Equilibrium Ratios
The equilibrium ratio (Kvalue), defined in equation \eqref{eq:def}, is one of the most important thermodynamic quantities for petroleum systems. The reason for this is the direct connection it has on the flash calculation, which is a key phase behavior calculation. Furthermore, the Kvalues have an intuitive and simple physical meaning. Given a Kvalue for a specific component at certain conditions, the magnitude will determine the affinity the component has to be in the vapor phase (Kvalue greater than 1) or the liquid phase (Kvalue smaller than 1). Theoretically, the Kvalues can be measured in the lab and an argument can be made that accurate Kvalue measurements is the best source of PVT data used in equation of state model development.
The shape of the Kvalue is typically divided into the lowpressure region and the highpressure region. For the low pressure region the Kvalues tend to be inversely proportional to the pressure (\(K_i \propto 1/p\)) yielding a 1 slope on a loglog plot and is said to follow Raoult's law and Dalton’s law
where \(p_{vi}(T)\) is the vapor pressure of component \(i\). In the highpressure region, all the Kvalues appear to converge to unity at the convergence pressure.
Figure 1: An example of pressure dependent Kvalues at a given temperature and composition.
Definition
The formal definition of the Kvalues are given by
Areas of Use
The main area of use for the Kvalue is, as mentioned above, that it is a key quantity in the isothermal flash calculation. The Kvalue is some sense the driving mechanism behind most PVT calculations. The shape of the Kvalues (especially for highpressures) drives the phase behavior of the fluid. For this reason, the Kvalue consistency check is essential as part of the equation of state model development.
Correlations
A wide range of correlations have been developed to estimate the Kvalues for hydrocarbon systems. However, as noted above, the only sure way to determine the true Kvalue is to solve the flash routine. Below, some correlations are given for reference.
Hoffman Correlation
where
By rearranging equation \eqref{eq:hoffman} and take the base10 logarithm of both sides yield
such that \(A_0\) and \(A_1\) become the intercept and slope of a straight line on a loglog plot. This is the basis for the Hoffman quality check plot.
Wilson Correlation
Note that the Wilson correlation is identical to the Hoffman correlation for \(A_0=\log{(p_{sc})}\) and \(A_1=1\) in equation \eqref{eq:hoffman} and the Edmister correlation ^{1} is used for the acentric factor.
Modified Wilson Correlation
A modified version of the Wilson equation was proposed by Whitson and Torp in their 1981 paper ^{2}.
where \(A_2\) is a constant sometimes used defined as 0.7, \(p_{ci}\) is the component critical pressure, \(p_k\) is the convergence pressure, \(\omega_i\) is the acentric factor, \(p_{ri}\) is the reduced pressure defined by \(p_r=p/p_c\) and \(T_{ri}\) is the reduced temperature defined by \(T_r=T/T_c\). The units for equation \eqref{eq:mod_wilson} and \eqref{eq:A1} are psia for the pressure and R in temperature.
Standing Correlation
The Standing correlation for Kvalues was developed using the Hoffman correlation and tuned for low pressure Kvalue data^{3}. The specified conditions where the correlation applies was described to be at \(p<1000\)psia and \(T<200^\circ\)F. The Standing correlation is often used to estimate the Kvalues for separator processes, specifically for the Hoffman plot quality check used for separator samples.
The equation for the Standing correlation is given by
where
Unlike the Hoffman correlation, Standing gives an expression for \(A_0\) and \(A_1\) as a function of pressure. The correlation also gives a method for calculating \(b_i\) and \(T_{bi}\) for \(C_{7+}\) based on the pressure and temperature.
In the Standing correlation given above, the units for pressure are in psia and the temperature is given in R.
Standing also gives a list of the component values for \(b_i\) and \(T_{bi}\) and they are given in the table below.
Component  \(b_i\)(cycle\(^\circ\)R)  \(T_{bi}\) (\(^\circ\)R) 

N\(_2\)  470  109 
CO\(_2\)  652  194 
H\(_2\)S  1136  331 
C\(_1\)  300  94 
C\(_2\)  1145  303 
C\(_3\)  1799  416 
iC\(_4\)  2037  471 
nC\(_4\)  2153  491 
iC\(_5\)  2368  542 
nC\(_5\)  2480  557 
C\(_6\)  2738  610 
Simplified Flash
In 1986 Michelsen introduced a new approach to solve the flash in a simplified manner for an EOS model where all the BIPs were set to zero^{4}. This simplified flash approach allows for a direct calculation of the Kvalues from the EOS parameters. The solution to Michelsen's procedure is given by
where \(b_i\) is the component EOS parameter and
where \(u_i\) can be the vapor molar amount (\(y_i\)) or liquid molar amount (\(x_i\)).
By using the fact that \(K_i = \phi_i^L/\phi_i^V\) and taking the natural logarithm of both sides yields
which can be calculated from equation \eqref{eq:simp_flash_coef}.

W. C. Edminster. Applied hydrocarbon thermodynamics, part 4: compressibility factors and equations of state. 1958. ↩

C. H. Whitson and S. B. Torp. Evaluating constant volume depletion data. In SPE Annual Technical Conference and Exhibition, paper SPE–10067–MS. Society of Petroleum Engineers, 1981. doi:https://doi.org/10.2118/10067MS. ↩

M.B. Standing. A set of equations for computing equilibrium ratios of a crude oil/natural gas system at pressures below 1,000 psia. Journal of Petroleum Technology, 31:paper SPE–7903–PA, 1979. doi:https://doi.org/10.2118/7903PA. ↩

M. L. Michelsen. Simplified flash calculations for cubic equations of state. Industrial & Engineering Chemistry Process Design and Development, 25:184–188, 1986. ↩