Reynolds Experiment
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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 2 low f Turbulent 4000 flow Transition 4000 2000 flow Laminar 2000 Re V h V h VD f f Laminar Turbul ent
description
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. Turbulent. Laminar. - PowerPoint PPT Presentation
Transcript of Reynolds Experiment
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
2lowfTurbulent4000
flowTransition40002000
flowLaminar2000
Re
Vh
VhVD
f
f
Laminar Turbulent
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
D
Lhhh
ds
dhD
zp
ds
dD
sDds
dzsAsA
ds
dp
sDWAsds
dpppAF
f
s
021
0
0
0
0
44
)]([4
)(0
)(sin)(0
• Since h is constant across the cross-section 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
)(Re,
)(Re,
;Re;
,,:variablesRepeating
),(
),,,,(
20
20
20
321
214
0
D
eFV
D
eF
V
VD
e
DV
F
eDVF
V D e
ML-1T-2 ML-3 LT-1 ML-1T-1 L L
)(Re,82
)(Re,82
)(Re,4
4
2
2
2
0
D
eFf
g
V
D
Lfh
D
eF
g
V
D
L
D
eFV
D
L
D
Lh
f
f
Darcy-Weisbach Eq. Friction factor
Laminar Flow in Pipes
• Laminar flow -- Newton’s law of viscosity is valid:
2
0
20
20
2
14
44
2
2
2
r
r
ds
dhrV
ds
dhrCC
ds
dhrV
drds
dhrdV
ds
dhr
dr
dV
dr
dV
dy
dV
ds
dhr
dy
dV
• Velocity distribution in a pipe (laminar flow) is parabolic with maximum at center.
2
0max 1
r
rVV
Discharge in Laminar Flow
ds
dhD
ds
dhrQ
rr
ds
dh
rdrrrds
dhVdAQ
rrds
dhV
r
r
128
8
2
)(
4
)2()(4
)(4
4
40
0
220
2
022
0
220
0
0
ds
dhDV
A
QV
32
2
Head Loss in Laminar Flow
2
21
12212
2
2
2
32
)(32
32
32
32
D
VLh
hhh
ssD
Vhh
dsD
Vdh
DV
ds
dh
ds
dhDV
f
f
Re
64
2
2/)(Re
64
2/))((64
2/
2/32
32
2
2
2
2
2
2
2
fV
D
Lfh
VD
L
VD
L
DV
V
V
D
VL
D
VLh
f
f
Nikuradse’s Experiments
• In general, friction factor
– Function of Re and roughness
• Laminar region
– Independent of roughness
• Turbulent region– Smooth pipe curve
• All curves coincide @ ~Re=2300
– Rough pipe zone• All rough pipe curves flatten
out and become independent of Re
Re
64f
Blausius
Re 4/1k
f Rough
Smooth
Laminar Transition Turbulent
Blausius OK for smooth pipe
)(Re,D
eFf
Re
64f
2
9.010Re
74.5
7.3log
25.0
D
ef
Moody Diagram
Pipe Entrance
• Developing flow
– Includes boundary layer and core,
– viscous effects grow inward from the wall
• Fully developed flow
– Shape of velocity profile is same at all points along pipe
flowTurbulent 4.4Re
flowLaminar Re06.01/6D
LeeL
Entrance length Le Fully developed flow region
Region of linear pressure drop
Entrance pressure drop
Pressure
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 – 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
g
VKh
g
V
hK L
L
2or
2
2
2
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
• Where,
m
mm
mfL KD
Lf
g
Vhhh
2
2
lossheadTotalLh
lossheadFrictionalfh
mhm fittingfor lossheadMinor
mKm fittingfor tcoefficienlossheadMinor
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
