Hazen–Williams equation


The Hazen–Williams equation is an empirical relationship that relates the flow of water in a pipe with the physical properties of the pipe and the pressure drop caused by friction. It is used in the design of water pipe systems such as fire sprinkler systems, water supply networks, and irrigation systems. It is named after Allen Hazen and Gardner Stewart Williams.
The Hazen–Williams equation has the advantage that the coefficient C is not a function of the Reynolds number, but it has the disadvantage that it is only valid for water. Also, it does not account for the temperature or viscosity of the water, and therefore is only valid at room temperature and conventional velocities.

General form

discovered that the velocity of a fluid was proportional to the square root of its head in the early 18th century. It takes energy to push a fluid through a pipe, and Antoine de Chézy discovered that the hydraulic head loss was proportional to the velocity squared. Consequently, the Chézy formula relates hydraulic slope S to the fluid velocity V and hydraulic radius R:
The variable C expresses the proportionality, but the value of C is not a constant. In 1838 and 1839, Gotthilf Hagen and Jean Léonard Marie Poiseuille independently determined a head loss equation for laminar flow, the Hagen–Poiseuille equation. Around 1845, Julius Weisbach and Henry Darcy developed the Darcy–Weisbach equation.
The Darcy-Weisbach equation was difficult to use because the friction factor was difficult to estimate. In 1906, Hazen and Williams provided an empirical formula that was easy to use. The general form of the equation relates the mean velocity of water in a pipe with the geometric properties of the pipe and the slope of the energy line.
where:
  • V is velocity
  • k is a conversion factor for the unit system
  • C is a roughness coefficient
  • R is the hydraulic radius
  • S is the slope of the energy line
The equation is similar to the Chézy formula but the exponents have been adjusted to better fit data from typical engineering situations. A result of adjusting the exponents is that the value of C appears more like a constant over a wide range of the other parameters.
The conversion factor k was chosen so that the values for C were the same as in the Chézy formula for the typical hydraulic slope of S=0.001. The value of k is 0.001−0.04.
Typical C factors used in design, which take into account some increase in roughness as pipe ages are as follows:
MaterialC Factor lowC Factor highReference
Asbestos-cement140140-
Cast iron new130130
Cast iron 10 years107113
Cast iron 20 years89100
Cement-Mortar Lined Ductile Iron Pipe140140–
Concrete100140
Copper130140
Steel90120–
Galvanized iron120120
Polyethylene140140
Polyvinyl chloride 150150
Fibre-reinforced plastic 150150

Pipe equation

The general form can be specialized for full pipe flows. Taking the general form
and exponentiating each side by gives
Rearranging gives
The flow rate, so
The hydraulic radius for a full pipe of geometric diameter is ; the pipe's cross sectional area is, so

U.S. customary units (Imperial)

When used to calculate the pressure drop using the US customary units system, the equation is:
where:
  • Spsi per foot = frictional resistance in psig/ft
  • Sfoot of water per foot of pipe
  • Pd = pressure drop over the length of pipe in psig
  • L = length of pipe in feet
  • Q = flow, gpm
  • C = pipe roughness coefficient
  • d = inside pipe diameter, in

    SI units

When used to calculate the head loss with the International System of Units, the equation will then become
where:
  • S = hydraulic slope
  • hf = head loss in meters over the length of pipe
  • L = length of pipe in meters
  • Q = volumetric flow rate, m3/s
  • C = pipe roughness coefficient
  • d = inside pipe diameter, m