Flow rate equation
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12.21: PROBLEM DEFINITION
Situation: Oxygen flows from a reservoir with temperature of 200oCand300kPa,
abs. through conduit with Mach number of 0.9.
Find:(a)Velocity.
(b) Pressure.
(c) Temperature.
Properties:FromTableA.2(EFM10e),k=1.4,R
O2=260J/kg-K
SOLUTION Total properties
Speed of sound
Velocity
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12.22: PROBLEM DEFINITION
Situation: HighMachnumberflow of air from a reservoir at 300 K. Condensation
occurs when T=50 K.
Find: Mach number where condensation will occur.
Properties:FromTableA.2(EFM10e),k=1.4
PLAN Apply total temperature equation setting the Tto 50 K and Ttto 300 K.
SOLUTION Total temperature equation
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12.23: PROBLEM DEFINITION
Situation:Hydrogenflow from a reservoir with temperature 20oC and pressure 500
kPa, abs through a 2-cm diameter duct where velocity is 250 m/s.
Find:
(a) Temperature.
(b) Pressure.
(c) Mach number.
(d) Mass flow rate.
Properties:FromTableA.2(EFM10e),k=1.41,R
H2= 4127 J/kg-K.
SOLUTION Evaluate cpfrom
Stagnation conditions
Energy equation
Speed of sound
Mach number
Total properties (pressure)
Ideal gas law
Flow rate equation
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12.24: PROBLEM DEFINITION
Situation:A3cm diameter sphere in a Mach-2.5 wind tunnel with total pressure of
547 kPa,abshasdragcoefficient of 0.95.
Find: Drag on the sphere.
Properties:FromTableA.2(EFM10e),k=1.4
SOLUTION Find static pressure and then dynamic pressure.
Drag force
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12.25: PROBLEM DEFINITION
Situation: Application of equation for total pressure to calculate pressure coefficient,
Cp=(pt−p)/(1
2ρV 2)
Find: (a) Expression for pressure coefficient.
(b) Values for pressure coefficient
SOLUTION
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12.26: PROBLEM DEFINITION
Situation: With low velocities, the pressure ratio can be approximated with pt/p =
1+ε.
Find: Show that Mach number goes to zero as goes to zero, and that Eq. (12.31)
in EFM 10e reduces to M=[(2/k)(pt/p −1)]1/2
SOLUTION
Neglecting higher order terms
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12.27: PROBLEM DEFINITION
Situation:Trueandfalsestatementsconcerningshockwaves.
SOLUTION Shock waves occur only in supersonic flows (T) Static pressure increases
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12.28: PROBLEM DEFINITION
Situation: Occurrence or normal shock waves in subsonic flows
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12.29: PROBLEM DEFINITION
Situation: A normal shock wave occurs in 500 m/s stream of nitrogen with static
pressure of 70 kPa and static temperature of -50oC.
Find:(a)Machnumber.
(b) Pressure downstream of wave.
(c) Temperature downstream of wave.
(d) Entropy increase.
Properties:FromTableA.2(EFM10e),k=1.41,R
N2= 297 J/kg-K.
SOLUTION Speed of sound
Mach number
Normal shock wave (Mach number)
Normal shock wave (Pressure ratio)
Temperature ratio
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Entropy increase
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12.30: PROBLEM DEFINITION
Situation: A normal shock wave occurs in a Mach 3flow of air with static temperature
of 35 ◦Fand static pressure of 30 psia.
Find: (a) Mach number downstream of shock wave.
(b) Pressure downstream of shock wave.
(c) Temperature downstream of shock wave.
Properties:FromTableA.2(EFM10e),k=1.4.
SOLUTION
Mach number (downstream)
Temperature ratio
Pressure ratio
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12.31: PROBLEM DEFINITION
Situation: Pitot-static tube used to measure Mach number in supersonic flow. Total
pressure downstream of shock is 150 kPa and static pressure upstream of shock is 40
kPa..
Find:Machnumber.
Properties:FromTableA.2(EFM10e),k=1.41.
PLAN Find pressure ratios and apply the compressible flow tables.
SOLUTION
Using compressible flow tables, Table A.1 (EFM 10e),:
p
t2
p
1
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12.32: PROBLEM DEFINITION
Situation: A shock wave occurs in a Mach 3 stream of methane where static pressure
is 89 kPa and temperature is 20oC.
Find:(a)ThedownstreamMachnumber.
(b) Static pressure.
(c) Static temperature.
(d) Density.
Properties:FromTableA.2(EFM10e),k=1.31,R
methane=518 J/kg-K.
PLAN Apply the normal shock wave equations to find Mach number, pressure, and
temperature. Apply the ideal gas law to find density.
SOLUTION Normal shock wave
Mach number
Pressure ratio
Temperature ratio
Ideal gas law
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12.33: PROBLEM DEFINITION
Situation: A shock wave occurs in helium where downstream Mach number is 0.85
and static temperature is 110oC.
Find:Velocityupstreamofwave
Properties:FromTableA.2(EFM10e), k=1.66; RHe =2,077 J/kg-K.
SOLUTION Normal shock wave
Mach number
Temperature ratio
Speed of sound
Velocity upstream
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12.34: PROBLEM DEFINITION
Situation: Expressions for minimum Mach number and largest density downstream
of normal shock wave.
Find: (a) Lowest Mach number possible downstream of shock wave
(b) Largest density ratio possible
(c) Limiting values of M2and ρ2/ρ1for air.
SOLUTION
Because
So in limit
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12.35: PROBLEM DEFINITION
Situation: Mach number downstream of weak shock wave.
Find: (a) Approximation for Mach number downstream of wave.
(b) Compare M2computed with equation from (a) with values in Table A.1 (EFM
10e) for M1=1,1.05,1.1,and 1.2.
SOLUTION
M1M2M2(Table A.1)
1.0 1.0 1.0
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12.36: PROBLEM DEFINITION
Situation: Meaning of back pressure.
SOLUTION Back pressure is the pressure of the surroundings or environment to
which an nozzle exhausts.
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12.37: PROBLEM DEFINITION
Situation: Computer program for flow through a truncated nozzle. Inputs: total
pressure, total temperature, back pressure, ratio of specific heats, gas constant, and
nozzle diameter.
Find: (a) Develop a computer program for calculating the mass flow.
(b) Compare program with Example 12.12 with back pressures of 80,90,100,110,
120,and 130 kPa and make a table.
SOLUTION The computer program shows the flow is subsonic at the exit and the
Back pressure, kPa Flow rate, kg/s
80 0.243
REVIEW
One notes that the mass flow rate begins to decrease more quickly as the back pressure
approaches the total pressure.
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