One Dimensional Flow of Blissful Fluid -III
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Always Start with simplest Inventions……..
Differential Form of Momentum Equation
One dimensional steady inviscid flow :
du dp
u 
 0  udu  dp
dx dx
udu  dp
du
1

dp
u
The relation between pressure and velocity is continuous.
Differential Form of Energy Equation
One dimensional steady inviscid Adiabatic flow :
dh  udu  0
du
1

dh
u
The relation between enthalpy and velocity is continuous.
Summary
dp u
1

2
dA
A 1 M
du u
1

2
dA A M  1
2


du
1

dp
u
du
1

dh
u


du
1
dA

2
u
M 1 A


du
1 dp
M

u
 p
2
Subsonic Nozzle
Subsonic Diffuser
dA < 0 & M <1
So, du > 0 & dp <0
dp
d

p

M dA
d

2
M 1 A

2
M 2 dA
1 dp

2
M 1 A
 p
dA > 0 & M <1
So, du < 0 & dp>0
  1M 2
dA
dT

2
M 1 A
T
du
1
dA

2
u
M 1 A


du
1 dp
M

u
 p
2
Supersonic Diffuser
M 2 dA
1 dp

2
M 1 A
 p
Supersonic Nozzle
dA < 0 & M >1
dA > 0 & M >1
So, du < 0 & dp >0
So, du >0 & dp<0
dp
d

p

M 2 dA
d

2
M 1 A

  1M 2
dA
dT

2
M 1 A
T
Generation of High Pressure from Supersonic velocity
dp   du
 2
p M u
dA 1  M 1 dp

A
M2  p
2
d
1 dp

  p
dT   1 dp

T
 p
An Ideal Diffuser at Design Conditions
p1
pthroat
p2d
p1
p2d
p
p*
Generation of Supersonic Velocity from Rest
du  M 2 dp

u

p
dA 1  M 2 1 dp

A
M2  p
d
1 dp

  p
dT   1 dp

T
 p
RAMJET Engine
Capacity of A Cross Section : An implicit Model
Mass flow rate through any cross section of area A
m   ( x) A( x)u ( x)
With a condition that sonic velocity occurs at throat !
  throat AthroatCthorat
m
pthroat
Cthorat 
 RTthroat
throat
Stagnation Temperature for the
Adiabatic Flow of a Calorically Perfect Gas
• Consider an adiabatic flow field with a local gas Temperature T(x),
pressure p(x), and a velocity V(x)
T(x)
x
p(x)
V(x)
• Since the Flow is adiabatic
u ( x) 2
u ( x) 2
h( x ) 
 C pT ( x) 
 constant
2
2
Introduce an obstruction in the inviscid flow field :
T o (y)
T(x)
x
p(x)
V(x)
y
p o (y)
V(y)=0
• This obstruction generates a location y, within this flow field
where the gas velocity is reduced to zero.
• Since the Flow is adiabatic
u ( x) 2
u ( x) 2
C pT ( x ) 
 C pT0  T0  T(x) 
2
2C p
2
u ( x)
T0  T ( x) 
 R 
2

   1
u ( x)
 1
2
T0  T ( x) 
 T0  T ( x) 
u ( x)
2R
 R 
2

   1
2
   1 u ( x) 
  1
2
T0  T ( x)1 
M ( x) 
  T ( x)1 
2 RT ( x) 
2



2
Holds anywhere within an adiabatic Flow field
In general for an adiabatic Flow Field the Stagnation Temperature
is defined by the relationship
  1 2
T0

 1
M
T
2
Stagnation Temperature is constant throughout an adiabatic flow
field.
• T0 is also sometimes referred to at Total Temperature
• T is sometimes referred to as Static Temperature
• Stagnation temperature is a measure of the Kinetic Energy of the
flow Field.
• Largely responsible for the high Level of heating that occurs on
high speed aircraft or reentering space Vehicles …
  1 2
T0

 1
M
T
2

 T0   1    1 2   1
    1 
M 
2
T 


“stagnation” (total) pressure : Constant throughout Isentropic flow
field.
p0
p
• Similarly Stagnation density for isentropic flow field is
0

1
  1 2 

 T0   1 
 
 1 
M 
2
T 


1
 1
Stagnation Properties of Isentropic Flow
  1 2
T0

 1
M
T
2
p0
p
0



 T0   1    1 2   1
    1 
M 
2
T 


1
 T0   1    1 2 
    1 
M 
2
T 


1
 1
What was Stagnation Temperature At Columbia Breakup
  1 2
T0

 1
M
T
2
Loss Of Signal at:
61.2 km altitude
~18.0 Mach Number
T∞ ~ 243 K
Ideal gas
Variable Properties
Real gas
Capacity of A Cross Section
Mass flow rate through any cross section of area A
m   ( x) A( x)u ( x)
With a condition that sonic velocity occurs at throat !
m  thoat AthroatCthorat
pthoat
Cthorat 
 RTthoat
thorat
Calorically perfect gas:
 T0 
0


 ( x) T ( x) 
1
 1

 1
  1
2
 1 
M ( x) 
2






1
m   0 

1    1 M ( x) 2 
2






1
  ( x)   0 



1
1 
M ( x) 2 
2


1
 1
A( x)u ( x)
1
 1




1
m   0 

1    1 M ( x) 2 
2






1
m   0 

1    1 M ( x) 2 
2


  1 2
T0

 1
M
T
2
1
 1
A( x) M ( x)c( x)
1
 1
A( x) M ( x) RT
T0
 T ( x) 
  1
2
M ( x) 
1 
2






1
m   0 

1    1 M ( x) 2 
2


1
 1
T0
A( x) M ( x) R
  1
2
M ( x) 
1 
2






1
m  RT0  0 

1    1 M ( x) 2 
2


m  RT0  0
1
1 1

 1 2
 1
2  1
2
  1
M ( x) 
1 
2


A( x) M ( x)
A( x) M ( x)
m  RT0  0
A( x) M ( x)
  1
M ( x) 
1 
2


p0
m  RT0
RT0
m 
 p0
R
 1
2  1
2
T0
A( x) M ( x)
 1
2  1
2
  1
M ( x) 
1 
2


A( x) M ( x)
 1
2  1
2
  1
M ( x) 
1 
2


Specific Mass flow Rate
Mass flow rate per unit area of cross section:
m
 p0

A( x)
R T0
M ( x)
 1
2  1
2
  1
M ( x) 
1 
2


Design of Converging Diverging Nozzles
P M V Subbarao
Associate Professor
Mechanical Engineering Department
I I T Delhi
From the Beginning to the Peak or Vice Versa….
Quasi-One-Dimensional Flow
Distinction Between True 1-D Flow and Quasi 1-D Flow
• In “true” 1-D flow Cross sectional
area is strictly constant
• In quasi-1-D flow, cross section
varies as a Function of the
longitudinal coordinate, x
• Flow Properties are assumed
constant across any cross-section
• Analytical simplification very
useful for evaluating Flow properties
in Nozzles, tubes, ducts, and diffusers
Where the cross sectional area is
large when compared to length
Specific Mass flow Rate
Mass flow rate per unit area of cross section:
m
 p0

A( x)
R T0
M ( x)
 1
2  1
2
  1
M ( x) 
1 
2


Maximum Capacity of A Nozzle
m
 p0

A( x)
R T0
M ( x)
 1
2  1
2
  1
M ( x) 
1 
2


• Consider a discontinuity at throat “choked-flow” Nozzle … (I.e.
M=1 at Throat)
• Then comparing the massflow /unit area at throat to some other
station.
m
 p0

A( x)
R T0
m
Athroat

M ( x)
  1
M ( x) 
1 
2


 p0
R
 1
2  1
2
T0
1
 1
 1 2 1
 
1 

2 

Take the ratio of the above:
 p0
R
Athroat

A( x)
M ( x)
T0
 1
2  1
2
  1
M ( x) 
1 
2


 p0
1
 1
R T0
   1 2 1
1 

2 

M ( x)
 1
2  1
2
  1
M ( x) 
1 
Athroat 
2


1
A( x)
 1
 1 2  1
 
1 

2 

 1
 1 2  1
Athroat
 M ( x)
A( x)



 2 
 1
2  1
2
  1
M ( x) 
1 
2


 1
 2 1
A( x)
1  2    1
2

M ( x ) 
1 

Athroat M ( x)    1
2

 1
 2 1
A( x)
1  2    1
2

M ( x) 
1 

*
A
M ( x)    1
2

Design Analysis
 1
 2 1
A( x)
1  2    1
2

M ( x) 
1 

*
A
M ( x)    1
2

For a known value of Mach number, it is easy to calculate
area ratio.
Throat area sizing is the first step in the design.
If one needs to know the Mach number distribution for a
given geometric design!
Find the roots of the non-linear equation.
Typical Design Procedure
• The Space Shuttle Main Engines burn LOX/LH2 for
propellants with A ratio of LOX:LH2 =6:1
• The Combustor Pressure, p0 is 20.4 Mpa, combustor
temperature, T0 is 3300 K.
• Decide throat diameter based on the requirement of
thrust.
• What propellant mass flow rate is required for
choked flow in the Nozzle?
• Assume no heat transfer through Nozzle no
frictional losses.
• Combustion product is water vapor.
Space Shuttle Main Engines
m
Athroat

 p0
R
T0
1
 1
 1 2 1
 
1 

2 

Specifications of SSME
• Specific Impulse is a commonly used measure of performance
For Rocket Engines,and for steady state-engine operation is defined
As:
1
I sp 
g0
Fthrust
•
m
m
 g0  9.806 2 (mks)
sec
propellant
• At 100% Throttle a SSSME has the Following performance
characteristics
Fvac
=
2298 kNt
Fsl
=
1600 kNt
Ispvac
=
450 sec.
SEA Level Performance
One needs to know the Mach number
distribution for a given geometric design!
Find the roots of the non-linear equation.
Numerical Solution for Mach Number Caluculation
• Use “Newton’s Method” to extract numerical solution
• Define:
1
F(M ) 
M
 2  
  1 2  

1
M 



2

   1 
 1
2  1
• At correct Mach number (for given A/A*) …
F(M )  0
• Expand F(M) is Taylor’s series about some arbitrary Mach
number M(j)
A
 *
A
 F 
F(M )  F(M ( j ) )  
M  M( j) 
 M  ( j )


 2 F 
 M 2  M  M ( j )
( j)

2

2

 ...O M  M ( j )
• Solve for M
  2 F 
M  M( j)

2
 M  ( j )

F(M )  F(M ( j ) ) 

2


M  M( j) 
 F 


M  ( j )


2

 ...O M  M ( j )


3






3
• From Earlier Definition
F(M )  0 , thus
  2 F 
M  M( j)

2
 M  ( j )

F(M ( j ) ) 

2


M  M( j) 
 F 


M  ( j )
Still exact expression

• if M(j) is chosen to be “close” to M




3




2

 ...O M  M ( j )
M  M( j)

  M  M 
2
( j)
And we can truncate after the first order terms with “little”
Loss of accuracy
• First Order approximation of solution for M
F(M ( j ) )
M  M( j) 
 F 


M ( j )
^
“Hat” indicates that solution is no longer exact
• However; one would anticipate that
^
M  M  M  M( j)
“estimate is closer than original guess”
^
• If we substitute M back into the approximate expression
^
^
^
F(M )
M  M
 F 

 ^
M |M
^
^
^
• And we would anticipate that
^
MM  MM
“refined estimate” …. Iteration 1
• Abstracting to a “jth” iteration
^
^
M ( j 1)
F(M ( j ) )
 M ( j) 
 F 


M |( j )
^
Iterate until convergence
j={0,1,….}
• Drop from loop when
2
 2  

  1 ^

1
M ( j 1)  

^


2

M ( j 1)    1  
1
A
A*
 1
2  1
A
 *
A

Plot Flow Properties Along Nozzle Length
• A/A*
• Mach Number
^
^
^
M ( j 1)  M ( j ) 
F(M ( j ) )
 F 


M  |( j )
• Temperature
T0 = 3300K
T (x) 
Tthroat = 2933.3 K
T0
 1
1
M (x)2
2
• Pressure
P(x) 
P0 = 20.4Mpa
P0

 1
 1

2
1

M
(x)


2
Pthroat = 11.32 MPa
Nozzle at Off Design Exit Pressure
pthroat
p2d
p1
p2a > p2d
1u1 A1   2 au2 a A2 d
p1 A1  1u1 A1  p2a A2d  2u 2 a A2d
2
u 12
u

 c p T1  T2 , a 
2
2
2
2a
2
Nozzle at Off Design Exit Pressure
pthroat
p1
p1
p1 > p2a > pthroat
p2a
P*
p
2a
p*
p2d