Incompressible Flow in

Pipes and Channels

By

Farhan Ahmad

farhanahmad@uet.edu.pk

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 τ and r

At r =0 , τ = 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

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

𝑺 = 𝝅𝒓𝒘𝟐

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Kinetic energy correction factor

For Laminar Flow α = 2

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Momentum correction factor

For Laminar Flow β = 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

Transition 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.