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8/9/2019 Internal Flow Presentation
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!"#$%"&' )'*+,)-./0 123456/37 !!
8/9/2019 Internal Flow Presentation
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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
8/9/2019 Internal Flow Presentation
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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)
General Concept of Flows in Pipe
Dr. M. Khosravy
8/9/2019 Internal Flow Presentation
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General Concept of Flows in Pipe
" )-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
8/9/2019 Internal Flow Presentation
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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
General Concept of Flows in Pipe
5
Dr. M. Khosravy
8/9/2019 Internal Flow Presentation
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General Concept of Flows in Pipe
" +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
8/9/2019 Internal Flow Presentation
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General Concept of Flows in Pipe
" )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
8/9/2019 Internal Flow Presentation
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General Concept of Flows in Pipe
" +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
8/9/2019 Internal Flow Presentation
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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
General Concept of Flows in Pipe
8/9/2019 Internal Flow Presentation
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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
8/9/2019 Internal Flow Presentation
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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
8/9/2019 Internal Flow Presentation
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" #42>2=8>2H
9426 /7 3867A56A
8/9/2019 Internal Flow Presentation
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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
8/9/2019 Internal Flow Presentation
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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
8/9/2019 Internal Flow Presentation
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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>
8/9/2019 Internal Flow Presentation
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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
8/9/2019 Internal Flow Presentation
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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
!
8/9/2019 Internal Flow Presentation
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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
8/9/2019 Internal Flow Presentation
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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
!
8/9/2019 Internal Flow Presentation
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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
8/9/2019 Internal Flow Presentation
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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
8/9/2019 Internal Flow Presentation
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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
! " #
$ % &
8/9/2019 Internal Flow Presentation
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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
!
" #
$
% &
8/9/2019 Internal Flow Presentation
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Moody Diagram
Dr. M. Khosravy
24
8/9/2019 Internal Flow Presentation
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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
"
# $
%
& '
8/9/2019 Internal Flow Presentation
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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
8/9/2019 Internal Flow Presentation
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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
8/9/2019 Internal Flow Presentation
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Minor Loss
Dr. M. Khosravy
! Source of minor loss:
1. Inlets & Outlets
2. Enlargements & Contractions3. Valves & Fittings
4. Pipe Bends
8/9/2019 Internal Flow Presentation
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1/68> '87727
" #8A5- 4250 -877 /6 5 7?7A2@ /7 38@;>/720 8= @5f8>-87727 -87727
8/9/2019 Internal Flow Presentation
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30
Dr. M. Khosravy
8/9/2019 Internal Flow Presentation
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31
Dr. M. Khosravy
8/9/2019 Internal Flow Presentation
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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
+
=
+
=
8/9/2019 Internal Flow Presentation
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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
8/9/2019 Internal Flow Presentation
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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
8/9/2019 Internal Flow Presentation
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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
8/9/2019 Internal Flow Presentation
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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 $$,h
" "8A2H A42 565-8:? 9/A4 2-23A>/35- 3/>3./A7 748.-0 E2 8ER/8.7
" )-89 F89 >5A2 >26A
" S>277.>2 :>50/26A /35- ;8A26J5-
8/9/2019 Internal Flow Presentation
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S/;/6: "2A98>b7 560 S.@; ,2-23J86
" +426 5 ;/;/6: 7?7A2@ /6R8-R27 ;.@;7 560c8> A.>E/627C;.@; 560 A.>E/62 4250 @.7A E2 /63-.020 /6 A42 262>:?2\.5J86
" #42 .72=.- 4250 8= A42 ;.@; A42 4250 2BA>53A20E? A42 A.>E/62 E/62C2DC 5>2 =.63J867 8= R8-.@2 F89 >5A2C/G2GC A42? 5>2 68A 3867A56A7G
" *;2>5J6: ;8/6A 8= 7?7A2@ /7 942>2 A42 7?7A2@ /7 /6 E5-5632C2G:GC 942>2 ;.@; 4250 /7 2\.5- A8 A42 4250 -87727G
37
Dr. M. Khosravy
8/9/2019 Internal Flow Presentation
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S.@; 560 7?7A2@7 3.>R27
" ,.;;-? 3.>R2 =8> 4;.@;C.H
02A2>@/62 2B;2>/@26A5--? E?@56.=53A.>2>G +426 .7/6: $$,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
8/9/2019 Internal Flow Presentation
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Dr. M. Khosravy
39
8/9/2019 Internal Flow Presentation
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40