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Cubic Equations 01 State and Their Milin! Rulnn
Hasan Orbey and Stanley I. Sandler
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Modeling Vapor-Liquid Equilibria
Cubic equations of state are frequently used in the chemical and petroleum industries to model complex phase behavior and to design chemical processes. Recently developed mixing rules have greatly increased the accuracy and range of applicability of such equations. This book presents a state-of-the-art review of this important topic and discusses the use of cubic equations of state to model the vapor-liquid behavior of mixtures of all degrees of nonideality. A special feature of the book is that it includes a disk of computer programs for all the models discussed along with tutorials on their use. With the programs and tutorials, readers can easily reproduce the results reported and test all the models presented with their own data to decide which will be most useful in their own work. This book will be an invaluable tool for chemical engineers, research chemists, and those involved in the simulation and design of chemical processes.
C A M B R I D G E SERIES I N C H E M I C A L E N G I N E E R I N G
Series Editor: Afvind Varma, University of Notre Dame Editorial Board: Alexis T. Bell, University of California, Berktzley John Bridgwater, University of Cambridge L. Gary Leal, University of California, Santa Barbara Massimo Morbidelli, ETH, Zurich Stanley I . Sandier, University o f Delaware Michael L. Schuler, Cornell Univer.~ity Arthur W. Westerberg, Camegie-Mellon University Titles in the Series: Diffusion: Mass Transfer in Fluid Systems, second edition, E. L. Cussler Principles of Gas-Solid Flows, Liang-Shih Fan and Chao Zhu Modeling Vapor-Liquid Equilibria: Cubic Equations of State and Their Mixing Rules, Hasan Orbey and Stanley I. Sandler
Modeling Vapor-Liquid Equilibria Cubic Equations o f State and Their Mixing Rules
Hasan Orbey*
Stanley I. Sandler
University of Delaware
University of Delaware
*Current address: Aspen Technology Inc. Ten Canal Park Cambridge, MA 02/41-2201 U.S.A.
CAMBRIDGE UNIVERSITY PRESS
PUBLISHEDBY T H E P R E S S SYNDICATE O F T H E UNIVERSITY OF CAMBRIDGE
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@Cambridge University Press 1998 This book is in copyright. Subject to statutory exception and to the orovisions of relevant collective licensine u aereements no reproduction of any part may take place without the written permission of Cambridge University Press. L
First published 1998 Printed in the United States of America
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Typeset in Gill Sans and Times Roman
OM0124330
Libra? of Congrers Cataloging-in-Publication Dafa Orbey, Hasan. Modeling vapor-liquid equilibria: cubic equations of state and their mixing rules 1 Hasan Orbey, Stanley I. Sandler. cm. - (Cambridge series in chemical engineering) p. Includes bibliographical references and index. ISBN 0-521-62027-9 (hb) 1. Vapor-liquid equilibrium. I. Sandler Stanley I., 194011. Title. 111. Series. TP156E65073 1998 6601.2963-dc21 97-43340 CIP
A catalog recuril j"r fhis book is avuilahle from fhe British Library ISBN 0 521 62027 9 hardback
Contents
List of Symbols
xv
Preface I
Introduction
2
Thermodynamics of Phase Equilibrium
3
page xi
2.1
Basic Thermodynamics o f Phase Equiibr~um
2.2
Vapor-Liquid Phase Equilibrium
2.3
Gamma-Phi Method for Vapor-Liquid Phase Equilibrium
2.4
Several Activity Coefficient (Excess Free-Energy) Models
2.5
Equation o f State Models for Vapor-Liquid Phase Equilibrium Calculatons
Vapor-Liquid Equilibrium Modeling with Two-Parameter Cubic Equations o f State and the van der Waals Mixing Rules
3.1
Cubic Equations o f State and Their Modifications for Phase Equilibrium Calculations o f Nonideal Mixtures
3.2
General Characterirtics o f Mixing and Combining Rules
3.3
Conventional van der Waals Mix~ngRules with a Single Binary Interaction Parameter ( I PVDW Model)
3.4
Vapor-Liquid Phase Equilibrium Calculations with the I PVDW Model
3.5
Nonquadratic Combining Rules for the van der Waals One-Fluid Model (2PVDW Model)
4
5
Mixing Rules that Combine an Equation o f State with an Activity Coefficient Model 4.1
The Combination o f Equation o f State Models with Excess Free-Energy (EOS-Gex) Models: An Overview
4.2
The Huron-Vidal (HVO) Model
4.3
The Wong-Sandler (WS) Model
4.4
Approximate Methods o f Combining Free-Energy Models and Equations o f State: The MHVI , MHV2, LCVM, and HVOS Models
4.5
General Comments on the Correlative and Pred~ct~ve Capablltles o f Varlous Mxlng Rules wlth Cubc Equations o f State
Completely Predictive EOS-GeXModels
5.1
Completely Predictive EOS-Ge" Models for Mixtures of Condensable Compounds
5.2 Predcton of Infinite D i u t o n Adivity Coefficients with the EOS-G" Approach
5.3 Completely Predictive EOS-GeXModels for Mixtures of Condensable Compounds with Supercritical Gases 6
Epilogue 6.1
Systematic Investigation o f EOS Plus Mxing Rule Combinations for the Thermodynamic Modeling of Mixture Behavior at High Dilution
6.2
Simultaneous Correlation and Prediction o f VLE and Other Mixture Properties such as Enthapy, Entropy, Heat Capacity, etc.
6.3
Representation of Polymer-Solvent and Polymer-Supercritical Fluid VLE and LLE with the EOS Models
6.4
Simultaneous Representaton of Chemlca Reactlon and Phase Equ~lbriumand the Evaluaton o f Phase Envelopes of Reactwe Mixtures
6.5
Correlation of Phase Equilbrium for Mixtures that Form Microstructured Micellar Solutions
6.6
Systematic Investigation o f LLE and VLLE for Nonelectrolyte Mixtures with an EOS
Appendixes Appendix A: Bibliography of General Thermodynamics and Phase Equilibria References Appendix B: Summary o f the Algebraic Details for the Varous Mixing Rules and Computational Methods Using These Mixing Rules Appendix C: Derivation o f Helmholtz and G~bbsFree-Energy Departure Functions from the Peng-Robinson Equation o f State at Infinite Pressure Appendix D: Computer Programs for Binary Mixtures Appendix E: Computer Programs for Multicomponent Mxtures
References Index
List of Symbols
a A A"" A"" -FOS Aex -Y AIG -
A.B, . . b B B, C, . . . C(!L)
equation of state constant reduced equation of state constant, a P / R 2 T 2 molar excess Helmholtz free energy molar excess Helmholtz free energy from an equation of state molar excess Helmholtz free energy from an activity coefficient model molar excess Helmholtz free energy for ideal gas constants in Redlich-Kister expansion (eqn. 2.4.1) equation of state constant reduced equation of state constant, b P / R T virial coefficients (second, third, . . . ) a molar-volume-dependent function specific to the equation of state (eqn. 4.1.5) value of C(V) at infinite pressure (eqn. 4.1.6 and Appendix C ) term used in Wong-Sandler mixing rule (Appendix B) fugacity (of the mixture or of pure component) fugacity of species i in a mixture partial molar excess Gibbs free energy molar Gibbs free energy of a mixture (or of pure component) molar excess Gibbs free energy of mixture molar excess Gihbs free energy of mixture from an equation of state molar excess Gibbs free energy of mixture from a liquid activity coefficient model NRTL liquid activity model parameter (eqn. 2.4.1 1) binary interaction parameter binary interaction parameter UNIQUAC model parameter (eqn. 2.4.14) binary interaction parameter
R; qfi R T Tc T,.
binary interaction parameter total mole number of a mixture mole number of species i in a mixture pressure critical pressure reduced pressure, PIP, pure component saturation pressure term used in Wong-Sandler mixing rule (Appendix B) surface area parameter volume parameter parameter in EOS models (eqn. 4.1.9 and 4.4.4 to 4.4.7) gas constant absolute temperature critical temperature reduced temperature, T I T , internal energy change of vaporization of pure colnponent packing fraction, V / b (eqn. 4.4.1 1) volume molar volume of mixture (or of pure component) cxcess molar volume of a mixture partial molar volume of species i in a mixture reduced volume V / Vc constants of equation (2.4.15) group mole fraction in the UNIFAC modcl (eqn. 2.4.19) mole fraction of species i (in liquid) mole fraction of species i (in vapor) compressibility factor, P Y I R T critical compressibility factor P,V,/RT,. coordination number in the UNIQUAC model (eqn. 2.4.13) mole fraction of species i (generic)
Greek Letters residual group contribution to activity coefficient in the UNIFAC model (eqn. 2.4.18) Wilson model parameter (eqn. 2.4.9) surface area fraction of gronp m in the UNIFAC model (eqn. 2.4.19) UNIFAC model parameter (eqn. 2.4.20) UNIFAC model parameter (eqn. 2.4.20) volume fraction in regular solution model (eqn. 2.4.16) temperature-dependent equation of state parameter (eqn. 3.1.3) Redlich-Kister equation parameter (eqn. 2.4.3) solubility parameter (eqn. 2.4.16)
E
@ -
4; Yi y."
'I
fii
u;) Vi
'Pi K
a/bRT fugacity coefficient (of the mixture or of pure component) fugacity coefficient of species i in a mixture activity coefficient activity coefficient of species i at infinite dilution in species j chemical potential number of k groups present in species i (eqn. 2.4.21) constants of Antoine equation (eqn. 2.3.1 1) volume fraction in the UNIQUAC model (eqn. 2.4.14) Peng-Robinson equation parameter (eqn. 3.1.5) PRSV equation parameter (eqn. 3.1.8) PRSV equation parameter (eqn. 3.1.9) LCVM model parameter (eqn. 4.4.10) surface area fraction in the UNIQUAC model (eqn. 2.4.13) NRTL model parameter (eqn. 2.4.11) binary interaction parameter in liquid activity coefficient models (eqn. 2.4.11) Pitzer's acentric factor (eqn. 3.1.7) Redlich-Kister equation parameter (eqn. 2.4.3)
Preface
S
EPARATION and purification processes account for a large portion of the design, equipment, and operating costs of a chemical plant. Further, whether ornot amixture forms an zeotrope or two liquid phases may determine the process flowsheet for the separations section of a chemical plant. Most separation processes are contact operations such as distillation, gas absorption, gas stripping, and the like, the design of which requires the use of accurate vapor-liquid equilibrium data and correlating models or, in the absence of experimental data, of accurate predictive methods. Phase behavior, especially vapor-liquid equilibria, is important in the design, development, and operation of chemical processes. Because the modeling of vapor-liquid equilibria is a mature subject, one might thhk that the available activity coefficient models for low-pressure-phase equilibria and various equations of state (EOS) with the simple van der Waals one-fluid mixing rules for high-pressure applications provide sufficient tools for its treatment. In fact, the reality is different from this perception. For example, activity coefficient models for highly nonideal mixtures are applicable only to the liquid phase but, even then, with temperature-dependent parameters. Cubic equations of state with the classic mixing rules can be used over wide ranges of temperature and pressure, although only for hydrocarbons and the inorganic gases. Also, the use of an activity coefficient model for the liquid phase and an equation of state for the vapor phase is very inaccurate near and above the critical conditions. Therefore, until recently it was very difficult to model nonideal mixtures of organic chemicals adequately over large ranges of temperature and pressure. This limitation was a significant problem, for in the chemical industty some 30,000 finished products are produced, and they are obtained from approximately 500 basic or commodity chemicals such as acetone, methanol, water, and so forth. The end products are usually complex molecules for which the conventional modeling methods mentioned are not always adequate. The phase behavior of the molecules in the basic chemicals category is simpler to model; however, these chemicals are produced
in large quantities with significant global market competition, and thus more accurate modeling of them can have a significant economic impact. Recently it has also been recognized that emissions and waste products of any sort can pose severe environmental problems and must be minimized. Consequently, the design requirements for the manufacture of chemicals are becoming ever more stringent, and any improvements through better modeling that can be made in processes involving even the basic chemicals are important. As a result, throughout the chemical manufacturing spectrum, there is the need for vapor-liquid equilibrium (VLE) models of good accuracy. This is one of the reasons for the considerable recent activity devoted to the development of mixing rules for cubic EOS models to describe the vapor-liquid equilibria of ever more complex mixtures. The purpose of this monograph is to present a summary and evaluation of the current state of modeling the vapor-liquid equilibria of nonideal mixtures using cuhic equations of state. The emphasis is on the use of recently developed mixing rules that combine EOS models with excess free-energy (or liquid activity coefficient) models, that is, the new class of EOS-G"" models. However, other models for VLE correlation, such as the use of a cuhic EOS with the conventional van der Waals onefluid mixing rules and the direct use of activity coefficient models, are also included for comparison and to stress the strengths and weaknesses of these traditional methods when compared with the new EOS-Ge" models. This monograph is written for the practicing engineer. In recent years, at conferences on phase equilibriummodeling and through contacts with colleagues in industry, we have noted two incorrect ideas about the use of equations of state. One is that many practicing engineers have not yet recognized the potential of the recently developed mixing rules that connect equations of state with liquid activity coefficient or excess Gibhs free-energy models (EOS-Grxmodels); they incorrectly believe that these models are too complex to use easily and, in spite of the added complexity, that they do not provide any significant improvement over the older conventional methods. The second is that some engineers believe that only the conventional method of using the van der Waals mixing rule with a cuhic EOS, though perhaps less accurate, is always reliable and thus best for simulation studies where robustness and ease of convergence of the VLE model is an important factor. Both these misconceptions need to be corrected. and this is what we are attempting to do in this monograph. We hope that the material presented in this monograph and, equally important, the computer programs we provide, will demonstrate in a practical way the current state of VLE modeling with cuhic equations of state and reveal how far this field has advanced in recent years. First, we show that the various EOS-GCxmodels available are not really very much more complicated than the simple van der Waals mixing rule and that they are almost as easy to implement and program. In fact, we provide the programs here. We will also demonstrate that the EOS-G'" models offer much greater flexibility, extrapolation capability, and reliability of predictions than the conventional EOS models that use the van der Waals mixing rule or than through the direct use of
activity coefficient models. It will also be shown that there are serious deficiencies in the simple van der Waals one-fluid mixing model that are not widely recognized and that may cause difficulties when used in simulations. For example, as we show, the van der Waals one-fluid model is so inaccurate for the description of the vapor-liquid equilibria of some relatively simple and common binary mixtures, such as acetone and water, that computer programs for this'model may not converge. Several EOS-GCX models are considered in detail in this monograph. At first glance some of these models may appear to be conceptually and algebraically similar; however, as we show; there can be significant differences in their performance. Thesc differences are especially evident in the extrapolation capabilities of each model over a range of temperatures, in the representation of phase behavior in the dilute concentration regions, and in the predictive capabilities of each model when used with group contribution methods. All these differences become important when one must choose the most useful model for a particular application. In this monograph we have selected for study some mixtures of industrial interest to show and eluphasize the differences among various EOS-G" models. An important part of this monograph is the included computer programs that allow the reader to reproduce all of the results wereport and to test all of the presented models for the correlation and prediction of the vapor-liquid equilibria of mixtures that he or she may wish to investigate. Two tutorial appendixes, one for the binary mixture programs and another for the multicomponent mixture programs, are included to facilitate the use of the programs provided on the accompanying disk. These programs should significantly enhance the usefulness of this monograph to the reader. We have tried to explain the basis for each of the models used here in a coherent fashion. However, we assume that the reader is knowledgeable in the basic principles of college-level classical solution thermodynamics, and thus in Chapter 2 we only summarize the basic principles of VLE thermodynamics. Excellent texts are available for further study of fundamental thermodynamics, some of which are listed in the bibliography (Appendix A). Although this book has been written as a monograph, the material presented here should also be useful in advanced thermodynamics courses in chemical engineering. What might be viewed as missing from this monograph is a clear recommendation as to the best EOS-Ge" model to use. This is intentional. First, we would have a clear prejudice towards some of the models we have developed. Second, for some cases there is no obvious best model; several of the EOS-Gex models perform quite well. Also, by providing programs for all the models presented, we want to encourage the reader to examine several models to determine which i$ best for the system he or she is studying. We would like to acknowledge the U.S. Department of Energy and the U.S. National Science Foundation for financial support of the research that led to this monograph. Also, we could not have completed this work without the emotional support of our families.
Introduction
S
EPARATION and purification processes account for a major portion of both the design and operating costs of chemical plants. Even though some novel separation processes, such as membrane separation, pervaporation, and others, are now being implemented on the commercial scale, contact phase separations such as distillation, gas absorption, extraction, and the like remain the major separation processes. Consequently, the thermodynamic modeling of phase equilibrium is a core concern in chemical process design. The design and operation of contact phase separators are based, to a great extent, on a knowledge of phase equilibria between the coexisting phases. Phase equilibrium information is also essential in the development of new chemical processes because the occurrence of an azeotrope or a liquid-liquid phase split may require reconsideration of the whole process flow scheme. Equations of state (EOS), the volumetric relations between pressure, molar volume, and absolute temperature, have played a central role in the thermodynamic modeling of the vapor-liquid equilibrium (VLE) of hydrocarbon fluids, especially at moderate and highpressures. With recent developments inEOS modeling, which is the subject of this monograph, equations of state are becoming useful tools also for the correlation and prediction of vapor-liquid equilibria of highly nonideal mixtures over broad ranges of pressure and temperature. In addition, equations of state are also becoming standard tools for the description of the liquid-liquid equilibrium (LLE) and of vapor-liquidliquid equilibrium (VLLE) of mixtures - areas that were traditionally the domain of liquid activity coefficient (excess free energy) models, but this topic is not considered here. The transformation of phase equilibriummodeling from activity coefficient models to equations of state is largely the result of the recently developed class of mixing rules that allows the use of liquid activity coefficient models in the EOS formalism. These mixing rules and their application to VLE are the central theme of this monograph. The implications of this transformation are far-reaching, for an EOS offers a unified approach in thermodynamic property modeling. In contrast to the use of liquid excess free-energy or activity coefficient models (with the EOS phase transitions are smooth),
Modelng Vapor-Ljquid Equilbria
including in the critical region, there is no need to choose an arbitrary reference state (which is especially troublesome when hypothetical states are used), and other thermodynamicproperties, suchas volumetric and calorimetric properties of mixtures, can also be obtained from the same EOS model. In this monograph we limit ourselves to the vapor-liquid equilibria because so far only VLE has been studied to a significant extent with the recent EOS mixing rules discussed here. This monograph is organized in the following fashion: Chapter 2 summarizes the underlying thermodynamic framework of phase equilibrium. In Sections 2.1 and 2.2 we present the basic thermodynamics of phase equilibrium with emphasis on VLE. In Sections 2.3 and 2.4, the applicationof liquid activity coefficient models to the VLE calculations is summarized, and some of the excess free-energy models also used in this monograph are presented. In Section 2.5 the basis of EOS modeling of VLE is briefly discussed. The extension of equations of state from pure fluids to the correlation and prediction of the phase behavior of mixtures is done using mixing and combining rules. Among these mixing rules, the combination of two-parameter cubic equations of state with thc classical van der Waals mixing rules is probably the most extensively used modeling tool for the VLE of hydrocarbon mixtures and of hydrocarbons with organic gases, and is the method best known to practicing engineers. Therefore, Chapter 3 is devoted to VLE modeling with cubic equations of state and the conventional van der Waals onefluid mixing rules. After a brief review of recent modifications of cubic equations of state to improve Lheir accuracy in representing pure fluids (Section 3. I), we discuss the capabilities and limitations of the van der Waals mixing rules and recent modifications. In Section 3.2 we present an overview of the general characteristics of mixing and combining rules, and the role of the van der Waals mixing rule in this general picture. Sections 3.3. and 3.4 are devoted to a discussion of the van der Waals one-fluid mixing rules with a single bina~yinteraction parameter (IPVDW). In Section 3.3 the lPVDW mixing rule and the rationale behind it is presented. Vapor-liquid equilibrium correlations and predictions with the lPVDW mixing rule are presented in Section 3.4. It is important to recognize the capabilities and limitations of this mixing rule; it is these limitations that have led investigators to develop the more sophisticated and widely applicable mixing rules that we consider here. Also, this simple mixing rule semes as a reference to judge the capabilities of the new class of mixing rules we present here. Some limitations of the lPVDW mixing rule can he removed by an empirical approach that adds further composition dependence and parameters to this mixing rule. Section 3.5 is devoted to such modifications of the van der Waals mixing rules and to VLE correlations with these two-or-more-parameter van der Waals mixing rules (2PVDW). Many mixtures of interest in the chemical industry exhibit strong nonidealities that can not be described by the EOS with any form of the van der Waals mixing rules. Mixing rules that combine equations of state with liquid excess Gibhs free-energy (or, equivalently, activity coefficient) models are more suitable for the thermodynamic
description of such mixtures. Chapter 4 is devoted to such mixing rules. In Section 4.1 we present an overview of these EOS-G" models. In recent years, especially since 1990, there has been a tremendous increase in the number and application of such models, and it is not possible to include all the variations in this monograph. We selected some of the most used models for study here. The original Huron-Vidal (referred to in this monograph as HVO) model (1979), which is the pioneering work in the area, is presented in Section 4.2. The HVO model, though mathematically rigorous, had limitations inherent from its development. Most of these limitations were eliminated by the model proposed by Wong and Sandler (1992). The WongSandler (referred to here as WS) model is the subject of Section 4.3. The HVO and WS models are mathematically rigorous, for there are no ad hoc approximations in their development. As discussed in Sections 4.1 and 4.2, this is largely due to the fact that these mixing rules establish the relation between a liquid activity coefficient model and an EOS at liquid densities (found in the limit of infinite pressure, where there is always a high-density, liquid-like solution to the equation of state). Recently, some approximate but successful mixing rules have been developed as alternatives to these infinite pressure-based models. The approximate nature of these models stems from their inclusion of some form of approximation to overcome a mathematical difficulty that arises in their development. Some of these approximate models are collectively presented in Section4.4. Section 4.5 is devoted to a general analysis of the capabilities and limitations of the EOS-G'hixing rules presented in Chapter 4, and a comparison is made with the van der Waals mixing rules of Chapter 3. An important characteristic of the EOS-GeXcombinations is their use as predictive, rather than only correlative, models for phase equilibria. Predictive liquid activity coefficient models based on the group contribution concept, such as UNIFAC or ASOG, are well developed. Some of the EOS-Gexmodels discussed in this monograph successfully incorporate these group contribution activity coefficient methods into the EOS formalism and thereby extend their application to high pressures and temperatures. Such combinations are considered in Chapter 5. In Section 5.1 the treatment of nonideal mixtures of condensable compounds by EOS-GL' models is considered. An extension of such models is to use them to predict infinite dilution activity coefficients; such information is of great value in chemical and environmental engineering in several ways. One application is the determination of the parameters in thermodynamic models, and another use is the study of the fate of chemicals in the environment. In Section 5.2 the prediction of infinite dilution activity coefficients with EOS-GeX models is considered. Section 5.3 deals with the application of EOS-GeXmodels to mixtures of condensable compounds with supercritical gases. This area is the subject of ongoing research, and some of the progress to date is discussed. In Chapter 6, an epilogue, we present our subjective view of the current state of EOS-Gexmodels and areas for future study. The development of EOS models is an active area of academic and industrial interest; the number of articles appearing in scientific journals grows steadily. Several
books and monographs are available on various types of phase equilibrium modeling, including the use of equations of state. Some of these sources are listed in the bihliography (Appendix A). A general bibliography citing all the references mentioned in the monograph is also included. Appendixes B and C contain some of the algebraic details for the various mixing rules and discuss computational methods. Appendix D is devoted to the description of the computer programs includcd with this monograph to describe vapor-liquid equilibria in binary mixtures. These programs can be used to obtain the pure component parameters needed for VLE calculations, to correlate data with all of the mixing rules discussed in this monograph, and to make EOS predictions of VLE in the absence of experimental data by the methods described here. Our aim has been to provide the reader with the tools for testing and comparing the various models presented in this monograph in order to judge their capabilities and limitations better. Tutorials are presented in this appendix to facilitate the use of the computer programs. The final part of this monograph, Appendix E, describes programs that we include for the calculation of vapor-liquid equilibria in multicomponent mixtures. As in Appendix D, we also provide tutorials on the use of these programs.
Thermodynamics of Phase Equilibrium
2.1.
Basic Thermodynamics of Phase Equilibrium The starting point for a phase equilibrium calculation is the thermodynamic reqnirement that the temperature, pressure, and partial molar Glhbs free energy of each species be the same in all phases in which that species is present. That is,
where the partial molar Gibbs free energy of species i in phase J , '??: (x:, T,P ) , is defined as
Here T and P are temperature and pressure, respectively, x, represents all the mole fractions in phase J , GJ is the molar Gibbs free energy of the phase, and the subscript N&, indicates that the derivative is to be taken with respect to the number of moles of species i in phase J with all other mole numbers held constant. The partial molar Gibbs free energy of a species is equal to its chemical potential w ; , and this is shown as the last equality in eqn. (2.1.2). Equation (2.1.1) is an exact relation from thermodynamics. However, in chemical engineering design and process simulation, what is needed is interrelations between the compositions of the phases in equilibrium rather than among the chemical potentials. Consequently, considerable effort in applied thermodynamics is devoted to converting therelationof eqn. (2.1. I), together with the definitionof the chemical potential in eqn. (2.1.2), into interrelations between the compositions of the equilibrium phases. In an idcal homogeneous mixture
Modelng Vapor-Liquid Equilibria
where G, is the pure component molar Gibbs free energy of species i, the superscript IM indicates an ideal mixture, and G, is partial molar Gibbs free energy of species i. Few mixtures are ideal mixtures, and real mixture behavior may be described in terms of departures from eqn. (2.1.3). For liquidmixtures, this is usually done by introducing the activity coefficient, y,, of component i in a solution where
For the ideal solution the activity coefficients of the constituents are unity, and for the real solutions they are defined with respect to a suitable reference state with the limitation that the temperature of the reference state must be that of the solution. We will return to the activity coefficient concept later when we discuss models for the liquid mixtures. With equations of state, real mixture behavior is described by introducing the fugacity, f .The fugacity of a species in a real mixture is
where f,(T, P) is the pure component fugacity of the species at the temperature and pressure of the mixture. A fugacity coefficient defined as
is sometimes more convenient to use. Using the fugacity, the equilibrium relation of eqn. (2.1.1) becomes
7,'(xi,T, P ) = 7; (xi'.
T, P ) =
fy(xi1', T ,P ) = . . .
(2.1.7)
There still remains the problem of reducing these fugacity expressions to equations explicit in temperature, pressure, and composition. This is done in different ways for different phase-equilibrium problems. For VLE we present the method in some detail in the next section.
2.2.
Vapor-Liquid Phase Equilibrium The starting point for VLE calculations is eqn. (2.1.7) rewritten as 7 f ( x , , T , P ) = 7y(yi, T , P)
(2.2.1)
where superscripts L and V represent liquid and vapor, respectively, and y, is the mole fraction of species i in the vapor. An EOS is always used to obtain the fugacity in the vapor phase in terms of temperature, pressure, and composition. There are, however, two different methods for the description of the liquid phase; either the same EOS
Therrnodynarncs of Phase Equilbrium
used for the gas phase is also used for the liquid phase or the activity coefficient method is employed. This latter approach has been referred to as the y -@ method (here y indicates that an activity coefficient is used for the liquid phase and @ that an EOS is used to compute the for vapor-phase fugacity coefficient). The procedure of using an EOS to calculate the fugacity of species in both phases is sometimes referred to as the @-@ method. Each route is examined in some detail in the remainder of this chapter.
2.3. Gamma-Phi Method for Vapor-Liquid Phase Equilibrium From the relation between the fugacity, the Gibbs free energy, and an equation of state, the fugacity in a vapor can be computed from
In this equation V is total volume, and Z = P I f J R T is the compressibility factor computed from an equation of state, and si! the molar volume of the mixture. Most equations of state used in engineering are pressure explicit, that is, thcy are in a form in which the pressure is explicit and the volume dependence is more complicated. One such example is the virial equation
Here B, C, D, and so forth, are the second, third, fourth, and so forth, virial coefficients, which in a pure fluid are only a function of temperature, and in a mixture are functions of only temperature and mole fraction. Another class of commonly used equations of state is based on the van der Waals equation. One member of this class is the Peng-Robinson (1976) equation
which we use in this monograph as the prototype for this class of EOS. In this and many other two-parameter cubic equations for a pure fluid, a is a function of temperature and b is a constant. As discussed later, both are also functions of composition in a mixture. For a given temperature and pressure, equations of state may have more than one solution for the volume. When using these equations of state to compute the vapor-phase fugacity, it is the vapor-phase (largest) solution for the volume that is to be used. Other members of this class of equations include the Redlich-Kwong,
Modeling Vapor-Lquid Equlibra
Redlich-Kwong-Soave, and others that are reviewed elsewhere (Sandler, Orhey, and Lee 1994, Chap. 2). The simplest EOS for a mixture, valid only at low pressures and for nonassociating species, is the ideal gas equation Z = 1, which, for a mixture, is
For such a mixture it is easily shown that
It should be emphasized that this expression is only valid for gases at low pressure (below several atmospheres) and provided that the compounds do not associate. Hydrogen fluoride and acetic acid are two examples of species that associate in the vapor phase and for which eqn. (2.3.5) would not he correct, even at low pressure. To calculate VLE at low pressures with the y - @ approach, we need to solve the equation
For the description of fugacities in the liquid phase in terms of composition, one uses an activity coefficient or, equivalently, an excess Gihbs free-energy model. The molar excess Gibbs free energy of mixing p,the partial molar excess Gihbs free and the activity coefficient yi are interrelated by energy of a species,
c,
G(T, P, x,) = Gex(T,P , xi)
+ GIM(T,P, xi)
= G ' ~ ( TP, , xi)
+ CX~G:"(T,
= @(T,
+ RT E x i lny,(T, P , xi)
P,x;)
P, xi) (2.3.7)
With this definition, the fugacity of a species in a liquid mixture is
where ~,I.(T,P ) is the fugacity of pure component i as aliquid at the temperature and pressure of the mixture. If a volumetric equation of state (that is, the relationship among the molar volume, temperature, and pressure) is applicable to pure liquid i at T and P, the purecomponent fugacity can be computed from
Thermodynamci of Phase Equiibrtum
where @i is the fugacity coefficient for the pure component. In this case it is the highdensity, smallest real, positive volume root to the EOS that is used to obtain the liquid density. For a pure liquid at its saturation pressure (vapor pressure), PVdP(T),to a very good approximation we have that
provided the saturation pressure is low. The pure component vapor pressures can be computed from the Antoine or other vapor pressure equations that can be found in some of the data references provided in the bibliography of this monograph. The Antoine equation is
where qk denotes temperature-independent equation constants. At higher vapor pressures we need to include the fugacity coefficient computed from eqn. (2.3.9). and if the liquid is at a pressure higher than its vapor pressure we need to add a Poynting correction, as shown below.
%
P'*(T)@i(T, P"-(T)) exp
[
Yb[P ;;""P(T)]
(2.3.12)
The last term, written assuming the liquid molar volume, J",is independent of pressure, is usually small enough to be neglected unless the total pressure is high or the temperature is low, as in cryogenic processing. One complication with this description is that a species can be present in a liquid mixture, though at the temperature and pressure of the mixture the substance would be a vapor or a solid as a pure component. This is especially troublesome if the compound is below its melting point, so that it is the solid sublimation pressure rather than the vapor pressure that is known, or if the compound is above its critical temperature, so that the vapor pressure is undefined. In the first case one frequently ignores the phase change and extrapolates the liquid vapor pressure from higher temperatures down to the temperature of interest using, for example, the Antoine equation, eqn. (2.3.1 1). For supercritical components it is best to use an EOS and compute the fugacity of a species in a mixture, as described in Section 2.5. The more difficult problem is deciding upon the appropriate choice of activity coefficient model and values of the model parameters. Numerous models are available, some of which are presented in Section 2.4. A valuable reference for choosing an appropriate model is Volume I - Vapor-Liquid Equilibriunz Data Collection of the DECHEMA Chemistry Data Series (Gmehling and Onken 1977). This "volume"
Modeling Vapor-Lquid Equiibra
currently consists of thirteen separate books reporting measured VLE data for binary (and some multicomponent) mixtures; the thermodynamic consistency of the data; Antoine vapor pressure constants for the pure components; fits of the mixture data with the two-constant Margules, van Laar, Wilson, UNIQUAC, and three-constant NRTL models discussed in Section 2.4; and a recommendation for the model that provides the most accurate correlation of each data set. On the basis of an analysis of 3,563 data sets in 7 of the DECHEMA Chemistry Series books, Walas (1985) found that the NRTL model most often gave the best fit of aqueous organic mixtures; that the Wilson model provided the best fit of systems containing alcohols, phenols, and aliphatic hydrocarbons; and that the Margules model provided the best fit for mixtures containing aldehydes, ketones, esters, and aromatic components. Generally, the activity coefficient is considered to be a function of only temperature and composition, as described in Section 2.4. At low and slightly above ambient pressures, eqn. (2.3.8) reduces to (2.3.13)
F)(xi, T, P) = x i y i ( ~ , x i ) ~ y ( ~ )
If the mixture is ideal, then Grx= 0 and yi = 1 in eqn. (2.3.13). Generally mixtures are not ideal in either the vapor or liquid phase, and the pressure may not necessarily be low. However, if the pressure is low, the liquid is an ideal mixture, the vapor is an ideal gas mixture, and we have that nip,"" = y i p , then, summing over all species yields ~ x ~ P , ' " P = ~ ~P , P =
(2.3.14)
0.9 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.6 0.9 1.0
B
molar composition, xn
A
Figure 2.3.1. Pressure versus molar composition for
various nonideal binary mixtures with respect to Raoult's law (solid line).
Thermodynamics of Phase Equilibrium
and
Equations (2.3.14 and 15), which only apply for the rare case of a low-pressure, ideal mixture, are known as the Raoult's law. Although Raoult's law applies only to a very limited group of real solutions owing to severe restrictions for it to he applicable, it represents a reference against which the behavior of real fluids is compared. A constant temperature P-x diagram, as obtained from Raoult's law, is shown in Figure 2.3.1. The P versus x line is straight. Any binary mixture that exerts a total pressure higher than predicted by Raoult's law results in a curved line above this straight line and is said to show positive deviations from Raoult's law. Similarly, any mixture that exerts total pressures less than predicted by Raoult's law is said to show negative deviations.
2.4.
Several Activity Coefficient (Excess Free-Energy) Models The starting point in using activity coefficient models for the liquid phase is eqn. (2.3.8). To proceed one needs to formulate expressions or models for p and, especially, for its composition dependence. Onc boundary condition that must be satisfied is that, in the limit of a pure component, the molar Gibbs free energy of a binary mixture G, must be equal to the molar Gihhs free energy of that component. Since GIM(T,P, x, + 1) = G,(T, P), this boundary condition means that p ( T , P, x, + 1) = 0. That is, the excess Gibbs free energy of mixing at constant temperature and pressure must be the zero in the limit of a pure component. By a similar argument, P ( T , P , x2 + 1) = 0. A simple function of composition (in a binary mixture) satisfying this boundary condition is the Redlich-Kister equation
where A, IB,@, D, . . . are temperature-dependent parameters. In most engineering applications it is common to use only two adjustable parameters for each pair of components in the mixture. Therefore, if we assume that @ = D = . . . = 0, using
gives RT in y, = (,x:
+ blx;
and
RT iny2 = (2x:
+B~X:
(2.4.3)
Modeling Vapor-Lquid Equilibr~a
where A
-
1
and
Bi=4(-1)'B
These are the so-called two-constant Margules equations. If it is assulned that B is also zero, the following even simpler results are obtained:
which leads to R T In y, = AX;
and
RT in yz = AX:
(2.4.4)
This last result, with only one adjustable parameter, is too simple to be useful but does show that, to a first approximation, the Margules model is symmetric in mole fraction. This is evident because the activity coefficients are mirror images of each other, and the excess Gibbs free energy is symmetric around x , = 0.5. The higher-order terms in eqn. (2.4.1) lead to more realistic, unsymmetric behavior. It has been found experimentally that for most mixtures theexcess Gibbs freeenergy of mixing is not a symmetric function of mole fraction. In fact, the excess Gibbs free energy for many mixtures is closer to being a symmetric function of volume fraction than mole fraction. For generality we define a new composition variable zi as follows:
where, if Q, is a measure of the molecular or molar volume, then zi is a volume fraction; however, the Q i may also be adjustable parameters. Wohl, on the basis of the mixture virial equation of state, used the following expansion:
Again, to have only a two-constant activity coefficient model, we assume that a,,, = a122= . . . = 0, and obtain
where 4 = 2Qla12 and ?j = 2Q2aI2;al2is a constant, and Q l . Q2 are adjustable parameters. These are well-known and commonly used van Laar equations. The activity coefficient models mentioned above depend on the overall spaceaveraged composition of the solution. On the other hand the range of intermolecular forces acting in an ordinary liquid mixture is rather short and is limited to a few molecular diameters. Consequently, it has been proposed that one use a local composition around the molecules that could he different from the overall composition of the solution. A thorough analysis of the local composition concept can be found
Thermodynamics of Phase Equlibrum
elsewhere (see, for example, Chapter 1 by Abbott and Prausnitz in Sandler 1994). The theoretical basis for the local composition concept is rather weak, and models based on the local composition idea should be regarded as empirical. Nevertheless, numerous successful excess Gibbs free-energy models have been proposed based on this concept. Most notable among them are the Wilson (1964) model, the UNIQUAC model of Abrams and Prausnitz (1975), and the NRTL model of Renon and Prausnitz (1968). With the same number of adjustable parameters these latter models usually represent the properties of the nonideal mixtures better than the models based on the overall composition. Wilson presented the following expressions for the molar excess Gibbs free energy of a binary solution:
where Aij are binary interaction parameters. Equation (2.4.8) leads to the following relations for the activity coefficients:
In the Non-Random-Two-Liquid (NRTL) model of Renon and Prausnitz (1968), the molar excess Gibbs free energy for a binary mixture is given as
with In Gij = -ori;. The parameters, o, r,,, and rji are adjustable. The activity coefficient expression from this model is
and in y2 is obtained by interchanging the indices 1 and 2. The UNIQUAC model of Abrams and Prausnitz (1975) is
Modeling Vapor -Liqud Equibria
with
where R, and Qi are volume and surface area parameters for species i, respectively, z is the coordination number, which is taken to be 10;li = ( R i - Q i ) z / 2- (Ri - I), 8, = x ; Q i / ~ x j Qis, the surface area fraction of species i ; pi = x i R i / C X , ~ Ris ,the ~ volume fraction of species i ; and In sii = -(uij - u i i ) / R T ,where r;, is a moleculemolecule interaction parameter with in T;, = 0. In each of these models two or more adjustable parameters are obtained, either from data compilations such as the DECHEMA Chemistry Data Series mentioned earlier or by fitting experimental activity coefficient or phase equilibrium data, as discussed in standard thermodynamics textbooks. Typically binary phase behavior data are used for obtaining the model parameters, and these parameters can then be used with some caution for multicomponent mixtures; such a procedure is more likely to be successful with the Wilson, NRTL, and UNIQUAC models than with the van Laar equation. However, the activity coefficient model parameters are dependent on temperature. and thus extensive data may be needed to use these models for multicomponent mixtures over a range of temperatures. The van Laar, Wilson, and NRTL models require only binary mixture information to obtain values of the parameters, whereas the UNIQUAC model also requires pure component molar volumes as well as surface area and volume parameters. These latter parameters for the UNIQUAC model are usually obtained using a group contribution method in which a molecule is considered to be a collection of functional groups and the surface area and volume of the molecule are the sum of like quantities over all groups in the molecule. The Wilson equation will not result in the prediction of liquid-liquid phase splitting and therefore can not be used in such applications or for vapor-liquid-liquid equilibria. The NRTL model has the advantage of having three adjustable parameters that allow it to be used for fitting the phase behavior of highly nonideal mixtures, though sometimes a is set to a fixed value (usually 0.2 for liquid-liquid equilibria and 0.3for vapor-liquid equilibria). In any design, engineers are unlikely to have phase behavior data for all mixtures and at all the conditions of interest. Therefore, extrapolation or prediction methods, may be needed. To extrapolate the values of the model parameters over a range of
Thermodynamcs of Phase Equilbrium
temperatures, it is common to use expressions such as
w!.
w 11. .-- w p . + 2 + . . . t,
(2.4.15)
where wij represents any of the binary parameters in the models mentioned above. Because of the nonlinearity of the phase equilibrium and activity coefficient relations, at a given temperature generally more than one set of binary parameters will fit the experimental data to the same degree of accuracy. The parameters obtained can depend upon the initial guesses, the objective function, and the minimization procedure used. Therefore, if it is desired to fit parameters to experimental phase behavior over a range of temperatures, it is best initially to consider the model parameters to be temperature dependent and fit all the data simultaneously rather than to fit each isotherm separately and then attempt to correlate the temperature dependence of the parameters using eqn. (2.4.15). Amore serious problem for the engineer is what to doin the absence of experimental data from which to obtain model parameters. In this case it is necessary to make complete predictions. In the past, this was done using the regular solution model, which required only pure component properties. The regular solution model for a binary mixture is
which results in
where 41, is the molar volume of pure liquid i, @, is the volume fraction.
6, is the solubility parameter defined by the cxprersion
and A P P is the internal energy change upon vaporization at the normal boiling point. Among the advantages of the regular solution model are that it requires only pure component property information and is easily extendable to multicomponent mixtures. One disadvantage of this model is that it is only applicable to mixtures of hydrocarbons and other nonpolar components and frequently is not veIy accurate even for those mixtures. Also, the regular solution model is not accurate for polar or hydrogen-bonding components and by its form can only lead to positive deviations from Raoult's law. The most accurate prediction methods are based on the mixture group contribution concept. The idea behind such models is that each molecule is considered to be a
Modeling Vapor-Liquid Equilibria
collection of functional groups and that thus the behavior of a mixture can be predicted based on known functional group-functional group interactions (or interaction parameters). When one regresses available experimental data containing these functional groups, a matrix of functional group-functional group interaction parameters is obtained that is then used for predictions involving mixtures for which data are not available. The UNIFAC (Fredenslund, Gmehling, and Rasmussen 1977) and ASOG (Kojima and Tochigi 1979) models are the two most important group contribution models for mixtures, hut the UNIFAC model, being applicable to the largest number of compounds, is the most commonly used. The UNIFAC model is the group contribution version of the UNIQUAC model, whereas the ASOG model is based upon the Wilson equation. The UNIFAC model is presented briefly in the next paragraph. When using the UNIFAC model one needs to identify the functional subgroups present in each molecule by means of the UNIFAC group table. Next, similar to the UNIQUAC model, the activity coefficient for each species is written as eqn. (2.4.14), except for the the residual term, which is evaluated by a group contribution method in UNIFAC. The residual contribution of the logarithm ol' the activity coefficient of group k in the mixture, In T h ,is obtained from
where
in which X, is the mole fraction of group m in the mixture, and $mn
= exp(-QmnlT)
(2.4.20)
where a,,is a measure of interaction energy between groups m and n and the sums are over all groups in the mixture. The residual contribution to the activity coefficient of species i is then computed from
In eqn. (2.4.21) vf'is the number of k groups present in species i , and In rf)is the residual contribution to the activity coefficient of group k in a pure fluid of species i molecules. The purpose of the last term is to ensure that, in the limit of pure species i (which is still a mixture of groups unless of course the molecules of species i consist of a single functional group), the residual term is zero. It needs to be stressed that any extrapolation method, and more seriously any prediction method, can be of limited and unproven accuracy. Therefore, it is always
Thermodynamcs of Phase Equilbrium
desirable to base engineering design work on experimental data. It is only in the absence of experimental data, and then only for preliminluy design work, that predictive methods should be used. An advantage of the y-@ method is that very nonideal mixtures can be described because an activity coefficient model, with suitable values of its parameters, can give very large excess Gibbs free energies of mixing. However, there are also important disadvantages of the y-@ method. In particular, because a different model is being used for the vapor and liquid phases, this method is incapable of properly describing critical region behavior. Indeed, a critical point will not be predicted, as the mathematical conditions for its occurrence can not be satisfied. Also, excess Gibbs free-energy (or activity coefficient) models are based on the mixing of pure liquids at a specified temperature and pressure (the standard state) to form a liquid mixture at these same conditions. This poses a problem when one (or more) of the components in the mixture is not a liquid at the standard-state temperature and pressure, and especially when the mixture temperature is above the critical temperature of one or more components. Therefore, the y -@ method is not useful for the description of, for example, supercritical extraction and other mixtures containing supercritical components. Finally, the y-$ method can be used for the calculation of VLE, but in contrast to equations of state, other thermodynamic properties such as densities, enthalpies, and heat capacities can not also be computed from the same model unless the Gibhs free-energy of the mixture is known as a function of temperature and pressure, as well as composition, which is not generally the case. In this monograph we use activity coefficient models in two ways. First we use them in the traditional y-$ method to correlate and predict VLE behavior at low to moderate pressures. Second, we also incorporate these models into equations of state for the description of the VLE of nonideal mixtures at high pressures, as will be discussed in Chapter 4. Programs for binary VLE calculations with activity coefficient models are provided on the diskette included with this monograph. The programming details are given in Appendix D.
2.5.
Equation of State Models for Vapor-Liquid Phase Equilibrium Calculations If an equation of state is used to describe the liquid phase, the fugacity of a species in a liquid mixture is computed from
Model~ngVapor-Lquid Equibria
where the compressibility factor Z is computed from an EOS. This equation differs from eqn. (2.3.1) only in that the liquid phase (smallest volume) solution to the EOS is used in calculating the fugacity. Most equations of state used in engineering are either extended forms of the virial equation [eqn. (2.3.2)], or variations of the classic van der Waals equation, such as the Peng-Robinson, eqn. (2.3.3). There are also lesser-used empirical equations fitted to experimental data and equations derived from theory such as the equations from the perturbed hard chain theory (PHCT), the statistical associating fluid theory (SAFT), the chain of rotators model (COR), and others. The central theme of this monograph is phase equilibrium calculations and predictions with two-parameter cubic equations of state coupled with novel mixing and combining mles; consequently, we will limit our discussion to this subject in the following chapter. A more thorough discussion of these and other equations may be found in the recent review of Sandler et al. (Sandler 1994, Chap. 2).
Vapor-Liquid Equilibrium Modeling with Two-Parameter Cubic Equations o f State and the van der Waals Mixing Rules
T
WO-PARAMETER cubic equations of state coupled with the classical van der Waals mixing rules are probably the most extensively used modeling tool for the VLE of hydrocarbon mixtures and of hydrocarbons with organic gases. In this chapter, after a brief review of recent modifications of cubic EOS for pure compounds (Section 3.1), we discuss the capabilities and limitations of the van der Waals mixing rules and their modifications.
3.1.
Cubic Equations of State and Their Modifications for Phase Equilibrium Calculations of Nonideal Mixtures Many cubic equations of state are available in the literature; some recent and comprehensive reviews are available (Anderko 1990; Sandler 1994). For the purposes of illustration, here we use the Peng-Robinson (1976) equation of state
though the principles and models discussed are general and thus applicable to all other two-parameter cubic equations of state. The form of this equation (and others in this categoty) was originally chosen to give a reasonable representation of the volumetric behavior of hydrocarbons in the gasoline carbon-number range. The parameters in the equation were then determined. First, to ensure that the equation of state predicts the correct crit~caltemperature, T,, and pressure, PC, of the mixture, the following conditions were invoked at the critical point:
For the Peng-Robinson equation this leads to
Modeng Vapor-Liquid Equlibra
and
These relations ensure that a critical point is obtained from the EOS. The term a ( T ) in eqn. (3.1.3) is temperature dependent, is unity at the critical temperature, and has been chosen to ensure that the vapor pressure calculated from the EOS at other temperatures is acceptably accurate. One such representation is
which is applicable to hydrocarbons and organic gases with the following form for K :
where w is the Pitzer acenhic factor defined as
where T, = T / T,. is the reduced temperature. In this form, the equation is completely predictive once the three constants (critical temperature, critical pressure, and acentric factor) are given. Consequently, this equation is a two-parameter EOS ( a and b ) that depends upon the three constants (T,, PC,and w). The same temperature dependence of the a term can he used for nonhydrocarbon fluids; however, as we discuss below, the completely generalized form of the a function given in eqn. (3.1.5) does not give accurate vapor pressures in this case. Other expressions have to be used for nonhydrocarbons for the dependence of a on temperature, usually with one or more parameters that are specific to the fluid of interest, rather than completely generalized in terms of the critical properties and the acentric factor. Several investigators (Mathias and Copeman 1983; Stryjek and Vera 1986a,h) have introduced ways of adding further species-specific constants to provide accurate vapor pressure correlations, especially at lower temperatures and for nonhydrocarbon fluids, that are needed for a better description of vapor-liquid equilibrium. Here we use the temperature dependence of the a term proposed by Stryjek and Vera (1986a). In their approach, eqn. (3.1.6) is replaced by the relation
with K,,
= 0.378893
+ 1.4897153~- 0.17131848w2 + 0.0196554w3
(3.1.9)
and where the constant K , is specific for each pure compound and is used to fit low-temperature saturation pressures accurately. This version of the Peng-Robinson EOS is referred to as the PRSV equation. The pure component constants of the
Vapor-L~qud Equl~br~um Modeling
Table 3.1.1. Pure component parameters for PRSV equation of state Compound
T,,K
P,,bar
o
KI
Acetone Benzene Carbon dioxide Cyclohenane
508.10 562.16 304.21 553.64 513.92 190.55 512.58 506.85 562.98 617.50 540.10 507.30 469.70 369.82 508.40 647.29
46.96 48.98 73.82 40.75 61.48 45.95 80.96 46.91 44.13 21.03 27.36 10.12 33.69 42.50 47.64 220.90
0.30667 0.20929 0.22500 0.20877 0.64439 0.01045 0.56513 0.32027 0.59022 0.49052 0.35022 0.30075 0.25143 0.15416 0.66372 0.34380
-.00888 0.07019 0.04285 0.07023 -.03374 -.00159 -.I6816 0.05791 0.33431 0.04510 0.04648 0.05104 0.03946 0.03136 0.23264 0.06635
Ethanol
Methane Methanol Methyl acetatc n-Butanol n-Decane n-Heptane n-Hexane n-Pentanc Propane 2-Propanol Water
PRSV equation for the substances considered here are given in Table 3.1.1; KI of the PRSV equation is obtained by fitting pure component saturation pressure (Pvap) versus temperature data. A computer program to optimize K, for a set of T versus P"" data is provided on the diskette accompanying this monograph, and the program details are presented in Appendix D. The effect of this parameter on the accuracy of vapor pressure correlations for several fluids is shown in Figure 3.1.1. Before leaving this subject we should note several other important points. The Stryjek-Vera modification of a takes care of the inaccuracies in temperature dependence of the a term at low temperatures. However, since the a term is based on vapor pressure, it is not well defined at temperatures above the critical temperature of a component. The behavior of this function for the PRSV EOS used here is shown in Figure 3.1.2 for various values of its parameters. For fluids for which both w and KI have nonzero values, the value of the a function extrapolated from subcritical conditions tends to increase very rapidly with temperature at supercritical conditions. This is a problem when dealing with a fluid whose critical temperature is very low, such as hydrogen. In these cases the use of other a functions becomes necessary. Twu et al. (1991; 1995a,b) have made a thorough analysis of this problem and have proposed a new a function that avoids extrema in the supercritical region and smoothly goes to zero at infinite temperature. In this monograph the function given in eqn. (3.1.8) will be used, but readers interested in applications to mixtures that contain fluids in the highly supercritical state, such as hydrogen-containing mixtures, may wish to consider alternative forms such as those presented by Twu et al.
Modeling Vapor-Liqud Equil~bria
260 280 300 320 340 360 380 400 temperature, K
Figure 3.1.1. Effect of the KI parameter on the pure component saturation pressure calculated with the PRSV equation of state. Points denote experimental saturation pressures of methanol (0) and hutanol (0)(Vargaftik 1975). Dashed lines represent results calculated with KI = 0, and solid lines are results calculated with KI values reported in Table 3.1. I.
reduced temperature
Figure 3.1.2. The parameter rr (see eqn. 3.1.3) as a function of reduced temperature (TIT,). Points represent ol values required to reproduce experimentally reported compressibilities tbr various fluids, and lines signify calculatedol values from the PRSV equation of state with different values o f ~and i acentricfactor (w).
Vapor-Liquid Equibrium Model~ng
Mathias and Klotz (1994) have shown that utilizing multiproperty titting (that is, simultaneously fitting the parameters of the a function to data such as the enthalpy of vaporization and heat capacity in addition to vapor pressure) greatly improves the overall performance of an EOS. This should be remembered when saturation pressure versus temperature information is not sufficiently accurate for good parameter estimation and when the EOS is intended for calculation of other properties, such as excess enthalpies, along with phase equilibrium. One shortcoming of two-parameter cubic equations of state is their inaccuracy in predicting liquid density; there is typically a 5- to 10-percent error, and it is greater as the critical point is approached. Further, when the two parameters in a cubic EOS are adjusted to give the correct critical pressure and critical temperature, the critical volume (or equivalently the critical compressibility Z , = P,Y,/RT,) will he in error. One indication of this is that for all fluids, the Soave-Redlich-Kwong (SRK) (Soavc 1972) equation gives Z , = 113, and the PR equation gives Z,. = 0.3074, whereas the critical compressibilities for common fluids range from 0.12 for hydrogen fluoride, 0.229 for water, around 0.27 for many hydrocarbons, and from 0.286 to 0.31 1 for the noble gases. One way to improve the poor liquid density predictions of cubic equations of state is to allow the hard core parameter b to be temperature dependent. Xu and Sandler (1987) did this with the original PR equation using fluid-specific a and b parameters to obtain good accuracy in both densities and vapor pressures. Others have introduced various temperature dependencies into two-parameter cubic equations (Fuller 1976; Heyen 1980). A problem with such approaches is that if the temperature dependencies of the a and b parameters are not carefully coordinated, the isotherms may cross over in certain regions of the pressure-volume-temperature and pressure-enthalpy-temperature space, leading to negative heat capacities and other anomalies, as shown by Trebble and Bishnoi (1986). A clever method of improving the saturated liquid molar volume predictions of a cubic EOS was introduced by Peneloux, Rauzy, and Freze (1982) by translating the calculated volumes without changing the predicted phase equilibria. For example, the volume translations + Y + c and b + b c applied to the PR equation give the correct liquid saturation volume at some temperature depending upon the choice of the parameter c and lead to an improvement in the liquid densities overall while, because of their small magnitude, having a negligible effect on the vapor densities. For the Peng-Robinson EOS used here, a more detailed volume translation was introduced by Mathias, Naheiri, and Oh (1989).
+
3.2.
General Characteristics of Mixing and Combining Rules In order to use equations of state for the correlation and prediction of the phase behavior of mixtures, a compositional dependence has to he introduced. This is done by devising mixing and combining rules for the EOS parameters. All extensions of
Modeling Vapor-Liqud Equilbria
equations of state to mixtures are, at least partially, empirical in nature, because there is no exact statistical mechanical solution relating the properties of dense fluids to their intermolecular potentials, nor is detailed information available on such intermolecular potential functions. One of the exact results we do have from statistical mechanics is the virial equation of state
where Z is the compressibility factor, P is the pressure, T is the temperature, J! is the molar volume, and B and C are second and third virial coefficients, respectively. The virial coefficients are related to the intermolecular potential between molecules, and for pure fluids they are functions of tempcratnre only. Also, for mixtures the only composition dependence of the virial coefficients is given by
C=
~ X ~ X ~ X ~ C and ~ ~so~forth. ( T ) , i
j
(3.2.3)
k
Since eqns. (3.2.2 and 3.2.3) are exact, these. equations can be considered low-density boundary conditions that should be satisfied for mixtures by other equations of state when expanded into the virial form. At present, there is no exact high-density boundary condition for mixture equations of state. However, there is the observation that, at liquid densities, the empirical activity coefficient models (such as those of van Laar, Wilson, NRTL, UNIQUAC, etc.) discussed earlier provide a good representation of the excess or nonideal part of the free energy of mixing. Therefore, another boundary condition that could be imposed is as follows: excess free energy of mixing calculated from an EOS
excess free energy of mixing = calculated from an activity
(3.2.4)
coefficient model Equations (3.2.2) and (3.2.4) have been used to develop EOS mixing rules in recent years, and they will be considered here. Because so many such models have been proposed over the years, we discuss only those that have stood the test of time or are relatively new and appear to be important advances for EOS modeling of industrial mixtures. We start here with the traditional van der Waals mixing rules that are based on eqn. (3.2.2). These have been used for decades with success to describe the high-pressure VLE of hydrocarbon mixtures and of hydrocarbons with inorganic gases. They are, however, not adequate for mixtures involving organic chemicals. The capabilities and limitations of the van der Waals mixing rules are discussed in Sections 3.3 and 3.4 together with the reasons for their failure when applied to highly
Vapor-L~quid Equlibrium Modeling
nonideal mixtures. The next step in the development of mixing rules for the cubic equations was the introduction of more than one binary interaction parameter in the conventional van der Waals mixing rules. Section 3.5 is devoted to the analysis of such multiparameter van der Waals mixing rules. In recent years equation of state mixing rules have been developed that combine excess free-energy models with a cubic EOS (Dahl and M~chelsen1990; Michelsen 1990a, h; Holderbaum and Gmehling 1991; Wong and Sandler 1992; Bonkouvalas et al. 1994; Orbey and Sandler 1995~).Various methods based on this concept will be considered in detail in Chapters 4 and 5
3.3.
Conventional van der Waals Mixing Rules with a Single Binary Interaction Parameter ( I P V D W Model)
The first successful method of generalizing a pure fluid EOS to mixtures was the one-fluid model proposed by van der Waals. The underlying assumption of this model is that the same EOS used for pure fluids can be used for mixtures if a satispactory way is found of obtaining the mixture EOS parameters. The common method for doing this is based on expanding the EOS in virial form, that is, in powers of (l/Y). For the Peng-Robinson equation one obtains
To satisfy the boundary condition of eqn. (3.2.2), the composition dcpendence of two-parameter cubic equations of state of the van der Waals family must conform to the relation
It should also be noted that
c ( x ~ T) , =
2a b C C C x ~ x ; ~ ~ c , . ~=~b2( T+) RT i
j
(3.3.3)
k
Clearly, it is not possible to set a composition dependence of the b parameter to satisfj eqns. (3.3.2 and 3.3.3) simultaneously for two-parameter cubic equations of state. Until recently, themost common way of choosing mixture parameters was to satisfy only eqn. (3.3.2) with the van der Waals one-fluid mixing ~ u l e sas follows:
Modeling Vapor-Liqu~d Equlbria
The following combining rules are frequently used to obtain the cross coefficients aij and b,j from the corresponding pure component parameters:
where kij and I , are the binary interaction parameters obtained by fitting EOS predictions to measured phase equilibrium and volumetric data. Generally lij is set to zero, in which case
We will refer to this one-binary-interaction-parametel--per-pair version of the van der Waals mixing rules (eqns. 3.3.4, 3.3.6, and 3.3.8) as the 1PVDW model. There is only aqualitative justification for eqns. (3.3.6 and 3.3.7). The a parameter is related to attractive forces, and, from intermolecular potential theory, the parameter in the attractive part of the intermolecular potential for a mixed interaction is given by a relation like eqn. (3.3.6). Similarly, the excluded volume parameter b would be given by eqn. (3.3.7) if the molecules were hard spheres. However, there is no direct relation between the attractive part of the intermolecular potential and the u parameter in a cubic EOS, and real molecules are not hard spheres. The fugacity coefficient of species i in a one-phase mixture is obtained from the PR-EOS using lPVDW as follows:
with
Also for futnre reference, note that the pure component fugacity coefficient from the PR equation is
Vapor-Liquid Equibrum Modeling
3.4.
Vapor-Liquid Phase Equilibrium Calculations with the I PVDW Model The performance of two-parameter cubic equations of state with the conventional van der Waals mixing rules (IPVDW model) is relatively well known and is presented here mainly for reference, but also to indicate certain misconceptions about this method. The results presented in this section were obtained using the computer program VDW provided on the accompanying diskette. The program details are presented in Section D.3 of Appendix D. When fitting VLE data with the lPVDW model, it is round that the binary interaction parameter kij is approximately zero for relatively simple mixtures, such as alkane mixtures, whereas for some other mixtures such as hydrocarbons with industrial gases like carbon dioxide and organic solvents, it is not only nonzero but will also change in value with temperature. For highly nonideal mixtures, which are our main concern here, accurate correlation of VLE is not possible by this method. In Figure 3.4.1 the results for the methane and n-pentane (Knapp et al. 1982) binary system are presented. This is a typical mixture for which the van der Waals one-fluid mixing rules with a single constant binary interaction parameter performs very well
d
v VLE data at 377 K
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of methane Figure 3.4.1. VLE correlation of the methane and n pentane binary system at 310, 377, and 444 K with the IPVDW mixing rule and the PRSV equation of state. The lines represent VLE results calculated with the binary interaction parameter k u = 0.0215. (Data are from the DECHEMA Chemistry Series, Gmehling and Onken 1977, Vol. 6, p. 445; data files for this system on the accompanying disk are ClC5310.DAT, ClC5377.DAT, and CIC5444.DAT.)
Modelng Vapor-Liquid Equl~br~a
0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of carbon dioxide
Figure 3.4.2. VLE correlation of the carbon dioxide md propane binary system at 277, 310, and 344 K
with the IPVDW mixing rule and the PRSV equation of state. Solid lines represent VLE results calculated with the binary interaction parameter k l 2 = 0.121, and the dashed lines denote results calculated with A.12 = 0.0. (Data are from DECHEh4A Chemistly Series, Gmehling and Onken 1977, Vol. 6, p. 589; data files for this system on the accompanying disk are C02C3277.DAT, C02C3310.DAT. and C02C3344.DAT.) (solid lines in the figure) over a wide range of temperature (from 310 to 444 K). Moreover, thc binary interaction parameter is small (kij = 0.0215), and, even if it were set to zero, a good description of the VLE behavior is possible (dashed lines in the figure). Another application of this mixing rule is for the description of mixtures of inorganic gases with hydrocarbons. An example is shown in Figure 3.4.2 for the carbon dioxide and propane (Knapp et al. 1982) mixture. Here, setting the binary interaction parameter to zero leads to unsatisfactory results, which are shown as dashed lines in the figure. However, the use of a nonzero but constant binary interaction parameter ( k i i = 0.1210 in this case) leads to very good correlation of VLE at all temperatures considered. Consequently, for such cases some experimental VLE data are necessaty to fit this parameter. The binary interaction parameter obtained at temperatures and pressures away from the design conditions may be used to extrapolate phase equilibrium information to the actual design case. Our main concern here is with mixtures significantly more nonideal than those discussed above. Many industrial mixtures fallinto this category, and for these systems the lPVDW model is not adequate. To see this we first consider the n-pentane and ethanol (Gmehling andonken 1977) binary mixture (Figure 3.4.3). Alkane and alcohol
Vapor-Liqurd Equlibrium Modeling
mole fraction of n-pentane
Figure 3.4.3. VLE correlation of the n-pentane and ethanol binary system at 373, 398, and 423 K with the lPVDW mixing rule and the PRSV equation of state. Solid lines are the VLE results calculated with adifferent binary interaction parameter, k l z , for each temperature. (The points are the data of Campbell, Wilsak, and Thodos 1987; data files for this system on the accompanying disk are PE373.DAT, PE398.DAT. and PE423.DAT.)
mixtures are industrially important and are also a stringent test of EOS models. There are many studies in the literature for which the van der Waals one-fluid mixing rules lead to predictions of false liquid-liquid splits. The example selected for study here is of amixture that is not very asymmetric in the size or vapor pressure of its components and therefore should not be v e v difficult to model. Still, in the temperature range of 373 to 423 K, there is not a single isotherm that can be accurately correlated with the lPVDW mixing rule, even when using a different binary interaction parameter for each isotherm. For example, the azeotropic point pressure is always underestimated. The results for the more asymmetric propane and methanol (Galivel-Solastiuk, Laugier, and Richon 1986) mixture at 313 K is shown in Figure 3.4.4. In this case the correlation with the 1PVDW model is poor, giving a false liquid-liquid split and underpredicting the pressure over the whole concentration range. Similar results are obtained at other temperatures for this system with this mixing rule. Difficulties are also encountered when water and alcohol mixtures are considered. The correlation of the 2-propanol and water binary system at 353 K is shown in Figure 3.4.5. Here we see that at a temperature of 353 K, and also at lower temperatures, the IPVDW mixing rule gives a false liquid split and poorly represents the VLE data. At higher temperatures, the results for this system improve somewhat, as shown in Figure 3.4.6 for 523 K, but the correlation is still not acceptable for industrial design.
Modelng Vapor-Lqu~d Equlibr~a
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of propane
Figure 3.4.4. VLE correlation of the propane and methanol binary system at 313 K with the lPVDW mixing mle and the PRSV equation of state; symbols are the experimental data. A binary interaction parameter ktz = 0.0451 was used. (Points are the data of Galivel-Solastiouket al. 1986; the data file for this system on the accompanying disk is PM313.DAT.) 1.2
.
.
1.1 1.0
,
#
,
8
,
s
r VLE data at 353 K
-
-
0.4
" " " " " " " " "
0.0 Od
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of 2-propanol
Figure 3.4.5. VLE correlation of the 2-propanol and water binary system at 353 K with the IPVDW mixing rule and the PRSV equation of state. A binary interaction parameter klz = -0.1621 was used. (Points are the data of Wu, Hagewiessche, and Sandler 1988; the data file for this system on the accompanying disk is 2PW80.DAT.)
Vapor-Liqu~d Equlibrium Modeling
v VLE dataat 523 K
3
0 ~ ~ " ' ~ ' ~ 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 mole fraction of 2-propanol
'
~
'
"
~
Figure 3.4.6. VLE correlation of the 2-propanol and water binary system at 523 K with the lPVDW mixing rule and the PRSV equation of state; r and v are the experimental data. A binary interaction parameter klz = 0.1120 was used. (Points are the data of BarrDavid and Dodge 1959; the data file for this system on the accompanying disk is 2PW250.DAT.)
Similarly poor results are obtained with the lPVDW mixing rule for the acetone and water (Griswold and Wong 1952) system. The correlation of the experimental data for this binary system is shown in Figure 3.4.7 and 3.4.8. The correlation results are very poor at low (Figure 3.4.7) and at high (Figure 3.4.8) temperatures. The overall conclusion is that even though the conventional van der Waals mixing rules are simple to use andconfom to the second virial coefficient boundary condition, they arc very limited in their application and are not useful for either the correlation or the prediction of the VLE of complex mixtures. Finally, we would like to point out an important but overlooked point about the van der Waals one-fluid mixing rules. It is frequently assumed that when the binav interaction parameters of some of the pairs in a multicomponent mixture are not available as a result of insufficient data, it is acceptable to set the binary interaction coefficient of these pairs to zero (usually the binary pairs for which data are missing are the more nonideal pairs). The results of doing this are presented in Figure 3.4.9 for the acetone and water binary system at 423 K, which shows that such an assumption may causeproblems. For this case, when k 1 2is setto zero, thereis arangeof concentrationin the water-rich region for which the EOS model leads to unrealistic phase equilibrium predictions. Indeed, a computer program using this model may not even converge unless the programmer has taken many precautions. Why the van der Waals one-fluid mixing rules cannot describe highly nonideal mixtures can be understood by starting with the relation between the molar excess
Modeling Vapor-Lqu~d Equibra
0.3
2
$ 3
0.2
7' VI
arn 0.1
0.0
-
.
VLE data at 298 K
& -
-0
-
0.0 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of acetone
Figure 3.4.7. VLE correlation of the acetone and water binary system at 298 K with the lPVDW mixing rule and the PRSV equation of state. A binary interaction parameter k l 2 = -0.256 was used. (Points are the data of Griswold and Wong 1952; the data file for this system on the accompanying disk is AW25.DAT.)
mole fraction of acetone
Figure 3.4.8. VLE correlation of the acetone and water binary system at 523 K with the IPVDW mixing mle and the PRSV equation of state. A binary interaction parameter klz = -0.093 was used. (Points are the data of Griswold and Wong, 1952; data file for this system on the accompanying disk is AW250.DAT.)
Vapor-L~quid Equlibrium Modeling
mole fraction of acetone
Figure 3.4.9. VLE prediction of the acetone and water binary system at 423 K with the lPVDW mixing rule and the PRSV equation of state. The binary interaction
parameter k l 2 was set to zero. In this figure and denote experimental data, and 0 and signify calculated values. There is a region where convergence is not obtained for water-rich mixtures. (Data are from Griswold and Wong, 1952; the data file for this system on the accompanying disk is AW15O.DAT.) Gihhs free energy of mixing, p,and fugacity coefficients ohtained from an EOS:
where q5 and @, are the fugacity coefficients of the mixture and of the pure component i calculated from eqn. (2.3.9) [or eqn. (3.3.1 1) for the PR EOS] using the mixture and pure component parameters, respectively. For the van der Waals cuhic EOS, this leads to the following expres~ionfor the excess Gihhs free energy of a binary mixture:
This equation shows that the excess Gihbs free energy computed from a cuhic EOS of the van der Waals type and the one-fluid mixing rules contains three contributions. The first, which is the Flory free-volume term, comes from the hard core repulsion terms and is completely entropic in nature. The second term is very similar to the excess free-energy term in the regular solution theoly, and the third term is similar to a term that appears in augmented regular solution theory. Consequently, one is led
to expect that the combination of a cubic EOS with the van der Waals mixing rules can only represent those mixtures that are describable by augmented regular solution theory. This excludes polar and hydrogen-bonding fluids. Equation (3.4.2) also shows that setting the binary interaction parameter ki2 equal to zero does not result in an ideal solution, for even with ki2 = 0, GeXis not equal to zero; indeed, none of the three terms in the equation vanish, nor do the terms cancel.
3.5.
Nonquadratic Combining Rules for the van der Waals One-Fluid Model (2PVDW Model) An empirical approach to overcome the shortcomings of the van der Waals one-fluid model for a cubic EOS has simply been to add an additional composition dependence and parameters to the combining rule for the a parameter, generally leaving the b parameter rule unchanged. Some examples are the combining mles of Panagiotopoulos and Reid (1 986)
Adachi and Sugie (1986) kij = K;, +&,(xi - xj)
(3.5.2)
Sandoval, Wilseck-Vera, and Vera (1989)
and Schwartzenhuher and coworkers (1986, 1989)
where in the last equation K i i = K,;, lji = - I j i , mii = 1 - m i , , and Kjj = 1;; = 0. It should be pointed out that these combining rules do not satisfy the boundary condition of eqn. (3.2.2). By an appropriate choice of their parameters, these combining rules reduce to one another and to eqn. (3.3.6). For the binary systems considered here, all these models reduce to
k;, = K Z j x ;+ Kj,xj
(3.5.5)
The combination? of eqns. (3.3.4, 3.3.6, 3.3.8, and 3.5.5) will he referred to as the 2PVDW model. This model has been shown to provide a good correlation of VLE data of highly nonideal systems that previously could he correlated only with activity coefficient models.
Vapor-Liquid Equilibrium Modelng
Note that with this modification to the van der Waals one-fluid model, the fngacity coefficient expression for species i given in eqn. (3.3.9) will change because an additional compositional dependence has been introduced to the a term of the EOS. For the PR EOS, with the van der Waals one-fluid model, a more general form of the fugacity coefficient expression of species i in a mixture is
where the derivative tern depends on the form chosen for the composition dependence of k,,. For example, for eqn. (3.5.5) used here the derivativc term for a binary system becomes
For more details see Stryjek and Vera (1986b). Some binary VLE correlations with the 2PVDW model are presented in the remainder of this section. The calculations with this model were performed with the program VDW, and the computational details are presented in Section D.3 of Appendix D. The results for the n-pentane and ethanol binary system with this model are given in Figure 3.5.1, where we show the correlation of the individual isotherms with the 2PVDW model (solid lines) and also the prediction of VLE behavior at 423 and at 373 K with the parameters obtained from fitting VLE data at 398 K (short dashed lines). Also included in the figure is the correlation of each isotherm with the IPVDW model (medium dashed lines). Several conclusions can he drawn from these results. First, for this binary system, the VLE at any temperature can accurately he correlated with the two-parameter 2PVDW model. Second, the predictions with this model are generally not very good when one attempts to use the parameters obtained at one temperature to predict the VLE at other temperatures, even over a small range of temperature. We will see this in other examples which follow. Finally it is interesting to note that these less-than-accurate predictions are still better than the correlations of the individual isotherms of this system with the IPVDW model. The more asymmetric alkane-alcohol systems, however, can not successfnlly be represented by the 2PVDW model. The results for the propane and methanol system are shown in Figure 3.5.2. There the 2PVDW model (solid line) is more accurate than the IPVDW model (dashed lines); however, the 2PVDW mixing rule also predicts a false liquid-liquid split. Alcohol-hydrocarbon systems are generally difficult to describe, and more complex mixing rules that we will discuss later can overcome this
Madel~ngVapor-Liqud Equl~bra
.
VLE data at 42 *VLE dataat 398 A VLE dataal373
0.0 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of n-pentane Figure 3.5.1. VLE correlation of the n-pentane and ethanol binary system with the 2PVDW mixing rule and the PRSV equation of state. Solid lines are the results of correlation with k l ~ l k 2=~0.19510.049 at 373 K, 0.205610.073 at 398 K, and 0.20710.096at 423 K. Short dashed lines are the results of VLE predictions with k l ~ l = k 0.20010.073 ~ ~ at all temperatures, and themediumdashed lines are IPVDW modelcorrelations presented earlierinFigure 3.4.3. (Data are from Campbell et al., 1987; data files for this system on the accompanying disk are PE373.DAT, PE398.DAT and PE423.DAT.)
problem, but only with a constraint fit of their parameters. However, with the 2PDVW mixing mle it is not possible to obtain a satisfactory fit even in this manner. The results of correlation and prediction with the 2PVDW model of the 2-propanol and water system are presented inFigure 3.5.3 and 3.5.4, respectively. InFigure.3.5.3, the VLE correlation with the 2PVDW model (solid line) at 353 K is shown. For comparison we have also included the correlated results for this system with the lPVDW model. The two-parameter model can correlate the data almost within experimental accuracy in contrast to the lPVDW model that fails seriously. The results at 523 K are even more interesting. Here the correlation with the 2PVDW mixing rule (solid line) is very accurate and much better than correlation with the IPVDW mixing rule (medium dashed lines), but the prediction with the 2PVDW model with parameters obtained at 353 K (short dashed lines) is the least accurate. This shows that the use of temperature-independent parameters is not good for the extrapolation of VLE information with the 2PVDW model. That is, the model is good for correlation but not extrapolation, and then for only moderately nonideal systems. The results for the acetone and water system are similar. In Figure 3.5.5 we present the results of correlating VLE data for this binary mixture at 298 K. The use of the
mole fraction of propane
Figure 3.5.2. VLE correlation of the propane and methanol binary system at 313 K with the 2PVDW mixing rule and the PRSV equation of state. Solid lines denote correlation results with k12/k21 = 0.0953/0.0249. Dashed lines show IPVDW model correlations presented earlier in Figure 3.4.4. (Points are the data of Galivel-Solastiouk et al., 1986; data file for this system on the accompanying disk is PM313.DAT.l
I
0 . 4 ~ " " " " " " " ' 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of 2-propanol
Figure 3.5.3. VLE correlation of the 2-propanol and water binary system at 353 K with the 2PVDW mixing rule and the PRSV equation of state. Solid lines denote correlation results with k l z / k z l = 0.0953/0.0249. Dashed lines show IPVDW model correlations presented earlier in Figure 3.4.5. (Points are the data of Wu et al., 1988; data file for this system on the accompanying disk is 2PWXO.DAT.)
30 0.0
~
" 0.1
"
'
~
"
'
~
'
~
'
0.2 0.3 0.4 0.5 0.6 0.7 0.8
mole fraction of 2-propanol
Figure 3.5.4. VLE correlation of the 2-propanol and water binary system at 523 K with the 2PVDW mixing rule and the PRSV equation of state. Solid lines denote correlation results with k12/k21 = 0.0239/-0.1378. Short dashed lines show results of predictions with binary interaction parameters klzlk21 = -0.0911/-0.1766 obtained at 353 K, medium dashed lines represent IPVDW model correlations presented earlier in Figure 3.4.6. (Points are the data of Barn-David and Dodge, 1959; data file for this system on the accompanying disk is 2PW250.DAT.)
mole fraction of acetone
Figure 3.5.5. VLE correlation of acetone and water binary system at 298 K with the 2PVDW mixing rule and the PRSV equation of state. Solid lines denote correlationresults with k i 2 / k ~ ,= -0.1416/-0.2822, and the medium dashed lines represent IPVDW model correlations presented earlier in Figure 3.4.7. (Points are the experimental data of Griswold and Wong 1952; dala file fix this system on the accompanying disk is AW25.DAT.)
'
Vapor-Liquid Equibrium Modelng
mole fraction of acetone
Figure 3.5.6. VLE correlation of acetone and water binary system at 523 K with the 2PVDW mixing rule
and the PRSV equation of state. Solid lines denote correlation results with k l ~ l k 2 1= 0.0445/-0.1521 Short dashed lines reflect predictions with k l z l k z l = 0.1416/-0.2822obtainedat 298 K, andthemedium dashed lines represent 1PVDW model correlations presented earlier in Figure 3.4.8. (Points are the data of Griswold and Wong, 1952: data file for this system on the accompanying disk is AW250.DAT.) two-binary-interaction-parameter 2PVDW model provides a significant improvement in correlation over the one-parameter model. The difference between the results from the two models is less at 523 K, as shown in Figure 3.5.6. Moreover the predictions of the 2PVDW model with parameters from the fit of the 298 K (sho1-t dashed lines) experimental data are very poor, again indicating the need for the temperaturedependent parameters in the 2PVDW model. The two-binary-interaction-per-pair (2PVDW) approach allows one to use activity coefficient data for the prediction of VLE of moderately nonideal mixtures. The relationship between an activity coefficient and an EOS is
where&(^, P,x i ) is the fngacity coefficient of species i in a mixture, and &(T, P ) is the pure component fugacity coefficient, both obtained from the EOS at the temperature and pressure of the mixture. Consequently, the two interaction parameters per binary in this class of mixing rules can be related to the activity coefficients over the whole composition range or to the values at a specified composition, such as the two infinite dilution activity coefticients of a mixture (Torress-Marchal, Catalino, and De Brito 1989; Pividal et al. 1992), thus eliminating the need for VLE data over a
Modeling Vapor-Liqud Equilibria
range of compositions. In this latter case one can write eqn. (3.5.8) for a cubic EOS for the two infinite dilution limits, obtaining two equations for the two parameters in eqn. ( 3 . 5 3 , in terms of the pure component EOS parameters and infinite dilution acrivity coeficients. For the PR equation we have
with
and
where y,? is infinite dilution activity coefficient of species j in i , Zi is the pure component liquid phase compressibility at saturation, and $?(T, P ) is the fugacity coefficient of species j in a mixture at infinite dilution in species i, with K j i being obtained by index rotation. In this case it is possible to predict the phase behavior of some moderately nonideal systems successfully using only infinite dilution activity coefficient information at the same temperature. Experimentally measured infinite dilution activity coefficients of the binary pairs in one another are used in eqn. (3.5.9). However, this kind of data may not always be available, so that a predictive group contribution method such as UNIFAC may have to be used to obtain the necessary infinite dilution activity coefficients, and thus the method becomes completely predictive. An example of the use of this method is given in Figure 3.5.7 for the methyl acetate(1) and cyclohexane(2) binary system (Pividal et al. 1992) at 313 K. The infinite dilution activity coefficient of each component in the other is available for this binary pair, the mixture is nearly symmetric and deviates only moderately from ideal solution behavior (yim/y,? = 4.81/4.54). The solutions of eqns. (3.5.9 to 3.5.11) give values of the binary interaction coefficients of k,2 = 0.0905 and k2, = 0.1 167, and predictions with these parameters are shown in the figure as dashed lines. Direct correlation of VLE with the 2PVDW model gives the binary interaction parameters as k I 2 = 0.0944, and k2, = 0.1027, and the results obtained with these parameters are shown as the solid line in the same figure. For this case, predictions based on infinite dilution activity coefficients are good, and a reasonable representation of the VLE data over the whole concentration range is possible with this information. However, this approach becomes less dependable as the nonideality of the mixiure increases. For example, in Figure 3.5.8, the results for the ethanol and heptane binary system are shown. This mixture is more nonideal, (y,"/y,? = 16.27114.21), and the predictions obtained (long dashed lines) with parameters k12 = 0.0565 and k2, = 0.1621 based on only infinite dilution activity coefficients are much worse than those obtained by
Vapor-Liquid Equibrum Modeling
,
,
,
,
,
,
,
,
,
,
,
,
.
,
VLE data at 31 3 K
, , , , , , , . , ,
,
,
,
,
,
,
-
,
,
mole fraction of methylacetate Figure 3.5.7. VLE correlation of the methyl acetate and cyclohexane binary system at 313 K with the 2PVDW mixing rule and the PRSV equation of state. Solid lines are model predictions obtained by direct correlation of VLE data, and the dashed lines are predictions using infinite dilution activity coefficient
Onken 1977, Vol. I , Pt. 5 , p. 392; the data file for this system on the accompanying disk is MAC640.DAT.)
correlating data over the whole composition range (solid line) to obtain the parameters k I 2 = 0.0201 and k2, = 0.1376. There is another, albeit more cumbersome, way of using activity coefficient information. This is described by Pividal et al. (1992) as follows: First, a two-parameter excess free-energy model is selected, and its parameters are obtained from the measured activity coefficients at infinite dilution. Next, by use of these parameters the activity coefficients are computed from the Gibhs free-energy model at a selected mole fraction (Pividal et al. used x = 0.5). Finally the two EOS mixing rule parameters are determined by equating these midconcentration activity coefficient values to the activity coefficient expressions from the EOS. This procedure requires no optimization and leads to predictions shown inFigure 3.5.8 as short dashed lines for the ethanol and heptane binary system. These results are considerably better than the predictions obtained by solving eqns. (3.5.9 to 3.5.1 1) (long dashed lines) but still inferior to the direct fit of the data (solid lines). Moreover, a liquid two-phase split is erroneously predicted. There are several problems associated with these multiparameter combining rules that limit their use in process design for mixtures containing many compounds, or to mixtures which contain some species with similar characteristics (such as with mixtures of isomers, etc.). The first of these problems is the so-called dilution effect in
Modelng Vapor-Lqutd Equ~libra
0.2 ' I " ' ~ " ' " " ' " ' 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0,s 1.0 mole fraction of ethanol
Figure 3.5.8. VLE correlation of the ethanol and nheptane binary system at 333 K with the 2PVDW mixing rule and the PRSV equation of state. Solid lines are model predictions obtained by direct correlation of VLE data, and the dashed lines are predictions using intinite dilution activity coefficient data. See text for details. (Points are experimental data from the DECHEMA Chemistry Series, Gmehling and Onken 1977, Vol. 1, Pt. 5, p. 392; the data file for this system on the accompanying disk for this system is ETC7333DAT.)
which, as the number of components in a mixture increases, the mole fraction of any one component in the system becomes smaller. This leads to smaller contributions from the terms with the higher-order composition dependence and the added parameter(~),thus effectively reducing the mixing rule to the quadratic one-Ruid mixing mle of van der Waals (IPVDW model) as the number of components in the mixture increases. The second difficulty withmixing rules in this categoty, as pointedout by Michelsen and Kistenmacher (1990), is that they do not result in the correct treatment of multicomponent mixtures containing two or more identical subcomponents. Consider a mixture of three components, and for the sake of demonstration allow two of these components, such as 2 and 3, to become identical. (This is not a trivial thought experiment, for there are multicomponent mixtures of industrial interest in which two or more components have very similar EOS parameters such as the isomers 1-methylnaphthalene and 2-methyl-napthalene.) In this case, to be internally consistent, the mixing rule for the parameter a should reduce to that for a binary mixture with the composition ( ~ 2 x3) for the new component 2 3. The one-fluid combining mle of van der Waals satisfies this criterion, whereas the combining rules being considered in this section do not.
+
+
Vapor-Liqud Equl~brumModeng
Finally, a difficulty with these mixing rules is that, because of the added composition dependence of the a parameter, they fail to satisfy the theoretical quadratic composition dependence of the second virial coefficient given in eqn. (3.2.2). This fact is usually ignored on the grounds that pure component second virial coefficients calculated from a cubic EOS are in poor agreement with experimental data. However, the expression for the fugacity coefficient of a species in a mixture (eqn. 2.3.1) shows that its evaluation from the EOS involves an integral of a term that includes a partial derivative with respect to composition from zero density to the density of intere~t. Therefore, an error in the composition dependence of the EOS at low densities will affect the fugacity coefficient calculated from eqn. (2.3.1) at all densities. It should be stressed that the correct composition dependence of the second virial coefficient and correct numerical values of the pure component second virial coefficients are two different problems, and both will affect the computed fugacity coefficient. Models that have the correct second virial composition dependence eliminate one of these sources of error.
Mixing Rules that Combine an Equation of State with an Activity Coefficient Model
ANY mixtures of interest in the chemical industry exhibit strong nonideality, greater than that describable by regular solution theory, and have traditionally been described by activity coefficient (or free-energy) models lor the liquid phase and an equation of state for the vapor phase. However, as discussed earlier, there are numerous problems with the activity coefficient description. For example, there are difficulties in defining standard states (especially for supercritical components), the parameters in these models are very temperature dependent, and critical phenomena are not predicted because a different model is used for the vapor and liquid phases. Also, other thermodynamic properties (densities, enthalpies, entropies, ctc.) can not usually be obtained from the same model because the Gibhs energy is rarely known as a function of temperature and pressure. Therefore, interest exists in mixture EOS models that are capable of describing greater degrees of nonideality than is possible with the van der Waals one-fluid model and its variations. A very attractive route for developing better mixing rules is to combine an EOS with activity coefficient models, and this approach is the subject of this chapter.
M
4.1.
The Combination of Equation of State Models with Excess Free-Energy (EOS-GeX)Models: An Overview It is possible to obtain the activity coefficient of a species i in a mixture from an EOS where 4,is the fugacity coefficient using the relation given in eqn. (3.5.81, yi = of component i in the mixture and d, is the pure component fugacity, both of which are computed from an EOS at the temperature and pressure of the mixture. The molar excess Gihbs free-energy is
Mixng Rules that Combine an Equation of State with an Activ~tyCoefficient Model
The combination of eqns. (3.5.8) and (4.1.1) leads to
or, equivalently, for the excess Helmholtz free energy of mixing
For the two-parameter cubic equations considered here, eqns. (4.1.2 and 4.1.3) become
Here C(V) is a molar-volume-dependent function specific to the EOS chosen. For In [V+"JSJb example, for the Peng-Robinson equation C(v) = ~JZ Equations (4.1.4 or 4.1.5) use one degree of freedom in establishing a relation between the EOS parameters a and b and the mixture composition with activity coefficient or excess Gibbs free-energy models. Coupled with another independent piece of information that connects a or b (or both) to composition, one can solve for the EOS parameters a and b in terms of the mole fractions. There are several alternative ways of implementing such a procedure, and they are reviewed here and in the following sections.
~+(I+JsJ~I.
Modeling Vapor-Lquid Equilbria
For later reference we note that in the limit of infinite pressure, V ; + bi and Vmi, + bmi, so that C(V,,, = bmi,) = C(J.I; = b,) = C*; for the Peng-Robinson equation C* = ln(& - I)/& = 0 . 6 2 3 2 3 . Then eqns. (4.1.2 and 4.1.3) become
and
If&os or e;o",,s equated to those from an activity coefficient model, then eqns. (4.1.6 and 4.1.7) are a mathematically rigorous combination of an EOS and an activity coefficient model; they have been obtained in the limit of infinite pressure. From the definition of an excess property change upon mixing, it is necessary that the pure components and the mixture be in the same state of aggregation; therefore, to use eqns. (4.1.6 and 4.1.7) the same must be true. It is obvious from these equations that the excess Gihbs and Helmholtz free energies of mixing computed from an EOS are a function of pressure, whereas activity coefficient models are independent of pressure or density. Therefore, the equality between Gex(or p)from an EOS and from an activity coefficient model can he made at only a single pressure. Models that combine equations of state and activity coefficient models can be categorized into two groups: those that make this link at infinite pressure (Huron and Vidal 1979; Wong and Sandler 1992) and those that make this link at low or zero pressure (Dahl and Michelsen 1990; Holderbaum and Gmehliug 1991; Michelseu 1990b, among others). However, in the zero pressure limit there is no mathematically rigorous solution applicable to all phase equilibrium problems. This is because, above some temperature, the cubic EOS will not have the necessary real root for the liquid phase at zero pressure. Such a case is schematically shown in Figure 4.1.1 in which several isotherms calculated from the reduced PR EOS [eqn. (4.1.8)l
are shown. In eqn. (4.1.8) the reduced pressure, temperature, and volume are defined as P, = PIP,, T,. = TIT,, and If. = VP,/(0.307RT,.) respectively. The PR EOS with w set to zero can give a real positive root for the liquid molar volume at the limit of zero pressure at a reduced temperature of 0.8 ( P = 0 in the figure) and yet fail to give a root at a reduced temperature of 0.9. The limiting reduced temperature for obtaining a real root for liquid molar volume is about 0.85 for this EOS with o = 0. Consequently, an approximation must he introduced for the temperatul-e range T, > 0.85 to obtain a pseudoliquid root. In spite of this shortcoming, these
Mixng Rules that Comb~nean Equat~onof State w ~ t han Arilv~tyCoefic~entModel
reduced volume
Figure 4.1.1. Reduced pressure (PIP,) versus re-
duced volume [~P,l(0.307RTc)] as a function of reduced temperature (TIT,.) from the Peng-Robinson equation of state wlth acentric factor set to zero. See text for details. approximate models are successful in many cases and are therefore considered in this monograph. Most of these approximate zero pressure models take the form
where q, is a model-dependent approximate parameter (or function). When using a zero pressure model, the approximate eqn. (4.1.9) is used instead of the exact relation given by eqn. (4.1.7), which was obtained at the limit of infinite pressure. By relaxing mathematical rigor in establishing the connection between excess freeenergy models and EOS, several successful approximate models have been developed in the limit of infinite pressure. One such model that uses excess Helmholtz free energy was introduced by Orbey and Sandler (1995~)and is as follows:
Equation (4.1.10) is algebraically very similar to eqn. (4.1.9) except that the approximate parameter q,, is now replaced with the parameter C*, whlch, as pointed out earlier, is dependent on the EOS used. This seemingly small difference, however, has a nontrivial effect on the results obtained, as will be shown later. In this approach, the approximate eqn. (4.1 .lo) replaces the exact expression given by eqn. (4.1.6).
Modeling Vapor-Liqud Equil~bria
4.2.
The Huron-Vidal (HVO) Model Vidal (1978) and later Huron and Vidal(1979) proposed the first successful combination of an EOS and activity coefficient models by requiring that the mixture EOS at liquid densities should behave like an activity coefficient model. To ensure a liquid density at all temperatures, this equality was made using eqn. (4.1.7) and the relation
where Q,(T, P = co,x) and Ge,',,(T, P = co,x) are the excess Gibhs free energies at infinite pressure (i.e., at liquid-lie densities) calculated from an activity coefficient model and the EOS, respectively. Because Gex = Ae' P F ,with AeXin the liquid state being almost independent of pressure, to use eqn. (4.1.7) it is necessary that
+
since at infinite pressure from an equation of state yi = hi. That is, to keep g;, finite in order to use eqn. (4.1.7), eqn. (4.2.2) must also be used. Equations (4.1.7 and 4.2.2) provide the two equations necessary to determine the two EOS constants. The resulting mixing rule for the a parameter is
is an excess free energy of mixing expression appropriate for the mixture where of interest. Equations (4.2.3 and 3.3.8) constitute the original Huron-Vidal mixing rule. We will refer to this mixing rule as the HVO model in this monograph. Some results obtained with the HVO mixing rule are presented in the following paragraphs. These results were obtained with the computer program HV provided on the accompanying diskette; the computational details are presented in Section 0 . 4 of Appendix D. The correlation of data for the methane and pentane binary system is shown in Figure 4.2.1. In this case the van Laar excess Gibhs free-energy model has been used in the HVO model; the two model parameters were fitted to VLE data on the 277 K isotherm, and the vapor-liquid equilibria at higher temperatures were predicted with the same temperature-independent parameters. The results are very good in this case and similar to those obtained with the lPVDW and 2PVDW models. The results for the carbon dioxide and propane binary system, shown as dashed lines in Figure 4.2.2, on the other hand are not as good. When compared with the performance of the IPVDW model (solid lines in Figure 4.2.2), the use of the same parameters for all isotherms leads to inferior results at higher temperatures despite the use of an extra parameter in the Huron-Vidal model. This indicates that, for the mixtures containing supercritical components, the HVO mixing rule, when combined
Mixng Rules that Cornbne an Equation of State w t h an Adiv~tyCoefficient Model
.
r VLE data at 31 0 K I VLE data at 377 K o VLE data at 4 A
o
mole fraction of methane Figure 4.2.1. VLE col~elationof the methane and pentane binary system at 310, 377, and 444 K with the Huron-Vidal original (HVO) mixing mle with the van Laar excess free-energy model and the PRSV equation of state. The van Laar model parameters used are t/fl = 1\12/1121 = 0.1201/0.1430. Points are experimental data from the DECHEMA Chemistry Data Series, Gmehling and Onken 1977, Vol. 6, p. 445; data file? for this Fystem on the accompanying disk are CIC5310.DAT, CIC5377.DAT and CIC5444.DAT. with the conventional excess free-energy models (such as that of van Laar) is inferior to the van der Waals one-fluid model. This was observed earlier by Shibata and Sandler (1989). It should be noted, however, that by choosing specific algebraic forms for the excess free-energy term it is possible to reduce the HVO model to the I PVDW model. See, for example, Huron and Vidal (1979) and Orbey and Sandler (1995a,c) for the details of such modified excess free-energy models. For more nonideal mixtures, the HVO model shows -good correlation capabilities but is not satisfactory for extrapolation over a range of temperatures. See, for example, the results presented inFigures 4.2.3 and 4.2.4 for the acetone and water binary system and those in Figures 4.2.5 and 4.2.6 for the 2-propanol and water binary system. In these figures, the dashed lines are direct correlations of the isothermal VLE data with the HVO mixing rule. The solid lines are predictions with model parameters obtained from the DECHEMA Chemistry Series at or near room temperature for the excess free-energy model. In each case the model was observed to be superior to both the one-parameter (IPVDW) and the two-parameter (2PVDW) van der Waals models for the correlation of VLE data. On the other hand, poor predictive performance observed in these figures indicates that, even though the Huron-Vidal approach allows the ute of Ge%odels with an EOS, one cannot use excess Gibbs free-energy model parameters obtained from the y - @ method at low pressure (for example those in the
Modelng Vapor-Liquid Equilibia
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of carbon dioxide
Figure 4.2.2. VLE correlation (dashed lines) of the carbon dioxide and propane binary system at 277, 310, and 344 K with the Huron-Vidal original (HVO) mixing rule with the van Laar excess free-energy model and the PRSV equation of state. The van Laar model parameters used are (/,B = A12/A21 = 1.1816/1.6901. Solid lines represent the IPVDW model correlations presented earlier in Figure 3.4.2. (Points are experimental data from the DECHEMA Chemistry Data Series, Cmehling and Onken 1977, Vol. 6, p. 589; data files for this system on the accompanying &sk are CO2C3277.DAT, CO2C3310.DAT and C02C3344,DAT.)
DECHEMA Chemistry Data Series) with this EOS model. This is because the excess Gibbs free energy of mixing from experiment and as calculated from an EOS is very pressure dependent; therefore, the excess Gibbs free energies at infinite pressure and at the pressure at which experimental data were obtained can be vely different. Consequently, a fundamental shoncoming of the Huron-Vidal approach is the use of the pressure-dependent Gibbs excess free-energy in the EOS rather than Helmholtz excess free-energy, which is much less pressure dependent. This shortcoming was corrected by Wong and Saudler (1992), and their work is discussed next.
4.3.
The Wong-Sandler (WS) Model Wong and Sandler (1992) have developed a mixing rule that combines an EOS with a free-energy model but produces the desired EOS behavior at both low and high densities without being density dependent, uses the existing table of Gexparameters. allows extrapolation over wide ranges of temperature and p r e s ~ r e and , provides a
Mrxng Rules that
Comb~nean Equaton
of State with an Actvity
Coefic~entModel
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of acetone
Figure 4.2.3. VLE correlation of the acetone and water binary system at 298 K with the Huron-Vidal original (HVO) mixing rule with the van Laar excess freeenergy model and the PRSV equation of state. The dashed lines denote results calculated with van Laar model parameters c/p = 1\12/1121= 3.512112.2227 obtained from fitting the experimental data, and the solid lines represent the results obtained with model parameters (/p = Al~/A21 = 1.9399/1.8022 obtained at the same temperature from the DECHEMA Chemistry Series (Gmehling and Onken 1977, Vol. I , Pt. I , p. 238).
conceptually simple method of accurately extending UNIFAC or other low-pressure VLE prediction methods to high temperature and pressure. This mixing rule is based on several observations. The first is that, although eqns. (3.3.4 and 3.3.5) are sufficient conditions to ensure the proper composition dependence of the second virial coefficient, they are not necessary conditions. In particular, the van der Waals one-fluid mixing rules of eqns. (3.3.4 and 3.3.5) place constraints on the two functions a and b to satisfy the single relation
The original version of thc Wong-Sandler mixing rule (Wong and Sandler 1992) uses the last equality of eqn. (4.3.1) as one of the restrictions on the parameters together with the combining rule
which introduces a second virial coefficient binary interaction parameter k i i . Note that eqn. (4.3.1) does not provide relations for the parameters rr and b separately but only
Modeng Vapor-L~qu~dEqul~br~a
mole fraction of acetone
Figure 4.2.4. VLE correlation of the acetone and water binary system at 523 K with the Huron-Vidal orig-
inal (HVO) mixing rule combined with the van Laar excess free-energy model and the PRSV equation of state. The dashed lines represent results calculated with van Laar model parameters (/,Cl = A I ~ / A = ~I 4.2206/1.7264 obtained from fitting the experimental data,and thesolid linesdenotethe resultsobtained with model parameters (/,Cl = A12/A21 = 2.1700/1.7264 from the DECHEMA Chemistry Series at 353 K (Gmehling and Onken 1977, Vol. 1, Pt. 1, p. 334). for the sum (b - a / R T ) , and thus an additional equation is needed. Also, by using eqn. (4.3.1) as one of the relations to determine the EOS parameters, the proper composition dependence of the second virial coefficient is assured, regardless o i which additional equation is used. The second equation in their mixing rule is based on the observation that the excess Helmholtz free-energy of mixing calculated from a cubic EOS is much less sensitive to pressure than the Gibbs free-energy, as can be seen in Figure 4.3.1. Consequently, to an excellent approximation:
G X ( TP , = 1bar, x) = e -
( T , P = 1 bar, x) = K Y T . high pressure, .r)
(4.3.3)
The first of these equalities results from the relation p = A" + P p and that the P T term at low pressures. The second of these equalities is a result ofthe essential pressure independence of A'" at liquid densities. Therefore, the second equation for the u and b parameters comes from eqn. (4.3.3) in the form of
A,h,(T, P
= oo,x) = _ A ( T , P = oo,x) = A Y ( T , low P , x ) = G Y ( T , low P, x)
Mixing Rules that Combne an Equation of State w~than Activity Coeficent Model
mole fraction 2-propanol Figure 4.2.5. VLE correlation of the 2-propanol and water binary system at 353 K with the Huron-Vidal original (HVO) mixing rule combined with the NRTL excess free-energy model and the PRSV equation of state. The dashed lines represent results calculated with a =0.2893 and t , 2 / t 2 1 =0.7882/3.9479 obtained from fitting the experimental data, and the solid lines denote results calculated with u = 0.2893 and r12/~21=0.1509/1.8051 obtained from the DECHEMA Chemistry Series at 303 K (Gmehling and Onken 1977, Vol. 1, Pt. 1, p. 325).
Combining eqns. (4.3.1,4.3.4, and 4.1.6) gives the following mixing rules:
where the cross term in eqn. (4.3.5) is obtained from eqn. (4.3.2). Any excess Gibbs free-energy model may be used in eqn. (4.3.6). The model parameters are the parameter k i j in eqn. (4.3.2) and the parameters of the excess Gibhs free-energy model used in eqn. (4.3.6). This mixing rule is referred to as the WS mixing rule in the remainder of this monograph. The WS mixing rule satisfies the low-density boundluy condition that the second virial coefficient be quadratic in composition and the high-density condition that excess free energy be produced like that of currently used activity coefficient models, whereas the mixing rnle itself is independent of density. This model provides a correct alternative to the earlier ad hoc density-dependent mixing rules (Copeman and
Modeling Vapor-Liquid Equlibra
mole fraction of 2-propanol
Figure 4.2.6. VLE correlation of the 2-propanol and water binary system at 523 K with the Huron-Viiidal original (HVO) mixing rule combined with the NRTL excess free-energy model and the PRSV equation of state. The dashed lines represent results calculated with or=0.2893 and rlrlrzl =0.3952/4.1518 obtained by fitting the experimental data, and the solid lines denote results calculated with a = 0.2893 and rlz/rzl =0.1019/1.2185 obtained from the DECHEMA Chemistry Series (Gmehling and Onken 1977, Vol. I, Pt. 1, p. 325). 0.450 Gei at 1 bar, Aex at 1 &lo00 bar
0.350
-
.
mole fraction of methanol Figure 4.3.1. The excess Gibbs and Helmholtz energies of mixing for the methanol and benzene binary system at 373 Kcalculated with the Woug-Sandlel.(WS) mlxingrule and the PRSV equation of state at 1 and 1000 bar.
Mixing Rules that Cornbne an Equation of State w ~ t han AdiviQ Coefficient Model
Mathias 1986; Michel, Hooper, and Prausnitz 1989; Sandler et al. 1986) and retains an important feature of the one-fluid model: that the EOS for the pure fluids and the mixture have the samz density dependence. This mixingrule has been successful in several ways. First, when combined withany cubic EOS that gives the correct vapor pressure and an appropriate activity coefficient model for the p term, it has been shown to lead to very good correlations of vaporliquid, liquid-liquid, and vapor-liquid-liquid equilibria, indeed generally comparable to those obtained when the same activity coefficient models are used directly in they -@ method. Consequently, this mixing rule extends the range of application of equations of state to mixtures that previously could be correlated only with activity coefficient models. Second, because low-pressure p information has been used in developing this mixing rule, Wong, Orbey, and Sandler (1992) found that activity coefficient parameters reported in data banks, such as the DECHEMA Data Series, could be used directly and with good accuracy in the WS mixing rule without the need of refitting any experimental data. In this case, however, one point about the binary interaction parameter kij needs to be made. The excess Helmholtz free-energy calculated from an EOS is not necessarily independent of pressure for an arbitrary value of k,,. Therefore, the curve is reproduced binary interaction parameter kij should be chosen so that the as closely as possible by BAS at the pressure at which the activity coefficient model parameters are reported. Thus, the parameter k , contains no further information than that already included in The correlative and predictive capabilities of the WS mixing rule arc shown for some systems in Figures 4.3.2 to 4.3.9. The results obtained were calculated using the computer program WS on the diskette provided with this monograph, and the programming details are given in Section D.5 of Appendix D. Results obtained for the methane and n-pentane binary mixture are presented in Figure 4.3.2. In this case the model parameters are fitted to VLE data at 277 K, and the VLE behavior at higher temperatures is predicted with those values of parameters. The results are good but slightly less accurate than those obtained by the lPVDW model correlated to each isotherm. In contrast, the correlation of the carbon dioxide and propane binary system shown in Figure 4.3.3 is superior to both the HVO and the IPVDW models. However, the tme advantage of the WS model is in predicting the vapor-liquid equilibria of highly nonideal mixtures containing condensable components for which activity coefficient model parameters are available only at or near room temperature. In Figures 4.3.4 and 4.3.5, the results for the 2-propanol and water binary system are presented at 353 and 523 K, respectively. In these figures the dashed lines are obtained from the direct fit of the model parameters to the experimental data, whereas the solid lines are predictions with model parameters that have been obtained from the DECHEMA correlation from djta at 303 K. The results show that the correlations are excellent, hut more importantly the predictions at temperatures as much as 200 K above the correlation temperature are almost as accurate as the correlations. Similar results, shown
e;.
e:
,
,
,
,
,
,
,
,
,
A A VLE data at 444K 0
0.0 0.1
VLE data at St 0 K
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of methane
Figure 4.3.2. VLE correlation of the methane and npentane binary system at 310, 377, and 444 K with
the Wong-Sandler (WS) mixing rule combined with the van Laar excess free-energy model and the PRSV equation of state. The van Laar model parameters used are(/B = AI2/A2,= 0.1924/0.6719,andtheWongSandler mixing rule parameter kl2 = 0.5216. (Points are experimental data from the DECHEMA Chemistry Data Series, Gmehling and Onken Vol. 6 , p. 445; data filesfor this system on the accompanying disk are ClC5310.DAT, ClC5377.DAT and ClC5444.DAT.l in Figures 4.3.6 to 4.3.8, are observed for the acetone and water binary system. This illustrates the good accuracy of extrapolations of vapor-liquid phase behavior to high temperatures that can be obtained with the WS mixing rule and a simple EOS. To demonstrate the differences between the WS and the HVO models, the results of VLE predictions for the 2-propanol and water binary system at 353 K with the parameters obtained from the DECHEMA tables at 303 K are shown in Figure 4.3.9 in which the solid line is the prediction with the WS mixing rule and the dashed line describes the results of the HVO model. The significant advantage of the WS model over the HVO model in predictions is clearly visible in this figure. Experience has shown that the choice of activity coefficient model to be coupled with the EOS has some effect on the predictive performance of hoth the WS and HVO models. For example, as the complexity of the mixture increases, the NRTL and the UNIQUAC models, which have a temperature dependence of their parameters, usually give better results over a range of temperatures than the simple van Laar model. A desirable characteristic of an excess free-energy-based mixing rule is that it gee$ smoothly to the conventional van der Waals one-fluid mixing rule for some values of its parameters. This is useful because in multicomponent mixtures only some of the binary pairs may form highly nonideal mixtures requiring mixing rules such as
Mxing Rules that Combne an Equation of State with an Activity Coefficient Model
0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of carbon dioxide
Figure 4.3.3. VLE correlation of the carbon dioxide andpropane binary systemat 277,310, and344 K with the Wong-Sandler (WS) mixing rule combined with the van Laar excess free-energy model and the PRSV equation of state. The van Laar model parameters used are t/,5= A12/A21 = 0.7897/0.7928, and the Wong-Sandler mixing rule parameter klz = 0.3565. (Points are experimental data from the DECHEMA Chemistry Data Series, Cmehling and Onken, Vol. 6, p. 589; data files for this system on the accompanying disk are COZC3277.DAT. COZC3310.DAT and COZC3344.DAT.) those described in this section, whereas other binary pairs in the same mixture can be adequately described using the van der Waals mixing rule of eqns. (3.3.4 and 3.3.8). Orbey and Sandler (1995a) have proposed a slightly reformulated version of the WS mixing rule to accomplish this. They retained the mixing mle of eqns. (4.3.5 and 4.3.6) but replaced the combining rule of eqn. (4.3.2) by
This equation introduces the binary interaction parameter in amanner similar to that of eqn. (3.3.6) of the van der Waals one-fluid mixing mle. Next, the following modified form of the NRTL equation was used for the excess free-energy term:
with
G j i = hj exp(-rr,,)
Modeling Vapor-L~quid Equilibria
VLE data at 353 K
0.4
" "
" " " " " " "
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction 2-propanol
Figure 4.3.4. VLE correlation (solid lines) of the 2-propanol and water binary system at 353 K with the Wong-Sandler (WS) mixing rule combined with the NRTL excess free-energy model and the PRSV equation of state. The dashed lines are calI culated with ci = 0.2529, T , ~ / T ~=0.1562/2.7548, and with the Wong-Sandler mixing rule parameter k i z = 0.2529 obtained by fitting the experimental data. The solid lines represents results calculated with w =0.2893 and r l z / r z l =0.1509/1.8051 obtained from the DECHEMA Chemistry Series at 303 K (Gmehling and Onken 1977, Vol.1, Pt. I, p. 325) and with the Wong-Sandler mixing rule parameter k l z = 0.3659 obtained by matching the excess Gihhs free-energy from the equation of state and from the NRTL model at 303 K. Experimental data ( 0 .0) are from Wu, Hagewiessche, and Sandler 1988. where b j is the volume parameter in the EOS for species j . This modified NRTL form was suggested earlier by Huron and Vidal(1979) for use in their model. There are four dimensionless parameters in eqns. (4.3.7 to 4.3.9): u, r,,, rji, and k i j . This version of the WS mixing rule can be used as a four-parameter model to correlate the behavior of complex mixtures or in several ways that have fewer adjustable parameters. For example, one can solve the two relations obtained from eqn. (4.3.8) in the infinite dilution limit
for selected values of u and k,, where the yi;* is the infinite dilution activity coefficient of species i in j . Orbey and Sandler (1995a) found that setting a = 0.1 and klj = 0 worked quite well for several nonideal mixtures of organic chemicals, and thus one could use the model to predict the complete phrdse behavior over large ranges of
M ~ x n gRules that Combine an Equaton of State w~than Actv~tyCoeficient Model
mole fraction of 2-propanol Figure 4.3.5. VLE correlation (solid lines) of the 2propanol and water binary system at 523 K with the Wong-Sandler (WS) mixing rule combined with the NRTL excess free-energy model and the PRSV equation of state. The dashed lines represent results calculated with a = 0.2893, T I ~ / T ~=I - 0.430212.5280, and with the Wong-Sandier mixing rule parameter k12= 0.3 159 obtained by fitting the experimental data. The solid lines denote results calculated with a = 0.2893, rlz/rz, = 0.1019/1.2185 obtained from the DECHEMA Chemistry Series at 303 K (Gmehling and Onken 1977,Vol. I , Pt. 1,p. 325) with the WongSandler mixing rule parameter klz = 0.3659 obtained by matching the excess Gibbs free energy from the equation of state and from the NRTL model at 303 K. Points are the experimental data from Barr-David and Dodge 1959.
temperature and pressure from the two infinite dilution activity coefficients at a single temperature. This model could also be used in a completely predictive fashion, in the absence of any experimental data, with infinite dilution activity coefficients obtained from the UNIFAC model. Also, by setting rr = 0 and solving for r12and r 2 from ~
one recovers the van der Waals one-fluid mixing rule with a single binary interaction parameter kij that can he used for only slightly nonideal binary pairs in a multicomponent mixture. It should he noted that eqn. (4.3.1 1) is not unique, and that other expressions also lead to the van der Waals mixing rule. Further details of this approach are available elsewhere (Orbey and Sandler 1995a,c).
Modelng Vapor-Liquid Equi~bria
mole fraction of acetone
Figure 4.3.6. VLE correlation of the acetone and water binary system at 298 K with the Wong-Sandler -,
~
~-~
dashed lines represent results calculated with van Laar model parameters c / p = A12/A21 = 1.7724/2.0291 and the Wong-Sandler mixing rule parameter k l 2 = 0.2529 obtained by fitting the experimental data. The solid lines denote the results using model parameters t/,4= A12/A21 = 1.9399/1.8022 obtained at the same temperature from the DECHEMA Chemistry Data Series (Gmehling and Onken 1977, Vol. 1, Pt. 1, p. 238, which also lists the experimental data shown as points) and with the Wong-Sandler mixing rule parameter k12 = 0.2417 obtained by matching the excess Gibbs free-energy from the equation of state and from the van Laar model at 298 K.
Because the WS mixing rule uses VLE information only at low pressure, it can also be used to make predictions at high pressure based on low-pressure prediction techniques such as UNIFAC and other group contribution methods (Orhey, Sandler, and Wong 1993). This completely predictive method using the WS and other excess free-energy-based EOS models is discussed in Chapter 5. The WS mixing rule has also been applied to some asymmetric mixtures such as hydrogen and hydrocarbon binaries (Huang, Sandler, and Orbey 1994). However, in this case there is a potential problem, depending on the temperature, if the usual temperature dependence of the EOS a parameter is used. The problem arises because the temperature-dependent a term in equations of state is usually obtained by fitting vapor pressure and other subcritical data, as pointed out earlier, and is poorly defined at supercritical, high reduced temperatures (such as can be encountered with hydrogen ~. owing to its low critical temperature or with other asymmetric mixtures containing
Mixing Rules that Combine an Equation of State w ~ t han Adiviiy Coefirent Model
VLE dais at 373 K 1 .o
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of acetone Figure 4.3.7. VLE correlation of the acetone and water binary system at 373 K with the Wong-Sandler (WS) mixing rule combined with the van Laar excess free-energymodel and thePRSV equationof state. The dashed lines represent results calculated with van Laar model parameters ( / B = A 1 2 / A ~ i= 2.0287/1.6009 and the Wong-Sandler mixing rule parameter klz = 0.2779 obtained by fitting the experimental data. The solid lines denote the results using model parameters c / p = AlzlA2, = 1.9399/1.8022 obtained at the same temperature from the DECHEMA Chemistry Series (Gmehling and Onken 1977, Vol. I, Pt. I , p. 238) and with the Wong-Sandler mixing rule parameter k12= 0.2417 obtained by matching the excess Gihbs free-energy from the equation of state and from the van Laar model at 298 K. Points are the experimental data of Griswold and Wong 1952. very light components). The mixture b parameter in the WS model can be written as
and the denominator of this equation contains three terms. The excess free-energy term can be negative or positive and should vanish at high temperatures. For simplicity this term is neglected in the discussion here. In this case, unless the ( a , / b , R T )term is larger than unity for all the components of the mixture, there will he a composition of the (liquid or vapor) mixture at which the denominator becomes zero and then changes sign from negative to positive. The first of these possibilities is theoretically not allowed because of the discontinuity that results in the value of b of the mixture,
mole fraction of acetone Figure 4.3.8. VLE correlation of the acetone and water binary system at 523 K with the WongSandler (WS) mixing rule combined with the van Laar excess free-energy model and the PRSV equation of state. The dashed lines represent results calculated with the van Laar model parameters 5//3 = A l z / A 2 1 = 1.9520/1.3812 and the Wong-Sandler mixing rule parameter k12 = 0.2641 obtained by fitting the experimental data. The solid lines denote the results using model parameters ( / B = A , ~ / A ? I= 19399/1.8022 obtained at the same temperature from the DECHEMA Chemistry Series (Gmehling and Onken 1977, Vol. I , Pt. 1, p. 238) and with the Wong-Sandler mixing rule parameter k12~ 0 . 2 4 1 7obtained by matching the excess Cibbs free-energy fromthe equation of state and from the van Laar model at 298 K. Points are the experimental data of Criswold and Wong 1952. and the second case is not allowed because it leads to meaningless negative values for the b parameter. In order to avoid these possibilities it is necessary at all temperatures to have
for all the components of the mixture, which will ensure that the (ui/biRT)term will always b e larger than unity. For cubic equations of state this term can b e written in a generalized form; for example, for the Peng-Robinson equation of state we have
Mix~ngRules that Combine an Equation of State with an Activity Coefic~entModel
0.4 0.0 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of 2-propanol Figure 4.3.9. Comparison of VLE predictions of the ?-propano1 and water binary system at 353 K from thc
Wong-Sandler (solid lines) and Huron-Vidal original (dashed lines) models with both model parameters obtained by fitting the experimentaldata at 303 K. Points are the experimental data of Barr-David and Dodge 1959. Thus, the requirement of eqn. (4.3.13) becomes a ( T ) 1 Tr/5.87712 for the PengRobinson EOS. For most cubic equations of state at reduced temperatures below 2 the requirement of eqn. (4.3.13) is met, but, depending upon the temperature dependence of a(T), it may not be satisfied at higher reduced temperatures. A simple solution for mixtures with one or more highly supercritical components is to use the equation a ( T ) = T,/M for the highly supercritical component, with M being an EOSdependent constant, for example, 5.87712 for the Peng-Robinson EOS. This removes the singularity and does not cause any numerical problems in the calculation of phase behavior or thermodynamic properties; however, in this case, for highly asymmetric mixtures such as those containing hydrogen with heavy hydrocarbons, the VLE correlations and predictions may not he as accurate as those that have been discussed here.
4.4.
Approximate Methods of Combining Free-Energy Models and Equations of State: The MHVI, MHV2, LCVM, and HVOS Models
The problem with the HVO model resulting from the pressure dependence of the excess Gibbs free-energy of mixing led a number of investigators to propose EOS mixing rules based on the idea of combining activity coefficient models and equations of state at low (or zero) pressures, as originally suggested by Mollerup (1986). This approach requires solving for the liquid density at zero (or low) pressure from the
Modeling Vapor-Liqud Equilibria
volumetric EOS for each species in the mixture. Consequently, the complication that arises in this category of methods is that there may not be a liquid density solution for one or more of the pure components at the temperature of the mixture and at zero (or low) pressure (see Figure 4.1.1). Avoiding this complication requires some sort of approximate extrapolation technique. Several approximate models of this kind are reviewed below. The modified Huron-Vidal (MHV) mixing rule of Michelsen (1990h) is one of the most used of this class. The idea behind this mixing rule is to use eqn. (4.1.4) at P = 0 to obtain
where superscript 0 denotes the value of the volume at P = 0. Equation (4.4.1) looks more like the Michelsen form when rewritten as
where F
= a/bRT
and q { ~is)the function
These equations use one of the degrees of freedom in choosing the mixture EOS parameters. The second equation that is used to define the mixing rule completely is eqn. (3.3.8). However, note that, because it is no longer necessaty to be concerned with divergence of the excess Gibhs free energy at infinite pressure, it is not necessary to impose eqn. (3.3.8); other choices, such as the last equality of eqn. (4.3.1), could be used as well. At temperatures at which there is no liquid root of the EOS to use in the right-hand side of eqn. (4.4.3), Michelsen (1990b) chose an approximate linear extrapolation
that is used for all values of E , resulting in
This relation, combined with eqn. (3.3.8), is known as the MHVl model and will be referred to as suchin this monograph. It isinteresting to note that an alternatederivation = lJl/bi = of the MHVl model is to start with eqn. (4.4.1) and assume that l'$',,/b,,,
Mixng Rules that Combine an Equaton of State with an Ariiv~tyCoeficient Model
constant, so that C(pm,,) = C ( e ) = q l , which immediately gives eqn. (4.4.5). In this derivation it is seen that an assumption inherent in the MHVl model is that the ratio of the zero-pressure liquid molar volume to the close-packing parameter b is the same for the mixture and for each of its pure components. Later, for better accuracy, a quadratic extrapolation was proposed for q { & ]
and parameters were chosen to ensure continuity of the function q and its derivatives. The relation between the excess Gibbs free energy from an EOS and from an activity coefficient model takes the following quadratic form in this approach:
This equation, together with eqn. (3.3.8), is known as MHV2 model (Dahl and Michelsen 1990). The MHVl and MHV2 models are considered here for col~elation and prediction of the VLE of various mixtures, and computer programs that use these methods are provided in the accompanying disk. Further details are given later in this section. As indicated above, eqn. (3.3.8) is not the only option that can be used with eqn. (4.4.1). lndeed Tochigi et al. (1994) have proposed a MHV mixing rule consistent with the second virial coefficient boundary condition by combining eqns. (4.3.1 and 4.3.2 and 4.4.5). In their implementation they have eliminated the binary interaction parameter in eqn. (4.3.2), leading to the mixing rule
This mixing rule model has been tested for five binary and three ternary systems by the authors and shown to be approximately equivalent in accuracy to the MHVl model. Therefore, we do not consider it further. Boukouvalas et al. (1994) proposed a new mixing rule by forming the following linear combination of the HVO and Michelsen models (LCVM)
Modeling Vapor-Liquid Equilibria
This is a hybrid model and lacks a firm theoretical basis because it combines p,of the HVO model, which is evaluated at infinite pressure, and Q; of the Michelsen model, which is evaluated at zero pressure. Nevertheless the authors have shown that this model can be used to obtain successful correlation and prediction of the VLE of various nonideal systems. As we will discuss in Section 5.1, part of this success is due to a cancellation of errors. This model is also included in the computer programs on the disk accompanying this monograph and is tested below in this section for VLE correlations and predictions. The EOS-GeXmodels that are based on the zero pressure limit are mathematically approximate because of the lackof liquid density roots of the EOS at zero pressure and some temperatures. If mathematical rigor in establishing the tie between excess freeenergy models and EOS is to be sacrificed, successful approximate models can also be developed at the limit of infinite pressure. One such model was introduced by Orbey and Sandler (1995~)and is also tested in this monograph. In this model it is assumed that there is a universal linear algebraic function that relates the liquid molar volumes to their hard core volumes, such as V = ub, where u is a positive constant larger than unity. This assumption is similar to the concept of constant packing fraction introduced by Peneloux, Abdoul, and Rauzy (1989) and also to the assumption inherent in the MHVl model discussed earlier. Using the Helmholtz free-energy function for the reasons explained earlier, and assuming that at infinite pressure, both for mixtures and for the pure components, u approaches a unique value, one obtains
Further, since u approaches unity at both infinite pressure and at very low temperatures, for simplicity, as an approximation, one can take u = 1 at all conditions (so C{b} = C*) and obtain
In this model, eqns. (4.4.12 and 3.3.8) are used to obtain the EOS parameters a and h. This model is referred to as Huron-Vidal as modified by Orbey-Sandler (HVOS) model in this monograph and is also included in the programs supplied on the accompanying disk. It is an approximate model but is in agreement with the spirit of the van der Waals hard core concept, and it is algebraically very similar to several of the commonly used zero-pressure models mentioned in this section. Yet it does not
Mixing Rules that Combine an Equation of State with an Aaivity Coeficient Model
0.0
0.2
0.4
0.8
0.8
1.0
0.0
mole fraction of methane
n
180
180
180
160
400
m
80
?
so
n
0.8
0.8
1.0
140
120
g
OA
mole fraction of methane
l40
2
02
2
120
-
loo
0
g
40 20
80 60 40
20 0
0
00
02
0.1
0.6
0.8
mole fraction of methane
1.0
0.0
0.2
0.4
0.8
0.8
1.0
mole fraction of methane
Figure 4.4.1. VLE correlations of the methane and n-pentane binary system with various approximate EOS-GeXmodels. Clockwise from top left: HVOS, MHV2, MHV1, and LCVM mixing rules combined with the van Laar excess free-energy model and the PRSV equation of state. Solid lines are model predictions. The points aremeasured VLE dafa at 444 K ( A , A), 377 K (W, 0) and 310 K (e,0 ) from the DECHEMA Chemistry Data Series (Gmehling and Onken 1977, Vol. 6, p. 445).
contain any arbitraty constant or function because no hypothetical liquid volumes are needed. It is simpler than those zero-pressure models, and, as will be shown below, it is as successful. Some comparisons of VLE correlations with the approximate EOS-Gex models discussed in this section (MHVI, MHV2, LCVM and HVOS) follow. These results have been calculated using the computer program HV, which is described in Section D.4 of Appendix D and included on the disk accompanying this monograph. To make the comparison a fair one, the same activity coefficient model was used in all cases. For the methane and n-pentane binaty system shown in Figure 4.4.1, the lowesttemperature isotherm data were fitted with two van Laar parameters in each model, and the vapor-liquid equilibria at other temperatures were predicted. The parameters are reported in Table 4.4. I . All the models perform almost identically, albeit with different parameters, as seen in the table, and each provides an excellent description of the system. However, one must also keep in mind that, for this binary mixture, the 1PVDW model performs comparably with only one parameter, whereas all the excess free-energy-based models are multiparameter models.
Table 4.4.1. Van Laar model parameters (A12/A2,) of some binary mixtures for variou5 approximate EOS-Gex mixing rules EOS-GeXModel Binary Mixture
MHVl
0.4321-0.677 Methane + n-pentane (310 K) CO2 + propane (278 K) 0.61111.01 Acetone + water (373 K) 1.97911.245
MHV2
LCVM
HVOS
-0.1541-1.066
-0.2681-0.350
-0.4281-0.632
0.552/0.906 2.25311.300
0.72811.227 2,61411,524
0.77611.284 2.65211.577
mole fraction of carbon dioxide
mole fraction of carbon dioxide
mole fraction of carbon dioxide
mole fraction of carbon dioxide
Figure 4.4.2. VLE correlations of thecarbon dioxide and propane binary system with various approximate EOS-Ge' models. Clockwise from top left: HVOS, MHV2, MHV1, and LCVM mixing rules combined with the van Laar excess free-energy model and the PRSV equation of +), state. Solid lines are model predictions. The points are measured VLE data at 343 K (0, 310 K ( A , A) and277 K ( 0 ,0 ) from the DECHEMA Chemishy Data Series (Gmehling and Onken 1977, Vol. 6, p. 589).
Mxing Rules that Cornb~nean Equaton of State w ~ t han A c t ~ v ~Coefic~ent ty Model
The results for the carbon dioxide and propane binary system are shown in Figure 4.4.2. Again the lowest-isotherm data were used to obtain the model parameters, and the hehavior at other temperatures is predicted. In this case again, all the approximate EOS-GeX models are comparable, and there is a slight deterioration of the performance in all models along the highest-temperature isotherm. Similar results were ohtained with the HVO model (see Figure 4.2.2). Only the lPVDW (Figure 3.4.2) and the WS (Figure 4.3.3) models can correlate all isotherms with a single set of parameters to the desired high accuracy. However, the IPVDW model accomplishes this with only one parameter. The overall conclusion we reach from these results is that the real value of EOS-GeXmodels is in the correlation and prediction of the vapor-liquid phase hehavior of highly nonideal mixtures; for mixtures of hydrocarbons and hydrocarbons with inorganic gases, the results of the simpler lPVDW model are very good, and more complicated models are not needed. The performance of the approximate EOS-Ge" models for the more nonideal acetone and water system is shown in Figures 4.4.3 to 4.4.5. In these figures the results were ohtained in two ways. First, each isotherm has been separately correlated with each model (solid lines). The results indicate that all these models are very good for correlation, though each with a different set of parameters (see Table 4.4.1). Next, the VLE behavior was predicted with the parameters of the excess free-energy term (van Laar expression in this case) ohtained from the DECHEMA tables at 298 K (dashed lines). Now we see that the performance of the different models in predicting the VLE behavior with the parameters from the DECHEMA tables is significantly different. At 298 K, among the approximate models considered in this section, all except the MHVl model gave very good predictions (see Figure 4.4.3). The MHVl model, however, predicted an incorrect liquid-liquid split. At 373 K, (Figure 4.4.4), the best predictions are ohtained with the MHV2 model, followed by the MHVl model. The HVOS and LCVM models performed comparably, and they were inferior to MHV-type models because they underestimated the saturation pressure. Most importantly, if one compares these results with the predictions ohtained with the WS model (Figure 4.3.7). it can he seen that the WS model performs better than all of the approximate models considered here. For the 523 K isotherm of this system, shown in Figure 4.4.5, the predictions of the HVOS and the LCVM models, which are comparable, are superior to those ohtained using the MHVl and MHV2 models (the reverse of the previous case), which overpredict the saturation pressure. However, looking hack at Figure 4.3.8, we see that the WS model again leads to predictions for this isotherm that are superior to all of the approximate models considered here. Another nonideal binary mixture investigated here is the 2-propanol and water system, the results for which are presented in Figures 4.4.6 and 4.4.7. Again, two types of calculations were carried out. First, at each temperature the model param-
Modelng Vapor-L~qutd Equl~br~a
mole fraction of acetone
mole fraction of acetone
mole fractlon of acetone
mole fraction of acetone
Figure 4.4.3. VLE correlations of the acetone and water binary system at 298 K with various approximate EOS-GeXmodels. Clockwise from top left:
HVOS, MHV2, MHVI, andLCVM mixing rules combined with the van Laar excess free-energy model and the PRSV equation of state. The solid lines represent correlations with the van Laar parameters fit to experimental data, and the dashed lines show predictions with the van Laar parameters obtained from the DECHEMA tables at 298 K. The points are measured VLE data at 298 K from the DECHEMA Chemistry Data Series, Gmehling and Onken 1977, Val. I, Pt. 1, p. 238. eters were separately fitted to the experimental data (dashed lines). Second, predictions were made at the higher temperatures with the parameters of the excess free-energy model (NRTL in this case) obtained from the DECHEMA correlation at 303 K. For this system, all four models successfully correlated the data on the 353 K isothenn with parameters reported in Table 4.4.2. However, the predictive performance of the approximate models was different. Best results were obtained with the MHV2 model. The MHVl model overpredicted the saturation pressure, whereas the HVOS and LCVM models again behaved very similarly and underpredicted the pressure. None of these approximate models, however, was able to predict the phase behavior as accurately as the WS model (see Figure 4.3.5) at 353 K. At 523 K (Figure 4.4.7), the correlations were less accurate than those achieved at 353 K, and the predictions with all models were rather poor, especially when compared with the good predictions from the WS model (Figure 4.3.4) at this temperature.
Mix~ngRules that Combine an Equaton of State w~than Actvity Coeficient Model
Table 4.4.2. NRTL model parameters (in callmol) of ?-propano1 and water binary mixture for various approximate EOS-GeXmixing rules EOS-CeXModel
Temperature (K) MHVl
0.D
0.2
0.1
0.5
MHV2
0.8
1.0
LCVM
0.0
0.2
HVOS
0.1
0.8
0.8
mole traction of acetone
mole traction 01 acetone
mole fraction of acetone
mole fraction of acetone
1.0
Figure 4.4.4. VLE correlations of the acetone and water binary system at 373 K with various approximate EOS-GeXmodels. Clockwise from top left: HVOS, MHV2, MHV1, andLCVM mixingrules combined with the van Laar excess free-energy model andthe PRSV equation of state. Solid lines represent correlations with the van Laar parameters fit to experimental data, and the dashed lines show predictions with the van Laar parameters reported in the DECHEMA tables for 298 K. Points are VLE data at 373 K of Griswold and Wong 1952.
4.5.
General Comments on the Correlative and Predictive Capabilities of Various Mixing Rules with Cubic Equations of State In this and the preceding chapter we examined the correlative and semipredictive capabilities of various mixing rules. We use the term semipredictive in the sense that
Modeling Vapor-Liquid Equilibria
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
mole fraction of acetone
mole fraction of acetone
mole fraction of acetone
mole fractlon 01 acetone
Figure 4.4.5. VLE correlations of the acetone and water binary system at 523 K with various approximate EOS-GeXmodels. Clockwise from top left: HVOS, MHV2, MHV 1, and LCVM mixing rules combined with the van Laar excessfree-energymodelandthePRSVequationof state. Solidlinesrepresent correlations with the van Laar parameters fit to experimental data, and the dashed lines show predictions with the van Laar parameters reported in the DECHEMA tables for 298 K. Points are VLE data at 373 K of Griswold and Wong 1952.
a minimum amount of experimental binary VLE data, such as those used to obtain the parameters reported in the DECHEMA tables, are used in one way or another to obtain the model parameters. In Chapter 5, we investigate completely predictive procedures that are based on group contribution methods. However, before that, we present a summary of the correlative performance we have observed withmodels so far considered. In general, the behavior of a model in the correlation (and extrapolation with respect to temperature) of VLE information is an indicator of its predictive potential. The mixing rules considered are the IPVDW model, the 2PVDW model, two rigorous EOS-Ge" models (namely, the HVO and the WS mixing rules), and four approximate EOS-Gexmodels (the modified MHVl and MHV2 models based on the zero pressure limit, the HVOS model, and the hybrid LCVM model, which combines the MHV I model developed at zero pressure with the HVO model developed at infinite pressure). For moderately asymmetric mixtures of nonpolar components, such as the mixture of methane with n-pentane, the conventional van der Waals mixing rules with no interaction parameter provide a sufficient description, and the EOS-G" models
M~xngRules that
Comb~nean Equaton of State w~than Actv~tyCoefficent Model
mole fraction of 2-propano1
0.0
0.2
0.4
0.6
0.8
mole fraction of 2-propanol
mole fraction of Ppropanol
1.0
0.0
02
0.4
0.6
0.8
1.0
mole fractlon of Z-prapanol
Figure 4.4.6. VLE correlations of the 2-propanol and water binary system at 353 K with various approximate EOS-G" models. Clockwise from top left: HVOS, MHV2, MHVI, and LCVM mixing rules combined with thc van Laar excess free-energy model and the PRSV equation of state. Solid lines represent correlations with van Laar parameters fit to experimental data, the dashed lines represent predictions with the van Laar parameters obtained from the DECHEMA tables for data at 303 K. Points are data from Wu, Hagewiesche, and Sandler 1988.
introduce an unnecessary number of parameters and complexity. Furthermore, they do not necessarily lead to better results. However, as the asymmetry and difference in chemical functionality of the components of a mixture increase, and when gases such as carbon dioxide are involved, then the lPVDW and 2PVDW methods require binary interaction parameters and lose their predictive capability. In such cases, as we discuss in Chapter 5, the EOS-G"" methods are useful. If such mixtures do not contain highly supercritical components, the WS model is the most accurate of the EOS-GeX models for extrapolations in temperature and pressure. Some of the EOS-Ge"odels discussed have been combined with the UNIFAC group contribution methods for asymmetric systems, resulting in completely predictive VLE models for mixtures, and these will be discussed in Chapter 5. For highly nonideal and polar mixtures of organic compounds, the 1PVDW model is inadequate, and thevarious forms of the 2PVDW model havelimitations as acorrelalive method and suffer from computational problems such as the dilution effect and the Michelsen-Kistenmacher syndrome mentioned in Section 3.5. Such models should only be used with caution as semipredictive methods, and they have little utility as
Modelng Vapor-Liquid Equilibria
mole fractlon of Ppropanol
mole fraction of Ppropanol
70
b
P
b
60
P
g
@
80
"I "I
"I n
50
'y
a 10
50
/
0,
, ,'
10
0.0
0.1 0.2 0 1
0.4
0.5 0.6 0 7 0 8
mole fraction of Z-propanol
0.0 0.1 0.2 0.3 0.1 0.5 0.8 0.7 0.8
mole fraction of P-propanal
Figure 4.4.7. VLE correlations of the 2-propanol and water binary system at 523 K with various approximate EOS-G"" models. Clockwise from top left: HVOS, MHV2, MHVI, and LCVM mixing rules combined with the van Laar excess free-energy model and the PRSV equation of state. Solid lines represent correlations with the van Laar parameters fit to experimental data, and the dashed Lines are predictions using van Laar parameters obtained from the DECHEMA tables at 303 K. Points are the VLE data at 523 K from Barr-David and Dodge 1959.
completely predictive methods. On the other hand, the combined EOS-GeXmodels are of real value in these cases. Among such models analyzed here, only the HVO and the WS models are mathematically rigorous, and, as shown earlier, of the two, only the WS model has predictive capabilities. All of the approximate methods (MHVI, MHV2, HVOS, and LCVM) demonstrate good correlative capabilities, and some predictive capabilities, though they are generally less accurate than the WS method for extrapolation. This is especially obvious when extended ranges of temperature are considered. Although the quality of predictions of the WS mixing mleremains about the same over wide temperature ranges, predictions from the other more approximate methods can deteriorate considerably. Among the approximate models considered here, not one is - consistently superior to the others. Overall, the behavior of the MHVl and MHV2 models were similar, and the performance of the LCVM and HVOS methods were also comparable in most cases. All of the EOS-GeXmodels considered here can be used in a completely predictive mode if the excess free-energy term in the mixing rule is obtained from a predictive model like UNIFAC. This is considered in Chapter 5.
Completely Predictive EOS-GexModels
I
5.1.
N the last several years mixing rules that combine predictive excess free-energy methods, such as the UNIFAC model and equations of state, have been a subject of intense interest. Several mixing-combining rnles have been proposed (Dahl and Michelsen 1990; Holderbaum and Gmehling 1991; Orbey et al. 1993; Boukouvalas et al. 1994; Kalospiros et al. 1995; Voutsas et al. 1995) and used to predict the vaporliquid phase behavior of highly nonideal mixtures. A brief discussion of the nature of these models and their capabilities is presented in this chapter. More details on this subject can be found elsewhere (Orhey and Sandler 1995~).In this monograph we will use the original UNIFAC group contribution method. The principles involved, however, are general and are applicable to any group contribution method.
Completely Predictive EOS-GeXModels for Mixtures of Condensable Compounds Here we consider the prediction of the VLE behavior of mixtures, the constituents of which exist as pure liquids at the temperature andpressure at which the UNZFAC modelparameters are evaluated. The reason for this restriction is that the excess free energy in most models is defined with respect to the pure liquid state. However, it is also possible to treat mixtures of noncondensable gases with condensable compounds by means of such predictive models, and this is the subject of Section 5.3. The goal of predictive phase equilibrium models is to provide reliable and accurate predictions of the phase behavior of mixtures in the absence of experimental data. For low and moderate pressures. this has been accomplished to a considerable extent by using the group contribution activity coefficient methods, such as the UNIFAC or ASOG models, for the activity coefficient term in eqn. (2.3.8). The combination of such group contribution methods with equations of state is very attractive because it makes the EOS approach completely predictive and the group contribution method
Modelng Vapor-Liquid Equi~bria
for activity coefficients suitable for use at high pressures and temperatures at which the y - b method is not applicable. Selecting a proper EOS mixing rule to be used with a group contribution excess free-energy model is an interesting problem for the design engineer because in such a combination the characteristics of the two types of models are superimposed, and the overall accuracy of the combined method is a result of that choice. If the temperature and pressure are near ambient and the constituents of a mixture exist as pure liquid over the whole range of temperature and pressure, it may be most convenient to use the group contribution activity coefficient method directly in the conventional way (that is, in a y -@ model), for there would be little advantage to coupling it with an EOS. (Of course if one needs to compute other properties such as excess volumes in addition to phase equilibrium properties from the same model simultaneously, then an EOS should be used.) The real potential of the predictive EOS-G" methods is realized when the temperature and pressure of application are considerably higher than those of the ambient, thus rendering one or more of the constituents of the mixture supercritical. In such circumstances, application of the conventional activity coefficient approach becomes difficult owing to lackof apurecomponent vaporpressurc, or thereliability of extrapolating the group contribution method parameters is questionable. Most group contribution methods, such as the original UNIFAC model, were designed to beused at near room temperature, and their extrapolation capability over a range of temperature is more reliable when they are used in an EOS-G"" mixing rule than in a y - 4 model, as we will show in the following paragraphs. In the remainder of this section we examine several EOS-GeXmodels using three prototype binary mixtures that form strongly nonideal solutions. For comparison, we also include the predictions of the UNIFAC model used directly in the y-@ method wherever applicable. The systems considered are the methanol and benzene (Butcher and Medani 1968), the acetone and water (Gmehling and Onken 1977), and the 2-propanol and water (Barn-David and Dodge 1959) binary mixtures. Note rhat there are many systems with small to moderate solution nonideality for which all or most of the methods mentioned above work reasonably well. We are not concerned with such systems here because the method selection would not be a problem in such cases. Rather we consider only those systems that are more nonideal and for which the differences between the models discussed here are clearly evident. We use the following mixing rules combined with the original UNIFAC excess freeenergy model: the rigorous HVO and WS models and four approximate models (the HVOS, MHVI, MHV2, and LCVM mixing rules). The programs used for the VLE predictions, named WSUNF and HVUNF, are provided on the diskette accompanying this monograph, and some of the computational details are described in Sections D.6 and D.7 of Appendix D, respectively. Other computational details may be found in Appendixes B and C. Before proceeding, it is necessary to stress once again a characteristic difference between the WS model and the other five models used here. All the EOS-Gexmodels.
Completely Prediaive EOS-Gex Models
mole fraction of methanol
Figure 5.1.1. VLE prediction forthe methanol and henzene binary system at 293 K by various methods. Circles represent experimental data, the solid line with crosses shows the UNIFAC predictions, and the smooth solid line denotes the results of the WS model. The large, medium, and short dashed lines are from the HVOS, HVO, and MHVl models, respectively; the dotted line is from the MHV2 model; and the dot-dash line reflects the results of the LCVM model. (Points are VLE data from the DECHEMA Chemistry Series, Gmehling and Onken 1977, Vol. I, Pt. 2a, p. 220; the data file name on the accompanying disk for this system is MB20.DAT.)
including the WS mixing rule, require as input the excess free-energy model (that is, the G,! term and the two activity coefficients calculated from it). (See Appendixes B and C for a summary of the relations.) Only the WS model also requires a value for the additional parameter k i j ;however, this parameter is chosen so that the G,!cuwe from the UNIFAC model is matched as closely as possible by Kcoscalculated from the EOS at the low pressure (and temperature) at which the UNIFAC parameters were obtained (see Section 4.3). Thus, the parameter kij does not contain any further information than that already included in the Gy term from the UNIFAC model; indeed, it merely provides the additional flexibility to better match theexcess free-energy curve from the UNIFAC model with that from EOS(WS)-Gex(UNIFAC) model. As we discuss later, this additional flexibility, unique to the WS model, is useful in some circumstances. The first system we consider is the methanol and benzene binary system. The results of predictions at 293 K are shown in Figure 5.1.1. (In the case of the WS mixing rule, kii was set equal to 0.2808 using the procedure described above.) Although none of the models is highly accurate, there are considerable differences among.the performance of the various methods. The original UNIFAC model used in the y - 4 approach with an ideal vapor phase predicts the VLE behavior reasonably; however, it is not the best ~
Modeling Vapor-Liqud Equlibria
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of methanol Figure 5.1.2. Excess Gibbs free-energy predictions for the methanol and benzene binary system at 293 K.
The circles represent the excess Gibbs energy calculated using experimental data (see text), the line - . . - reflects the UNIFAC predictions. and the smooth solid line denotes the results of the WS model. The large, medium, and short dashed lines are from the HVOS, HVO, and MHV1 models, respectively; the dotted line is from the MHV2 model: and the dot-dash line represents the results of the LCVM model. model. The WS model that uses the UNIFAC model in the EOS-Gexis more accurate than the original UNTFAC model for this mixture. All the other EOS-GeXmodels show deviations from the experimental behavior to different extents, and the MHVl model even shows a false liquid-liquid split. Some insight into the hehavior of various EOS-Gexmodels can be obtained if one examines the excess free-energy curves derived from these models and compares them with the excess Gibbs free energy computed from the measured data (we used the measured P - x - y data and the ideal gas assumption to obtain activity coefficients and hence the "experimental" excess Gibbs free-energy information through eqns. 2.3.6, 8, and 10). The results are shown in Figure 5.1.2. In the figure the points represent the excess Gibbs free energy computed from experimental data, and the lines are the model predictions. None of the models can represent the experimental excess Gihhs free-energy behavior exactly, and consequently their VLE predictions deviate from the experimental behavior. Of course there is a direct correlation between the capability of a model in representing the experimental excess Gihbs free energy and its prediction of VLE hehavior. Models that overpredict the excess free energy also overpredict the equilibrium total pressure and conversely for models that underpredict the Gex.
Completely Predictive EOS-GeXModels
.
VLE data at 453 K
mole fraction of methanol Figure 5.1.3. VLE prediction for the methanol and benzene binary system at 453 K by various methods. The circles denote experimental data, the solid line with crosses represents the UNIFAC predictions, and the smooth solid line reflects the results of the WS model. The large, medium, and short dashed lines are from the HVOS, HVO, and MHVl models, respectively; thedottedline is from the MHVZmodel; andthe dot-dash line denotes the results of the LCVM model. (Points ( 0 ,e)are VLE datafromButcher and Medani 1968; the data file name on the accompanying disk for this system is MB I80.DAT.) In Figure 5.1.3 the prediction of the VLE behavior of the various models for the methanol and benzene binary system at 453 K is shown. The direct use of the UNIFAC activity coefficient model in the y - @model qualitatively behaves difrerently than the other models and performs relatively poorly. The various EOS-GeX models show similar behavior that is in qualitative agreement with the experimental data; however, quantitatively the WS model is the most accurate. The results for the acetone and water binary system at 298 K are shown in Figure 5.1.4. Here the direct use of the UNIFAC activity coefficient model in the y-q? method gives the best results. The WS mixing rule with k j j = 0.2458 obtained by matching its excess free-energy curve to that of the conventional UNIFAC model gives the best results among the EOS-Gex models. The MHVl model yields the largest positive deviations, and the HVO model gives the greatest negative deviations from the experimental data. Interestingly, the linear combination of these two models, the LCVM model, performs better than both owing to a cancellation of errors. The LCVM, MHV2, and HVOS models all perform comparably and with slightly less accuracy than the WS model. For this mixture the comparison of the excess free-energy predictions from various models at 298 K with the excess free-energy
Modelng Vapor-Liquid Equilibra
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of acetone
Figure 5.1.4. VLEpredictions forthe acetone and water binary system at 298 K by various methods. Solid line with crosses represents the UNIFAC predictions, and the smooth solid line reflects the results of the WS model. The large, medium, and short dashed lines are from the HVOS, HVO, and MHVl models, respectively; the dottedlineis from the MHV2 model; and the dot-dash line denotes the results of the LCVM - - model. - - - ~ ~ (The points ( 0 , e) are VLE data for this system at 298 K from Griswold and Wong 1952; the data file for this system on the accompanying disk is AW25.DAT.) ~
~ -
values calculated from measured VLE data is shown in Figure 5.1.5. Among the EOS-G" models, the WS model agrees most closely with the UNIFAC curve, as a result of the matching procedure described earlier. (That there is scatter in the experimental values in the dilute acetone region indicates some inaccuracy in the measurements.) A comparison of the VLE predictions for the acetone and water system at 473 K using the 298 K parameters is revealing (Figure 5.1.6) The most accurate results are obtained with the WS mixing mle. The extrapolations using the HVOS and LCVM models are reasonably good, whereas the MHV2 model, which produced results similar to those of the HVOS and LCVM models at 298 K, now shows the largest overprediction of the saturation pressure. The MHVl model also overpredicts the saturation pressure, whereas the HVO model fails in representing the azeotrope and by underpredicting saturation pressure over the whole concentration range. As before, the LCVM model is a combination of the worst two models for extrapolating VLE information, but because of a cancellation of errors is better than both. In this case the direct use of the UNIFAC activity coefficient model gave poor results, indicating that the extrapolation of VLE predictions in the y-q5 formalism is less reliable than using the UNIFAC model in the EOS-Ge" formalism.
mole fraction of acetone Figure 5.1.5. Excess Gibbs free-energy predictions for the acetone and water h i n w system at 298 K. The circles are calculated from experimental data (see text), the solid line with crosses reflects the UNIFAC predictions, and the smooth solid line denote3 the results of the WS model. The large. medium, and short dashed lines are from the HVOS. HVO, and MHVl models, respectively; the dotted line is from the MHV2 model; and the dot-dash line represents the results of the LCVM model.
15
-
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of acetone Figure 5.1.6. VLE prediction for the acetone and water binary system at 473 K by various methods. The solid line with crosses represents the UNIFAC predictions, and the smooth solid line reflects the results of the WS model. The large, medium, and shon dashed lines are from the HVOS, HVO, and MHVl models, respectively; the dotted line is from the MHV2 model; and the dot-dash line denotes the results of the LCVM model. (The points ( 0 ,0 )are the experimental data of Griswold and Wong 1952; the data file for this system on the accompanying disk is AW200.DAT.)
Modelng Vapor-Liqu~d Equilibria
0.5
,
,
,
I
,
,
,
,.~.... ..
,
,
,
,
,
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of 2-propanol Figure 5.1.7. Excess Gibbs free-energy predictions for the 2-propanol and water hinary system at 298 K. The circles represent the energy calculated using experimental data (see text), the solid line with crosses reflects the UNIFAC predictions, and the smooth solid line denotes the results of the WS model. The large, medium, and short dashed lines are from the HVOS, HVO, and MHV1 models, rehpectively; the dotted line is from the MHV2 model; and the dot-dash line represents the results of the LCVM model.
The last example we consider is the 2-propanol and water system, which was selected to show how inaccuracies in the activity coefficient model used affect the performance of the EOS-GeXmodel. In Figure 5.1.7, the excess free-energy predictions for this system are shown. By setting krj of the WS model to 0.3814, we can represent the UNIFAC behavior very closely, though not exactly. All of the other EOS-Gcx models deviate from the UNIFAC behavior to varying extents. especially in the midconcentration range. Note also that the excess Gibbs free-energy behavior derived from the experimental data can not be represented by any one of the models over the whole concentration range. Examining the VLE behavior of this binary system at 298 K shown in Figure 5.1.8, we see that not one of the models is satisfactory. The direct use of the UNIFAC model does not provide an accurate description of this system and even gives a false liquid-liquid split in the water-rich region. An EOS-GeX model such as the WS model that can reproduce the UNIFAC excess free-energy behavior closely also inherits its problems, and therefore also shows an erroneous phase split. This poor low-temperature description also causes a problem when EOS-GeX models are used to predict phase behavior at elevated temperatures and pressures. In Figure 5.1.9, the VLE predictions for the 2-propanol and water binary system at 523 K are shown. The MHV2 model gives the largest overprediction of PI-essnre,
Completely Preddive EOS-GexModels
0 . 0
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.Q 1.0
mole fraction of 2-propanol
Figure 5.1.8. VLE prediction for the 2-propanol and water binary system at 298 K. Thesolidline withcrosses represents the UNIFAC predictions, and the solid line reflectsthe results of the WS model. The large, medium, and short dashed lines are from the HVOS, HVO, and MHVl models, respectively; the dotted line is from the MHV2 model; and the dot-dash line denotes the results of the LCVM model. (The points ( 0 ,e) are measured data reportedin the DECHEMA Chemistry Data Series, Gmehling and Onken 1977, Val. 1, Pt. lh, p. 220; the data file for this system on the accompanying disk is 2PW25.DAT.)
followed by the MHVl and WS models, whereas the HVO model underpredicts the pressure. The best results are obtained with the HVOS and LCVM models, which behave almost identically. However, because these two models underpredicted the experimental saturation pressures at 298 K, the agreement here must be viewed as fortuitous. The 2-propanol and water system is considered again now using the UNIQUAC model. which is the correlative model closest to the UNIFAC model, to examine the effect of activity coefficient model choice. Because the UNIQUAC model is correlative, it is possible to fit the parameters of each EOS-GeXmodel to VLE data at 298 K. The fitted parameters are given in Table 5.1.1. The VLE correlations at 298 K and the predictions at 523 K are shown in Figures. 5.1.10 and 5.1.1 I , respectively. In this case all of the models are able to provide a very accurate correlation of the low-pressure data, as shown in Figure 5.1.10. However, when the same parameters are used to predict VLE behavior of this system at 523 K, the performance of various models differs, as shown in Figure 5.1.11. The WS model once again gives the best prediction, followed by the HVOS and LCVM models, both of which somewhat underpredict the saturation pressure. The HVO model underpredicts the pressure significantly, and
Modelng Vapor-Lquid Equ~lbr~a
5.1.1. UNIQUAC model parameters (callmol) for the 2propanol and water system" at 298 K for various mixing rules Table
HVO
MHVl
MHV2
LCVM
"UNIQUAC pure component parameters are r 2-propanol; r = 0.92, q = y' = 1.4 for water. k,, = O 15
3
I
wSb
HVOS
= 2.78. q = q' = 2.51
VLE dataat 523 K
I
0 1 " ' ~ ' ' ~1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
'
for
~
mole fraction of 2-propanol
Figure 5.1.9. VLE prediction for the 2-propanol and water binary system at 521 K. The solid line reflects the results of the WS model. The large, medium, and short dashed lines are from the HVOS, HVO, and MHVl models, respectively; the dotted line is from the MHV2 model; and the dot-dash line denotes the results of the LCVM model. (The points ( 0 ,*)are the measured VLE data of Barr-David and Dodge 1959; the data file for this system on the accompanying disk is 2PW25O.DAT.)
both the M H V l and M H V 2 models overpredict the pressure, the MHV2 model being more seriously in error. From the analysis of various mixing rules for the prediction of mixtures of condensable compounds, w e reach the following conclusions: 1. At low pressures, for which the UNIFAC method was developed, theHelmholtz free energy andGibbs free energy calculated from anEOS arevirtually indistinguishable
Completely Predict~veEOS-GeXModels
.
VLE dataat 298 K
0.03
-
.
-
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 mole fraction of 2-propanol
Figure 5.1.10. VLE correlations forthe 2-propdnol and water binary system at 298 K using UNIQUAC method with the EOS-G" models. The solid line reflects the results of the WS model. The large, medium, and short dashed lines which almost coincide are from the HVOS, HVO, and MHV 1models, respectively; the dotted line is from the MHV2 model; and the dot-dashlinedenotes the results of the LCVM model. (The points ( 0 ,0 ) are the measured VLE data reported in the DECHEMA Chemistry Data Series, Gmehling and Onken 1977, Vol. 1, Pt. lb, p. 220.)
and can be used interchangeably. With this in mind, there are two conditions that must be satisfied for successful VLE predictions by such models: a. The excess free energy function obtained from the direct use of the UNIFAC model should be in agreement with the experimental excess Gibbs free energy at low pressure. b. The excess free energy computed from the EOS-G" (UNIFAC) model at low pressure should match the excess free energy obtained from the direct use of the UNIFAC model in the y - @method as closely as possible. If either one of these conditions is not satisfied, the combined EOS-Gex model may fail to represent the low-pressure equilibria or lead to inaccurate extrapolations at high pressures and temperatures with the parameters obtained at low pressures, or both. 2. Among the mixing rules tested here in the predictive EOS-GeXformalism, only the WS model can bemade to match the excess free energy from aconventional activity coefficient model closely by varying the model parameter k i j . This flexibility can also be used to incorporate infinite dilution activity coefficient information into this model, as discussed in the next section.
Modeling Vapor-Liquid Equllibra
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of 2-propanol Figure 5.1.11. VLE predictions for the 2-propanol and water binary system at 523 Kusing the UNIQUAC method with the EOS-Gexmodels, and parameters ohPained at 298 K. The solid line reflects the results of the WS model. The large, medium, and short dashed lines are from the HVOS, HVO, and MHVl models, respectively; the dotted line is from the MHV2 model; and the dot-dash line denotes the results of the LCVM model. (The points ( 0 ,). are measured VLE data from Bar-David and Dodge 1959.)
5.2.
Prediction of Infinite Dilution Activity Coefficients with t h e EOS-GeXApproach An extension of completely predictive EOS-Gexmodels for modeling mixtures of condensable compounds is to use them to obtain infinite dilution activity coefficients. The capability of EOS models in predicting infinite dilution activity coefficients is important for describing dilute solution behavior. Among the EOS mixing rules discussed in this monograph, the conventional lPVDW model is not suitable for the correlation of the infinite dilution activity coefficient behavior because it is a one-parameter mode!, whereas there are two infinite dilution activity coefficients per pair of components to estimate. Consequently, this model can be made to fit one, but not both components. The two-parameter 2PVDW model can be made to fit both infinite dilution activity coefficients in a binary mixture but is then not capable of also replicating the excess free energy over the whole composition range (Pividal et al. 1992). The use of infinite dilution activity coefficient information in obtaining parameters in the 2PVDW mode! was presented earlier in Section 3.5.
Completely Predidive EOS-GexModels
There are several recent studies analyzing theutility ofEOS-G" models (Kalospiros et al. 1994; Orbey and Sandler 1996a) for the prediction of infinite dilution activity coefficients. We present a brief analysis of these studies in the following paragraphs. Among the EOS-G" models considered in this monograph the WS mixing rule can exactly reproduce the infinite dilution activity coefficient of one species in the other if one selects an appropriate value for its k,, parameter, hut not both infinite dilution activity coefficients. All of the other EOS-G" models can give only approximate estimates because they have no additional degrees of freedom once the predictive G y model (here UNIFAC) is specified. As an example, we consider the predictions of the infinite dilution activity coefficients of some heavy alkanes in n-hexane. The results are presented in Figure 5.2.1. The dots are data taken from Kniaz (1991), and the lines are predictions using various EOS-Gex models. As has been explained above, with the WS model a value of kij can be selected for each binary to represent one of the infinite dilution activity coefficients exactly, and thus the results for the WS model are not shown because they are coincident with the experimental data. All other EOS-Gex models show various degrees of deviation from the measured infinite dilution data. Of the other models, the best predictions are obtained with the HVOS model (long dashed line). Only the MHVl model (short dashed line) generates positive deviations (activity coeflicients larger than unity); all the other models, including the use of the UNIFAC model in the y - $ formalism (solid line), give negative deviations from ideal behavior. The HVOS model slightly overpredicts the activity coefficients; the remaining
alkane number
Figure 5.2.1. Infinite dilution activity coefficients of n-alkanes in hexane. The dots denote experimental data (Kniaz 1991); the solid line represents the UNIFAC predictions; the large, medium, and short dashed lines are from HVOS, HVO, and MHVl models, respectively; the dotted line is from MHV2 model; and the dot-dash line reflects results from the LCVM model.
Modeling Vapor-Liquid Equilibria
EOS-G'" models underpredict the experimental activity coefficients, and the largest error comes from the MHV2 model (dotted line). The LCVM (dot-and-dash line) model, which is alinearcombination of the HVO (medium dashed line) and theMHVl models, gives better results than both again as a result of the cancellation of errors.
5.3.
Completely Predictive EOS-GeX Models for Mixtures of Condensable Compounds with Supercritical Gases In many industrially important processes, mixtures of supercritical gases with pnre or mixed solvents are encountered. These are highly asymmetric systems that are usually difficult to measure with high accuracy at the desired design conditions, and which are generally above (or below) room temperature (See, for example, the compilation of Fogg and Gerard, 1991 for details of the experimental accuracy in gas solubility measurements in liquids, etc.). Therefore, it is important to have reliable predictive models for the vapor-liquid phase equilibrium and other properties such as the Henry's constant, activity coefficient, and so forth, of gases in liquids for such systems. There are many empirical methods for the prediction of the dilute solution phase behavior for such systems (Catte et al. 1993); however, the best candidates for this type of modeling are equations of state. An EOS can simultaneously provide the phase behavior, activity coefficients, the Henry's constants, and volumetric and calorimetric properties. The prediction of the phase behavior of mixtures of solvents with supercritical gases using an EOS is an area of ongoing research, and there are several emerging methods, such as those recently proposed by Dahl, Fredenslund, and Rasmnssen (1991), Apostolon, Kalospiros, and Tassios (1 9 9 3 , and Fischer and Gmehling (1995) that are all based on the EOS-Ge" combination and use a predictive excess free-energy model such as UNIFAC for the G y term. Here we use two such EOS-Ce" models to demonstrate the capabilities of these models and to address several issues for future development in this area. As mentioned earlier, because the present excess free-enerev -. models used in the EOS-Gexformalism were developed for mixtures of pure liquids, in principle the EOSG" approach is applicable to mixtures in which all constituents exist as pnre liquids at the temperature and pressure at which temperature-independent model parameters of the excess free-energy model (here UNIFAC) have been evaluated. Using such models to describe the mixture of a supercritical gas dissolved in a liquid is an approximation that has certain consequences, as we discuss later in this section. The EOS-G" models used so far to describe gas solubility are the approximate mixing rules such as the MHV2, HVOS, or LCVMmodels, and we will restrict our discussions to these models. When the UNIFAC group contribution method is used for the prediction of phase equilibrium properties of a mixture, two types of input parameters are used in the prediction. One set is made up of the volume ( R i ) and shape or surface area ( Q , )
Completely Pred~dveEOS-GeXModels
parameters of the groups that compose the pure constituents of the mixture, and the other set consists of two group interaction parameters per each pair of groups in the mixture. Thus, the first problem to be addressed is to choose size and shape parameters for the groups that form the gases involved. Some of the gases treated so far by the methods discussed here are light industrial gases, C 0 2 ,CO, H2S,N2 and the like, and light paraffins from methane to hutane(s). All of the present models treat these gases as individual groups, for each is relatively small, and define the volume and surface area parameters for each gas. In some cases these structural parameters are estimated using semitheoretical methods like those of Bondi (1968) and Apostolou et al. (1995), hut their values are essentially arbitrary; for example Dahl et al. (1991) use Ri = 1.7640 and Qi = 1.9100foroxygen, whereas Apostolouet al. (1995) report 0.8570 and0.940 for these parameters; a similar trend is observed for other gases. Consequently, the gas parameters in Dahl's correlation are about twice the size of the gases of Apostolou et al. There has not yet been a thorough analysis of which values are the best. In addition, these values must be set before the optimum values of the group interaction parameters of the UNIFAC model can be found; thus, any deficiency that exists in the selection of these structural parameters will affect the group interaction parameters. With the treatment of gases as individual groups, some binary (or multicomponent) gas-liquid mixlures are reduced to mixtures of only two groups. For example, the carbon dioxide and methanol mixture considered at the conclusion of this section is actually a molecular mixture because hoth molecules are treated as groups by the UNIFAC approach. Similarly, mixtures of carbon dioxide with benzene or with paraffinic hydrocarbon liquids contain only two groups. The results for such systems are remarkably successful, as will he discussed in this section. The description of mixtures with more than two groups is possible for some of the present models, and the results look promising (Apostolou et al. 1995). Here we consider only two of the approximate models discussed in this monograph coupled with the PRSV EOS to estimate the VLE of gas-solvent mixtures. These are the HVOS (Orbey and Sandler 1995c) and LCVM models (Boukouvalas et al. 1994), each with parameters for gases reported by Apostolou et al. (1995) in hoth cases. To test the capabilities of these models, we used the original UNIFAC model with two temperature-independent parameters per pair of groups (instead of four in the modified UNIFAC models) for simplicity. We optimized the binary group interaction parameters of the UNIFAC model in the EOS formalism at a selected temperature for various mixtures and then estimated the vapor-liquid phase behavior at other temperatures. We first investigated the behavior of mixtures of the normal paraffinic solvents pentane, heptane, and decane with gaseous methane. These mixtures consist of two main UNTFAC groups, methane and the main methyl group CH2; thus, there are only two binary interaction parameters to evaluate. We used the VLE data for the 377 K isotherm of the methane and n-heptane mixture to obtain these parameters for hoth the HVOS and LCVM models; the parameter values are reported in Table 5.3.1. We then estimated the VLE at all other temperatures of the three mixtures. The results
Modeng Vapor-Lqu~d Equ~libra
Table 5.3.1. UNIFAC interaction parameters ( Z I I ~ / U Z I ) for various groups used in VLE calculations with the LCVM and HVOS EOS-Gex models Group Pair
+
CH4 CH2 C02 + CHIOH C02 + C H =
HVOS Model
LCVM Model
-100.811109.09 251.391100.01 140.961-11.16
-32.28197.17 209.64194.94 30.62198.62
mole fraction of n-heptane
Figure 5.3.1. VLE predictions for the methane and heptane binary system using the HVOS (solid lines) and LCVM (dashed lines) models. See text for details. (Measured data ( 0 . and n, m) reported in the DECHEMA Chemistry Data Series, Knapp et al. 1982, pp. 471472. The data files for this system on the accompanying disk are ClC7377.DAT and ClC7477.DAT.)
are presented in Figures 5.3.1 to 5.3.3. In these figures the solid lines result from the HVOS model, and the dashed lines are from the LCVM model. The performance of the two models is essentially comparable, and both are very good considering the change in size of the paraffinic hydrocarbon and the ranges of temperature and pressure involved. Undoubtedly even better results would be obtained if all the data were used in obtaining the binary parameters instead of only the data for one binary mixture at one temperature, as we have done here. Another mixture containing only two UNIFAC groups is the carbon dioxide and benzene binary system. Forthis system we fitted the binary group parameters of carbon dioxide with the a r ~ m a t i c-CH= group at 313 K and predicted the behavior at 393 and 273 K. These results are presented in Figure 5.3.4. The results are vely good for
Completely Prediitive EOS-GeXModels
mole fraction of n-pentane
F i y r e 5.3.2. VLE predictions for the methane and n-pentane binary system using the HVOS (solid lines) and LCVM (dashed Lines) models. See text for details. (The points ( 0 , for 310 K, and 0, rn for 410 K) are from the DECHEMA Chemistry Data Series. Knapp et al. 1982, p. 445. The data files for this system on the accompanying disk are CIC5310.DAT and ClC54 10.DAT)
VLE data at 377 K VLE data at 542 K
300
"
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 . 0
mole fraction of n-decane
Figure 5.3.3. VLE predictions for the methane and n-decane binary system using the HVOS (solid lines) and LCVM (dashed lines) models. See text for details. (The points ( 0 , for 377 K, and 0, rn at 542 K) are measured VLE data reported in DECHEMA Chemistry Data Series Knapp et al. 1982, p. 486 and 489. The data files for this system on the accompanying disk are ClC10377.DAT and ClC10542.DAT.)
Modelng Vapor-L~quid Equilibria
16
-
. A
l4
12
VLE at 393 K VLEat313K VLEat 273K
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 . 0
mole fraction of carbon dioxide Figure 5.3.4. VLE predictions for the carbon dioxide and benzene binary system using the HVOS (solid lines) and LCVM (dashed lines) models. See text for details. (Data are from Gupta et al. 1982and Kaminishi et al. 1987; data file names on the accompanying disk for this system are COZBZ273,DAT,C02BZ3I3.DAT and C02BZ393.DAT.) the 273 and 313 Kisothenns and, in spite of some overprediction of pressure, are still acceptable at 393 K. The last binary system we considered was the carbon dioxide and methanol binary mixture, the results for which are presented inFigure 5.3.5. In this case the model could be considered a molecular one because both carbon dioxide and methanol are treated as groups; nevertheless, the mixture represents a stringent test in as much as most EOS models predict false liquid-liquid splits at lower temperatures for this mixture even when correlation rather than prediction is used (Schwartzentruber et al. 1986; Orbey and Sandler 1995b). In this case we fitted the group parameters for the carbon dioxide and methapol system to data at 394 K and then predicted the behavior at 273 and 477 K. The results at 394 and 273 K are reasonably good except in the vicinity of the critical point for the mixture at 394 K, and they deteriorate noticeably around the critical point at 477 K. One noteworthy finding, however, is that the typical false liquid-liquid split that has been observed with several EOS-mixing rule combinations at the 273 K isotherm is not found with the models considered here. Moreover, some of the previous EOS models were very sensitive to the parameter values (that is, with some sets of parameters they gave false phase splits, whereas with other sets they did not). The models used here were much less sensitive because, even with group parameters other than those that give the most accurate VLE predictions, false liquid-liquid splits were still avoided. Overall, we consider the approximate group contribution-EOS-based models considered here to be promising for the prediction of gas-liquid phase equilibrium. Future 92
Completely Predictive EOS-Gex Models
VLE at 477 K
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
mole fraction of carbon dioxide
Figure 5.3.5. VLE predictions for the carbon dioxide and methanol biniuy system using the HVOS (solid lines) and LCVM (dashed lines) models. See text for details. (Data are from Hong and Kohayashi 1988 and Kaminishi, Yokoyama and Takahashi 1987. Data file names on the accompanying disk for this system are CO2ME273.DAT, C02ME394.DAT. and CO2ME477.DAT.)
efforts should be concentrated on selecting the best alternative among these models l as well as on developing and obtaining the best pure component s t ~ c t u r aparameters, a more inclusive group interaction parameter base. It is also important to determine whether the simultaneous representation of both dilute solution (Henry's coefficients) and bulk solution (VLE at finite compositionranges) behavior is possible with a single set of parameters.
Epilogue
T
HE main theme in this monograph has been the application of various recent mixing rules for cubic equations of state to the calculation of VLE in mixtures. In principle, these models are not restricted to VLE and are applicable to all other phase equilibrium problems. Vapor-liquid phase equilibrium has been considered hcrc because only VLE has thus far been studied in detail with these mixing rules. Indeed, the significant improvement obtained with these models in such a mature area as VLE correlation is an encouraging indicator for their extension to other types of phase equilibrium. We have tried to present the models currently used in systematic, critical, but unprejudiced manner. Consequently, we have not declared any one of these models to be better than all others for every mixture. Indeed, several EOS-GCxmodels perform quite well in some circumstances. Further, it can be difficult to determine in those cases in which the EOS-Gexdescription is not accurate whether the problem is with the mixing rule, the EOS, or the excess free-energy model used. More importantly, we hope that by providing programs for all the models presented, the reader will be encouraged to examine some or all of the models to determine which is best for the system he or she is studying. In closing, we would like to present our thoughts about future areas for study and possible application for these EOS-GeXmixing rules. However, before proceeding, it might be worthwhile to comment on the argument that cubic equations of state are too simple to be useful for phase equilibrium calculations more complex than basic vapor-liquid equilibria. Van Konynenburg and Scott (1980) have shown in a comprehensive fashion that the van der Waals cubic EOS, combined with the simple van der Waals one-fluid mixing rule, is capable of describing a wide range of complex phase behavior, albeit qualitatively. Indeed, the use of simple models to represent complex behavior in thermodynamics is not new; for example many simple, twoparameter liquid solution models are capable of describing complex liquid solution behavior, such as liquid-liquid phase separation. Cubic equations of state, in spite of their simplicity, have been shown to be very versatile thermodynamic models. They are capable of representing the continuity
between the fluid phases and consequently can he used to represent liquid-liquid equilibrium (LLE) and vapor-liquid-liquid equilibrium (VLLE), as well as VLE. Many such examples have appeared in the literature. See, for example, the application of PRSV EOS to the LLE of ternary mixtures using various mixing rules considered here (Ohta 1989; Wong and Sandler 1992). We consider the areas discussed in the sections below in no particular order to be promising for future investigation with the models considered in this monograph. These choices are subjective, and in some cases even speculative. However. by mentioning them here we hope to stimulate research into these challenging areas of thermodynamic modeling. In some or these areas significant progress has been made, and in these cases we cite some of the recent work.
6.1.
Systematic Investigation of EOS and Mixing Rule Combinations for the Thermodynamic Modeling of Mixture Behavior a t High Dilution The evaluation of Henry's constants of gases in pure and mixed solvents for gas solubility calculations and of the infinite dilution activity coefficients of solutes in pure and mixed solvents are in this categoly. Some of the models considered here have already been used in this problem area (see Section 5.2 for examples). It is useful to point out that similar models, such as the LCVM and HVOS models, which perform comparably for VLE calculations in the finite concentration range, give significantly different predictions for dilute solution behavior. Such a case is shown in Figure 5.2.1. A thorough explanation and comprehension of these differences is essential for the development of predictive EOS models that can be applied with a single set of parameters over the whole concentration range.
6.2.
Simultaneous Correlation and Prediction of VLE and O t h e r Mixture Properties such as Enthalpy, Entropy, Heat Capacity, etc. The accurate representation of phase behavior by any model requires an accurate correlation of the excess Gibbs free energy of the solution, whereas the representation of calorimetric properties of solutions requires an accurate correlation of the excess enthalpy. For most liquid mixtures, the composition dependence of enthalpy is a more complicated function than that of the Gibbs free energy. See, for example, Figure 6.2. I in which the excess Gibbs free energy and excess enthalpy of the acetone and watermixture at ambient temperatureobtainedfrommeasureddata are shown. The EOS-Gebodels considered in this monograph make use of analytic expressions that were developed solely for the representation of excess Gibbs (or Helmholtz) free energy of liquid mixtures and have not been very successful for the simultaneous representation of the excess enthalpy of solutions with parameters that have been tuned
Modelng Vapor-L~qud Equilibria
mole fraction of acetone
mole fraction of acetone
Figure 6.2.1. Excess Gibbs free energy and excess enthalpy of the acetone and water binary mixture at 293 K. The excess Gibbs free energy was calculated from VLE data as described in Section 5.1. The excess enthalpy data are as reported in the DECHEMA Chemistry Data Series Heat of Mixing Collection, Christiansen et al. 1984, Vol. 1, PI. lb, pp. 148-9.
to the Gibbs free energy. See, for example, the representation of excess enthalpy of the benzene and cyclohexane binary mixture at 393 K (Figure 6.2.2) with temperatureindependent model parameters fitted to room temperature VLE and excess enthalpy data (Orbey and Sandler 1996b). In this case all models fit the excess enthalpy at 293 K within experimental accuracy; however, theEOS-G" models using the van Laar excess free-energy expression predicted an incorrect trend for the excess entbalpy at
~ ~ 0 8 enlhaipy 8 s data et
293 K
mole fraction of benzene Figure 6.2.2. Excess enthalpy for the benzene and cyclohexane system at 293 K (dots) and at 393 K (trangles). The lines denote correlations at 293 K and predictions at 393 K using various models. The solid line reflects predictions using the 2PVDW model, the dotted line represents the predictions using the van Laar activity coefficient model, the short dashed lines signify predictions using the HVOS model, and the long dashed line denotes predictions made with the WS model. Data are from the DECHEMA Chemistry Series (Gmehling and Onken 1977, Vol. 3, Pt. 2, p. 992).
393 K as did the van Laar activity coefficient model. Only the 2PVDW model, which is not an EOS-Gex model, predicted a qualitatively correct trend. Consequently, the development of better liquid excess property models that can simultaneously account for enthalpy and phase equilibrium behavior is needed, not only for general use but also for use with the EOS-G" mixing rules. Indeed, a general comment that can be made is that in some cases, such as this one, the EOS-Ge" model inherits the shortcomings of the excess free-energy model that it includes. Consequently, it is the inadequacy of the underlying GeXmodel that is at fault when one tries to obtain the simultaneous representation of both the excess free energy and the excess enthalpy.
6.3.
Representation of Polymer-Solvent and Polymer-Supercritical Fluid VLE and LLE with the EOS Models Polymeric materials, both as end products and intermediates, are an ever-increasing segment of the chemical industry. Representation of polymer mixtures by equations of state especially developed for this task is a fairly mature area (see the review of
Modeling Vapor-Lquid Equilibr~a
Sanchez and Panayiotou in Sandler 1994, Chap. 3). Recently the phase equilibria of mixtures of polymers in organic liquid solvents and in supercritical fluids have became very important (Folie and Radosz 1995). The EOS-G'" models considered here have the potential of describing such mixtures, and some work has been done on EOS-Gexmodels that are applicable to polymer-solvent systems (Harismaidis et al. 1994; Kontogeorgis et al. 1994a,h; Orbey and Sandler 1994; Kalaspiros and Tassios 1995; Xiong and Kiran 1995). However, the results so far indicate that there is a need for more work, especially in developing accurate predictive, rather than correlative, models. One point to stress is that the results of these early studies seem to indicate that the EOS parameters used for the pure polymers are not v e v critical to the success of these models, but how the solvent is described appears to he more important to the final results. However, this needs to be investigated further.
6.4.
Simultaneous Representation of Chemical Reaction and Phase Equilibrium and the Evaluation of Phase Envelopes of Reactive Mixtures In some cases of industrial interest, chemical reaction may occur together with phase equilibrium. Reactive distillation is a good example. In such cases it is important to he able to predict the phase envelope as a function of temperature and pressure so that the design engineer will know whether a second liquid phase will form during the progress of the reaction, which may affect the reaction kinetics and other design factors (Wu et al. 1991a,b). It has already been shown that a cubic EOS with the simple van der Waals mixing rules can be used with acceptable accuracy in predicting the phase envelope for many cases in which only phase equilibria of simple mixtures are encountered. See, for example, the recent review of Sadus (1994). Reactive systems usually consist of mixtures of molecules with very different functional groups, which consequently are veIy nonideal, and the types of mixing rules discussed here may he necessary for their representation. A systematic investigation of recent mixing rules with cubic equations of state in representing the phase behavior of such reactive mixtures would be very useful.
6.5.
Correlation of Phase Equilibrium for Mixtures that Form Microstructured Micellar Solutions Liquid-liquid emulsions and other microstructured fluids have been the subject of much academic and industrial interest, for they offer a new area for scientific research, and their behavior influences many engineering technologies; indeed they represent the backbone of several emerging chemical and biochemical processes. Colloids are in a transition domain between macroscopic and microscopic regimes of matter and
are not well understood. One can classify the types of phase behavior observed in such liquid-liquid emulsions into two general categories, depending on the scale involved. The first type is conventional, macroscopic phase equilibrium. The second type is the formation of microstructures in some (and sometimes in each) of the separatedphases. Many important characteristics of emulsions are dictated by their microstructured phases. However, it is not possible to investigate the microstructure of such systems without an understanding of the macroscopic phase behavior. A knowledge of the macroscopic phase behavior is a prerequisite for identifying the phase boundaries in systems within which the microstructures are formed. Only a few recent attempts have been made to describe the macroscopic phase behavior of such systems quantitatively (Kahlweit et al. 1988; Sassen et al. 1992; Kao et al. 1993; Knudsen, Stenby, and Andersen 1994), hut all have been limited in their scope and success. Kao et al. used the Peng-Robinson EOS and a phenomenological multiparameter mixing rule to describe ternary. phase behavior of the water, carbon dioxide and C4E, (2-butoxyethanol) . system. For the C4E, and water system, Kao et al. were able to correlate isothermal VLE data successfully with two binary parameters, but those parameters could not be used to represent the LLE between these two species at higher pressures. The closedloop LLE exhibited by these two species at higher pressures could be correlated with their model hut only with a set of two binary interaction parameters that were a function of temperature. This study supported the findings of van Pelt, Peters, and de Swaan Arons (1991), which showed that, when coupled with nonquadratic mixing rules, equations of state can represent closed-loop liquid-liquid miscibility gaps characteristic of so-called type VI systems that cannot he represented by the conventional van der Waals mixing rules. However, it was also shown that the parameters of such phenomenological mixing rules provide little or no extrapolation capability. Knudsen et al. (1994) studied surfactant systems us in^- the MHV2 model considered in this monograph with a modified Soave-Redlich-Kwong equation of state. They investigated the same surfactant and water binary rystem previously investigated by Kao et al. and found that a reasonably successful correlation could be obtained with the MHV2 equation coupled with the UNIQUAC model by fitting two strongly temperature-dependent parameters per binary pair to the data. Even though not very successful, the results of that work, and that of Kao et al., are somewhat encouraging. First, they show that equations of state can correlate the phase behavior of the binary pairs in a ternary micellar system. Second, they were able to predict, albeit only qualitatively, ternary phase behavior on the basis of these correlations, which is an important goal in modeling such systems. However, it was clear from the results that there is still much to be done to develop accurate extrapolations with respect to temperature and pressure with these EOS models for such systems. The challenge of quantitatively predicting ternary phase behavior using only data on binary systems remains for these systems, and indeed more generally. It should be noted that even when using activity coefficient models directly, temperature dependent parameters are needed. There is no excess Gibbs free energy model
Modelng Vapor-Liquid Equiibrta
with temperature-independent parameters that can describe such behavior. The development of such a model would be an important contribution to applied engineering thermodynamics.
6.6.
Systematic Investigation of LLE and VLLE for Nonelectrolyte Mixtures with an EOS
A thorough investigation of the use of cubic equations of state in the EOS-GrXformalism for the description of LLE and VLLE needs to be undertaken. As indicated above, the prediction of phase transitions from VLE to VLLE and to LLE is smooth with an EOS. This is a significant advantage in computer simulations because no a priori knowledge of thenumber of phases present may be available, and, consequently, the applicability of a single model to all possible situations would be an important advantage. Among the systems that should be considered in such an analysis are fluid mixtures near the solvent critical point. Supercritical extraction, the production of liquefied natural gas or gas condensates, and enhanced recovery of hydrocarbon resources with carbon dioxide and methane are a few examples of such systems. It is in the vicinity of their critical points that supercritical solvents have their largest extractive powers, and such mixtures can exhibit transitions from VLE to LLE and VLLE. Although experimental data for the analysis of such systems are available (see, for example, recent works of Patton and Luks 1995 and Peters et al. 1995), most EOS models are not satisfactory for quantitative description of such systems. Computational tools are also needed for the analysis of azeotropic separations (see, for example, the work of Bossen, Jorgensen, and Gani 1993 and Coats, Mullins, and Thies 1991). The recent review of Dohm and Bmnner (1995) contains much information on additional systems that can be studied with the models presented here. The computational aspects of EOS modeling of systems that exhibit LLE and VLLE behavior are also somewhat more complicated than for VLE; the works of Michelsen and his colleagues (Heidemann and Michelsen 1995; Michelsen 1986, 1987, 1993, 1994; Mollerup and Michelsen 1992) provide an excellent discussion of algorithms that can be used.
Bibliography of General Thermodynamics and Phase Equilibria References
A
VARIETY of good reference sources are available for those who wish to learn more about phase equilibrium calculations and the recent advances in the subject. A partial list of source books is given below. Some of them are recent and provide up-to-date developments, and some dated sources introduce the basic principles in a coherent and easy-to-understand fashion.
Malanowski, S., and Anderko, A. Modeling Phase Equilibria. J. Wiley and Sons, New York, 1992. Null, R. H. Phase Equilibrium in Process Design. Wiley-Interscience, New York, 1970. Prausnitz, J. M., Lichtenthaler, R. N., and de Azevedo, E. G. Molecular Thermodjanamics of Fluid Phase Equilibria. 2d ed. Prentice-Hall, Englewood Cliffs, New Jersey, 1986. Reid, R. C., Prausnitz, J. M., and Poling, B. E. The Properties of Gases and Liquids. 4th ed. McGraw-Hill, New York, 1987. Sandler, S. I. Models for Thermodynamic and Phase Equilibria Calculations. MarcelDekker, New York, 1994. Sandler, S. I. Chemical and Engineering Thermodynamics, 2d ed. I. Wiley and Sons, New York, 1988. Van Ness, H. C. Classical Thermodynamics qfNon-Electrolyte Solutions. Pergamon Press, Oxford, 1964. Wdlas, S. M. Phase Equilibria in Chemical Engineering. Butterworth-Heinemann, Boston, 1985.
Summary o f the Algebraic Details for the Various Mixing Rules and Computational Methods Using These Mixing Rules
I
I.
N this appendix we present the algebraic expressions for the EOS parameters a and b and for the hlgacity coefficient expressions for mixtures f a each of the various mixing rules considered in this monograph. These are the basic relations needed to do VLE calculations. Comments concerning the activity coefficient models and programming details are also included. Although some of the material in this appendix has been presented in the preceding chapters, it is repeated hwe for the purposes of clarity and completeness. This appendix is not intended to provide exhaustive mathematical or thermodynamic details; for those the interested reader should refer to the various hooks and papers given in the reference section of this monograph. It should also he noted that by following the derivations provided here one can develop the expressions that can he used to modify the programs included with this monograph to accommodate new mixing rules that are proposed in the future.
Activity Coefficient Models
The general constraint for VLE is
where f is the fugacity of species i in a homogeneous liquid or vapor mixture. We use the overbar to indicate a property of a species in a mixture, and the superscripts L and V represent the liquid and vapor phases, respectively. Also T and P are absolute temperature and pressure and x and y are mole fractions (of species i ) in the liquid and vapor, respectively. In the y - 4 method, the equilibrium constraint in eqn. (B.I.1) is rewritten as
where yi is the activity coefficient, Ti is the fugacity coefficient of component i in the homogeneous vapor mixture, and fiL is the fugacity of pure component i as a liquid.
Appendix 0. Summary of the Algebraic Details for the Various Mixng Rules
The pure component fugacity can be computed from
where qbi is the fugacity coefficient for the pure component, and the exponential term, written here assuming that the pure liquid molar volume, is independent of pressure, is the Poynting correction. In general the pure component fugacity is obtained from an EOS using the relation
v),
in which Ni is the number of molcs of species i. V is the total volume, Z = P V / N iRT is the compressibility factor, and R is the gas constant. For a pure liquid at its saturation vapor pressure, PVap,as an approximation we have that
provided the saturation pressure is low. If the saturation pressure is low but the liquidis at a pressure higher than its vapor pressure, we need to add the Poynting correction in eqn. (B.1.3.) In most cases, however, this term is usually small enough to be neglected unless the total pressure is very high. With these in mind eqn. (B.I.2) becomes
Furlhermore, for most vapor mixtures at low pressure, & 1s very close to unity (there are exceptions to this assumption; for example, associating gases such as hydrogen fluoride or acetic acid), and that leads to the equilibrium relation we used in this monograph to calculate the vapor-liquid phase equilibrium by the direct use of activity coefficient methods:
In eqn. (B.I.7), the activity coefficient is obtained from an excess Gibbs fi-ee-energy model, which provides an expression for the molar excess Gibbs free energy of a mixture, using the rigorous thermodynamic relation
(37,
In eqn. (B.1.8), the activity coefficient is obtained by multiplying the molar excess Gihhs free-energy expression with total number of moles of the mixture, N , and then differentiating the resultant total excess Gihbs free-energy term with respect to the
Modeling Vapor-Liquid Equlibra
mole number of species i , keeping all other mole numbers and T and P constant. Note that all conventional excess free-energy models are pressure independent with temperatnre-dependent parameters. Thus the model parameters are constant when the derivative with respect to mole number in eqn. (B.I.8) is taken.
II.
Equation of State Models
In the EOS approach the equilibrium constraint of eqn. (B.I.1) is again used, except in this case the same EOS is used for both phases:
Here the same expression is used for the fngacity coefficients of species i in both homogeneous liquid and vapor phases:
where here z, is used as a generic mole fraction term; when eqn. (B.IT.2) is applied to the liquid phase, x, is substituted; for the vapor phase y, is used instead. The pressure and the compressibility factor terms appearing in this equation must be obtained using the liquid and vapor molar volumes as appropriate, as will be explained next. In the EOS approach, an equation, like the Peng-Robinson equation below, is used to obtain J! or, equivalently, Z in eqn. (B.II.2). RT p=-v-b -
a(T) V(V+b)+b(Vb) NRT N[Na(T)I or P = V - (Nb) V[V (Nb)l (Nb)[V - (Nb)l
+
+
When a homogeneous liquid phase is in equilibrium with a vapor, T and P are the same in both phases. In such a case when the EOS is solved at a selected T, and P , and composition, three volume roots are obtained at temperatures less than the critical temperature. For the liquid phase, the liquid composition must be used, and the smallest of the volume roots is taken as the solution. The compressibility factor Z is then calculated with that root. If the vapor composition is used, the largest root for the volume is used.
Appendix B: Summary ofthe Algebraic Deta~lsforthe Various Mixlng Rues
The eqns. (9.11.2 and 9.11.3) are general and applicable to any mixing and combining rule included in this monograph. Different mixing rules, however, result in different composition dependencies for the EOS parameters a and b in eqn. (B.II.3), and because of this, each model is different. For the mixing-combining rules discussed in this monograph, the expressions for the mixture a and b parameters and for the fugacity coefficient of species i in a one-phase mixture are given next.
I1.A.
One-Parameter van der Waals One-Fluid Model ( I PVDW)
In this case the mixing rules for the two parameters of the PR EOS are
a=
xx
ziz,ai,
1
i
and
b=
x
zibi
(9.11.4)
with the combining rule
With these mixing and combining rules, the fugacity coefficient of species i in a homogeneous mixture obtained from eqn. (9.11.2) is
where B = h P / R T . A = a P/(RT)', and Z = PVIRT. Note that throughout this appendix in the double index notation mi; = mi, where m is any indexed variable. in this and all cases that follow, the compressibility factor Z is computed from the EOS. B
Two-Parameter van der Waals One-Fluid Model (2PVDW)
In this case eqns. (B.II.4 to B.11.6) are used with the exception that the binary parameter employed in Section 1I.A is now a composition-dependent two-parameter term
and the fugacity expression then takes the form
Modeling Vapor-Lquid Equilibria
with
for binary mixtures.
1I.C.
Wong-Sandler Mixng Rule (WS) In the Wong-Sandler mixing rule the EOS parameters for a homogeneous liquid or vapor mixture is computed from
with
or, alternatively,
and
In these equations, GY, the molar excess Gibbs free-energy obtained from any excess free-energy model, is a function of temperature and composition only. Even though the Wong-Sandler derivation involves the Helmholtz free-energy of the mixture, this substitution is due to the assumption that G Y ( T , z i ) = A&,(T, P. zi). See Section 4.3 for details. The C* term is the EOS-dependent constant, as explained in Section 4.1. For the PR EOS C* = [ln(fi - I)]/& = -0.62323. Either of the combining rules of eqns. (B.II.ll or B.II.12) can be used, yielding slightly different results; again, see Section 4.3 for details.
Append~xB: Summary ofthe Algebrac D e t a s for the Varous Mxing Rules
The fugacity coefficient expression for species i in a mixture for the Wong-Sandler mixing rule is In & ( T , P , z,) =
(aNb/aN,)~ N,,, b
(Z-1)-In(Z-B)
The partial derivative terms are
with
and
i [aNGYtl(T. zO] Iny - ' - RT aNA T N,i, I.D.
Huron-Vidal (Original) M x n g Rule (HVO)
In the Huron-Vidal mixing rule the mixture EOS parameters are given as
which is identical to the mixing rule for the b parameter in IPVDW and 2PVDW rules, and
Model~ngVapor-Lquid Equilibria
In this case the fugacity coefficient of species i in a mixture becomes
where again the compressibility factor is computed from the EOS. 1I.E.
Modified Huron-Vida First-Order M~xingRule (MHVI) In this mixing rule the expression for the b parameter is the same as the HVO (eqn. B.II.22) lPVDW and 2PVDW mixing rnles in Section 1T.D, and for the a parameter we have
where ql is an empirical parameter obtained by fitting pure component information. The fugacity coefficient of species i in a homogeneous solution is
1I.F
Modified Huron-Vidal Second-Order Mixng Rule (MHV2) In this mixing rule eqn. (B.II.22) again (b = x , z i b i )is used for the b parameter. The parameter a is obtained by solving the following quadratic expression for E = a/bRT
and choosing the larger of the two real roots for E to obtain a in terms of b. In eqn. (B.IL27), q, and q2 are empirical constants obtained by fitting pure component properties. See Section 4.4 and Dahl and Michelsen (1990) for details. The fugacity expression for i in a homogeneous mixture is
Appendix B: Summary o f t h e Algebrac Details for the Varous Mixing Rules
where
1I.G.
Linear Combination o f Huron-Vida and Michelsen Models (LCVM) As the name implies, this model is a combination of HVO and MHVl models, and once again eqn.(B.II.22) ( b = C,z,b,)is used for the b parameter. The a parameter is obtained from
where A is an arbitrary parameter that has been selected to give the best results for the particular system under consideration once the excess free-energy model is chosen. Note that A = 1 gives the HVO model, and A = 0 gives the MHVl model. The fugacity coefficient of species i in a homogeneous mixture is
1I.H.
Orbey-Sandler Modification o f t h e Huron-Vidal Mxing Rule (HVOS) Again b = C,zibi is used, and the a parameter relation is
The fugacity coefficient of species i in a mixture becomes b, In 4, = -(Z b
-
I) - ln ( Z - B )
Modeling Vapor-Ltqud Equilibria
HI.
The Programming Details for the VLE Calculations The VLE calculations presented here were done using an isothermal bubble point algorithm. A flow diagram of the algorithm is presented in Figure B.l for the EOS methods. When optimizing the model parameters, we used the objective function
for the minimization in a simplex formalism Spdfy liquid mole fractions x, and T
GUeSs bubble-point
pressure P
Guess set of K, = y; I y for all commnents
I
calmlate i l ( ~ , ~ , x , , ) f oallr mmponents using zL
calculate I"(T,P,~,)for ail mmponents using ZV
y; = y,
f, tor a11components
IS y;' = y, for all components?
Yes computed bubble-point pressure and vapor compositions are correct
Figure B.1.
Appendx B: Summary ofthe Agebrac Detais forthe Various M x ~ n gRules
When the WS model is used in the predictive mode, the excess Helmholtz free energy calculated from the EOS at each liquid composition in a given data set was matched, as closely as possible, with the excess Gibbs free energy calculated from the UNIFAC model at the same liquid compositions by adjusting the value of the k i j ~ d rameter in the model. The ki,parameter obtained this way was used in eqn. (B.TI.l l). The UNlFAC model was also used to calculate the Gy term in eqns. (B.II.lO, B.II.13, B.II.18, and B.II.21). For all other predictive EOS models, the only input requirement for the mixing rule was the G;" term (and the activity coefficients calculated from it using eqn. [B.1.8]), which was obtained from the UNlFAC model. Thus. in the predictive mode, no experimental VLE data are needed in any of the excess free-energy-based EOS models for VLE calculations. Note, however, that thc programs provided on the accompanying disk require the experimental P - x - y data only if one wants to calculate the deviations from those data.
-
Derivation o f Helmholtz and Gibbs Free-Energy Departure Functions from the Peng-Robinson Equation o f State at Infinite Pressure
T
HE Helmholtz free-energy departure function (from ideal gas behavior) for the
Peng-Robinson equation at a given temperature, pressure, and composition is (Wong and Sandler 1992)
The first logarithmic term in eqn. (C.1) is zero in the limit of infinite pressure because for the PR EOS
and as P + m, V + b; thus, the right-hand side in eqn. (C.2) becomes unity, and its logarithm is zero. In the second term in eqn. (C.l), the argument of the logarithmic expression at the infinite pressure limit becomes
thus
Because the excess Helmholtz free energy is closely related to this departure term (see Wong and Sandler 1992 for details) the same C* term appears in the excess Helmholtz free-energy term obtained from the PR EOS. If a similar analysis is done for the Gibbs free energy, the analogue to eqn. (C.l) is
Appendix C: Dervation of Helmholtr and Gibbs Free-Energy Departure Functions
At infinite pressure the last term is infinite; however, whrn the excess Gibbs free energy of the mixture is evaluated, this tern cancels out between the pure component and mixture terms, but only if one assumes b = xihifor mixtures. In that case, the relation between excess Helmholtz and Gibbs free energy becomes
xi
See the work of Huron and Vidal(1979) and Fischer and Gmehling (1995) for further details. Note that for other cubic equations of state a similar analysis holds, but the numerical value of the C*term is different, apoint that has sometimes been overlooked in literature.
Computer Programs for Binary Mixtures
T
H E disk that accompanie~this monograph contains the programs and samplc data file5 that can be used to correlate and prcdict vapor-liquid equilibria using various equations of state and activity coefficient models. All programs are coded in FORTRAN using MICROSOFT FORTRAN Version 5.1 and are also supplied as standalone executable modules (EXE files) that run on DOS or WINDOWS-based personal computers. The files are compressed and must be decompressed, preferably into a directory in the hard disk of your personal computer. There are four compressed files on the disk: EXEFILES.ZIP, DATFILES.ZIP, FORFILES.ZIP, and MAKFILES.ZIP. EXEFILES contains the ten executable programs. seven of which are described helow (the remaining three are for prediction of the VLE of multicomponent mixtures, and they are described in Appendix E). DATFILES contains all thc data files (a total of fifty-six files: forty with the DAT extension; two with the DTA extension; four with the ACT extension; two with the VDW extension; two with the HVN extension; two with the WSN extension; and one each with the WSU. HVW, HVU, and WSW extensions [see also Appendix El) used in this monograph. FORFILES contains the FORTRAN subroutine programs (a total of seventy-nine), and MAKFILES contains nine MAKE files that are used to build nine multimodule FORTRAN cxecutahle programs, the subroutines for which are provided. The tenth FORTRAN executable program, VDWMIX, is a single-module program and does not need a MAK file, (see Appendix E). All that follows applies to operating in DOS or in a DOS window of a computer using any version of the WINDOWS operating system. If you are interested in only using stand-alone executable modules, only EXEFILES.ZIP and DATFILES.ZIP need to be decompressed. To do this, inse~t the accompanying disk into the floppy drive designated drive A. Next create a subdirectory in your root (C:>) directory (for example, a directory called TEST). To create a subdirectory called 'test' type the following command: At C:\>type MD TEST and press RETURN. (This results in creation of a subdirectory named TEST under the root directory C.) Type the following commands:
Appendx D: Computer Programs for Binary Mixtures
At C:\> type CD TEST and press RETURN. At C:\TEST> type A: and press RETURN. At A:\> type PKUNZIP EXEFILES C:\TEST and press RETURN. (This results in the decompression of the ten executable files and one auxiliary file [PKZIP.EXE] into the subdirectory TEST). At A:\> type PKUNZIP DATFILES C:\TEST and press RETURN to decompress the data files and place them in the TEST subdirectory. (However, in this case an overwrite warning message will appear for the auxiliary file PKZ1P.EXE. Type "n" to proceed.) To decompress all the files the following commands are used: At C:\> type CD TEST and press RETURN. At C:\TEST> type A: and press RETURN. At A:\> type PKUNZIP * C:\TEST and press RETURN. This results in decompression of all the files into the subdirectory TEST. In this caw you can use FORTRAN and MAKE files with the Microsoft FORTRAN package to change or rebuild the executable modules, or both. For further details on this mode, refer to the Microsoft FORTRAN manuals. The EXE files can he run directly from the DOS prompt. To do this, the directory where the EXE files reside (for example TEST directory) is selected, and the name of the EXE filc is typed at the prompt. Eachprogramis separately describedin the following sections. and a tutorial section is included to facilitate the use of each program. In these tutorials, the output that will appear on the screen is indicated in bold and in a smaller font. The information the user is to supply is shown in the normal font.
D. I. Program AC: VLE by Direct Use of Activity Coefficient Models The Program AC can be used to correlate or predict VLE using activity coefficients models directly, without an EOS, that is, using the y - $ method. There are five activity coefficient models available in this program: UNIQUAC, the Nan-Random Two Liquid (NRTL), the van Laar, UNIFAC, and the Wilson models. The gas phase is assumed to be ideal in this program. The instmctions that appear on the screen must be followed to execute the program. See the tutorial given later in this section. The program can be used in two ways. If no experimental T-P-x-yinformation is available, the user only needs to supply the temperature and saturation pressure of each compound at the temperature of interest as input. These data are entered following the commands that appear on the screen. In this mode the program will return isothermal x - y - P predictions at the temperature entered in the composition range x, = 0 to 1 at intervals 0.1, 0.2, 0.3, and so on. In the second mode, available isothermal VLE data can he correlated. The data needed are the temperature, the measured mole fractions (of species 1) in the liquid and
Wodelng Vapor-Liquid Equibria
vapor phases, and the pressure. The program reads previously stored data or accepts new data entered from the keyboard. Again, the activity coefficient models require experimentalpure component saturation pressures as input information. Consequently, if new data are entered from the keyboard, the first data point must be x,,, = 0 , ye,, = 0, and p,,, = P Y P ,where P,""' is the pure component vapor pressure of the second component, and the last data point must be x,,, = 1, ye,, = 1, and p,, = P y , where P y is the pure component vapor pressure of the first component (see Example D. I .B in the tutorial). On the accompanying disk, the extension ACT, such as MW25.ACT, has been used for the sample data files employed with this program. The UNIFAC model is predictive; hence, its use leads to the direct prediction of VLE without any parameter optimization. In this option. however, theuser must supply information as to the groups constituting themolecules required in theUNlFAC model. For the other activity coefficient models in the correlative mode, the program uses a simplex optimization routine to optimize the activity coefficient model parameters, thus minimizing the absolute enor between the experimental and calculatedpressures. Owing to the nature of the simplex optimization routine, a local minimum, rather than a global minimum. may he obtained. Therefore, the final results of the optimization may depend on the the initial guess for the parameters. Also, an inappropriate choice of values for the initial parameters may result in a divergence, in which case calculations . with new initial guesses should be attempted. The results from the program AC can be sent to a printer, to a disk file, or both. To make this choice, the commands that appear on the scrcen upon the completion of calculations should he followed. Please see the following tutorial for further details. ~
~
Tutorial on the Use of AC.EXE Example D. I .A: Fitting Activity Coefficient Model Parameters t o VLE Data Change to the directory containing AC.EXE (e.g., A>, or C>, etc.). Start the program by typing AC at the DOS prompt. A program introduction message appears on the screen. Press ENTER (or press RETURN). The following appears: AC:
VLE CALCULATIONS WITR VARIOUS ACTIVITY COEFFICIENT MODELS
YOU CAN USE THIS PROGRAM FOR VLE CALCULATION IN TWO WAYS. MODE (1): IF NO T-P-x-Y DATA RRE AVAILABLE TO COMPARE RESULTS WITH YOU MUST SUPPLY THE T E M P E R A T m , AND SATURATION PRESSURE OF EACH COMPOUND AT THAT TEMPERATURE.
Appendix D: Computer Programs for Binary Mixtures
IN THIS MODE THE PROGRAM WILL RETURN ISOTHERElAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED, IN THE COMPOSITION RANGE X1=0 TO 1 AT INTERVALS OF 0.1. NODE (2): IF YOU HAVE ISOTHEFMAL x-y-P DATA, YOU CAN ENTER THESE DATA FOLLOWING C O W S THAT APPEAR ON THE SCREEN (OR USE PREVIOUSLY ENTERED DATA) TO FIT THE MODEL PAPAMETERS TO THE VLE DATA. ALTERNATIVELY, YOU CAN CALCULATE VLE WITH PREVIOUSLY SELECTED PARAMETERS AND COKPARE THE RESULTS WITH THE VLE DATA.
TERMINATE THE P R O G ~ type " 2 and pres? RETURN. (With this entry it will he possible either to use a previously stored data file or enter data available and store it in a file.) At "DO YOU WANT TO USE A N EXISTING DATA FILE (Y/N)?" type Y (or y ) and press RETURN.
A~"INPUTEXISTING
DATA FILE NAME (for example,
a:mwzs.act) :"typeA:MW25.ACT.
(This selection results in the use of an existing data file stored in the disk in drive A with the name MW25.ACT.)
A~~~SELE ANC T ACTIVITY
COEFFICIENT MODEL
O=EXIT 1=NRTL 2=VAN LAAR 3 = W I F A C 4=WILSON 5=WIQUACr'
type 2 and press RETURN. (This selection results in the use of the van Laar equation as the activity coefficient model.) Atr,INPUT INITIAL GUESSES FOR VAN LAAR PARAMETERS P12, P21 (PIJ ARE DIMENSIONLESS KAPPA PARAMETERS OF THE VAN
LAAR MODEL) : type
At,,DO
1, 1 and press RETURN.
YOU WANT TO FIT THE PARAMETERS TO VLE DATA (1)
OR DO YOU WANT TO DO A CALCULATION OF VLE WITH
THE PARAMETERS JUST ENTERED (2)?" type 1 and press RETURN.
(At this point the program starts the optimization. When the calculations are completed, the final results appear on the screen as shown below.) AC-VLE FROM ACTIVITY COEFFICIENT MODELS THE VAN
LAAR
MODEL
PARAMETERS Pl2, P21: .5853
.3458
METHANOL WATER TEST DATA FOR PROGRAM AC 25 C TEMPERATURE (K)
:
298.15
PRESSURE IS IN THE UNITS OF THE DATA. XEXP
YEXP
YCAL
PEXP
PCAL
ACT1
ACT2
SUM
(The first, second and fourth columns are the experimental liquid and vapor mole fractions of species I and the total pressure, respectively, and columns three, five. six, and seven are the calculated vapor mole fractions o l species 1, pressure, and activity coefficients of species 1 and 2, respectively. Column eight lists the sum of vapor phase mole fractions that are calculated separately and printed as a check; values should be unity or very close to unity.) At "DO YOU WANT A PRINT-OUT (YIN) ?" type Y (or y) and press RETURN. (This command sends the results on the screen to your printer.) At "DO Y O U WANT T O S A V E T H E R E S U L T S T O AN O U T P U T F I L E (Y/N) ?" type Y (or y) and press RETURN. At "INPUT A NAME FOR THE OUTPUT FILE:" type A:MW25.OUT and press RETURN. (This command saves the results above in your disk in drive A under the name MW25.OUT in ASCII file code. ) At "DO YOU WANT TO DO ANOTHER VLE CALCULATION (YIN) ?" type N (or n) and press RETURN. Example D. I .B:Use of UNIFAC to Predict VLE Data
Change to the directory containing AC.EXE (e.g., A> or C>, etc.). Start the program by typing AC at the DOS prompt. Press ENTER (or press RETURN). The program introduction message appears on the screen. Press ENTER (or press RETURN). The following appears: AC: VLE CALCULATIONS WITH VARIOUS ACTIVITY COEFFICIENT MODELS YOU CAN USE THIS PROGRAM FOR VLE CALCULATION IN TWO WAYS. MODE (1): IF NO T-P-x-y DATA ARE AVAILABLE TO COMPARE RESULTS WITH YOU MUST SUPPLY THE TEMPERATURE, AND SATURATION PRESSURE OF EACH COMPOUND AT THAT TEMPERATURE. IN THIS MODE THE PROGRAM WILL RETURN ISOTHERl&&L X-y-P PREDICTIONS AT THE TEMPERATURE ENTERED, IN THE COMPOSITION RANGE X1=0 TO 1 AT INTERVALS OF 0.1.
Append~xD Computer Programs for Btnary Mxiures
MODE ( 2 ) : IF YOU HAVE ISOTHERMAL x-y-P DATA, YOU CAN ENTER THESE DATA FOLLOWING COMMANDS THAT APPEAR ON THE SCREEN (OR USE PREVIOUSLY ENTERED DATA) TO FIT THE MODEL PARAMETERS TO THE VLE DATA. ALTERNATIVELY, YOU CAN CALCUIATE VLE WITH PREVIOUSLY SELECTED PARAMETERS AND COMPARX THE RESULTS WITH THE VLE DATA.
At "ENTER 1 FOR MODE (1) , 2 FOR MODE (2) , OR 0 TO TERMINATE THE PROGRAM" type 2 and press RETURN. At ,,DO YOU WANT TO USE AN EXISTING DATA FILE (YIN)?" typen (or N) and press RETURN. At "INPW NEW DATA FILE NAME:" type A:TEMP.ACT and press RETURN. (The preceding command will lead to saving a data file to the disk in the A drive under the name TEMP.ACT. A disk must be in that drive.) At "INPUT A TITLE FOR THE NEW FILE:" type 'temporary data file for methanol-water at 25°C' and press RETURN. (You can enter any title composed of up to forty alphanumeric characters for the title statement given above to describe your file for later reference.) At "INPUT ~ E OF DATA R POINTS:', type 4 and press RETURN. At ,,INPUT TEMPERATURE in K:" type 298.15 and press RETURN. At "INPUT FACTOR TO CONVERT EXPERIMENTAL PRESSURE DATA TO BAR BY DIVISION" (ex: if original data in mm Hg, type 750 if original data in psia, type 14.5 etc. ) :" type 750 and press RETURN.
At "INPUT EXPERIMENTAL DATA IN LIQUID MOLE FRACTION (XEXP) OF SPECIES 1, VAPOR MOLE FRACTION (XEXP) OF SPECIES 1, AND BUBBLE POINT PRESSURE (PEXP) IN THE UNITS OF THE ORIGINAL DATA" (three in a row, segarated by commas) REMINDER: FIRST DATA POINT MUST BE X=O AND P=SATURATION P OF PURE SPECIES 2 LAST DATA POINT MUST BE X=l AND P=SATURATION P OF PURE SPECIES 1
At "INPUT XEXP, YExP, PEXP:" type O,0, 23.7, and press RETURN. At "INPUT XEXP, YEXP, PEXP:,, type 0.19, 0.6187, 53, and press RETURN. At "INPUT XEXP, YEXP, PEXP:" type 0.849,0.9384,112, and press RETURN. Atj'reminder: this entry is the last enter X=Y=l and P=Psat. of pure species 1 INPUT XEXP, YEXP, PEXP:' type 1, 1, 127.7, and press RETURN. (When the number of sets of data specified by NP, here four, is entered, the program writes the data to the file under the name TEMP.ACT specified above and then continues. This data file now is an existing data file and can be used if
the program is run again. The data file appears as shown below if callcd by an editor program.) temporary data file for methanol water system at 25C
A~-,SELECTAN
COEFFICIENT
ACTIVITY
MODEL
O = E X I T l=NRTL 2-VAN LAAR 3sUNIFAC 4=WILSON 5=UNIQUACr'type 3 and press RETURN. (This choice results in the use of UNIFAC for the activity coefficient model.)
At "THE UNIFAC MODEL R E Q U I R E S
GROUP
DATA FRON A D I S K .
THESE DATA ARE STORED I N TWO FILES NAMED UNFI1.DTA AND UNFI2.DTA.
UNFI1.DTA
CONTAINS UNIFAC
UNFI2.DTA
CONTAINS UNIFAC
GROUP PARAElETER INFORMATION. BINARY GROUP INTERACTION PARAMETER
INFORMATION.
I F YOU ALREADY HAVE THESE DATA FILES I N THE CURRENT DIRECTORY, ENTER 1, OTHERWISE ENTER
THEN
2" type 2 and press RETURN.
(The data files UNFII.DTA and UNFI2.DTA are provided on the disk that accompanies this monograph. It is better if these data files are copied to the hard disk directory that is used to run the programs. In this case 1 should be entered. An entry of 2, as above, indicates that these files are not present in the current directory. In this case the user must provide the directory and file names as below.) At "TYPE
THE
DIRECTORY & THE
NAME
OF THE
FILE
WHERE
UNIFAC
GROUP PARAMETER
INFORMATION I S STORED (default = a:UNFIl.DTA)"
type a:UNFIl.DTA and press RETURN. At "TYPE
THE DIRECTORY k THE NAME O F THE F I L E WHERE UNIFAC
PAPAMETER INFORMATION ( d e f a u l t = a:UNFI2.DTA)" type a:UNFI2.DTA and press RETURN. At "ENTER COMPONENT INFORMATION
BINARY INTERACTION
I S STORED
ENTER COMPONENT NAME (max. 1 2 Characters) FOR
COMPONENT
1
Appendix D: Computer Programs for Bnary M~xtures
OR ENTER <press RETURN> TO TERMINATE ENTRIES"
type METHANOL and press RETURN. (Following the preceding comment, a group selection table will appear on the screen. The user must follow the instructions at the top of the tahle to choose one CH,OH group for methanol and enter press RETURN.) At "ENTER COMPONENT INFORMATION ENTER COMPONENT NAME (max. 12 Characters ) FOR COXPONENT 2 OR ENTER <press RETURN> TO TERMINATE ENTRIES"
type WATER and press RETURN. Following the preceding comment the group selection table will again appear on the screen. The user should follow the instructions at the top of the tahle to choose one H 2 0 group for water and then press RETURN. After the last entry, a summary of the parameter input appears on the screen. Press RETURN to continue. The following results will appear on the screen: AC-VLE FROM ACTIVITY COEFFICIENT MODELS THE UNIFAC MODEL temporary data file for methanol water, 25 C TEMPERATURE (R): 298.15 PRESSURE IS IN THE UNITS OF THE DATA. PEXP
PCAL
ACT1
ACT2
SUM
2.2446
1.0000
1.0000
XEXP
YEXP
YCAL
,0000
.OOOO
.OOOO
23.7000
23.7000
.I900
,6188
.6446
53.0000
56.1636
1.4921
1.0398
1.0000
.a490
.9384
.9542
112.0000
114.4886
1.0076
1.4659
1.0000
1.0000
1.000
1,000
127.7000
127.700
1.0000
1.6046
1.0000
(This is a part of the methanol-water binary system data used in Example D.1 .A. As before, in this tahle the first, second, and fourth columns are the measured liquid and vapor mole fractions of species 1 and the pressure, respectively, and columns three, five, six, and seven are the calculated vapor mole fractions of species 1, pressure, and the activity coefficients of species 1 and 2, respectively. Column eight is the calculation confirmation line described earlier.) At -DO YOU WANT A PRINT-OUT (Y/N)?" type Y (or y) and press RETURN. (This command sends the results on the screen to printer.) At "DO YOU W A N T TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?"
type Y and press RETURN. At ,,INPUT A NAME FOR THE OUTPUT FILE:" type A:TEMP.OUT and press RETURN. (This command saves the results given above on the disk in drive A with the name TEMP.OUT as an ASCII file.)
Modeling Vapor-Lqud Equilibr~a
A~"DoYOU
WANT T O W ANOTHER VLE CALCULATION
(Y/N)?"typeN
(or n) and press RETURN.
Example D. I .C: Direct Use of Activity Coefficient Model to Predict VLE
Change to the directory containing AC.EXE (e.g., A > or C>, etc.) Start the program by typing AC at the DOS prompt. The program introduction message appears on the screen. Press RETURN. The following appears: AC: VLE CALCULATIONS WITH VARIOUS ACTIVITY COEFFICIENT MODELS YOU CAN USE THIS PROGRAM FOR VLE CALCULATION IN TWO WAYS. MODE (1): IF NO T-P-x-y DATA ARE AVAILdBLE TO COMPARE RESULTS WITH YOU MUST SUPPLY THE TEMPERATURE, AND SATURATION PRESSURE OF EACH COMPOUND AT THAT TEMPERATURE. IN THIS MODE THE PROGRAM WILL RETURN ISOTHERMAL X-y-P PREDICTIONS AT THE TEMPERATURE ENTERED, IN THE COMPOSITION RANOE X1=0 TO 1 AT INTERVALS OF 0.1. MODE ( 2 ) : IF YOU HAVE ISOTHERMAL x-Y-P DATA, YOU CAN ENTER THESE DATA FOLLOWING C O W S THAT APPEAR ON THE SCREEN (OR USE PREVIOUSLY ENTERED DATA) TO FIT THE MODEL PARAMETERS TO THE VLE DATA. ALTERNATIVELY, YOU CAN CALCULATE VLE WITH PREVIOUSLY SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA.
A ~ ~ ~ E N T1 EFOR R
MODE (I), 2 FOR MODE (2), OR
o
TO TERMINATE
THE PROGRAM,, type 1 and press RETURN.
(This results in the selection of the predictive mode of the program. In this mode no experimental VLE data can he entered to, or accessed from, the disk. The user must supply a temperature and the pure component vapor pressures following the commands on the screen. In addition, the user must select a model and provide the model parameters. The program returns x-y-Ppredictions at the temperature entered in the liquid mole fraction range xl = 0 to 1 at intervals 0.1, 0.2, 0.3, etc.)
A~"YOUMAY
ENTER A TITLE (25 CHARACTERS MAX.) FOR THE
MIXTURE TO BE PREDICTED (OR Y O U MAY PRESS RETURN T O SKIP THE TITLE) :"enter
methanol-water 25 C and press RETURN. (The title entry is optional.) At "INPUT TEMPERATURE in K:" enter 298.15 and press RETURN. At "INPUT VAPOR PRESSURE O F COMPONENT 1 ( I N ANY UNIT) :"enter 127.698 and press RETURN.
Appendix D: Computer Programs for Binary Mixtures
.
A ~ ~ T I N P UVAPOR T PRESSURE OF COMPONENT 2 ( I N ANY UNIT) :"enter 23.70 and press RETURN. A~~JINPUT FACTOR TO CONVERT PRESSURE INTO BAR BY DIVISION" (type 1 if you e n t e r e d vapor p r e s s u r e s i n bar type 750 if you e n t e r e d t h e m i n mm Hg. e t c . ) :"enter750 and press RETURN A ~ ~ ~ S E L EAN C TACTIVITY COEFFICIENT MODEL O=EXIT l=NRTL 2=VAN LAAR 3=UNIFAC 4=WILSON 5=UNIQUAC"type 1 and press RETURN. At ,*INPUT ALPHA OF THE NRTL MODEL:" enter 0.35 and press RETURN. AtmINPUT REDUCED NRTL PARAMETERS PI2 AND P 2 1 [ P I J = A I J / ( R T ) WHERE AIJ IS IN CALIMOL. I :"enter 1,l and press RETURN. The following results appear on the screen:
AC-VLE FROM ACTIVITY COEFFICIENT MODELS THE NRTL MODEL PARAbETERS P12, P2l 1.0000 1.0000 RLPHA=.350 methanol - water 25C TEMPERATURE (K):298.15 PRESSURE IS I N THE UNITS OF THE DATA. XEXP
YEXP
YCAL
PEXP
.OOOO
-
.OOOO
.lo00
-
.6944
.ZOO0
-
.7827
-
.3000
-
.a153
.4000
-
5.4997
71.0950
3.8660
1.0000
93.6455
2.8698
1.0000
-
104.8255
2.2308
1.0000
,8320
-
110.7753
1.8044
1.0000
.a434
-
114.4529
1.5119
1.0000
117.3457
1.3083
1.0000
120.1946
1.1672
1.0000
123.2530
1.0733
1.0000
126.2193
1.0185
1.0000
127.6980
1.0000
1.0000
,8000
-
.8896
.go00
-
.9274
1.0000
-
1.0000
.
SUM
23.7024
.7000
.SO00
ACT1
1.0000
-
.5000
PCAL
.a542 .a680
-
At -DO YOU WANT A PRINT-OUT ( Y / N ) type n (or N) and press RETURN. At "DO YOU W A N T TO SAVE THE RESULTS TO AN OUTPUT F I L E ( Y / N ) ? " enter N (or n) and press RETURN. At -DO YOU WANT TO DO ANOTHER CALCULATION ( Y / N ) ? " typeN (or n) and press RETURN
D.Z.
Program KOPT: Evaluation of the Equation of State
(KI)
Parameter for the PRSV
The program KOPT is used for the evaluation of the K , constant of pure flulds in the PRSV equation (see Section 3.1). The data required for this program are critical temperature (in Kelvin), critical pressure (in bar), and acentric factor of the fluid as well as data for the temperature (in Kelvin) versus vapor pressure (in any units). The program returns the K I value, which minimizes the average difference between the estimated and experimental vapor pressures. A simplex optimization routine is used in the calculations. The program reads previously stored data or accepts new data entered from the keyboard. The extension DAT, such as ACETONEDAT (one of the sample data sets included on the accompanying disk), was used for the data files for this program on the accompanying disk. A tutorial is provided below to demonstrate the use of the KOPT program. As a requirement of the simplex minimization procedure, an initial guess for K I must he provided. The initial guess can be a positive or a negative number, usually in the range from zero to one. The results from KOPT can be sent to a printer, to a disk file, or both. To make this choice, the commands that appear on the screen at the completion of calculations must be followed. Please see the following tutorial for further details. Tutorial on the Use of KOPT.EXE Example D.2.A: Determnation o f Optimum o f State with Existing Data
"KOPT:
EO YOU
K!
in the PRSV Equaton
Change to the directory containing KOPT.EXE (e.g., A> or C>, etc.). Start the program typing KOPT at the DOS prompt. Press RETURN (or ENTER). An introductory message appears on the screen. Press RETURN. At OPTIMIZES PURE COMPONENT KAPPA-1 PARAMETER I N THE PRSV EOS WANT TO USE AN E X I S T I N G DATA F I L E ( Y / N ) ?" type Y (or y) and press RETURN. (for example: a: acetone.dat :,' type a:acetone.dat. At "INPUT AN I N I T I A L GUESS FOR THE KAPPA-1 PARAMETER:" type 0.1 and press RETURN. (Following the preceding command, the results of the intermediate iterations are graphically shown in the form of an error bar on the screen so that the user can follow the convergence of the calculations. Next, a message showing the results of the optimization appears on the screen. To proceed, press RETURN.)
Appendix D: Computer Programs far B~naryMxtures
At ,,DO YOU WANT A PRINT-OUT (YIN)?" type y (or Y) and press RETURN. (With this command the results shown below are sent to the printer.) KOPT: RAPPA-1 OPTIMIZATION FOR THE PRSV EQUATION
acetone.dat ACETONE VARGAFTIK 427
PEXP (BAR)
-
DECXEMA COWBTNED
RRD
VL(CM3/MOL)
W
.0935
.0937
,1563
81.5939
241384.2
.I552
.I552
.0001
82.4861
150681.4
.2473
.2471
.0660
83.4442
97672.6
.2666
.2784
4.4392
83.7150
87394.8
.3801
.3798
.0823
84.4751
65452.8
.5333
.5548
4.0296
85.5295
45984.8
,5660
.5656
.0756
85.5872
45168.9
1.0270
1.3477
87.6269
25885.4
.a187
.0372
86.7905
31990.6
1.0133 ,8190
PCAL
2.0200
2.0677
2.3600
90.8483
13415.4
5.0600
5.1506
1.7907
97.4385
5593.1
1.7840
10.1300
10.3107
106.2131
2786.3
20.2600
20.3683
,5343
123.1467
1316.7
30.3900
30.1622
.7495
144.0444
792.8
40.5200
..no convergence for this data point..
PERCENT RRD (OVERALL), SUM~~S(PEXP-PCAL)/PEXP)alOO/NP: 1.497
At "Do YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (YIN)?" type y and press RETURN. At "INPUT A NAME FOR THE OUTPUT FILE:" type a:acetone.out and press RETURN. (The last two commands save the results above in the disk in the A drive under the name acetone.out in ASCII code.)
A ~ " D oYOU WANT
TO DO ANOTHER KAPPA-1 CALCULATION (Y/N)?"
type n (or N) and press RETURN. Example D.2.B: Determination of Optimum K in the PRSV Equation of State Entering N e w Data
Change to the directory containing KOPTEXE (e.g., A > or C>, etc.) Start the program typing KOPT at the DOS prompt. Press RETURN (or ENTER).
Modeng Vapor-Liquid Equl~br~a
The program introduction message appears on the screen. Press RETURN. At "KOPT :OPTIMIZES PURE COMPONENT KAPPA-1 PARAMETER IN THE PRSV EOS DO YOU WANT TO USE AN EXISTING DATA FILE (Y/N)?" type N
(01 n)
and press
RETURN. At "INPUT NAME OF THE DATA FILE TO BE CREATED:" type a:telnpl.dat and p r e s ~RETURN. (The preceding command will lead to saving a data file named templ.dat on the disk in drive A. If you choose to do this, a disk must bc present in the A drive.) At "INPUT A TITLE FOR THE NEW DATA FILE:" type "I' VS P DATA FOR PURE ACETONE' and press RETURN. (For the title statement above you can enter any title of up to forty alphanumeric characters to describe your file for later reference.) At "INPUT TC (K), PC (BAR), ACENTRIC FACTOR W: " type 508.1,46.96,0.30667, and press RETURN. At "INPUT NUMBER OF DATA POINTS :" type 3 and press RETURN.
A~"INPUTFACTOR
TO CONVERT PRESSURE INTO BAR BY DIVISION,,
(ex: if source data are in mm Hg, type 750 if source data are i n psia type 14.5 etc.) :"type l and
press RETURN. At "INPUT T(K). PSAT:" type 283.15.0.155189, and press RETURN. At "INPUT T(K). PSAT:" type 313.15,0.56598, and press RETURN. At "INPUT T(K), PSAT:" type 478.15,30.39, and press RETURN. (When the specified number of sets of data, here three, has been entered, the program writes the data to the file under the name templ.pur and continues. This data file becomes an existing data file and can be used when the program is run again. The data file appears as shown below if called by an editor program.) T VS P DATA FOR
508.1
46.96
PURE ACETONE .30667
3 1.0000 283.15
,155189
313.15
.56598
At "INPUT AN INITIAL GUESS FOR KAPPA-1 PARAMETER:" type 0.1 and press RETURN. (Following the preceding command, the results of intermediate iterations are displayed on the screen as an error bar for the user to follow the convergence of the calculations. Next, an intermediate message summarizing the results appears on the screen. Press RETURN to continue.)
Appendix D: Computer Programs for B~naryMxtures
At ,,DO YOU WANT A PRINT-OUT (Y/N)?I, type y (or Y) and press RETURN. (With this command the results, like those shown below, are sent to the printer.) KOPT: KAPPA-1 OPTIMIZATION FOR THE PRSV EQUATION t-.out
T VS P DATA FOR PURE ACETONE m P P A - 1 = -.0100 TIK)
PEXPIBAR)
PCAL
AALI
W
VL(CN3/NOL)
a83.1500
,1552
.I552
.0001
82.4861
150681.4
313.1500
.5660
.5656
.0756
85.5872
45168.9
478.1500
30.3900
30.1622
.7495
144.0444
792.8
PERCENT AALI (OVERALL), SUMlABS(PEXP-PCAL)/PEXP)-IOOINP:
.275
At"D0 YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?" type y (or Y) and press RETURN. At "INPUT A NAME FOR THE OUTPUT FILE:" type a:templ.oUt and press RETURN. (With these commands the rcsultr cited above are saved on the disk in drive A under the name templ.out in ASCII code.) &"DO
YOU WANT TO W ANOTHER KAPPA-1 CALCULATION (Y/N)?"
type n (or N) and prcss RETURN.
D.3. Program VDW: Binary VLE with the van der Waals One-Fluid Mixing Rules (I PVDW and ZPVDW) The program VDW can be used to calculate binary VLE using the PRSV EOS and the van der Waals one-fluid mixing rules (either IPVDW or ZPVDW, see Sections 3.3 to 3.5). The program can be used in two ways. If experimental isothermal VLE data are available, the program can be used with user-provided binary interaction parameter(s) to calculate VLE at measured liquid mole fractions, and the calculated and experimental bubble pressures and vapor mole fractions can be then compared. Alternatively, the program can be used to optimize the values of the binary interaction parameter(s) by fitting them to experimental composition versus total pressure data using a simplex algorithm. In this mode, the program reads previously stored data or accepts new VLE data entered from the keyboard. The data needed are critical temperatures (K), pressures (bars), acentric factors, the K , constants of thePRSV EOS forboth pure components, isothermal VLE data in the form of measured liquid and vapor mole fractions of the first component (that is, x and y in the liquid and vapor phases, respectively), and the total pressure, P, (in any units) at a given temperature. The datafile structure qf this and all the remainingprograms that use an EOS to calculate VLE is the same, and the ~
Modelng Vapor-L~qud Equltbria
data file createdfor VDW2 can also be used with the other EOSprograms that are described in Sections 0.4 to D. 7.To help convergence, it is advantageous to designate the component with the lower critical temperature as the first component in thesedata files. The sample data files on the accompanying disk are identified with the DAT extension, such as MW25.DAT, etc. A tutorial is provided below (seeExamples D.3.A andD.3.B). During program execution, as a requirement of the simplex approach, initial guess(es) for the binary interaction parameter(s), (k12 for IPVDW or k l z and k,, for 2PVDW model) must be provided by the user. The initial guess(es) can be positive or negative number(s). Depending upon the nonideality of the system, an initial guess may have to be significantly different from zero (such as 0 . 1 5 for the acetone-water binary system, as shown in Example D.3.B below) to achieve convergence. If convergence cannot be obtained with a (set of) initial guess(es), the user should try again with different choices. When no experimental VLE information is available, the user only needs to supply the critical temperature, the critical pressure, acentric factor, PRSV K , parameter for each compound, and a temperature as input following the directions that appear on the screen. In this mode the program will return isothermal x - y - P predictions at the temperature entered in the composition range xl = 0 to 1 at intervals of 0.1. Several temperature values can be selected successively. A tutorial is provided below (see Example D.3.C). The results from the program VDW can be sent to a printer, to a disk file, or both. The commands that appear on the screen upon the completion of calculations must be followed to make this choice. Please see the following tutorial for further details.
Tutorial on the Use of VDW.EXE Example D.3.A: Fitting Binary VLE Data with the van der Waals One-Fluid Mixng Rule
Change to the directory containing VDW.EXE (e.g., A> or C>, etc.). Start the program by typing VDW at the DOS prompt and then press RETURN (or ENTER). A program introduction message appears on the screen. Press RETURN to continue. At
, v ~ w :BINARY
VLE WITH VAN DER WAALS ONE-FLUID
MIXING
RULES
1: CONVENTIONAL (1PVDWI 2: 2-PAPAMETER COMPOSITION DEPENDENT MIXING RULE (2PVDW) DO YOU WANT TO USE 1-PARAMETER W W MODEL OR 2-PAPAMETER W W MODEL (1/2)?"
type I and press RETURN.
Appendx D: Computer Programs for Bnary Mxtures
(This results in the selection of lPVDW model for the VLE calculations. The message below appears on the screen.) W W : BINARY VLE CALCULATIONS WITH VAN DER WAALS ONE-FLUID MIXING RULES
YOU CAN USE THIS PROGRAM FOR VLE CALCULATION IN TWO WAYS. MODE (1): IF NO T-P-x-y DATA TO COMPARE RESULTS WITH ARE AVAILABLE, YOU MUST SUPPLY CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR, PRSV KAPPA-1 P-TER
FOR EACH COMPOUND.
AND
A TEMPERATURE
ALONG WITH A (PAIR OF) PREVIOUSLY SELECTED MODEL PARANETER(S). IN THIS MODE THE PROORAM WILL RETURN I S O T H E m X-y-P PREDICTIONS AT THE TEMPERATURE ENTERED, IN THE COMPOSITION RANGE X1=0 TO 1 AT INTERVALS OF 0.1. MODE (2): IF YOU HAVE ISOTHERMAL x-y-P DATA, YOU CAN ENTER THESE DATA FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN (OR USE PREVIOUSLY ENTERED DATA) TO FIT MODEL PARAMETERS TO THAT VLE DATA. ALTERNATIVELY, YOU CAN CALCULATE VLE WITH PREVIOUSLY SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA.
At "ENTER
1 F O R MODE (1), 2 F O R MODE (2) , O R 0 T O TERMINATE
THE PROGRAM" enter 2 and press RETURN.
(With this selection, the user can enter new VLE data from the keyboard, or use previously entered VLE data for correlation with lPVDW or 2PVDW models.) At -DO YOU WANT TO USE AN EXISTING DATA FILE (YIN)?" typeY (or y) and press RETURN. At "INPUT NAME OF T H E EXISTING DATA FILE (for example, a:pe373.dat):"
type a:pe373.dat and press RETURN. (This results in theuseof anexistingdatafilepe373,dat on the diskin your A drive.) At "INPUT INITIAL GUESS F O R BINARY INTERACTION PARAMETER K12:"
type 0.1 and press RETURN. At ,,DO YOU WANT T O F I T Kij T O V L E DATA (1) O R D O YOU WANT T O CALCULATE V L E W I T H VALUE OF Kij ENTERED (Z)?"
type 1 and press RETURN. (At this stage the program is run to optimize the kij parameter and intermediate results will be displayed on the screen as an error bar. Next a message summarizing the results of parameter optimization appears on the screen. Press RETURN to continue after inspecting the results. The information below appears on the screen.)
PENTANE ETHANOL 372.7 K
PKRSE VOLVPlES ARE IN CCIMOL, PRESSURE I S I N UNITS OF THE DATA.
X-EXP
P-EXP
Y-EXP
Y-CAL
.oooo
220.000
.00000
.0830
422.600
.49100
.I710
537.400
.3030
618.800
.4410 .6260
VL-CAL
W-CAL
.00003
69.10
13444.1
.59494
74.41
5037.3
.62900
.64564
80.23
4365.9
.69000
.64764
89.23
4338.9
654.300
.72400
,64948
98.73
4330.2
678.100
.74700
.68772
111.11
4206.9
.7360
684.300
.76800
.73800
118.02
4156.0
.a390
682.600
,80300
.a1050
124.08
4178.2
,9370
658.100
,86000
.91147
129.43
4305.0
.9999
591.000
,99990
.99984
132.65
4466.8
Dress return to continue.
.
(The first, second, and third columns are the measured liquid mole fi-actionof species 1, total pressure, and vapor mole fraction of species I, respectively. The third, fifth, sixth, and seventh columns are total pressui-e, vapor mole fraction, and liquid and vapor saturated phase volumes, respectively, calculated at the experimental liquid mole fractions.) Press RETURN to continue. At ,,DO YOU WANT A PRINT-OUT ( Y I N ) ?" type y (or Y) and press RETURN. (This command sends the results, similar to those shown above, to the printer.) At"D0 YOU WANT TO SAVE THE RESULTS TO AN OUTPUT F I L E ( Y I N ) ? " type y (or Y) and press RETURN At "INPUT A NAME FOR THE OUTPUT FILE:" type A:TEMP2.0UT and press RETURN. (With this command the results shown above are saved in the disk in drive A under the name TEMP2.0UT in ASCII code.) Atr'DO YOU W A N T TO DO ANOTHER VLE CALCULATION ( Y / N ) ? " enter n (or N) and press RETURN.
Example D.3.B: Fittng Binary VLE Data with the Two-parameter van der Waals One-fluid Mixng Rule
Change to the directory containing VDW.EXE (e.g., A> or C>, etc.). Start the program by typing VDW at the DOS prompt and then press RETURN (or ENTER). The program introduction message appears on the screen. Press RETURN to continue.
Append~xD. Computer Programs for B~naryM~xtures
At "VTW:BINARY VLE WITH VAN DER WAALS ONE-FLUID MIXING RULES 1: CONVENTIONAL (1PVDW) 2: 2-PARAMETER COMPOSITION DEPENDENT MIXING RULE (2PVDW) DO YOU WANT T O USE 1-PARAMETER VDW MODEL OR 2-PARAMETER VDW MODEL (1/21?"
type 2 and press RETURN. (This results in the selection of 2PVDW model for the VLE calculations. Next the message below will appear on the screen.) VDW: BINARY VLE CALCULATIONS WITH VAN DER WAALS ONE-FLUID MIXING RULES YOU CAN USE THIS PROGRAM FOR VLE CALCULATION IN TWO WAYS. MODE (1): IF NO T-P-x-y DATA TO COMPARE RESULTS WITH ARE AVAILABLE, YOU m S T SUPPLY CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR, PRSV KAPPA-1 PARAMETER FOR EACH COMPOUND, AND A TEMPERATURE ALONG WITH A (PAIR OF) PREVIOUSLY SELECTED MODEL PARAMETER(S). IN THIS MODE THE PROGRAM WILL RETURN ISOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOSITION RANGE X1=0 TO 1 AT INTERVALS OF 0.1. MODE (2): IF YOU HAVE ISOTHERMAL x-y-P DATA, YOU CAN ENTER THESE DATA FOLLOWING CO-S
TKXT WILL APPEAR ON THE SCREEN
(OR USE PREVIOUSLY ENTERED DATA) TO FIT MODEL PARAMETERS TO THAT VLE DATA. ALTERNATIVELY, YOU CAN CALCULATE VLE WITH PREVIOUSLY SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA.
A ~ ~ ~ E N T1 EFOR R
MODE (1). 2 FOR MODE (2), OR 0 TO TEFXINATE
THE PROGRAM" enter 2 and press RETURN. At "W YOU WANT T O USE AN EXISTING DATA FILE (Y/N)?" type
ll
(or N) and press RETURN.
At "INPUT
.
NEW DATA FILE NAME:" type a:temp3.dat and press RETURN.
(The preceding command will lead to saving a data file named temp3.dat on the disk in the A drive. You must have a disk in the A drive, or choose another directory, by typing c:temp3.out, for example, to save the file on the hard drive.) AtrrINPUTA TITLE FOR THE NEW DATA FILE:" type 'acetone-water temporary file at I00 C' and press RETURN. (You can enter any title up to forty alphanumeric characters to describe your file for later reference.)
A~S~CRITICAL PARAMETERS:
TC=CRITICAL TEMP, K PC=CRITICAL PRESSURE, BAR W= ACENTRIC FACTOR KAP= PRSV KAPPA-1 PARAMETER
INPUT T C ~ ,P C ~ ,w1, KAP~:,,type 508.1,46.96, 0.30667, -0.00888, and
press RETURN.
Modeling Vapor-Lquld Equilbria
-
(These are the pure component constants of acetone for the PRSV EOS from Table 3.1.1.) At "INPUT TC2, PC2, W2, KAP2 : type 647.286,220.90,0.3438, -0.06635, and press RETURN. (These are the pure component constants of water for the PRSV EOS from Table 3.1.1.) At "INPUT NUMBER OF DATA POINTS:" type 3 and press RETURN. At "INPUT TEMPERATURE in K:" type 373.15 and press RETURN. At
"INPUT FACTOR TO CONVERT EXPERIMENTAL PRESSURE DATA TO BAR BY DIVISION (e.g. if original data in mm Hg, type 750 if original data in psia, type 14.5 etc. ) :"type 750 and press RETURN.
At "INPUT EXPERIMENTAL DATA IN LIQUID MOLE FRACTION (XlEXP), 'VAPOR MOLE FRACTION (YlEXP) OF SPECIES 1, BUBBLE POINT PRESSURE (PEXP) (three in a row, separated with commas)" ,,INPUT XIEXP, YIEXP, PEXP:" type 0,108.0.632, 2089.28, and then press
RETURN. n~~~~~
X ~ E X P ,YIEXP, PEXP:" type 0.480,0.747,2606.43, and then press
RETURN. "INPUT XlEXP, YlEXP, PEXP:" type 0.771,0.837,2761.58, and then press
RETURN. When the number of sets of data, specified by NP, here three is entered, the program writes the data to the file with the name temp3.dat and continues. These data now become an existing data file and can be used when this program is run again or when using the other programs described below. This data file looks as shown below if called by an editor program. acetone-water trial file at 100 C
The program then continues as shown below. At "INPUT INITIAL GUESSES FOR BINARY INTERACTION PARAMETERS K12, K21:",
type 0.1,0. 1 and press RETURN.
Appendix D. Computer Programs for B~naryMxtures
(Because the 2PDW model was selected at an earlier stage, here values of two parameters are needed.) At -DO YOU WANT TO FIT Kij TO VLE DATA (1) OR DO YOU WANT CALCULATION OF VLE WITH Kij ENTERED ( 2 ) ? "
type 1 and press RETURN. (With the entry given above, the data fit process is initiated. However, the initial guesses 0.110.1 are not suitable for the acetone-water binary system; therefore, the message below appears on the screen.) INITIAL GUESS(ES) YOU SELECTED
.loo .loo LEADS TO XEGATIW LOG VALUES ENTER 1 TO SELECT ANOTHER (SET OF) Kij VALUE(S) ENTER 2 FOR ANOTHER VLE CALCULATION:
Type 1 and press RETURN. (The preceding entry will give the user the opportunity to tly new initial guesses, as shown below.) At "INPUT INITIAL GUESSES FOR BINARY INTERACTION PARAMETERS K12, K21:",
type 0 . 1 5 , 0 . 1 5 and press RETURN. At "DO YOU WANT TO FIT Kij TO VLE DATA (1) OR DO YOU WANT TO CALCULATE VLE WITH VALUE OF Ki j ENTERED ( 2 ) ?"
type 1 and press RETURN. (At this stage the program runs to optimize ki,. Intermediate results are displayed on the screen in the form of an error bar. Next a summary of optimization results appears on the screen. Press RETURN to continue.) The following results appear on the screen: VDW : VAN DER WAALS MODEL(S) + PRSV EOS VLE PROGRAM
acetone-water temporary file at 100 C K121 -.0728 K21= -.2351 T(K)= 373.15 PHASE VOLUhlES ARE IN CC/MOL, PRESSURE IS IN UNITS OF THE DATA. X-EXP
P-EXP
P-CAL
Y-EXP
Y-CAL
VL-CAL
W-CAL
.lo80
2089.280
2090.522
.63200
,65198
29.60
10604.7
.a800
2606.430
2601.658
.74700
,74949
55.71
8377.3
.7710
2761.580
2788.980
.a3700
.a3700
77.54
7744.4
press return to continue.
Modelng Vapor-Liquid Equ~libr~a
Press RETURN to continue. At ,,DO YOU WANT A PRINT-OUT (Y/N)?" type n (or N) and press RETURN. At "DO YOU WANT T O SAVE THE RESULTS TO A N OUTPUT FILE (Y/N)?" type y (or Y) and press RETURN. At "INPUT A NAME FOR THE OUTPUT FILE:" type A:TEMP3.0UT and press RETURN. (With this com~nandtheresultssimilar to those shown above are saved on the diskin drive A under the name TEMP3.0UT as an ASCII file.)
A ~ " D oYOU
WANT TO DO ANOTHER VLE CALCULATION (YIN)?",
enter n (or N) and press RETURN.
Example D.3.C: Bnary VLE Predctions Using the van der Waals One-Flu~dModel
Change to the directory containing VDW.EXE (e.g., A> or C>, etc.). Start the program by typing VDW at the DOS prompt. Press RETURN (or ENTER). The program introduction message appears on the screen. Press RETURN to continue. At "VDW:BINARY
VLE WITH VAN DER WAALS ONE-FLUID MIXING RULES
1: CONVENTIONAL (1PVDW) 2: 2-PARAMETER COMPOSITION DEPENDENT MIXING RULE (2PVDW) D O YOU WANT T O USE 1-PARAMETER VDW MODEL OR 2-PARAMETER VDW MODEL (1/2)?"
type 2 and press RETURN. The following message will appear on the screen: W W : BINARY VLE CALCULATIONS WITH VAN DER WARLS ONE-FLUID MIXING RULES YOU CAN USE THIS PROGRAM FOR VLE CALCULATION IN TWO WAYS. MODE (1): IF NO T-P-x-y DATA TO COMPARE RESULTS WITH ARE AVAILABLE, YOU MUST SUPPLY CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR, PRSV KAPPA-1 PARAMETER FOR EACH COMPOUND, AND A TEMPERATURE ALONG WITH A (PAIR OF) PREVIOUSLY SELECTED MODEL PARAMETER(S). IN THIS MODE THE PROGRAM WILL RETURN ISOTHERMAL X-y-P PREDICTIONS AT THE TEMPEFATURE ENTERED, IN THE COMPOSITION RANGE X1=0 TO 1 AT INTERVALS OF 0.1. MODE ( 2 ) : IF YOU HAVE ISOTHERMAL x-y-P DATA, YOU CAN ENTER THESE DATA FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN (OR USE PREVIOUSLY ENTERED DATA) TO FIT MODEL PARAMETERS
Appendx D Computer Programs for Binary M~xtures
TO THAT VLE DATA. ALTERNATIVELY, YOU CAN CALCULATE VLE WITH PREVIOUSLY SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA.
At "ENTER 1 FOR MODE ( 1), 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE P R O G ~ enter " 1 and press RETURN. (This example is presented to demonstrate a case for which no experimental VLE data are available, so that no data are entered to, or accessed from, the disk. The user should provide, following the commands that appear on the screen, T,., PC, the acentric factor and K , parameter of the PRSV equation of state For cach compound in addition to a temperature, and the mixing rule paraineter(s) k , j . The program returns isothermal x - y - P predictions at the temperature selected. Repeated temperature entries are allowed.) At "YOU MAY ENTER A TITLE (30 CHAFACTERS
m.) FOR
THE MIXTURE TO BE PREDICTED
(OR YOU m y PRESS RETURN TO SKIP THE TITLE) :"enter acetone-water 100 C
and press RETURN. Atr,TC=CRITICAL TEMPERATURE PC=CRITICAL PRESSURE W=ACENTRIC FACTOR KAPPA=KAPPAl PARAMETER OF THE PRSV EOS INPUT TCI, P C ~ ,WI, KAPPA-I:~enter 508.1,46.96, ,30667, 0 . 0 0 8 8 8 ,
and press RETURN. At "INPUT T C ~ ,P C ~ ,w2, KAPPA-2 :"enter 647.286,220.8975,0.3438, -.06635, and press RETURN. At ,,INPUT TEMPERATURE in K:" type 373.15, and press RETURN.
A ~ ~ ~ I N FACTOR PUT
TO CHOOSE UNITS OF REPORTED PRESSURE
DEFAULT IS BAR, TYPE 1 IF YOU WANT PRESSURE IN BAR. (type 750 if you want pressure i n run Hg, e t c . ) :"enter 750
and press RETURN. At "INPUT BINARY INTERACTION PARAMETERS K12, K21: enter -0.0716, -0.2356, and press RETURN. (At this stage, the program runs and a summary of results appears on the screen. In this case percent error in total pressure is not reported because there is no experimental information. Press RETURN to continue.) The following results appear on the screen: "
W W : VAN DER
WAUS MODEL(S) + PRSV EOS
acetone-water 100 C K12= -.0716 K21= -.2356
T(K)= 373.15
VLE PROGRAM
PHASE VOLUMES ARE IN CC/NOL. FACTOR YOU ENTERED TO CONVERT PRESSURE FROM BIUl IS: 750.00 X-EXP
.oooo .I000 .2000 .3000 .do00 .5000 .GOO0 .7000 ,8000 .goo0 1.0000
P-EXP .
-
-
P-CAL
Y-EXP
Y-CAL
W-CAL
761.381
-
.00004
30295.7
.
.64497
10809.0
.69616
9434.4
.71942
8899.2
.73674
8568.7
.75374
8311.1
.77401
8081.3
2053.378
2461.554
-
2548.672
.
2331.920
2833.493
-
2799.596
-
2620.572 2687.615 2751.583 2806.006
,80135
7868.8
.a4103
7688.7
.go169
7583.4
1.0000
7640.9
prese return t o continue.
Press RETURN to continue. At ,,DO YOU WANT A PRINT-OUT (Y/N)?" type y (or Y) and press RETURN. At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?" type y (or Y) and press RETURN. At "INPUT A NAME FOR THE OUTPUT FILE:" type C: temp4.0ut and press RETURN. (With this command results similar to those reported above are written to a file named temp4.out in the C directory.) At "DO YOU WANT TO DO A VLE CALCULATION AT ANOTHER TEMPERATURE (Y/N)?"
type Y (or y) and press RETURN. (This entry allows the user to calculate VLE at another temperature for the binary mixture under consideration.) At "INPUT TEMPERATURE i n K:" type 425.15 K and press RETURN. At "INPUT BINARY INTERACTION PARAMETERS K12, K21: " type -0.0716, -0.2356 and press RETURN. (A summary of results appear on the screen. Press RETURN to continue.) The following results appear on the screen: VDW - VAN DER WAALS MODEL(S1 + PRSV EOS YLE PROOFAM acetone + water 100
c
Append~xD: Computer Programs for Bnary Mxtures
PHASE V O L W S ARE IN CCIMOL
.
FACTOR YOU ENTERED TO CONVERT PRESSURE FROM BAR IS: 750.00
X-EXP
P-EXP
-
.oooo .I000 .zoo0
P-CAL
Y-EXP
W-CAL
7145.493
-
3010.2
3768.422 6212.298
6818.6 3903.6 3313.4
.3000
-
7734.740
.
.4000
-
8171.392
-
2810.6
.5000
-
8516.838
.
2663.8
,6000 ,7000 ,8000 .goo0 1.0000
-
8796.600 9014.079 9151,503 9167.150 8989.156
-
2549.2 2459.0 2393.6 2361.1 2378.9
press return to continue.
.
Press RETURN to continue. At ,*no YOU WANT A PRINT-OUT (Y/N)?" type N (or n) and prcss RETURN. At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?" type y (or Y) and press RETURN. At "INPUT A NAME FOR THE OUTPUT FILE:" type c: temp4.out and prcss RETURN. (With this command the results above are appended to the file temp4.out opened earlier to save the results of previous VLE predictions at 373.15 K.) At
'DO YOU WANT TO DO A VLE CALCULATION AT ANOTHER TEMPERATURE (Y/N)?"
type N (or n) and press RETURN. At "W YOU WANT TO W ANOTHER VLE CALCULATION (Y/N)?",type n (or N) and press RETURN.
D.4. Program HV: Binary VLE with the Huron-Vidal Mixing Rule (HVO) and Its Modifications (MHVI,MHV2, LCVM,and HVOS) The program HV can be used to calculate VLE using the PRSV EOS and one of the following Gibbs excess free-energy-based mixing rules: HVO, MHVI, MHV2, LCVM, or HVOS. This program allows the NRTL, van Laar. Wilson, or the UNIQUAC excess free-energy models to be used in the EOS formalism. Any mixing rule and excess free-energy model combination can be chosen during the program execution following the directions that appear on the screen. A tutorial is provided in this
section. The program can be used in two ways. When isothermal VLE data are available, the program can be used to calculate VLE with model parameters provided by the user at measured liquid mole fractions and to compare the calculated bubble pressures and vapor mole fractions with the measured values. Alternatively, the program can be used to obtain parameters of a selected model by fitting them to measured liquid composition versus bubble pressure data with a simplex algorithm. In this mode the program reads previously stored data or accepts new data entered from the keyboard. The data structure is identical to that described in Section D.3 for the program VDW, and details concerning the data input requirements can be found there. The data files that can be used by this program are those on the disk with the DAT extension. In this mode, initial guesses for model parameters must be provided by the user for the parameter optimization by the simplex method. The initial guesses can be positive or negative numbers. The input parameters required are in reduced forin, and a value between zero and one for each is usually satisfactory. If convergence is not achieved with a set of initial guesses, the user should t ~ again y with a different choice of parameters. In the absence of any experimental VLE data, the program can he used to calculate VLE at a given temperature using internally generated liquid mole fractions of component 1 from 0 to 1 at intervals of 0.1. In this case the user only needs to supply the critical temperature, critical pressure, acentric factor, and PRSV K , parameter for each compound, and a temperature as input following the directions that appear on the screen. In this mode the program will return isothermal x-y-P predictions at the temperature entered in the composition range x, = 0 to 1 at intervals of 0. I . Several temperature values can be selected successively. A tutorial is provided below (sce Example D.4.C). The results from the program HV can be sent to a printer. to a disk file, or both. To make this choice, the commands that appear on the screen upon the completion of calculations must be followed. Please see the following tutorial for further details.
Tutorial on the Use of HV.EXE Example D.4.A:Use o f the HVO Model t o Correlate Experimental Data
Change the directory containing HV.EXE (e.g., A> or C>, etc.). Start the program by typing HV at the DOS prompt. Press RETURN (or ENTER). A program introduction message appears. Press RETURN to continue. The following appears on the screen: HV: BINARY VLE CALCULATIONS WITH THE HURON-VIDPL NODEL AND ITS VARIATIONS
YOU CAN USE THIS PROGRAM FOR VLE CALCULATION IN TWO WAYS. MODE (1): IF NO T-P-x-y DATA TO COMPARE RESULTS WITH ARE AVAILABLE,
Appendix D: Computer Programs for Binary Mixtures
YOU MUST SUPPLY CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR, PRSV mPPA-1 PRRAMETER FOR EACH COMPOUND, AND A TEMPERATrmE ALONG WITH A SET OF PREVIOUSLY SELECTED MODEL PRRAMETERS. IN THIS MODE THE PROGRAM WILL RETURN ISOTHERhlAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED, IN THE COMPOSITION RANGE X 1 = 0 TO 1 AT INTERVALS OF 0.1. MODE (2): IF YOU HAVE ISOTHERMAL x-y-P DATA, YOU CAM ENTER THESE DATA FOLLOWING C O W S THAT WILL APPEAR ON THE SCREEN (OR USE PREVIOUSLY ENTERED DATA) TO FIT MODEL PAXAMETERS TO THE VLE DATA. ALTERNATIYELY, YOU CAN CALCULATE VLE WITH PREVIOUSLY SELECTED P-TERS
AND COMPARE THE RESULTS WITH THE VLE DATA.
At "ENTER 1 FOR MODE (1). 2 FOR MODE ( Z ) ,
OR 0 T O TERMINATE THE PROGRAM"
type 2 and press R E T U R N . (This selection allows the entry of new VLE data from the keyboard or use of previously entered VLE data.) Atr'SELECT A MIXING RULE MODEL: WV-O= HURON-VIDAL ORIGINAL MHV1= MODIFIED HURON-VIDAL 1ST ORDER MHV2= MODIFIED HURON-VIDAL 2ND ORDER LCVM= LINEAR COMBINATION OF HURON-VIDAL AND MICHELSEN W O S = HURON-VIDAL MODIFIED BY ORBEY AND SANDLER
O=EXIT I=HV-o Z = M H V ~3=MHV2 ~ = L C V M5 = W O S u type5 andpress RETURN.
(This results in the selection of the HVOS model for the mixing rule.) At "SELECT AN EXCESS FREE ENERGY MODEL: l=NRTL 2=VAN LAAR 3=UNIQUAC 4=WILSONrrtype 3 and press R E T U R N . (This results in the selection of the UNIQUAC model for the excess free-energy term in the HVOS mixing rule.) At "THE UNIQUAC MODEL WILL REQUIRE PURE COMPONENT PARAMETERS R, Q, Q'. IF YOU DO NOT HAVE THEM PLEASE SELECT ANOTHER MODEL."
At "DO YOU WANT T O SELECT ANOTHER ACTIVITY COEFFICIENT MODEL (Y/N)?"
type n (or N) and press R E T U R N .
At
"DO YOU WANT TO USE AN EXISTING DATA FILE (Y/N)?"type Y
(or y) and press R E T U R N . At "INPUT THE NAWZ OF EXISTING DATA FILE (for example a:pe373.dat)"
type a:awlOO.dat. (Both pe373.dat and aw100.dat are data files provided on the accompanying disk.
Modeng Vapor-L~qud Equlibr~a
The aw100.dat file contains isothermal VLE data for the acetone-water binary system at 100" C.) At
"EHER UNIQUAC
PURE COMPONENT SURFACE AND VOLUME PARAMETERS
INPUT W I Q U A C PARAMETERS R1, Q1, Ql' :"type 2.57, 2.34, 2.34, and press
RETURN. (Thcse are the surface and volume parameters for component 1, acetone, obtained from Prausnitz et al. 1980, p.145.) At "INPUT W I Q U A C PARAMETERS R2, Q2, Q2 ' :'' type 0.92,1.4,1.0, and press RETURN. (These are the surface and volume parameters for component 2, water.) At "INPUT INITIAL GUESSES FOR PI2 AND P21 OF THE W I Q U A C MODEL [PIJ=EXP(-AIJ/ (RT)) , AIJ IN CAL/MOLl :"type 1, 1 and press RETURN.
Atr'DO YOU WANT TO FIT THE PARAMETERS TO VLE DATA (1) OR DO YOU WANT TO CALCULATE VLE WITH THE VALUES OF THE PARAMETERS JUST ENTERED (2)?" type I and press
RETURN. (At this stage the program attempts to optimize the two model parameters of the UNIQUAC model, and intermediate results will be continuously displayed on the screen as an error bar. When the optimization is completed, a summary of the results appears on the screen. Press RETURN to continue. The results given below appear on the screen.)
W:
BINARY VLE CALCUT.AT1ONS WITH THE HURON-VIDAL MODEL ANE ITS VARIATIONS
ACETONE WATER 100 C EOS MODEL = HVOS; EXCESS ENERGY MODEL = W I Q U A C UNIQUAC PI2 (=EXPI-A12IRT)l .2900 UNIQUAC P21(=EXP(-A2lIRT)l ,9947 TEMPERATURE in K: 373.15 PHASE VOLUMES ARE IN CCIMOL, PRESSURE IS IN UNITS OF THE DATA X-EXP
P-EXP
P-CAL
Y-EXP
Y-CAI.
VL-CAL
W-CAL
.0033
832.610
845.123
.09020
,10260
22.72
27219.5
.0040
848.120
862.565
.lo900
.I2111
22.77
26654.6
,0045
879.150
874.893
.11800
.I3376
22.80
26269.0
,0080
977.410
958.267
.20700
.a1077
23.03
23922.8
.0480
1680.730
1630.24
,54500
.54615
25.63
13786.0
.0820
1835.880
1938.15
.61300
.62416
27.87
11488.2
.0980
2001.370
2034.17
.63700
,64413
28.93
10913.5
.lo80
2089.280
2083.01
.63200
.65372
29.59
10641.3
.2200
2301.310
2334.17
.70500
.69929
37.18
9420.4
Appendx D: Computer Programs for Binary Mixtures
press return to see more results on the soreen.
Press RETURN to continue
press return to continue.
Press RETURN to continue. At IDO YOU WANT A PRINT-OUT (Y/N)?" type y (or Y) and press RETURN. At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?" type y (or Y) and press RETURN. At nINPUT A FILE N?XE FOR THE OUTPUT FILE:" typeA:TEMP4.0UT and press RETURN. (With this command the results shown above are saved on the disk in drive A under the name TEMP4.0UT in ASCII code.) At "DO YOU WANT TO DO ANOTHER VLE CALCULATION (YIN)?" type n (or N) and press RETURN. Example D.4.B: Use o f the HV Model t o Correlate New Data
Change the directory containing HV.EXE (e.g., A> or C>, etc.). Start the program by typing HV at the DOS prompt. Press RETURN (or ENTER). A program introduction message appears on the screen. Press RETURN. The following message appears on the screen: HV; BINARY VLE CALCYLATIONS WITH THE IWRON-VIDAL MODEL AND ITS VARIATIONS YOU CAN USE THIS PROGRAM FOR VLE CALCULATION IN TWO WAYS. MODE (1): IF NO T-P-x-y DATA TO COMPARE RESULTS WITH ARE AVAILRBLE, YOU MUST SUPPLY CRITICAL TEMPERA=,
CRITICAL PRESSURE, ACENTRIC FACTOR,
PRHV KAPPA-1 PARAMETER FOR EACH COMPOUND, AND A TEMPERATURE
Modeling Vapor-Liquid Equ~lbria
ALONG WITH A SET OF PREVIOUSLY SELECTED MODEL PARAWETERS. IN THIS MODE THE PROGRAM WILL RETURN ISOTHERMAL X-Y-P PREDICTIONS AT THE TEMPERAT=
ENTERED IN THE COMPOSITION RANGE X1=0 TO 1
AT INTERVALS OF 0.1. MODE ( 2 ) : IF YOU HAVE ISOTHERMAL x-y-P DATA, YOU CAN ENTER THESE DATA FOLLOWING COMMANDS TRAT WILL APPEAR ON THE SCREEN (OR USE PREVIOUSLY ENTERED DATA) TO FIT MODEL PARAMETERS TO THE VLE DATA. ALTERNATIVELY, YOU CAN CALCULATE VLE WITH PREVIOUSLY SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA.
"ENTER 1 FOR MODE (I), 2 FOR MODE (2), OR 0 TO TERMINATE THE PROGRAM"
type 2 and press RETURN. (This selection allows the entry of ncw VLE data from the keyboard or use of previously entered VLE data.)
A~"SELECTA
MIXING RULE MODEL:
EN-O= HURON-VIDAL ORIGINAL MINI= MODIFIED HURON-VIDAL 1ST ORDER MIN2= MODIFIED HURON-VIDAL 2ND ORDER LCVM=LINEAR COMBINATION OF HURON-VIDAL AND MICHELSEN W O S = HURON-VIDAL MODIFIED BY ORBEY AND SANDLER O=EXIT i = m - o 2 = m ~ 1 3 = m 2 ~=LCVM:5=EN0Srrtype l and press
RETURN. (This results in the selection of the original Huron-Vidal model, HVO, for the mixing rule model.)
At
"SELECT AN EXCESS FREE ENERGY MODEL: l=NRTL 2 = V m WLR 3=UNIQUAC 4=WILSON?" type
2 and press
RETURN. (This results in the selection of the van Laar model for the excess free-energy term in the HVO mixing rule.) Atlroo YOU WANT TO USE AN EXISTING DATA FILE (YIN)?" typeN (or n) and press RETURN. At ,,INPUT NEW DATA FILE N-:" type a:temp4.dat and press RETURN. (The preceding command results in saving a data file named temp4.dat 011 the disk in drive A. You must have a disk in the A drive. or select another drive by typing, for example, c:temp4.ont, to save the results in the hard drive.) At "INPUT A TITLE FOR THE NEW DATA FILE:" type 'methanol - water trial data at 100 C' and press RETURN. (Yon can enter any title composed of up to forty alphanumeric characters to describe your file for later reference.)
Appendx D: Computer Programs for Binary Mixtures
. A~~~CRITICAL
PARAMETERS: TC=CRITICAL TEMP, K PC=CRITICAL PRESSURE, BAR W=ACENTRIC FACTOR KAP=THE PRSV EOS KAPPA1 PARAMETER
-
INPUT TC1, PC1, W1, KAP1:" type512.58,80.9579,0.56533, -0.16816, and press RETURN. (These are constants of methanol for the PRSV EOS from Table 3.1.1.) At "INPUT TC2, PC2, W2, KAP2 :,,type 647.286,220.8975,0.3438, -0.06635, and press RETURN. (These are constants of water for the PRSV EOS from Table 3.1.1 .) At "INPUT N ~ B E ROF DATA POINTS:', type 3 and press RETURN. At nINPUT TEMPERATURE in K:" type 373.15 and press RETURN. At
"INPUT FACTOR TO CONVERT EXPERIMENTAL PRESSURE DATA TO BAR BY DIVISION (e.g. if original data in nun Hg, type 750 if original data in gsia, type 14.5 etc. ) " type
750 and press RETURN.
At "INPUT EXPERIMENTAL DATA IN LIQUID MOLE FRACTION (XlEXP) OF SPECIES 1, VAPOR MOLE FRACTION (YIEXP) OF SPECIES 1, BUBBLE POINT PRESSURE (PEXP) (three in a row, segarated by commas)". "INPUT X1EXP.YlEXP.PEXP:~~type 0.035,0.191,931, and press RETURN. "INPUT X1EXP.YlEXP. PEXP: " type 0.28 1, 0.619, 1535.96, and press
RETURN. ,,INPUT X~EXP,YIEXP,PEXP:'~type 0.826,0.911,2337.76, and press RETURN. (When the number of items of data, specified by NP, here three, is entered, the program writes the data to a file under the name temp4.dat as specified above and continues. This data file becomes an existing data file and can be used when this program or other EOS programs are run again. This data file appears as shown below if called by an editor program.)
methanol-water trial data at 100 C
Modeling Vapor-Liquid Equlibra
(Note that this format is exactly the same as that of the input data created following the tutorialgiveninExampleD.3.B as all EOS programs use the same datastructure. When the data entry process is complete, the program continues as below.)
AI"INPUT INITIAL
GUESSES FOR VAN LMR P A R A ~ ~ ~ E T ~ ER 1 S2 ,~ 2 1
(PIJ ARE DIMENSIONLESS PARAMETERS OF THE VAN LMR MODEL)"
type 1, 1 and press RETURN. At "DO YOU WANT TO FIT THE PARAMETERS TO VLE DATA (1) OR DO YOU WANT TO CALCULATE VLE WITH THE VALUES OF THE PARAMETERS JUST ENTERED (2) ?" type 1 and press RETURN.
(At this stage the program attempts to optimize the two model parameters of the van Laar model, and the intermediate results are continuously displayed on the screen in the form of an error bar. When the optimization is complete, a message displaying summary of results appears on the screen for inspection. Press RETURN to continue. The results given below appear on the screen.)
W : BINARY VLE CALCULATIONS WITH THE HURON-VIDAL MODEL AND ITS VARIATIONS methanol-water trial data at 100 C EOS MODEL = W O ; EXCESS ENERGY MODEL = VAN LAAR PlZ(=DIMENSIONLESS KAPPA12 OF VAN LAAR) 1.4468 P21(=DIMENSIONLESS KAPPA21 OF VAN LAAR) .6705 TEMPERATURE T(K) 373.15 PHILSE VOLUMES RRE IN CCIMOL, PRESSURE IS IN UNITS OF THE DATA. X-EXP
P-EXP
P-CAL
Y-EXP
Y-CAL
VL-CAL
W-CAL
.0350
931.000
933.475
.I9100
.21068
23.50
24622.4
.2810
1535.960
1535.978
.61900
.60611
30.58
14778.0
.a260
2337.760
2368.482
.91100
.92059
47.26
9405.2
press return to continue.
Press RETURN to continue. At "DO YOU WANT A PRINT-OUT (Y/N)?" type y (or Y) and press RETURN. (With this command the results shown above are sent to the printer.) At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?" type y (or Y) and press RETURN. At "INPUT A NAME FOR THE OUTPUT FILE:" typeA:TEMP4,0UT and press RETURN. (With these commands the results shown above are saved on a disk in drive A under the name TEMP4.0UT in ASCII code.) At "DO YOU WANT TO DO ANOTHER VLE CALCULATION (YIN)?" type n (or N) and press RETURN.
Appendx D: Computer Programs for Binary M~xtures
Example
D.4.C: Binary VLE Predictions Using the Huron-Vidal Model
Change to the directory containing HV.EXE (e.g., A> or C>, etc.). Start the program by typing HV at the DOS prompt. Press RETURN (or ENTER). The program introduction message appears on the screen. Press RETURN. The following message appears on the screen: MI: BINARY VLE CALCULATIONS WITH THE HURON-VIDAL MODEL AND ITS VARIATIONS
YOU CAN USE THIS PROGRAM FOR VLE CALCULATION IN TWO WAYS. MODE (1): IF NO T-P-X-y DATA TO COMPARE RESULTS WITH ARE AVAILABLE, YOU MUST SUPPLY CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR, PRSV KAPPA-1 PARAMETER FOR EACH COMPOUND, AND A TEMPERATURE ALONG WITH A SET OF PREVIOUSLY SELECTED MODEL PARAMETERS. IN THIS MODE THE PROGRAM WILL RETURN ISOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOSITION RANGE X1=0 TO 1 AT INTERVALS OF 0.1. MODE ( 2 ) :
IF YOU
h7AVE
ISOTHERMAL x-y-P DATA, YOU CAN ENTER THESE DATA
FOLLOWING COMMRNDS THAT WILL APPEAR ON THE SCREEN (OR USE PREVIOUSLY ENTERED DATA) TO FIT MODEL PARAMETERS TO THE VLE DATA. ALTERNATIVELY, YOU CAN CALCULATE VLE WITH PREVIOUSLY SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA.
At "ENTER 1 F O R MODE (1), 2 F O R MODE ( 2 ) , O R 0 T O TERMINATE T H E PROGRAM"
type 1 and press RETURN. (This results in the program being used in the predictive mode. This example is presented to demonstrate a case for which no experimental VLE data are available. In this case no data are entcred to, or accessed from, the disk. The user must provide, following the commands that appear on the screen, T., P,,the acentric factor, and the K I parameter of the PRSV equation of state for each compound in addition to a temperature and model parameter(s) for the selected model. The program returns isothermal x-y-Ppredictions at the temperature selected. Repeated temperature entries are allowed.)
A ~ ~ ~ S E L AE CMIXING T
RULE MODEL:
HV-O= HURON-VIDAL ORIGINAL MHV1= MODIFIED HURON-VIDAL 1ST ORDER M W 2 = MODIFIED HURON-VIDAL 2 N D ORDER LCVM=LINEAR COMBINATION OF HURON-VIDAL A N D MICHELSEN ?NOS=
HURON-VIDAL MODIFIED BY ORBEY AND SANDLER
O=EXIT 1 = m - o 2
=
~ 3=MHV2 ~ 1 4=LCVM 5=HVOSr' type
5 and press
RETURN. (This commands results in the use of the HVOS model for the mixing rule model.)
Modeling Vapor-Lquid Equiibra
A ~ T ~ S E L EAN C TEXCESS
FREE ENERGY MODEL:
1=NRTL 2=VAN I.AAR 3=UNIQUAC 4=WILSON" type
1 and press RETURN.
(This command results in the selection of the NRTL model to be used as the excess free-energy term in the HVOS mixing rule.) At "YOU MAY ENTER A TITLE (30 CHARACTERS MAX. ) FOR THE MIXTURE TO BE PREDICTED (OR YOU MAY PRESS RETURN TO SKIP THE TITLE) :" entermethan~l-water 100 C and
press RETURN. A~"TC=CRITICAL T E ~ E R A T U R E PC=CRITICAL PRESSURE W=ACENTRIC FACTOR KAP=KAPPAl PARAMETER OF THE PRSV EOS INPUT T C ~ ,P C ~ ,w1, KAP-1:" enter 512.58,80.9579,0.56533,
-0.16816, and press RETURN.
At n~~~~
T C ~ ,P C ~ ,w2, ~ ~ p - enter 2 : 647.286,220.8975, ~ ~ 0.3438,-.06635, and prets RETURN. At ,,INPUT TEMPERATURE in K:" type 373.15 and press RETURN.
At "INPUT FACTOR TO CHOOSE UNITS OF REPORTED PRESSURE DEFAULT IS BAR, TYPE 1 IF YOU WANT PRESSURE IN BAR. (type 750 if you want pressure in rmn. Hg, etc. ) * enter 750 and press
RETURN. At "INPUT ALPHA OF THE NRTL MODEL:" type 0.35 and press RETURN. Atr'INPUT REDUCED NRTL PARAMETERS P12, P21: [PIJ =AIJ/ (RT) AND AIJ IN CAL/MOLl" enter 0.5, 0.5 and press RETURN. (The program then runs, and a summary of the results appears on the screen. In this case the percent error in total pressure is not reported because there is no experimental information. Press RETURN to continue.) The following results appear on the screen:
HV: BINARY VLE CALCULATIONS WITH THE HURON-VIDAL MODEL AND ITS VRRIATIONS
methanol-water 1 0 0 C EOS MODEL = HVOS; EXCESS ENERGY MODEL= NRTL ALPHA= . 3 5 0 0 NRTL PlZ(=AIZIRT) . 5 0 0 0 NRTL PZl(=AZlIRT) .SO00 TEMPERATURE in
K:
373.15
PHLSE VOLUMES ARE IN CC/MOL. FACTOR YOU ENTERED TO CONVERT PRESSURE FROM BAR IS: 7 5 0 . 0 0
Appendix D: Computer Programs for B~naryMxtures
X-EXP .OOOO
P-EXP
P-CAL
Y-EXP
Y-CAL
VL-CAL
W-CAL
-
760.002
-
.00001
22.51
30351.2
~ r e s sreturn to continue.
.
Press RETURN to continue. At -DO YOU WANT A PRINT-OUT At"D0 YOU WANT TO SAVE THE type n (or N) and press RETURN. At
( Y IN) ?"
type N (or n) and press RETURN.
RESULTS TO AN OUTPUT F I L E ( Y I N ) ? "
"DO YOU WANT TO DO VLE CALCULATION AT ANOTHER TEMPERATURE
.
type N (or n) and press RETURN. A~,,Do YOU WANT TO W ANOTHER (or N) and presr RETURN.
VLE CALCULATION
(Y/N)?"
(Y/N)?"Lypen
D.5. Program WS: Binary VLE from Wong-Sandler Mixing Rule The program WS is used to calculate VLE using the PRSV EOS and the WongSandler Mixing rule. One of four (UNIQUAC, van Laar, Wilson, or NRTL) excess free-energy models can be ured wlth this mixing rule following the instructions that appear on the screen during program execution. This program can be used in two ways, as shown in the tutorial that follows. If measured isothermal VLE data are available, the program can he used to calculate VLE at the measured liquid mole fractions with user-provided model parameterr and to compare the calculated bubble pressures and vapor mole fraction5 with the experimental ones. Alternatively, the program can he used to opti~nizeparameters of a selected model by fitting parameters to measured liquid mole fraction versus bubble pressure data. Parameter optimization is done using a simplex algorithn~.In this mode the program scads previously stored data or accepts new data entered from the keyboard. The input data structure is identical to that described in
Modeling Vapor-Lquid Equilibria
Section D.3 for the program VDW, and the details concerning the input data can be found there. During parameter optimization, as a requirement of the simplex method, an initial guess must be provided for each parameter. The initial guesses may be positive or negative numbers; they are in reduced form and thus a value of between zero and one is a useful choice in many cases. However, depending on the nonideality of the mixture, an initial guess may need to be significantly different from unity in order for the program to converge. If convergence cannot be achieved with a set of initial guesses, the user should try again with different initial guesses (see Examples D.5.A and D.5.B). If no experimental VLE data are available, the program can he used for predictions using internally generated liquid mole fractions of species 1 in the range from 0 to 1 at intervals of 0.1. In this case the user must provide all model parameters and temperature in addition to pure component critical temperature and pressure, acentric factor, and the K , parameter of the PRSV equation of state for each compound. An example is given below (Example D.5.C) for this mode of operation of the program. The results from the program can be scnt to a printer, to a disk filc, or both. This choice is made following the commands that appear on the screen upon completion of the calculations. Please see the following tutorial for further details.
Tutorial on the Use of WS.EXE Example D.5.A: Use o f the WS Model t o Correlate Data Change the directory containing WS.EXE (e.g., A> or C>,). Start the program by typing WS at the DOS prompt. Press RETURN (or ENTER). A message introducing the program appears on the screen. Press RETURN to continue. The following appears on the screen: WS: BINARY VLE CALCULATIONS WITH THE WONG-SANDLER MIXING RULE YOU CAN USE THIS PROGRAM FOR VLE CALCULATION IN TWO WAYS. MODE (1): IF NO T-P-x-y DATA TO COMPARE RESULTS WITH ARE AVAILABLE, YOU MUST SUPPLY CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR, PRSV KAPPA-1 PARAMETER FOR EACH COMPOUND, AND A TEMPERATURE ALONG WITH A SET OF PREVIOUSLY SELECTED MODEL PARAMETERS. IN THIS MODE THE PROGRAM WILL RETURN ISOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOSITION RANGE X1=0 TO 1 AT INTERVALS OF 0.1. MODE (2): IF YOU HAVE ISOTHERMU x-y-P DATA, YOU CAN ENTER THESE DATA FOLLOWING C O W S THAT WILL APPEAR ON THE SCREEN
Appendx
D.Computer Programs for Elnary Mxtures
(OR USE PREVIOUSLY ENTERED DATA) TO FIT MODEL PARAXETERS TO THE VLE DATA. ALTERNATIVELY, YOU CAN CALCULATE VLE WITH PREVIOUSLY SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA.
.
At "ENTER 1 FOR MODE (1), 2 FOR MODE ( 2 ) , OR 0 TO TERMINATE THE PROGRAM" type 2 and press RETURN. (This selection allows the entry of new VLE data from the keyboard or use of previously entered data.)
At "DO YOU WANT TO DO A PARAMETER FIT (ENTER A) OR CALCULATION WITH
PREVIOUSLY FITTED PARAMETERS (ENTER B)?,I type
A and press RETURN.
At "ENTER NVMBER OF PARAMETERS TO BE FIT (2 OR 3) (2): TWO PARAMETERS OF EXCESS FREE ENERGY MODEL ARE FIT (3): IN ADDITION K12 PARAMETER OF THE WS MODEL IS FIT (other parameters such as alpha of the NRTL model, or UNIQUAC pure component parameters must be supplied by user.):"
-
type 3 and press RETURN. (With this command, all three parameters in the WS mixing mle, the two exces, free-energy model parameters, and the hinary interact~onparameter, k,, , are optimized.) At -DO YOU WANT TO USE AN EXISTING DATA FILE (YIN)?" type y (or Y) and press RETURN.
At
"INPUT NAWE OF THE EXISTING DATA FILE (for example a:pe373.dat):"
type a:pe373.dat.
A~"SELECTAN
EXCESS FREE ENERGY MODEL:
O=EXIT ~ ~ W I Q U A C 2 = v m LAAR 31WILSON ~ = N R T L "type 2 and press
RETURN. (This results in selection of the van Laar model for the cxcess-energy term in the WS mixing rule.)
At "PARAIIIETERS PI2 AND P21 ARE REDUCED AS DESCRIBED BELOW. AIJ ARE PARAMETERS AS TABULATED IN THE DECHEMA TABLES.
.
FOR UNIQUAC, PIJ=EXP (-AIJ/RT)
FOR VAN -, PIJ=AIJ. FOR WILSON, PIJ=(VLPJ/VLPI)*EXP(-AIJIRT). FOR NRTL, PIJ=AIJ/RT. WITH THIS REDUCTION, IT IS POSSIBLE TO USE INITIAL GUESSES IN THE W G E OF ZERO TO ONE. INITIAL VALUES RECOMMENDED FOR PI2 AND P21 ARE 0.1. INPUT INITIAL GUESSES FOR P12, P21:,, typeO.l, 0.1 and press RETURN
Modeling Vapor-Liqud Equilbria
At "INPUT INITIAL GUESS FOR THE WS MIXING-RULE PARAMETER K12:"
type 0.3 and press RETURN. (At this stage the program is run to optimize the two parameters of the NRTL model and the binary interaction parameter, k;,. Intermediate results will be continuously displayed on the screen in the form of an error bar. When the optimization is complete, a message summarizing the results appears on the screen for inspection. Press RETURN to continue.) The following results appear on the screen: WS: THE WONG-SANDLER MIXING RULE FOR BINRRY VLE CALCUIATIONS PENTANE ETHANOL 372.7 K EXCESS ENERGY MODEL = VAN LAAR K12= .3084 PlZ(=DIMENSIONLESS KAPPA12 OF VAN LAARI 1.2172 PZl(=DIMENSIONLESS KAPPA21 OF VAN LAAR) 2.9709 TEMPERATURE in K: 372.70 PHASE VOLUMES ARB IN CCIKOL, PRESSURE IS IN UNITS OF THE DATA. X-EXP
P-EXP
P-CAL
Y-EXP
Y-CAL
.oooo
220.000
220.608
.00000
.0830
422.600
392.650
,49100
.I710
537.400
517.326
,3030
618.800
618.967
.4410
654.300
.6260
678.100
.7360
684.300
.a390
682.600
.9370 .9999
VL-CAL
W-CAL
.00011
69.10
13461.1
.46706
70.22
7335.6
,62900
.61874
72.21
5396.2
.69000
.70420
77.00
4376.9
660.413
,72400
.73733
84.65
4045.8
678.399
,74700
.75796
99.44
3911.1
683.632
.76800
.77313
110.22
3869.2
682.601
,80300
.80117
120.53
3864.0
658.100
657.648
.a6000
.a6940
128.94
4006.8
591.000
591.174
.99990
,99964
132.65
4470.6
press return to continue.
Press RETURN to continue. At -DO YOU WANT A PRINT-OUT (Y/N)?" type y (or Y) and press RETURN. (With this command the results, shown above, are sent to the printer.) At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)? " type y (or Y) and press RETURN. At "INPUT A NAME FOR THE OUTPUT FILE": type A:TEMPS.OUT and press RETURN. (With this command the results shown above are saved on the disk in drive A under the name TEMP5.OUT in ASCII code.)
Appendix D: Computer Programs for Bnary Mxtures
At "DO YOU WANT T O W (or N) and press RETURN.
ANOTHER VLE CALCULATION (Y/N)?"
type ll
Example D.5.B: Use of the WS Model t o Correlate Data
Change the directoty containing WS.EXE (e.g., A> or C>, etc.). Start the program by typing WS at the DOS prompt. Press RETURN (or ENTER). The message introducing the WS program appears on the screen. Press RETURN to continue. The following appears on the screen: WS: BINARY VLE CALCULATIONS WITH THE WONG-SANDLER MIXING RULE YOU CAN USE THIS PROGRAM FOR VLE CALCULATION IN TWO WAYS. MODE (1): IF NO T-P-x-y DATA TO COMPARE RESULTS WITH ARE AVAILABLE, YOU MUST SUPPLY CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR, PRSV KAPPA-1 PARAlG3TER FOR EACH COMPOUND, AND A TEMPERATURE ALONG WITH A SET OF PREVIOUSLY SELECTED MODEL PARAEIETERS. IN THIS MODE THE PROGRAM WILL RETURN ISOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOSITION RANGE X1=0 TO 1 AT INTERVALS OF 0.1. MODE (2): IF YOU HAVE ISOTHERMAL x-y-P DATA, YOU CAN ENTER THESE DATA FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN (OR USE PREVIOUSLY ENTERED DATA1 TO FIT MODEL PARAMETERS TO THE VLE DATA. ALTERNATIVELY, YOU CAN CALCULATE VLE WITH PREVIOUSLY SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA.
At (I), 2 FOR MODE (2). O R 0 TO TERMINATE THE PROGRAM" type 2 and press RETURN. (This selection allows the entry of new VLE data from the keyboard or use of previously entered data.) At
"ENTER 1 FOR MODE
"DO YOU WANT T O DO A PARAWETER FIT (ENTER A ) OR CALCULATION WITH
PREVIOUSLY FITTED PARAWETERS (ENTER B) ?" type A and press RETURN.
At "ENTER
NUMBER OF PARAMETERS T O B E F I T (2 OR 3)
(2) : TWO PARAMETERS OF EXCESS FREE ENERGY MODEL ARE FIT
(3) : I N ADDITION K12 PARAWETER OF THE WS MODEL IS FIT (other parameters such as alpha of the NRTL model, or UNIQUAC gure component parameters must b e supplied
by user. ) :"type 2 and press RETURN. (With this command, the excess free-energy parameters in the WS mixing rule are optimized. The initlal value of the binary interaction parameter, k,,, is used in computations.)
Modeling Vapor-Lquld Equibria
At "DO YOU WANT TO USE AN EXISTING DATA FILE (Y/N)?" typen (or N) and press RETURN. At "PROVIDE THE FOLLOWING INPUT INFORMaTION INPUT NEW DATA FILE NAME:"
type tempo6.DAT and prev RETURN. At "INPUT A TITLE FOR THE NEW DATA FILE:" type methanol-water at 373 K and prcss RETURN At "CRITICAL PARAMETERS: TC=CRITICAL TEMPERATURE, K PC=CRITICAL PRESSURE, BAR W=ACENTRIC FACTOR KAP=THE PRSV EOS KAPPA-1 PARAMETER INPUT TCl, PC1, W1, KAP1:" type 512.58,80.9579,0.56533,
-0,16816, and press RETURN. (These are parameters of methanol for the PRSV EOS from Table 3.1.1 .) At ,,INPUT TC2, PC2, WZ, RAP2 :" type 647.286,220.8975,0.3438, -0.06635, and press RETURN. (These are parameters of water for the PRSV EOS from Table 3.1.1 .) At "INPUT NUMBER OF DATA POINTS:" type 3 and press RETURN. At "INPUT TEMPERATURE T in K:" type 373.15 and press RETURN. At "INPUT FACTOR TO CONVERT EXPERIMENTAL PRESSURE DATA TO BAR BY DIVISION (e.g. if original data in mm Hg, type 750
-
if original data in gsia, type 14.5 etc. )
: j r
type 750 and press RETURN.
At
"INPUT EXPERIMENTAL DATA IN LIQUID MOLE FRACTION (XlEXP) OF SPECIES 1, VAPOR MOLE FRACTION (YlEXP) OF SPECIES 1, AND BUBBLE POINT PRESSURE (PEXP) (three in a row, separated by c o m a s ) INPUT XlEXP, YlEXP, PEXP:" type 0.035.0.191,931.
At "INPUT XlEXP, YlEXP, PEXP:" type 0.281,0.619, 1535.96. At "INPUT XlEXP, YlEXP, PEXP: ,,type 0.826,0.911,2337.76. (When the number of items of data, specified by NP, here three, is entered, the program writes the data to a file under the name tempo6.dat as specified above and continues. This data file becomes an existing data file and can he used when this program or other EOS programs are run again.)
A ~ S ~ S E L EAN C TEXCESS
FREE ENERGY MODEL:
O=EXIT 1-UNIQUAC 2=VAN LAAR 3=WILSON 4=NRTLr,type 2 and
-
press RETURN. (This results in selection of the van Laar model for the excess energy term in the WS mixin&rule.) At
"PARAMETERS PI2 AND P21 ARE REDUCED AS DESCRIBED BELOW. AIJ ARE PARAMETERS AS TABULATED IN THE D E C H E n TABLES.
Appendix D: Computer Programs for Blnary Mxtures
.
FOR UNIQUAC, PIJ=EXP(-AIJ/RT) FOR VAN
-,
PIJ=AIJ.
FOR WILSON, PIJ=(VLPJ/VLPI)*EXP(-AIJ/RT). FOR NRTL, PIJ=AIJ/RT. WITH THIS REDUCTION, IT IS POSSIBLE TO USE INITIAL GUESSES IN THE RANGE OF ZERO TO ONE. INITIAL VALUES RECOMMENDED FOR PI2 AND P21 ARE 0.1. INPUT INITAL GUESSES FOR P12, P21:" type 0.1,o.l and pres? RETURN.
At "INPUT THE WS MIXING-RULE PARAMETER K12 :"type 0.2and press RETURN. (At this stage the program optimizes the two parameters of the van Laar model. Intermediate results will continuou~lyhe displayed on the screen in the form of an error bar. When the optimization is completed, a message summarizing the results appears on the screen. Press RETURN to continue.) The following results appear on the screen: WS: THE WONG-SANDLER MIXING RULE FOR BINARY VLE CALCULATIONS methanol - water at 373 K EXCESS ENERGY MODEL = VAN LILAR Kl2= .a001 Pl2(=DIMENSIONLESS KAPPA12 OF VAN TJUIR) .6358 P21l=DIMENSIONLESS KAPPA21 OF VAN LRAR) .I095 TEMPERATURE in K: 373.15 PHASE VOLUMES ARE IN CCIMOL, PRESSURE IS IN W I T S OF THE DATA. X-EXP
P-EXP
P-CAL
Y-EXP
Y-CAL
VL-CAL
W-CAL
.0350
931.000
931.024
.I9100
.20817
22.90
24717.6
.2810
1535.960
1535.929
.61900
.61062
26.45
14820.6
.8260
2337.760
2375.031
.91100
.92047
43.50
9391.6
press return to continue.
Press RETURN to continue. At "DO YOU WANT A PRINT-OUT (Y/N)?" type y (or Y) and press RETURN. (With this command the results, shown above, are sent to the printer.) Atr'DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?"
type n (or N) and press RETURN. At "DO YOU WANT TO DO ANOTHER VLE CALCULATION (Y/N)?" typen (or N)and press RETURN. Example D.5.C:Use of the WS Mixing Rule in the Predictive Mode Change to the directory containing WS.EXE (e.g., A> or C>, etc.) Start the program by typing WS at the DOS prompt. Press RETURN (or ENTER).
Modelng Vapor-Liquid Equlibrla
the message introducing the program appears on the screen. Press RETURN to continue. The following appears on the screen: WS: BINARY YLE CALCULATIONS WITH THE WONG-SANDLER MIXING RULE YOU CAN USE THIS PROGRAM FOR VLE CALCULATION IN TWO WAYS. MODE (1): IF NO T-P-x-y DATA TO COMPARE RESULTS WITH ARE AVAILABLE, YOU MUST SUPPLY CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR. PRSV KAPPA-1 PABAMETER FOR EACH COMPOUND, AND A TEMPERATURE ALONG WITH A SET OF PREVIOUSLY SELECTED MODEL PARAMETERS. IN THIS MODE THE PROGRAM WILL RETURN ISOTHERMRL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOSITION RRNGE X1=0 TO 1 AT INTERVALS OF 0.1. MODE (2): IF YOU HAVE 1SO'rHEXma.L x-y-P DATA, YOU CAN ENTER THESE DATA FOLLOWING CO-S
THAT WILL APPEAR ON THE SCREEN
(OR USE PREVIOUSLY ENTERED DATA) TO FIT MODEL PARAMETERS TO THE VLE DATA. ALTERNATIVELY, YOU CAN CALCULATE VLE WITH PREVIOUSLY SELECTED PARAMETERS AND COMPARE THE RESULTS WITH THE VLE DATA.
(I), 2 FOR MODE (2). O R 0 T O TERMINATE THE PROGRAM" type 1 and press RETURN. (This example serves to demonstrate the predictive mode of the program WS, which is selected with the preceding entry. This mode is used in the absence of VLE data, and therefore no data are entered to, or can be accessed from the disk in this mode. Instead, the user provides the critical temperature, critical presssure, acentric factor, and the PRSV k , parameter for each pure component, selects an excess free-energy model; provides model parameters and a temperature. The program will return isothermul x-y-P predictions at the temperature entered, in the composition range x, = 0 to 1, at intervals of 0.1 .) At
"ENTER 1 FOR MODE
"YOU MAY ENTER A TITLE (25 CHARACTERS W . ) FOR THE MIXTURE T O BE PREDICTED (OR YOU MAY PRESS RETURN TO SKIP THE TITLE) :" enter
'meoh-water binary system'
and press RETURN. At"TC=CRITICAL
TEMPERATURE
PC=CRITICAL PRESSURE W=ACENTRIC FACTOR KAP=KAPPA-1 PARAMETER O F THE PRSV EOS
INPUT TCI, P C ~ .w1, KAPI:" enter 512.58,80.9579,0.56533,
-0.16816, and press RETURN.
At "INPUT
TC2, P C ~ .~ 2 ,KAPZ :"enter 647.286,220.8975,0.3438,
-0.06635, and press RETURN.
Appendix
D:Computer Programs for
Binary Mixtures
At "INPUT TEMPERATURE i n K:" type 373.15 and press RETURN. At "INPUT FACTOR TO CHOOSE UNITS OF REPORTED PRESSURE. DEFAULT IS BAR, TYPE 1 IF YOU WANT PRESSURE IN BAR. ( t m e 750 if you want pressure in nun ~ g ,etc. ) :',enter 750 and press RETURN. EXCESS FREE ENERGY MODEL: O=EXIT l=UNIQUAC 2=VAN LAAR 3=WILSON 4=NRTLn type 2 and
A~"SELECTAN
-
press RETURN. (This results in selection of the van Laar model for the excess energy tcrm in the WS mixing rule.) At
"INPUT REDUCED PARAMETERS PI2 AND P21. BELOW AIJ ARE PARAMETERS AS TABULATED IN THE DECHEMA TABLES. FOR UNIQUAC, PIJ=EXP(-AIJ/RT). FOR VAN LAAR, PIJ=AIJ. FOR WILSON, PIJ=(VLPJ/VLPI)*EXP(-AIJIRT). FOR NRTL, PIJ=AIJ/RT. WITH THIS REDUCTION, IT IS POSSIBLE TO USE INITIAL GUESSES IN THE W E OF ZERO TO ONE. INPUT PARAMETERS PIZ, ~21:" type 0.7727,0.3088, and press RETURN.
At ,,INPUT THE WS MIXING-RULE PARAMETER K12 : type 0.1 and press RETURN. (At this stage the program mns, and a message summarizing results appears on the screen. No average ahsolute deviation in bubble pressure is reported in that message because measured bubble pressure information is not available. Press RETURN to continue.) The following results appear on the screen: "
WS: THE WONG-SANDLER MIXING RULE FOR BINARY VLE CALCULATIONS
meoh-water binary system EXCESS ENERGY MODEL = VAN LAAR K12= .lo00 P12(=DIMENSIONLESS KAPPA12 OF VAN LAAR) .7727 PZl(=DIMENSIONLESS KAPPA21 OF VAN LAAR) .3088
PHASE VOLUMES ARE I N CClMOL. FACTOR YOU ENTERED TO CONVERT PRESSURE FROM BAR IS: 750.00 X-EXP .OOOO .lo00 .ZOO0 ,3000
P-EXP
P-CAL
-
760.051
Y-EXP
1369.104
-
1555.544
-
1132.200
Y-CAL .00008
VL-CAL
W-CAL
22.51
30349.3
.38471
24.29
20242.6
.53202
26.25
16663.5
.62420
28.43
14610.4
Modeling Vapor-Lqud Equ~lbr~a
press return to continue.
Press RETURN to continue. At "DO YOU WANT A PRINT-OUT (Y/N)? ' type n (or N) and press RETURN.
A~"DoYOU
WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?"
type y (or Y) and press RETURN. At "INPUT A N
m FOR THE OUTPUT FILE:''
type A:TEMPX.OUT and
press RETURN. At "DO YOU WANT TO DO A VLE CALCULATION AT ANOTHER TEMPERATURE (Y/N)?"
type y (or Y) and press RETURN. At "INPUT TEMPERATURE T in K: " enter 393.15 and press RETURN.
A ~ ~ ~ S E L AN E C TEXCESS
FREE ENERGY MODEL:
O=EXIT l=WIQUAC 2=VAN LAAR 3=WILSON 4=NRTLn type 2 and press
RETURN. (This command results in the selection of the van Laar model to be used as the excess free-energy tern in the WS mixing rule.) At "INPUT REDUCED PARAMETERS PI2 AND P21. BELOW AIJ ARE PARAMETERS AS TABULATED IN THE DECHEMA TABLES. FOR WIQUAC, PIJ=EXP(-AIJ/RT). FOR VAN LAAR, PIJ=AIJ. FOR WILSON, PIJ=(VLPJ/VLPI)*EXP(-AIJIRT). FOR NRTL, PIJ=AIJ/RT. WITH THIS REDUCTION, IT IS POSSIBLE TO USE INITIAL GUESSES IN THE RANGE OF ZERO TO ONE. P12, P21:" type 0.7727,0.3088, and press RETURN. INPUT PARA~~IETERS At "INPUT THE WS MIXING-RULE PARAMETER K12 :"type 0.1 and press
RETURN. (At this stage the program runs, and the message summarizing the results appears again on the screen. Press RETURN to continue.) The following results appear on the screen: WS: THE WONG-S?XL%ER MIXING RULE FOR B I W Y VLE CALCULATIONS
meoh-water binary system
Appendx D: Computer Programs for Binary Mxtures
EXCESS ENERGY MODEL = VAN LAAR K121 .lo00 Pl2(=DIMENSIONLESS IVLPPAl2 OF VAN LAAR) ,7727 P21(=DIMENSIONLESS KAPPA21 OF VAN LAAR) .3088 TEMPERATURE in K: 393.15 PHASE VOLUMES AP.E IN CC/MOL. FACTOR YOU ENTERED TO CONVERT PRESSURE FROM BAR IS: 750.00 X-EXP
P-EXP
P-CAL
Y-EXP
.OOOO
-
1488.881
.lo00
-
2145.238
.ZOO0
-
2564.061
-
Y-CAL
VL-CAL
W-CAI
.00007
22.95
16221.6
.36245
24.78
11147.8
.50879
26.81
9261.0
Press return to continue.
Press RETURN to continue At ,,no YOU WANT A PRINT-OUT ( Y / N ) ? " type n (or N) and press RETURN. AtrrDO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE ( Y / N ) ? " type y (or Y) and press RETURN. At "INPUT A NAME FOR THE OUTPUT FILE:" type A:TEMPX.OUT and press RETURN. (With this entry the results above are appended to the file TEMPR.OUT, which already contains the predictions for this binary system at 373.15 K). At "DO YOU WANT TO W A VLE CALCULATION AT ANOTHER TEMPERATURE ( Y / N ) ? "
type n (or N) and press RETURN. At "DO YOU WANT TO DO ANOTHER VLE CALCULATION (Y/N)?'type n (or N)and press RETURN.
D.6. Program WSUNF: Binary VLE Predictions Using the Wong-Sandler Mixing Rule Combined with the UNIFAC Excess Free-Energy Model The program WSUNF is used to predict VLE by means of the PRSV EOS coupled with the Wong-Sandler mixing rule and the UNIFAC group contribution method
Modelng Vapor-Liquid Equlfbr~a
without using any measured VLE data. To use the Wong-Sandler mixing rule this way, it is necessary to determine the value of the kij parameter of this mixing rule to match the excess Gibbs free energy from the EOS with the excess free energy of the UNIFAC activity coelficient model as closely as possible at or near 25°C (see Section 5.1 for details). Thus, one task of the WSUNF program is the evaluation of the optimum kii parameter of the WS model by matching the excess free-energy functions mentioned above. This is accomplished by entering the appropriate commands during execution (a tutorial is provided below). The program can be used in two ways. If measured isothermal VLE data are available, the program can be run to predict VLE at the measured liquid mole fractions; then the calculated and measured bubble pressures and vapor mole fractions are compared. In this mode the program reads previously stored data or accepts new data entered from the keyboard. The input data structure is identical to that used for all other EOS mixture programs, and the details of the input data have been given in Section D.3. If no experimental data arc available, bubble pressures and vapor mole fractions are calculated at liquid mole fractions x , = 0 to 1 at intervals of 0.1. In this mode no data are entered to, or accessed from, the disk. Instead, the user provides critical temperature, critical pressure, acentric factor, and the PRSV k , parameter for each pure component in addition the Wong-Sandler mixing-rule parameter k12and a temperature. The program then returns isothermal bubble pressure and vapor molc fraction predictions at the temperature entered. In either mode, during the matching of excess energy functions from the equation of state and from UNIFAC, the k i 2 parameter is varied to minimize thc objective function F = C - EzLrAC/ using a simplex algorithm. As a requirement of the simplex approach, an initial guess must be provided for the k L z. The initial guess may be a positive or a negative number; usually between zero and one. If convergence cannot be achieved with the selected initial guess, the user should t ~ yagain with different choices. The results from WSUNF can be sent to a printer, to a disk file. or both. This selection is made from the commands that appear on the screen at the completion of the calculations. See the following tutorial for further details.
icFos
Tutorial on the Use of WSUNF.EXE Example D.6.A: Use of the Wong-Sandler Mixing Rule and UNIFAC for Binary VLE Predictions Using an Exstng Data File Change to the directory containing WSUNEEXE (e.g., A>). Start the program by typing WSUNF at the DOS prompt. Press RETURN (or ENTER).
Appendix D: Computer Programs for Binary Mixtures
A program introduction message appears on the screen. Press RETURN to continue. At "THE UNIFAC MODEL REQUIRES GROUP DATA FROM A DISK. THESE DATA ARE STORED I N TWO FILES NAMED UNFI1.DTA AND UNFI2.DTA. UNFI1.DTA
CONTAINS UNIFAC GROUP PARAMETER INFORMATION
UNFI2.DTA
CONTAINS W I F A C BINARY GROUP INTERACTION PARAMETER INFORMATION.
IF YOU ALREADY HAVE THESE DATA FILES IN THE CURRENT DIRECTORY
THEN ENTER 1, OTHERWISE ENTER 2 :" type 2 and press RETURN.
(The data files UNFII .DTA and UNFI2.DTA are provided on the disk that accompanies this monograph. The program is easier to use if these data files are copied to the hard disk directory used to run the programs. In this case, 1 must be entered. An entry of 2, as above, indicates that these files are not present in the current directory. In this case the user must provide the directory and file names as indicated below.) At "TYPE THE DIRECTORY AND NAME OF THE FILE WHERE UNIFAC GROUP PARAMETER INFORMATION IS STORED (default = a:UNFIl.DTA)"
type a:UNFIl.DTA and press RETURN. Atr,TYPE THE DIRECTORY AND NAME OF THE FILE WHERE UNIFAC BINARY INTERACTION PARAMETER INFORMATION IS STORED
type a:UNFIZ.DTA and press RETURN. The following message appcars on thc screen: (default= a:UNFI2 .DTA)"
WSUNF: BINARY VLE CALCULATIONS WITH THE WONG-SANDLER MIXING RULE AND THE UNIFAC MODEL YOU CAN USE THIS PROGRAM FOR VLE PREDICTION IN TWO WAYS. MODE (1): IF NO T-P-x-y DATA TO COMPARE RESULTS WITH ARE AVAILABLE, YOU MUST SUPPLY CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR AND PRSV KAPPA-1 PARANIETER FOR EACH COMPOUND AND A TEMPERATURE. IN THIS MODE THE PROGRAM WILL RETURN ISOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOSITION RANGE X1=0 TO 1 AT INTERVALS OF 0.1. MODE (2): IF YOU HAVE ISOTHERMAL x-y-P DATA, YOU CAN ENTER THESE DATA FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN (OR USE PREVIOUSLY ENTERED DATA) TO PREDICT VLE BEHAVIOR AND COMPARE RESULTS WITH THE VLE DATA.
At "ENTER 1 FOR MODE (1). 2 FOR MODE ( 2 ) ,
enter 2 and press RETURN.
OR 0 T O TERMINATE THE PROGRAM:"
At "DO YOU WANT TO USE AN EXISTING DATA FILE (Y/N)?" type y (or Y) and press RETURN. At "INPUT THE NAME OF EXISTING DATA FILE (for example: a:am25.dat):"
type a:am25.dat and press RETURN. (This results in the use of am25.dat, the existing isothermal VLE data for the acetone plus methanol binary system at 25°C.) Atr'ENTER UNIFAC GROUP PARAMETER INFORMATION ENTER NAME OF THE COMPONENT 1"
type 'acetone' and press RETURN. (Following the preceding comment a group selection table will appear on the screen. The user must follow the instructions at the top of the table to choose one CH3 and one CH3CO group for acetone and press RETURN.) At "ENTER UNIFAC GROUP PARAMETER INFORMATION ENTER NAME OF THE COMPONENT 2"
type 'methanol' and press RETURN. (The group selection table will again appear on the screen. In this example the user should choose one CH3OH group for methanol and then press RETURN. Following this a summary of group selections will appear on the screen. Press RETURN to continue.) At "INPUT INITIAL GUESS FOR THE WS MIXING-RULE PARAMETER K12:"
type 0.1 and press RETURN.
A t n ~ oYOU
WANT TO FIT THE K12 TO Gex OF UNIFAC (1) OR DO YOU WANT TO CALCULATE VLE WITH K12 ENTERED (2)?"
type 1 and press RETURN. (At this point the program calculates a value of kij that matches the excess Cibhs free-energy values from the EOS and from the UNlFAC model. Intermediate results will continuously be displayed on the screen in the form of an error bar. When the optimization is completed a message summarizing the results appears on the screen. Press RETURN to continue.) Calculated results are then displayed on the screen as shown below:
WSUNF: THE WONG-SANDLER MIXING RULE FOR BINARY VLE CALCULATIONS WITH THE UNIFAC MODEL ACETONE mTHANOL 25 C Kl2= .lo20 TEMPERATURE (K) = 298.15 PHASE VOLUMES AR3 IN CCIMOL, PRESSURE IS IN UNITS OF THE DATA.
Appendx D: Computer Programs for Bnary Mxturei
X-EXP
P-EXP
Y-EXP
Y-CAL
VL-CAL
W-CAL
.0001
127.700
.00010
,00037
47.72
145935.7
.0610
146.200
.21600
.I7852
49.13
127134.2
.0860
153.200
.26800
.23233
49.74
121637.0
.0940
156.000
.29000
,24788
49.94
120069.9
.2040
178.600
.43500
.40774
52.85
104762.7
.2360
183.400
.46800
.44178
53.76
101763.7
.do20
205.200
.59800
.57628
58.95
91364.5
.4600
211.200
.62300
,61433
60.95
88968.7
.5820
220.800
.69500
.68933
65.47
85099.6
.6610
224.700
.74100
,73771
68.62
83219.0
.7860
231.000
.a0700
,81983
73.94
81046.8
.a120
231.200
.81000
.83841
75.10
80714.1
press return to see more results on the screen.
( T h i ~message appears when the number of data points exceeds twelve. Press RETURN to continue.)
press return to continue.
Press RETURN to continue. At ,'DO YOU WANT A PRINT-OUT (Y/N)?" type y (or Y) and press RETURN. (With this command the results shown above are sent to the printer.) At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?" type y (or Y) and press RETURN. At "INPUT A NAME FOR THE OUTPUT FILE:" type A:TEMP6.0UT and press RETURN. (With this command the results shown above are saved on the disk in drive A under the name TEMP6.0UT in ASCII code.) At ,,DO YOU WANT TO DO ANOTHER VLE CALCULATION (Y/N)?" type n (or N) and press RETURN.
Example D.6.B: Use of the WS Mixing Rule and UNIFAC for B~naryVLE t Existing Data File Predictions W ~ t h o u an
Change to the directoty containing WSUNEEXE (e.g., A>). Start the program by typing WSUNF at the DOS prompt. Press RETURN (or ENTER).
Modeling Vapor-Llquid Equlibr~a
A program introduction message appears on the screen. Press RETURN to continue. At "THE UNIFAC MODEL REQUIRES GROUP DATA FROM A DISK. THESE DATA ARE STORED IN TWO FILES N-D
UNFI~.DTA AND UNFI2.DTA.
UNFI1.DTA CONTAINS UNIFAC GROUP PARAMETER INFORMATION. UNFI2.DTA CONTAINS UNIFAC BINARY GROUP INTERACTION PARAMETER INFORMATION. IF YOU ALREADY HAVE THESE DATA FILES I N THE CURRENT DIRECTORY, THEN ENTER 1; OTHERWISE ENTER 2 :"type
1 and press R E T U R N . (The data files UNFII . D T A and UNF12.DTA are provided on the disk that accompanies this monograph. The program is easier to use if these data files are copied to the hard disk directory used to run the programs. In this case, an entry of 1 must be used. An entry of 2, as discussed earlier in Example D.6.A,indicates that these files are not present in the current directory. In that case the user must provide the directory and file names.) The following message appears on the screen:
WSUNF: BINARY VLE CALCULATIONS WITH THE WONG-SANDLER MIXING RULE AND THE UNIFAC MODEL YOU CAN USE THIS PROGRAM FOR VLE PREDICTION IN TWO WAYS. MODE (1): IF NO T-P-x-y DATA TO COMPARE RESULTS WITH ARE AVAILABLE, YOU MUST SUPPLY CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR AND PRSV KAPPA-1 P-TER
FOR EACH COMPOUND AND A TEMPERATVRE.
IN THIS MODE THE PROGRAM WILL RETURN ISOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATITRE ENTERED IN THE COMPOSITION RANGE X1=0 TO 1 AT INTERVALS OF 0.1. MODE ( 2 ) :
IF YOU HAVE ISOTHERMAL J-y-P DATA, YOU CAN ENTER THESE DATA
FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN (OR USE PREVIOUSLY ENTERED DATA) TO PREDICT VLE BEHAVIOR AND COMPARE RESULTS WITH THE VLE DATA.
At "ENTER 1 FOR MODE (I), 2 FOR MODE (2), OR 0 TO TERMINATE THE PROGRAM"
enter 2 and press R E T U R N .
At"~oYOU
WANT TO USE AN EXISTING DATA FILE (YIN)?" typen
(or N) and press RETURN.
A~"PROVIDE THE
FOLLOWING INPUT INFORMATION:
type a:tempo8.dat and press R E T U R N . (The preceding command will lead to saving a data file named tempo8.dat on the disk in drive A. You must have a disk in the A drive or select another directory by typing c:tempoE.dat, for example, to save the file on the hard drive.) INPUT NEW DATA FILE NAME:"
Appendix D: Computer Programs for Binary M~xiures
-
At "INPUT A TITLE FOR THE NEW DATA FILE:" type methanol water 25C and press RETURN. A~,,CRITICAL PARAMETERS: TCICRITICAL TEMP, K PC=CRITICAL PRESSURE, BAR W=ACENTRIC FACTOR KAPPA=THE PRSV EOS KAPPA-1 PARAlrIETER INPUT TC1, PC1, W1, KAPPA-1 :"type 512.80,80.9579,0.56533, 0 . 1 6 8 1 6 , and press RETURN. (These are EOS constants for methanol.) At "INPUT TC2, PC2, W2, KAPPA-2 :" type 647.286,220.8975,0.3438, -0.06635. (These are EOS constants for water.) At "INPUT NUMBER OF DATA POINTS :" type 3 and press RETURN. At "INPUT TEMPERATURE in K:" type 298.15 and press RETURN. At
"INPUT FACTOR TO CONVERT EXPERIMENTAL PRESSURE DATA TO BAR BY DIVISIOX (e.g. if original data in nun Hg, type 750 if original data in psia, type 14.5 etc. ) :"enter 750 and press RETURN.
At "INPUT EXPERIMENTAL DATA IN LIQUID MOLE FRACTION (XIEXPI OF SPECIES 1, VAPOR MOLE FRACTION (YlEXP) OF SPECIES I, AND BUBBLE POINT PRESSURE (PI?:,::) (three in a row, separated by commas) INPUT XlEXP. YlEXP, PEXP:,, type 0.19,0.6187,53, and press RETURN.
At "INPUT XlEXP, YlEXP, PEXP:" type 0.4943,0.7934, 82.3, and press RETURN. At "INPUT XlEXP, YlEXP, PEXP:" type 0.8492,0.9384, 1 1 2, and press RETURN.
(When the number of items of data specified by NP, here three, has been entered, the program writes the data to a file with the name tempo8.dat as specified above and continues. This data file becomes an existing data file and can be used when this program or other EOS programs are run again.)
A ~ T ~ E N TUNIFAC ER
GROUP PARAMETER INFORNATION
ENTER NAME OF THE COMPONENT 1"
type 'methanol' and press RETURN. (Following the preceding comment a group selection table will appear on the screen. The user must follow the instructions at the top of the table to choose one CH30H for methanol and press RETURN.) At ',ENTER UNIFAC GROUP PARAMETER INFORMATION ENTER NAME OF THE COMPONENT 2"
type 'water' and press RETURN. (The group selection table will again appear on the screen. In this example the user should choose one H20 group for water and then press RETURN. Following
Modeling Vapor-Liquid Equilbria
this a summary of group selections will appear on the screen. Press RETURN to continue.) At "INPUT INITIAL GUESS FOR THE WS MIXING-RULE PARAMETER K12:"
type 0.1 and press RETURN. A t " w YOU WANT TO FIT THE K12 TO Gex OF W I F A C (1) OR DO YOU WANT TO CALCULATE VLE WITH K12 ENTERED ( Z ) ? "
type 1 and press RETURN. (At this stage the program obtains a value of k,, that matches the excess Gibbs free-energy values from the EOS and from the UNIFAC model. Intermediate results will continuously be displayed on the screen in the form of an error bar. When the optimization is completed a message summarizing the results appears on the screen. Press RETURN to continue.) Calculated results are displayed on the screen as shown below. WSUNF: THE WONG-SANDLER MIXING RULE FOR BINARY VLE CALCUIJLTIONS WITH THE UNIFAC MODEL E3THANOL WATER 25C K12= .0877
TEMPERATURE (K)
1
298.15
PHASE VOLVl5S ARE IN CCIMOL, PRESSURE IS IN UNITS OF THE DATA Y-CAL
YL-CAL
.61870
,64050
24.66
333363.3
.79340
.a2829
31.50
220964.8
,93840
.95350
42.16
163964.9
X-EXP
P-EXP
P-CAL
Y-EXP
.I900
53.000
55.682
.4943
82.300
83.919
,8492
112.000
112.963
W-CAI.
press return to continue.
Press RETURN to continue. At "w YOU WANT A PRINT-OUT (YIN)?" type y (or Y) and press RETURN. (With this command the results shown above are sent to the printer.) At"W YOU WANT TO SAVE THE RESULTS TO A N OUTPUT FILE (Y/N)?" type y (or Y) and press RETURN. At ,,INPUT A NAME FOR THE OUTPUT FILE:" type A:TEMPS.OUT and press RETURN. (With this command the results shown above are saved on the disk in drive A under the name TEMP8.OUT in ASCII code.) At "DO YOU WANT TO DO ANOTHER VLE CALCULATION (YIN)?" type y (or Y) and press RETURN. In the first part of this example, we matched excess Gibbs energy from the PRSV equation of state with excess Gibbs energy from UNIFAC at 25°C and obtained
Appendx D: Computer Programs for Bnary Mxturez
the Wong-Sandler mixing rule binary interaction parameter, k12, as 0.0869. Also we compared predictions at 25°C with k12 = 0.0869 to experimental data entered from the keyboard. In the second part of this example, shown below, we use the same k l z value to predict isothermal VLE data at 100"C, this time using internally generated liquid mole fractions x, = 0,0. 1,0.2, etc. This mode is implemented as described below. The following message reappears on the screen: WSUNF: BINARY VLE CALCULATIONS WITH THE WONG-SANDLER M I X I N G RULE AND THE UNIFAC MODEL
YOU CAN USE THIS PROGRAM FOR VLE PREDICTION I N TWO WAYS.
MODE (1): I F NO T-P-x-y
DATA TO COMPllRE RESULTS WITH ARE AVAILABLE,
YOU MUST SUPPLY CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR AND PRSV KAPPA-1 PARAXETER FOR EACH COMPOUND AND A TEMPERATURE.
I N THIS MODE THE PROGRAM WILL RETURN ISOTHERMAL x-y-P
PREDICTIONS
AT THE TEMPERATURE ENTERED I N THE COMPOSITION RANGE X1=0 TO 1 AT INTERVALS OF 0.1. MODE ( 2 ) :
I F YOU HAYE ISOTHERMAL x-y-P DATA, YOU CAN ENTER THESE DATA
FOLLOWING C O W S THAT WILL APPEAR ON THE SCREEN (OR USE PREVIOUSLY ENTERED DATA) TO PREDICT VLE BEHAVIOR AND COMPARE THE RESULTS WITH THE VLE DATA.
At "ENTER 1 FOR MODE ( 1), 2 FOR MODE ( 2 )
, OR
0 TO TEMINATE THE PROGRAM: ''
enter 1 and press RETURN. At "YOU MAY ENTER A TITLE ( 2 5 CHARACTERS MAX.)
(OR YOU MAY PRESS RETURN TO SKIP
FOR THE MIXTURE TO BE PREDICTED
THE TITLE) :"enter meoh-water
100 C and
press RETURN. A~"TC=CRITICAL TEMPERATURE PC=CRITICAL PRESSURE W=ACENTRIC FACTOR KAPPA=KAPPA-1 PARAMETER OF THE PRSV EOS INPUT TC1,
PC1,
W1,
KAPPA-1:"
enter 512.58,80.9579,0.56533,
-0.168 16, and press RETURN. At m~~~~~ TC2, P C ~ ,WZ, KAPPA-2:" enter 647.286,220.8975,0.3438, -0.06635, and press RETURN. At "INPUT TEMPERATURE i n K:" type 373.15 and press RETURN. A~"INPUTFACTOR TO CHOOSE UNITS OF REPORTED PRESSURE DEFAULT I S BAR,
TYPE 1 I F YOU WANT PRESSURE I N BAR.
Modelng Vapor-Liquid Equilibria
(type 750 if you want pressure in mm Hg, etc.) :"enter 750
and press RETURN.
A~"ENTERUNIFAC
GROUP PARAMETER INFORMATION
ENTER NAME OF THE COMPONENT 1" type 'methanol' and press RETURN. (Following the preceding comment a group selection table will appear on the screen. The user must follow the instructions at the top of the table to choose one CH30H for methanol and press RETURN.)
At "ENTER
W I F A C GROUP PARAMETER INFORMATION
ENTER NAME OF THE COMPONENT 2" type 'watcr' and press RETURN.
(The group sclection table will again appear on the screen. In this example the user should choose one H 2 0 group for water and then press RETURN. Following this a summary of group selections will appear on the screen. Press RETURN to continue.) At "INPUT INITIAL GUESS FOR THE WS MIXING-RULE PARAMETER K12:"
type 0.086 and press RETURN. At "DO YOU WANT TO FIT THE K12 TO Gex OF UNIFAC (1) OR DO YOU WANT TO CALCULATE VLE WITH K12 ENTERED (Z)?"
type 2 and press RETURN. (At this stage the program runs with kl? = 0.086, and a summary of intermediate results appears on the screen for inspection. Because no experimental data are entered in this case, no average absolute deviation in pressure is reported. Press RETURN to continue.) The following results appear on the screen: WSUNF: THE WONG-SANDLER MIXING RULE FOR BINARY VLE CALCULATIONS WITH THE m I F A c MODEL ETHANOL WATER 100 C
TEMPERATURE (K) = 373.15 PHASE V O L W S ARE IN CClMOL. FACTOR YOU ENTERED TO CONVERT PRESSURE FROM BAR IS: 750.00 X-EXP .OOOO ,1000
P-EXP
-
P-CAL
Y-EXP
760.510
-
,00076
22.51
30330.7
1160.189
-
,40078
24.40
19742.1
Y-CAL
VL-CAL
W--
Appendx D: Computer Programs for Binary Mxtures
press return to continue.
Press RETURN to continue. At -DO YOU WANT A PRINT-OUT (Y/N)?" type y (or Y) and press RETURN. (With this command the results shown above are sent to the printer.) At"D0 YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?"
type y (or Y) and press RETURN. At "INPUT A NAKE FOR THE OUTPUT FILE:" type A:TEMPg.OUT and press RETURN. (With this command the results shown above are appended to the file temp8.out, which already exists on the disk in drive A,) At "DO YOU WANT TO DO A VLE CALCULATION AT ANOTHER TEMPERATURE (Y/N)?"
type n (or N) and press RETURN. At "DO YOU WANT TO DO ANOTRER VLE CALCULATION (YIN)?" type n (or N) and press RETURN.
D.7. Program H V U N F : Binary VLE Predictions from the Huron-Vidal Mixing Rule (HVO)and Its Modifications (MHVI,MHV2, LCVM,
and HVOS) The program HVUNF can be used to predict VLE using the PRSV EOS and one of the Gibhs excess-energy-based mixing rules, HVO, MHVI, MHV2, LCVM, and HVOS coupled with the UNIFAC group contribution method without the use of any measured VLE data. The program can be used in two ways. If experimental VLE data (isothermal) are available, the program can be run to calculate VLE at the measured liquid mole fractions; then, the calculated and measured bubble pressures and vapor mole fractions are compared. In this mode the program reads previously stored data or accepts new data entered from the keyboard, The input data structure is identical to that used for all other EOS mixture programs, and the details of the input data have been described in Section D.3. If no experimental data are available, bubble pressures and vapor mole fractions are calculated over the liquid mole fraction range of x, = 0 to I at intervals of 0.1. In this mode no data are entered to, or accessed from, the disk. Instead, the user provides critical temperature, critical presssure, the acentric factor and PRSV K , parameter for each pure component, and temperature. The program then returns
Modeng Vapor-L~qu~dE q u l ~ b r ~ a
isothermal bubble pressure and vapor mole fraction predictions at the tempwalure entered in the composition range x, = 0 to 1, at intervals of 0.1. The results from the program HVUNF can be sent to a printer, to a disk file, or both. To make this choice, follow the commands that appear on the screen upon the completion of calculations. Please see the following tutorial for further details.
Tutorial on the Use of HVUNF.EXE
Example D.7.A: Use of the Huron-Vida Class of Mixing Rule, Here HVOS, with UNIFAC t o Predict Binary VLE Data
Change to the directory containing HVUNEEXE (e.g., A> or C>, etc.). Start the program by typing HVUNF at the DOS prompt. Press RETURN (or ENTER). A program introduction message appears. Press RETURN to continue. At "THE W I F A C MODEL REQUIRES GROUP DATA FROM A DISK. UNFI1.DTA AND WFI2.DTA. THESE DATA ARE STORED IN TWO FILES N-D 1JNFIl.DTA CONTAINS UNIFAC GROUP PARAMETER INFORMATION UNFI2.DTA CONTAINS W I F A C BINARY GROUP INTERACTION PARAMETER INFORMATION. IF YOU ALREADY HAVE THESE DATA FILES IN THE CURRENT DIRECTORY;
THEN ENTER 1; OTHERWISE ENTER 2 :,,type I and press RETURN.
(The data files UNFI1.DTA and UNFI2.DTA are provided on the disk that accompanies this monograph. The program is easier to run if these data files are copied to the hard disk directory used to run the programs. In this case 1, as above, must be entered. An entry of 2, as shown earlier in Example D.6.A, indicates that these files are not present in the current directory. In that case the user must provide the directory and file names.) At "HVUNF:
BINARY VLE CALCULATIONS WITH KURON-VIDAL TYPE MIXING RULES
AND THE UNIFAC EXCESS FREE ENERGY MODEL SELECT A MIXING RULE MODEL WV-O=HURON-VIDAL ORIGINAL MHVl=MODIFIED HURON-VIDAL 1ST ORDER EIKV2=MODIFIED HURON-VIDAL 2ND ORDER LCVM=LINEAR COMBINATION OF HURON-VIDAL AND MICHELSEN HVOS=HURON-VIDAL MODIFIED BY ORBEY AND SANDLER OsEXIT 1=HV-0 2 = W 1 3=MHV2 4=LCVM 5=HVOS"
type 5 and press RETURN. e (This results in using the HVOS model for the mixing ~ u l model.)
Appendrx D. Computer Programs for Binary Mixtures
The following message appears on the screen: XWJNF:
BINARY VLE CALCUIATIONS WITH HURON-VIDAL TYPE MIXING RULES
AND THE UNIFAC MODEL
YOU CAN USE THIS P R O G W FOR VLE PREDICTION IN TWO WAYS. MODE I : IF NO T-P-X-Y DATA TO COMPARE RESULTS WITH ARE AVAILABLE, YOU MUST SUPPLY CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR AND PRSV KAPPA-1 PARAMETER FOR EACH COMPOUND AND A TEMPERATURE. IN THIS MODE THE PROGRAM WILL RETURN ISOTIiEE3l&L X-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOSITION RANGE X1=0 TO 1 AT INTERVALS OF 0.1. MODE ( 2 ) :
IF YOU XAVE ISOTHEPXAL x-y-P DATA, YOU CAN ENTER THESE DATA
FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN (OR USE PREVIOUSLY ENTERED DATA) TO PREDICT VLE BEHAVIOR AND COMPARE RESULTS WITH THE VLE DATA.
At "ENTER 1 FOR MODE ( 1 ) , 2 FOR MODE ( 2) , O R 0 TO TERMINATE THE PROGRAM"
enter 2 and press RETURN. At"D0 YOU WANT T O U S E AN EXISTING DATA FILE (Y/N)?" typey
(or Y) and press RETURN.
At "INPUT THE NAME OF EXISTING DATA FILE (for example, a:ab25.dat):"
type a:ab25.dat and press RETURN.
A~-ENTERUNIFAC
GROUP PARAMETER INFORElATION
NAME OF COMPONENT 1"
type 'acetone' and press RETURN. (Following the preceding comment a group selection table will appear on the screen. The user must follow the instructions at the top of the table and for this example choose one CH3 and one CH,CO group for acetone and enter press RETURN.)
A~"ENTERUNIFAC
GROUP PARAMETER INE'ORElATION
NAME OF COMPONENT 2"
type 'benzene' and press RETURN. (The group selection table will again appear on the screen. For this example the user must choose six ACH groups for benzene and then press RETURN. Following this, a summary of group selections will appear on the screen for inspection of the entries. After inspection you can press any key to continue. At this stage the program runs, and, when point-to-point calculations for each data point in the ab25.dat data file are completed, the results are displayed on the screen as shown below.)
Modeling Vapor-Liqud Equilibria
HYITNF: VLE CALCULATIONS WITH m 0 N - V I D A L TYPE MODELS AND UNIFAC
ab25.dat ACETONE-BENZENE AT 25C FROM DECHEMA-1-38-163 TEMPEmTURE (Kl= 298.15 MIXING RULE: HYOS PL%SE VOLUMES AK3 IN CC/MOL, PRESSURE IS IN UNITS OF THE DATA. X-EXP
P-EXP
P-CAL
Y-EXP
Y-CAL
VL-CAL
.0001
95.600
95.040
.00010
.00035
87.09
.0500
106.400
106.678
.I4600
.I5257
86.96
,1000
116.600
117.525
.26000
.a6900
86.83
.I500
126.300
127.618
.35300
.36118
86.70
.zoo0
135.400
137.018
.42900
.43632
86.57
.2500
144.000
145.780
.49400
.49911
86.44
,3000
152.100
153.958
,54900
.55269
86.30
.3500
159.900
161.606
.59800
.59928
86.16
.4000
167.200
168.768
.64100
,64051
86.02
.4500
174.200
175.493
.68000
.67757
85.87
,5000
180.800
181.824
.71500
.71141
85.72
.5500
187.000
187.802
,74700
.74278
85.57
Dress return to see more results on the screen.
(This message appears when the number of data points exceeds twelve. Press RETURN to continue.)
press return to continue.
Press RETURN to continue. At -DO YOU WANT A PRINT-OUT (Y/N)?"type y (or Y ) and press RETURN. (The results above will be sent to the printer.) .&"DO
YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?n
type y (or Y) and press RETURN.
Appendx D. Computer Programs for Bnary M~xtures
At "INPUT A NAME FOR THE OUTPUT FILE:" type A:TEMP9.OUT and press RETURN. (With this command the results shown above are saved in the disk in drive A under the name TEMP9.0UT in ASCII code.) At -DO YOU WANT TO DO ANOTHER VLE CALCULATION (Y/N)?" type n (or N) and press RETURN. Example D.7.B:Use of the Huron-Vidal Class of Mixing Rule. Here HVOS, with UNIFAC t o Predict Binary VLE
Change to the directory containing HVUNEEXE (e.g., A> or C>, etc.). Start the program by typing HVUNF at the DOS prompt. Press RETURN (or ENTER). The program introduction message appears. Press RETURN to continue. At "THE UNIFAC MODEL REQUIRES GROUP DATA FROM A DISK. THESE DATA ARE STORED IN TWO FILES NAMED LlNFI1.DTA AND UNFI2.DTA. UNFI1.DTA CONTAINS UNIFAC GROUP PARAMETER INFORM&TION UNFI2.DTA CONTAINS UNIFAC BINARY GROUP INTERACTION PAPAMETER INFORMATION. IF YOU ALREADY HAVE THESE DATA FILES IN THE CURRENT DIRECTORY, THEN ENTER 1; OTHERWISE ENTER 2 :"type I and press RETURN.
(The data files UNFII .DTA and UNFI2.DTA are provided on the disk that accompanies this monograph. The program is easier to run if these data files are copied to the hard disk directory used to run the programs. In this case an entry of 1 is required. An entry of 2, as shown earlier in Example D.6.A, indicates that these files are not present in the current directory. In that case the user must provide the directory and file names.) At "HVUNF:
BINARY VLE CALCULATIONS WITH HURON-VIDU TYPE MIXING RULES
AND THE UNIFAC EXCESS FREE ENERGY MODEL SELECT A MIXING RULE MODEL HV-O=IIURON-VIDAL ORIGINAL MHVl=MODIFIED HURON-VIDAL 1ST ORDER MHVZ=MODIFIED HURON-VIDAL 2ND ORDER LCVM=LINEAR COMBINATION OF HURON-VIDAL AND MICHELSEN HVOS= HURON-VIDAL MODIFIED BY ORBEY AND SANDLER
OZEXIT 1 = w - o 2 = m 1 3 = m 2 ~ = L C V M5 = w O S " type
5 and press RETURN. (This results in using the HVOS model for the mixing rule model.) The following message appears on the screen:
HYUNF: BINARY VLE CALCULATIONS WITH HURON-VIDAL TYPE MIXING RULES
AND THE UNIFAC NODEL
Modeling Vapor-Liquid Equibr~a
YOU CAN USE THIS PROGRAM FOR VLE PREDICTION IN TWO WAYS. MODE (1): IF NO T-P-x-y DATA TO COMPARE RESULTS WITH RRE AVAILdBLE, YOU MUST SUPPLY CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR RND PRSV KAPPA-1 PRRAMETER FOR EACH COMPOUND
A TEMPERA-.
IN THIS MODE THE PROGRAM WILL RETURN ISOTHERMAL X-Y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOSITION RMIGE X1=0 TO 1 AT INTERVALS OF 0.1. MODE (2): IF YOU HAVE ISOTHERMAL x-y-P DATA, YOU CAN ENTER THESE DATA FOLLOWING CO-S
THAT WILL APPEAR ON THE SCREEN
(OR USE PREVIOUSLY ENTERED DATA) TO PREDICT VLE BEHAVIOR, AND COMPARE RESULTS WITH THE VLE DATA.
At "ENTER 1 FOR MODE (1). 2 FOR MODE (2). OR 0 TO TERMINATE THE PROGRAM"
enter 2 and press RETURN. At ''DO YOU WANT TO USE AN EXISTING DATA FILE (Y/N)1" type n (or N) and press RETURN.
A~"PROVIDEFOLLOWING
INPUT INFORNFLTION:
INPUT NEW DATA FILE NAME:" type a:temp09.dat and press RETURN.
(The preceding command will lead to saving a data file named temp09.dat on the disk in drive A.You must have a disk in the A drive, or select another directory, by typing c:temp09.dat, for example, to save the file on the hard drive.) At "INPUT A TITLE FOR THE NEW DATA FILE:" type acetone-benzene 25°C. At "CRITICAL PARAMETERS: TC=CRITICAL TEMP, K PC=CRITICAL PRESSURE, BAR W=ACENTRIC FACTOR KAPPA=KAPPA-1 P-TER
OF THE PRSV EOS
INPUT TC1, PC1, Wl, KAPPA-1:" type 508.1,46.96, 0.30667, -0.00888, and press
RETURN. (These are EOS constants for acetone.) At "INPUT TC2, PC2, W2, KAPPA-2:" type 562.16,48.98,0.20929,
0.07019, and press RETURN. (These are EOS constants for benzene.) At "INPUT NUMBER OF DATA POINTS :"type 3 and press RETURN. At "INPUT TEMPERATURE in K: " type 298.15 and press RETURN. At "INPUT FACTOR TO CONVERT EXPERIMENTAL PRESSURE DATA TO BAR B Y DIVISION (e.g. if original data in m Hg, type 750 if original data in psia, type 14.5 etc. ) :"enter 750 and press RETURN.
Appendx D: Computer Programs for B~naryMixtures
At "INPUT EXPERIMENTAL DATA IN LIQUID MOLE FRACTION (XlEXP) SPECIES 1, VAPOR MOLE FRACTION (YlEXP) OF SPECIES 1, AND BUBBLE POINT PRESSURE (PEXL? (three in a row, separated by commas) INPUT XlEXP, YlEXP, PEXP:" type 0.1, 0.26, 116.6, and press RETURN. At "INPUT XlEXP, YlEXP. PEXP:" type 0.5, 0.7150, 180.8, and press
RETURN. At "INPUT XlEXP, YlEXP, PEXP:" type 0.9,0.965. 224.8, and press RETURN. (When the number of items of data specified by NP, here three, is entered, the program writes the data to a file under the name temp09.dat as specified above and continues. This data file becomes an exisling data file and can he used when the program is run again. The data used here are part of those from the data file ab25.dat used in the previous example.)
A~"ENTERUNIFAC
GROUP PARAMETER I N F O ~ T I O N
ENTER NAME OF THE COMPONENT 1" type 'acetone' and press RETURN. (Following the preceding comment a group selection table will appear on the screen. The user must follow the instructions a1 the top of the table to choose one CH3and one CHiCOfor acetone and press RETURN.)
A ~ S ~ E N TUNIFAC ER
GROUP P-ETER INFORMATION ENTER NAKE OF THE COMPONENT 2" type 'benzene' and press RETURN.
(The group selection table will again appear on the screen. For this example the user should choose six ACH groups for benzene and then press RETURN. Following this a summary of group selections will appear on the screen. Press RETURN to continue.) HVUNF: VLE CALCULATIONS WITH HURON-VIDAL TYPE MODELS AND UNIFAC
a:t-09.dat
ACETONE BENZENE 25C TEMPERATURE (K)= 298.15 MIXING RULE: HVOS PHASE VOLUMES ARE IN CC/MOL, PRESSURE IS IN W I T S OF THE DATA. X-EXP
P-EXP
P-CAL
Y-EXP
Y-CAI
VI-CAI.
W-CAL
.lo00
116.600
117.515
.26000
.26899
86.83
157160.1
.SO00
180.800
181.805
.71500
,71140
85.72
101290.3
.go00
224.800
222.427
.96500
.93779
84.35
82642.5
press return to continue
Press RETURN to continue. At ,,DO YOU WANT A PRINT-OUT (Y/N)?" type y (or Y ) and press RETURN. (The results above will be sent to the printer.)
Modeling Vapor-Lquid Equilibria
A ~ " D oYOU
WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?"
type y (or Y) and press RETURN. At "INPUT A NAME FOR THE OUTPUT FILE:" type A:TEMPlO.OUT and press RETURN. (With this command the results shown above are saved on the disk in drive A under the name TEMP10.OUT in ASCII code.) At IW YOU WANT TO DO ANOTHER VLE CALCULATION (YIN)?" type y (or Y) and press RETURN.
A~"SELECTA
MIXING RULE MODEL
HV-O=HURON-VIDAL ORIGINAL 2UIHVl=MODIFIED HURON-VIDAL 1ST ORDER MHV2=MODIFIED HURON-VIDAL 2ND ORDER LCVM=LINEAR COMBINATION OF HURON-VIDAL AND MICHELSEN WVOS=IIURON-VIDAL MODIFIED B Y ORBEY AND SANDLER
O=EXIT I=HV-O ~ = M H V I 3=MHV2 ~ = L C V M 5=HVOSr'type
5 and press
RETURN. The following message appears on the screen: IWTNF: BINARY VLE CALCULATIONS WITH HURON-VIDAI. TYPE MIXING RULES AND THE UNIFAC MODEL
YOU CAN USE THIS PROGRAM FOR VLE PREDICTION IN TWO WAYS. MODE (1): IF NO T-P-x-y DATA TO COMPARE RESULTS WITH ARE AVAIIABLE, YOU MUST SUPPLY CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR AND PRSV KAPPA-1 PARAElETER FOR EACH COMPOUND AND A TEMPERATURE. IN THIS MODE THE PROGRAM WILL RETURN ISOTHERMAL x-y-P PREDICTIONS AT THE TEMPERATURE ENTERED IN THE COMPOSITION RANGE Xl=O TO 1 AT INTERVALS OF 0.1. MODE (2): IF YOU HAVE ISOTHERDIAL x-y-P DATA, YOU CAN ENTER THESE DATA FOLLOWING COMMANDS THAT WILL APPEAR ON THE SCREEN (OR USE PREVIOUSLY ENTERED DATA) TO PREDICT VLE BEHAVIOR, AND COMPARE RESULTS WITH THE VLE DATA.
At "ENTER 1 FOR MODE (1). 2 FOR MODE (2), OR 0 T O TERMINATE THE PROGRAM"
enter 1 and press RETURN. (In the first part of this example, we compared VLE predictions for the acetone-benzene binary mixture at 25°C with experimental data entered from the keyboard. In the second part, shown below, we use the same model (HVOS) to predict isothermal VLE data at 100"C, this time using internally generated liquid
Appendx D. Computer Programs for Bnary M ~ u r e s
mole fractions x, = 0. 0.1, 0.2, etc. This mode is implemented by entering 1 above.) At "YOU MAY ENTER A TITLE (25 CHARACTERS MAX.) FOR THE MIXTURE TO BE PREI (OR YOU m y PRESS RETURN TO SKIP THE TITLE) :" enter acetone-water I00 C
and press RETURN. At "TC=CRITICAL TEMPERATURE PC=CRITICAL PRESSURE W=ACENTRIC FACTOR KAPPA=KAPPA-1 PARAMETER OF THE PRSV EOS INPUT TCI, P C ~ ,w1, KAPPA-1:"
enter 508.1,46.96, 0.3067, -0.0089,
and press RETURN. At "INPUT TCZ, PCZ, WZ, KAPPA-2 :"enter 562.16, 48.98,0.2093, 0.0702,
and press RETURN.
At -INPUT TEMPERATURE in K:" type 373.15 and press RETURN. A~"INPUTFACTOR TO CHOOSE W I T S OF REPORTED PRESSURE DEFAULT IS BAR, TYPE 1 IF YOU WANT PRESSURE IN BAR. (type 750 if you want pressure in mm Hg, etc.) :"enter750
and press RETURN. AtrrENTER W I F A C GROUP PARAMETER INFORMATION ENTER NAME OF THE COMPONENT 1" type acetone' and press RETURN.
(Following the preceding comment a group selection table will appear on the screen. The user must follow the instructions at the top of the table to choose one CH, and one CHzCO for acetone and press RETURN.)
A~"ENTERUNIFAC
GROUP PAFIAMZTER INFORMATION
ENTER NAME OF THE COMPONENT 2"
type 'benzene' and press RETURN. (The group selection table will again appear on the screen. In this example the user should choose six ACH group for benzene and then press RETURN. Following this a summary of group selections will appear on the screen. Press RETURN to continue.) The following results appear on the screen:
HYUNF:VLE CALCULATIONS WITH HURON-VIDAL TYPE MODELS AND UNIFAC
acetone-benzene lOOC TEMPERATURE (K)=373.15 REMINDER: XEXP VALUES ARE INTERNALLY GENERATED NO ACTUAL EXPERIMENTAL DATA ARE AVAILABLE MIXING RULE: W O S PHASE VOLUMES ARE IN CCIMOL. FACTOR YOU ENTERED TO CONVERT PRESSURE FROM BAR IS 750.00
Modeng Vapor-L~qud Equlibria
X-EXP
P-EXP
P-CAL
Y-EXP
W-CAL 16499.6
1569.039
-
-
1767.497
.
.3000
.
1944.526
.4000
-
2103.222
.5000
.
2246.362
.6000
.
2376.377
.oooo
-
1345.799
.I000
.
.zoo0
.7000 .8000 .goo0 1.0000
.
14075.6 12432.7
2495.281
-
2604.568
.
8253.4
2705.021
.
7926.6
2796.386
-
7650.5
11249.1 10357.0 9660.3 9100.3 8639.5
press return to continue
Press RETURN to continue. At ,,DO YOU WANT A PRINT-OUT (Y/N)?" type y (or Y) and press RETURN. (The results above will he sent to the printer.) A ~ " D oYOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?" type y (or Y) and press RETURN. At "INPUT A NAME FOR THE OUTPUT FILE?" type A:TEMPlO.OUT and press RETURN. (With this command the results shown above are appended to the file templ0.out in the disk in drive A in ASCII code.) At "DO YOU WANT TO
M)
A VLE CALCULATION AT ANOTHER TEMPERATURE (Y/N)?"
type n (or N) and press RETURN. At "DO YOU WANT TO DO ANOTHER VLE CALCULATION (Y/N)?" type n (or N) and press RETURN.
Computer Programs for Multicomponent Mixtures
The accompanying disk contains the programs and sample data files that can be used to predict vapor-liquid equilibria of multicomponent mixtures using the EOS models discussed in this monograph. All the programs coded in FORTRAN using MICROSOFT FORTRAN Version 5.1 and are also supplied as stand-alone executable modules (EXE files) that run on DOS or WINDOWS-based personal computers. For more details, see the introduction section of Appendix D. Each program is separately described in the following sections, and tutorials are included to facilitate the use of each program. In these tutorials, the output that will appear on the screen is indicated in bold and in a smaller font. The information the user is to supply is shown here in the normal font.
E. I. Program VDWMIX: Multicomponent VLE Calculations with van der Waals One-Fluid Mixing Rules The program VDWMIX is used to calculate multicomponent VLE using the PRSV EOS and the van der Waals one-fluid mixing rules (either IPVDW or 2PVDW, see Sections 3.3 to 3.5 and Appendix D.3). The program can be used to create a new input file for a multicomponent liquid mixture and then to calculate the isothermal bubble point pressure and the composition of the coexisting vapor phase for this mixture. In this mode the information needed is the number of components (up to a maximum of ten), the liquid mole fractions, the temperatures at which the calculations are to be done (for as many sets of calculations as the user wishes, up to a maximum of fifty), critical temperatures, pressures (bar), acentric factors, the K , constants of the PRSV equation for each compound in the mixture, and, if available, the ex~erimental bubble point pressure and the vapor phase compositions (these last entries are optional and are used for a comparison between the experimental and calculated results). In addition, the user is requested to supply binary interaction parameter(s) for each pair of components in the multicomponent mixture. These interaction parameters can be
Modeling Vapor-Lqud Equilbria
obtained using the program VDW (see Appendix D.3) if experimental data are available for each of the bina~ypairs. Alternatively, the user can select an already existing data file (we use extension VDW for these files, and some examples of such data files are provided on the accompanying disk) to calculate multicomponent VLE for the mixture of that input file. The results from the program VDWMIX can be sent to a printer, to a disk file, or both. The commands that appear on the screen upon the completion of the calculations must be followed to make this choice. Please sec the following tutorial for further details. Tutorial o n the Use of VDWMIX.EXE
Example E. I .A: Creating a New Input File and Calculation o f Muticomponent VLE
. "VDWMTX:
Change to the directory containing the program VDWM1X.EXE (e.g., A> or C>, etc.). Start the program by typing VDWMIX at the DOS prompt. A program introduction message appears on the screen. Press ENTER (or press RETURN). The following appears: At MULTICOMPONENT VLE CAIACULATIONS WITH THE VAN DER
WAALS ONE-FLUID MIXING RULES. T H I S PROGRAM CAN BE USED FOR ISOTHERM?& A NEW INPUT F I L E ,
BUBBLE POINT CALCULATIONS CREATING
OR USING A PREVIOUSLY STORED INPUT F I L E .
YOU MUST SUPPLY NUMBER O F COMPONENTS, C R I T I C A L TEMPEPATURE,
L I Q U I D MOLE FRACTION,
C R I T I C A L PRESSURE,
ACENTRIC FACTOR,
PRSV KAPPA-1 PARAMETER FOR EACH COMPOUND, TEMPERATURE,
AND MODEL PARAMETER(S) FOR EACH P A I R O F COMPONENTS. ENTER 1 TO CREATE A NEW INPUT F I L E , STORED INPUT F I L E ,
2 TO USE A PREVIOUSLY
OR 0 TO TETUXINATE THE PROGRAM.
O/lI2?"
type 1 and RETURN (With this selection a new input file will be created.) At "ENTER A NAME FOR THE NEW INPUT FILE ( * * * * * * * .MW) :rr enter a name for the new file (such as A:TEST.VDW) and press RETURN. At "ENTER A TITLE FOR THE NEW INPUT FILE:" enter a descriptive title for the file (for example "ACETONE-METHANOL-WATER AT 523 K") and press RETURN.
Appendx E: Computer Pmgrams for Multicomponent Mixtures
At "INPUT At
m E R OF COMPONENTS:" enter 3 and press RETURN.
"HOW MANY SETS OF ISOTHERNIAL BUBBLE POINT CALCULATIONS DO YOU WANT TO W ? (FOR EACH SET YOU MUST PROVIDE A NEW LIQUID COMPOSITION AND TEMPERATURE):" enter 3 and press RETURN. At "INPUT PURE COMPONENT PARAMETERS: TC=CRITICAL TEMPERATURE, K PC=CRITICAL PRESSURE, BAR W=PITZER'S ACENTRIC FACTOR KAP=KAPPA-1 PARAMETER OF THR PRSV EOS INPUT TC,PC. W,KAP OF COMPONENT 1: " type 508.1,46.96,0.30667, -0.0088, and press RETURN. At "INPUT TC,PC,W,KAP OF COMPONENT 2 :" type 512.58,80.96,0.56533, -0.1 6816, and press RETURN. At "INPUT TC,PC,W, KAPl OF COMPONENT 3 :"type 647.29,220.90, 0.3438, -0.06635, and press RETURN. At "INPUT TEMPERATURE (K) OF SET NO. 1:" enter 523.15 and press
RETURN.
A~"INPUT LIQUID
MOLE FRACTION OF COMPONENT 1 IN SET 1:"
enter 0.05 and press RETURN. A~"INPUTLIQUID MOLE FRACTION OF COMPONENT 2 IN SET 1:" enter 0.05 and press RETURN. At"1NPUT LIQUID MOLE FRACTION OF COMPONENT 3 IN SET 1:" enter 0.90 and press RETURN. At "INPUT TEMPERATURE (K) OF SET NO. 2 :" enter 523.15 and press
RETURN.
A~-INPUTLIQUID
MOLE FRACTION OF COMPONENT 1 IN SET 2:"
enter 0.1 and press RETURN. A~-INPUT LIQUID MOLE FRACTION OF COMPONENT 2 IN SET 2 : " enter 0.1 and press RETURN. A~"INPUT LIQUID MOLE FRACTION OF COMPONENT 3 IN SET 2:" enter 0.8 and press RETURN. At "INPUT TEMPERATURE (K) OF SET NO. 3 :"enter 523.15 and press
RETURN.
A~"INPUTLIQUID
MOLE FRACTION OF COMPONENT 1 IN SET 3:" enter 0.15 and pres? RETURN.
AI~~INPUT LIQUID
MOLE FRACTION OF COMPONENT 2 IN SET 3 : "
enter 0.15 and press RETURN.
A~"INPUTLIQUID
MOLE FRACTION OF COMPONENT 3 IN SET 3:"
enter 0.7 and press RETURN.
Modeling Vapor-Lquid Equiibrta
"DO YOU WANT TO INPUT EXPERIMENTAL VALUES FOR VAPOR MOLE FRACTION AND PRESSURE FOR COMPARISON WITH THE CALCULATED VALUES (Y/N)?"
type n (or N) and RETURN (The entry of experimental vapor mole fractions and bubble point pressures is optional. In this example no entry is made, because no experimental data were available.) At "PROVIDE BINARY INTERACTION PARAMETER(S) FOR EACH PAIR OF COMPONENTS IN THE MIXTURE. THERE ARE TWO OPTIONS: 1=ONE PARAlrlETER W W MODEL (1PVDW) 2=TWO PARAMETER VDW MODEL (2PVDW) ENTER 1 FOR THE l P W W MODEL, OR 2 FOR THE 2PVDW MODEL. l/2?"
type 2 and press RETURN. (With this selection the user will be prompted to provide two binary interaction parameters for each pair in the m~xture.) At " 2 P W W OPTION: TWO PARAMETERS PER PAIR ARE REQUIRED. FOR THE PAIR 1 2 :"type 0.026 INPUT INTERACTION P-TER and RETURN. At "INPUT INTERACTION PARAMETER FOR THE PAIR 1 3:" type 0.0461 and RETURN. At "INPUT INTERACTION PARAMETER FOR THE PAIR 2 1 : type 0.0076 and RETURN. At "INPUT INTERACTION PARAMETER FOR THE PAIR 2 3:" type -0.0429 and RETURN. Atr,INPUT INTERACTION PARAMETER FOR THE PAIR 3 1:" type -.I56 and RETURN. At "INPUT INTERACTION PARAMETER FOR THE PAIR 3 2:" type -0.0845 and RETURN. (These binary interaction parameters were obtained using the program VDW.EXE described in Appendix D.3, and the data files am200.dat, mw250.&at, and aw250.dat, respectively, for acetone-methanol, methanol-water, and acetone-water binary pairs.) After the last of the binary interaction parameters is entered, the program calculates the VLE and the following appears on the screen: "
VDWMIX: MULTICOMPONENT VLE CALCULATIONS WITH THE VAN DER WAALS ONE FLUID NIXING RULES.
Appendix E: Computer Programs for Multicomponent Mixtures
INPUT PILE NAME: TEST.WW ACETONE-METHANOL-WATER AT 523 K SET NO.
TEMP(K)
PEXP(BAR)
PCAL
VLIQ(CM3/MOL).
WAP
press return for phase c-ositions.
Press RETURN to continue. The following appears: PHASE COMPOSITIONS (IN MOLE FmCTION) SETNO.
COMPONENT
XEXP
YEXP
YCAL
press return for the binary parameter matrix.
Press RETURN to continue. The binary parameter matrix for the VDW mixing rule appears: THE BINARY P-TER 1
MATRIX FOR THE VDW MIXING RULE 2
3
At "DO YOU WANT A PRINT-OUT (Y/N)?" type n (or N) and press RETURN. At"D0 YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?" type n (or N) and press RETURN. At -w YOU WANT TO START A NEW CALCUL~TION (YIN)?" type n (or N) and press RETURN.
Modeling Vapor-Liquid Equibr~a
Example E. I .B:Calculation o f Multicomponent VLE Using an Existing Input File
Change to the directory containing the program VDWMIX.EXE (e.g., A> or C>, etc.). Start the program by typing VDWMIX at the DOS prompt. A program introduction message appears on the screen. Press ENTER (or press RETURN). The following appears: At "VDWMIX: MULTICOMPONENT VLE CALCULATIONS WITH THE VAN DER WAALS ONE-FLUID MIXING RULES. THIS PROGRAM CAN BE USED FOR ISOTHER*1ILZI BUBBLE POINT CALCULATIONS CREATING A NEW INPUT FILE, OR USING A PREVIOUSLY STORED INPUT FILE. YOU MUST SUPPLY NUMBER OF COMPONENTS, LIQUID MOLE FRACTION. CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR, PRSV KAPPA-1 P-TER
FOR EACH COMPOUND, TEMPERA-,
AND MODEL PIUULIWTER(S1 FOR EACH PAIR OF COMPONENTS. ENTER 1 TO CREATE A NEW INPUT FILE, 2 TO USE A PRFYIOUSLY STORED INPUT FILE, OR 0 TO TERMINATE T W PROGRAM.
01112?"
type 2 and RETURN. (This results in the selection of an already existing input file.) A~"ENTERTHE NAME OF EXISTING INPUT FILE (for example, a : m 2 5 0 . M W ) :"enter a:AMW250.VDW and RETURN. The following appears on the screen: VDWIX: MULTICOMPONENT VLE CALCULATIONS WITH THE VAN DER WAALS ONE FLUID MIXING RULES. INPUT FILE NANE: amw250.vdw ACETONE-METHANOL-WATER 250 C SETNO.
TEMP(K)
PEXP(=)
PCAL
VLIQ(CM3/MOL)
WAP
1
523.15
62.060
61.015
48.417
462.8
2
523.15
58.480
58.023
41.567
515.3
3
523.15
52.890
52.378
34.576
616.9
press return for phase compositions.
Press RETURN to continue
Appendx E. Computer Programs for Multcomponent Mxiures
PHASE COMPOSITIONS (IN MOLE FRACTION) SET NO.
1
2
3
COBSPONENT
XEXP
YEXP
YCAL
1
.I370
.2370
.2390
2
.0940
.I480
.I407
3
.7690
.6150
,6203
1
.0880
.l920
,1941
2
.0840
.I450
,1433
3
.8280
.6630
.6626
1
.0430
.I430
.I442
2
.0490
.0950
.lo61
3
.9080
,7620
.7497
press return for the binaey parameter matrix.
Press RETURN to continue. The binary parameter matrix for the VDW mixing rule appears: BINARY PAF2AMETER MATRIX FOR THE MW MIXING RULE
At ,,DO YOU WANT A PRINT-OUT ( Y / N ) ?" enter y (or Y) and RETURN. (This command sends the results, similar to those shown above, to the printer.) Atr'DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT F I L E ( Y / N ) ? n enter y (or Y) and press RETURN. At "ENTER A NAME FOR THE OUTPUT FILE:" type a file name of your choice (for example A:OUTPUTl.OUT) and press RETURN. (With this command the results shown above are saved on the disk in drive A with the name OUTPUT1.OUT as an ASCII file.) At "DO YOU WANT TO START A NEW CALCULATION ( Y / N ) ?" type n (or N) and press RETURN to terminate the program.
E.2.
Program WSMMAIN: Multicomponent VLE Calculations with Wong-Sandler Mixing Rules The program WSMMAIN can be used to calculate multicomponent VLE using the PRSV EOS and the Wong-Sandler mixing rule. One of the three (the UNIQUAC, Wilson, or NRTL) excess free-energy models can be used with this mixing rule by following the instructions that appear on the screen during program execution
Modeling Vapar-Lqu~d Equilibria
This program can be used to create a new input file for a multicomponent liquid mixture and then to calculate the isothermal bubble point pressure and the composition of the coexisting vapor phase for this mixture. In this mode the information needed is the number of components (up to a maximum of ten), the liquid mole fractions, the temperatures at which calculations are to be done (for the number of sets of calculations, as the the user wishes, up to a maximum or fifty), critical temperatures, pressures (bar), acentric factors, the K , constants of the PRSV eauation for each compound in the mixture, and, if available, the experimental buhble point pressure and vaporphase compositions (theselast entries areoptional, and areused for a comparison between the experimental and calculated results). In addition, the user is requested to supply model parameters for each pair of components in the multicomponent mixture. These model parameters can be obtained using the program WS (see Appendix D.5) if experimental data are available for each of the binary pairs. Alternatively, the user can select an already existing tile (for these files we use the extensions WSN, WSW, and WSU, respectively, for the WS-NRTL, WS-WILSON, and WS-UNIQUAC options, and some examples are provided on the accompanying disk) and calculate the multicomponent VLE for the mixture of that input file. The results from the program WSMMAlN can be sent to a printer, to a disk tile, or both. The commands that appear on the screen upon the completion of the calculations must be followed to make this choice. Please see the following tutorial for further details. Example E.2.A: Creating a New Input File and Calculation o f Muticomponent VLE
Change to the directory containing the program WSMMAIN.EXE (e.g., A> or C>, etc.). Start the program by typing WSMMAIN at the DOS prompt. A program introduction message appears on the screen. Press ENTER (or press RETURN). The following appears: WSM: MULTICOMPONENT VLE CALCULATIONS WITH THE WONG-SANDLER MIXING RULE. YOU HAVE TO SELECT AN EXCESS ENERGY MODEL TO BE USED IN THE MIXING RULE. THE SELECTIONS ARE:
Type 1 and press RETURN. (This results in the selection of the NRTL model.) At "WSM: MULTICOMPONENT VLE CALCULATIONS WITH THE WONG-SANDLER-NRTL MIXING RULE THIS PROGPAM CAN BE USED FOR ISOTHERDIAL BUBBLE POINT CALCULATIONS CREATING
A NEW INPUT FILE, OR USING A PREVIOUSLY STORED INPUT FILE.
Appendx E: Computer Programs for Multicomponent Mixtures
YOU MUST SUPPLY THE NUMBER OF COMPONENTS, LIQUID MOLE FRACTION. CRITICAL TEMPERATURE, PRSV KAPPA-1
CRITICAL PRESSURE, ACENTRIC FACTOR,
P W T E R FOR EACH COMPOUND, TEMPERATURE,
AND MODEL PARAMETERS FOR EACH PAIR OF COMPONENTS. ENTER 1 TO CREILTE A NEW INPUT F I L E , STORED INPUT F I L E ,
2 M SELECT A PREVIOUSLY
OR 0 TO TERMINATE THe PROGRAM.
0/112?"
Type 1 and press RETURN. (With this selection the user is prompted to create a new input file.) At "ENTER A NAME FOR THE NEW INPUT FILE (format: ********.WSN):"
.
type a:testl.wsn and press RETURN. (This will result in the creation of an input file named TESTI.WSN that will be stored on the disk on drive A.) At "ENTER A TITLE FOR THE NEW INPUT FILE:" type 'acetone-methanol-water at 250 C by WS+NRTL model' and press RETURN (The title is a descriptive statement, with a maximum 60 characters, about the input file to be created.) At "INPUT NUMBER OF COMPONENTS:" type 3 and press RETURN. At
"HOW MANY SETS OF I S O T H E W L BUBBLE POINT CALCULATIONS DO YOU WANT TO DO? (FOR EACH SET YOU PROVIDE A NEW LIQUID COMPOSITION AND TEMPERATURE):"
type 2 and press RETURN. At "ENTER PURE COMPONENT PAFIAMETERS: TC=CRITICAL TEMPERATURE, K PC=CRITICAL PRESSURE, BAR W=PITZER'S ACENTRIC FACTOR KAP=THE KAPPA-1 PARAMETER OF THE PRSV EOS INPUT TC, PC. W, KAP OF COMPONENT 1:" type 508.1,46.96,0.10667, -0.0088, and press RETURN. At "INPUT TC, PC,W, KAP OF COMPONENT 2 :',type 512.58,80.96, 0.56533, -0.16816, and press RETURN. At "INPUT TC, PC,W. KAPl OF COMPONENT 3:" type 647.29,220.90, 0.3438, -0.06635, and press RETURN. At "INPUT TEMPERATURE (K) OF SET 1:" enter 523.15 and press RETURN.
A~"INPUTLIQUID
MOLE FRACTION OF COMPONENT 1 IN SET 1:"
enter 0.05 and press RETURN.
A~"INPUTLIQUID
MOLE FRACTION OF COMPONENT 2 IN SET 1:"
enter 0.05 and press RETURN.
Modelng Vapor-Liqud Equilbria
AtrrINPUTLIQUID MOLE FRACTION OF COMPONENT enter 0.90 and press RETURN. At "INPUT TEMPERATURE (K) OF SET 2:" enter 523.15 and press RETURN. A~"INPUT LIQUID MOLE FRACTION OF COMPONENT enter 0.15 and press RETURN. A~"INPUTLIQUID MOLE FRACTION OF COMPONENT enter 0.15 and press RETURN. A~~TINPUT LIQUID MOLE FRACTION OF COMPONENT enter 0.7 and press RETURN.
3 IN SET 1:"
1 IN SET 2:"
2 IN SET 2:" 3 IN SET 2:"
At "DO YOU WANT TO INPUT EXPERIMENTAL VALUES FOR VAPOR MOLE FRACTION AND PRESSURE FOR COMPARISON WITH CALCULATED VALUES (Y/N)?"
type n (or N) and press RETURN.
At "INPUT MODEL PARAMETERS. THEY ARE: KIJ=THE WONG-SANDLER MODEL BINARY INTERACTION PARAMETER. ALPHAIJ=THE NRTL MODEL ALPHA PARAMETER. AIJ=THE NRTL MODEL ENERGY PAPAMETERS, TWO FOR EACH PAIR OF COMPONENTS, IN CAL/MOLE. [AIJ=TAUIJ*RTI INPUT KIJ, ALPHAIJ FOR THE PAIR 1 2:"
type 0.05, 0.35, and press RETURN.
At "INPUT
KIJ AND ALPHAIJ FOR THE PAIR 1 3:" type 0.35,0.35, and press RETURN.
At "INPUT
KIJ AND ALPHAIJ FOR THE PAIR 2 3:" type 0.05,0.35, and press RETURN.
At "INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 1 2:" type 451.58 and press RETURN.
At "INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 1 3:" type 452.77 and press RETURN.
At "INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 2 1:" type 95.0 and press RETURN.
At "INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 2 3:" type 197.52 and press RETURN.
At "INPUT THE NRTL MODEL ENERGY PAFAMETER AIJ FOR THE PAIR 3 1:" type 1042.88 and press RETURN.
Appendix E: Computer Programs for Multicomponent Mixtures
At A I J FOR THE PAIR 3 2:"
"INPUT THE NRTL MODEL ENERGY P-TER
type 520.60 and presy RETURN. (These binary interaction parameters were ohtained using the program WS.EXE described in Appendix. D.5, and the data files am200.dat, mw250.dat, and aw250.dat for acetone-methanol, methanol-water, and acetone-water binary mixtures, respectively.) Following these entries of the model parameters, the information is written to the disk in drive A, and the program then calculates the multicomponent VLE. The following results appear on the screen: WSM: MULTICOMPONENT VLE WITH THE WONG--SANDLER MIXING RULE INPUT FILE NAlm: teatl.wsn
acetone-methanol-water at 250 C by WS+NRTL model SET NO.
TEMP(K)
1
523.15
PEXP(BAR) .
PCAL
VLIQ(CM3/MOL)
WAP
52.527
31.481
643.3
press return for the phase compositions.
Press RETURN to see phase compositions. The following appears: PXASE COMPOSITIONS (IN MOLE FRACTION) SETNO. 1
2
COMPONENT
XEXP
YEXP
-
1
.0500
2
.0500
3
.go00
1
.I500
2
.I500
3
.7000
-
YCAL .1611
.I133 .7256 .2536 .2230 ,5234
press return for parameter matrices.
Press RETURN to continue. The following parameter matrix for the k , parameter of the Wong-Sandler mixing rule appears: P U T E R MATRIX FOR THE KIJ PAEZAMETER
press return for the alpha parameter matrix.
Press RETURN to continue. The following parameter matrix for the NRTL model parameter appears:
Modellng Vapor-Liquid Equibria
MATRIX FOR TXE 1 U m . PARAMETER
P-rn
1
2
3
press return for the NRTL model energy parameter matrix.
Press RETURN to continue. The following parameter matrix for the NRTL model energy parameters appears: P W T E R MATRIX FOR THE NRTL ENERGY PARAMETER, AIJ
At ,,DO YOU WANT A PRINT-OUT (YIN)?" type y (or Y) and press RETURN. At "DO YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?" type y (or Y) and RETURN. At "ENTER A NAME FOR THE OUTPUT FILE: type a:outputl.out. At "DO YOU WANT TO START A NEW CALCULATION WITH THE NRTL MODEL (Y/N)?"
type n (or N)and press RETURN. A t r w o YOU WANT TO SELECT A NEW EXCESS ENERGY MODEL (YIN)?" type n (or N) and press RETURN.
E.2.B. Calculat~onof Mult~componentVLE U s ~ n gan Exlsting Input F~le Change to the directory containing the program WSMMAIN.EXE (e.g., A> or C>, etc.). Start the program by typing WSMMAIN at the DOS prompt. A program introduction message appears on the screen. Press ENTER (or press RETURN). The following appears: WSU: WJLTICOMPONENT VLE CALCULATIONS WITH THE WONG-SANDLER NIXING RULE. YOU HAVE TO SELECT AN EXCESS ENERGY MODEL M BE USED IN THE MIXING RULE. THE SELECTIONS ARE:
Appendix E: Computer Programs for Muticomponent Mlxtures
Type 1 and press RETURN. (This results in the selection of the NRTL model.) At "WSM: MULTICOMPONENT VLE CALCULATIONS WITH WONG-SANDLER-NRTL MIXING RULE. THIS PROGRAM CAN BE USED FOR ISOTHERMAL BUBBLE POINT CALCULATIONS, CREATING
A NEW INPUT FILE, OR USING A PREVIOUSLY STORED I N P m FILE. YOU MUST SUPPLY THE NUMBER OF COMPONENTS, LIQUID MOLE FRACTION, CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR, PRSV KAPPA-1 P-TER
FOR EACH COMPOUND, TEMPERATURE,
AND MODEL PARAMETERS FOR EACH PAIR OF COMPONENTS. ENTER 1 TO CREATE A NEW INPUT FILE, 2 TO SELECT A PREVIOUSLY STORED INPUT FILE, OR 0 TO TERMINATE THE PROGRAM. 0/1/2?"
type 2 and press RETURN. (This results in the selection of an already existing input file.) A~"ENTERTHE NAWE OF EXISTING INPUT FILE (for examgle, a:PE423.WSN):"
type a:amw250.wsn and press RETURN. The following appears on the screen: WSM: MULTICOMPONENT VLE WITH THE WONCI-SANDLER MIXING RULE INPUT FILE NAbE: AMW250.WSN INPUT FILE: ACETONE-METXANOL-WATER 250 C SET NO.
TEMP(K)
PEXP(BAR)
PCAL
VLIQ(CMS/MOL)
WAP
1
523.15
62.060
62.226
38.088
485.0
press return for the phase compositions.
Press RETURN to see phase compositions. The following appears: PHASE COMPOSITIONS (IN MOLE FRACTION) SET NO.
COMPONENT
XEXP
YEXP
YCAL
Modeling Vapor-Liqud Equilbria
Dress return for parameter matrices
Press RETURN to continue. The following parameter matrix for the k , parameter of the Wong-Sandler mixing mle appears: PllRRDdETER MATRIX FOR THE K I J PARAMETER
Dress return for the alpha parameter matrix.
Press RETURN to continue. The following parameter matrix for the NRTL model w parameter appears: PllRRDdETER MATRIX FOR THE ALPHA PARAMETER
press return for the NRTL model energy parameter matrix.
Press RETURN to continue. The following parameter manix for the NRTL model-energy parameters appears: PARAKETER MATRIX FOR THE NRTL ENERGY PARAKETER, A I J
1
2
3
At *,DO YOU WANT A PRINT-OUT (Y/N)?" type n (or N) and press RETURN. At"W YOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?" type n (or N) and RETURN. At "DO YOU WANT TO START A
NEW CALCULATION WITH THE NRTL MODEL (Y/N)?"
type n (or N) and press RETURN.
Appendlx E: Computer Programs for Multicomponent
A t " m YOU WANT TO SELECT A type n (or N) and press RETURN.
E.3.
Mxtures
NEW EXCESS
ENERGY
MODEL ( Y / N ) ? "
Program HVMMAIN: Multicomponent VLE Calculations with Modified Huron-Vidal (HVOS) Mixing Rule The program HVMMAIN can he used to calculate multicomponent VLE using the PRSV EOS and the HVOS mixing rule (see Section D.4). One of the three (the UNIQUAC, Wilson, or NRTL) excess free-energy models is selected for use with this mixing rule by following the instmctions that appear on the screen during execution of the program. This program can be used to create a new input file for a multicomponent liquid mixture and then to calculate the isothermal bubble point pressure and the composition of the coexisting vapor phase for this mixture. In this mode the information needed is number of components (up to a maxi~numof ten), the liquid mole fractions, the temperatures at which calculations are to be done (for the number of sets of calculations the nser wishes, up to a maximum of fifty), critical temperatures, pressures (bar), acentric factors, the K , constants of the PRSV equation for each compound in the mixture, and, if available, the experimental bubble point pressure and vapor phase compositions (these last entries are optional and are used for a comparison between the experimental and calculated results). In addition, the user is requested to supply model parameters for each pair of components in the multicomponent mixture. These model parameters can he obtained using the program HV (see Section D.4) if experimental data are available for each of the binary pairs. Alternatively, the nser can select an already existing file (for these files we use the extensions HVN, HVW, and HVU, respectively, for the HVOS-NRTL, HVOS-WILSON, and HVOS-UNIQUAC options, and some examples are provided on the accompanying disk) and"calcu1ate the multicomponent VLE for the mixture of that input file. The results from the program HVMMAIN can be sent to a printer, to a disk file, or both. The commands that appear on the screen upon the completion of the calculations must be followed to make this choice. Please see the following tutorial for further details. Tutorial on the Use of HVMMAIN.EXE Example E.3.A: Calculation of Multicomponent VLE Creating
a New Input File Change to the directory containing the program HVMMAIN.EXE (e.g., A> or C>, etc.). Start the program by typing HVMMAIN at the DOS prompt. A program introduction message appears on the screen. Press ENTER (or press RETURN).
Modelng Vapor-Llqu~d E q u l ~ b r ~ a
The following appears: "HVX: MULTICOMPONENT VLE CALCULATIONS WITH THE HVOS MIXING RULE. YOU HAVE TO
SELECT AN EXCESS ENERGY MODEL TO BE USED IN THE HVOS MIXING RULE. THE SELECTIONS ARE: l=NRTL ZIWILSON 3=UNIQUAC 1/2/3?"
Type 1 and press RETURN (This results in selection of the NRTL model.) At MIM: MULTICOMPONENT VLE CALCULATIONS WITH THE HVOS
+
NRTL MODEL.
THIS PROGRAhl CAN BE USED FOR ISOTHEIWU BUBBLE POINT CALCULATIONS, CREATING A NEW INPUT FILE, OR USING A PREVIOUSLY STORED INPUT FILE. YOU MUST SUPPLY THE NUMBER OF COMPONENTS, LIQUID MOLE FRACTION. CRITICAL TEMPERATURE, CRITICAL PRESSURE, ACENTRIC FACTOR, PRSV KAPPA-1 P-TER
FOR EACH COMPOUND, TEMPERATURE,
AND MODEL PARAMETERS FOR EACH PAIR OF COMPONENTS. ENTER 1 TO CREATE A NEW INPUT FILE, 2 TO SELECT A PREVIOUSLY STORED INPUT FILE, OR 0 TO TERMINATE THE PROGRAM. 0/1/2?"
type 1 and press RETURN. (With this selection the user is prompted to create a new input file.) At "ENTER A NAME FOR THE NEW INPUT FILE (format: * * * * * * * * . W V N ) : "
type a:testl.hvn and press RETURN. (This results in the creation of an input file named testl.hvn that will he stored on the disk in drive A. ) At "ENTER A TITLE FOR THE NEW INPUT FILE:" type ACETONE-METHANOL-WATER AT 523 K and press RETURN. (The title is a descriptive statement, maximum 60 characters, for the input file to be created.) At "INPUT NUMBER OF COMPONENTS: type 3 and press RETURN. At "HOW MANY SETS OF ISOTHERCiIAL BUBBLE POINT CALCULATIONS DO YOU WANT TO DO? (FOR EACH SET YOU PROVIDE A NEW LIQUID COMPOSITION AND TEMPERATURE):"
type 2 and press RETURN. At "ENTER PURE COMPONENT PARAIUIETERS: TC=CRITICAL TEMPERATURE, K PClCRITICAL PRESSURE, BAR W=PITZER'S ACENTRIC FACTOR
Appendix E: Computer Programs for Muticomponent Mixtures
KAP=THE KAPPA-1
"DO
PARAMETER
OF THE PRSV EOS
I N P U T TC, PC, W, KAP O F COMPONENT 1:" type 508.1,46.96,0.30667, -0.0088, and press RETURN. At "INPUT TC, PC,W. KAP OF COMPONENT 2:- type 512.58,80.96, 0.56533, -0.1681 6, and press RETURN. At "INPUT TC, PC,W. KAP OF COMPONENT 3:" type 647.29,220.90, 0.3438, -0.06635, and press RETURN. At "INPUT TEMPERATURE ( K ) OF SET I:" enter 523.15 and press RETURN. A~"INPUT LIQUID MOLE FRACTION OF COMPONENT 1 IN SET 1:" enter 0.05 and press RETURN. A~"INPUT LIQUID MOLE FRACTION OF COMPONENT 2 IN SET 1:" enter 0.05 and press RETURN. A t S r 1 ~ LIQUID ~ u ~ MOLE FRACTION OF COMPONENT 3 IN SET 1:" enter 0.90 and press RETURN. At I N P U T TEMPERATURE ( K ) O F SET 2 :" enter 523.15 and press RETURN A~"INPUT L I Q U I D MOLE FRACTION OF COMPONENT 1 I N SET 2 : " enter 0.10 and press RETURN. A~"INPUT LIQUID MOLE FRACTION OF COMPONENT 2 IN SET 2:" enter 0.10 and press RETURN. AtrrINPUT L I Q U I D MOLE FRACTION OF COMPONENT 3 I N SET 2:" enter 0.80 and press RETURN. At YOU WANT TO ENTER EXPERIMENTAL VALUES FOR VAPOR MOLE FRACTION
AND PRESSURE FOR COMPARISON WITH THE CALCULATED VALUES ( Y I N ) ? "
type n (or N) and press RETURN. At "ENTER EXCESS GIBBS ENERGY
MODEL
PARAMETERS.
THEY ARE:
ALPHAIJ= THE NRTL MODEL ALPHA PARAMETER. A I J = THE NRTL MODEL ENERGY PARAMETERS, TWO FOR EACH P A I R OF COMPONENTS.
I N CAL MOLE.
[AIJ=TAUIJ*RT] PAIR 1 2:"
I N P U T A L P H A I J FOR THE
type 0.35 and press RETURN. At ,,INPUT ALPHAIJ FOR THE At "INPUT ALPHAIJ FOR THE At "INPUT
"INPUT
PAIR
1 3 :"type 0.35 and press RETURN.
PAIR
2 3 :"type
0.35 and press RETURN.
THE NRTL MODEL ENERGY PARAMETER A I J FOR THE P A I R 1 2:"
type - 103.0 and press RETURN. At THE NRTL MODEL ENERGY PARAMETER type 278.86 and press RETURN.
A I J FOR THE P A I R 1
3:"
At "INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 2 1:"
type 476.29 and press RETURN. At "INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 2 3:"
type -1 15.58 and press RETURN. At "INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 3 1:"
type 2322.80 and press RETURN. At "INPUT THE NRTL MODEL ENERGY PARAMETER AIJ FOR THE PAIR 3 2:"
type 1019.48 and press RETURN. These binary interaction parameters were obtained using the program HV.EXE described in Appendix. D.4, and the data files am200.dat, mw250.dat, and aw250.dat for acetone-methanol, methanol-water, and acetone-water hinary pairs, respectively. Following the entry of the model parameters, the information is written to a disk in drive A, and the program calculates the multicomponent VLE. The following results appear on the screen: HVM: MULTICOMPONENT VLE WITH THE WYOS MIXING RULE INPUT FILE w :test1.hACETONE-METKPNOL-WATER AT 523 K SET NO.
TEMP(K)
PEXP(BAR)
PCAL
VLIQ(CM3/MOL)
WAP
1
523.15
-
53.045
35.352
603.2
press return for phase comgositions.
Press RETURN to see phase compositions. The following appears: PHASE COMPOSITIONS (IN MOLE FRACTION) SETNO.
COMPONENT
XEXP
YEXP
YCAL
press return for the NRTL model alpha parameter matrix.
Press RETURN to continue. The following parameter matrix for the NRTL model or parameter appears:
Appendix E. Computer Programs for Multcornponent Mixtures
PARAMETER MATRIX FOR THE ALPHA PARAMETER
press return for the NRTL rnodel energy parameter matrix.
Press RETURN to continue. The following parameter matrix for the NRTL model energy parameters appears: PARAMETER MATRIX FOR THE NRTL MODEL ENERGY PARAMETER AIJ (CALIMOLE)
At YOU WANT A PRINT-OUT (Y/N)? v type y (or Y) and press RETURN. A ~ " D oYOU WANT TO SAVE THE RESULTS TO AN OUTPUT FILE (Y/N)?" type y (or Y )and RETURN. At "ENTER A NAME FOR THE OUTPUT FILE:" type a:outp~tl.OUt. At "W YOU WANT TO START A
NEW CALCULdTION WITH THE NRTL MODEL (Y/N)?"
type n (or N) and press RETURN. A t " W YOU WANT TO SELECT A NEW EXCESS ENERGY MODEL (Y/N)?" type n (or N) and press RETURN. Example E.3.B: Calculation of Multicomponent VLE Using an Existing Input File
Change to the directory containing the program HVMMAIN.EXE (e.g., A> or C>, etc.) Start the program by typing HVMMAIN at the DOS prompt. A program introduction message appears on the screen. Press ENTER (or press RETURN). The following appears: "HVM: MULTICOMPONENT VLE CALCULATIONS WITH THE W O S MIXING RULE. YOU HAVE TO SELECT AN EXCESS ENERGY MODEL TO 8E USED IN THE W O S MIXING RULE. THE SELECTIONS ARE: l=NRTL 2lWILSON 3=UNIQUAC 1/2/3?"
Modeling Vapor-Liquid Equilbria
Type 1 and press RETURN. (This results in the selection of the NRTL model.)
At "m:MULTICOMPONENT
VLE CALCULATIONS WITH THE HYOS + NRTL MODEL.
THIS PROGRADl CAN BE USED FOR ISOTHEIWAL BUBBLE POINT CALCULATIONS, CREATING A NEW INPUT FILE, OR USING A PREVIOUSLY STORED INPUT FILE. YOU MUST SUPPLY THE NUMBER OF COMPONENTS, LIQUID MOLE FRACTION, CRITICAL TEMPERAlWtE, CRITICAL PRESSSURE, ACENTRIC FACTOR, PRSV KAPPA-I PRRAMETER FOR EACH COMPOUND, TEMPERATEXE, AND MODEL P M T E R S FOR EACH PAIR OF COMPONENTS.
ENTER 1 TO CFZATE A NEW INPUT FILE, 2 TO SELECT A PREVIOUSLY STORED INPUT FILE, OR 0 TO TEWINATE THE PROGRAM. 011127
type 2 and press RETURN. (This results in the use of an existing input file.)
Atr'ENTER
EXISTING INPUT FILE (for example, a:PE423 .m):"type AMW250.HVN and press RETURN. NAME OF THE
The followillg appears on the screen: Mild:
MULTICOMPONENT VLE WITH THE HYOS MIXING RULE
INPUT FILE N m : AMW250.HVN ACETONE-METHANOL-WATER 250 C SET NO.
TEMP(K)
PEXP(W)
PCAL
VLIQ(CN3/MOL)
WAP
1
523.15
62.060
61.734
48.421
452.7
2
523.15
58.480
58.477
41.539
508.4
3
523.15
52.890
52.141
34.546
621.6
press return for phase conwositions.
Press RETURN to see phase compositions. The following appears: PHASE COMPOSITIONS (IN MOLE FRACTION) SET NO. COMPONENT 1
3
XEXP
YEXP
YCAL
1
.I370
.2370
.2455
2
.0940
.I480
.I436
3
.7690
.6150
.GI09
1
,0430
.I430
.I416
2
.0490
.0950
.lo87
3
.9080
.7620
.7496
Dress return for the NRTL model alpha parameter matrix.
Appendix E Computer Programs for Multicomponent M d u r e s
Press RETURN to continue. The following parameter matrix for the NRTL model a parameter appears: PARAMETER NATRIX FOR THE ALPHA PARAMETER
1
2
3
1)
.OOOO
,3500
.35001
21
.3500
.OOOO
.35001
31
.3500
.3500
.OOOOl
press return for the NRTL model energy Barameter matrix.
Press RETURN to continue. The following parameter matrix for the NRTL model energy parameters appears: P-TER
NATRIX FOR THE NRTL ENERGY PARAMETER A I J (CALIMOLE)
At -DO YOU WANT A PRINT-OUT (Y/N)?" type n (or N) and press RETURN. At "DO YOU WANT TO SAVE THESE RESULTS TO AN OUTPUT FILE (Y/N)?"
type n (or N) and RETURN. At "DO YOU WANT TO START A NEW CALCULATION WITH THE NRTL MODEL (Y/N)?"
type n (or N) and press RETURN. Attwo YOU WANT TO SELECT A NEW EXCESS ENERGY MODEL (YIN)?,, type n (or N) and press RETURN.
References
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+
+
+
+
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+
+
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Index
lPVDW model, 26-27.35. 105 2PVDW model, 34, 105 Acentric factor, 20.21 Acetone PRSV parameters, 21 VLE with waar,31, 36.49, 56, 69, 79-81 Activity coefficient, 6, 11, 103 at infinite dilution, 86 from an equation of state. 39 Activity coefficient models, 8, 11-17, 102 Margules. 13 NRTL, 13 UNIFAC, 16 UNIQUAC, 13 Wilson, 13 van Laar. 12 Alpha (a)parameter. Peng-Robinson equation. 20 Antoine equation, 9 ASOG model, I6 Benzene PRSV parameters, 21 VLE with carbon dioxide, 92-93 VLE with methanol, 77-79 Binary interaction parameters, 26, 34.40, 51, 57 Carbon dioxide PRSV parameters, 21 VLE with benzene, 90-92 VLE with methanol, 92-93 VLE with propane, 28,48,69 Chemical potential, 5 Chemical reaction. 98 Combining mle(s), 23 Compressibility factor, 7, 18 Computational methods
Computer program(s) for binary mixtures, 114 for multicc,mponent mixtures. 177 Critical compressibility, 23 temperature, 19, 21 pressure, 19, 21 Cubic equations of state, 19 Cyclohrnane PRSV parameters. 21 VLE with methyl acetate, 41 DECHEMA. 9,14 Dcnsity dependent mining mle, 53 Enthalpy, 95 Entropy, 95 Ethanol PRSV parameters, 21 VLE with n-heptane, 28,35,40-42 Equation of state, 7 Models, 17, 104 Excess free energy Gibbs, of mixing, 33 Helmholtz, mixing. 44 Molar, Gibbs. 44 FORTRAN, 1 14 Fugacity, 6, 17, 103 Fugacity coefficient, 6. 7. 8, 104 Gamma-Phi method, 7 Gibbs excess energy. of mixing, 4 4 free energy departure function, 112
Gibbs (continued) frce energy in ,deal mixture, 5 pattial molar free energy, 5 Heat capacity, 95 Henry's constants, 95 Helmholtz excess free energy, of mixing, 44 free energy deparmrc function, 112 Huron-Vidal (HVO) model, 48, 107 HVO model, see Huron-Vidal model HVOS model, 63.66 Ideal gas equation, 8 Infinite dilution activity coefficient, 86.95 Infinite pressure, limit, 46 Kappa ( K ) Parameter Peng-Robinson, 20 PRSV, 20.21 LCVM model, 63.65, 109 Liquid-liquid equilibrium (LLE), 95,97, 100 LLE, see liquid-liquid equilibrium Margulea equation, 12 Methane PRSV parameters, 21 VLE with n-decane, 89-91 VLE with n-heptane, 89-90 VLE with n-pentane, 27.48.67.89-91 Methanol PRSV parameters, 21 VLE with benzene, 77-79 VLE with propane, 29, 35 Methyl acetate PRSV parameters, 2 1 VLE with cyclohexanr, 41 MHVI model, 63.64, 108 MHV2 model, 63,65, 108 Micellar solutions, 98 Michelsen-Kistenmacher (syndrome), 42 Mixing mle(s), 23, 25, 44 HVOS, 63.66, 109 LCVM, 63.65, 109 MHVI, 63,64, 108 MHV2,63,65,108 van der Waals, 26.34 Wong and Sandler, 50, 106 Multicomponent, computer programs for, mixtures, 177 n-Buwnol, PRSV parameters, 21 n-Decnne
PRSV parameters, 21 VLE with methane, 89-91 n-Heptane PRSV parameters, 21 VLE with ethanol, 40-42 VLE with methane, 89-91 n-Hexane, PRSV parameters, 21 Nonelectrolyte mixtures, 100 Non-quadratic combining rules, 34 n-Pentane PRSV parameters, 21 VLE with ethanol, 28, 35 VLE with methane, 27,48, 67, 89-90 NRTL model, 13 Modified form of, 57 One-Huid model, 25, 105 Partial molar Gibbs free energy, 5 Peng-Robinson equation of state, 7, 19, 104 virial form for the, 25 reduced form, 44 Pitzer's acenmc factor, 20,21 Polymer, 97 Poynting correction, 9 Predictive models, 75 for mixtures of condensable compounds, 75 for mixtures with supercritical gases, 88 Propane PRSV parameters, 21 VLE with carhon dioxide, 28.48, 69 VLE with methanol, 29, 35 Propanol, 2PRSV parameters, 21 VLE with water, 29, 36.49, 56, 69, 82-84 PRSV equation, 20 Raoult's law, 11 Redlich-Kister equation, 1 1 Reduced Peng-Robinson equation of state. 46 temperature, 22,46 pressure, 46 Regular solution model. 15 Saturation pressure, pure component. 9, 21 Simplex formalism, l I 0 Solubility parameter, 15 Supercritical Ruid, 97 gases, 88 UNIFAC model, 16 UNIQUAC model, 13
van der Waals mixing mles, 26, 34 van Laar equation, 12 Vapor-liquid equilibrium, 6, 7, 19 of, acetone with water. 31, 36,49, 56, 69, 79-81 of. benzene with carbon dioxide, 9 G 9 2 of, benzene with methanol, 77-79 of, carbon dioxide with benzene, 90-92 of, carbon dioxide with methanol, 92-93 of, carbon dioxide with propane, 28.48, 69 of, cyclohexane with methyl acetate, 41 of, ethanol with n-pentane, 28, 35 of, ethanol with n-heptane, 40112 of, methane with n-decane. 89-91 of, methane with n-heatane. 89-90 of, methane with n-pentane, 27.48.67.89-91 of, methanol with henzene, 77-79 of, methanol with propane, 29,35 of, methyl acetate with cyclohexane. 41 of, n-decane with methane, 89-91 of. n-hcptane with ethanol, 40112 of, n-heptane with methane, 89-90 of, n-pentanc with ethanol, 28, 35 of, n-pentane with methane, 27,48,67, 89-91 of, propane with carbon dioxide, 28, 48, 6 9 of, propane with methanol, 29.35 of, 2-propanol with water, 29, 36.49, 56, 69, 82-84
of, water with acetone, 31, 36.49, 56, 69, 79-81 of. water with 2-propanal, 29, 36.49, 56, 69, 82-84 programming, 110 Vapor-liquid-liquid equilibrium, 95, 100 Vapor pressure, pure liquid, 9 Virial equation of state, 7, 24 second coefficient, 7 , 2 4 third coefficient, 7 , 2 4 VLE, see vapor-liquid equilibrium VLLE, see vapor-liquid-liquid equilibrium Vl,lume, reduced. 46 Water PRSV parameters, 2 1 VLE with acetone, 31, 36,49, 56,69, 79-8 1 VLE with 2-propanol, 30, 36,49, 56, 69.82-84 Wilson equation, 13 Wohl expimiion. 12 Wong-Sandlcr model, 50, I06 WS model, see Wong-Sandler model
Zero pressure, limit, 46
1 re frequently used in :hemica1 and petrc industriesto model complex phase behavior andct ;ign chemical proce Recently developed mixing rules have greatly increased the accuracy and range of applicability of such equations. This book presents a state-of-the-art review of this important topic and discusses the use of cubic equations of state to model the vapor-liquid behavior of mixtures of all degrees of nonideality. A special feature of the book is that it includes a disk of computer programs for all fhe'models discussec' along with tutorials on their use. With the programs and'tutorials, readers cal easily reproduce the results reported and test all the models presented with their own data to decide which will be most useful in their own work. This book will be an invaluable tool for chemical engineers, research che~ and those involved in the simulation and design of chemical process. L,
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