The head loss due to friction in Pipes with uniform run off. Also covered are Tapered Pipes
In the supply of water for Domestic, Commercial and Irrigation purposes, it is common for water to be taken from the pipe line, in many places along the length. Due to the viscosity of a fluid in motion, energy is dissipated as the fluid circulates through the pipe. As a result,the pressure of the fluid at various points along the pipeline will vary.
This section analyses the head loss in uniform pipe.
Head Loss (or friction head or resistance head) is due to the frictional forces acting against a fluid's motion by the container.
23287/uniform-run-off-and-tapered-pipes-001.png cannot be found in /users/23287/uniform-run-off-and-tapered-pipes-001.png. Please contact the submission author.
Let the rate of flow at the pipe entry be
and that at the pipe exit be
Consider a short length of pipe
at a distance
from the entry point.
The velocity at this section is:
The frictional head lost over:
Thus the Total Head Lost over the pipe length is given by:
It can be seen that the above equation assumes that not all the water is discharged over the length
. If however this is not the case,
And the Head Lost:
i.e. A third of the head loss which would have been lost with a uniform velocity of flow
Example - Example 1
A horizontal water-main comprises 5000 ft. of 6 in pipe followed by 3000 ft. of 4 in. pipe (
=0.007 for both). All the water is drawn off at a uniform rate per ft. length of pipe.
If the total input is
, find the total pressure
drop along the main, neglecting all losses except friction. Also draw the Hydraulic Gradient diagram
taking the pressure head at inlet as 180 ft.
23287/uniform-run-off-and-tapered-pipes-003.png cannot be found in /users/23287/uniform-run-off-and-tapered-pipes-003.png. Please contact the submission author.
The velocity of flow at
As the water is drawn off at a steady rate, the rate of flow at the end of the 6 in. pipe is given by :
The quantity flowing
And the rate of flow
Substituting the above values into equation (1) of Total Head Lost:
For the 4 in length of pipe
= 0 since all the water is used up and nothing flows out of the end of the pipe. The velocity of flow is now given by:
Using equation (1) of Total Head Lost again:
Hence the total head lost
The Hydraulic Gradient
Three points on the graph are already known. The inlet pressure of 180 ft. and consequently the pressures at the end of the 6 in. pipe and the 5 in. pipe. It is now necessary to establish the pressure varies between these points.
enters the pipe and it is all drawn off at a uniform rate over the complete length of the pipe. Thus at any point distant
from the start of the pipe the quantity flowing will be:
is in the 6 in. diameter section of the pipe:
Hence the head lost due to friction between the inlet and the point
being in the 6 in. section of the pipe) is given by Equation (1) of Total Head Lost:
Thus the pressure at
ft. from the
is given by:
At the start of the 4 in. pipe the velocity of flow is 3.88 ft./sec. and at the end of the pipe the velocity is zero. Hence the velocity at any point
is given by:
Hence the frictional head lost over the distance
is given by:
Which can be written as:
Hence, using equations (1
) and (4
) the following graph can be drawn.
23287/uniform-run-off-and-tapered-pipes-002.png cannot be found in /users/23287/uniform-run-off-and-tapered-pipes-002.png. Please contact the submission author.
This is the required Hydraulic Gradient. The values between the known points are functions of