Critical phenomena in binary fluid mixturesKritische verschijnselen in binaire fluïde mengsels: klassifikatie van fasenevenwichten met de simplified-perturbed-hard-chain theory: Classification of phase equilibria with the simplified-perturbed-hard-chain t

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(2) TR diss 2168 CRITICAL PHENOMENA IN BINARY FLUID MIXTURES. Classification of Phase Equilibria with the Simplified-Perturbed-Hard-Chain Theory.

(3) Cover: p,T-Projection of a binary mixture showing a closed liquid-liquid critical curve and a high-pressure critical curve. Copyright © 1992, by A. van Pelt, Rotterdam, the Netherlands. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the written prior permission of the author.. Printed in the Netherlands.

(4) CRITICAL PHENOMENA IN BINARY FLUID MIXTURES Classification of Phase Equilibria with the Simplified-Perturbed-Hard-Chain Theory. KRITISCHE VERSCHIJNSELEN IN BINAIRE FLUÏDE MENGSELS Klassifikatie van fasenevenwichten met de Simplified-Perturbed-Hard-Chain Theory. PROEFSCHRIFT. ter verkrijging van de graad van doctor aan de Technische Universiteit Delft op gezag van de Rector Magnificus Prof. drs. P.A. Schenck, in het openbaar te verdedigen ten overstaan van een commissie aangewezen door het College van Decanen op donderdag 17 december 1992 te 19.00 uur, door. Antonius van Pelt. Chemisch Technoloog Geboren te Rotterdam.

(5) Dit proefschrift is goedgekeurd door de promotor Prof. dr. ir. J. de Swaan Arons. en de toegevoegd promotor Dr. ir. C.J. Peters. De promotiecommissie bestaat uit:. Prof. P.A. Schenck ,. Rector Magnificus. Prof. J. de Swaan Arons ,. promotor. Dr. C.J. Peters ,. toegevoegd promotor. Prof. P.H.E. Meijer ,. The Catholic University of Washington , V.S.. Prof. M.D. Donohue ,. Johns Hopkins University , V.S.. Prof. G.M. Schneider ,. Ruhr Universität Bochum , Duitsland. Dr. U.K. Deiters ,. Ruhr Universität Bochum , Duitsland. Prof. A.H.M. Levelt ,. Katholieke Universiteit Nijmegen , Nederland.

(6) De natuur gedoogt dat gij haar bespiedt, niet dat gij haar ontraadselt. Pythagoras.

(7) DANKWOORD / ACKNOWLEDGEMENT / DANKSAGUNG Op deze plaats wil ik iedereen bedanken die heeft bijgedragen aan het totstandkomen van dit proefschrift. In de eerste plaats zijn dat natuurlijk mijn promotor Prof.dr.ir. Jakob de Swaan Arons en mijn begeleider Dr.ir. Cor Peters. Zij hebben het mij mogelijk gemaakt dit promotieonderzoek in de sektie "Toegepaste thermodynamika en fasenleer" te verrichten, het manuscript van dit proefschrift gelezen en dit bekommentarieerd. Dr.ir. Theo de Loos bedank ik voor zijn suggestie berekeningen rond het Van Laar punt uit te voeren, zie paragraaf 8.4. Verder bedank ik alle, niet met name genoemde, (ex-)leden van onze sektie voor hun collegialiteit. Prof. Paul Meijer (Catholic University of America, Washington DC.) dank ik voor vele interessante discussies, suggesties en adviezen. Herrn Priv. Doz. Dr. Ulrich Deiters bin ich dankbar für seine Einladung fünf Monate an der Ruhr-Universität Bochum, Deutschland, zu verbringen, für die Bereitstellung der Programme zur Berechnung globaler Phasendiagramme und für die Einweisung in ihre Benutzung. Selbstverständlich danke ich auch den anderen Mitarbeitern des "UKD-Teams": Andreas Bolz, Thomas Kraska, Gereon Hintzen und Martin Bluma. Uiteraard bedank ik ir. Jaap Kleimeer en Kees Ravesteyn van de systeemgroep voor het operationeel houden van de computers en installeren van software. Dr. Tony Wells is gratefully acknowledged for reading the manuscript of this PhD-thesis and correcting my (ab)use of the english language. Tenslotte ben ik dank verschuldigd aan de stichting Scheikundig Onderzoek in Nederland (SON), de tweede-geldstroom organisatie voor de chemie die deel uitmaakt van de Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO), voor de financiële ondersteuning van dit projekt (projektnummer: 700-343-024). En 'last but not least' bedank ik Henriëtte 'Piep' Sla, die, door haar voortdurende aanmoediging, er mede voor gezorgd heeft dat dit proefschrift en het hieraan voorafgegane werk tot een goed einde is gebracht. Rotterdam, 18 oktober 1992, Ton van Pelt.

(8) i. Contents 1.. INTRODUCTION 1.1 1.2 1.3 1.4. 2.. 3.. The role of thermodynamics in process development . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Supercritical fluid extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Relevance of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. GENERAL DISCUSSION OF FLUID PHASE EQUILIBRIA 2.1 2.2 2.3 2.4 2.5 2.6 2.7. 1. 13. Classification of fluid phase behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Type I phase behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Type II phase behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Type III phase behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Type IV phase behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Type V phase behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Type VI phase behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24. THE SIMPLIFIED-PERTURBED-HARD-CHAIN THEORY: DEVELOPMENT 27 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.1.1 The Van der Waals type equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.2 The Benedict-Webb-Rubin type equations of state . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1.3 Reference fluid equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1.4 Augmented hard body equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3.2 The Perturbed Hard Chain Theory and its family members . . . . . . . . . . . . . . . . . . . . . 31 3.3 The partition function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4 The free volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.5 The mean potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.6 Mixing rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.

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Contents THE SIMPLIFIED-PERTURBED-HARD-CHAIN THEORY: PERFORMANCE 56 4.1 Pure component parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.1.2 The optimization procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.1.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2 The temperature dependence of the attractive term . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.3 Virial coefficients from the SPHCT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.4 The isochoric heat capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.5 Binary flash calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.6 Calculation of Henry coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.7 Calculation of critical curves in binary mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.7.2 The Hicks-Young algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.7.3 Stability tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.7.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.7.5 Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110. 5.. TRANSITION STATES IN THE GLOBAL PHASE DIAGRAM. 113. 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.2 Thermodynamic conditions for the various transition states . . . . . . . . . . . . . . . . . . . . 116 5.2.1 The tricritical state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.2.2 The double critical endpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2.3 The mathematical double point . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . 126 5.2.4 The azeotropic boundary curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.2.5 The azeotropic critical endpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.2.6 The zero temperature endpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.. COMPUTATIONAL TECHNIQUES. 143. 6.1 Evaluation of the equations and unknowns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6.2 Optimization methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.2.1 Gradient methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.2.2 Method of steepest descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.2.3 Newton-Raphson method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 6.2.4 Gauss method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.2.5 The Marquardt method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.3 Calculation of the tricritical curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.4 The use of MAPLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.4.2 Derivation of expressions for Gnx with Maple . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.4.3 Derivation of expressions for AiVjx with Maple . . . . . . . . . . . . . . . . . . . . . . . . . . 158 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.

(10) Contents 7.. THE GLOBAL PHASE DIAGRAM FROM THE SPHCT EQUATION. iii 163. 7.1 The global phase behaviour of equal-sized and equally flexible molecules . . . . . . . 163 7.2 The global phase behaviour of molecules with equal flexibilities but slightly different sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.3 The global phase behaviour of molecules with equal flexibilities but large size differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 7.4 The global phase behaviour of molecules with equal sizes but unequal flexibilities 183 7.5 The global phase behaviour of molecules with unequal flexibilities and sizes . . . . . . 186 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190. 8.. SPECIAL PHENOMENA FROM THE GLOBAL PHASE DIAGRAM. 191. 8.1 The type VI/VII-region in the global phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 191 8.2 Critical curves in the binary mixture ethane(1)+n-butane(2) . . . . . . . . . . . . . . . . . . . . 198 8.3 Calculation of type VIII phase behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 8.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 8.4 The Van Laar point in the T,x-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 8.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 8.4.2 The neighbourhood of the Van Laar point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 8.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 8.4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226. 9.. CONCLUSIONS AND PROSPECT. 227. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 SAMENVATTING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 CURRICULUM VITAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240.

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(12) 1. CHAPTER 1 INTRODUCTION This thesis is mainly concerned with the prediction of phase equilibria in the critical region and critical phenomena in binary fluid mixtures with a semi-theoretical equation of state. With our investigations we hope to contribute to the understanding of these phenomena and relate them to the structure of the individual molecules. The predictive capabilities, as far as critical phenomena are concerned, of the used semi-theoretical model are explored in a general way, e.g., we did not focus on systems of practical importance but we investigated all possible variations in critical phase behaviour. However, in the various chapters, it will be shown that the connection between the model and the behaviour of real systems has not been lost. In section 1.1 it is explained why theoretical investigations of thermodynamic models are important. In section 1.2, a separation process from the chemical industry that might benefit from our research is introduced. Section 1.3 gives a general overview of previous investigations that have been made in this field of research. This chapter ends with a summary of the scope and structure of the thesis in section 1.4.. 1.1. The role of thermodynamics in process development. Phase equilibrium thermodynamics is one of the most important fundamental sciences for process development in the chemical industry. The results achieved by thermodynamics in the last decades have led to a profound change in working methods for the development of all processes in which phase equilibria play an important role. Whereas in the past the development of these processes was dominated by experimental methods, e.g., laboratory experiments, the working methods have changed significantly, becoming model-supported calculation methods [1]. This holds in particular for unit operations like distillation and absorption. With the aid of process simulators that contain physical property models, chemical processes can be designed and/or optimized with a computer. We want to emphasize that for these computations one is restricted to systems with thermodynamic properties that can accurately be described by a model. This does not mean that basic data have become superfluous. On the contrary, investigations on physical property models have shown that thermodynamic properties of many mixtures, like phase equilibria, especially if the mixtures contain polar components or electrolytes, are hard to predict from a model alone. In this respect basic data still are indispensible..

(13) Chapter 1. 2. In general, the use of physical property models in process development can lead to a considerable reduction of costs with respect to process development that is based on experiments only. However, it may be understood that this strategy depends on the availability of physical property models. In this work, a physical property model, the Simplified-Perturbed-Hard-Chain Theory (SPHCT), is thoroughly investigated with respect to the prediction of critical phenomena in binary fluid mixtures. On the one hand, predictions of thermodynamic properties for pure components and binary mixtures from the SPHCT are compared with experimental data. On the other hand, a general overview is presented of the critical phenomena that can be obtained from the SPHCT. This part is the most important aspect of this work. This general overview is extremely useful for obtaining insight into the complicated world of phase behaviour. The investigations that have been done are purely theoretical. It is not difficult to find a practical example, to which the knowledge of critical phenomena is applied. Supercritical fluid extraction is a good illustration of a practical application in the chemical industry where knowledge about the critical region of mixtures is necessary. By no means do we want to focus on the theoretical or practical implications of the process development of supercritical fluid extraction. The next section only serves to show a link between our theoretical investigations and one possible practical application. Our work might be seen as a theoretical support of the description of all processes in which fluid phase equilibria play an important role.. 1.2. Supercritical Fluid Extraction. Supercritical fluid extraction processes use the unique dissolving properties of solvents near their critical point [2-4]. In the Propane Deasphalting Process lube oils are refined with near-critical propane. Propane is also used in the Solexol Process for the purification and separation of vegetable and fish oils. The Rose Process uses near-critical butane or pentane for the extraction of high molecular weight components from oil residues. Supercritical carbon dioxide at 160 bar and 70°C is used for the extraction of nicotine from tobacco [2-4]. A simplified flowsheet of a supercritical extraction plant is shown in Fig.1.1. Gaseous solvent from the separator (5) and fresh solvent is condensed by a cooling machine (6) and led to a solvent buffer vessel (7). The pressure of the liquified solvent is increased to the extraction pressure by a pump (8). At this pressure, the solvent is heated to the desired temperature (9). Thereafter, the solvent enters the extraction column (3) at the bottom. At the same time the heated feed stream enters the extraction column at the top of the extraction column (3). After the countercurrent extraction, the loaded solvent leaves the extraction column at the top. The pressure of the loaded solvent is usually reduced (4) below the critical pressure of the solvent to ensure a good separation of the solvent and the extract in the separator (5). The raffinate that leaves the extraction column at the bottom can contain up to 50 wt.% solvent. The pressure of this stream is reduced (10) to separate the solvent (12) from the raffinate in a separator (11)..

(14) Introduction. Figure 1.1:. 3. Simplified flowsheet of a supercritical extraction plant.. This solvent stream (12) may be recycled. To compensate for the losses of solvent in the raffinate that may still occur, fresh solvent is added before the solvent recycle stream is condensed again. The increased solubility phenomenon of the solute in the supercritical solvent in the extraction column forms the basis of supercritical fluid processing. The enhanced solvating power of a supercritical solvent is illustrated in Fig.1.2. In this figure the solubility of solid naphthalene, yi, in supercritical carbon dioxide is shown as a function of the reduced pressure at various temperatures [5,6]. At the saturated vapour pressure of solid napthalene for the three reduced temperatures, i.e., pr≈0, the mole fraction of naphthalene in the vapour phase must be 1. Increasing the pressure above the saturated vapour pressure of naphthalene leads to a sharp decrease of the solubility. It is shown in Fig.1.2 that if the pressure is raised above the critical pressure of carbon dioxide (the solvent), the mole fraction of naphthalene (the solute) increases.

(15) Chapter 1. 4 strongly.. Figure 1.2:. Solubility of naphthalene in supercritical CO2 at various temperatures. Experimental data were taken from [6].. Figure 1.3:. Isotherms for pure CO2, Tc=304 K, pc=7.38 MPa, ρc=0.01064 mole.cm-3. The reduced temperatures are: a: 0.79, b: 0.89, c: 0.99, d: 1.05, e: 1.20, f: 2.01. Experimental data were taken from [7].. This large solubility enhancement in the vicinity of the critical pressure of carbon dioxide is mainly caused by the rapid increase in solvent density, see Fig.1.3. For reduced temperatures (T/Tc) ranging from 1.0 to 1.2 small changes in the reduced pressure (p/pc) can change the reduced density (ρ/ρc) of a solvent from about 0.1, a gas-like density, to about 2.5, a liquid-like density. By adjusting the pressure and/or temperature, the properties of a supercritical solvent can be changed from a gas, with little solvating power, to a liquid, with good solvating power [8]. Besides the density of the solvent, there are more thermodynamic properties that influence the solubility of a solute in a solvent, like the saturated vapour pressure of the solute and the chemical structure of the solute and the solvent. The effect of the vapour pressure is visible at high reduced pressures (pr>3) in Fig.1.2. The increase in solubility of naphthalene with temperature at a specified reduced pressure, can be explained with the increased saturated vapour pressure at higher temperatures. The effect of the chemical structure on the solubility can easily be explained with the 'like-dissolves-like' rule. In general it can be stated that the solubility in non-polar solvents, such as ethane and propane, will be smaller with decreasing vapour pressures and increasing polarity of the solute. This solubility behaviour may change when more polar solvents are used. The solubility effect of chemical structure differences can be identified by defining an enhancement factor, Ei, as a measure for the affinity of a solvent for a solute:.

(16) Introduction. Ei =. y .p actual mole fraction solute i = i sat ideal mole fraction solute i xi . pi. 5 (1.1). The factor Ei represents the enhancement of the actual mole fraction of the solute, yi, over the value xi.pisat/p, that can be calculated according to Raoult's law, if gas and liquid phase nonidealities and pressure effects on the liquid or solid phase are neglected. If pure supercritical solvents are used the choice is usually limited to non-polar solvents such as carbon dioxide, ethane and propane. Higher solubilities can sometimes be obtained by using mixed solvents. For more polar compounds a higher solubility can be obtained by adding small amounts of polar entrainers such as methanol, ethanol, acetone or water to the non-polar solvent. It is obvious that knowledge of the phase equilibria that occur in the supercritical fluid extraction process is necessary for the design of this process. Basic data obtained from experimental research are of the highest importance to acquire this knowledge. However, it can be useful to have some knowledge of the densities and heat capacities of the various streams, compositions of coexisting phases, such as raffinate and loaded solvent, etc., before one even starts to do experimental research. These rough predictions obtained from a physical property model can then be used to direct this experimental research, or if they are accurate enough, to support process development. It should be stressed again that the example of the supercritical extraction process only shows a link between our theoretical investigations on critical phenomena and a practical application.. 1.3. Relevance of this work. In the previous section an example was discussed of the application of supercritical solvents in chemical engineering. It is obvious that for the design of such a process a comprehensive knowledge of the thermodynamic properties of the solvent and the solute at nearcritical conditions of the solvent is necessary. It also has been mentioned in section 1.1, that model-based calculations become more and more important for the design of chemical processes, even though experimental data are still necessary. Equations of state provide chemical engineers with useful tools for modelling the phase equilibria. However, more research has to be done to improve the accuracy of the equations of state currently used by many chemical engineers in process simulators. In this work a semi-theoretical equation of state was tested in the critical region of binary fluid mixtures. However, the emphasis of this work is not on the applicability of this equation of state to the development of a particular process, but on its significance for a general and systematic investigation of phase behaviour and critical curves in binary fluid mixtures. Research in this field of science is not new but started more than two decades ago. Van Konynenburg and Scott [9,10] showed that almost all known types of fluid phase equilibria - vapour-liquid, liquidliquid and gas-gas - can be generated, at least qualitatively using the Van der Waals equation of state (VDW equation) and the Van der Waals mixing rules. They also suggested a classification.

(17) Chapter 1. 6. of binary phase diagrams. Their classification is based on the absence or presence of three-phase curves, and if present, on the way critical curves are connected to these three-phase curves. They distinguished six different classes of binary fluid phase behaviour. However, due to the limitations of the equation of state they were able to generate only five of these classes with the VDW equation of state. Because the classification of Van Konynenburg and Scott [9,10] forms the basis of our work, this classification will be discussed in the next chapter. Furman et al. [11] made similar studies of lattice-gas mixtures. The resultant lattice-gas equation of state was extended by Furman and Griffiths [12] to the Van der Waals lattice-gas equation of state and these investigators discovered new types of phase behaviour. Van Konynenburg and Scott completed their earlier work with the findings of Furman and Griffiths [12] in an extensive review on fluid phase behaviour [13]. Furman and coworkers [11,12] introduced another classification procedure. Their classification does not distinguish between the various types of phase behaviour in a pressure-temperature-projection (p,T-projection) of the critical curves in a binary system, but distinguishes between the various phenomena within one mixture. They called a homogeneous phase: A. Examples are liquid (L), solid (S) and vapour (V). A two-phase equilibrium is called A2, like L+V, S+L, S+V and L1+L2. When both coexisting phases become identical, the critical state is called: B, e.g. L=V. Equilibrium between three phases is called A3. Examples are L1+L2+V, S+L+V and S1+S2+L. When, in case of a two-phase equilibrium, both phases have the same composition, the mixture is said to be azeotropic. Furman and coworkers called this state: A2az. A critical endpoint, i.e., a critical phase that is in equilibrium with an alternate (noncritical) phase, is called BA, e.g., L1=L2+V, and a critical azeotropic point is called Baz. A quadruple point, a point in the p,T-plane where four phases are in equilibrium, is denoted by A4, e.g., S1+S2+L+V. Even phenomena that are rare in binary mixtures can be classified according to the classification of Furman and coworkers. A tricritical point (unsymmetric) is denoted by C, e.g., L1=L2=V. A tricritical point is the point where two critical endpoints on the same three-phase curve coincide. When a critical azeotropic phase (Baz) is in equilibrium with an another, noncritical, phase this phase is denoted by BazA. A tricritical phase that is in coexistence with an alternate noncritical phase, is denoted by CA. Although this classification gives a detailed insight into the phase behaviour of a certain binary mixture, it does not give a general overview of possible phase behaviour in binary mixtures. Clancy et al. [14] focussed on the influence of polar forces on the phase behaviour. They used an equation of state that was derived from perturbation theory. However, they did not include an investigation of all possible phase diagrams. Mazur and coworkers [15,16] and Boshkov and coworkers [17,18] used the Ree equation of state for Lennard-Jones molecules. They did extensive research on the global phase diagrams for binary mixtures that are composed of equal-sized molecules. New classes were added by them to the classification scheme of Van Konynenburg and Scott [9,10]. B. B. B. Meijer and coworkers [19-27] investigated the global phase behaviour of the symmetric lattice-gas [28], the Van der Waals lattice-gas and the Tompa-model [29]. The latter model is a variant of the lattice gas model for compressible binary mixtures in which the possibility of size.

(18) Introduction. 7. difference between the components is included. They also used the classification of Van Konynenburg and Scott [9,10] to distinguish between the various types of fluid phase behaviour. Most of their research focussed on mathematical double points and on the phase behaviour around the Van Laar point in the global phase diagram. Systematic investigations of global phase behaviour were undertaken recently by Deiters and Pegg [30] and Kraska and Deiters [31]. Deiters and Pegg [30] investigated all possible phase behaviour that can be predicted with the Redlich-Kwong equation of state (RK equation). Their investigations were based on the classification according to Van Konynenburg and Scott. Kraska and Deiters [31] did similar calculations on the Carnahan-Starling-Redlich-Kwong equation of state (CSRK equation). A new nomenclature for phase diagram classes of binary fluid mixtures has recently been proposed by Bolz [32]. His classification scheme is a combination of the classification of Van Konynenburg and Scott and the classification according to Furman and coworkers. The new nomenclature primarly describes the topology and connectivity of critical curves. Moreover, information about the critical curves is contained in this nomenclature: the basic idea of Bolz was that the construction of a p,T-projection of the critical curves of a binary system should be possible from the name. The critical point of a pure component is always the starting point of either a critical curve or a sequence of critical curves connected by three-phase curves. One starts with the critical curve originating at the critical point of the component with the higher critical temperature. All segments of the critical curve, that are connected by three-phase lines, are taken into account. The target of this critical curve is indicated with a superscript. Possible targets are: C for critical curves going upwards to a compact state at infinitely high pressures; P for critical curves going to the other pure critical point; Z for critical curves going to an endpoint, from where a three-phase curve goes to absolute zero; Q for curves going to an endpoint, from where a three-phase curve goes to a quadruple point (four fluid states in equilibrium). Subsequently, the critical curve originating at the critical point with the lower critical temperature is described, in the same way as mentioned above. Then, additional critical curves are classified as follows: l for critical curves coming from infinitely high pressures and going to an endpoint; n for critical curves with two endpoints; u for critical curves coming from and going to infinitely high pressures, going through a pressure minimum; these curves have no endpoints; o for critical curves that form completely stable closed loops.. Finally, additional information can be added directly behind the critical curve to which it refers. The following abbreviations should be used: H for heteroazeotropic behaviour;.

(19) Chapter 1. 8 A Q M. for azeotropic behaviour; for a quadruple point; for maxima, the number of maxima may be indicated by a superscript.. Several common examples of the new nomenclature are: 1P, 1Pl, 1C1Z. These examples correspond to types I, II and III phase behaviour according to the classification of Van Konynenburg and Scott [9,10]. Rare phase behaviour, like the phase behaviour of binary systems from the shield region, have longer names like: 1PAnlQ. An advantage of this nomenclature is the fact that it is very descriptive. It is obvious that a lot of work has already been done in the field of classification of phase behaviour. However, most of the work that has been done sofar is based on two-parameter equations of state that are more or less empirical. Initially we intend to approach the field of classification of phase behaviour armed with a semi-theoretical equation of state, that has primarily been designed to account for the nature of chainlike molecules. The equation of state is chosen in such a way that new types of binary phase behaviour are to be expected, although this cannot be ensured in advance. The equation of state should correspond to the objectives of our research group at the University of Technology in Delft, one of which is the study of phase behaviour of chain-like molecules. It is also hoped that our research will compliment the efforts made previously by other investigators. Following the trend of investigations on the VDW equation of state towards more advanced equations of state like the RK and CSRK equation, it is logical to explore the ability of the Simplified-Perturbed-Hard-Chain Theory to predict and describe critical curves and phase behaviour in binary fluid mixtures. The equation of state that results from this theory, the SPHCT equation, satisfies all three conditions that are mentioned above.. 1.4. Outline of this thesis. It is impossible to cover all subjects that are related to the calculation of high pressure phase equilibria in one PhD-thesis and so it is necessary to restrict oneself to a couple of subareas of this field of research. It is not our objective to make improvements on the way the various types of phase behaviour in binary systems are classified. We will stay close to the classification system of Van Konynenburg and Scott [9,10,13]. We have chosen not to emphasize investigations of types of phase behaviour that are unlikely to be found experimentally, like the types of phase behaviour found in the so-called shield-region [11,12]. We also will not pay much attention to possible numerical optimization procedures, that would be less time-consuming or mathematically more elegant. However, we will emphasize the global phase behaviour that can be predicted from the SPHCT equation and special attention is paid to particular critical phenomena that can be calculated from this equation like high-pressure critical curves and liquidliquid immiscibility loops. Liquid-liquid immiscibility loops cannot be calculated from the.

(20) Introduction. 9. VDW, RK and CSRK equations. High-pressure critical curves have been calculated from the RK and CSRK equation but this type of critical curve has not gained much attention in this respect. This chapter has given a short introduction on the subject. The role critical phenomena can play in the chemical industry has been explained by means of an example: supercritical fluid extraction. The various types of critical curves in binary fluid mixtures and the classification of these critical curves according to Van Konynenburg are discussed in chapter 2. An extensive derivation of the SPHCT equation of state that has been used to calculate critical curves in binary mixtures, is given in chapter 3. In chapter 4 we try to justify the choice of the SPHCT equation of state for this investigation. In order to do so, a number of experimentally determined thermodynamic properties such as saturated vapour pressures of pure components, temperature dependence of the attractive term of the equation of state, second and third virial coefficients, the molar isochoric heat capacity, binary isothermal phase diagrams, Henry coefficients and critical curves in binary systems, have been compared with results calculated from the SPHCT equation of state. Chapter 5 deals with the global phase diagram. All phenomena represented in a global phase diagram (tricritical points, Van Laar points, shield region, etc.), including the mathematical conditions to calculate these phenomena from an equation of state, are discussed. In chapter 6, the calculation procedures for the different curves in the global phase diagram are discussed. The numerical algorithm for solving a set of non-linear equations, the Marquardt-algorithm, is discussed. Most of the mathematical conditions have been elaborated with the symbolic algebra program MAPLE. We will digress on the use of this symbolic computation program in this chapter. Chapter 7 describes the global phase diagrams that have been calculated from the SPHCT equation of state. It turns out that from this equation of state ordinary types of phase behaviour, as well as some additional interesting features, like closed-loop liquid-liquid immiscibility, can be calculated. Another exceptional phenomenon observed is the appearance of two tricritical points on the same critical curve, with the possibility that these tricritical points coincide. This gives rise to new areas in the global phase diagram, and additionally, to new types of binary phase behaviour. A comparison is made with global phase diagrams predicted from other equations of state, like the VDW and RK equation of state. In chapter 8, some peculiarities in the calculated global phase diagrams are discussed, such as the already mentioned closed loop liquid-liquid immiscibility, the Van Laar point and high-pressure immiscibility. Finally, in chapter 9 an outlook on the possible development of the subject of global phase diagrams is given..

(21) Chapter 1. 10. NOMENCLATURE E L p S T V x y. enhancement factor liquid phase pressure solid phase temperature vapour phase mole fraction of a component in the liquid phase mole fraction of a component in the vapour phase. Greek letters ρ. density. Superscripts and subscripts az c i r sat. azeotropic critical property component index reduced property saturated property. REFERENCES 1. S. Zeck. Thermodynamics in process development in the chemical industry. Importance, benefits, current state and future development. Fluid Phase Equilibria 70, (1991), 125-140. 2. M.A. McHugh and V.J. Krukonis. Supercritical fluid extraction: principles and practice. Butterworths. Stoneham. (1986). 3. M.E. Paulaitis, V.J. Krukonis, R.T. Kurnik and R.C. Reid. Supercritical fluid extraction. Rev. Chem. Eng. 1, (1983), 179-250. 4. D.F. Williams. Extraction with supercritical gases. Chem. Eng. Sci. 36, (1981), 1769-1788. 5. Y.V. Tsekhanskaya, M.B. Iomtev and E.V. Mushkina. Solubility of naphthalene in ethylene and carbon dioxide under pressure. Russ. J. Phys. Chem. 38, (1964), 1173-1176. 6. M.A. McHugh and M.E. Paulaitis. Solid solubilities of naphthalene and biphenyl in supercritical carbon dioxide. J. Chem. Eng. Data 25, (1980), 326-329. 7. IUPAC, International Thermodynamic Tables of the Fluid State. Vol. 3: Carbon dioxide. Butterworths, London, (1977). 8. S.R. Allada. Solubility parameters of supercritical fluids. Ind. Eng. Chem. Process Des. Dev. 23, (1984), 344-348..

(22) Introduction. 11. 9. P.H. van Konynenburg. Critical lines and phase equilibria in binary mixtures. PhD-thesis University of California Los Angeles. (1968). 10. R.L. Scott and P.H. van Konynenburg. 2. Static properties of solutions: van der Waals and related models for hydrocarbon mixtures. Discuss. Faraday Soc. 49, (1970), 87-97. 11. D. Furman, S. Dattagupta and R.B. Griffiths. Global phase diagram for a three-component model. Phys. Rev. B, 15, (1977), 441-464. 12. D. Furman and R.B. Griffiths. Global phase diagram for a Van der Waals model of a binary mixture. Phys. Rev. A, 17, (1978), 1139-1148. 13. P.H. van Konynenburg and R.L. Scott. Critical lines and phase equilibria in binary van der Waals mixtures. Phil. Trans. 298A, (1980), 495-540. 14. P. Clancy, K.E. Gubbins and C.G. Gray. Thermodynamics of polar liquid mixtures. Discuss. Faraday Soc. 66, (1978), 116-129. 15. V.A. Mazur, L.Z. Boshkov and V.B. Fedorov. Phase behaviour in two component LennardJones systems. Dokl. Akad. Nauk SSSR. 282, (1985), 137-140. 16. V.A. Mazur, L.Z. Boshkov and V.G. Murakhovsky. Global phase behaviour of binary mixtures of Lennard-Jones molecules. Physics Letters 104, (1984), 415-418. 17. L.Z. Boshkov and V.A. Mazur. Phase equilibria and critical lines of binary mixtures of Lennard-Jones molecules. Russ. J. Phys. Chem. 60, (1986), 16-19. 18. L.Z. Boshkov. On the description of phase diagrams of two component mixtures with a closed domain of demixing based on the one-fluid model of an equation of state. Dokl. Akad. Nauk. SSSR. 294, (1987), 901-905 19. P.H.E. Meijer and M. Napiorkowski. The three-state lattice gas as model for binary gasliquid systems. J. Chem. Phys. 86, (1987), 5771-5777. 20. P.H.E. Meijer. Binary gas-liquid systems classification. Soc. Fr. de Chimie, Int. Symp. on Supercritical Fluids, Oct. 17-19, Nice. M. Perrut ed., Vol. 1, (1988), 239-244. 21. P.H.E. Meijer. The influence of the chain length of long molecules on the equation of state in binary gas liquid mixtures. J. Stat. Phys. 53, (1988), 543-548. 22. P.H.E. Meijer. Study of the critical line and its double point in the intermediate model. Physica A 152, (1988), 359-364. 23. P.H.E. Meijer, M. Keskin and I.L. Pegg. Critical lines for a generalized three state binary gas-liquid lattice model. J. Chem. Phys. 88, (1988), 1976-1982. Erratum: J. Chem. Phys. 90, (1989), 3408. 24. P.H.E. Meijer. The van der Waals equation of state around the van Laar point. J. Chem. Phys. 90, (1989), 448-456. 25. P.H.E. Meijer, I.L. Pegg, J. Aronson and M. Keskin. The critical lines of the van der Waals equation for binary mixtures around the van Laar point. Fluid Phase Equilibria 58, (1990), 65-80. 26. P.H.E. Meijer and I.L. Pegg. Structure of the critical lines for the lattice gas model. Physica A 174, (1991), 391-405. 27. M. Keskin, M. Gençaslan and P.H.E.Meijer. Evaluation and comparison of critical lines for various models of gas-liquid binary systems. J. Stat. Phys. 66, (1992), 885-896. 28. H. Tompa. Phase relationships in polymer solutions. Trans. Faraday Soc. 45, (1949), 11421152. 29. J.A. Schouten, C.A. ten Seldam and N.J. Trappeniers. The two-component lattice-gas model. Physica 73, (1974), 556-572..

(23) 12. Chapter 1. 30. U.K. Deiters and I.L. Pegg. Systematic investigation of the phase behavior in binary fluid mixtures. I. Calculations based on the Redlich-Kwong equation of state. J. Chem. Phys. 90, (1989), 6632-6641. 31. Th. Kraska and U.K. Deiters. Systematic investigation of the phase behavior in binary fluid mixtures. II. Calculations based on the Carnahan-Starling-Redlich-Kwong equation of state. J. Chem. Phys. 96, (1992), 539-547. 32. A. Bolz. Vergleichende Untersuchung globaler Phasendiagramme. PhD-thesis, RuhrUniversität Bochum, Germany, (1992)..

(24) 13. CHAPTER 2 GENERAL DISCUSSION OF FLUID PHASE EQUILIBRIA Van Konynenburg has shown that the Van der Waals equation of state when applied to binary mixtures, predicts qualitatively most types of fluid phase equilibria that have been observed experimentally. In this chapter these experimentally determined phase diagrams are discussed, starting with an introduction in section 2.1. Special attention is paid to fluid phase equilibria and critical phenomena. Van Konynenburg established a classification of the different types of fluid phase behaviour in binary mixtures in which he distinguished six main types: type I to VI, see sections 2.2-2.7.. 2.1. Classification of fluid phase behaviour. In everyday life three states of matter exist: solid, liquid and vapour. When a pure liquid is heated, it usually transforms into a vapour. The temperature at which this happens depends on the applied pressure. At its boiling temperature, the liquid phase is said to be in equilibrium with a vapour phase. The graphical representation in the pressure-temperature-plane (p,T-plane) of these phases and their mutual coexistence is called a phase diagram. The phase diagrams, describing all kinds of phase equilibria between vapour, liquid and/or solids in a binary system, are called complete phase diagrams. Because of the great variety of phenomena, not all complete phase diagrams can be discussed in this chapter. Valyasko [1] focussed on complete phase diagrams showing all kinds of critical phenomena. Complete phase diagrams with phenomena like polymorphism, azeotropy, formation of solid solutions and new compounds were not discussed by Valyashko [1]. According to his classification, complete phase diagrams fall into two groups, as far as critical phenomena are concerned: Group I:. Complete phase diagrams in which critical phases are not in equilibrium with a solid phase. Group II:. Complete phase diagrams in which critical phases occur that are in equilibrium with a solid phase. It should be noted that critical phenomena play an important role in this classification and we will continue with a more precise definition of critical phenomena. The critical point of a fluid is the point in pressure-temperature-composition-space (p,T,x-space), at which the coexisting liquid.

(25) 14. Chapter 2. and vapour phases become indistinguishable. All physical properties, such as density, refractive index, etc., of the two coexisting phases will become identical. For a pure component this occurs at a fixed temperature and pressure. For a binary system critical points may exist over a range of pressures, temperatures and compositions, resulting in a critical curve. Van Konynenburg and coworkers [2-4] extensively discussed a number of binary phase diagrams that later turned out to belong to the group I phase diagrams according to the classification of Valyashko [1]. In the studies of Van Konynenburg and coworkers [2-4] the solid phase equilibria were not taken into account. Fig. 2.1 shows the classification of binary phase diagrams suggested by Van Konynenburg and coworkers [2-4]. Their classification is based on the presence or absence of three-phase lines and, when present, on the way critical curves are connected to them. In type I mixtures, only vapour-liquid separation occurs, whereas in type II to type VI mixtures liquid-liquid immiscibility also occurs in some regions of the phase diagram. For type II and type VI mixtures, the vapour-liquid and liquid-liquid critical curves are quite distinct. In type III, type IV and type V mixtures, the critical curve is separated into two or more parts. These branches of the critical curve can no longer be clearly identified as being vapourliquid or liquid-liquid. A detailed discussion of the experimentally observed phase diagrams can be found in several reviews, e.g., in Rowlinson and Swinton [5]. We will restrict ourselves here to a short overview of the various types of experimentally determined phase diagrams by briefly discussing these types according to the classification of Van Konynenburg and Scott. Van Konynenburg [2] introduced the concept "class" as well as "type". Class 1 mixtures (types I and II) possess a continuous critical curve from one pure fluid critical point to the other, whereas class 2 mixtures (types III, IV and V) do not. Strictly speaking, Van Konynenburg and Scott [24] assign type VI to a third class, because type VI mixtures show a closed liquid-liquid immiscibility loop. Moreover, those authors did not use the term "type VI", but this term has been used by subsequent authors [5] and is now in standard usage. Each type of phase behaviour will be illustrated with one or more examples of binary mixtures that show this phase behaviour. A detailed overview of experimentally determined critical data for binary systems can be found in [6,7].. 2.2. Type I phase behaviour.. Type I mixtures have a continuous vapour-liquid critical curve without any liquid-liquid immiscibility. Often, in real mixtures, crystallization of one of the components masks a low temperature liquid-liquid phase separation. Type I behaviour occurs in binary systems with two substances that are chemically similar and/or have critical properties that are comparable. Typical examples are ethane + 2-methylpropane [8], CO2 + O2 [9,10] and benzene + cyclohexane [11,12]. Mixtures of substances belonging to a homologous series deviate from simple type I behaviour only when their size difference, and thus the values of their critical properties, exceeds a certain ratio..

(26) Fluid phase equilibria. Figure 2.1:. 15. The six main types of binary fluid phase behaviour, according to the classification of Van Konynenburg and Scott [2-4].. For example, in the n-alkane series, with methane as one component a change from type I to type V occurs first with n-hexane [13], whereas, with ethane as the lighter component, n-octadecane is the first to exhibit partial miscibility in the liquid state [14]. With propane, this behaviour occurs when the second component is n-C30 [15]. Type I mixtures can be conveniently distinguished first by considering the shape of the continuous critical curve that connects the critical points of the two pure components. In addition, the presence or absence of azeotropy should be noted. Fig.2.2 represents the various shapes of the p,T-projection of the critical curve. In this diagram the pure vapour pressures are represented as solid curves and critical curves are shown as dotted curves. The pure component critical points are shown as open circles..

(27) 16. Chapter 2. Mixtures corresponding to curve c, where the critical curve is almost linear in both the p,Tand the T,x-planes, are usually formed from substances with very similar critical properties, e.g., n-decane + benzene [16]. Mixtures corresponding to curve b where the critical curve is convex upwards and frequently exhibits a maximum in both the p,T- and p,xplanes are extremely common among type I systems and occur whenever there are moderately large differences between the critical temperatures or volumes of the pure components, e.g., propane + n-octane [17]. Mixtures whose critical curve conforms to the shape of curve a are extremely rare. In this case there is partial immiscibility at temperatures higher than the critical temperatures of both components. By common defintion the Figure 2.2: Six possible types of continuous critical curves for type I mixsubstances should thus be considered to be in tures. The solid phases are not the gaseous state. Phase behaviour like this, is shown. an example of so-called 'gas-gas immiscibility of the third kind'. A binary system that exhibits this phase behaviour is cycloheptane + tetraethylsilane [18]. Type I mixtures, whose critical loci are everywhere concave upwards, frequently show a pressure minimum as in curve d. A typical example is propane + hydrogen sulphide [19]. The critical curve e extends through a temperature minimum in the p,Tprojection. This phase behaviour is observed for several mixtures and is usually associated with the occurrence of a maximum-pressure azeotrope extending up to the critical curve. A typical example is acetone + n-pentane [20]. The final type of critical locus, shown in Fig.2.2, is critical curve f. The critical curve has a temperature minimum and is concave upwards. This type of phase behaviour is found in mixtures that are composed of an n-alkanol and a nalkane [21]. The calculational work of Van Konynenburg and coworkers [2-4] have shown that if a binary mixture of equal-sized molecules with a continuous vapour-liquid critical curve exhibits maximum-pressure azeotropy, there is always a liquid-liquid critical curve at relatively low temperatures. This means that the binary system belongs to type II. In other words: computationally a type I mixture with maximum-pressure azeotropy does not exist for equal-sized molecules. This is schematically shown in Fig.2.3. It is shown in Fig.2.3 that the azeotropic curve is tangent to the p,T-projection of the vapour-liquid critical curve. Consequently, the azeotropic curve does not meet the critical curve at the temperature minimum of the critical curve. This type of azeotropy is commonly called "positive", "maximum-pressure" or "minimum-boiling" azeotropy. The latter term refers to an isobar on a T,x-cross-section along which the azeotrope is the two-phase point of minimum.

(28) Fluid phase equilibria. 17. temperature. The term maximum-pressure azeotropy is preferred by us and will be used in this work. An example of a mixture that forms a maximum-pressure azeotrope is the system CO2 + ethane [22].. Figure 2.3: Type II phase behaviour with a maximum-pressure azeotrope.. Figure 2.4: Type I phase behaviour with a minimum-pressure azeotrope.. "Negative", "minimum-pressure" or "maximum-boiling" azeotropy is also possible. Here, in contrast to Fig.2.3, the critical curve displays a maximum in the critical temperature. The minimum-pressure azeotrope in a p,T-projection lies to the right of both pure-fluid vapour pressure curves, see Fig.2.4. If the bubblepoint temperature of a mixture is higher than that of either pure fluid, it follows that the cross (1-2) interaction is more strongly attractive than either the (1-1) or (2-2) interaction. This suggests the idea that the unlike molecules will undergo a chemical reaction. According to this concept the mixture is composed of three components, instead of two. A binary system that shows minimum-pressure azeotropy is dimethyl ether + SO2 [23]. Another possibility, occurring in the binary ethanol + benzene [24], is that an azeotrope exists at lower pressures; however, as the pressure is increased it terminates on a pure fluid vapour pressure curve and does not reach the critical curve. In almost all binary systems showing minimum-pressure azeotropy, the required interaction between the unlike molecules is lost at temperatures that correspond to the vapour-liquid critical region. The vapour pressure curves of pure components sometimes intersect in the p,T-plane. The point of intersection is called a Bancroft point. Binary mixtures with a Bancroft point usually show maximum-pressure azeotropy. However, systems that contain a Bancroft point can show a special form of azeotropy: double azeotropy, see Fig.2.5. An example of a binary system that exhibits double azeotropy is benzene + hexafluorobenzene [25-27]. More recently double azeotropy was found in the system 2,6-dimethylpyridine + 4-methylpyridine [27]. Both azeotropic curves end at an azeotropic endpoint. This is illustrated in Fig.2.5..

(29) 18. Chapter 2. Figure 2.5: Type I mixture with a Bancroft point (B) and showing double azeotropy. Both azeotropic curves end at an endpoint.. Figure 2.6: Type I phase behaviour, including high-pressure immiscibility. This type of phase behaviour is often considered to be a special form of type VI phase behaviour.. Fig.2.6 illustrates the phase diagram for water + 3-methylpyridine [28]. Both components are fully miscible in the liquid and vapour phase at low pressures, which suggests that this mixture belongs to the type I mixtures. At higher pressures, i.e., above 1400 bar, liquid-liquid phase separation occurs. This phenomenon is called 'high-pressure immiscibility'. Because this type of phase behaviour shows close resemblance with special forms of type VI phase behaviour, see section 2.7, it is often considered as a special form of type VI phase behaviour.. 2.3. Type II phase behaviour. As with type I mixtures, type II mixtures have a continuous vapour-liquid critical curve. In addition there is a three-phase line, L1L2V in the p,T-projection between the vapour pressure curves, ending at an upper critical endpoint (UCEP). From this UCEP a second critical curve of liquid-liquid nature goes to infinite pressures where the fluid mixture approaches closest packing. The liquid-liquid critical curve can also end at an endpoint S+(L1=L2). Actually the liquid-liquid critical curve can be extended to pressures lower than the pressure of the UCEP. In this case the extended part of the critical curve is metastable until it reaches the point of thermodynamic instability. Generally the liquid-liquid critical curve has one of the three shapes in the p,T-plane, that are shown in Fig.2.7. In this figure the dash-dotted curve represents the three-phase line L1L2V. The three possible shapes of the liquid-liquid critical curve can be described as follows:.

(30) Fluid phase equilibria. 19. a the liquid-liquid critical curve has a negative slope in the p,T-projection, (∂ p / ∂ T)c < 0. b the liquid-liquid critical curve has a positive slope in the p,T-projection, (∂ p / ∂ T)c > 0. c the critical curve has a negative slope in the UCEP and changes via a temperature minimum to a liquid-liquid critical curve with a positive slope at higher pressures. An example of a type II system is carbon dioxide + n-octane [29]. It is possible that a type II system exhibits maximum- or minimum-pressure azeotropy. Van Konynenburg and Scott denoted type II phase behaviour with maximum-pressure azeotropy as type II-A, see Fig.2.3. Type II phase Figure 2.7: Three possible types of critical behaviour with minimum-pressure azeotropy could not be calculated from the Van der curves in type II mixtures. Waals equation of state for equal-sized molecules. In the binary system water + phenol [30] positive azeotropy occurs as well as liquidliquid immiscibility. Minimum-pressure azeotropy together with liquid-liquid immiscibility is extremely rare. The occurrence of this behaviour at low vapour pressures implies that the unlike molecules weakly attract each other in a certain composition range, i.e., the composition range where the demixing occurs. On the other hand they form a negative azeotrope outside this composition range, which implies strong attractive forces between the unlike molecules. According to [5], there is strong evidence that the system acetic acid + triethylamine exhibits this kind of phase behaviour. This type of phase behaviour is schematically shown in Fig.2.8. If the three-phase line is above the vapour pressure curve of the most volatile component, a heterogeneous azeotrope or heteroazeotrope is formed. This type of phase behaviour, type IIHA, is illustrated in Fig.2.9. Type II-HA is probably the most common form of type II phase behaviour and is found if two liquids of similar vapour pressure are only partially miscible. Heteroazeotropy occurs in the mixtures methanol + n-hexane and methanol + n-heptane [21,31]..

(31) 20. Chapter 2. Figure 2.8: Binary mixture showing type II phase behaviour, including a minimum-pressure azeotrope. 2.4. Figure 2.9: Binary mixture showing type IIHA phase behaviour.. Type III phase behaviour. Type III mixtures have two distinct critical curves, one starting at the critical point of the component with the higher critical temperature that goes to infinite pressures. The other critical curve starts at the critical point of the component with the lower critical temperature and meets a three-phase line L1L2V at an UCEP. Four possible kinds of type III behaviour are shown in Fig.2.10 in p,T-projections. A typical example of a system that has a pressure maximum and a pressure minimum in its critical curve that goes to infinite pressures is the system ethane + methanol [32]. This corresponds to curve a in Fig.2.10. This type of phase behaviour is also called type IIIm, where the subscript m refers to the pressure-minimum. A binary mixture that corresponds to curve b in Fig.2.10 is ethane + nitromethane [33]. The curves c and d in Fig.2.10 show different kinds of gas-gas immiscibility. In gas-gas immiscibility of the first kind the critical curve starting at the critical point of the component with the higher critical temperature moves to higher pressures and temperatures on increasing the mole fraction of the other component, see curve d in Fig.2.10. A typical example of a binary system that shows gas-gas immiscibility of the first kind is He + Xe [34]. In gas-gas immiscibility of the second kind the critical curve starting at the critical point of the component with the higher critical temperature moves to higher pressures but initially passes through a temperature minimum on increasing the mole fraction of the other component, see curve c in Fig.2.10. Binary mixtures that display gas-gas immiscibility of the second kind are NH3 with CH4, Ar or N2. These binary systems are discussed by Schouten [35]..

(32) Fluid phase equilibria. Figure 2.10: Four possibilities for type III mixtures.. 21. Figure 2.11: Type III phase behaviour with heteroazeotropy.. Binary mixtures have been investigated where the three-phase line L1L2V lies everywhere above the vapour pressure curve of the more volatile component. This phenomenon is called heteroazeotropy and this type of phase behaviour, type III-H, is illustrated in Fig.2.11. The binary systems H2O + lower n-alkane up to n-C24H50 [36] are known to exhibit type III-H behaviour. Binary mixtures that are composed of H2O + n-alkane with a carbon number that is higher than 24, still show type III-H phase behaviour [36] although the connectivity of the critical curves differs from the one that occurs at lower carbon numbers (i.e., n < 24).. 2.5. Type IV phase behaviour. In type IV phase behaviour three distinct critical curves exist, as is shown in Fig.2.12. One of them is a liquid-liquid critical curve that starts at the UCEP of a three-phase line L1L2V and rapidly goes to infinite pressures. The second critical curve starts at the critical point of the component with the lower critical temperature and ends in an UCEP of a second three-phase line L1L2V. The third critical curve starts at the critical point of the component with the higher critical temperature and ends at a Lower Critical End Point (LCEP) of the second three-phase line L1L2V. The third critical curve continuously changes its character. Near the critical point of the pure component with the higher critical temperature the critical curve has a vapour-liquid character, whereas near the LCEP the critical curve has a liquid-liquid character. Type IV phase behaviour is known to occur in the binary systems methane + 1-hexene [37] and CO2 + n-tridecane [38]..

(33) 22. Chapter 2. Figure 2.12: Type IV phase behaviour. 2.6. Figure 2.13: Type V phase behaviour.. Type V phase behaviour. Type V mixtures, see Fig.2.13, have two distinct critical curves and correspond to type IV mixtures without the steep liquid-liquid critical curve. However, in mixtures that are known to belong to the type V mixtures the distinct liquid-liquid critical curve may be obscured by the appearance of the solid phase. Mixtures of n-alkanes with large size differences display type V phase behaviour. While the system methane + n-pentane still displays type I phase behaviour, type V behaviour occurs in the system methane + n-hexane [13], whereas, with ethane as the lighter component, n-octadecane is the first to exhibit partial miscibility in the liquid state [14]. With propane, this behaviour occurs when the second component is n-C30 [15]. A well-studied example of a mixture showing type V behaviour is the previously mentioned system methane + n-hexane [13]. In this system immiscibility is found: at temperatures below the LCEP (182.46 K, [39,40]) methane and n-hexane are fully miscible in the liquid state. 2.7. Type VI phase behaviour. Binary mixtures showing type VI phase behaviour have a continuous vapour-liquid critical curve between the critical points of the pure components. In addition, type VI mixtures have a closed liquid-liquid critical curve that starts at a LCEP of a three-phase line L1L2V and ends at an UCEP on the same three-phase line. The liquid-liquid critical curve can be extended below both endpoints in the p,T-projection. Each of the metastable parts of the critical curve ends.

(34) Fluid phase equilibria. 23. at a point where local instability begins. These two points are connected by an unstable critical curve that runs more or less parallell to the three-phase line L1L2V. Together the stable, metastable and unstable parts of the liquid-liquid critical curve form one closed critical curve. Type VI phase behaviour is schematically shown in diagram a of Fig.2.14. The maximum on this liquid-liquid critical curve is called a 'hypercritical point'. Type VI phase behaviour is found in mixtures where strong intermolecular bonding, such as hydrogen-bonding, can occur. A common example of type VI phase behaviour can be found in the system H2O + 2-butoxyethanol [41]. Schneider [41] has written an extensive review on type VI phase behaviour.. Figure 2.14: Four possibilities of type VI phase behaviour. Diagram c can also be seen as a special form of type I phase behaviour..

(35) 24. Chapter 2. In addition to the relatively low-pressure liquid-liquid immiscibility, a quite distinct liquid-liquid critical curve can appear at higher pressures, see diagram b of Fig.2.14. This type of phase behaviour occurs in the binary system heavy water + 2-methylpyridine [41]. Note the resemblance with type I phase behaviour with a high pressure immiscibility region, shown in diagram c of Fig.2.14. If the high-pressure critical curve interferes with the low-pressure critical curve, a 'tube of immiscibility' appears in the p,T,x-space. An example of this type of phase behaviour can be found in the binary mixture water + 3-methylpyridine [27], see diagram d in Fig.2.14. The various types of phase behaviour considered in this section and the previous sections do not cover all experimentally determined phase diagrams. However, most of the qualitatively different types of phase behaviour have been discussed. Van Konynenburg [2] evaluated the various types of phase equilibria that could be calculated with the Van der Waals equation of state in a systematic way. In the following chapters, we will also perform a systematic investigation of fluid phase behaviour and critical curves in binary mixtures. However, we have chosen a model that has been developed primarly to account for the nature of chainlike molecules: the Simplified-Perturbed-Hard-Chain-Theory (SPHCT). The SPHCT and the equation of state that results from it, will be thoroughly discussed in the next chapter.. NOMENCLATURE L LCEP p S T UCEP V x. liquid phase Lower Critical EndPoint pressure solid temperature Upper Critical Endpoint vapour mole fraction of the heavier component. REFERENCES 1. V.M. Valyashko. Complete phase diagrams of binary systems with different volatility components. Z. Phys. Chemie 267, (1986), 481-493. 2. P.H. van Konynenburg. Critical lines and phase equilibria in binary mixtures. PhD-thesis University of California Los Angeles. (1968). 3. R.L. Scott and P.H. van Konynenburg. 2. Static properties of solutions: van der Waals and related models for hydrocarbon mixtures. Discuss. Faraday Soc. 49, (1970), 87-97. 4. P.H. van Konynenburg and R.L. Scott. Critical lines and phase equilibria in binary van der Waals mixtures. Phil. Trans. 298A, (1980), 495-540. 5. J.S. Rowlinson and F.L. Swinton. Liquids and Liquid Mixtures Butterworths Monographs in Chemistry, London, Chap. 6, (1982)..

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