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09/10/2019 Dr. Munzer Ebaid 1
CHAPTER (12)
COMPRESSIBLE FLOW
DR. MUNZER EBAID
MECH. DEPT.
SUMMARY
09/10/2019 Dr. Munzer Ebaid 2
Application of Compressible Flows
a) High speed flight through air. (Concorde plane)
b) Flow of air through a Compressor.
c) Flow of steam through a Turbine.
d) Natural gas piped from producer to consumer.
e) Compressed gas tanks.
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Speed of Sound (Sonic Velocity)
Section of a Pressure Wave Propagate at a Velocity (c)
2. The process of compression is like a wave traveling through the medium is called sonic velocity.
1. The process of compression of gases layers at a finite velocity is called sonic velocity.
Definition of Speed of Sound
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Sound Waves by definition can be regarded as:
1) Reversible.
2) Adiabatic.
Reversible and Adiabatic means Isentropic Process, therefore
kRTc
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Mach Number
Mach Number
1) M < 1 Subsonic Flow:
2) M = 1 Transonic Flow or Sonic flow:
3) M > 1 Supersonic Flow: V > c
V < c V = c
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Mach Number Relationships Summary
2
2
11 Mk
TTt(1) Temperature
(2) Pressure
(3) Density
Tables can be used to find the above properties
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Total Temperature at upstream and downstream are the same across
a shock wave as the process is Adiabatic.
Total Pressure at upstream and downstream are not the same
across a shock wave in an Adiabatic flow.
Total Density at upstream and downstream are not the same
across a shock wave in an Adiabatic flow.
Total Pressure, Total Density and Total Temperature at upstream and downstream are the same
across a shock wave in an Isentropic flow.
Isentropic Flow
Adiabatic Flow
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Kinetic Pressure
By definition
222
22
1
2
1PMk
kRT
VkP
RT
PVq
Validation of Bernoulli's Equation for Compressible Flow
Thus applying Bernoulli equation would have led one to say that the flow was Supersonic, whereas the flow was actually
Subsonic.
In the limit of low velocities the Bernoulli
equation is valid for very low (M<<1) Mach numbers 1ppt
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Normal ShockWaves (NSW)
(NSW) are wave fronts normal to the flow across which a supersonic flow (M>1) is decelerated to subsonic flow (M<1) with an increase in static temp., pressure and density.
Control volume enclosing a normal shock wave in
Inviscid, Adiabatic, Steady Flow
NSW 1 2
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For means we have a Sound Wave and in this case, no change
in pressure and temperature as the change by definition is
Infinitesimal.
121 MM
Values of Mach Numbers can also be found from tables
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Total pressure at upstream and downstream
Total Temperature at upstream and downstream are the same across
a shock wave as the process is Adiabatic.
Total Pressure at upstream and downstream are not the same
across a shock wave in an Adiabatic flow.
Total Density at upstream and downstream are not the same
across a shock wave in an Adiabatic flow.
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(1) Subsonic Flow (M<1)
negativeisM 12
.sin,.,sin gdecreaisdx
dVresultthepositiveis
dx
dAeigincreais
dx
dAWhen
.sin,.,sin gincreaisdx
dVresultthenegativeis
dx
dAeigdecreais
dx
dAWhen
(2) Supersonic Flow (M>1)
positiveisM 12
.sin,.,sin gdecreaisdx
dVresultthenegativeis
dx
dAeigdecreais
dx
dAWhen
.sin,.,sin gincreaisdx
dVresultthepositiveis
dx
dAeigincreais
dx
dAWhen
(1)
(1)
(2)
(2)
12 VV
12 VV
12 VV
12 VV
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(3) Transonic Flow
(1) Subsonic Flow (M<1) Velocity at the throat is maximum
(2) Supersonic Flow (M>1) Velocity at the throat is minimum
1M
At (a): Sonic conditions can be attained for both M<1 & M>1
At (b): Sonic conditions can not be attained for both M<1 & M>1
Area is Min.
Area is Max.
M<1
M>1
M<1
M>1
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Laval Nozzle (Swedish Engineer)
Flow in Laval nozzle is assumed to be Isentropic, hence Total Temperature, Total Pressure and Total Density are constant throughout the nozzle.
The Laval nozzle is a duct of varying area that produces Supersonic Flow
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Area Ratio in Laval Nozzle
Area Ratio in Laval Nozzle can be found from tables for both subsonic and supersonic flows.
Mass Flow Rate Through Laval Nozzle
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Distribution of Static Pressure and Mach Number in a Laval nozzle
Classification of Nozzle Flow by Exit Condition
M<1 M=1
M>1
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The Static Pressure to Total Pressure at the Throat is given as
Critical Pressure ratio=
Critical pressure ratio at the throat
M=1
Throat
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Under Expanded Slightly Over Expanded Over Expanded
Classification of Nozzle Flow by Exit Conditions
Back Pressure is the pressure outside the Nozzle
Back Pressure = Exit Pressure (Ideally Expanded)
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Example (12.11)
Air
M>1 M<1
?exitP
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Mass Flow through a Truncated Nozzle
A Truncated Nozzle is a Laval Nozzle cut off at the throat as shown in
the figure.
It is important to engineers because it is used as a Flow metering device
for Compressible flow.
Truncated Nozzle
Throat
09/10/2019 Dr. Munzer Ebaid 21
Case (1):
exitatconditionssonicthatleadsMMppthatmeansp
p
p
pIf bb
tt
b 1
Case (2):
exitatconditionsSubsonicthatleadsMMppthatmeansp
p
p
pIf bb
tt
b 1
09/10/2019 Dr. Munzer Ebaid 22
END OF SUMMARY