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Hazen Williams Equation Assumptions And Critical Thinking

The Hazen–Williams equation is an empirical relationship 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[1] such as fire sprinkler systems,[2]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.[3]

General form[edit]

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.[4] Consequently, the Chézy formula relates hydraulic slope S (head loss per unit length) to the fluid velocity V and hydraulic radiusR:

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.[5]

The Darcy-Weisbach equation was difficult to use because the friction factor was difficult to estimate.[6] 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.


  • 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 hf/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.[7]

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.[8] The value of k is 0.001−0.04.[9]

Typical C factors used in design, which take into account some increase in roughness as pipe ages are as follows:[10]

Pipe equation[edit]

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

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

Rearranging gives

The flow rate Q = VA, so

The hydraulic radiusR (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

U.S. customary units (Imperial)[edit]

When used to calculate the pressure drop using the US customary units system, the equation is:[11]


Note: Caution with U S Customary Units is advised. The equation for head loss in pipes, also referred to as slope, S, expressed in "feet per foot of length" vs. in 'psi per foot of length' as described above, with the inside pipe diameter, d, being entered in feet vs. inches, and the flow rate, Q, being entered in cubic feet per second, cfs, vs. gallons per minute, gpm, appears very similar. However, the constant is 4.73 vs. the 4.52 constant as shown above in the formula as arranged by NFPA for sprinkler system design. The exponents and the Hazen-Williams "C" values are unchanged.

SI units[edit]

When used to calculate the head loss with the International System of Units, the equation becomes:[12]


  • S = Hydraulic slope
  • hf = head loss in meters (water) over the length of pipe
  • L = length of pipe in meters
  • Q = volumetric flow rate, m3/s (cubic meters per second)
  • C = pipe roughness coefficient
  • d = inside pipe diameter, m (meters)
Note: pressure drop can be computed from head loss as hf × the unit weight of water (e.g., 9810 N/m3 at 4 deg C)

See also[edit]


  1. ^"Hazen–Williams Formula". Archived from the original on 22 August 2008. Retrieved 6 December 2008. 
  2. ^"Hazen–Williams equation in fire protection systems". Canute LLP. 27 January 2009. Archived from the original on 2013-04-06. Retrieved 2009-01-27. 
  3. ^Brater, Ernest F.; King, Horace W.; Lindell, James E.; Wei, C. Y. (1996). "6". Handbook of Hydraulics (Seventh ed.). New York: McGraw Hill. p. 6.29. ISBN 0-07-007247-7. 
  4. ^Walski, Thomas M. (March 2006), "A history of water distribution", Journal of the American Water Works Association, American Water Works Association, 98 (3): 110–121 , p. 112.
  5. ^Walski 2006, p. 112
  6. ^Walski 2006, p. 113
  7. ^Williams & Hazen 1914, p. 1, stating "Exponents can be selected, however, representing approximate average conditions, so that the value of c for a given condition of surface will vary so little as to be practically constant."
  8. ^Williams & Hazen 1914, p. 1
  9. ^Williams & Hazen 1914, pp. 1–2
  10. ^ abcdefghijklHazen-Williams Coefficients, Engineering ToolBox, retrieved 7 October 2012 
  11. ^2007 version of NFPA 13: Standard for the Installation of Sprinkler Systems, page 13-213, eqn
  12. ^"Comparison of Pipe Flow Equations and Head Losses in Fittings"(PDF). Retrieved 2008-12-06. 
  • 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 

External links[edit]

The Darcy-Weisbach equation with the Moody diagram is considered to be the most accurate model for estimating frictional head loss for a steady pipe flow. Since the Darcy-Weisbach equation requires iterative calculation an alternative empirical head loss calculation like the Hazen-Williams equation may be preferred:

h = 0.2083 (100 / c)1.852 q1.852 / dh4.8655                          (1)


h = friction head loss in feet of water per 100 feet of pipe (fth20/100 ft pipe)

c = Hazen-Williams roughness constant

q = volume flow (gal/min)

dh = inside hydraulic diameter (inches)

Note that the Hazen-Williams formula is empirical and lacks a theoretical basis. Be aware that the roughness constants are based on "normal" conditions with approximately 1 m/s (3 ft/sec).

teh Related Mobile Apps from The Engineering ToolBox

- free apps for offline use on mobile devices.

Online Hazens-Williams Calculator

Imperial Units

The calculators below can used to calculate the specific head loss (head loss per 100 ft (m) pipe) and the actual head loss for the actual length of pipe:

SI Units

The Hazen-Williams equation is not the only empirical formula available. Manning's formula is commonly used to calculate gravity driven flows in open channels.

The flow velocity can be calculated as

v = 0.408709 q / dh2                                (2)


v = flow velocity (ft/s)


The Hazen-Williams equation is assumed to be relatively accurate for water flow in piping systems when

For hotter water with lower kinematic viscosity (example 0.55 cSt at 130 oF (54.4 oC)) the error will be significant.

Since the Hazen-Williams method is only valid for water flow - the Darcy Weisbach method should be used for other liquids or gases.

  • 1 ft (foot) = 0.3048 m
  • 1 in (inch) = 25.4 mm
  • 1 gal (US)/min =6.30888x10-5 m3/s = 0.227 m3/h = 0.0631 dm3(liter)/s = 2.228x10-3 ft3/s = 0.1337 ft3/min = 0.8327 Imperial gal (UK)/min

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