Karthik C. ManiDarnell Engineering Corp. HoustonMichael A. Matthews,Henry W. Haynes Jr. University of Wyoming Laramie
A new implementation of continuous vapor-liquid equilibrium (VLE) calculations eliminates the arbitrariness of traditional methods. In addition, the procedure selects mathematically optimal "pseudo-components," called quadrature components.
Calculations by the quadrature method generally are more accurate than calculations using an equal number of arbitrarily chosen pseudo-components. The method takes full advantage of the continuous nature of petroleum feed and product streams.
The quadrature method uses normal boiling point as the characterization variable. A true boiling point (TBP) distillation curve and the relationship between boiling point and specific gravity are also needed.
This relationship may be determined empirically by measuring the specific gravity of each distillation cut. Alternatively, if constancy of the Watson characterization factor can be assumed, only the overall mixture specific gravity is required.
A CONTINUOUS APPROACH
A continuous mixture is one containing hundreds, or even thousands, of difficult-to-characterize chemical compounds. Engineers have long recognized the continuous nature of petroleum-based streams in the refinery.
The large number of compounds, along with the approximate methods used for chemical analysis, makes it impossible to set up phase-split calculations using discrete chemical species. Instead it is common practice to divide the continuous mixture into a collection of narrow-boiling-range fractions called pseudo-components.
Often the selection of pseudo-components is rather arbitrary, and extensive trial-and-error may be needed to give the "best" set. Physical and thermodynamic properties are assigned to the pseudo-components so that they can be treated as individual components in vapor-liquid equilibrium calculations.
Using only a few arbitrarily chosen components may introduce a great deal of uncertainty into the results of a phase-split calculation. Such calculations are performed repeatedly in simulating processes such as petroleum fractionation (distillation) and it is vital that the process engineer's calculations be as accurate as possible.
The continuous thermo-dynamics approach was developed so that the continuous nature of complex mixtures would be recognized and maintained throughout the calculations.1 The composition of the continuous mixture is obtained experimentally from well-known methods such as true boiling point distillation analysis.
Such an analysis gives the engineer a "continuous" representation of the mixture composition (i.e., cumulative weight percent distilled vs. boiling point). The boiling point temperature thus becomes the variable characterizing the mixture on a continuous basis.
WORKING EQUATIONS
In the petroleum industry, it is common to use cubic equations of state, such as the Peng-Robinson equation, to describe phase behavior.2 The traditional development of these equations of state requires composition data for use in the so-called "mixing rules." These rules always involve summations over discrete mole fractions.
In continuous thermo-dynamics, summations that appear in the equation of state (EOS) formulation are replaced with integrals. The range of integration is over some characteristic variable such as normal boiling point or molecular weight.
The boiling point or molecular weight distribution of the complex mixture can be obtained using well-known methods such as true boiling point distillation or simulated distillation.
During phase-equilibrium calculations, the integral expressions in the EOS are approximated by computationally efficient Gaussian quadrature formulas, with the net result that the discrete form of the EOS is recovered and at the same time, a set of "quadrature" components is identified.
Quadrature components are really pseudo-components that approximate the integrals in a mathematically optimal sense. Thus fewer quadrature components are required to approximate a given mixture by comparison to arbitrarily chosen pseudo-components.34
The authors demonstrated how the integrals in a continuous EOS formulation may be rearranged and evaluated by Gauss-Legendre quadrature to obtain a system of equations identical to the discrete EOS.5 In the process, a set of quadrature components was identified, and equations for calculating the composition and properties of the quadrature components were derived.
Suppose the TBP distillation curve is in the form of temperature vs. cumulative vol % (Fig. 1) and that one wishes to approximate the continuous distillation curve with n quadrature components (pseudo-components). The quadrature components are identified by interpolation on this curve.
First the cumulative volume fractions corresponding to a chosen set of quadrature points, qi (where: 1 i n), are calculated using Equation 1 (see equations and nomenclature).
The qi are the zeros of the Legendre polynomial of order n. Values of qi and the associated weight factors, Wi, are tabulated for various values of n in most mathematical handbooks. A selection is presented in Table 1.
The larger the value of n, the more accurate the Gaussian approximation but as few as two values will sometimes give acceptable results.
Next, values of Ti corresponding to Fvi must be obtained by interpolation on a TBP distillation curve, such as Fig. 1. When using a computer, a cubic spline interpolation is normally suitable for this task.
The next step is to calculate the molecular weights, critical properties, and acentric factors corresponding to the Tis. Several authors have developed relations for these properties in terms of normal boiling point and specific gravity.6-9
One such set of relations, given by Riazi and Daubert, is Equations 2-4.6 The acentric factor may be calculated from the reduced saturation pressure curve of Lee and Kesler (Equation 5).10
To utilize these equations for calculating the quadrature component properties, an additional expression is needed, relating specific gravity to normal boiling point. This information may be available experimentally if specific gravities have been determined on the distillation cuts and if the cuts are sufficiently narrow. Thus, specific gravity can be defined as in Equation 6.
Alternatively, it can usually be assumed to a very good approximation that the Watson characterization factor, Kw, is constant for all species in the mixture.11 12
When the temperature is expressed in Kelvin, Watson characterization factor is defined by Equation 7.5 When Kw is unknown, it can be calculated from the mixture specific gravity using Equation 8.
Using either of the relations (Equations 6 or 7), SGi is readily eliminated from Equations 2-5, giving M, Pc, Tc, and w as functions of T only.
At this point, a set of quadrature components has been identified, and their corresponding molecular weights and critical properties have been calculated. As shown in an earlier paper, the Gauss-Legendre approximation also provides the component mole fractions (Equation 9). Equation 10 can be used in calculating Equation 9.
Now a complete set of quadrature components with their compositions and physical properties have been developed. This information can be substituted into any discrete EOS with parameters expressible in terms of the critical properties and acentric factor. Bubble point, dew point, and flash calculations can be performed in the usual manner.
A modification to some of these equations is required if the TBP distillation is available in terms of weight percent overhead (the usual case for a gas chromatograph-simulated distillation).
Equations 1, 8, 9, and 10 must be replaced with the analogous Equations 11, 12, 13, and 14, where Fwi is the cumulative weight fraction overhead.
EXAMPLE CALCULATIONS
Several examples have been designed to illustrate various applications of the quadrature components calculations.
NAPHTHA/KEROSINE BLEND
Edmister and Pollock report TBP distillation data for a 47.8 API, light naphtha/kerosine blendstock (Table 2).13 These data are plotted in Fig. 1. The stock contained essentially no nondistillable residue.
The refinery engineer might typically use some arbitrary lumping scheme to define pseudo-components from these analytical data. Mathematical quadrature, however, can be used to calculate the optimum quadrature component compositions and the associated properties. This will be done for the case of four quadrature components.
Solution: The qi for n = 4 are selected from Table 1 and entered into the first column of Table 3. The second column contains values of the corresponding Fvi calculated from Equation 1.
Boiling point temperatures are obtained by interpolation on the Fvi (expressed as a percentage) from Fig. 1. These values are entered into the last column of Table 3. They constitute the pseudo-component, or more appropriately the quadrature component, boiling points.
An API gravity of 47.8 is equivalent to a mixture specific gravity (60/60 F.) of 0.7891. Substitution of this value into Equation 8, along with the Ti from Table 3 and the Wi from Table 1, provides a Watson characterization factor of 11.89.
This value is assumed to be constant and the specific gravity corresponding to each quadrature component is calculated from Equation 7. These values are entered into the second column of Table 4. In the first column is the temperatures from Table 3, converted to Kelvin.
The corresponding molecular weights, critical temperatures, critical pressures, and acentric factors are calculated from Equations 2-5 and entered into Table 4.
Finally, the quantity SG/M is calculated from Equation 10 and the quadrature component mole fractions are computed from Equation 9. These values are tabulated in the last column.
COMMENTS
Quadrature component information such as that generated in Table 4 makes up the discrete input to a computer program for VLE computations.
While the quadrature component calculations can be done by hand, they are easily programmed for computer.
The subsequent iterative phase-split calculations realistically require computer solution, and so it is convenient to program the quadrature component selection with subsequent phase-split programs.
If desired, the quadrature components can be combined with truly discrete components and calculations can be performed on the resulting semicontinuous mixture.
EFV CALCULATIONS
Edmister and Pollock reported the results of equilibrium flash vaporization (EFV) experiments on the 47.8 API naphtha/kerosine feedstock described in the first example.13 The experiments were performed over a pressure range of 0-200 psig.
The refinery engineer might typically use such EFV curves to "tune" an EOS to use in a distillation column simulation, for example. The Peng-Robinson EOS and the continuous approach will be used to construct EFV curves for comparisons with the experimental data. Good agreement is obtained with no adjustable parameters (no binary interaction parameters).
Solution: Equilibrium flash calculations were performed using ChemE Computations' HPVLE program from its Thermopak 2.1 collection of computer programs for thermodynamics calculations. No binary interaction parameters were employed in this calculation.
The results of 4 and 8-point quadrature computations are plotted in Fig. 2 for comparison with the experimental data. Although the 8-point quadrature seems to track the experimental points best, the 4-point quadrature is also quite accurate. The errors are largest in the 0-10% vaporization region.
COMMENTS
Any product distribution involving a continuous mixture will be expressed in terms of quadrature component compositions. Such information is of limited utility for product characterization, especially when n is small.
A preferred procedure is to construct the continuous product distributions on a molar, weight, or volume basis over temperature intervals of interest. This involves integrations on the converged problem solution over selected intervals Ti to Ti+1. No iteration is involved. The reader is referred to Matthews, et al., for further details.14
EFV PRODUCT DISTR.
Edmister and Pollock reported vapor and liquid-phase composition distributions, as represented by ASTM distillations, for the feedstock from the first example at several flash conditions.13
For a traditional calculation involving arbitrary pseudo-components, the output of the phase-split calculation gives only pseudo-component mole fractions. A more accurate and informative calculation is to reconstruct the complete TBP distribution of the resultant liquid and vapor product.
In this example, the ASTM distillation curves will be constructed and the results for the 50 psig, 400 F. EFV will be compared with continuous Peng-Robinson predictions.
Solution: Using the procedure of Edmister and Okamato, the ASTM distillations were converted to TBP distillations and compared with the computed results (Fig. 3).15 Clearly the agreement is quite good.
COMMENTS
The previous examples were based on a light fraction where there was no nondistillable residue. In some processes, such as crude fractionation or petroleum reservoir management, the fluids will have a wide boiling range, and a complete TBP analysis is difficult or impossible to obtain experimentally.
Much effort, however, has been devoted to methods of estimating the complete composition distribution for heavy oils from partial distillation analyses. After performing such an extrapolation, continuous thermodynamic methods can be employed for VLE calculations.
BUBBLE POINTS
Pederson, et al., report bubble point pressures for several North Sea crudes.16 Distillation analyses up to C20 were reported, along with some additional characterizations of the nondistillable residue (specific gravity, average molecular weight).
Knowledge of bubble point conditions is often used by the refinery engineer to help tune an EOS and to anticipate necessary refinery operating conditions.
In this example, the continuous method will be used to calculate the bubble points and compare them with experimental values.
Solution: As shown by Pederson, et al., calculations are sensitive to both the estimated composition distribution and the choice of critical property correlations. They also summarize the estimated composition distributions for the various oils in their appendices. When these data are converted to TBP distributions, the continuous procedure can be invoked.
The Soave-Redlich-Kwong EOS was used with the critical property correlations of Cavett.8 16 Only three quadrature components were used to represent the C7+ fraction. The calculated vs. experimental bubble points for the reported oils are as shown in Fig. 4.
While the continuous approach can provide economy in the calculations, the method is only as good as the composition information and the thermodynamic model.
COMMENTS
For calculational efficiency, quadrature components can be used in the iterative portion of a phase equilibrium calculation to represent the continuous mixture. Thus, existing computational algorithms based on discrete components can be used to calculate the amount of vapor and liquid from the flash.
An essential step, however, is to reconstruct the continuous vapor and liquid-phase compositions after the iterative routine converges. Thus, the continuous nature of the process is recognized and maintained. This is important for multiple-step processes such as distillation, where the vapor and liquid distributions vary widely from the top to the bottom of the column.
Care must be taken to get as accurate a feed distribution as possible. The method is applicable to any popular EOS that requires Tc, Pc, and w as parameters. As with existing approaches, the availability of a few VLE data is useful in tuning the continuous EOS.
REFERENCES
1. Cotterman, R. L., Bender, R., and Prausnitz, J. M., "Phase Equilibria for Mixtures Containing Very Many Components, Development and Application of Continuous Thermodynamics for Chemical Process Design," Ind. Eng. Chem. Proc. Des. Dev., Vol. 24, 1985, pp. 194-203.
2. Peng, D.-Y., and Robinson, D. B., "A New Two-Constant Equation of State," Ind. Eng. Chem. Fund., Vol. 15, 1976, pp. 59-64.
3. Shibata, S. K., Sandler, S. I., and Behrens, R. A., "Phase Equilibrium Calculations for Continuous and Semicontinuous Mixtures," Chem. Eng. Sci., Vol, 42, 1987, pp. 1977-88.
4. Behrens, S. A., and Sandler, S. I., "The Use of Semicontinuous Description to Model the C7+ Fraction in Equation of State Calculations," SPE Reservoir Eng., August 1988, pp. 1041-47,
5. Haynes, H. W., Jr., and Matthews, M. A., "Continuous-Mixture Vapor-Liquid Equilibria Computations Based on True Boiling Point Distillations," Ind. Eng. Chem., Vol. 30, 1991, pp. 1911-15.
6. Riazi, M. R., and Daubert, T. E., "Simplify Property Predictions," Hydrocarbon Processing, March 1480, pp. 115-16.
7. Kesler, M. G., and Lee, B. I., "Improve Prediction of Enthalpy of Fractions," Hydrocarbon Processing, Vol. 55, 1976, pp. 153-58.
8. Cavett, R. H., "Physical data for distillation calculations, Vapor-liquid equilibria," American Petroleum Institute, 27th mid-year meeting, Division of Refining, 1964.
9. Winn, F. W., "Physical Properties by Nomogram," Pet. Refiner, Vol. 36, 1957, pp. 157-59.
10. Lee, B. I., and Kesler, M. G., "A Generalized Thermodynamic Correlation Based on Three Parameter Corresponding States," AICHE journal, Vol. 21, 1975, pp. 510-27.
11. Watson, K. M., and Nelson, E. F., "Improved Methods for Approximating Critical and Thermal Properties of Petroleum Fractions," Ind. Eng. Chem., Vol. 25, 1933, pp. 880-87.
12. Watson, K. M., Nelson, E. F., and Murphy, G. B., "Characterization of Petroleum Fractions," Ind. Eng. Chem., Vol. 27, 1935, pp. 1460-64.
13. Edmister, W. C., and Pollock, D. H., "Phase Relations for Petroleum Fractions," Chem. Eng. Prog., Vol. 44, 1948, pp. 905-26.
14. Matthews, M. A., Mani, K. C., and Haynes, H. W. Jr., Kinetic and Thermodynamic Lumping of Multicomponent Mixtures, "Continuous Phase Equilibrium Thermodynamics for Sequential Operations," G. Astarita and S.I. Sandler, Eds., proceedings of ACS Spring Meeting, 1991, Atlanta, Elsevier Press, New York.
15. Edmister, W. C., and Okamoto, K. K., "Applied Hydrocarbon Thermodynamics-Part 12, Equilibrium Flash Vaporization Curves for Petroleum Fractions," Pet. Refiner, Vol. 38, 1959, pp. 117-29.
16. Pederson, K. S., Thomassen, P., and Fredenslund, A., "Thermodynamics of Petroleum Mixtures Containing Heavy Hydrocarbons, I. Phase Calculations by Use of the Soave-Redlich-Kwong Equation of State," Ind. Eng. Chem. Proc. Des. Dev., Vol. 23, 1994, pp. 163-70.
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