flow through pipes 1

8/6/2019 Flow Through Pipes 1

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In general the flow of liquid along a pipe can be determined by the use of The BernoulliEquation and the Continuity Equation. The former is an algebraic expression of the

conservation of energy and the latter is a statement that what goes into one end of a pipe, comesout at the other end! It must be stressed that in this paper only Incompressible liquids are being

considered.

Bernoulli's Equation

A pipe is a tubular section or hollow cylinder, usually but not necessarily of circular cross-section, usedmainly to convey substances which can flow-liquids and gases (fluids), slurries, powders, masses of small

solids.

This is usually expressed as:

(1)

Where is the head lost due to friction.

Proof

Consider the motion of a fluid down a steamtube.


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Consider the motion of an isolated volume ,, , . After a time it is position ', ', ',

',and since the volume ', ', , , is to both.

The net change is equivalent to moving themass of fluid from , , ', ', to , , ', '.

The work done by the pressure on in time

=

All the work is expended in:

a) Doing work against the pressure at

i.e.

b) Raising the weight of , , ', ', to

, , ', ', i.e.

c) Increasing the kinetic energy of , ,', ', to that of , ', ',


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This equation can be expressed in terms of thequantity flowing per second.

Using the Imperial system of measurements

.

Hence

and

(4)

Darcy's Equation For Circular Pipes

(5)

where Wetted area/Wetted perimeter

The Chezy Equation

(6)

where

and

Reynolds Number

Which equals 2,300 at the point where the flow

changes from Laminar to Turbulent. This isknown as the Critical Velocity


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If is plotted against


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The critical velocity is the velocity at whichthe change over from laminar to turbulent flow

takes place.

Frictional resistance is proportional to the

wetted area

( Where is the perimeter of the pipe.)

Where:

Hydraulic mean Depth

the slope of the hydraulic gradient Note: is not a constant since is a functionof Reynolds number

This formula is known as the " CHEZY

FORMULA"Consider a circular pipe running full


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Which can be written as The DARCY

EQUATIONi.e. Head lost

Note: in the Darcy formula is not an empirical

coefficient

For Laminar flow

The change over from laminar flow to turbulentflow occurs when = 2300 and is independent

of whether the pipe walls are smooth or rough.

Laminar Flow

When the flow is laminar it is possible to use thefollowing equation to find the head lost.

The Poiseuilleequation states that for a round

pipe the head lost due to friction is given by:

Flow occurs because the force across ismore than that on and hence it is possible

to write down the following equation.


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(7)

Integrating

But when

Thus:

But:

(8)

Now the mean velocity is given by:-


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(9)From equation (7)

The head lost due to friction is given by:

Substituting from equation (8) for

Sudden Changes To Pipe Size Or Shape.

The sudden changes or the introduction of bends or valves disturbs the flow pattern along the

pipe. The result is an increase in turbulence and this increases the frictional losses.

Sudden Enlargement


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Note: The pressure just inside the larger pipe isthe same as that in the small pipe! Momentum

at 1 = mass Velocity ==

Momentum at 2 =

Drop in momentum

for this to happen there must be a force

opposing motionThe magnitude of the force =

=

Applying Bernoulli:

Sudden Contraction

As can be seen from the following diagram, eddy

currents reduce the effective diameter of thesmaller pipe. This causes the loss of head.


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The coefficient of Contraction is defined as :

where is the area of the choke and is thearea of the pipe

The head lost is given by:

(10)

is a constant and in this case is approximately0.5. The value of varies with the application

and a selection are shown in the table at the endof this section.

Loss of head, is given by:

But:

And:

Head Lost At An Obstruction (e.g. A Valve)

fig of valve


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Bendradius, which is measured to the inside curvature, is the minimum radius one can bend a pipe, tube,sheet, cable or hose without kinking it, damaging it, or shortening its life.

An elbow is a pipe fitting installed between two lengths of pipe or tubing to allow a change of direction,

usually a 90 degrees or 45 degrees angle, though 22.5 degrees elbows are also made.

A gatevalve, also known as a sluicevalve, is a valve that opens by lifting a round or rectangulargate/wedge out of the path of the fluid.

The formula from above hold good but with differing values for . The following table give

some typical values:

Example: 1234

[imperial]

Example - Example 1Problem

Water is siphoned out of a tank by means of a bent pipe , 80 ft. long and 1 in. in diameter.is below the water surface and 6 in . above the base sof the tank. is vertical and 30 ft long;

is 50 ft. long with the discharge end 5ft. below the base of the tank.

If the barometer is 34 ft. of water and the siphon action at ceases when the absolute pressure is

6 ft. of water, find the depth of water in the tank when the siphon action ceases. is 0.008 andthe loss of head at entry to the pipe is:

Where is the velocity of water in the pipe.


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flow through pipes 1

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