Before we discuss how to analyze pipes in parallel, it is important to briefly mention
how one analyzes the hydraulics of different pipes in a series.
The figure shown here is a section view of three pipes laid in series.
The first pipe has a different Diameter, roughness, and length from t the second, which is in
turn different from the third pipe in the series.
These pipes are join by reducers that allow the transition between each diameter.
By the way – because water can flow in both directions we call all transitions between
different diameters a “reducer”.
The headloss between point A and B shown here is simply the summation of the headlosses
in each pipe and the minor losses
between points A and B. That is it.
Pipes in Parallel, however, present a different challenge.
For example, we have a system, shown in Plan view here, where there is a flow rate entering
a junction that splits the flow into two pipes, Pipes A and B.
If the diameters, lengths, and pipe roughness are the same between the two pipes, then
We can easily estimate the flowrate in each pipe as being half of the flowrate, Q, entering
the system.
However it is more often the case that the diameter, length, and roughness of the pipelines
are not the same.
In this case the flowrates in each pipe are not assumed to be equal which means we need
an approach to determine the flowrate in each pipe.
… The first tool that we have to find a solution is conservation of mass.
In other words, the sum of the flowrates in both pipelines is equal to the Q that entered
the system.
The second tool we have is conservation of energy.
We express this as the headloss in Pipe A being equal to the headloss in Pipe B. In
other words, the amount of energy lost in pipe A, regardless of the diameter, roughness
or length of pipe A, is going to be the same as the headloss in pipe B. How is this possible?
Well, water will travel the path of least resistance.
In this case let’s say that water passing through A arrives at the end junction with
greater energy than the water in B.
This means that the water in A will push back against the flowrate in B until the flowrate
decreases enough to make the energy losses equal.
Another way to think about it is if the headloss in pipes A and B were not equal, then the
results would be that two different pressures trying to abide in the same space when the
two fluid streams join up again.
That is not possible unless there were a barrier of some sort to prevent the two flowrates
from interacting.
Ultimately there will be only one pressure head at the junction as the two flowrates
join back together.
Calculating the headloss we can use one of the headloss formulae previously covered – in
this case I will use the Hazen-Williams equation.
Because the headloss in A is equal to the headloss in B we can set the two headloss
calculations equal to each other.
Looking at it this way we can see that the pipe characteristics are given for both pipes
A and B, and I will represent this as r(a) and r(b).
Therefore the unknowns in these expressions are the flowrates, which written this way
allows us to solve for one Q in terms of the other.
With one Q in terms of the other, we can substitute this expression into the conservation of mass
equation written earlier and then solve for Q in pipe B. Once we’ve done that it is
pretty simple to calculate the flowrate in pipe A and then determine the headloss in
both pipes.
If we introduce a third pipe to the system, then we can approach the solution in the same
way by using conservation of mass … and by setting the headlosses in each pipe
equal to each other.
The unknowns are still the flowrates in each pipe.
In other words we have 4 equations and 3 unknowns.
Also doable, but the math gets a little long winded.
Now consider the reality of pipe networks where there are not merely 2 or 3 pipes but
many pipes and many analysis points in the system.
Therefore the approach of simultaneously solving this many equations with this many unknowns
is not feasible from a commercial standpoint.
We need a better method, to simplify the process for us.