Convergence pressure

The Kellogg and DePriester charts and their subsequent extensions and generahzations use the molar average boiling points of the liquid and vapor phases to represent the composition effect. An alternative measure of composition is the convergence pressure of the system, which is defined as that pressure at which the Kvalues for aU the components in an isothermal mixture converge to unity. It is analogous to the critical point for a pure component in the sense that the two... 

The widespread availabihty and utihzation of digital computers for distillation calculations have given impetus to the development of analytical expressions for iregression equation and accompanying regression coefficients that represent the DePriester charts of Fig. 13-14. Regression equations and coefficients for various versions of the GPA convergence-pressure charts are available from the GPA. 

The Gas Processors Suppliers Association [79] provides a more detailed background development of the K-factors and the use of convergence pressure. Convergence pressure alone does not represent a system s composition effects in hydrocarbon mixtures, but the concept does provide a rather rapid approach for systems calculations and is used for many industrial calculations. These are not well adapted for very low temperature separation systems. 

Step 5. Read the convergence pressure (ordinate) at the temperature (abscissa) corresponding to that of the desired flash conditions, from Figure 8-3A [79]. 

Step 6. Using the convergence pressure determined in Step 5, together with the system temperature and system pressure, obtain K-values for the components from the appropriate convergence-pressure K-charts. 

Step 8. Repeat Steps 2 through 7 until the assumed and calculated convergence pressures check within an acceptable tolerance, or until the two successive calculations for the same light and pseudo heavy components agree within an acceptable tolerance. 

For a temperature of 100°F, the convergence pressure is approximately 2,500 psia (dotted line) for the pseudo system methane-n-pentane (see Figure 8-3C). For n-pentane at convergence pressure of 3,000 psia (nearest chart) the K-value reads 0.19. The DePriester charts [80] check this quite well (see Figures 8-4A and B, and Figure 8-3D). 

Interpolation between charts of convergence pressure can be calculated, depending on how close the operating pressure is to the convergence pressure. The K-factor (or K-values) do not change rapidly with convergence pressure, (psia) [79]. 

Figure 8-3A. Convergence pressures for hydrocarbons (critical iocus). Used by permission, Gas Processors Suppliers Association Data Book, 9th Ed. V. 1 and 2 (1972-1987), Tulsa, Okla.

Brown, see Cajander et al. (1960), developed a method which relates the equilibrium constant K to four parameters component, pressure, temperature, and the convergence pressure. The convergence pressure is the pressure at which all K values tend to 1. The Brown Kw equation is limited to low pressure and its use is generally restricted to vacuum... 

The Benedict (4) equation of state has opened a new avenue to the correlation of the phase behavior of hydrocarbons. However, the extent of the iterative calculations required to determine the properties of the coexisting phases has prevented its widespread adoption. Recently the concept of convergence pressures (58, 76) has gained acceptance in the industry. This pressure is presumed to be that at which the equilibrium ratios approach unity. However, for a given mixture a true convergence pressure exists only at the critical temperature of the mixture. 

Later we will see that K-factors are correlated with a property termed convergence pressure. Convergence pressure will be defined in Chapter 14. The set of K-factor correlations in Appendix A is for use with petroleum mixtures having convergence pressures close to 5000 psia. 

Typical black oils have convergence pressures near 10,000 psia. Typical retrograde gases and wet gases have convergence pressures of about 5000 psia, The convergence pressures of volatile oils are between 5000 and 10,000 psia. 

A K-factor correlation based on a convergence pressure of 5000 psia can be used in separator calculations for all of these reservoir systems. Surface separators for black oils typically operate at low pressures. Convergence pressure has little effect on K-factors at low pressure. 

Attempts have been made to define this third property in several different ways.1 We will look only at convergence pressure, which appears to be the most convenient for the types of calculations required of petroleum engineers. 

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 correlation (not reproduced in Appendix A) are for use with mixtures with other convergence pressures. A value of convergence pressure applicable to the fluid of interest must be estimated in order to select the correct set of graphs. 

The value 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... 

Although the convergence pressure is not equal to the critical pressure of the mixture, to a certain extent it does characterize the properties of the mixture. Convergence pressure is useful in the correlation of K-factor data. 

Several methods of estimating convergence pressure have been proposed. These methods have been evaluated using laboratory data of petroleum reservoir samples.4 The method described below is as accurate as any and is the easiest to apply. 

Convergence pressures 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... 

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 from5... 

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. 

Fig. 14-3. Convergence pressures for hydrocarbons. (Engineering Data Book, 10th Ed., GPSA, Tulsa, 1987, with permission.)...

The estimate 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 of convergence pressure can be used as die new trial 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 Refactors. What value of convergence pressure should have been used for this mixture at 160°F ... 

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 less than the convergence pressure, 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. 

Black oils usually have convergence pressures of about 10,000 psia, retrograde gases about 5000 psia, and volatile oils about 7000 psia. 

Fig. 14-4. Convergence pressures for hydrocarbons (part of solution to Example 14-1).

The slopes and intercepts of the straight lines through K-factor data at several different pressures, such as lines 2 through 7 on Figure 14-8, can be correlated linearly against pressure.4 This allows interpolation and some extrapolation of the data. For instance, the upper limit of the data is a horizontal line at pKj equal to the convergence pressure. 

Third, remember that at convergence pressure all components have K = 1.0. Thus the slope of the pKj line is zero. Extrapolation of Figure 14-10 to slope = 0 gives a convergence pressure of 6400 psia for this mixture. 

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