# Benedict–Webb–Rubin equation

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 equation of state and increased the number of experimentally determined constants to eight.[1][2]

## The original BWR equation

${\displaystyle P=\rho RT+\left(B_{0}RT-A_{0}-{\frac {C_{0}}{T^{2}}}\right)\rho ^{2}+\left(bRT-a\right)\rho ^{3}+\alpha a\rho ^{6}+{\frac {c\rho ^{3}}{T^{2}}}\left(1+\gamma \rho ^{2}\right)\exp \left(-\gamma \rho ^{2}\right)}$,

where ${\displaystyle \rho }$ is the molar density.

## The BWRS equation of state

A modification of the Benedict–Webb–Rubin equation of state by Professor Kenneth E. Starling of the University of Oklahoma:[3]

${\displaystyle P=\rho RT+\left(B_{0}RT-A_{0}-{\frac {C_{0}}{T^{2}}}+{\frac {D_{0}}{T^{3}}}-{\frac {E_{0}}{T^{4}}}\right)\rho ^{2}+\left(bRT-a-{\frac {d}{T}}\right)\rho ^{3}+\alpha \left(a+{\frac {d}{T}}\right)\rho ^{6}+{\frac {c\rho ^{3}}{T^{2}}}\left(1+\gamma \rho ^{2}\right)\exp \left(-\gamma \rho ^{2}\right)}$,

where ${\displaystyle \rho }$ is the molar density. The 11 mixture parameters (${\displaystyle B_{0}}$, ${\displaystyle A_{0}}$, etc.) are calculated using the following relations

{\displaystyle {\begin{aligned}&A_{0}=\sum _{i}\sum _{j}x_{i}x_{j}A_{0i}^{1/2}A_{0j}^{1/2}(1-k_{ij})\\&B_{0}=\sum _{i}x_{i}B_{0i}\\&C_{0}=\sum _{i}\sum _{j}x_{i}x_{j}C_{0i}^{1/2}C_{0j}^{1/2}(1-k_{ij})^{3}\\&D_{0}=\sum _{i}\sum _{j}x_{i}x_{j}D_{0i}^{1/2}D_{0j}^{1/2}(1-k_{ij})^{4}\\&E_{0}=\sum _{i}\sum _{j}x_{i}x_{j}E_{0i}^{1/2}E_{0j}^{1/2}(1-k_{ij})^{5}\\&\alpha =\left[\sum _{i}x_{i}\alpha _{i}^{1/3}\right]^{3}\\&\gamma =\left[\sum _{i}x_{i}\gamma _{i}^{1/2}\right]^{2}\\&a=\left[\sum _{i}x_{i}a_{i}^{1/3}\right]^{3}\\&b=\left[\sum _{i}x_{i}b_{i}^{1/3}\right]^{3}\\&c=\left[\sum _{i}x_{i}c_{i}^{1/3}\right]^{3}\\&d=\left[\sum _{i}x_{i}d_{i}^{1/3}\right]^{3}\end{aligned}}}

where ${\displaystyle i}$ and ${\displaystyle j}$ are indices for the components, and the summations go over all components. ${\displaystyle B_{0i}}$, ${\displaystyle A_{0i}}$, etc. are the parameters for the pure components for the ${\displaystyle i}$th component, ${\displaystyle x_{i}}$ is the mole fraction of the ${\displaystyle i}$th component, and ${\displaystyle k_{ij}}$ is an interaction parameter.

Values of the various parameters for 15 substances can be found in Starling's Fluid Properties for Light Petroleum Systems..[3]

## The Modified BWR equation (mBWR)

A further modification of the Benedict–Webb–Rubin equation of state by Jacobsen and Stewart:[4] · [5]

${\displaystyle P=\sum _{n=1}^{9}a_{n}\rho ^{n}+\exp \left(-\gamma \rho ^{2}\right)\sum _{n=10}^{15}a_{n}\rho ^{2n-17}}$

where:

${\displaystyle \gamma =1/\rho _{c}^{2}}$

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.[6] Other fluids then are described by using reduced variables for temperature and density.[7]

## References

1. ^ Benedict, Manson; Webb, George B.; Rubin, Louis C. (1940), "An Empirical Equation for Thermodynamic Properties of Light Hydrocarbons and Their Mixtures: I. Methane, Ethane, Propane, and n-Butane", Journal of Chemical Physics, 8 (4): 334–345, Bibcode:1940JChPh...8..334B, doi:10.1063/1.1750658, ISSN 0021-9606
2. ^ Gramoll, Kurt; Huang, Meirong, "The Ideal-gas Equation of State", Multimedia Engineering Thermodynamics, retrieved May 16, 2012
3. ^ a b Starling, Kenneth E. (1973), Fluid Properties for Light Petroleum Systems, Gulf Publishing Company, p. 270, ISBN 978-0872012936
4. ^ Reid, Robert C.; Prausnitz, John M.; Poling, Bruce E. (April 1987), The Properties of Gases & Liquids (4th ed.), New York: McGraw-Hill, p. 741, ISBN 978-0070517998
5. ^ Jacobsen, Richard T.; Stewart, Richard B. (1973), "Thermodynamic Properties of Nitrogen Including Liquid and Vapor Phases from 63 K to 2000 K with Pressures to 10,000 Bar" (pdf), Journal of Physical and Chemical Reference Data, National Institute of Standards and Technology, 2 (4): 757–922, Bibcode:1973JPCRD...2..757J, doi:10.1063/1.3253132, ISSN 0047-2689
6. ^ Younglove, B. A.; Ely, J. F. (1987), "Thermophysical Properties of Fluids II Methane, Ethane, Propane, Isobutane, and Normal Butane", Journal of Physical and Chemical Reference Data, 16 (4): 577, Bibcode:1987JPCRD..16..577Y, doi:10.1063/1.555785, ISSN 0047-2689
7. ^ Widia, B. S. (August 2003), Variation of Density with Composition for Natural Gas Mixtures in the Supercritical Region (thesis), Texas A&M University, p. 11