Equilibrium-ratio correlationsThe equations for gas-liquid equilibria calculations and the definition of equilibrium ratio (K-factor) were discussed in chapter 12. This chapter will examine k-factor correlations.The equation which defines k-factor:

Kj=y ix iAnd the equation resulting from the combination of Raoult’s and Dalton’s equations:

y ix i

=Pvjp

Can be combined to give an equation for calculating the –factor of a component, j, in a mixture which behaves like an ideal solution.K j=

Pvjp

The vapor pressure of component j, pvj, is solely a function of temperature. Thus equation 14-1 shows that if ideal solution behavior exists, the k-factor of each component simply depends on pressure and temperature.Remember that one the principal properties used to define an ideal solution is that the intermolecular forces of attraction and repulsion are the same between unlike as between like molecules. This property does not exist in real solutions. Molecular behavior in a real solution depends on the types and sizes of the molecules which are interacting.

Therefore, the k-factor for a component of a real solution depends not only on pressure and temperature but also on the types and quantities of other substances present. This means that any correlation of k-factor must be based on at least three quantities: pressure, temperature and a third property which describes nonideal solution behavior. This property represent both the types of molecules present and their quantities in the gas and liquid.Attempts have been made to define this third property in several different ways. We will look only a convergence pressure, which appears to be the most convenient for the types of calculations required of petroleum engineers.A large number of k-factor correlations have been proposed. We will present only two: one graphical, the other in equation form. A partial set of k-factor curves is given in appendix A. only 14 charts out of a total of 90 of this correlation are given. These should suffice for illustrative exercises in this text.Notice that the k-factor charts in appendix A apply to petroleum mixtures which have convergence pressure of 5000 psia. The other charts of this correlations (not reproduced in appendix A) are for use with mixtures with other convergence pressure. A value of convergence pressure applicable to the fluid of interest must be estimated in order to select the correct set of graphs.Convergence pressureExperimentally determined k-factors normally are plotted against pressure on a log-log sacale. Figures 14-1 and 14-2 show equilibrium ratios of two typical petroleum mixture for several temperatures.The shapes of the curves in these figures are characteristic of most multicomponent mixtures. At low pressures, the slope of each curve is approximately – 1.0. A slope of 1.0 for ideal k-factors is predicted by equation 14-1. Each curve passes throught a value of unity at a pressure very close to the vapor pressure of the component at the temperature of the graph. This also is predicted for ideal-solution behavior.At higher pressures, the effects of nonideality are seen. The curve for each component departs from a slope of 1.0. The curves tend to converge toward a k-factor of 1.0.

Definition of convergence pressureThe values of pressure for which the k-factors appear to converge to unity is known as the convergence pressure. If a mixture is at its true critical temperature, the curves in reality will converge to a value of 1.0 at the critical pressure. At any temperature other than the critical temperature, the equilibrium-ratio curves actually do not extend past the bubble-point or dew-point pressure of the mixture. However, the curves can be extrapolated to determine the point of apparent convergence.

Sometimes convergence pressure is called apparent convergence pressure.Although the convergence pressure is not equal to the critical pressure of the mixture, to a certain extent it does characterize the properties of mixture. Convergence pressure is useful in the correlation of k-factor data.Estimation of convergence pressureSeveral methods of estimating convergence pressure have been proposed. These methods have been evaluated using laboratory data of petroleum reservoir samples. The method described below is as accurate as any and is the easiest to apply.Convergence pressure of binary hydrocarbon mixtures may be estimated from the critical locus curves given in figure 2-16. A similar curve which includes multicomponent mixtures is presented in figure 14-3.

This method of estimating convergence pressure involves a trial and error procedure in which a first trial value of convergence pressure is used to obtain k-factors. A reasonable estimate of convergence pressure for a first trial value may be obtained from:pk=60MC 7+¿−4200 ¿Where Mc7+ is the molecular weight of the heptanes plus fraction.The compositions and quantities of the equilibrium gas and liquid are computed in the usual manner. Then the liquid is equated to a pseudobinary mixture consisting of the lightest component of the liquid and a hypothetical heavy component.The lightest component of petroleum mixtures always is taken to be methane. The hypothetical heavy component is represented by a critical temperature and a critical pressure calculated as the weighted average of the true critical properties of all components of the liquid except the lightest.The critical properties of methane and the weighted-average critical properties of the hypothetical heavy component are plotted on figure 14-3, and a locus of convergence pressures for that pair is interpolated using the adjacent critical loci as guides.

The estimated of convergence pressure is taken from the point this locus crosses the temperature at which calculations are desired. If this convergence pressure is not reasonably close to the first trial value of convergence pressure, the procedure should be repeated. The estimated value to convergence pressure can be used as the new trail value.EXAMPLE 14-1: the compositions of the gas and liquid from a black oil at 1300 psia and 160˚F calculated with equation 12-17 are given in the table below. A convergence pressure of 5000 psia was used to obtain the k-factors. What value of convergence pressure should have been used for this mixture at 160˚F?

SOLUTIONFirst, the liquid composition must be expressed in weight fraction.

Second, delete methane and adjust the weight fraction of the hypothetical heavy component. Then calculate weight-average critical properties.

Third, plot weight-averaged critical point on figure 14-3, interpolate a locus of convergence pressures, and read pk at 160˚F.A convergence pressure of 10,000 psia should have been used for this mixture. See figure 14-4. Obtain k-factors at pk=10,000 and repeat the gas-liquid equilibrium calculation.When the operating pressure is considerably lees than the convergence prssure, an error in the estimate of convergence pressure has little effect on the resulting calculations. As operating pressure approaches convergence pressure, however, equilibrium ratios become very sensitive to the convergence pressure used and care must be taken in the selection of the correct value.

The correlations from which the equilibrium ratio data in appendix A were taken include charts for convergence pressures of 800, 1000, 1500, 2000, 3000, 5000, and 10,000 psia. When the convergence pressure for the mixture is between the values for which charts are provided, interpolate between charts. Interpolation is necessary when the operating pressure is near the convergence pressure. At low pressure, simply use the chart with convergence pressure nearest the value for the mixture.Bloack oil usually have convergence pressure of about 10,000 psia, retrograde gases about 5000 psia and volatiles oils about 7000 psia.