Aeronautical Engineering Chapter 9 Homework Compute The Mach Number Section

Air flows through a duct as in Fig. P9.84, where A1 = 24 cm2, A2 = 18 cm2, and A3 = 32 cm2. A

normal shock stands at section 2. Compute (a) the mass flow, (b) the Mach number, and (c) the

stagnation pressure at section 3.

Solution 9.84

We have enough information at section 1 to compute the mass flow:

Problem 9.85

A typical carbon dioxide tank for a paintball gun holds about 12 oz of liquid CO2. The tank is

filled 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 flow and exit velocity result? (b) Repeat the calculation for

helium.

Solution 9.85

For CO2, from Table A.4, R = 189 J/kg-K and k = 1.30. By “room temperature” we assume

Problem 9.86

Air enters a 3–cm diameter pipe 15 m long at V1 = 73 ms, p1 = 550 kPa, and T1 = 60C.

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 flow to be sonic?

Solution 9.86

First compute the inlet Mach number and then get (fLD)1:

Problem 9.87

Problem C6.9 gives data for a proposed Alaska-to-Canada natural gas (assume CH4) pipeline. If the

design flow rate is 890 kg/s and the entrance conditions are 2500 lbf/in2 and 140F, determine the

maximum length of adiabatic pipe before choking occurs.

Problem 6.C9

A pipeline has been proposed to carry natural gas 1715 miles from Alaska’s North Slope to

Calgary, Alberta, Canada. The (assumed smooth) pipe diameter will be 52 inches. The gas will

be at high pressure, averaging 2500 lbs/in2. (a) Why? The proposed flow rate is 4 billion cubic

feet per day at sea-level conditions. (b) What volume flow rate, at 20C, would carry the same

mass at the high pressure? (c) If natural gas is assumed to be methane (CH4), what is the total

pressure drop? (d) If each pumping station can deliver 12,000 hp to the flow, how many stations

are needed?

Solution 9.87

For CH4, from Table A.4, R = 518 m2/s2-K, k = 1.32, and



= 1.03E-5 kg/m-s. Convert to SI

Problem 9.88

Air flows adiabatically, with

f

= 0.024, down a long 6-cm-diameter pipe. At section 1,

conditions are T1 = 300 K, p1 = 400 kPa, and V1 = 104 m/s. At section 2, V2 = 233 m/s.

(a) How far downstream is section 2? Estimate (b) Ma2 ; (c) p2 and (d) T2.

Solution 9.88

First establish the Mach number at section 1:

1

104 104 0.30

347

1.4(287)(300)

V

Ma kRT

= = = =

Problem 9.89

Carbon dioxide flows through an insulated pipe 25 m long and 8 cm in diameter. The friction

factor is 0.025. At the entrance, p = 300 kPa and T = 400 K. The mass flow is 1.5 kg/s. Estimate

the pressure drop by (a) compressible; and (b) incompressible (Sect. 6.6) flow theory. (c) For what

pipe length will the exit flow be choked?

Solution 9.89

For CO2, from Table A.4, take k = 1.30 and R = 189 J/kgK. Tough calculation, no appendix

tables for CO2, should probably use the Gas Tables. Find inlet density, velocity, Mach number:

Problem 9.90

Air flows through a rough pipe 120 ft long and 3 inches in diameter. Entrance conditions are

p = 90 lbf/in2, T = 68F, and V = 225 ft/s. The flow 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?

Solution 9.90

Find the Mach number at the entrance:

Problem 9.91

Air flows steadily from a tank through the pipe in Fig. P9.91. There is a converging nozzle on

the end. If the mass flow is 3 kgs and the flow is choked, estimate (a) the Mach number at

section 1; and (b) the pressure inside the tank.

Solution 9.91

For adiabatic flow, T* = constant = To1.2 = 3731.2 = 311 K. The flow chokes in the small exit

nozzle, D = 5 cm. Then we estimate Ma2 from isentropic theory:

Problem 9.92

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.

Solution 9.92

First find the conditions at the entrance, which we will call section 1:

Problem 9.93

Air flows adiabatically in a 3-cm-diameter duct, with

0.018.f=

At the entrance, T1 = 323 K,

p1 = 200 kPa, and V1 = 72 m/s. (a) What is the mass flow? (b) For what tube length will the

flow choke? (c) If the tube length is increased to 112 m, with the same inlet pressure and

temperature, what will be the new mass flow?

Solution 9.93

(a) We have enough inlet information to calculate the mass flow:

Problem 9.94

Compressible pipe flow with friction, Sec. 9.7, assumes constant stagnation enthalpy and mass

flow but variable momentum. Such a flow is often called Fanno flow, and a line representing all

possible property changes on a temperature-entropy chart is called a Fanno line. Assuming a

perfect gas with k = 1.4 and the data of Prob. 9.86, draw a Fanno curve of the flow 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.

Problem 9.86

Air enters a 3–cm diameter pipe 15 m long at V1 = 73 ms, p1 = 550 kPa, and T1 = 60C.

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 flow to be sonic?

Solution 9.94

Recall from Prob. 9.86 that, at Section 1 of the pipe, V1 = 73 ms, p1 = 550 kPa, and

T1 = 60C = 333 K, with f  0.018. We can then easily compute Ma1  0.20,



1 = 5.76 kgm3, Vmax = 822 ms, and To = 336 K. Our basic algebraic equations are:

Problem 9.95

Helium (Table A.4) enters a 5-cm-diameter pipe at p1 = 550 kPa, V1 = 312 ms, and T1 = 40C.

The friction factor is 0.025. If the flow is choked, determine (a) the length of the duct and (b) the

exit pressure.

Solution 9.95

For helium, take k = 1.66 and R = 2077 JkgK. We have no tables for

k = 1.66, have to do our best anyway. Compute the Mach number at section 1:

Problem 9.96

Methane (CH4) flows through an insulated 15-cm-diameter pipe with f = 0.023. Entrance

conditions are 600 kPa, 100C, and a mass flow of 5 kg/s. What lengths of pipe will (a) choke

the flow; (b) raise the velocity by 50 percent; (c) decrease the pressure by 50 percent?

Solution 9.96

For methane (CH4), from Table A.4, take k = 1.32 and R = 518 J/kgK. Tough calculation, no

appendix tables for methane, should probably use Excel. Find inlet density, velocity, Mach

number:

Problem 9.97

By making a few algebraic substitutions, show that Eq. (9.74) may be written in the density form

2 2 2 1

12

2

* 2ln

1

k fL

kD



   



= + +



+



Why is this formula awkward if one is trying to solve for the mass flow when the pressures are

given at sections 1 and 2?

Solution 9.97

This much less laborious algebraic derivation is left as a student exercise. There are two awkward

bits: (1) we don’t know



1 and



2; and (2) we don’t know



* either, and preliminary computations

are necessary.

Problem 9.98

Compressible laminar flow, f  64Re, may occur in capillary tubes. Consider air, at stagnation

conditions of 100C and 200 kPa, entering a tube 3 cm long and 0.1 mm in diameter. If the

Problem 9.99

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 200C. The compressor data for pressure rise versus mass flow are

shown in the figure. Using the Moody chart to estimate

,f

compute the resulting mass flow.

Solution 9.99

The compressor performance is approximate by the parabolic relation

2

compressor

p 250 1563 m , with p in kPa and m in kg/s  − 

Problem 9.100

Natural gas, approximated as CH4, flows 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 flow at 68ºF, with

f ≈ 0.019, estimate the mass flow if the pressure is 5 bargs in Providence and 1 barg in

Narragansett.

Solution 9.100

For CH4, from Table A.4, R = 518 m2/s2 and k = 1.32. Both towns are at sea-level, ambient

Problem 9.101

How do the compressible-pipe-flow formulas behave for small pressure drops? Let air at 20C

enter a tube of diameter 1 cm and length 3 m. If

=0.028f

with p1 = 102 kPa and p2 = 100 kPa,

estimate the mass flow in kgh for (a) isothermal flow, (b) adiabatic flow, and (c) incompressible

flow (Chap. 6) at the entrance density.

Solution 9.101

For a pressure change of only 2%, all three estimates are nearly the same. Begin by noting that

Problem 9.102

Air at 550 kPa and 100C enters a smooth 1-m-long pipe and then passes through a second

smooth pipe to a 30-kPa reservoir, as in Fig. P9.102. Using the Moody chart to compute f,

estimate the mass flow through this system. Is the flow choked?

Solution 9.102

Label the pipes “A” and “B” as shown. Given (L/D)A = 20 and (L/D)B = 40. Label the relevant

sections 1, 2, 3, 4 as shown. With po1/pe = 550/30 = 18.3, these short pipes are sure to be

Problem 9.103

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 60C, the

mass flow is 20 kgs, and

=0.024f

, estimate the required entrance pressure for (a) isothermal

flow and (b) adiabatic flow.

Solution 9.103

The gas constant is Rgas = 831416  520 JkgK. First use Eq. 9.73:

2

223

12

212

pp

m 20

(a) Isothermal: A RT[fL/D 2ln(p /p )]

( /4)(0.81)



 −



==



 +

 

Problem 9.104

A tank of oxygen (Table A.4) at 20C 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 flow is

90 kg/h and f = 0.025, what should be the air pressure in the tank?

Solution 9.104

For oxygen, from Table A.4, take k = 1.40 and R = 260 J/kgK. Given To = 293 K and

fL/D = (0.025)(12 m)/(0.015 m) = 20. Use isothermal flow, Eq. (9.73), as a first estimate:

Problem 9.105

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?