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# Phase behavior of mixtures

The interaction of the different molecules in a mixture causes behavior not observed in pure fluids. Phase diagrams describe the volumetric behavior of mixtures. This article presents the basic procedure to predict the equilibrium phase behavior of mixtures by a cubic equation of state (EOS). More detailed information can be found in many sources, including Firoozabadi[1] and Whitson and Brule. [2]

## EOS of a mixture

The thermodynamic properties of a mixture can be calculated with the same EOS for a pure fluid, with some modifications. The primary difference is that the mixture molar volume for a phase is calculated with EOS constants and temperature-dependent functions of the phase molar composition, either xi or yi. For example, the Soave Redlich-Kwong EOS written for a mixture is

where subscript m indicates a mixture property. The mixture properties are calculated with mixing rules that are often linear or quadratic functions of the phase mole fractions. For example, for the liquid phase, the mixing rule for the product, , is often the quadratic equation,

where . The parameters kij are called binary interaction parameters. Binary interaction parameters are constants that are determined by fitting the cubic EOS to experimental PVT data. The mixing rule for is theoretically justified from virial EOS, which are discussed in several sources.[1][3][4][5][6][2] For bm, the linear relationship, , is often used.

For equilibrium calculations, the fugacity of every component in each phase must be calculated. Eq. 1 is used for this purpose. For example, substitution of the Soave Redlich-Kwong EOS into Eq. 1 gives the fugacity of a component in the liquid phase, which is written as

....................(2)

where . A similar equation is written for the vapor phase, where xi is replaced by yi, and superscript L is replaced by V.

## Procedure for equilibrium calculations of a mixture

The procedure for equilibrium calculations of a potential two-phase mixture is more complex than that of a pure fluid. For an equilibrium flash calculation, the pressure and temperature and overall mole fractions are specified (i.e., pressure and temperature are now independent, as specified by the Gibbs phase rule). The general procedure for a flash calculation is discussed next.

1. Make an initial guess of the K-values, where . When the guess of the K-values is near the equilibrium solution, the procedure will converge rapidly. If the guess is not good, the procedure might not converge at all. Most EOS programs use some empirical correlation to estimate the phase mole fractions based on K -values. The Wilson equation[7] is often used, where .
2. Calculate xi and yi with the Rachford-Rice procedure.[8] Once the K -values for each component are specified, the Rachford-Rice procedure is used to estimate the phase mole fractions. A material balance on each component gives zi = Lxi + (1 - L)yi, where L is the mole fraction liquid (see Eq. 2). Substitution of yi = Kixi into the material balance equation gives, upon rearrangement, or alternatively . Substitution of these expressions into the function gives . This is a nonlinear equation that can best be solved by a simple Newton-Raphson iteration, where for each iteration the new value of the liquid mole fraction is found from . For the first iteration, choose L = 0.5 and iterate until .
....................(2)
3. Calculate the cubic EOS parameters (e.g., am and bm). This step is very straightforward and depends on the selected EOS and its associated mixing rules. The critical temperatures, pressures, and acentric factors for each component are needed to calculate the EOS parameters.
4. Solve the cubic EOS for the phase molar volumes VmL and VmV. This step requires solution of the cubic EOS for the compressibility factor, Z, of the vapor and liquid (or alternatively for VV and VL). Because the compositions of the vapor and liquid are different, two separate solutions for the roots of the cubic EOS are required. A cubic equation-solver or iteration method should be used to obtain the roots of the EOS.
The procedure for this step is more complex than for a pure fluid because six roots of the cubic EOS are calculated (i.e., three roots for the liquid and three for the vapor). The middle root for the vapor and liquid are discarded because that solution leads to unstable phases, similar to pure fluids. One of the remaining two liquid roots is paired with one of the other vapor roots to calculate component fugacities and equilibrium. If the wrong root pairing is selected, the solution could be false in that an unstable or metastable solution could be obtained. The correct equilibrium solution is the one that minimizes the total Gibbs energy compared with the other possible root-pairings. Firoozabadi[1]and Whitson and Brule[2] provide a good description of how to select the liquid and vapor roots so that the total Gibbs energy of the two-phase mixture is minimized. For most cases, the correct root for the liquid is the one that gives the smallest molar volume, and the correct root for the vapor is the one that gives the largest molar volume.
Firoozabadi[1]and Whitson and Brule[2] also examine using stability analyses to determine whether a mixture will form three phases instead of just one or two phases. Phase diagrams for EOR processes discusses the formation of three equilibrium phases in CO2/crude oil systems.
5. Calculate the component fugacities of each component in each phase, and . The selected cubic EOS is used to determine an expression for the fugacity of a component in a phase (see Eq. 2 for example).
6. Check to see if equilibrium has been reached. A good criterion is for all components. If the criteria are satisfied, equilibrium has been obtained. The correct equilibrium solution is found when for each component. Because the solution is never found exactly, we accept the solution if for each component. The tolerance of 10-5 can be decreased if better accuracy is required.
7. If the criteria have not been satisfied, the K-values should be updated and steps two through six repeated. This step is also very important; it affects both the rate of convergence and whether the iteration converges at all. One procedure that works well is the simple successive substitution scheme that relies on the fact that and for each component. Therefore, . At equilibrium, the component fugacities are equal so that . We can use this ratio to estimate new K-values from the old ones. That is, . Once the new K-values are determined, steps two through six are repeated until convergence in step six is achieved. Convergence from successive substitutions can be slow near the critical region. Other methods may be required when convergence is slow.[1]

## Nomenclature

 = fugacity of a component in a mixture, mole2-pressure/mole2, Pa k = binary interaction parameter, dimensionless Ki = K-value of ith component, yi/xi, dimensionless L = liquid mole fraction, moles liquid/total moles, dimensionless p = pressure, force/area, Pa R = gas constant, pressure-volume/temperature/mole, Pa-m3/(Kelvin-mole) t = time, seconds T = temperature, Kelvin V = vapor mole fraction, moles vapor/total moles, dimensionless or molar volume of fluid, volume/mole, m3/mole x = x-coordinate, length, m xi = mole fraction of ith component in liquid, moles ith component in liquid/total moles liquid, dimensionless = fugacity coefficient for a component in a mixture, mole2-pressure/mole2-pressure, dimensionless

## References

1. Firoozabadi, A. 1999. Thermodynamics of Hydrocarbon Reservoirs. 355. New York City: McGraw-Hill Book Co. Inc Cite error: Invalid `<ref>` tag; name "r1" defined multiple times with different content Cite error: Invalid `<ref>` tag; name "r1" defined multiple times with different content Cite error: Invalid `<ref>` tag; name "r1" defined multiple times with different content Cite error: Invalid `<ref>` tag; name "r1" defined multiple times with different content
2. Whitson, C.H., and Brule, M.R. 2000. Phase Behavior, Vol. 20. Richardson, Texas: Monograph Series, SPE.
3. Prausnitz, J.M., Lichtenthaler, R.N., and de Azevedo, E.G. 1999. Molecular Thermodynamics of Fluid-Phase Equilibria, third edition. New Jersey: Prentice Hall.
4. Sandler, S.I. 2000. Chemical and Engineering Thermodynamics, third edition. New York City: John Wiley & Sons.
5. Smith, J.M., Van Ness, H.C., and Abbott, M.M. 2001. Chemical Engineering Thermodynamics, sixth edition. 787. New York City: McGraw-Hill Book Co. Inc.
6. Walas, S.M. 1985. Phase Equilibria in Chemical Engineering. 671. Boston, Massachusetts: Butterworth Publishings.
7. Wilson, G.M. 1969. A Modified Redlich-Kwong Equation-of-State, Application to General Physical Data Calculations. Paper 15c presented at the AIChE Natl. Meeting, Cleveland, Ohio, 4–7 May.
8. Rachford Jr., H.H. and Rice, J.D. 1952. Procedure for Use of Electronic Digital Computers in Calculating Flash Vaporization Hydrocarbon Equilibrium. J Pet Technol 4 (10): 19, 3. SPE-952327. http://dx.doi.org/10.2118/952327-G