Transcript 3. Friction and Head Losses

VIII. Viscous Flow and Head Loss
Contents
1.
Introduction
2.
Laminar and Turbulent Flows
3.
Friction and Head Losses
4.
Head Loss in Laminar Flows
5.
Head Loss in Turbulent Flows
6.
Head Loss of Steady Pipe Flows
7.
Minor Losses
8.
Examples
1. Introduction
Shear stress due to fluid viscosity
¶u
t = m
¶y
D’Alembert Paradox
r
F= 0
éaV 2 p
ù éaV 2 p
ù
ê
+ + zú = ê
+ + zú
êë 2g
úû êë 2g
úû
g
g
1
2
V2
2g
p
g
V2
2g
hw
hw
hw
2
V
2g
p
g
V
2
2g
V2
2g
p
g
p g
For real fluid flows
æaV 2
ö
æaV 2
ö÷
p
p
÷
çç
ç
+
+ z÷
- ç
+
+ z÷
= hw > 0
÷
÷
ç 2g
÷
÷
çè 2g
rg
r
g
ø
è
ø
upstream
downstream
Head Loss
Head Loss:
 Losses due to friction
 Minor Losses

entrance and exit

sudden change of cross sections

valves and gates

bends and elbows

……
2. Laminar and Turbulent Flows
Reynolds’ Experiment
Laminar Flows:
Movement of any fluid particle is regular
Path lines of fluid particles are smooth
Turbulent Flows:
Movement of any fluid particle is random
Path lines of fluid particles are affected by mixing
Transition from Laminar to Turbulent Flow:
 for different fluid
 for different diameter of pipe
Head Loss due to laminar and turbulent flows
hf
logh f
m
=
m
m=
2
.7
1
=
5
1
logV
Laminar Flows:
hf µ V
Turbulent Flows:
hf µ V
µV
2
(Rough wall)
1.75
(Smoot h wall)
Critical Condition
r Ud
Ud
R=
=
= 2300
n
m
Reynolds Number
3. Friction and Head Losses
V2
2g
hw
p1
g
z1
z2
p2
g
Momentum Equation
p1A - p2A - gA L sin a - t PL = 0
A : area of the cross-section
P: wetted perimeter
z 2 - z1
sin a =
L
p1A - p2A - gA L sin a - t PL = 0
p1 p2
PL
- (z 2 - z 1 ) = t
g
g
gA
hf
t P
t
=
=
L
gA
gRh
Hydraulic radius
rV 2
t = Cf
2
hf
t
=
L
gRh
L V2
hf = C f
R h 2g
LV2
hf = f
D 2g
Darcy-Weisbach equation
4. Head Loss in Laminar Flows
x
x= x
u = u (r )
y = r cos q
v= 0
z = r sin q
w= 0
¶u ¶v ¶w
+
+
= 0
¶x ¶y
¶z
æ¶ 2u ¶ 2u ¶ 2u ÷
ö
¶u
¶u
¶ u 1 ¶ p%
ç
u
+v
+w
+
= nç 2 +
+
÷
2
2÷
ç
¶x
¶y
¶z
r ¶x
¶y
¶z ø
è¶ x
2
2 ö
æ¶ 2v
¶v
¶v
¶ v 1 ¶ p%
¶
v
¶
v÷
u
+v
+w
+
= n çç 2 +
+
÷
2
2÷
ç
¶x
¶y
¶z r ¶y
¶y
¶z ø
è¶ x
æ¶ 2w ¶ 2w ¶ 2w ÷
ö
¶w
¶w
¶ w 1 ¶ p%
ç
u
+v
+w
+
= nç 2 +
+
÷
2
2 ÷
ç
¶x
¶y
¶z
r ¶z
¶y
¶z ø
è¶ x
¶u
= 0
¶x
æ¶ 2u ¶ 2u ö÷
1 ¶ p%
= n çç 2 +
÷
2÷
ç
r ¶x
¶z ø
è¶ y
¶ p%
= 0
¶y
¶ p%
= 0
¶z
y = r cos q
æ¶ r ¶
¶2
¶2
¶ q ¶ öæ
¶r ¶
¶ q ¶ ö÷
÷
ç
ç
+
= ç
+
+
÷
÷
çç
2
2
÷
ç
¶y
¶z
è¶ y ¶ r ¶ y ¶ q øè¶ y ¶ r ¶ y ¶ q ø÷
æ¶ r ¶
öæ¶ r ¶
ö
¶q ¶ ÷
¶q ¶ ÷
ç
ç
+ç
+
+
÷
÷
ç
÷
è¶ z ¶ r
øè
ø
¶ z ¶ q ¶ z ¶ r ¶ z ¶ q÷
æ
¶
sin q ¶ öæ
¶
sin q ¶ ö÷
÷
ç
ç
= çcos q
+
+
÷
÷
çcos q
÷
è
øè
¶r
r ¶q
¶r
r ¶ q ø÷
+
æ
öæ
¶
cos q ¶ ö÷
ççsin q ¶ - cos q ¶ ÷
ç
sin
q
֍
÷
è
øè
¶r
r ¶ q÷
¶r
r ¶ q ø÷
ö÷ 1 ¶ 2
1 ¶ æ
¶
ççr
=
+ 2 2
÷
÷
r ¶r è ¶r ø r ¶q
z = r sin q
æ¶ 2u ¶ 2u ö÷
dp%
= mçç 2 +
÷
2÷
ç
dx
¶z ø
è¶ y
é1 ¶ æ ¶ u ö 1 ¶ 2u ù
÷
ççr
= mê
+ 2 2ú
÷
êër ¶ r è ¶ r ÷
ø r ¶q ú
û
ö
m ¶ æ
¶u÷
ç
=
÷
çèr
ø
r ¶r ¶r ÷
1 dp% 2
u=
r + A log r + B
4m dx
u= 0
(r
u ® finit e
1 dp% 2
gJ 2
2
u= (a - r ) = (a - r 2 )
4m dx
4m
= a)
(r ® 0)
p%= p + gz
ù
é¶ u ù
¶ égJ 2
1
2
ê
ú
t = mê ú = m
a - r )
= - gJa
(
ê
ú
êë¶ r úûr = a
¶ r ë4m
2
ûr = a
hf
L
=
t
gRh
t = gJR h
a
Q = 2p ò ur dr = 2p ò
0
a
0
4
égJ 2
ù
pg
JD
ê (a - r 2 )úr dr =
êë4m
úû
128m
128mQ
32mV
J =
=
4
gp D
gD 2
J =
hf
L
=
V =
32mV
gD 2
g = rg
64m L V 2
hf =
rV D D 2g
f =
Q
pD 2 4
64
R
5. Head Loss in Turbulent Flows
Mean flow and fluctuation
B
B
1
B =
T
ò
t +T
B dt
t
t
Mean flow and fluctuation
B = B + B¢
B= B
BB = BB
B1 + B 2 = B1 + B 2
B ¢= 0
BB ¢= 0
æ¶ B ö
¶B
çç ÷
=
÷
è¶x ÷
ø ¶x
B ¢B ¢¹ 0
BB = BB + B ¢B ¢
Basic Equations of Turbulent Flows:
¶u ¶v ¶w
+
+
= 0
¶x ¶y
¶z
æ¶ 2u ¶ 2u ¶ 2u ÷
ö
¶u
¶u
¶u
¶u
1 ¶p
ç
+u
+v
+w
= fx + nç 2 +
+
÷
2
2÷
ç
¶t
¶x
¶y
¶z
r ¶x
¶y
¶z ø
è¶ x
2
2 ö
æ¶ 2v
¶v
¶v
¶v
¶v
1 ¶p
¶
v
¶
v÷
+u
+v
+w
= fy + n çç 2 +
+
÷
2
2÷
ç
¶t
¶x
¶y
¶z
r ¶y
¶y
¶z ø
è¶ x
æ¶ 2w ¶ 2w ¶ 2w ÷
ö
¶w
¶w
¶w
¶w
1 ¶p
ç
+u
+v
+w
= fz + nç 2 +
+
÷
2
2 ÷
ç
¶t
¶x
¶y
¶z
r ¶z
¶y
¶z ø
è¶ x
æ¶ 2u ¶ 2u ¶ 2u ÷
ö
¶u
¶u
¶u
¶u
1 ¶p
ç
+u
+v
+w
= fx + nç 2 +
+
÷
2
2÷
ç
¶t
¶x
¶y
¶z
r ¶x
¶y
¶z ø
è¶ x
¶u ¶v ¶w
+
+
= 0
¶x ¶y
¶z
¶u
¶u
¶u
¶u
¶v
¶u
¶w
+u
+u
+v
+u
+w
+u
¶t
¶x
¶x
¶y
¶y
¶z
¶z
2
2 ö
æ¶ 2u
1 ¶p
¶
u
¶
u÷
= fx + n çç 2 +
+
÷
2
2÷
ç
r ¶x
¶y
¶z ø
è¶ x
æ¶ 2u ¶ 2u ¶ 2u ÷
ö
¶ u ¶ uu ¶ uv ¶ uw
1 ¶p
ç
+
+
+
= fx + nç 2 +
+
÷
çè¶ x
¶t
¶x
¶y
¶z
r ¶x
¶y2 ¶z2 ÷
ø
Basic Equations of Turbulent Flows:
¶u ¶v ¶w
+
+
= 0
¶x ¶y
¶z
æ¶ 2u ¶ 2u ¶ 2u ÷
ö
¶ u ¶ uu ¶ uv ¶ uw
1 ¶p
ç
+
+
+
= fx + nç 2 +
+
÷
2
2÷
ç
¶t
¶x
¶y
¶z
r ¶x
¶y
¶z ø
è¶ x
2
2 ö
æ¶ 2v
¶ v ¶ vu ¶ vv ¶ vw
1 ¶p
¶
v
¶
v÷
+
+
+
= fy + n çç 2 +
+
÷
2
2÷
ç
¶t
¶x
¶y
¶z
r ¶y
¶y
¶z ø
è¶ x
æ¶ 2w ¶ 2w ¶ 2w ÷
ö
¶ w ¶ wu ¶ wv ¶ ww
1 ¶p
ç
+
+
+
= fz + nç 2 +
+
÷
2
2 ÷
ç
¶t
¶x
¶y
¶z
r ¶z
¶y
¶z ø
è¶ x
Reynolds’ Average
¶u ¶v ¶w
+
+
= 0
¶x ¶y
¶z
¶u ¶v
¶w
+
+
= 0
¶x
¶y
¶z
Reynolds’ Average
æ¶ 2u ¶ 2u ¶ 2u ÷
ö
¶ u ¶ uu ¶ uv ¶ uw
1 ¶p
ç
+
+
+
= fx + nç 2 +
+
÷
çè¶ x
¶t
¶x
¶y
¶z
r ¶x
¶y2 ¶z2 ÷
ø
¶u
¶ uu
¶ u ¢u ¢ ¶ uv
¶ u ¢v ¢ ¶ uw ¶ u ¢w ¢
+
+
+
+
+
+
¶t
¶x
¶x
¶y
¶y
¶z
¶z
2
2 ö
æ¶ 2u
1 ¶p
¶
u
¶
u÷
= fx + n çç 2 +
+
÷
çè¶ x
r ¶x
¶ y 2 ¶ z 2 ø÷
Reynolds Stresses
R xx = - r u ¢u ¢
R yx = - r u ¢v ¢
R zx = - r u ¢w ¢
y
v¢
u = u (y )
Reynolds Stresses
x
Mean flux of horizontal momentum:
Equivalent Shear Stress:
- R yx
Fx = v ¢(r u ) = r u ¢v ¢
Reynolds Equations:
¶u
¶u
¶u
¶u
+u
+v
+w
¶t
¶x
¶y
¶z
æ¶ 2u
ö
1 ¶p
¶ 2u ¶ 2u ö÷ 1 æ
ç
çç¶ R xx + ¶ R yx + ¶ R zx ÷
= fx + nç 2 +
+
+
÷
÷
çè¶ x
r ¶x
¶ y 2 ¶ z 2 ø÷ r èç ¶ x
¶y
¶ z ø÷
¶v
¶v
¶v
¶v
+u
+v
+w
¶t
¶x
¶y
¶z
æ¶ 2v
ö
1 ¶p
¶ 2v
¶ 2v ö÷ 1 æ
ç
çç¶ R xy + ¶ R yy + ¶ R zy ÷
= fy + nç 2 +
+
+
÷
÷
çè¶ x
r ¶y
¶ y 2 ¶ z 2 ø÷ r èç ¶ x
¶y
¶ z ø÷
¶w
¶w
¶w
¶w
+u
+v
+w
¶t
¶x
¶y
¶z
æ¶ 2w ¶ 2w ¶ 2w ö÷ 1 æ¶ R xz
¶ R yz
1 ¶p
¶ R zz ö÷
çç
= fz + n çç 2 +
+
+
+
+
÷
÷
çè ¶ x
r ¶z
¶y2
¶ z 2 ø÷ r èç ¶ x
¶y
¶ z ø÷
¶u
¶u
¶u
¶u
+u
+v
+w
¶t
¶x
¶y
¶z
2
2 ö
æ¶ 2u
ö
1 ¶p
¶
u
¶
u÷ 1æ
çç¶ R xx + ¶ R yx + ¶ R zx ÷
= fx + n çç 2 +
+
+
÷
÷
2
2÷
ç
ç
r ¶x
¶y
¶z ø r è ¶x
¶y
¶ z ø÷
è¶ x
ö
1æ
çç¶ s xx + ¶ t yx + ¶ t zx ÷
÷
r çè ¶ x
¶y
¶ z ø÷
s xx = r n
t yx
¶u
¶x
1 æ
¶ v ¶ u ö÷
ç
= rnç +
÷
2 çè¶ x
¶ y ø÷
t zx =
ö
1 æ
¶w ¶u÷
r n çç
+
÷
ø
2 è¶ x
¶z ÷
R xx = r ne
R yx
¶u
¶x
æ¶ v
ö
1
¶u ÷
ç
= r ne ç
+
÷
çè ¶ x
2
¶ y ø÷
R zx =
æ¶ w ¶ u ö÷
1
r ne çç
+
÷
è¶ x
2
¶ z ø÷
Theory of Mixing Length
y
t = - r u ¢v ¢
l¢
2
u = u (y )
æ ö
2 çdu ÷
= r cl ¢ ç ÷
çèdy ÷
ø
x
du
u ¢= l ¢
dy
t yx = r l 2
v ¢µ u ¢
du du
dy dy
Logarithmic Velocity Distribution
y
2
æ ö
2 2 çdu ÷
t = t 0 = rk y ç ÷
çèdy ÷
ø
u = u (y )
du
ky
=
dy
l = ky
t = t0
(k = 0.4)
t0
º v*
r
v*
u=
log y + C
k
6. Head Loss of Steady Pipe Flows
Logarithmic Velocity Distribution
v*
u=
log y + C
k
u
1
= log y + C ¢
v*
k
u
1
v*y
= log
+ C ¢¢
v*
k
n
R*
y
Logarithmic Overlap Layer
t
l
0

Logarithmic Velocity Distribution in a Pipe
y
30
u
v*
Viscous
Turbulent
20
u
vy
= 2.5 log * + 5.5
v*
n
10
u
vy
= *
v*
n
0
1
10
100
10000
1000
v*y n
Viscous sublayer:
v*y
0<
£ 5
n
Transition zone:
v*y
5<
< 70
n
Turbulent zone:
v*y
³ 70
n
Velocity Distribution in Viscous Sublayer
du
t = m
dy
t0
r v*2
u=
y=
y
m
m
u
v *y
=
v*
n
Velocity Distribution in a Pipe
y
Blasius’ 7th-root law
u
u max
1
7
æy ÷
ö
= çç ÷
÷
çèr ÷
ø
0
Valid for R = 3000 - 105
Wall Roughness
ks
Hydraulically smooth wall:
Roughness height is smaller than the
thickness of the viscous sublayer
v*ks
£ 5
n
Hydraulically rough wall:
Roughness height is larger than the lower
boundary of the turbulent zone
v*ks
³ 70
n
Hydraulically smooth pipe:
u
v*y
= 2.5 log
+ 5.5
v*
n
Hydraulically rough pipe:
u
y
= 2.5 log + 8.5
v*
ks
Velocity Distribution in a Pipe
Mean velocity in hydraulically smooth pipe:
u
v*y
v* (a - r )
= 2.5 log
+ 5.5 = 2.5 log
+ 5.5
v*
n
n
v*
V =
pa 2
2pv*
òS u dS = pa 2
ò
a
0
v* (a - r )
é
ù
2.5
log
+
5.5
ê
úrdr
êë
úû
n
V
v*a
= 2.5 log
+ 1.75
v*
n
Mean velocity in hydraulically rough pipe:
(a - r )
u
y
= 2.5 log + 8.5 = 2.5 log
+ 8.5
v*
ks
ks
v
V = *2
pa
2p v*
u
dS
=
òS
pa 2
ò
a
0
é
ù
(a - r )
ê2.5 log
+ 8.5úrdr
ê
ú
ks
ë
û
V
a
= 2.5 log + 4.75
v*
ks
Relation among mean velocity, friction velocity
and friction factor:
LV2
hf = f
D 2g
hf
t0
=
=
L
gRh
t0 1
8v*2
f = 8
= 2
2
r V
V
V = v* 8 f
t0
1
4 gD
Friction factor in hydraulically smooth pipe:
V
v*a
= 2.5 log
+ 1.75
v*
n
æV D
1
= 0.884 log çç
è n
f
ö
÷
f÷
- 0.91
÷
ø
= 2.04 log10 (R f ) - 0.91
Friction factor in hydraulically rough pipe:
V
a
= 2.5 log + 4.75
v*
ks
f =
=
1
2
é0.884 log a k + 1.68ù
s
êë
úû
( )
1
2
é2.04 log a k + 1.68ù
10
s
êë
úû
( )
Experiment of Nikuradse
ks
Modified friction factor in hydraulically smooth pipe:
æR f ö÷
1
÷
= 2 log10 (R f ) - 0.8 = 2 log10 ççç
÷
f
è 2.51 ÷
ø
æR ÷
ö
1
= 1.8 log10 çç ÷
è6.9 ÷
ø
f
0.316
f = 0.25
R
(3000 £
f = f (R )
8
4000
£
R
£
10
(
)
R £ 105 )
Modified friction factor in hydraulically rough pipe:
f =
=
1
2
é2 log a k + 1.74ù
10
s
êë
úû
( )
1
2
é2 log 3.7D k ù
10
s ú
êë
û
(
)
æks ö÷
f = f çç ÷
èa ÷
ø
Colebrook Equation:
æ
1
ks
2.51 ö÷
ç
÷
= - 2 log10 çç0.27 +
÷
f
D R f÷
è
ø
æ2.51 ö÷
÷
® - 2 log10 ççç
÷
èR f ÷
ø
æ 1 ks ö÷
® - 2 log10 çç
÷
è3.7 D ø÷
æks
ö
ç ® 0, smoot h ÷
÷
÷
çèD
ø
(R ®
¥ , rough
)
Head loss in hydraulically smooth pipe:
LV2
0.316 L V 2
1.75
hf = f
=
µ
V
0.25
D 2g
D 2g
VD n
(
)
Head loss in hydraulically smooth pipe:
æks ö÷ L V 2
LV2
2
hf = f
= f çç ÷
µ
V
èD ø÷D 2g
D 2g
Practical pipe: equivalent roughness
ks
7. Minor Losses
Head Loss due to Sudden Expansion
V2
2g
z+
hlx
p
g
Head Loss due to Sudden Expansion
1
2
æp1 V 12 ö÷
hlx = çç +
÷çè g
2g ø÷
p1¢
p1 - p2 V 12 - V 22
=
+
rg
2g
2
V 1A1 = V 2A2
=
(V 1 - V 2 )
2g
2
2
¢
r (A2V 22 - AV
1 1 ) = p1A1 + p1 (A2 - A1 ) - p2A2
2
2
p1 - p2 (A2V 2 - AV
1 1 )
=
= V 22 - V 1V 2
r
A2
æp2 V 22 ö÷
çç +
÷
çè g
2g ø÷
æ A1 ö÷ V 12
÷
= çç1 ÷ 2g
çè
A2 ø÷
V 12
= Vx
2g
Head Loss due to Sudden Contraction
V 22
hlc = Vc
2g
(
Vc = Vc D2 D1
hc
p V2
z+ +
g 2g
)
V2
hle = Ve
2g
Head Loss at Entrance
he
p V2
z+ +
g 2g
Head Loss at Bell-Mouthed Entrance
V2
he = Ve
2g
Head Loss in Bend
V2
hb = Vb
2g
( )
Vb = Vb r D
8. An Example
H = 20 m
L1 = 80 m
D1 = 0.2 m
ks = 0.05 mm
L2 = 50 m
Q= ?
D2 = 0.4 m
ks = 0.05 mm
æ p V 2 ö÷ æ p V 2 ö÷
ççz + +
ççz + +
= H =
÷
÷
÷
÷
çè
g 2g øA èç
g 2g øB
å
hw
V 12
l1 V 12
V 12
l2 V 22
V 22
hw = Ve
+ f1
+ Vx
+ f2
+ Vo
2g
D1 2g
2g
D2 2g
2g
Ve = 0.5
2
æ A1 ö÷
÷
Vx = çç1 = 0.5625
÷
çè
A2 ÷
ø
Vo = 1.0
f1 =
f2 =
1
2
= 0.01438
2
= 0.01250
é2 log 3.7D k ù
10
1
s ú
êë
û
(
)
1
é2 log 3.7D k ù
10
2
s ú
êë
û
(
)
æ l1
ö÷V 12 æ l2
ö÷V 22
ççf
H = ççf1
+ Ve + Vx ÷
+
+ Vo ÷
2
÷
÷
ç
÷
çè D1
ø 2g è D2
ø÷ 2g
æ
80
ö÷V 12 æ
50
ö÷V 22
= çç0.01438 ´
+ 0.5 + 0.5625÷ + çç0.01250 ´
+ 1.0÷
ø 2g
è
ø 2g è
0.2
0.4
V 12
V 22
= 6.8145
+ 2.5625
2g
2g
V 1A1 = V 2A2
V1 =
Þ
V 2 = 0.25V 1
2gH
= 7.50 m s
2
6.8145 + 0.25 ´ 2.5625
V 1 = 7.50 m s
V 2 = 0.25V 1 = 1.875 m s
V 2D 2
1.875 ´ 0.4
5
R2 =
=
=
7.5
´
10
n
10- 6
ks D2 = 0.05 ´ 10- 3 0.4 = 1.25 ´ 10- 4
Q = V 1A1 = V 2A2 = 0.2356 m 3 s