04(1)-Nozzle Flow Theory

8/9/2019 04(1)-Nozzle Flow Theory

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MAE 5347/4322

Rocket Propulsion

04-Nozzle Flow Theory


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• ASSUMPTIONS

– Steady, one-dimensional flow

– Adiabatic

– Frictionless

– Chemical equilibrium, established in combustion

chamber & does not change through nozzle

– Ideal gas model(TPG+CPG) (Thermal and Caloric)

– Axial exhaust velocity

Isentropic


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1. ENERGY EQUATION

h

p

V

h

S

2

2

V

0 0h p0

p

p

( , )h p

0

0

0

. 0

constant

constant

CS

h V ndA Q W

mmh const

h

2 21 2

0 1 2

2

2 2 1 1

2 2

2( )

V V h h h

V h h V

hopoV = 0

First Law with

Kinetic Energy

Term


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2

21 01 02 21

0

0

0

2 0 2

0,

and if we assume perfect gas with constant specific heats

( )

( )

2 ( )

2 (1 )

for perfect gas

)

:

2

2(

p

e p e

e p

p

NORMALLY V

p RT TPG

h C T CPG

V C

V

T T

T C T

T

R

h

C

h h h

V h h

0

0

1

8314.3univ.gas constant

mol. weight 1544 - f m

R J KG MOL K R R

M FT LB LB mol R

=

Exit


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Isentropic

• From First Law: (No K.E. in this derivation)dE = dQ + dW

• dW = -PdV (reversible work)

– Negative due to the convention that work done onthe system means that V2 < V1 in order for dW > 0,

must have a “-”

• dQ = TdS (reversible, constant T)dE = TdS – PdV and dH = TdS +VdP

– Prove dH formula by dH = dU + d(PV)


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Isentropic Relationship

• Ideal Gas Law PV = RT, P = RT/VdS = dE/T + PdV/T

dS = dH/T - RdP/P (R/P = V/T)

• CvdT = dE and CpdT = dHdS = CvdT/T + RdV/V (R/V = P/T)

dS = CpdT/T – RdP/P

• dS = 0:

Cv dT/T = -RdV/V

Cp

dT/T = RdP/P

Forgive my use of

Capitals vs. Lower

Case


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Isentropic (Cont)

R = CP - CV

= CP/CV

R/CV = ( - 1)• More Relationships, use Cp relationship

Cp ln(T2/T1) = Rln(P2/P1)

• Divide by Cv and use ln(T2/T1) = ( -1)ln(P2/P1)

• The rest can be proven with calculus….


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1

0 0

1

0

0

for isentropic flow in nozzle

T

T

21

1

e e

ee

and

p

p

p RV T

M p

From First Law and Isentropic

Assumptions


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Derive cp based on

• Algebra……prove it…and exercise for you to do

• cp = R/( - 1)

• cv = R/( - 1)

• cp and do not vary greatly with temperature

for air• Some exhaust gases can be modeled with the

proper


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2. ISENTROPIC FLOW RELATIONS

– Energy equation

– Assume CPG (Caloric)

2

02

V h h

2

0

2 2

0

2

2

2

1 12

21

-1 =1+

2

p p

p

V C T C T

T V V

T C T RT a

M

Cp Relationship for In Denominator

Important

Relationship

(Eq 3-12)


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1

1

1

1 2

0 0

1

1

0

2

0

isentropic flow

T -11+T 2

T -11+T 2

Reference: "Gas Tables", Keenan & Kaye

"Gasdynamics", Zucrow & Hoffman

e

e e

e

For

p M p

M

Tabulation for = 1.1,1.2,1.3,1.4 & 1.66


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Figure 3-9

Over expansion =

shocks

Under expansion =

expansion waves


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3. NOZZLE MASS FLOW RATE

– MASS FLOW RATE PARAMETER

ASSUME

– Steady,Quasi-one-dimensional flow

– Isentropic flow

– Perfect gas

0

0

p

T

,TH e A A

0

0

0 0

m VA

p M RT A

RT T p

p A p RT T


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1

1

2 1

2 2

0

0

0

20 0

-1 -11+ 1+

2 2

1

-11+2

m M p M M A RT

or

m T

M Ap RT M

0

0

m T

Ap

21 3 M

0 *

0 max

@ 1m T

M A A

Ap

In order to be

supersonic at

the exit, thethroat Mach

number must

be 1, choked

flow


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1

2 1

0

0

2 mass flow rate parameter =

1

*Nozzle Discharge Coefficient

C , ideal mass flow rated i

i

m T ideal

Ap R

mm

m

1

d C d

C

Re

.6

1

1

~.99

~1.15

W

T T

Low T

Used to

determine mass

flow rate


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4. AREA RATIO FUNCTION

– Consider a nozzle flow with fixed 0 0,m T and p

0

0

p

T m

1 2

0

02

0

01

and calculate

m T Ap

m T

Ap


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0

from the mass flow parameter

m T

0 A p2

0m T

0 A p1

R

2 M

R

1

2 1

1

2 1

22

2

11

2

1

2

22 2

21 11

-11+2.-1

1+2

solving for

-11+2.-1

1+2

M

M M

A A

M A M

A M M


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1

2 1

2

1

*

1 1

2

*

Now let () ~any point in nozzle

() ~sonic point in nozzle

~M =1; A =A

1 2 -1. 1+

1 2

A M

A M


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1

*

A

A

M>1

0 M1

M


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M1

0 p p

NOTE: recall from ideal compressible flow theory,

a C-D nozzle shape is required for continuous

acceleration from subsonic to supersonic

flow.

2

* Euler Eq.

* Velocity/area DE

11

dp V V

dx x

dV dAdx M dx


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5. THRUST

1

-1

1

-1

-1

0

0

*

0

0

1

2-12

*

0

0 0 0 *

-

2 * 1--1

2 *

1

2 21- -

-1 1

e e e a

ee

e e a e

F mV A P P

P RV T M P

A P M m

RT

P P P A F A P

P P P A

F f

* 00 0 *

0

, , , , ,

,

e a e

related

P P A A P

P P A

Independent of T M


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6. SPECIFIC IMPULSE

0 0 0

0 0 = ,

S

a

F mc c

I mg mg g

T P f

M P


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• THRUST COEFFICIENT ~IDEAL NOZZLE

0

0

p

T

TH e A A

a

e

p

p

*0

F

thrust coefficient

FC ( nozzle)

G e e e e a

G F TH

TH

Thrust m V A p p

Define

A A choked p A


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0 e a

0 0 0 0

0 e a

0 0 00

0 0

'

0

0

'

00

P P. -

P P

P P. -

P P

from nozzle energy eqation

V 2 2

= 2 1

2 1

e e e

TH TH

e e e

TH TH

e e p e

e N p

e e N p

m V T A

p A T A

m T V A

p A AT

h h C T T

T C T

T

V T C

T T


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1

2 1

1

2 1

1

2 1

0

0

1

e a

0 0 0

1

0

for choked isentropic flow

2

1

P P22 1 -

1 P P

and since .2 .21

2 21

1 1

TH

e e F N p

TH

p

e F

m T

A p R

p AC C

R p A

RC

R R

pC

p

e a

0 0

P P-

P P

e

TH

A

A


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F C

e a p p

e

TH

A

A

underexpanded

e a p p

overexpanded

e a p p

constante

a

p

p

Complete

expansion

Typical variation


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For constant occurs for complete

expansion

– Derivation•

• Physical argument

0 , MAX F

a

p NPR C

p

0 F

e

TH

C

A A

a

i

p

p


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Underexpanded e a p p

Loose available thrust

Overexpanded e a p p

Gain additional drag(+added weight)

Note: CF ~ thrust amplification by nozzle expansion as

compared to thrust produced by total pressure

exerted over throat area


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Ref; NAVWEPS Report 1488 “Handbook of Supersonic Aerodynamics”


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• OS/SEPARATED BL-

SUPERSONIC EXHAUST

OVEREXPANDED NOZZLE PERFORMANCE LOSS

(BL SEPARATION IN SUPERSONIC NOZZLE)

INVISCID FLOW

• NS IN NOZZLE WITH

SUBSONIC EXHAUST

VISCOUS CLOW

M>1 M


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FLOW MODEL

p p0

pS

x

CORE

MIXING REGION


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Ref; NAVWEPS Report 1488 “Handbook of Supersonic Aerodynamics”


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111 2

1

a

2

Thrust - Separated flow

2 2

C 11

coef

P

1

P

fic

+ -

P

ien

P

t

F

c

TH c c

s

s s

p

p

A

A


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04(1)-Nozzle Flow Theory

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