Hazen–Williams equation

The Hazen–Williams equation is an empirical formula which 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.

General form

The general form of the equation relates the mean velocity of water in a pipe with the geometric properties of the pipe and slope of the energy line.

$V = k\, C\, R^{0.63}\, S^{0.54}$

where:

• V is velocity
• k is a conversion factor for the unit system (k = 1.318 for US customary units, k = 0.849 for SI units)
• C is a roughness coefficient
• R is the hydraulic radius
• S is the slope of the energy line (head loss per length of pipe or h_f/L)

Typical C factors used in design, which take into account some increase in roughness as pipe ages are as follows:

Material	C Factor low	C Factor high	Reference
Asbestos-cement	140	140	-
Cast iron	100	140	-
Cement-Mortar Lined Ductile Iron Pipe	140	140	-
Concrete	100	140	-
Copper	130	140	-
Steel	90	110	-
Galvanized iron	120	120	-
Polyethylene	140	140	-
Polyvinyl chloride (PVC)	130	130	-
Fibre-reinforced plastic (FRP)	150	150	-

Pipe equation

The general form can be specialized for full pipe flows. Taking the general form

$V = k\, C\, R^{0.63}\, S^{0.54}$

and exponentiating each side by $1/0.54$ gives (rounding exponents to 2 decimals)

$V^{1.85} = k^{1.85}\, C^{1.85}\, R^{1.17}\, S$

Rearranging gives

$S = {V^{1.85} \over k^{1.85}\, C^{1.85}\, R^{1.17}}$

The flow rate Q = V A, so

$S = {V^{1.85} A^{1.85}\over k^{1.85}\, C^{1.85}\, R^{1.17}\, A^{1.85}} = {Q^{1.85}\over k^{1.85}\, C^{1.85}\, R^{1.17}\, A^{1.85}}$

The hydraulic radius R (which is different from the geometric radius r) for a full pipe of geometric diameter d is d/4; the pipe's cross sectional area A is $\pi d^2 / 4$, so

$S = {4^{1.17}\, 4^{1.85}\,Q^{1.85}\over \pi^{1.85}\,k^{1.85}\, C^{1.85}\, d^{1.17}\, d^{3.70}} = {4^{3.02}\,Q^{1.85}\over \pi^{1.85}\,k^{1.85}\, C^{1.85}\, d^{4.87}} = { 4^{3.02} \over \pi^{1.85}\,k^{1.85}} {Q^{1.85}\over C^{1.85}\, d^{4.87}} = { 7.916 \over k^{1.85}} {Q^{1.85}\over C^{1.85}\, d^{4.87}}$

U.S. customary units (Imperial)

When used to calculate the pressure drop using the US customary units system, the equation is:

$P_d=\frac{4.52\quad L\quad Q^{1.85}}{C^{1.85}\quad d^{4.87}}$

where:

P_d = pressure drop over a length of pipe, psig (pounds per square inch gauge pressure)

L = length of pipe, ft (feet)

Q = flow, gpm (gallons per minute)

d = inside pipe diameter, in (inches)

SI units

When used to calculate the pressure drop with the International System of Units, the equation becomes:

$h_f = \frac{10.67\quad L \quad Q^{1.85}}{C^{1.85}\quad d^{4.87}}$

where:

• h_f = pressure loss over a length of pipe, m (head pressure)
• L = length of pipe, m (meters)
• Q = volumetric flow rate, m³/s (cubic meters per second)
• d = inside pipe diameter, m (meters)

See also

• Darcy–Weisbach equation
• Fluid dynamics
• Friction
• Pressure
• Volumetric flow rate
• Water pipe

References

• Hazen, A.; Williams, G. S. (1920), Hydraulic Tables (3rd ed.), New York: John Wiley and Sons 
• Watkins, James A. (1987), Turf Irrigation Manual (5th ed.), Telsco 
• Finnemore, E. John; Franzini, Joseph B. (2002), Fluid Mechanics (10th ed.), McGraw Hill 
• Mays, Larry W. (1999), Hydraulic Design Handbook, McGraw Hill 

External links

• Engineering Toolbox reference
• Engineering toolbox Hazen–Williams coefficients
• Online Hazen–Williams calculator for gravity-fed pipes.
• Online Hazen–Williams calculator for pressurized pipes.