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| The '''Benedict–Webb–Rubin equation''' ('''BWR'''), named after [[Manson Benedict]], G. B. Webb, and L. C. Rubin, is an [[equation of state]] used in [[fluid dynamics]]. Working at the research laboratory of [[M. W. Kellogg Limited]], the three researchers rearranged the [[Beattie–Bridgeman model|Beattie-Bridgeman Equation of State]] and increased the number of experimentally determined constants to eight.<ref name=bwr1940>{{citation|last1=Benedict|first1=Manson|last2=Webb|first2=George B.|last3=Rubin|first3=Louis C.|title=An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures: I. Methane, Ethane, Propane, and n-Butane|journal=[[Journal of Chemical Physics]]|volume= 8|number=4|page= 334–345|year=1940|issn=0021-9606|doi=10.1063/1.1750658|bibcode = 1940JChPh...8..334B }}</ref><ref>{{citation|url=http://www.ecourses.ou.edu/cgi-bin/ebook.cgi?doc=&topic=th&chap_sec=02.4&page=theory|contribution=The Ideal-gas Equation of State|title=Multimedia Engineering Thermodynamics|last2=Huang|first2=Meirong |first1=Kurt|last2= Gramoll|accessdate=May 16, 2012}}</ref>
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| ==The original BWR equation==
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| :<math>P=\rho RT + \left(B_0 RT-A_0 - \frac{C_0}{T^2} \right) \rho^2 + \left(bRT-a\right) \rho^3 + \alpha a \rho^6 + \frac{c\rho^3}{T^2}\left(1 + \gamma\rho^2\right)\exp\left(-\gamma\rho^2\right)</math>
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| ==The BWRS equation of state==
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| A modification of the Benedict–Webb–Rubin equation of state by Professor Kenneth E. Starling of the University of Oklahoma:<ref name=starling>{{citation|title=Fluid Properties for Light Petroleum Systems|last=Starling|first=Kenneth E.|publisher=Gulf Publishing Company|year=1973|isbn=978-0872012936|pages=270}}</ref>
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| :<math>P=\rho RT + \left(B_0 RT-A_0 - \frac{C_0}{T^2} + \frac{D_0}{T^3} - \frac{E_0}{T^4}\right) \rho^2 + \left(bRT-a-\frac{d}{T}\right) \rho^3 + \alpha\left(a+\frac{d}{T}\right) \rho^6 + \frac{c\rho^3}{T^2}\left(1 + \gamma\rho^2\right)\exp\left(-\gamma\rho^2\right)</math>
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| :''ρ'' = the [[molar density]].
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| Values of the various parameters for 15 substances can be found in Starling's ''Fluid Properties for Light Petroleum Systems.''.<ref name=starling/>
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| ==The Modified BWR equation (mBWR){{anchor|mbwr}}==
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| A further modification of the Benedict–Webb–Rubin equation of state by Jacobsen and Stewart:<ref name=jacobsen>{{citation|first1=Robert C. |last1=Reid|first2=John M.|last2= Prausnitz|first3=Bruce E.|last3=Poling|title=The Properties of Gases & Liquids| edition=4th |publisher=McGraw-Hill|location=New York|date=April 1987|isbn=978-0070517998|pages=741}}
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| *{{citation|first1=Richard T. |last=Jacobsen|first2=Richard B.|last2=Stewart|title=Thermodynamic Properties of Nitrogen Including Liquid and Vapor Phases from 63 K to 2000 K with Pressures to 10,000 Bar|journal= Journal of Physical and Chemical Reference Data|doi=10.1063/1.555785|volume=2|number=4|page=757–922 |year=1973|publisher=[[National Institute of Standards and Technology]]|url=http://www.nist.gov/data/PDFfiles/jpcrd39.pdf|format=pdf|issn=0047-2689|doi=10.1063/1.3253132|bibcode = 1973JPCRD...2..757J }}</ref>
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| :<math>P=\sum_{n=1}^{9}a_n\rho^n+\exp\left(-\gamma\rho^2\right)\sum_{n=10}^{15}a_n\rho^{2n-17}</math>
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| where:
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| :<math>\gamma=1/\rho_c^2</math> | |
| <br>
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| The mBWR equation subsequently evolved into a 32 term version (Younglove and Ely, 1987) with numerical parameters determined by fitting the equation to empirical data for a reference fluid.<ref name=younglove>{{citation|last1=Younglove|first1=B. A.|first2= J. F. |last2=Ely|title=Thermophysical Properties of Fluids II Methane, Ethane, Propane, Isobutane, and Normal Butane|journal= Journal of Physical and Chemical Reference Data|issn=0047-2689|doi=10.1063/1.555785|volume=16|page=577|year=1987}}</ref> Other fluids then are described by using reduced variables for temperature and density.<ref>{{citation|url=http://repository.tamu.edu/bitstream/handle/1969.1/1094/etd-tamu-2003B-2003062110-Widi-1.pdf|page=11|title=Variation of Density with Composition for Natural Gas Mixtures in the Supercritical Region |date=August 2003|last=Widia|first=B. S.|format=thesis|publisher=[[Texas A&M University]]}}</ref>
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| ==See also==
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| * [[Real gas]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| *{{citation|last1=Benedict|first1=Manson|author1-link=Manson Benedict|last2=Webb|first2=George B.|last3=Rubin|first3=Louis C.|title=Mixtures of Methane, Ethane, Propane and ''n''-Butane|journal=[[Journal of Chemical Physics]]|volume= 10|number=12|page= 747–758|year=1942|issn=0021-9606|doi=10.1063/1.1723658|bibcode = 1942JChPh..10..747B }}
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| *{{citation|last1=Benedict|first1=Manson|last2=Webb|first2=George B.|last3=Rubin|first3=Louis C.|title=An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures. '''Constants for Twelve Hydrocarbons'''| journal=Chemical Engineering Progress (CEP)|volume=47|number=8|page= 419–422 |year=1951}}
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| *{{citation|last1=Benedict|first1=Manson|last2=Webb|first2=George B.|last3=Rubin|first3=Louis C.|title=An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures '''Fugacities and Liquid-Vapor Equilibria'''| journal=Chemical Engineering Progress (CEP)|volume=47|number=9|page= 449–454 |year=1951}}.
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| {{DEFAULTSORT:Benedict-Webb-Rubin equation}}
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| [[Category:Equations of fluid dynamics]]
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| {{fluiddynamics-stub}}
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