Elementary Mechanics of Fluids
Elementary Mechanics of Fluids
CE 319 F
CE 319 F
Daene McKinney
Daene McKinney
Flow in
Flow in
Pipes
Pipes
Reynolds Experiment
Reynolds Experiment
••
Reynolds Number
Reynolds Number
••
Laminar flow:
Laminar flow: Fluid m
Fluid moves in sm
oves in smooth
ooth
streamlines
streamlines
••
Turbulent flow: Violent mixing, fluid
Turbulent flow: Violent mixing, fluid
velocity at a point varies randomly with time
velocity at a point varies randomly with time
••
Transition to turbulence in a 2 in. p
Transition to turbulence in a 2 in. pipe is at
ipe is at
V
V
=2 ft/s, so most pipe flows are turbulent
=2 ft/s, so most pipe flows are turbulent
2 2 low low f f Turbulent Turbulent 4000 4000 flow flow Transition Transition 4000 4000 2000 2000 flow flow Laminar Laminar 2000 2000 Re Re V V h h V V h h VD VD f f f f Laminar
Turbulent
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 L h h h ds dh D z p ds d D s D ds dz s A sA ds dp s D W A s ds dp p pA F f s 0 2 1 0 0 0 0 4 4 )] ( [ 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
Darcy-Weisbach Equation
) (Re, ) (Re, ; Re; , , : variables Repeating ) , ( ) , , , , ( 2 0 2 0 2 0 3 2 1 2 1 4 0 D e F V D e F V V D e D V F e D V F
0
V
D
e
ML
-1T
-2ML
-3LT
-1ML
-1T
-1L
L
) (Re, 8 2 ) (Re, 8 2 ) (Re, 4 4 2 2 2 0 D e F f g V D L f h D e F g V D L D e F V D L D L h f f
Laminar Flow in Pipes
•
Laminar flow --
Newton’s law of viscosity is valid:
2 0 2 0 2 0 2 1 4 4 4 2 2 2 r r ds dh r V ds dh r C C ds dh r V dr ds dh r dV ds dh r dr dV dr dV dy dV ds dh r dy dV
•
Velocity distribution in a pipe (laminar flow)
is parabolic with maximum at center.
2 0 max 1 r r V VDischarge in Laminar Flow
ds dh D ds dh r Q r r ds dh rdr r r ds dh VdA Q r r ds dh V r r 128 8 2 ) ( 4 ) 2 ( ) ( 4 ) ( 4 4 4 0 0 2 2 0 2 0 02 2 2 2 0 0 0
ds dh D V A Q V 32 2
Head Loss in Laminar Flow
2 2 1 1 2 2 1 2 2 2 2 32 ) ( 32 32 32 32 D V L h h h h s s D V h h ds D V dh D V ds dh ds dh D V f f
Re 64 2 2 / ) ( Re 64 2 / ) )( ( 64 2 / 2 / 32 32 2 2 2 2 2 2 2
f V D L f h V D L V D L D V V V D V L D V L h f f Nikuradse’s Experiments
•
In general, friction factor
– Function of Re androughness
•
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 64
f
Re1 / 4 Blausius k f
Rough SmoothLaminar Transition Turbulent
Blausius OK for smooth pipe ) (Re, D e F f
Re 64
f 2 9 . 0 10 Re 74 . 5 7 . 3 log 25 . 0
D e fPipe 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
flow Turbulent 4.4Re flow Laminar Re 06 . 0 1/6 D Le e LEntrance 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 V K h g V h K L L 2 or 2 2 2
Elbow Loss in a Pipe
•
A piping system may have many minor losses
which are all correlated to V
2/2g
•
Sum them up to a total system loss for pipes
of the same diameter
•
Where,
m m m m f L K D L f g V h h h 2 2 loss head Total
L h loss head Frictional
f h m hm Minorheadlossforfittingm K m
MinorheadlosscoefficientforfittingEGL & 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
Ex(10.2)
Given: Liquid in pipe has = 8 kN/m3. Acceleration = 0.D= 1 cm, = 3x10-3N-m/s2.
Find: Is fluid stationary, moving up, or moving down?
What is the mean velocity?
Solution: Energy eq. from z = 0 to z = 10 m
s m V * x * ) . *( . V μL γD h V γD μLV h m h h h z p g V h z p g V - L L L L L L / 04 . 1 10 10 3 32 01 0 8000 25 1 32 32 upward) (moving 25 . 1 10 8 90 10 8000 000 , 110 8000 000 , 200 2 2 3 2 2 2 2 2 2 2 2 1 1 2 1 1
1 2Ex (10.4)
•
Given: Oil (S = 0.97, m = 10
-2lbf-s/ft
2) in 2
in pipe, Q = 0.25 cfs.
•
Find: Pressure drop per 100 ft of horizontal
pipe.
•
Solution:
ft psi/100 7 91 12 2 46 11 100 2 10 32 32 (laminar) 360 10 ) 12 / 2 ( * 46 . 11 * 94 / 1 * 97 . 0 Re / 46 . 11 4 / ) 12 / 2 ( 25 . 0 2 2 2 2 . ) / ( . * * -* D μLV Δp VD s ft A Q V
Ex. (10.8)
Given: Kerosene (S=0.94,=0.048 N-s/m2). Horizontal5-cm pipe.Q=2x10-3m3 /s.
Find: Pressure drop per 10 m of pipe. Solution: cf s s f t V V V V V g V γD μL V g α g V α γD μLV γD μLV h z γ p g V α h z γ p g V α L L 3 2 5 2 2 2 5 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 2 1 1 10 * 23 . 1 4 / (0.25/12) * * 1.6 A * V Q (laminar) 1293 10 * 4 ) 12 / 25 . 0 ( * 6 . 1 * 94 . 1 * 8 . 0 Re / 60 . 1 0 1 . 16 45 . 8 0 5 . 0 ) 32 / 1 ( * 4 . 62 * 8 . 0 10 * 10 * 4 * 32 2 2 0 5 . 0 32 2 0 0 2 32 5 . 0 0 0 32 2 2
Ex. (10.34)
2 / 62 . 0 , / 300 , 12 N m
Ns m
Given: Glycerin@ 20
oC flows commercial steel
pipe.
Find: h
Solution:
m γD μLV h h VD VD h z γ p z γ p h z γ p h z γ p z γ p g V α h z γ p g V α L L L L 42 . 2 ) 02 . 0 ( * 300 , 12 ) 6 . 0 )( 1 )( 62 . 0 ( 32 32 (laminar) 5 . 23 10 * 1 . 5 02 . 0 * 6 . 0 Re ) ( 2 2 2 2 4 2 2 1 1 2 2 1 1 2 2 2 2 2 1 1 2 1 1 Ex. (10.43)
Given: FigureFind: Estimate the elevation required in the upper reservoir to produce a water discharge of 10 cfs in the system. What is the minimum pressure in the pipeline and what is the pressure there?
Solution:
ft z s ft A Q V D L f K K K g V D L f K K K h z h z z γ p g V α h z γ p g V α E b e E b e L L L 133 2 . 32 * 2 73 . 12 75 . 10 0 . 1 4 . 0 * 2 5 . 0 100 / 73 . 12 1 * 4 / 10 75 . 10 1 430 * 025 . 0 ; 0 . 1 ; (assumed) 4 . 0 ; 5 . 0 2 2 0 0 0 0 2 2 2 1 2 2 2 1 2 2 2 2 2 1 1 2 1 1 5 5 2 2 2 1 2 1 2 1 1 2 1 1 10 * 9 10 * 14 . 1 1 * 73 . 12 Re 59 . 0 ) 53 . 1 ( * 4 . 62 35 . 1 2 . 32 * 2 73 . 12 1 300 025 . 0 4 . 0 5 . 0 0 . 1 7 . 110 133 2 2 2 * 1 0 0 2 2 VD psig p ft g V D L f K K g V z z γ p z γ p g V h z z γ p g V α h z γ p g V α b b e b b b b b b L b b b b LEx. (10.68)
ft x k s ft x10 5 2 / ; s 1.5 10 4 22 . 1
f t D D g D Q D g V D L f h f 06 . 8 984 , 33 1 2 ) ) 4 / /( ( 1000 * 015 . 0 1 2 5 2 2 2
Given: Commercial steel pipe to carry 300 cfs of water at 60oF with a head loss of 1 ft per 1000 ft of pipe. Assume pipe sizes are available in even sizes when the diameters are expressed in inches (i.e., 10 in, 12 in, etc.). Find: Diameter. Solution: Assume f = 0.015 Relative roughness: 0.00002 06 . 8 10 5 . 1 4
x D k sGet better estimate of f
6 5 2 10 9 . 3 10 22 . 1 ) 06 . 8 )( 4 / ( 300 Re ) 4 / ( ) 4 / ( Re x x D Q D D Q VD
f=0.010 . 89 43 . 7 656 , 22 1 5 in ft D D
Use a 90 in pipeEx. (10.81)
m h h z γ p g V α h z γ p g V α g Q h fL D g D Q D L f g V D L f h f f L f f 45 . 14 10 9810 000 , 60 9810 000 , 300 2 2 8 2 ) ) 4 / /( ( 2 2 2 2 2 2 1 1 2 1 1 5 / 1 2 2 2 2 2
Given: The pressure at a water main is 300 kPa gage. What size pipe is needed to carry water from the main at a rate of 0.025 m3/s to a factory that is 140 m from the main? Assume galvanized-steel pipe is to be used and that the pressure required at the factory is 60 kPa gage at a point 10 m above the main connection.
Find: Size of pipe. Solution: Assume f = 0.020 m g Q h fL D f 100 . 0 81 . 9 ) 025 . 0 ( 140 45 . 14 02 . 0 8 8 5 / 1 2 2 5 / 1 2 2
Relative roughness: 022 . 0 0015 . 0 100 15 . 0
f D k s Friction factor: m D 0.102 020 . 0 022 . 0 100 . 0 5 / 1
Use 12 cm pipeEx. (10.83)
Given: The 10-cm galvanized-steel pipe is 1000 m longand discharges water into the atmosphere. The pipeline has an open globe valve and 4 threaded elbows;h1=3 m andh2= 15 m.
Find: What is the discharge, and what is the pressure at A, the midpoint of the line?
Solution: 0 0 2 ) 4 1 ( 12 0 0 2 2 2 2 2 2 2 2 1 1 2 1 1
g V D L f K K K z γ p g V α h z γ p g V α b v e L D = 10-cm and assume f = 0.025 4 6 3 2 2 2 10 7 10 31 . 1 1 . 0 * 942 . 0 Re / 0074 . 0 ) 10 . 0 )( 4 / ( 942 . 0 / 942 . 0 1 . 265 24 ) 1 . 0 1000 025 . 0 9 . 0 * 4 10 5 . 0 1 ( 24 x x VD s m VA Q s m V g V V g
So f = 0.025 kP a p m g γ p g V D L f K γ p z γ p g V α h z γ p g V α A A b A L A A A A 8 . 90 ) 26 . 9 ( * 9810 6 . 9 15 2 ) 942 . 0 ( ) 1 . 0 500 025 . 0 9 . 0 * 2 ( 2 ) 2 ( 15 2 2 2 2 2 2 2 2 2 2
Ex. (10.95)
Given: If the deluge through the system shown is 2 cfs,what horsepower is the pump supplying to the water? The 4 bends have a radius of 12 in and the 6-in pipe is smooth. Find: Horsepower Solution: 5 5 2 2 2 2 2 2 2 2 2 2 1 1 2 1 1 10 17 . 4 10 22 . 1 ) 2 / 1 ( * 18 . 10 Re 611 . 1 2 / 18 . 10 ) 2 / 1 )( 4 / ( 2 ) 4 5 . 0 1 ( 2 60 0 30 0 0 2 2 x x VD f t g V s f t A Q V D L f K g V h h z γ p g V α h z γ p g V α b p L p
