Download pdf - Internal Flow Presentation

8/9/2019 Internal Flow Presentation

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!"#$%"&' )'*+,)-./0 123456/37 !!


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General Concept of Flows in Pipe

! As a uniform flow enters a pipe, the velocity atthe pipe walls must decrease to zero (no-slipboundary condition). Continuity indicates that thevelocity at the center must increase.

! Thus, the velocity profile is changing continuouslyfrom the pipe entrance until it reaches a fullydeveloped condition. This distance, L, is called the

entrance length.

Dr. M. Khosravy


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! For fully developed flows ( x>>L), flows becomeparallel, , the mean pressure remainsconstant over the pipe cross-section

v = (u(y),0,0)


Dr. M. Khosravy


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" )-897 /6 5 -86: ;/;2 595? =>8@ ;/;2 26A>5632 560 2B/A

>2:/86C !""#D 5>2 A42 -/@/A >27.-A7 8= E8.605>? -5?2> F897G

#42>2 5>2 A98 A?;27 8= ;/;2 F897H -5@/65> 560 A.>E.-26A

Dr. M. Khosravy


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" I>/J35- %2?68-07 6.@E2> D=8> F89 /6 5 >8.60 ;/;2

%2 K LMNN H -5@/65>

LMNN O %2 O PNNN H A>567/J865-

%2 Q PNNN H A.>E.-26A

" "8A2 A45A A4272 R5-.27 5>25;;>8B/@5A2G

" )8> 5 :/R26 5;;-/35J86C %23> 02;2607.;86

" S/;2 >8.:46277" T/E>5J867

" U;7A>25@ F.3A.5J867C 0/7A.>E56327EA42 F89D


5

Dr. M. Khosravy


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" +42A42> A42 F89 /7 -5@/65> 8> A.>E.-26A 02;2607 86 A42

%2?68-07 6.@E2>C 942>2 $% /7 A42 3>877V723J865- @256

R2-83/A? 02W620 E?

" #>567/J86 =>8@ -5@/65> A8 A.>E.-26A =8> F897 /6 3/>3.-5> ;/;2 8=

0/5@2A2> & 833.> 5A '(XLMNN

U m =

1

Au dA

A

!!

Dr. M. Khosravy

U m

U m


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" )8> ;/;27 9/A4 R5>/5E-2 0/5@2A2>C % /7 7J-- A42 [email protected] A8 38672>R5J86 8= @577C E.A ) * , ) -

D2

V2

2

1

V1

D1

m m

7

Dr. M. Khosravy


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" +426 ;/;2 F89 /7 A.>E.-26AG #42 R2-83/A? /7 .67A250/-? >5608@

5608@-? 9/A4 J@2DC A42 F89 /7 345>53A2>/Y20 E? A42

@256 5:20D R2-83/A? 02W620 57H

" Z.2 A8 A.>E.-26A @/B/6:C A42 R2-83/A? ;>8W-2 8= A.>E.-26A ;/;2

F89 /7 @8>2 .6/=8>@ A426 A45A 8= -5@/65> F89G

v ( y) = limT !"

12T

v( y, t )dt #T

T

$

Dr. M. Khosravy


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Dr. M. Khosravy

" [2632C A42 @256 R2-83/A? :>50/26A 5A A42 95-- =8> A.>E.-26A

F89 /7 -5>:2> A456 -5@/65> F89G

" #42 95-- 7425> 7A>277C C/7 5 =.63J86 8= A42 R2-83/A? :>50/26AG

#42 :>25A2> A42 3456:2 /6 9/A4 >27;23A A8 ? 5A A42 95--C A42

4/:42> /7 A42 95-- 7425> 7A>277G #42>2=8>2C A42 95-- 7425> 7A>277

560 A42 =>/3J865- -87727 5>2 4/:42> /6 A.>E.-26A F89G

w!

u



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Poiseuille Flow

Dr. M. Khosravy

" I867/02> A42 7A250?C =.--? 02R2-8;20 -5@/65> F89 /6 5 7A>5/:4A

;/;2 8= 3/>3.-5> 3>877 723J86 9/A4 3867A56A 0/5@2A2>C &G

" #42 388>0/65A2 /7 348726 7.34 A45A ! /7 5-86: A42 ;/;2 560 . /7 /6

A42 >50/.7 0/>23J86 9/A4 A42 8>/:/6 5A A42 326A2> 8= A42 ;/;2G

!

Db

y

x1 p 2 p


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Dr. M. Khosravy

" )8> 5 386A>8- R8-.@2 8= 5 3?-/602> 625> A42 ;/;2 326A2>C A42

E5-5632 8= @8@26A.@ /6 /6A2:>5- =8>@ /6 ! V0/>23J86 >2\./>27

A45A A42 ;>277.>2 =8>32C

53J6: 86 A42 =5327 8= A42 3?-/602> E2 2\.5- A8 A427425> 7A>277 53J6: 86 A42 3/>3.@=2>26J5- 5>25C 42632

" !6 5338>05632 9/A4 A42 -59 8= =>/3J86


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" #42>2=8>2H

9426 /7 3867A56A


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Poiseuille Flow

Dr. M. Khosravy

" #42 R2-83/A? 0/7A>/E.J86 /7 ;5>5E8-/3 8R2> A42 >50/.7C 560 A42 @5B/@.@

R2-83/A? 86 A42 ;/;2 5B/7 E238@27H

" #42>2=8>2C

" #42 R8-.@2 F89 >5A2 /7H

um = ! 1

4 µ

dp

dx R

2

u

um

=1! y

2

R2

Q = u( y)2! ydy" = # 14µ " dp

dx R2 # y

2( )2! ydy

Q = # !

8µ

dp

dx R4 =

1

2um! R

2


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Poiseuille Flow

Dr. M. Khosravy

" #42 F89 >5A2 /7 ;>8;8>J865- A8 A42 W>7A ;892> 8= A42 ;>277.>2 :>50/26A 560A8 A42 =8.>A4 ;892> 8= A42 >50/.7 8= A42 ;/;2G

"

Z2W62 @256 R2-83/A? 57

" #42>2=8>2C

" #4/7 78-.J86 833.>7 /6 ;>53J32 57 -86: 57C

U m =

Q

! R2

U m =1

2um =

1

8 µ ! dp

dx

" # $

% & ' R

2

Re =U m

D

v


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Poiseuille Flow

Dr. M. Khosravy

" #42 >2-5J86 E2A9226 A42 62:5JR2 ;>277.>2 :>50/26A 560 A42 @256 R2-83/A?8= A42 F89 /7 >2;>2726A20 /6 26:/622>/6: 5;;-/35J86 E? /6A>80.3/6: 5>27/7A5632 382]3/26A 8= ;/;2 F89C 5 G

" #4/7 382]3/26A /7 5 686V0/@267/865- 62:5JR2 ;>277.>2 :>50/26A .7/6: A420?65@/3 4250 57 ;>277.>2 735-2 560 A42 ;/;2 0/5@2A2> 57 -26:A4 735-2C /G2GC

" !6A>80.3/6: A42 5E8R2 2B;>277/86 =8>


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Poiseuille Flow

Dr. M. Khosravy

" &A A42 95--C

" ,8C

" &7 5 >27.-AC A42 95-- =>/3J86 382]3/26A /7H

! w = !

µ du

dy y= R

= ! µ dp

dx

2 y

4 µ y= R

= !

R

2

dp

dx

C f =! w

1

2 " U m

2

=

f

4=

16

Re

2! w

R= !

dp

dx=

f

D

1

2 " U m

2


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Head Loss in Pipe

Dr. M. Khosravy

"

)8> F897 /6 ;/;27C A42 A8A5- 262>:? ;2> .6/A 8= @577 /7

:/R26 E? 942>2 A42 38>>23J86 =53A8> /7

02W620 57C

9/A4 E2/6: A42 @577 F89 >5A2 560 & /7 A42 3>877 723J865-

5>25G

p

! +" U

m

2

2+ gy

! " #

$ % &

! = ! u2

" udA A" !mU

m

2

!m = ! Q

!


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Head Loss in Pipe

Dr. M. Khosravy

"

,8 A42 A8A5- 4250 -877 E2A9226 723J86 ^ 560 L 8= ;/;27 /7H

" 4-X4250 -877 0.2 A8 =>/3J865- 2_23A7 /6 =.--? 02R2-8;20 F89 /6 3867A56A

5>25 3860./A7

" 4-@X@/68> -87727 0.2 A8 26A>56327C W`6:7C 5>25 3456:27C 2A37G

hlt =

p1

! +"

1

U m12

2+ gy

1

#

$

%%

&

'

( ( )

p2

! +"

2

U m22

2+ gy

2

#

$

%%

&

'

( (

hlt = h

l + h

lm


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Head Loss in Pipe

Dr. M. Khosravy

"

,8C =8> 5 =.--? 02R2-8;20 F89 A4>8.:4 5 3867A56AV5>25 ;/;2C

" &60 /= . *1. -C

hl = p

1 ! p

2

!

+ g( y1 ! y2 )

hl = p1 ! p2

! = " p

!


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Head Loss in Pipe

Dr. M. Khosravy

" )8> -5@/65> F89C

" [2632

!dp

dx=

" p

L= !

32 µ U m

D2

hl =! p

! =

32 L µ U m

D2 =

64 µ

! U m D

L

D

U m2

2=

64

Re

L

D

U m2

2

hl = f L

D

U m2

2


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Turbulent Pipe Flow

Dr. M. Khosravy

" )8> A.>E.-26A F897a 92 35668A 2R5-.5A2 A42 ;>277.>2 0>8;565-?J35--?G +2 @.7A .72 2B;2>/@26A5- 05A5 560 0/@267/865-565-?7/7G

" !6 =.--? 02R2-8;20 A.>E.-26A ;/;2 F89C A42 ;>277.>2 0>8;C C 0.2 A8=>/3J86 /6 5 48>/Y86A5- 3867A56AV5>25 ;/;2 /7 b689 A8 02;260 86H

" S/;2 0/5@2A2>C &

" S/;2 -26:A4C #

" S/;2 >8.:46277C (" &R2>5:2 F89 R2-83/A?C $%

" )-./0 0267/A?C

" )-./0 R/7387/A?C

p!

!

µ


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Turbulent Pipe Flow

Dr. M. Khosravy

"

#42>2=8>2C

" Z/@267/865- 565-?7/7C

" $B;2>/@26A7 7489 A45A A42 686V0/@267/865- 4250 -877 /7 0/>23A-?;>8;8>J865- A8 'cZC 42632

! p ! U m

2 ="

1 µ

! U m D, L D

, e D" # $ % & '

hl =! p !

( h

l

! U m2 ="

1 Re ,

L

D, e

D

" # $

% & '

! p = ! p D, L, e,U m, ! , µ ( )

hl

U m

2/ 2

=

L

D!

2Re,

e

D

! " #

$ % &


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Turbulent Pipe Flow

Dr. M. Khosravy

"

Z2W6/6: A42 =>/3J86 =53A8> 57C C 42632

942>2 5 /7 02A2>@/620 2B;2>/@26A5--?G

" #42 2B;2>/@26A5- >27.-A 5>2 .7.5--? ;-8d20 /6 5 345>A 35--20 1880?

Z/5:>5@G

f =! 2 Re , e

D! " #

$ % &

hl = f L

D

!

" #

$

% &

U m2

2

!

" #

$

% &


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Moody Diagram

Dr. M. Khosravy

24


25/40

Turbulent Pipe Flow

Dr. M. Khosravy

"

!6 8>02> A8 78-R2 A42 ;/;2 F89 ;>8E-2@7 6.@2>/35--?C 5

@5A42@5J35- =8>@.-5J86 /7 >2\./>20 =8> A42 =>/3J86 =53A8>C 5 C /6

A2>@7 8= A42 %2?68-07 6.@E2> 560 A42 >2-5JR2 >8.:46277G

" #42 @87A 9/02-? .720 =8>@.-5 =8> A42 =>/3J86 =53A8> /7 A45A 0.2 A8

I8-2E>88bC

"

#4/7 56 /@;-/3/A 2\.5J86C 78 /A2>5J86 ;>8320.>2 /7 622020 A8

02A2>@/62G

1

f =

!2log

e / D

3.7

+2.51

Re f

"

# $

%

& '


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Turbulent Pipe Flow

Dr. M. Khosravy

" 1/--2> 7.::27A20 A8 .72 =8> A42 /6/J5- 27J@5A2C

" #45A ;>80.327 >27.-A7 9/A4/6 ^e /6 5 7/6:-2 /A2>5J86

f o = 0.25 log e / D

3.7 +

2.51

Re 0.9

!

" #

$

% &

'

()

*

+,

-2


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Minor Loss

Dr. M. Khosravy

" #42 @/68> 4250 -877 @5? E2 2B;>27720 57C

942>2 A42 -877 382]3/26AC 9 C @.7A E2 02A2>@/620 2B;2>/@26A5--? =8> 2534

3572G

" 1/68> 4250 -877 @5? E2 2B;>27720 57

942>2 #( /7 56 2\./R5-26A -26:A4 8= 7A>5/:4A ;/;2

hlm =

f Le

D

! " #

$ % & U

m

2

2

! " #

$ % &

hlm

=

KU m

2

2


28/40

Minor Loss

Dr. M. Khosravy

! Source of minor loss:

1. Inlets & Outlets

2. Enlargements & Contractions3. Valves & Fittings

4. Pipe Bends


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1/68> '87727

" #8A5- 4250 -877 /6 5 7?7A2@ /7 38@;>/720 8= @5f8>-87727 -87727


30/40

30

Dr. M. Khosravy


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31

Dr. M. Khosravy


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!

Pipe flow results sometimes can be used for non-circular ducts or open channel flows to estimate thehead loss

! Use Hydraulic Diameter,

A - Cross section area; P - Wetted perimeter

! For a circular duct,

! For rectangular duct,

where Ar =b/a is the geometric aspect ratio

Non-Circular Ducts

Dr. M. Khosravy

P

A Dh

4=

D

D

Dh ==!

! 4/ 42

1(2

4

)(2

4 a

ba

abh

+

=

+

=


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Non-Circular Ducts

Dr. M. Khosravy

! Effect of Aspect Ratio (b/a):# For square ducts:

# For wide rectangular ducts with b>>a:

Thus, flows behave like channel flows#

However, pipe flow results can be used with goodaccuracy only when:

a=b Ar=1 Dh=a

Ar ! Dh! 2a!

1/3


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S/;/6: "2A98>b7 560 S.@; ,2-23J86

" #98 :262>5- A?;27 8= 62A98>b7

" S/;27 /6 72>/27

" T8-.@2 F89 >5A2 /7 3867A56A

" [250 -877 /7 A42 7.@@5J86 8= ;5>A7

" S/;27 /6 ;5>5--2-

" T8-.@2 F89 >5A2 /7 A42 7.@ 8= A4238@;8626A7

" S>277.>2 -877 53>877 5-- E>563427 /7 A42 75@2

34

Dr. M. Khosravy


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S/;/6: "2A98>b7 560 S.@; ,2-23J86

"

)8> ;5>5--2- ;/;27C ;2>=8>@ IT 565-?7/7 E2A9226 ;8/6A7& 560 g

" ,/632 !; /7 A42 75@2 =8> 5-- E>563427C 4250 -877 /6 5--E>563427 /7 A42 75@2

35

Dr. M. Khosravy


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S/;/6: "2A98>b7 560 S.@; ,2-23J86

" [250 -877 >2-5J8674/; E2A9226 E>563427 5--897 A42 =8--89/6: >5J87 A8 E202R2-8;20

" %25- ;/;2 7?7A2@7 >27.-A /6 5 7?7A2@ 8= 686V-/625> 2\.5J867G T2>? 257? A878-R2 9/A4 $$,h

" "8A2H A42 565-8:? 9/A4 2-23A>/35- 3/>3./A7 748.-0 E2 8ER/8.7

" )-89 F89 >5A2 >26A

" S>277.>2 :>50/26A /35- ;8A26J5-


37/40

S/;/6: "2A98>b7 560 S.@; ,2-23J86

" +426 5 ;/;/6: 7?7A2@ /6R8-R27 ;.@;7 560c8> A.>E/627C;.@; 560 A.>E/62 4250 @.7A E2 /63-.020 /6 A42 262>:?2\.5J86

" #42 .72=.- 4250 8= A42 ;.@; A42 4250 2BA>53A20E? A42 A.>E/62 E/62C2DC 5>2 =.63J867 8= R8-.@2 F89 >5A2C/G2GC A42? 5>2 68A 3867A56A7G

" *;2>5J6: ;8/6A 8= 7?7A2@ /7 942>2 A42 7?7A2@ /7 /6 E5-5632C2G:GC 942>2 ;.@; 4250 /7 2\.5- A8 A42 4250 -87727G

37

Dr. M. Khosravy


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S.@; 560 7?7A2@7 3.>R27

" ,.;;-? 3.>R2 =8> 4;.@;C.H

02A2>@/62 2B;2>/@26A5--? E?@56.=53A.>2>G +426 .7/6: $$,C

/A /7 257? A8 E./-0 /6 =.63J865-

>2-5J8674/; =8> 4;.@;C.G

"

,?7A2@ 3.>R2 02A2>@/620 =>8@565-?7/7 8= F./0 0?65@/37

2\.5J867

" * ; 2 > 5 J 6 : ; 8 / 6 A / 7 A 4 2

/6A2>723J86 8= 7.;;-? 560

02@560 3.>R27

" != ;25b 2]3/263? /7 =5> =>8@

8;2>5J6: ;8/6AC ;.@; /7 9>86:

=8> A45A 5;;-/35J86G

38

Dr. M. Khosravy


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Dr. M. Khosravy

39


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