Chapter 12 Fluid Flow

12-1 The Basic Equation of Steady-Flow

12-1-1 The Conversation of Mass

A ： Cross section area of flow duct

c ： Velocity of fluid

v ： Specific volume of fluid

v

Acm

v

dv

c

dc

A

dA

Differential form:

const

12-1-2 The Conversation of Energy

2

2

1ch

)2

1( 2cddh

cdcdh

Differential form:

12-1-3 Process EquationUsually the fluid flows too fast in a duct to exchange heat with its surroundings, so it undergoes an adiabatic process, then:

pv k= const

0v

dvk

p

dp

Differential form:

12-1-4 Velocity of Sound and Mach Number

From

Then

p

av

pv

2

0v

dvk

p

dp

v

dvk

p

dp

v

pk

v

p

kpva

The Velocity of Sound is denoted by a

Mach Number Ma is defined as ：

a

cMa

kpv

c

Ernst Mach was born on Feb. 18, 1838, d. Vaterstetten. He found an experimental proof for the Doppler Effect (Christian Doppler) and by inspection of fast moving projects proposed the Mach's principle. The "Mach Number" named after him describes the relation of a body's velocity to sonic speed. His attitude was characterized by empirical thinking based on scientific findings. One of his strictest opponents and critics was M. Planck. After retiring from the University in 1901 he was appointed to the upper chamber of the Austrian Parliament, a post he held for 12 years. He died in 1916

12-2 The Fluid Flow in DuctDuring the fluid flow in a duct, the properties of fluid changes along the stream line. In most situations this can be treated as a one-dimensional flow

12-2-1 The Pressure and VelocityFrom conservation of energy

dh= - cdc

du +d(pv)= - cdc

du+pdv+vdp= -cdc

From the first law of thermodynamics:

δq=du+pdv

=0

Then: du+pdv+vdp= -cdc

vdp= - cdc

- vdp= cdc

To increase the velocity of fluid(dc>0), the pressure must be decreased. This kind of duct is called Nozzle

Or to decrease the velocity (dc<0) of fluid to obtain a high pressure in a duct flow. This kind of duct is called diffuser

12-2-2 Velocity and The Cross Section Area of Duct

From conservation of mass ：

v

dv

c

dc

A

dA

c

dc

v

dv

A

dA

0v

dvk

p

dp

kp

dp

v

dv

kpv

vdp

v

dv

v

dvk

p

dp

From the process equation

c dc- vdp

kpva

2a

cdc

v

dv

2a

cdc

v

dv

c

dc

a

c

v

dv2

2

c

dcMa

v

dv 2

c

dc

v

dv

A

dA c

dc

c

dcMa

A

dA 2

c

dcMa

A

dA)1( 2

As to a nozzle dc>0

c

dcMa

A

dA)1( 2

(1)If the fluid velocity is subsonic, then (Ma2-1)<0

Therefore: dA<0

The nozzle’s shape should be as following:

Subsonic flow

Subsonic flow

convergent nozzle

0)( dcc

dcMa

A

dA)1( 2

(2)If the fluid velocity is ultrasonic, then (Ma2-1)>0

Therefore: dA>0

The nozzle’s shape should be as following:

Ultrasonic flow

Ultrasonic flow

divergent nozzle

0)( dcc

dcMa

A

dA)1( 2

(3)If the nozzle’s inlet velocity is subsonic, but outlet velocity ultrasonic, then:

dA<0 → dA=0 → dA>0

The nozzle’s shape should be as following:

Subsonic flow Ultrasonic flow

convergent-divergent nozzle

throat

As to a diffuser dc<0

c

dcMa

A

dA)1( 2

(1)If the fluid velocity is subsonic, then (Ma2-1)<0

Therefore: dA>0

The diffuser’s shape should be as following:

Subsonic flowSubsonic flow

0)( dcc

dcMa

A

dA)1( 2

(2)If the fluid velocity is ultrasonic, then (Ma2-1)>0

Therefore: dA<0

The diffuser’s shape should be as following:

Ultrasonic flow Ultrasonic flow

0)( dcc

dcMa

A

dA)1( 2

(2) If the diffuser’s inlet velocity is ultrasonic, but outlet velocity subsonic, then:

dA<0 → dA=0 → dA>0

The diffuser’s shape should be as following:

Ultrasonic flow

Subsonic flow

12-2-3 Applications

Ram-jet engine

Diffuser(compressor)

combustion chamber

nozzle

Rocket

Space Shuttle

12-3 The Calculation of Nozzle

12-3-1 The Flux of Subsonic Nozzle

)(2 212 hhc

k

k

p

p

k

kRT1

1

21 11

2

We define P2 /P1 as compression ratio ε

k

k

k

kRTc

11

2 11

2

k

k

k

kRT

v

A

v

Acm

11

2

2 11

2

12-3-2 The Critical Compression Ratio For convergent-divergent nozzle, the velocity at throat keeps as sound-velocity. This state is called critical state. The flux of this kind of nozzle is depend on that of throat.

The Compression Ratio which is low enough to make the air flow at sound-velocity at the exit of nozzle is called the critical compression Ratio.

It is denoted by εc

1

1

2

k

k

c k

εc=

0.528 for air

0.546 for saturated steam

0.577 for superheated steam

1

11

2

1

2

12

v

p

kk

kAm

k

throat

The End of This Book

Thank You