Compressible Flow
CBE 150A – Transport
Spring Semester 2014
Goals
• Describe how compressible flow differs from
incompressible flow
• Define criteria for situations in which compressible flow
can be treated as incompressible
• Provide example of situation in which compressibility
cannot be neglected
• Write basic equations for compressible flow
• Describe a shape in which a compressible fluid can be
accelerated to velocities above speed of sound
(supersonic flow)
CBE 150A – Transport
Spring Semester 2014
Basic Equations
Five changeable quantities are important in
compressible flow:
1. Cross-sectional area, S
2. Velocity, u
3. Pressure, p
4. Density, r
5. Temperature, T
CBE 150A – Transport
Spring Semester 2014
Basic Equations
Restrict focus to those systems in which properties
are only changing in flow direction.
Generally, cross-sectional area S is specified as a
function of x. (S=S(x))
Need four equations to describe the other four
variables.
CBE 150A – Transport
Spring Semester 2014
Basic Equations
1. Mass Balance relates r, u, S
2. Mechanical Energy Balance relates r, u, S, p
3. Equation of State relates T, p, r
4. Total Energy Balance relates Q, T
What is different about compressible flow?
r, u, p all change with position.
Need to use differential form of equations.
CBE 150A – Transport
Spring Semester 2014
Mass Balance
m  r uS  constant
In differential form
d r uS   r udS  r Sdu  uSdr
Divide both sides by ruS
d  m  dS du d r



0
m
S
u
r
CBE 150A – Transport
Spring Semester 2014
Mechanical Energy Balance
p
2





u
dp
Wˆ  
 gZ       h f
 2
 p r 
2
1
Differentiate and assume Ŵ = 0
 u2
 d 
 2
CBE 150A – Transport

dp
  gdz 
 dh f  0
r

Spring Semester 2014
Viscous Dissipation
2
4f L u
hf 
D 2
Assumes only
wall shear (no
fittings)
For a short section of pipe:
2
4 f dL u
dh f 
D 2
2
 u2 
dp
f dL u
 d    gdz 
4
0
r
D 2
 2 
CBE 150A – Transport
Spring Semester 2014
Equation of State
pV  zRT
M PM
r

V
zRT
For simplicity it is assumed that z is either 1 (ideal) or a
constant
Volume:
dp dV dT


0
p V
T
Density:
dp dr dT


0
p
r T
CBE 150A – Transport
Spring Semester 2014
Total Energy Balance
For gases thermodynamics allows a better calculation of
the heat transfer Q and changes in internal energy. These
were terms that were previously included in the viscous
dissipation term.
The temperature of a flowing gas depends on:
• Rate of heat transfer Q from environment.
• Rate of viscous dissipation (significant in compressors).
Included in work term Ŵc
• Thermodynamic changes H.
CBE 150A – Transport
Spring Semester 2014
Total Energy Balance
 u2
 Q ˆ

 gZ  H    Wc
 2
 m
Q is the rate of heat addition along the entire length of the
channel and Ŵc is the total rate of energy input into the
system and includes efficiency to account for viscous
dissipation.
For Ŵc to be in the correct units use:
1 B T U  7 7 8 f t  lb f
CBE 150A – Transport
Spring Semester 2014
Compressible vs. Incompressible
When can simpler incompressible equations be used?
• Density change is not significant (<10%)
• Fans, airflow through packed beds
Mach number is a measure of the importance of density
changes for compressible fluids.
N Ma 
velocity fluid
velocitysound
Rule of Thumb: NMa < 0.3 assume incompressible
CBE 150A – Transport
Spring Semester 2014
Isentropic Flow
Adiabatic (Q = 0) and Reversible
Isentropic (ΔS = 0)
Venturi meter, Rocket propulsion
CBE 150A – Transport
Spring Semester 2014
Adiabatic Flow
Adiabatic (Q = 0), Frictional
Mathematically more difficult
Short Insulated Pipes
CBE 150A – Transport
Spring Semester 2014
Isothermal Flow
Isothermal, Frictional
Long Uninsulated Pipes
CBE 150A – Transport
Spring Semester 2014
Compressible Flow Through Pipes
CBE 150A – Transport
Spring Semester 2014
Goals
• Describe equations useful for analyzing isothermal,
compressible flow through a constant diameter pipe.
• Describe how Mach number and L are related for flow
in a constant diameter pipe.
• Use equations for isothermal flow to compute the flow
rate of compressible fluids in constant diameter pipes.
CBE 150A – Transport
Spring Semester 2014
Isothermal Flow
Constant Diameter Pipe
P1, r1
P2, r2
Goal is to analyze the friction section. Flow through
pipes is irreversible so viscous dissipation is important.
CBE 150A – Transport
Spring Semester 2014
Mass Balance
r uS 1  r uS 2
S is constant
r u 1  r u 2
G1  G2
Mass velocity constant
Differential Balance
1 dr 1 du

0
r dx u dx
CBE 150A – Transport
Spring Semester 2014
Mechanical Energy Balance
 du   dz  1  dp   dh f
 u   g       
 dx   dx  r  dx   dx
turbulent
horizontal
 ˆ
  W

no compressor
 du  1  dp  4 f u
u     
0
 dx  r  dx  D 2
2
CBE 150A – Transport
Spring Semester 2014
Total Energy Balance
du
dz
dT  dQ ˆ

m  u
g
 Cp
 Wc

dx
dx
dx  dx

turbulent
horizontal isothermal
no work
du 1 dQ
u

dx m dx
Note: This indicates that there must be heat transfer
because dT = 0. This is the heat required to keep T
constant.
CBE 150A – Transport
Spring Semester 2014
Equation of State
1 dp 1 dr 1 dT


0
p dx r dx T dx
isothermal
1 dp 1 dr

0
p dx r dx
CBE 150A – Transport
Spring Semester 2014
Isothermal Flow
Combining Mass, MEB and EOS

p2
p1
dp
r1

2
p
p1G

p2
p1
2f
p dp 
D

L
0
dx  0
Assume friction factor f is constant and integrate:
 p2 
L
r1
2
2
4f
 p1  p2
 ln  
2
D
p1G
 p1 

CBE 150A – Transport

2
Spring Semester 2014
Constant f ?
G  r u  constant
T  constant
Re 
Dr u

 constant
CBE 150A – Transport
  constant
f  constant
Spring Semester 2014
Isothermal Flow
 p2 
L
r1
2
2
4f
 p1  p2
 ln  
2
D
p1G
 p1 

P1, r1
CBE 150A – Transport

2
P2, r2
Spring Semester 2014
Isothermal Flow


M
2
2
p1  p2
G 2  zRT
2
 p2 
L
4 f  ln  
D
 p1 
For a fixed P1 this expression has a maximum at:
G
CBE 150A – Transport
2
max

r1 p1
 p1 r1 
4f L
1
 ln  2 
D
 G max 
Spring Semester 2014
Maximum Flow
Gmax  r2 p2
umax 
p2
r2

zRT  u
S ,T
M
Ernst Mach (1838-1916)
Thus for a constant cross-section pipe the maximum obtainable velocity is Mach
one for any receiver pressure. This is said to be choked flow.
CBE 150A – Transport
Spring Semester 2014
“Choked” Flow
PCritical
Vsonic
GMax
P1
GMax  U Sonic r End of Pipe 
 P Mwt 
G  Ur  U 

R
T


CBE 150A – Transport
G
0
P
Unattainable
Flows
Sonic Velocity
Attainable Flows
G
PCritical
P1
Spring Semester 2014
Example Problem (maximum flow)
An astronaut is receiving breathing
oxygen at 10 C from his space
capsule through a 7 meter long, 1.7
cm diameter, hose. The capsule
supply pressure is 200 kPa and the
suit pressure is 100 kPa. What is the
flow rate of the oxygen to the suit ? If
the hose breaks off at the suit, what
is the flow rate of oxygen ? What is
the pressure at the end of the hose ?
The hose is “smooth”.
CBE 150A – Transport
Spring Semester 2014
Calculation Approach (subsonic flow)
Given P1, P2, and T
Assume
subsonic flow
at the end of
the pipe.
Assume G
Calculate
NRE
Calculate
f
Calculate
G
Iterate
Calculate
V at end
of pipe
If V > V sonic - flow is unattainable - got to next page
Calculate
V sonic at
end of
pipe
CBE 150A – Transport
Spring Semester 2014
Calculation Approach (sonic flow)
Given P1, P2, and T
Assume sonic
flow at the
end of the
pipe.
Assume
GMax
Calculate
Assume
FDTF
f
Calculate
GMax
Iterate
Check FDTF
assumption
Calculate
P2 (sonic)
Calculate
NRE
If P2 (sonic) > P2 - flow is sonic at end of pipe and G = GMax
CBE 150A – Transport
Spring Semester 2014
10 Minute Problem
Nitrogen ( = 0.02 cP ) is fed from a high pressure cylinder through
¼ in. ID stainless steel tubing ( k = 0.00015 ft) to an experimental
unit. The line ruptures at a point 10 ft. from the cylinder. If the
pressure in the nitrogen in the cylinder is 3000 psig and the
temperature is constant at 70 F, what is the mass flow rate of the
gas through the line and the pressure in the tubing at the point of the
break ?
10 ft
P = 3014 psia
P = 1 atm
CBE 150A – Transport
Spring Semester 2014
Reversible Adiabatic Flow
CBE 150A – Transport
p0
pr
T0
Tr
Spring Semester 2014
Converging/Diverging Nozzle
CBE 150A – Transport
Spring Semester 2014
Isentropic Flow of Inviscid Fluid
Q  0 S  0
In this case The mass balance and MEB are the same as
that for the isothermal case.
Now though the total energy balance will give a relation
between the velocity and temperature
CBE 150A – Transport
Spring Semester 2014
Total Energy Balance
dZ
dT  dQ
 du
m  u
g
 Cp

0

dx
dx  dx
 dx
1
horizontal
adiabatic
du
dT
u
Cp
0
dx
dx
CBE 150A – Transport
Spring Semester 2014
Equation of State
1 dp 1 dr 1 dT


0
p dx r dx T dx
Given the normal equation of state, the TEB, MEB, and the
thermodynamic relation Cp – Cv = zR/M, isentropic flow gives
the following useful values.
CBE 150A – Transport
Spring Semester 2014
Useful Relationships
Given the normal equation of state, the TEB, MEB, and the
thermodynamic relation Cp – Cv = zR/M, isentropic flow gives
the following useful values.
pV   p0V0
p
r

p0
p T 
  
p0  T0 
CBE 150A – Transport
 
r 0
  1
Cp
Cv
Spring Semester 2014
From Mechanical Energy Balance
du 1 dp
u

0
dx r dx
or
udu 
1
1
r
dp  0
1
  p 
udu   dp    r 0    dp
r
  p0  
1
Integrating
 1



2 p0   p  
2
2
u  u0 
1   
r 0   1   p0  


CBE 150A – Transport
u↔p
Spring Semester 2014
Isentropic Flow
2 zRT0    T 
u u 
1   
M   1   T0 
2
2
0
 1




2
p

r
0
1    
u 2  u02 
r0   1   r0  


CBE 150A – Transport
u↔T
u↔r
Spring Semester 2014
Velocity, NMa, and Stagnation
For isentropic flow the definition of the speed of sound is:
uS , S
 dp 
p
 RT
   

r
M
 dr  S
It is also convenient to express the relationships in terms of
a reference state where u0 = 0. This is called the stagnation
condition (u0 = 0) and P0 and T0 are the stagnation pressure
and temperature.
CBE 150A – Transport
Spring Semester 2014
Velocity – Mach Relationships
The previous relationships now become:
N
2
Ma
 1



2  p0 
   1


  1  p 


and
N
CBE 150A – Transport
2
Ma
2  T0  

   1

  1  T  
Spring Semester 2014
Cross-Sectional Area
for Sonic Flow
Application of the continuity (mass balance) equation gives:
2

S
1 2    1N Ma 



*
S
N Ma 
 1

 1
2  1
S* is a useful quantity. It is the cross-sectional area that
would give sonic velocity (NMa = 1).
CBE 150A – Transport
Spring Semester 2014
Summary of Equations for Isentropic Flow
2

S
1  2    1N Ma



*
S
N Ma 
 1

T    1 2 
 1 
N Ma 
T0 
2

1
 1
2  1
p    1 2 
 1 
N Ma 
p0 
2


r    1 2 
 1 
N Ma 
r0 
2

1
1 
1 
p0, T0, r0, are at the stagnant (reservoir) conditions.
These ratios are often tabulated versus NMa for air ( = 1.4).
One must use the equations for gases with  ≠ 1.4.
CBE 150A – Transport
Spring Semester 2014
Maximum Mass Flow Rate
Since the maximum velocity at the throat is NMa = 1, there is
a maximum flow rate:
m max  S *
 2 
r0 p0 

   1
 1
 1
Increase flow by making throat larger, increasing stagnation
pressure, or decrease stagnation temperature. Receiver
conditions do not affect mass flow rate.
CBE 150A – Transport
Spring Semester 2014
Drug Injection via Converging / Diverging
Nozzle
Supersonic jet
Helium cylinder
Contour Shock Tube
CBE 150A – Transport
Powdered drug cassette
Spring Semester 2014
Shock Behavior
CBE 150A – Transport
Spring Semester 2014
Shock Behavior
Isentropic Paths
Po
Pt
PR
PR = P c
PR = P e
PR = P f
Non-Isentropic
Paths
Pe< PR < Pf
Sonic Flow at throat (maximum mass flowrate)
CBE 150A – Transport
Spring Semester 2014
10 Minute Problem
Air flows from a large supply tank at 300 F and 20 atm (absolute) through
a converging-diverging nozzle. The cross-sectional area of the throat is 1
ft2 and the velocity at the throat is sonic. A normal shock occurs at a point
in the diverging section of the nozzle where the cross-sectional area is
1.18 ft2. The Mach number just after the shock is 0.70.
What would be the pressure (P1) at S = 1.18 ft2 if no shock occurred ?
What are the new conditions (T2 and P2 ) after the shock ?
What is the Mach number and pressure at a point in the diverging section of the
nozzle where the cross-sectional area is 1.8 ft2 ?
CBE 150A – Transport
Spring Semester 2014
CFD Simulation of Nozzle Behavior
CBE 150A – Transport
Spring Semester 2014