Nozzles and Diffusers and converging diverging .... Example 1: Speed of sound
calculation. Determine the ... Example 2: a needle nose projectile traveling at a.
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