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The head lost due to friction in Pipes with uniform run off. Also covered are Tapered Pipes

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Introduction

In the supply of water for Domestic, Commercial and Irrigation, it is common for water to be taken from the pipe line, in many places along the length. This paper analysis the head lost in a pipe with uniform head lost. Also included is the frictional loss in tapered pipes.

Pipe Flow with Uniform Run-off

23287/uniform-run-off-and-tapered-pipes-001.png

Let the rate of flow at the pipe entry be Q and that at the pipe exit be mQ

Consider a short length of pipe $\delta x$ at a distance x from the entry point.

The velocity at this section is:-

$v_x=v-\frac{x}{l}(v-mv)$

(1)

$Or\;\;\;\;\;v_x=v\left [ 1-\frac{x}{l}\left ( 1-m \right ) \right ]$

(2)

$\text{The frictional head lost over}\;\delta x=\frac{4f{v_{x}}^{2}\;\delta x}{2dg}$

(3)

$=\frac{4fv^2}{2gd}\left [ 1-\frac{x}{l}\left ( 1-m \right ) \right ]^2\;dx$

(4)

Thus the Total Head Lost over the pipe length is given by:-

$=\frac{4fv^2}{2dg}\int_{0}^{l}\left [ 1-\frac{2x}{l}(1-m)+\frac{x^2}{l^2}(1-m)^2 \right ]\;dx$

(5)

$=\frac{4fv^2}{2dg}\left [ x-\frac{x^2}{l}(1-m)+\frac{x^3}{3l^2}(1-m)^2 \right ]_{0}^{l}$

(6)

$=\frac{4fv^2}{2dg}\left [ 1-\frac{l^2}{l}(1-m)+\frac{l^3}{3l^2}(1-m)^2 \right ]$

(7)

$=\frac{4fv^2l}{2dg}\left [ 1-1+m+\frac{1}{3}(1-2m+m^2) \right ]$

(8)

$=\frac{4fv^2l}{2dg}\times \frac{1}{3}(1+m+m^2)$

(9)

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It can be seen that the above equation assumes that not all the water is discharged over the length l. If however this is not the case m = 0 and:-

$\text{And}\;\;\;\;\;\;\text{The head Lost}\;=\frac{1}{3}\times \frac{4flv^2}{2dg}$

(10)

i.e. A third of the head loss which would have been lost with a uniform velocity of flow v

Example 1

A horizontal water-main comprises 5000 ft. of 6 in pipe followed by 3000 ft. of 4 in. pipe (f=0.007 for both) All the water is drawn off at a uniform rate per ft. length of pipe.

If the total input is $0.9\;ft.^3/sec$ , find the total pressure drop along the main, neglecting all losses except friction. Also draw the Hydraulic Gradient taking the pressure head at inlet as 180 ft. (B.Sc. part 2)

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To view the workings please click on the red button

Frictional Loss in a Tapered Pipe

23287/uniform_run_off_and_tapered_pipes_004.png

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Let the velocity and diameter at the sections 1 and 2 be as shown on the diagram. Consider the head lost over a short length dx which is a distance x from position 1 and let the velocity and diameter associated with this short length be v and d.