I get a lot of questions from students about the Hazen-Williams equation.
The challenge is that there are so many different forms to the equation and so students often
make mistakes when they use a form but not the correct units for the input values.
So I thought it would be helpful to make a video showing where each of the different
forms derive from the original equation.
It is important to note that Alan Hazen and Gardner Williams created this equation after
observing the velocity flowing through installed water pipelines ranging from 2 inches to 90
inches in diameter.
Their goal was to provide an equation where the friction coefficient was easier to estimate
than the Darcy-Weisbach friction factor.
In this equation, roughness coefficient is a constant that is not a function of the Reynolds
number.
It is often scorned by researchers because the equation does not apply to other fluids,
nor does it account for temperature variation.
However, it is used extensively in the United States and is therefore an acceptable tool
when analyzing water distribution networks and pipelines.
The original empirical equation estimates the velocity equal to a conversion factor
k, times the roughness coefficient C, times the hydraulic radius R to the 0.63 power,
times the friction slope, or S (f) to the 0.54 power.
The conversion factor for US common units is 1.318 and for metric units is 0.849.
Since this equation is really only used in the US, so I'm not going to develop the equation
for the metric approach unless I hear from someone in another country where you want
to see that.
Working with US custom units, the velocity is in feet per second the hydraulic radius
is in feet and the friction slope is unitless but for our example we will assume it is in
feet per feet.
Despite the fact that this equation started off as a solution for fluid velocity, we rarely
care about solving for velocity in a pipeline.
What is really important is how much energy we are losing,
Which is accounted for in the friction slope term.
To make this equation easier to use, we need to plug in all the constant values and also
use the pipe diameter as one of the inputs, so we bring in the 1.318 for k and D/4 for
the hydraulic radius.
Typically, we have the design flowrate for the pipeline Q, so instead of V we substitute
in Q over A then substitute in pi D squared over four for the area.
After a little math we combine all the constants to get a version of the equation where we
have the flow rate in cubic feet per second, the pipe diameter in feet, and the friction
slope in feet per feet.
Since the friction slope is essentially the head loss measured in feet divided by the
length of pipe we can separate that out and solve for the head loss in the pipeline.
This provides us with our first useful stopping point.
This equation will allow you to solve for the total head loss in any pipeline with the
flow rate of Q, in cubic feet per second, and a pipe diameter of D, measured in feet,
and the unitless roughness coefficient, C. The headloss in this equation is the length
times the quantity – open parentheses - 2.31 times Q divided by C times D to the 2.63 power
– close parentheses - to the 1.85 power.
To further make this equation easier to use we need the flow rate to be in gallons per
minute and the diameter in inches because those are the typical units used in water
systems in the United States.
Substituting those conversion values into our equation and then combining these constants
once again provides us with a new equation where the conversion constant changes from
2.31 to, 3.55, and that allows us to use the flow rate in
gpm and the diameter in inches.
So essentially have the same equation as before except the conversion constant is 3.55 and
the flow rate that you input should be in gallons per minute and the diameter that you
input should be in inches.
There are many water agencies across the US that use the Hazen-Williams equation to assess
pipeline performance based upon the unit head loss.
The unit headloss allows us to compare pipes of different diameters and flow rates based
upon how much energy they are losing per 1000 feet of pipe.
To create the unit headloss version of this equation, we insert 1000 for the pipe length
and then do some math to bring it inside the parentheses, which gives us a new conversion
constant of 148.
Substituting this into the equation we get a final version
open parentheses – 148 times Q, divided by the roughness coefficient times D to the
2.6 three power – close parentheses – to the 1.85 power.
For instance, water agencies can say a pipeline in their system cannot have more than 10 feet
per thousand feet of pipe.
With this form of the equation, you can easily input your flowrate in gallons per minute
and diameter in inches and check if it meets that criteria using this equation.