Area, Mass Flow Functions C-D Nozzles

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• Area, Mass Flow Functions

• C-D Nozzles

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Compressible Flow Relationships

• Mass Flow parameter

0

0

m VA

dm VdA AdV VAd

dm dA dV d

m A V

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• Area-Mach number differential relation

• Area-Mach number integral relation

222

11

MdA dV dpM

A V M p

1

2 121 2 1

11 2

AM

A M

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• What does subscript * mean?– For all flow variables it means when that

variable when M=1 [sonic]

– For area A* this is reference area for choking flow [M=1]

• Note this area is a minimum or throat

1

2 121 2 1

11 2

AM

A M

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Compressible Flow Relations

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222

11

MdA dV dpM

A V M p

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222

11

MdA dV dpM

A V M p

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Compressible Flow Examples

0 00 0

: 450 1890 1.5

3.671 6938 1.45 652.5

1.4 1716 450 1040

1.5 1040 1560

s s

s s

s s

Given T R p psf M

p Tp psf T R

p T

a a RT fps

V fps

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Compressible Flow Examples01 01 *

*

0 0

*

0

10 300 / 6

/ 6 0.097

1.006 9.94 1.002 299.4

1.4 287 299.4 346.8

33.6

/ 6 3.368

63.13 0.

s ss s

s s

ss

Consider isentropic flow in C D nozzle

p atm T K A A

Subsonic A A M

p Tp atm T K

p T

a a RT mps

V mps

Supersonic A A M

pp

p

0

1 /2 1

0 *

0

1584 3.269 91.77

192 646.7

2

1

ss

s

Tatm T K

T

a a mps V mps

p Am VA if choked

RT

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Compressible Flow

• Mass Flow Parameters:

VRT

P

A

m

AVm

cos

cos

1/ 2

0

0cos

Tm V g

PA RT TgRT

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Compressible Flow Relationships• Mass flow parameters

0 0

0 0

00 1

0 2 12

1/ 2

0 2

( , )

11

2

11

2ss

m VA

m p V pV RT M

A RT a RT

m T TpM f M

p A p T R

m T R MFP

p AM

m T RFP M M

p A

Note: FPo, FPs are similar, but different f[M] powers

0

( , ).79

F MFP p

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Compressible Flow Relationships• Mass flow parameters

00 1

0 2 12

01

0 2 12

11

2

11

2

m T R MFP

p AM

p A Mm

RTM

How to get more mass flow, i.e. greater thrust, more power?

1

2 10

0

, 1

2

1

if choked at throat M

p Am

RT

00

0

0

0

0

0

0

0

1

1716 /1.4

32.2

1.0888

RTmFP

p A g

m T gR

p A g

m T

p A

m T

p A

For air

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Compressible Flow Examples

01 01 *

*

0 0

1 /2 1

0 *

0

10 300 / 6

/ 6 3.368

63.13 0.1584 3.269 91.77

192 646.7

2

1

s ss s

s

Consider isentropic flow in C D nozzle

p atm T K A A

Supersonic A A M

p Tp atm T K

p T

a a mps V mps

p Am VA if choked

RT

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Example2

0 0

2*

*

00

0

2 ,

0.5 1 300

1.4, 0.5 1.340 1.49

( , ) 353.6 / sec

Air in duct of A m has flow such that

M p atm T K

AFor M A m

A

area to choke

p Am FP M kg

T R

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Static Pressure Mass Flow Parameter

• Defining: FP = Flow parameter=f(M)

• For Air

1/ 220 1

1cos 2s

RTmFP M M

PA g

01.0883coss

m TFP

PA

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Static Pressure Mass Flow Parameter

• Remember that

• can be inverted:

2/12

2

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MMFPs

2/1

2

1

1211

sFPM

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Total Pressure Mass Flow Parameter• Introducing P0:

• No explicit solution for M

• FPs is single valued, FPo is not

1/ 2

00

0 0cos

Tm P gPM

A P RT T

1/ 2

0 00

0 0cos

RT Tm PFP M

P A g P T

1

2 12

0

11

2FP M M

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Total Pressure Mass Flow Parameter

• FP0 is double valued

• FP0 maximizes at M = 1.0

• Maximum FP0 is 0.5787 for = 1.4

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Mass Flow Parameters

Be careful: FPs single valued, FPo double values

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Other Parameters

• Example:

m = 50 lb/sec A = 200 sq.in.

P0 = 14.7 psia = 30

T0 = 519 R

00

0

0.4869cos

RTmFP

P A g

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Example Solution to Mass Flow Parameter

• Rearrange FP0 equation:

1 / 2 / 12

0

11

2calc guessM FP M

0.5997M

Area, Mass Flow Functions C-D Nozzles

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