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Thermodynamic modelling and data evaluation for life sciences applications

Ruszczynski, Lukasz

Publication date:2019

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Citation (APA):Ruszczynski, L. (2019). Thermodynamic modelling and data evaluation for life sciences applications. Kgs.Lyngby: Technical University of Denmark.

https://orbit.dtu.dk/en/publications/aedd9021-75d8-450f-80e4-92213cf53e74

DTU Chemical EngineeringDepartment of Chemical and Biochemical Engineering

Thermodynamic modelling and data evaluation for life sciences applications Łukasz Ruszczyński

TECHNICAL UNIVERSITY OF DENMARK

PH.D. THESIS

Thermodynamic modelling and data evaluationfor life sciences applications

Łukasz Ruszczynski

Process and Systems Engineering Centre (PROSYS)Department of Chemical and Biochemical Engineering

April 2019

i

“The fight itself towards the summits suffices to fill a heart of man; it is necessary to imagineSisyphus happy.”

Albert Camus

iii

Preface

This thesis serves as partial fulfilment of the requirements for a PhD degree in Chem-ical Engineering. The work presented here was supervised by Assoc. Prof. JensAbildskov, and co-supervised by Dr. Alexandr Zubov and Assoc. Prof. GürkanSin, and performed from May 2016 to April 2019 at the Process and Systems En-gineering Centre (PROSYS) within the Department of Chemical and BiochemicalEngineering at the Technical University of Denmark (DTU). The research for thiswork was funded by the European Union (EU) project ModLife: Advancing Mod-elling for Process-Product Innovation, Optimization, Monitoring and Control in LifeScience Industries (grant agreement no. 675251) under Horizon 2020 FrameworkProgramme of the European Union H2020-MSCA-ITN-2015 call.

Kgs. Lyngby, April 2019Łukasz Ruszczynski

v

Acknowledgements

In the course of my PhD studies, I have been lucky to work together with many dif-ferent people across universities and industry. First and foremost, I would like toexpress my sincere gratitude to all my supervisors Associate Professor Jens Abild-skov, Dr. Alexandr Zubov and Associate Professor Gürkan Sin for giving me theopportunity to do this project and for the guidance along the way. Moreover, Asso-ciate Professor Jens Abildskov and Dr. Alexandr Zubov are highly acknowledgedfor all their patience, discussions and feedback. Thank you for challenging me to dothis PhD and believing in me.

I would like to express my warmest thanks to Professor John P. O’Connell for hiscontributions and discussions regarding the liquid-liquid equilibrium model valida-tion.

In addition, I would like to thank Guoping Lian for welcoming me in Unileverfor the secondment. Special thanks goes to Mattia Turchi for nice discussions andtime in Unilever.

Thanks to all friends, I have met at DTU, for an encouraging working atmo-sphere, great discussions, not only work-related, and for all the great fun together,which has created beautiful memories. I would like to thank all people involved inthe ModLife project, in particular colleagues at DTU: Ergys, Frederico, Héctor, Mark,Sasha and Resul. Thanks for being great travel partners, the support in overcomingthe problems and for all the fun we had.

Special thanks to all my former and current office mates: Andreas, Anders, Han-nah, Jess and Nikolaus for wonderful company.

Rasmus Fjordbak Nielsen is thanked for the help in translating the abstract intoDanish.

Last, but not least, I would like to thank to all my friends and family for theirsupport over the years.

Dziekuje Ci Mamo za niewysłowiona miłosc i ogromne wsparcie okazane wtrakcie doktoratu. Zawsze byłas dumna z moich sukcesów, jak równiez dodawałasotuchy w najgorszych momentach.

Łukasz Ruszczynski,Kgs. Lyngby, April 2019

vii

Abstract

The knowledge of reliable thermodynamic properties of multicomponent systemsis of central importance in process systems engineering and process-product design.Thermodynamic models very often pose an alternative to experimental data, whenthese are too expensive to collect or data are too limited and/or even unreliable.Moreover, many thermodynamic data are published and new continue to appear atthe significant rate. It means that the currently used methods of data evaluation areno longer capable of supporting this data growth. Therefore, new data validationmethods and predictive models designed to identify questionable data and preventerroneous data in modelling and engineering applications are needed.

This thesis addresses the issue of generating and using thermodynamic modelsarising from a statistical mechanical framework called fluctuation solution theory(FST). This theory of solutions is a foundation for reliable, but unusually simple inthe form models, describing liquid and solid-liquid equilibria (LLE and SLE) in thebinary and ternary systems. These models are reliable to describe systems, whereminimum deviation from rigor is to be expected i.e. close to infinitely diluted so-lutions. Two new models describing liquid-liquid equilibria in binary and ternarysystems (both with molecular and ionic liquids) used in the processes of separationsand formulations of biologically active compounds are developed. Extrapolation ofthe model outside the proven range is also demonstrated. The models for solid-liquid equilibria in binary and ternary systems are refined and applied to systemswith active pharmaceutical ingredients.

Next, these models are used in the development of criteria for the validation ofbinary and ternary LLE and solubility data of solids in mixed solvents. This is doneby the combination of the uncertainty and sensitivity analysis of the models andmodel predictions, outliers detection techniques, evaluation of the model parame-ters and finally the employment of auxiliary, derivative solution experimental data.A new and first comprehensive methodology, based on a nearly rigorous models, forbinary and ternary LLE data validation is proposed. The criteria for the validationof solubility data of solids in mixed solvents are formulated. This becomes a sig-nificant contribution in the almost unexplored field of the ternary condensed phaseequilibria data validation.

ix

Resumé på dansk

Kendskab til pålidelige termodynamiske egenskaber ved multikomponentsystemerer af største betydning inden for processystemteknik og proces-produktdesign. Ter-modynamiske modeller udgør ofte et alternativ til eksperimentelle data, når disse erfor dyre at indsamle eller data er for begrænsede og/eller endda upålidelige. Desu-den udgives mange termodynamiske data, og nye fortsætter med at fremkomme ien betydelig hast. Dette betyder, at de nuværende metoder til dataevaluering ikkelængere er i stand til at understøtte denne datavækst. Derfor er der brug for nyedata-validerings metoder og prædiktive modeller designet til at identificere tvivl-somme data og forhindre fejlagtige data i modellering og tekniske applikationer.

Denne afhandling omhandler generering og anvendelse af termodynamiske mod-eller, der stammer fra en statistisk mekanisk teori kaldet fluctuation solution the-ory (FST). Netop denne løsningsteori er et fundament for pålidelige modeller, somer usædvanligt simple i form, der beskriver væske- og fast stof-væske ligevægte ibinære og ternære systemer. Disse modeller er pålidelige til at beskrive systemer,hvor minimal afvigelse fra teoretiske formler kan forventes, dvs. tæt på uendeligtfortyndede opløsninger. To nye modeller, der beskriver væske-væske ligevægte ibinære og ternære systemer (med både molekylære og ioniske væsker), der anven-des i processen med separationer og formuleringer af biologisk aktive forbindelser,bliver her udviklet. Anvendelsen af modellen udefor dens teoretiske gyldighed-sområde testes også. Modellerne for fast stof-væske ligevægte i binære og ternæresystemer anvendes på systemer med aktive farmaceutiske stoffer.

Dernæst anvendes disse modeller til udvikling af kriterier for validering af binæreog ternære væske-væske ligevægte og opløselighedsdata for fastformige stoffer iblandede solventer. Dette gøres ved at kombinere usikkerheds- og sensitivitetsanal-yse af modellerne og model-forudsigelserne, outlier detektionsteknikker, evaluer-ing af modelparametre og endelig ved brug af supplerende eksperimentelle data forsamme opløsninger. En ny metode, som er den første af sin slags, der er baseret påteori, til validering af ligevægts-data for binære og ternære væske-væske udvikles.Kriterierne for validering af opløselighedsdata for fastformige stoffer i blandede sol-venter formuleres. Dette resulterer i et væsentligt bidrag til det næsten uudforskedeområde for datavalidering af ternære kondenserede faser.

xi

Contents

Preface iii

Acknowledgements v

Abstract vii

Resumé på dansk ix

List of Figures xv

List of Tables xix

List of Abbreviations xxiii

List of Symbols xxv

1 Introduction 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Thermodynamic data evaluation and consistency tests 52.1 Vapour-liquid equilibria data evaluation . . . . . . . . . . . . . . . . . . 5

2.1.1 Thermodynamic consistency test methods . . . . . . . . . . . . 52.1.2 NIST-TDE quality assessment algorithm for VLE data . . . . . . 11

Cross-check between VLE data and other types of data . . . . . 122.1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Solid-liquid equilibria data validation . . . . . . . . . . . . . . . . . . . 132.2.1 SLE consistency tests . . . . . . . . . . . . . . . . . . . . . . . . . 14

Ruckenstein and Shulgin test . . . . . . . . . . . . . . . . . . . . 14Cunico test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Kang test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.2 Solubility data project . . . . . . . . . . . . . . . . . . . . . . . . 172.2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Liquid-liquid equilibria data validation . . . . . . . . . . . . . . . . . . 182.4 Data collections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Overall conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Activity coefficients from Fluctuation Solution Theory 213.1 General relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2 Expansion method – the expression for the unsymmetrically normal-

ized activity coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Model for the excess solubility in binary solvents based on FST . . . . 24

xii

4 Binary liquid-liquid equilibria modelling 294.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Models used in the modelling of LLE . . . . . . . . . . . . . . . . . . . . 29

4.2.1 FST-based model . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Parameter/property connection . . . . . . . . . . . . . . . . . . 32

4.2.2 Non-random Two Liquids (NRTL) . . . . . . . . . . . . . . . . . 324.2.3 COSMO-SAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Model uncertainty analysis . . . . . . . . . . . . . . . . . . . . . . . . . 364.4 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.4.1 Graphical analysis of experimental data . . . . . . . . . . . . . . 384.4.2 Regression of aα, aβ, bα, bβ with fixed values of cα and cβ . . . . 394.4.3 Stability analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.5 Results for different classes of LLE binary systems . . . . . . . . . . . . 444.5.1 Hydrocarbon + water systems . . . . . . . . . . . . . . . . . . . 444.5.2 Ionic liquids + water systems . . . . . . . . . . . . . . . . . . . . 454.5.3 Hydrocarbon + nitroethane systems . . . . . . . . . . . . . . . . 464.5.4 Comparison of the Unsymmetric model with NRTL . . . . . . . 484.5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.6 Model modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Ternary liquid-liquid equilibria modelling 595.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.2 Modelling framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.2.1 Binary VLE and LLE data . . . . . . . . . . . . . . . . . . . . . . 60Binary 1+2 (VLE) . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Binary 1+3 (VLE) . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Binary 2+3 (LLE) . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.2.2 Ternary LLE model . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2.3 Stability conditions of the ternary LLE model . . . . . . . . . . . 645.2.4 Parameters regression . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 Application of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3.1 Model performance on the selected cases . . . . . . . . . . . . . 66

Case study 1: benzene (1) + cyclohexane (2) + acetonitrile (3)at 298.15 and 318.15 K . . . . . . . . . . . . . . . . . . . 67

Case study 2: propan-2-ol (1) + benzene (2) + water (3) widegap and different temperatures . . . . . . . . . . . . . 69

Case study 3: ethanol (1) + acetonitrile (2) + n -octane (3) T=298.15K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3.2 Comparison with the NRTL model . . . . . . . . . . . . . . . . . 715.3.3 Identification of c (A

′23 and A

′′23) parameters based on the ternary

LLE data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.5 Parameter tables for binary systems . . . . . . . . . . . . . . . . . . . . 77

6 Validation of binary and ternary LLE 816.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 Binary Tx LLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.2.1 General methodology . . . . . . . . . . . . . . . . . . . . . . . . 826.2.2 Selection of the FST-based LLE model . . . . . . . . . . . . . . . 85

xiii

6.2.3 Outliers detection techniques . . . . . . . . . . . . . . . . . . . . 86Outlier detection with Cook’s distance . . . . . . . . . . . . . . 86COVRATIO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.2.4 Connection between LLE model parameters and infinite dilu-tion partial molar excess properties . . . . . . . . . . . . . . . . 88

6.2.5 Quality factor for LLE data . . . . . . . . . . . . . . . . . . . . . 906.2.6 Selected examples . . . . . . . . . . . . . . . . . . . . . . . . . . 90

octan-1-ol (1) + water (2) system . . . . . . . . . . . . . . . . . . 90[hmim][BF4] (1) + water (2) . . . . . . . . . . . . . . . . . . . . . 91[bmim][PF6] (1) + butan-1-ol (2) . . . . . . . . . . . . . . . . . . 92LLE systems parameters . . . . . . . . . . . . . . . . . . . . . . . 93

6.3 Ternary liquid-liquid data validation . . . . . . . . . . . . . . . . . . . . 1016.3.1 Selected examples . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.3.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7 Solubility of solids in mixed solvents 1117.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1117.2 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.2.1 Excess solubility model . . . . . . . . . . . . . . . . . . . . . . . 1137.2.2 Obtaining model parameters . . . . . . . . . . . . . . . . . . . . 1147.2.3 Extension of the model to solubility of solids forming poly-

morphs in mixed solvents . . . . . . . . . . . . . . . . . . . . . . 1157.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1177.3.1 Parameters from COSMO-SAC . . . . . . . . . . . . . . . . . . . 118

4-nitrobenzonitrile (1) + ethyl acetate (2) + methanol (3) at T=298.15K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

Caffeine (1) + water (2) + 1,4-dioxane (3) at 298.15 K . . . . . . . 1207.3.2 Parameters from Modified Margules model . . . . . . . . . . . . 121

Cholesterol (1) + hexane and ethanol at 293.15 K . . . . . . . . . 121Cholesterol (1) + benzene (2) + ethanol (3) at 293.15 K . . . . . . 121Naphthalene (1) + acetone (2) + water (3) at T=298.15 K . . . . 122Naphthalene (1) + ethylene glycol (2) + water (3) at T=298.15

K [205] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

7.4 Systems with polymorphs . . . . . . . . . . . . . . . . . . . . . . . . . . 1287.5 Excess solubility data validation . . . . . . . . . . . . . . . . . . . . . . 132

7.5.1 Solute-solvent interaction parameters . . . . . . . . . . . . . . . 1327.5.2 Solvent-solvent term . . . . . . . . . . . . . . . . . . . . . . . . . 1337.5.3 Goodness-of-fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337.5.4 Data quality factors . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.5.5 Thermodynamic data consistency test . . . . . . . . . . . . . . . 135

7.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

8 Overall conclusions and outlook 1398.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Bibliography 143

xiv

A Reduction of the ternary LLE model to the binary 159A.0.1 Reduction of the ternary model to the LLE binary system . . . . 159A.0.2 Model reduction to VLE binary pairs 1+2 and 1+3 . . . . . . . . 159

Pair 1+2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159Pair 1+3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

B Tables 163B.0.1 Thermophysical properties of solutes . . . . . . . . . . . . . . . 163B.0.2 NRTL model parameters for ternary systems . . . . . . . . . . . 163

C Ruckenstein-Shulgin consistency test 167

D Publication activity of the author 175

xv

List of Figures

1.1 Current and projected growth of thermodynamic data; grey: stati-cally evaluated data; green: erroneous experimental data; red: non-evaluated experimental values. Figure adapted from Frenkel [10]. . . . 2

2.1 VLE (γivs.xi) in the system of methyl ethyl ketone (1) + p-xylene (2) at91.3 kPa, reported by Chandrashekara and Sechadri [19]. Both com-ponents deviate in opposite direction from Raoult’s law. This set isinconsistent. Figure extracted from [15]. . . . . . . . . . . . . . . . . . . 6

2.2 Demonstration of consistency with the Kang’s tests for SLE for thebinary system of benzene and 1,2-dibromoethane. Figure extractedfrom [36]. The green points should match with grey ones close topure component limits. The red line (slope) should be the same as theslope of the experimental data (grey points). . . . . . . . . . . . . . . . 16

2.3 Demonstration of the proposed by Diky composition-stretched scaleLLE phase diagram in the aniline (1) + water (2) system. Figure bor-rowed from [53]. The stretching has revealed that one of the data-setsis inconsistent. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.1 Liquid-liquid equilibria in toluene (1) with water (2); results of un-certainty analysis. Note the confidence intervals are similar in bothphases, but scaling of the axes is different. The error bars show exper-imental uncertainty in the molar fraction. Lower bound of confidenceinterval has negative values. . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.2 Methodology workflow involved in data correlation. . . . . . . . . . . 384.3 ln xi as a function of inverse temperature data for the toluene (1) /

water (2) system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4 Contour plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.5 Liquid-liquid equilibrium in n-hexane (1) with water (2) including un-

certainty analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.6 Liquid-liquid equilibrium in [hmim][BF4] (1) with water (2) including

uncertainty analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.7 Contour plots for [hmim][BF4] with water; red point indicates mini-

mum of the objective function; green square - COSMO-SAC predic-tion. Upper figure: Full range of c provided by COSMO-SAC model;Lower figure: zoom for smaller range. . . . . . . . . . . . . . . . . . . . 46

4.8 Liquid-liquid equilibrium of hydrocarbon (1)/nitroethane (2) systemsincluding uncertainty analysis: n- octane . . . . . . . . . . . . . . . . . 47

4.9 Liquid-liquid equilibrium of hydrocarbon (1)/nitroethane (2) systemsincluding uncertainty analysis: n- decane . . . . . . . . . . . . . . . . . 47

4.10 Sample of liquid-liquid equilibrium correlation by unsymmetric model(with confidence intervals) and NRTL in the systems [hmim][BF4]/water(top), octane/nitroethane (middle) and n-hexane/water. . . . . . . . . 49(a) hmimBF4/water . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

xvi

(b) n-octane/nitroethane . . . . . . . . . . . . . . . . . . . . . . . . . 49(c) n-hexane/water . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.11 Partial molar excess enthalpies at infinite dilution of 72 binary sys-tems at 298.15 K determined experimentally and predicted by theCOSMO-SAC model. Root mean square deviation from RMSD =[

1N ∑N

i=1 (hE,∞calc − hE,∞

exp )2]1/2

is equal to 2.6 kJ/mol. Full circles repre-

sent results for nitromethane/nitroethane and hydrocarbons (includ-ing n-alkanes) systems, full square corresponds to the toluene/watersystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.12 Liquid-liquid equilibrium of cyclohexane (1)/acetonitrile (2) system(left); zoom on the critical region (right);* denotes critical point, xc=0.477, Tc= 349.79, β=0.3262, A1=0.9962, A2=-0.439 . . . . . . . . . . . . 55

4.13 LLE in methylvinylketone (1) + water (2) system . . . . . . . . . . . . . 56

5.1 Liquid-liquid equilibria phase diagram for cyclohexane (1) + acetoni-trile (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2 Isothermal vapor-liquid equilibria in the system of benzene (1) + ace-tonitrile or cyclohexane (2). Circles are experimental points, line iscorrelation by Porter model. . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.3 Ternary phase diagrams of benzene (1) + cyclohexane (2) + acetoni-trile (3) system (top at T=318.15 K, bottom T=298.15 K). Circles - ex-perimental data; solid line - experimental tie-lines; dashed lines arecalculated tie-lines; blue line - calculated binodal curve. . . . . . . . . . 69

5.4 Ternary phase diagrams propan-2-ol (1) + benzene (2) + water (3) (topleft: 323 K, top right 313 K), bottom 303 K. Legend is the same as inthe previous figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.5 Ternary LLE in ethanol (1) + acetonitrile (2) + n-octane (3) system atT=298.15 K; experimental data vs prediction by the FST-based modelproposed in this chapter. Points are experimental data, the proposedmodel calculates blue solid line, solid black lines are experimental tielines and dashed lines are calculated tie lines. . . . . . . . . . . . . . . . 71

5.6 NRTL correlation with six parameters for benzene (1) + acetonitrile(2) + n-heptane (3) at 318.15 K. . . . . . . . . . . . . . . . . . . . . . . . . 73

5.7 NRTL prediction from binaries with six parameters for benzene (1) +acetonitrile (2) + n-heptane (3) at 318.15 K. . . . . . . . . . . . . . . . . . 73

5.8 LLE in benzene (1) + acetonitrile (2) + n-heptane (3) at T=318.15 KUpper figures: A

′23, A

′′23 fixed as zero (left) ; set_1 of parameters (as

in the Table 5.6), Lower figure: set_2 of A′23, A

′′23 (left); 4 parameters

regressed from ternary data (right). . . . . . . . . . . . . . . . . . . . . . 755.9 Ternary LLE in the system of ethanol (1) + acetonitrile (2) + n-heptane

(3) at 298.15 K for different sets of A′23, A

′′23 parameters. Upper figures:

interaction parameters between 2 and 3 are equal to zero (left), set_1(right) Lower figures: set_2 (left), four parameters regressed fromternary data (right).In the acetonitrile + n-heptane system (A

′23, A

′′23)

are significant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.10 LLE in the binary system of acetonitrile and n-heptane calculated with

the interaction parameters A′23 and A

′′23 regressed from ternary LLE

data. Points are experimental data, lines model prediction. . . . . . . . 76

6.1 General binary LLE validation methodology . . . . . . . . . . . . . . . 84

xvii

6.2 Initial regression of the model parameters for the system octan-1-ol(1) + water (2). Points - experimental data, Line - fitted curve. Upperfigure: Phase α (rich in octan-1-ol), c was found to be -0.01; Lowerfigure: Phase β (rich in water), c = -0.0004. . . . . . . . . . . . . . . . . . 86

6.3 Initial regression of the model parameters for the system [hmim][BF4](1) + water (2). Points - experimental data, Line - fitted curve. Upperfigure: Phase α (rich in ionic liquid), c was found to be -72; Lowerfigure: Phase β (rich in water), c was found to be -5841. . . . . . . . . . 87

6.4 LLE in the octan-1-ol (1) and water (2). . . . . . . . . . . . . . . . . . . . 926.5 Liquid-liquid equilibrium in [hmim][BF4] (1) with water (2). . . . . . . 936.6 Liquid-liquid equilibrium in [bmim][PF6] (1) with butan-1-ol (2). . . . . 936.7 Liquid-liquid equilibrium in [bmim][PF6] (1) with butan-1-ol (2). Pos-

sible outliers in data - experimental points in black circles. . . . . . . . 946.8 Quality factor distribution for studied systems. . . . . . . . . . . . . . . 956.9 LLE in γ-valerolactone (1) + cyclohexane (2) . . . . . . . . . . . . . . . 966.10 LLE in heptylamine (1) + water (2) . . . . . . . . . . . . . . . . . . . . . 966.11 Binaries to ternary system LLE data validation general framework . . 1016.12 Ternary LLE phase diagram benzene (1) + acetonitrile (2) + n-heptane

(3) at T=318.15 K [102] along with the corresponding binary LLE inacetonitrile (2) + n-heptane (3) [103]. Orange lines, correlation withFST-binary model, red and green lines are 95 % confidence intervals,lower and upper bounds, respectively. Blue lines ternary FST-basedmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.13 Example of suspicious ternary LLE data acetonitrile (1) + chloroben-zene (2) + water (3) at 304.15 K. . . . . . . . . . . . . . . . . . . . . . . . 105

6.14 LLE in the binary system of chlorobenzene (1) and water (2) . . . . . . 1056.15 LLE in the binary system of nitromethane (1) and water (2) (lower fig-

ure) and ternary ethanol (1) + nitromethane (2) + water (3) LLE data(upper figure). Blue lines are model predictions, circles are experi-mental points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

7.1 The activity coefficient derivative with respect to the composition inethanol (1) + benzene (2) system at 298.15 K. . . . . . . . . . . . . . . . 118

7.2 Solubility of the 4-nitrobenzonitrile in ethyl acetate and methanol bi-nary solvent at 298.15 K. Left figure: solvent-solvent term from COSMO-SAC, solute-solvent estimated from binary or regression to the ternary(mixed solvent) data ; right figure: solvent-solvent term from modi-fied Margules model, solute-solvent estimated from binary or regres-sion to the ternary (mixed solvent) data . . . . . . . . . . . . . . . . . . 119

7.3 Solubility and excess solubility in the system of caffeine (1) + water (2)+ 1,4-dioxane (3) at 298.15 K. Left figure: solvent-solvent term fromCOSMO-SAC, solute-solvent estimated from binary or regression tothe ternary (mixed solvent) data; right figure: solvent-solvent termfrom modified Margules model, solute-solvent terms from COSMO-SAC or regression to the ternary (mixed solvent) data. . . . . . . . . . . 120

7.4 Cholesterol (1) in hexane (2) + ethanol (3) (left, dataset #1) and ethanol(2) + hexane (3) (right, dataset #2) at 293.15 K. . . . . . . . . . . . . . . 121

7.5 Solubility and excess solubility in cholesterol (1) + benzene (2) + ethanol(3) system at 293.15 K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.6 Solubility of the naphthalene in acetone and water binary at 298.15 K. . 123

xviii

7.7 Solubility of naphthalene in ethylene glycol and water binary solvent298.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.8 Solubility of meroponem (1) in water (2) + methanol (3) solvent sys-tem at different temperatures: 273.15, 278.15 (upper plots), 283.15 and288. 15 K (bottom plots). Black dotted lines are predictions, yellowline is solubility, when ideal mixing is assumed. . . . . . . . . . . . . . 130

7.9 Solubility of carbamazepine (1) in water (2) and ethanol (3). There isno dramatic change in the solubility between forms as for the meropenemcase. However, model overestimates the solubility of the solid aroundthe transition point. Dashed line represents ideal mixing. . . . . . . . . 131

7.10 Sodium fusidate solubility in water + acetone binary solvent at 298.15K. Dashed line represents ideal mixing. . . . . . . . . . . . . . . . . . . 131

7.11 Solubility of fusidic acid + water (2) + ethanol (3) at T=294.15 K. Dashedline denotes ideal mixing. . . . . . . . . . . . . . . . . . . . . . . . . . . 132

xix

List of Tables

4.1 Parameters of the COSMO-SAC model (2010 version) . . . . . . . . . . 344.2 Sample of parameter initial guesses and optimized values (in paren-

theses) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.3 Sample of the lowest values of c-s . . . . . . . . . . . . . . . . . . . . . 424.4 Estimated parameters for all considered systems. In all tables phase

α is rich in component 1 and β rich in component 2. Numbers inbold are expected to be similar for water in alkanes and nitroethanein alkanes. Numbers for cα and cβ in parentheses are positive, whichis inconsistent with phase stability, as discussed below along with un-certainties in these parameters. . . . . . . . . . . . . . . . . . . . . . . . 43

4.5 Estimated parameters for NRTL model (α = 0.2) for all considered sys-tems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.6 Comparison of average absolute relative deviation (AARD) in molefraction using unsymmetric formulation (temperature dependence with2 or 3 parameters) and NRTL. + denotes reference temperature Tre fequal to 298.15 K, in other cases 308.15 was used. . . . . . . . . . . . . . 50

4.7 Comparison of parameter values estimated with COSMO-SAC (at T= 298.15 K) and regressed values. . . . . . . . . . . . . . . . . . . . . . . 51

4.8 Regressed parameters (all six) for considered systems. Parameter val-ues with confidence ranges from eq. (4.49). In all cases phase α is richin component 1 and β rich in component 2. . . . . . . . . . . . . . . . . 54

4.9 Activity coefficient derivatives with respect to composition with dif-ferent models: COSMO–SAC, NRTL, Modified Margules and Wilsonequation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.1 Modelling expressions of the isothermal or isobaric binary VLE . Theresiduals in the objective function are the differences between exper-imental (temperature or pressure) values and calculated i.e. δxi =xexp

i − xcalci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2 Modelling expressions of the isothermal or isobaric binary VLE . Theresiduals in the objective function are the differences between exper-imental (temperature or pressure) values and calculated i.e. δxi =xexp

i − xcalci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.3 Selected systems shown in the Section 5.3 and vapour-liquid and liquid-liquid equilibria for binary systems and liquid-liquid equilibria forternary systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.4 Parameters regressed from ternary data along with the root meansquare deviation (RMSD) for the three case studies elaborated in Sec-tion 5.3. Results for other investigated systems are provided in Chap-ter 6. TL denotes number of tie lines. . . . . . . . . . . . . . . . . . . . . 72

xx

5.5 Sample of NRTL model parameters along with the root mean squaredeviation (RMSD) both from correlation of only ternary LLE data andprediction by the parameters obtained from binary pairs. Referencesand temperatures for the systems are the same as in Table 5.4 . . . . . . 72

5.6 Values of the different parameter estimates and objective function val-ues in the system {benzene or ethanol} (1) + acetonitrile (2) + n-heptane(3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.7 Model parameters obtained from VLE data, asterisk means that insome cases isobaric VLE were used to regress parameters. . . . . . . . 78

5.8 Parameters obtained from the LLE binary data. . . . . . . . . . . . . . . 79

6.1 Comparison between experimental and calculated partial molar en-thalpy at infinite dilution HE,∞

i for selected systems . . . . . . . . . . . 896.2 Outlier detection in the system of octan-1-ol/water. Suggested ouliers

are marked by asterisk. Values of the partial molar properties at infi-nite dilution (at 298.15 K) with i-th point omitted are calculated. . . . . 91

6.3 Partial molar enthalpy at infinite dilution of butan-1-ol in 1-butyl-3-methyl-imidazolium hexafluorophosphate. . . . . . . . . . . . . . . . . 94

6.4 FST-based model parameters along with their uncertainties for theconsidered systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.5 FST model parameters for solubility data, number of possible outliers,model deviations and quality factors. #P -no. of experimental points,#O - no. of possible outliers. . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.6 FST model parameters for solubility data, number of possible outliers,model deviations and quality factors. Systems with closed loop orhourglass phase diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.7 Examples of results for the test FtLLE,2. . . . . . . . . . . . . . . . . . . . 1026.8 Ternary LLE FST model parameters along with quality factors. . . . . . 103

7.1 Systems of solutes and two solvents used to test the model . . . . . . . 1267.2 Systems of solutes and two solvents used to test the model. Parame-

ters from COSMO-SAC model. . . . . . . . . . . . . . . . . . . . . . . . 1287.3 The regressed parameters of model together with the absolute aver-

aged relative deviation in meroponem (1) + water (2) + methanol (3)system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7.4 Systems of solute in binary solvent along with quality factors. . . . . . 1347.5 D values obtained via Eq. (2.35) for solubility data of cholesterol (1)

in benzene (2) + ethanol (3) mixture . . . . . . . . . . . . . . . . . . . . 1357.6 D values obtained via Eq. (2.35) for solubility data of naphthalene (1)

in acetone (2) + water (3) mixture . . . . . . . . . . . . . . . . . . . . . . 136

B.1 Thermophysical properties of solutes . . . . . . . . . . . . . . . . . . . . 163B.2 NRTL model parameters for ternary systems, α=0.2. . . . . . . . . . . . 164B.3 NRTL binary systems (VLE or LLE) parameters . . . . . . . . . . . . . . 166

C.1 Paracetamol (1) + methanol (2) + ethyl acetate (3) . . . . . . . . . . . . . 167C.2 Paracetamol (1) + dioxane (2) + water (3) . . . . . . . . . . . . . . . . . 167C.3 Paracetamol (1) + ethanol (2) + ethyl acetate (3) . . . . . . . . . . . . . . 168C.4 Phenacetin (1) + ethanol (2) + ethyl acetate (3) . . . . . . . . . . . . . . . 168C.5 Phenacetin (1) + water (2) + dioxane (3), 298.15 K, 298 K and 313 K

(each next three columns, respectively). . . . . . . . . . . . . . . . . . . 168

xxi

C.6 Cholesterol (1): hexane (2) + ethanol (3); ethanol (2) +hexane(3); hex-ane (2) + benzene (3), all at 293.15 K, (each next three columns, respec-tively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

C.7 4-nitrobenzonitrile (1) + ethyl acetate (2) + methanol (3): 318.15; 298.15and 278.15 K (each next three columns, respectively). . . . . . . . . . . 169

C.8 Naphthalene (1) + propan-2-ol (2) + water (3); naphthalene (1) + ethanol(2) + water (3); naphthalene (1) + methanol (2) + water (3); (each nextthree columns, respectively). . . . . . . . . . . . . . . . . . . . . . . . . . 169

C.9 Naphthalene (1) + benzene (2) + hexane (3); naphthalene (1) + acetone(2) + water (3); naphthalene (1) + DMSO (2) + water (3); (each nextthree columns, respectively). . . . . . . . . . . . . . . . . . . . . . . . . . 170

C.10 Benzoic acid (1) + cyclohexane (2) + hexane (3) 303.15 K;benzoic acid(1) + cyclohexane (2) + hexane (3) 298.15 K; benzoic acid (1) + hexane(2) + CCl4 (3) 298.15 K; (each next three columns, respectively). . . . . 170

C.11 Mefenamic acid (1) + ethanol (2) + ethyl acetate (3) . . . . . . . . . . . . 170C.12 Testosterone (1) + ethanol (2) + ethyl acetate (3) . . . . . . . . . . . . . . 171C.13 Caffeine (1) + water (2) + dioxane (3) . . . . . . . . . . . . . . . . . . . . 171C.14 Pregabalin (1) + methanol (2) + water (3) 338.15 K; Pregabalin (1) +

methanol (2) + water (3) 298.15 K; (each next three columns, respec-tively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

C.15 Anthracene (1) + dibutyl ether (2) + hexane (3) 298.15 K;Anthracene(1) + dibutyl ether (2) + heptane (3) 298.15 K; Anthracene (1) + dioxane(2) + propan-1-ol (3) 298.15 K; (each next three columns, respectively). 172

C.16 Anthracene (1) + propan-1-ol (2) + hexane (3) 298.15 K; Anthracene (1)+ propan-1-ol (2) + heptane (3) 298.15 K; Anthracene (1) + propan-1-ol(2) + octane (3) 298.15 K; (each next three columns, respectively). . . . . 172

C.17 Sulphamethoxypyridazine (1) + ethanol (2) + water (3) 298.15 K . . . . 172C.18 Naphthalene (1) + ethylene glycol (2) + water (3) 298.15 K . . . . . . . . 173

xxiii

List of Abbreviations

AAD Absolute average deviationAARD Absolute average relative deviationACN AcetonitrileAPI Active pharmaceutical ingredientCOSMO-RS Conductor-like screening model for real solventsCOSMO-SAC Conductor-like screening model segment activity coefficientCOV Covariance matrixDMF DimethylformamideDMSO Dimethyl sulphoxideFST Fluctuation solution theoryGC-EOS Group-contribution equation of stateIDAC Infinite dilution activity coefficientIL Ionic liquidsIUPAC International Union of Pure and Applied ChemistryLLE Liquid-liquid equilibriumMeOH MethanolMSE Mean square errorNIST National Institute of Standards and TechnologyNRTL-SAC Non-random two-liquid segment activity coefficient modelobj Objective functionPC-SAFT Perturbed chain- statistical association fluid theoryRMSD Root mean squared deviationSD Standard deviationSLE Solid-liquid equilibriumTDE ThermoData EngineTHF TetrahydrofuranUNIFAC Universal functional activity coefficientVLE Vapour-liquid equilibrium

xxv

List of Symbols

Roman symbols

a Parameter of the FST-based modelaij Parameter in the NRTL modelakj Parameter in Wilson equationAi Area of the segment iA12, A21 Constants appearing in Margules equationAij Interaction parameter in ternary LLE FST-based model or Porter modelb Parameter of the FST-based modelbij Parameter in the NRTL modelc Parameter of the FST-based modelcp Heat capacity at constant pressured Parameter of the FST-based modelD Determinant of the matrix or quantity in Ruckenstein-Shulgin testDi Cook’s distance of the point ifi Fugacity of the component ifij Difference in total correlation function integrals or solute-solvent interaction parameterF Quality factor or objective function in LLE ternary modelF(k) Variable in modified Margules equationgij Interaction energy parameters between i and j in NRTL modelG Total Gibbs energyGi Variable in the modified Margules equationGij Integral of the radial distribution function or variable in the NRTL modelh Molar enthalpyH EnthalpyHi Henry’s law constant of iHij Henry’s law constant of i in jHij Total correlation function integral between i and jJ Jacobian matrixkB Boltzmann’s constant 1.38 · 10−23 J · K−1

M Number of points in set in Total number of moles n = ∑i nini Number of molecules of species iN Total number of moles N = ∑i Ni or number of points in setNi Number of molecules of species iP PressureQ Overall quality factorR Universal gas constant 8.314 J ·mol−1· K-1S EntropyT TemperatureU Internal energyv Molar volume

xxvi

V Total system volumeVi Volume of the molecule iW Multiple pair products of correlation function integralsxi Liquid-phase mole fraction of iyi Vapour-phase mole fraction of iz Coordination number

Greek symbols

α Non-randomness parameter in the NRTL modelαij Parameter in the modified Margules equationβ Inverse thermal energy β ≡ (kBT)−1

γi Activity coefficient of species iΓi Segment activity coefficient of species i∆ Denotes the difference in the property that follows this symbolεki Variable in the Wilson equationη Parameter in the modified Margules equationηk Variable in the Wilson equationθ Arbitrary parameterΛ Variable in the Wilson equationµi Chemical potential of species iρi Molar density of iσm Screening charge density on the segment mτij Dimensionless interaction parameters in the NRTL modelφi Fugacity coefficient of species i

Subscripts

c Critical propertyi, j, k Chemical speciesm Melting or mixture propertyf us Connected to enthalpy of fusion

Superscripts

′ Solute-free composition or phase′′ Phase− Mean or partial value∗ Unsymmetrical convention (unsymmetrical γ∗i )∞ Infinite dilution0 Pure species property or infinite dilution in single solvent+ Infinite dilution in mixed solventα, β Phasesb Property in binary systembin Property in binary systemcalc Calculated (predicted) valueE Excess quantityexp Experimental valueid Ideal solution property

xxvii

L Liquid-phase propertylim Limit valuemax Maximum valuesat Saturated propertyt Property in ternary systemV Vapour-phase propertyω Phase

Miscellaneous symbols

〈〉 Average valueT Transpose of a matrix

1

Chapter 1

Introduction

1.1 Background

The knowledge of reliable thermodynamic properties of multicomponent systemsis of central importance in process systems engineering and process-product design.There are in general two approaches for obtaining such kind of data: experiment andmodelling. Despite the fact that the experimental approach is the most trusted, it istypically time-consuming and satisfactory data are often not available for desiredprocess conditions (e.g. temperature, pressure and composition) for a given designproblem. Therefore, it is usually necessary to predict the missing phase equilibriawith the help of thermodynamic models for screenings and exploration and thenprioritise measurements for final validation.

Over past few decades much research has led to useful thermodynamic modelsfor prediction of properties. There are several widely used models with differenttheoretical foundations: activity coefficient models, equations of state, models basedon the quantum mechanical calculations as well as classical mechanical moleculardynamics (MD) and Monte Carlo methods [1]. The latter ones are usually used forsystems with small molecules as they are computationally expensive. The widelyused group of activity coefficients in engineering applications are Group Contribu-tion Models (GCM). Examples are models such as UNIFAC [2], and Modified UNI-FAC [3], [4]. These models are very good in the prediction of fluid phase equilibria,especially vapour-liquid equilibrium, but only at low pressures. Unfortunately, theyrequire many interaction and group parameters obtained from the fitting to experi-mental data.

Another group of models is the so-called Equation of State (EoS) Models. Theseare typically multi-term expressions of the residual Helmholtz energy, where asmany terms are included as can be justified for a system and the data available. Oneof the EoS models with broad range of application is the family of SAFT-based mod-els [5], for instance PC-SAFT (Perturbed Chain-Statistical Association Fluid Theory)[6] or CPA (Cubic plus Association) [7]. Their advantage is that can model systemsat high pressures (very important in petroleum engineering). They also require pa-rameters fitted to the experimental data.

Although the models listed above provide results in the qualitative agreementwith experiment, they often are not able to predict phase equilibria accurately, i.e.in a good quantitative agreement (comparable to a measurement error) with experi-mental data in a wide range of physical conditions. Moreover, some of them requirea lot of parameters obtained from experimental input data which are sometimeshardly available for molecules with many functional groups such as pharmaceuti-cals, peptides or organometallic compounds.

2 Chapter 1. Introduction

Besides the models mentioned, there are also quantum-based solvation modelswhich provide an alternative means of predicting activity coefficients and other ther-modynamic properties. In these models quantum mechanics is used to describe en-ergy states of the components in the mixture. One of the members within this groupis the COSMO-SAC model [8]. This model is an excess Gibbs energy model basedon quantum mechanical calculations. It was first developed by Lin and Sandler [8],with inspiration from COSMO-RS method as developed by Klamt [9]. In this modeleach considered molecule is described by a screening charge density which is rep-resented by a σ-profile connecting a segment of the molecular surface to its chargedensity. Over the last 15 years several improvements of this model have been made.Due to that fact, the possibilities for prediction of properties in systems containingcomplex molecules, especially bio-functional compounds, have been increased.

Nonetheless, all these models have set of parameters, which needs to be re-gressed from experimental accurate data.

FIGURE 1.1: Current and projected growth of thermodynamic data;grey: statically evaluated data; green: erroneous experimental data;red: non-evaluated experimental values. Figure adapted from

Frenkel [10].

Here is, where the evaluation of the experimental data comes into play. A sig-nificant growth in published data per year has been the general trend over the pastfew years. This is clearly noticeable for thermodynamic data [10], where the esti-mated growth is 7% per year, which means a doubling of the consolidated databaseevery 10 years (Fig. 1.1). Such a massive growth is caused by the advancementsin the measurement science as well as the development of new processes and prod-ucts. Unfortunately, this trend seems accompanied by a decrease in the quality of theexperimental data. Approximately 30% of the articles reporting new data contain er-rors either in numerical values or their uncertainties [10], [11]. The currently usedmethods of data evaluation are no longer capable of supporting this data growth.

To tackle this problem, data validation methods and predictive models designedto identify questionable data and prevent erroneous data in modelling and engi-neering applications are needed. For example, in the development/testing of newmodels, in the model parameters estimation and in the comparison between modelpredictions and experimental values only evaluated data should be used. Or when

1.2. Objectives of the thesis 3

a researcher studies the properties of compounds with applications to life sciences(e.g. the active pharmaceutical ingredients - APIs), minimization of errors in predic-tion models for compounds is a task that has to be accomplished.

Validation is an essential part in the critical data evaluation process [10],[12],[13].It strengthens quality, reliable, and consistent data as well as data reporting. Pre-dictive and correlative models are essential tools in data validation, enabling for thequality assessment of the obtained experimental data. These models very often posean alternative to experimental data, when these are too time-consuming and/or ex-pensive to collect or data are too limited and/or even unreliable.

1.2 Objectives of the thesis

The main focus of this thesis is the thermodynamic modelling of binary and ternarysystems containing compounds used in life sciences applications or in general or-ganic compounds whose structure includes at least one functional groups as wellas liquid systems (both with molecular and ionic liquids) used in the processes ofseparations and formulations of these compounds.

The main objectives of this thesis are given:

1. Development and testing of models based on the Fluctuation Solution Theory(FST). This exact theory of solutions is a strong basis for reliable, but simple in theform models describing liquid and solid-liquid equilibria in the binary and ternarysystems. These models are reliable to describe systems, where minimum deviationfrom rigor is to be expected i.e. close to relevant standard states (infinitely dilutedsolutions).

2. The use of these models in the development of criteria of validation method-ologies of the above mentioned data. This is done with the combination of theuncertainty and sensitivity analysis of the models and model predictions, outliersdetection techniques and finally the employment of auxiliary, derivative solutionexperimental data.

Since two types of data are considered: solubility of solids in liquids and liquid-liquid equilibria, the thesis has been divided into two parts: I. Liquid-liquid equilib-ria and II. Modelling and data validation of solubility of solids.

1.3 Thesis outline

The thesis is organized in the following manner:

Chapter 1. The general problem of the modelling of thermodynamic propertiesin chemical and process systems engineering and data validation is given. The mo-tivation of the work is briefly introduced.

Chapter 2. This chapter gives a literature overview on the thermodynamic evalu-ation and consistency tests used in the fluid phase equilibria data quality assessment.

4 Chapter 1. Introduction

Chapter 3. The chapter introduces briefly concepts of the fluctuation solutiontheory, which constitutes a foundation for the models used in this work. The deriva-tion of the activity coefficient expressions used further in the models is provided.

Chapter 4. With this chapter Part I of the thesis is initiated. The binary liquid-liquid model framework is presented along with the procedure of obtaining its pa-rameters, uncertainty analysis, modifications and application to the several binarysystems.

Chapter 5. This chapter introduces a model for ternary liquid-liquid systems ofType I. The procedure for obtaining model parameters from corresponding binaryliquid-liquid or vapour-liquid equilibrium data is described.

Chapter 6. A methodology for the liquid-liquid equilibrium evaluation in binaryand ternary mixtures is proposed and applied to selected case studies. This chapterconcludes Part I devoted to liquid-liquid equilibria.

Chapter 7. This chapter belongs to Part II of the thesis. Chapter 7 shows themodelling and data validation of the solubility of solids in binary solvents.

Chapter 8. The overall conclusions of the thesis, and future perspectives arediscussed.

Each of the chapter 4 to 7 is based either on articles already published in peer-reviewed journals (or conference proceedings) or manuscripts submitted or intendedfor later submission. The actual text of the paper (or manuscript) was revised accord-ingly to the needs when writing this thesis.

5

Chapter 2

Thermodynamic data evaluationand consistency tests

This chapter gives a literature overview of methods used in vapour-liquid (VLE),solid-liquid (SLE) and liquid-liquid (LLE) data validation as well as thermodynamicdata consistency tests.

2.1 Vapour-liquid equilibria data evaluation

Vapour-liquid equilibrium (VLE) is one crucial type of data used in the design ofthe separation processes of fluid mixtures. As a consequence, VLE data have beenintensively measured and reported for many binary and multicomponent systems.VLE data are not the theme of this thesis. Yet, the common methods used in the fieldof VLE data validation are described, also to point out why SLE/LLE data validationhas received less attention for many years.

2.1.1 Thermodynamic consistency test methods

Over the decades many techniques have been developed for the validation and thethermodynamic consistency of VLE data [14], [15], [16]. The data evaluation ap-proaches are of a dual nature. One involves the application of various thermody-namic consistency tests for considered VLE data-sets and screening among thembased on the fail/pass criteria. The main disadvantage of this approach is that itrequires personal (subjective) judgment of the evaluator. The second, more generalapproach evaluates a single numerical quality factor for each of the reported VLEdata-set [17]. This prevents from eliminating of any experimental data points fromfurther considerations e.g. in the validation of thermodynamic models.

For the assessment of the quality of VLE data, many consistency tests have beenderived, mostly on the basis of the Gibbs-Duhem (GD) equation, which can be writ-ten in general form for m component systems as

m

∑i=1

xid ln γi −VE

RTdP +

HE

RT2 dT = 0 (2.1)

or on the form applicable to data at constant pressure (p) and temperature (T)

m

∑i=1

xid ln γi = 0 (2.2)

Here xi is the liquid mole fraction, γi is the activity coefficient, VE and HE areexcess volume and enthalpy, respectively, and R is the universal gas constant.

6 Chapter 2. Thermodynamic data evaluation and consistency tests

Here the most commonly used tests in the assessment of thermodynamic consis-tency of VLE data and their advantages and shortcomings are briefly described.

1.Point-to-point test (slope test) [15], [18]

For a binary system, Eq. (2.2) can be written as (at constant T and p)

x1d ln γ1

dx1+ x2

d ln γ2

dx1= 0 (2.3)

When one plots the activity coefficients vs. composition and draw tangents tothe curves at any given composition, measure the slopes and calculate the lhs of Eq.(2.3), then the data for that chosen point is consistent, if it is equal to zero. Thisreasoning can be applied to each point of the data-set.

The main disadvantage of this method is the difficulty of measuring the slopeof the curve with sufficient accuracy, which makes this test inconvenient. Never-theless, it could be used as a helpful tool of visualisation of outliers in data-set orinconsistencies in the deviation from ideal behaviour as shown in Fig. 2.1. If, at agiven composition, d ln γ1/dx1 is negative, then d ln γ2/dx2 must be also negative,and if d ln γ1/dx1 is zero, then d ln γ2/dx2 must also be zero.

FIGURE 2.1: VLE (γivs.xi) in the system of methyl ethyl ketone (1) +p-xylene (2) at 91.3 kPa, reported by Chandrashekara and Sechadri[19]. Both components deviate in opposite direction from Raoult’s

law. This set is inconsistent. Figure extracted from [15].

2. Area test [20]The Gibbs-Duhem relation might as well be integrated from the limits of xi from 0to 1 and written as (at constant pressure and temperature):∫ 1

0ln

γ1

γ2dx1 = 0 (2.4)

The areas under the curve (ratio of the activity coefficients vs. composition) mustbe equal, if the data are consistent. The test seems easily applicable, but there aresome underlying problems with the curves determination using measured valuessuch as the measurement uncertainty, error in plotting curves, extrapolation to infi-nite dilution or finally the way of calculating the areas. Moreover, when the ratio of

2.1. Vapour-liquid equilibria data evaluation 7

γ1 to γ2 is calculated, the total pressure cancels out. Therefore, the area test does notutilize the most accurate measurement, which is the total pressure P in the system.

Finally, this test as the slope test are restricted to data obtained at constant tem-perature and pressure, which make them not general.

3. The Redlich-Kister test [20], [15]

This test is based on Gibbs-Duhem equation (Eq. (2.1)), therefore it can be ap-plied to the data obtained when only one of the variables (pressure or temperature)is constant. This method has three modifications. First is for cases of constant pres-sure and temperature. In this situation, the method looks similar to the area test.The second modification is for the isothermal data. For this case∫ x1=1

x1=0ln

γ1

γ2dx1 = −

∫ p(x1=1)

p(x1=0)

vE

RTdP (2.5)

In this version, not only the areas under curves (as in the area test) need to beconsidered, but also the effect of change in volume of mixing [15]. In fact, it turnsout that the change in volume is very small in the comparison with area error, soit can often be omitted and the consistency test reduces to the isobaric-isothermalRedlich-Kister version.

The third modification is for isobaric values. In this case∫ x1=1

x1=0ln

γ1

γ2dx1 =

∫ T(x1=1)

T(x1=0)

hE

RT2 dT (2.6)

Here, the effect of the heat of mixing needs to be taken into account. It has beenshown that this effect has the same order as the allowed value of the area difference[15]. The problem is that, to apply this test one needs to know the heat of mixing atdifferent temperatures and compositions, which is reported only for a limited num-ber of binary systems.

4. The Herington test [21]The Herington test belongs to one of the oldest tests for checking the consistency ofVLE data. Since, the heat of mixing data are not widely available for range of tem-peratures and compositions, Herington estimated the upper limit of the right handside (named J) of the Eq. (2.6) using thermodynamic relations and data available athis time and he proposed the criteria for consistent data.

∫ T01

T02

hE

RT2 dT <hE

max

RT0i |θ|

(2.7)

where

100|I|Σ

< 50∣∣∣∣hE

maxgE

max

∣∣∣∣ ∣∣∣∣ θ

Ti

∣∣∣∣ (2.8)

where |hEmax| was the largest value of the excess enthalpy that occurred in the

range of 0 ≤ x1 ≤ 1, |θ| is the difference between the highest and the lowest boilingpoint of the isobaric data, |gE

max| is the maximum value of excess Gibbs energy in thesame range of x1, I is the difference between areas of the ln γ1

γ2plot, and Σ denotes

the total area of this plot. The following relation for determination of the consistencyis used

|D| = 100|I|Σ

< 150∣∣∣∣ θ

Ti

∣∣∣∣ = |J| (2.9)

According to Herington, if |D − J| < 10 the data are consistent and if |D − J| >10, not.

This test is very often used in checking the thermodynamic consistency of VLEdata, because it is simple. However, permissible value of J were obtained a long timeago, only on the limited experimental data and has never changed. Many new datahave been published and J should be updated. Besides, Wisniak [22] found errorsin the assumptions made by Herington i.e. limited amount of data used to estimateJ. This test is rather recommended not to use to evaluate data sets or at least theconsistency of data cannot be checked only by the means of this test. Additionally,the J value should be updated.

5. Fredenslund’s test [23], [15]This consistency test of VLE is based on the ability of the vapour composition tobe predicted from T, x1 data only. One can correlate either isobaric or isothermaldata with a GE model being a polynomial function (of the liquid phase composi-tions only). The GE expression allows the calculation of the activity coefficient andconsequently the predicted values of vapour composition yi,predicted

yi,predicted =xiγi f 0

ipΦi

(2.10)

where f 0i is the fugacity of the component i and Φi fugacity coefficient in the

vapour phase, at the equilibrium pressure and temperature.Fredenslund proposed that the set of (p, T, x1, y1) data is consistent, if the average

absolute deviation of the experimental and predicted vapour compositions are lessthan 0.01

n

∑i=1

|yi,exp − yi,predicted|n

≤ 0.01 (2.11)

This test is in general recommended to perform on the VLE data-set, but shouldbe accompanied with the examination of both pressure and vapour compositionresiduals, as this should be random.

6. The test of Wisniak [24]Wisniak has proposed a new thermodynamic consistency test. The basis for the testis the fundamental relation

gE = RTcomponents

∑i=1

xi ln γi (2.12)

One can replace the activity coefficient in this equation by γi =yi pxi p0

i

gE = RTcomponents

∑i=1

xi

(ln

yi

xi+ ln

pp0

i

)(2.13)

d ln p0i

dT=

∆hvap

RT2 (2.14)


One can use the Clausius-Clapeyron equation (Eq. (2.14) to obtain the value ofln p

p0i

by the integration between the limits (p0i , T) and (p, T0

i ) where the p0i is the satu-

ration vapour pressure of the component i at the boiling temperature of the solutionT, T0

i denotes the boiling point of the same component at the equilibrium pressureof the system, and ∆hvap is the enthalpy of vaporisation.

Finally,

lnpp0

i= −

∆h0i

R

(1

T0i− 1

T

)=

∆s0i

RT(T0

i − T) (2.15)

where ∆s0i and ∆h0

i are the entropy and heat of vaporisation of the component iat the pressure p.

The final expression for gE would be

gE = RT ∑i=1

xi

(ln

yi

xi

)+ ∑

i=1xi∆s0

i (T0i − T) (2.16)

Based on this equation Wisniak [24] established a following relation (valid alsofor multicomponent systems)

Li =

[∑ xi

∆s0i

∑ xi∆s0i

T0i − T

]=

[gE

∑ xi∆s0i− RT

∑ xi∆s0i∑ xi

(ln

yi

xi

)]= Wi (2.17)

According to Eq. (2.17), the experimental point i is consistent if Li = Wi (point-to-point test).

For a binary system the test can be converted to area test by integrating over thewhole composition range to give∫ 1

0Lidxi =

∫ 1

0Widxi. (2.18)

Wisniak defined a parameter D, which if the binary data are consistent, is lessthan 5

D = 100|L−W|L + W

. (2.19)

In the derivation he used additional properties such as the heat and entropy ofvaporisation, which is not always available and can influence the results of test.This test is both an area and a point-to-point test. It can be also applied to multi-component systems.

7. The Van Ness point-to-point test [25]Van Ness proposed a practical and rigorous test based on the Gibbs-Duhem equa-tion. The final form of the GD equation used in this test is the following (for a binarycase and assuming the errors of the data to be random)

ln(

γ1

γ2

)− ln

(γ1,exp

γ2,exp

)= x1

(d ln γ1,exp

dx1

)+ x2

(d ln γ2,exp

dx1

)(2.20)

where the activity coefficient with the subscript exp are determined from the data di-rectly, whereas these without are obtained from the regression model such as Wilson[26] or NRTL [27].

If the data are consistent, the rhs of the above equation becomes small (consistentwith the Gibbs-Duhem equation). Further, the residuals from the lhs of the sameequation must be random.

To perform this, firstly activity coefficients for each data point in the set must becalculated, next fitted to the model and finally residuals calculated.

The weak point of the method is that for some thermodynamic models the resid-uals would be random and not random for others. This has to do with the adequacyof the used model. Therefore, Wisniak [15] suggested that this test could be also per-formed as a procedure for deciding which model is more adequate to fit VLE data.

8. Test of Kojima [28], [29]

This test is based on a combination of three tests: Point test or differential test ofthe Gibbs free energy (a), area test (b) and an infinite dilution test (c).

a) Point test (differential test)This test is based on the relations

δ∗k =

[d(GE/RT)

dx1− ln

γ1

γ2

]k= −

(hE

RT2

)(∂T∂x1

)p= ε (2.21)

and

δ =100 ∑N

k=1 δ∗kN

(2.22)

where δ∗k represents the deviation for individual experimental point k; δ repre-sents an overall deviation in percent and N number of experimental data points.

The value of ε cannot be neglected for isobaric data sets. One can avoid the com-plexity by not applying it to isobaric data sets. [29]. Firstly, the values of activitycoefficients for both components are calculated from the experimental T− p− x− ydata. Next, the calculated values of GE/RT are fitted by the use of a Padé approxi-mation for the activity coefficient

GE

RT= x1x2

a0 + ∑N=3n=1 an(x1 − x2)n

1 + b(x1 − x2)(2.23)

δ∗k is determined then from the slope of the Padé approximation and calculatedvalues of the ln γ1

γ2[29]. Kojima has established a criterion, that if δ < 5 the VLE data

set passes the test; otherwise, it fails.

b) Area test (explained earlier)

c) Infinite dilution testThis test considers the limiting behaviour of GE/(RTx1x2) and the activity coef-

ficients γ1 and γ2.The percent deviations in both limits are calculated from relations

I1 = 100∣∣∣∣GE/(x1x2RT)− ln(γ1/γ2)

ln(γ1/γ2)

∣∣∣∣x1=0

(2.24)

I1 = 100∣∣∣∣GE/(x1x2RT − ln(γ1/γ2

ln(γ1/γ2

∣∣∣∣x2=0

(2.25)

Kojima proposed that, if I1 < 30 and I2 < 30, the VLE data-set passes the test. Toobtain GE/(x1x2RT) and the activity coefficients, the Padé approximation given byEq. (2.23) is used.

The disadvantage of the Kojima’s test is certainly the complexity. Besides, it isbased partially on the diluted region, where the data are in general less accurate.Moreover, the heat of mixing in the wide range of temperature and composition re-quired in the area test is limited [15].

9. McDermott-Ellis test applicable to ternary systems [30]

This test is based on the isothermal-isobaric form of Gibbs-Duhem equation givenby Eq. (2.2). Integrating this equation directly along the loop of points a, b, ..., y, z bythe trapezoidal rule gives

∑componentsi=1

[ xib+xia2 (ln γib − ln γia) +

xib+xic2 (ln γic − ln γib) + ...

+xiy+xiz

2 (ln γiz − ln γiy)(2.26)

This method suggests that two neighbouring points (when ignoring the factor of12 in Eq. (2.26)) named a and b, are thermodynamically consistent if the deviation Dis lower than Dmax, where D is defined by

D =components

∑i=1

(xib + xia)(ln γib − ln γia) (2.27)

The authors proposed a certain global value of Dmax and later, Wisniak and Tamir[31] recommended to use the maximum value deviation evaluated based on the er-rors in measurements of the liquid composition, temperature and pressure as wellas coefficients of the Antoine equation describing the temperature dependence ofthe vapour pressure of the pure components. The test is easily applicable to ternarysystems.

It is worth mentioning, that test of Wisniak and McDermott-Ellis test are the onlytests used for ternary VLE data.

A detailed description of these consistency tests are given e.g. by Wisniak etal. [15], the same paper proposed a set of actions required in the thermodynamicvalidation of VLE data. These include the following

1. All papers delivering VLE data should contain experimental values obtaineddirectly from measurements (p, T, x, y).

2. The quality of the data should be checked by applying several consistencytests.

3. As pointed by Wisniak [15] and Marcilla et al. [32] the Herington test shouldbe avoided in data validation due to the explained earlier limitations.

2.1.2 NIST-TDE quality assessment algorithm for VLE data

A more general methodology for VLE data validation has been implemented as analgorithmic framework for the quality data assessment in the ThermoData Enginesoftware (TDE) developed at NIST (National Institute of Standards and Technology)by Frenkel et al. [33]. It is a part of the dynamic data evaluation concept, which


requires the development of the huge databases capable of storing all known exper-imental values. These databases with the combination of a data expert-system soft-ware allows an automatic, dynamic generation of the recommended values basedboth on the experimental and predicted results.

In particular, the procedure used there for the VLE data evaluation is based onthe subset of consistency tests, which determine the quality of a VLE data set on apass/fail basis by evaluating a single quality factor QVLE ranging from 0.1 to 1. Itmeans that the methodology does not reject any VLE data sets as inconsistent. In-stead it assigns a quality factor for each set of VLE data. This quality factor QVLE fora VLE data set introduced by Kang et al. [29] is calculated as a sum of the individualF factors formulated from five different consistency tests which form in overall theKang test.

QVLE = Ftest,pure(Ftest1 + Ftest2 + Ftest3 + Ftest4), QVLE ≤ 1 (2.28)

In this equation Ftest,pure is evaluated from the pure component test in which theconsistency between the limits on the VLE curve (mole fractions 0 or 1) and the purecomponent vapour pressures (saturation pressures) must be imposed. The rest ofFtesti factors are evaluated from: van Ness point-to-point (test 1), Herington (test 2),point or differential test (test 3) and infinite dilution test (test 4).

Each of Ftesti has a value between 0.025 and 0.25, resulting in the sum of the allfour factors to be in the range from 0.1 to 1. When all tests are passed the sum ofthem is equal to 1. If at least one of the tests fail, the sum is less than 1 with the lowerlimit equal to 0.1. [29]

Ftest1,max + Ftest2,max + Ftest3,max + Ftest4,max = 1 (2.29)

Regarding the pure component test, the quality factor associated with it is de-fined as

Fpure =2

100(∆p01 + ∆p0

2), 1 ≤ ∆p0

1, ∆p02 ≤ 10 (2.30)

where ∆p01 is the absolute relative (to pure component vapour pressure) error

between limiting value of the pressure and pure component vapour pressure.If the vapour pressures agree within 0.01∆p0 for both components, the pure com-

ponent factor is equal to 1 [29]. For higher deviations, the factor becomes smaller,resulting in the lower value of the overall factor QVLE.

Cross-check between VLE data and other types of data

A very convenient way to support the particular VLE data set is to use the auxiliaryinformation from other types of data such as infinite dilution activity coefficients(IDAC) and excess enthalpies (HE). Van Ness et al. [34] proposed methods for VLEdata quality assessment, which are based on the extrapolation of the excess Gibbsenergy at fixed temperature to the infinite dilution region. Next, if the experimen-tal IDAC for the considered system are available one can compare these with theextrapolated values obtained from VLE experimental data. The results of such acomparison might be used for a detection of highly inconsistent data, which wouldbe visible when the compared values do not agree even qualitatively. However, thiscross-check should be applied with caution due to large uncertainties in the extrap-olations. Definitely, it should not be the only approach used to validate a VLE dataset, because it might be often inconclusive.

2.2. Solid-liquid equilibria data validation 13

The other cross-check can be done between VLE data and excess enthalpies asan excess enthalpy data consistency check (Gibbs-Helmholtz test) Eq. (2.6). Sucha procedure was proposed by Olson [35]. In brief, the temperature dependence ofthe excess Gibbs energy (GE) should be the same as the excess enthalpy at a giventemperature. The excess enthalpy can be determined as a slope from the plot ofthe dimensionless Gibbs energy (GE/RT) vs. reciprocal of the temperature for aequimolar composition and mean temperature. A criterion for the consistent data-set was established as

0.5HEexperimental ≤ HE

VLE ≤ 1.5HEexperimental ; x1, x2 = 0.5 (2.31)

Again, this test can be used to support a consistency/inconsistency of the VLEdata set. Definitely, it should not be applied when there are small deviations fromideal behaviour in the considered system. In such cases, the test might be unreliabledue to the variations in the excess Gibbs energy larger than the absolute value of thecalculated Gibbs energy [36].

The cross-check methods are implemented as well in the TDE software devel-oped at NIST.

2.1.3 Conclusions

There are several options, which can be applied to the VLE data evaluation. As ithas been said in the book of Prausnitz, Lichtenthaler and de Azevedo [37]: “Theliterature is rich with articles on testing for thermodynamic consistency since it is somuch easier to test someone else’s data than to obtain one’s own in the laboratory.”Unfortunately, there is no the best method. Although almost all of them are basedon the Gibbs-Duhem equation, each of them give different results. Some of them areeven claimed to be wrong e.g. Herington test [22]. It seems that the combinationof few tests is the current trend, in which a single quality factor is assigned to thedata-set rather than a rejection of the inconsistent set. Certainly, one should notmake a final judgment on the quality of the data by performing only one test. It isalso important to note that, methods presented in this section are mainly applied tobinary systems.

2.2 Solid-liquid equilibria data validation

As far as there is a broad spectrum of methods for the validation of vapour-liquidequilibria data available in the literature/software, the validation methods for solu-bility systems have been less explored. Consistency tests based on the Gibbs-Duhemequation cannot be easily applied. In vapour-liquid equilibria, if the nature of thevapour phase is known then the corresponding activities in the liquid phase can beobtained. For solid-liquid equilibria there are no states where both component activ-ities can be obtained simultaneously. Furthermore, pressure has a negligible effecton phase diagrams, but there is a strong temperature dependence of the data. Inaddition, pure component melting point properties are less unambiguous than purecomponent vapour pressures and models used to describe non-idealities in VLE maynot be reliable for solubility of solids [17].


2.2.1 SLE consistency tests

One of the earliest attempts to develop a consistency test for the solid-liquid sol-ubility data was work by Null in 1965 [38]. The author has discussed a methodfor validation of experimental SLE data based on the Gibbs-Duhem equation. Sev-eral possible test equations between solid and liquid activity coefficients (based onthe van Laar type of equation) were presented and applied to metal/alloy systems,where the data from both liquid and solid phase are reported. That is not a case forthe eutectic solubility data-sets found in most chemical engineering applications,where only one solid component precipitates.

Ruckenstein and Shulgin test

Ruckenstein and Shulgin [39] proposed a consistency test for solubility data in theternary systems of drugs in aqueous solutions. The authors have developed a ther-modynamic consistency test based on the Gibbs-Duhem equation, which is similarto the McDermott–Ellis consistency test [30] for ternary VLE data (described earlier).

For an N-component system, McDermott and Ellis have obtained that

N

∑i=1

(x(c)i + x(d)i )(ln γ(d)i − ln γ

(c)i ) = 0 (2.32)

for a pair of experimental points named c and d.For a ternary mixture this equation has the following form

(x(c)1 + x(d)1 )(ln γ(d)1 − ln γ

(c)1 ) + (x(c)2 + x(d)2 )(ln γ

(d)2 − ln γ

(c)2 )

+(x(c)3 + x(d)3 )(ln γ(d)3 − ln γ

(c)3 ) = 0

(2.33)

If one considers a poorly soluble solid in binary mixed solvents Eq. (2.33) resultsin the form

(x(c)1 + x(d)1 )(ln γ(d)1 − ln γ

(c)1 ) + (x(c)2 + γ

(d)2 )(ln x(d)2 − ln γ

(c)2 )

+(x(c)3 + x(d)3 )(ln γ(d)3 − ln γ

(c)3 ) = 0

(2.34)

If the solubility in the mixed solvent is so low, one can assume that the activitycoefficients of the solvents (1) and (3) are equal to these in the solute-free binarysolvent (ln γ

′1 and ln γ

′3).

Knowing that xi = x′i(1− x2) for each solvent i and by further manipulations,

Eq. (2.34) becomes

D = (x′(c)1 x(c)2 + x

′(d)1 x(d)2 )(ln γ

′(d)1 − ln γ

′(c)1 ) + (x(c)2 + x(d)2 )(ln x(d)2 − ln x(c)2 )+

(x′(c)3 x(c)2 + x

′(d)3 x(d)2 )(ln γ

′(d)3 − ln γ

(c)′3 )

(2.35)where D is the local deviation and ’ means the solute-free molar fraction.According to this method, two points c and d are thermodynamically consistent

if a deviation D ≤ Dmax = 0.0001 (the procedure of finding its value is described inthe paper of Ruckenstein and Shulgin).

The suggested test is applicable to only ternary systems with very low solubil-ities as well as sets, in which the above two compositions of the mixed solvent are


close enough to each other (the intervals of 0.025 in mole fractions). This require-ment is not always a case, since in the substantial number of data-sets, there areonly few experimental points given and moreover, these are rarely evenly spaced.In addition, the suggested method of finding the maximum value of Dmax was donearbitrarily on the few selected data-sets for the solubilities of poorly soluble sub-stances in mixed solvents. The threshold value might be then easily changed, whendifferent solubility data (e.g. non-aqueous mixtures with higher solubility) are used.Moreover, some solutes can form different polymorphic forms in mixed solvents,which should be taken into account as an incorrect interpretation of data might leadto incorrect conclusions about the data consistency.

Cunico test

Solid-liquid equilibrium (SLE) data validation was explored by Cunico et al. [40],[17].They have developed two tests for checking the quality of the binary SLE data-sets.

Test 1 developed resembles the VLE pure component test available in the Ther-moData Engine program. It checks if the mixture data goes to the pure componentmelting points, when xi → 1, where xi is the mole fraction of the component i. Shehas proposed the quality factor QSLE,test1 associated with this test

QSLE,test1 =

(2

1000(∆t01 + ∆t0

2)−U

)(2.36)

where

∆t0i =

∣∣∣∣∣T0mi − t0

i

t0i

∣∣∣∣∣ (2.37)

andU = 0.1(θ1 + θ2) (2.38)

In Eqs. 2.36 to 2.38, T0mi is the melting point of the mixture extrapolated to the

limit xi → 1, t0i is the measured melting point of the i-th component and θi is the

uncertainty of t0i .

For the second test (named Test 2), Cunico et al. have developed additionally analternative activity coefficient model. They used an expansion of non-ideality aboutinfinite dilution based on Fluctuation Solution Theory (FST). This theory of solutionswas first introduced by Kirkwood and Buff in 1951 [41]. It was later expanded byO’Connell [42], to establish a form of the equations equivalent to the Porter modelfor unsymmetrical convention activity coefficients. This model is reliable to describesystems, where minimum deviation from rigor is to be expected i.e. close to relevantstandard states. It has a reliable composition dependence provided that the data areaccurate, which gives criteria for consistent data.

In the next step, a relation for the quality factor QSLE associated with the FST-based model has been developed

QSLE,test2 =

(1

1 + AAD(%)

)(2.39)

where AAD (%) is the deviation measure for the selected objective function usedin the regression of parameters.


Kang test

Kang et al. [36] have also addressed the consistency tests for binary SLE data. Theyhave proposed a consistency check of data which is implemented in the TDE soft-ware.

The first criterion utilizes the fact, that the limiting intercept of the equilibriumline (liquidus) should match the melting point of the pure component i (Eq. (2.41))(Fig. 2.2), and the second implies that experimental limiting slope is in the agreementwith the solubility equation (2.40) and (2.42)

xL1 =

1γL

1exp

[∆H f us,1

RTm,1

(1− Tm,1

T

)](2.40)

T → Tm,1 if x2 → 0 (2.41)

limx1→0

(dx1

dT

)(2.40)

= limx1→0

(dx1

dT

)(experimental)

(2.42)

where γL1 is the activity coefficient of the component 1 in the liquid phase, x1 is

the mole fraction, Tm,1 is the melting temperature and ∆H f us,1 is the enthalpy of fu-sion (melting enthalpy) of the solute 1. The experimental limiting slope is estimatedwith the SLE temperatures for the mole fractions ranges 0 to 0.2 and 0.8 to 1. Tocorrelate the experimental solubility, the following equation is used

ln x1 = A + B · T + C ln T, (2.43)

where A, B, C are fitted parameters.

FIGURE 2.2: Demonstration of consistency with the Kang’s tests forSLE for the binary system of benzene and 1,2-dibromoethane. Figureextracted from [36]. The green points should match with grey onesclose to pure component limits. The red line (slope) should be the

same as the slope of the experimental data (grey points).

Another consistency requirement is that:


(∂ ln γ1

∂x1

)T= 0 at x1 = 1 (2.44)

where the γ1 is the activity coefficient of the component 1 calculated by the com-mon mixture models (e.g. NRTL, UNIFAC).

Based on these tests, a single quality factor of the SLE data-set QSLE is proposedas an analogy to the test for VLE data-sets

QSLE = 0.25F1,pure + 0.25F2,pure + 0.25F1,slope + 0.25F2,slope (2.45)

where 1 and 2 is the component number and F is a function of two quantitiesdescribing: the difference between experimental melting point and evaluated fromthe mixture solubility data,

∆Tm,i = |Tm,exp − Tm,(2.40)| (2.46)

and the deviation between the slopes calculated based on the Eq. (2.40)

∆slopei =

∣∣∣∣∣ limx1→0

(dx1

dT

)(2.40)

− limx1→0

(dx1

dT

)(experimental)

∣∣∣∣∣ . (2.47)

The paper of Kang et al. presents the way how each of the four quantities ap-pearing in the Eq. (2.45) are evaluated

Fi,pure = 1 when ∆Tm,i ≤ 0.5K, andFi,pure = 0.5/∆Tm,i when ∆Tm,i > 0.5K

(2.48)

and

Fi,slope = 1 when ∆slopei ≤ 0.2, andFi,slope = 0.2/∆slopei when ∆slopei > 0.2

. (2.49)

2.2.2 Solubility data project

Another very ambitious and still on-going work in the field of data validation is thatinitiated by A. S. Kertes in 1978 in a project called Solubility Data Project (SDP) underIUPAC and later also NIST auspices [43]. The aim of the project can be summarizedas preparing a comprehensive critical compilation of data on solubilities of gases,liquids and solids from secondary sources and its critical evaluation. This includes:(i) quality of data after consideration of the purity of the materials, the experimentalmethod employed and the uncertainties in control of physical parameters, the re-producibility of data, the agreement of the experimenter’s results on accepted testsystems with standard values; (ii) recommended numerical data including standarddeviations and a set of smoothing equations derived from the experimental data.The main limitations of the methods used in this project are that they require anabundance of data to make a final judgement of the quality of the certain data-set.Moreover, the equations used to fit the data are usually generalized mathematicalforms (e.g. polynomials) [44],[45],[46] containing many constants/parameters onlyvalid for the certain type of solubility systems e.g. solubility of polycyclic aromatichydrocarbons in water. [45]

Up to now, 105 volumes of IUPAC-NIST Solubility Data Series have been issued[47].


2.2.3 Conclusions

Over the few past years, there have been several tests proposed for the validationof the SLE data. Some of them are more universal i.e. applicable to broad rangeof SLE data-sets and based on the general, more fundamental models, like tests im-plemented by Cunico or Kang. The general shortcomings of all of them would bean application mainly to binary cases, lack of the options to treat systems with apolymorphic transitions in the solid phase. Finally, other possibilities like statisti-cal methods for the outliers identification as well as critical assessment of the modelparameters should be addressed.

2.3 Liquid-liquid equilibria data validation

Liquid-liquid equilibrium (LLE) is a thermodynamic phenomenon of a great impor-tance used in a wide range of downstream separations and formulated products.The already existing huge databases and new data coming along at a significant rateonly confirms the importance of this property in chemical engineering. Althoughmany relevant data exists, there is a need for a comprehensive review of publisheddata, which results in reliable data. To make it harder, a thermodynamic data con-sistency tests based on the Gibbs-Duhem relation as for vapour-liquid equilibria arenot possible in the evaluation of the quality of binary LLE Tx data. A true thermo-dynamic consistency test results from the fact that a subset of the experimental vari-ables can be calculated from the other experimental variables [48]. For binary VLE,one may calculate the vapour mole fraction y1 from T, p and liquid mole fraction x1.If y1 is measured, it is possible to apply a consistency test. Unfortunately, the samemethod is not possible for binary LLE. At moderate conditions, the pressure has anegligible effect on mutual solubilites. Therefore, there is no basis for incorporatingpressure in the models describing non-idealities in the liquid phase. Moreover, thecorrelation and prediction of LLE using models for liquid phase nonidealities stillhas significant issues: different sets of parameters giving the same satisfactory fit,the fulfillment of only isoactivity condition, no guarantee of parameter consistencyin all the composition space, to just name few [49]. This becomes one of the reasonsfor lack of sufficient methods enabling critical evaluations of the reported data.

One of the earliest attempts to LLE data validation was the publication of theDECHEMA volumes on LLE by Sørensen and Arlt [48]. They somewhat have ad-dressed the issue of erroneous of published data and provided some criteria basedon which incorrect data were excluded from the compilations (e.g. in the situationwhen when the sum of concentrations in a given phase differed significantly from100 mol percent).

The substantial part of the Solubility Data Project, mentioned in the connectionto SLE, concerns of liquid-liquid solubility e.g. [50],[51]. The Tx LLE data are col-lected and if more than one set of data for the considered system is available, acritical evaluation is made, which results in the distinction between recommended,tentative, doubtful or rejected experimental data.

When each system is evaluated separately, the estimation of the data quality canbe difficult. Therefore, the current trend in data validation within SDP incorporatesinformation from many systems, which results in the concept of the reference data.The calculation of the reference data consists of usually two steps and it will beexplained on the example of the mutual solubility of hydrocarbons and water. [52].

First step of the standard procedure is the approximation of the solubilites ofhydrocarbons in water with a smoothing equation, which contains hydrocarbons

2.3. Liquid-liquid equilibria data validation 19

properties and empirical coefficients. In the second step, the LLE calculation is per-formed, which yields in the solubility of water in hydrocarbons. As an input forthese calculations, solubility of a hydrocarbon in water calculated by the smoothingequation in the first step is required. As a result, the solubilities of hydrocarbons inwater calculated by the smoothing equation and water solubility in hydrocarbonscalculated by solving the isofugacity criterion are assumed to be the reference data.

In the critical evaluation, it is assumed that the measured data are in a goodagreement with the reference data when the difference between them does not ex-ceed two to three times the standard deviation (expressed as the root mean squareddifference between experimental and calculated mole fractions). Data are considereddoubtful when experimental data deviate more than two to three times the standarddeviation from the reference data. If at least two experimental points from differentsources (at similar temperatures) agree with the reference data within the specifiedlimit, these points are recommended. If the experimental points for a given systemare measured by one laboratory, they are considered as tentative, even if they agreewith the reference data.

In the work of Kang et al. [36] it has been indicated that Gibbs-Duhem relationcannot be used to develop a consistency test, since for most of the systems there areno excess enthalpy data available in both liquid phases. Furthermore, Gibbs-Duhemequations do not connect compositions of the liquid phases being at the equilibrium.

FIGURE 2.3: Demonstration of the proposed by Diky composition-stretched scale LLE phase diagram in the aniline (1) + water (2) sys-tem. Figure borrowed from [53]. The stretching has revealed that one

of the data-sets is inconsistent.

The authors suggested that, the analysis of the LLE data-sets should be madeon the particular data scenario (e.g. abundance of experimental data points) ratherthan trough the generic consistency test. It is very often, that the analysis of thedata for a given system leads to automatic rejection of an inconsistent data-set orneed to be further investigated by the expert (evaluator). Nevertheless, Chirico etal. [54], Frenkel et al. [33] have developed a general algorithm for the automatizeddeletion of the inconsistent data set. The major drawback in this procedure is that theanalytical representation used for LLE modelling was a flexible polynomial functionT = f (xi), with no theoretical background.


A very helpful, in the data evaluation, way of representation of LLE and SLEdata in binary and ternary systems was developed by Diky [53]. He has proposedthat stretching of the phase diagram over the whole composition range by the trans-formation of mole fraction to logarithmic scale, might reveal very interesting detailsespecially in the regions of low solubility (Fig. 2.3). This could be a convenient toolfor assessing the data consistency.

2.4 Data collections

An important part of data validation is to have them collected in one place. Alsoto ensure that the full judgment of the data quality can be facilitated e.g. by a sim-ple comparison of different data sets for the same system. That is why all kindsof the data collections or compilations are of the utmost importance. Very often,they consist of available experimental data divided into recommended or doubtfulcategories, provide parameters for common correlation models e.g NRTL and un-certainties of the experimental data.

The most significant are DECHEMA Chemistry Data Series [55], which provideevaluated data, including fluid phase equilibria. So far, 15 volumes containing sev-eral parts were prepared.

Tremendous work has been initiated at NIST together with IUPAC to preparecritically evaluated data and reference data on the solubility of solids, liquids andgases. The VLE data are also considered in the IUPAC-NIST compilations and pub-lished in Journal of Chemical and Physical Reference Data e.g. [56].

2.5 Overall conclusions

The brief literature survey has revealed that the available methods used in the evalu-ation of data are still not sufficient. Many models used as an inherent part of the eval-uation methodologies are either specific for the certain type of systems, are empiricaland contain many constants/parameters. Some of the features, such as outliers andinfluence statistics, uncertainty analysis of the models and theirs parameters evalu-ation as well as more emphasis on the ternary systems (including SLE polymorphicsystems) were not thoroughly explored. It is also very important to note that thevalidation of LLE in multicomponent (in particular) ternary systems data has beenthe least explored type of data. Therefore, new methods supporting validation of theSLE and LLE data, which include these missing parts are focus points of this thesis.

21

Chapter 3

Activity coefficients fromFluctuation Solution Theory

The Fluctuation Solution Theory (FST) (known also as fluctuation theory of solutionsor Kirkwood-Buff theory) has its origins in statistical thermodynamics. It relatesderivatives of thermodynamic, macroscopic properties to the microscopic structureof liquids via integrals of molecular distribution functions or particle number fluc-tuations.

This chapter provides a brief coverage of some fundamental concepts of the fluc-tuation solution theory introduced by Kirkwood and Buff in 1951 [41] and later ex-panded by O’Connell in 1971 [42]. The expression for the unsymmetrical activitycoefficient derived from FST and used in this thesis for modelling of liquid-liquidequilibria and solid solubility is outlined. To learn more about the general conceptsand theoretical basis of FST as well as applications in the chemistry, biophysics andchemical engineering, readers of the thesis are referred to the book [57].

3.1 General relations

The Fluctuation Solution Theory is based on the grand canonical ensemble, in whichthe independent variables (maintained constant) are chemical potentials of the com-ponents i to N, {µi}i=1,...,N , volume V and temperature T. To apply then the FST oneneeds first to understand, how to relate the the fluctuations of the properties, de-fined as the instantaneous differences between the value of the property X and theirtime averages in the ensemble 〈X〉, to the macroscopic properties of an isothermal-isobaric system. This type of the system is the most experimentally studied one.

The statistical thermodynamics, in the grand canonical ensemble, gives the ex-pressions for the number of particles of each molecules and internal energy,

Ni =

(∂β pV∂βµi

){βµ}′,V,β

= 〈Ni〉 (3.1)

U = −(

∂β pV∂β

){βµ},V

= 〈Ei〉 (3.2)

where the β= (kBT)−1 and kB is the Boltzmann constant.Second derivatives of the number of particles and energy play a key role in the

FST. Of the utmost important is the derivative of the number of particles with respectto chemical potential, which involves particle fluctuations and is a crucial to themain applications of FST,

22 Chapter 3. Activity coefficients from Fluctuation Solution Theory

(∂〈Ni〉∂βµj

){βµ}′,V,β

= 〈δNiδNj〉 (3.3)

Derivatives of the chemical potential µ with respect to composition are a basis ofthe theory. The primary derivative is defined as

µij =

(∂βµi

xj

)p,T,Nk 6=j

(3.4)

Introducing a distribution/correlation function (to replace the number fluctua-tions), FST gives a connection between radial distribution function (abbreviated asRDF) and the chemical potential µ in the grand canonical ensemble (µVT).

For a binary system with subscripts indicating indices of involved species, thiscan be written as follows(

∂µ1

∂x1

)T,P,N2

= kBT(

1x1− c2(G11 + G22 − 2G12)

1 + x1c2(G11 + G22 − 2G12)

)(3.5)

where x1 is the mole fraction of the component 1, ci is the number density ofthe molecule type i, Gij is the integral of the radial distribution function taken fromgrand canonical ensemble, defined by

Gij =∫ ∞

0

(gµVT

ij (r)− 1)

r2dr (3.6)

where gµVTij is a grand canonical distribution function, r is the scalar interparticle

distance between centres of mass, r = |r2− r1|. This equation is a basis of the statisti-cal mechanical theory of solutions derived by Kirkwood and Buff. Later, O’Connellhas continued the development and has found very useful connection between thederivative of natural logarithm of activity coefficient γi with respect to the composi-tion xi and Total Correlation Function Integrals (TCFI) or Hij. In the binary systemthis is expressed by (

∂ ln γ1

∂x1

)T,P,N2

= −(

x2 f12

1 + x1x2 f12

)(3.7)

wheref12 = H11 + H22 − 2H12 (3.8)

Hij = ρGij (3.9)

where ρ stands for the total number density and fij is the difference in total corre-lation function integrals, which describes deviations from ideal correlation betweenpairs.

In general, the composition dependence of the elements in H matrix is unknownand usually complicated, which makes the integration impossible. O’Connell [42]has obtained a useful expression for calculation of the fij elements by expanding thevalues of the ln γi about the reference state.

3.2. Expansion method – the expression for the unsymmetrically normalizedactivity coefficient

23

3.2 Expansion method – the expression for the unsymmetri-cally normalized activity coefficient

For systems with low solubilities (dilute systems) i.e. if the solubility (in terms ofmolar fraction) xi ≤ 0.01, it is convenient to express the nonidealities in terms of theunsymmetrically normalized activity coefficients, γ∗i . The connection between thesymmetrical (also called Lewis-Randall) and unsymmetrical standard state is thefollowing

limxi→0

ln γi = ln (γi/γ∗i ) = − limxi→1

ln γ∗i with limxi→0

ln γ∗i = 0. (3.10)

In general, Taylor expansion of the activity coefficient about a reference state ofinfinite dilution is

ln γi = (ln γi)xi=0 + ni

(∂ ln γi

∂ni

)T,P,nj 6=i ,xi=0

+ni

2

2

(∂2 ln γi

∂ni2

)T,P,nj 6=i ,xi=0

+

+ni

3

6

(∂3 ln γi

∂ni3

)T,P,nj 6=i ,xi=0

(3.11)

The well-known relation between the activity coefficient and the chemical poten-tial is

µi (T, P, x) = µi (T, P, xi = 1) + RT ln xiγi (T, P, x) . (3.12)

By subtracting the first term on the right-hand side, and rearranging Eq. (3.10),Eq. (3.11) becomes,

ln γ∗i = xi

(n(

∂µi/RT∂ni

)− 1

xi+ 1)

T,P,nj 6=i ,xi=0+

x2i

2

(n(

∂2µi/RT∂n2

i

)+

+1x2

i− 1)T,P,nj 6=i ,xi=0 +

x3i

6

(n

(∂3µi/RT

∂n3i

)− 2

x3i+ 2

)T,P,nj 6=i ,xi=0

.(3.13)

By further manipulations the unsymmetrical activity coefficient becomes

ln γ∗i =

(xi −

xi2

2

)(n(

∂µi/RT∂ni

)T,P,nj 6=i

− 1xi

+ 1

)xi=0

+

+

(xi

2

2− xi

3

3

)(n2(

∂2µi/RT∂ni

2

)T,P,nj 6=i

+1

xi2 − 1

)xi=0

+ . . .

(3.14)

If the real system state (dilute solution) does not deviate strongly from the stan-dard state (usually less than 0.01 in terms of molar fraction), one can truncate theexpansion series in Eq. (3.14) after the first-order term. This results in the final ex-pression for the activity coefficient expressed in the unsymmetrical convention

ln γ∗i =

(xi −

xi2

2

)(∂ ln γi

∂xi

)T,P,nj 6=ixi=0

. (3.15)


The expression resembles the Porter model for the activity coefficient expressedin the unsymmetrical convention [37]. This means that dilute Porter model is rigor-ous in the first order expansion from FST at low concentrations of i.

When one considers the expression(

∂ ln γi∂xi

)at the infinite dilution, then Eq. (3.7)

becomes

limx1→0

(∂ ln γ1

∂x1

)T,P

= − limx1→0

(x2 f12

1 + x1x2 f12

)= − f12. (3.16)

Substituting this result in Eq. (3.15), we obtain

ln γ∗i = −(

xi −xi

2

2

)fij, (3.17)

which is the expression for the activity coefficient in the unsymmetrical conven-tion.

This composition dependence seems to be reliable to describe the system nonide-alities outside the critical region as long as the experimental data are accurate. Thus,reliable data at high dilution should adhere to Eq. (3.17). Moreover, when one dealswith diluted systems, the more appropriate choice is the unsymmetrical convention.This ensures, that components in the liquid phases are closer to their reference states(infinitely dilute solutions).

3.3 Model for the excess solubility in binary solvents basedon FST

In the next step, a useful relation for the derivative in Eq. (3.15) needs to be estab-lished. Combining Eq. (3.15) with the definition of the chemical potential yields(

∂ ln γi

∂xj

)T,P,nk 6=j

=1

RT

(∂µi

∂xj

)T,P,nk 6=j

− 1xi

δji − xi

1− xj. (3.18)

O’Connell [58] established connections between the derivative in Eq. (3.18) andintegrals of pair correlation functions.

Here, the final expression given by O’Connell in 1971 is presented. The fullderivation of the model can be also found in the PhD dissertation of Ellegaard [59].

(∂ ln γ1

∂x1

)T,P,n2,n3

=

x2x31−x1

f23 − (x2 f12 + x3 f13)− x2x3W1 + x1x2 f12 + x1x3 f13 + x2x3 f23 + x1x2x3W

. (3.19)

Unfortunately, W (which contains the multiple pair products of correlation func-tion integrals) cannot be factorized into f ′ijs, and therefore contains explicit values ofHij. Seeking simplicity, the expression has been limited to binary distribution func-tion integrals only. The triplet distribution function integrals were discarded. Asexplained by O’Connell [58] these are less significant since W is multiplied by twomole fractions. Moreover, it is unrealistic to approximate W within reasonable ac-curacy without obtaining all values of Hij. If one disregards W and consider onlybinary mixture (x3 = 0), the equation (3.19) turns into(

∂ ln γ1

∂x1

)T,P,n2

= − x2 f12

1 + x1x2 f12. (3.20)

3.3. Model for the excess solubility in binary solvents based on FST 25

Combining Eqs. (3.15) and (3.19) gives the activity coefficient of 1 in mixture of 2and 3 (disregarding W)

ln γ1∗ = −(

x1 − x12

2

){limx1→0

x2x31−x1

f23−(x2 f12+x3 f13)−x2x3W1+x1x2 f12+x1x3 f13+x2x3 f23+x1x2x3W

}≈ −

(x1 − x1

2

2

)x2x3 f+23−(x2 f+12+x3 f+13)

1+x1x2 f+12+x1x3 f13+x2x3 f+23

(3.21)

Here, superscript + is used to denote the infinite dilution of the component 1 inthe mixed solvent.

The excess solubility of the solute in the mixed solvent is defined by the followingexpression

sEi = ln xE

i = ln xi,m −solvents

∑j 6=i

x′j ln xi,j, (3.22)

where the x′j denotes the solute-free molar fraction of the solvent j and xi,m, xi,j is the

solubility of the solute i in the mixed solvent m and pure solvent j, respectively.If the solute is at infinite dilution, the excess solubility can be written as

sEi = lim

xi→1ln γ∗i,m −

solvents

∑j 6=i

x′j

(limxi→1

ln γ∗i,j

), (3.23)

provided that melting point properties are composition independent.Forming the difference in Eq. (3.23) to obtain the excess solubility yields

sE1 = lim

x1→1

{(x1 − x1

2

2

)x2x3 f+23−(x2 f+12+x3 f+13)

1+x1x2 f+12+x1x3 f13+x2x3 f+23

}−∑solvents

j 6=i x′j

(limxi→1

(x1 − x1

2

2

)f 01j

)= 1

2x2x3 f+23−(x2 f+12+x3 f+13)

1+x2x3 f+23− 1

2

[x2 f 0

12 + x3 f 013

],

(3.24)which combined with Eq. (3.20) for the solvent binary results in the general form

of the excess solubility model

sE1 = − x3

2

(∂ ln γ3

∂x3

)+

T,P,n2

− x2

2

[f+12

1 + x2x3 f+23− f 0

12

]− x3

2

[f+13

1 + x2x3 f+23− f 0

13

].

(3.25)The superscript 0 is used to denote the infinite dilution of the component 1 in

pure solvent 2 or 3.Unfortunately, f+ij changes with the composition of the binary solvent, which

makes it not practical to use. Molecular correlation functions are strong functions ofdensity [59]. Assuming that the solvent density variations are small (which is truefor most organic solvents mixtures), one can equate the mixture term f+ij with thepure solvent term f 0

ij.With this approximation, the final model has a form of

ln xE1 ≈ −

x3

2

(∂lnγ3

x3

)+

T,P,n2

[1 + x2 f 0

12 + x3 f 013]

. (3.26)


The derivative in Eq. (3.26) is independent of the solute (solvent-solvent term).It can be computed from binary vapour-liquid equilibrium data for the solvent mix-ture, when the certain model for Gibbs energy GE is assumed.

Ellegaard [60] has shown, by application to an extensive database, that the excesssolubility equation in this simplified form gives good accuracy. The model is theo-retically well-formulated in the region of infinite dilution of the solute if multi-bodycorrelation functions are ignored, but also can reliably describe solubility at higherconcentrations.

27

PART ILiquid-liquid

equilibria

29

Chapter 4

Binary liquid-liquid equilibriamodelling

This chapter is based on the paper:Ł. Ruszczynski, A. Zubov, J. P. O’Connell, J. Abildskov, Reliable Correlation forLiquid-Liquid Equilibria Outside the Critical Region in J. Chem. Eng. Data, 2017,62 (9), 2842–2854andŁ. Ruszczynski, A. Zubov, J. P. O’Connell, J. Abildskov, Reply to “Comment on‘Reliable Correlation for Liquid–Liquid Equilibria outside the Critical Region’” inJ. Chem. Eng. Data, 2017, 62 (11), 4043–4044.

4.1 Introduction

Chapter 2 shows that the correlation and prediction of LLE using models for liquidphase nonidealities still has significant issues. In addition to model inadequacies – aproblem which extensive research efforts have addressed – unreliable and conflict-ing data often prevent conclusions about system and model behavior. Sørensen andArlt [48] used NRTL to calculate liquid-liquid equilibria, since it gives reasonablygood results. Data validation requires a rigorous model - even though this may putlimitations on its range of validity. This new model should be nearly rigorous. Oneneeds to make sure that data with a poor fit are inaccurate. In this way regressioncan be used for discrimination of high-accuracy data.

Here, the initial steps for validating the temperature dependence of binary LLEbased on the expansion of nonideality about infinite dilution based on fluctuationsolution theory (FST) are reported. A new LLE model is formulated with the un-symmetric convention for normalizing activity coefficients of dilute species. A verysimple form of this model is compared to a benchmark model of LLE correlation,such as NRTL.

In Chapter 6, the extended form of the model is used for validation of binary LLETx data, together with a form for ternary LLE.

4.2 Models used in the modelling of LLE

Here is the outline of the modeling approaches of this chapter. Since the treatmentof the equilibrium relations may be unfamiliar, the relations are developed in somedetail; additional details can be found in the books by O’Connell/Haile [61] andPrausnitz et al. [37]

30 Chapter 4. Binary liquid-liquid equilibria modelling

4.2.1 FST-based model

Theory

The binary LLE problem (with two phases α and β) consists of solving a set of twoequilibrium equations, expressed as follows

f α1 (T, p, xα

1) = f β1

(T, p, xβ

1

)f α2 (T, p, xα

2) = f β2

(T, p, xβ

2

) (4.1)

The pressure p can be neglected, since it has negligible effect on liquids, exceptat hundreds of bars. In fact, normally pressure is not even reported within LLEdata-sets. It is common to employ pure component standard state fugacities for thecomponents in both phases α and β. Here different standard states for the compo-nents in the phases will be used. Thus, if phase α is rich in component 1 and phaseβ is rich in component 2, then the phase equilibrium relationships are the following

xα1γ1 (T, xα

1) f1 (T, x1 = 1) = xβ1

γ1

(T,xβ

1

)γ1(T,x1=0) limx1=0

f1(T,x1=0)x1

(1− xα1)

γ2(T,xα1)

γ2(T,x1=1) limx2=0f2(T,x2=0)

x2= (1− xβ

1 )γ2

(T, xβ

1

)f2 (T, x2 = 1)

(4.2)

The Poynting factor for pressure dependence, in these forms, can be omitted,since pressure is rarely of concern in liquid-liquid equilibria. These equations are tobe solved for two of the three variables: T, xα

1 , xβ1 . Introducing Henry’s law constants,

Hij(T) = limxj=1

fi (T, xi = 0)xi

(i 6= j) (4.3)

and taking logarithms gives:

ln [xα1γ1 (T, xα

1)] = ln

[xβ

1γ1

(T,xβ

1

)γ1(T,x1=0)

]+ ln H12(T)

f1(T,x1=1)

ln[(1− xα

1)γ2(T,xα

1)γ2(T,x1=1)

]+ ln H21(T)

f2(T,x2=1) = ln[(1− xβ

21)γ2

(T, xβ

1

)] (4.4)

Activity coefficients at infinite dilution are related to Henry’s law constants asfollows:

γ∞1 (T) = γ1 (T, x1 = 0) =

H12

f1 (T, x1 = 1)(4.5)

γ∞2 (T) = γ2 (T, x1 = 1) =

H21

f2 (T, x2 = 1)(4.6)

Introduction of the unsymmetrically normalized activity coefficients, γ∗i , gives:

ln [xα1γ1 (T, xα

1)] = ln[

xβ1 γ∗1

(T, xβ

1

)]+ ln γ∞

1 (T) (4.7)

ln [(1− xα2) γ∗2 (T, xα

1)] + ln γ∞2 (T) = ln

[(1− xβ

1 )γ2

(T, xβ

1

)](4.8)

FST provides the following expansion of the unsymmetrical activity coefficientfor the components in phase β (cf. (3.21))

4.2. Models used in the modelling of LLE 31

ln γ∗1

(T, xβ

1

)= −h0β

2 (T)[

2xβ1 −

(xβ

1

)2]− h0β

3 (T)[

32

(xβ

1

)2−(

xβ1

)3]− . . . (4.9)

ln γ2

(T, xβ

1

)= h0β

2 (T)(

xβ1

)2+ h0β

3 (T)(

xβ1

)3+ . . . (4.10)

This combination of expressions, employed for 1 and 2, satisfies the Gibbs-Duhemequation. In fact, the form employed for species 2 (4.10) is derived from the form em-ployed for species 1 (4.9), using the Gibbs/Duhem equation. These relations are alsoused for phase α where the component identities are switched. Here h0

2 and h03 are

related to integrals of infinite dilution molecular direct correlation functions.The infinite dilution activity coefficient is a function only of temperature and is usu-ally modelled with 2 parameters per phase, a and b, using a simple form such as

ln γ∞,β1 = aβ +

bβ

T, (4.11)

ln γ∞,α2 = aα +

bα

T, (4.12)

though more terms can be added if justified.For cases of very dilute phases (0.01 > xi) the composition effect can be ignored

(γ∗i ≈ 1). For dilute compositions (0.01 < xi < 0.10) the non-ideality should be small,so the expansion of eq. (4.9) can be truncated to one term (h0

3 . . . = 0) as long as thedata are remote from the critical region. Cunico et al. found that the temperaturedependence could be h0α

2 = − cα

T , and h0β2 = − cβ

T .

The modeling expressions are thus

ln γα1 = − cα

T(xα

2)2 = − cα

T(1− xα

1)2, (4.13)

ln γα∗2 = − cα

T

[(xα

2)2 − 2xα

2

]=

cα

T

[1− (xα

1)2]

, (4.14)

ln (Hα21/ f 0

2 ) = aα +bα

T, (4.15)

ln γβ∗1 =

cβ

T

[2xβ

1 − (xβ1 )

2]=

cβ

T

[1− (xβ

2 )2]

, (4.16)

ln γβ2 = − cβ

T(xβ

1 )2= − cβ

T(1− xβ

2 )2, (4.17)

ln (Hβ12/ f 0

1 ) = aβ +bβ

T. (4.18)

The final equilibrium relations for correlation of LLE in dilute phases are

ln xα1 − cα

T (1− xα1)

2 = ln xβ1 +

cβ

T

[2xβ

1 − (xβ1 )

2]+ aβ + bβ

T

ln (1− xα1) +

cα

T

[1− (xα

1)2]+ aα + bα

T = ln (1− xβ1 )−

cβ

T (xβ1 )

2(4.19)


The total number of parameters in here is six. The method will be to estimatethe significant parameters and then solve equation (4.19) to find compositions ofboth components in each phase. The Methodology section below fully describes astrategy for parameter estimation. Briefly, one systematically chooses values of cα

and cβ independently. Then with (cα, cβ) fixed, regress a and b, update (cα, cβ) untilthe minimum of an objective function (defined later) is found. The ranges of c-valuesto be explored can be predicted by a model such as COSMO-SAC as outlined later.

Parameter/property connection

Properties with a straightforward connection to the model parameters are the firstcomposition derivative of the natural logarithm of the activity coefficient

(d ln γ1

dx1

)T,P

,

which can be connected to the c-parameter expression in the model by:

2cβ

T=

(∂ ln γ

β1

∂x1

)∞

T,P

(4.20)

2cα

T=

(∂ ln γα

2∂x2

)∞

T,P(4.21)

Values of the parameter b can also be estimated using COSMO-SAC. It is propor-tional to the first temperature derivative of an infinite dilution activity coefficient,i.e. the partial molar excess enthalpy at infinite dilution,

bα ∼=[

d ln γ∞,α2

d (1/T)

]T

(4.22)

bβ ∼=[

d ln γ∞,β1

d (1/T)

]T

(4.23)

These relationships are only to formulate initial guesses for parameters usingpredictive models. Using (b, c) values from auxiliary data in the unsymmetric modelformulation will be addressed more completely in the Discussion section.

4.2.2 Non-random Two Liquids (NRTL)

One model widely used in the calculation or correlation of liquid-liquid equilib-ria is the NRTL model. It is an activity coefficient model developed by Renon andPrausnitz in 1968 [27] and is based on the hypothesis of Wilson that the local concen-trations around molecules are different from these in the bulk concentration. This isa theory of local compositions. This difference is caused by the difference betweeninteraction energy of the central molecule with the molecules of the same type, Uii,Ujj and that with the molecules of the other types Uij.

For a multicomponent mixture, the NRTL model has the following Gibbs excessenergy

gE

RT=

N

∑i=1

xi∑N

j=1 τjiGjixj

∑Nk=1 Gkixk

(4.24)

τij =∆gji

RT(4.25)


Gji = exp(−αjiτji

)(4.26)

where τji 6=τij and it was assumed that αji=αij = α=0.2.For a binary mixture the NRTL model is used on the following form

GE = x1x2

[∆g21 exp (−α∆g21/RT)

x1 + x2 exp (−α∆g21/RT)+

∆g12 exp (−α∆g12/RT)x2 + x1 exp (−α∆g12/RT)

](4.27)

Sørensen and Arlt [48] were quite explicit about the inadequacy of the temperaturedependence of the NRTL model with temperature independent parameters. In fact,they tabulated different NRTL parameters at different temperatures. Here tempera-ture dependent parameters will be used

∆g21 = g12 − g11 = a21 + b21T∆g12 = g12 − g22 = a12 + b12T (4.28)

This gives 4 parameters in total. These are : a12, a21, b12 and b21 in addition tothe non-randomness parameter, α. The first four are regressed, with α fixed at therecommended values.

The NRTL model was used to compare the performance of the developed un-symmetrical LLE model.

4.2.3 COSMO-SAC

Another activity coefficient model is COSMO-SAC (COnductor-like Screening MOdelfor Segment Activity Coeffcient) [8] based on the COSMO-RS model originally de-veloped by Klamt [9], where RS stands for Real Solvents. Contrary to the modelssuch as NRTL [27], Wilson [26] or Margules [62], COSMO-SAC can be parametrizedfor systems where no data exist. Thus, it is recognised that it represents data lessaccurately, but has more predictive power. The model arises from a combination ofthe statistical thermodynamics and quantum mechanics.

The molecular structure optimization is done using density functional theory(DFT) with the hybrid exchange-correlation functional B3LYP (Becke, 3-parameter,Lee-Yang-Parr) at the 6-311G (d,p) basis set. The quantum-chemical calculationsare conducted using Gaussian09 Revision D.01. Based on the equilibrium geometryof molecule, a quantum-chemical calculation (COSMO) is performed in the idealconductor to calculate the screening charge densities on the cavities of the molecules.The resulting screening charge densities (expressed in the so-called sigma profile)are then used, together with general interaction energy terms, in a model of pairwiseindependently interacting surface segments. These are then used to obtain activitycoefficients of all components in a mixture.

The screening charge densities from the .cosmo file output are averaged to give’apparent’ charge density on a standard surface segment. This is done by using thefollowing expression

σm =∑n σ∗n

r2nr2

e f f

r2n+r2

e f fexp(−3.57 d2

mnr2

n+r2e f f)

∑nr2

nr2e f f

r2n+r2

e f fexp(−3.57 d2

mnr2

n+r2e f f)

(4.29)


with rn =√

anπ and re f f =

√ae f f (=7.25)

π and ∗ denotes charge densities before averag-ing process.

To distinguish between segments which can form hydrogen-bond and that whichcannot, sigma profile is divided into 3 parts (with the help of the one additionalparameter σ0 = 0.007 e/Å

2). All universal parameters of the COSMO-SAC 2010

model [63] with its values are shown in table below.

TABLE 4.1: Parameters of the COSMO-SAC model (2010 version)

Parameter with unit Value

ae f f [Å2] 7.25

fdecay 3.57σ0[e/Å

2] 0.007

r[Å3] 66.69

q[Å2] 79.53

AES[(kcal ·Å4)/(mol · e2)] 6525.69

BES

[(kcal ·Å4 · K2)/(mol · e2)

]1.4859·108

cOH−OH([kcal ·Å4)/(mol · e2)] 4013.78

cOT−OT ([kcal ·Å4)/(mol · e2)] 932.31

cOH−OT ([kcal ·Å4)/(mol · e2)] 3016.43

The COSMO-SAC model computes the activity coefficient in the following way

lnγi/S =∆G?res

i/S − ∆G?resi/i

RT+ lnγC

i/S = lnγRi/S + lnγC

i/S (4.30)

Where, the combinatorial term is expressed by the well-known Stavermann-Guggenheim expression

ln γCi/S = ln

φi

xi+

z2

qilnθi

φi+ li −

φi

xi∑

jxjlj (4.31)

θi =xiqi

∑j xjqj(4.32)

φi =xiri

∑j xjrj(4.33)

li =z2[(ri − qi)− (ri − 1)] (4.34)

The sigma profile of the pure component i is a histogram showing the probabil-ity of finding a surface segment with the screening charge density σ. This can beexpressed as

pi (σ) =Ai(σ)

Ai(4.35)

where Ai is a total surface area of molecule i and Ai(σ) is the summation of thesurface areas of all segments with charge density σ.

The sigma profile for mixture is computed as follows

pS (σ) =∑i xi Ai pi (σ) |≡ Ai(σ)

∑i xi Ai(4.36)

where i is a i-th component and S stands for the solution (mixture).To sum up, the above equation is a summation of the sigma profile of each sub-

stance in the system weighted by its surface and molar fraction.The expression for segment activity coefficient (activity coefficient for surface

segment σm) is defined as

Γtj(σt

m)=

{hb,nhb

∑s

∑œn

psj (σ

sn) Γt

j (σsn) exp

[−∆W

(σt

m, σsn)

RT

]}−1

(4.37)

where the exchange energy between segment m and n is expressed by the follow-ing form

∆W(σt

m, σsn)=

(AES +

BES

T2

)(σt

m + σsn)

2 − chb(σt

m, σsn)(σt

m − σsn)

2 (4.38)

with j being either mixture or pure component and superscripts s and t mean eithernhb or hb segments; m and n denote m-th and n-th segment on the molecule and chbis defined by Eq. (4.39)

chb(σt

m, σsn)=

cOH−OH if s = t = OH and σt

m · σsn < 0

cOT−OT if s = t = OT and σtm · σs

n < 0cOH−OT if s = OH, t = OT and σt

m · σsn < 0

0 otherwise

(4.39)

For species i in the solution j the restoring free energy is given by the followingform

∆G?resi/j

RT=

Ai

ae f f

nhb,hb

∑s

∑σm

psi (œ

sm) lnΓs

j (sm) (4.40)

And the final expression for the residual part of the activity coefficient can beobtained from

lnγRi/S =

(∆G?res

i/S − ∆G?resi/i

RT

)=

Ai

ae f f

nhb,hb

∑s

∑σm

psi (œ

sm) ln(Γs

j (sm) (4.41)

The c parameters of the FST-based model are connected to the first derivative ofactivity coefficient with respect to the composition. The derivative has been calcu-lated numerically using central differences.(

∂ln γi/S

∂xi

)T,p,xi 6=j

≈(

∆ln γi/S

∆xi

)∆xi→0

(4.42)

In order to calculate the value of the c parameter in the FST-based model fromCOSMO-SAC, one needs to calculate a partial derivative of the activity coefficient atinfinite dilution i.e.


limxi→0

(∂ln γi/S

∂xi

)T,p,xi 6=j

=

(∂ln γi/S

∂xi

)∞

T,p

(4.43)

From the FST model we obtain(∂ln γi/S

∂xi

)∞

T,p=

2cT

, (4.44)

therefore, the final value of c parameter can be calculated as follows

c =T2

(∂lnγi/S

∂xi

)∞

T,p(4.45)

For the calculation of the initial guesses, the temperature for the computationwas chosen as an average temperature from experimental Tx LLE data-set.

Similarly, the COSMO-SAC model can be applied to calculate value of b. Inthe FST-based model, it has been assumed that infinite dilution activity coefficient(IDAC) is modelled as

ln γ∞i = a +

bT

(4.46)

The value of the first derivative of IDAC over reciprocal of T is equal to b param-eter, which can be written as

∂ln γ∞i

∂1/T= −T2 ∂ln γ∞

i∂T

=HE,∞

iR

= b (4.47)

Note that b is closely related to the partial molar enthalpy at infinite dilutionHE,∞

i .

4.3 Model uncertainty analysis

In order to calculate the 95% confidence intervals of estimated parameters, the co-variance matrix COV(θ) of the parameter estimates is used [64]

COV(θ) =ssed f

(J (θ)T · J (θ)

)−1(4.48)

Here sse is the value of the objective function. The degrees of freedom calculatedas the difference between the number of experimental data points and number ofestimated parameters is identified as df, while J is the Jacobian matrix (or local sen-sitivity matrix). The confidence interval of the parameters vector at αt significancelevel is given as:

θ1−α,t = θ ±√

diag (COV(θ)) · t(d f , αt/

2) (4.49)

Here, t(d f , αt/

2)is the t-distribution value corresponding to the αt/2 percentilewith df degrees of freedom. The pairwise correlation between parameters is givenas follows:

COR(θk, θl) =COV(θk, θl)√

σ2θk

σ2θl

(4.50)

4.4. Methodology 37

To estimate the uncertainty of predicted compositions expressed in molar frac-tions by the model, the covariance matrix of the parameters as well as the sensitivitymatrix of the model are used. To calculate 95% confidence intervals of the predictedcompositions the following equation is used:

xi,1−α,t = xi ±√

diag(

J (θ) · COV(θ) · J (θ)T)· t(d f , αt

/2) (4.51)

Fig. 4.1 shows an example of uncertainty plots for the toluene/water binary. The fitto the data is good with the 95% confidence limits of similar widths. Note that thedata point as x1 = 0.99 seems inconsistent with the others.

FIGURE 4.1: Liquid-liquid equilibria in toluene (1) with water (2);results of uncertainty analysis. Note the confidence intervals are sim-ilar in both phases, but scaling of the axes is different. The error barsshow experimental uncertainty in the molar fraction. Lower bound

of confidence interval has negative values.

4.4 Methodology

The following steps are involved in data correlation using the FST-based model de-scribed above as summarized in Fig. 4.2 .


FIGURE 4.2: Methodology workflow involved in data correlation.

4.4.1 Graphical analysis of experimental data

For the binary LLE, experimental data are usually binodal compositions in bothphases (e.g. expressed most often in molar fractions) as a function of temperature.If the component is nearly pure in one phase and dilute in the other phase, equation(4.19) reduces to

ln xα1γα

1 ≈ 0 = ln xβ1 +

(aβ + bβ

T

)ln xβ

2 γβ2 ≈ 0 = ln xα

2 +(

aα + bα

T

) (4.52)

With this approximation, lnxi will be a linear function of inverse temperature.This is often seen in practice, even at moderate mole fractions. An example is shownin Fig. 4.3 , where toluene/water data for the dilute components have been fitted toestimate the (a,b) parameters (in both phases)

ln xβ1 = −

(aβ + bβ

T

)ln xα

2 = −(

aα + bα

T

) (4.53)

The values obtained in this way can provide initial estimates for further regressions.

4.4. Methodology 39

FIGURE 4.3: ln xi as a function of inverse temperature data for thetoluene (1) / water (2) system

4.4.2 Regression of aα, aβ, bα, bβ with fixed values of cα and cβ

This step regresses the (a,b) parameters in both phases, with different combinationsof fixed cα and cβ parameters. With cα and cβ fixed at a given set of values, we min-imize an objective function defined as the sum of the squared differences betweenthe experimental and calculated phase mole fractions:

obj = minN

∑i=1

β

∑ω=α

(xω,expi − xω,calc

i )2

(4.54)

The objective function value, of course, depends on the values of cα and cβ. Min-imizing obj for different combinations of fixed cα and cβ allows us to plot in twodimensions obj as a function of cα and cβ. Such an objective function surface plot, asa function of some range of cα and cβ values, will yield a minimum that locates thepair of c parameters (cα and cβ) that produces the best agreement with experimentaldata.Fig.4.4 shows an example of this procedure for the toluene/water system. In Fig.4.4a, the ranges of c-values explored are:

− 106 < cα < 106 , −104 < cβ < 104 (4.55)

The colors indicate the calculated objective function with blue being smallest. COSMO-SAC predicts the values (as indicated by the green squared symbol)

cα ∼ −0.2 · 106 , cβ ∼ −0.75 · 104 (4.56)

However, the minimization of the (a,b) parameters gives the lowest objective func-tion with c values that cannot be distinguished from zero on this plot. Fig.4.4b showsobjective function values for narrower ranges of c-values

− 102 < cα < 102 , −102 < cβ < 102 (4.57)

Since the precise optimal c-values also cannot be determined with the scale of thisplot, zooming closer (Fig. 4.4c) finally shows the optimum parameters are about

cα ∼ −0.4 , cβ ∼ 1 (4.58)


(A)

(B)

(C)

FIGURE 4.4: (A) Contour map of objective function with varying val-ues of cα and cβ parameters for toluene (1)/water (2). Ranges of bothc parameters are provided by COSMO-SAC (green square). Red dotindicates minimum of objective function.(B) Zoom on contour mapof objective function with varying values of cα and cβ parameters fortoluene (1)/water (2). Red dot indicates minimum of objective func-tion. (C) Further zoom on contour map of objective function withvarying values of cα and cβ parameters for toluene (1) with water (2).

Red dot indicates the minimum of objective function.

4.4. Methodology 41

TABLE 4.2: Sample of parameter initial guesses and optimized values(in parentheses)

System (1)/(2) aα bα aβ bβ

toluene/water -3.415 (-3.340) 2775.9 (2784.8) 5.530 (5.028) 1068.8 (1079.9)n-hexane/water -5.248 (-7.040) 3802.7 (4322.3) 16.888 (17.144) -1172.7 (-1248.8)

[omim][BF4] / water -1.348 (-1.258) 481.66 (501.1) -18.939 (-11.15) 4027.5 (4717.2)n-hexane/nitroethane -13.602 (-19.553) 4467.9 (6140) -13.462 (-18.939) 4392.0 (5931.7)

These are nearly zero (and indeed small compared to the COSMO-SAC predic-tions) and the minimum is quite flat. This suggests that the c-parameters are not veryimportant to the representation of the toluene/water binary. This is not unexpected,since the mutual solubilities are very low. If, in addition to regression of 4 (a,b) pa-rameters - with the c’s fixed, regression by mathematical optimization is done forall 6 parameters (a,b,c), then the confidence intervals for the c-parameters from thetwo analyses can be compared, to evaluate if the c-parameters can be determinedfrom a given data set. For the present toluene/water set, the c-parameters cannotbe identified in this way, since (as shown later) the parameter confidence intervalsgreatly exceed the parameter values found by optimization. Thus 4 parameters canbe identified for this system.

Furthermore, liquid-liquid equilibrium systems with strong positive deviationfrom Raoult’s law should in general not produce positive c-values. The above merelyshows the results of an optimization study. In the following we have put in paren-theses the cases where c is positive by optimization. For low concentrations, the in-fluence of c is not very strong, and its impact on the objective function is negligible.This is also the basis for our method of determining c (Fig.4.2). When the contri-bution from the nonideality term is not significant, or c has nonphysical values, cshould not be used.

The methodology described in this section was later refined. The modificationincludes the simultaneous regression of all six parameters, combined with the ini-tial guesses found from the graphical representation of experimental data and con-straints imposed on c parameters to ensure the phase stability. All the details of itare described in Chapter 6.

4.4.3 Stability analysis

The proposed model has in total six parameters (named a, b and c) when applied tobinary systems (3 for each coexisting phases). So far, the methodology was to regressonly 4 (with initial guesses taken from experimental data) and c was modifying inranges until the minimum of objective function was found.

The regression of all of six parameters (as shown later) gives very uncertain c,especially when a miscibility gap for considered system is very wide.

Therefore, ways to determine a more realistic order of magnitude for the c pa-rameter have been addressed. This was done by providing reliable constraints inwhich c should be regressed. As suggested by Glass and Mitsos [65], if the Gibbsfree energy model of each liquid phase is partially convex and minGα, Gβ is locallynon-convex with respect to mole fraction, isofugacity is both necessary and sufficientcriterion for thermodynamic stability.

Having expressions for activity coefficient for both components in both phases,one can express Gibbs free energy of each phase Gα and Gβ and based on the stabilitycriterion derive constraints to be enforced on c parameter


∂2Gα,β

∂(xα,β

)2 =2cα,β

(xα,β −

(xα,β)2

)+ T

Txα,β(1− xα,β)> 0, (4.59)

where Gα,β stands for Gα or Gβ.After some manipulations, the constraints on the c parameters are obtained

xα,β(

1− xα,β) [

2cα,βTxα,β(

1− xα,β)+ T2

]> 0 (4.60)

[xα,β

(1− xα,β

)] (2cα,βT + T2

)> 0 (4.61)

T(

2cα,β + T) [

xα,β(

1− xα,β)]

> 0 (4.62)

cα,β > − T2xα,β

(1− xα,β

) . (4.63)

In order to guarantee favourable convexity properties of Gibbs free energy forminimization, Glass and Mitsos [65] have proposed to limit cα,β to be not less than−2T (c.a. 600), since into the expression above they substitute xα,β = 0.5. This appliesto systems where mole fraction covers the whole range from [0, 1], but in fact forsystems where mole fractions are close to pure component limits we can expect thatrealistic values of c are more negative. Therefore we should investigate each systemseparately and then realistic value of cα,β is calculated by exact Eq. (4.63) with xα,β

equal to the 110% of maximum value of the solubility in each phase (xα,β1,lim).

Most of the considered systems in this work are in fact systems with wide mis-cibility gaps. Therefore the lower bound that ensures the split is less than −2T. Asample of cα,β values calculated from Eq. (4.63) is shown in the Table 4.3.

TABLE 4.3: Sample of the lowest values of c-s

System xα1,lim cα xβ

1,lim cβ

toluene/water 0.985 -1.3·104 2.5·10−4 -7.18·105

n-heptane/water 0.998 -8.2 ·104 5.6·10−7 -2.9·108

n-hexane/nitroethane 0.699 -714 0.33 -680octan-1-ol/water 0.715 -793 7.9·10−5 -2.05·106

[hmim][BF4]/water 0.137 -1410 0.021 -8100

The upper bound should be set to 0 as it was pointed out by Ruszczynski et al.[66] Mole fraction derivatives of activity coefficient are negative, since LLE exists insystems characterized by highly positive deviations from ideality.

4.4. Methodology 43

TAB

LE

4.4:

Esti

mat

edpa

ram

eter

sfo

ral

lcon

side

red

syst

ems.

Inal

ltab

les

phas

eα

isri

chin

com

pone

nt1

and

βri

chin

com

pone

nt2.

Num

bers

inbo

ldar

eex

pect

edto

besi

mila

rfo

rw

ater

inal

kane

san

dni

troe

than

ein

alka

nes.

Num

bers

for

cαan

dcβ

inpa

rent

hese

sar

epo

siti

ve,w

hich

isin

cons

iste

ntw

ith

phas

est

abili

ty,a

sdi

scus

sed

belo

wal

ong

wit

hun

cert

aint

ies

inth

ese

para

met

ers.

Syst

em(1

)/(2

)aα

bαaβ

bβcα

cβ

tolu

ene/

wat

er-3

.340±

0.00

227

84.8±

0.5

5.02

8±

0.00

210

79.9±

0.5

-0.4

(1)

n-pe

ntan

e/w

ater

-10.

82±

0.00

0255

14.3±

0.1

15.3

082±

0.00

02-1

152±

0.1

00

n-he

xane

/wat

er-7

.040±

0.00

243

22.3±

0.4

17.1

44±

0.00

2-1

248.

8±

0.4

(0.1

)-0

.05

n-he

ptan

e/w

ater

-4.7

643±

0.00

2335

92.8±

0.7

17.1

00±

0.00

23-7

86.2

7±

0.69

00

n-oc

tane

/wat

er-5

.149±

0.00

137

37±

0.3

22.9

5±

0.00

1-2

032±

0.3

00

[hm

im][

BF4]

/wat

er-1

.241±

0.02

346

3.1±

7.1

-14.

55±

0.02

5525

.9±

7.1

-4.7

-0.1

[om

im][

BF4]

/wat

er-1

.258±

0.03

050

1.1±

7.4

-11.

15±

0.03

4717

.7±

7.4

-11.

30

n-he

xane

/nit

roet

hane

-19.

553±

0.17

761

40±

51.9

-18.

939±

0.17

759

31.7±

51.9

(2.5

)-8

.2n-

octa

ne/

nitr

oeth

ane

-17.

039±

0.15

955

77.6±

46.4

-13.

799±

0.15

945

61.2±

46.4

(60)

-40

2,2,

4-tr

imet

hylp

enta

ne/

nitr

oeth

ane

-18.

659±

0.20

658

95.8±

59.9

-12.

501±

0.20

640

20.9±

59.9

(10)

-5n-

deca

ne/

nitr

oeth

ane

-8.4

89±

0.14

131

52.3±

41.3

-9.4

86±

0.14

134

44.1±

41.3

(11)

-8


4.5 Results for different classes of LLE binary systems

The developed model has been applied to 88 different binary systems of differenttypes, exhibiting different widths of the miscibility gap. All considered systems aredescribed in Chapter 6.

In this chapter three types of systems (with different widths of the miscibilitygap) have been chosen for testing this model with experimental data. Except forsystems of hydrocarbons and water, all data were taken from a single work. Incase of hydrocarbons plus water systems, data were taken from different researchgroups, mainly from the compilations of data of Maczynski and Shaw [67] who dis-tinguished among reliable, tentative and doubtful data. Here only data marked asrecommended and tentative were used.

4.5.1 Hydrocarbon + water systems

The first examples consist of systems with very wide miscibility gaps. Such LLEphase diagrams appear for water and a non-polar organic compound with a longalkyl chain such as hydrocarbons. Since this case deals with very dilute phases(mole fractions less than 0.01) one may expect that there will be two parametersper coexisting phase (4 in total), though six parameters are regressed. Examples inthis category cover toluene + water and n-alkanes (C5-C8) + water [68], [52], [69].

Optimized parameters along with their uncertainty calculated using equation(4.49) for all systems are shown in Table 4.4 (row 1-5). We follow the strategy ofFig.4.2 by regressing four parameters (a and b for both phases) with fixed values oftwo c parameters and initial estimates of a and b from a graph of the experimentaldata as in Fig. 4.3. Table 4.2 compares optimized values with initial guesses fromgraphical analysis. As can be seen, agreement is quite good in all cases. Then thevalues of (cα, cβ) are varied systematically, and objective function contour plots asin Fig. 4.4 for water/toluene are prepared. Very shallow minima are found in thevicinity of (cα, cβ)∼ (0, 0). Thus, each phase can be described with only 2 parametersand the c–parameters are irrelevant.

FIGURE 4.5: Liquid-liquid equilibrium in n-hexane (1) with water (2)including uncertainty analysis.

Simultaneous regression of all six parameters was also done for each system. Aset of values is listed in Table 4.8 for the toluene, n-hexane and n-heptane systems.It turns out that both c parameters have very wide confidence intervals indicatingagain that composition effects play an insignificant role, and activity coefficients are

4.5. Results for different classes of LLE binary systems 45

equivalent to infinite dilution activity coefficients. Fig. 4.5 presents the experimentaldata (water/n-hexane) and graphs correlated with the model and confidence inter-vals. Most of the points are included in the 95% confidence interval calculated byEq. (4.51).

FIGURE 4.6: Liquid-liquid equilibrium in [hmim][BF4] (1) with water(2) including uncertainty analysis.

4.5.2 Ionic liquids + water systems

Systems with ionic liquids (ILs) have received much attention in recent years, es-pecially those with water where there may be significant miscibility [70],[71]. Twosystems with imidazolium-based ionic liquids, namely hexyl-3-methylimidazoliumtetrafluoroborate ([hmim][BF4]) and octyl-3-methylimidazolium tetrafluoroborate([omim][BF4]), were studied [72]. The miscibility gap is not so wide and the varia-tions of composition with temperature are great, so the c-parameter for compositioneffects in the model can be expected to be more important.


FIGURE 4.7: Contour plots for [hmim][BF4] with water; red pointindicates minimum of the objective function; green square - COSMO-SAC prediction. Upper figure: Full range of c provided by COSMO-

SAC model; Lower figure: zoom for smaller range.

The COSMO-SAC estimates gave very large values, with cα � 0 and cβ � 0. Onthe scale of the upper part of Fig. 4.7, the objective function minimum is near (0,0).Zooming in, the minimum is actually (cα, cβ) ∼ (-5,0). This happens because the βphase is dilute, though the α phase is not. Of importance is that all data points arewithin the 95% confidence interval (Fig. 4.6).

Independent regression for all six parameters was conducted for the [hmim][BF4]system (Table 4.8). It turns out that both c parameters have non-zero values andquite wide confidence intervals, though smaller than for the hydrocarbon systems.This confirms that the nonideality term has more effect here than in the hydrocarbonsystems, but it is still not very significant.

4.5.3 Hydrocarbon + nitroethane systems

The case studies above had at least one of the phases being dilute. Systems with rela-tively wide miscibility gap and strong dependence of temperature and compositionare now considered. Four systems containing nitroethane plus an alkane (n-hexane;n-octane; 2,2,4-trimethylpentane; n-decane) were selected [73].


FIGURE 4.8: Liquid-liquid equilibrium of hydrocarbon (1)/ni-troethane (2) systems including uncertainty analysis: n- octane

As seen from Fig. 4.8, the alkane solubilities in nitroethane vary from about0.05 to 0.3 in mole fraction. Similarly, the nitroethane solubilities vary from about0.35 to 0.05 in n-octane and n-decane. For all the systems, an upper critical solutiontemperature was observed. While the model can produce an upper critical solutiontemperature, it will not produce non-classical behavior near the critical region. Wehave not tested whether the assumed temperature dependence will be reliable withthe current formulation. Thus, since the model is limited in its development to non-critical situations, the 2-4 points at the highest temperatures were excluded from theanalysis.

In the n-alkanes + water systems the b-value is near 4000 in three alkanes, but is5514.3 in n-pentane. b parameters obtained for nitroethane are similar (near 6000) inthe three shorter alkanes, but not in n-decane. This may be useful to discriminatecases, as we will discuss later. COSMO-SAC predicted negative c parameters for allsystems, but the fitted c values had negative signs only in the β phase (rich in ni-troethane). The contour plots shows that the objective function minimum is locatedaway from (0,0) and most data points are within the 95% confidence interval (Fig.4.8 and 4.9). Independent regression for all six parameters was also conducted forall n-alkane systems (Table 4.8). The nonideality term has more effect than in thehydrocarbon systems and it is slightly more significant than for the aqueous ionicliquids system.

FIGURE 4.9: Liquid-liquid equilibrium of hydrocarbon (1)/ni-troethane (2) systems including uncertainty analysis: n- decane


4.5.4 Comparison of the Unsymmetric model with NRTL

The performance of the unsymmetric model for LLE correlation can be compared tothe NRTL model. All estimated NRTL parameters are listed in Table 4.5, and a set ofcorrelation results for water + [hmim][BF4], n-octane + nitroethane and n-hexane +water are shown in Fig. 4.10c.

Table 4.6 compares results obtained with both methods. As can be seen there arecases where the models are very similar in AARD (less than 1.5 % difference), andcases where one or the other model produces the lower value, with a tendency tobetter overall performance of NRTL. Also, the prediction intervals are similar for thetwo methods.

It has been chosen to seek simplicity in the formulation of the temperature de-pendence of infinite dilution activity coefficients. Analysis of the infinite dilutionexpressions of NRTL and the unsymmetric model, shows that the NRTL form withtemperature dependent parameters has both the zero order and inverse temperatureterms that the unsymmetric model has as well as terms with exponential functions:

ln γ∞1 =

a21

RT+

b21

R+

a12 + b12

RTexp

(−α

a12 + b12

RT

), (4.64)

ln γ∞2 =

a12

RT+

b12

R+

a21 + b21

RTexp

(−α

a21 + b21

RT

). (4.65)

Thus, NRTL has more terms. However, the NRTL form must use the same pa-rameters in both phases, so it is not obvious which model will provide the better fit.At this point, NRTL does perform well for most cases investigated.

A deeper analysis of this issue would involve simultaneous incorporation of tem-perature derivatives, such as partial molar excess enthalpies and heat capacities (atinfinite dilution) for the respective phases. The current FST form gives a partial mo-lar heat capacity at infinite dilution of zero, whereas NRTL allows it to be non-zero,though there is no guarantee that the NRTL will be closer to the experimental excesspartial molar heat capacity than our value of zero.

One important point of difference between the two models is that, parameters inthe unsymmetrical model are related to other thermodynamic properties, i.e. theirvalues can be compared to measured data and/or be interpreted in terms of quanti-ties that are well-defined molecular physics concepts.


(A) hmimBF4/water

(B) n-octane/nitroethane

(C) n-hexane/water

FIGURE 4.10: Sample of liquid-liquid equilibrium correlation by un-symmetric model (with confidence intervals) and NRTL in the sys-tems [hmim][BF4]/water (top), octane/nitroethane (middle) and n-

hexane/water.


TABLE 4.5: Estimated parameters for NRTL model (α = 0.2) for allconsidered systems.

System (1)/(2) a12 /J·mol−1 a21 /J·mol−1 b12 /J·K−1·mol−1 b21 /J·K−1 ·mol−1

toluene / water 21722.3 (±2973.1) 3310.97 (±279.71) -37.855 (±9.125) 48.697 (±8.637)n-pentane / water 44538.7 (±8192.0) -6039.78 (±538.99) -97.015(±28.437) 100.704(±18.872)n-hexane / water 34877.8(±4546.74) -4092.98(±1402.39) -65.676(±26.718) 106.811 (±7.305)n-heptane / water 28728.9(±12298.7) 2496.62 (±989.89) -46.250(±40.420) 113.682(±32.518)n-octane / water 35820.5 (±1680.6) -3024.11 (±513.86) -65.323(±97.900) 129.695(±29.933)

[hmim][BF4] / water 12629.1 (±1026.4) 4517.63 (±386.8) -63.499 (±3.329) 49.864 (±1.197)[omim][BF4] / water 14934.6 (±1057.5) 16316.3 (±335.6) -70.318 (±3.726) 17.289 (±0.939)

n-hexane / nitroethane 13244.9 (±2858.1) 15499.6 (±2665.9) -32.908 (±9.639) -42.699 (±8.995)n-octane / nitroethane 17549.8 (±1483.4) 7345.34 (±1323.8) -45.489 (±4.935) -14.240 (±4.429)

2,2,4-trimethylpentane/nitroethane 20568.4 (±2722.9) 5002.12 (±222.03) -56.436 (±9.217) -7.2483 (±0.7545)n-decane / nitroethane 13568.6 (±212.8) 13295.9 (±2051.6) -31.446 (±6.939 -31.356 (±6.700

TABLE 4.6: Comparison of average absolute relative deviation(AARD) in mole fraction using unsymmetric formulation (tempera-ture dependence with 2 or 3 parameters) and NRTL. + denotes ref-erence temperature Tre f equal to 298.15 K, in other cases 308.15 was

used.

System (1)/(2) AARD*/%Unsymmetric model 2p NRTL Unsymmetric model 3p

Toluene / water 4.19 5.71 1.31+

n-pentane / water 3.98 4.07 2.90+

n-hexane / water 4.60 3.20 4.15+

n-heptane / water 6.97 9.83 3.13+

n-octane / water 8.48 1.16 5.23+

[hmim][BF4] / water 7.00 3.54 2.11[omim][BF4] / water 22.69 24.49 21.22n-hexane / nitroethane 4.31 2.97 2.45n-octane / nitroethane 5.43 1.51 2.032,2,4-trimethylpentane / nitroethane 3.37 0.84 1.29n-decane / nitroethane 5.99 5.25 4.91

∗AARD = 100N ∑N

i=1

∣∣∣∣ xcalci −xexp

ixexp

i

∣∣∣∣For example, a is connected to the partial molar excess entropy at infinite dilu-

tion, b to the partial molar excess enthalpy at infinite dilution, and c to the composi-tion derivative of the activity coefficient (connected to correlation function integrals).Parameters in the NRTL model are not explicitly connected to other properties. Thusvalidation of their values is not easily assessed by simple comparison with othermeasured thermodynamic properties.

Improvement of the FST form can be obtained by extending a temperature de-pendence. That will retain the rigor in form for data outside the critical region, ifrequired.

4.5.5 Discussion

Models based on theoretical concepts such as molecular thermodynamics allow bet-ter determination and justification of parameter values, both for establishing initialvalues for regression and assessing final values for extension. Further, the expectedreliability of such models would allow for implementation in validating measureddata.

Here, initial parameter estimates for temperature dependence and dilute solu-tion non-ideality have been made with COSMO-SAC. The estimates have proven to


be only qualitatively reliable. Additional databases have been analysed to examinethis situation further.

First, COSMO-SAC calculations of infinite dilution partial molar excess enthalpieswere made for 72 binary systems from Sherman et al. [74] and other literature values[75], [76]. The mixtures covered are mostly mixtures of organic species - with andwithout LLE, including data on water/toluene and nitroalkane/n-alkanes. Fig. 4.11shows the results where data are in the range from -5 to 25 kJ/mol. The sign is cor-rect in the majority of cases and only a few of the discrepancies are larger than threekJ/mol.

FST gives solution non-ideality as a MacLaurin series in mole fraction [42]. The cparameter is proportional to the infinite dilution derivative of the activity coefficient,

h0β2 = − cβ

T= −1

2limx1→0

[∂ ln γ

β1

∂xβ1

]T,P

(4.66)

The behaviour of these derivatives has been reported [77],[78], [79], [80], [81].COSMO–SAC can also be evaluated for this property. Note that all values of thederivative must be negative for thermodynamic stability. Table 4.9 compares pre-dicted COSMO-SAC values with those derived from parameters regressed to VLEwith the Wilson equation, NRTL and modified Margules models. The three excessGibbs energy models were chosen, because descriptions of the derivatives can dif-fer in their reliability [77]. Here, the same negative signs and similar magnitudesdemonstrates the variability in fitting VLE data with such models. The differencesfor the COSMO-SAC model are considerable, though often the sign is correct andsome magnitudes are similar. The data in Table 4.9 are obtained from fits to VLEdata, so the derivatives for LLE systems are expected to be much larger than thosein Table 4.9 (in the end of this chapter). Thus, while employing COSMO-SAC forselecting initial guesses might be satisfactory, it is unlikely that the method can pro-vide reliable final results.

TABLE 4.7: Comparison of parameter values estimated withCOSMO-SAC (at T = 298.15 K) and regressed values.

System (1)/(2) bα (COSMO-SAC) bα (regressed) bβ (COSMO-SAC) bβ (regressed)toluene/water 4319 2784.8 778.3 1079.9n-pentane/water 6141.2 5514.3 1620.1 -1152n-hexane/water 6134.9 4322.3 1768.7 -1248.8n-heptane/water 6166.3 3592.8 2085.5 -786.27n-octane/water 6161.7 3737 2287.4 -2032[hmim][BF4] / water -15.167 463.1 -4967.1 5525.9[omim][BF4] / water -500.9 501.1 -4528.6 4717.7n-hexane/nitroethane 1363.9 6140 1112.9 5931.7n-octane/ nitroethane 1383.4 5577.6 1366.4 4561.22,2,4-trimethylpentane/ nitroethane 1173.7 5895.8 1055.1 4020.9n-decane/ nitroethane 1490.7 3152.3 1818.5 3444.1

It would be expected that the parameters in the present model would show sys-tematic variations with molecular structure. In fact, group contribution methodssuggest that b values in the systems of the same solute in a series of hydrocarbonsshould be identical [82]. Trends in calorimetric data confirm this expectation [83],[84]. Thus, regressed b values for water or nitroethane in a series of hydrocarbonsshould be quite similar. However, within the ranges of uncertainty, the b values arenot identical, neither for water nor for nitroethane (Table 4.4). If one trusts the resultof group contribution analysis that any solute has the same partial molar excess en-thalpy in all alkanes, then one conclusion could be that the data sets are inconsistent


with this idea and should be suspicious. Alternatively, the temperature dependenceof the present model could be inadequate to represent the partial molar excess en-thalpies of the data. The current form of the model implies that the partial molarexcess enthalpy at infinite dilution for any solute/solvent pair is a constant, inde-pendent of temperature. Data on partial molar excess enthalpies and partial molarheat capacities at infinite dilution are however not temperature-independent [85].The impact of these effects requires more studies of accurate data on these proper-ties.

Table 4.4 shows that regressed b- parameters for water have similar values forsolutes in three out of four n-alkanes. The COSMO-SAC predictions at 298.15 K arevirtually identical (Table 4.7). On the other hand, for the aqueous ionic liquids, theCOSMO SAC predicts negative values, not positive as from regression, indicatingthat the method can be unreliable.

FIGURE 4.11: Partial molar excess enthalpies at infinite dilution of72 binary systems at 298.15 K determined experimentally and pre-dicted by the COSMO-SAC model. Root mean square deviation from

RMSD =

[1N ∑N

i=1 (hE,∞calc − hE,∞

exp )2]1/2

is equal to 2.6 kJ/mol. Full

circles represent results for nitromethane/nitroethane and hydrocar-bons (including n-alkanes) systems, full square corresponds to the

toluene/water system.

Reliable models must be developed to validate LLE data, including degree of ac-curacy, identifying outliers, making connections to other properties, and discerningsystematic errors.

Simplicity has been emphasized in initial exploration, but the temperature de-pendence of infinite dilution properties is not in all cases as good as that of NRTL,though it is also not always correct, but a sufficiently reliable and general temper-ature dependence has not yet been found with 2 parameters only. Thus, furtherstudies are needed. Our form of FST is not directly applicable to multicomponentsystems, though rigorous extensions are possible. The form of the application willdepend upon the kind of multicomponent system. Multicomponent liquid-liquid


systems differ from those of the present chapter (which essentially deals with tem-perature variation), since temperature is often unchanging. Again, further data-based study is required.


TAB

LE

4.8:

Reg

ress

edpa

ram

eter

s(a

llsi

x)fo

rco

nsid

ered

syst

ems.

Para

met

erva

lues

wit

hco

nfide

nce

rang

esfr

omeq

.(4.

49).

Inal

lcas

esph

ase

αis

rich

inco

mpo

nent

1an

dβ

rich

inco

mpo

nent

2.

Syst

em(1

)/(2

)aα

bαaβ

bβcα

cβ

tolu

ene/

wat

er-3

.304

2736

.65.

718

1000

.9-0

.002

3(0

.099

9)±

0.00

8±

2.22

±0.

006±

1.71

6±

23.7

±22

57.2

n-pe

ntan

e/w

ater

-10.

821

5514

.315

.307

-115

1.3

0.00

5-0

.000

7±

0.00

1±

0.2

±0.

001

±0.

3±

21.8

71±

1452

n-he

xane

/wat

er-7

.042

4322

.317

.144

-124

8.8

3.11·1

0−17

0±

0.00

2±

0.6

±0.

003±

1.13

±29

.6±

3425

6n-

hept

ane/

wat

er-4

.764

3592

.817

.1-7

86.2

72.

74·1

0−15

0±

0.00

3±

0.9

±0.

004±

1.27

±31

.7±

2151

98n-

octa

ne/w

ater

-5.1

4937

37.1

23.9

-230

4.1

00

±0.

002

±0.

5±

0.00

3±

0.7

±34

.4±

2405

69[h

mim

][BF

4]/w

ater

-1.2

3046

0.99

-12.

225

4945

.3-4

.7-0

.1±

0.21

8±

73.4

9±

0.30

3±

64.3

±14

6.3

±18

4.1

[om

im][

BF4]

/wat

er-1

.402

409.

87-1

5.71

958

29.5

-10.

39(0

.03)

±0.

135±

45.7

7±

0.18

9±

40.0

4±

91.1

6±

114.

64n-

hexa

ne/n

itro

etha

ne-1

9.55

861

37.6

-18.

926

5931

.7(3

)-7

.998

±0.

905±

241.

1±

0.84

1±

225.

6±

77.6

±80

.8n-

octa

ne/n

itro

etha

ne-1

1.98

540

45.8

-8.8

5030

93.9

(60)

-40.

00±

0.20

3±

54.7

±0.

161±

43.1

±23

.13

±28

.78

2,2,

4-tr

imet

hylp

enta

ne/n

itro

etha

ne-1

2.35

340

32.8

-8.3

5428

03.4

(10)

-5.0

05±

0.43

7±

114.

6±

0.77

1±

190.

3±

64.7

1±

132.

18n-

deca

ne/n

itro

etha

ne-9

.049

3309

.1-9

.973

3594

.3(1

0)-8

.01

±0.

187±

48.9

±0.

318±

85.5

±32

.07

±63

.91

4.6. Model modifications 55

4.6 Model modifications

The developed unsymmetrical model is devoted to use in the correlation of LLE dataoutside the critical region. It has been found that the current form of the model canbe applied up to usually 1-2 K (but is dependent on the system) below the upper crit-ical solution temperature (UCST) or 2 K above in the case of LLE with lower criticaltemperature (LCST). The correlation of the LLE data close to the critical point, re-quires combination (cross-over) of the unsymmetrical model with scaling law equa-tion (including first Wegner correction) [86], [87].

|x− xc| = A1tβ

|x− xc| = A1tβ(1 + A2t0.5)(4.67)

In the above equations, t = 1−T/Tc; x and T are mole fraction and temperature,xc and Tc the corresponding critical composition and temperature; A1,A2 and β thecritical amplitudes and critical exponent, respectively. The critical exponent accord-ing to theory should be equal to 0.327 [88]. The LLE data-set is divided into twoparts: outside the critical region and the critical region. In the first region data arecorrelated by the FST model. In the critical region, scaling law equations were usedto correlate data (xc, Tc, β, A1, A2 regressed). The example of that appraoch is shownin the Fig. 4.12

FIGURE 4.12: Liquid-liquid equilibrium of cyclohexane (1)/acetoni-trile (2) system (left); zoom on the critical region (right);* denotes crit-

ical point, xc= 0.477, Tc= 349.79, β=0.3262, A1=0.9962, A2=-0.439

Probably adding the second order term in the Taylor expansion of the unsym-metrical activity coefficient given by Eqs. (4.9) and (4.10) could be an another option,although this expression is an expansion about the infinite dilution.

However, LLE phase diagrams with critical regions were not the focus of thisthesis and additional investigations should be performed.

So far, the unsymmetrical model was applied in the systems with classical be-haviour, i.e. outside critical region and with a monotonic solubility, either increas-ing or decreasing with the temperature. There has been found that phase diagramscould be of different types, such as the most complicated with both UCST and LCST(e.g. binary butan-1-ol + water system or polymers binary solutions), minimum ofthe solubility at certain temperature (e.g. furfural + water, tetrahydrofuran + water,methylvinylketone + water (Fig.4.13) or hourglass type (found in polymer solutionsand ionic liquids mixtures). In order to apply the unsymmetrical model to such


systems one needs to modify the expression for the activity coefficient at infinitedilution in the following manner


bβ

τ+ dβ ln τ (4.68)


bα

τ+ dα ln τ (4.69)

where τ = TTre f

, Tre f is a reference temperature e.g. from the middle of the dataset or 298.15 K.

With this modification the linear temperature dependence of the excess enthalpyat infinite dilution is enforced (Eq. (4.70)), while in the basic form of the model, theexcess enthalpy at infinite dilution is constant.

∂ln γ∞i

∂1/T= −T2 ∂ln γ∞

i∂T

=HE,∞

iR

= Tre f (b− dτ) (4.70)

FIGURE 4.13: LLE in methylvinylketone (1) + water (2) system

4.7. Conclusions 57

4.7 Conclusions

A model has been constructed and tested for the correlation of binary LLE data overranges of temperature. At least four parameters are needed, which is similar to cur-rent models. Depending upon the relative solubilities, additional parameters canbe needed. A procedure has been given for parameter estimation, including ini-tial guesses from graphical representation of data and from COSMO-SAC estimates,and for uncertainty analysis of the parameters and estimated mole fractions fromthe regressions. Three types of systems, covering different ranges of solubility andtemperature have been examined.

Comparisons with results from the NRTL model indicate mostly similar reliabil-ity within 95% confidence limits. Comparisons of regressed parameters with esti-mates from COSMO-SAC show mostly similar temperature dependence, but largediscrepancies for solution non-idealities. The model parameter values suggest sys-tematic variations with structure that could be exploited in data validation.


TA

BL

E4.9:A

ctivitycoefficientderivatives

with

respecttocom

positionw

ithdifferentm

odels:CO

SMO

–SAC

,NR

TL,Modified

Margules

andW

ilsonequation.

System(1)/(2)

T/K

CO

SMO

-SAC

NR

TLM

odM

argulesW

ilson(∂ln

γ1 /

∂x1 )T

,P(∂ln

γ2 /

∂x2 )T

,P(∂ln

γ1 /

∂x1 )T

,P(∂ln

γ2 /

∂x2 )T

,P(∂ln

γ1 /

∂x1 )T

,P(∂ln

γ2 /

∂x2 )T

,P((∂ln

γ1 /

∂x1 )T

,P*

x1 =

0benzene/aniline

298.15-1.09

-0.694-0.0581

-0.858m

ethanol/water

313.05-4.23

-2.01-2.24

acetone/benzene298.15

-0.183-1.32

-1.37acetone/w

ater298.15

-8.89-6.01

-5.71-7.29

ethanol/benzene298.15

-318-14.1

-25.8-20.7

1-propanol/water

303.15-20.1

-17.1-12.9

-40.7aniline/toluene

293.15-2.17

-9.36-7.48

-6.65C

Cl4 /m

ethanol293.15

-4.13-6.55

-4.74-4.99

CC

l4 /ethanol293.15

-1.79-3.53

-1.95-2.91

CC

l4 /acetone313.15

-0.385-0.972

-1.06-1.02

CC

l4 /aniline298.15

-3.19-1.81

-2.29-2.11

x1 =

1benzene/aniline

298.15-3.30

-2.80-2.72

-3.07m

ethanol/water

313.050.0793

-0.680-0.715

acetone/benzene298.15

0.317-0.646

-0.672acetone/w

ater298.15

-17.6-4.51

-4.69-5.35

ethanol/benzene298.15

-1.31-3.57

-3.12-3.11

1-propanol/water

303.15-0.775

-2.93-2.38

-2.59aniline/toluene

293.15-1.16

-7.50-5.80

-6.14C

Cl4 /m

ethanol293.15

-2930-22.5

-60.5-53.5

CC

l4 /ethanol293.15

-734-19.7

-31.4-37.7

CC

l4 /acetone313.15

-1.23-2.92

-3.37-3.14

CC

l4 /aniline298.15

-8.2-5.61

-7.40-5.2

∗denotes

(∂lnγ

2 /∂x

2 )T,P

59

Chapter 5

Ternary liquid-liquid equilibriamodelling

5.1 Introduction

Many combinations of liquids have limited mutual solubilities. Particularly of tech-nical relevance are 3-component LLE systems. These data are of the utmost impor-tance to describe formulated products and in the wide range of downstream separa-tions such as liquid-liquid extraction or extractive distillation.

The importance of this phenomenon implies the fact that there is an abundanceof data measured for the multicomponent systems and new data are constantly com-ing along. Therefore, new models and methods for data validation are needed. Asfar as binary LLE data validation has been addressed to some extent in differentworks, focus on the ternary LLE data has been minor [48], [43], [36], [53]. Moreover,a good prediction of ternary LLE from data on constituent binary pairs data is a dif-ficult problem. Several attempts [89], [90], [91], based on activity coefficient modelshave been made so far for correlation and prediction of ternary liquid-liquid systemsbehavior [27], [92]. Generally, in the prediction, the parameters in the activity coeffi-cient models need to regressed from vapor-liquid equilibria (VLE) for a completelymiscible binary pairs and liquid-liquid equilibria for partially miscible binary.

Here, a new model intended for a three-component mixture of fluids where max-imum two pairs exhibit miscibility gap is proposed. The development is in principlesuitable for modeling type I liquid-liquid systems in the Treybal classification, of onepartially miscible binary (components 2 and 3) and two completely miscible binaries(1+2 and 1+3) when the liquid phases are rich in one component (’ is rich in 2 and” is rich in 3). The model is proposed for distribution of the components amongin the liquid phases, based on Henry’s law standard states in the liquids as in thepreviously explored model for binary LLE modelling [66] and solid-liquid equilib-rium validation [40], [93]. LLE is formulated with unsymmetrical convention fornormalizing activity coefficient of the diluted species. For very low concentrations(mole fraction less than 0.01) only two parameters for the temperature dependenceof the Henry’s law constants of the dilute component in each coexisting phase wererequired for fitting to obtain nearly as good results as NRTL. At higher concentra-tions, an additional parameter for each phase was added. In total 4-6 parameterswere used to correlate Tx binary LLE data.

Here the excess Gibbs energy models for the both liquid phases (’ and ”) arebased on binary additivity of the dilute solution Porter model [37], which is rigor-ous in the first-order expansion from infinite dilution of Fluctuation Solution The-ory [58] at least for mole fractions up to 0.01. This model requires experimentalVLE data for the completely miscible binaries and LLE data for partially misciblebinaries to obtain most of the model parameters, i.e. binary interaction parameters

60 Chapter 5. Ternary liquid-liquid equilibria modelling

and temperature-dependent Henry’s constants. The remaining binary interactionparameters are obtained from ternary LLE data. The proposed model is a naturalextension of the binary model; it means the ternary LLE model reduces to the binarycase when the mole fraction of the one single component vanishes.

The model can be used for data validation. Primarily, in low concentrationsregions the developed model is nearly rigorous. Therefore, the experimental datashould follow the assumed concentration dependence, provided that they are accu-rate.

5.2 Modelling framework

In this section, the thermodynamic framework for the proposed model of type I LLEsystems of one partially miscible binary (2+3) and two completely miscible binaries(1+2 and 1+3) is presented. The phase denoted by a single prime,′ is rich in compo-nent 2 and the double prime phase, ′′ is rich in component 3. The model describesdistribution of the components in the liquid phases, based on Henry’s law standardstates in the liquids.The Henry’s constants for components i in phases ′ and ′′ are defined as follows:

ln[

H′i (T)

]= ln

[f 0i (T)

]+ lim

x2→1lnγ

′i (5.1)

ln[

H′′i (T)

]= ln

[f 0i (T)

]+ lim

x3→1lnγ

′′i (5.2)

5.2.1 Binary VLE and LLE data

As a first step, binary phase equilibrium data are used to obtain Henry’s law con-stants and their temperature dependence as well as four of six interaction parame-ters.

Binary 1+2 (VLE)

Since in the ternary model, phase ′ is rich in component 2, the fugacity of component2 is expressed in the symmetrical convention. For the component 1, the unsymmet-rical convention is used, taking as a reference state infinitely diluted component 1instead of pure component 1.

ln f′1

(T, x

′1

)= ln x

′1 + ln

[H′1 (T)

]+ A

′12

((x′1)

2 − 2x′1

)= ln (y1φ1P) = ln f V

1 (T, P, y1)

ln f′2

(T, x

′1

)= ln x

′2 + ln

[Psat

2 (T)]+ A

′12

((x′1)

2)= ln (y2φ2P) = ln f V

2 (T, P, y1)

(5.3)Alternatively, one can write

ln f′1

(T, x

′1

)= ln x

′1 + A

′12 + ln

[Psat

1 (T)]+ A

′12

(x′1

2 − 2x′1

)= ln (y1φ1P) =

ln f V1 (T, P, y1)

ln f′2

(T, x

′1

)= ln x

′2 + ln

[Psat

2 (T)]+ A

′12

((x′1)

2)= ln (y2φ2P) = ln f V

2 (T, P, y1)

(5.4)since the Henry’s law constant for component 1 is defined as follows

5.2. Modelling framework 61

H1 =γ1Psat

1 (T)γ∗1

= γ∞1 Psat

1 = Psat1 eA

′12 (5.5)

.If one assumes an ideal gas behavior of vapour phase and negligible Poynting

factors (as with modified Raoult’s law), i.e. φ1=1 and express the temperature de-

pendence of Henry’s constant as ln[

H fi (T)

]= a f

i +b f

iT .

The saturation pressure is calculated using an Antoine form equation with threeconstants (A, B, C):

log10

(Psat

i1bar

)= Ai −

Bi

T/K + Ci(5.6)

Depending upon which type of data (isobaric or isothermal) are used, the modelequations follow those shown in Table 5.1. As it is in line with the recommendationsof Van Ness [25], [94] in the case of isothermal set, one should use the experimentalpure component vapor pressures, since often these are forming an integral part ofthe data-set. In general, we would like the pure component vapor pressures to be re-produced. Note that this means different parameter sets will need to be determineddepending upon the type of binary VLE data available, isothermal of isobaric.

TABLE 5.1: Modelling expressions of the isothermal or isobaric binaryVLE . The residuals in the objective function are the differences be-tween experimental (temperature or pressure) values and calculated

i.e. δxi = xexpi − xcalc

i

Data type Isothermal VLE data Isobaric VLE dataEquilibrium

conditions{

ln x′1 + A

′12 + ln[Psat

1 (T)] + A′12(x

′1

2 − 2x′1) = ln(y1P)

ln x′2 + ln[Psat

2 (T)] + A′12(x

′1

2) = ln(y2P)

{ln x

′1 + a

′1 +

b′1

T+ A

′12(x

′1

2 − 2x′1) = ln(y1P)

ln x′2 + ln[Psat

2 (T)] + A′12(x

′1

2) = ln(y2P)Psat

i (T) Experimental vapor pressure of the pure components. The vapor pressure is calculated via Antoine equation.F-objective function min ∑n

i=1 (δpi)2 min ∑n

i=1 (δTi)2

Parameters to be obtained 1 parameter: A′12 3 parameters: A

′12, a

′1,b

′1

Binary 1+3 (VLE)

The same assumptions as for system 1+2 have been made for the binary system 1+3(activity coefficient unsymmetrically normalized for component 1, symmetrically for3). Similarly, one can consider the isothermal as well as the isobaric case cf. Table5.2.

TABLE 5.2: Modelling expressions of the isothermal or isobaric binaryVLE . The residuals in the objective function are the differences be-tween experimental (temperature or pressure) values and calculated

i.e. δxi = xexpi − xcalc

i

Data type Isothermal VLE data Isobaric VLE data

Equilibrium conditions{

ln x′′1 + A

′′13 + ln[Psat

1 (T)] + A′13(x

′′1

2 − 2x′′1) = ln(y1P)

ln x′3 + ln[Psat

3 (T)] + A′′13(x′1

′2) = ln(y3P)

{ln x

′′1 + a

′′1 +

b′′1

T+ A

′′13(x

′′1

2 − 2x′′1) = ln(y1P)

ln x′′3 + ln[Psat

3 (T)] + A′′13(x

′′1

2) = ln(y3P)Psat

i (T) Experimental vapor pressure of the pure components The vapor pressure is calculated via Antoine equation.F-objective function min ∑n

i=1 (δpi)2 min ∑n

i=1 (δTi)2

Parameters 1 parameter: A′′13 3 parameters to be obtained: A

′′13, a

′′1,b

′′1

The fugacity expressions become


ln f′′1

(T, x

′′1

)= ln x

′′1 + ln

[H′′1 (T)

]+ A

′′13

(x′′1

2 − 2x′′1

)= ln (y1φ1P) = ln f V

1 (T, P, y1)

ln f′′3

(T, x

′′1

)= ln x

′′3 + ln

[Psat

3 (T)]+ A

′′13

(x′′1

2)= ln (y3φ3P) = ln f V

3 (T, P, y1)

(5.7)And with the assumptions (ideal gas vapor, Poynting factor of unity) inserted

ln x′′1 + a

′′1 +

b′′1T + A

′′13

(x′′1

2 − 2x′′1

)= ln (y1P)

ln x′′3 + ln

[Psat

3 (T)]+ A

′′13

(x′′1

2)= ln (y3P)

(5.8)

Binary 2+3 (LLE)

LLE in the binary system 2-3 is modelled by the unsymmetrical model exploredpreviously with A

′23 = − cα

T and A′′23 = − cβ

T .

ln f′2

(T, x

′2

)= ln x

′2 + ln

[f 02 (T)

]+ A

′23

(1− x

′2

)2= ln x

′′2 + ln

[H′′2 (T)

f 02 (T)

]+ ln

[f 02 (T)

]+ A

′′23

(x′′2

2 − 2x′′2

)= ln f

′′2

(T, x

′′2

)ln f

′3

(T, x

′2

)= ln (1− x

′

2) + ln[

H′3(T)

f 03 (T)

]+ ln

[f 03 (T)

]− A

′23

(x′2

2 − 2x′2

)= ln (1− x

′′

2)+

ln[

f 03 (T)

]+ A

′′23

(x′′2

2)= ln f

′′3

(T, x

′′2

)(5.9)

or

ln x′2 + A

′23

(1− x

′2

)2= ln x

′′2 + a

′′2 +

b′′2T + A

′′23

(x′′2

2 − 2x′′2

)=

ln f′′2 (T, x

′′2)

ln f′3

(T, x

′2

)= ln(1− x

′2) + a

′3 +

b′3

T − A′23

(x′2

2 − 2x′2

)= ln(1− x)

′′2) + A

′′23

(x′′2

2)

= ln f′′3

(T, x

′′2

)(5.10)

The parameters to be obtained from LLE data are A′23, (or cα) , A

′′23 (or cβ) , a

′′2 ,

b′′2 , a

′3,b

′3.

To obtain these model parameters an objective function defined as the sum of thesquared differences between the experimental and calculated phase mole fractionsis minimized:

F=minN

∑i=1

∑j=1,2

(xj,expi −xj,calc

i )2

. (5.11)

5.2.2 Ternary LLE model

For the ternary LLE system, we consider a partially miscible binary (2-3) and twocompletely miscible binaries (1-2 and 1-3). Phase ′ is rich in component 2 and phase′′ in component 3.

The equality of the fugacities of the components in either binaries or ternarymixture is expressed as follows


(1) ln x′1 + ln[H

′1(T)] + ln γ1(T, x

′1, x

′2)− limx2→1 ln γ

′1 = ln x

′′1 + ln[H

′1′(T)]+

ln γ1(T, x′′1 , x

′′2)− limx3→1 ln γ

′′1

(2) ln x′2 + ln[ f 0

2 (T)] + ln γ2(T, x′1, x

′2) = ln x

′′1 + ln[H

′2′(T)

f 02 (T)

]+

ln[ f 02 (T)] + ln γ2(T, x

′′1 , x

′′2)− limx3→1 ln γ

′′2

(3) ln(1− x′1 − x

′2) + ln[H

′3(T)

f 03 (T)

] + ln[ f 03 (T)] + ln γ3(T, x

′1, x

′2)− limx2→1 ln γ

′3 =

ln(1− x′′1 − x

′′2) + ln[ f 0

3 (T)] + ln γ3(T, x′′1 , x

′′2)

(5.12)The excess Gibbs energy models for both phases are based on the Porter model,

which is rigorous in the first order expansion of the infinite dilution according to thefluctuation solution theory (FST)

GE(T, x′1, x

′2)

RT= A

′12(T)x

′1x′2 + A

′13(T)x

′1x′3 + A

′23(T)x

′2x′3 (5.13)

GE(T, x′′1 , x

′′2)

RT= A

′′12(T)x

′′1x′′2 + A

′′13(T)x

′′1x′′3 + A

′′23(T)x

′′2x′′3 (5.14)

If the symmetric convention is employed, the expression for the activity coeffi-cients in phase ′ should be

ln γ′1 = A

′12(x

′2

2) + A′13x

′3

2 − A′23x

′2x′3 + (A

′12 + A

′13)x

′2x′3 (5.15)

ln γ′2 = A

′12(x

′1

2) + A′23x

′3

2 − A′13x

′1x′3 + (A

′23 + A

′12)x

′1x′3 (5.16)

ln γ′3 = A

′23(x

′2

2) + A′13x

′1

2 − A′12x

′1x′2 + (A

′23 + A

′13)x

′1x′2 (5.17)

and similarly for the phase ′′

ln γ′′1 = A

′′12(x

′′2

2) + A′′13x

′′3

2 − A′′23x

′′2x′′3 + (A

′′12 + A

′′13)x

′′2x′′3 (5.18)

ln γ′′2 = A

′′12(x

′′1

2) + A′′23x

′′3

2 − A′′13x

′′1x′′3 + (A

′′23 + A

′′12)x

′′1x′′3 (5.19)

ln γ′′3 = A

′′23(x

′′2

2) + A′′13x

′′1

2 − A′′12x

′′1x′′2 + (A

′′23 + A

′′13)x

′′1x′′2 (5.20)

The expression for an unsymmetrical activity coefficient, which is shown as anexample for the expression named a below has the following form. First the standardstate

limx2 → 1

x1, x3 → 0

ln γ′1 = A

′12 + A

′13x

′3

2 − A′23x

′2x′3 + (A

′12 + A

′13)x

′2x′3 = A

′12 (5.21)

Having this, the unsymmetrical activity coefficient becomes:

a = ln γ1(T, x′1, x

′2)− lim

x2→1ln γ

′1 = A

′12(x

′2

2) + A′13x

′3

2− A′23x

′2x′3 + (A

′12 + A

′13)x

′2x′3−

A′12 = A

′12(x

′2

2 − 1) + A′13x

′3

2 − A′23x

′2x′3 + (A

′12 + A

′13)x

′2x′3(5.22)

The remaining expressions for activity coefficients are as follows


a = ln γ1(T, x′1, x

′2)− lim

x2→1ln γ

′1 = A

′12(x

′2

2− 1)+ A′13x

′3

2−A′23x

′2x′3 +(A

′12 + A

′13)x

′2x′3

(5.23)b = ln γ1(T, x

′′1 , x

′′2)− lim

x3→1ln γ

′′1 = A

′′12x

′′2

2 + A′13(x

′′3

2− 1)−A′′23x

′′2x′′3 +(A

′′12 + A

′′13)x

′′2x′′3

(5.24)c = ln γ

′2 = A

′12x

′1

2 + A′23x

′3

2 − A′13x

′1x′3 + (A

′23 + A

′12)x

′1x′3 (5.25)

d = ln γ2(T, x′′1 , x

′′2)− lim

x3→1ln γ

′′2 = A

′′12x

′′1

2 + A′′23(x

′′3

2− 1)−A′′13x

′′1x′′3 +(A

′′23 + A

′′12)x

′′1x′′3

(5.26)e = ln γ3(T, x

′1, x

′2)− lim

x2→1ln γ

′3 = A

′13x

′1

2 + A′23(x

′2

2− 1)−A′12x

′1x′2 +(A

′23 + A

′13)x

′1x′2

(5.27)f = ln γ

′′3 = A

′′13x

′′1

2 + A′′23x

′′2

2 − A′′12x

′′1x′′2 + (A

′′23 + A

′13′)x′′1x′′2 (5.28)

In the ternary LLE calculations three isoactivity equations (Eq. (5.12)) and threemolar balances (Eq. (5.29)) are solved to obtain six molar amounts in total (each ofthe three components in both phases).

ni = n′i + n

′′i ∀i ∈ {1, 2, 3} (5.29)

In total there are 14 parameters in the model, 12 of them (A′23, A

′′23,A

′′13, A

′12, a

′′2,

b′′2 ,a

′3,b

′3,a

′′1,b

′′1 , a

′1,b

′1) come from binary systems. The two remaining binary interac-

tion parameters, i.e. A′13, A

′′12 need to be determined from ternary data. It goes

without saying that binary parameters should be - if possible - regressed at the sameor similar temperature as the ternary LLE data.

5.2.3 Stability conditions of the ternary LLE model

In this section, the necessary and sufficient conditions imposed on the model param-eters to ensure a thermodynamic stability of the phases are explored.

In general, Gibbs free energy of the phase is the sum of ideal part and the Gibbsexcess energy

G = GE + Gid = GE + RT3

∑i=3

xi ln xi (5.30)

where Gibbs excess energies are expressed by the Eqs. (5.13) and (5.14).The Gibbs energies of the proposed ternary model are thus (for both phases):

G′

RT= A

′12x

′1x′2 + A

′13x

′1x′3 + A

′23x

′2x′3 + x

′1 ln x

′1 + x

′2 ln x

′2 + x

′3 ln x

′3 (5.31)

G′′

RT= A

′′12x

′′1x′′2 + A

′′13x

′′1x′′3 + A

′′23x

′′2x′′3 + x

′′1 ln x

′′1 + x

′′2 ln x

′′2 + x

′′3 ln x

′′3 (5.32)

(x′1 = 1− x

′2 − x

′3 since components 2 and 3 are chosen to be independent).

The stability criterion for a ternary system is that the Hessian matrix of the G′

function must be positive definite, which corresponds to the G′ surface being con-vex [95]. In fact, the Gibbs free energy of each submodel: G′, G′′ must be partiallyconvex and min{G′ , G′′} locally nonconvex to ensure that the isopotential criterionis a necessary and sufficient to satisfy thermodynamic stability [65], i.e.


D =

∣∣∣∣∣∣∂2G′

∂x′22

∂G′

∂x′2∂x′3∂G′

∂x′3∂x′2

∂2G′

∂x′23

∣∣∣∣∣∣ > 0 (5.33)

with ∂2G′

∂x′22> 0 and ∂2G′

∂x′23> 0.

By inserting the expressions for derivatives into Eq. (5.33) we obtain the expres-sions (Eqs. (5.34) to (5.36)), which guarantee a thermodynamic stability of the singlephase

D = ((1x′1

+1x′2

)− 2A′12)((

1x′3

+1x′1

)− 2A′13)− (

1x′1

+ A′23 − A

′13 − A

′12)

2 > 0 (5.34)

∂2G′

∂x′22= (

1x′1

+1x′2

)− 2A′12 > 0 (5.35)

∂2G′

∂x′23= (

1x′3

+1x′1

)− 2A′13 > 0 (5.36)

The same considerations apply to the phase ′′.Additionally, according to Section 4.4.3, interaction parameters A

′23 and A

′′23 must

fullfill the stability condition.The stability check has been done after the parameter regression. The regressed

parameters have been inserted to expressions given by Eqs. (5.34) to (5.36) and checkif these are greater than zero for a given phase compositions.

5.2.4 Parameters regression

To regress the binary interaction parameters, two types of objective function are usu-ally explored [96]

Fa =M

∑k=1

3

∑i=1

(x′ikγ

′ik − x

′′ikγ

′′ik)

2 (5.37)

and

Fx =3

∑i=1

2

∑j=1

M

∑k=1

(xj,calcik − xj,exp

ik )2 (5.38)

Here i denotes a component, j denotes a phase and the sum is over k experimentaltie lines.

To obtain the binary interaction parameters A′13, A

′′12, the following approach

was taken. First, the objective function in terms of activities (Fa) was used to pre-regress parameters for the next step, in which minimization of distances betweenthe experimental and calculated mole fractions (Fx) was done.Minimization of Fx involves the following workflow:

1. Getting total experimental amount of moles in both phases for each experi-mental tie-line: nk∀k ∈ {1, 2, . . . , M} by assuming 1 mole of phase ′ and ′′,to obtain a set of nj,exp

ik . Assuming there is 1 mole of each phase, a lever armprinciple will give a feed composition coordinate.


2. Solving the set of equations Eq. (5.12) + Eq. (5.29) to get the mole numbers cor-responding to the current set of parameters from experimental mole numbers.

3. Conversion of the evaluated mole numbers to molar fractions xj,expik .

4. Evaluation of the objective function value.

5. If the convergence criteria are not satisfied, update parameters and repeat fromstep 2. The fminsearch solver (based on the simplex search method) has beenused in the parameter estimation problem.

The quality of the regression is evaluated by the RMSD (root mean square devi-ation) defined as

RMSD = (∑3

i=1 ∑2j=1 ∑M

k=1 (xj,calcik − xj,exp

ik )2

M)1/2 (5.39)

where M is the number of experimental tie-lines.An important issue is the quality of the set of parameters regressed from binary

data. While parameters (especially binary interaction parameters) regressed frombinary VLE are readily transferable, this is not always the case for parameters ob-tained from binary LLE. As has been previously found [66], several sets of (cα,cβ)-parameters give similar goodness of fit to LLE binary data. It has been observed thatc can vary significantly, without affecting the objective function strongly, especiallyfor systems showing wide miscibility gaps. However, the values of the interactionparameters A

′23, A

′′23 are important to the ternary systems since they appear in the

composition-dependence terms. To avoid such ambiguity in parameter values, theternary data can be used to determine four parameters (including A

′23, A

′′23) instead

of only two. Parameters obtained from binary LLE can then be used as initial guessfor the estimation routine.

Afterwards, interaction parametersA′23, A

′′23 regressed from ternary LLE data might

be tested in binaries, in order to help with their identification (which is difficult todo based on binary data alone).

This issue is demonstrated later in this chapter, cf. section 5.3.3.

5.3 Application of the model

In this section three case studies (with different widths of the miscibility gap) havebeen chosen for testing this model with experimental data. In principle, the modelis dedicated to systems in which the concentrations of the diluted species are low(less than 0.01), such as propan-2-ol/benzene/water system. It has been also tested,whether the model can be applied to systems with wider miscibility gap (2) + (3)and with higher concentrations of the component (1) in the phases. Also, the tem-perature dependence on the ternary LLE has been explored. The comparison withthe benchmark model (NRTL) is presented.

5.3.1 Model performance on the selected cases

In order to benchmark the developed model, 46 different experimental data sets cov-ering 27 distinct three-component systems were selected. The range of systems waslimited mainly to type I systems and to those for which the available data in binarysystems at similar temperature as the ternary LLE data were reported. In the col-lected literature data sets, experimental points close to the plait point were excluded

5.3. Application of the model 67

since the model does not describe the critical behavior similarly as in binary Tx LLEdiagrams. The characteristics of the selected case studies in this section are presentedin Table 5.3.

TABLE 5.3: Selected systems shown in the Section 5.3 and vapour-liquid and liquid-liquid equilibria for binary systems and liquid-

liquid equilibria for ternary systems.

No. System(1/2/3) T/K T/KVLE VLE LLE(1+2) (1+3) (2+3)

1 benzene/cyclohexane/acetonitrile 318.15 313.14 318.15 276.50-349.042 benzene/cyclohexane/acetonitrile 298.15 313.14 318.15 276.50-349.043 propan-2-ol/benzene/water 303 318.15 303.15 303.15-373.154 propan-2-ol/benzene/water 313 318.15 303.15 303.15 -373.155 propan-2-ol/benzene/water 323 318.15 303.15 303.15 -373.156 ethanol/acetonitrile/n-octane 298.15 313.15 313.15 293.20-333.20

Case study 1: benzene (1) + cyclohexane (2) + acetonitrile (3) at 298.15 and 318.15 K

1. Binary data

In order to estimate the model parameters isothermal VLE data has been used forthe systems of benzene/cyclohexane at T=313.14 K [97] and benzene/acetonitrile atT=318.15 K [98]. The comparison of predicted and experimentally determined phasediagram of liquid-liquid equilibria for cyclohexane-acetonitrile system is shown inFigure 5.1 [99]. The reported range of temperatures is 276.5 to 349.04 K. Parametersobtained from binary data were used in the regression of the LLE in the ternarysystem of benzene (1) + cyclohexane (2) + acetonitrile (3) measured at 298.15 K and318.15 K. All interaction parameters regressed from 2-component system are shownin Table 5.7 and Table 5.8, in the end of this chapter.


FIGURE 5.1: Liquid-liquid equilibria phase diagram for cyclohexane(1) + acetonitrile (2)

FIGURE 5.2: Isothermal vapor-liquid equilibria in the system of ben-zene (1) + acetonitrile or cyclohexane (2). Circles are experimental

points, line is correlation by Porter model.

1. Ternary data

Experimental tie lines for benzene (1) + cyclohexane (2) + acetonitrile (3) systemat T=298.15 and 318.15 K were collected from [100], but 3 points close to the plaitpoint (critical point) were excluded from the parameter regression. The reason isthat the proposed model will not deal with non-classical behavior near the criticalregion, since the assumed composition dependence is not suitable close to the criticalpoint.

As mentioned earlier, ideally the parameters of the model should be regressedfrom binary data at similar temperature as the reported ternary data. That is the case


FIGURE 5.3: Ternary phase diagrams of benzene (1) + cyclohexane(2) + acetonitrile (3) system (top at T=318.15 K, bottom T=298.15 K).Circles - experimental data; solid line - experimental tie-lines; dashed

lines are calculated tie-lines; blue line - calculated binodal curve.

only for the ternary data-set reported at 318.15 K. However, the binary parameterswere used also at the lower temperature giving fairly sufficient fit with the RMSDonly two times greater than for the higher temperature. Parameters show systematicvariation (decreasing) with the increasing temperature.

Case study 2: propan-2-ol (1) + benzene (2) + water (3) wide gap and differenttemperatures

The ternary system studied in this section exhibits wide two-phase region and verywide miscibility gap between components 2 and 3 in the binary system, which isa typical characteristics of water-hydrocarbons systems. The ternary LLE tie lineswere reported at three different temperatures ranging from 303 to 323 K [101]. Thebinary data used to obtain some of the model parameter were isothermal VLE in


propan-2-ol + {benzene or water} at 318.15 and 308.15 K, respectively. Tx binaryLLE data in the system of benzene and water was reported in the range of 293.2-333.2 K. The best fit was obtained at 313 K. (Fig. 5.4).

Interaction parameters between water and benzene turned out to have negligibleimpact on the model predictions with estimated values numerically close to zero.

FIGURE 5.4: Ternary phase diagrams propan-2-ol (1) + benzene (2) +water (3) (top left: 323 K, top right 313 K), bottom 303 K. Legend is

the same as in the previous figure.

Case study 3: ethanol (1) + acetonitrile (2) + n -octane (3) T=298.15 K.

Next example is a LLE in the ternary system: ethanol (1) + acetonitrile (2) + n-octane(3) at T=298.15 K [102]. The binary fluid phase equilibria data used to obtain param-eters of the model are collected in the Table 5.3. The miscibility gap is less wide thanin the case of hydrocarbons and water system, but mole fractions of the componentbeing diluted in either phase is less than 0.1. The interaction parameters betweencomponents 2 and 3 were found to be zero, even though the miscibility gap is wider.The regressed parameters of the proposed model are: A

′13 = -4.65 and A

′′12= -3.13.

The model calculated compositions and tie lines along with experimental values areshown in the Fig. 5.5. The RMSD is equal to 0.054.


FIGURE 5.5: Ternary LLE in ethanol (1) + acetonitrile (2) + n-octane(3) system at T=298.15 K; experimental data vs prediction by the FST-based model proposed in this chapter. Points are experimental data,the proposed model calculates blue solid line, solid black lines are

experimental tie lines and dashed lines are calculated tie lines.

5.3.2 Comparison with the NRTL model

The performance of the developed FST-based model can be compared to the NRTLmodel with the following form of Gibbs excess energy of a ternary mixture [27]:

gERT = ∑N=3i=1 xi

∑3j=1 τjiGjixj

∑3k=1 Gkixk

τij =∆gjiRT

Gji = exp(−αjiτji),

(5.40)

where τji 6= τij and it was assumed that αji=αij=0.2.This gives in total six interaction parameters:∆g12,∆g21, ∆g13,∆g31,∆g23,∆g32 in

addition to non-randomness parameter αij.Two ways of determining the adjustable energy parameters in the NRTL model

were applied. One was the correlation of binary VLE or LLE and then use of the pa-rameters to predict the ternary behavior. In the second approach, NRTL parameterswere regressed from ternary data using those obtained from binaries as an initialguesses. All estimated parameters are listed in the Table 5.5. In all cases the ternarycorrelation gives the best results in terms of RMSD, which is not surprising sinceonly ternary data are used to obtain NRTL model parameters. Those parametersare estimated to give best accuracy, but do not contain information of binaries. Thesecond approach, on the contrary uses only binary pairs interaction parameters. Inthe proposed model, parameters obtained from binaries are fixed, but ternary dataare used to obtain remaining ones. Therefore, pure prediction by the NRTL model ofthe ternary diagram using binary pair parameters will be usually less accurate thanthe application of present model. The formulation of the proposed model require atleast two parameters to be regressed from ternary LLE data. This can help also in avalidation of data. Parameters obtained from corresponding binary systems shouldbe easily transferable to ternary system. A prediction of the ternary LLE from bina-ries by the means of NRTL seems often less accurate than the FST model, cf. Tables5.4 and 5.5. The example for benzene (1) + acetonitrile (2) + n-heptane is shownin Figures 5.6 and 5.7. A comparison with the FST-based model (Fig. 5.8, lower


left) confirms this observation. The NRTL model parameters for all studied ternarysystems in this thesis can be found in Appendix B.

TABLE 5.4: Parameters regressed from ternary data along with theroot mean square deviation (RMSD) for the three case studies elabo-rated in Section 5.3. Results for other investigated systems are pro-

vided in Chapter 6. TL denotes number of tie lines.

No. System Ref. T/K A′13 A

′′12 RMSD TL

1 benzene/cyclohexane/acetonitrile [100] 318.15 -3.44± 0.25 -1.65±0.18 0.035 72 benzene/cyclohexane/acetonitrile [100] 298.15 -3.60±0.42 -1.73±0.34 0.056 63 propan-2-ol/benzene/water [101] 303 -31.61±1.81 -17.54±0.82 0.057 64 propan-2-ol/benzene/water [101] 313 -30.09±0.00 -17.93±0.00 0.052 65 propan-2-ol/benzene/water [101] 323 -26.29±0.001 -17.33±0.001 0.071 56 ethanol/acetonitrile/n-octane [102] 298.15 -4.65±0.53 -3.13±0.53 0.054 77 benzene/acetonitrile/n-heptane [102] 318.15 -5.608±0.009 1.246±0.001 0.016 98 ethanol/n-hexane/acetonitrile [102] 313.15 -1.62±0.38 2.45±0.19 0.031 79 ethanol/acetonitrile/n-heptane [102] 298.15 -2.62±0.34 5.58±2.47 0.096 710 acetonitrile /chlorobenzene/water [102] 304.15 -7.76±1.67 -26.79±0.00 0.108 8

TABLE 5.5: Sample of NRTL model parameters along with the rootmean square deviation (RMSD) both from correlation of only ternaryLLE data and prediction by the parameters obtained from binarypairs. References and temperatures for the systems are the same as

in Table 5.4

.

No. System ∆g12 ∆g21 ∆g13 ∆g31 ∆g23 ∆g32 RMSD RMSD(corr.) (pred.)

1 benzene/cyclohexane -3429.5 -23699 -4785.3 -19962 4894.2 3946.5 0.003 0.056acetonitrile

2 benzene/cyclohexane/ -2774.6 6344.6 -3396.5 8912.5 5980.0 3898.9 0.006 0.059acetonitrile

3 propan-2-ol/benzene -2111.44 6928.54 1334.51 1312.66 33945 11969 0.021 0.555water

4 propan-2-ol/benzene/ 3230.6 220.65 -5629.5 12938 9283.5 8022.2 0.011 0.529water

5 propan-2-ol/benzene/ 5666.3 -422.15 -6531.9 14047 6987.6 7992.9 0.013 0.656water

6 ethanol/acetonitrile/ -7822.66 15293 4983.7 314.55 12501 3045.2 0.015 0.068n-octane

7 benzene/acetonitrile/ -2813.2 9652.4 -359.4 5164.5 5855.8 3516.0 0.017 0.048n-heptane

8 ethanol/n-hexane/ 11209 -3242.5 -9095.3 8282.4 2280.3 7183.7 0.008 0.032acetonitrile

9 ethanol/acetonitrile/ -6737.7 7346.7 1023.7 -2201.4 6693.5 3290.8 0.021 0.149n-heptane

10 acetonitrile /chlorobenzene -2988.743 7298.6 13433 96.662 3494.3 15960 0.011 n/awater

5.3.3 Identification of c (A′23 and A

′′23) parameters based on the ternary

LLE data.

In some cases, the model parameters - especially those regressed from binary LLEdata - are not easily transferable to ternaries. In other words, it means that evenif different sets of c parameters (regressed from binary Tx LLE data) result in simi-lar objective function values (correlation of binary LLE), they would not reproduceequally well the ternary phase diagram. As an example, two systems were consid-ered: benzene (1) + acetonitrile (2) + n-heptane (3) and ethanol (1) + acetonitrile (2)+ n-heptane (3). In both systems the same binary pair 2+3 exhibits a miscibility gap.From two binary LLE data-sets there has been found two sets of A

′23, A

′′23 which give

similarly good fit in the binary pair 2+3 at constant T=318.15 K (in terms of RMSD).However, only one (marked as set_2) give lower RMSD, when using parameters tocalculate LLE in a ternary system. (Table 5.6)


FIGURE 5.6: NRTL correlation with six parameters for benzene (1) +acetonitrile (2) + n-heptane (3) at 318.15 K.

FIGURE 5.7: NRTL prediction from binaries with six parameters forbenzene (1) + acetonitrile (2) + n-heptane (3) at 318.15 K.

For the system, benzene (1) + acetonitrile (2) + n -heptane (3) two sets of A′23, A

′′23

parameters were used. Both of them give low and similar of the objective function:2.20·10−5 and 5.99·10−5, while applied to calculate binary LLE at T=318.15 K [103].One would expect similar accuracy in the reproducing the ternary LLE, but it is notthe case here. Set_2 gives lower (about four times lower) objective function value,than set_1. In such a case, it has been decided to regress these 2-3 interaction pa-rameters from ternary data. The parameters regressed in that way are close to thesefrom set_2 and objective function value is similar (0.00233).

Regarding ethanol (1) + acetonitrile (2) + n -heptane (3) system only one dataset (set_1) gives lower objective function value in 2+3 binary LLE (the difference is2 orders of magnitude). Surprisingly, both sets give similar accuracy reproducingternary LLE diagram, slightly lower for the set_1.

In the next step, the regressed values of A′23, A

′′23 from ternary data were used to

calculate the LLE in binary system of acetonitrile + n-heptane. As it can seen from the


TABLE 5.6: Values of the different parameter estimates and objectivefunction values in the system {benzene or ethanol} (1) + acetonitrile

(2) + n-heptane (3).

System A′13 A

′′12 A

′23 A

′′23 Objective function

valueacetonitrile/n-heptane(binary at 318. 15 K)_set 1 0.115 1.107 2.20·10−5

(binary at 318. 15 K)_set 2 0.719 2.766 5.99·10−5

Benzene/ acetonitrile/n-heptane with set 1 318.15 K -8.483 -3.079 0.115 1.107 0.008489with set 2 -5.608 1.246 0.719 2.766 0.002249All 4 parametersregressed from ternary LLE -5.649 1.2196 0.705 2.759 0.00233acetonitrile/n-heptane(binary at 298.15 K)_set 1 0.1227 1.1813 7.64·10−6

(binary at 298. 15 K)_set 2 0.7669 2.9518 4.57·10−4

ethanol/ acetonitrile/n-heptane with set 1 298.15 K -3.976 6.348 0.1227 1.1813 0.0612with set 2 -2.617 5.577 0.7669 2.9518 0.0648All 4 parametersregressed from ternary LLE -7.431 5.071 -1.9813 2.9091 0.05624

Figure 5.10 it gives accurate match to the experimental data not close to the criticalregion, since the binary LLE model has not sufficient temperature- and compositiondependence.

This confirms that if the ternary LLE data are accurate, they can also help in theidentification of the c- parameters in the binary LLE model.


FIGURE 5.8: LLE in benzene (1) + acetonitrile (2) + n-heptane (3) atT=318.15 K Upper figures: A

′23, A

′′23 fixed as zero (left) ; set_1 of pa-

rameters (as in the Table 5.6), Lower figure: set_2 of A′23, A

′′23 (left); 4

parameters regressed from ternary data (right).


FIGURE 5.9: Ternary LLE in the system of ethanol (1) + acetonitrile(2) + n-heptane (3) at 298.15 K for different sets of A

′23, A

′′23 parame-

ters. Upper figures: interaction parameters between 2 and 3 are equalto zero (left), set_1 (right) Lower figures: set_2 (left), four parametersregressed from ternary data (right).In the acetonitrile + n-heptane sys-

tem (A′23, A

′′23) are significant.

FIGURE 5.10: LLE in the binary system of acetonitrile and n-heptanecalculated with the interaction parameters A

′23 and A

′′23 regressed

from ternary LLE data. Points are experimental data, lines modelprediction.

5.4. Conclusions 77

5.4 Conclusions

A new model has been derived and tested for the correlation of ternary LLE datatype I. Only two interaction parameters are obtained from ternary data, the remain-ing model parameters are regressed from only VLE and LLE binary data in the con-stituting pairs. The presented model is an extension of the binary LLE model tothree-component systems. Different types of systems, covering different ranges oftemperatures and compositions have been examined.

The FST-based model in most cases does better than NRTL, when both modelsfit binary parameters to binary data. If all six NRTL parameters are regressed fromternary data, NRTL performs better, but the predictive power is none.

The ternary data correlated with the proposed activity coefficient model can alsohelp in the identification of the interaction parameters A

′23, A

′′23 in binaries, which

has been flagged as difficult only from binary data alone. The model may be appliedin data validation. The criterion used for data validation is based on a form of themodel, that allows ternary diagram information to be obtained from mostly binarydata. The experimental points at low concentrations, if accurate, should follow themodel. Finally, the model may be applied to show inconsistency between reportedbinary LLE and ternary data, which is elaborated more in Chapter 6.

5.5 Parameter tables for binary systems


TABLE 5.7: Model parameters obtained from VLE data, asteriskmeans that in some cases isobaric VLE were used to regress parame-

ters.

No. VLE Ref. A′12(orA

′′13)∗ T∗

1 propan-2-ol/water [104] 2.024 303.152 benzene/n-heptane [105] 0.463 308.193 benzene/propan-2-ol [106] 1.34 298.154 benzene/propan-2-ol [107] 1.570 318.155 2-propanol/n-hexane [108] 1.8038 328.216 benzene/acetonitrile [98] 1.0109 318.157 ethanol/acetonitrile [109] 1.289 313.158 ethanol/n-hexane [108] 1.947(1.18;1423.21) 101.33*9 ethanol/n-hexane [56] 1.972 313.1510 benzene/cyclohexane [97] 0.4428 313.1411 benzene/n-hexane [110] 0.4454 328.212 ethanol/water [111] 1.3626 298.1513 ethanol/dibutyl ether [112] 1.5583 333.1514 ethyl acetate/acetonitrile [113] 0.4677 333.1515 ethyl acetate/n-hexane [114] 1.1649 353.1516 methanol/water [115] 0.6014 312.9117 methanol/ethyl acetate [116] 1.2025 313.1518 ethanol/ethyl acetate [116] 1.101 313.1519 benzene/sulfolane [117] 0.9024 303.1520 benzene/nitromethane [118] 1.228 298.1521 acetonitrile/water [109] 2.214953 333.1523 methanol/benzene [119] 2.199 298.1524 methanol/toluene [119] 2.143 318.1525 acetone/toluene [120] 0.7259 308.1526 methanol/propan-2-ol [121] 0.0549 328.1527 toluene/cyclohexane [122] 0.3746 318.1529 acetone/cyclohexane [123] 1.7432 298.1530 acetone/methanol [124] 0.72591 298.1531 ethanol/benzene [119] 1.808 303.1532 methanol/acetonitrile [125] 1.0696 303.1533 ethanol/cyclohexane [126] 2.2227 293.1534 nitromethane/ethanol [127] 1.93 298.1835 nitromethane/ethanol [127] 1.416 348.1836 benzene/n-octane [128] 0.286 328.1537 acetonitrile/chlorobenzene [129] 1.0987 328.1538 acetonitrile/toluene [113] 1.274 293.1540 ethanol/cyclohexane [126] 2.223 303.1541 ethanol/n-heptane [108] 2.342 303.1542 ethanol/n-heptane [108] 2.4248(-0.3395; 2506.6) 101.32*43 ethanol/n-octane [108] 2.287 313.15

5.5. Parameter tables for binary systems 79

TAB

LE

5.8:

Para

met

ers

obta

ined

from

the

LLE

bina

ryda

ta.

No.

Syst

em(2

/3)

Ref

.a′ 3

a′′ 2

b′ 3b′′ 2

c′(−

A′ 23

.T)

c′′(−

A′′ 23

.T)

1cy

cloh

exan

e/ac

eton

itri

le[1

03]

-7.0

15-2

.201

3068

.15

1553

.79

-528

-889

2ac

eton

itri

le/n

-hex

ane

[103

]-2

.59

-3.5

616

66.9

719

21.7

0-8

22.7

8-6

45.3

43

acet

onit

rile

/n-h

epta

ne[1

03]

-1.6

6-2

.22

1387

.57

1560

.18

-228

.65

-880

.10

4di

buty

leth

er/w

ater

[130

]-7

.71

-18.

4134

5278

050

05

n-he

xane

/wat

er[1

31]

4.35

17.1

411

21.1

0-1

248.

2-1

4902

2-6

9798

6he

xane

/sul

fola

ne[1

32]

-6.4

2-3

.18

4095

.09

2405

.78

-115

.99

-516

.76

7w

ater

/eth

ylac

etat

e[1

33]

5.64

31.

103

-516

.03

507.

318

-237

.526

-953

.793

8be

nzen

e/w

ater

[134

]-5

.043

3.22

932

38.9

614

02.4

80

09

tert

-am

ylm

ethy

leth

er/w

ater

[135

]-1

6.83

1-2

2.89

359

47.4

487

66.7

20

010

cycl

ohex

ane/

wat

er[6

7]-6

.67

-6.2

943

32.7

243

77.3

0-2

031.

56-2

022.

7011

nitr

omet

hane

/cyc

lohe

xane

[136

]-2

.82

-3.9

918

58.7

921

92.6

4-9

14.8

4-7

43.1

112

met

hano

l/cy

cloh

exan

e[1

37]

-0.7

6-4

.55

970.

6827

9.55

-805

.60

-601

.18

13ni

trom

etha

ne/w

ater

[136

]-3

.89

0.76

1985

.79

926.

57-3

64.8

7-2

524.

5714

nitr

omet

hane

/hep

tane

[136

]-6

.05

-3.7

531

63.5

821

08.3

5-1

023.

05-6

18.8

915

hept

ane/

met

hano

l[1

38]

-4.6

7-2

.91

2042

.12

1706

.89

-443

.80

-974

.27

16to

luen

e/w

ater

[69]

-3.3

45.

028

2784

.80

1079

.90

-0.4

-117

n-he

ptan

e/w

ater

[69]

-4.7

6417

.135

92.8

0-7

86.2

70

018

acet

onit

rile

/oct

ane

[103

]-1

3.55

-10.

3952

54.7

739

82.2

90.

003

0.00

819

chlo

robe

nzen

e/w

ater

[139

]-6

.884

2.15

138

41.2

3121

61.5

41-1

5.40

3-2

20.2

35

81

Chapter 6

Validation of binary and ternaryLLE

This chapter is based on the conference paper Liquid-liquid equilibria data validationpresented at 20th Symposium on Thermophysical Properties in Boulder, Colorado,USA, June 24-29, 2018.

6.1 Introduction

As mentioned in Chapter 2, there is a need for new methods, which can supportexisting ones in the field of data validation. Liquid-liquid equilibrium data accuracyare critical for the design and operation of separation processes of mixtures, and forunderstanding the behaviour of several products.

Compared to other fluid phase equilibria, liquid-liquid equilibria are difficult torepresent (quantitative description). This fact is usually explained by a strong tem-perature dependence of activity coefficients which plays a key role in the equilib-rium equations and generally is not well described by any thermodynamic model.Furthermore, the validation of the liquid-liquid equilibria is not straightforward asthe Gibbs-Duhem equations (a common basis for VLE consistency tests) do not re-late the compositions of both liquid phases being at equilibrium. Sørensen and Arlt[48] also pointed out the reason for non-existence of consistency test for LLE. Unlikefor the VLE, the pressure has extremely low impact on mutual solubilites at mod-erate conditions. Therefore it is not possible to calculate one of the experimentallymeasured variables from other experimentally measured variables.

In this chapter, a systematic methodology for the validation of the binary LLEdata and steps towards ternary LLE validation are discussed. This is done by theapplication of the rigorous model arising from FST. Such a model can help in thediscrimination of inconsistent data. The parameters regressed from binary LLE dataas well as from VLE binary data are used to model ternary LLE systems. The ad-vantage of this approach is the fact, that the LLE model based on FST has a strongtheoretical basis. The benchmark model, such as NRTL, is not rigorous, even thoughit can be made to represent certain data better.

6.2 Binary Tx LLE

A temperature dependent binary LLE data set here is one for which the compositionsof both coexisting liquids phases are reported at the same or similar temperatures.Such data can be visualized in phase diagrams at constant pressure (isobars). Onlysystems at atmospheric pressure or in general low pressure (up to 10 bar) have beenconsidered, although LLE exhibit low pressure dependence. Moreover, the current

82 Chapter 6. Validation of binary and ternary LLE

version of the model is not applicable to a reliable correlation in the critical region,therefore only the experimental data outside the critical region have been consid-ered.

The LLE models used in the validation procedure are based on fluctuation solu-tion theory (FST) and have been explained in the previous two chapters. The FSTmethod has the advantage of having a reliable concentration dependence of the ac-tivity coefficient especially at low concentrations, provided that the data used formodel parameters estimation are measured with minimal error (accurate).

6.2.1 General methodology

The methodology consists of two parts (I. Data collection and LLE modelling, II.Inter-property consistency check). It is summarised in Figure 6.1. The first partstarts with the collection of the binary LLE data from different sources, directly frompapers reporting data and/or from existing data collections/compilations. The con-solidated database is built of 104 different binary systems of different compounds,both molecular, commonly used in the chemical engineering organic solvents (hy-drocarbons, alcohols, ketones, esters, ethers etc.), water as well as ionic liquids. Intotal 2538 experimental data points have been collected. There is at least one dataset for each binary system with no less than 5 experimental points for each of thephases coexisting at the equilibrium.

Next step is the correlation of the LLE data with the adequate FST-based modeland uncertainty analysis of the model output. For binary mixtures, up to three pa-rameters (a, b and c) for each coexisting phase were explored for data sets with pointsnot-too-close to the critical region. For closed-loop or hourglass-like phase diagrams,one additional parameter for each phase in the expression for the infinite dilutionactivity coefficient is needed beyond the two (named a and b) already present. Thismeans that 4- ,6- or even 8-parameters can be applied. The output from this step is:

1. A set of the model parameters along with their uncertainty based on the parame-ters’ covariance matrix for the binary system (1)/(2).

2. Predicted molar fractions with 95% confidence interval (CI) and the evaluated ab-solute average relative deviation (AARD), which indicates the goodness-of-fit. TheAARD is later used to calculate the LLE data quality factor as described later in thissection.

Next feature of the methodology focuses on detecting possible outliers in thedata sets. Two types of outliers detection and/or influence statistics (Cook’s dis-tance [140] and COVRATIO [141]) are applied and compared. Application of thesetechniques requires the removal of one experimental point from the set and the per-formance of the previous steps, i.e. model parameters regression and uncertaintyanalysis associated with it. The number of the possible outliers also is indicative ofthe quality of the data set.

In general, the preliminary recommended data would be those which are not out-liers and have as low as possible deviation measure (or the quality factor QLLE,test asclose as possible to unity). The intention is not to explicitly remove these question-able data. However, the rejection procedure could be considered for these points,where the diagnostic measure exceeds three times its threshold value, which is sim-ilar to the rejection procedure implemented for LLE in the TDE software [36]. Thewhole validation procedure helps its user rather to realise the problems with some

6.2. Binary Tx LLE 83

data and to be careful while choosing the data for a certain application or in a certaincontext.

This summarises the first part (Data collection and LLE modelling) of the method-ology. Identification of derivatives of solution properties, directly from measureddata, requires high quality data. This suggests that discrimination of high-accuracydata from others based on identification of derivative properties could be also a cri-terion in data validation. Thus, the consistency with values of auxiliary solutionproperty data obtained from independent data – not the set considered – has beenexplored in the part II of the validation procedure.

The FST-based model has also the advantage that the a and b parameters aredirectly connected to calorimetric properties (they have a physical interpretation),which could be measured independently from LLE. Present efforts have been re-stricted to the comparison of the experimental excess partial molar enthalpy at in-finite dilution with that calculated based on the model b parameters. It is worthmentioning that this step can be done additionally because the experimental valuesare not always available. The same could be done for infinite dilution activity coef-ficients or even for a parameters as these are related to excess partial molar entropyat infinite dilution, but the latter kind of data are seldomly reported. In general, oneexpects consistent values when they have at least the same sign and order of mag-nitude, which is usually satisfied. A more empirical test assumes that they shouldfulfill the following criterion

0.5HE,∞i,experimental ≤ HE,∞

i,LLE ≤ 2HE,∞i,experimental . (6.1)

The criterion is similar to HE at equimolar composition obtained from VLE data(cf. [36]). In that test, values of 0.5 and 1.5HE,∞

i,experimental have been established. How-ever, based on the calculated values of partial molar excess enthalpy at infinite dilu-tion for several systems (cf. Table 6.1) and the fact that in general data in the dilutedregions are measured with lower accuracy, the range of calculated HE,∞

i is wider.Therefore, factors 0.5 and 2 have been established. This constitutes additional qual-ity test for LLE binary data.

Nonetheless, the comparison of the calculated values of calorimetric propertiesand infinite dilution activity coefficient obtained from LLE with these measured ex-perimentally need to be done carefully. Similarly as for LLE data, the experimentalexcess properties and infinite dilution activity coefficients might also not be so reli-able. The excess enthalpy or entropy as well as IDAC are measured at infinite dilu-tion and usually the accuracy of their values is lower and their values may naturallyvary. Therefore the inter-property check might not be conclusive. Hence, methodsfor the validation of these derivative properties should be developed, but this prob-lem has not been addressed in this thesis. However, if there are few experimentallydetermined values of the single property, one can verify them. If at least two valuesfrom different measurements agree with each other, it can be assumed that these aretentatively reliable. One would expect, that if there is more than one source of exper-imental IDACs and excess enthalpies for a particular system, they need to have thesame sign and at least the same order of magnitude to be reliable. Therefore, in thecomparison of excess properties (calculated from LLE and experimental), the firstcheck in the validation methodology developed has been done with the sign andorder of magnitude of the derivative properties. Moreover, it is an additional stepin the validation methodology and it is not always performed as data simply mightnot be available. Again, the methods for evaluation of experimental calorimetricproperties such as the excess enthalpy and activity coefficient at infinite dilution has


FIG

UR

E6.1:G

eneralbinaryLLE

validationm

ethodology

not been considered. Unfortunately reliability of data concerns all thermodynamicproperties.

In the next few subsections, the validation features are described in more detail.

6.2.2 Selection of the FST-based LLE model

Having Tx LLE data collected for a certain binary system (1)/(2), now one can corre-late these with the adequate LLE model developed in Chapter 4. As it was shown, ingeneral up to 6 parameters for the data sets with points not-too-close to the criticalregion are needed. In this validation procedure, the parameter estimation strategyhas been slightly modified. Previously, the regression strategy was to map the val-ues of cα and cβ in the predefined range and for each of their combination regress forthe values of aα, bα, aβ, bβ, modifying cα and cβ until the minimum objective functionvalue was found. The range for allowed values of cα and cβ was provided firstlyfrom COSMO-SAC, later directly from the stability criterion.

Here, the methodology has been slightly refined as additional constraints on themodel parameters have been added. It has been decided to use either the 4- or 6-parameter model with a simultaneous regression of all parameters. The selection ofthe model is based on the the graphical representation of experimental data, plottedas ln xω

1 = f (1/T) for phase ω = α, β.First step is to plot the experimental data in ln xω

1 = f (1/T) coordinates. It isoften found in practice that for low solubilities (xi <0.01 or ln xi -4.6 ), ln xω

1 is a nearlylinear function of the inverse temperature. For higher concentrations the deviationfrom the straight line is observed, which is corrected by the c parameter (responsiblefor the curvature of the line). The initial regression of the model parameters foreach phase separately is done (by simple fitting the ln xω

1 = f (1/T) dependency),assuming that the other phase is pure. This approach provides initial guesses forthe a, b, c in both liquid phases, for a proper parameter estimation, where the LLEisoactivity condition is solved.

The shape of the dependency then creates two possibilities:

a) 4-parameter model

This model is chosen when the initial values of c’s are close to 0 (c ≈ 0) andthere is no improvement in the goodness-of-fit, when c’s are added. If oneregresses also c although it was inconsequential, c would have a high uncer-tainty, definitely higher than a and b. This was shown in Table 4.8. The examplefor the octan-1-ol (1)/ water(2) is shown in the Fig. 6.2.

b) 6-parameter model

The 6-parameter model is used, if the c obtained from the plot ln xω1 = f (1/T)

in pre-regression step is much lower than zero c� 0. This means that the com-position effects become significant and the parameters c cannot be neglected.Moreover, use of 6 parameters improves the goodness-of-fit. Also, c must sat-isfy the stability condition, as it has been outlined in the Section 4.4.3. Anexample of a system, in which the c in both phases was found to be significantis presented in Fig. 6.3.


FIGURE 6.2: Initial regression of the model parameters for the systemoctan-1-ol (1) + water (2). Points - experimental data, Line - fittedcurve. Upper figure: Phase α (rich in octan-1-ol), c was found to be

-0.01; Lower figure: Phase β (rich in water), c = -0.0004.

6.2.3 Outliers detection techniques

An often overlooked tool in the thermodynamic data validation is an outlier detec-tion, which has been used in statistics for many years [142]. The outlier detection isespecially relevant for models dependent on the quality of experimental data used toregress model parameter estimates, such as the FST-based model. Several statisticalmethods and diagnostics have been developed to detect possible outliers.

One group of statistics used for this purpose is the so-called influence statistics,determining how influential a data point is, by measuring the effect of excludinga single data point. A highly influential data point can signify a potential outlier.According to Belsey et al. [141]: “An influential observation is one which, eitherindividually or together with several other observations, has a demonstrably largerimpact on the calculated values of various estimates (...) than is the case for mostof the other observations”. This section looks at possible influence diagnostics. Animportant aspect is that the outlier detection should be simple and mathematicallyconsistent.

An outlier detection procedure can be integrated in the data validation. Here,two ways of outlier detection are applied separately to identify the possible outliers.

Outlier detection with Cook’s distance

In 1977, R. D. Cook [140] introduced a diagnostic to estimate the influence of a singledata point by measuring the change in model values (parameter estimation) after the


FIGURE 6.3: Initial regression of the model parameters for the system[hmim][BF4] (1) + water (2). Points - experimental data, Line - fittedcurve. Upper figure: Phase α (rich in ionic liquid), c was found to be

-72; Lower figure: Phase β (rich in water), c was found to be -5841.

exclusion of data point i. A particular Cook’s distance Di can be assigned to everypoint i according to the Eq. (6.2).

Di =∑n

k=1(ypredk − ypred

k(i) )2

p ·MSE(6.2)

where ypredk is the model prediction (molar fractions) if all points were used to

regress parameters, ypredk(i) is the prediction if the i-th point is removed from the pa-

rameter estimation. MSE denotes the mean square error and p is the number ofmodel parameters.

By the definition data points having Cook’s distance larger than 4 divided by thenumber of data points n are considered as an outlier (cutoff value) [143]

Di >4n

. (6.3)

The Cook’s distance Di can be interpreted as the distance the estimated modelvalue moves within a confidence ellipsoid that represents a region of plausible val-ues for the parameters. A large value of Di indicates that the particular data point ihas a strong influence on the fit of the model and model parameter values.

COVRATIO

COVRATIO [141] is a statistic that measures the impact of each data point i on thecovariance matrix of the parameter estimates

COVRATIOi =det COV[(β(i))]

det[COV(β)](6.4)

where COVβ is the covariance matrix of the parameters if all experimental pointsare used, COVβ(i) is the covariance matrix for parameters from refitted regressionwhere observation i is excluded, det stands for determinant of a matrix.

COVRATIOi should be close to unity, i.e. COVRATIOi ≈ 1. If not, it meansthat the observation i has strong influence. When COVRATIOi < 1, data point idegrades the precision of the parameter estimation. When COVRATIOi > 1, thedata point i improves the precision of parameter estimation. The cutoff [141] is usedto determine if the point is a possible outlier

|COVRATIOi − 1| ≥ 3pn

(6.5)

This cutoff is only recommended when the number of experimental points n ismore than 3p, where p denotes the number of model parameters.

6.2.4 Connection between LLE model parameters and infinite dilutionpartial molar excess properties

The FST-based model (or unsymmetrical model) has two important advantages.Firstly, the activity coefficient has quite reliable concentration dependence providedthat the experimental data are accurate.

Secondly, the model parameters (a’s and b’s) are directly connected to the partialmolar properties at the infinite dilution. This advantage has been used in the inter-property cross-check test (Table 6.1).

From the model explored in the Chapter 4, we obtain the expressions for thelimiting activity coefficients of both components. As a reminder, the α phase is richin component 1 and β in component 2.

For the basic model with either 4 or 6 parameters:


bβ

T(6.6)


bα

T(6.7)

Hence, the partial molar enthalpy at the infinite dilution can be calculated asfollows:

∂ln γ1∞,β

∂1/T= −T2 ∂ln γ1

∞

∂T=

HE,∞1R

= bβ (6.8)

∂ln γ2∞,α

∂1/T= −T2 ∂ln γ2

∞

∂T=

HE,∞2R

= bα (6.9)

And finally, the partial molar enthalpy HE,∞i calculated based on the LLE is equal

to b[α,β] · R.


TABLE 6.1: Comparison between experimental and calculated partialmolar enthalpy at infinite dilution HE,∞

i for selected systems

No. System (1)/(2) Ref. HE,∞1 /kJ ·mol−1 HE,∞

2 /kJ ·mol−1

exp. calc. exp. calc.1 toluene/water [144], [145] 2.39 8.98 26.2 23.152 octan-1-ol/water [146] 8.86 3.5 1.183 dimethyl carbonate/water [147] -3.83 3.54 6.254 2-methylpentane/DMF [148] 8.027 14.21 25.25 aniline/water [149] 1.799 18.21 8.716 benzene/water [144] 2.365 11.66 26.937 diethyl ether/water [150] -19.54 -17.54 2.38 methyl-vinylketone/water [151] -6.92 -18.37 -9.269 water/furfural (288.15 K) [151] 0.897 -0.438 -9.459 water/furfural (298.15 K) [151] 2.39 0.697 -3.949 water/furfural (308.15 K) [151] 3.88 1.834 1.579 water/furfural (318.15 K) [151] 5.37 2.97 7.0710 tetrahydrofuran/water [150] -14.96 -109.72 -29.1611 butan-1-ol/water [152] -9.318 -0.312 1.4612 pentan-1-ol/water [152] -7.99 -19.68 5.1713 cyclohexane/acetonitrile (ACN) [153] 9.8 13.72 15.00 23.9814 ACN/pentane [153] 22.69 8.95 30.1115 ACN/hexane [153] 15.98 10.33 13.8616 ACN/heptane [153] 12.97 13.43 11.5417 ACN/water [85] -1.61 31.48 44.4618 nitroethane/cyclohexane [153] 12.7 35.82 37.0519 nitromethane/cyclohexane [153] 16.53 18.23 15.4520 methanol/cyclohexane [153] 20.9 17.29 4.98 8.0721 ethyl acetate/water [154] -7.95 -1.94 7.2922 methyl metacrylate/water (288.15 K) [147] -7.95 -7.3 11.6422 methyl metacrylate/water (298.15 K) [147] -5.35 -4.19 11.7822 methyl metacrylate/water (303.15 K) [147] -4.08 -2.64 11.8522 methyl metacrylate/water (308.15 K) [147] -2.93 -1.09 11.9222 methyl metacrylate/water (318.15 K) [147] -0.58 2.02 12.0623 ethyl propanoate/water (298.15 K) [154] -10.23 -3.04 23.6824 diethyl carbonate/water [147] -6.649 -47.48 4.1825 propyl acetate/water [154] -9.65 -9.32 8.3326 2-methylbutan-1-ol [155] -8.29 -4.8927 nitromethane/water [153] 3.52 7.7 14.00 16.5128 nitromethane/heptane [153] 17.53 14.5 26.329 octane/methanol [153], [84] 6.63 15.52 22.49 20.0630 heptane/methanol [153], [84] 5.73 14.19 22.93 16.9831 ACN/octane [153] 33.11 13.43 43.69HE,∞

i are reported at 298.15 K, otherwise temperatures are provided in the parentheses


From the well-known thermodynamic relation, one can obtain the expressionfor parameters aα and aβ, which are related to the partial molar entropy at infinitedilution, SE,∞

i

RT ln γ∞i = GE,∞

i = HE,∞i − TSE,∞

i (6.10)

After dividing both sides of the Eq. (6.10) by RT, the final relation between a andSE,∞

i is found

SE,∞1 = −R · aβ

SE,∞2 = −R · aα

(6.11)

It is noticeable, that the partial molar properties calculated in this way are con-stant i.e. temperature independent, which is not correct. This means that for thewhole Tx LLE data-set, one value of the partial molar enthalpy and entropy at in-finite dilution is assigned. Primarily, the simplicity in the model has been sought.However, to have a correct temperature dependence, one should consider the modelfrom the section 4.6. With that model, the enthalpy and entropy will have at leastlinear temperature dependence.

In the comparison between the calculated partial molar properties one needs toremember that the parameters should be regressed (i.e for LLE reported) at similartemperatures as the reported value of the partial molar property. Otherwise, it maylead to wrong conclusions.

As explained earlier, the reliable calorimetric data should be used for compari-son, whenever it is possible.

For example, the partial molar enthalpy at infinite dilution of acetonitrile (ACN)in water (system #17, Table 6.1) has different sign, but the miscibility gap in thissystem is below 273.15 K, whereas the partial molar value was measured at 298.15K. Similarly, the LLE for the system #10 (Table 6.1) was reported at the temperaturesbetween 353.15 and 403.15 K, whereas the partial molar enthalpy at infinite dilutionwas measured at 298.15 K.

6.2.5 Quality factor for LLE data

Similarly as for VLE and SLE data-sets, the quality factor based on the FST model iscalculated.

QLLE,test =

(1

1 + AARD

), (6.12)

where the AARD is the average absolute relative deviation

AARD(%) =100N

N

∑i=1

∣∣∣∣∣ xcalci − xexp

i

xexpi

∣∣∣∣∣. (6.13)

6.2.6 Selected examples

octan-1-ol (1) + water (2) system

The LLE data for this system have been taken from [156]. To correlate the data, a 4-parameter model (cf. Table 6.4 and Table 6.5) has been used, since it has been found,that the initial values of cα and cβ are close to 0. The experimental data, together withthe model fit and confidence intervals (CI) are shown in the Fig. 6.4. The AARD


was found to be 0.61% and the objective function value 4.14 · 10−6. Some pointsof are lying outside the CI. In the next step the outlier detection has been applied.According to the Cook’s distance (Table 6.2), there are two possible outliers, the firstand the last point in the α phase (Di for these points exceeds the cut-off value equalto 0.5). According to COVRATIO the most influential points are also 1st and thelast in the α phase, cf. Table 6.2. Although COVRATIO do not exceed the cut-off(|COVRATIOi-1| ≥ 0.75) value for these points, theirs COVRATIO is more remotefrom unity than for other points. It is often found, that the first and the last pointare identified as possible outliers in the data-set, as the peripheral points have thebiggest influence on the fit. Removing the possible outliers decreases an objectivefunction value two times (1.71 · 10−6 and AARD is equal to 0.54%).

TABLE 6.2: Outlier detection in the system of octan-1-ol/water. Sug-gested ouliers are marked by asterisk. Values of the partial molarproperties at infinite dilution (at 298.15 K) with i-th point omitted are

calculated.

No. (T/K) Cook’s COVRATIO Partial molar propertiesdistance SE,∞

2 / HE,∞2 / SE,∞

1 / HE,∞1 /

J · (mol−1 · K−1) kJ ·mol−1 J · (mol−1 · K−1) kJ ·mol−1

1 (288.15) 0.5592* 1.11* -6.56 1.25 -47.01 9.342 (293.15) 0.1264 1.01 -6.7 1.21 -48.39 8.93 (298.15) 0.1003 0.99 -6.83 1.16 -48.61 8.844 (303.15) 0.1089 0.98 -6.75 1.20 -48.61 8.835 (308.15) 0.1426 1.00 -6.76 1.19 -48.43 8.886 (313.15) 0 1.01 -6.78 1.18 -48.27 8.947 (318.15) 0.0001 0.97 -6.78 1.18 -48.86 8.778 (323.15) 0.7814* 0.91* -7.05 1.20 -49.72 8.51All -6.78 1.18 -48.48 8.86

In the case of octan-1-ol (1) / water (2) system, there are available experimentalvalues of the partial molar enthalpy at infinite dilution of water in octan-1-ol (3.5kJ · mol−1) [146] as well as the partial molar entropy at infinite dilution of waterin octan-1-ol (-3.68 J · (mol−1 · K−1)) [146], measured at 298.15 K. Both values areconsistent with the calculated ones (based on the model parameters regressed withall data points), which are 1.18 kJ · mol−1 and -6.78 J · (mol−1 · K−1), respectively.This data set is considered as a provisionally reliable, however few possible outliershave been recognized.

[hmim][BF4] (1) + water (2)

In this case, the FST model with 6 parameters (cf. Table 6.4) was used to correlate theLLE data in the system of 1-hexyl-3-methylimidazolium tetrafluoroborate (1) andwater (2) [72]. The experimental data, together with the model fit and confidenceintervals (CI) are shown in the Fig. 6.5. Most of the points are within the 95 %confidence interval. Cook’s distance and COVRATIO do not exceed the cut-off valuefor any point in the data set. Therefore, no outliers have been recognized.

The second part of the validation methodology, i.e. inter-property check withthe calorimetric properties is not possible, since there are no reported partial molarproperties at the infinite dilution neither for the ionic liquid (IL) in water nor waterin IL.


FIGURE 6.4: LLE in the octan-1-ol (1) and water (2).

[bmim][PF6] (1) + butan-1-ol (2)

The LLE data for the system of 1-butyl-3-methyl-imidazolium hexafluorophosphate(1) and butan-1-ol (2) were measured independently by four groups [157], [158],[159], [160]. In the 6-parameter FST LLE model regression (parameters are reportedin Table 6.4) all reported data are used to obtain parameters. Additionally, the re-gression of the model parameters separately in each data set was done as well. Theexperimental data along with the model fit using complete data-set and confidenceintervals are shown in Fig. 6.6.

The next step was the outlier detection. It has been found that some of the points,based on the Cook’s distance and COVRATIO, were highly influential on the modelfit, therefore are suspected as outliers. These are marked by black symbols in Fig.6.7.

Not unexpectedly, most of the points from the data-set # 3 are the most influ-ential. After removing the possible outliers the objective function value decreasedfrom 0.0211 about 10 times, to the value of 0.003. The removal of possible outliers


FIGURE 6.5: Liquid-liquid equilibrium in [hmim][BF4] (1) with water(2).

FIGURE 6.6: Liquid-liquid equilibrium in [bmim][PF6] (1) with butan-1-ol (2).

always increase the quality factor of the data-set.The comparison between experimental and calculated partial molar enthalpy at

the infinite dilution of butan-1-ol in [bmim][PF6] has been done, both based on theFST parameters regressed from all data and for each data set separately (Table 6.3).The experimental data have been reported at 288.15, 298.15, 308.15, 318.15 and 328.15K [161], therefore the range of the experimental HE,∞

2 is provided. However, oneshould bear in mind that the partial molar enthalpy at infinite dilution calculatedform the model parameters is constant, independent of a temperature. The com-parison shows a very good agreement with the experimental data. When the HE,∞

2is calculated by the FST model parameters, regressed for each set separately, onevalue (9.78 kJ ·mol−1) is off the measured values (set no. 3). This, together with thefact, that points from this set are found to be outliers, indicates that the set is lessconsistent than others.

LLE systems parameters

The LLE data validation methodology was applied to 92 binary systems. The param-eters for all considered binary LLE systems along with the quality measures (model’s


TABLE 6.3: Partial molar enthalpy at infinite dilution of butan-1-ol in1-butyl-3-methyl-imidazolium hexafluorophosphate.

Setno.of points

HE,∞,exp2 /kJ ·mol−1 HE,∞,calc

2 /kJ ·mol−1

1 10

10.66-13.41 [161]

10.402 28 11.093 8 9.734 32 12.79

All 78 12.36

FIGURE 6.7: Liquid-liquid equilibrium in [bmim][PF6] (1) with butan-1-ol (2). Possible outliers in data - experimental points in black circles.

deviation and number of potential, identified outliers) are presented in Tables 6.5and 6.6.

The evaluated binary LLE data (and regressed from these FST model parameters)along with the parameters obtained from VLE binary data for the constituting pairs


TABLE 6.4: FST-based model parameters along with their uncertain-ties for the considered systems

System (1)/(2) aα aβ bα bβ cα cβ

octan-1-ol/water 0.816 5.838 142.00 1065.55 0 -0.0004± 0.012 ± 0.012 ±3.68 ± 3.68 - -

[hmim][BF4] / water -1.157 -8.458 452.39 3784.20 -20.19 -7577.9±0.140 ±0.195 ±47.19 ±41.3 ±94.001 ±118.2

[bmim][PF6] /butan-1-ol -3.361 -4.244 1487.2 3248.45 -0.0403 -14364.7(all) ± 1.044 ± 1.006 ± 244.7 ± 233.97 ± 186.8662 ± 402.6[bmim][PF6] /butan-1-ol -2.557 -2.167 1364.6 1251.3 -278.304 -28572.9[157] ± 0.484 ± 1.479 ± 119.6 ± 352.08 ± 75.217 ± 2285.7[bmim][PF6] /butan-1-ol -2.808 -6.024 1334.75 3983.12 -2.169 -20470.3[162] ± 0.975 ± 0.910 ± 220.15 ± 202.18 ± 204.943 ± 1083.4[bmim][PF6] /butan-1-ol -2.493 -3.382 1170.46 823.336 -0.148 -32473.3[159] ± 5.621 ± 5.621 ± 1185.83 ± 1183.809 ± 1165.249 ± 491.3[bmim][PF6] /butan-1-ol -3.5361 -7.202 1537.88 4224.9 -0.173 -11120.9[158] ± 1.0513 ± 1.02 ± 249.59 ± 239.9 ± 178.4 ± 245.7

FIGURE 6.8: Quality factor distribution for studied systems.

are then used in the ternary LLE model. Figure 6.8 shows the quality factor QLLErange distribution of all systems studied. Approximately 80% of all compiled data-sets have quality factors greater than 0.9. Only 4 systems are characterised by thequality factor lower than 0.75. For example, for the data in the system of heptylamine(1) + water (2) a very low quality factor has been found (0.719), cf. Fig. 6.10. This iscaused by the fact, that the solubility curve for water in heptylamine has an unusualshape, due to the formation of hydrates in this liquid phase [163]. The current FST-based model is not suitable for such phase diagrams of unusual shape, therefore thegoodness of fit of a model is rather poor. The moderate value (0.926) is assigned ine.g. γ-valerolactone + cyclohexane (Fig. 6.9 and high value (> 0.99) for octan-1-ol +water system, which has been studied in the previous section.


FIGURE 6.9: LLE in γ-valerolactone (1) + cyclohexane (2)

FIGURE 6.10: LLE in heptylamine (1) + water (2)


TAB

LE

6.5:

FST

mod

elpa

ram

eter

sfo

rso

lubi

lity

data

,num

ber

ofpo

ssib

leou

tlie

rs,m

odel

devi

atio

nsan

dqu

alit

yfa

ctor

s.#P

-no.

ofex

peri

men

talp

oint

s,#O

-no.

ofpo

ssib

leou

tlie

rs.

Syst

em(1

)/(2

)R

ef.

#Paα

aβbα

bβcα

cβA

AR

D%

QLL

E#O

tolu

ene/

wat

er[6

9]16

-3.3

405.

028

2784

.80

1079

.90

-0.4

-14.

190.

960

2n-

pent

ane/

wat

er[5

2]12

-10.

820

15.3

0855

14.3

0-1

152.

000

03.

980.

962

4n-

hexa

ne/w

ater

[131

]18

-7.0

4017

.144

4322

.30

-124

8.80

0.1

-0.0

54.

610.

956

2n-

hept

ane/

wat

er[6

9]18

-4.7

6417

.100

3592

.80

-786

.27

00

6.79

0.93

62

n-oc

tane

/wat

er[6

8]16

-5.1

4922

.950

3737

.00

-203

2.00

00

8.48

0.92

20

[hm

im][

BF4]

/w

ater

[72]

20-1

.157

-8.4

5845

2.39

3784

.20

-20.

19-7

577.

905.

870.

945

0[o

mim

][BF

4]/

wat

er[7

2]28

-1.3

06-1

0.87

449

5.10

4794

.95

0.00

-113

84.2

022

.14

0.81

90

n-he

xane

/nit

roet

hane

[73]

14-7

.265

-8.4

5727

81.4

030

63.8

0-6

07.0

1-4

39.7

44.

310.

959

0n-

octa

ne/n

itro

etha

ne[7

3]20

-11.

492

-4.4

0939

88.5

020

09.9

0-1

12.0

2-6

92.1

25.

560.

947

42,

2,4-

trim

ethy

lpen

tane

/ni

troe

than

e[7

3]16

-5.4

42-1

.866

2262

.70

1216

.50

-601

.17

-677

.21

3.35

0.96

80

n-de

cane

/nit

roet

hane

[73]

14-4

.209

-7.6

9420

67.7

029

95.4

0-7

94.6

7-2

77.5

05.

010.

952

2oc

tan-

1-ol

/wat

er[1

56]

160.

816

5.83

814

2.00

1065

.55

0-0

.000

40.

610.

994

21-

met

hylim

idaz

ole/

cycl

ohex

ane

[164

]44

-5.3

23-7

.517

2393

.80

3707

.70

-0.0

9-1

375.

501.

540.

997

0[C

2mim

][O

Ac]

/[P

66

614

][O

Ac]

[165

]14

6.10

42.

046

-437

.87

-340

.14

-811

.00

3.32

0.28

0.99

74

[C2m

im][

NTf

2]/[

P66

614

][N

Tf2]

[166

]16

-3.2

47-2

.268

1946

.92

1104

.84

-118

4.75

-111

.63

3.12

0.97

02

[C2m

im][

NTf

2]/[

P66

614

][N

Tf2]

[166

]16

-8.9

72-3

.137

4158

.27

1405

.19

0.00

0.00

5.25

0.95

02

C2m

imC

l/P6

6614

Cl

[165

]12

6.81

63.

923

0.00

-592

.90

0.00

0.00

0.85

0.99

20

[HO

CH

2MIM

][BF

4]/b

enze

ne[1

67]

180.

153

-7.1

0749

6.56

3683

.40

-496

.33

-158

36.3

33.

480.

966

0n-

hept

ylam

ine/

wat

er[1

63]

30-8

.321

8.51

233

31.7

9-3

19.4

80.

000.

0039

.08

0.71

96

dibu

tyla

min

e/w

ater

[163

]20

4.67

813

.170

-925

.22

-176

0.40

-209

.00

-663

41.7

624

.66

0.80

20

tert

Oct

ylam

ine/

wat

er[1

63]

202.

082

21.7

39-2

40.7

2-4

687.

50-2

97.2

1-4

99.0

525

.66

0.79

60

2-m

ethy

lbut

an-1

-ol/

wat

er[1

68]

160.

265

6.76

429

9.90

-588

.16

-75.

51-1

05.7

73.

830.

963

2di

met

hylc

arbo

nate

/wat

er[1

33]

160.

135

2.31

975

2.14

425.

59-7

81.3

5-1

492.

801.

380.

986

2cy

cloh

exan

e/su

lfol

ane

[169

]28

1.49

0-2

.622

1022

.01

2032

.87

-452

2.53

-577

.57

3.09

0.97

00

cycl

ooct

ane/

sulf

olan

e[1

69]

26-5

.449

-2.9

3234

39.1

322

51.7

8-9

06.7

2-8

59.3

43.

120.

970

0n-

pent

ane/

sulf

olan

e[1

32]

28-1

2.26

0-0

.881

5673

.62

1238

.53

-118

3.86

-939

.48

0.59

0.99

40

n-he

xane

/sul

fola

ne[1

32]

26-6

.415

-3.1

8240

95.0

924

05.7

8-1

15.9

9-5

16.7

64.

260.

959

0n-

hept

ane/

sulf

olan

e[1

32]

28-4

.567

-0.7

2533

72.9

215

79.5

4-4

51.3

4-8

07.7

75.

770.

945

0n-

octa

ne/s

ulfo

lane

[132

]26

-7.3

33-1

.402

4179

.75

1853

.59

-107

.78

-817

.54

2.56

0.97

50

tert

-am

ylm

ethy

leth

er(T

AM

E)/w

ater

[135

]12

-19.

607

-9.9

5868

23.6

347

86.3

4-1

00.0

0-1

.15

15.2

00.

868

2di

buty

leth

er(D

BE)/

wat

er[1

30]

14-7

.710

-18.

407

3451

.68

7805

.32

0.00

0.00

21.7

50.

821

0C

onti

nued

onne

xtpa

ge


Tabl

e6.

5–

cont

inue

dfr

ompr

evio

uspa

geSy

stem

(1)/

(2)

Ref

.#P

aαaβ

bαbβ

cαcβ

AA

RD

%Q

LLE

#ON

-met

hylf

orm

amid

e(N

MF)

/hex

adec

ane

[170

]12

2.01

7-5

.913

1435

.49

3711

.49

0.00

0.00

8.82

0.91

90

2-m

ethy

lpen

tane

/DM

F[1

48]

18-7

.444

-3.1

1430

33.1

617

09.3

4-2

79.9

2-6

50.6

30.

980.

990

0n-

pent

ane/

NM

F[1

71]

36-5

.614

-0.5

6331

50.2

812

61.8

283

7.29

-120

3.40

2.09

0.98

00

n-he

xane

/NM

F[1

71]

26-0

.220

-0.0

2413

61.6

911

79.9

8-1

917.

57-1

484.

903.

840.

963

0n-

hept

ane/

NM

F[1

71]

28-4

.227

0.84

026

26.5

310

19.5

2-4

74.9

8-2

506.

552.

700.

974

0n-

octa

ne/N

MF

[171

]26

-1.4

08-0

.742

1658

.37

1553

.22

-113

6.18

-179

7.49

3.02

0.97

10

gam

mav

aler

olac

tone

(GV

L)/h

epta

ne[1

72]

16-2

.436

-3.4

7115

89.8

521

11.6

8-6

49.1

7-1

012.

763.

220.

969

0G

VL/

n-de

cane

[172

]20

-1.9

12-6

.064

1705

.30

2898

.00

-139

9.01

-422

.38

3.76

0.96

42

GV

L/n-

dode

cane

[172

]18

-2.4

93-6

.570

2018

.03

3031

.50

-170

1.50

-174

.95

1.23

0.98

80

GV

L/cy

cloh

exan

e[1

72]

12-0

.867

-6.5

5596

7.43

2906

.53

-660

.89

-784

.77

8.00

0.92

60

GV

L/22

4-tr

imet

hylp

ent-

1-en

e[1

72]

16-0

.911

-4.6

9797

6.22

2125

.33

-716

.50

-602

.09

0.86

0.99

10

met

hano

l/pi

nene

[173

]18

-1.1

88-2

.868

1229

.49

1531

.35

-112

1.16

-508

.56

1.42

0.98

62

etha

nol/

pine

ne[1

73]

16-5

.141

-1.2

3822

26.6

110

31.2

4-4

51.3

5-6

34.3

51.

720.

983

3pr

opan

e-1,

3-di

ol/[

bmim

][PF

6][1

59]

16-6

.309

-4.0

5032

16.3

118

06.0

3-1

99.9

6-9

2.50

1.52

0.98

56

prop

ane-

1,3-

diol

/[em

im][

BF4]

[159

]12

-14.

258

-9.3

6949

98.1

931

03.9

5-9

81.7

2-4

2.81

1.15

0.98

96

[bm

im][

PF6]

/but

an-1

-ol

[162

]78

-3.3

61-4

.244

1487

.232

48.4

5-0

.040

3-1

4364

.736

.12

0.73

58

[bm

im][

PF6]

/but

an-1

-ol

[157

]10

-2.5

572.

167

1364

.612

51.3

-278

.304

-285

72.9

1.25

0.98

71

[bm

im][

PF6]

/but

an-1

-ol

[162

]28

-2.8

08-6

.024

1334

.75

3983

.12

-2.1

69-2

0470

.36.

900.

935

5[b

mim

][PF

6]/b

utan

-1-o

l[1

59]

8-2

.493

3.38

211

70.4

682

3.33

6-0

.148

-324

73.3

8.71

0.91

90

[bm

im][

PF6]

/but

an-1

-ol

[158

]32

-3.5

361

-7.2

0215

37.8

842

24.9

-0.1

73-1

1120

.97.

600.

929

2[b

mim

][PF

6]/p

ropa

n-1-

ol[1

62]

28-3

.257

-9.4

6213

26.6

646

03.2

4-1

0.02

-500

.05

20.6

80.

829

2[b

mim

][PF

6]/p

enta

n-1-

ol[1

62]

28-3

.793

-1.9

2917

68.3

525

67.4

7-2

0.25

-500

.03

6.52

0.93

94

[hm

im][

PF6]

/but

an-1

-ol

[162

]28

-3.2

03-1

2.60

813

05.1

252

66.4

0-3

5.46

-499

.85

20.3

60.

831

0[h

mim

][PF

6]/p

ropa

n-1-

ol[1

62]

22-1

.960

-17.

883

1019

.45

6638

.10

-276

.27

-599

.21

22.3

30.

817

2[h

mim

][PF

6]/p

enta

n-1-

ol[1

62]

28-3

.457

-4.6

7214

72.2

631

68.3

5-1

4.13

-500

.29

14.5

10.

873

0[o

mim

][PF

6]/b

utan

-1-o

l[1

62]

20-4

.231

-15.

381

1452

.29

5873

.32

-0.2

9-5

01.1

87.

800.

928

0[o

mim

][PF

6]/p

ropa

n-1-

ol[1

62]

16-4

.290

-20.

151

1372

.93

6928

.00

-0.1

8-5

00.5

18.

310.

923

0[o

mim

][PF

6]/p

enta

n-1-

ol[1

62]

24-1

.277

-16.

252

909.

4965

52.6

0-3

94.6

9-5

69.5

713

.93

0.87

80

anili

ne/w

ater

[163

]36

-1.9

60-2

.527

1047

.63

2190

.09

-0.0

7-9

94.6

213

.04

0.88

52

benz

ene/

wat

er[1

34]

16-5

.05

3.23

3239

.48

1402

.13

00

2.10

0.97

90

phen

ol/w

ater

[174

]12

-1.2

4-8

.02

478.

1332

86.0

30

05.

530.

948

0di

ethy

leth

er/w

ater

[175

]26

3.00

311

.260

276.

84-2

110.

15-2

936.

13-4

54.9

43.

150.

969

0cy

cloh

exan

e/w

ater

[67]

24-6

.669

-6.2

8943

32.7

243

77.3

0-2

031.

56-2

022.

7016

.50

0.85

84

Con

tinu

edon

next

page


Tabl

e6.

5–

cont

inue

dfr

ompr

evio

uspa

geSy

stem

(1)/

(2)

Ref

.#P

aαaβ

bαbβ

cαcβ

AA

RD

%Q

LLE

#Oac

eton

itri

le/p

enta

ne[9

9]28

-9.5

25-6

.206

3621

.74

2728

.60

-138

.91

-670

.54

1.84

0.98

20

acet

onit

rile

/hex

ane

[99]

12-2

.590

-3.5

5816

66.9

719

21.7

0-8

22.7

8-6

45.3

410

.92

0.90

20

acet

onit

rile

/hep

tane

[99]

10-1

.663

-2.2

2113

87.5

715

60.1

8-2

28.6

5-8

80.1

09.

490.

913

0ac

eton

itri

le/w

ater

[99]

32-1

3.33

4-1

8.94

237

86.8

453

47.3

2-9

0.74

-40.

708.

450.

922

5ni

troe

than

e/cy

cloh

exan

e[1

76]

16-1

4.26

4-1

2.82

344

56.6

043

08.1

7-3

.29

-599

.96

1.80

0.98

20

nitr

omet

hane

/cyc

lohe

xane

[136

]18

-2.8

20-3

.989

1858

.79

2192

.64

-914

.84

-743

.11

0.89

0.99

10

met

hano

l/cy

cloh

exan

e[1

37]

16-0

.758

-4.5

5097

0.68

2079

.55

-805

.60

-601

.18

0.28

0.99

70

nitr

omet

hane

/wat

er[1

36]

38-3

.889

0.75

819

85.7

992

6.57

-364

.87

-252

4.57

3.29

0.96

80

nitr

omet

hane

/n-h

epta

ne[1

36]

36-6

.052

-3.7

5331

63.5

821

08.3

5-1

023.

05-6

18.8

91.

250.

988

0n-

octa

ne/m

etha

nol

[138

]16

-11.

341

-11.

169

4019

.341

69.5

-0.0

08-0

.005

3.25

0.96

90

n-he

ptan

e/m

etha

nol

[138

]12

-4.6

68-2

.909

2042

.12

1706

.89

-443

.80

-974

.27

2.29

0.97

82

met

hano

l/n-

hexa

ne[1

38]

10-1

.749

-5.5

7712

39.4

522

44.9

2-8

21.8

4-4

57.5

80.

140.

999

0ch

loro

benz

ene/

wat

er[1

39]

226.

884

2.15

138

41.2

3121

61.5

41-1

5.40

3-2

20.2

353.

550.

966

0ac

eton

itri

le/n

-oct

ane

[177

]10

-13.

549

-10.

394

5254

.77

3982

.29

0.00

0.01

2.62

0.97

50


TAB

LE

6.6:

FST

mod

elpa

ram

eter

sfo

rso

lubi

lity

data

,num

ber

ofpo

ssib

leou

tlie

rs,m

odel

devi

atio

nsan

dqu

alit

yfa

ctor

s.Sy

stem

sw

ith

clos

edlo

opor

hour

glas

sph

ase

diag

ram

s.

No.

Syst

em(1

)/(2

)R

ef.

No.

ofT r

efaα

aβbα

bβcα

cβdα

dβ#O

AA

RD

%Q

LLE

poin

ts1

met

hylv

inyl

keto

ne/w

ater

[178

]22

321.

1547

.084

91.7

65-4

6.36

1-8

9.74

5-1

06.1

60-9

65.5

-46.

202

-89.

257

02.

100.

982

wat

er/f

urfu

ral

[179

]40

325.

1565

.895

17.1

36-6

2.18

2-1

5.54

5-2

84.0

80-2

56.5

-66.

224

-17.

915

01.

960.

983

tetr

ahyd

rofu

ran

/wat

er[4

8]12

373.

1548

.311

172.

754

-47.

654

-171

.067

-105

.962

-162

.8-4

7.88

0-1

69.8

350

0.98

0.99

4bu

tan-

1-ol

/wat

er[1

68]

3230

8.15

6.50

437

.726

-5.8

63-3

3.73

90.

000

-4,9

08.4

-6.6

48-3

4.74

10

0.36

0.99

65

pent

an-1

-ol/

wat

er[1

68]

5430

3.15

8.08

666

.268

-7.4

85-6

1.37

70.

000

-1,6

50.7

-9.6

95-6

9.99

40

10.5

20.

96

buta

n-2-

one/

wat

er[1

78]

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0.99

6.3. Ternary liquid-liquid data validation 101

6.3 Ternary liquid-liquid data validation

Having a methodology for the validation of binary data, the next step is the inte-gration of this methodology and binary and ternary LLE models developed in thistowards a ternary LLE validation framework. The criterion used as basis in theternary LLE data validation is such that the activity coefficient model is rigorous forlow mole fractions, similarly as in the binary LLE model. In this way the regressioncan be used in the discrimination of inconsistent data. Moreover, the validation forternary LLE can be done by the cross-checking between ternary LLE data and datafor binary subsystems, either LLE or VLE. It should be possible, provided that bi-nary data are reliable, by using the parameters regressed from them, to reproducethe ternary diagram, which has been already addressed in the previous subsection5.3.3. The whole framework applied towards the ternary LLE data validation is pre-sented in Fig. 6.11.

FIGURE 6.11: Binaries to ternary system LLE data validation generalframework

Two tests for quality of the ternary LLE data sets were developed here. The firsttest is a relation for quality factor FtLLE,1 based on the deviation measure between ex-perimental data and data predicted by ternary FST-based model. This test is similarto the binary LLE quality factor proposed earlier in this chapter.

FtLLE,1 =

(1

1 + RMSD

), (6.14)

where the RMSD is a root mean square deviation used here to describe the errorbetween the experimental and calculated data defined by Eq. (5.39) in Chapter 5.

The second test is an analogy to the Test 1 developed by Cunico et al. It evaluateswhether the ternary data asymptote to the binary points (binary partially misciblepair 2 + 3), as the mole fraction of the component 1 approaches 0, xi → 0. The qualityfactor associated with this test is calculated as

FtLLE,2 =

(2

100(∆x′,bin2 + ∆x

′′,bin3 )

)(6.15)


TABLE 6.7: Examples of results for the test FtLLE,2.

Compounds(1) (2) (3) T/K x

′ ,bin2 x

′′ ,bin3 ∆x

′ ,bin2 ∆x

′′ ,bin3 FtLLE,2

ACN chlorobenzene water 304.15 0.9 0.9999 0.097 0.000008 0.206x′b,exp2 =0.997 x

′′b,exp3 = 0.99991

cyclohexane ACN n-heptane 298.15 0.9589 0.9286 0.00305 0.0121 1x′b,exp2 = 0.956 x

′′b,exp3 = 0.9399

cyclohexane ACN n-heptane 318.15 0.9126 0.8834 0.0067 0.00555 1x′b,exp2 = 0.9065 x

′′b,exp3 = 0.8883

propan-2-ol benzene water 323 0.9901 0.9922 0.0103 0.00219 1x′b,exp2 =0.98 x

′′b,exp3 = 0.99

ethanol ACN n-heptane 318.15 0.9489 0.9306 0.0137 0.00777 0.929x′b,exp2 = 0.9622 x

′′b,exp3 = 0.9379

where

∆x′,bin2 =

∣∣∣∣∣ x′,bin2 − x

′b,exp2

x′b,exp2

∣∣∣∣∣ , (6.16)

and

∆x′′,bin3 =

∣∣∣∣∣ x′′,bin3 − x

′′b,exp3

x′′b,exp3

∣∣∣∣∣ . (6.17)

In Eqs. (6.15) to (6.17), x′,bin2 is the mole fraction of the component 2 in the phase

’ measured or extrapolated to the limit x1 → 0 (at ternary diagram), x′b,exp2 is the

measured mole fraction of the component 2 in phase ’ in the binary (2 + 3) system(taken from a binary LLE phase diagram). The same applies to the mole fractionof the component 3 in phase ”. In this study, the values of ∆x

′′,bin3 and ∆x

′,bin2 have

lower limit of 0.01. If then, the molar fraction agrees within 0.01∆xi, the qualityfactor FtLLE,2 is equal to 1 (the maximum value). If the discrepancy is higher, thisfactor becomes smaller.

The use of the two tests forms the overall quality factor QtLLE for the ternary LLEdata-set

QtLLE = 0.5FtLLE,1 + 0.5FtLLE,2. (6.18)

The overall quality factor QtLLE is equal or less to unity.Table 6.7 gives the examples of results for the FtLLE,t2 factor. There are cases in

which the factor is very close to unity or equal to one (e.g. propan-2-ol + benzene+ water), which means that data are consistent according to the used method. Forsome cases, the quality factors become lower e.g. FtLLE,t2 is very low (acetonitrile+ chlorobenzene + water), which indicates the inconsistencies between binary com-positions at both ternary and binary phase diagrams. In Table 6.8 the values of theFST model parameters and quality factors for several ternary systems treated with acurrent methodology are presented.

6.3.1 Selected examples

Here, three examples, which illustrate data with different overall quality factor arepresented.


TABLE 6.8: Ternary LLE FST model parameters along with qualityfactors.

System (1)/(2)/(3) T/K Ref. A′13 A

′′12 RMSD TL* FtLLE,1 FtLLE,2 QtLLE

benzene/acetonitrile/n-hexane 318.15 [181] -11.94 -11.28 0.072 4 0.933 1.00 0.966benzene/acetonitrile/n-hexane 308.15 [181] -9.72 -10.60 0.056 5 0.947 0.98 0.963benzene/acetonitrile/n-hexane 298.15 [181] -1.44 -3.98 0.106 4 0.905 1.00 0.952benzene/acetonitrile/n-heptane 318.15 [102] -5.61 -1.25 0.016 7 0.984 1.00 0.992ethanol/dibutyl ether/water 298.15 [130] -8.47 -0.85 0.245 6 0.803 1.00 0.902ethanol/n-hexane/acetonitrile 313.15 [109] -1.62 2.45 0.031 7 0.969 0.46 0.715ethyl acetate/ACN/n-hexane 318.15 [182] -4.81 -9.15 0.082 6 0.924 0.42 0.670ethyl acetate/ACN/n-hexane 308.15 [182] -4.77 -8.90 0.103 6 0.907 1.00 0.953ethyl acetate/ACN/n-hexane 298.15 [182] -4.33 -9.17 0.148 6 0.871 0.51 0.693propan-2-ol/hexane/water 303 [101] -11.13 -9.75 0.063 8 0.941 1.00 0.970propan-2-ol/n-hexane/water 313 [101] -5.22 -5.10 0.070 8 0.934 1.00 0.967propan-2-ol/n-hexane/water 323 [101] -5.07 -5.03 0.067 8 0.937 1.00 0.969benzene/n-hexane/sulfolane 348.15 [183] -8.88 0.22 0.165 4 0.859 1.00 0.929benzene/n-hexane/sulfolane 373.15 [183] -10.41 2.82 0.144 4 0.874 1.00 0.937ethanol/water/ethyl acetate 298.15 [184] -38.16 -15.72 0.081 5 0.925 0.25 0.587ethanol/water/ethyl acetate 308.15 [184] -35.34 -12.58 0.095 5 0.913 0.27 0.591ethanol/water/ethyl acetate 318.15 [184] -35.70 -10.52 0.088 5 0.919 0.26 0.591methanol/water/ethyl acetate 343.15 [185] -20.42 -20.13 0.159 5 0.863 0.33 0.598propan-2-ol/benzene/water 303 [101] -31.61 -17.54 0.057 6 0.946 1.00 0.973propan-2-ol/benzene/water 313 [101] -30.09 -18.54 0.052 6 0.951 1.00 0.976propan-2-ol/benzene/water 323 [101] -26.29 -17.33 0.071 5 0.934 1.00 0.967methanol/t-amyl methyl ether/ 298.15 [135] -3.64 -11.02 0.100 6 0.909 1.00 0.954waterpropan-2-ol/cyclohexane/water 303 [101] -17.58 19.54 0.064 7 0.940 1.00 0.970propan-2-ol/cyclohexane/water 313 [101] -25.71 11.78 0.085 6 0.921 1.00 0.961propan-2-ol/cyclohexane/water 323 [101] -31.90 32.02 0.078 6 0.928 1.00 0.964benzene/nitromethane/ 298.15 [186] -9.48 -12.30 0.069 4 0.935 1.00 0.968cyclohexanebenzene/MeOH/cyclohexane 298.15 [187] -4.36 -3.25 0.031 3 0.970 1.00 0.985toluene/MeOH/cyclohexane 298.15 [187] -3.82 -1.82 0.022 3 0.979 1.00 0.989acetone/MeOH/cyclohexane 298.15 [187] -4.50 -3.38 0.050 4 0.952 1.00 0.976propan-2-ol/MeOH/cyclohexane 298.15 [187] -0.01 -2.36 0.063 7 0.940 1.00 0.970ethanol/nitromethane/water 303.15 [186] -1.64 -4.58 0.048 4 0.954 1.00 0.977ethanol/nitromethane/water 313.15 [186] -1.98 -4.22 0.041 4 0.96 0.75 0.756ethanol/nitromethane/water 333.15 [186] -1.25 -4.44 0.023 4 0.977 0.61 0.796benzene/nitromethane/n-heptane 298.15 [186] 10.59 -4.09 0.120 6 0.892 1.00 0.946benzene/nitromethane/n-heptane 303.15 [186] 16.31 -4.93 0.076 6 0.930 1.00 0.965ACN/chlorobenzene/water 304.15 [99] -7.76 -26.79 0.108 8 0.903 0.21 0.554ACN/benzene/water 298.15 [99] -15.58 -17.27 0.099 8 0.910 1.00 0.955ACN/benzene/water 318.15 [99] -4.93 -46.14 0.025 7 0.975 1.00 0.988ACN/n-hexane/water 298.15 [99] -5.32 -2.80 0.070 5 0.935 1.00 0.967ACN/toluene/water 303.15 [99] -5.19 -28.46 0.083 4 0.923 1.00 0.962ethanol/ACN/cyclohexane 298.15 [99] -1.67 -4.52 0.081 4 0.925 0.91 0.917ethanol/ACN/n-heptane 298.15 [99] -2.62 -5.58 0.096 7 0.912 1 0.955ethanol/ACN/n-heptane 313.15 [99] -6.24 -6.73 0.089 5 0.918 0.93 0.924ethanol/ACN/n-octane 298.15 [99] -4.65 -3.13 0.054 7 0.949 1.00 0.975benzene/cyclohexane/ACN 318.15 [99] -3.44 -1.65 0.035 7 0.966 1.00 0.983benzene/cyclohexane/ACN 298.15 [99] -3.60 -1.73 0.056 6 0.947 1.00 0.973* TL- number of tie lines


• benzene (1) + acetonitrile (2) + n-heptane (3) The ternary and binary LLE datafor the system benzene (1) + acetonitrile (2) + n-heptane (3) at T=318.15 K [102]are presented in Fig. 6.12. In principle, the compositions of both phases, whenmole fraction of the component (1) becomes zero at the ternary LLE phasediagram (which might be part of the ternary data set) should correspond tothose reported in the binary phase diagram for the given temperature.

FIGURE 6.12: Ternary LLE phase diagram benzene (1) + acetonitrile(2) + n-heptane (3) at T=318.15 K [102] along with the correspond-ing binary LLE in acetonitrile (2) + n-heptane (3) [103]. Orange lines,correlation with FST-binary model, red and green lines are 95 % con-fidence intervals, lower and upper bounds, respectively. Blue lines

ternary FST-based model.

From Fig. 6.12 it is visible that both the experimental points and model predic-tions are in agreement between both types of data (binary or ternary LLE). Thisis also confirmed by the high value of the overall quality factor equal to 0.983.Moreover, this example shows also that the FST-model (in principle restrictedto diluted systems, mole fractions less than 0.01) can reliably correlate data athigher concentrations as long as they are away from the plait point.

• acetonitrile (1) + chlorobenzene (2) + water (3)

The second example is the ternary system of acetonitrile (1) + chlorobenzene(2) + water (3) [102] presented at the phase diagram along with the binary com-positions for the phase ’ (rich in chlorobenzene) and ” (rich in water). Thesedata in the ternary phase diagram seem not to be consistent with those mea-sured in the binary system of chlorobenzene and water [139] and presented atFig. 6.14. Both data sets – binary (chlorobenzene + water) and ternary – areinternally consistent, but when inserting experimental points in the binary 2+3edge, some discrepancies can be revealed (Fig. 6.13). The quality factor FtLLE,2is very low (0.21), which makes also the overall quality factor QtLLE = 0.554.It is often difficult to say unambiguously, which data are wrong, but in this


case phase ′ seems to be inconsistent. This is also supported by the fact thatthe mutual solubility of benzene or halogenated benzenes in water is very low[188]. Such and observation cannot be made based on this ternary diagram.

FIGURE 6.13: Example of suspicious ternary LLE data acetonitrile (1)+ chlorobenzene (2) + water (3) at 304.15 K.

FIGURE 6.14: LLE in the binary system of chlorobenzene (1) and wa-ter (2)

• ethanol (1) + nitromethane (2) + water (3)

Here, the example of the LLE data in the system of ethanol (1) + nitromethane(2) + water (3) at two temperatures 303.15 and 313.15 K is described. In this casethe LLE data (nitromethane/water) of the binary system are very well fittedby the 6- parameter FST model, even though the solubility are higher than 0.01(cf.6.15). Unfortunately, when transferring LLE binary parameters to ternarymodelling, the model is able to fit quite well the data in the nitromethane-rich phase, the water-rich phase reproduction is much worse. It might be thatsolubility data for water-rich phase are less accurate. It might be also thatthe VLE (nitromethane/ethanol) parameters were regressed to data at muchhigher temperature 348.15 K. At the lower temperature, the ternary model fitis worse than at 313.15 K. However, the quality factor FtLLE,2 describing the setat 313.15 K is almost two times lower than at 303.15 K. The reason for that is


a bigger difference between the corresponding binary compositions at the bi-nary phase diagram (x

′′3=0.9574, x

′2 =0.89) vs. compositions taken from ternary

phase diagram (x′′3=0.968, x

′2=0.876).

FIGURE 6.15: LLE in the binary system of nitromethane (1) and water(2) (lower figure) and ternary ethanol (1) + nitromethane (2) + water(3) LLE data (upper figure). Blue lines are model predictions, circles

are experimental points.

6.3.2 Summary

Based on the results, there is a clear connection between the goodness of FST-ternariesmodel fit and data quality in most cases. However, sometimes data-set can be char-acterised by an accurate goodness-of-fit (for both ternary and binary LLE data) andsubsequently the high value of the quality factor FtLLE,1, but the quality factor con-nected to the cross-check between ternary and binary phase diagram is lower than1. Therefore, this two factors should be taken into account and the overall weighted

6.4. Conclusions 107

factor should be assigned to the data-set. This procedure is in the spirit of the vali-dation techniques developed by other researchers for binary SLE and VLE data.

Here, the whole LLE data validation framework utilizes a (nearly) rigorous model,which originally was restricted to low solubilities. Despite that fact, its applicationwas extended successfully for data outside the diluted regions, but away from crit-ical (plait) point, i.e. experimental points close to plait point were excluded fromconsideration. Moreover, the LLE ternary model is an extension of the binary model.Thus, it relies on the parameters obtained from binary LLE (and VLE) data, whichshould be reliable. It is unambiguous that binary parameters should be regressedfrom VLE and LLE data at the same or similar range of temperature as the reportedternary phase diagrams. The ternary LLE data can also provide an additional helpin the binary data LLE validation. The parameters obtained from the binary data,which are accurately reproducing the ternary LLE suggests the high quality of bi-nary data-set.

The current validation methodology for LLE data has been applied to in total140, both binary and ternary, systems. The systematic combination of the informa-tion obtained from validated LLE, regression of FST-model parameters and a crosschecking between different kind of data leads to the final statement about the dataquality, expressed for LLE ternary data as overall quality factor.

6.4 Conclusions

The framework for the validation of the experimental LLE ternary data has beendeveloped. The framework consists of the developed binary and ternary LLE mod-els based on the fluctuation solution theory, validation methodology for binary LLEdata and finally combination of these in the ternary LLE data validation.

A methodology for the validation of the experimental LLE data in binary systemshas been set up. The methodology is divided into two parts: LLE modelling andinter-property consistency check. The first part includes a development of a newLLE model, estimation of its parameters along with the uncertainty analysis andevaluation of the regressed parameters. The LLE model, for both binary and ternarycases, is based on the statistical theory of solutions and rigorous in the region ofinfinite dilution (close to reference states of components). Moreover, a very practicaltool such as influence statistics has been used for the detection of possible outliers.The quality factor, based on the FST model for each of the LLE data-set is proposed.

The second part of the methodology is a comparison with other solution prop-erties, values of which might be obtained directly from the FST-based model pa-rameters. Moreover, the parameters of the model can be directly interpreted as thepartial molar excess (enthalpy and entropy) at the infinite dilution as well as to thecorrelation functions integrals from FST.

Having the binary LLE model, the initial steps towards the validation of ternaryLLE have been addressed. These are based on the identification of the developedternary model parameters from the corresponding binary pairs, i.e. vapour-liquidand liquid-liquid equilibria data. The FST-based model is nearly rigorous when theapplication is restricted to dilute solutions. The model has been used outside itsproven range. This aspect is essential to data validation.

It is important to note here, that this systematic methodology for validation ofternary LLE based on a (nearly) rigorous model is the only one available in literature.Nonetheless, it has some limitations. The main limitation of the model is that itcannot treat the experimental points close to critical point in the both binary and


ternary phase diagram. This is due to the model formulation, which includes therigorous expression for the unsymmetrical activity coefficient in the dilute region.However, it has been shown that the FST model can be applied for solubilites greaterthan 0.01 when expressed in mole fractions. Moreover, the calorimetric propertiesused in Part II of the binary LLE validation methodology are available only for thelimited number of binaries and almost not available partial molar excess propertiesin ternary systems. Finally, the ternary LLE model is restricted only to the systemsfor which the LLE and VLE data for the constituting sub-systems exist. Finally, onlytype I (one partially miscible pair) of ternary diagrams was considered.

109

PART IIModelling

and datavalidation ofsolubility of

solids

111

Chapter 7

Solubility of solids in mixedsolvents

This chapter presents a thermodynamic method for validation of solid solubilites inmixed solvents. The derivation of the excess solubility model used in this frame-work has been already shown in Chapter 3. Here, results for modelling, parameterestimation and validation of the solubility of solids data as well as limitations of thisapproach are presented.

7.1 Introduction

In the pharmaceutical industry, a commonly employed separation method is crys-tallization. The driving force in crystallization is supersaturation, which is a state ofa solution that contains more dissolved solute than can be dissolved at equilibrium.Hence, a crucial piece of information is the phase diagram of the system includinga saturation (solubility) curve. There are two main modes of performing a crystal-lization: cooling and antisolvent. In the latter mode a secondary solvent, called aprecipitant or anti-solvent is added to the solution causing a decrease in solubilityof the solute of interest. In this way, the solubility of solids in mixed (binary) sol-vents are relevant to separation process design. Experimental solubilities of solidsin single solvents have been reported extensively. Those in mixed solvents are alsoreported extensively in terms of numbers, but due to the large number of possiblecombinations of solvent mixtures, knowledge of such experimental data sets is fre-quently incomplete. Therefore, methods for solubility prediction of solids in mixedsolvents are desirable.

In the past few years, several models have been examined for predicting solubil-ity of more or less complex solutes in single and mixed solvents e.g. UNIFAC [189],PC-SAFT [190], COSMO-RS [191], COSMO-SAC [192], [193], [194] and NRTL-SAC[195], [196]. In spite of numerous efforts a great deal of data fitting is required tomake these models represent experimental behaviour.

Another other group of models is based on the excess solubility concept [197].One example is the model explored by Ellegaard et al. [60]. The model has a simpleformulation, but a strong foundation arising from statistical mechanical fluctuationsolution theory for composition derivatives of activity coefficients. The main con-cept is to obtain the derivative of the activity coefficients in the mixture of solvents inwhich the solute is dissolved. This is usually done via correlation of vapour-liquidequilibria (VLE) data in solvent-solvent systems with GE models such as Wilson,

112 Chapter 7. Solubility of solids in mixed solvents

NRTL or Modified Margules equation. To make this model less dependent upon ex-perimental data one could use for this purpose a predictive model such as COSMO-SAC if satisfactory results in the prediction of VLE in the binary solvent-solventsystem could be achieved.

An additional challenge to modelling of crystallization of active pharmaceuti-cal ingredients (APIs), is that very often the solute can form different polymorphs[198]. Either as pure (single component) crystals or mixed crystals where solventmolecules are incorporated into the crystal structure, thus forming solvates (pseudo-polymorphs). This transformation among different forms results in irregularities ofthe chemical and physical properties including solubility. The formation of the poly-morphs depends usually on the type of the solvent used for crystallization and thecomposition of the binary solvent and a number of other things. This phenomenonwould also be considered in the solubility calculations, but it is not straightforwardsince thermophysical properties of the solid phase at equilibrium are not usuallyavailable due to the instability of solvates and the transition point (which could beat certain temperature or solvent composition) is not reported in the literature data.In this chapter, the effect of polymorphic transformations is incorporated into themodel of Ellegaard. Moreover, the usage of the COSMO-SAC model for the solventmixture representation has been explored.

Furthermore, in general (but not least) solubility data should be reliable. There-fore, methods for validation of the experimental/predicted data are of utmost im-portance. The rate of publication of measured data has continued to increased inrecent decades.

This issue has been addressed by developing some criteria allowing preliminaryscreening of reliable data. Existing works on the evaluation of solid solubility dataare available from Cunico et al. [40] and from Kang et al. [36]. They have developedtests for quality of SLE data in particular for binary, eutectic SLE systems. In thework of Cunico et al. two validation tests have been proposed. The first data qualitytest evaluates whether the binary data asymptote to the pure components meltingpoints. The second one uses the FST-based activity coefficient model to evaluate aquality SLE factor based on the deviation measure (absolute average deviation) fromthe model. Kang et al. [36] proposed a quality factor for binary SLE data (QSLE) as ananalogy to those shown for VLE data [29], that can be calculated based on differencesbetween values calculated from solubility data and the pure component properties.Regarding ternary mixtures, there is only one existing method to test the quality ofdata suggested by Ruckenstein and Shulgin [39]. It has been described in details inChapter 2, section 2.2.1. Briefly, this thermodynamic consistency test is based on theGibbs-Duhem equation. It has a form of an equation, which connects the solubilitesof the solid and the activity coefficients of the constituents of the mixed solvent intwo mixed solvents of close compositions.

Here, a new procedure consisting of a set of rules for preliminary screening forreliable SLE data in ternary systems (solubility of solid substances in binary solvents)has been developed. The methodology is based on the model arising from statisticalthermodynamics, which requires very limited input data.

7.2 Modelling

This section deals with the modelling of the solubility of solids in binary solventmixtures by means of a model based on Fluctuation Solution Theory. The modelrequires a GE model for the binary solvent mixture, values of two (solute-solvent)

7.2. Modelling 113

f 01j parameters for either solute-solvent pair. If the ideal solubility is required, ther-

mophysical properties of the solid species are also required.The model has been already derived in Chapter 3. Here only the final form of

the model is presented.

7.2.1 Excess solubility model

The excess solubility model (assumes same solid form solidifies) of interest is of theform

lnxE1 = lnx1 −

(x′2 ln x1,2 + x

′3 ln x1,3

)≈ − x3

2

(∂lnγ3

x3

)+

T,P,n2

[1 + x

′2 f 0

12 + x′3 f 0

13

],

(7.1)with two solute-solvent interaction parameters f = [ f 0

12, f 013].

The derivative in Eq. (7.1) is independent of the solute 1 (and the product withx3 is symmetric with respect to solvent index, in accordance with the Gibbs-Duhemequation). The superscript ’ denotes to the solute-free mole fraction.

To calculate the product x3

(∂lnγ3

x3

)+T,P,n2

= x2

(∂lnγ2

x2

)+T,P,n3

one needs a GE model.

This could be one of the correlation equations such as Wilson, Modified Margules,NRTL or more predictive e.g. UNIFAC or COSMO-SAC. The latter model can calcu-late, in theory, the activity coefficient of all species in multicomponent mixture withminimum input obtained from quantum mechanical calculations. Several options toobtain the solvent-solvent term have been explored. First, the COSMO-SAC modelhas been used to calculate the derivative of the activity coefficient with respect tomole fraction. To test this approach, the derivatives obtained by the COSMO-SACmodel were compared with those obtained from correlation of Wilson, Margulesor NRTL to the VLE data. It has been found that COSMO-SAC reproduces thederivatives with qualitative accuracy. However, especially at infinite dilution, thederivative is highly underestimated. This approach therefore should be only usedwhen there are no VLE data available for the solvent (2) + (3) system. Second ap-proach was the use of the VLE data. These were correlated by the 5-parameters(Aji, Aij, αij, αji, η) Margules model (binary only) according to Eq. (7.2) or the Wilsonequation given by Eq. (7.5) to obtain the parameters. Then, the derivative of theactivity coefficient with respect to the mole fraction has been calculated.

ln γi = Gi(1− xi)2, (7.2)

where

Gi = Aij + 2(Aji − Aij)xi − 2F(1)xixj + F(2)(αij + ηx2j )x2

i (7.3)

and F(k) is defined as follows

F(k) =αijαjixixj

(αijxi + αjixj + ηxixj)k (7.4)

The 2-parameter (aij, aji) Wilson activity coefficient model is

ln γi = 1− ln ηi −∑k

xkεk (7.5)

where


εki =Λkiηi

ηk = ∑j xjΛkj

Λkj =v0

j

v0k

exp(− akj

T

) (7.6)

where v0j denotes the molar volume of the pure component i.

7.2.2 Obtaining model parameters

There are three ways of obtaining the solute–solvent interaction parameters f 01j. f 0

1jis connected to the derivative of the activity coefficient at the infinite dilution withrespect to mole fraction. Considering the equation for the case of 1 in 2

f12 = − limx1→0

(∂ ln γ1

∂x1

)T,P,n2

. (7.7)

One method of obtaining values of the solute-solvent parameters can be from thederivatives of activity coefficient in pure solvents. A single experimental solubilitypoint allows to determine a single parameter of a GE model. The simplest modelapplied here could be the Porter equation (Eq. (7.8)). The expression for the activ-ity coefficient and the further relation between the Porter model parameter and themolecular integral f 0

1j are as follows

lnγ1 = A1j (T) (1− x1)2, (7.8)(

∂lnγ1

∂x1

)0

T,P,n2

= −2A1j = − f 01j. (7.9)

The value of A1j can then be found by rearranging Eq. (7.10)

x1,j =1

γ1,jexp

[∆H f us,1

RTm,1

(1− Tm,1

T

)], (7.10)

to obtain

f 01j (T) = 2

ln xid1

(T, ∆H f us, Tm

)− ln x1,j(T)

(1− x1,j (T))2 . (7.11)

It is important to note here, that determination of the Porter parameter requiresalso the thermophysical properties of the solute (1): melting point Tm and enthalpyof fusion ∆H f us. It is worth noting, that the thermophysical properties data of puresolid must be as accurate as possible to make the model reliable. The enthalpy offusion and melting point of the solute are measured using calorimetry. It happensvery often, that the values vary for the same solute. This discrepancies are caused bymany factors such as the purity and source of the sample, type of the measurementand its accuracy or even a different form of the solid present in the sample usedin measurements. One should be careful while choosing the sets of thermophysi-cal data for the considered solute. In this chapter, whenever it was possible (morethan one reported value), an average of the reported values was used to calculatethe ideal solubility. Ideally, one should check what is the impact of different sets ofthermophysical properties on the calculated ideal solubility and subsequent interac-tion parameters as well as which set leads to the best model-data agreement. Thisapproach has been used by Ellegaard et al. [60], but in general has been found not

7.2. Modelling 115

feasible and the average value was used. It is clear that, if there is only one mea-sured value available, one should trust it. Methods for prediction the melting pointand enthalpy of fusion, like group contribution methods, are not accurate enough tobe a reference. The problem with choosing the correct value of the pure componentproperty is unfortunately still not yet fully resolved.

An application of a more accurate GE model would obviously provide better re-sults. However, such a model requires more parameters, which increases the modelinput and would be less accurate.

The second approach utilizes a model, where the necessary parameters are avail-able for estimating the activity coefficients. For that purpose, a predictive modelsuch as COSMO-SAC has been applied

f 0ij = −

(∂lnγ1j

∂x1j

)0

T,P,n2

= − limxj→1

(∂lnγ1j

∂x1j

)COSMO−SAC

T,P,n2

. (7.12)

The third approach would be regression from ternary data, i.e. solubility in bi-nary solvents data. This is done by adjusting the values of f 0

1j to minimize an objec-tive function Q(f) written as

Q(f) = minM

∑n=1

(ln xE,calc1 − ln xE,exp

1 )2n, (7.13)

where M is the number of experimental data points.Throughout this chapter, the solute-solvent parameters regressed from ternary

data (mixed solvent data) are denoted as f 0,t1j , predicted from pure solvent data (bi-

nary system) f 0,b1j . Those obtained from COSMO-SAC are identified as f 0,CS

1j .The accuracy of the prediction or fitting is expressed by two deviation measures:

AAD (absolute average deviation) and AARD (absolute average relative deviation)

AARD =1n

n

∑i

∣∣∣∣∣ xcalc1 − xexpr

1

xexpr1

∣∣∣∣∣, (7.14)

AAD =1n

n

∑i

∣∣∣ln xE,calc1 − ln xE,expr

1

∣∣∣, (7.15)

where n is the number of experimental points.

7.2.3 Extension of the model to solubility of solids forming polymorphsin mixed solvents

Very often, the solute (1) can form two different, stable polymorphic structures: Onein solvent (2) and one in solvent (3). If this is the case, Eq. (7.1) needs a modification.The existing model is appropriate for the situation where the solute forms the samepolymorphic form from all solvents, i.e. the ideal solubility term is the same (thesame thermophysical properties) and it cancels out in the contribution to the excesssolubility. When one considers the solute, which can be present in different poly-morphic forms in (2) and (3) (and their mixtures), the ideal solubility term in (7.11)does not cancel out. This can be accounted for using a simple correction term Eq.(7.16). One assumes that the transition between different pseudo-polymorphs oc-curs in the solvent mixture at a certain composition, denoted as xtransition. The idealsolubility of the solute (1) in the mixture (2) + (3) would be the ideal solubility of (1)


in (3), when the solute-free mole fraction of the solvent (2) is less than xtransition andwhen mole fraction of the solvent (2) is greater than xtransition.(

ln xE1

)ideal= ln xideal

1,(2+3) −(

x′2 ln xideal

1,2 + x′3 ln xideal

1,3

)(7.16)

ln xE1 ≈ −

x3

2

(∂ ln γ3

x3

)+

T,P,n2

[1 + x2 f 0

12 + x3 f 013]+(

ln xE1

)ideal(7.17)

x′2 ≤ xtransition ln xideal

1,(2+3) = ln xideal1,3 (7.18)

x′2 > xtransition ln xideal

1,(2+3) = ln xideal1,2 (7.19)

To obtain parameters of the extended model, the following approach can bemade. Typically one does not know the thermophysical properties (melting pointand the enthalpy of fusion) of the polymorphic form existing in the solution sincethe solvate (hydrate) is formed. Therefore, the parameters of the model would bean ideal solubility of the solute in solvent (2) and the ideal solubility of the solutein solvent (3): ln xideal

1,3 and ln xideal1,2 respectively. These are in principle different since

polymorphs have different thermophysical properties even if these are hypothetical.In the general case, there are two unknown parameters of the model: ideal solu-

bilities of each polymorphic form in either solvent (2) or (3). It could be that one orboth of them are known, but this is rare. The other adjustable parameter is xtransition.This may be indicated in the data-set. If not, it could be estimated from the solubilitycurve as a visible discontinuity in the solubility plot.

The regression strategy is to fix the value of xtransition (reported or estimated fromthe data-set) and regress for ln xideal

1,j , modifying xtransition until the minimum of theobjective function value Q is found

Q = minM

∑n=1

(ln xE,calc1 − ln xE,exp

1 )2n, (7.20)

where M is the number of experimental data points.

7.2.4 Summary

Before continuing with examples of modelling results, the main characteristics andassumptions made in the derivation of the excess solubility model need to be re-peated as these are essential for later application of the model in data validation.

The form of the excess solubility model given by Eq. (7.1) has a rigorous basis.However, it has a limited set of assumptions, which need to be highlighted here.First of all, the model is rigorous in the region of infinite dilution of the solute i.e. forsolubilities less than 0.01 expressed in molar fractions. The composition dependenceof the unsymmetrical activity coefficient derived from FST seems to be reliable todescribe the system non-idealities in the diluted region. Nevertheless, it has beenshown for LLE [66] and SLE [59] that the model can be used in spite of the deviationsfrom rigor and this will be also demonstrated in the next section.

Secondly, it has been assumed that the f+ij (i at infinite dilution in the mixedsolvent) do not depend on the solvent composition and one can equate this to f 0

ij (i inthe pure solvent j). Because the molecular correlation functions are strong functionsof density [199], this assumption seems reasonable for mixtures of organic solvents,where the variations of the density are often small. But as it was shown by Ellegaard

7.3. Results and discussion 117

et al. [200], [60], these assumptions seem to be valid in aqueous mixtures as well.This is important since water is present as solvent in many crystallization processes,therefore it is desired to be able to model the solubility of solids in aqueous solutionswith the current model.

Next aspect pertains to the solvent-solvent term, which needs to be describedaccurately. The best approach would be to regress the parameters of a given GE

model from vapour-liquid equilibria data of the mixed solvent system (2) + (3).However, when there is a lack of VLE data in the solvent system, one may use apredictive model such as COSMO-SAC. This will be addressed in the next section.Even though, the VLE is very well represented by different models and the devi-ations from Raoult’s law in the solvent mixture (positive or negative) identified, itmay be that the deviations will be different when comparing with excess solubilities.This will be illustrated for one example of the solubility of naphthalene in ethyleneglycol and water mixed solvent in the subsequent section. This is not a commonsituation, but does flag a general issue. It seems there is not an obvious solution tothis problem. Certainly it is not a matter of changing to more compatible VLE set toregress parameters of a GE model. A possible solution leads to the next assumptionmade in the derivation of the excess solubility model.

This limitation is connected to the exclusion from consideration a multi-body (inparticular triplet) interactions in the Eq. (3.19). Only binary distribution functionintegrals were included. It might be very well that in some mixtures each tripletcontribution of solute and two solvent species play bigger role and a new parameter,(W), being a product of fij’s should be additionally regressed from mixed solventsolubility data. Triplet contributions were not analysed here as the simplicity of themethod has been sought.

Last assumption made in the derivation of the excess solubility model was suchthat no solid phase transition occurs in the mixed solvent as its composition is chang-ing. Giron [198] has shown that the phenomenon of polymorphism applies to manysolutes, but it is rarely reported in the sets of solubility in mixed solvents. Therefore,the model has been modified in a way to take into account this effect. Unfortunately,in this situation one can rely only on the parameters regressed to mixed solvent sol-ubility data as thermophysical properties of polymorphs are sparsely reported. Theprediction of the polymorph-solvent interaction parameters is not possible with themodels with more predictive power (e.g. COSMO-SAC or UNIFAC). The polymor-phic structures of the same solute may consist of the identical functional groups, buttheir thermophysical properties would be completely different. Nonetheless, whenthe deviations from rigor are small (low solubilities), the excess solubility modelwith the additional correction term is expected to be reliable.

7.3 Results and discussion

This section looks at different aspects of the described solubility model in the previ-ous section. First and foremost the model was applied primarily to the the systems,in which the solubility of the solute in the mixed solvent is low (lower than 0.01),but the use of the model outside this region is also tested. Promising results couldcertainly expand the area of the application to more soluble substances in mixturesof both organic solvents and aqueous solutions.

Focusing on the model itself, different approaches to obtain solute-solvent inter-action parameters f 0

ij and solvent-solvent term are tested. The aim is to find the bestmethod, which as accurately as possible and with the minimum of the binary data


can fit the solubility data in mixed solvents. This needs to be done to ensure that themodel is rigorous enough to be used in the screening of the unreliable data i.e. toelaborate a benchmark model used in the validation of data.

To address this questions, the excess solubility model has been first tested inalmost 40 data sets at different temperatures ranging from 278.15 to 338.15 K with amajority of systems for which solubility was determined at 298.15 K (Table 7.1 andTable 7.2).

7.3.1 Parameters from COSMO-SAC

The possibility of obtaining both, solvent-solvent term and parameters f0 for solute-solvent pairs from COSMO-SAC model was explored. For more details of the model,including the way of computing the derivatives of the activity coefficients, the readeris referred to Section 4.2.3.

To evaluate the accuracy in the prediction of the derivatives by the COSMO-SACmodel, the database of concentration derivatives for a variety of binary liquids fromWooley and O’Connell [77] has been used. They obtain this property from the corre-lation of the reliable vapour-liquid equilibria data by the means of common activitycoefficient models such as NRTL, Wilson and modified Margules. Since the activ-ity coefficients and their concentration derivatives are sensitive to the used model,some discrepancies between different models have been found, especially at the in-finite dilution. Here, the results from COSMO-SAC are compared to all three modelsas well as an averaged value. It means that the arithmetic average of the value of thederivative from each model at given concentration was calculated. Derivatives ofthe activity coefficients with respect to mole fractions are only in qualitative agree-ment with those obtained from VLE data (cf. Fig. 7.1). The values of the derivativesfor Margules, NRTL and Wilson equation were taken directly from [77].

FIGURE 7.1: The activity coefficient derivative with respect to thecomposition in ethanol (1) + benzene (2) system at 298.15 K.

Next, the values of the solute-solvent interaction parameters f0 were computedby means of COSMO-SAC. The values of these calculated from COSMO-SAC (Eq.(7.12) (and similarly UNIFAC) are always overestimated by at least one or two orders


of magnitude, with makes them unreliable to use. Here few examples of COSMO-SAC model performance are shown.

In the case of the solubility of caffeine and cholesterol, the excess solubility modelwith the interaction parameters f0 obtained from COSMO-SAC highly overestimatethe solubility, giving almost nonphysical results. The reason is that COSMO-SACpoorly predicts the activity coefficient (and consecutively derivative) for big, com-plex molecules with more than three different functional groups such as cholesterol,caffeine or paracetamol. All values of the f0 computed by the COSMO-SAC modelfor the considered solute-solvent pairs are given in Table 7.2.

4-nitrobenzonitrile (1) + ethyl acetate (2) + methanol (3) at T=298.15 K

Solubility data of 4-nitrobenzonitrile (1) in ethyl acetate (2) and methanol (3) mixedsolvent are reported at three different temperatures [201]. The model, with parame-ters regressed from binary solubility data shows systematic trends and the solubilityincreases with increasing temperature. Both methods (from either COSMO-SAC orthe Margules equation) of getting the solvent-solvent representation gives similarresults, slightly better when the Margules model is used to correlate experimentalVLE data in the system of ethyl acetate (2) + methanol (3).

FIGURE 7.2: Solubility of the 4-nitrobenzonitrile in ethyl acetate andmethanol binary solvent at 298.15 K. Left figure: solvent-solvent termfrom COSMO-SAC, solute-solvent estimated from binary or regres-sion to the ternary (mixed solvent) data ; right figure: solvent-solventterm from modified Margules model, solute-solvent estimated from

binary or regression to the ternary (mixed solvent) data

The application of the COSMO-SAC model to estimate the solute-solvent interac-tion parameters gives similar results to those, if f 0

1j are obtained from the regressionto solubility data in mixed solvent systems. The reason for that, could be that theconsidered molecule has relatively simple structure (in comparison to other activespecies) and COSMO-SAC produces more accurate results.

This case shows, that COSMO-SAC could be applied to obtain both terms of themodel. One needs to be also very careful, because it might be that the inaccuraciesin the description of solvent-solvent term, generated by the COSMO-SAC are com-pensated when the parameters are regressed from the ternary (mixed solvent) SLEdata.


Caffeine (1) + water (2) + 1,4-dioxane (3) at 298.15 K

FIGURE 7.3: Solubility and excess solubility in the system of caffeine(1) + water (2) + 1,4-dioxane (3) at 298.15 K. Left figure: solvent-solvent term from COSMO-SAC, solute-solvent estimated from bi-nary or regression to the ternary (mixed solvent) data; right figure:solvent-solvent term from modified Margules model, solute-solventterms from COSMO-SAC or regression to the ternary (mixed solvent)

data.

Caffeine solubility data in water/1,4-dioxane binary solvent are measured at298.15 K [202]. Estimation of the interaction parameters f0, either from the binarySLE or ternary SLE, give similar results with very low absolute deviation. The ratioof f (0,b)

1j / f (0,t)1j for both solvents is almost equal to one. The experimental data are

very well described by the proposed excess solubility model and are considered asreliable. On the other hand, the estimation of the f 0

1j for solute-solvent pairs from

COSMO-SAC was not successful. The interaction parameter f (0,b)12 is approximately

10 times lower than estimated by the COSMO-SAC, 4.90 and 42.8, respectively. Dueto that fact, the excess solubility of the caffeine is highly overestimated (Fig. 7.3).

By performing the calculations, it has been found that the COSMO-SAC is morereliable for calculations of the solvent-solvent terms in the model rather than com-putation of the solute-solvent interaction parameters f0. However, the qualitativeresults from COSMO-SAC are not sufficient in the excess solubility in order to makeit reliable for the data validation. It may be that good modelling results generatedwith the help of COSMO-SAC model are only coincidental.

Therefore, fitting of the model parameters to experimental VLE binary data,whenever possible, should be done. The excess solubility model will be less pre-dictive, but it will gain reliability. Thus, VLE data are used to fit the parameters ofModified 5-parameter Margules equation. Next, the activity coefficients of solventsin the mixed solvent system as well as the analytical (or numerical) derivative of theactivity coefficient are calculated.

With the established way of obtaining the solvent-solvent term, one can lookat other aspects of the excess solubility model such as solvent-solute parameters(regression to mixed solvent solubility or estimation from binary SLE), the use of themodel outside its proven range: x1 > 0.01 and aqueous solutions.


7.3.2 Parameters from Modified Margules model

In the most cases, the 5-parameter Margules equation with some exceptions for theWilson model, when the parameters of the first were not reported, was applied. Pa-rameters of the models were taken, whenever is possible, from the paper of Wooleyand O’Connell [77].

Cholesterol (1) + hexane and ethanol at 293.15 K

Two data-sets for the solubility of cholesterol in the binary solvent composed of hex-ane and ethanol [203] (data-set #1), [204] (data-set #2) were investigated. This sys-tem is an example, where the solubility is slightly higher than 0.01. In addition, themixed solvent is the mixture of two organic solvents. Therefore, the model shouldreliably correlate data if these are accurate. Moving to the modelling results, if thesolute-solvent interaction parameters are estimated from binary SLE data, the modelis in slightly better agreement with the experimental data reported in data-set #2.This is indicated by the lower absolute average relative deviation for the data-set #2(0.18). The model predictions (solute-solvent terms estimated from binary SLE) arein similarly good agreement with experimental data with AARDb equal to 1.4 and2.1, for set #1 and #2, respectively. The analysis of ratios of the solute-solvent terms:f (0,b)12 / f (0,t)

12 (-3.72 for set #1, 3.91 for set #2) and f (0,b)13 / f (0,t)

13 (0.72 for set #1, 1.17 forfor set #2) (cf. Table 7.4), obtained either from the regression or estimation from thebinary SLE data can also be the evidence of the data quality. One expect this ratio tobe as close as possible to unity. In this example, the ratios are slightly closer to unityand positive for the data-set #2. Based on this, it seems that the experimental datareported in data-set #1 are less consistent.

The excess solubility model, with the parameters regressed to solubility data inbinary solvent correlates data very well, although the solubilities of the cholesterolare slightly outside very the diluted region (0.01).

FIGURE 7.4: Cholesterol (1) in hexane (2) + ethanol (3) (left, dataset#1) and ethanol (2) + hexane (3) (right, dataset #2) at 293.15 K.

Cholesterol (1) + benzene (2) + ethanol (3) at 293.15 K

Solubility of cholesterol (1) in benzene (2) + ethanol (3) binary solvent was measuredat 293.15 K [203]. The model, with the interaction parameters estimated from binary


FIGURE 7.5: Solubility and excess solubility in cholesterol (1) + ben-zene (2) + ethanol (3) system at 293.15 K.

SLE is in a very good agreement with the experimental data (AARDb=0.25), Fig.7.5 The sign of interaction parameters f 0 is conserved regardless the approach usedto estimate them. In fact, the ratios f (0,b)

12 / f (0,t)12 and f (0,b)

13 / f (0,t)13 are very close to

unity, 1.37 and 1.19 respectively. The AARD and AAD for both approaches is withinthe considered range. The experimental data shows very good agreement with thepredictive model (estimation from binary SLE) and a correlation of the solubilitydata in the mixed solvent.

Solubility of cholesterol in this case in even higher (max. mole fraction is 0.056).Despite that fact, the excess solubility is predicted very well.

Naphthalene (1) + acetone (2) + water (3) at T=298.15 K

Here we examine the case, in which the solubility is very low, but the mixed solventconsists of one organic solvent and water. The solubility of naphthalene in acetoneand water mixture was reported at 298.15 K, Fig. 7.6. It is worth mentioning thatthe naphthalene in both water and acetone exhibits high deviations from ideality,since both solvents are considered as polar, especially water. The excess solubilitymodel overestimates the solubility of naphthalene in mixed solvent, more than 150%in AADb, when the solute-solvent parameters are estimated from binary SLE data.The reason for that could be the approximation made in the derivation of the model,that the solute-solvent parameters are independent of the solvent compositions. Thismeans that, one can equate the mixture term f+ij with the pure solvent term f 0

ij. Here,the solvent is an aqueous solution of acetone, therefore this assumptions might beweak. However, closer to the pure solvents ends on the diagram, the predictionis better. Nonetheless, with the model parameters estimated from ternary data theAAD decreases to 17% and what is important, the sign of the f 0 is also conserved.In spite of the deviations from rigor the model performs surprisingly good.

Naphthalene (1) + ethylene glycol (2) + water (3) at T=298.15 K [205]

This system is an example of the system, in which the solubility of the solute is verylow and solvents form a mixture with negative deviations from Raoult’s law [59].

However, the excess solubility of naphthalene in this binary solvent is clearlypositive. This means that the sign of the excess solubility is opposite to the solvent


FIGURE 7.6: Solubility of the naphthalene in acetone and water bi-nary at 298.15 K.

FIGURE 7.7: Solubility of naphthalene in ethylene glycol and waterbinary solvent 298.15

gE. For this particular system, a high positive values of the solute-solvent interac-tion parameters (10.62 and 20.42) obtained by the estimation from binary solubilitydata is inconsistent with negative values of the parameters when regressed from theternary solubility data. The excess solubility predicted by the model is in the oppo-site of the experimental data. The model shows a small region of negative excesssolubility. It is not possible to match this behaviour also with other GE models suchas Wilson. The problem of incompatible signs of GE of the solvent (2)+ (3) systemsand excess solubilities occurs as well in the naphthalene (1) + DMSO (2) + water (3)system. Similarly, these solvents form a mixture with negative deviations from theideality.

This example illustrates a limitation of the model. There is no obvious cure forthis problem. It may be that one should include in the model term connected totriplet interactions. But on the present form the model cannot deal with such sys-tems. Fortunately, this problem is not common, therefore the model in the existingform is sufficient to describe the substantial number of systems.

The results for all tested systems: parameters with their deviations and model


accuracy measures are shown in Table 7.1. Table 7.2 presents the predicted solute-solvent terms by the COSMO-SAC model.

Discussion

The excess solubility model has been applied to number of systems. Even thoughthe model has limited set of assumptions can be considered as rigorous for infinitelydiluted solutes xi ≤ 0.01. However, very often, many of the bio-molecules solute-solvent mixtures show experimental solubilities at higher concentrations. This isparticularly visible for the mixture of organic solvents. It seems, that the methoddoes well for such cases, too. Nevertheless, systems with mixture of solvents exhibit-ing negative deviations from Raoult’s law are not well described with the currentmodel. The results obtained for these systems by the model should be interpretedwith caution.

Regarding the parameter estimation procedure one can conclude, that this isdone the best when they are regressed from the solubility of the solute in mixedsolvent. But very often, the estimation from binary SLE gives similar, satisfactory re-sults. Ellegaard et al. [60] have decided to perform a global regression of the solute-solvent parameters using all of the available ternary solubility data for a particularsolute. By doing this they ensured that all of the solute 1 -solvent j interaction pa-rameters have the same value regardless the second solvent present in the ternarysystem. However, this also means that the interaction parameters are temperatureindependent, which is not true by virtue of their definition connected to correlationfunctions. He has supported his reasoning by the fact that most of the solubilitysystems treated there were reported at 298.15 K.

However, in this thesis it has been decided to regress the interaction parametersto ternary data separately for each system. This results in the conservation of thetemperature dependence, as number of solubility systems studied here are reportedat temperatures different than 298.15 K. The parameters reported in Table 7.1 merelysatisfies the mathematical constraint of minimizing the objective function. Hence,the intention is to match the experimental data as accurately as possible by adjustingthe solute-solvent parameters. This approach informs about the data quality. First,since the model is nearly rigorous, accurate experimental data should follow theassumed form of the model. Secondly, the solute-solvent interaction parametersshould have similar numerical values despite the method used to regress them. Itshould be that the ratios f 0,b

12 / f 0,t12 ∼ 1 and f 0,b

13 / f 0,t13 ∼ 1. The discrepancies between

them can inform of the quality of ternary solubility data. For example, there aretwo data-sets of the solubility of phenacetin in water/dioxane mixtures at 298.15 K.Clearly one of them is of lower quality, as the regressed from ternary data interactionparameters are significantly different from those estimated based on the solubilityin pure solvents.

The example of the solubility of pregabalin in methanol/water mixtures showsthe situation, in which the solute-solvent interaction parameters have different signs,depending which data are used to obtain these parameters. Estimation of the param-eters from the solubility of the pregabalin in pure solvents, indicated negative devi-ations from Raoult’s law. However, parameters regressed from ternary data showsslightly positive deviations from ideal mixing. Negative values of the parametersobtained from binary SLE data are caused by the fact that the enthalpy of fusion ofpregabalin is very high (122.32 kJ · mol−1), significantly higher than for the rest ofstudied solutes. Thus, the ideal solubility is very low and further values of the inter-action parameters are negative. The enthalpy of fusion of pregabalin seems rather


suspicious. The sign of the ratio of f 0,b1j / f 0,t

1j is one of the criteria in the validation ofsolubility data later in this chapter.

Regarding the solvent-solvent term present in the model, these must be obtainedfrom the correlation of the VLE binary data by the modified Margules (or equivalent)model.

Nonetheless, results for the systems tested in this section confirm that the methodarising from FST and simple in the form is successful to describe the excess solubilitybehaviour in many systems. Generally, the errors from regressing solute-solventparameters are less than 0.12 (AARD). The rigor behind the model will be a basisfor the data validation criteria developed in section 7.5.


TAB

LE

7.1:

Syst

ems

ofso

lute

san

dtw

oso

lven

tsus

edto

test

the

mod

el

solu

te(1

)so

lven

t(2)

solv

ent(

3)T

/Kf0,

b12

f0,b

13A

AR

Db

AA

Db

f0,t

12SD

f0,t

13SD

AA

RD

tA

AD

tph

enac

etin

wat

erdi

oxan

e29

81.

9612

.59

0.23

0.27

0.93

0.83

13.9

30.

490.

212

0.22

5ph

enac

etin

wat

erdi

oxan

e29

8.15

1.97

12.2

04.

121.

256.

730.

555.

670.

380.

147

0.15

4ph

enac

etin

wat

erdi

oxan

e31

31.

9712

.56

0.25

0.41

0.96

0.01

14.4

90.

570.

246

0.26

2ph

enac

etin

etha

nol

ethy

lace

tate

298.

152.

312.

192.

561.

020.

600.

080.

630.

160.

031

0.03

2pa

race

tam

olm

etha

nol

ethy

lace

tate

298.

15-1

.10

2.73

1.23

0.28

-1.5

30.

584.

731.

210.

178

0.23

para

ceta

mol

diox

ane

wat

er29

8.15

-1.4

25.

200.

691.

40-2

.24

1.15

14.2

70.

930.

296

0.33

7pa

race

tam

olet

hano

lm

etha

nol

298.

15-1

.42

-1.1

00.

010.

01-0

.98

0.00

-1.0

00.

000.

000

0.00

para

ceta

mol

etha

nol

ethy

lace

tate

298.

15-1

.70

2.53

0.38

0.51

-0.5

50.

454.

060.

870.

154

0.17

3ch

oles

tero

lhe

xane

etha

nol

293.

154.

466.

121.

430.

64-1

.20

0.73

5.48

0.82

0.18

00.

188

chol

este

rol

etha

nol

hexa

ne29

3.15

6.12

4.46

2.06

0.92

5.23

0.71

1.14

0.56

0.25

00.

291

chol

este

rol

benz

ene

etha

nol

293.

150.

706.

120.

250.

220.

510.

415.

160.

530.

144

0.15

4ch

oles

tero

lhe

xane

benz

ene

293.

154.

460.

700.

290.

3710

.63

0.37

4.04

0.23

0.02

40.

025

4-ni

trob

enzo

nitr

ileet

hyla

ceta

tem

etha

nol

278.

151.

336.

343.

831.

100.

710.

222.

040.

130.

045

0.05

4-ni

trob

enzo

nitr

ileet

hyla

ceta

tem

etha

nol

298.

151.

285.

982.

470.

980.

570.

491.

610.

230.

114

0.12

14-

nitr

oben

zoni

trile

ethy

lace

tate

met

hano

l31

8.15

0.86

5.50

2.00

0.88

0.36

0.45

1.52

0.21

0.10

90.

115

naph

thal

ene

prop

an-2

-ol

wat

er29

8.15

4.49

22.5

319

6.68

3.60

7.67

1.06

6.06

0.47

0.24

60.

262

naph

thal

ene

etha

nol

wat

er29

8.15

4.48

22.5

37.

671.

6511

.05

0.72

6.45

0.40

0.16

90.

161

naph

thal

ene

benz

ene

hexa

ne29

8.15

0.10

2.44

0.36

0.27

-0.0

10.

04-0

.76

0.03

0.00

50.

005

naph

thal

ene

acet

one

wat

er29

8.15

1.60

22.5

37.

121.

503.

190.

4212

.66

0.32

0.16

80.

171

benz

oic

acid

cycl

ohex

ane

hexa

ne29

8.15

5.47

5.73

0.06

0.06

2.03

0.24

1.16

0.30

0.00

30.

003

benz

oic

acid

cycl

ohex

ane

hexa

ne30

3.15

5.26

5.54

0.05

0.05

2.71

0.04

1.32

0.05

0.00

10.

001

benz

oic

acid

hexa

nete

trac

hlor

omet

hane

298.

155.

732.

690.

180.

16-0

.87

0.09

-0.2

00.

060.

002

0.00

2m

efen

amic

acid

etha

nol

ethy

lace

tate

298.

15-0

.12

-1.6

30.

410.

61-0

.23

0.74

5.32

1.16

0.17

10.

201

test

oste

rone

tric

hlor

omet

hane

cycl

ohex

ane

298.

15-5

.88

10.2

00.

561.

0756

6.63

25.4

893

7.45

27.0

70.

072

0.07

1an

thra

cene

dibu

tyle

ther

hexa

ne29

8.15

2.04

4.12

0.02

0.02

6.27

0.25

4.06

0.27

0.00

20.

002

anth

race

nedi

buty

leth

erhe

ptan

e29

8.15

2.04

3.70

0.07

0.07

-2.9

71.

081.

101.

010.

023

0.02

3an

thra

cene

diox

ane

prop

an-1

-ol

298.

150.

365.

651.

210.

68-0

.68

0.05

1.34

0.05

0.01

70.

017

anth

race

nepr

opan

-1-o

lhe

xane

298.

155.

654.

120.

140.

130.

980.

094.

060.

130.

006

0.00

6an

thra

cene

prop

an-1

-ol

hept

ane

298.

155.

653.

700.

160.

140.

590.

135.

050.

230.

009

0.01

0an

thra

cene

prop

an-1

-ol

octa

ne29

8.15

5.65

3.39

0.07

0.07

0.92

0.30

13.5

30.

670.

015

0.01

5ca

ffei

new

ater

diox

ane

298.

154.

902.

290.

380.

529.

980.

55-3

.54

0.88

0.23

90.

256

naph

thal

ene

met

hano

lw

ater

298.

155.

4122

.53

1.60

0.73

16.7

30.

752.

350.

490.

071

0.07

5C

onti

nued

onne

xtpa

ge


Tabl

e7.

1–

cont

inue

dfr

ompr

evio

uspa

geso

lute

(1)

solv

ent(

2)so

lven

t(3)

T/K

f0,b

12f0,

b13

AA

RD

bA

AD

bf0,

t12

SDf0,

t13

SDA

AR

Dt

AA

Dt

naph

thal

ene

DM

SOw

ater

298.

152.

3622

.53

0.86

7.56

-8.9

10.

544.

270.

251.

630

0.17

9na

phth

alen

eet

hyle

negl

ycol

wat

er29

8.15

10.6

220

.42

0.62

1.33

-23.

523.

23-3

.18

1.01

0.04

70.

049

preg

abal

inm

etha

nol

wat

er29

8.15

-21.

71-2

4.32

0.73

2.23

3.88

1.12

1.16

0.69

0.23

0.04

9pr

egab

alin

met

hano

lw

ater

338.

15-1

2.25

-13.

720.

671.

638.

270.

461.

070.

280.

016

0.03

3su

lpha

met

hoxy

pyri

dazi

neet

hano

lw

ater

298.

154.

7612

.07

0.66

0.47

0.82

0.69

12.8

61.

060.

094

0.1

subs

crip

tor

supe

rscr

iptb

refe

rsto

the

quan

tity

obta

ined

from

bina

rySL

Eda

ta,t

from

mix

edso

lven

tsol

ubili

tyda

ta


TABLE 7.2: Systems of solutes and two solvents used to test themodel. Parameters from COSMO-SAC model.

solute (1) solvent (2) solvent (3) Ref. f 0,CS12 f 0,CS

13phenacetin water dioxane [206] 110.35 -2.21phenacetin water dioxane [207] 110.03 -2.21phenacetin water dioxane [206] 107.28 -2.05phenacetin ethanol ethyl acetate [207] -1.43 -1.29paracetamol methanol ethyl acetate [208] -9.27 -1.46paracetamol dioxane water [209] -2.17 50.80paracetamol ethanol methanol [208] -8.45 -9.27paracetamol ethanol ethyl acetate [210] -8.45 -1.46cholesterol hexane ethanol [203] 75.34 20.14cholesterol ethanol hexane [204] 20.14 75.34cholesterol benzene ethanol [203] 67.71 20.14cholesterol hexane benzene [203] 75.34 67.714-nitrobenzonitrile ethyl acetate methanol [201] -1.06 1.754-nitrobenzonitrile ethyl acetate methanol [201] -0.81 1.984-nitrobenzonitrile ethyl acetate methanol [201] -0.63 2.12napthalene propan-2-ol water [211] 1.44 91.54napthalene ethanol water [211] 2.00 91.54napthalene benzene hexane [212] 0.003 1.36napthalene acetone water [211] -0.94 91.54benzoic acid cyclohexane hexane [213] 86.63 80.01benzoic acid cyclohexane hexane [213] 82.48 75.89benzoic acid hexane CCl4 [213] 80.01 71.69mefenamic acid ethanol ethyl acetate [214] -3.56 -4.86testosterone CHCl3 cyclohexane [215] 54.67 346.75anthracene dibutyl ether hexane [216] -0.58 2.02anthracene dibutyl ether heptane [216] -0.58 1.88anthracene dioxane propan-1-ol [217] -2.28 1.78anthracene propan-1-ol hexane [218] 1.78 2.02anthracene propan-1-ol heptane [218] 1.78 1.88anthracene propan-1-ol octane [218] 1.78 1.74caffeine water dioxane [202] 42.80 -0.40naphthalene methanol water [211] 4.17 91.54naphthalene DMSO water [211] -1.02 91.54pregabalin methanol water [219] n/a n/apregabalin methanol water [219] n/a n/asulphamethoxypyridazine ethanol water [220] n/a n/a

7.4 Systems with polymorphs

As mentioned earlier, formation of the polymorphs is a common phenomenon in thepharmaceutical downstream processing and attention should be paid to it to ensurethat the unexpected solid phase change does not happen during the production orstorage of the final product. Although this aspect has been reported in open litera-ture many times, ternary solubility data including solid phase change in the mixtureof solvent have not been widely reported. The ideal data-set would consist of thesolubility of the respective chemical compound at a different solvent compositions

7.4. Systems with polymorphs 129

with the indication, when the actual transformation in the residual solid phase oc-curs confirmed by analytical method such as X-ray powder diffraction (XRPD) ordifferent scanning calorimetry (DSC), and the stable form. The data containing fullinformation are very scarce [221], [222], [223]. Some data-sets provide the infor-mation of the transition in the solid phase, but no numerical values of solubilitiesare provided [224], [210]. Nonetheless, few case studies were selected to test theextended model on fusidic acid [225], sodium fusidate [225], meropenem [223], car-bamazepine [222].

In this section, the performance of the extended excess solubility model is an-alyzed on the selected cases of solubility data exhibiting polymorphic transitions.Furthermore, the possibility to include the data with polymorphic transitions as aspecial case under the data validation method using the modified excess solubilitymodel is presented. To describe the solvent-solvent system, Margules model hasbeen employed, except for the methanol-water, where the derivative of the activitycoefficient was obtained via Wilson equation. Similarly as for the classical behaviourof solutes in the mixed solvent (with no phase transitions), the particular GE modeldoes should not strongly affect results.

Solubility of meroponem in water and methanol

The solubilities (Fig. 7.8) of meropenem (1) in water (2) and methanol (3) havebeen reported at four different temperatures: 273.15, 278.15, 283.15 and 288.15 K[223]. For all systems the solid phase at each measured concentration has been takenbefore and phase (pseudo-polymorph) characterized. In water, meropenem formsa trihydrate, whereas in pure methanol it is present as a methylate. The transitionpoint has been found to be between x

′2 = 0.19 and 0.21 depending upon the temper-

ature.There is a visible polymorphic transition in the solution of water and methanol,

while the composition of the solvent is changing. The transition concentrations arereported as a part of the data-sets, and are confirmed by XRPD measurements. Themodel also correctly capture the trend in the solubility data and give the best fit,when the xtransition has been assumed to be equal to that reported in the paper. More-over, the solubility data have been reported at different temperatures. The regressedparameters, which can be interpreted as the ideal solubilities of the solute, exhibit acorrect trend. It means that, while temperature is increasing, ideal solubility shouldincrease as well.

TABLE 7.3: The regressed parameters of model together with the ab-solute averaged relative deviation in meroponem (1) + water (2) +

methanol (3) system.

T/K ln xideal1,2 ln xideal

1,2 xtransition AARD273.15 -9.68 -12.78 0.191 0.3608278.15 -9.52 -12.73 0.195 0.3104283.15 -9.25 -12.58 0.202 0.2640288.15 -8.97 -12.30 0.204 0.2188

Solubility of carbamazepine (1) in water (2) + ethanol (3) binary solvent at T= 293.15 K

The solubility of carbamazepine (1) in water (2) and ethanol (3) has been reportedat 293.15 K together with the phase characterization [222], Fig. 7.9. The transition


FIGURE 7.8: Solubility of meroponem (1) in water (2) + methanol (3)solvent system at different temperatures: 273.15, 278.15 (upper plots),283.15 and 288. 15 K (bottom plots). Black dotted lines are predic-

tions, yellow line is solubility, when ideal mixing is assumed.

point has been found to be near x′2= 0.4. The estimated parameters are ln xideal

1,2 = -0.66and ln xideal

1,3 = -4.63. The absolute averaged relative deviation is equal to 0.485.

Solubility of sodium fusidate (1) in water (2) + acetone (3) at 298.15 K

Solubility of sodium fusidate (NaFA) has been measured (Fig. 7.10) in the binarysolvent of water and acetone and the samples of the solid phase at each measuredcomposition of the solvent were analyzed [225]. It has been found that the transfor-mation occurs at very low concentration of the acetone, near x

′2= 0.03. In this case,

it has been found that in the pure water the stable form is sodium fusidate, in ace-tone the form is unknown. Knowing this, the only one parameter ln xideal

1,3 = -2.29 hasbeen estimated, since it is possible to calculate the ideal solubility of the NaFA. TheAARD is equal to 0.797.

Solubility of fusidic acid (1) in water (2) + ethanol (3) at T=294.15 K

In this case, the ideal solubility of the form of fusidic acid in methanol (anhydrateI) was reported and hemihydrate form in water [225].

The transition was estimated to be between concentrations of 0.45 and 0.58, al-though it is not visible from the solubility plot, but X-ray characterization of samplesat each composition has confirmed the the change in the solid phase occurs. Themodel (with the solvent-solvent term calculated from 5-parameter Margules equa-tion) gives a good agreement with the experimental data (Fig. 7.11). On the left a

7.4. Systems with polymorphs 131

FIGURE 7.9: Solubility of carbamazepine (1) in water (2) and ethanol(3). There is no dramatic change in the solubility between forms as forthe meropenem case. However, model overestimates the solubility ofthe solid around the transition point. Dashed line represents ideal

mixing.

FIGURE 7.10: Sodium fusidate solubility in water + acetone binarysolvent at 298.15 K. Dashed line represents ideal mixing.

prediction from binary solubility data, with the assumed ideal solubility of the FAforms: hemihydrate in water and anhydrate I in ethanol is shown. The right partof the Fig. 7.11 presents the correlation of the experimental data, assuming the un-known solubility of the FA form present in water (ln xideal

1,2 = -3.16).In all tested cases, the modified model is able to capture the transition in the solid

phase, while the composition of the mixed solvent is changing. The simple correc-tion to the original version of the model does not remove a rigor from its form. Itis important that the model is reliable in the low concentration region (solubility x1< 0.01), which is the case in the considered examples. Having a powerful solubilitymodel, a simple data validation criterion based on the goodness-of-fit of the datato model is established. The further criteria of the validation method could be cer-tainly elaborated from testing the model to more systems with solutes, which showthe polymorphism depending from which solvent they solidify. Unfortunately, the


FIGURE 7.11: Solubility of fusidic acid + water (2) + ethanol (3) atT=294.15 K. Dashed line denotes ideal mixing.

experimental data are rather not sufficient to address the phenomenon more thor-oughly. New experimental data for the common solutes and mixed solvents both or-ganic and aqueous, at different compositions and temperatures could be measured.

Nevertheless, a provisional extension of the excess solubility model was appliedfor excess solubilities, where the phase transition in the solid phase occurs.

7.5 Excess solubility data validation

In the first place, the main assumptions of the excess solubility model used here forthe validation need to be emphasized. The excess solubility model is powerful inthe region of infinite dilution of the solute, but also can reliably describe solubilityat higher concentrations. The solute-solvent terms in the model are assumed to beindependent of the composition of mixed solvent. This assumption is usually truein the mixture of common organic solvents and water. Next, the triplet distribu-tion function integrals were neglected, as in the substantial number of systems theseare less significant. Finally, originally no polymorphic phase transition occurs inthe solid. This has been addressed in the previous section and is a special case ofthe solubility systems. Therefore such systems should be treated separately in thevalidation methodology.

The model has been used outside its proven range. Based on the deviations in thevalues of the estimated model parameters, deviation between the model predictionsand experimental data AAD (absolute average deviation), AARD (absolute averagerelative deviation), and reliability of the selected GE model in the reproduction ofthe VLE in a binary solvent mixture, several criteria for data validation have beenformulated.

7.5.1 Solute-solvent interaction parameters

Interaction parameters regressed from fitting to ternary data or estimated from bi-nary solubility data should have similar numerical values. This means that ratiosf 0,b12 / f 0,t

12 ∼ 1 and f 0,b13 / f 0,t

13 ∼ 1. This also implies that they should have the samesign, i.e. solute-solvent pair exhibits either positive ( f 0

1j>0) or negative deviation

7.5. Excess solubility data validation 133

( f 01j<0) from ideality: f 0,b

1j / f 0,t1j > 0. The quality factor associated with this observa-

tion has been developed. Ideally, if the ratio between parameters is one, the qualityfactor Ftest1/2 is equal to 1. As as trade-off the following values of the quality factorhas been assigned. If 0.8 ≤ f 0,b

1j / f 0,t1j ≤ 1.25, the Ftest1/2 is equal to unity, for the value

of the ratio between 0.2 and 0.8 or 1.25 and 5 Ftest1/2 is equal to 0.5. Ftest1/2 is equalto 0.1, when 0.01≤ f 0,b

1j / f 0,t1j < 0.2 or 5 < f 0,b

1j / f 0,t1j ≤ 10. For values greater than 10 or

negative values, a quality factor equal to zero is assigned.Although, the factor can be assigned to the given data-set, one needs to remem-

ber that the solute-solvent parameters might have different signs due to the modellimitations: f 0

1j independent of solvent composition and neglect of multi-body in-teractions. It has been explained earlier, that no matter how accurate VLE data andmodel used to correlate them are, the deviations from Raoult’s law in the experi-mental excess solubility may be incompatible with the behaviour in (2) + (3) solventsystem.

7.5.2 Solvent-solvent term

The solvent-solvent term in the excess solubility model must be obtained from reli-able VLE data at the same or similar temperature as the reported solubility data-set.These data are fitted with the correlation activity coefficient model such as ModifiedMargules to obtain its parameters. Then the composition derivative of the activity

coefficient and the product x3

(∂ ln γ3

x3

)+T,P,n2

is computed. The correct description of

the VLE is a basis for the validity of the model. But, again one needs to bear in mindthat deviations from the ideality might differ between GE of solvents and excesssolubility data.

7.5.3 Goodness-of-fit

Two deviation measures, the average absolute deviation AAD and the relative de-viation AARD are used. The advantage of the latter is that, all data points areweighted equally, regardless of their absolute value.

Regressing the solute-solvent parameters from mixed solvent solubility data, theaverage value of AARD and AAD is equal to 0.12 and 0.09, respectively (basedon number of data-sets indicated in Table 7.1). When estimating f 0 from binarysolubility data, so including only the solute pure solvent solubilities, the values are5.76 and 0.82, for AARD and AAD respectively.

When removing from consideration systems, for which the standard deviationexceeds two times the average values, AARD and AAD decrease to 1.10 and 0.75,respectively. Such high deviation suggest that the model was not able to describecorrectly the behaviour of some systems (probably due to the opposite deviationsfrom Raoult’s law in the solvents mixture and solubility data). Based on this fact, anadditional criterion for reliable data has been constructed. If one takes the averagevalues and add standard deviation, it will result in the following constrains of thedeviation measures:

AARDt<0.38; AARDb<2.9andAADt<0.2; AADb<2

Data, which satisfy these conditions, are considered as preliminarily recommended.


TABLE 7.4: Systems of solute in binary solvent along with qualityfactors.

solute (1) solvent (2) solvent (3) f 0,b12 / f 0,t

12 f 0,b13 / f 0,t

13 Ftest3 Ftest12 Ftest12 QSLE,mixedphenacetin water dioxane 2.10 0.90 0.816 0.5 1 0.764phenacetin water dioxane 0.29 2.15 0.866 0.5 0.5 0.616phenacetin water dioxane 2.05 0.87 0.792 0.5 1 0.756phenacetin ethanol ethyl acetate 3.83 3.50 0.969 0.5 0.5 0.650paracetamol methanol ethyl acetate 0.72 0.58 0.814 0.5 0.5 0.599paracetamol dioxane water 0.63 0.36 0.748 0.5 0.5 0.577paracetamol ethanol methanol 1.45 1.10 1.000 0.5 1 0.825paracetamol ethanol ethyl acetate 3.10 0.62 0.852 0.5 0.5 0.611cholesterol hexane ethanol -3.72 0.72 0.842 0 1 0.608cholesterol ethanol hexane 1.17 3.91 0.775 1 0.5 0.751cholesterol benzene ethanol 1.37 1.19 0.867 0.50 1 0.781cholesterol hexane benzene 0.42 0.17 0.976 0.5 0.1 0.5204-nitrobenzonitrile ethyl acetate methanol 1.87 3.11 0.955 0.5 0.5 0.6454-nitrobenzonitrile ethyl acetate methanol 2.25 3.71 0.892 0.5 0.5 0.6244-nitrobenzonitrile ethyl acetate methanol 2.38 3.63 0.897 0.5 0.5 0.626napthalene propan-2-ol water 0.59 3.72 0.792 0.5 0.5 0.591napthalene ethanol water 0.41 3.49 0.862 0.5 0.5 0.614napthalene benzene hexane -16.07 -3.19 0.995 0 0 0.328napthalene acetone water 0.50 1.78 0.854 0.5 0.5 0.612benzoic acid cyclohexane hexane 2.69 4.93 0.997 0.5 0.5 0.659benzoic acid cyclohexane hexane 1.94 4.18 0.999 0.5 0.5 0.660benzoic acid hexane CCl4 -6.57 -13.44 0.998 0 0 0.329mefenamic acid ethanol ethyl acetate 0.53 -0.31 0.833 0.5 0 0.440testosterone CCl4 cyclohexane -0.01 0.01 0.934 0 0.1 0.341anthracene dibutyl ether hexane 0.33 1.01 0.998 0.5 1 0.824anthracene dibutyl ether heptane -0.69 3.37 0.977 0 0.5 0.487anthracene dioxane propan-1-ol -0.53 4.23 0.984 0 0.5 0.490anthracene propan-1-ol hexane 5.77 1.01 0.994 0.1 1 0.691anthracene propan-1-ol heptane 9.50 0.73 0.991 0.1 0.5 0.525anthracene propan-1-ol octane 6.12 0.25 0.985 0.1 0.5 0.523caffeine water dioxane 0.49 -0.65 0.796 0.5 0 0.428naphthalene ethylene glycol water -0.45 -6.42 0.953 0 0 0.314naphthalene methanol water 0.32 9.57 0.930 0.5 0.1 0.505naphthalene DMSO water -0.26 5.28 0.848 0 0.1 0.313pregabalin methanol water -5.60 -20.96 0.953 0 0 0.275pregabalin methanol water -1.48 -12.78 0.969 0 0 0.309sulphamethoxy- ethanol water 5.81 0.94 0.909 0.1 1 0.663pyridazine

7.5.4 Data quality factors

Assuming that the solubility model is well founded on theory, one can simply con-nect the goodness-of-fit with a quality of the given data-set. For each solubility data-set the quality factor Ftest3 based on AADt value can be assigned. It is the analogy tothe factor proposed by Cunico et al. [40], [17] for binary SLE data-sets. Ftest3 calcu-lated by the following expression

Ftest3 =1

1 + AADt. (7.21)

Finally, the overall quality factor, being a weighted sum of all three quality factorsdeveloped here would be

QSLE,mixed =13

Ftest1/2 +13

Ftest1/2 +13

Ftest3. (7.22)

This section has seen the application of the FST-based model in the validation ofthe excess solubility data. An examination showed that the model, even with its lim-itations can be successfully used to assess the quality of data. Quality factor basedon the goodness-of-fit as well as values of the solute-solvent parameters is proposed.

7.5. Excess solubility data validation 135

Usually for low values of Ftest3, also the overall quality factors is low e.g. naphtha-lene in DMSO and water binary solvent. This is due to earlier explained limitations.However, one needs to be very careful is some cases, because very good fit to ternarydoes not follow the correct behaviour of the solvent-solvent system, e.g. solubilityof naphthalene in ethylene glycol and water. Therefore, one should consider thesetwo factors separately, although these were combined into one overall factor. Veryhelpful would be for the given system draw a diagram with experimental data andmodel predictions. With few exceptions this method can be used as a benchmark inthe validation of the solubility data in mixed solvents. Especially due to the fact thatthe developed method is one of only two existing for ternary SLE data. The secondone is a thermodynamic consistency test developed by Ruckenstein and Shulgin,which seems to have two important limitations, which makes that not directly ap-plicable to most data-sets. This is discussed in the next section.

7.5.5 Thermodynamic data consistency test

In Chapter 2, a thermodynamic consistency test for ternary mixtures proposed byRuckenstein and Shulgin [39] has been discussed. In the beginning, the essentialsteps of this method are repeated.

The Gibbs-Duhem equation for ternary system is used to analyse the qualityof experimental data. The trapezoidal rule for the integration of the Gibbs-Duhemequation is used. Based on the thermodynamic relation between the solubility ofthe solute in mixed solvent at two different, very close solvent compositions andactivity coefficients of the mixed solvent components calculated by Wilson model,the authors have derived an expression Eq. (2.35), which value needs to be lowerthan the limit value Dmax=0.0001, if the two neighbouring points are consistent.

This test has been applied to the systems considered in this chapter, cf. AppendixC. The values of D for each experimental point in the data-sets have been calculated.In general, according to the test a thermodynamic consistency for most systems inthe diluted region and for small changes in the composition of the mixed solvent hasbeen found. However, some systems considered in this chapter report solubiliteshigher than 0.01 and with sparsely distributed points, which lead to the situationthat only part of the data-set can be treated.

For these systems, the calculated D values are significantly higher than the estab-lished limit Dmax= 0.0001. The representative example is the solubility of cholesterolin benzene and ethanol mixtures at 293.15 K. It has been found that the FST modelmatch very accurately the experimental data (Fig. 7.5), but according to the consis-tency test all points have D values above the proposed threshold, cf. Table 7.5. Atleast according to this method, the data points are less thermodynamically consis-tent, since D’s are higher than the maximum value.

TABLE 7.5: D values obtained via Eq. (2.35) for solubility data ofcholesterol (1) in benzene (2) + ethanol (3) mixture

x1 x′2 D

0.00762 0.139 0.01340.0249 0.302 0.03850.0359 0.396 0.022220.0439 0.493 0.016020.0536 0.659 0.01940.0562 0.722 0.00510.0477 0.853 0.0171


However, in fact, this limit was calculated based on only two data sets, exhibitingvery low solubilites (less than 0.002 in mole fractions) and it could be easily updated(probably to higher value), when using other solubility data (even slightly higherthan 0.01). Nonetheless, the validation method developed here gives similar resultsfor systems of interest (with very low solubilities, parameters of the Margules orWilson model regressed from validated VLE data). The example of D values forsolubility data of naphthalene (1) in acetone (2) + water (3) mixture is presentedin Table 7.6. Only data below the composition of the solvent equal to 0.3 can beconsidered in this method.

Furthermore, some solutes can form different polymorphs in the mixed solvent.This sometimes implies dramatic change in the solubility observed at the phase di-agram. This should be taken into account in checking the thermodynamic consis-tency of the solubility points, since the incorrect interpretation of data might lead toincorrect conclusions about the data consistency. Finally, the Ruckenstein-Shulginmethod is used for the internal consistency test of the ternary solubility data re-ported at the one constant temperature. It might be very well, that data-sets arefound to be consistent at different temperatures, but when compared together willnot follow the common temperature dependence (increase of the temperature willincrease the solubility).

TABLE 7.6: D values obtained via Eq. (2.35) for solubility data ofnaphthalene (1) in acetone (2) + water (3) mixture

x1 x′2 D

7.83E-06 0.018 1.07E-053.37E-05 0.056 6.07E-051.09E-04 0.091 1.67E-043.96E-04 0.134 6.52E-046.15E-04 0.182 4.43E-042.91E-03 0.226 5.47E-036.55E-03 0.279 7.66E-030.0191 0.375 2.75E-020.0325 0.445 2.71E-020.0501 0.527 3.54E-020.0813 0.649 6.29E-020.0957 0.701 2.79E-020.1257 0.812 5.90E-02

The interesting question to ask here would be, if the systems, which Ruckensteinand Shulgin have found to be most consistent, are consistent also according to themethod presented here. First of all, it needs to be said that authors considered solu-bility of drugs only in aqueous systems, so the solubilities are rather low, especiallytowards the pure water. Moreover, only parts of the data-set were used, as the exper-imental determinations were made with small changes in the solvent compositionin these parts. They have tested four systems (all at 298.15 K):

• solubility of naphthalene in acetone-water mixtures (6 points) [211],

• solubility of naphthalene in ethanol-water mixtures (6 points) [211],

• solubility of naphthalene in ethylene glycol-water mixtures (16 points) [205],

• solubility of sulphamethoxypyridazine in ethanol-water mixtures (7 points)[220].

7.6. Conclusions 137

They have found that almost all points are thermodynamically consistent. In thesystem of naphthalene (1)/ethylene glycol (2)/ water(3) two points at the x

′2 ≈ 0.6

are less accurate.Comparing these results to the FST method one can make similar conclusions.

For all systems the quality factor Ftest3 is higher than 0.85. Especially in the sameranges of solvent compositions as considered by authors, the experimental data arein a very good agreement with the FST-model (cf. Fig. 7.6 and 7.7). Interestingly, thetwo points in the system of naphthalene (1)/ethylene glycol (2)/ water(3) at the x

′2 ≈

0.6 also match less accurately the FST model. However, one needs to remember, thatthe behaviour in this system is difficult to capture, at least by the estimation frombinary SLE data alone.

To sum up, the FST-based model agrees with the thermodynamic consistencytest of Ruckenstein and Shulgin in the diluted region. For higher deviations from adiluted region the latter method becomes inaccurate and it cannot be used, with theproposed criteria. FST model, even though rigorous in the diluted region, can alsoreliably correlate experimental data of higher solubility.

7.6 Conclusions

A simple, but powerful model, derived from fluctuation solution theory by Elle-gaard et al. has been applied to describe the solubility of solids in mixed solvents.Different approaches of obtaining the model parameters, including predictive modelsuch as COSMO-SAC have been explored. The COSMO-SAC based values are usu-ally much larger than those found from binary SLE data or those found from re-gressions to ternary data. It has been found that the best approach would be anregression of the excess solubility model parameters from solubility of the solute inmixed solvents and solvent representation (derivative) taken from VLE data. Therigorous nature of the model is the basis for the criteria used in validation of theexperimental data. A new validation procedure developed here can become a pow-erful alternative/support to the only one method existing for the quality assessmentof the solubility data in mixed solvent.

Additionally, the excess solubility model was extended for systems with solutes,which can form different polymorphs and it can be provisionally used in the valida-tion of data as a special case.

139

Chapter 8

Overall conclusions and outlook

Fluctuation solution theory (FST) is an exact theory of solutions and connects deriva-tive thermodynamic properties to correlation function integrals. This thesis has de-livered an application of this rigorous and simple in form models for thermody-namic properties, such as liquid-liquid equilibria in binary and ternary systems aswell as solubility of solids in solvents. Based on these rigorous models, criteria forthe validation of data has been formulated.

Below the main conclusions from the thesis are outlined along with theirs signif-icance. Based on the insights into the data validation few topics are recognized asparticularly worth studying in the future.

8.1 Conclusions

The brief literature overview on the methods used in the fluid phase equilibria datavalidation as well as short coverage of some fundamental expressions from fluctua-tion solution theory have been presented.

The first part of the thesis has focused on the modelling and validation of LLEdata. First, in Chapter 4, the binary LLE model has been developed and tested. Ex-pressing an unsymmetrical component activity coefficient using fluctuation solutiontheory has led to simple expressions for a model for binary liquid-liquid equilibriaover ranges of temperature. At least four parameters are needed, which is similar tobenchmark models such as NRTL. Additional parameters can be added, if the solu-bilities in liquid phases are higher. A procedure for parameter estimation has beengiven. This includes the graphical representation of the data and phase stabilitycriteria. The uncertainty analysis of the model parameters and estimated mole frac-tions from the regressions has been performed. Comparisons with results from thebenchmark model i.e. NRTL, indicated mostly similar reliability. The main advan-tage of the model is its rigorous form, therefore it can be used in the discriminationof unreliable data.

In Chapter 5, the ternary LLE expressions based on the fluctuation solution the-ory suitable for the modelling of Type I phase diagrams (and not too close to plaitpoint) has been developed. The model relies on the parameters regressed from bi-nary subsystems data: vapour-liquid equilibria and liquid-liquid equilibria. Com-parisons with results from the NRTL model have shown that the proposed modeldoes better than NRTL with parameters regressed to binary data alone. However,the correlation of the ternary data only is outperformed by the NRTL model. Eventhough, it seems that in general NRTL is more accurate, the FST-based model hasbetter transferability of the parameters. These should be regressed from the binaryVLE and LLE data at the same or similar temperature as the reported ternary LLEdata-set. It has been found that parameters obtained from binary VLE are readily

140 Chapter 8. Overall conclusions and outlook

transferred to ternary systems, whereas the interaction parameters obtained fromLLE might cause some issues.

In Chapter 6, the validation methodology for binary and ternary LLE has beenproposed. The basis for the validation criteria are rigorous models developed in pre-vious chapters and additional derivative solution properties. The use of the reliableactivity coefficient model allows for quality factors to be established. The additionalcriteria based on the auxiliary properties calculated from the binary LLE model pa-rameters has been formulated.

The second part of the thesis has addressed the validation of solubility of solidsin mixed solvents data.

In Chapter 7, the model for the excess solubility of solids in mixed solvents basedon the FST, developed by Ellegaard has been considered. This model is rigorous,within a limited set of assumptions. Different approaches of obtaining the solvent-solute and solvent-solvent parameters has been tested, prior to a development ofthe criteria for the validation of the solubility data in mixed solvents. Quality factorsbased on the accuracy of the model and parameters values have been established.Finally, the model has been extended to the solubilities of solids, which form pseudo-polymorphs in the mixtures of solvents.

Overall this thesis shows the successful development and application of nearlyrigorous models in the modelling and data evaluation of the thermodynamic prop-erties: LLE and SLE in the systems with the potential use in life sciences applications.

8.2 Outlook

By no means have all aspects of the validation of LLE and SLE been addressed in thethesis. Future aspects of data validation related to the studies and findings in thisthesis are discussed here.

• Software implementation of the proposed LLE models

The LLE data validation methodology could be automatized in a software con-taining integrated database of relevant thermodynamic data (e.g. NIST-TDE orDECHEMA), options for data analysis, model selection and parameter estima-tion.

• Models extensions

The proposed LLE models are rigorous outside the critical region. The naturalextension would be to combine the existing model with a reliable correlationmodel, which is able to describe a critical region. In Chapter 4 it has beenindicated that scaling law equation could be considered. It is essential, thatrigor in the model must be conserved in order to use it in the discrimination ofunreliable data.

Furthermore, the modelling expressions in the ternary LLE model are suitableonly for Type I phase diagrams. Therefore, an extension to other types, inwhich more than one binary pair is partially miscible should be also consid-ered.

Finally, the modelling of LLE systems with more than 3 components can beanalyzed.

• Experimental work

8.2. Outlook 141

Despite the fact that this thesis is only about modelling there are some sug-gestions for the experimental work. First, it is desirable to expand the litera-ture with the experimental data on solubility data of solids having tendencyto form polymorphs in mixed solvents. This is due to not only validating thesolubility model but also for further crystallization process design. An idealdata-set should contain the solubility values and the identified solid form ateach composition of the mixed solvent and temperature.

In order to make a use of the whole LLE validation methodology, measure-ments of excess partial molar properties, especially partial molar excess en-tropies and partial molar excess heat capacities as well as partial molar excessproperties in ternary systems are desirable. Moreover, the methods for evalua-tion of the infinite dilution properties data should be developed to ensure thatthe experimental data can serve as a reference in the inter-property compari-son.

• Models improvements

Finally, it would be worth studying whether the use of validated LLE/SLEdata in the estimation of the global set of parameters of the models such asUNIFAC or COSMO-SAC, will improve their prediction accuracy.

143

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159

Appendix A

Reduction of the ternary LLEmodel to the binary

A.0.1 Reduction of the ternary model to the LLE binary system

It is worth to analyze if the ternary model reduces to the binary models in case of asingle component disappearing. This is tested in this section. When only two com-ponents (2 and 3) are left, all that remains is the partially miscible pair; the ternarymixture gE reduces to, in the two respective phases:

gE(T, x′2, x

′3)

RT= A

′23(T)x

′2x′3 (A.1)

gE(T, x′′2 , x

′′3)

RT= A

′′23(T)x

′′2x′′3 (A.2)

c = ln γ′2 = A

′23x

′3

2 (A.3)

d = ln γ2(T, x′′1 , x

′′2)− lim

x3→1ln γ

′′2 = A

′′23(x

′′3

2 − 1) (A.4)

e = ln γ3(T, x′′1 , x

′′2)− lim

x2→1ln γ

′′3 = A

′23(x

′2

2 − 1) (A.5)

f = ln γ′′3 = A

′′23x

′′2

2 (A.6)

Whence

{ ln x′2 + A

′23x

′3

2 = ln x′′2 + ln[

H′2′(T)

f 02 (T)

] + A′′23(x

′′3

2 − 1)

ln(x′3) + ln[

H′3(T)

f 03 (T)

] + A′23(x

′2

2 − 1) = ln(x′′3) + A

′′23x

′′2

2

(A.7)

which are exactly the same modelling expressions as in the binary unsymmetri-cal model, when one inserts A

′23 = − cα

T and A′′23 = − cβ

T .

A.0.2 Model reduction to VLE binary pairs 1+2 and 1+3

Pair 1+2

When x′3 goes to zero the excess Gibbs energy expression reduces to:

160 Appendix A. Reduction of the ternary LLE model to the binary

gE(T, x′1, x

′2)

RT= A

′12(T)x

′1x′2 (A.8)

Therefore, symmetrically normalized activity coefficients are expressed

ln γ1 = A′12(x2

2)and ln γ2 = A′12(x2

1) (A.9)

The left hand side the below equation will remain unchanged. Since in theternary model we assume, that phase ′ is rich in component 1, we use unsymmetricalconvention for component 1 and symmetrical for component 2. The right hand sideof the equation needs to express fugacity of both components in the vapor phase.The standard state fugacity of the pure component 2 as a liquid is equal to satura-tion pressure of pure liquid 1 at temperature T.

{ ln x′1 + ln[H

′1(T)] + ln γ1(T, x

′1, x

′2)− lim

x2→1ln γ

′1 = ln(y1P) = ln f V

1 (T, P, y1)

ln x′2 + ln[ f 0

2 (T)] + ln γ2(T, x′1, x

′2) = ln(y2P) = ln f V

2 (T, P, y1)(A.10)

Substituting the activity coefficient by the model expression, we obtain:{ ln f′1(T, x

′1) = ln x

′1 + ln[H

′1(T)] + A

′12(x

′1

2 − 2x′1) = ln(y1P)

ln f′2(T, x

′1) = ln x

′2 + ln[Psat

2 (T)] + A′12(x

′1

2) = ln(y2P)(A.11)

This gives the final VLE model.

Pair 1+3

When the concentration of x′′2 goes to zero the excess Gibbs energy expression re-

duces:

gE(T, x′′1 , x

′′3)

RT= A

′′13(T)x

′′1x′′3 (A.12)

Therefore,ln γ1 = A

′′13(x2

3)and ln γ3 = A′′13(x2

1) (A.13)

Applying the same reasoning (in this case component 3 appears in excess) weobtain

{ ln x′1 + ln[H

′1(T)] + ln γ1(T, x

′1, x

′2)− lim

x2→1ln γ

′1 = ln(y1P) = ln f V

1 (T, P, y1)

ln(x′′3) + ln[ f 0

3 (T)] + ln γ3(T, x′′1 , x

′′3) = ln(y3P) = ln f V

2 (T, P, y1)(A.14)

Substituting the activity coefficient by the model expression, we obtain the 1-3pair VLE expressions:

{ ln f′′1 (T, x

′′1) = ln x

′′1 + ln[H

′′1 (T)] + A

′′13(x

′′1

2 − 2x′′1) = ln(y1P)

ln f′′3 (T, x

′′1) = ln x

′3 + ln[Psat

3 (T)] + A′′13(x

′′1)

2 = ln(y3P)(A.15)

i.e.

Appendix A. Reduction of the ternary LLE model to the binary 161

{ ln f′′1 (T, x

′′1) = ln x

′′1 + ln[H

′′1 (T)] + A

′′13(x

′′1

2 − 2x′′1) = ln(y1ϕ1P) = ln f V

1 (T, P, y1)

ln f′3(T, x

′′1) = ln x

′′3 + ln[Psat

3 (T)] + A′′13(x

′′1)

2 = ln(y3ϕ3P) = ln f V3 (T, P, y1)

(A.16)

163

Appendix B

Tables

B.0.1 Thermophysical properties of solutes

TABLE B.1: Thermophysical properties of solutes

Solute ∆H f us /(kJ/mol) Tm / K ∆Cp,m / (J/(mol · K))phenacetin 28.75 [60] 407.4 [60]paracetamol 27.85 [60] 443.01 [60]cholesterol 26.63 [60] 421.43 [60] 8.8 [60]4-nitrobenzonitrile 17.73 [226] 420.6 [226]naphthalene 18.98 [227] 353.35 [227]benzoic acid 18.07 [227] 395.55 [227]mefenamic acid 38.7 [227] 503.65 [227]testosterone 26.179 [59] 427 [59]anthracene 29.37 [228] 488.15 [227]caffeine 21.6 [227] 512.15 [227]pregabalin 122.32 [219] 464.32 [219]citric acid 26.7 [229] 426.15 [229]malonic acid 25.48 [229] 407.95 [229]malic acid 25.3 [229] 403.15 [229]3-nitrophthalic acid 32.47 [230] 596.49 [230]naproxen 34.2 [231] 428.75 [231]aspirin 29.8 [232] 416.15 [232]pyrene 17.36 [227] 424.35 [227]thiamphenicol 46.91 [233] 437.76 [233]2-acetyl-naphth-1-ol 22.52 [234] 371.75 [234]beta-alanine 23.399 [235] 450.3 [235]L-glutamic acid 97.577 [236] 592.1 [236]L-aspartic acid 76.709 [236] 597.6 [236]valsartan 31.647 [237] 380.65 [237]glyphosate 31.59 [238] 487.8 [238]phthalimide 28.6 [239] 507.2 [239]sulphamethoxypyridazine 30.07 [220] 454 [220]

B.0.2 NRTL model parameters for ternary systems

164 Appendix B. Tables

TAB

LE

B.2

:NR

TLm

odel

para

met

ers

for

tern

ary

syst

ems,

α=0

.2.

Syst

emT

/K∆

g 12

∆g 2

1∆

g 13

∆g 3

1∆

g 23

∆g 3

2R

MSD

(cor

r.)R

MSD

(pre

d.)

benz

ene/

acet

onit

rile

/hex

ane

318.

1533

36.1

7-6

13.3

615

56.8

5-6

46.9

150

40.8

336

51.3

30.

0021

0.05

9be

nzen

e/ac

eton

itri

le/h

exan

e30

8.15

6284

.95

-238

3.07

32.9

944

6.65

5496

.23

3573

.62

0.00

300.

009

benz

ene/

acet

onit

rile

/hex

ane

298.

1523

19.2

9-2

38.2

829

70.0

4-1

956.

9148

57.1

238

10.6

90.

0058

0.01

2be

nzen

e/ac

eton

itri

le/h

epta

ne31

8.15

-281

3.16

9652

.38

-359

.41

5164

.47

5855

.81

3515

.98

0.00

580.

041

etha

nol/

dibu

tyle

ther

/wat

er29

8.15

3420

.18

-333

4.09

-169

2.59

-103

9.51

5495

.75

2925

5.41

0.00

450.

045

etha

nol/

hexa

ne/a

ceto

nitr

ile31

3.15

1120

9.12

-324

2.51

-909

5.26

8282

.42

2280

.26

7183

.73

0.00

830.

033

ethy

lace

tate

/AC

N/h

exan

e31

8.15

6812

.54

-496

5.75

1001

.41

293.

2449

52.3

037

26.1

30.

0030

0.06

1et

hyla

ceta

te/A

CN

/hex

ane

308.

1532

31.1

8-3

321.

1396

0.15

310.

3749

23.5

537

56.9

30.

0040

0.02

7et

hyla

ceta

te/A

CN

/hex

ane

298.

1557

02.8

1-4

638.

6691

0.31

357.

0649

37.1

637

35.5

50.

0039

0.03

2pr

opan

-2-o

l/he

xane

/wat

er30

361

6.97

5365

.20

-825

.03

6973

.37

8734

.47

2752

1.79

0.03

540.

050

prop

an-2

-ol/

hexa

ne/w

ater

313

567.

9731

13.4

7-2

653.

4610

002.

5785

85.0

616

802.

420.

0777

0.07

6pr

opan

-2-o

l/he

xane

/wat

er32

357

5.90

3194

.47

-272

9.34

9580

.71

8543

.88

1762

2.17

0.09

30n/

abe

nzen

e/he

xane

/sul

fola

ne34

8.15

87.0

6-9

17.9

550

52.2

5-2

323.

1311

324.

2756

25.1

50.

0037

0.02

6be

nzen

e/he

xane

/sul

fola

ne37

3.15

5929

.78

1468

.61

3674

.06

4110

.90

1252

4.04

5067

.09

0.10

300.

011

etha

nol/

wat

er/e

thyl

acet

ate

298.

15-2

803.

7735

69.8

4-2

740.

2798

31.8

016

720.

4647

592.

730.

0041

0.19

4et

hano

l/w

ater

/eth

ylac

etat

e30

8.15

-610

7.49

1333

2.55

-211

5.92

1825

8.95

1527

1.34

5088

8.79

0.00

850.

283

etha

nol/

wat

er/e

thyl

acet

ate

318.

1560

31.2

6-3

801.

14-4

029.

1382

92.6

815

370.

1450

228.

730.

0095

0.24

6m

etha

nol/

wat

er/e

thyl

acet

ate

343.

15-4

043.

6883

45.8

9-2

029.

3610

531.

1413

051.

4593

648.

940.

0129

n/a

prop

an-2

-ol/

benz

ene/

wat

er30

3-2

111.

4469

28.5

413

34.5

113

12.6

633

944.

5411

969.

300.

0213

n/a

prop

an-2

-ol/

benz

ene/

wat

er31

332

30.5

722

0.65

-562

9.47

1293

7.69

9283

.46

8022

.19

0.01

08n/

apr

opan

-2-o

l/be

nzen

e/w

ater

323

5666

.29

-422

.15

-653

1.99

1404

7.39

6987

.57

7992

.89

0.01

27n/

am

etha

nol/

t-am

ylm

ethy

leth

er/w

ater

298.

1525

.30

3673

.37

1138

0.52

-398

2.25

3716

.48

2058

9.42

0.02

050.

166

prop

an-2

-ol/

cycl

ohex

ane/

wat

er30

332

98.6

251

59.0

610

02.2

878

77.1

072

18.1

267

79.8

30.

0411

0.17

6pr

opan

-2-o

l/cy

cloh

exan

e/w

ater

313

3701

.04

4836

.20

968.

2479

80.7

571

69.4

684

88.0

90.

0322

0.16

7pr

opan

-2-o

l/cy

cloh

exan

e/w

ater

323

4485

.73

3184

.98

895.

0472

65.9

468

95.9

486

26.9

90.

0247

0.16

8be

nzen

e/ni

trom

etha

ne/c

yclo

hexa

ne29

8.15

1671

4.70

-507

3.88

-401

3.33

3274

.15

4223

.54

6732

.47

0.02

510.

146

AC

N/w

ater

/eth

ylac

etat

e33

3.15

-344

7.60

9464

.11

4359

.77

-257

5.60

1213

2.55

5608

4.70

0.01

16n/

abe

nzen

e/M

eOH

/cyc

lohe

xane

298.

15-2

773.

6390

81.7

024

20.1

1-3

34.3

328

45.6

244

626.

240.

0088

0.15

5to

luen

e/M

eOH

/cyc

lohe

xane

298.

15-3

97.0

4-4

406.

01-7

025.

0222

46.7

527

47.5

645

20.4

20.

0101

0.15

7ac

eton

e/M

eOH

/cyc

lohe

xane

298.

15-4

889.

23-2

046.

33-1

597.

92-2

4490

.81

3068

.80

4179

.34

0.01

630.

113

prop

an-2

-ol/

MeO

H/c

yclo

hexa

ne29

8.15

-561

7.38

6086

.84

3602

.34

-572

6.93

3493

.11

3605

.59

0.05

230.

256

etha

nol/

nitr

omet

hane

/wat

er30

3.15

-380

2.87

4210

.84

782.

15-3

247.

8026

98.4

478

47.8

10.

0122

0.17

2C

onti

nued

onne

xtpa

ge

Appendix B. Tables 165

Tabl

eB

.2–

cont

inue

dfr

ompr

evio

uspa

geSy

stem

T/K

∆g 1

2∆

g 21

∆g 1

3∆

g 31

∆g 2

3∆

g 32

RM

SD(c

orr.)

RM

SD(p

red.

)et

hano

l/ni

trom

etha

ne/w

ater

313.

15-3

874.

09-2

6344

3-3

39.0

8-2

6901

7.24

2531

.87

7625

.38

0.02

990.

150

etha

nol/

nitr

omet

hane

/wat

er31

3.15

1818

.93

6320

.98

-112

2.96

1024

2.60

2638

.25

6186

.92

0.05

150.

133

etha

nol/

nitr

omet

hane

/wat

er32

3.15

-449

6.42

-425

7.23

-619

3.04

-283

.70

2446

.49

7389

.80

0.00

970.

113

etha

nol/

nitr

omet

hane

/wat

er33

3.15

-691

1.46

-198

67.8

6-2

979.

91-2

5049

.72

2291

.99

7426

.78

0.00

700.

107

benz

ene/

nitr

omet

hane

/cyc

lohe

xane

298.

15-3

88.0

118

30.0

1-4

276.

1440

53.3

351

22.5

660

40.5

00.

0162

0.08

4be

nzen

e/ni

trom

etha

ne/h

epta

ne29

8.15

-219

3.42

-171

5.23

-278

0.46

-214

5.71

8629

.89

4092

.44

0.00

63n/

abe

nzen

e/ni

trom

etha

ne/h

epta

ne30

3.15

3969

.62

182.

59-1

19.5

773

9.42

7617

.26

4099

.37

0.00

720.

078

AC

N/c

hlor

oben

zene

/wat

er30

4.15

-298

8.74

7298

.61

1343

2.92

96.6

634

94.3

315

960.

260.

0374

0.00

0A

CN

/ben

zene

/wat

er29

8.15

8833

.57

-454

4.54

557.

6157

39.0

614

895.

1912

308.

750.

0153

0.05

5A

CN

/ben

zene

/wat

er31

8.15

375.

4511

53.4

070

8.96

5480

.56

1475

8.59

1323

8.10

0.01

250.

040

AC

N/h

exan

e/w

ater

298.

1561

01.1

035

58.6

316

16.5

018

77.9

496

88.7

828

40.8

30.

0076

0.03

8A

CN

/tol

uene

/wat

er30

3.15

556.

2119

27.0

721

77.6

739

83.3

684

53.2

314

302.

240.

0073

n/a

AC

N/h

epta

ne/w

ater

298.

1598

16.9

419

65.3

6-2

900.

0476

33.0

311

35.8

128

521.

890.

0036

0.02

0et

hano

l/A

CN

/cyc

lohe

xane

298.

15-6

534.

3252

21.0

744

90.3

4-2

40.4

848

25.5

946

47.6

90.

0214

0.11

4et

hano

l/A

CN

/hep

tane

298.

15-7

266.

4984

77.1

351

11.1

0-8

8.73

7994

.12

3074

.79

0.02

140.

133

etha

nol/

AC

N/h

epta

ne31

3.15

-673

7.73

7346

.71

1023

.68

-220

1.35

6693

.45

3290

.83

0.02

150.

149

etha

nol/

AC

N/o

ctan

e29

8.15

-782

2.66

1529

3.21

4983

.74

314.

5512

501.

3930

45.2

00.

0149

0.06

8

166 Appendix B. Tables

TABLE B.3: NRTL binary systems (VLE or LLE) parameters

System (i)/(j) aij aji bij bjibenzene/cyclohexane 1445.54 -160.54benzene/acetonitrile 1457.27 1410.83cyclohexane/acetonitrile 21272.29 2505.47 -52.10 4.02benzene/ACN 1457.27 1410.83benzene/hexane 3613.83 -1588.27ACN/hexane 5216.33 16976.28 -2.97 -40.86benzene/acetonitrile 1457.27 1410.83benzene/heptane 4373.95 -2095.74acetonitrile/heptane 7288.40 16212.87 -4.94 -40.55EtOH/DBE 596.20 4298.94EtOH/water -458.79 4120.48DBE/water 27171.20 22335.38 -72.69 -25.99EtOH/hexane 1712.27 5600.67EtOH/acetonitrile 2101.42 1538.87hexane/ACN 16976.28 5216.33 -40.86 -2.97ethyl acetate/ACN -795.50 2277.77ethyl acetate/hexane 253.63 3350.25ACN/hexane 5216.33 16976.28 -2.97 -40.862-propanol/water -723.44 6610.562-propanol/hexane 1269.39 4554.30hexane/water 34877.80 -4092.98 -65.68 106.81benzene/water 25406.71 5994.30 -51.53 30.76water/ethyl acetate 59374.37 -24144.73 -104.07 267.05benzene/2-propanol 4328.83 575.812-propanol/cyclohexane 3298.62 5159.06EtOH/ethyl acetate 639.73 2524.67ACN/water 980.73 5803.09MeOH/t-amyl methyl ether(TAME) 125.30 3673.37benzene/nitrometahne 2385.43 1004.35nitromethane/cyclohexane 6738.01 17621.07 -5.35 -41.00benzene/cyclohexane 1404.87 -135.81methanol/benzene 577.05 6135.07methanol/toluene 1620.99 4746.22acetone/methanol 1197.81 709.25methanol/isopropanol -1355.51 1615.90cyclohexane/toluene 1029.33 11.99acetone/cyclohexane 2888.12 2074.43methanol/cyclohexane -1081.44 21693.98 13.39 -58.47benzene/sulfolane 5293.89 -1392.58hexane/sulfolane 18490.44 14320.46 -21.12 -24.29methanol/water -391.29 1725.05methanol/ethyl acetate 344.97 3038.57TAME/water 45228.80 437.86 -139.12 40.63ethanol/benzene 108.26 5717.952-methylpropanol/benzene -532.93 5334.622-methylpropanol/water 8232.93 -17455.46 -34.38 98.02ACN/octane 17529.09 17832.57 -33.88 -46.70ethanol/octane 2248.00 4744.00ethanol/nitromethane 2154.57 3197.71nitromethane/water 18401.89 -2334.52 -49.04 28.60heptane/methanol 25568.91 -7820.00 -76.31 42.16ethanol/cyclohexane 1265.06 5719.58ethanol/heptane 1822.00 5412.90methanol/hexane 3394.90 5376.90

167

Appendix C

Ruckenstein-Shulgin consistencytest

Here, the values of D from consistency test of Ruckenstein-Shulgin for all studiedsystems are tabulated.

TABLE C.1: Paracetamol (1) + methanol (2) + ethyl acetate (3)

x1 x′2 D

0.0206 0.1 0.03100.0338 0.2 0.02700.0487 0.4 0.03020.0506 0.5 0.00390.0519 0.55 0.00270.0519 0.6 0.00010.0471 0.65 0.00950.0466 0.7 0.00090.0458 0.75 0.00150.0471 0.8 0.00270.0457 0.85 0.00270.042 0.9 0.0073

TABLE C.2: Paracetamol (1) + dioxane (2) + water (3)

x1 x′2 D

0.005 0.02295 0.0070.0129 0.05019 0.0170.0261 0.08306 0.0270.0525 0.1235 0.0550.0772 0.1745 0.0500.1081 0.2407 0.0620.1318 0.2819 0.0470.1527 0.3303 0.0410.1468 0.388 0.0120.1559 0.4581 0.0180.1684 0.4905 0.0240.1841 0.545 0.0300.1575 0.5858 0.0540.1325 0.6554 0.0510.0982 0.8006 0.070

168 Appendix C. Ruckenstein-Shulgin consistency test

TABLE C.3: Paracetamol (1) + ethanol (2) + ethyl acetate (3)

x1 x′2 D

0.0226 0.1575 0.03380.0386 0.296 0.03280.0474 0.4189 0.01770.0625 0.5286 0.03050.068 0.6227 0.01110.0863 0.7969 0.03690.0833 0.8706 0.00590.0753 0.905 0.01590.0757 0.938 0.00090.0764 0.9572 0.00150.0688 0.9697 0.01510.0542 1 0.0292

TABLE C.4: Phenacetin (1) + ethanol (2) + ethyl acetate (3)

x1 x′2 D

0.0002 0.023 0.000210.0004 0.0503 0.000420.0006 0.0833 0.000400.001 0.1238 0.000820.0015 0.1749 0.001010.0035 0.2413 0.004230.0084 0.3309 0.010390.0178 0.3888 0.019590.0291 0.4589 0.022890.0438 0.5458 0.029500.0471 0.6562 0.006150.0172 1 0.06517

TABLE C.5: Phenacetin (1) + water (2) + dioxane (3), 298.15 K, 298 Kand 313 K (each next three columns, respectively).

x1 x′2 D x1 x

′2 D x1 x

′2 D

0.0002 0.0230 0.00021 0.000248 0.0230 0.00037 0.000461 0.0230 0.000700.0004 0.0503 0.00042 0.000404 0.03606 0.00032 0.000737 0.0361 0.000560.0006 0.0833 0.00040 0.000648 0.0503 0.00050 0.00128 0.0503 0.001110.001 0.1238 0.00082 0.00102 0.0660 0.00076 0.00204 0.066 0.001550.0015 0.1749 0.00101 0.00155 0.0833 0.00107 0.00319 0.0833 0.002340.0035 0.2413 0.00423 0.00351 0.1238 0.00413 0.00702 0.1236 0.008040.0084 0.3309 0.01039 0.00835 0.1749 0.01026 0.016 0.1749 0.018940.0178 0.3888 0.01959 0.0178 0.2413 0.01975 0.0309 0.2413 0.030790.0291 0.4589 0.02289 0.0291 0.33099 0.02294 0.052 0.3309 0.042950.0438 0.5457 0.02950 0.0438 0.4589 0.02956 0.071 0.4589 0.037900.0471 0.656 0.00615 0.0471 0.6562 0.00618 0.0764 0.6562 0.010120.0172 1 0.06517 0.0172 1 0.06517 0.0307 1 0.09832

Appendix C. Ruckenstein-Shulgin consistency test 169

TABLE C.6: Cholesterol (1): hexane (2) + ethanol (3); ethanol (2) +hex-ane(3); hexane (2) + benzene (3), all at 293.15 K, (each next three

columns, respectively).

x1 x′2 D x1 x

′2 D x1 x

′2 D

0.00826 0.1007 0.0151 0.01863 0.1249 0.0336 0.0386 0.149 0.02140.0223 0.2326 0.0303 0.02615 0.2341 0.0150 0.0435 0.2174 0.00980.0373 0.4096 0.0306 0.03137 0.36 0.0101 0.0422 0.3178 0.00270.0403 0.6496 0.0058 0.03219 0.4575 0.0011 0.033 0.5135 0.01860.00427 1 0.1002 0.02983 0.5534 0.0053 0.0164 0.7379 0.0346

0.0271 0.6632 0.0061 0.00427 1 0.02790.01853 0.8014 0.01800.00953 0.905 0.01910.00185 1 0.0189

TABLE C.7: 4-nitrobenzonitrile (1) + ethyl acetate (2) + methanol (3):318.15; 298.15 and 278.15 K (each next three columns, respectively).

x1 x′2 D x1 x

′2 D x1 x

′2 D

0.02542 0.0388 0.0247 0.01271 0.0388 0.0128 0.00652 0.0388 0.00690.03913 0.0833 0.0278 0.0199 0.0833 0.0146 0.0105 0.0833 0.00810.05281 0.1348 0.0274 0.02709 0.1348 0.0144 0.01456 0.1348 0.00810.06526 0.1951 0.0247 0.03356 0.1951 0.0128 0.01825 0.1951 0.00720.07642 0.2667 0.0218 0.03929 0.2667 0.0112 0.02152 0.2667 0.00630.08771 0.3529 0.0218 0.04699 0.3529 0.0150 0.02301 0.3529 0.00260.1007 0.459 0.0248 0.05407 0.459 0.0135 0.02654 0.459 0.00660.1163 0.5926 0.0294 0.05947 0.5926 0.0099 0.03318 0.5926 0.01260.1291 0.766 0.0230 0.06943 0.766 0.0186 0.03415 0.766 0.00100.1422 1 0.0224 0.07205 1 0.0033 0.04044 1 0.0115

TABLE C.8: Naphthalene (1) + propan-2-ol (2) + water (3); naphtha-lene (1) + ethanol (2) + water (3); naphthalene (1) + methanol (2) +

water (3); (each next three columns, respectively).

x1 x′2 D x1 x

′2 D x1 x

′2 D

0.000007952 0.0302 8.52E-06 0.00000795 0.0438 8.51E-06 0.00000647 0.0411 5.31E-060.00005802 0.0821 0.00013 0.0000125 0.0672 9.24E-6 0.0000116 0.0926 1.05E-050.0001532 0.1024 0.00020 0.0000285 0.1024 3.38E-05 0.0000242 0.1483 2.63E-50.0008686 0.16 0.00177 0.0000597 0.1308 6.51E-05 0.000057 0.2128 6.96E-050.001652 0.2123 0.00161 0.000214 0.1826 0.00035 0.000173 0.2947 0.000260.003397 0.2831 0.00362 0.000373 0.2101 0.00033 0.000782 0.4376 0.001440.005113 0.3405 0.00344 0.00105 0.2808 0.00147 0.00129 0.4932 0.001040.00807 0.4281 0.00593 0.00232 0.3587 0.00266 0.00336 0.6238 0.004460.01416 0.5615 0.01232 0.00375 0.4213 0.00290 0.00772 0.7684 0.009240.02392 0.761 0.01955 0.0063 0.5022 0.00518 0.0103 0.8196 0.005240.03781 1 0.02736 0.0134 0.6721 0.01477 0.0133 0.8715 0.00609

0.0175 0.7399 0.00808 0.0167 0.9237 0.006910.0296 0.9064 0.02444 0.0227 1 0.012210.0381 1 0.01659


TABLE C.9: Naphthalene (1) + benzene (2) + hexane (3); naphthalene(1) + acetone (2) + water (3); naphthalene (1) + DMSO (2) + water (3);

(each next three columns, respectively).

x1 x′2 D x1 x

′2 D x1 x

′2 D

0.1313 0.0743 0.029 7.83E-06 0.018 1.07E-05 0.000006039 0.0144 4.42E-060.159 0.2121 0.055 3.37E-05 0.056 6.07E-05 0.00000895 0.0254 5.90E-060.2028 0.4109 0.088 1.09E-04 0.091 1.67E-04 0.00002042 0.0555 2.42E-050.24 0.596 0.074 3.96E-04 0.134 6.52E-04 0.00002706 0.0732 1.34E-050.2744 0.8219 0.068 6.15E-04 0.182 4.43E-04 0.00005334 0.1054 5.47E-050.2905 0.9506 0.031 2.91E-03 0.226 5.47E-03 0.00007605 0.1239 4.61E-050.2946 1 0.006 6.55E-03 0.279 7.66E-03 0.0001598 0.1656 0.0002

0.0191 0.375 2.75E-02 0.0003225 0.2081 0.00030.0325 0.445 2.71E-02 0.0005233 0.2401 0.00040.0501 0.527 3.54E-02 0.001408 0.3092 0.00190.0813 0.649 6.29E-02 0.004155 0.3977 0.00610.0957 0.701 2.79E-02 0.01226 0.5116 0.01790.1257 0.812 5.90E-02 0.02614 0.6156 0.02940.1759 1.000 9.89E-02 0.06097 0.7696 0.0748

0.1204 1 0.1261

TABLE C.10: Benzoic acid (1) + cyclohexane (2) + hexane (3) 303.15K;benzoic acid (1) + cyclohexane (2) + hexane (3) 298.15 K; benzoicacid (1) + hexane (2) + CCl4 (3) 298.15 K; (each next three columns,

respectively).

x1 x′2 D x1 x

′2 D x1 x

′2 D

0.0136 0.2341 0.0020 0.0108 0.2573 0.0016 0.0395 0.155 0.019490.0138 0.2845 0.0004 0.0113 0.4699 0.0010 0.0297 0.3419 0.019760.0146 0.5059 0.0016 0.0115 0.5342 0.0004 0.0251 0.4408 0.009250.015 0.667 0.0008 0.0118 0.6566 0.0006 0.0195 0.5951 0.011290.0151 0.7821 0.0002 0.0117 0.8044 0.0002 0.0137 0.8028 0.011750.0146 1 0.0010 0.0115 1 0.0004 0.01 1 0.00749

TABLE C.11: Mefenamic acid (1) + ethanol (2) + ethyl acetate (3)

x1 x′2 D

0.00632 0.0813 0.005060.00678 0.158 0.000920.00873 0.296 0.003920.0085 0.419 0.000460.00796 0.529 0.001080.00755 0.627 0.000820.00477 0.797 0.005660.00291 0.938 0.003800.00182 1 0.00222


TABLE C.12: Testosterone (1) + ethanol (2) + ethyl acetate (3)

x1 x′2 D

0.000426 0.0287 0.000350.000618 0.0533 0.000390.000957 0.0795 0.000690.00141 0.105 0.000920.00178 0.131 0.000740.00649 0.253 0.010700.0211 0.367 0.032530.0577 0.474 0.079270.102 0.575 0.090980.163 0.67 0.124220.18 0.759 0.034020.204 0.844 0.048050.217 0.924 0.025990.233 1 0.03200

TABLE C.13: Caffeine (1) + water (2) + dioxane (3)

x1 x′2 D

0.0226 0.3446 0.030370.027 0.5419 0.008630.0261 0.612 0.002060.0282 0.6697 0.003900.0265 0.7181 0.003720.0263 0.7593 0.000730.0243 0.7947 0.004340.0214 0.8255 0.006120.02 0.8526 0.003090.0162 0.8765 0.007890.0147 0.8978 0.003230.012 0.9169 0.005620.0078 0.9498 0.008680.00453 0.9771 0.006800.00229 1 0.00471

TABLE C.14: Pregabalin (1) + methanol (2) + water (3) 338.15 K;Pregabalin (1) + methanol (2) + water (3) 298.15 K; (each next three


x1 x′2 D x1 x

′2 D

0.00711 0.059 0.00141 0.003383 0.059 0.000690.007818 0.123 0.00142 0.003046 0.123 0.000670.008947 0.194 0.00227 0.002787 0.194 0.000520.009099 0.273 0.00032 0.00281 0.273 0.000050.010494 0.36 0.00281 0.003005 0.36 0.000400.010416 0.457 0.00013 0.002905 0.457 0.000190.009658 0.568 0.00149 0.002584 0.568 0.000630.007807 0.694 0.00368 0.001899 0.694 0.001370.003229 1 0.00972 0.001085 1 0.00166


TABLE C.15: Anthracene (1) + dibutyl ether (2) + hexane (3) 298.15K;Anthracene (1) + dibutyl ether (2) + heptane (3) 298.15 K; An-thracene (1) + dioxane (2) + propan-1-ol (3) 298.15 K; (each next three


x1 x′2 D x1 x

′2 D x1 x

′2 D

0.00294 0.2515 0.00134 0.00309 0.233 0.00104 0.00102 0.0939 0.0008790.00245 0.4505 0.00098 0.00268 0.4186 0.00082 0.00153 0.1829 0.0010320.00218 0.5676 0.00054 0.00245 0.5379 0.00046 0.00278 0.3461 0.0025670.00199 0.6493 0.00038 0.00226 0.6243 0.00038 0.00369 0.4717 0.0018170.00157 0.8402 0.00084 0.00189 0.813 0.00075 0.00448 0.5753 0.0015610.00127 1 0.00060 0.00157 1 0.00065 0.00621 0.7832 0.003449

0.00718 0.8978 0.0018800.00833 1 0.002222

TABLE C.16: Anthracene (1) + propan-1-ol (2) + hexane (3) 298.15 K;Anthracene (1) + propan-1-ol (2) + heptane (3) 298.15 K; Anthracene(1) + propan-1-ol (2) + octane (3) 298.15 K; (each next three columns,

respectively).

x1 x′2 D x1 x

′2 D x1 x

′2 D

0.00129 0.1751 3.98E-05 0.00157 0.1719 1.6738E-07 0.00178 0.1908 0.000120.00124 0.3162 0.0001005 0.00146 0.3272 0.00022 0.00163 0.3483 0.000300.00107 0.5447 0.0003 0.00118 0.5674 0.00056 0.0013 0.5892 0.000660.00099 0.6331 0.0002 0.00108 0.6622 0.00020 0.00115 0.6806 0.000300.000898 0.7297 0.0002 0.000953 0.7444 0.00026 0.00101 0.7661 0.000280.00074 0.8712 0.0003 0.000762 0.8824 0.00038 0.000784 0.8939 0.000450.000661 0.9411 0.0002 0.000681 0.9398 0.00016 0.000682 0.9477 0.000200.000591 1 0.0001 0.000591 1 0.00018 0.000591 1 0.00018

TABLE C.17: Sulphamethoxypyridazine (1) + ethanol (2) + water (3)298.15 K

x1 x′2 D

0.000285 0.17 0.000640.001100 0.33 0.001830.001645 0.43 0.001100.002092 0.48 0.000900.002467 0.54 0.000750.002651 0.6 0.000370.002713 0.66 0.000120.0025972 0.74 0.000230.0021330 0.82 0.000930.0021101 0.87 0.000050.0019127 0.9 0.000390.0014468 1 0.00094


TABLE C.18: Naphthalene (1) + ethylene glycol (2) + water (3) 298.15K

x1 x′2 D

0.00001175 0.008 1.26E-060.00001273 0.017 1.96E-060.00001371 0.026 1.96E-060.00001501 0.035 2.60E-060.00001638 0.044 2.74E-060.00001735 0.054 1.94E-060.00002031 0.075 5.93E-060.00002435 0.097 8.11E-060.00003058 0.122 1.25E-050.00003749 0.148 1.39E-050.0000474 0.177 1.99E-050.00007722 0.244 6.096E-050.0001283 0.327 0.00010.0002021 0.43 0.00020.0003273 0.564 0.00030.0003726 0.59 0.0001

175

Appendix D

Publication activity of the author

Here, the publications resulting from this PhD project are listed.

Articles in Peer-Reviewed Journals

1. L. Ruszczynski, A. Zubov, J. P. O’Connell, J. Abildskov, Reliable Correlationfor Liquid-Liquid Equilibria outside the Critical Region. Journal of ChemicalEngineering Data, 2017, 62, 2842-2854.

2. L. Ruszczynski, A. Zubov, J. P. O’Connell, J. Abildskov, Reply to “Comment on‘Reliable Correlation for Liquid-Liquid Equilibria outside the Critical Region’”,Journal of Chemical Engineering Data, 2017, 62, 40434044.

3. L. Ruszczynski, A. Zubov, J. P. O’Connell, J. Abildskov, Liquid-Liquid Equilib-ria in ternary systems - modelling and data validation, in preparation.

4. L. Ruszczynski, M. D. Ellegaard, A. Zubov, J.P. O’Connell, J. Abildskov, Mod-elling and data validation of the solubility of solids in mixed solvents, in prepa-ration.

Contributions in Peer-Reviewed Conference Proceedings

1. L. Ruszczynski, A. Zubov, G. Sin, J. Abildskov, Data Validation and Modellingof Thermodynamic Properties of Systems with Active Pharmaceutical Ingre-dients (APIs) in Complex Media for Skin Absorption Processes, Proceedings ofthe 27th European Symposium on Computer Aided Process Engineering - ESCAPE27 in Computer Aided Chemical Engineering, October 1st-5th 2017, Barcelona,Spain, Espuña A., Graells M., Puigjaner L. (Editors), p.247-252. ISBN 978-0-444-639653-0.

2. G. Molla, L. Ruszczynski, J. Abildskov, G. Sin, Property Prediction of Pharma-ceuticals for Designing of Downstream Separation Processes, Proceedings of the28th European Symposium on Computer Aided Process Engineering – ESCAPE 28,June 10th -13th, 2018 A. Friedl, J. J. Klemeš, S. Radl, P. S. Varbanov, T. Wallek(Editors), p.287. ISBN 987-0-444-64237-0

Presentations at International Conferences

1. L. Ruszczynski, A. Zubov, G. Sin, J. Abildskov, Data Validation and Modellingof Thermodynamic Properties of Systems with Active Pharmaceutical Ingre-dients (APIs) in Complex Media for Skin Absorption Processes, oral presenta-tion, 27th European Symposium on Computer Aided Process Engineering –ESCAPE27, October 1st-5th 2017, Barcelona, Spain.

176 Appendix D. Publication activity of the author

2. G. S. Molla, L. Ruszczynski, J. Abildskov, G. Sin, Property Prediction of Phar-maceuticals for Designing of Downstream Separation Processes, oral presenta-tion, 28th European Symposium on Computer Aided Process Engineering – ESCAPE28, June 10th-13th 2018, Graz, Austria.

3. L. Ruszczynski, A. Zubov, J. P. O’Connell, J. Abildskov, Liquid-liquid equilibriadata validation, oral presentation, 20th Symposium on Thermophysical Properties,June 24th-29th 2018, Boulder, USA.

4. L. Ruszczynski, A. Zubov, J. Abildskov, G. Sin, Data validation and mod-elling of solubility of pharmaceuticals for crystallization process development,poster, Nordic Workshop on Pharmaceutical Process Development and Scale-up, 12thOctober 2018, LEO Pharma, Ballerup, Denmark.

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