Bernoulli
Bernoulli
Bernoulli
Buoyancy
Every submerged object has a buoyancy force and a
weight force
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CTC 261
Flow in Pipes
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Objectives
Know how to characterize flow
Know how to apply the continuity equation
Know how to apply the Bernoulli’s equation
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Flow types
Uniform Flow: Velocity does not change from point to
point within the channel reach
Space criterion
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Turbulent and Laminar Flow
Turbulent – mixed flow; random movement
http://freshgasflow.com
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Reynold’s Number: Pipe Flow
Re=(Velocity*Diameter)/Kinematic viscosity
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Reynold’s # Example: Pipe Flow
Given:
Velocity=5 fps
Diameter=1 foot
Kinematic Viscosity @ 50F= 1.41E-5 (ft2/sec)
Re=354,610
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Reynold’s Number
Open Channel Flow
Re=(Velocity*D)/Kinematic viscosity
Where D=the hydraulic radius (wetted area/wetted
perimeter)
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Reynold’s # Example:
Open Channel Flow
Given:
Channel Width=5” (0.417 ft)
Channel Depth=2” (0.167 ft)
Velocity=0.2 ft/sec
Kinematic Viscosity @ 50F= 1.41E-5 (ft2/sec)
Hydraulic Radius=(0.417)(0.167)/(0.417+2(0.167))=.0927ft
Re=(0.2ft/sec)(.0927ft)/1.41E-5 ft2/sec=1300 (transitional)
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Calculating Average Velocity
V=Q/A
Q=V*A
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Example
A 24” diameter carries water having a velocity of 13 fps.
What is the discharge in cfs and in gpm?
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Continuity
Q=A1*V1=A2*V2
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Continuity Example
A 120-cm pipe is in series with a 60-cm pipe. The rate
of flow of water is 2 cubic meters/sec.
V60=Q/A60=7.1 m/s
V120=Q/A120=1.8 m/s
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Storage-Steady Flows
Q in=Qout+(Storage/Discharge Rate)
Qin=20 cfs
Qout=15 cfs
Storage or discharge?
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Storage-Steady Flows
Storage
Qin=20 cfs
Qout=15 cfs
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Storage Example
A river discharges into a reservoir at a rate of 400,000
cfs. The outflow rate through the dam is 250,000 cfs.
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Storage Example
Answer 11.5 ft/day
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Break
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Bernoulli’s Equation
http://www.rcuniverse.com/magazine/article_display.cfm?article_id=455
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Assumptions
Steady flow (no change w/ respect to time)
Incompressible flow
Constant density
Frictionless flow
Irrotational flow
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3 Forms of Energy
Kinetic energy (velocity)
Potential energy (gravity)
Pressure Energy (pump/tank)
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Kinetic Energy (velocity head)
V2/2g
Resulting units?
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Pressure Energy (pressure head)
Pressure / Specific weight
Resulting units?
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Potential Energy
Height above some datum
Units?
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Units
Energy (ft or meters)
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Example
Calculate the total energy in a pipeline with an elevation
head of 10 ft, water pressure of 50 psi and a velocity of 2
fps?
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Bernoulli’s equation
Energy @ section 1 = Energy @ section 2
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Reservoir Example
Water exits a reservoir through a pipe. The WSE
(water surface elevation) is 125’ above the datum (pt A)
The water exits the pipe at 25’ above the datum (pt B).
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Reservoir Example
Point A:
KE=0
Pressure Energy=0
Potential Energy=125’
Point B:
KE=v2/2g
Pressure Energy=0
Potential Energy=25’ (note: h=100’)
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Reservoir Example
Bernoulli’s: Set Pt A energy=Pt B energy
v2/2g=h
v=(2gh).5
Velocity=80.2 ft/sec
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Energy Grade Line (EGL)
Graphical representation of the total energy of flow of
a mass of fluid at each point along a pipe. For
Bernoulli’s equation the slope is zero (flat) because no
friction loss is assumed
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Hydraulic Grade Line (HGL)
Graphical representation of the elevation to which
water will rise in a manometer attached to a pipe. It
lies below the EGL by a distance equal to the velocity
head.
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Hints for drawing EGL/HGL graphs
EGL=HGL+Velocity Head
EGL=Potential+Pressure+Kinetic Energies
HGL=Potential+Pressure Energies
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Reducing Bend Example (1/5)
Water flows through a 180-degree vertical reducing
bend. The diameter of the top pipe is 30-cm and
reduces to 15-cm. There is 10-cm between the pipes
(outside to outside). The flow is 0.25 cms. The
pressure at the center of the inlet before the bend is
150 kPa. What is the pressure after the bend?
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Reducing Bend Example (2/5)
Find the velocities using the continuity equation
(V=Q/A):
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Reducing Bend Example (3/5)
Use Bernoulli’s to solve for the pressure after the bend
Kinetic+Pressure+Potential Energies before the bend =
the sum of the energies after the bend
Potential energy before bend = 0.325m
Potential energy after bend=0m (datum)
The only unknown is the pressure energy after the bend.
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Reducing Bend Example (4/5)
The pressure energy after the bend=60 kPA
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Reducing
Bend
Example
(5/5)
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Next Lecture
Energy equation
Accounts for friction loss, pumps and turbines
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