fluid equations figures tables

20
!"#$%&%$ (#%)*+ !"#$%&%$ ,-./%+0 1!,2 3%4#5.+%$ 6%7$87%+0 !  = " g  SG = "  H 2 O @4 ! C "  v = µ "  9-#77:-# 6.-%.+%84 ;<#.= ,.7 >.(  p = !  h + p o   p = "  RT  ?0<-87+.+%$ @8-$# 84 . 9=.4# !:-&.$# F  R = !  h c  A   y = y c +  I  ˆ  x  ˆ  x  Ay c    x = x c +  I  ˆ  x  ˆ  y  Ay c  A#-48:==% BC:.+%84 D.77 @=8( E.+# 68=:5# @=8( E.+# F/#-.)# 6#=8$%+0  p !  + V 2 2g + z = const  ˙ m =  " V  #d A  A $  Q =  V  #d A  A $  V = " V  #d A  A"  E#048=<7 G-.47"8-+ G*#8-#5  DB SYS  Dt = ! ! t "# d $ +  " CS % CV %  # V d A  H84+%4:%+0 BC:.+%84 0 = ! ! t " d # +  " V d A CS $ CV $  >%4#.- D85#4+:5 BC:.+%847 %4 H.-+#7%.47 H88-<%4.+#7 F  x = F Sx + F  Bx = ! ! t u" d # +  u" V d A CS $ CV $  F  y = F Sy + F  By = ! ! t v" d # +  v" V d A CS $ CV $  F  z = F Sz + F  Bz = ! ! t w" d # +  w" V d A CS $ CV $  >%4#.- D85#4+:5 BC:.+%84 &8- . H84+-8= 68=:5# 58/%4) (%+* $847+.4+ /#=8$%+0 F S + F  B = ! ! t V  xyz " d # +  V  xyz " V  xyz d A CS $ CV $  

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Page 1: Fluid Equations Figures Tables

8/11/2019 Fluid Equations Figures Tables

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!"#$%&%$ (#%)*+ !"#$%&%$ ,-./%+0 1!,2 3%4#5.+%$ 6%7$87%+0

!  = " g   SG =

 H 2O@4!

C " 

  v =µ 

"  

9-#77:-# 6.-%.+%84 ;<#.= ,.7 >.(

 p = !  h +  po 

 p =  "  RT  

?0<-87+.+%$ @8-$# 84 . 9=.4# !:-&.$#

F  R= !  h

c A   !  y = yc +

 I  ˆ x ˆ x

 Ayc

  !  x = xc +

 I  ˆ x ˆ y

 Ayc

 

A#-48:==% BC:.+%84 D.77 @=8( E.+# 68=:5# @=8( E.+# F/#-.)# 6#=8$%+0

 p

!  +

V 2

2g + z = const   m =   " V  #dA A

$    Q =   V  #dA A

$    V  =

" V  #dA

 A"   

E#048=<7 G-.47"8-+ G*#8-#5

 DBSYS 

 Dt =

! t "# d $+   " 

CS 

% CV 

%    # VdA  

H84+%4:%+0 BC:.+%84

0 =! 

! t  " d #+   " VdACS $ CV $   

>%4#.- D85#4+:5 BC:.+%847 %4 H.-+#7%.47 H88-<%4.+#7

F  x= F 

Sx+ F 

 Bx=

! t u" d #+   u" VdA

CS 

$ CV 

$   

F  y= F Sy + F  By =! 

! t v" d #+   v" VdA

CS 

$ CV 

$   

F  z= F 

Sz+ F 

 Bz=

! t w" d #+   w" VdA

CS 

$ CV 

$   

>%4#.- D85#4+:5 BC:.+%84 &8- . H84+-8= 68=:5# 58/%4) (%+* $847+.4+ /#=8$%+0

F S  + F  B =! 

! t V  xyz" d #+   V  xyz" V  xyzdA

CS 

$ CV 

$   

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D8/%4)I J84K<#&8-5%4) H84+-8= 68=:5#

"# $%& '()$*(+ ,(+-.& .(,&/ 0$ 0 '()/$0)$ ,&+('1$2 then! 

VCV  !V  xyz =V  "

VCV 

 

3%&*&4 V CV 5 '()$*(+ ,(+-.& ,&+('1$26 V  xyz 5 *&+0$1,& ,&+('1$2 31$% *&/7&'$ $( $%& !" 0)8 V 5 09/(+-$&

,&+('1$2: ;%& *&+0$1,& ,&+('1$2 V  xyz 1/ $%& #+-18 ,&+('1$2 /&&) 92 0) (9/&*,&* .(,1)< 31$% $%& '()$*(+

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F Sx + F  Bx !   arefx" d # =$ 

$ t u xyz" d #+   u xyz" V  xyzdA

CS 

% CV 

% CV 

%   

F Sy + F  By !   arefy" d # =$ 

$ t v xyz" d #+   v xyz" V  xyzdA

CS 

% CV 

% CV 

%   

F Sz + F  Bz !   arefz" d # =$ 

$ t w xyz" d #+   w xyz" V  xyzdA

CS 

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r ! F S + (r ! g)CV 

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  ! 

V# VdACS 

" CV 

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!@A

 p1

!  + V 1

2

2g+ z

1+ H  p =  p2

!  + V 2

2

2g+ z

2+ H t  + H  L  

98(#- )#4#-.+#< M0 +*# +:-M%4#

WT  = mgH T !T    3%&*& !T 

4 &##1'1&)'2

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98(#- -#C:%-#5#4+ M0 . ":5"

WP =mgH P

!P

=

Q"   H P

!P

  3%&*& !P

4 &##1'1&)'2

D8+%84 8& . @=:%< 9.-+%$=# 13%4#5.+%$72

V=

u( x, y, z,t ) ˆi+

v( x, y, z,t ) ˆ j+

w( x, y, z,t ) ˆk

a = D

V

 Dt = u

! ! 

V

!  x+ v

! ! 

V

!  y+ w

! ! 

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!  z+

! ! 

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! t  

N%&&#-#4+%.= H84+%4:%+0 BC:.+%84

N%&&#-#4+%.= D85#4+:5 BC:.+%847 &8- . J#(+84%.4 @=:%<O J./%#-K!+8P#7 BC:.+%847

N%5#47%84.= F4.=07%7 Q N#+#-5%4.+%84 8& 9% G#-57

?:  B1/$ 0++ ,0*109+&/ $%0$ 0*& 1),(+,&8 1) $%& 7*(9+&.:

@: 

C=7*&// &0'% ,0*109+& 1) $&*./ (# 90/1' 81.&)/1()/:

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P: 

Q%&'M 0++ *&/-+$1)< G1 $&*./ $( .0M& /-*& $%&2 0*& 81.&)/1()+&//:

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;4+#-4.= ;4$85"-#77%M=# 6%7$8:7 @=8( 1@=8( %4 9%"#2

D.R8- ?#.< =877#7

 H loss =   f   l

 D

!  "  # 

$  %  &  V 

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2g  E0*'2 S T&1/90'% &F-0$1()

#(* +0.1)0* #+(36  f  =!(Re)   #(* $-*9-+&)$ #+(36  H l =   f  = !   Re, "  D

 

$  % 

'  (  ;  Re = ! 

VD

µ   

CF-0$1()LU0/&8 Q(**&+0$1()/ #(* $%& E0*'2 K*1'$1() K0'$(*4

D%48- ?#.< >877#7

 H loss =   K i!  V 

2

2g  3%&*& K 

i LV +(// '(&##1'1&)$  

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 H lossm =   f   Le

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BS+#-4.= @=8(

A8:4<.-0 >.0#- &=8( 8/#- . &=.+ "=.+#

$%&'(%) *+,-

 x

=

5.48

Re x

; C  f  =0.730

Re x

  (using a velocity profileu

= 2  y

"  

#  

$ %  

&  

'  (  y

"  

#  

$ %  

&  

'  2

./)0/+1(2 *+,-   >

 

5!105

< Re x 

<107A

 x=

0.382

Re x1 5

; C  f  =0.0594

Re x1 5

  (using a velocity profileu

U =

 y

"  #  $ 

%  &  '  

1 7

)

 

3%&*& $%& IM1) #*1'$1() '(&##1'1&)$ !  *  1/ <1,&) 92 C  f  =! w

1

2" U 2

 

N-.) H8#&&%$%#4+  

C  D=

F  D

1

2! V 

2 A

=

" wdA

PS 

# 1

2! V 

2 A

; where PS: plate surface 

9:-# @-%$+%84 N-.)O @=8( 8/#- . &=.+ "=.+# ".-.==#= +8 +*# &=8(

$%&'(%) *+,- O C  D=

1.33

Re L

T

./)0/+1(2 *+,-3  C  D=

0.0742

Re L

1 5  (valid for 5 !10

5<Re

 L<10

7)  

C  D=

0.455

(logRe L

)2.58  (valid for Re

 L<109) 

4,) % 5,/(6%)7 +%71) 28%2 '9 '('2'%++7 +%&'(%) %(6 /(61):,19 2)%(9'2',( %2 9,&1 ;,'(2 ,( 281 ;+%21 

C  D=

0.0742

Re L

1 5  !

1740

Re L

  (valid for 5 "105<Re

 L<10

7)  

C  D=

0.455

(logRe L

)2.58

  !1610

Re L

  (valid for 5 "105<Re

 L<10

9)  

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W&(.&$*1' G*(7&*$1&/ (# /(.& '(..() /%07&/

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;09+& X:Y G*(7&*$1&/ (# X1* 0$ X$.(/7%&*1' G*&//-*& >Z:I: Q-/$(.0*2 Z)1$/A

;09+& X:?[ G*(7&*$1&/ (# X1* 0$ X$.(/7%&*1' G*&//-*& >I" Z)1$/A

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;09+& X:P G*(7&*$1&/ (# T0$&* >Z:I: Q-/$(.0*2 Z)1$/A

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;09+& X:R G*(7&*$1&/ (# T0$&* >I" Z)1$/A

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