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{{Short description|Equation on water flow in pipes}}
The '''Hazen–Williams equation''' is an [[Philosophy:Empirical relationship|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<ref>{{Cite web |url=http://docs.bentley.com/en/HMFlowMaster/FlowMasterHelp-06-05.html |title=Hazen–Williams Formula |access-date=2008-12-06 |url-status=dead |archive-url=https://web.archive.org/web/20080822051759/http://docs.bentley.com/en/HMFlowMaster/FlowMasterHelp-06-05.html |archive-date=22 August 2008  }}</ref> such as [[Engineering:Fire sprinkler system|fire sprinkler system]]s,<ref>{{Cite web|url=http://www.canutesoft.com/index.php/Basic-Hydraulics-for-fire-protection-engineers/Hazen-Williams-formula-for-use-in-fire-sprinkler-systems.html|title=Hazen–Williams equation in fire protection systems|date=27 January 2009|publisher=Canute LLP|access-date=2009-01-27 |archive-url=https://web.archive.org/web/20130406095047/http://www.canutesoft.com/index.php/Basic-Hydraulics-for-fire-protection-engineers/Hazen-Williams-formula-for-use-in-fire-sprinkler-systems.html |archive-date=2013-04-06}}</ref> [[Engineering:Water supply network|water supply network]]s, and [[Earth:Irrigation|irrigation]] systems. It is named after [[Biography:Allen Hazen|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 [[Chemistry:Water|water]].  Also, it does not account for the temperature or [[Physics:Viscosity|viscosity]] of the water,<ref>{{Cite book |last1=Brater |first1=Ernest F. |last2=King |first2=Horace W. |last3=Lindell |first3=James E. |last4=Wei |first4=C. Y. |title=Handbook of Hydraulics |publisher=McGraw Hill |location=New York |year=1996 |edition=Seventh |pages=6.29 |chapter=6 |isbn=0-07-007247-7}}</ref> and therefore is only valid at room temperature and conventional velocities.<ref>{{Cite book|url=https://www.worldcat.org/oclc/144609617|title=Pumping station design|date=2006|publisher=Butterworth-Heinemann|others=Jones, Garr M.|isbn=978-0-08-094106-6|edition=3rd|location=Burlington, MA|pages=3.4|oclc=144609617}}</ref>

==General form==
[[Biography:Henri Pitot|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 [[Physics:Hydraulic head|hydraulic head]] loss was proportional to the velocity squared.<ref>{{Citation |last=Walski |first=Thomas M. |title=A history of water distribution |journal=Journal of the American Water Works Association |publisher=American Water Works Association |volume=98 |issue=3 |pages=110–121 |date=March 2006 |doi= 10.1002/j.1551-8833.2006.tb07611.x|s2cid=108604497 }}, p.&nbsp;112.</ref>  Consequently, the [[Physics:Chézy formula|Chézy formula]] relates hydraulic slope ''S'' (head loss per unit length) to the fluid velocity ''V'' and [[Hydraulic radius|hydraulic radius]] ''R'':
:<math>V=C\sqrt{RS}=C\, R^{0.5}\, S^{0.5}</math>

The variable ''C'' expresses the proportionality, but the value of ''C'' is not a constant. In 1838 and 1839, [[Biography:Gotthilf Hagen|Gotthilf Hagen]] and [[Biography:Jean Léonard Marie Poiseuille|Jean Léonard Marie Poiseuille]] independently determined a head loss equation for [[Physics:Laminar flow|laminar flow]], the [[Hagen–Poiseuille equation]]. Around 1845, [[Biography:Julius Weisbach|Julius Weisbach]] and [[Biography:Henry Darcy|Henry Darcy]] developed the [[Darcy–Weisbach equation]].<ref>{{Harvnb|Walski|2006|p=112}}</ref>

The Darcy-Weisbach equation was difficult to use because the friction factor was difficult to estimate.<ref>{{Harvnb|Walski|2006|p=113}}</ref>  In 1906, Hazen and Williams provided an [[Philosophy:Empirical relationship|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.

:<math>V = k\, C\, R^{0.63}\, S^{0.54}</math>

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|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<sub>f</sub>/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.<ref>{{Harvnb|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."</ref>

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.<ref>{{harvnb|Williams|Hazen|1914|p=1}}</ref>  The value of ''k'' is 0.001<sup>−0.04</sup>.<ref>{{harvnb|Williams|Hazen|1914|pp=1–2}}</ref>

Typical ''C'' factors used in design, which take into account some increase in roughness as pipe ages are as follows:<ref name="ET">{{Citation |url=http://www.engineeringtoolbox.com/hazen-williams-coefficients-d_798.html |title=Hazen-Williams Coefficients |publisher=Engineering ToolBox |access-date=7 October 2012 }}</ref>

{| class="wikitable sortable"
|-
! Material !! C Factor low !! C Factor high !! Reference
|-
| [[Chemistry:Eternit|Asbestos-cement]] || 140 || 140 || -
|-
| [[Chemistry:Cast iron|Cast iron]] new || 130 || 130 || <ref name="ET"/>
|-
| [[Chemistry:Cast iron|Cast iron]] 10 years || 107 || 113 || <ref name="ET"/>
|-
| [[Chemistry:Cast iron|Cast iron]] 20 years ||  89 || 100 || <ref name="ET"/>
|-
| Cement-Mortar Lined Ductile Iron Pipe || 140 || 140 || –
|-
| [[Chemistry:Concrete|Concrete]] || 100 || 140 || <ref name="ET"/>
|-
| [[Chemistry:Copper|Copper]] || 130 || 140 || <ref name="ET"/>
|-
| Steel || 90 || 110 || –
|-
| Galvanized iron || 120 || 120 || <ref name="ET"/>
|-
| [[Chemistry:Polyethylene|Polyethylene]] || 140 || 140 || <ref name="ET"/>
|-
| [[Chemistry:Polyvinyl chloride|Polyvinyl chloride]] (PVC) || 150 || 150 || <ref name="ET"/>
|-
| [[Physics:Fibre-reinforced plastic|Fibre-reinforced plastic]] (FRP) || 150 || 150 || <ref name="ET"/>
|}

==Pipe equation==
The general form can be specialized for full pipe flows. Taking the general form
:<math>V = k\, C\, R^{0.63}\, S^{0.54}</math>
and exponentiating each side by {{math|1/0.54}} gives (rounding exponents to 3–4 decimals)
:<math>V^{1.852} = k^{1.852}\, C^{1.852}\, R^{1.167}\, S</math>
Rearranging gives
:<math>S = {V^{1.852} \over k^{1.852}\, C^{1.852}\, R^{1.167}} </math>
The flow rate {{math|1=''Q'' = ''V'' ''A''}}, so
:<math>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}} </math>
The [[Hydraulic radius|hydraulic radius]] {{mvar|R}} (which is different from the geometric radius {{mvar|r}}) for a full pipe of geometric diameter {{mvar|d}} is {{math|''d''/4}}; the pipe's cross sectional area {{mvar|A}} is {{math|&pi; ''d''<sup>2</sup> / 4}}, so
:<math>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}}
</math>

===U.S. customary units (Imperial)===
<!-- rho g h_f = P_d; find refs that use h_f (Mays?) and P_d (NFPA) forms -->
When used to calculate the pressure drop using the US customary units system, the equation is:<ref>2007 version of NFPA 13: Standard for the Installation of Sprinkler Systems, page 13-213, eqn 22.4.2.1</ref>
:<math>S_{\mathrm{psi\ per\ foot}} = \frac{P_d}{L} = \frac{4.52\ Q^{1.852}}{C^{1.852}\ d^{4.8704}}</math>

where:
*''S''<sub>psi per foot</sub> = frictional resistance (pressure drop per foot of pipe) in psig/ft (pounds per square inch gauge pressure per foot)
*''S''<sub>foot of water per foot of pipe</sub>
*''P<sub>d</sub> = pressure drop over the length of pipe in psig (pounds per square inch gauge pressure)
*''L'' = length of pipe in feet
*''Q'' = flow, gpm (gallons per minute)
*''C'' = pipe roughness coefficient
*''d'' = inside pipe diameter, in (inches)

:<small>'''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.</small>

===SI units===
When used to calculate the head loss with the [[International System of Units]], the equation will then become
:<ref name="urlrpitt.eng.ua.edu">{{Cite web |url=http://rpitt.eng.ua.edu/Class/Water%20Resources%20Engineering/M3e%20Comparison%20of%20methods.pdf |title=Comparison of Pipe Flow Equations and Head Losses in Fittings |access-date=2008-12-06 |archive-date=21 January 2022 |archive-url=https://web.archive.org/web/20220121124350/http://rpitt.eng.ua.edu/Class/Water%20Resources%20Engineering/M3e%20Comparison%20of%20methods.pdf |url-status=dead }}</ref>

:<math>S = \frac{h_f}{L} = \frac{10.67\ Q^{1.852}}{C^{1.852}\ d^{4.8704}}</math>

where:

* ''S'' = Hydraulic slope
* h<sub>f</sub> = [[Physics:Hydraulic head#Head loss|head loss]] in meters (water) over the length of pipe
* ''L'' = length of pipe in meters
* ''Q'' = volumetric flow rate, m<sup>3</sup>/s (cubic meters per second)
* ''C'' = pipe roughness coefficient
* ''d'' = inside pipe diameter, m (meters)
:<small>Note: pressure drop can be computed from head loss as ''h<sub>f</sub>'' &times; the unit weight of water (e.g., 9810 N/m<sup>3</sup> at 4 deg C)</small>

==See also==
*[[Darcy–Weisbach equation]] and [[Prony equation]] for alternatives
*[[Physics:Fluid dynamics|Fluid dynamics]]
*[[Physics:Friction|Friction]]
*[[Physics:Minor losses in pipe flow|Minor losses in pipe flow]]
*[[Engineering:Plumbing|Plumbing]]
*[[Physics:Pressure|Pressure]]
*[[Physics:Volumetric flow rate|Volumetric flow rate]]

==References==
{{reflist|30em}}

== Further reading ==

*{{Citation
 |last1= Finnemore
 |first1= E. John
 |last2= Franzini
 |first2= Joseph B.
 |title= Fluid Mechanics
 |edition= 10th
 |year= 2002
 |publisher= McGraw Hill
 }}
*{{Citation
 |last= Mays
 |first= Larry W.
 |title= Hydraulic Design Handbook
 |year= 1999
 |publisher= McGraw Hill
 }}
*{{Citation
 |last= Watkins
 |first= James A.
 |title= Turf Irrigation Manual
 |edition= 5th
 |year= 1987
 |publisher= Telsco
 }}
* {{Citation
 |last2= Hazen
 |first2= Allen
 |last1= Williams
 |first1= Gardner Stewart
 |title= Hydraulic tables: showing the loss of head due to the friction of water flowing in pipes, aqueducts, sewers, etc. and the discharge over weirs
 |edition=first
 |year=1905
 |location= New York
 |publisher= John Wiley and Sons
 |url=https://books.google.com/books?id=dE9DAAAAIAAJ
 }}
*[http://catalog.hathitrust.org/Record/005745859 Williams and Hazen, Second edition, 1909]<!-- 1st 1905, 2nd 1908 -->
* {{Citation
 |last2= Hazen
 |first2= Allen
 |last1= Williams
 |first1= Gardner Stewart
 |title= 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.
 |edition= 2nd revised and enlarged
 |year=1914
 |location= New York
 |publisher= John Wiley and Sons
 |url=https://books.google.com/books?id=7m4gAAAAMAAJ&pg=PR1
 }}
* {{Citation
 |last2= Hazen
 |first2= Allen
 |last1= Williams
 |first1= Gardner Stewart
 |title= 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.
 |edition= 3rd
 |year=1920
 |location= New York
 |publisher= John Wiley and Sons
 |oclc=1981183
 }}

==External links==
*[http://www.engineeringtoolbox.com/hazen-williams-water-d_797.html Engineering Toolbox reference]
*[http://www.engineeringtoolbox.com/hazen-williams-coefficients-d_798.html Engineering toolbox Hazen–Williams coefficients]
*[http://www.calctool.org/CALC/eng/civil/hazen-williams_g Online Hazen–Williams calculator for gravity-fed pipes.]
*[http://www.calctool.org/CALC/eng/civil/hazen-williams_p 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).
{{DEFAULTSORT:Hazen-Williams Equation}}
[[Category:Eponymous equations of physics]]
[[Category:Equations of fluid dynamics]]
[[Category:Piping]]
[[Category:Plumbing]]
[[Category:Hydraulics]]

{{Sourceattribution|Hazen–Williams equation}}