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Branched Pipes

The application of the Bernoulli equation to Branched pipes
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Key Facts

Gyroscopic Couple: The rate of change of angular momentum (\inline \tau) = \inline I\omega\Omega (In the limit).
  • \inline I = Moment of Inertia.
  • \inline \omega = Angular velocity
  • \inline \Omega = Angular velocity of precession.


It is common for a pipeline to be branched and for the system to be feeding more than one reservoir. This page examines this situation.

Bernoulli's principle states that for an inviscid flow, an increase in the speed of the fluid occurs simultaneously with a decrease in pressure or a decrease in the fluid's potential energy. Bernoulli's principle can be applied to various types of fluid flow, resulting in what is loosely denoted as Bernoulli's equation.

Branched Pipes


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Applying Bernoulli's equation to the whol system but neglecting both the entry head and the junction head.

H_2=\frac{4fl_1{v_{1}}^{2}}{2d_1g}+\frac{4fl_3{v_{3}}^{2}}{2d_3 g}+\frac{{v_{2}}^{2}}{2g}

It most cases it is possible to neglect the last terms of \inline \displaystyle\frac{v^2}{2g}.

Applying the continuity equation:

Example - Example 1
Water is pumped from a river to two reservoirs and . The water surface in reservoir is at the same hight as the river whilst that in reservoir is 20 ft. higher.

Pumping from the river takes place by means of a centrifugal pump, the equation relating flow (in cubic ft./sec.) and ft. at a constant speed being given by

From the river to a junction is a common pipe is used of 8 in. diameter and 500 ft. long. The branch to the reservoir is 5 in. in diameter and 200 ft. long. The branch from to reservoir is 6 in. in diameter and 200 ft. long.

Neglecting all losses other than pipe friction, calculate the discharge to and . Take as 0.007 throughout.

Darcy's equation can be rewritten as follows:

Applying Bernoulli at the river and reservoir :


But by continuity:

From equation (1)

Subtracting equation (2) from (1)

Substituting into equation (3) squared with values for from equation (4) gives:

Rearranging and collecting terms:

Squaring gives:

Treating this as a quadratic in

From equation (4)