Lect-30 - NPTELnptel.ac.in/courses/101101002/downloads/Lect-30.pdf · Lect-30. In this lecture... • Subsonic and supersonic nozzles ... • For supersonic Mach numbers, ... than

1

Lect-30

Prof. Bhaskar Roy, Prof. A M Pradeep, Department of Aerospace, IIT Bombay2

Lect-30

In this lecture...

• Subsonic and supersonic nozzles• Working of these nozzles• Performance parameters for nozzles


Lect-30

Variation of fluid velocity with flow area

P0, T0

M=1(sonic)

P0, T0

M<1(subsonic)

M=1(sonic)

Sonic velocity will occur at the exit of the converging extension, instead of the exit of the original nozzle, and the mass flow rate through the nozzle will decrease because of the reduced exit area.


Lect-30

Variation of fluid velocity with flow area

M<1

Subsonic nozzle

M>1M>1

M<1

Subsonic diffuser

Supersonic nozzle Supersonic diffuser

P, T decreasesV, M increases

P, T increasesV, M decreases

P, T decreasesV, M increases

P, T increasesV, M decreases


Lect-30

Governing equations

( ){ }

{ }

{ } )1(2/)1(20

0

)1/(2

2/12

0

0

00

0

M)2/)1((1M

RTAPm

toreducesonsimplicationThisM)2/)1((1M)2/)1((1MA

RTP

TT

RTP

PP)MA(ARTM

RTPuA m

is nozzle the through flow mass Thenozzle. a through

flow gas perfect ycaloricall aconsider us Let

−+

−

−+=

−+

−+=

=

==

γγ

γγ

γγ

γγγ

γγρ


Lect-30

Isentropic flow through converging nozzles

• Converging nozzle in a subsonic flow will have decreasing area along the flow direction.

• We shall consider the effect of back pressure on the exit velocity, mass flow rate and pressure distribution along the nozzle.

• We assume flow enters the nozzle from a reservoir so that inlet velocity is zero.

• Stagnation temperature and pressure remains unchanged in the nozzle.


Lect-30


ReservoirP0, T0

Pe

Pb: back pressure

Pb = P0

Pb > P*

Pb = P*

Pb < P*

Pb = 0

P/P0

x

1

2

34

5

P*/P0

1

0

Choked flow


Lect-30


Pb/P0

•

m

P*/P0

1

2

345

1.0

max

•

m

345

1

2

P*/P0

1.0Pe/P0

The effect of back pressure Pb on the mass flow rate and the exit pressure Pe.


Lect-30


• From the above figure,

• For all back pressures lower that the critical pressure, exit pressure = critical pressure, Mach number is unity and the mass flow rate is maximum (choked flow).

• A back pressure lower than the critical pressure cannot be sensed in the nozzle upstream flow and does not affect the flow rate.

<

≥=

∗∗

∗

PPPPPP

Pb

bbe for

for


Lect-30

Nozzle efficiency

s

h

Pe

es

P0i

e

P0eT0i =T0t =T0e


Lect-30

Converging nozzles

i0es

e0e

esi0

ei0n

es

e

i0esi0

ei0n

T/T1T/T1

TTTT

,estemperaturingcorrespondtheoftermsIn.exitnozzlethe

atenthalpystaticisentropictheish,exitnozzletheatenthalpystaticactualtheish,inletnozzletheat

enthalpystagnationtheishwhere,hhhh

asdefinedisnozzleaofefficiencyThe

−−

=−−

=

−−

=

η

η


Lect-30

Converging nozzles

.conditionchokedunderoperatingisnozzlethe

,PP

PPIf

)))1/()1)((/1(1(1

PP

,thereforeisratiopressureThe)P/P(1

))1/(2(1,1M,flowchokedFor

a

i0

C

i0

)1/(nC

i0

/)1(i0C

n

<

+−−=

−+−

=

=

−

−

γγ

γγ

γγη

γη


Lect-30

Converging nozzles• If a convergent nozzle is operating under

choked condition, the exit Mach number is unity.

• The exit flow parameters are then defined by the critical parameters.

• To determine whether a nozzle is choked or not, we calculate the actual pressure ratio and then compare this with the critical pressure ratio.

• If the actual pressure ratio > critical pressure ratio, the nozzle is said to be choked.


Lect-30

Isentropic flow through converging-diverging nozzles

• Maximum Mach number achievable in a converging nozzle is unity.

• For supersonic Mach numbers, a diverging section after the throat is required.

• However, a diverging section alone would not guarantee a supersonic flow.

• The Mach number at the exit of the converging-diverging nozzle depends upon the back pressure.


Lect-30

xExitM

PAABCD

PB

PC

PDPE

PFPG

Pb

Pb

Pe

P0

P*

P0

P

Throat

Inlet Throat

Sonic flow at throat

Shock in nozzle

AB

CD

1.0

Subsonic flow at nozzle exitNo shockSubsonic flow at nozzle exitShock in nozzleSupersonic flow at nozzle exitNo shock in nozzle

Supersonic flow at nozzle exitNo shock in nozzle

Subsonic flow at nozzle exitShock in nozzle

Subsonic flow at nozzle exitNo shock


Lect-30

Converging-diverging nozzles

• The flow through nozzles is normally assumed to be adiabatic as the heat transfer per unit mass is much smaller than the difference in enthalpy between the inlet and outlet.

• The flow from the inlet to the throat can be assumed to be isentropic, but the flow from the throat to exit may not be due the possible presence of shocks.


Lect-30


)1/(/)1(

i0

en

e

i0

e0

i0

e

i0

e0

e

e0

i0

)1/(/)1(

i0

en

e0

e

/)1(i0e

/)1(e0e

i0es

e0e

esi0

ei0

esi0

ei0n

PP11

PP

PP

PP

PP

PP,Since

PP11

PP,Therefore

)P/P(1)P/P(1

T/T1T/T1

TTTT

hhhh

asdefinedisnozzleaofefficiencyThe

−−

−−

−

−

−−=⇒=

−−=

−−

=

−−

=−−

=−−

=

γγγγ

γγγγ

γγ

γγ

η

η

η


Lect-30


−

−=

−=−=

−=−=

−

−

γγ

γγ

ηγγ

ηη

η

/)1(

i0

ei0n

/)1(

i0

ei0npesi0np

esi0nei0e

PP1T

)1(R2

PP1Tc2)TT(c2

)hh(2)hh(2u

fromcalculatedbecanvelocityexitThe


Lect-30


{ }{ }

−−

−−

=

−

−

+−

=

=−

+=

==

−

−

−

γγ

γγ

γγ

ηη

γ

γγη

γγ

/)1(i0en

/)1(i0en

/)1(

i0

i2i

n2e

e

i02e

e

e0

e

2e

2e

2e2

e

)P/P(11)P/P(1

12

PP1M

211

12M

TTM

211

TT,Since

RTu

auMisnumberMachexitThe


Lect-30


))1(2/1(

2e

*i0

ee0

e

*

t

))1(2/1(

2e

2t

t

e

i0

e0

))1(2/1(

2e

2t

t

e

ot

e0

e

t

M)2/)1((1)2/)1(

PMP

AA

,1M,chokedisthroattheIfM)2/)1((1M)2/)1((1

MM

PP

M)2/)1((1M)2/)1((1

MM

PP

AA

,isareaexittheandareathroatthebetweenratiothe,throatuptoflowisentropicgminassu

alsoand,earlierdiscussedequationgoverningtheFrom

−+

−+

−+

−++

=

=

−+−+

=

−+−+

=

γγ

γγ

γγ

γγ

γγ

γγ


Lect-30


expansion. incomplete :pressure ambient the from different be to pressure exit nozzle the in

result may thisHowever control.under drag external the keep to is Thisunity. to possible as close as

/AA ratio area the keep to like would one design Byarea.

throat and etemperatur pressure, stagnationinlet the of function a is rate flow mass The

)2/)1((1

RTPAm

,bethereforewillrateflowmassThe

ei

)1(2/)1(*i0

*i0

*

−++= γγγ

γ


Lect-30


• Underexpanded nozzle:– Pe > Pa– The flow is capable of additional

expansion.– Expansion waves originating from the

lip of the nozzle.• Overexpanded nozzle:

– Pe < Pa– Shock waves originate from the nozzle

lip.


Lect-30


• Fully expanded nozzle:– Pe = Pa

– No shock waves/expansion waves.• If Pe << Pa

– Shock waves will occur within the divergent section of the nozzle.


Lect-30


Fully expanded: Pe=Pa Underexpanded: Pe=Pa

Overexpanded: Pe=Pa Pe<<Pa

Oblique shocks

Expansion waves


Lect-30

In this lecture...

• Subsonic and supersonic nozzles• Working of these nozzles• Performance parameters for nozzles


Lect-30

In the next lecture...

• Tutorial on intakes and nozzles

Documents

Lect-30 - NPTELnptel.ac.in/courses/101101002/downloads/Lect-30.pdf · Lect-30. In this lecture... • Subsonic and supersonic nozzles ... • For supersonic Mach numbers, ... than