Hazen–Williams equation

Henri Pitot 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.[5] Consequently, the Chézy formula relates hydraulic slope S (head loss per unit length) to the fluid velocity V and hydraulic radius R:

${\displaystyle V=C{\sqrt {RS}}=C\,R^{0.5}\,S^{0.5}}$

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.[6]

The Darcy-Weisbach equation was difficult to use because the friction factor was difficult to estimate.[7] 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 slope of the energy line.

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

where:

• V is velocity (in ft/s for US customary units, in m/s for SI units)
• 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 (in ft for US customary units, in m for SI units)
• S is the slope of the energy line (head loss per length of pipe or h_f/L)

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.[8]

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.[9] The value of k is 0.001^−0.04.[10]

Typical C factors used in design, which take into account some increase in roughness as pipe ages are as follows:[11]

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

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

and exponentiating each side by 1/0.54 gives (rounding exponents to 3–4 decimals)

${\displaystyle V^{1.852}=k^{1.852}\,C^{1.852}\,R^{1.167}\,S}$

Rearranging gives

${\displaystyle S={V^{1.852} \over k^{1.852}\,C^{1.852}\,R^{1.167}}}$

The flow rate Q = V A, so

${\displaystyle S={V^{1.852}A^{1.852} \over k^{1.852}\,C^{1.852}\,R^{1.167}\,A^{1.852}}={Q^{1.852} \over k^{1.852}\,C^{1.852}\,R^{1.167}\,A^{1.852}}}$

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 π d² / 4, so

${\displaystyle S={4^{1.167}\,4^{1.852}\,Q^{1.852} \over \pi ^{1.852}\,k^{1.852}\,C^{1.852}\,d^{1.167}\,d^{3.7034}}={4^{3.019}\,Q^{1.852} \over \pi ^{1.852}\,k^{1.852}\,C^{1.852}\,d^{4.8704}}={4^{3.019} \over \pi ^{1.852}\,k^{1.852}}{Q^{1.852} \over C^{1.852}\,d^{4.8704}}={7.8828 \over k^{1.852}}{Q^{1.852} \over C^{1.852}\,d^{4.8704}}}$

• Darcy–Weisbach equation and Prony equation for alternatives
• Fluid dynamics
• Friction
• Minor losses in pipe flow
• Plumbing
• Pressure
• Volumetric flow rate

• Finnemore, E. John; Franzini, Joseph B. (2002), Fluid Mechanics (10th ed.), McGraw Hill
• Mays, Larry W. (1999), Hydraulic Design Handbook, McGraw Hill
• Watkins, James A. (1987), Turf Irrigation Manual (5th ed.), Telsco
• Williams, Gardner Stewart; Hazen, Allen (1905), Hydraulic tables: showing the loss of head due to the friction of water flowing in pipes, aqueducts, sewers, etc. and the discharge over weirs (first ed.), New York: John Wiley and Sons
• Williams and Hazen, Second edition, 1909
• Williams, Gardner Stewart; Hazen, Allen (1914), Hydraulic tables: the elements of gagings and the friction of water flowing in pipes, aqueducts, sewers, etc., as determined by the Hazen and Williams formula and the flow of water over sharp-edged and irregular weirs, and the quantity discharged as determined by Bazin's formula and experimental investigations upon large models. (2nd revised and enlarged ed.), New York: John Wiley and Sons
• Williams, Gardner Stewart; Hazen, Allen (1920), Hydraulic tables: the elements of gagings and the friction of water flowing in pipes, aqueducts, sewers, etc., as determined by the Hazen and Williams formula and the flow of water over sharp-edged and irregular weirs, and the quantity discharged as determined by Bazin's formula and experimental investigations upon large models. (3rd ed.), New York: John Wiley and Sons, OCLC 1981183

• Engineering Toolbox reference
• Engineering toolbox Hazen–Williams coefficients
• Online Hazen–Williams calculator for gravity-fed pipes.
• Online Hazen–Williams calculator for pressurized pipes.
• https://books.google.com/books?id=DxoMAQAAIAAJ&pg=PA736
• https://books.google.com/books?id=RAMX5xuXSrUC&pg=PA145 States pocket calculators and computers make calculations easier. H-W is good for smooth pipes, but Manning better for rough pipes (compared to D-W model).