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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 ms, p1 = 550 kPa, and T1 = 60C.
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 (fLD)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 140F, 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 20C, 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
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/kgK. 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 = 68F, 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 kgs 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 = To1.2 = 3731.2 = 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 ms, p1 = 550 kPa, and T1 = 60C.
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 ms, p1 = 550 kPa, and
T1 = 60C = 333 K, with f 0.018. We can then easily compute Ma1 0.20,
1 = 5.76 kgm3, Vmax = 822 ms, 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 ms, and T1 = 40C.
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 JkgK. 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, 100C, 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/kgK. 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
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 64Re, may occur in capillary tubes. Consider air, at stagnation
conditions of 100C 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 200C. 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 20C
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 kgh 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 100C 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 60C, the
mass flow is 20 kgs, and
=0.024f
, estimate the required entrance pressure for (a) isothermal
flow and (b) adiabatic flow.
Solution 9.103
The gas constant is Rgas = 831416 520 JkgK. 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 20C 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/kgK. Given To = 293 K and
fL/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?
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 140F, 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 20C, 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 9.105
From Prob. P9.87, we found the following data. For CH4, from Table A.4, R = 518 m2/s2-K,
k = 1.32, and
= 1.03E-5 kg/m-s. Convert to SI units: p1 = 2500 psi = 1.72E7 Pa,
