Introduction to Compressible Flow Notes

Introduction to Compressible Flow Dρ ≠0 Dt The density of a gas changes significantly along a streamline

Compressible Flow Definition of Compressibility: the fractional change in volume of the fluid element per unit change in pressure p + dp

p

p

p

v

p + dp

v − dv

p + dp

p + dp

p

Compressible Flow 1. Mach Number:

M =

V local velocity = c speed of sound

2. Compressibility becomes important for High Speed Flows where M > 0.3 • M < 0.3 – Subsonic & incompressible • 0.3 0: indicating an increase in pressure in a converging channel. Subsonic Flow: M < 1 and dA > 0, then dP > 0 : indicating an increase in pressure in a diverging channel. Supersonic Flow: M > 1 dA > 0, then dP < 0 : indicating a decrease in pressure in a diverging channel.

P

P

P

P

P

P

P P

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Steady Isentropic Duct Flow – Nozzles Diffusers and Converging Diverging Nozzles Describes how the pressure behaves in nozzles and diffusers under various flow conditions

dA dp = (1 − M 2 ) ρV 2 A

(††)

Recall, the momentum equation here is: 0=

dp

ρ

dp

+ VdV

ρ

= −VdV (**)

Now substitute (**) into (††) : dA dV = (M 2 − 1) A V

Or,

(

dA A = M 2 −1 dV V

)

Nozzle Flow Characteristics

(

)

dA dV = M 2 −1 A V 1.

2.

3.

4.

Subsonic Flow: M < 1 and dA < 0, then dV > 0: indicating an accelerating flow in a converging channel. Supersonic Flow: M > 1 and dA < 0, then dV < 0: indicating an decelerating flow in a converging channel. Subsonic Flow: M < 1 and dA > 0, then dV < 0 : indicating an decelerating flow in a diverging channel. Supersonic Flow: M > 1 dA > 0, then dV > 0 : indicating an accelerating flow in a diverging channel.

Converging-Diverging Nozzles Amin Subsonic

Supersonic

M=1

Amax Subsonic Supersonic

M1

Subsonic Supersonic

Flow can not be sonic

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Choked Flow – The maximum possible mass flow through a duct occurs when it’s throat is at the sonic condition Consider a converging Nozzle: receiver po

pr

pe

To

Ve

ρo

plenum Mass Flow Rate (ideal gas): m& = ρ VA =

m& =

p M RT

p VA RT

M =

kRT A = p

V = c

V kRT

k MA RT

k MA RT

m& = p

Choked Flow Mass Flow Rate (ideal gas): m& = p

k MA RT

Recall, the stagnation pressure and Temperature ratio and substitute: k

po  k − 1 2  k −1 = M + 1 p  2  m& = p o

To k − 1 2 = M +1 T 2 k +1

k k − 1 2  2 (1− k )  MA  1 + M  RT o 2  

If the critical area (A*) is where M=1:

k +1

m& = p o A *

k  k + 1  2 (1− k )   RT o  2 

The critical area Ratio is: A 1 = A* M

 2 + (k − 1)M 2  k +1 

k +1

 2 ( k −1 )  

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