/h?

At 3008C and 1 atm, estimate the speed of sound of (a) nitrogen, (b) hydrogen, (c) helium, (d ) steam, and (e) 238UF6 (k < 1.06).

Assume that water follows Eq. (1.19) with n < 7 and B < 3000. Compute the bulk modulus (in kPa) and the speed of sound (in m/s) at (a) 1 atm and (b) 1100 atm (the deepest part of the ocean). (c) Compute the speed of sound at 208C and 9000 atm and compare with the measured value of 2650 m/s (A. H. Smith and A. W. Lawson, J. Chem. Phys., vol. 22, 1954, p. 351)

Consider steam at 500 K and 200 kPa. Estimate its speed of sound by two different methods: (a) assuming an ideal gas from Table B.4, or (b) using fi nite differences for isentropic densities between 210 kPa and 190 kPa.

Benzene, listed in Table A.3, has a measured density of 57.75 lbm/ft3 at a pressure of 700 bar. Use this data to estimate the speed of sound of benzene.

The pressure-density relation for ethanol is approximated by Eq. (1.19) with B 5 1600 and n 5 7. Use this relation to estimate the speed of sound of ethanol at 2000 atmospheres.

A weak pressure pulse Dp propagates through still air. Discuss the type of refl ected pulse that occurs and the boundary conditions that must be satisfi ed when the wave strikes normal to, and is refl ected from, (a) a solid wall and (b) a free liquid surface.

A submarine at a depth of 800 m sends a sonar signal and receives the refl ected wave back from a similar submerged object in 15 s. Using Prob. P9.12 as a guide, estimate the distance to the other object

Race cars at the Indianapolis Speedway average speeds of 185 mi/h. After determining the altitude of Indianapolis, fi nd the Mach number of these cars and estimate whether compressibility might affect their aerodynamics.

In 1976, the SR-71A, fl ying at 20 km standard altitude, set a jet-powered aircraft speed record of 3326 km/h. Estimate the temperature, in 8C, at its front stagnation point. At what Mach number would it have a front stagnation-point temperature of 5008C?

Air fl ows isentropically in a channel. Properties at section 1 are V1 5 250 m/s, T1 5 330 K, and p1 5 80 kPa. At section 2 downstream, the temperature has dropped to 08C. Find (a) the pressure, (b) velocity, and (c) Mach number at section 2.

N2O expands isentropically through a duct from p1 5 200 kPa and T1 5 2508C to a downstream section where p2 5 26 kPa and V2 5 594 m/s. Compute (a) T2; (b) Ma2; (c) To; (d ) po; (e) V1; and (f ) Ma1

Given the pitot stagnation temperature and pressure and the static pressure measurements in Fig. P9.22, estimate the air velocity V, assuming (a) incompressible fl ow and (b) compressible fl ow. Air V 100C 120 kPa 80 kPa

A gas, assumed ideal, fl ows isentropically from point 1, where the velocity is negligible, the pressure is 200 kPa, and the temperature is 3008C, to point 2, where the pressure is 40 kPa. What is the Mach number Ma2 if the gas is (a) air, (b) argon, or (c) CH4? (d ) Can you tell, without calculating, which gas will be the coldest at point 2?

For low-speed (nearly incompressible) gas fl ow, the stagnation pressure can be computed from Bernoullis equation: p0 5 p 1 1 2 V2 (a) For higher subsonic speeds, show that the isentropic relation (9.28a) can be expanded in a power series as follows: p0 < p 1 1 2 V2 a1 1 1 4 Ma2 1 2 2 k 24 Ma4 1 pb (b) Suppose that a pitot-static tube in air measures the pressure difference p0 2 p and uses the Bernoulli relation, with stagnation density, to estimate the gas velocity. At what Mach number will the error be 4 percent?

If it is known that the air velocity in the duct is 750 ft/s, use the mercury manometer measurement in Fig. P9.25 to estimate the static pressure in the duct in lbf/in2 absolute. P9.25 Air at 100F Mercury 8 in

Show that for isentropic fl ow of a perfect gas if a pitotstatic probe measures p0, p, and T0, the gas velocity can be calculated from V2 5 2cpT0 c 1 2 a p p0 b (k21)/k d What would be a source of error if a shock wave were formed in front of the probe?

A pitot tube, mounted on an airplane fl ying at 8000 m standard altitude, reads a stagnation pressure of 57 kPa. Estimate the planes (a) velocity and (b) Mach number.

Air fl ows isentropically through a duct. At section 1, the pressure and temperature are 250 kPa and 1258C, and the velocity is 200 m/s. At section 2, the area is 0.25 m2 and the Mach number is 2.0. Determine (a) Ma1; (b) T2; (c) V2; and (d ) the mass fl ow.

Steam from a large tank, where T 5 4008C and p 5 1 MPa, expands isentropically through a nozzle until, at a section of 2-cm diameter, the pressure is 500 kPa. Using the steam tables [15], estimate (a) the temperature, (b) the velocity, and (c) the mass fl ow at this section. Is the fl ow subsonic?

When does the incompressible-fl ow assumption begin to fail for pressures? Construct a graph of p0/p for incompressible fl ow of a perfect gas as compared to Eq. (9.28a). Plot both versus Mach number for 0 # Ma # 0.6 and decide for yourself where the deviation is too grea

Air fl ows adiabatically through a duct. At one section V1 5 400 ft/s, T1 5 2008F, and p1 5 35 lbf/in2 absolute, while farther downstream V2 5 1100 ft/s and p2 5 18 lbf/in2 absolute. Compute (a) Ma2, (b) Umax, and (c) p02/p01.

The large compressed-air tank in Fig. P9.32 exhausts from a nozzle at an exit velocity of 235 m/s. The mercury manometer reads h 5 30 cm. Assuming isentropic fl ow, compute the pressure (a) in the tank and (b) in the atmosphere. (c) What is the exit Mach number? P9.32 30C Air ptank? pa ? 235 m/s Mercury h

Air fl ows isentropically from a reservoir, where p 5 300 kPa and T 5 500 K, to section 1 in a duct, where A1 5 0.2 m2 and V1 5 550 m/s. Compute (a) Ma1, (b) T1, (c) p1, (d ) m # , and (e) A*. Is the fl ow choked?

Air in a large tank, at 3008C and 400 kPa, fl ows through a converging-diverging nozzle with throat diameter 2 cm. It exits smoothly at a Mach number of 2.8. According to onedimensional isentropic theory, what is (a) the exit diameter, and (b) the mass fl ow?

Helium, at T0 5 400 K, enters a nozzle isentropically. At section 1, where A1 5 0.1 m2 , a pitot-static arrangement (see Fig. P9.25) measures stagnation pressure of 150 kPa and static pressure of 123 kPa. Estimate (a) Ma1, (b) mass fl ow m # , (c) T1, and (d ) A*.

An air tank of volume 1.5 m3 is initially at 800 kPa and 208C. At t 5 0, it begins exhausting through a converging nozzle to sea-level conditions. The throat area is 0.75 cm2 . Estimate (a) the initial mass fl ow in kg/s, (b) the time required to blow down to 500 kPa, and (c) the time at which the nozzle ceases being choked.

Make an exact control volume analysis of the blowdown process in Fig. P9.37, assuming an insulated tank with negligible kinetic and potential energy within. Assume critical fl ow at the exit, and show that both p0 and T0 decrease during blowdown. Set up fi rst-order differential equations for p0(t) and T0(t), and reduce and solve as far as you can. Insulated tank p0 (t) T0(t) Volume V Ae, Ve, me  Measurements of tank pressure and tempera

Prob. P9.37 makes an ideal senior project or combined laboratory and computer problem, as described in Ref. 27, Sec. 8.6. In Bober and Kenyons lab experiment, the tank had a volume of 0.0352 ft3 and was initially fi lled with air at 50 lb/in2 gage and 728F. Atmospheric pressure was 14.5 lb/in2 absolute, and the nozzle exit diameter was 0.05 in. After 2 s of blowdown, the measured tank pressure was 20 lb/in2 gage and the tank temperature was 258F. Compare these values with the theoretical analysis of Prob. P9.37.

Consider isentropic fl ow in a channel of varying area, from section 1 to section 2. We know that Ma1 5 2.0 and desire that the velocity ratio V2/V1 be 1.2. Estimate (a) Ma2 and (b) A2/A1. (c) Sketch what this channel looks like. For example, does it converge or diverge? Is there a throat?

Steam, in a tank at 300 kPa and 600 K, discharges isentropically to a low-pressure atmosphere through a converging nozzle with exit area 5 cm2 . (a) Using an ideal gas approximation from Table B.4, estimate the mass fl ow. (b) Without actual calculations, indicate how you would use real properties of steam to fi nd the mass fl ow.

Air, with a stagnation pressure of 100 kPa, fl ows through the nozzle in Fig. P9.41, which is 2 m long and has an area variation approximated by A < 20 2 20x 1 10x 2 with A in cm2 and x in m. It is desired to plot the complete family of isentropic pressures p(x) in this nozzle, for the range of inlet pressures 1 , p(0) , 100 kPa. Indicate which inlet pressures are not physically possible and discuss briefl y. If your computer has an online graphics routine, plot at least 15 pressure profi les; otherwise just hit the highlights and explain. P9.41 A(x) p (x)? p0 p 0 0 1 m 2 m x

A bicycle tire is fi lled with air at an absolute pressure of 169.12 kPa, and the temperature inside is 30.08C. Suppose the valve breaks, and air starts to exhaust out of the tire into the atmosphere (pa 5 100 kPa absolute and Ta 5 20.08C). The valve exit is 2.00 mm in diameter and is the smallest cross-sectional area of the entire system. Frictional losses can be ignored here; one-dimensional isentropic fl ow is a reasonable assumption. (a) Find the Mach number, velocity, and temperature at the exit plane of the valve (initially). (b) Find the initial mass fl ow rate out of the tire. (c) Estimate the velocity at the exit plane using the incompressible Bernoulli equation. How well does this estimate agree with the exact answer of part (a)? Explain.

Air fl ows isentropically through a variable-area duct. At section 1, A15 20 cm2 , p15 300 kPa, 15 1.75 kg/m3 , and Ma15 0.25. At section 2, the area is exactly the same, but the fl ow is much faster. Compute (a) V2, (b) Ma2, (c) T2, and (d ) the mass fl ow. (e) Is there a sonic throat between sections 1 and 2? If so, fi nd its area.

In Prob. P3.34 we knew nothing about compressible fl ow at the time, so we merely assumed exit conditions p2 and T2 and computed V2 as an application of the continuity equation. Suppose that the throat diameter is 3 in. For the given stagnation conditions in the rocket chamber in Fig. P3.34 and assuming k 5 1.4 and a molecular weight of 26, compute the actual exit velocity, pressure, and temperature according to one-dimensional theory. If pa 5 14.7 lbf/in2 absolute, compute the thrust from the analysis of Prob. P3.68. This thrust is entirely independent of the stagnation temperature (check this by changing T0 to 20008R if you like). Why?

It is desired to have an isentropic airfl ow achieve a velocity of 550 m/s at a 6-cm-diameter section where the pressure is 87 kPa and the density 1.3 kg/m3 . (a) Is a sonic throat needed? (b) If so, estimate its diameter, and compute (c) the stagnation temperature and (d ) the mass fl ow

A one-dimensional isentropic airfl ow has the following properties at one section where the area is 53 cm2 : p 5 12 kPa, 5 0.182 kg/m3 , and V 5 760 m/s. Determine (a) the throat area, (b) the stagnation temperature, and (c) the mass fl ow

In wind tunnel testing near Mach 1, a small area decrease caused by model blockage can be important. Suppose the test section area is 1 m2 , with unblocked test conditions Ma 5 1.10 and T 5 208C. What model area will fi rst cause the test section to choke? If the model cross section is 0.004 m2 (0.4 percent blockage), what percentage change in test section velocity results?

A force F 5 1100 N pushes a piston of diameter 12 cm through an insulated cylinder containing air at 208C, as in Fig. P9.48. The exit diameter is 3 mm, and pa 5 1 atm. Estimate (a) Ve, (b) Vp, and (c) m # e.

Consider the venturi nozzle of Fig. 6.40c, with D 5 5 cm and d 5 3 cm. Stagnation temperature is 300 K, and the upstream velocity V1 5 72 m/s. If the throat pressure is 124 kPa, estimate, with isentropic fl ow theory, (a) p1, (b) Ma2, and (c) the mass fl ow.

Methane is stored in a tank at 120 kPa and 330 K. It discharges to a second tank through a converging nozzle whose exit area is 5 cm2 . What is the initial mass fl ow rate if the second tank has a pressure of (a) 70 kPa or (b) 40 kPa?

The scramjet engine is supersonic throughout. A sketch is shown in Fig. C9.8. Test the following design. The fl ow enters at Ma 5 7 and air properties for 10,000 m altitude. Inlet area is 1 m2 , the minimum area is 0.1 m2 , and the exit area is 0.8 m2 . If there is no combustion, (a) will the fl ow still be supersonic in the throat? Also, determine (b) the exit Mach number, (c) exit velocity, and (d ) exit pressure

A convergingdiverging nozzle exits smoothly to sealevel standard atmosphere. It is supplied by a 40-m3 tank initially at 800 kPa and 1008C. Assuming isentropic flow in the nozzle, estimate (a) the throat area and (b) the tank pressure after 10 s of operation. The exit area is 10 cm2

Air fl ows steadily from a reservoir at 208C through a nozzle of exit area 20 cm2 and strikes a vertical plate as in Fig. P9.53. The fl ow is subsonic throughout. A force of 135 N is required to hold the plate stationary. Compute (a) Ve, (b) Mae, and (c) p0 if pa 5 101 kPa. P9.53 Air 20C 135 N Plate Ae = 20 cm2

The airfl ow in Prob. P9.46 undergoes a normal shock just past the section where data was given. Determine the (a) Mach number, (b) pressure, and (c) velocity just downstream of the shock.

Air, supplied by a reservoir at 450 kPa, fl ows through a convergingdiverging nozzle whose throat area is 12 cm2 . A normal shock stands where A1 5 20 cm2 . (a) Compute the pressure just downstream of this shock. Still farther downstream, at A3 5 30 cm2 , estimate (b) p3, (c) A*3 , and (d ) Ma3

Air from a reservoir at 208C and 500 kPa fl ows through a duct and forms a normal shock downstream of a throat of area 10 cm2 . By an odd coincidence it is found that the stagnation pressure downstream of this shock exactly equals the throat pressure. What is the area where the shock wave stands?

Air fl ows from a tank through a nozzle into the standard atmosphere, as in Fig. P9.57. A normal shock stands in the exit of the nozzle, as shown. Estimate (a) the pressure in the tank and (b) the mass fl ow. P9.57 10 cm2 14 cm2 Sea-level air Air at 100C Shock

Downstream of a normal shock wave, in airfl ow, the conditions are T2 5 603 K, V2 5 222 m/s, and p2 5 900 kPa. Estimate the following conditions just upstream of the shock: (a) Ma1; (b) T1; (c) p1; (d ) po1; and (e) To1.

Air, at stagnation conditions of 450 K and 250 kPa, fl ows through a nozzle. At section 1, where the area is 15 cm2 , there is a normal shock wave. If the mass fl ow is 0.4 kg/s, estimate (a) the Mach number and (b) the stagnation pressure just downstream of the shock.

When a pitot tube such as in Fig. 6.30 is placed in a supersonic fl ow, a normal shock will stand in front of the probe. Suppose the probe reads p0 5 190 kPa and p 5 150 kPa. If the stagnation temperature is 400 K, estimate the (supersonic) Mach number and velocity upstream of the shock

Air fl ows from a large tank, where T 5 376 K and p 5 360 kPa, to a design condition where the pressure is 9800 Pa. The mass fl ow is 0.9 kg/s. However, there is a normal shock in the exit plane just after this condition is reached. Estimate (a) the throat area and, just downstream of the shock, (b) the Mach number, (c) the temperature, and (d ) the pressure

An atomic explosion propagates into still air at 14.7 lbf/in2 absolute and 5208R. The pressure just inside the shock is 5000 lbf/in2 absolute. Assuming k 5 1.4, what are the speed C of the shock and the velocity V just inside the shock?

Sea-level standard air is sucked into a vacuum tank through a nozzle, as in Fig. P9.63. A normal shock stands where the nozzle area is 2 cm2 , as shown. Estimate (a) the pressure in the tank and (b) the mass fl ow.

Air, from a reservoir at 350 K and 500 kPa, fl ows through a convergingdiverging nozzle. The throat area is 3 cm2 . A normal shock appears, for which the downstream Mach number is 0.6405. (a) What is the area where the shock appears? Calculate (b) the pressure and (c) the temperature downstream of the shock.

Air fl ows through a convergingdiverging nozzle between two large reservoirs, as shown in Fig. P9.65. A mercury manometer between the throat and the downstream reservoir reads h 5 15 cm. Estimate the downstream reservoir pressure. Is there a normal shock in the fl ow? If so, does it stand in the exit plane or farther upstream? 100C 300 kPa At = 10 cm2 h Mercury Ae = 30 cm

In Prob. P9.65 what would be the mercury manometer reading h if the nozzle were operating exactly at supersonic design conditions?

A supply tank at 500 kPa and 400 K feeds air to a converging diverging nozzle whose throat area is 9 cm2 . The exit area is 46 cm2 . State the conditions in the nozzle if the pressure outside the exit plane is (a) 400 kPa, (b) 120 kPa, and (c) 9 kPa. (d ) In each of these cases, fi nd the mass fl ow.

Air in a tank at 120 kPa and 300 K exhausts to the atmosphere through a 5-cm2 -throat converging nozzle at a rate of 0.12 kg/s. What is the atmospheric pressure? What is the maximum mass fl ow possible at low atmospheric pressure?

With reference to Prob. P3.68, show that the thrust of a rocket engine exhausting into a vacuum is given by F 5 p0Ae(1 1 k Mae 2 ) a1 1 k 2 1 2 Mae 2 b k/(k21) where Ae 5 exit area Mae 5 exit Mach number p0 5 stagnation pressure in combustion chamber Note that stagnation temperature does not enter into the thrust.

Air, with po 5 500 kPa and To 5 600 K, fl ows through a convergingdiverging nozzle. The exit area is 51.2 cm2 , and mass fl ow is 0.825 kg/s. What is the highest possible back pressure that will still maintain supersonic fl ow inside the diverging section?

A converging-diverging nozzle has a throat area of 10 cm2 and an exit area of 28.96 cm2 . A normal shock stands in the exit when the back pressure is sea-level standard. If the upstream tank temperature is 400 K, estimate (a) the tank pressure and (b) the mass fl ow.

A large tank at 500 K and 165 kPa feeds air to a converging nozzle. The back pressure outside the nozzle exit is sealevel standard. What is the appropriate exit diameter if the desired mass fl ow is 72 kg/h?

Air fl ows isentropically in a convergingdiverging nozzle with a throat area of 3 cm2 . At section 1, the pressure is 101 kPa, the temperature is 300 K, and the velocity is 868 m/s. (a) Is the nozzle choked? Determine (b) A1 and (c) the mass fl ow. Suppose, without changing stagnation conditions or A1, the (fl exible) throat is reduced to 2 cm2 . Assuming shock-free fl ow, will there be any change in the gas properties at section 1? If so, compute new p1, V1, and T1 and explain

Use your strategic ideas, from part (b) of Prob. P9.40, to actually carry out the calculations for mass fl ow of steam, with po 5 300 kPa and To 5 600 K, discharging through a converging nozzle of choked exit area 5 cm2 .

A double-tank system in Fig. P9.75 has two identical converging nozzles of 1-in2 throat area. Tank 1 is very large, and tank 2 is small enough to be in steady-fl ow equilibrium with the jet from tank 1. Nozzle fl ow is isentropic, but entropy changes between 1 and 3 due to jet dissipation in tank 2. Compute the mass fl ow. (If you give up, Ref. 9, pp. 288290, has a good discussion.) P9.75 Air 100 lbf/in2 abs 520 R 10 lbf/in2 abs 1 23

A large reservoir at 208C and 800 kPa is used to fi ll a small insulated tank through a convergingdiverging nozzle with 1-cm2 throat area and 1.66-cm2 exit area. The small tank has a volume of 1 m3 and is initially at 208C and 100 kPa. Problems 671 Estimate the elapsed time when (a) shock waves begin to appear inside the nozzle and (b) the mass fl ow begins to drop below its maximum value

A perfect gas (not air) expands isentropically through a supersonic nozzle with an exit area 5 times its throat area. The exit Mach number is 3.8. What is the specifi c-heat ratio of the gas? What might this gas be? If p0 5 300 kPa, what is the exit pressure of the gas?

The orientation of a hole can make a difference. Consider holes A and B in Fig. P9.78, which are identical but reversed. For the given air properties on either side, compute the mass fl ow through each hole and explain why they are different. 0.2 cm2 0.3 cm2 p1 = 150 kPa, T1 = 20C p2 = 100 kPa A B mB m ?

A large tank, at 400 kPa and 450 K, supplies air to a converging-diverging nozzle of throat area 4 cm2 and exit area 5 cm2 . For what range of back pressures will the fl ow (a) be entirely subsonic; (b) have a shock wave inside the nozzle; (c) have oblique shocks outside the exit; and (d ) have supersonic expansion waves outside the exit?

A sea-level automobile tire is initially at 32 lbf/in2 gage pressure and 758F. When it is punctured with a hole that resembles a converging nozzle, its pressure drops to 15 lbf/in2 gage in 12 min. Estimate the size of the hole, in thousandths of an inch. The tire volume is 2.5 ft2 .

Air, at po 5 160 lbf/in2 and To 5 3008F, fl ows isentropically through a convergingdiverging nozzle. At section 1, where A1 5 288 in2 , the velocity is V1 5 2068 ft/s. Calculate (a) Ma1, (b) A*, (c) p1, and (d ) the mass fl ow, in slug/s

Air at 500 K fl ows through a convergingdiverging nozzle with throat area of 1 cm2 and exit area of 2.7 cm2 . When the mass fl ow is 182.2 kg/h, a pitot-static probe placed in the exit plane reads p0 5 250.6 kPa and p 5 240.1 kPa. Estimate the exit velocity. Is there a normal shock wave in the duct? If so, compute the Mach number just downstream of this shock

When operating at design conditions (smooth exit to sealevel pressure), a rocket engine has a thrust of 1 million lbf. The chamber pressure and temperature are 600 lbf/in2 absolute and 40008R, respectively. The exhaust gases approximate k 5 1.38 with a molecular weight of 26. Estimate (a) the exit Mach number and (b) the throat diameter.

Air fl ows through a duct as in Fig. P9.84, where A1 5 24 cm2 , A2 5 18 cm2 , and A3 5 32 cm2 . A normal shock stands at section 2. Compute (a) the mass fl ow, (b) the Mach number, and (c) the stagnation pressure at section 3. 1 2 3 Air Normal Ma shock 1 = 2.5 p1 = 40 kPa T1 = 30C

A typical carbon dioxide tank for a paintball gun holds about 12 oz of liquid CO2. The tank is fi lled no more than one-third with liquid, which, at room temperature, maintains the gaseous phase at about 850 psia. (a) If a valve is opened that simulates a converging nozzle with an exit diameter of 0.050 in, what mass fl ow and exit velocity results? (b) Repeat the calculations for helium.

Air enters a 3-cm-diameter pipe 15 m long at V1 5 73 m/s, p1 5 550 kPa, and T1 5 608C. The friction factor is 0.018. Compute V2, p2, T2, and p02 at the end of the pipe. How much additional pipe length would cause the exit fl ow to be sonic?

Problem C6.9 gives data for a proposed Alaska-to-Canada natural gas (assume CH4) pipeline. If the design fl ow rate is 890 kg/s and the entrance conditions are 2500 lbf/in2 and 1408F, determine the maximum length of adiabatic pipe before choking occurs.

Air fl ows adiabatically, with f 5 0.024, down a long 6-cm-diameter pipe. At section 1, conditions are T1 5 300 K, p1 5 400 kPa, and V1 5 104 m/s. At section 2, V2 5 233 m/s. (a) How far downstream is section 2? Estimate (b) Ma2, (c) p2, and (d ) T2

Carbon dioxide fl ows through an insulated pipe 25 m long and 8 cm in diameter. The friction factor is 0.025. At the entrance, p 5 300 kPa and T 5 400 K. The mass fl ow is 1.5 kg/s. Estimate the pressure drop by (a) compressible and (b) incompressible (Sec. 6.6) fl ow theory. (c) For what pipe length will the exit fl ow be choked?

Air fl ows through a rough pipe 120 ft long and 3 in in diameter. Entrance conditions are p 5 90 lbf/in2 , T 5 688F, and V 5 225 ft/s. The fl ow chokes at the end of the pipe. (a) What is the average friction factor? (b) What is the pressure at the end of the pipe?

Air fl ows steadily from a tank through the pipe in Fig. P9.91. There is a converging nozzle on the end. If the mass fl ow is 3 kg/s and the nozzle is choked, estimate (a) the Mach number at section 1 and (b) the pressure inside the tank. L = 9 m, D = 6 cm f = 0.025 De = 5 cm Nozzle Pa = 100 kPa

Air enters a 5-cm-diameter pipe at 380 kPa, 3.3 kg/m3 , and 120 m/s. The friction factor is 0.017. Find the pipe length for which the velocity (a) doubles, (b) triples, and (c) quadruples.

Air fl ows adiabatically in a 3-cm-diameter duct, with f 5 0.018. At the entrance, T1 5 323 K, p1 5 200 kPa, and V1 5 72 m/s. (a) What is the mass fl ow? (b) For what tube length will the fl ow choke? (c) If the tube length is increased to 112 m, with the same inlet pressure and temperature, what will be the new mass fl ow?

Compressible pipe fl ow with friction, Sec. 9.7, assumes constant stagnation enthalpy and mass fl ow but variable momentum. Such a fl ow is often called Fanno fl ow, and a line representing all possible property changes on a temperatureentropy chart is called a Fanno line. Assuming a perfect gas with k 5 1.4 and the data of Prob. P9.86, draw a Fanno curve of the fl ow for a range of velocities from very low (Ma ! 1) to very high (Ma @ 1). Comment on the meaning of the maximum-entropy point on this curve

Helium (Table A.4) enters a 5-cm-diameter pipe at p1 5 550 kPa, V1 5 312 m/s, and T1 5 408C. The friction factor is 0.025. If the fl ow is choked, determine (a) the length of the duct and (b) the exit pressure

Methane (CH4) fl ows through an insulated 15-cm-diameter pipe with f 5 0.023. Entrance conditions are 600 kPa, 1008C, and a mass fl ow of 5 kg/s. What lengths of pipe will (a) choke the fl ow, (b) raise the velocity by 50 percent, or (c) decrease the pressure by 50 percent?

By making a few algebraic substitutions, show that Eq. (9.74) may be written in the density form 1 2 5 2 2 1 *2 a 2k k 1 1 fL D 1 2 ln 1 2 b Why is this formula awkward if one is trying to solve for the mass fl ow when the pressures are given at sections 1 and 2?

Compressible laminar fl ow, f < 64/Re, may occur in capillary tubes. Consider air, at stagnation conditions of 1008C and 200 kPa, entering a tube 3 cm long and 0.1 mm in diameter. If the receiver pressure is near vacuum, estimate (a) the average Reynolds number, (b) the Mach number at the entrance, and (c) the mass fl ow in kg/h

A compressor forces air through a smooth pipe 20 m long and 4 cm in diameter, as in Fig. P9.99. The air leaves at 101 kPa and 2008C. The compressor data for pressure rise versus mass fl ow are shown in the fi gure. Using the Moody chart to estimate f , compute the resulting mass fl ow. D = 4 cm L = 20 m 250 kPa p Parabola Te = 200C m m 0.4 kg/s

Natural gas, approximated as CH4, fl ows through a Schedule 40 six-inch pipe from Providence to Narragansett, RI, a distance of 31 miles. Gas companies use the barg as a pressure unit, meaning a bar of pressure gage, above ambient pressure. Assuming isothermal fl ow at 688F, with f < 0.019, estimate the mass fl ow if the pressure is 5 bargs in Providence and 1 barg in Narragansett.

How do the compressible pipe fl ow formulas behave for small pressure drops? Let air at 208C enter a tube of diameter 1 cm and length 3 m. If f 5 0.028 with p1 5 102 kPa and p2 5 100 kPa, estimate the mass fl ow in kg/h for (a) isothermal fl ow, (b) adiabatic fl ow, and (c) incompressible fl ow (Chap. 6) at the entrance density.

How do the compressible pipe fl ow formulas behave for small pressure drops? Let air at 208C enter a tube of diameter 1 cm and length 3 m. If f 5 0.028 with p1 5 102 kPa and p2 5 100 kPa, estimate the mass fl ow in kg/h for (a) isothermal fl ow, (b) adiabatic fl ow, and (c) incompressible fl ow (Chap. 6) at the entrance density.

Natural gas, with k < 1.3 and a molecular weight of 16, is to be pumped through 100 km of 81-cm-diameter pipeline. The downstream pressure is 150 kPa. If the gas enters at 608C, the mass fl ow is 20 kg/s, and f 5 0.024, estimate the required entrance pressure for (a) isothermal fl ow and (b) adiabatic fl ow

A tank of oxygen (Table A.4) at 208C is to supply an astronaut through an umbilical tube 12 m long and 1.5 cm in diameter. The exit pressure in the tube is 40 kPa. If the desired mass fl ow is 90 kg/h and f 5 0.025, what should be the pressure in the tank?

5 Modify Prob. P9.87 as follows: The pipeline will not be allowed to choke. It will have pumping stations about every 200 miles. (a) Find the length of pipe for which the pressure has dropped to 2000 lbf/in2 . (b) What is the temperature at that point?

Air, from a 3 cubic meter tank initially at 300 kPa and 2008C, blows down adiabatically through a smooth pipe 1 cm in diameter and 2.5 m long. Estimate the time required to reduce the tank pressure to 200 kPa. For simplicity, assume constant tank temperature and f < 0.020. t = 0: 200 C 300 kPa 3 m3 pa = 100 kPa

A fuelair mixture, assumed equivalent to air, enters a duct combustion chamber at V1 5 104 m/s and T1 5 300 K. What amount of heat addition in kJ/kg will cause the exit fl ow to be choked? What will be the exit Mach number and temperature if 504 kJ/kg are added during combustion?

What happens to the inlet fl ow of Prob. P9.107 if the combustion yields 1500 kJ/kg heat addition and p01 and T01 remain the same? How much is the mass fl ow reduced?

A jet engine at 7000-m altitude takes in 45 kg/s of air and adds 550 kJ/kg in the combustion chamber. The chamber cross section is 0.5 m2 , and the air enters the chamber at 80 kPa and 58C. After combustion the air expands through an isentropic converging nozzle to exit at atmospheric pressure. Estimate (a) the nozzle throat diameter, (b) the nozzle exit velocity, and (c) the thrust produced by the engine.

Compressible pipe fl ow with heat addition, Sec. 9.8, assumes constant momentum (p 1 V2 ) and constant mass fl ow but variable stagnation enthalpy. Such a fl ow is often called Rayleigh fl ow, and a line representing all possible property changes on a temperatureentropy chart is called a Rayleigh line. Assuming air passing through the fl ow state p1 5 548 kPa, T1 5 588 K, V1 5 266 m/s, and A 5 1 m2 , draw a Rayleigh curve of the fl ow for a range of velocities from very low (Ma ! 1) to very high (Ma @ 1). Comment on the meaning of the maximum-entropy point on this curve.

Add to your Rayleigh line of Prob. P9.110 a Fanno line (see Prob. P9.94) for stagnation enthalpy equal to the value associated with state 1 in Prob. P9.110. The two curves will intersect at state 1, which is subsonic, and at a certain state 2, which is supersonic. Interpret these two states vis--vis Table B.2.

Air enters a duct at V1 5 144 m/s, p1 5 200 kPa, and T1 5 323 K. Assuming frictionless heat addition, estimate (a) the heat addition needed to raise the velocity to 372 m/s; and (b) the pressure at this new section 2

Air enters a constant-area duct at p1 5 90 kPa, V1 5 520 m/s, and T1 5 5588C. It is then cooled with negligible friction until it exits at p2 5 160 kPa. Estimate (a) V2, (b) T2, and (c) the total amount of cooling in kJ/kg.

The scramjet of Fig. C9.8 operates with supersonic fl ow throughout. Assume that the heat addition of 500 kJ/kg, between sections 2 and 3, is frictionless and at constant area of 0.2 m2 . Given Ma2 5 4.0, p2 5 260 kPa, and T2 5 420 K. Assume airfl ow at k 5 1.40. At the combustion section exit, fi nd (a) Ma3, (b) p3, and (c) T3.

5 Air enters a 5-cm-diameter pipe at 380 kPa, 3.3 kg/m3 , and 120 m/s. Assume frictionless fl ow with heat addition. Find the amount of heat addition for which the velocity (a) doubles, (b) triples, and (c) quadruples.

An observer at sea level does not hear an aircraft fl ying at 12,000-ft standard altitude until it is 5 (statute) mi past her. Estimate the aircraft speed in ft/s.

A tiny scratch in the side of a supersonic wind tunnel creates a very weak wave of angle 178, as shown in Fig. P9.117, after which a normal shock occurs. The air temperature in region (1) is 250 K. Estimate the temperature in region (2).

A particle moving at uniform velocity in sea-level standard air creates the two disturbance spheres shown in Fig. P9.118. Compute the particle velocity and Mach number. P9.118 V Particle 3 m 8 m

The particle in Fig. P9.119 is moving supersonically in sea-level standard air. From the two given disturbance spheres, compute the particle Mach number, velocity, and Mach angle. P9.119 V Particle 3 m 8 m 8 m

The particle in Fig. P9.120 is moving in sea-level standard air. From the two disturbance spheres shown, estimate (a) the position of the particle at this instant and (b) the temperature in 8C at the front stagnation point of the particle. P9.120 3 m 6

A thermistor probe, in the shape of a needle parallel to the fl ow, reads a static temperature of 2258C when inserted into a supersonic airstream. A conical disturbance cone of half-angle 178 is created. Estimate (a) the Mach number, (b) the velocity, and (c) the stagnation temperature of the stream.

2 Supersonic air takes a 58 compression turn, as in Fig. P9.122. Compute the downstream pressure and Mach number and the wave angle, and compare with smalldisturbance theory. P9.122 Ma1 = 3 p1 = 100 kPa 5 Ma2, p2

The 108 defl ection in Example 9.17 caused a fi nal Mach number of 1.641 and a pressure ratio of 1.707. Compare this with the case of the fl ow passing through two 58 defl ections. Comment on the results and why they might be higher or lower in the second case.

When a sea-level fl ow approaches a ramp of angle 208, an oblique shock wave forms as in Figure P9.124. Calculate (a) Ma1, (b) p2, (c) T2, and (d ) V2. 40 1 2 20

We saw in the text that, for k 5 1.40, the maximum possible defl ection caused by an oblique shock wave occurs at infi nite approach Mach number and is max 5 45.588. Assuming an ideal gas, what is max for (a) argon and (b) carbon dioxide?

Airfl ow at Ma 5 2.8, p 5 80 kPa, and T 5 280 K undergoes a 158 compression turn. Find the downstream values of (a) Mach number, (b) pressure, and (c) temperature.

Do the Mach waves upstream of an oblique shock wave intersect with the shock? Assuming supersonic downstream fl ow, do the downstream Mach waves intersect the shock? Show that for small defl ections the shock wave angle lies halfway between 1 and 2 1 for any Mach number.

Air fl ows past a two-dimensional wedge-nosed body as in Fig. P9.128. Determine the wedge half-angle for which the horizontal component of the total pressure force on the nose is 35 kN/m of depth into the paper. P9.128 Ma = 3.0 p = 100 kPa 12 cm

Air fl ows at supersonic speed toward a compression ramp, as in Fig. P9.129. A scratch on the wall at point a creates a wave of 308 angle, while the oblique shock created has a 508 angle. What is (a) the ramp angle and (b) the wave angle caused by a scratch at b? P9.129 a b Ma > 1 30 50

A supersonic airfl ow, at a temperature of 300 K, strikes a wedge and is defl ected 128. If the resulting shock wave is attached, and the temperature after the shock is 450 K, (a) estimate the approach Mach number and wave angle. (b) Why are there two solutions?

The following formula has been suggested as an alternate to Eq. (9.86) to relate upstream Mach number to the oblique shock wave angle and turning angle : sin2 5 1 Ma1 2 1 (k 1 1) sin sin 2 cos ( 2 ) Can you prove or disprove this relation? If not, try a few numerical values and compare with the results from Eq. (9.86)

Air fl ows at Ma 5 3 and p 5 10 lbf/in2 absolute toward a wedge of 168 angle at zero incidence in Fig. P9.132. If the pointed edge is forward, what will be the pressure at point A? If the blunt edge is forward, what will be the pressure at point B? P9.132 Ma = 3 p = 10 lbf/in2 abs A B 16 16

Air fl ows supersonically toward the double-wedge system in Fig. P9.133. The (x, y) coordinates of the tips are given. The shock wave of the forward wedge strikes the tip of the aft wedge. Both wedges have 158 defl ection angles. What is the free-stream Mach number? Shocks (1 m, 1 m) (0, 0) Ma P9.133

When an oblique shock strikes a solid wall, it refl ects as a shock of suffi cient strength to cause the exit fl ow Ma3 to be parallel to the wall, as in Fig. P9.134. For airfl ow with Ma1 5 2.5 and p1 5 100 kPa, compute Ma3, p3, and the angle . P9.134 Ma1 = 2.5 Ma2 Ma3 40

A bend in the bottom of a supersonic duct flow induces a shock wave that reflects from the upper wall, as in Fig. P9.135. Compute the Mach number and pressure in region 3. 676 Chapter 9 Compressible Flow Ma1 = 3.0 2 3 10

Figure P9.136 is a special application of Prob. P9.135. With careful design, one can orient the bend on the lower wall so that the refl ected wave is exactly canceled by the return bend, as shown. This is a method of reducing the Mach number in a channel (a supersonic diffuser). If the bend angle is 5 108, fi nd (a) the downstream width h and (b) the downstream Mach number. Assume a weak shock wave. P9.136 1 m Ma = 3.5 Shock Shock

A 68 half-angle wedge creates the refl ected shock system in Fig. P9.137. If Ma3 5 2.5, fi nd (a) Ma1 and (b) the angle . 3 2 1

The supersonic nozzle of Fig. P9.138 is overexpanded (case G of Fig. 9.12b) with Ae/At 5 3.0 and a stagnation pressure of 350 kPa. If the jet edge makes a 48 angle with the nozzle centerline, what is the back pressure pr in kPa?

Airfl ow at Ma 5 2.2 takes a compression turn of 128 and then another turn of angle in Fig. P9.139. What is the maximum value of for the second shock to be attached? Will the two shocks intersect for any less than max? P9.139 Ma1 = 2.2 2 3 12 max

The solution to Prob. P9.122 is Ma2 5 2.750 and p2 5 145.5 kPa. Compare these results with an isentropic compression turn of 58, using Prandtl-Meyer theory

Supersonic airfl ow takes a 58 expansion turn, as in Fig. P9.141. Compute the downstream Mach number and pressure, and compare with small-disturbance theory. P9.141 Ma1 = 3 p1 = 100 kPa 5 Ma2, p

A supersonic airfl ow at Ma1 5 3.2 and p1 5 50 kPa undergoes a compression shock followed by an isentropic expansion turn. The fl ow defl ection is 308 for each turn. Compute Ma2 and p2 if (a) the shock is followed by the expansion and (b) the expansion is followed by the shock

Airfl ow at Ma 5 3.4 and 300 K encounters a 288 oblique shock turn. What subsequent isentropic expansion turn will bring the temperature back to 300 K?

The 108 defl ection in Example 9.17 caused the Mach number to drop to 1.64. (a) What turn angle will create a Prandtl-Meyer fan and bring the Mach number back up to 2.0? (b) What will be the fi nal pressure?

Air at Ma1 5 2.0 and p1 5 100 kPa undergoes an isentropic expansion to a downstream pressure of 50 kPa. What is the desired turn angle in degrees?

Air fl ows supersonically over a surface that changes direction twice, as in Fig. P9.146. Calculate (a) Ma2 and (b) p3. 170 168 Ma1 = 2.0 p1 = 200 kPa p3 Ma2

A convergingdiverging nozzle with a 4:1 exit-area ratio and p0 5 500 kPa operates in an underexpanded condition (case I of Fig. 9.12b) as in Fig. P9.147. The receiver pressure is pa 5 10 kPa, which is less than the exit pressure, so that expansion waves form outside the exit. For the given conditions, what will the Mach number Ma2 and the angle of the edge of the jet be? Assume k 5 1.4 as usual. P9.147 pa = 10 kPa Jet edge Jet edge

Air fl ows supersonically over a circular-arc surface as in Fig. P9.148. Estimate (a) the Mach number Ma2 and (b) the pressure p2 as the fl ow leaves the circular surface. 32 Ma1 = 2.0 p1 = 150 kPa Ma2 p2

Air fl ows at Ma 5 3.0 past a doubly symmetric diamond airfoil whose front and rear included angles are both 248. For zero angle of attack, compute the drag coeffi cient obtained using shock-expansion theory and compare with Ackeret theory.

A fl at-plate airfoil with C 5 1.2 m is to have a lift of 30 kN/m when fl ying at 5000-m standard altitude with U`5 641 m/s. Using Ackeret theory, estimate (a) the angle of attack and (b) the drag force in N/m

Air fl ows at Ma 5 2.5 past a half-wedge airfoil whose angles are 48, as in Fig. P9.151. Compute the lift and drag coeffi cient at equal to (a) 08 and (b) 68. P9.151 Ma = 2.5 4 4

The X-43 model A scramjet aircraft in Fig. C9.8 is small W 5 3000 lbf, and unmanned, only 12.33 ft long and 5.5 ft wide. The aerodynamics of a slender arrowhead-shaped hypersonic vehicle is beyond our scope. Instead, let us assume it is a fl at plate airfoil of area 2.0 m2 . Let Ma 5 7 at 12,000 m standard altitude. Estimate the drag, by shockexpansion theory. Hint: Use Ackeret theory to estimate the angle of attack.

A supersonic transport has a mass of 65 Mg and cruises at 11-km standard altitude at a Mach number of 2.25. If the angle of attack is 28 and its wings can be approximated by fl at plates, estimate (a) the required wing area in m2 and (b) the thrust required in N

The F-22 supersonic fi ghter cruises at 11,000 m altitude, with a weight of 50,000 lbf and thrust of 10,000 lbf. Its wing area is 840 ft2 . Assume the wing is a 6-percent-thick diamond shape and provides all lift and thrust. Use Ackeret theory to estimate the resulting Mach number.

The F-35 airplane in Fig. 9.30 has a wingspan of 10 m and a wing area of 41.8 m2 . It cruises at about 10 km altitude with a gross weight of about 200 kN. At that altitude, the engine develops a thrust of about 50 kN. Assume the wing has a symmetric diamond airfoil with a thickness of 8 percent, and accounts for all lift and drag. Estimate the cruise Mach number of the airplane. For extra credit, explain why there are two solutions.

Consider a fl at-plate airfoil at an angle of attack of 68. The Mach number is Ma` 5 3.2 and the stream pressure p` is unspecifi ed. Calculate the predicted lift and drag coeffi cients by (a) shock-expansion theory and (b) Ackeret theory.

The Ackeret airfoil theory of Eq. (9.104) is meant for moderate supersonic speeds, 1.2 , Ma , 4. How does it fare for hypersonic speeds? To illustrate, calculate (a) CL and (b) CD for a fl at-plate airfoil at a 5 58 and Ma` 5 8.0, using shock-expansion theory, and compare with Ackeret theory. Comment.