Transcript CE319F Elementary Fluid Mechanics - I

Reynolds Experiment
•
•
•
•
Reynolds Number
Laminar flow: Fluid moves in
smooth streamlines
Turbulent flow: Violent mixing,
fluid velocity at a point varies
randomly with time
Transition to turbulence in a 2 in.
pipe is at V=2 ft/s, so most pipe
flows are turbulent
Laminar
Turbulent

 2000

 4000
Laminarflow
VD 
Re 
2000 4000 T ransitionflow
 
hf V
T urbulentf low h f  V 2
Shear Stress in Pipes
•
Steady, uniform flow in a pipe:
momentum flux is zero and pressure
distribution across pipe is hydrostatic,
equilibrium exists between pressure,
gravity and shear forces
dp
s ) A  W sin    0 (D )s
ds
dp
dz
0   sA  As   0 (D )s
ds
ds
D d
p
 0  [  (  z )]
4 ds 
D dh
0  
4 ds
4 L 0
h1  h2  h f 
D
 Fs  0  pA  ( p 
•
•
Since h is constant across the crosssection of the pipe (hydrostatic),
and –dh/ds>0, then the shear stress
will be zero at the center (r = 0)
and increase linearly to a maximum
at the wall.
Head loss is due to the shear stress.
•
•
Applicable to either laminar or
turbulent flow
Now we need a relationship for the
shear stress in terms of the Re and
pipe roughness
Darcy-Weisbach Equation
0

V

D
e
ML-1T-2
ML-3
LT-1
ML-1T-1
L
L
 0  F (  , V ,  , D, e)
 4  F ( 1 ,  2 )
Repeat ingvariables:  , V , D

e
; 3  0
D
V 2
0
e
 F (Re, )
D
V 2
e
 0  V 2 F (Re, )
D
 1  Re;  2 
hf 

4L
0
D
4L
e
V 2 F (Re, )
D
D
L V2

D 2g
e 

8
F
(Re,
)

D 
L V2
hf  f
D 2g
Darcy-Weisbach Eq.
f  8F (Re,
Friction factor
e
)
D
Laminar Flow in Pipes
• Laminar flow -- Newton’s law of
viscosity is valid:
dV
r dh

dy
2 ds
dV
dV

dy
dr
dV r dh

dr 2  ds
r dh
dV 
dr
2  ds
 
r 2 dh
V 
C
4  ds
r02 dh
C
4  ds
2
r02 dh   r  
1    
V 
4  ds   r0  


  r 2 
V  Vmax 1    
  r0  


• Velocity distribution in a pipe
(laminar flow) is parabolic with
maximum at center.
Discharge in Laminar Flow
 dh 2 2
( r0  r )
4  ds
 dh 2 2
Q   VdA  0r0 
( r0  r )( 2rdr)
4  ds
V 
 dh ( r 2  r02 ) 2

4  ds
2
r04 dh
Q
8 ds
D 4 dh

128 ds
V 
Q
A
D 2 dh
V 
32 ds
r0
0
Head Loss in Laminar Flow
D 2 dh
V 
32 ds
dh
32
ds
 V
dh  V
h2  h1  V
D 2
32
D
32
2
ds
( s  s1 )
2 2
D
h1  h2  h f
hf 
32LV
D 2
hf 

32LV
D 2
32LV V 2 / 2
D 2 V 2 / 2

L
 64(
)( ) V 2 / 2
V D D

64 L
( ) V 2 / 2
Re D
L V 2
hf  f
D 2
f 
64
Re
Nikuradse’s Experiments
•
In general, friction factor
f  F (Re,
–
•
e
)
D
Function of Re and roughness
f 
Laminar region
64
f 
Re
–
•
k
Re1 / 4
Rough
Blausius
Independent of roughness
Turbulent region
–
Smooth pipe curve
•
–
f 
64
Re
Rough pipe zone
•
f 
All curves coincide @
~Re=2300
All rough pipe curves flatten
out and become independent
of Re
Smooth
Blausius OK for smooth pipe
0.25

5.74 
 e


log10 
 3.7 D Re 0.9 

2
Laminar
Transition
Turbulent
Moody Diagram
Pipe Entrance
• Developing flow
– Includes boundary layer and
core,
– viscous effects grow inward
from the wall
• Fully developed flow
Pressure
– Shape of velocity profile is Entrance
pressure drop
same at all points along
pipe
Le  0.06 Re Laminarflow

D 4.4Re1/6 T urbulentflow
Fully developed
flow region
Entrance length Le
Region of linear
pressure drop
Le
x
Entrance Loss in a Pipe
•
In addition to frictional losses, there are
minor losses due to
–
–
–
–
•
•
Entrances or exits
Expansions or contractions
Bends, elbows, tees, and other fittings
Valves
Losses generally determined by experiment
and then corellated with pipe flow
characteristics
Loss coefficients are generally given as the
ratio of head loss to velocity head
K
•
hL
V2
2g
or
hL  K
V2
2g
K – loss coefficent
–
–
–
K ~ 0.1 for well-rounded inlet (high Re)
K ~ 1.0 abrupt pipe outlet
K ~ 0.5 abrupt pipe inlet
Abrupt inlet, K ~ 0.5
Elbow Loss in a Pipe
• A piping system may have
many minor losses which are all
correlated to V2/2g
• Sum them up to a total system
loss for pipes of the same
diameter
V2
hL  h f   hm 
2g
m
 L

f

K

m
 D
m


• Where,
hL  T otalheadloss
h f  Frictionalheadloss
hm  Minorheadlossfor fittingm
Km  Minorheadlosscoefficient for fittingm
EGL & HGL for Losses in a Pipe
• Entrances, bends, and other
flow transitions cause the EGL
to drop an amount equal to the
head loss produced by the
transition.
• EGL is steeper at entrance than
it is downstream of there where
the slope is equal the frictional
head loss in the pipe.
• The HGL also drops sharply
downstream of an entrance