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Stagnation Properties
P M V Subbarao Professor
Mechanical Engineering DepartmentI I T Delhi
Capacity of A Resource…..
Stagnation Properties of Isentropic Flow
T0
T1
1 2
M 2
p0
p
T0
T
1
1 1
2M 2
1
0
T0
T
1
1 1
2M 2
1
11
What was Stagnation Temperature At Columbia Breakup
Loss Of Signal at:61.2 km altitude~18.0 Mach Number
T∞ ~ 243 K
T0
T1
1 2
M 2
Ideal & Calorically perfect Gas
Ideal Gas with Variable Properties
Real Gas with Variable Properties
Capacity of A Cross Section
Mass flow rate through any cross section of area A
)()()( xuxAxm
Maximum Capacity is obtained when sonic velocity occurs at throat !
thoratthroatthoat CAm
thoatthorat
thoatthorat RT
pC
Calorically perfect gas:
1
1
20
12
1
1
00
)(2
11
1)()(
2
11
)()(
xMxxM
xT
T
x
)()()(
21
1
11
1
20 xuxA
xMm
T0
T1
1 2
M 2
2
0
)(2
11
)(xM
TxT
)()()()(
21
1
11
1
20 xcxMxA
xMm
RTxMxAxM
m
)()()(
21
1
11
1
20
2
0
1
1
20
)(2
11
)()()(
21
1
1
xM
TRxMxA
xMm
)()()(
21
1
12
1
1
1
200 xMxA
xMRTm
)()(
)(2
11
1
12
1
2
00 xMxA
xM
RTm
12
1
2
00
)(2
11
)()(
xM
xMxARTm
12
1
20
00
)(2
11
)()(
xM
xMxA
RT
pRTm
12
1
20
0
)(2
11
)()(
xM
xMxA
T
p
Rm
Specific Mass flow Rate
Mass flow rate per unit area of cross section:
12
1
20
0
)(2
11
)(
)(
xM
xM
T
p
RxA
m
Design of Supersonic Intake / Nozzle
P M V SubbaraoAssociate Professor
Mechanical Engineering DepartmentI 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 assumedconstant 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:
12
1
20
0
)(2
11
)(
)(
xM
xM
T
p
RxA
m
Maximum Capacity of An Intake/Nozzle
12
1
20
0
)(2
11
)(
)(
xM
xM
T
p
RxA
m
• 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.
12
1
20
0
)(2
11
)(
)(
xM
xM
T
p
RxA
m
12
10
0
21
1
1
T
p
RA
m
throat
Take the ratio of the above:
12
10
0
12
1
20
0
21
1
1
)(2
11
)(
)(
T
p
R
xM
xM
T
p
R
xA
Athroat
12
1
12
1
2
21
1
1
)(2
11
)(
)(
xM
xM
xA
Athroat
12
1
2
12
1
)(2
11
21
)()(
xM
xMxA
Athroat
12
1
2)(2
11
1
2
)(
1)(
xMxMA
xA
throat
12
1
2*
)(2
11
1
2
)(
1)(
xMxMA
xA
Design Analysis 12
1
2*
)(2
11
1
2
)(
1)(
xMxMA
xA
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 we know the details of the resource/requirements, we can calculate the size of throat.
12
10
0
21
1
1
T
p
RA
m
throat
Cryogenic Rocket Engines
12
10
0
21
1
1
T
p
RA
m
throat
A ratio of LO2:LH2 =6:1
T0 = 3300K.
P0 = 20.4 Mpa
Specifications of A Rocket Engine
• Specific Impulse is a commonly used measure of performanceFor Rocket Engines,and for steady state-engine operation is definedAs:
I sp 1
g0
Fthrust•
m propellant
g0 9.806m
sec2(mks)
• At 100% Throttle a RE has the Following performance characteristics
Fvacuum = 2298 kNt
Ispvacuum = 450 sec.
Fsea level = 1600 kNt
Specific impulse of various propulsion technologies
Engine"Ve" effective exhaust velocity(m/s, N·s/kg)
Specific impulse(s)
Energy per kg(MJ/kg)
Turbofan jet engine 300 3000 43
Solid rocket 2500 250 3.0
Bipropellant liquid rocket 4400 450 9.7
Plasma Rocket 29 000 3000 430
VASIMR 290 000 30 000 43 000
Design Procedure
I sp 1
g0
Fthrust•
m propellant
g0 9.806m
sec2(mks)
Select a technology : Isp & Fthrust
SEA Level Performance
One needs to know the Mach number distribution for a given geometric design!
Find the roots of the non-linear equation.
12
1
2*
)(2
11
1
2
)(
1)(
xMxMA
xA
Numerical Solution for Mach Number Caluculation
• Use “Newton’s Method” to extract numerical solution
• At correct Mach number (for given A/A*) …
F(M ) 0
F(M ) 1
M
2
1
1 1
2M 2
1
2 1
A
A*
• Define:
• Expand F(M) is Taylor’s series about some arbitrary Mach number M(j)
F(M ) F(M ( j ) ) F
M
( j )
M M ( j ) 2F
M 2
( j )
M M ( j ) 2
2 ...O M M ( j ) 3
• Solve for M
M M ( j )
F(M ) F(M ( j ) )
2F
M 2
( j )
M M ( j ) 2
2 ...O M M ( j ) 3
F
M
( j )
• From Earlier Definition , thusF(M ) 0
M M ( j )
F(M ( j ) )
2F
M 2
( j )
M M ( j ) 2
2 ...O M M ( j ) 3
F
M
( j )
• if M(j) is chosen to be “close” to M M M ( j ) 2 M M ( j )
And we can truncate after the first order terms with “little”Loss of accuracy
Still exact expression
• First Order approximation of solution for M
• However; one would anticipate that
“Hat” indicates that solution is no longer exact
M^
M ( j ) F(M ( j ) )
F
M
( j )
M M^
M M ( j )
“estimate is closer than original guess”
• And we would anticipate that
“refined estimate” …. Iteration 1
M^^
M^
F(M
^
)F
M
|M^
M M^^
M M^
• If we substitute back into the approximate expressionM^
• Abstracting to a “jth” iteration
Iterate until convergencej={0,1,….}
M^
( j1) M^
( j ) F(M
^
( j ) )F
M
|( j )
1
M^
( j1)
2
1
1 1
2M
^
( j1)
2
1
2 1
A
A*
A
A*
• Drop from loop when
Plot Flow Properties Along Nozzle Length
• A/A*
• Mach NumberM
^
( j1) M^
( j ) F(M
^
( j ) )F
M
|( j )
• Temperature T (x) T0
1 1
2M (x)2
T0 = 3300KTthroat = 2933.3 K
• Pressure
P0 = 20.4Mpa Pthroat = 11.32 MPa
P(x) P0
1 1
2M (x)2
1
Operating Characteristics of Nozzles
P M V SubbaraoProfessor
Mechanical Engineering DepartmentI I T Delhi
Realizing New Events of Physics…….
Converging Nozzle
p0
pb
pb = Back Pressure
Design Variables: 00 ,, Tpm
Outlet Condition:
exitexitb MorAorp
Designed Exit Conditions
121
00
2
11
exit
exitexit
MT
T
p
p
12
1
20
0
21
1
exit
exitexit
M
MA
T
p
Rm
Under design conditions the pressure at the exit plane of the nozzle is applied back pressure.
121
00
2
11
exit
exitb
MT
T
p
p
Profile of the Nozzle 12
1
2
2
21
1
)(2
11
)(
)(
exit
exit
exit M
xM
xM
M
A
xA
1
2
2
)(12
12)(
xM
M
p
xp exit
b
1
2
2
)(12
12)(
xM
M
p
xp exit
exit
At design Conditions:
Full Capacity Convergent Nozzle
1
2, )(12
1)(
xMp
xp
criticalb
12
1
21
)(2
11
)(
1)(2
,
xM
xMA
xA
criticalexit
Remarks on Isentropic Nozzle Design
• Length of the nozzle is immaterial for an isentropic nozzle.
• Strength requirements of nozzle material may decide the nozzle length.
• Either Mach number variation or Area variation or Pressure variation is specified as a function or arbitrary length unit.
• Nozzle design attains maximum capacity when the exit Mach number is unity.
Converging Nozzle
p0
Pb,critical
1
,
0
2
1
criticalbp
p
1
0, 1
2
pp criticalb
Operational Characteristics of Nozzles
• A variable area passage designed to accelerate the a gas flow is considered for study.
• The concern here is with the effect of changes in the upstream and downstream pressures
• on the nature of the inside flow and • on the mass flow rate through a nozzle.• Four different cases considered for analysis are:• Converging nozzle with constant upstream conditions.• Converging-diverging nozzle with constant upstream conditions.• Converging nozzle with constant downstream conditions.• Converging-diverging nozzle with constant downstream
conditions.
Pressure Distribution in Under Expanded Nozzle
p0
Pb,critical
pb=p0
pb,critical<pb1<p0
pb,critical<pb2<p0
pb,critical<pb3<p0
At all the above conditions, the pressure at the exit plane of nozzle, pexit = pb.
Variation of Mass Flow Rate in Exit Pressure
0p
pb
0p
pe
1
1
0
,
p
p criticalb
0
,
p
p criticale
Variation of in Exit Pressure
0p
pb
0p
pe
1
1
0
,
p
p criticalb
0
,
p
p criticale
Variation of in Mass Flow Rate
0p
pb
m
10
,
p
p criticalb
chokedm
Low Back Pressure Operation
0
)(
p
xp
0
*
p
p
00 p
p
p
p bexit
Convergent-Divergent Nozzle Under Design Conditions
Convergent-Divergent Nozzle with High Back Pressure
p*< pb1<p0
pthroat> p*
Convergent-Divergent Nozzle with High Back Pressure
• When pb is very nearly the same as p0 the flow remains subsonic throughout.
• The flow in the nozzle is then similar to that in a venturi.
• The local pressure drops from p0 to a minimum value at the throat, pthroat , which is greater than p*.
• The local pressure increases from throat to exit plane of the nozzle.
• The pressure at the exit plate of the nozzle is equal to the back pressure.
• This trend will continue for a particular value of back pressure.
Convergent-Divergent Nozzle with High Back Pressure
At all these back pressures the exit plane pressure is equal to the back pressure.
pthroat> p*
0
2
2
)()( TC
xuxTC pp
12
)(
1
)( 20
22
cxuxc
12
1
0
12
0
2
1
00
)()()()(
x
p
xp
T
xT
c
xc
)(1
2
1
)(
12)( 22
0
2202 xcc
xccxu
20
2202 )(
11
2)(
c
xccxu
1
0
202 )(
11
2)(
p
xpcxu
1
0
02 )(1
1
2)(
p
xpRTxu
1
00
02 )(1
1
2)(
p
xppxu
2/1
1
00
0 )(1
1
2)(
p
xppxu
2/1
1
00
0
00
)(1
1
2)(
)(
p
xppxA
xm
)()()( xuxAxm
)()()(
00 xuxA
xm
2/1
1
00
0
1
00
)(1
1
2)(
)(
p
xppxA
p
xpm
At exit with high back pressure pb
2/1
1
00
0
1
00 1
1
2
p
ppA
p
pm exit
exitb
At throat with high back pressure pb
2/1
1
00
0*
1
00 1
1
2
p
ppA
p
pm tt
2/1
1
0
*
1
0
2/1
1
0
1
0
11
p
pA
p
p
p
pA
p
p ttbexit
b
•For a given value of high back pressure corresponding throat pressure can be calculated. •As exit area is higher than throat area throat pressure is always less than exit plane pressure.•An decreasing exit pressure produces lowering throat pressure
Variation of Mass Flow Rate in Exit Pressure
0p
pb
0p
pe
1
1
0
,
p
p criticalb
0
,
p
p criticale
Variation of in Mass Flow Rate
0p
pb
m
10
,
p
p criticalb
chokedm
