In 1936 Hardy Cross published a method to analyze water distribution networks.
This method remained one of the primary approaches until the use of computer software replaced
the hand analysis.
In order to understand how to use this method, suppose we have the following water distribution
network, shown here in plan view.
All of the pipes are labeled as P-1, P-2 and so forth, while all of the network junctions,
or nodes, are labeled as J-1, J-2 and so on.
In this water system the water demands leaving the distribution system at each of the junctions
are labeled as Q J-1, Q J-2 and so forth.
This completes our generic system schematic.
When using the Hardy Cross method, the first step is to identify the loops in the system,
in other words, the series of pipes that begin and end at the same point and have no interior
pipes.
For the network here, there are two loops, and we arbitrarily assign the clockwise direction
as the positive direction in the loop.
The next step is to assume a flow direction for the water in each of the pipelines.
I represent the direction of flow with this small chevron symbol applied to each of the
pipes in our network schematic.
This is also arbitrary, but it helps make the calculations easier to understand if the
directions that you assign are based in some reasoning and not purely random.
The third step is to make an initial guess for the flow rate that is being sent down
each pipeline.
This also can be arbitrary so long as you satisfy the condition of continuity.
In other words, at each junction, the amount of water coming into the junction is the same
as the amount of water leaving the junction.
For example – at Junction 4the flow rate from pipe 3 and pipe 4 are coming into the
junction, while the demand at junction 4 is leaving
the system.
The summation of these flow rates need to be 0.
So you may make a guess for the flow rate in Pipes 3 and 4 whatever you want, so long
as their sum is equal to the demand leaving at junction 4.
The fourth step is the calculate the headlosses in each pipeline based upon the flow rate
you guessed in step 3.
The headloss equation you use does not matter much.
Hardy Cross simplified both Darcy-Weisbach and Hazen-Williams to the following general
form: headloss is equal to r times Q to the n power.
Within the US the Hazen-Williams equation is the standard approach to network analysis,
so the n value is 1.85 and the r value is a characteristic of the pipeline’s physical
constants – its length, diameter, and roughness – as seen here.
Using this format of the equation allows us to use Q in gallons per minute and D in inches
– very handy!
The fifth step is to calculate the capital R value, which is given by Hardy Cross in
this expression.
With a little algebra, we can use this simplified expression to make it a little easier on our
calculator fingers.
Just remember that despite the direction you assigned the flow rate earlier,
this value must remain positive.
In the next step you calculate the summation of the headlosses as well as the summation
of the capital R values within a loop.
Remember here that the headloss has a direction assigned to it based upon our sign convention
we chose earlier.
Therefore in a completely balanced loop, the headlosses should sum to zero.
Until that is achieved, however, you can calculate both positive and negative values for the
summation of headlosses in each loop.
In step 7 calculate the amount of correction needed for the flow rate guesses you made
in step 3.
The correction factor is an invention of Hardy Cross.
It is given from the following expression which is simply the negative summation of
the headlosses divided by the summation of the capital R values.
For eighth step – and I promise we are nearing the end – you calculate the adjusted flow
rate to start the next iteration.
The adjusted flow rate is equal to the flow rate that you guessed in step 3
plus the correction factor calculated for the loop – the result from step 7.
Simple right?
The only complication comes when pipes are shared between two loops.
For shared pipes the adjusted flow rate is going to be the flow rate guessed in step
3 plus the loop adjustment from the loop in which you are working MINUS the correction
from the shared loop.
I know.
This is a little confusing, so let me give an example.
So Pipe 5 is shared between both loop1 – the triangular loop – and loop 2.
In Loop 1 the adjusted flow rate is the guess you made plus the correction factor obtained
for loop 1 MINUS the correction factor obtained for loop
2 – which is acting as the shared loop in this instance.
However, when working in loop 2, the adjusted flow rate is the guessed flow rate in step
3 plus the correction value for loop 2 MINUS the correction value for loop 1.
The key to understand is that once you have completed applying the correction factor the
values of flow rate in the shared pipe should be the same but have opposite sign convention.
Once you have corrected all of your guesses from step 3, now you are set to start the
next iteration.
So the last step, in the Hardy Cross method, is to repeat steps 4 through 8 until the summation
of headloss in each loop is very near to zero.
When that happens, the correction value should be so small that the change in flow rate between
iterations should no longer be significant.
The result is that you finally have a better idea as to how much water is flowing in each
of the pipelines of your network.
I realize that this process is not so easily learned from a generic example such as this,
so I have also prepared a series of videos that walk you through setting up the analysis
of a slightly more complicated network using Excel.