What Is the Hazen-Williams Equation?

The Hazen-Williams equation is a formula for calculating how much the ambient pressure drops in a fluid as it flows through a pipe due to friction with the interior surface of the pipe, the pipe's interior diameter, and the velocity of flow rate for the fluid. This flow rate reduction has been commonly used by engineers in the past when the fluid flow was turbulent, as it gave a good approximation of loss of velocity. The formula is relatively simple, but it has several limiting factors to its effective use, and the advent of personal computers has made it largely obsolete.

Water pipe systems for transferring fluid all have what is known as head loss, which is a sum of the elevation, velocity, and pressure loss of the fluid as it moves due to friction within the fluid, as it interacts with the pipe wall and other pipe obstructions, and as a side-effect of the turbulence that these interactions cause. Head loss is also based on the friction factor present, which is calculated from the type of pipe material used and the velocity of fluid flow. Friction factors can range from 80 to 130 or more, and this variability makes the Hazen-Williams equation only a rough calculation for pressure drop.

Typical limiting factors in calculating volumetric flow rate using the Hazen-Williams equation are accepted by engineers. These factors include the limitation that the fluid must have a viscosity of at least 1.13 centistokes, which is what water displays at a room temperature of 60° Fahrenheit (15.5° Celsius). The pipe must also be larger than 2 inches (5.08 centimeters) in diameter, and the flow rate cannot exceed 10 feet per second (3.05 meters per second).

There are two formulas typically used in the Hazen-Williams equation, one based on empirical or experimental data and imperial units, and one that uses standard international units. The imperial formula is written as h_f = 0.002083 L (100/C)^1.85 x (gpm^1.85/d^4.8655) where "h_f" equals the head loss being determined in feet, "L" represents the length of pipe in measurement by feet, and "C" is the friction coefficient for the type of pipe material. "Gpm" represents gallons per minute calculated as US measured gallons of flow through the pipe, and "d" represents the interior initial diameter of the pipe before build-up or corrosion on the pipe wall takes place. Here, the value of 100 in the formula represents a dimensionless Hazen-Williams factor.

The standard international units formula is just another way of calculating head loss, also known as a frictional pressure drop, with metric units. It is stated as ΔP = 1.1101 x 10¹⁰ (Q/C)^1.85 1/D^4.87 where "ΔP" is the pressure drop in kiloPascals per meter, "Q" is the flow rate of fluid in meters cubed per hour, "D" is the internal pipe diameter, and "C" here is the dimensionless Hazen-Williams factor. While using the standard of 100 for the Hazen-Williams factor is routine, if the pipe is 10 to 15 years old, often a value of 75 may be substituted instead due to mineral deposits and corrosion in the pipe that increase the friction level and turbulence.

The use of the Hazen-Williams equation in the absence of more accurate computer-based calculations is still possible for many types of liquid flow rate systems. It can be used for fire sprinkler systems to irrigation systems, or water supply networks for buildings or municipalities. This is because several established friction factors now exist for pipe material types that are input into the formula, such as brass and copper tubing at 130, poly-vinyl chloride (PVC) pipe at 150, steel pipe at 120, and more. Each value also has some leeway in that it is an approximation based on the presence of deformations in the internal pipe surface that build up over time as the pipe ages. Where more accurate values are necessary for head loss or a fluid flow for a substance other than water is being measured, the Darcy-Weisbach equation is employed, which uses a friction coefficient more accurately calculated from a Moody diagram incorporating Reynolds numbers.