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C & CD Nozzles for Jet Propulsion
P M V Subbarao
Professor
Mechanical Engineering Department
Design Vs Off-Design Characteristics…..
Operational Characteristics of Isentropic C Nozzles
• A converging 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 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
in
b
p
p
in
e
p
p
1
1
in
criticalb,
p
p
in
criticale,
p
p
Variation of in Exit Pressure
1
1
in
b
p
p
in
e
p
p
in
criticalb,
p
p
in
criticale,
p
p
Variation of in Mass Flow Rate
0
b
p
p
m
10
criticalb,
p
p
chokedm
Low Back Pressure Operation
Frictional Adiabatic Flow in A Variable Area Duct
2
0p
'''
h
22
2
2
2
M1TC
q
A
dA2
D
fdxM
M1
M2
11
M
dMγ
δγ
γ0
A
dA2
D
fdxM
M1
M2
11
M
dM
h
22
2
2
2
γ
γ
Sonic Point : M=1
0dxD
4f
22
11
A
dA
2
11
h
γγγ
04
22
1
2
1
dxD
f
A
dA
h
dxD
4f
2A
dA
h
γ
Throat Conditions
02
1
2
2
2
2
h
2
M2
11
M1
M
dM
D
fdxM
A
dAγ
γ
2
2
h
3
M1
M12
4D
fdxMdM
γ
γ
The capacity of Frictional throat is always lower than ideal throat!!!
The Real Nozzle for Sonic Flow
• It is impossible to get a sonic flow with real converging nozzle.
• The flow is always subsonic (transonic) at the throat.
• A compact real converging nozzle can produce transonic jet.
• A real nozzle for sonic exit is a CD Nozzle.
0p
p(x)
0
*
p
p
0
b
0
exit
p
p
p
p
Ideal Convergent-Divergent Nozzle Under Design Conditions
Convergent-Divergent Nozzle with High Back Pressure
pb1< p0 but > 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*
1/2
in0,
exit
in0,
in0,exit
in0,
exitin0, p
p1
ρ1
2A
ρ
ρρm
1γ
γ
γ
γp
exitexitexit uAρm
exitexitin0,
exit0.in uA
ρ
ρρm
1/2
0
exit
0
0exit
1
0
exit0 p
p1
ρ1
2A
p
pρm
1γ
γ
γ
γ
γp
in
in
RT
p
,0
,0in0,ρ
Mass Flow Rate at Higher Back Pressures
11
γ
γγ
γ
γ 2exit
exit
in0,
exit
in0, M2
11
T
T
p
p
Mass flow rate at high back pressure pb
1/2
0
b
0
0exit
1
0
b0 p
p1
ρ1
2A
p
pρm
1γ
γ
γ
γ
γp
At throat with high back pressure pb
1/2
0
t
0
0*
0
t0 p
p1
ρ1
2A
p
pρm
1
1
γ
γ
γ
γ
γp
1/2
0
t*
0
t
1/2
0
bexit
1
0
b
p
p1A
p
p
p
p1A
p
p
1
1
1 γ
γ
γγ
γ
γ
•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.•A decreasing exit pressure produces lowering throat pressure
Variation of Mass Flow Rate in Exit Pressure
0
b
p
p
0
e
p
p
1
1
0
criticalb,
p
p
0
criticale,
p
p
Variation of in Mass Flow Rate
0
b
p
p
m
1
0
criticalb,
p
p
chokedm
Further Decrease the Back Pressure till the throat is just choked….
Define this back pressure as critical back pressure, pb.critical.
1/2
0
**
1
0
*
1/2
0
criticalb,exit
1
0
criticalb,
p
p1A
p
p
p
p1A
p
p
11 γ
γ
γγ
γ
γ
1/2
0
*1
0
*
exit
*
1/2
0
criticalb,
1
0
criticalb,
p
p1
p
p
A
A
p
p1
p
p
11 γ
γ
γγ
γ
γ
This generates choked condition at the throat.This is the back pressure which produces maximum flow rate Through the nozzle.Is this back pressure same as design back pressure?If not??????
Pthroat = p*
Pexit = pb,critical
Variation of in Exit Pressure
0
b
p
p
0
e
p
p
1
1
0
criticalb,
p
p
0
criticale,
p
p
Variation of in Throat Pressure
0
b
p
p
0
throat
p
p
1
1
0
criticalb,
p
p0
*
p
p
Variation of in Mass Flow Rate
0
b
p
p
m
10
criticalb,
p
p
chokedm
What Next?
What happens if back pressure is further reduced?
Still the pressure at the exit plane is equal to the throat pressure?
Further lower pressure at throat !?!?!
Convergent-Divergent Nozzle with High Back Pressure
Pthroat< p*????
pb< pb,critical <p0
1/2
0
throat
0
throat
exit
*
1/2
0
criticalb,
1
0
criticalb,
p
p1
p
p
A
A
p
p1
p
p
1
1
1 γ
γ
γγ
γ
γ
It is impossible to have a pressure lower than p* at the throat. However at any downstream location the pressure can reach p*.
Pthroat = p*
1/2
0
*1
0
*
exit
1/2
0
criticalb,
1
0
criticalb,
p
p1
p
p
A
A(?)
p
p1
p
p
11 γ
γ
γγ
γ
γ
Find out the downstream location, x where p* can be achieved with lower back pressures.
pb < pb,critical <p0
?
M<1M<1
A=?
Feasible solutionInfeasible solution
0p
p(x)
0
*
p
p
0
designb,
0
exit
p
p
p
p
Convergent-Divergent Nozzle Under Off-Design Conditions
0
criticalb,
0
exit
p
p
p
p
0
criticalb,
0
exit
p
p
p
p
M>1M<1 M<1
Normal Shock : A large discontinuity
Moving Shock Towards Exit
1/2
0
*1
0
*
exit
1/2
1γ
γ
0
criticalb,
1
0
criticalb,
p
p1
p
p
A
A(?)
p
p1
p
p
1γ
γ
γγ
1/2
0
*1
0
*
exit
exit
1/2
0
exitshockb,
1
0
exitshockb,
p
p1
p
p
A
A
p
p1
p
p
11 γ
γ
γγ
γ
γ
0
exitshockb,
0
b
p
p
p
p
Flow Visualization Studies
0.66p
p
designb,
b
0.85p
p
designb,
b
Design Back Pressure
1.0p
p
designb,
b
Steady Cruising Design Conditions
Back Pressure Lower than the design conditions
0
designb,
0
b
p
p
p
p
Back Pressure Lower than the design conditions
Back Pressure Lower than the design conditions
