UNIT - I BASIC CONCEPTS AND ISENTROPIC FLOW IN VARIABLE AREA DUCTS

1. ME 6604 - GAS DYNAMICS AND JET PROPULSION
UNIT – I
BASIC CONCEPTS AND FUNDAMENTALS OF
COMPRESSIBLE FLOW
T.SURESH
ASSISTANT PROFESSOR
DEPT OF MECHANICAL ENGG
KAMARAJ COLLEGE OF ENGINEERING

2. PART - A
• FUNDAMENTALS OF COMPRESSIBLE FLOW
• Energy and momentum equations for compressible
fluid flows, various regions of flows, reference
velocities, stagnation state, velocity of sound,
critical states, Mach number, critical Mach number,
types of waves, Mach cone, Mach angle, effect of
Mach number on compressibility.

3. •
PART – B
• Flow through variable area duct
• Isentropic flow through variable area ducts, T-s
and h-s diagrams for nozzle and diffuser flows,
area ratio as a function of Mach number, mass
flow rate through nozzles and diffusers, effect
of friction in flow through nozzles

4. FLOW THROUGH VARIABLE AREA DUCTS

5. FLOW THROUGH VARIABLE
AREA DUCTS
• As a gas is forced through a tube, the gas molecules are deflected by
the walls of the tube. If the speed of the gas is much less than the
speed of sound of the gas, the density of the gas remains constant
and the velocity of the flow increases. However, as the speed of the
flow approaches the speed of sound we must
consider compressibility effects on the gas. The density of the gas
varies from one location to the next. Considering flow through a
tube, as shown in the figure, if the flow is very gradually
compressed (area decreases) and then gradually expanded (area
increases), the flow conditions return to their original values. We say
that such a process is reversible. From a consideration of the second
law of thermodynamics, a reversible flow maintains a constant value
of entropy. Engineers call this type of flow an isentropicflow; a
combination of the Greek word "iso" (same) and entropy.

6. FLOW THROUGH VARIABLE
AREA DUCTS

7. FLOW THROUGH VARIABLE
AREA DUCTS
• The conservation of mass is a fundamental concept of
physics. Within some problem domain, the amount of
mass remains constant; mass is neither created or
destroyed. The mass of any object is simply the volume
that the object occupies times the density of the
object. For a fluid (a liquid or a gas) the density,
volume, and shape of the object can all change within
the domain with time and mass can move through the
domain.
• The conservation of mass (continuity) tells us that the
mass flow rate mdot through a tube is a constant and
equal to the product of the density r, velocity V, and
flow area A:

8. Conservation of mass

9. Conservation of mass
• Solid Mechanics
• The conservation of mass is a fundamental concept of physics along with the conservation of energy and
theconservation of momentum. Within some problem domain, the amount of mass remains constant--mass
is neither created nor destroyed. This seems quite obvious, as long as we are not talking about black holes
or very exotic physics problems. The mass of any object can be determined by multiplying the volume of
the object by the density of the object. When we move a solid object, as shown at the top of the slide, the
object retains its shape, density, and volume. The mass of the object, therefore, remains a constant between
state "a" and state "b."
• Fluid Statics
• In the center of the figure, we consider an amount of a static fluid , liquid or gas. If we change the fluid
from some state "a" to another state "b" and allow it to come to rest, we find that, unlike a solid, a fluid may
change its shape. The amount of fluid, however, remains the same. We can calculate the amount of fluid by
multiplying the density times the volume. Since the mass remains constant, the product of the density and
volume also remains constant. (If the density remains constant, the volume also remains constant.) The
shape can change, but the mass remains the same.
• Fluid Dynamics
• Finally, at the bottom of the slide, we consider the changes for a fluid that is moving through our domain.
There is no accumulation or depletion of mass, so mass is conserved within the domain. Since the fluid is
moving, defining the amount of mass gets a little tricky. Let's consider an amount of fluid that passes
through point "a" of our domain in some amount of time t. If the fluid passes through an area A at
velocity V, we can define the volume Vol to be:
• Vol = A * V * t

10. Conservation Laws for a Real Fluid
  0. 


V
t



    wqVe
t
e







.
    gVV
t
V
ij
ˆ.. 



 

iiij pij
 '

    gpVV
t
V
ij ˆ.. '




 


11. Conservation of Mass Applied to 1 D Steady Flow
  0. 


V
t



Conservation of Mass:
Conservation of Mass for Stead Flow:
  0.  V


Integrate from inlet to exit :
  onstant. CVdV
V




12. One Dimensional Stead Flow
A,
V

A+dA,
V+dV d
dl
  onstant.. CdxdAV
V



  onstant. Cdx
dx
VAd


  0VAd 
0
A
dA
V
dVd



13. Conservation of Momentum For A Real Fluid Flow
  pVV ij
 '
.. 

  VdpVdVdVV
VVV
ij   '
.. 

No body forces
One Dimensional Steady flow
A,
V

A+dA,
V+dV d
dl

14.   dAdxpdxdAdAdxVV
V
w
VV
ij   '
. 

     dx
dx
pAd
dx
dx
Ad
dx
dx
AVd ww
 
 2
     pAdAdAVd ww   2

15. Conservation of Energy Applied to 1 D Steady Flow
    wqVe
t
e







.
Steady flow with negligible Body Forces and no heat transfer is
adiabatic real flow
  wVe 

 .
For a real fluid the rate of work transfer is due to viscous stress and
pressure. Neglecting the the effect of viscous dissipation.
  VdAnpVe

.ˆ.  

16. For a total change from inlet to exit :
   
AV
VdAnpVdVe

.ˆ. 
Using gauss divergence theorem:
One dimensional flow
   
VV
VdVpVdVe

.. 
   
VV
dAdxVpdAdxVe

.. 

17.    
  dx
dx
pAVd
dx
dx
eVAd 
   pAVdeVAd 
2
2
V
ue 




























 AV
p
d
V
uVAd


2
2
0
2
2























Vp
uVAd

 0
2
2















V
hVAd 

18. Summary of Real Fluid Analysis
0
A
dA
V
dVd


     pAdAdAVd ww   2
0
2
2















V
hVAd 

19. Further Analysis of Momentum equation
     pAdAdVAdVVAVd ww  
   pAdAdVAdV ww  
   pAdPxddVm w  
   pAdAddVm ww  
pdAAdpPdxxdPPxddVm www  

20. Frictional Flow in A Constant Area Duct
0
V
dVd


AdpPdxPxddVm ww  
0
2
2















V
hVd 

21. Frictional Flow in A Constant Area Duct
AdpPdxdVm w  
w
The shear stress is defined as and average
viscous stress which is always opposite to the
direction of flow for the entire length dx.
AdpPdxPxddVm ww  
AdpPdxAVdV w  

22. AdpPdxAVdV w  
Divide by AV2
22
V
dp
dx
A
P
VV
dV w



0
V
dVd


00
2
2






 VdVdTC
V
hd p

23. One dimensional Frictional Flow of A Perfect Gas
0
V
dVd


0VdVdTCp
2
V
dp
dx
A
P
f
V
dV


T
dT
V
dV
p
dp
T
dTd
p
dp




24. Sonic Equation
2
22
2
2
2 2
2
RT
dTV
RT
VdV
MdM
RT
V
c
V
M


Differential form of above equation:
T
dT
V
dV
M
dM
2

T
dT
V
dV
p
dp

T
dT
M
dM
p
dp
2


25.  
  M
dM
M
M
T
dT














2
2
2
1
1
1


Energy equation can be modified as:
T
dT
M
dM
p
dp
2

 
  M
dM
M
M
M
dM
p
dp














2
2
2
1
1
1
2
1



26. 1D steady real flow through constant area duct : momentum equation
022

V
dp
dx
A
P
VV
dV w


022

p
dp
V
p
dx
A
P
VV
dV w


022

p
dp
V
p
dx
A
P
VV
dV w



27. 022

p
dp
V
p
dx
A
P
VV
dV w




022

p
dp
V
p
dx
A
P
VV
dV w




0
1
22

p
dp
M
dx
A
P
VV
dV w



28. 0
1
22

p
dp
M
dx
A
P
VV
dV w


 
  M
dM
M
M
T
dT














2
2
2
1
1
1


 
  M
dM
M
M
M
dM
p
dp














2
2
2
1
1
1
2
1


T
dT
V
dV
M
dM
2

Differential Equations for Frictional Flow Through
Constant Area Duct
T
dT
M
dM
p
dp
2


29. 0
1
22

p
dp
M
dx
A
P
VT
dT
M
dM w


 
 
 
  0
2
1
1
1
2
11
2
1
1
1
2
2
22
2
2









































M
dM
M
M
M
dM
M
dx
A
P
VM
dM
M
M
M
dM w
















 

 dx
A
P
VM
MM
M
dM w
22
22
1
2
1
1





30.   








 dx
A
P
VM
M
T
dT w
22
4
1
1


  








 dx
A
P
VM
MM
p
dp w
22
22
1
11












 

 dx
A
P
VM
MM
M
dM w
22
22
1
2
1
1





31. Second Law Analysis
vdpdTCTds p 
dp
T
v
T
dT
Cds p 
p
dp
R
T
dT
Cds p 

32. p
dp
R
T
dT
Cds p 
p
dp
T
dT
C
ds
p 
 1









 2
1
1
V
TC
T
dT
T
dT
C
ds p
p 

 









TT
T
T
dT
T
dT
C
ds
p 02
1
1



33. TT
dT
T
dT
C
ds
p 


02
11



 


T
T
T
T
s
s p iii
TT
dT
T
dT
C
ds
02
11

































2
1
0
0
/1
ln
iip
i
TT
TT
T
T
C
ss
  








 dx
A
P
VM
M
T
dT w
22
4
1
1



34. Fanno Line
Adiabatic flow in a constant area with friction is termed as Fanno flow.

35. Isentropic Nozzle and Adiabatic Duct

36. C Nozzle Discharge Curve

37. CD Nozzle + Discharge Curve

38. Nature of Real Flow
Entropy of an irreversible adiabatic system should always increase!
 






 dx
A
P
V
MCds w
p 2
2
1













 

 dx
A
P
VM
MM
M
dM w
22
22
1
2
1
1




  








 dx
A
P
VM
M
T
dT w
22
4
1
1


  








 dx
A
P
VM
MM
p
dp w
22
22
1
11



39. M dM dp dT dV
<1 +ve -ve -ve +ve
>1 -ve +ve +ve -ve

40. Compressible Real Flow
),(Re, M
d
k
functionf 
Effect of Mach number is negligible….
)(Re,
d
k
functionf 

1
Re 
n
T
T







2
1
2
1



41. Pressure drop in Compressible Flow
Laminar Flow
Turbulent Flows



 









22
2
2
1
1
1
MM
M
M
dM
dx
A
P
f


Re
16
f
2
9.0
Re
74.5
7.3
log
0625.0














hD
k
f

42. Moody Chart

43. Compressible Flow Through Finite Length Duct
Integrate over a length l



 









22
2
2
1
1
14
MM
M
M
dM
D
fdx
h


M
dM
MM
M
D
fdx e
i
M
M
l
h




 



22
2
0
2
1
1
14



44. 


 









22
2
2
1
1
14
MM
M
M
dM
D
fdx
h



















 






 









22
22
22
2
1
1
2
1
1
ln
2
11114
ie
ei
eih
MM
MM
MM
l
D
f





is a Mean friction factor over a length l .
f

45. Maximum Allowable Length
• The length of the duct required to give a Mach number of
1 with an initial Mach number Mi
Similarly

















 






 









2
2
2max
2
1
1
1
2
1
1
ln
2
1
1
114
i
i
ih
M
M
M
l
D
f





 
 
















1
2
2
*
2
1
1
2
1
1
*
iM
p
p
M
dM
M
M
p
dp
p
p



46.  
 














1
2
2
2
1
1
1
*
ii M
T
T
M
dM
M
M
T
dT


2/1
2
*
2
1
1
2
1
1














i
i MMp
p
















2
*
2
1
1
2
1
iMT
T


2
5.1
5 .
111
1.384
293
103.3
m
sN
T
T






 


47.  1/
2
2
1
1






 



i
o
M
p
p
*
0
*
0
*
0
0
p
p
p
p
p
p

  1
2
**
0
0
2
1
2
1
1



















iM
p
p
p
p
   12
2
*
0
0
2
1
2
1
1
1



















i
i
M
Mp
p

48. Compressible Frictional Flow through
Constant Area Duct
HD
fL *4
0
*
0
p
p
p
p*
T
T*
V
V *
M

49. Frictional Flow in A Variable Area Duct
0
A
dA
V
dVd


A,
V

A+dA,
V+dV d

50. 0
4
2
2
2

V
dV
Mdx
D
fM
p
dp
h


T
dTd
p
dp



0
A
dA
V
dVd


A
dA
V
dV
T
dT
p
dp


51. 0
4
2
2
2

V
dV
Mdx
D
fM
A
dA
V
dV
T
dT
h


T
dT
M
dM
V
dV
T
dT
V
dV
M
dM
22

0
2
4
22
2
2















T
dT
M
dM
Mdx
D
fM
A
dA
T
dT
M
dM
T
dT
h


 
  M
dM
M
M
T
dT














2
2
2
1
1
1



52. 0
4
2
2
1
1
1 2
2
2




 dx
D
fM
M
dM
M
M
A
dA
h


dx
D
fM
M
M
A
dA
M
M
M
dM
h
4
21
2
1
1
1
2
1
1 2
2
2
2
2






























Constant Mach number frictional flow
hD
fM
dx
dA 2
2


53.   dx
D
fM
M
A
dA
M
M
dM
M
h
4
22
1
1
2
1
11
2
222 





 






 

Sonic Point : M=1
0
4
22
1
1
2
1
1 





 






 
 dx
D
f
A
dA
h

0
4
22
1
2
1






 






 
 dx
D
f
A
dA
h

dx
D
f
A
dA
h
4
2



54. 54
Stagnation Properties
Consider a fluid flowing into a diffuser at a velocity , temperature T, pressure P, and
enthalpy h, etc. Here the ordinary properties T, P, h, etc. are called the static properties; that
is, they are measured relative to the flow at the flow velocity. The diffuser is sufficiently
long and the exit area is sufficiently large that the fluid is brought to rest (zero velocity) at the
diffuser exit while no work or heat transfer is done. The resulting state is called the
stagnation state.

V
We apply the first law per unit mass for one entrance, one exit, and neglect the potential
energies. Let the inlet state be unsubscripted and the exit or stagnation state have the
subscript o.
q h
V
w h
V
net net o
o
    
 2 2
2 2

55. 55
Since the exit velocity, work, and heat transfer are zero,
h h
V
o  
2
2
The term ho is called the stagnation enthalpy (some authors call this the total enthalpy). It is
the enthalpy the fluid attains when brought to rest adiabatically while no work is done.
If, in addition, the process is also reversible, the process is isentropic, and the inlet and exit
entropies are equal.
s so 
The stagnation enthalpy and entropy define the stagnation state and the isentropic
stagnation pressure, Po. The actual stagnation pressure for irreversible flows will be
somewhat less than the isentropic stagnation pressure as shown below.

56. 56