A correlation is presented for the thermodynamic properties of pure fluids containing small or large molecules. The residual Helmholtz energy is given in terms of perturbed-hard-chain (PHC) theory, extended to polar fluids with a multipolar expansion. The novel feature of this correlation is a separation of the Helmholtz energy into low-density and high-density contributions. The low-density contribution follows from a virial expansion and the high-density contribution from a perturbation expansion. For intermediate densities, a continuous function is used to interpolate between the two density limits. This modification of PHC theory improves agreement with experimental second virial coefficients, vapor pressures, and saturated liquid densities. Since all molecular parameters used here have a well-defined physical significance, they can be reliably estimated for high-molecular-weight-fluids where experimental data are scarce. More important, separation into low-density and high-density contributions allows separate mixing rules for each density region; this flexibility in mixing rules significantly improves representation of mixture properties, as discussed in Part II.

For some multicomponent mixtures, where detailed chemical analysis is not feasible, the composition of the mixture may be described by a continuous distribution function of some convenient macroscopic property such as normal boiling point or molecular weight. To attain a quantitative description of phase equilibria for such mixtures, this work has developed thermodynamic procedures for continuous systems; that procedure is called continuous thermodynamics. To illustrate, continuous thermodynamics is here used to calculate dew points for natural-gas mixtures, solvent loss in a high-pressure absorber, and liquid-liquid phase equiBbria in a polymer fractionation process. Continuous thermodynamics provides a rational method for calculating phase equilibria for those mixtures where complete chemical analysis Is not available but where composition can be given by some statistical description. While continuous thermodynamics is only the logical limit of the well-known pseudo-component method, it is more efficient than that method because it is less arbitrary and It often requires less computer time.

This work describes two procedures for performing flash calculations using continuous thermodynamics. The first procedure, called the mejthod of moments, provides only an approximation because it does not strictly satisfy all material balances; however, In some cases this approximation can be very good. A second procedure, called the quadrature method, uses efficient Gaussian integration; it does not use an algebraic form for the distribution function but provides exact solutions to the flash problem at selected values of the distribution variable. Both procedures are Illustrated with realistic examples, including fluid mixtures where a selected component (e.g., C02) is considered as a discrete component while all others are considered as continuous components; this Is the semicontinuous case. Calculations are also given for fluid mixtures containing several homologous series (or ensembles) as found, for example, in pefroleum mixtures where the ensembles may be paraffinic, naphthenic, and aromatic hydrocarbons. Calculated results using the quadrature method are compared to experimental data for phase equilibria in a natural-gas mixture. Agreement Is very good for compositions of coexisting phases and for liquid yield during retrograde condensation.

Sir: Kvaalen's remarks contain two misconceptions regarding my paper. The first is that the paragraph in question refers to the Deming method. While that method is mentioned, the subject of the paragraph is methods that are similar but not identical with Deming's.The second and more important is the interpretation of the phrase "linear Constraints". I agree that my paper is ambiguous on this point. As Kvaalen shows, the various methods can give different parameter estimates for constrainta that would normally be considered linear. However, as stated in my paper and covered in detail by the authors to whom I referred (e.g., Reilly and Patino-Leal, 1981), the maximum likelihood method treats both the measured variables, z, and the parameters, 8, as estimates to be adjusted in the optimization procedure. Therefore, in this case the constraints must be linear with respect to both z and 8, i.e.where A, B, and c are constant matrices. Kvaalen's example does not satisfy this requirement.His example does, however, illustrate the main point of my paper, which was that one should regard the output of any parameter estimation method with healthy skep-f(z,e) = o = AZ + Be + c ticism. It is true that computing time should not be overly emphasized, but the fact that the maximum likelihood fit gives a lower sum of squares does not mean that the maximum likelihood fit is "correct" or that the Deming method "does not work". It is clear from his figure that the (hypothetical) data and/or the model are suspect, as are the assumptions implicit in the maximum likelihood approach. I would not put much faith in the parameter estimates produced by either method in this case. If, on the other hand, the (hypothetical) experimentalist had spent less time on the computer but had obtained more precise data covering a wider range of conditions (and had derived the true theoretical relationship between x and y ) the difference in the parameter estimates would have been insignificant. Literature CitedRellly, P. M.; Patlndeal, H. Technometrlca 1981, 23(3), 221.

The molecular-thermodynamic correlation presented in Part I for pure fluids is extended t o mixtures; this extension is particularly useful for mixtures of components with large differences in molecular size or potential energy. The novel feature of this correlation is a separation of the high-density and low-density contributions to the residual Helmholtz energy; this separation allows use of separate mixing rules for each density regime, guided by theoretically defined boundary conditions. Such flexibility is necessary to represent phase equilibria in highly asymmetric fluid mixtures such as water-hydrocarbon systems. Phase equilibria are correlated for a variety of binary and multicomponent mixtures, including a petroleum reservoir fluid containing carbon dioxide and including a mixture of hydrogen and coal-derived liquids.

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Sir: In our recent article (Cotterman and Prausnitz, 1985), we report a numerical procedure using Gaussian quadrature to calculate phase equilibria for a continuous or semicontinuous mixture. We describe the composition of a continuous mixture by a continuous distribution function F(Z) where the distributed variable Z is some characterizing property such as molecular weight. The phase-equilibrium equations contain integrals of this distribution function.In Appendix I11 of our article, we discuss Laguerre-Gauss quadrature for integrating semiinfinite distribution functions and we present a relation (eq 111-8) to scale tabulated quadrature points to match the range of the distributed variable. Equation 111-8 was developed empirically for simple, unimodal distribution functions.A more appropriate integration procedure for semiinfinite distributions, called "generalized Laguerre-Gauss quadrature" (see, e.g., Davis and Rabinowitz, 1984; Stroud and Secrest, 1966)) has been suggested by Vincent Van Brunt, Department of Chemical Engineering, University of South Carolina. This procedure eliminates the need for eq 111-8 and provides a rational method to determine quadrature points and weighting factors for integrating semiinfinite distributions. We discuss briefly here the implementation of generalized Laguerre-Gauss quadrature for calculating phase equilibria for a continuous or semicontinuous mixture.Generalized Laguerre-Gauss quadrature approximates a semiinfinite integral of an arbitrary function f(u) by a weighted sum of function evaluations with the weighting function ua-le-u. The n-point quadrature formula is

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