Finite Control Volume Analysis

Chap. 5

Finite Control Volume

Analysis

5.1 Conservation of Mass – The Continuity Equation

 Law of Conservation of Mass:

 To express the control volume basis, Reynolds transport equation with B=mass and b=1 will result in

Dt 0 Dm



“The amount of mass in a system is constant.”

 

  















 

 







 

 



 

 

1 cv cs

b

cs cv

sys sys

dA n V t dV

dA n V b t bdV

0 t bdV

Dt DB

 

 

 Conservation of Mass (Continuity Equation):



 



 



 



 



 





surface control

e through th

mass of

flow of

rate net volume

control the

of mass the

of change of

rate time system

the of mass the

of change of

rate time

  

 ^{   }



 

1 cv cs

b

sys

dV V n dA

t Dt

DB  

  ^{V n dA 0}

t

 dV 

  



   ^{ }

0 m

m

t

 dV 







   ^  ^

or

0 m

m

t

 dV 







   ^  ^

m 



 







 



 

















volume control

the leaving

flow mass

of Rate volume

control the

entering

flow mass

of Rate volume

control the

of inside mass

of

on accumulati of

Rate

Here, means mass flowrate(flux, [kg/s]) across the control volume surface.

  ^{V n dA A V Q}

m ^  

 ^  ^    

where is the average value normal to the control surface.

Q is called volume flowrate(flux, [m

/s]):

V  

A

dA n V V ^cs









V

A

Q 

 When the flow is steady, the continuity equation becomes

(if the flow has only one stream)

 If the steady flow is also incompressible,

(if the flow has only one stream) 0

m m

 



 ^{ }

2 2 2 1

1

1

A V A V

m     

0 Q

Q

 





2 2 1

1

V A V

A

Q  

Ex. 5.1

0 m

m

t ^cvdV ^out  ⁱⁿ 





 



zero in steady flow

Ex. 5.2

Severe changes in

pressure and temperature The flow should be

treated as compressible.

Ex 5.3

0 m

m

t

 dV 







   ^  ^

zero

Ex. 5.4

0 m

m

t ^cvdV ^out  ⁱⁿ 





 



 

t ^cvdV ^cs  





 

 

 

t ^cvdV ^out   ⁱⁿ  





 



0

 

A

dA n V V ^cs









Ex. 5.5

0 m

m

t ^cvdV ^out  ⁱⁿ 





 



0 m

t  ^cvdV ⁱⁿ 





 

0 V

dt A ) Vol ( d

j

j









Neglecting this column of water

0

Incompressible:

0 V dt A

) h A ( d

j j

B  

A_B 

j j

j

B

A V Q

dt

A dh  

B j

A Q dt

dh 



5.1.3 Moving, Nondeforming Control Volume

  ^{V n dA 0}

t

 dV 

  



   ^{ }

V 

W 

  

 ^{   }



 



sys

dV W n dA

0 t Dt

Dm  

  ^{W n dA 0}

t

 dV 

  



   ^{ }

The Reynolds transport equation for a moving control volume

where

V 

W 

V 

V

V

W   





Ex. 5.6

2021 )

971 ( 1050 V

V

W₂  ₂  _cv     Vcv

V W  





absolute velocity

V₂ V₁0

971 )

971 ( 0 V V

W₁  ₁  _cv     V_cv

₁

₂

Ex. 5.7

Ex. 5.8

0 m

m

t

 dV 







   ^  ^

0 )

m m

(

t dV

t ) Vol (

cv

   











 ^{ }







0

  ^{( Q Q ) 0}

A t

V 1

p







 

 





 





 ^{Q Q}  ⁰

V

A

 









1

2 2

1 leak 2

p

A

Q 1 . 0 Q

A Q

V Q 

 



  ^{A ( Q Q ) 0}

t

 







  

5.2 Newton’s Second Law

– The Linear Momentum and Moment of Momentum Equations

 Derivation of the Linear Momentum Equation If we let b= ,

That is,

V 

 

  



 







 

 

 

 

 

 

cs cv

sys sys

sys sys

dA n V V dV

t V

t F V dV m

t V Dt

DB

 





 



  

 ^{F  } _ ^ _t 

^{ V  dV }

^{ V  V   n  dA}

Linear momentum equation

    



 ^ _ ^  ^{    }

in

out A n

A n

cv

V dV V V dA V V dA

F  t   

or

  M

M

M

F  t   



 

 



where denotes the momentum flow.M

 

 



 

 

 



 

 

 



 

 







 







CV entering

flow

momentum of

Rate

CV leaving

flow

momentum of

Rate CV

of momentum of

n accumultio of

Rate volume

control the

on acting

forces

external of

Sum

 Looking into the momentum flux terms:

  ^{   }

V m V

m M

M

V A V V

A V dA

n V V M

M

in out

in out

in

n out

CS n in

out

 

 







 



 









 























x-component: M M









in out

in

n out

x n in

out ^{  }











y-component:

z-component:

Writing in components,

 V A  v  V A  v m v m v M

M

in out

in

n out

y n in

out ^

^





 ^

 ^











M

in out

in

n out

z n in

out ^{  }











  M

M

M

F  t   



 

 



  _ ^  ^ 

 

in out

cv

y

m v m v

t

F Mv  

x-component:

^{ }   _ ^  ^ 

in out

cv

x

m u m u

t

F Mu  

y-component:

z-component:

Therefore, the linear momentum equation becomes

 _  ^  ^ 

 

in out

cv

z

m w m w

t

F Mw  

Ex. 5.10

 

F t   



 

 









in out

x mu mu

F  

 

 

m 2

x

AV V cos AV V

F

2













 







0

p₁=p_atm=p₂ (Bernoulli equation across the straignt streamiles)

No net pressure force exerted on cv.

(Height difference was neglected.)

From the incompressible continuity equation, V₁=V₂



 



F_x   _{1 1} 







in out

z mw mw

F  

 ^  ^

 AV V sin

F

0 i

V V_{1 1}

 







V

V_{2 2}  









Ex. 5.11

 

F t   



 

 











 

in out

cv

z mw mw

t

F Mw  

in in out

out

w 2

2 1 1 A z

w m w

m

A p A p F F



 













P₂=0 p₁





w 2

2 1 1

A p A p A m w w

F      





w 2

2 1 1

A p A p A m w w

F       

k w w

; k w

w_{out 2} _{in 1}









2 1 2

1 1

1 A

w Q

; A

w  Q 

0

since

where

Thrust

Turbo-Jet 엔진 : 전투기용

압축기 연소실 배기 노즐

초음속용

소음 과다

연료 과소비

Turbo-fan 엔진 : 여객기용

아음속 용

소음 감소

효율 증가

Turbo-prop 엔진 : 프로펠러기용

Turbo-shaft 엔진 :헬리콥터용

헬기 로터

비행기에 작용하는 힘

Thrust Vectoring

Cobra

Bell 끄루크

Vertical Take-off of Harrier

JSF F-35 Joint Strike Fighter

JSF F-35

Vertical Take-off

Ship

V

 60 Knots (108km/h)

Waterjet

Waterjet Propulsion for Amphibian Tank

Waterjet Propulsion for Amphibian Tank

PNU ME CFD LAB.

Configuration of Waterjet

Waterjet Propulsion

Front oblique view Side view Near the rotor blade

Streamlines past the rotor of waterjet

Flow Analysis of Waterjet

Front oblique view Side view Near the rotor blade

Streamlines past the stator of waterjet

Flow Analysis of Waterjet

Pressure

Flow Analysis of Waterjet

Streamline

PNU ME CFD LAB.

PNU ME CFD LAB.

Armored Vehicle (Amphibian)

Armored Vehicle (Amphibian)

Armored Vehicle (Amphibian)

Ex. 5.12

in in out

out

2 2 1

1 A

y

v m v

m

A p A

p F

F













j v v

; j v

v_{out 2} _{in 1}









 



A

m v v A p p

F      





A

m v v p A p A

F



since

 





 

in out

cv

y mv mv

t

F Mv  

0

P_1,abs

Gage pressure

EX. 5.13

in in out

out

2 2 1 1 x

x

u m u

m

A p A p R

F



 











in in out

out 2

2 1

1

x

p A p A m u m u

R







 



x A p p m u u

R     



 



x

A p p m v v

R     

j v u

; j v

u_{out 2} _{in 1}





 





 

in out

cv

x mu mu

t

F Mu  

Using

RT AV AV p

m 



 



 







If R_x has negative value, R_x acts  direction

Ex. 5.14









 

in out

cv

z

m w m w

t

F Mw  

 

     ^{ }

   

0 2 2

1 1

CS 2 2

in in out

CS out out

in in out

out

2 2 1 1 z z

w R dr

r 2 w

w Aw dA

j j w j w

w m dA n

w w

w m w

m

A p A p R F





















































 









   

    







2 2 z

2

1 R w 2 r dr R w

A p 1

p

0 velocityconst across the cross-section

Q: p₁-p₂=?

2 2 1R 3 w

4

Ex. 5.15

 





 

in out

cv

x mu mu

t

F Mu  

0

in in out

out

2 2 1 1 th x

u m u

m

A p A p F

F



 













2 2 1

1

in in out

out 2

2 1

1 th

u u

m A

p A

p

u m u

m A

p A

p F



























1 1 1 1 1

1

1 A V

RT V p

A

m 



 











where

Ex. 5.16

 





 

in out

cv

x mu mu

t

F Mu  

0 R

b 2 H

F_x  1  ²  _x 

b 2 H

R_x  1 ²



(a) when the gate is closed

0 0 0

no flow

(b) when the gate is open ^{ }

 





in out

cv

x mu mu

t

F Mu  









f 2

x 2

x h b F u hb u u Hb u

2 R 1

b 2 H

F  1         









f 2

2

x h b F u hb u u Hb u

2 b 1 2 H

R  1       

If H>>h, u₁<<u_{2 x} ² h²b F_f





b 1 2 H

R  1      

Galilean transformation

Moving Control Volumes

If a control volume is moving with a constant velocity of , V 

  

 ^{   }



 

cs

cv A

V

sys

b dV b W n dA

t Dt

DB  

V

V

W   





V 

V 

where

: Relative velocity of the fluid relative to control volume

: Absolute velocity of the fluid : Velocity of the control volume

Vcv

V  V 

W 

W 

For an inertial (no accelaration),

moving, nonderofming control volume

 For steady linear constant translating, nondeforming control volume

  

 ^{   }



 

cs

cv A

V

sys

b dV b W n dA

t Dt

DB  

steady and b=

W 

  

 ^{F  }

^{ W  W   n  dA}

  

 ^F

^

^{ W}

^W

^{dA    }

out x

in

n x

out

nA W W A W mW mW

W

  



  

In component

  

 ^F

^

^{ W}

^W

^dA

^

^

^

^

^

^

^

^

  

 ^F

^

^{ W}

^W

^dA

^

^

^

^

^

^

^

^

Ex. 5.17

5.2.3 Moment-of-Momentum (Angular Momentum) Equation

 obtained by setting b  r V 





       

  _    



 













 

 







 

 



 

 

cs cv

sys sys sys sys

sys

dA n V V r dV

V t r

M F

r V

r t m

dV V t r

Dt DB

 

 

 



 

 

 

  _   _    





r

V dV

r

V V

n dA M t

F

r        



  ^{ }   ^{ }  

 

 

 

t

V r A V V

r A V dV

V t r

M

in cv out

in

n out

cv n sys

 

 

 

 

 

 

 













 

 















 

 



 



 



 If in steady flow,

    ^{ }  

 

 

m M

is that

V r A V V

r A V M

in out

sys

in

n out

n sys

 

 

 



 

 































,





Application of Angular Momentum Eq.

Euler (Pump and Turbine) Equation

  ^{ }   ^{ }  

 

 

 

t

k V r A V k

V r A V k

dV V t r

M

in cv out

in

n out

cv n z

 

 

 

 

 



 

 

 

 

















 

  



 



















 

  



 



 



 



   









2

in out

shaft

V r m V

r m

k V r

m k

V r

m T



















  





 

 

 

 

Torque:

   





m m m

1 1 1 2

2 2 shaft

V r V

r m

V r m V

r m T

2 1



 



























for steady state

Power transferred to the fluid by the rotor :





shaft

shaft

T m r V r V

W    



  

Since U=r,





shaft

m U V U V

W   



Euler (Pump and Turbine) Equation





shaft

m U V U V

W   



For a pump, , so is positive.

For a turbine, , so is negative.

1 1 2

2V U V

U _  _ ^W _shaft

1 1 2

2V U V

U  W _shaft

Velocity diagram:

V  U  W 





Rotor

velocity Relative velocity viewed from a coordinate

system fixed to the rotor Absolute

velocity





shaft

m U V U V

W   







shaft

m U V cos U V cos

W      

   





2 2

2

1 1 1

1 1 1

1 1

n 1

b r 2 sin

V

b r 2 sin

V b

r 2 V m

























Mass flowrate:

Ex. 5.18





shaft

m r V r V

T  



0

Ex. 5.19





shaft

m U V U V

W   



Since V₁=radial V_1=0





shaft

m U V U V

W   



0

5.3 First Law of Thermodynamics – The Energy Equation

 setting b=e (total energy per unit mass)

  







 

cs cv

sys

sys

e dV e V n dA

dV t t e

Dt

DB  

e=internal energy + kinetic energy + potential energy = ^{u }

^V

^{/ 2}

^gz

.

Reynolds transport eq:

in net in

net

W

Q   

 

in net in

cs net

cv

e dV e V n dA Q W

t

 

 











 

  

Hence,

 If considering the flow work by pressure in ,

 If the flow is steady and are uniform,

in

W 

 

in net shaft in

cs net

2

cv

gz V n dA Q W

2 V u p

dV t e



 







 



  

 



 



 

gz and 2 ,

, V , p u

2





 

in net shaft in

cs net 2

W Q

dA n V

2 gz V

u p         

 



  

 



in net shaft in

net in

2

out

2

(enthalpy) h

W Q

m 2 gz

V u p

m 2 gz

V

u p    















 



 



  

 





















 









in net shaft in

net in

2

out

2

W Q

m 2 gz

h V m

2 gz

h V    

 



 



 



  

 



 



   



in net shaft in

net in

2

out

2

(enthalpy) h

W Q

m 2 gz

V u p

m 2 gz

V

u p    















 



 



  

 



 

 





 

 







 

 





Ex. 5.20

in net shaft in

net in

2

out

2

(enthalpy) h

W Q

m 2 gz

V u p

m 2 gz

V

u p     













 



 



  

 





















 









Ex. 5.21

ve ' W

, Q

in net in

net   



when heat and work are added into the system

ve ' W

, Q

in net in

net   



in net shaft in

net in

2

out

2

W Q

m 2 gz

h V m

2 gz

h V    

 



 



 



  

 



 



  





when heat and work are extracted from the system

Ex. 5.22

5.3.3 Comparison of the Energy Equation with the Bernoulli Equation

in net in

2

out 2

Q 2 gz

V u p

2 gz V u p

m    















 



  

 

 



 



  

 

Energy equation:

m Q u

u 2 gz

V gz p

2 V

p _in^net

in out

quation Bernoullie

in 2

out 2



















 











 









 



 



  

 



 



  



If the flow is incompressible and inviscid (i.e., frictionless), there is no changes in internal energy and no heat transfer occurs.

Assumptions for Bernoulli eq: zero -Steady state

-Inviscid flow

-Incompressible flow -Along the streamline

 Energy equation:

loss energy

w 2 gz

V gz p

2 V p

in net shaft

equation

Bernoulli

in 2

out

2

  



 



  

 

 

 



  





        







 



m W w ^{net in}

shaft

in net shaft







m / Q u

u loss energy

in net in

out   

  



L s

head unit in the equation

Bernoulli

in 2

out 2

h h

g z 2 V z p

g 2 V

p   



 



  

 

 

 



  





          



 



W g

m W g

w

h ^{net in}

shaft in

net shaft in

net shaft

s    







g loss Energy

h_L  :head loss

Ex. 5.24

Ex. 5.25