Download - Fluid Mechanics - Chap5
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Incompressible Flow in
Pipes and Channels
By
Farhan Ahmad
Department of Chemical Engineering,
University of Engineering & Technology Lahore
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Industrial processes - flow of fluids through pipes, conduits,
and processing equipment.
Circular cross-section
Non-circular cross-section
Flow of fluids in
Totally or partially filled pipes,
Layers down vertically inclined surfaces,
Through beds of solids, and
Agitated vessels.
Significance
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Consider the steady flow of a viscous fluid at constant
density in fully developed flow through a horizontal tube.
Visualize a disk-shaped element of fluid, concentric with the
axis of the tube.
Flow of Incompressible Fluids in Pipe
Shear-Stress Distribution
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Flow of Incompressible Fluids in Pipe
Shear-Stress Distribution
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At wall
After subtraction
Relation between and r
At r =0 , = 0
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Pressure Drop
Apply the balance
Relation between Skin Friction and Wall Shear
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ratio of the wall shear stress to the product of the density and
the velocity head.
Friction Factor
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Relations between Skin Friction Parameter
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Laminar
Turbulent
Fluid may be
Newtonian
Non-Newtonian
Flow in Pipe
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Velocity Distribution
Average velocity
Momentum and Kinetic energy correction factors
Laminar Flow of Newtonian Fluids
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Circular cross-section
Local velocity u depends on radius r
Consider a thin ring of radius r and width dr
According to Newton's law of viscosity
Laminar Flow of Newtonian Fluids
Velocity Distribution
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Using both equations
Integrate with boundary condition u = 0, at r = rw
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Maximum velocity is at the center of pipe i.e., at r = 0
Relation of local to maximum velocity
Maximum velocity
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Graphical representation
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Average Velocity
=
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Kinetic energy correction factor
For Laminar Flow = 2
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Momentum correction factor
For Laminar Flow = 4/3
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Hagen-Poiseuille Equation
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Velocity variations with radius for power law fluids
The pressure difference for power law fluids
Laminar Flow for Non-Newtonian Liquids
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Laminar Flow for Non-Newtonian Liquids
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Bingham-plastic fluids:
The general shape of the curve of u versus r in case of Bingham-plastic fluids is;
In the central portion - no velocity variation with the radius
the velocity gradient is confined to an annular space between the central portion and tube wall.
The center portion is moving in plug flow.
Laminar Flow for Non-Newtonian Liquids
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The velocity distribution is;
The shear diagram is;
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Laminar Flow for Non-Newtonian Liquids
Bingham-plastic fluids:
For the velocity variation in the annular space between the tube wall and the plug, the following equation applies;
and
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Turbulent Flow in Pipes and Closed Channels
Viscous Sublayer
Buffer layer
Turbulent core
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Velocity Distribution for Turbulent Flow
Newtonian fluid
Turbulent flow at Reynolds No 10000
Smooth pipe
Velocity gradient is zero at centerline
Turbulent core eddies large but of low intensity
Transition zone eddies small but intense
Kinetic energy
At centerline - isentropic turbulence anisotropic in turbulence core
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Velocity Distribution for Turbulent Flow
It is customary to express the velocity distribution in turbulent flow not as velocity vs. distance but in terms of dimensionless parameters defined by the following eqns;
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Velocity Distribution for Turbulent Flow
For the velocity distribution in the laminar sublayer;
An empirical equation for the so-called buffer layer is;
An equation proposed by Prandtl for the turbulent core is;
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Velocity Distribution for Turbulent Flow
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Flow Quantities for Turbulent Flow
Average Velocity:
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Flow Quantities for Turbulent Flow
Reynolds Number Friction Factor Law for Smooth Pipe:
Von Karman equation
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Flow Quantities for Turbulent Flow
Kinetic Energy and Momentum Correction Factors:
For turbulent flow f is of the order of 0.004, and for this value
and both are assumed to be unity in case of turbulent flow.
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Flow Quantities for Turbulent Flow
Relation between Maximum velocity and Average Velocity:
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Flow Quantities for Turbulent Flow
Effect of Roughness:
In turbulent flow, a rough pipe leads to a larger friction factor for
a given Reynolds number than a smooth pipe does.
If a rough pipe is smoothed, the friction factor is reduced.
When further smoothing brings about no further reduction in
friction factor for a given Reynolds number, the tube is said to be
hydraulically smooth.
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Flow Quantities for Turbulent Flow
Effect of Roughness:
Roughness parameter k
f is a function of both NRe and the relative roughness k/D, where D is the diameter of the pipe.
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Flow Quantities for Turbulent Flow
Effect of Roughness:
All clean, new commercial pipes seem to have the same type of
roughness.
Each material of construction has its own characteristic
roughness parameter.
Old, foul and corroded pipe can be very rough, and the character
of the roughness differs from that of clean pipe.
Roughness has no appreciable effect on the friction factor for
laminar flow unless k is so large that the measurement of the
diameter becomes uncertain.
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Friction Factor Chart
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Friction Factor Chart
For Laminar flow straight line with slope -1
For turbulent flow the lowest line represents the friction factor
for smooth tubes. A much more convenient empirical equation
for this line is the relation;
Over a range of Reynolds number from about 50,000 to 1 106
Over a range of Reynolds number from about 3000 to 3 106
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Friction Factor Chart
Comparing the above two equations
For Power Law Fluids
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Friction Factor Chart
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Drag Reduction in Turbulent Flow
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Effect of Heat transfer / Non-isothermal flow
When the fluid is either heated or cooled by a conduit wall hotter or colder than the fluid, the velocity gradient is changed.
The effect on the velocity gradients is especially pronounced with liquids where viscosity is a strong function of temperature.
1. The Reynolds number is calculated on the assumption that the fluid temperature equals the mean bulk temperature, which is defined as the arithmetic average of the inlet and outlet temperatures.
2. The friction factor corresponding to the mean bulk temperature is divided by a factor
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Effect of Heat transfer / Non-isothermal flow
<
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Flow through Channels of Non-Circular cross-sections
Equivalent Diameter:
It is four times the hydraulic radius.
Hydraulic Radius:
It is the ratio of the cross-sectional area of the channel to the wetted perimeter of the channel.
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Friction Factor in Flow through Channels of Non-
Circular Cross-Sections
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Friction Factor in Flow through Channels of Non-
Circular Cross-Sections
For circular cross-section = 1.0
For Parallel planes = 1.5
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Friction from changes in velocity and direction
Change in velocity direction or magnitude
Additional resistance to skin friction
Boundary layer separation
Sudden expansion
Sudden contraction
Fittings and valves
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Friction loss from sudden expansion (1)
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Friction loss from sudden expansion (2)
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Friction loss from sudden contraction
Vena contracta
Kc is contraction loss coefficient
For laminar flow, this coefficient < 0.1
For turbulent flow
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Effect of fitting and valves
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Form friction losses in Bernoullis equation
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Separation from Velocity Decrease
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Minimizing Contraction Losses
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Minimizing Expansion Losses
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Couette Flow
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Layer Flow with Free Surface
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Layer Flow with Free Surface
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Layer Flow with Free Surface
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Layer Flow with Free Surface
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Reynolds Number