1911
Znd. Eng. Chem. Res. 1991,30,1911-1915 Umarcq, M. C. (Prod. Chim. Ugine Kuhlmann), Fr. Pat. 2,187,799, appl. June 7, 1972. DBmarcq, M. C. Redox Dissociation of Melted Tetraphosphorus Decasulfide. Phosphorus Sulfur 1981,11,65-69. DBmarcq, M. C. Novel Molecular Phosphorus Sulfides with S / P Higher than 2.5. I. Genesis and Characterization by Physical and Chemical Methods. Phosphorus Sulfur 1987,33,127-134. DBmarcq, M. C. Scrambling of P-SP and P-SC Bonds between Tetraphosphorus Decasulfide and Phosphorotetrathioic Esters. J. Chem. Soc., Dalton Trans. 1988,2221-2224. DBmarcq, M. C. Kinetic and Mechanistic Aspects of the Redox Dissociation of Tetraphosphorus Decasulfide in Solution. J. Chem. Soc., Dalton Trans. 1990a,35-39. DBmarcq, M. C. NMR Spectroscopy of Phosphorus-Sulfur Glasses. J. Phys. Chem. 1990b,94,7330. Doi, T. Physico-Chemical Properties of Sulfur. Rev. Phys. Chem. Jpn. 1965,35,1-10, 11-17, 18-24. Eisenberg, A. The Viscosity of Liquid Sulfur. A Mechanistic Reinterpretation. Macromolecules 1969,2,44-48. Fairbrothers, F.; Gee, G.; Merral, G. T. The Polymerization of Sulfur. J. Polym. Sei. 1955,16,459-469. Forthmann, R Schneider, A. The System Sulfur-Phosphorus. 2. Phys. Chem. 1966,49,22-37. Knapsack, A.-G. Fr. Pat. 1,459,478,appl. Nov 30,1965. Larkin, J. A,; Katz, J.; Scott, L. Phase Equilibria in Solutions of Liquid Sulfur. 11. Experimental Studies in Ten Solvents: Carbon Disulfide, Benzene, Toluene, o-Xylene, Naphthalene, Biphenyl, Triphenylmethane, cis-Decalin, and trans-Decalin. J. Phys. Chem. 1967,71,352-358. Lecher, H. 2.;Greenwood, R.A.; Whitehouse, K. C.; Chao, T. H. The Phosphonation of Aromatic Compounds with Phosphorus Penta1956, 78,5018-5022. sulfide. J. Am. Chem. SOC. Meisel, M.; Grunze, H. New Phosphorus Sulfide Phase, P4SW 11. Different Preparative Methods, Properties, and Constitution of P4SW2.Anorg. Allg. Chem. 1970,373,265-278. Meyer, B. Elemental Sulfur. Chem. Reu. 1976,76,367-388. Moedritzer, K.; Van Wazer, J. R. Studies on Phosphorus Sulfides. J. Znorg. Nucl. Chem. 1963,25,683-690.
Mondain-Monval, P.; Schneider, P. The Temperature of Transformation of Liquid Sulfur to Viscous Sulfur. C. R. Hebd. Seances Acad. Sci. 1928,186,751-753. Neela, J.; Meisel, M.; Wolf, G.-U.; Erfurt, G. Ger. (East) Pat. 111,909, appl. Apr 23,1974. Niermann, H.; Reichert, G.; Ebert, H.; Neumann, F. (Hoechst A.-G.) U.S. Pat. 4,419,104,appl. Aug 30, 1979. Pohl, K.-D.; Briickner, R. Structural Viscosity of Sulfur and Arsenic Sulfide Melts: Density and Refractive Index. Phys. Chem. Classes 1981,22,150-157. Robota, S. (Hooker Chem. Corp.), US. Pat. 3,023,086;3,146,069, appl. May 27,1959;U.S. Pat. 3,282,653,appl. Feb 25, 1963. Roth, R. F.; Taylor, J. A. (The Amer. Agricult. Chem. Co.) Fr. Pat. 1,293,326,U.S. prior. June 28,1960. Schenk, J. Some Properties of Liquid Sulfur and the Occurrence of Long Chain Molecules. Physica 1957,23,325-337. Scott, R. L. Phase Equilibria in Solutions of Liquid Sulfur. J. Phys. Chem. 1965,69,261-270. Snyder, L. G.; Chesluk, R. P. (Texaco Inc.) US.Pat. 3,528,916,appl. Jan 23, 1968. Specker, H.Plastic-Elastic Properties of the System Sulfur-Phosphorus. Angew. Chem. 1949,61,439. Specker, H. The Molecular Weight of r-Sulfur. 2. Anorg. Chem. 1950,261,116-120. Specker, H. Inorganic High Polymers. Angew. Chem. 1953, 65, 299-303. Tullius, M.; Lathrop, D.; Eckert, H. Glasses in the System Phosphorus-Sulfur: A slP Spin-Echo and High-speed MAS-NMR Study of Atomic Distribution and Local Order. J. Phys. Chem. 1990,94,2145-2150. Vincent, H. Studies on Phosphorus Sulfides. T h h e de DocteurIngBnieur, Lyon, France, 1969. Vincent, H.; Vincent-Forat, C. Phosphorus Sulfides P4S9,PIS4,and P4S2 Bull. Soc. Chim. Fr. 1973,499-502. Received for reuiew September 6,1990 Accepted March 20, 1991
Continuous-Mixture Vapor-Liquid Equilibria Computations Based on True Boiling Point Distillations Henry W. Haynes, Jr.,* and Michael A. Matthews Chemical Engineering Department, Uniuersity of Wyoming, P.O. Box 3295, University Station, Laramie, Wyoming 82071
Continuous vapor-liquid equilibria computations are performed with normal boiling point temperature as the independent variable, and the distribution function is the experimental TBP distillation. Critical properties, acentric factor, and molecular weight are related to boiling point by well-established petroleum correlations. The procedure may be used with any equation of state whose parameters are expressed in terms of critical properties and acentric factor. Integrals are evaluated by the Gauss-Legendre quadrature formula. Examples illustrate that complex mixtures can sometimes be represented by as few as two quadrature points. Good agreement was observed between a discrete 20-component calculation and a continuous calculation whose distribution function was derived from the discrete composition. Application of the procedure to a semicontinuous mixture is also illustrated. Common practice in the petroleum industry is to divide
a continuous mixture of hydrocarbons into a collection of
narrow boiling range fractions or Ypseudocomponentsnand then to assign properties to these pseudocomponents so that they may be treated as individual discrete components in vapor-liquid equilibria (VLE)calculations. An alternative approach assumes that the mixture is a continuous function of some independent variable such as normal boiling point or molecular weight. The summations that
* T o whom correspondence should be addressed. 0888-5885/91/2630-1911$02.50/0
appear in discrete equation of state (EOS) formulations are then replaced with integrals over an appropriate distribution function in the independent variable. The continuous approach has been the subject of much discussion in recent literature (e.g., Cotterman and Prausnitz, 1985; Kehlen et al., 1985; Cotterman et al., 1986; Willman and Teja, 1986,Shibata et al., 1987;Behrens and Sandler, 1988; Wang and Whiting, 1988;Ying et al., 1989);however, no widely applicable procedure has yet evolved. Potentially, the continuous thermodynamics approach can result in considerable savings in computation time 0 1991 American Chemical Society
1912 Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991
since efficient numerical methods for evaluating integrals are known. However, all the methods thus far described in the literature suffer from one or more of the following deficiencies: (1)Some analytical form of the distribution function is assumed (such as a y function). This may severely limit the applicability of the method. (2) The EOS parameters are directly related to the independent variable. This restricts the computation to a particular class of compounds. In addition, this approach sacrifices consistency with discrete component calculations and removes or alters criticality constraints and other important features of the EOS. (3) Some formulations have not utilized the efficient Gaussian quadrature formulas for evaluating integrals in the continuous EOS. In this event the continuous formulation offers no computational advantage over the pseudocomponent method. The calculation procedure subject of this report does not suffer from any of these limitations. We take the normal boiling point as the independent variable. Normal boiling point is preferred by comparison to molecular weight because vapor pressure or boiling point is more directly associated with vapor-liquid equilibria. Also, by taking the normal boiling point as the independent variable, the distribution function is the experimental true boiling point (TBP) distillation curve. We use the cumulative distribution function directly to avoid the introduction of errors associated with numerical differentiation. Rather than relate the EOS parameters directly to boiling point, we employ well-established equations relating the critical properties to boiling point and specific gravity. The usual expressions for the EOS parameters in terms of their critical properties are thus retained. A n additional relation is needed giving specific gravity as a function of boiling point. The Watson characterization factor serves this purpose when such a relation is not available from experiment. Finally, the integrals are evaluated by using the Gauss-Legendre quadrature formulas, thus providing a very efficient computational algorithm. The method can be used with any EOS whose parameters are written in terms of critical temperature, critical pressure, and acentric factor. The formulation is also applicable to semicontinuous mixtures in which some of the components are discrete and others are continuous. Method Development Suppose that TBP data are available as a plot of temperature versus weight percent overhead. We define a weight fraction distribution function such that fw(T) d T = weight fraction of the mixture boiling between T and T + d T (1) This probability density function is related to the cumulative distribution function, Fw(T), according to f w = dFw/dT (2) and lOOFw(5") is the weight percent distilled at the temperature T. To aid the discussion, it is helpful to also define a mole fraction distribution function, f(T). The two functions are related by f = (M/M)fw (3) where M = M ( T ) is the molecular weight corresponding to a given T , and M is the average molecular weight of the mixture. For illustrative purposes let us consider the familiar mixing role for the parameter b in the discrete SoaveRedlich-Kwong (SRK) EOS: b = Cbixi 1
(4)
The continuous analogue to this equation is b = l T f b ( T )f(T) d T Ti
(5)
where the lower and upper limits on the integral are the initial and final boiling points, respectively. Upon substituting from eqs 2 and 3, we have b = M l l0b ( T ) / M ( T ) d F w
(6)
and we note that Fw(T)is a monotonic function of T. A second change of variables gets the integral in final form. Substituting Fu. = yg(q + 1)
(7)
provides (8)
A very efficient means of evaluating integrals of this type is by Gauss-Legendre integration. Thus n
b
E
72MC Wib(qi)/M(qi)
(9)
i=l
where the qi are zeros of the Legendre polynomial of order n (quadrature points), and the Wi are associated weight factors. The qi's and Wi's are tabulated for various n in most mathematical handbooks. The last equation can be written as n
b = Cb(qi)Xi
(10)
i=l
where xi
= YZMWi/M(qi)
(11)
This result could as well have been derived from the parameter a or from any of the other integral expressions in the EOS. By comparison with the discrete case, eq 4,we see that eq 11 defines a set of n pseudocomponent mole fractions that can be substituted into an EOS in precisely the same manner as a discrete component mole fraction. The continuous method is thus a pseudocomponent method; only the pseudocomponents are defined in a manner that approximates the integrals with as few terms, Le., as few pseudocomponents, as possible for a desired accuracy (e.g., Behrens and Sandler, 1988). The molecular weight of the mixture is obtained by integrating over the weight fraction distribution function according to
Upon substituting from eq 2 and performing operations similar to the above steps, we have
To implement these results, we need the boiling point temperature dependence of the molecular weight, the critical properties, and the acentric factor. Unfortunately it is not possible to develop a single set of such relations that is valid for all classes of compounds-paraffins, aromatics, etc. However, by introduction of a second parameter, the specific gravity, it is possible to obtain correlations for application to most hydrocarbon systems. The following relations were developed from pure-component
Ind. Eng. Chem. Res., Vol. 30, No. 8, 1991 1913 properties by Riazi and Daubert (1980): A4 = 1.66069 X 10"!P*1w2SG-1~0164
P, = 5.4580 X 107T2*3125SG2.3201
(14) (15)
(16) T, = 19.06210.wSG0.3696 where SG is the specific gravity at 60 O F relative to water at 60 OF, T is in kelvins, and P is in atmospheres. According to the authors, the predicted accuracy is reasonable over the boiling range 100-850 OF. However, Whitson (1983) observed that the correlations were acceptable at higher temperatures except for the critical pressure. At temperatures above 850 O F eq 15 should be replaced with
P, = 1.69345 X 1012T3.88618SG4.2448 (17) The acentric factor may be calculated from the reduced saturation pressure curve of Lee and Kessler (1975): w = (In P: - 5.92714 + 6.O9648/Tr + 1.28862 In T, 0.169347T,8)/(15.2518 - 15.6875/Tr 13.4721 In T, + 0.435772':) (18) where T, = T/ T, is the reduced temperature at the normal boiling point and p", = 1/P, with P, in atmospheres. Thus we see that the molecular weight and the EOS parameters through their dependence upon T,, P,, and w are functions of boiling point temperature, T, and specific gravity, SG. We need an additional expression relating specific gravity to boiling point. This relation is sometimes available from experiment when the specific gravity of each distillation cut is determined. Alternatively, it can usually be assumed to a very good approximation that the Watson Characterization factor (Watson and Nelson, 1933; Watson et al., 1935) is constant for all species in the mixture. When T is expressed in kelvins, the Watson characterization factor, K,, is defined by the relation
K, = 1.2164404F/3/SG
(19) With this relation, SG is readily eliminated from the above equations, giving T,, P,,M,and w as functions of T only. An extensive tabulation of characterization factors is available in the API Technical Data Book (Daubert and Danner, 1986). Values typically vary over a range from about 10 for highly aromatic species to about 13 for highly paraffinic components. When the characterization factor of the continuous mixture is not available, it can be calculated from the mixture specific gravity, ,%. By integrating over the weight fraction distribution function, we have 1
/. Tr
1
and upon eliminating SG between eqs 19 and 20 and evaluating the integral using Gauss-Legendre quadrature, we obtain
K, = 2.4329/&
i=l
WiT(qi)-1/3
(21)
Thus K , is calculated from a knowledge of and vice versa. In summary the calculation procedure proceeds as follows: the quadrature points and weight functions, qi and Wi,respectively, are obtained from the handbooks. The corresponding values of Fw(qi)are computed from eq 7, and the T(qJ are obtained from interpolations on the distillation curve. Interpolation via cubic splines is usually convenient. The SG(qJ are next calculated from eq 19
Table I. Discrete Components Mixture of Example 1 name Th, K K," l 3 i W ; X; 0.002 0.002 0.0034 333.4 12.32 2-methylpentane 341.9 12.78 0.011 0.009 0.0152 n-hexane 2,2-dimethylpentane 352.4 12.60 0.025 0.014 0.0203 363.2 12.72 0.061 0.036 0.0522 2-methylhexane 2-methyl-3-ethylpentane 388.8 (12.33) 0.106 0.045 0.0572 3-methylheptane 392.1 12.54 0.129 0.023 0.0292 2,2,4-trimethylhexane 399.7 12.45 0.208 0.079 0.0895 424.0 12.64 0.319 0.111 0.1257 n-nonane 3,3,5-trimethylheptane 428.8 (12.45) 0.427 0.108 0.1103 447.3 12.64 0.600 0.173 0.1767 n-decane n-undecane 469.1 12.70 0.732 0.132 0.1227 n-dodecane 489.5 12.74 0.814 0.082 0.0699 n-tridecane 508.6 12.75 0.865 0.051 0.0402 n-tetradecane 526.7 12.87 0.902 0.037 0.0271 n-pentadecane 543.8 12.86 0.930 0.028 0.0191 n-hexadecane 560.0 12.90 0.950 0.020 0.0128 n-heptadecane 575.2 12.98 0.967 0.017 0.0103 n-octadecane 589.5 13.05 0.984 0.017 0.0097 n-nonadecane 603.1 13.06 0.997 0.013 0.0070 n-eicosane 617.0 13.10 LOO0 0.003 0.0015 1.OO0 l.m "From Daubert and Danner (1986) except for values in parentheses, which were calculated from eq 19.
assuming constant K, These values when substituted into eqs 14-18 provide M(qi),Ps(qi), T,(qi),and o(qi). The average molecular weight, M, and the pseudocomponent mole fractions, xi, are calculated from eqs 13 and 11, respectively. The EOS VLE calculations then proceed in exactly the same manner as for discrete components. Computations based on the volume percent distillation curve proceed in an identical fashion; only eqs 11,13, and 21 are replaced with the analogous expressions WiSG(q i ) a (22) xi = =
y25WiSG(qi)/M(qi) ill
-
(23)
n
K, = (0.60822/SG)C WiT(qi)'13 ill
(24)
In the following section, this method is applied to several example problems formulated to illustrate the close agreement between the continuous and discrete computations for a complex mixture as well as to illustrate the generality of the procedure. Computations All the calculations performed here employed a commercially available computer program, PHSENV, available in the THERMOPAK 2.1 collection of programs for thermodynamics calculations (ChemE Computations, P.O.Box 4056, University Station, Laramie, WY 82071). Example 1. A pressure-temperature plot was constructed for the 20-componentdiscrete mixture of paraffii listed in Table I by using the SRK EOS. The Watson characterization factor for the mixture was estimated by using the weight fraction average of the pure components, K , = XwjKwi,as recommended by Daubert and Danner (19861, and the discrete distribution was approximated by a continuous cumulative weight fraction distribution as shown in Figure 1. The continuous curve was then used for the same calculation a t n = 4, 6, and 8 quadrature points. Results are plotted in Figure 2, where the solid line represents the discrete computation and the dashed lines represent the continuous computation for n = 4 and n = 6 as indicated. The curve for n = 8 quadrature points is superimposed upon the n = 6 point curve.
1914 Ind. Eng. Chem. Res., Vol. 30, No.8,1991 300
