two-phase flow in liquid chromatography - experimental and theoretical investigation

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Doctoral Thesis

Two-Phase Flow in Liquid Chromatography - Experimental andTheoretical Investigation

Author(s): Ortner, Franziska

Publication Date: 2018

Permanent Link: https://doi.org/10.3929/ethz-b-000270250

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Diss. ETH No. 25031

T W O - P H A S E F L O W I N L I Q U I DC H R O M AT O G R A P H Y

- E X P E R I M E N TA L A N D T H E O R E T I C A LI N V E S T I G AT I O N

A thesis submitted to attain the degree of

doctor of sciences of eth zurich

(Dr. sc. ETH Zurich)

presented by

Franziska Marei Ortner

M.Sc., Technical University Munich

born on March 17th, 1990

citizen of Germany

accepted on the recommendation of

Prof. Dr. M. Mazzotti, examinerProf. Dr. M. Morbidelli, co-examiner

Prof. Dr. F. M. Orr, co-examiner

Zurich, 2018

A C K N O W L E D G M E N T S

First and foremost, I would like to express my sincere gratitudeto Prof. Marco Mazzotti for first offering me a short but enlighte-ning semester project, for guiding, supporting and challenging methroughout my master and PhD thesis, and for opening the worldof mathematics and modeling to me. His passion for science is in-spiring and contagious, and he takes an honest interest not only inhis research, but also in the development and well-being of his PhDstudents, and of the world as a whole. I would like to thank him forhis constructive criticism and advice, as well as for his straightfor-ward, flexible and trustful manner of finding good solutions.I would also like to thank Prof. Massimo Morbidelli and Prof. Frank-lin Orr for taking an interest in my work and for accepting the taskof co-examiners of my thesis. I want to express special gratitudeto Professor Morbidelli for offering me a side project in collabo-ration with his group, hence allowing me to consider the field ofchromatography from a different perspective. Many thanks also toProf. Ronny Pini for interesting and instructive discussions and newideas on the fluiddynamic aspects of my work.I would like to thank my colleagues Lisa, Max and Ian, for their ad-vice on adsorption and thermodynamics, for inspiring discussionsand for patiently answering my questions. Furthermore, I wouldlike to express my gratitude to my students Chantal Ruppli (masterproject and research assistent) and Helena Wiemeyer (semester pro-ject), who contributed significantly to the work presented in chap-ters 4 and 9. I am very lucky to have worked with such motivatedand gifted students, who shared my enthusiasm, joy, and someti-mes desperation for the projects and who helped me to hopefullybecome a better supervisor. I would also like to acknowledge Mar-kus Huber and Daniel Trottmann, for their help in the lab and foradding unconventional devices to my HPLC setup.My time at the Separation Processes Laboratory was a very enri-ching experience not only from a professional, but also from a per-sonal point of view. In this context, I would like to thank all my col-

leagues for the joyful and relaxed atmosphere, and for shared coffeeand lunch breaks, as well as evening beers. I would like to point outFra, Lisa, Luca, Chantal, Hari, Paco and Elena for many eveningsat hot pasta, common sportive activities, summers at the Limmat,TV evenings, wedding celebrations and mountain trips, and for justbeing there. Thanks also to my friends outside SPL for making thelast four years unforgettable.Finally, I would like to cordially thank my parents and sisters, fortheir care and their continuous support and help, and Stephan, forhis stability, his friendship and his love.

A B S T R A C T

Liquid chromatography is a technology applied for challenging se-paration tasks, for example for separating or purifying sensitive orchemically similar products. When applying liquid chromatographyfor analytics, dilute solutions of the analytes are injected in shortpulses. In turn, in preparative applications, higher concentrationsof the components are injected for a longer feed duration. In thiscontext, it is possible that, upon an enrichment of one or multiplecomponents, the fluid phase becomes supersaturated, and a phasesplit and two-phase flow occurs. Such enrichment can be due toan interaction of two or more adsorbing components, or to a che-mical reaction occurring in the chromatographic column. Since thepresence of multiple liquid phases in chromatography is poorly un-derstood, it is commonly avoided by reducing fluid phase concen-trations. However, this imposes a limitation upon operating condi-tions, and potentially upon process performance. Hence, this thesisinvestigates experimentally and describes theoretically the physicalimplications of two-phase flow in liquid chromatography, and eva-luates its impact on process performance.The trigger to this project was the investigation of the delta-shock,a phenomenon which could not be evidenced experimentally in li-quid chromatography. Instead, the experimental investigation pro-vided the evidence of a liquid-liquid phase split and a two-phaseflow. In this spirit, the first part of this thesis presents two differentmathematical approaches to achieve identical results for conditionsand properties of the singular shocks, and hence consolidates exis-ting proof of the theoretical existence of the delta-shock in liquidchromatography. In turn, the challenge of finding experimental evi-dence remains unresolved.In order to investigate two-phase flow in liquid chromatography,material balance equations, accounting for multiple fluid phases inthermodynamic equilibrium, for adsorption, and for different velo-cities of the fluid phases, are derived. These equations are generic,but require algebraic relationships which describe the physical pro-

perties of the specific chromatographic system.For the determination of such relationships, a reversed-phase chro-matographic system is characterized. The thermodynamic proper-ties of the fluid phase(s) are determined by phase equilibrium expe-riments, and are described by a fitted UNIQUAC equation. Single-component and binary adsorption is investigated by Frontal Ana-lysis, and described in a thermodynamically consistent manner, as-suming a dependence on the liquid phase activities, and using theadsorbed solution theory. Fluiddynamic properties are assessed byimbibition/drainage experiments of phases in thermodynamic equi-librium, and are shown to behave according to a Brooks-Corey cor-relation, which is often applied in the context of multi-phase flowin natural reservoirs.The physical relationships are implemented in the material balanceequations, which are solved by two different methods, applying a fi-nite volume discretization scheme, or the method of characteristics.A comparison of simulation results and experimental profiles tes-tifies a quantitative agreement, which serves as a validation of theunderlying model assumptions and established physical relations-hips. In addition, the comparison of simulations and experimentaldata allows for an assessment of the impact of different physical as-pects on the shape of the elution profiles.Having gained a good confidence and understanding of the modelassumptions and physical properties, the model is applied to eva-luate the impact of two-phase flow on the performance of chroma-tographic processes. The performance of a chromatographic reactorwith an esterification reaction is assessed theoretically. It is shownthat the retention of the stronger adsorbing component, often enri-ched in the more wetting phase, can be additionally increased byfluiddynamic effects, i.e. by a slower moving wetting phase. Thiseffect can enhance the separation efficiency between the differentsolutes, hence increasing the amount of weakly adsorbing productpurified per chromatographic cycle, but it also involves the risk ofan increased cycle time due to the slow propagation of the wettingphase.

Z U S A M M E N FA S S U N G

Flüssigchromatographie ist eine Technologie, die insbesondere füranspruchsvolle Trennungen, unter anderem für die Trennung undAufreinigung sensitiver oder chemisch ähnlicher Produkte, Anwen-dung findet. Während in der Analytik kleine Volumina stark ver-dünnter Lösungen injiziert werden, werden in präparativen Anwen-dungen höhere Konzentrationen über eine längere Dauer appliziert.In diesem Kontext ist eine Übersättigung der fluiden Phase auf-grund der Anreicherung einer oder mehrerer Komponenten mög-lich, die zur Phasentrennung und Zweiphasenfluss führt. Die An-reicherung in der fluiden Phase kann durch die Interaktion ver-schiedener adsorbierender Komponenten, oder durch eine chemi-sche Reaktion hervorgerufen werden. Da ein Zweiphasenfluss inder Chromatographie kaum untersucht und verstanden ist, wird erhäufig durch eine Reduktion der Konzentrationen in der fluidenPhase vermieden. Dies beschränkt die Betriebsbedingungen, und li-mitiert möglicherweise die Prozessproduktivität. Ziel dieser Arbeitist daher, die mit dem Zweiphasenfluss einhergehenden physika-lischen Aspekte experimentell zu untersuchen und theoretisch zubeschreiben, sowie den Einfluss des Zweiphasenflusses auf das Ver-halten chromatographischer Prozesse zu evaluieren.Ursprung des Projektes ist die Untersuchung des Delta-Schocks, ei-nes Phänomens, dessen Existenz in der Flüssigchromatographie ex-perimentell nicht nachgewiesen werden konnte. Anstelle des Delta-Schocks wurde eine flüssig-flüssig Phasentrennung und ein Zwei-phasenfluss beobachtet. In diesem Kontext präsentiert der erste Teilder Arbeit zwei verschiedene mathematische Ansätze, die identi-sche Ergebnisse bezüglich der Bedingungen und Eigenschaften ei-nes singulären Schocks erzielen, und damit einen bestehenden Be-weis zur theoretischen Existenz des Delta-Schocks in der Chroma-tographie verfestigen. Im Gegensatz dazu verbleibt die Frage nachdem experimentellen Nachweis des Delta-Schocks ungelöst.Zur Untersuchung des Zweiphasenflusses in der Chromatographiewerden Massenbilanzen aufgestellt, die mehrere fluide Phasen im

thermodynamischen Gleichgewicht, Adsorption, sowie verschiede-ne Geschwindigkeiten der fluiden Phasen berücksichtigen. DieseGleichungen sind allgemeingültig, benötigen allerdings algebraischeGleichungen, die die physikalischen Eigenschaften eines spezifischenSystems beschreiben.Für die Bestimmung dieser physikalischen Eigenschaften wird einUmkehrphasen-System charakterisiert. Die thermodynamischen Ei-genschaften der fluiden Phase(n) werden durch die Messung vonPhasengleichgewichten bestimmt, und mit einer gefitteten UNI-QUAC Gleichung beschrieben. Die singuläre und binäre Adsorpti-on wird mit Durchbruchskurven untersucht, und thermodynamischkonsistent als Funktion der Flüssigphasen-Aktivitäten, unter Ver-wendung der Adsorbed Solution Theory, beschrieben. Fluiddynami-sche Eigenschaften werden durch die Verdrängung zweier Phasenim Gleichgewicht gemessen, und verhalten sich entsprechend derBrooks-Corey Korrelation, die oft zur Beschreibung von Mehrpha-senfluss in Naturreservoirs verwendet wird.Die ermittelten Gleichungen werden in die Massenbilanzen imple-mentiert, welche durch zwei verschiedene Ansätze gelöst werden,basierend auf einer Finite-Volumen Diskretisierung oder auf derCharakteristiken-Methode. Der Vergleich von Simulationen und Ex-perimenten zeigt eine quantitative Übereinstimmung, was Modell-annahmen und die implementierten Gleichungen validiert. Darüberhinaus ermöglicht dieser Vergleich eine Quantifizierung verschiede-ner physikalischer Einflüsse auf das Elutionsverhalten.Im Anschluss an die Modellvalidierung und das damit gewonneneVertrauen in die Modellannahmen, wird das Modell zur Evaluie-rung des Prozessverhaltens in Anwesenheit des Zweiphasenflussesverwendet. Das Verhalten eines chromatographischen Reaktors miteiner Esterifizierungsreaktion wird theoretisch untersucht. Es wirdgezeigt, dass die Retention der stärker adsorbierenden Komponente,die oft in der benetzenden Phase angereichert ist, zusätzlich durchfluiddynamische Effekte, insbesondere durch eine langsamer flies-sende benetzende Phase, erhöht werden kann. Dieser Effekt kanndie Trennleistung des Systems steigern, und damit die Menge anaufgereinigtem, schwächer adsorbierendem Produkt pro Trennzy-klus erhöhen. Allerdings besteht auch das Risiko einer verlängertenZykluszeit durch den langsamen Fluss der benetzenden Phase.

C O N T E N T S

1 introduction 1

1.1 Two-phase flow in porous media: background in na-tural reservoirs 1

1.2 Evidence of two-phase flow in liquid chromatography 3

1.3 Objectives 4

1.4 Structure of this thesis 6

2 mathematical analysis of the delta-shock 9

2.1 Introduction 9

2.2 Background 11

2.2.1 Characteristics in the physical and in the ho-dograph plane 11

2.2.2 Characteristic parameters 13

2.2.3 Riemann problems and shocks 14

2.2.4 Classical and singular solutions of Riemannproblems 15

2.2.5 Analysis of the delta-shock 16

2.2.6 Moving coordinate framework 18

2.2.7 Approximate solutions 19

2.3 Analysis based on Colombeau generalized functions19

2.3.1 Definitions 20

2.3.2 Derivation of the generalized solution 22

2.3.3 Application of the concept of association 24

2.4 Analysis using box approximations 26

2.4.1 Definitions 27

2.4.2 Equations 29

2.5 Exact results on velocity and strength of the delta-shock 31

2.6 Conclusion 32

3 modeling two-phase flow in liquid chromato-graphy 35

3.1 Introduction 35

3.2 Material balances 37

3.2.1 Equilibrium theory model 37

x contents

3.2.2 Lumped kinetic model 39

3.3 Algebraic equations 41

3.3.1 Thermodynamic equilibria between convectivephases 41

3.3.2 Adsorption equilibria 42

3.3.3 Fluiddynamics 43

4 characterization of physical properties 47

4.1 Introduction 47

4.2 Theoretical background 48

4.2.1 Adsorbed solution theory 48

4.2.2 Application of AST to the BET isotherm 52

4.3 Experimental 53

4.3.1 Material and basic methods 53

4.3.2 Density measurements 55

4.3.3 Phase equilibrium experiments 56

4.3.4 Single component Frontal Analysis 56

4.3.5 Binary Frontal Analysis 58

4.3.6 Determination of the column permeability K 59

4.3.7 Hydrodynamic two-phase flow behavior 59

4.4 Volume additivity 67

4.5 Thermodynamic equilibrium of the quaternary sy-stem 69

4.6 Single component adsorption behavior 75

4.7 Binary adsorption behavior 78

4.8 Two-phase flow behavior 91

4.8.1 Phase viscosities 91

4.8.2 Relative permeabilities 91

4.9 Conclusions 97

5 solution and evaluation of chromatographic

models 101

5.1 Introduction 101

5.2 Model system 102

5.3 Solution of the ET model assuming equal velocities 103

5.3.1 General Solution 104

5.3.2 Hodograph plane 107

5.3.3 Solution of exemplary cases 110

5.4 Solution of the ET model assuming different veloci-ties 113

contents xi

5.4.1 General solution 114

5.4.2 Hodograph plane 116

5.4.3 Solution of exemplary cases 122

5.5 Solution of the lumped kinetic model 131

5.6 Conclusions 133

6 experimental validation 135


6.2 Experimental 135

6.2.1 Material and basic methods 135

6.2.2 Dynamic column experiments involving ad-sorption and two-phase flow 136

6.3 Comparison of model predictions and experimentaldata 136

6.4 Conclusions 143

7 evaluation of two-phase flow in a chromato-graphic reactor 147


7.2 Equilibrium theory model 149

7.2.1 Derivation of model equations 149

7.2.2 Critical assessment of the model assumptions 152

7.3 Hexyl-Acetate system 153

7.3.1 Thermodynamic equilibria 153

7.3.2 Reaction equilibrium 156

7.3.3 Two-phase flow behavior 159

7.4 Elution properties and process performance 160

7.4.1 Elution profiles for single- and two-phase flowconditions 160

7.4.2 Process Evaluation 166

7.5 Conclusion 169

8 conclusions and outlook 171

8.1 Summary and Conclusion 171

8.2 Outlook 174

9 addendum :interconversion of carbohydrates 179


9.2 Theoretical Background 181

9.2.1 Mutarotation and adsorption of Glucose andFructose 181

xii contents

9.2.2 Mass balance model 183

9.2.3 Fitting procedure for rate constants: peak areamethod 186

9.3 Experimental procedure 188

9.4 Results 189

9.4.1 Experimental campaign 189

9.4.2 Apparent dispersion coefficient 192

9.4.3 Henry constants 194

9.4.4 Reaction rate parameters 199

9.4.5 Comparison of Simulated and ExperimentalProfiles 205

9.4.6 Reaction rates in the stationary phase 206

9.5 Conclusion 208

bibliography 210

a supplementary material for chapter 4 223

a.1 Uncertainty quantification of isotherm and RAST pa-rameters 223

a.2 Solution of the Buckley-Leverett Equation 224

a.3 Phase equilibria 229

a.3.1 Calculation of phase equilibria 229

a.3.2 Fitting of UNIQUAC parameters 230

a.3.3 Additional experimental data 237

b supplementary material for chapter 5 243

b.1 Thermodynamic properties in chromatographic co-des 243

b.1.1 Soluble region 243

b.1.2 Two-phase region in the numerical code 245

b.1.3 Two-phase region in the equilibrium theorycode 245

b.2 Analysis of the binodal and equivelocity curve 246

b.2.1 Binodal curve 247

b.2.2 Equivelocity curve 248

b.3 ET solution assuming different velocities (continued) 249

b.3.1 Connection of states in the miscible region tostates in the immiscible region 250

b.3.2 Case studies 255

contents xiii

c supplementary material for chapter 7 269

c.1 Solution of the equilibrium theory model 269

c.1.1 Solution in the miscible region 269

c.1.2 Solution in the immiscible region 271

c.1.3 Illustration of the solution in the hodographplane 276

c.1.4 Derivation of elution profiles 278

c.2 Interconversion of mole fractions and concentrations 280

c.3 Calculation of the Jacobian 281

d supplementary material for chapter 9 283

d.1 Profiles based on parameters obtained by the peakarea method 283

L I S T O F F I G U R E S

Figure 2.1 Hodograph plane and (ω2,ω1)-plane for abinary system with a competitive-cooperativegeneralized Langmuir isotherm 12

Figure 2.2 Typical representatives of the generalized He-aviside and delta functions 22

Figure 2.3 Illustration of the box function 27

Figure 4.1 Iteration procedure to solve the RAST modelin the lumped kinetic model. 54

Figure 4.2 Pressure drop measurements over the chro-matographic column to determine the per-meability K. 60

Figure 4.3 Densities of mixtures of the binary systems(a) methanol-water, (b) PNT-methanol, (c) TBP-methanol, (d) TBP-PNT. 69

Figure 4.4 Ternary diagrams: (a) PNT - methanol - wa-ter, (b) methanol - water - TBP, (c) PNT - met-hanol - TBP, (d) PNT - water - TBP. 71

Figure 4.5 Illustration of the qualitative behavior of thesystem PNT (component 1) - water (compo-nent 2) - TBP (component 3) 72

Figure 4.6 Simulation of phase equilibria and experimen-tal data in the quaternary system PNT - TBP- methanol - water 74

Figure 4.7 Feed compositions of single component bre-akthrough experiments plotted in the rele-vant ternary diagrams 76

Figure 4.8 Single component adsorption data and iso-therm of PNT. 77

Figure 4.9 Single component adsorption data and iso-therm of TBP. 77

Figure 4.10 Feed compositions of the binary breakthroughexperiments (red filled circles), plotted in theternary diagrams PNT - solvent - TBP 81

list of figures xv

Figure 4.11 Functions for the description of non-idealitiesin the adsorbed phase. 83

Figure 4.12 Comparison of simulated profiles with expe-rimental data of binary breakthrough experi-ments 84

Figure 4.13 Binary breakthrough experiments BT 1 to 6,experimental and simulated 87

Figure 4.14 Binary breakthrough experiments BT 7 to 10,experimental and simulated 88

Figure 4.15 Binary breakthrough experiments BT 11 to14, experimental and simulated 89



Figure 4.18 Viscosity estimation for the binary system me-thanol-water at 296 K 92

Figure 4.19 Flow and pressure profiles of the drainage/im-bibition experiment 93

Figure 4.20 Saturation and pressure profiles during pri-mary and secondary imbibition cycles. 94

Figure 4.21 Saturation and pressure profiles during se-condary drainage cycles. 94

Figure 4.22 Relative permeability and fractional flow cur-ves. 95

Figure 5.1 Characteristics in the Hodograph plane forthe ternary system PNT - methanol - water 109

Figure 5.2 Illustration of the qualitative behavior of Γ1characteristics in the hodograph plane. 110

Figure 5.3 Paths in the hodograph plane for exemplarycases with an initial state A and two differentfeed states in the soluble region. 111

Figure 5.4 Concentration profiles of the chromatographiccycles connecting the initial state A to thefeed state (a) B1 and (b) B2 112

Figure 5.5 Characteristics in the hodograph plane ac-counting for different velocities of the con-vective pahses 118

xvi list of figures

Figure 5.6 Propagation velocities λt and λnt along tielineand non-tieline characteristics 119

Figure 5.7 Mapping of Γ1 and Γ2 characteristics in theternary diagram. 121

Figure 5.8 Initial and feed conditions of the five diffe-rent exemplary cases solved by equilibriumtheory. 123

Figure 5.9 Solutions for exemplary case 1 with initialstate A and feed state B1. 124





Figure 6.1 Initial (Ai) and feed (Bi) states (mass fracti-ons), investigated in the validation experi-ments. Combinations of initial and feed sta-tes are connected by a continuous line. 137

Figure 6.2 Concentration (predicted) and flow (experi-mental and predicted) profiles of PNT withinitial state A1 and feed states B1 to B4. 138

Figure 6.3 Concentration (predicted) and flow (experi-mental and predicted) profiles of PNT withinitial state A2 and feed states B1 to B4. 139

Figure 7.1 Ternary diagrams of the esterification system,predicted by the mod. UNIFAC model. 155

Figure 7.2 Thermodynamic behavior of the quaternarymodel system, predicted by the mod. UNI-FAC model. 157

Figure 7.3 Liquid phases in thermodynamic and reactionequlibrium, illustrated in the quaternary di-agram and in the (C2, C1) plane. 158

Figure 7.4 Fractional flow function assumed for the es-terification system. 159

list of figures xvii

Figure 7.5 Solution paths in the (C2, C1) plane for ex-amples i = 1 - 4 with initial state A and feedstates Bi. 161

Figure 7.6 Adsorption profiles for examples i = 1 - 4

with initial state A and feed states Bi. 163

Figure 7.7 Desorption profiles for examples i = 1 - 4

with initial state A and feed states Bi. 164

Figure 7.8 Cyclic productivity P as a function of the over-all fractional feed flow FB

1. 167

Figure 9.1 Equilibrium composition of fructose over tem-perature. 183

Figure 9.2 Illustration of the fitting procedure based onpeak areas 187

Figure 9.3 Experimental and simulated elution profilesfor Glucose at different temperatures and flow-rates. 190

Figure 9.4 Experimental and simulated elution profilesfor Fructose at different temperatures and flow-rates. 191

Figure 9.5 Elution profiles resulting from Glucose pulseinjections performed at 38 C with varyingflowrate, plotted over elution time and elu-tion volume. 193

Figure 9.6 Van Deemter curves and apparent dispersioncoefficient over the interstitial velocity. 195

Figure 9.7 Van’t Hoff Plot for both structural forms ofGlucose and Fructose. 197

Figure 9.8 The logarithmic ratios ln(Nui /N

tot) vs. reten-tion time tRi for Glucose and Fructose. 200

Figure 9.9 Arrhenius plot for the interconversion of struc-tural forms 1 (blue) and 2 (red) for Glucoseand Fructose. 202

Figure 9.10 Reaction rate constants in the mobile phase,adsorbed phase and apparent reaction rateconstants between 15 and 45 C for both Glu-cose anomers. 208

xviii list of figures

Figure A.1 Exemplary fractional flow curve, being a functionof the phase saturation and exhibiting a typi-cal S-shape. 227

Figure A.2 Solution of the Buckley-Leverett equation fordisplacement a (displacement of phase 2 byphase 1) 228

Figure A.3 Solution of the Buckley-Leverett equation fordisplacement b (displacement of phase 1 byphase 2). 229

Figure A.4 Structure of the thermodynamic code 231

Figure A.5 XRD spectra of the solid phase resulting fromselected phase equilibrium experiments, com-pared to the XRD spectrum measured for pureTBP. 237

Figure B.1 Propagation velocities λ1 (blue) and λ2 (red)along selected tie-lines characteristics. 251

Figure B.2 Location of states S in the immiscible regions,connected through shocks to initial and feedstates A and B, respectively, in the miscibleregion. 254

Figure B.3 Paths in the hodograph plane for the dis-placement cycle A (initial state) to B1 (feedstate). (a) SR − ln(xL

1) plane. (b) ternary dia-gram (mass fractions). 257

Figure B.4 (a) Concentration and (b) flow profile for theadsorption step, i.e. the displacement of stateA by state B1. 258

Figure B.5 (a) Concentration and (b) flow profile for thedesorption step, i.e. the displacement of stateB1 by state A. 259


1) plane. (b) ternary dia-gram (mass fractions). For notation and linestyles, see Figure B.3 261

Figure B.7 (a) Concentration and (b) flow profile for theadsorption step, i.e. the displacement of stateA by state B2. 262



1) plane. (b) ternary dia-gram (mass fractions). For notation and linestyles, see Figure B.3 266


Figure C.1 Γt and Γnt characteristics in the entire immis-cible region, and propagation velocities λt andλnt. 275

Figure C.2 Γ1 and Γ2 characteristics of the model systemin the hodograph plane (zoom to relevant con-ditions). 277

Figure D.1 Experimental and simulated elution profilesfor glucose at different temperatures and flow-rates. Reaction kinetic parameters used inthe simulations are those obtained throughthe peak area method. 283

Figure D.2 Experimental and simulated elution profilesfor fructose at different temperatures and flow-rates. Reaction kinetic parameters used inthe simulations are those obtained throughthe peak area method. 284

L I S T O F TA B L E S

Table 4.1 Combinatorial parameters of PNT (P), TBP(T), methanol (M) and water (W) in the UNI-QUAC model. 72

Table 4.2 Fitted UNIQUAC parameters 74

xx list of tables

Table 4.3 Single-component isotherm parameters. 78

Table 4.4 Feed compositions of binary breakthrough ex-periments 80

Table 4.5 Parameters of the relative permeability functi-ons 96

Table 7.1 Isotherm parameters for the esterification sy-stem. 156

Table 9.1 Henry constants for Glucose and Fructose atdifferent temperatures 196

Table 9.2 Fitted Van’t Hoff parameters for the structu-ral forms 1 and 2 of both sugars. 198

Table 9.3 Apparent reaction rate constants for Glucoseand Fructose at different temperatures 201

Table 9.4 Arrhenius parameters for the apparent reactionrate constants of both forms 1 and 2 of thetwo sugars. 203

Table 9.5 Reaction rate constants for Glucose anomers1 and 2 in adsorbed and mobile phase as wellas apparent reaction rate constants. 207

Table A.1 Phase equilibrium conditions 232

Table A.2 Implementation of phase equilibrium condi-tions 233

Table A.3 Original quaternary compositions of phaseequilibrium experiments. 238

Table A.4 Original quaternary compositions of phaseequilibrium experiments - continuation. 239

Table A.5 Phase equilibria in the system PNT - metha-nol - water: Experimental and Calculated 240

Table A.6 Phase equilibria in the system TBP - metha-nol - water: Experimental and Calculated 241

Table A.7 Phase equilibria in the system PNT - TBP -methanol: Experimental and Calculated 242

Table A.8 Phase equilibria in the system PNT - TBP -water: Experimental and Calculated 242

Table A.9 Phase equilibria in the quaternary system PNT- TBP - methanol - water: Experimental andCalculated 242

1I N T R O D U C T I O N

This thesis was motivated by the unexpected evidence of a liquid-liquid phase split and two-phase flow in a chromatographic system,when actually looking for experimental evidence of another rarephenomenon called the “delta-shock”. A phase split and two-phaseflow have been rarely reported and are not understood to date inthe context of liquid chromatography, yet we expect them to be ob-served occasionally, but to be avoided by adjustment of the opera-ting conditions. We hence aim at elucidating these phenomena byboth experimental investigation and theoretical description, and toevaluate the impact of multiple convective phases on the processperformance.While the consideration of multi-phase flow is novel in the con-text of liquid chromatography, it has been extensively investiga-ted in the context of natural reservoirs, which, just like chromato-graphic columns, constitute porous media. The thorough understan-dinggained in that field will be exploited in this work. The introductionhence starts by providing a background of two-phase flow in naturalreservoirs, before describing its evidence in liquid chromatography,which constitutes the motivation of this work. Finally, we formulateobjectives and describe the structure of the thesis.

1.1 two-phase flow in porous media : background in

natural reservoirs

Two-phase flow in porous media has been studied and described fordecades in the context of applications in natural reservoirs. Typicalexamples are irrigation systems,1,2 or petrochemical applications,3,4

where oil or gas is recovered through the displacement with immis-cible phases. Apart from the classical displacement of oil by waterf-looding, more recent applications achieve an enhanced recovery ofoil4–7 or of natural gas from coal and shale.8–10 Enhanced oil/gas

2 introduction

recovery describes the reduction in oil or gas saturation in the reser-voir below the residual oil/gas saturation, i.e. the volume retaineddue to capillary forces or high viscosities (in the case of heavy oils).This is achieved by injecting components which are miscible or par-tially miscible with the oil/gas, or which reduce the viscosity of theheavy oil. For the recovery of oil, solutions of surfactants, CO2 (inmixture with water, surfactants or polymers) or steam are applied,whereas CO2 or N2 are injected for the displacement of natural gas.In the light of recent efforts to reduce CO2 levels in the atmosp-here, CO2 sequestration has become another interesting applicationin natural reservoirs,8,11 which is achieved by injection of CO2 intocoalbeds (often combined with oil/gas recovery) or into deep salineaquifers.As indicated by the different applications named above, a varietyof different gaseous and liquid phases has to be dealt with, such asliquid phases rich in oil or water (ground water or resulting fromwaterfloods), solutions of surfactants and polymers, CO2 foams, orgaseous phases consisting of CO2, N2, or natural gas. Likewise, po-rous media with a multitude of different properties have to be consi-dered, ranging from granular soils to fractured rocks, and forminga three-dimensional body which allows a fluid flow in multiple di-rections. The accurate physical description and prediction of two- oreven multiphase flow in such porous media is thus highly challen-ging, involving multiscale modeling from the pore-scale12 to the ma-croscopic scale,3 and accounting for three-dimensional flow.13 Ho-wever, it was shown that the macroscopic flow behavior of a majo-rity of experimental systems involving two or more fluid phases canbe satisfactorily described by an empirical power-law equation,1,2,14

and that basic physical properties of displacements in a natural re-servoir can be understood and described by a simple material ba-lance, accounting for the two-phase flow in a single dimension.4,6,15

Commonly, such simple models based on material balances accountonly for the fluiddynamic properties of multiple convective phasesin a porous medium, while neglecting an interaction with the solidphase through adsorption. However, it was found in several appli-cations that adsorption and desorption has a great impact on theprocess performance, such as in the recovery of methane, which canbe adsorbed to the porous medium to a great extent,16 or when

1.2 evidence of two-phase flow in liquid chromatography 3

using surfactants during enhanced oil recovery, which might parti-ally adsorb and thus cannot be fully recovered.17 In a few contribu-tions,6,10,18 the effect of adsorption has thus been considered inthe respective models, based on simplistic physical assumptionssuch as linear isotherms. Only in recent years, sophisticated expe-rimental methods, such as CT scans of multi-phase flow in poroussamples,19–21 provide a deeper insight of gas adsorption and will al-low a more accurate description of adsorption in natural reservoirsin the future.

1.2 evidence of two-phase flow in liquid chromato-graphy

Liquid chromatography is a separation process based on the adsorp-tion of solutes. It commonly involves a packed bed of adsorbentparticles (which themselves can be porous or non-porous), and onesingle liquid mobile phase, composed of inert solvent componentsand adsorbing solutes in dilute concentration. Due to different ad-sorption properties, solutes travel through the chromatographic co-lumn at different propagation velocities, which allows a separationof the solutes. Hence, the modeling of such process intrinsically re-quires an accurate description of adsorption properties.22 Standardchromatographic models are based on component material balan-ces. The simplest model is the equilibrium theory model,23,24 whichassumes thermodynamic equilibrium between the liquid and the ad-sorbed phase, and neglects any kind of kinetic limitations, such asdispersive effects or mass transfer resistances. This model, accoun-ting for adsorption and for a single convective phase only, can bewritten as

ε∂ci∂t

+ (1− ε)∂ni∂t

+ u∂ci∂z

= 0; i = 1...Nc (1.1)

Here, ci and ni are the liquid and the adsorbed phase concentrationof component i (number of components Nc), ε is the total porosity(inter- and intraparticle porosity), u is the superficial velocity, and tand z denote the time and space variables. The adsorbed phase con-centration ni is usually described by adsorption isotherms, beinga function of the liquid phase concentrations. A very common isot-

4 introduction

herm in a system with two adsorbing components 1 and 2 is thegeneralized Langmuir isotherm

ni =Hici

1+ p1K1c1 + p2K2c2; i = 1, 2 (1.2)

where Hi and Ki are the Henry constant and equilibrium constantof component i, and pi = ±1 is a pre-factor, determining whethercomponent i features Langmuir (+1) or anti-Langmuir (-1) behavior.The equilibrium theory model (equations 1.1), forming a system ofhomogeneous, first-order partial differential equations, can be sol-ved analytically by using the method of characteristics.23,24

The occurrence of a phase split and subsequent multiphase flowhas rarely been reported in the literature, and hence is not consi-dered in the standard chromatographic models. To the best of ourknowledge, the first and only reference to two-phase flow in liquidchromatography25,26 arose from the study of the delta-shock.8,27

This phenomenon has been found theoretically in the course of deri-ving the equilibrium theory solution for a binary system subject toa generalized Langmuir isotherm.28 More specifically, it is formedin theory when a high concentration of a component 1, featuringan anti-Langmuir behavior, is displaced by a high concentration ofcomponent 2, exhibiting a Langmuir behavior, and with H2 > H1.Under these specific and rare conditions, the mathematically correctsolution involves a singular shock, the “delta-shock”, observed as aspike of theoretically infinite concentration in the elution profiles.8

Details concerning the conditions and properties of a delta-shockwill be discussed in chapter 2. A supposed experimental evidenceof this phenomenon27 has recently been refuted.25 Rather than for-ming the expected spikes, it was shown that the liquid phase ofthe investigated experimental system becomes unstable and liquid-liquid phase separation occurs. This finding gave rise to a thoroughinvestigation and description of two-phase flow in liquid chromato-graphy, results of which are reported in this thesis.

1.3 objectives

Despite the little evidence of two-phase flow in the literature, it issuspected that this phenomenon is indeed observed, since the in-

1.3 objectives 5

teraction of multiple solutes in liquid and adsorbed phase, the im-pact of modifiers, or chemical reactions in chromatographic reactors,might well result in an enrichment of one or multiple components,which might lead to supersaturation and a subsequent phase split.However, being considered as an unwanted and poorly understoodeffect, a phase split is commonly avoided by changing operatingconditions (reducing concentrations) to obtain a single-phase flow.The limitation to operating conditions which assure a single-phaseflow can also involve a constraint of the process performance. Amain objective of this thesis is to investigate whether such limitationis necessary, or whether an efficient and robust chromatographicprocess is possible in the presence of multiple fluid phases.For a reliable evaluation of the effects of phase split and two-phaseflow on the performance of chromatographic processes, these phe-nomena first have to be understood thoroughly and described the-oretically. Chromatographic model equations, accounting for bothadsorption and two-phase flow, can be formally constructed on thebasis of standard chromatographic models accounting for adsorp-tion only,22–24 and of models describing multi-phase flow in porousmedia, originating from applications in natural reservoirs.4,10,18 Amajor challenge is the description of physical properties in the pre-sence of multiple convective phases, such as thermodynamic equili-bria between the different fluid and adsorbed phases, fluiddynamicproperties, or kinetic effects. Adsorption equilibria are commonlydescribed by isotherms, which are a function of the liquid phaseconcentration. How can adsorption properties be described in thepresence of multiple convective phases with different concentrati-ons, and with a thermodynamically nonideal behavior? To whichextent do the different fluid phases travel with different interstitialvelocities, and is it possible to describe the fluiddynamic propertiesin the context of liquid chromatography with models originatingfrom applications in natural reservoirs? Is it necessary to accountfor kinetic limitations, in particular concerning a spontaneous phasesplit, as well as mass transfer resistances between the different con-vective and adsorbed phases?To clarify these questions, an exemplary system, involving the adsor-bent Zorbax 300SB-C18, as well as the components water, methanol,phenetole (PNT) and 4-tert-butylphenol (TBP) shall be characteri-

6 introduction

zed experimentally, and mathematical relationships describing thedifferent physical properties are to be established based on the ex-perimental data. The chosen experimental system has already beeninvestigated in the context of the delta-shock,8,27 and was the firstsystem for which a phase split and two-phase flow was observedand reported in the literature.25,26

The chromatographic model with the algebraic relationships imple-mented shall be solved and results shall be compared to experimen-tal data obtained in the presence of adsorption and two-phase flow,in order to validate the underlying model assumptions. Finally, theestablished model and gained physical understanding can be usedto evaluate the impact of a phase split and two-phase flow on theperformance of chromatographic processes.

1.4 structure of this thesis

In the spirit of the delta-shock study,8,27 chapter 2 presents a tho-rough mathematical analysis of the conditions and the properties ofa delta-shock in liquid chromatography, and suggests two alterna-tive approaches to achieve the singular shock solutions. With diffe-rent approaches reaching identical results, this chapter can be regar-ded as further proof of the theoretical existence of singular shocksin the context of liquid chromatography. In contrast, the search forexperimental evidence of singular shocks has been unsuccessful todate; meanwhile it has lead to the discovery of phase split and two-phase flow in liquid chromatography,25 and hence constitutes thefoundation for the investigation of these phenomena, which followsin the subsequent chapters.In chapter 3, chromatographic models, accounting for adsorptionand two-phase flow with different velocities, are developed formally.These models are based on component material balances, whicheither neglect any kind of kinetic limitations (equilibrium theorymodel), or account for dispersive effects and mass transfer resistan-ces in one lumped term (lumped kinetic model). These models ad-ditionally require algebraic equations, which describe the physicalproperties of a specific system, concerning thermodynamic equili-bria of convective and adsorbed phases, as well as fluiddynamic

1.4 structure of this thesis 7

properties. A theoretical discussion of these properties is also in-cluded in chapter 3. For the determination of these physical relati-onships, an exemplary system is characterized experimentally anddescribed theoretically in chapter 4. The resulting algebraic equa-tions are implemented in the chromatographic models, which aresolved numerically based on finite volume discretization (lumpedkinetic model) or by applying the method of characteristics (equi-librium theory model) in chapter 5. Simulation results using thetwo different models at identical operating conditions allow a di-rect comparison and evaluation of the advantages and disadvanta-ges of the models. In chapter 6, elution profiles simulated with theequilibrium theory model are compared with experimental data tovalidate the underlying model assumptions and algebraic relations-hips. The model is extended to additionally account for equilibriumreactions, and used to theoretically evaluate the effect of phase splitand two-phase flow on the performance of a chromatographic re-actor (chapter 7), assuming an esterification reaction. Finally, con-clusions concerning the investigation of two-phase flow in liquidchromatography are drawn in chapter 8. As an addendum, a studyof kinetically limited interconversion reactions of carbohydrates insingle-phase chromatography is provided in chapter 9.

2M AT H E M AT I C A L A N A LY S I S O F T H ED E LTA - S H O C K

2.1 introduction

As discussed in section 1.2, the investigation of the delta-shock inthe context of liquid chromatography provided the basis for the dis-covery of two-phase flow in liquid chromatography. With referenceto the delta-shock study,8,27 we will in this chapter mathematicallyderive the conditions and properties of the delta-shock in nonlinearliquid chromatography. Furthermore, we will provide two alterna-tive approaches to derive the singular solution, which will add furt-her proof of the theoretical existence of the delta-shock. For thispurpose, let us consider the two-dimensional system of first orderPDEs, which are of the conservation laws type:(

ui +aiui

1− u1 + u2

)τ

+ (ui)ξ = 0 (i = 1, 2), (2.1)

where only the pairs (u1,u2) that fulfill the conditions u1 > 0, u2 >0 and d = 1−u1+u2 > 0 are allowed, and the inequality a2 > a1 >0 holds true.8 For convenience, we also define

vi =aiui

1− u1 + u2=aiuid

(i = 1, 2), (2.2)

For this system, the following Riemann problem will be studied:

at τ = 0, 0 6 ξ 6 1 : ui = uAi , (state A),

at ξ = 0, 0 < τ < +∞ : ui = uBi , (state B). (2.3)

Note that in this chapter, for the sake of consistency with the subse-quent chapters, the nomenclature of initial states A and inlet states

The results presented in this chapter have been reported in: Ortner F.; Mazzotti M.,Singular shock solutions in nonlinear chromatography, Nonlinear Anal.: Real WorldAppl. 2018, 41, 66-81

10 mathematical analysis of the delta-shock

B is inverted with respect to the nomenclature in literature contri-butions of this field8 (where initial states are denoted as B and inletstates as A).The equations above describe a system of great interest in nonli-near chromatography, which can be obtained from equations 1.1with isotherm equations 1.2 for the case of competitive (species 2)-cooperative (species 1) adsorption by defining ui = Kici, vi = Kiniand ai = νHi, with ν = (1 − ε)/ε. The dimensionless time andspace coordinates are defined as τ = ut/εL and ξ = z/L, respecti-vely, with L being the column length.28

Several researchers have recently studied equations 2.1,29–31 but inthe special case where a1 = a2 = 1, which leads admittedly to a rat-her different and possibly simpler mathematical solution. The con-dition a1 = a2 = 1 is non-generic from a physical point of view, asit represents a combination of values of physical parameters, whichis impossible in practice. Moreover, if a1 = a2, independently oftheir specific value, the two species have the same retention factor,i.e. they cannot be separated by chromatography, thus making itsuse pointless.Previously, equations 2.1 were proven to exhibit singular solutions,and exact criteria on the Riemann problem were derived for theoccurrence of the singular solution. A smooth approximation, con-stituting a solution to equation 2.1 in the sense of weak convergence,was introduced first and then used to derive closed expressions forthe rate of propagation of the delta-shock and for the rate of growthof its strength.8 The latter was achieved by following an approachsimilar to what Keyfitz and Kranzer used in section 2.2 of their1995 seminal paper.32 The expressions derived in Ref. [8] have beenconfirmed very recently by Tsikkou,33 who used the Geometric Sin-gular Perturbation Theory to show existence of a singular solutionto equations 2.1.Here, again in the same spirit of the previous paper,8 two alternativeways of reaching the same results are presented. The first approachis based on Colombeau generalized functions, whose representati-ves can be considered as the smooth approximations used in Ref.[8]. Although the presented approach is basically analogue to theone presented in Ref. [8], in this contribution we make rigorous useof the Colombeau algebra, and thus use that approach in more ge-

2.2 background 11

neral terms. The second approach, which is novel in the context ofthe considered system of PDEs, is based on box approximations (asin section 2.3 of Ref. [32]). The chapter is structured as follows: Firstthe key features of equations 2.1 are summarized in section 2.2, thenthe approaches based on Colombeau generalized functions (section2.3), and on box approximations (section 2.4), are presented anddiscussed. Section 2.5 finally brings the two approaches together toobtain exact results about the delta-shock solutions.

2.2 background

Equations 2.1 can be studied using the method of characteristics.24,28

The key results necessary to understand their singular solutions aresummarized below.

2.2.1 Characteristics in the physical and in the hodograph plane

Consider the Jacobian matrix V of the adsorption isotherm (2.2), i.e.the matrix whose elements are the partial derivatives vij = ∂vi/∂uj:

V =1

d2

[a1(1+ u2) −a1u1

a2u2 a2(1− u1)

](2.4)

and its eigenvalues θj, j = 1, 2; these are real and distinct when thefollowing inequality holds true:

[a1(1+ u2) + a2(1− u1)]2 − 4a1a2(1− u1 + u2) > 0, (2.5)

which implies that the system (2.1) is hyperbolic. This happens infour regions of the part of the hodograph plane (u1,u2) whereui > 0 and d > 0, whereas in a fifth region the system is ellip-tic (for a detailed description of the different regions, see Ref. [28];note that each of them is connected, though their union is discon-nected). Here, only the first region, namely the one which includesthe origin, the segments of the u1 and the u2 axes of coordinates upto u1 = 1− a1/a2 and up to u2 = a2/a1 − 1, respectively, and thatis bounded by the parabola defined by the discriminant (left-hand


side of equation 2.5) being equal to 0 (along which the system is pa-rabolic), is considered and shown in Figure 2.1a, together with theneighboring elliptic region.

0 0.2 0.4 0.6 0.8 1

u2

0

0.2

0.4

0.6

0.8

1

u1

reg. 1

Γ1

Γ2

elliptic region

(a)

a1 a2

ω1

a1

a2

ω2 u

1=0

envelope

u2=0

(b)

Figure 2.1: Hodograph plane and (ω2,ω1)-plane for the system specifiedby equations 2.1. (a) Region 1 and elliptic region in the hodo-graph plane, separated by the envelope, along which the left-hand side of equation 2.5 equals 0. Red and blue lines indicateΓ1 and Γ2 characteristics, respectively. (b) Mapping of region 1 inthe (ω2,ω1)-plane. The diagonal and the vertical and horizon-tal boundary of region 1 in the (ω2,ω1)-plane correspond to theenvelope and the horizontal and vertical axis in the hodographplane, respectively. Γ1 and Γ2 characteristics map on the red andblue dashed lines in the (ω2,ω1)-plane.

Where the system is hyperbolic there are two sets of characteris-tics, identified by the index j = 1, 2, i.e. the characteristics Cj in thephysical plane (ξ, τ), defined as:

Cj :dτdξ

= σj = 1+ θj (j = 1, 2), (2.6)

and the characteristics Γj in the hodograph plane (u1,u2):

Γj :du1du2

=θj − v22v21

(j = 1, 2). (2.7)

The families of characteristics Γj (j = 1, 2) in the hodograph planeform a network of straight lines .24,28

2.2 background 13

2.2.2 Characteristic parameters

The hyperbolic part of the hodograph plane can be one-to-one map-ped onto the characteristic plane spanned by the characteristic pa-rameters (ω1,ω2), which are the roots of the following quadraticequation:24,28

(1−u1 +u2)ω2 − [a1(1+u2) +a2(1−u1)]ω+a1a2 = 0. (2.8)

The discriminant of this equation is positive if the system (2.1) ishyperbolic. The four hyperbolic regions in the hodograph plane cor-respond to four different ranges of ω1 and ω2 values. The range ofinterest here, corresponding to region 1 in the hodograph plane, iswhere

a1 6 ω1 < ω2 6 a2, (2.9)

as shown in Figure 2.1b. Characteristics Γj in the hodograph planemap on segments parallel to the axes in the characteristic plane,namely characteristics Γ1 on segments with ω2 =const., and cha-racteristics Γ2 on segments with ω1 =const.. The segment whereω2 = a2 corresponds to u2 = 0, and that where ω1 = a1 mapsonto u1 = 0; therefore the point u1 = u2 = 0 maps onto (ω1 = a1,ω2 = a2); the curve where the system is parabolic maps ontoa1 6 ω1 = ω2 6 a2.28

Inversely, given a pair of values ω1 and ω2, ui, d, vi and θi can becalculated as follows:28

ui =a3−i(ω1 − ai)(ω2 − ai)

ω1ω2(a2 − a1), (2.10)

d =a1a2ω1ω2

, (2.11)

vi =(ω1 − ai)(ω2 − ai)

a2 − a1. (2.12)

θi =ω2iω3−i

a1a2=ωid

. (2.13)

Note that ωj and θj are positive, as d is always positive; since ω1 <ω2 then θ1 < θ2. Substituting the last equation into equation 2.6the slope of the characteristics Cj are obtained in terms of ω1 andω2, and one sees that σ1 < σ2 always.


2.2.3 Riemann problems and shocks

Determining the solution of a Riemann problem is very simplewhen one works in the characteristic plane, where the initial stateA and the inlet state B map onto (ωA1 ,ωA2 ) and (ωB1 ,ωB2 ), respecti-vely. The classical solution of a Riemann problem consists of threeconstant states, namely from left to right in space the inlet state B,then the intermediate state I, and finally the initial state A. The stateI is at the intercept of the characteristics of different index emana-ting from states A and B. Since along a characteristic Γj, the valueof ω3−j remains constant and since σ1 < σ2, it follows that the in-termediate state I maps onto (ωB1 ,ωA2 ) and the corresponding state(uI1,uI2) can be calculated using equations 2.10.The three states are connected by two transitions, namely one cor-responding to a Γ2 characteristic between states B and I, and onecorresponding to a Γ1 characteristic between states I and A. The j-thtransition is continuous (a simple wave) if the Cj characteristics inthe physical plane that are images of the Γj characteristic fan cloc-kwise, i.e. if θj and ωj decrease when moving from the state on theleft (state L, i.e. either state B or state I) to that on the right (state R,i.e. either states I or state A), namely when ωLj > ω

Rj .

When the opposite occurs, i.e. when ωLj < ωRj , Cj characteristicswould fan counterclockwise, which would be impossible. Therefore,in this case a discontinuous transition appears, whose image in thephysical plane is a straight line of slope:

σj = 1+ θj = 1+[v1]

[u1]= 1+

[v2]

[u2], (2.14)

where the symbol [·] denotes the jump across the discontinuity ofthe quantity enclosed (left quantity minus right one). The discon-tinuity is called a Sj shock, and its image in the hodograph planeis a Σj that coincides with the Γj because characteristics are straight.The last equation embodies the same conservation principle (massconservation in the case of nonlinear chromatography) as the sy-stem of PDEs (2.1) and incorporates the so called Rankine-Hugoniot

2.2 background 15

condition.8,28 The quantity θj can be expressed in terms of ω1 andω2 as follows:

θj =ωLjω

Rj ω3−j

a1a2=ωLj

dR=ωRj

dL. (2.15)

Note that ω3−j is constant along a Γj characteristic hence across aSj shock, therefore ω3−j carries no superscript in equation (2.15).Note also that across an S1 shock, due to the condition ωL1 < ωR1 ,the following inequalities hold true:

σL1 < σ1 < σR1 < σ

R2 , (2.16)

σ1 < σL2 < σ

R2 . (2.17)

Accordingly, across an S2 shock, where ωL2 < ωR2 , one has:

σL1 < σL2 < σ2 < σ

R2 , (2.18)

σL1 < σR1 < σ2 . (2.19)

The above inequalities imply that the characteristics of the sametype on both sides of a shock impinge on the shock itself, but notthose of different type.

2.2.4 Classical and singular solutions of Riemann problems

While the different possible combinations of shocks and simple wav-es for the general Riemann problem (2.3) have been discussed indetail elsewhere,8 here the focus is on the case where the classicalsolution exhibits two shock transitions, which according to the pre-vious section entails the conditions ωB1 < ω

A1 and ωB2 < ω

A2 . Based

on equations 2.14 and 2.15 the two shocks, i.e. S2 connecting the in-let state B and the intermediate state I (ωB1 ,ωA2 ) and S1 connectingI with the initial state A, have the slopes:

σ2 = 1+ωB1ω

B2ω

A2

a1a2, (2.20)

σ1 = 1+ωB1ω

A1 ω

A2

a1a2. (2.21)


Note that this solution is feasible if and only if σ2 > σ1, i.e. ifωB2 > ω

A1 ; this case is illustrated in Figure 7 of the earlier paper.8

However, the last condition can be easily violated by choosing inletand initial states so as ωB2 < ω

A1 ; in this case the two shocks above

would swap position in the physical plane, with S1 steeper than S2.This leads to a multivalued solution, which is physically unfeasible.On the other hand, no classical single transition can connect A andB directly, because they do not belong to the same characteristic.This is the situation yielding singular solutions of the delta-shocktype,8 namely when the following set of inequalities is fulfilled:

ωB1 < ωB2 < ω

A1 < ω

A2 , (2.22)

where the first inequality and the third inequality are a direct con-sequence of the definition of the ω values hence obvious. In thiscase the states A and B are connected by a single transition, i.e. adelta-shock, whose image in the physical plane is a straight linewith slope σδ. The following inequalities hold:

σB1 < σB2 < σ2 < σδ < σ1 < σ

A1 < σ

A2 ; (2.23)

these express a condition of over-compression that is typical of sin-gular solutions. This case is illustrated in Figures 9 and 11 of theearlier paper.8 Note that the reciprocal of the slope σδ is a dimen-sionless propagation rate that will be called s in the following, i.e.s = 1/σδ.

2.2.5 Analysis of the delta-shock

From the condition of over-compression (equation 2.23), it can beconcluded that characteristics of both types 1 and 2 of inlet and ini-tial state impinge on the delta-shock path, with slope σδ. This is incontrast to the case of a classical shock, where only characteristics ofthe same type i impinge on the shock path, with slope σi. Physically,this implies that, while for a classical shock a propagation velocitycan be found which fulfills the mass balance without accumulationof mass at the discontinuity, for the delta-shock matter is conveyedfrom the upstream side to the discontinuity at a rate which is higherthan the rate with which matter is removed from the discontinuity

2.2 background 17

downstream. The consequence is accumulation of matter at the sin-gularity.Since the principle of mass conservation has to be fulfilled also inthe case of a delta-shock, an integral mass balance over the singula-rity can be written:

[ui + vi]∆ξ+wi(τ+∆τ) −wi(τ) = [ui]δτ (i = 1, 2), (2.24)

Note that wi corresponds to the overall amount of component iaccumulated at the singularity in both liquid phase (ui) and ad-sorbed phase (vi), which is also denominated as Rankine-Hugoniotdeficit. Taking the limit ∆τ → 0, with ∆τ/∆ξ → 1/s, leads to thefollowing pair of ordinary differential equations:

dwidτ

= s[ui]

(1

s−

(1+

[vi]

[ui]

))(i = 1, 2), (2.25)

where the initial value of wi, i.e. at τ = 0 and ξ = 0, is obviously 0.Since these are two equations that contain three unknowns, namelyw1, w2 and s, they cannot be solved at this point. With reference toequation 2.14, note that the term (1+ [vi]/[ui]) would be the slopeof a classical shock between states A and B, if this were possible.Note also that in the case of a δ-shock solution [u1] < 0 < [u2],hence as (1+ [v2]/[u2]) < 1/s < (1+ [v1]/[u1]) both rates of changein equation 2.25 are positive.Based on these considerations, a solution to equation 2.1 underthe discussed conditions is postulated to exhibit a singularity, con-necting initial and inlet state, which propagates at a constant velo-city s, and with a strength increasing linearly over time. It is note-worthy that numerical solutions of a regularization of equations 2.1,including a dispersion term, are in line with these considerations,exhibiting spikes in the four variables ui and vi which separate ini-tial and inlet state, propagate at a constant velocity and feature anarea which increases linearly over time (compare Figure 11 of Ref.[8]).An expression for ui can be written formally as the following linearcombination of a Heaviside function, i.e. H, and of a Dirac’s deltafunction, i.e. δ, both centered at the delta-shock location:

ui(τ, ξ) = uLi − [ui]H(ξ− sτ) + gui τδ(ξ− sτ) (i = 1, 2), (2.26)


A similar formal relationship can be written for vi:

vi(τ, ξ) = vLi − [vi]H(ξ− sτ) + gvi τ δ(ξ− sτ) (i = 1, 2), (2.27)

where no assumption is made on the relationship between gu1 andgu2 , and between gui and gvi . The quantities gui τ and gvi τ correspondto the strength of the singularity in ui and vi. As these two quan-tities denote the amount of component i accumulated within thesingularity at time τ, it is intuitive that their sum equals the Rankine-Hugoniot deficit wi:

wi = (gui + gvi )τ (i = 1, 2); (2.28)

The latter statement will also be proved within the scope of thiswork.

2.2.6 Moving coordinate framework

In this context it is convenient to replace the (stationary) space coor-dinate ξ with a moving coordinate ξ, defined as

ξ = ξ− sτ . (2.29)

It follows that the conservation laws 2.1 can be recast in the movingcoordinate framework as:

(ui + vi)τ + ((1− s)ui − svi)ξ = 0 (i = 1, 2) (2.30)

whereas the Riemann problem (2.3) is re-defined in the (τ, ξ) spacein the following equivalent form:

at τ = 0, ξ > 0 : ui = uAi (state A), (2.31)

at τ = 0, ξ < 0 : ui = uBi (state B). (2.32)

Accordingly, ui and vi in equations 2.26 and 2.27 can obviously berecast as functions of τ and ξ:

ui(τ, ξ) = uLi − [ui]H(ξ) + gui τδ(ξ) (i = 1, 2), (2.33)

vi(τ, ξ) = vLi − [vi]H(ξ) + gvi τ δ(ξ) (i = 1, 2). (2.34)

2.3 analysis based on colombeau generalized functions 19

2.2.7 Approximate solutions

Since equations 2.33 are generalized functions including the Diracmeasure, and thus they do not allow for the use of nonlinear ope-rators, it is not possible to derive an expression of vi based on uiusing equation 2.2, and to substitute these expressions in the conser-vation laws in order to determine the five unknown quantities (gu1 ,gu2 , gv1, gv2 and the propagation rate s).In turn, we define two different types of approximate delta-shocksolutions. The first solution, provided in equations 2.38, is basedon the Colombeau algebra, which, besides being a multiplicativedifferential algebra, defines the concept of association as a genera-lization of the concept of equality. The generalized functions 2.38,represented by nets of smooth functions, are associated to the for-mal singular expressions given by equations 2.33, and satisfy theconservation laws 2.30 in the sense of association, provided theirparameters fulfill the conditions and constraints derived in section2.3.The second type of approximate solution, defined in equations 2.60,is based on box approximations, and satisfies equations 2.30 in thesense of weak convergence. In their limits, equations 2.60 approachequations 2.33 in a distributional sense. Since these approximationsare piecewise constant functions defined at every point, they allowthe use of the standard operational calculus and thus can also besubstituted in the conservation laws (see section 2.4). It is remarka-ble that the two approaches provide identical expressions for theunderlying parameters.

2.3 analysis based on colombeau generalized functi-ons

In the previous paper,8 the delta-shock was analyzed using so called“smooth approximations”. In this section, we reanalyze this appro-ach in a more general and sound manner, by rigorously applyingthe Colombeau algebra, and by correcting some minor errors in theprevious derivation. For a complete presentation of the Colombeaualgebra, the reader should refer to Ref. [34], for summaries to Refs.


[35, 36]. We use the notation introduced in Ref. [34], with an ex-ception for the representatives of generalized functions, which inthis contribution are indicated by small letters with superscript ε.Colombeau generalized functions G ∈ G are defined by representa-tives gε, which are nets of smooth functions. They can be associatedto distributions in the sense of weak convergence.Our derivation will proceed as follows: We set up a generalizedfunction Ui describing ui in the sense of association. The represen-tatives of this generalized function can also be considered as smoothapproximations of equations 2.33. With G being a multiplicative dif-ferential algebra, we can apply the generalized functions Ui to de-rive expressions for D and Vi, i.e. for the denominator of equation2.2 and for the adsorbed phase concentrations, respectively. Consi-dering joint and disjoint supports of the generalized delta functionsin Ui and Vi, substituting the resulting expressions in equations2.30 and exploiting the concept of association, we enforce conditi-ons which allow determining the unknown parameters u, u1, u2and s.

2.3.1 Definitions

Since the Colombeau algebra has been derived in detail and sum-marized in multiple contributions,34–36 we will provide here onlythe crucial definitions necessary for the derivation that follows.Let Ω be an open subset in Rn, n ∈ N. A generalized functionG ∈ G (Ω) describes an equivalence class, defined by its representa-tives gε, which constitute nets of smooth functions belonging to thesubset EM[Ω] of all smooth functions defined in Ω.34 Distributionsu ∈ D ′(Ω) can be imbedded in G (Ω). For u ∈ D ′(R), this imbed-ding in G (R) is achieved by convolution with a function ϕ ∈ D(R),with

∫∞−∞ϕ(ξ)dξ = 1. In the more general case that u ∈ D ′(Ω), the

imbedding in G (Ω) is also possible and canonical, however slightlymore elaborate.34 Elements of G (Ω) can be related to distributionsu ∈ D ′(Ω) in the sense of weak convergence, by defining the con-cept of association:A generalized function G ∈ G (Ω) is said to be associated to a distri-bution u ∈ D ′(Ω) (G ≈ u), if for some (hence for every) representa-tive gε of G, gε → u in D ′(Ω) as ε → 0. Two generalized functions


G and K are associated if G− K ≈ 0. One can further show that ifG,K ∈ G (Ω) and G ≈ K, then34

∂αG ≈ ∂αK; α ∈Nn0 . (2.35)

In the following, we are going to work in R. We define several typesof generalized functions, which will be of use for the further deriva-tion.The generalized functions F ∈ G (R) are called generalized Heavi-side functions, if they are bounded, and if their representatives aregiven by

fε(ξ) =

0, ξ < −ε

1, ξ > ε. (2.36)

Furthermore, the generalized delta functions B0 ∈ G (R) are gi-ven by their representatives bε0(ξ), for which bε0(ξ) > 0 ∀ξ ∈ R,supp(bε0(ξ)) ⊆ [−ε, ε], and

∫∞−∞ bε0(ξ)dξ = 1.36 Typical representa-

tives bε0 and fε are illustrated in Figure 2.2, corresponding to thesmooth sequences bε and hε defined in equations 53 and 58 of Ref.[8].Accordingly, B± ∈ G (R) are defined by the representatives bε±(ξ) =bε0(ξ∓ 2ε), which describe shifted model delta nets.35 Also for thesefunctions, typical representatives are illustrated in Figure 2.2. Notethat supp(bε+(ξ))∩ supp(bε−(ξ)) = ∅ and supp(bε±(ξ))∩ supp(bε0(ξ))= ∅. The latter condition was not guaranteed in Ref. [8], where thedelta sequences bε±(ξ) were only shifted by ∓ε.With the concept of association defined above, it is clear that F ≈ H

and B0 ≈ δ, and thus F ′ ≈ H ′ ≈ δ, and B ′0 ≈ δ′. It can further be

shown that also B± ≈ δ (and thus B ′± ≈ δ ′) by proving that:∣∣∣∣∣1ε∫ε(κ+1)ε(κ−1)

b(xε− κ)φ(x)dx−φ(0)

∣∣∣∣∣ =∣∣∣∣∣∫1−1b (y)

(φ(0) + ε (y+ κ)φ ′(χ)

)dy−φ(0)

∣∣∣∣∣6∣∣2ε (1+ |κ|) supbφ ′

∣∣(2.37)


-2ǫ -ǫ 0 ǫ 2ǫ

ξ

0

0.2

0.4

0.6

0.8

1

bǫ

0bǫ

+bǫ

-

f ǫ

Figure 2.2: Typical representatives for F and B(±).

where φ(x) ∈ D(R), κ ∈ R (specifically in this study, κ = ±2 or 0),bε(x) = 1

εb(xε ) and 0 < χ < ε(y+ κ) or ε(y+ κ) < χ < 0.

2.3.2 Derivation of the generalized solution

We formulate a solution (in the sense of association) to equations2.30, which is a combination of shifted generalized delta and Heavi-side functions.

Ui(τ, ξ) = uLi − [ui]F(ξ) + uiτB−(ξ) + uτB+(ξ) (2.38)

Representatives of the generalized function defined in equation 2.38

can also be considered as smooth approximations of the exact ex-pression given by equation 2.33. While the first two terms in equa-tion 2.38 are bounded, the third and fourth term, containing thepositive constants ui and u, are singular. The constants ui fulfill thekey assumption that u2 > u1 > 0 (guaranteeing non-negativity ofthe denominator of the function vi in the following derivation). Thespecial feature of this definition is that the singular part of uεi con-sists of two terms, which differ in their microscopic details (disjointsupport) but are both associated to the Dirac’s delta. As a conse-quence, comparing equations 2.38 and 2.33 shows that:

gui = ui + u . (2.39)


As we will see, it is of the uttermost importance that the left handside of the singular part of Ui, i.e. that proportional to B−(ξ), differsfor the two components, whereas the right hand side, i.e. uτB+(ξ),is exactly the same. The choice of introducing this asymmetry in thesingular terms is decisive in allowing for a successful determinationof the delta-shock properties.Since the expressions Ui, in contrast to the exact expressions ui, al-low the application of non-linear operations, we can use them toderive generalized functions D and Vi for the denominator d ofequation 2.2 and for the adsorbed phase concentrations vi, respecti-vely:

D(τ, ξ) = 1−U1(τ, ξ) +U2(τ, ξ)

= dL − [d]F(ξ) + (u2 − u1)τB−(ξ),(2.40)

Vi(τ, ξ) =aiUi(τ, ξ)D(τ, ξ)

= vLi − [vi]F(ξ) +aiuτB+(ξ)

D(τ, ξ)+

[vi][d]F(ξ)(1− F(ξ))

D(τ, ξ)

+τB−(ξ)

D(τ, ξ)

(aiui − (u2 − u1)(v

Li − [vi]F(ξ))

)(2.41)

with dX = 1−uX1 +uX2 ; X = L,R and [d] = dL−dR, and accordinglyvXi = aiu

Xi /d

X and [vi] = vLi − vRi . Note that the correspondingequation 63 in Ref. [8], defining vεi , contained a mistake in the verylast term that had however no consequences on the final result.As a consequence of the definition of the singular part of Ui, thesingular part of D contains only one positive term proportional toB−(ξ). In the developments below it will be proven that the thirdterm in equation 2.41 is the only term associated to the singularityin the exact expression (equation 2.34). With this, it will becomeobvious that the choice made about the two singular parts of Ui,due to the ensuing fact that D has only one singular term, is theonly choice that allows having a non-zero singular part of Vi. Thesingular behavior exhibited both by D and Vi is in line with thenumerical evidence provided and discussed in Figure 11 of Ref. [8].


2.3.3 Application of the concept of association

In the following, we will analyze the third, fourth and fifth term ofequation 2.41, with the intention to simplify Vi in the sense of asso-ciation. For this purpose, we introduce the following three lemmas.Lemma 1: Let A,C ∈ G (R2+), R2+ := R × (0,∞), with A beingof a bounded type, and s1, s2 : R+ → R+ be smooth functions.B1,B2 ∈ G (R) are generalized functions associated to the Diracdelta, with supp(bεi ) ⊂ [ηiε− ε,ηiε+ ε], ηi ∈ R. If bε1 = bε2 (andconsequently B1 = B2), then

A(τ, ξ)s1(τ)B1(ξ)C(τ, ξ) + s2(τ)B2(ξ)

≈ 0. (2.42)

Proof::∣∣∣∣∫∞0

dτ∫∞−∞

aε(τ, ξ)s1(τ)bε1(ξ)cε(τ, ξ) + s2(τ)bε2(ξ)

ψ(τ, ξ)dξ∣∣∣∣

<

∣∣∣∣∣∫T0

dτ∫η1ε+εη1ε−ε

aε(τ, ξ)s1(τ)s2(τ)

ψ(τ, ξ)dξ

∣∣∣∣∣<

∣∣∣∣2εT supaε(τ, ξ)s1(τ)

s2(τ)ψ(τ, ξ)

∣∣∣∣ = O(ε) ;

(2.43)

where ψ ∈ D(R2+), vanishing for τ > T and for |ξ| > X.Lemma 2: WithA,C,Bi and si as above, if supp(bε1)∩ supp(bε2) = ∅,then

A(τ, ξ)s1(τ)B1(ξ)C(τ, ξ) + s2(τ)B2(ξ)

≈ A(τ, ξ)s1(τ)C(τ, ξ)

δ(ξ). (2.44)

Proof:Keeping in mind that <mv,φ>=<v,mφ> for m ∈ C∞, v ∈ D ′

and φ ∈ D , and considering equation 2.37, the proof of the lasttheorem is straightforward:∫∞

−∞aε(τ, ξ)s1(τ)bε1(ξ)cε(τ, ξ) + s2(τ)bε2(ξ)

ψ(τ, ξ)dξ

=

∫η1ε+εη1ε−ε

aε(τ, ξ)s1(τ)bε1(ξ)cε(τ, ξ)

ψ(τ, ξ)dξ

ε→0−→ aε(τ, 0)s1(τ)cε(τ, 0)

ψ(τ, 0)

(2.45)


Note that the discussed lemmas correspond to Lemma 2 in Ref. [36].Lemma 3:

A(τ, ξ)F(ξ)(1− F(ξ)) ≈ 0, (2.46)

with A as above. The proof of this lemma is trivial, considering thatF2 ≈ F.34

We can thus conclude that the fourth and fifth term in equation2.41 are ≈ 0, due to equations 2.46 and 2.42, respectively. Applyingequation 2.44, the third term in equation 2.41 yields:

aiuτB+(ξ)

D(τ, ξ)≈ aiuτ

dRδ(ξ) (2.47)

Note for the latter conclusion that fε = 1 where bε+ 6= 0. Equation2.47 (specifically the denominator) deviates from the result in equa-tion 70 in Ref. [8], which is only correct for a symmetric functionhε. The deviation stems from the different supports of the shifteddelta nets (shifted by ±2ε in this work, and by ±ε in the previousone), which guarantee that supp(bε±(ξ)) ∩ supp(bε0(ξ)) = ∅ onlyfor this work. Although the current definition allows for a greatergenerality of the generalized Heaviside functions, the mentioneddifference does not affect the final result.With the previous results, we can simplify Vi as:

Vi(τ, ξ) ≈ vLi − [vi]F(ξ) +aiuτB+(ξ)

D(τ, ξ)

≈ vLi − [vi]H(ξ) +aiuτ

dR δ(ξ)

(2.48)


Substituting equations 2.38 and 2.48 into equation 2.30 and applyingthe concept of association yields:

(Ui + Vi)τ + ((1− s)Ui − sVi)ξ ≈

≈ (ui + u)δ+aiu

dRδ+ (1− s)(−[ui]δ+ (ui + u)τδ

′)

− s(−[vi]δ+aiuτ

dRδ ′)

= (ui + u+aiu

dR− (1− s)[ui] + s[vi])δ

+ ((1− s)(ui + u)τ− saiuτ

dR)δ ′

= kδ+ lδ ′ ≈ 0

(2.49)

For this last equation to hold true, the expressions k and l have tovanish independently, thus resulting in the following four algebraicequations for i = 1, 2, which allow determining the four unknownsu, u1, u2 and s:

(1− s)(ui + u) −aius

dR= 0 , (2.50)

ui + u+aiu

dR− (1− s)[ui] + s[vi] = 0 . (2.51)

2.4 analysis using box approximations

This second approach, motivated by Keyfitz and Kranzer,32 is ba-sed on the use of box approximations. The employed box functi-ons q(ξ) are piecewise constant functions and converge to the Di-rac delta in a weak (distributional) sense. Since these functions, asthe Heaviside function, are classical, piecewise constant distributi-ons, which according to our definition are defined at every pointin R, it is possible to apply the standard operational calculus ofaddition, multiplication and differentiation. We thus derive approx-imating expressions for ui, d and vi, consisting of a combinationof exact Heaviside functions and box functions, and apply these tothe conservation laws (equation 2.30). Enforcing the residuals of theconservation laws upon substitution of the box approximations toapproach zero yields a set of algebraic equations that allows deter-

2.4 analysis using box approximations 27

mining the unknown parameters introduced when defining the boxapproximations.

2.4.1 Definitions

In the following derivation, the Heaviside function Hη(ξ) = H(ξ−

η) will be defined as:

Hη(ξ) =

0 ξ < η

1 η 6 ξ, (2.52)

In the sense of distributions, one can say that dHη/dξ = H ′η = δη,where δη is the Dirac delta supported in ξ = η. With δ or H (wit-hout subscript) we denote the Dirac delta or the Heaviside function,respectively, with η = 0.We define the ε-square pulse (box function, illustration see Figure

η-ǫ/2 η η+ǫ/2

ξ

0

1/(2ǫ)

1/ǫqǫ

η

Figure 2.3: Box function qεη

2.3) centered in η as:

qεη(ξ) = qη(ξ) =1

ε

(Hη−ε/2(ξ) −Hη+ε/2(ξ)

)(2.53)

With∫+∞−∞ qη(ξ)dξ = 1 and supp(qη) = [η− ε/2,η+ ε/2), equation

2.37 applies also to shifted box functions qη+κε. We can thus con-clude that every family qη+κε(ξ) is a delta sequence. Furthermore,


in a distributional sense, the derivatives of the box functions can bedetermined as

q ′η(ξ) =1

ε

(δη−ε/2(ξ) − δη+ε/2(ξ)

). (2.54)

In principle, it would be convenient to express δη±ε/2 as Taylorexpansions in η. However, Taylor series are not convergent (but onlyasymptotic) for distributions on test functions φ ∈ D(R).37 Instead,we introduce the following functionals, employed in a similar wayin the derivation by Keyfitz and Kranzer:32

< Mrs,φ >=1

r− s

∫rsφ ′(ξ)dξ (2.55)

The introduced term is a bounded functional, which can also be in-terpreted as the first order difference quotient of φ over the interval[s, r]. Note that, as r → s, the functional approaches the first orderderivative:

< Mrs,φ >r→s−→ φ ′(s) =< −δ ′s,φ > (2.56)

We define M− =Mη,η−ε and M+ =Mη+ε,η. For the derivatives ofthe box functions, we can thus conclude:

q ′η− ε2=1

ε(δη−ε − δη) = −M− (2.57)

q ′η+ ε2=1

ε(δη − δη+ε) = −M+ (2.58)

We can further perform the following manipulations on these functi-onals:

< M+ −M−,φ > =1

ε

∫η+εη

(φ ′(ξ) −φ ′(η)

)dξ

−1

ε

∫ηη−ε

(φ ′(ξ) −φ ′(η)

)dξ

=1

ε

∫η+εη

(η+ ε− ξ

)φ ′′(ξ)

dξ

−1

ε

∫ηη−ε

(η− ε− ξ

)φ ′′(ξ)

dξ

(2.59)

2.4 analysis using box approximations 29

Both terms on the right-hand side of equation 2.59, and thereforealso their difference, are O(ε). This difference corresponds to ε timesthe second order difference quotient of φ over the interval [η− ε,η+ε]. We will make use of the defined functionals and their propertiesin the derivations to follow.

2.4.2 Equations

We set up a box approximation of ui in equation 2.33, which con-stitutes a delta-shock solution to equation 2.30 in the sense of weakconvergence:

uεi (τ, ξ) = uLi − [ui]H(ξ) + uiτq−ε/2(ξ) + uτq+ε/2(ξ) (2.60)

Analogue to the generalized functions established in the previousapproach, this box approximation is made up of a bounded part(first two terms in equation 2.60) and of a singular part (third andfourth term) consisting of two box functions with disjoint support,where the second term is equal, while the first term differs for u1and u2.The variable uεi (τ, ξ) can also be defined in intervals as:

uεi (τ, ξ) =

uLi ξ < −ε

uLi + uiτ/ε −ε 6 ξ < 0

uRi + uτ/ε 0 6 ξ < ε

uRi ε 6 ξ

(2.61)

For dε = 1− uε1 + uε2 and vεi = aiu

εi /d

ε, this gives:

dε(τ, ξ) =

dL ξ < −ε

dL + (u2 − u1)τ/ε −ε 6 ξ < 0

dR 0 6 ξ < ε

dR ε 6 ξ

(2.62)


and

vεi (τ, ξ) =

vLi ξ < −εai(εu

Li+uiτ)

εdL+(u2−u1)τ−ε 6 ξ < 0

vRi + aiuτεdR

0 6 ξ < ε

vRi ε 6 ξ

. (2.63)

Equations 2.62 and 2.63 can be recast as a combination of Heavisideand box functions:

dε(τ, ξ) = dL − [d]H(ξ) + (u2 − u1)τq−ε/2(ξ) , (2.64)

and

vεi (τ, ξ) = vLi − [vi]H(ξ) +(aiui − (u2 − u1)v

Li )τ

εdL + (u2 − u1)τεq−ε/2(ξ)

+aiuτ

dRq+ε/2(ξ) . (2.65)

In the following, we will abbreviate the third term of the last equa-tion as α7εq−ε/2, where α7, as well as (α7)τ, are bounded. Notethat this third term, applied to a smooth test function, vanishes asε → 0. Substitution of the box approximations (equations 2.60 and2.65) into the conservation equations 2.30 results in the residuals Ri:

Ri = (uεi + vεi )τ + ((1− s)uεi − sv

εi )ξ

= uiq− ε2+ uq+ ε

2+ (α7)τ εq− ε

2+aiu

dRq+ ε

2

+ (1− s) (−[ui]δ− uiτM− − uτM+)

− s

(−[vi]δ−α7εM− −

aiuτ

dRM+

)= uiq− ε

2+ (α7)τ εq− ε

2+ uq+ ε

2+aiu

dRq+ ε

2

− ((1− s) [ui] − s[vi]) δ

−

[(1− s) (uiτ+ uτ) − s

aiuτ

dR

]M+ + sα7εM−

+ (1− s) uiτ (M+ −M−)

(2.66)

2.5 exact results on velocity and strength of the delta-shock 31

For ε → 0, and considering equations 2.37 and 2.55 to 2.59, theresiduals approach the following expression in D ′:

Riε→0−→

(ui + u+

aiu

dR− (1− s) [ui] + s[vi]

)δ

+

((1− s) (ui + u) τ−

saiuτ

dR

)δ ′

(2.67)

Enforcing Ri → 0 as ε→ 0, one obtains the algebraic equations 2.50

and 2.51, determining the unknown parameters u, u1, u2 and s.

2.5 exact results on velocity and strength of the delta-shock

The algebraic equations 2.50 and 2.51 provide the parameters u, u1,u2 and s of the approximate solutions, which, in order to obtain ourexact expressions of the form defined in equations 2.26 and 2.27,have to be transformed into the parameters gui , gvi , and σδ. Thetransformation is straightforward and has already been presentedelsewhere.8 For the sake of completeness, we want to summarizehere the most important equations.Based on the coefficient of the singular part in the approximatedsolution of vi (compare equations 2.48 and 2.65), we define gvi as(i = 1, 2):

gvi = aigv =

aiu

dR(2.68)

Using equation 2.39, we can then recast equations 2.51 and 2.50 as:

(1− s)gui = saigv (2.69)

gui + aigv = [ui] − s([ui] + [vi]) (2.70)

Combining the four equations 2.69 and 2.70, we can easily deriveexpressions for the reciprocal σδ of the propagation velocity of thedelta-shock s, as well as for gui and gv:

σδ =1

s= 1+

a1[v2] − a2[v1]

a1[u2] − a2[u1](2.71)


gui = s[ui] − s2([ui] + [vi])

=ai(a1[u2] − a2[u1]) ([u1][v2] − [u2][v1])

(a1[u2] − a2[u1] + a1[v2] − a2[v1])2

(2.72)

gv =gui (σδ − 1)

ai=

(a1[v2] − a2[v1]) ([u1][v2] − [u2][v1])

(a1[u2] − a2[u1] + a1[v2] − a2[v1])2(2.73)

Note that [u1] < 0 < [u2] for the delta-shock to exist, hence boththe numerator and the denominator in equation 2.71 are positive.Finally, substituting equation 2.71 in equation 2.25, we can concludethat:

dwidτ

=ai([u1][v2] − [u2][v1])

a1[u2] − a2[u1] + a1[v2] − a2[v1]= gui + gvi (2.74)

With these relationships, we have derived all the quantities formingpart of the expressions defined in equations 2.26 and 2.27, as wellas the Rankine Hugoniot deficit (equation 2.25).

2.6 conclusion

In this paper, properties of the delta-shock in a chromatographicsystem were derived by two different approaches. The first appro-ach, presented previously in a similar manner,8 was reanalyzed andcorrected by rigorously applying the Colombeau algebra, while thesecond, new approach is based on box approximations. Note thatboth approaches presented in sections 2.3 and 2.4, as well as a third,perturbation method approach presented by Tsikkou33 are consis-tent, and thus confirm equations 2.50 and 2.51 and the expressionsderived in section 2.5.While with these three approaches, the properties of the delta-shockin a chromatographic system are well characterized from a mathe-matical perspective, finding experimental evidence for this pheno-menon remains challenging. The main problem is the strong enri-chment of both components during the delta-shock, such that, in

2.6 conclusion 33

a liquid chromatographic system, supersaturation of these compo-nents and a subsequent phase split is virtually unavoidable.25 Ac-counting for a phase split and a subsequent multi-phase flow inthe model equations results in a change of the model structure andthus requires a reanalysis of the model, including the existence ofsingular shocks. Higher chances of finding experimental evidenceare expected in a gas chromatographic system, where limitations tomaintain a single fluid phase are only set by the vapor pressures ofthe different components. This option has yet to be evaluated, andremains as a future task in the investigation of the delta-shock. Inturn, the evidence of phase split and two-phase flow in liquid chro-matography has raised a new field of interest, namely the under-standing and description of these phenomena in chromatographicprocesses, which the remainder of this thesis is dedicated to.

3M O D E L I N G T W O - P H A S E F L O W I N L I Q U I DC H R O M AT O G R A P H Y

3.1 introduction

The aim of this chapter is to derive model equations to describe li-quid chromatography in the presence of multiple fluid phases. Suchmodel, like the common chromatographic models (e.g. equations1.1), and also like several macroscopic models describing two-phaseflow in natural reservoirs,4,10,18 is based on one-dimensional com-ponent material balances, neglecting any kind of radial effects, andforming a system of partial differential equations. The three impor-tant physical aspects to be considered when deriving the componentmaterial balances are

• Thermodynamic equilibria between the different convectiveand adsorbed phases

• Fluiddynamic behavior of the convective phases in the porousmedium

• Kinetic effects, in particular expected during a spontaneousphase split and in the form of mass transfer resistances bet-ween different phases

While the first two aspects will be discussed in greater detail below(section 3.3), we shortly want to comment on the impact and descrip-tion of kinetic effects. In standard chromatography with one singleconvective phase, kinetic limitations can be observed in the formof axial dispersion, and of mass transfer limitations, with barriersconsidered in particular between the bulk and the pore space, and

The model equations presented in this section are derived and applied in: Ortner,F.; Mazzotti, M. Two-phase flow in liquid chromatography, Part 1: Experimental in-vestigation and theoretical description Ind. Eng. Chem. Res. 2018, 57(9), 3274-91 andOrtner, F.; Mazzotti, M. Two-phase flow in liquid chromatography, Part 2: ModelingInd. Eng. Chem. Res. 2018, 57(9), 3292-3307

36 modeling two-phase flow in liquid chromatography

between the liquid phase and the solid, adsorbed film. An exten-sive review is provided by Gritti and Guiochon.38 Chromatographicmodels discussed in the literature primarily differ by their treat-ment of these kinetic effects, reaching from equilibrium theory mo-dels which neglect any kind of kinetic limitations, to general ratemodels which account separately for axial diffusion, solid film dif-fusion and pore diffusion by considering separate material balancesfor the bulk and the pore space. Frequently, axial diffusion and masstransfer limitations are considered to have additive effects (namelyband broadening) on the elution profiles. Hence, they are often lum-ped in one term in the material balances.39 An accurate descriptionof kinetic limitations is primarily important in the presence of ma-cromolecules, which feature much higher mass transfer resistancesthan small molecules. In the presence of multi-phase flow, a spon-taneous phase split can be particularly prone to kinetic limitations,but also mass transfer limitations might increase, owing to additio-nal phase boundaries.This work is focused on chromatographic systems with small mole-cules. Two-phase flow is investigated experimentally by the displa-cement of immiscible phases, and a spontaneous phase split with-inthe column is not studied in detail. Hence, any kind of kinetic limi-tation is neglected for a major part of this thesis. An equilibriumtheory model is derived and used, assuming thermodynamic equi-librium between all convective and adsorbed phases at any timeand location in the column. In addition, a modification to the equili-brium theory model is considered, which accounts for kinetic limita-tions in a very simplistic manner, lumping axial diffusion and masstransfer effects into one term, while maintaining the assumption ofthermodynamic equilibrium between the convective phases. In thefollowing, the equilibrium theory model and the lumped kinetic mo-del, both based on component material balances, are introduced andalgebraic relationships required by these models to describe physi-cal properties, such as fluiddynamic and thermodynamic behavior,are discussed.

3.2 material balances 37

3.2 material balances

3.2.1 Equilibrium theory model

The equilibrium theory model for a multicomponent system in thepresence of multiple convective (liquid) phases, accounting for ad-sorption and neglecting dispersive effects and mass transfer resis-tances, is given by the component material balances

(∂εCi)

∂t+ (1− εref)

∂ni∂t

+∂ (uFi)

∂z= 0, i = 1...NC, (3.1)

where t and z are time and space variables, respectively, NC is thenumber of components and ni is the adsorbed phase concentra-tion of component i in thermodynamic equilibrium with the liquidphase(s) (mass of component i per volume of adsorbent in the com-pletely regenerated state). The overall liquid phase concentrations,Ci, and overall fractional flows, Fi, are defined as:

Ci =

NP∑j=1

wijρmixj Sj =

NP∑j=1

cijSj, Fi =

NP∑j=1

wijρmixj fj =

NP∑j=1

cijfj

(3.2)

where NP denotes the number of phases, wij and cij are the massfraction and concentration of component i in phase j, respectively,and ρmix

j is the density of phase j (note that ρi corresponds to thedensity of the pure component i). The volumetric fraction of phasej, also called phase saturation, is indicated by Sj, whereas fj is thevolumetric fractional flow of phase j. Accounting for the fact thatdifferent convective phases can move with different interstitial velo-cities, the fractional flow fj is not necessarily equal to Sj. Commonly,fj is expressed as an empirical (rational) function of Sj, as discussedin detail below. As a consequence of fj being in general not equal toSj, the values of Ci and Fi can differ, since they are averages of thephase concentrations cij, weighed by the phase saturations Sj or thefractional flow fj, respectively.The phase densities ρmix

j are given as follows, as a function of the


phase composition and of the component densities, by assumingadditivity of volumes:

ρmixj =

NC∑i=1

wij

ρi

−1

. (3.3)

We further assume thermodynamic equilibrium between the liquidphases (i.e. equal chemical potentials of each component in all liquidphases), which allows to calculate phase split and phase composi-tions based on a thermodynamic model. For the description of theequilibrium between the adsorbed phase and the liquid phases, ad-sorption isotherms have to be developed, which in some way relatethe adsorbed phase concentrations, ni, to the compositions of thefluid phases.Dealing with a non-dilute system, volumetric changes due to ad-sorption/desorption cannot be neglected. Thus, we define the stati-onary phase to be made up of the adsorbent (with reference volume(1− εref)Vc and reference porosity εref at the completely regenera-ted state, i.e. in the absence of any adsorbed component) and theadsorbed phase, and we assume every component to occupy thesame volume in the adsorbed as in the liquid phase (defined by thecomponent densities ρi), thus yielding the following mathematicalrelationship:

1− εref1− ε

1+ NC∑i=1

niρi

= 1, (3.4)

which can be recast as:

ε = εref − (1− εref)

NC∑i=1

niρi

. (3.5)

By normalizing each component material balance by the componentdensity ρi and summing up over all the component material balan-ces, the overall material balance reduces to

∂u

∂z= 0. (3.6)

3.2 material balances 39

Thus, under the previous assumptions of variable porosity, volumeadditivity in the liquid and the adsorbed phase, and equal compo-nent densities in the liquid as in the adsorbed phase, the superficialvelocity (overall flowrate divided by the cross-sectional area of thecolumn) remains constant.Finally, time and space coordinates are made dimensionless:

ξ =z

L; τ =

ut

L, (3.7)

where L is the column length. The final form of the componentmaterial balances is then

∂ (εCi)

∂τ+ (1− εref)

∂ni∂τ

+∂Fi∂ξ

= 0, i = 1...(NC − 1). (3.8)

TheNC−1 independent concentrations Ci are uniquely determinedby the system ofNC−1 component material balances, while the con-centration of component NC is a function of the other componentconcentrations, based on the assumption of volume additivity:

NC∑i=1

Ciρi

=

NC∑i=1

cij

ρi= 1 (3.9)

Since the conservation equations 3.8 form a system of first order,homogeneous, partial differential equations with unknowns Ci (i =1, ..., (NC − 1)), the model can be solved by applying the method ofcharacteristics, which will be explained in detail in chapter 5.

3.2.2 Lumped kinetic model

We further consider a modification to the above conservation laws,to be solved numerically, which lumps axial diffusion and masstransfer effects into a simple empirical mass transfer term. It shouldbe pointed out that, while this additional term accounts for possibledispersive effects and mass transfer resistances between liquid pha-ses and adsorbed phase, the assumption of thermodynamic equi-librium between the liquid phases is nevertheless maintained. Theresulting modified conservation equations read:

∂ (εCi)

∂τ+ (1− εref)

∂ni∂τ

+∂Fi∂ξ

= 0, i = 1... (NC − 1) , (3.10)


∂ni∂τ

= Sti (ni − ni) , i = 1, ...,NC (3.11)

The variable ni denotes the adsorbed phase concentration in equili-brium with the liquid phase(s) throughout the entire work, whereasits actual, non-equilibrium value, in the context of equation 3.11 iscalled ni. The dimensionless Stanton number Sti is defined as

Sti =akiL

u, (3.12)

where a is the specific area of the adsorbent and ki is the mass trans-fer coefficient of component i. The introduction of the mass transferterm results in band broadening effects in the elution profiles, re-ducing the sharpness of fronts, which is physically more realisticthan the very sharp transitions obtained under the assumption ofinstantaneous thermodynamic equilibrium. However, this lumpedkinetic term was not only introduced for physical, but mainly fornumerical reasons, as it circumvents the calculation of derivativesof ni with respect to C, which would have been necessary for anumerical solution of the equilibrium theory model. With n1 beinga function of the liquid phase activities, analytical expressions ofsuch derivatives would have been rather complex in the miscibleregion (involving a differentiation of the UNIQUAC equation), andimpossible in the immiscible region. The introduction of the masstransfer term instead of a numerical calculation of the derivatives ofn1 allowed a reduction in computational effort. A rather high Stan-ton number Sti = 200 (corresponding to minor kinetic limitations)was assumed for all components, so as the solution of the lumpedkinetic model is expected not to deviate excessively from that obtai-ned using the equilibrium theory model.Since in this work we are dealing with Riemann problems (i.e. step-wise constant initial value problems), the initial condition is definedas:

Ci(ξ, 0) = C0i ; ni(ξ, 0) = n0i , (3.13)

3.3 algebraic equations 41

where C0i and n0i define the initial state of the column at τ = 0. Forthe first-order lumped kinetic model used in this work a Dirichletboundary condition can be applied:

Fi(0, τ) = FFi (3.14)

where FFi denotes the feed overall fractional flow. For the solution

of the model equations, liquid and adsorbed phase concentrationswere normalized by the density of the corresponding componentρi. A semi-discrete finite volume scheme with a VanLeer flux limi-ter was applied as suggested in Ref. [40], discretizing the partialdifferential equations 3.10 in space and thus reducing them to ordi-nary differential equations with respect to time. The resulting ODEs(discretized equations 3.10 and equations 3.11) were solved with abuilt-in Matlab routine (ode113).

3.3 algebraic equations

The material balance equations presented above represent the chro-matographic model in general terms. However, in order to describea specific chromatographic system, these equations have to be com-bined with algebraic relationships describing (i) the thermodynamicequilibrium between the convective phases, (ii) the thermodynamicequilibrium between convective and adsorbed phases (i.e. an ad-sorption isotherm) and (iii) the fluiddynamic behavior of the con-vective phases. The models applied in this work to describe thesephysical properties are reviewed theoretically in this section.

3.3.1 Thermodynamic equilibria between convective phases

Different phases are in thermodynamic equilibrium, if the Gibb’senergy for the overall system (over all phases) is at a global minimum.41

In order to assess or enforce this condition, the system has to becharacterized by so-called excess functions, describing the excessGibb’s energy, i.e. the difference of the Gibb’s energy of a real so-lution with respect to its value assuming an ideal behavior. Hence,such excess functions evaluate the thermodynamic nonideality of asystem. There are a multitude of different excess functions based


on different physical or empirical assumptions; the most commonlyused models for liquids are the Wilson42, NRTL43 or UNIQUAC44–46

equation. Parameters of these models have to be determined basedon experimental phase equilibrium data. In turn, a predictive modelis provided by the UNIFAC equation.47,48 Apart from the determi-nation of the excess Gibb’s energy, these models allow the calcula-tion of activities of stable liquid phases, as well as the calculationof phase splits for unstable compositions, by minimization of theGibb’s energy.49,50

3.3.2 Adsorption equilibria

Thermodynamic equilibria between convective and adsorbed pha-ses in liquid chromatography are commonly described by adsorp-tion isotherms being a function of the liquid phase concentrations,and hence assuming ideal conditions in the liquid phase. Due tothe assumption of thermodynamic ideality, the applicability of suchisotherms is limited to dilute solute concentrations. At elevated con-centrations, these relationships are likely to fail, and even more soin the presence of multiple convective phases with different soluteconcentrations.25 A thermodynamically consistent approach is touse isotherm equations based on liquid phase activities, as it hasbeen done in the context of separating concentrated acids and carbo-hydrates on elastic ion exchange resins.51,52 Such isotherms are alsoexpected to be applicable in the presence of multiple fluid phasesin thermodynamic equilibrium, since these phases feature identicalactivities and chemical potentials.In the presence of multiple adsorbing components, competition orcooperation for adsorption sites between the different solutes has tobe accounted for. While binary isotherms are commonly constructedby an empirical combination of single-component isotherms,26 athermodynamically consistent approach, also based on single-com-ponent isotherms, is the adsorbed solution theory,53–55 assuming anideal behavior of the adsorbed phase (ideal adsorbed solution the-ory - IAST), or accounting for non-ideal behavior (real adsorbedsolution theory - RAST). A detailed explanation of these conceptswill be provided in section 4.2.1.


3.3.3 Fluiddynamics

In the case of two (or multiple) convective phases, interstitial velo-cities of these phases are commonly not identical. The macroscopicproperties of two-phase flow in porous media have been studied ex-tensively in the context of applications in natural reservoirs.56,57

For laminar flow (Reynolds numbers Re < 1; for the system consi-dered in chapters 4 to 6 Re is about 5×10−3) of a single convectivephase through a porous medium, Darcy’s law establishes a linearrelationship between pressure gradient and superficial velocity.3,58

This concept can be extended to multiphase flows, and simplifiedfor the assumption of one-dimensional, horizontal flow as follows:

uj =kr,jK

ηj

∂pj

∂z, (3.15)

where uj is the superficial velocity of phase j, describing the flowof that phase through the cross-sectional area of the column, K isthe permeability, i.e. a property of the porous medium, ηj is the

viscosity of phase j, and ∂pj∂z is the pressure gradient in phase j.

The relative permeability kr,j, with values ranging between 0 and1, accounts for the fact that not the entire void space of the porouspacking is available for phase j, since it has to be shared with otherconvective phases. As a consequence, the resulting superficial velo-city uj is lower than it would be at the same pressure gradient, ifphase j were the only convective phase.Neglecting capillary pressures between all convective phases, onecan conclude that ∂pj∂z = ∂pk

∂z = ... = ∂p∂z , and as a consequence

u =

NP∑n=1

uj =

NP∑n=1

(kr,nK

ηn

)∂p

∂z, (3.16)

and

fj =uj

u=

kr,jηj

NP∑n=1

kr,nηn

. (3.17)

At this point, we would like to underline that capillary effects arenot neglected in the two-phase flow description. In fact, they are


the key property determining the system-specific relationship of therelative permeabilities krj with respect to the phase saturations Sj,which we will establish in the following. However, for most systems,the pressure difference of the different fluid phases (correspondingto the capillary pressure) is negligible with respect to their absolutepressure levels.It was found that experimental breakthrough data for two-phase(nonadsorbing) systems in porous media (mostly in natural reser-voirs) is very often described accurately assuming a power-law rela-tionship between relative permeabilities kr,j and phase saturationsSj.1,2,14 In this contribution, we consider the following relationship:

SReff < 0 : kR

r = 0; kLr = 1 (3.18a)

0 6 SReff 6 1 : kR

r = kRr,max(S

Reff)mR

;

kLr = k

Lr,max(1− S

Reff)mL

(3.18b)

SReff > 1 : kR

r = 1; kLr = 0, (3.18c)

with

SReff =

SR − Si1− Si − Sr

. (3.19)

The superscripts R and L denote the PNT-rich (wetting) and -lean(non-wetting) phase in chapters 4 to 6, and the water-rich and -lean phase in the esterification system in chapter 7. Equations 3.18

and 3.19 include the following six parameters: the irreducible/ re-sidual saturation Si/r, determining the saturation of the wetting/non-wetting phase, respectively, at which this phase becomes hy-draulically disconnected and thus cannot be displaced further bythe corresponding (non-wetting/ wetting) phase in thermodynamicequilibrium; the maximum relative permeabilities kR/L

r,max, which de-termine the pressure drop at the residual/ irreducible saturation;the exponents mR/L, which determine the shape of the relative per-meability functions. In principle, these parameters are functions ofthe interfacial tension σ between the wetting and the non-wettingphase, and should thus change for different tielines connecting pha-ses in thermodynamic equilibrium. The parameters should behave


in such a way that an equal-velocity behavior (fj = Sj) is appro-ached as the phases in thermodynamic equilibrium approach theplait point of the binodal curve (since the two phases merge to onephase at that point). In this thesis, we simplify the description insuch a way that we assume the six parameters to be constant, re-gardless of the change in interfacial tension.

4C H A R A C T E R I Z AT I O N O F P H Y S I C A L P R O P E RT I E S

4.1 introduction

As demonstrated in the previous chapter, it is possible to derivegeneric material balance equations accounting for adsorption andmulti-phase flow (see section 3.2). In turn, the algebraic equationsto be implemented in the material balances, describing physical pro-perties such as thermodynamic and fluiddynamic equilibria, have tobe determined specifically for each system.In order to evaluate the physical implications of phase split andtwo-phase flow in liquid chromatography, an exemplary chromato-graphic system is characterized. The investigated system involvesthe components water and methanol (forming the eluent, and bothassumed to be inert), as well as two adsorbing components phene-tole (PNT) and 4-tert-Butylphenol (TBP), with the adsorbent Zorbax300SB-C18 and at a constant temperature of 23

C. Independent ex-perimental campaigns are carried out to investigate the mixing be-havior of the quaternary system (PNT - TBP - methanol - water), itsthermodynamic properties and phase equilibria, and in particularadsorption and hydrodynamic properties of the quaternary systemin contact with the adsorbent. Existing theoretical concepts for eachof these properties are considered, and underlying parameters arefitted to the experimental data.The chapter is structured as follows: In section 4.2, the adsorbedsolution theory, providing a thermodynamically consistent conceptfor the description of binary and multicomponent adsorption, is

Results presented in this chapter are summarized from two publications. Thermo-dynamic and fluiddynamic properties of the ternary system (PNT, methanol, water)have been reported in: Ortner, F.; Mazzotti, M. Two-phase flow in liquid chromato-graphy, Part 1: Experimental investigation and theoretical description Ind. Eng. Chem.Res. 2018, 57(9), 3274-91

Thermodynamic properties of the quaternary system (including TBP) have been re-ported in: Ortner, F.; Ruppli, C.; Mazzotti, M. Description of adsorption in liquidchromatography under non-ideal conditions Langmuir 2018, 34(19), 5655–71

48 characterization of physical properties

explained and applied to the case of a binary system with single-component BET isotherms. In addition, the theory introduced inthe previous chapter is relevant for this work. Experimental met-hods are described in section 4.3. The subsequent sections 4.4 to4.8 present the experimental and theoretical characterization of theinvestigated system, concerning additivity of volumes, thermodyna-mic equilibria between the convective phases, adsorption equilibria(single component and binary) and hydrodynamics. Finally, conclu-sions are drawn in section 4.9.

4.2 theoretical background

4.2.1 Adsorbed solution theory

The adsorbed solution theory treats the adsorbed phase as a solu-tion, which can be assumed to be ideal (IAST) or not (RAST). As-suming thermodynamic equilibrium between liquid and adsorbedphase and an adsorption behavior described by the Gibbs adsorp-tion isotherm, this theory allows for the thermodynamically consis-tent description of binary and multi-component adsorption equili-bria on the basis of single-component isotherms. The theory wasoriginally established in the context of gas adsorption,53 but was la-ter adjusted to be applicable for liquid chromatography.54 A recentreview article provides a good overview over model assumptions,equations and applications.55 In the following, we want to shortlysummarize the theory as it is applied in this work. In the context ofthe AST, we use liquid and adsorbed phase concentrations ci andni in moles per volume of adsorbed phase. For the implementationinto the chromatographic model, these can be easily transformedinto concentrations ci and ni in mass per volume, through multipli-cation by the corresponding molecular weight.The Gibbs isotherm provides the relationship between adsorbedphase concentrations ni and spreading pressure π at a constant tem-perature T :

Aa

Va dπ =

Nc−1∑i=1

nidµai + nsdµa

s (4.1)

4.2 theoretical background 49

The variables Aa and Va denote the surface area and the volumeof the adsorbent, while ni and ns are the adsorbed phase concen-trations of the Nc − 1 solutes and the solvent, respectively. Accor-dingly, µa

i and µas are the chemical potentials of solutes and solvent

in the adsorbed phase. In our system, we define water to be thesolvent component, while methanol (the modifier) in this contextis one of the Nc − 1 solutes, which however does not adsorb. TheGibbs-Duhem equation in the liquid phase can be written as

Nc−1∑i=1

cidµli + csdµl

s = 0, (4.2)

with ci and cs denoting the liquid phase concentrations of solutesand solvent, respectively, in moles per volume, and the superscriptl indicating the liquid phase. Substituting equation 4.2 in equation4.1, and assuming thermodynamic equilibrium between adsorbedand liquid phase (i.e. identical chemical potentials in both phases,dµai = dµl

i), yields:

Aa

Va dπ =

Nc−1∑i=1

ncori dµa

i, (4.3)

with

ncori = ni −

cicsns. (4.4)

The second term in equation 4.4 is negligible under the conditi-ons that ci << cs (low solute concentrations), or that ni >> ns(strongly adsorbing solutes),54 thus resulting in ncor

i ≈ ni. We con-sider both conditions valid for the components PNT and TBP, whileadsorbed phase concentrations of methanol (subscript M) and wa-ter are assumed to be zero, i.e. ncor

M = nM = 0. We would furtherlike to comment on the accessible surface area of the adsorbent, A,which might differ for adsorbing molecules of different sizes, but isassumed constant in this contribution (compare equations 4.1 and4.3). While adsorption (hence an impact on the spreading pressure)of both water and methanol is neglected, the two adsorbing com-ponents PNT and TBP feature similar molecular weights and UNI-QUAC parameters indicating similar molecular volume and exter-nal surface areas (compare combinatorial UNIQUAC parameters in


Table 4.1). We thus consider the assumption of a constant accessiblesurface area A as realistic and acceptable.The chemical potential of a solute in the liquid phase is given by:

µli(T , zl) = µi (T) + RT lnai (4.5)

where the effect of pressure p on liquid phase properties is neg-lected. Accordingly, the chemical potential of a solute in the adsor-bed phase can be written as:

µai(T ,π, za) = µi (T ,π) + RT ln (γa

izai)

= µi (T) + RT lna∗i (π) + RT ln (γaiz

ai)

(4.6)

In the upper two equations, reference states are denoted by the su-perscript , and zx

i and γxi (x = l, a) are the mole fractions and acti-vity coefficients in phase x, respectively. The liquid phase activitiesare designated as ai, while a∗i denotes the hypothetical activity ofthe single adsorbing component i in the liquid phase, at a spreadingpressure π identical to that attained by the multi-component systemat the liquid composition of interest. For the activity coefficients, thefollowing thermodynamic constraints must be fulfilled:41,59,60

(i) The activity coefficient of component i has to approach unityas the component’s mole fraction zxi in the corresponding phaseapproaches unity:

limzxi→1

γxi = 1, x = l, a (4.7)

(ii) As the spreading pressure approaches zero, i.e. at low surfacecoverages, ideal behavior must be approached in the adsorbedphase:

limπ→0

γai = 1 (4.8)

Equating µli (equation 4.5) and µa

i (equation 4.6) under the assump-tion of thermodynamic equilibrium yields:

ai = a∗iγ

aiz

ai. (4.9)


Therefore:

Nc∑i=1

zai =

Nc∑i=1

aia∗iγ

ai

= 1. (4.10)

The hypothetical activities a∗i can be determined by consideringequation 4.5 in the Gibbs isotherm (equation 4.1) and integratingthe resulting expression for a single component:

Π =Aaπ

VaRT=

a∗i∫0

n∗i (s)

sds (4.11)

with Π = Π∗i = Πmix, and n∗i denoting the hypothetical adsorbedphase concentration, obtained from the single component isothermof component i.The overall adsorbed phase concentration ntot =

∑ni is related

to the molar area αtot occupied in the adsorbed phase throughntot = Aa/(Vaαtot). The excess molar area of mixing αm is giventhrough:53,59

αm = αtot −

Nc∑i=1

zaiα∗i = RT

Nc∑i=1

zai

(∂ lnγa

i

∂π

)(4.12)

As a consequence, ntot can be determined from

1

ntot=

Nc∑i=1

zai

n∗i+

Nc∑i=1

zai

(∂ lnγa

i

∂Π

). (4.13)

Finally, the adsorbed phase concentration ni of component i can beobtained from

ni = zaintot. (4.14)

It is worth noting that in this derivation, both liquid and adsorbedphase are considered to be nonideal. In the case that an ideal ad-sorbed phase is assumed, the activity coefficients in the adsorbedphase γa

i equal 1 and are independent of Π, which results in a sim-plification of equations 4.9, 4.10 and 4.13. The case of an ideal liquid


phase is not considered in this work, because it is physically unrea-listic for our system, as deduced from the single component adsorp-tion behavior reported below (section 4.6). Furthermore, we wantto point out that the above derivation, being based on liquid phaseactivities, is also applicable in the presence of multiple fluid phasesin thermodynamic equilibrium (i.e., featuring identical liquid phaseactivities).Equations 4.9 to 4.11, together with a thermodynamic model rela-ting γa to za, have to be solved simultaneously in order to obtainzai and a∗i . Then, ntot and ni can be determined through equations

4.13 and 4.14. An explicit analytical solution of equations 4.9 to 4.11

is only possible under the assumption of an ideal adsorbed solution,and for rather simple single component isotherms with identical sa-turation capacities.61,62 In most cases, a numerical solution is requi-red, and a multitude of different numerical approaches is suggestedin the literature.63–66

4.2.2 Application of AST to the BET isotherm

The Brunauer, Emmett, and Teller (BET) model accounting for mul-tilayer adsorption was originally developed to describe gas adsorp-tion equilibrium,67 but it can also be applied in the context of li-quid chromatography.61 The BET isotherm equation applied in li-quid chromatography, and based on liquid phase activities, reads

n =qsatbSa(

1− bLa) (1− bLa+ bSa

) , (4.15)

where qsat is the saturation capacity in the monolayer in mole pervolume of adsorbent, and bS and bL are the adsorption-desorptionequilibrium constants for the first and all subsequent layers, re-spectively. Under the assumption of an ideal adsorbed solution (γa

i =

1), and of two solutes featuring an identical saturation capacity qsat,

4.3 experimental 53

an explicit binary isotherm can be obtained analytically by solvingequations 4.9 to 4.14:61

ni(a1,a2) =

qsat (bSi +

(bLib

S3−i − b

L3−ib

Si

)a3−i

)ai(

1− bL1a1 − b

L2a2

) (1− bL

1a1 − bL2a2 + b

S1a1 + b

S2a2

) (4.16)

for i = 1, 2. In turn, assuming a nonideal adsorbed solution, a ther-modynamic model of the adsorbed phase relating γa to za has to beimplemented. Combining equations 4.10 and 4.11 provides a relati-onship for a∗ in terms of a and γa:

a∗i =bS1a1γ

a2 + b

S2a2γ

a1(

bLib

Si−3 − b

Sib

Li−3

)ai−3γ

ai + b

Siγ

aiγ

a2

; i = 1, 2. (4.17)

Equations 4.9 and 4.17, and the relationship resulting from the ther-modynamic model γa

i = f(za) have to be solved numerically in or-der to determine za, a∗, γa, which are required for the calculationof ni based on equations 4.13 and 4.14. For the implementation intothe numerical chromatographic model introduced in chapter 3.2.2,we solve equations 4.9 and 4.17, together with the thermodynamicmodel equations iteratively, with the initial guess obtained from thevalue of the previous time step in the same spatial cell. The iterativeprocedure is outlined in Figure 4.1.

4.3 experimental

4.3.1 Material and basic methods

The chemicals used in this chapter are phenetole (PNT, purity >99%, Sigma-Aldrich), 4-tert-butylphenol, (TBP, purity 99%, Sigma-Aldrich), methanol (HPLC grade, purity > 99.9%, Sigma-Aldrich)and deionized water (Millipore). While PNT, methanol and waterare liquids at the experimental temperature of 23

C and ambientpressure, TBP is a solid. Mixtures of the four components were pre-pared gravimetrically, using analytical balances (DeltaRange AX205

and XP2003S, Mettler Toledo) and precision balances (DeltaRangePM4600, Mettler Toledo). Compositions of the mixtures are given


Initialization (Eq. 15)

From previous time step

on the same cell :

Iterative solution (Eq. 15, 23)

Adsorbed phase concentration (Eq. 19, 20)

Figure 4.1: Iteration procedure to solve the RAST model in the lumped ki-netic model.

4.3 experimental 55

in mass fractions wi, or were transformed into concentrations ci inmass of component per volume of the mixture. For the conversionof mass to volume, the mixture density ρmix was determined fromthe component densities ρi assuming additivity of volumes (com-pare equation 3.3).All dynamic column experiments and HPLC analyses were per-formed with the stationary phase Zorbax 300SB-C18, using a pre-packed stainless steel column (150 × 4.6 mm or 50 × 4.6 mm) fromAgilent Technologies. The column of 150 mm length was used forsingle component adsorption experiments of TBP, for binary adsorp-tion experiments, and for HPLC analysis, while the 50 mm columnwas applied for single component adsorption experiments of PNTand imbibition/drainage experiments. Total (inter- and intraparti-cle) porosities of the columns were determined through pulse injecti-ons of an analytical tracer (Uracil, purity > 99%, Sigma-Al-drich),yielding a porosity εref of 0.611 for the 150 mm column and of 0.615

for the 50 mm column.Chromatographic experiments were carried out on a modular HPLCunit (Agilent Technologies 1200 Series), equipped with a degasser,quaternary piston pump, autosampler, heated column compartment,and a UV/DAD detector. Furthermore, for the injection of large vo-lumes, a manual injector (Rheodyne 7725i) with injection loops ofvolumes 5, 9 and 20 mL was used. Due to backmixing effects insidethe loops, not the entire volume of the loops was used, but injectionvolumes were limited to 4, 7.2 and 16 mL, respectively. All expe-riments were performed at a flow rate Q = 1.2 mL/min and at atemperature T = 23C. Note that all the data obtained through dy-namic column experiments has been corrected for the system deadtime, measured by tracer injections through a column bypass. Forthe HPLC analysis of PNT and TBP, an eluent composition of metha-nol:water 57.42:42.58 (w:w) (corresponding to 63:37 (v:v)) was used.

4.3.2 Density measurements

The density ρmix of mixtures of TBP in methanol, PNT in methanoland TBP in PNT at 23

C was measured with a density meter (DMA48, Anton Paar).


4.3.3 Phase equilibrium experiments

Phase equilibrium experiments were performed at 23C with two

to four components of the quaternary system under investigation.Since at 23

C and 1 bar TBP is a solid, while the other three com-ponents (PNT, methanol and water) are liquids, the possible phaseequilibria to investigate are solid-liquid (SL), liquid-liquid (LL) andsolid-liquid-liquid (SLL) equilibria. Immiscible initial compositions(approx. 10 mL) were prepared gravimetrically and stirred (800 rpm)for at least 36 h at 23

C in a thermostated reactor (EasyMax 102

workstation, Mettler Toledo). The resulting phases (up to two liquidphases and one solid phase), assumed to be in thermodynamic equi-librium, were separated by centrifugation for up to 10 min (10,000

rpm, Rotina 420, Hettich). Subsequently, liquid phases were trans-ferred manually into separate vessels; possible dispersed solids stillpresent were removed by passing the liquid phases through syringefilters (PTFE, 0.22µm pore size). The remaining solid phase wasdried on a vacuum filter, crushed and analyzed by powder X-raydiffraction (Bruker AXS D8 Advance PXRD).To determine the composition of the liquid phase(s), the mass fracti-ons of the less concentrated components (1, 2 or 3) were analyzed,while the mass fraction of the remaining “bulk” component wascalculated from the mass fractions of the other components. Massfractions of PNT and TBP were determined by HPLC analysis, met-hanol fractions were analyzed by gas chromatography (Clarus 480,Perkin Elmer), on a 80/100 Porapak Q column (6’x1/ 8"x2.1 mm,Supelco Analytical), and the water content was measured by KarlFischer titration (titrator: 702 145 SM Titrino, Metrohm, Karl Fischerreagent: Hydranal-Composite 5, Sigma-Aldrich).

4.3.4 Single component Frontal Analysis

Single component Frontal Analysis experiments are dynamic co-lumn experiments carried out to determine the adsorbed phase con-centration of the investigated solute, which is in thermodynamicequilibrium with the liquid phase concentration applied to the co-lumn inlet. Performing such experiments at different feed concen-trations provides an experimental data set, which enables the deter-

4.3 experimental 57

mination of the single component adsorption isotherm.For such experiments, the chromatographic column (Zorbax 300SB-C18, 150x4.6 mm for TBP, 50x4.6 mm for PNT) is equilibrated withpure eluent (mixture of methanol and water) of a specific compo-sition, determined through the mass fraction r of methanol in theeluent. Subsequently, a certain feed concentration of the adsorbingsolute, in a solution featuring the same eluent composition r, is ap-plied to the column using the injection loop (start of the feeding ata time t0). Once the feed solution reaches the column outlet, indica-ting complete saturation of the column with the feed solution, theadsorption step is complete (at time tF) and the eluent is applied toreequilibrate the column (desorption step). The UV signal at 295 nmwas recorded throughout the experiment. The UV profile was trans-formed to a concentration profile via a rational calibration function(established from the relationship between the UV signal at the feedplateau and the feed concentration).The adsorbed phase concentration nF

i in equilibrium with the feedconcentration cF

i can be determined from the measured elution pro-file, applying the integral component material balance of compo-nent i:

tF∫t0

ucFiAc dt−

tF∫t0

uci(t,L)Ac dt =

= (εFcFi + (1− εref)n

Fi − ε

0c0i − (1− εref)n0i )AcL,

(4.18)

whereAc denotes the cross-sectional area and L is the column length.In equation 4.18, the first and second terms account for the fluxesinto and out of the column, respectively, whereas the term on theright-hand side describes the hold-up of component i in the liquidand the adsorbed phase.Substituting equation 3.5 into equation 4.18 with c0i = n0i = 0, theadsorbed phase concentration of component i, nF

i , for a specific li-


quid phase concentration, cFi , and mass fraction of methanol in the

solvent, r, can be determined as:

nFi =

u

(tF∫t0

(cFi − ci(t,L)

)dt− cF

itvoid

)

L (1− εref)

(1−

cFi

ρFi

) , (4.19)

where

tvoid =εrefL

u. (4.20)

Frontal Analysis experiments were performed for different solute(PNT and TBP) concentrations at the same eluent composition, inorder to obtain an experimental data set. Series of Frontal Analysisexperiments were performed at different eluent compositions.

4.3.5 Binary Frontal Analysis

Binary Frontal analysis experiments were performed to assess thebinary adsorption behavior of TBP and PNT. As in the single com-ponent Frontal Analysis experiment, the column (Zorbax 300SB-C18,150x4.6 mm) was equilibrated with pure eluent of a specific metha-nol:water ratio. A specified volume of feed solution, here containingboth TBP and PNT in eluent of the same ratio, was applied to thecolumn using one of the three injection loops available. By closingthe loop, the feeding period was terminated and the column wasreequilibrated with the eluent.With both PNT and TBP absorbing UV light at similar wavelengths,the UV signal cannot simply be converted into concentration pro-files. Instead, the eluate was diluted automatically with a diluentof composition methanol:water 57.42:42.58 (w:w), using an additi-onal HPLC pump (Waters 515 HPLC pump, Milford, MA, USA)connected to the setup after the UV detector. The dilution ratiofor the experiments performed in this work varied between 1.2:4.5and 1.2:10.0 (v:v) eluate:diluent. The diluted eluate was collected infractions at a sampling rate of 0.12 to 0.18 min per fraction, using afraction collector (Gilson FC 203B, Middleton, WI, USA). Note that

4.3 experimental 59

the dilution ratio and sampling rate were adjusted according to theexperimental conditions, in order to achieve a suitable dilution andsample volume for the subsequent analysis. The samples were thenanalyzed offline by HPLC, in order to obtain concentrations of TBPand PNT in the eluate and to reconstruct the elution profile.As for single component Frontal Analysis, multiple experimentswere performed, varying both the solute feed concentrations andthe eluent compositions.

4.3.6 Determination of the column permeability K

For the investigation of the two-phase flow behavior (determinedfor the ternary system PNT - methanol - water, in the absence ofTBP), the permeability of the column packing K had to be determi-ned. For this purpose, pure PNT, as well as different mixtures ofmethanol-water (with different viscosities in the liquid phase, esti-mated as described below, in section 4.8.1) were pumped throughthe 50×4.6 mm Zorbax 300SB-C18 column at a constant flowrateof 1.2 mL/min and at a temperature of 23

C. The stable pressuredrop (after equilibration) was measured for each mobile phase. Forthe latter purpose, the HPLC unit was additionally equipped withtwo pressure sensors, type 21 PY (Keller), located directly beforeand after the column, and the pressure signal was recorded onlinewith the Labview software. Based on Darcy’s law for single-phaseflow,3,58 the permeability K could be determined accurately througha linear fit between pressure drop and viscosity data (compare Fi-gure 4.2).

4.3.7 Hydrodynamic two-phase flow behavior

The hydrodynamic two-phase flow behavior of the ternary systemPNT - methanol - water through the porous medium (50×4.6 mmZorbax 300SB-C18 column) was determined by displacement expe-riments, displacing liquid phases which are in thermodynamic equi-librium with each other (i.e. two compositions on the binodal curveconnected by a tieline). Since the activities ai of each component iare identical for both phases, and with our assumption of the ad-


0 0.5 1 1.5 2

η [mPas]

0

20

40

60

80

∆p

[bar]

methanol-water

phenetole

K=1.554 (± 0.020) x10-14

m2

Figure 4.2: Pressure drop over the column for different mobile phases (purePNT and methanol-water mixtures), at a flowrate of 1.2 mL/minand a temperature of 23

C. A linear fit provides the permeabilityK of the column packing.

sorbed phase concentration ni being a function of the liquid phaseactivities a, neither adsorption nor desorption occurs during suchdisplacement. Considering equation 3.5, also the porosity ε remainsconstant. Hence, all the effects observed in the resulting elution pro-files are due to the hydrodynamic behavior of the two liquid phases.The two liquid phases in thermodynamic equilibrium were produ-ced by stirring an immiscible composition of 40.4 % (w:w) PNT, 34.6% (w:w) methanol, and 25.0 % (w:w) water for 36 h in the EasyMax102 workstation and separating the resulting phases in thermody-namic equilibrium as described in section 4.3.3. According to theUNIQUAC model established for the ternary system (see section4.5 below), the compositions of the resulting phases are PNT : met-hanol : water 2.95 : 55.73 : 41.32, and 97.11 : 2.58 : 0.31 (w:w:w),respectively.The column was equilibrated with the PNT-lean (non-wetting) phase,until reaching a constant plateau in the UV signal at a wavelengthof 295 nm (equilibration time ∼ 20 min). Subsequently, the PNT-rich(wetting) phase was fed to the column by opening the 20 mL in-jection loop, constraining the feed volume to 16 mL to avoid loss

4.3 experimental 61

of accuracy due to backmixing effects in the loop. The displace-ment of the non-wetting phase by the wetting phase, starting froma column completely equilibrated with the non-wetting phase, iscalled primary imbibition (PI). At the maximum displacement, i.e.when the eluate reaches the composition of the PNT-rich phase,the saturation of the non-wetting phase within the column is re-duced to the residual saturation, where the non-wetting phase be-comes hydraulically disconnected and remains trapped in the co-lumn. Upon closing the injection loop, the wetting (PNT-rich) phasewas re-displaced by the non-wetting (PNT-lean) phase. This seconddisplacement is called drainage, and as it starts from a bed inclu-ding trapped non-wetting phase, it is a secondary drainage cycle(SD). Maximum drainage is achieved when the volume fraction ofthe wetting phase in the column reaches the irreducible saturation.Drainage was carried out for 26 min (injection volume 31.2 mL), andwas followed by another secondary imbibition (SI) and drainage(SD) cycle (starting from the irreducible or residual saturation, re-spectively), which were performed in the same manner as the pre-vious two cycles.During the entire procedure, the UV signal at a wavelength of 295

nm, as well as the pressure before and after the column (to deter-mine the overall pressure drop ∆p), was recorded (see section 4.3.6).To determine the elution (flow) profiles of PNT, a second HPLCpump (515 HPLC pump, Waters) was attached after the UV de-tector to automatically dilute the eluate with methanol at a ratioeluate : diluent 1.2 : 10 (v:v) during imbibition and 1.2 : 4.5 (v:v)during drainage. The diluted eluate was sampled in fractions witha fraction collector (FC 203B, Gilson) and analyzed by HPLC (sam-ples collected during imbibition were further diluted manually at aratio sample : methanol 1 : 50 (v:v) before analysis).For the remainder of this section, the displaced phase will be indica-ted as phase 2 (corresponding to the wetting phase during drainageand to the non-wetting phase during imbibition), and the injectedphase will be indicated as phase 1 (corresponding to the non-wettingphase during drainage and to the wetting phase during imbibition).The breakthrough of the injected phase at the column outlet com-monly occurs as a shock-wave transition (see Ref. [15] and Appen-dix A.2), especially if the behavior of fractional flows fj (j = 1, 2)


with respect to phase saturations Sj can be described by S-shapedrelationships, as it is the case for most experimental systems. Fromthe wave parts of the experimentally determined flow and pressureprofiles, one can derive relationships for the relative permeabilitieskr,1 and kr,2 of both phases 1 and 2, respectively, as a function ofthe phase saturation S1 (S2 = 1− S1). The procedure is explainedin the following.We start from the component material balance presented in Equa-tion 3.1. With all (convective and adsorbed) phases being incom-pressible and in thermodynamic equilibrium, and thus ni =const.and ε =const., the component material balances, considering ourspecific case of two convective phases, simplify to:

ε∂

∂τ

2∑j=1

wijρmixj Sj

+∂

∂ξ

2∑j=1

wijρmixj fj

= 0; i = 1...NC (4.21)

Furthermore, as the two convective phases are in thermodynamicequilibrium, their compositions and thus wij and ρmix

j remain con-stant. The component material balance can therefore be further sim-plified:

2∑j=1

wijρmixj

(ε∂Sj

∂τ+∂fj

∂ξ

)= 0; i = 1...NC (4.22)

With S2 = 1− S1 and f2 = 1− f1, the equation can be transformedfurther to(

wi1ρmix1 −wi2ρ

mix2

)(ε∂S1∂τ

+∂f1∂ξ

)= 0; i = 1...NC (4.23)

We thus end up with the Buckley-Leverett equation,15 which, forthe sake of the further derivation and of the use for the evaluationof experimental data, is given in dimensional form:

ε∂S1∂t

+ u∂f1∂z

= 0. (4.24)

Note that the same result can be obtained with a very similar de-rivation, if multiple (more than two) convective phases in thermo-dynamic equilibrium are present. Obtaining the Buckley-Leverett

4.3 experimental 63

equation, which represents a “phase balance”, from the componentmaterial balances under the discussed conditions can physically beexplained by the fact that, since all phases maintain a constant com-position and are incompressible, the overall amount (mass and vo-lume) of each phase remains constant throughout the experiment.For the displacement of phase 2 by phase 1, the following initialand feed conditions apply:

S1(t, 0) = 1, thus f1(t, 0) = 1 for t > 0 (4.25a)

S1(0, z) = S01 for 0 6 z 6 Lc, (4.25b)

where S01 = 0 for primary imbibition/drainage, S01 = Si for secon-dary imbibition and S01 = Sr for secondary drainage. The solutionof the Buckley-Leverett equation by the method of characteristicsfor specific initial and feed conditions has been multiply discussedin the literature,4,15,68,69 but for the sake of completeness and under-standing is illustrated in Appendix A.2. The characteristic propaga-tion velocity for a state S1 in non-dimensionless form is given as:

(dzdt

)∣∣∣∣S1

=u

ε

df1dS1

=u

εf ′1(S1) (4.26)

With u =const. and ε =const., and f ′1 being independent of spaceand time, the propagation velocity is constant for a specific state S1,such that

z =u

εf ′1(S1)t (4.27)

Derivation of equation 4.27 with respect to S1 yields

dz =ut

εf ′′1 (S1)dS1 (4.28)

Furthermore, rearrangement of equation 4.27 and derivation withrespect to t yields:

∂S1∂t

= −εz

ut2f ′′1 (S1)= −

f ′1(S1)

tf ′′1 (S1)(4.29)


In a next step, we define the average saturation over the column:

Sav,1(t) =1

Lc

Lc∫0

S1(t, z)dz. (4.30)

Using equations 4.25 and 4.28 in equation 4.30 yields the followingexpression:68

Sav,1(t) =ut

εLc

S1(t,Lc)∫S1(t,0)

S1f′′1 (S1)dS1 =

=ut

εLc

[S1f ′1(S1)]S1(t,Lc)S1(t,0)

−

S1(t,Lc)∫S1(t,0)

f ′1(S1)dS1

=

= S1(t,Lc) +ut

εLc(1− f1(t,Lc)) (4.31)

Thus, the saturation at the column outlet S1(t,Lc) can be determi-ned from the average saturation Sav,1(t). In turn, the average satura-tion can be obtained from the fractional flow profile at the columnoutlet f1(t,Lc) by integrating equation 4.24 over space and time:

1

Lc

Lc∫0

(S1(t, z) − S1(0, z))dz−u

εLc

t∫t0

(f1(t, 0) − f1(t,Lc))dt = 0,

(4.32)

Using equations 4.25 and 4.30 in equation 4.32 yields:

Sav,1(t) = Sav,1(t = t0) +

1

tvoid

t∫t0

(1− f1(t,Lc))dt =

= S01 +1

tvoid

t∫t0

(1− f1(t,Lc))dt, (4.33)

Knowing the phase compositions of both phases, as well as the con-centration of PNT FP(t,Lc) in the eluate (measured concentrations

4.3 experimental 65

of the eluate fractions), the fractional flow f1(t,Lc) can be calcula-ted based on equation 3.2. From equation 4.33 one can obtain theaverage saturation over the column Sav,1(t) at any time t, based onwhich one can in a next step determine S1(t,Lc), applying equation4.31.In order to determine relative permeabilities kr,1(t,Lc) correspon-ding to the saturations S1(t,Lc) thus obtained, we consider the ex-tended Darcy’s law (equation 3.15), and consider its sum over allconvective phases (equation 3.16), which provides a relationship be-tween pressure drop and superficial velocity:

u = −

NP∑n=1

kr,n

ηn

K∂p∂z

(4.34)

For the sake of simplicity, in the following equations we express thesum in equation 4.34 as Y:

Y(S1) =

NP∑n=1

kr,n

ηn; (4.35)

where Y is a function of S1 only, since kr,n of each phase n is afunction of S1, and the dynamic viscosity ηn of phase n is constantas the phase composition of the liquid phases, which are in thermo-dynamic equilibrium, do not change during the displacement.Equation 4.34 can be integrated to obtain the overall pressure drop∆p:

∆p(t) =u

K

Lc∫0

1

Y(S1)dz. (4.36)


Differentiation with respect to t and application of equations 4.27 to4.29, as well as of the Leibnitz rule and of partial integration withrespect to S1 yields:70

d∆p(t)dt

=u

K

Lc∫0

dY−1(S1)dS1

∂S1∂t

dz =

=u2

εK

S1(t,Lc)∫S1(t,0)

Y ′(S1)

Y2(S1)f ′1(S1)dS1 =

= −uLc

Kt

1

Y(t,Lc)+u

Kt

Lc∫0

1

Y(S1)dz =

=∆p(t)

t−uLc

Kt

1

Y(t,Lc)

(4.37)

To describe the behavior of the pressure drop mathematically, a ra-

tional function ∆p(t) = h1t3+h2t

2+h3t+h4t+q was fitted to the expe-

rimental pressure data. Equation 4.37 can then be used to obtainY(t,Lc). Finally, the relative permeabilities (at a saturation S1(t,Lc))can be determined from equation 3.17 as:

kr,1(t,Lc) = f1η1Y(t,Lc) (4.38)

Finally, a kr,i − Si relationship can be established between the va-lues for S1(t,Lc), derived from experimental fractional flow pro-files through material balance considerations, and the values forkr,1(t,Lc), determined from measured pressure profiles by implyingthe extended Darcy’s law.Note that the presented approach can only be applied for the wavepart of the elution profile, since it is only in that part of the profilethat the states S1 travel at their characteristic propagation velocitydefined in equation 4.26. This propagation velocity is used in equa-tions 4.31 and 4.36 to 4.38, and thus constrains the validity of thecorresponding equations to wave transitions. As a consequence, re-lative permeabilities can only be determined for saturations S1(t,Lc)

reached during the wave part of the breakthrough profile.

4.4 volume additivity 67

4.4 volume additivity

The validity of the assumption of volume additivity in the liquidphase, made in the chromatographic model (see equation 3.3), wasassessed by measuring the density (or specific volume, vmix =

(ρmix)−1) of binary mixtures of methanol-water, PNT-methanol, TBP-methanol and TBP-PNT. The results are presented in Figure 4.3 (a)to (d), respectively. Experimental data for the first system was takenfrom the literature,71 while data for the other three binary systemswas determined experimentally. The systems methanol-water andPNT-methanol are completely miscible and thus allow an investi-gation over the entire range of compositions from one pure com-ponent to the other. In turn, TBP (solid) exhibits a solubility limitin methanol and PNT, and density measurements were performedthroughout the entire miscible range. The binary systems TBP-waterand PNT-water were not investigated due to the very low solubilityof the solutes in water (below 0.1% wt. for both TBP and PNT).The experimental data was compared to the linear behavior ex-pected from the assumption of volume additivity (compare equa-tion 3.3). In the systems methanol-water and PNT-methanol, bothcomponent densities are known from the literature or can be mea-sured directly (densities of methanol ρM = 788.07 g/L, water ρW =

997.54 g/L and PNT ρP = 961.63 g/L at 23C); hence equation 3.3

can be used in a predictive manner. With TBP being a solid at ourexperimental conditions, the apparent component density in solu-tion is unknown, and can be determined through a linear fit of themodel to the experimental data of the systems TBP-methanol andTBP-PNT (with the intercept of the vertical axis fixed by the specificvolume of methanol and PNT, respectively). For the two systems,an apparent specific volume of TBP of 1.014 L/kg in methanol andof 1.0165 L/kg in PNT was determined. Therefore in this work, anaverage specific volume of TBP of 1.0152 L/kg in solution, corre-sponding to ρT = 985.02 g/L, has been used.The good agreement of the experimental data to the linear model forthe systems PNT-methanol, TBP-methanol and TBP-PNT, as well asthe consistency in apparent specific volumes of TBP for the last twosystems, supports the assumption of volume additivity. A notabledisagreement with respect to the linear behavior can be observed


for the system methanol-water. However, moderate relative errorsbelow 4% are accepted for the sake of the model simplicity. Admit-tedly, the investigation, being constrained to binary systems, cannotclarify beyond doubt the mixing behavior of the multi-componentsystem. Nevertheless, it provides a strong indication that potentialerrors introduced by the assumption of volume additivity are minor.In this context, it is noteworthy that binary mixing data has beenused previously to estimate excess volumes of multi-componentmixtures, while neglecting multi-component interactions.72,73 Forour system, such method would clearly achieve negligible excessvolumes, given the binary mixing data provided in Figure 4.3. Wewould further like to mention that an estimation of excess volumesbased on the UNIQUAC model established below is not possible,since a pressure dependence is not considered in this model, as it iscommonly the case for liquid systems.41,72

Finally, we would like to comment on the assumption of volumeadditivity in the adsorbed phase, which was introduced by equa-tion 3.5 in the chromatographic model. This assumption is in linewith the concepts applied in the adsorbed solution theory (comparesection 4.2.1) if the excess area of mixing αm equals zero, i.e. inthe case of an ideal adsorbed solution, or of a real adsorbed solu-tion with a negligible dependence of γa on the spreading pressureπ. We are indeed going to consider a real adsorbed solution with aspreading pressure dependence later in this work, which is in con-tradiction with the assumption of volume additivity. However, weargue that the error introduced by the assumption of volume addi-tivity when estimating the adsorbed phase volume (equation 3.5) isminor, namely not large enough to justify the considerable increasein complexity of the model when accounting for an excess volumeof mixing. Introduction of a nonideal mixing behavior in the conser-vation laws would involve a nonconstant superficial velocity u dueto the volumetric changes, in order to fulfill the overall material ba-lance. In the case of a varying velocity u, an analytical solution of theequilibrium theory model is still possible,10,18,74 though more com-plex. The effects on u and ε are extremely challenging to quantifyexperimentally, and we consider them to be negligible compared tothe impact of adsorption and fluiddynamic effects on the elutionbehavior. In other words, the theoretical improvements in accuracy

4.5 thermodynamic equilibrium of the quaternary system 69

achieved by relaxing the volume additivity assumption in the adsor-bed phase would likely be cancelled by the intrinsic uncertainty inquantifying such non-ideal effect.

0 0.2 0.4 0.6 0.8 1

mass fraction wM

[-]

1

1.05

1.1

1.15

1.2

1.25

1.3

vm

ix [

L k

g-1

]

exp [22]

model

errmax

=3.7%

(a)

0 0.2 0.4 0.6 0.8 1

mass fraction wP [-]

1

1.05

1.1

1.15

1.2

1.25

1.3

vm

ix [

L k

g-1

]

exp [this work]

model

(b)

0 0.2 0.4 0.6 0.8

mass fraction wT [-]

1.05

1.1

1.15

1.2

1.25

1.3

vm

ix [

L k

g-1

]

exp [this work]

model

(c)

vmix= 1.27 - 0.256 wT

0 0.05 0.1 0.15 0.2

mass fraction wT [-]

1.034

1.035

1.036

1.037

1.038

1.039

1.04

vm

ix [L k

g-1

]

exp [this work]

model

(d)

vmix= 1.04 - 0.0235 wT

Figure 4.3: Densities of mixtures of the binary systems (a) methanol-water,(b) PNT-methanol, (c) TBP-methanol, (d) TBP-PNT. Experimen-tal data was taken from the literature for the system water-methanol,71 and was determined for the systems PNT-methanol,TBP-methanol and TBP-PNT in this study. Model predictions arebased on the assumption of volume additivity.

4.5 thermodynamic equilibrium of the quaternary sy-stem

In order to quantify thermodynamic nonidealities in the liquid phase,i.e. to calculate liquid phase activities for quaternary mixtures, athermodynamic model for the liquid phase has to be established.For this purpose, liquid-liquid (LL), solid-liquid (SL), and solid-


liquid-liquid (SLL) phase equilibrium experiments were carried outin the ternary systems resulting when one of the four components isabsent. The choice of experimental conditions to investigate was ba-sed on simulations using the predictive modified UNIFAC model,48,75

with the procedure of phase equilibrium calculations described inAppendix A.3.1. In total, 13 LL, 2 SLL, and 7 SL conditions wereinvestigated experimentally, most of which were replicated at leastonce. The compositions of the phases in equilibrium were measuredand are presented in Figure 4.4 as black circles, connected to eachother by black lines representing tie-lines. Due to the very goodreproducibility, replicates overlap and are hence barely distinguis-hable in Figure 4.4. The original (immiscible) compositions of all ex-periments are provided in Tables A.3 and A.4, while the measuredcompositions of the phases in equilibrium are provided in TablesA.5 to A.8 in the appendix.The experimental data indicates a very low miscibility of PNT withwater (solubility of PNT in water < 0.1% wt.), while PNT is comple-tely miscible with methanol. Accordingly, the ternary system PNT- methanol - water (Figure 4.4a) exhibits a big miscibility gap (LLregion) between PNT and water, which decreases as the content ofmethanol increases.The phase behavior of the ternary system TBP - methanol - wa-

ter (Figure 4.4b) is more complex, with TBP being a solid at theexperimental temperature and 1 bar. Similar to PNT, the solubilityof TBP in water is very low (< 0.1% wt.), whereas it is very highin methanol (≈ 75% wt.). Mixtures beyond these concentrations ofTBP form a solid-liquid equilibrium, with the solid being pure TBPof one crystal structure, determined from XRD spectra of the solidphases being in good agreement with the XRD spectrum of pureTBP. A selection of XRD spectra is provided in Appendix A.3.3, seeFigure A.5. However, the ternary system TBP - methanol - waterdoes not only feature SL phase equilibria. In a certain range of com-positions, the liquid phase is unstable, resulting in a region of LLphase equilibrium, and, beyond the solubility limit of TBP, a regionof SLL phase equilibrium.The solubility of TBP in PNT is rather high (≈ 25% wt.), and withboth TBP and PNT being well soluble/miscible in methanol, theternary system PNT - TBP - methanol features a large miscible re-


20

40

60

80

20 40 60 80

20

40

60

80

MeOH(l)

PNT(l)

H2O

(l)

(a)

20

40

60

80

20 40 60 80

20

40

60

80

H2O

(l)

MeOH(l)

TBP(s)

(b) Solid - Liquid

Liquid - Liquid

Solid - Liquid - Liquid

20

40

60

80

20 40 60 80

20

40

60

80

MeOH(l)

PNT(l)

TBP(s)

(c)

20

40

60

80

20 40 60 80

20

40

60

80

(d) H2O

(l)

PNT(l)

TBP(s)

Figure 4.4: Ternary diagrams: (a) PNT - methanol - water, (b) methanol - wa-ter - TBP, (c) PNT - methanol - TBP, (d) PNT - water - TBP. Blackcircles show the data points in thermodynamic equilibrium, con-nected by a black line. Thin colored lines are tie-lines, simulatedby the UNIQUAC model with parameters fitted to the experi-mental data. Thicker colored lines indicate phase boundaries.Green: Solid - Liquid equilibrium, Blue: Liquid - Liquid equili-brium, Red: Solid - Liquid - Liquid equilibrium


20

40

60

80

20 40 60 80

20

40

60

80

comp 2

comp 1comp 3

SL

LL

SLL

SL

Figure 4.5: Illustration of the qualitative behavior of the system PNT (com-ponent 1) - water (component 2) - TBP (component 3); the corre-sponding quantitative description is provided in Figure 4.4d.

Table 4.1: Combinatorial parameters of PNT (P), TBP (T), methanol (M) andwater (W) in the UNIQUAC model.

Comp. i ri qi q ′i

P 4.8411 3.7480 3.7480

T 6.3084 4.9440 4.9440

M 1.4311 1.4320 0.9600

W 0.9200 1.4000 1.0000

gion (Figure 4.4c), adjacent to a SL region. Since both TBP and PNTare hardly soluble/miscible in water, the system PNT - TBP - water(Figure 4.4d) exhibits very small miscible regions, and most compo-sitions result in a LL or SLL phase split. Since single-phase and SLregions are hardly recognizable in Figure 4.4d, for the sake of claritythe qualitative phase behavior of this ternary system is representednot in scale in Figure 4.5.The modified UNIFAC model provides a good qualitative descrip-

tion of the experimental behavior, which considerably simplifies thechoice of experimental conditions, though not achieving the quanti-tative accuracy required for its application to the remainder of thiswork. In order to achieve a quantitative description, a UNIQUACmodel45,46 was used. Combinatorial parameters of the UNIQUAC


model, namely the parameters qi, q ′i and ri, are summarized in ta-ble 4.1; note that subscripts P, T, M and W indicate the componentsPNT, TBP, methanol and water throughout this work. These para-meters were taken from the literature for water and methanol.45 ForPNT and TBP, the parameters qi and q ′i are identical, and qi andri were estimated from group volume and area parameters Rk andQk, as suggested for the UNIFAC model:47

ri =∑k

νikRk; qi =∑k

νikQk (4.39)

where νik is the number of chemical groups of type k in componenti (PNT or TBP). Group parameters Qk and Rk were taken from Fre-denslund et al.,47 apart from the parameters for group “AC”, whichwere taken from Larsen et al.48 The 12 interaction parameters (2 foreach pair of components in the quaternary system) were fitted tothe experimental data and are provided in Table 4.2 (column “Pa-rameters (quaternary)”). Previously, in the context of investigatingthe ternary system PNT - methanol - water in the absence of TBP,a UNIQUAC model for this ternary system had been fitted to theexperimental data (9 LL conditions) of this system only. The fittedUNIQUAC parameters for the ternary system are also provided inTable 4.2 (column “Parameters (ternary)”). Unless stated explicitly,references to the UNIQUAC model correspond to the parametersfitted for the entire quaternary system. A detailed description ofthe fitting procedure is provided in Appendix A.3.2.Phase equilibria in the ternary systems were calculated based on

the established UNIQUAC model (procedure of phase equilibriumcalculations see Appendix A.3.1), and are shown in Figure 4.4. Ex-perimental data and calculated results are in very good agreement;major discrepancies can only be observed in the water-rich corner,particularly concerning the solubility of TBP in water, which is con-siderably underestimated (compare experimental data and modelcalculations of conditions 1, 15 and 22 in Tables A.5 to A.8). Thesedifferences are not surprising, since experimental data located inthis corner was deliberately excluded from the parameter fit, in or-der to obtain a better description of the regions relevant for the ap-plication to chromatography (see Appendix A.3.2). As a water-richliquid phase is incompatible with the studied adsorbent, relevant


Table 4.2: Fitted UNIQUAC parameters

Components Parameters (quaternary) Parameters (ternary)

i j ∆uij/R [K] ∆uji/R [K] ∆uij/R [K] ∆uji/R [K]

P T 271.76 -166.28 - -

P M 744.76 -80.43 1096.76 -76.94

P W 1152.65 277.85 1164.51 270.68

T M 317.70 -298.69 - -

T W -1.88 2242.82 - -

M W 0.19 -276.00 -83.02 200.99

regions are located far from the water-rich corner.The established UNIQUAC model was then used to calculate phase

PNT4040

20

80

20

40

60

80

20

6080

60 20

40

60

80

MeOH

H2O

60

40

20

80 60 40 20

TBP

80

(b)

Figure 4.6: (a) Simulation of phase equilibria of the quaternary system PNT- TBP - methanol - water, based on the established UNIQUACmodel. (b) Experimental data in the quaternary system (black),and corresponding simulations with the established UNIQUACmodel (color code see Figure 4.4)

equilibria in the quaternary system, which are illustrated in Figure4.6a. Note that the ternary systems, which were investigated forthe parameter fit, form the sides of the prism representing the qua-ternary system in Figure 4.6a. To check the accuracy of the modelpredictions in the quaternary system, another four experiments (3

4.6 single component adsorption behavior 75

LL and 1 SLL) within the quaternary system were performed andcompared to the calculated phase equilibria, as illustrated in Figure4.6a. The original (immiscible) quaternary compositions of the expe-riments are included in Table A.4, while measured and calculatedcompositions of phases in equilibrium are provided in Table A.9in the Appendix. Comparison of the experimental and the calcula-ted data demonstrates a good accuracy also within the quaternarysystem, especially when considering that the application of the UNI-QUAC model within the quaternary system corresponds to an ex-trapolation from the phase behavior in the ternary systems.

4.6 single component adsorption behavior

The single component adsorption behavior of PNT and TBP wasinvestigated by Frontal Analysis, as described in section 4.3.4. Theinvestigated feed compositions are illustrated in the relevant ternarydiagrams as red dots in Figure 4.7. Frontal Analysis was performedat different eluent compositions (weight fraction of methanol r ran-ging from 0.4 to 0.7 for PNT and from 0.4 to 0.6 for TBP). Five toten different feed compositions were investigated per eluent com-position, spanning the entire miscible range; all conditions with thesame eluent composition are located on the same straight line in Fi-gure 4.7, connecting the pure eluent axis with the pure solute corner.Since the UNIQUAC prediction of phase equilibria is less accuratetowards the plait point of the LL region in the system TBP - met-hanol - water, a few miscible conditions are located inside the areapredicted to exhibit a LL phase split (compare Figure 4.7b). It isworth pointing out that in these experiments, operating conditionswere constrained to the single-phase region in order to prevent atwo-phase flow which at this stage could not be evaluated.Adsorbed phase concentrations determined from the elution pro-

files through equation 4.19 are plotted over the liquid phase con-centrations in the feed in Figures 4.8a and 4.9a for PNT and TBP,respectively. Data series of each component PNT or TBP at diffe-rent eluent compositions indicate a qualitatively similar behavior(similar isotherm shape), but cover very different ranges of liquidphase concentrations. However, plotting adsorbed phase concentra-tions over liquid phase activities, calculated from the liquid phase


20

40

60

80

20 40 60 80

20

40

60

80

MeOH(l)

PNT(l)

H2O

(l)

(a)

20

40

60

80

20 40 60 80

20

40

60

80

H2O

(l)

MeOH(l)

TBP(s)

(b)

Figure 4.7: Feed compositions of single component breakthrough experi-ments plotted in the relevant ternary diagrams (red dots). Blacklines indicate lines of constant ratio methanol : water. (a) PNT inmethanol - water , (b) TBP in methanol - water

concentrations using the UNIQUAC model established in section4.5, data series at different eluent compositions fall onto one curve(within a certain error margin). The observed scattering of the me-asured data is primarily attributed to the (slight) inaccuracy of theUNIQUAC model, which provides a good, but not a perfect descrip-tion of the nonidealities in the liquid phase.Since data series fall together when plotted over the liquid phase

activity of the corresponding solute, it is possible to describe thesingle component adsorption behavior of the solute (TBP or PNT)by an isotherm being a function of the solute activity in the liquidphase. This isotherm is then valid for a very broad range of eluentcompositions and of solute concentrations (reaching until the solu-bility limit). Single component BET isotherms (see equation 4.15)with an identical saturation capacity qsat for both solutes PNT andTBP were fitted simultaneously to the TBP and the PNT data, mini-mizing the objective function

φiso =1

NP

NP∑i=1

(nmodP −n

expP )2 +

1

NT

NT∑i=1

(nmodT −n

expT )2, (4.40)

where NP and NT are the number of data points of PNT and TBP,respectively. The estimated values of the parameters are reportedin Table 4.3, together with standard deviations (uncertainty estima-tion see Appendix A.1). The resulting isotherms are plotted in Figu-

4.6 single component adsorption behavior 77

0 20 40 60 80

cP [gL-1]

0

50

100

150

200

nP [g

L-1

]

(a)

0 0.2 0.4 0.6 0.8 1

aP [-]

0

50

100

150

200

nP [g

L-1

]

r=0.40r=0.50r=0.57r=0.60r=0.70

(b)

Figure 4.8: Single component adsorption data and isotherm of PNT. (a) Ad-sorption data over liquid phase concentrations of PNT. (b) Ad-sorbed phase concentrations over liquid phase activities of PNT,which are calculated from the liquid phase concentrations usingthe established UNIQUAC model. Black line: Fitted BET isot-herm, with parameters provided in Table 4.3.

0 100 200 300 400

cT [gL-1]

0

50

100

150

nT [g

L-1

]

(a)

0 0.05 0.1 0.15

aT [-]

0

50

100

150

nT [g

L-1

]

r=0.40r=0.43r=0.46r=0.50r=0.57r=0.60

(b)

Figure 4.9: As in Figure 4.8, but for the single-component adsorption beha-vior of TBP.


Table 4.3: Parameters of single component BET isotherms for PNT and TBP.Values in brackets denote the standard deviations, estimated bythe procedure explained in Appendix A.1

PNT TBP

qsat [mol L−1] 0.333 (0.020)

bS [-] 1.141 (0.149) 88.71 (24.20)

bL [-] 0.922 (0.013) 4.159 (0.166)

res 4.8b and 4.9b as solid lines. Previously, considering the ternarysystem PNT - methanol - water (in the absence of TBP), an anti-Langmuir isotherm for PNT of the form

ni =qsat

P KPaP

1−KPaP(4.41)

had been fitted to the experimental data of PNT. In this context, fit-ted parameter values were qsat

P = 0.394 mol/L and KP = 0.862, andliquid phase activities of PNT had been determined by the UNI-QUAC model describing the ternary system only (in the absence ofTBP, see Table 4.2). Note that the anti-Langmuir isotherm formallycorresponds to a BET isotherm (equation 4.15) with KP = bS = bL.

4.7 binary adsorption behavior

Having established a description for the single-component adsorp-tion behavior, let us focus on the binary adsorption behavior. Thismeans that, apart from the interaction of each solute with the ad-sorbent, also the competition and cooperation of different solutesfor adsorption sites, as well as the mixing behavior in the adsorbedphase (ideal or non-ideal), have to be accounted for. To investigatethe binary adsorption behavior of our system, binary breakthroughexperiments (binary Frontal Analysis) were performed as describedin section 4.3.5. In total, 19 different feed conditions with three dif-ferent eluent compositions (r =0.5742, 0.50 and 0.43) were investiga-ted. Again, conditions were constrained in such a way as to assurea single-phase flow, since hydrodynamic effects of a two-phase flow

4.7 binary adsorption behavior 79

were assessed only afterwards. The feed conditions are illustratedas red filled circles in the three ternary diagrams in Figure 4.10,which in their corners show pure PNT, pure TBP, and pure eluentof the relevant composition r. The exact feed compositions are re-ported in Table 4.4. Note that experiments BT1 to BT13 (r = 0.5742)had been presented previously,25 and will be re-analyzed here. Toextend the investigated range of conditions, one additional experi-ment at r = 0.5742 at higher concentrations (close to the solubilitylimit) was performed, as well as three experiments at r = 0.50 andtwo at r = 0.43. Experimental profiles for all conditions are presen-ted in Figures 4.13 to 4.17; the color of the experiment label indicatesthe eluent composition (r = 0.5742 in ocher, r = 0.50 in green andr = 0.43 in violet). Replicates were performed for a majority of theexperimental conditions, and demonstrate a very good reproducibi-lity.For an accurate description of the experimental profiles hence of thebinary adsorption behavior, both the IAST and the RAST have beenconsidered and implemented in the chromatographic model (lum-ped kinetic model, see equations and numeric solution in section3.2.2). Note that in this context, the chromatographic model is app-lied in the presence of one convective phase only, hence f1 = S1 = 1

and Fi = Ci = ci. Furthermore, in the presence of one fluid phaseonly, the UNIQUAC equation, providing liquid phase activities, canbe solved explicitly, such that the implementation of the thermodyn-amic model is straightforward, whereas implementation of the IASTand RAST is discussed in the following.The application of the IAST does not require any further parameterbesides the single-component isotherm parameters estimated above;in the case of two BET isotherms with identical saturation capacitiesqsati IAST provides the binary BET isotherm of equation 4.16. This

binary isotherm can be implemented in the chromatographic modelto simulate elution profiles at the relevant feed conditions. On thecontrary, application of the RAST requires a thermodynamic mo-del describing non-idealities in the adsorbed phase, thus providingactivity coefficients γa at different compositions za and spreading


Table 4.4: Feed compositions of binary breakthrough experiments

Exp. # r (eluent) PNT (% wt.) TBP (% wt.)

BT1 0.5742 0.61 2.06

BT2 0.5742 1.09 4.05

BT3 0.5742 1.52 5.22

BT4 0.5742 1.87 6.91

BT5 0.5742 2.06 7.63

BT6 0.5742 2.39 8.96

BT7 0.5742 0.34 0.11

BT8 0.5742 1.12 0.38

BT9 0.5742 1.58 0.53

BT10 0.5742 2.61 0.87

BT11 0.5742 1.12 1.12

BT12 0.5742 1.97 1.97

BT13 0.5742 2.56 2.56

BT14 0.5742 4.96 17.7

BT15 0.5000 1.35 0.75

BT16 0.5000 1.40 4.21

BT17 0.5000 1.45 7.97

BT18 0.4300 0.51 1.51

BT19 0.4300 0.45 2.32


20

40

60

80

20 40 60 80

20

40

60

80

PNTTBP

(a) solvent (r=0.57)

60

8020

40

PNTTBP

solvent (r=0.57)

20

40

60

80

20 40 60 80

20

40

60

80

PNTTBP

(b) solvent (r=0.50)

60

8020

40

PNTTBP

solvent (r=0.50)

20

40

60

80

20 40 60 80

20

40

60

80

PNTTBP

(c) solvent (r=0.43)

60

8020

40

PNTTBP

solvent (r=0.43)

Figure 4.10: Feed compositions of the binary breakthrough experiments(red filled circles), plotted in the ternary diagrams PNT - sol-vent - TBP; entire diagrams (left) and zoom to the experimen-tally relevant region (right). Weight fraction of methanol in theeluent: (a) r = 0.5742, (b) r = 0.50, (c) r = 0.43. Dots indi-cate compositions in LL (blue), SL (green) and SLL (red) equi-librium.

pressures Π. A two-parameter Redlich-Kister equation41 is assumed,with a sigmoidal spreading pressure dependence:

lnγaT = (aT(z

aP)2 + bT(z

aP)3)s(Π) (4.42a)

lnγaP = (aP(z

aT)2 + bP(z

aT)3)s(Π) (4.42b)


with

aT = A+ 3B; aP = A− 3B (4.43a)

bT = −4B; bP = 4B (4.43b)

s(Π) =exp

(Π−p1p2

)1+ exp

(Π−p1p2

) (4.43c)

The four parameters A, B, p1 and p2 were fitted by implementingthe RAST (based on the single-component BET isotherms) into thechromatographic model as described in section 4.2.2 and by mini-mizing the error between simulated and experimental profiles ofselected conditions (BT1, BT6, BT8, BT10, BT11, BT13, BT15, BT16,BT18, BT19), using a built-in Matlab genetic algorithm. The objectivefunction of this fitting can be written as:

φRAST =

Nexp∑j=1

Ndata,j∑i=1

1

Ndata,j

(∣∣∣∣∣cmodP,j,i − c

expP,j,i

cFP,j

∣∣∣∣∣+∣∣∣∣∣c

modT,j,i − c

expT,j,i

cFT,j

∣∣∣∣∣)

(4.44)

where Nexp is the number of binary Frontal Analysis experimentsused in the fit (10, only one replicate of each condition used), andNdata,j is the number of measured data points during the experi-ment j. The resulting parameter values are A = 0.42(0.14), B =

−0.63(0.32), p1 = 608.9(81.9) and p2 = 95.6(38.3); standard deviati-ons in brackets were estimated by the procedure explained in Ap-pendix A.1. The Redlich-Kister function (equations 4.42 with s(Π) =1) and spreading pressure dependence (equation 4.43c) are illus-trated in Figure 4.11. Note that the sigmoidal spreading pressuredependence is not identical to 0 at Π = 0, but falls below 0.002,such that condition 4.8 is fulfilled only approximately, though witha very good accuracy. The choice of a sigmoidal function instead ofthe commonly chosen exponential decay function59,60 was crucialfor an accurate description of the experimental data.The impact of interactions in the liquid phase and in the adsor-

bed phase on the simulated elution profiles shall be outlined ba-sed on four different types of simulations, which were performedfor exemplary feed conditions (BT1, BT6 and BT16) and are compa-red to experimental profiles in Figure 4.12. The first type of simula-tion (“S.C.”) accounts for the single component adsorption of only


0 0.2 0.4 0.6 0.8 1

zaT [-]

0.5

1

1.5

2

2.5

3γa i[-]

(a)

γaP

γaT

0 500 1000 1500

Π [mol·m−3]

0

0.2

0.4

0.6

0.8

1

s(Π)[-]

(b)

Figure 4.11: Functions for the description of non-idealities in the adsorbedphase. (a) Redlich Kister function (equation 4.42) with para-meters A = 0.42(±0.14), B = −0.63(±0.32) and s(Π) = 1; (b)Spreading pressure dependence function (equation 4.43c) withparameters p1 = 608.9(±81.9), p2 = 95.6(±38.3)

one component (PNT or TBP); separate simulations for each compo-nent were performed with the other component being absent. Thesecond type of simulation (“Act. LP”) is still based on the single-component BET isotherms, thus neglecting interaction in the adsor-bed phase, but it accounts for interaction in the liquid phase dueto the impact of the different components on the liquid phase acti-vities. Finally, simulations were performed based on the IAST andthe RAST, accounting for competition/ cooperation for adsorptionsites and assuming an ideal or real mixing behavior in the adsorbedphase, respectively. It is worth noting that the choice of the threefeed conditions (BT1, BT6 and BT16) is representative of the entireset of experiments, where the same qualitative observations as dis-cussed below for the selected conditions can be made.From the first row in Figure 4.12, a clear deviation of the experi-

mental profiles from the single component adsorption behavior canbe noted. Adsorption fronts of PNT are considerably steeper, indi-cating a reduction in the adsorbed amount of PNT in the presenceof TBP. Profiles of TBP indicate an intermediate state during adsorp-tion which is higher than the feed concentration at rather low feedconcentrations, but lower than the feed concentration at rather highfeed concentrations. Hence, the adsorbed amount of TBP is reduced


S.C.

0 2 4 6 8 10

t [min]

0

5

10

15

20

ci [

gL

-1]

exp1

exp2

sim

PNT

TBPBT1

0 2 4 6 8 10

t [min]

0

20

40

60

80

100

ci [

gL

-1]

exp1

exp2

sim

PNT

TBP

BT6

0 5 10 15

t [min]

0

10

20

30

40

ci [

gL

-1]

exp1

exp2

sim

PNT

TBP

BT16

Act. LP

0 2 4 6 8 10

t [min]

0

5

10

15

20

ci [

gL

-1]

exp1

exp2

sim

PNT

TBPBT1

0 2 4 6 8 10

t [min]

0

20

40

60

80

100

ci [

gL

-1]

exp1

exp2

sim

PNT

TBP

BT6

0 5 10 15

t [min]

0

10

20

30

40

ci [

gL

-1]

exp1

exp2

sim

PNT

TBP

BT16

IAST

0 2 4 6 8 10

t [min]

0

5

10

15

20

ci [

gL

-1]

exp1

exp2

sim

PNT

TBPBT1

0 2 4 6 8 10

t [min]

0

20

40

60

80

100

ci [

gL

-1]

exp1

exp2

sim

PNT

TBP

BT6

0 5 10 15

t [min]

0

10

20

30

40

ci [

gL

-1]

exp1

exp2

sim

PNT

TBP

BT16

RAST

0 2 4 6 8 10

t [min]

0

5

10

15

20

ci [

gL

-1]

exp1

exp2

sim

PNT

TBPBT1

0 2 4 6 8 10

t [min]

0

20

40

60

80

100

ci [

gL

-1]

exp1

exp2

sim

PNT

TBP

BT6

0 5 10 15

t [min]

0

10

20

30

40

ci [

gL

-1]

exp1

exp2

sim

PNT

TBP

BT16

Figure 4.12: Comparison of simulated profiles with experimental data ba-sed on (i) “S. C.”: single component adsorption in absence ofthe other adsorbing component, (ii) “Act. LP”: Interaction inthe liquid phase only, through liquid phase activities. Single-component adsorption isotherms assumed. (iii) “IAST”: Idealadsorbed solution theory with a non-ideal liquid phase (basedon activities). (iv) “RAST”: Real adsorbed solution theory witha non-ideal liquid phase. Experiments BT1, BT6 and BT16 arechosen as illustrative examples.


in the presence of PNT at low feed concentrations but enhanced athigh feed concentrations.Accounting for the interaction in the liquid phase (“Act. LP”), thepresence of TBP (PNT) in the liquid phase results in a decrease inthe activity of PNT (TBP), which in turn has a negative effect on theadsorbed amount of both components. At low feed concentrations(such as in BT1), these effects result in a good quantitative descrip-tion of the experimental behavior. In turn, at higher feed concen-trations (BT6 and BT16), experimental profiles of PNT exhibit aneven steeper adsorption front (indicating an additional competitionin the adsorbed phase), while profiles of TBP exhibit an intermedi-ate state lower than the feed concentration, which is an indicationfor a cooperative interaction.Accounting additionally for competition/cooperation in the adsor-bed phase, but assuming an ideal mixing behavior (IAST) yields aconsiderable effect on the elution profiles only at high feed concen-trations (BT6 and BT16), while profiles at low feed concentrations(BT1) remain unaltered. The interaction described by IAST is clearlyof a cooperative nature, flattening the adsorption fronts of PNT andreducing the concentration level of TBP at the intermediate state. Inthe case of TBP, this determines an improvement of the descriptionof the experimental behavior, while for PNT the description deteri-orates.It is obvious that an interaction in the adsorbed phase based on theassumption of ideal mixing does not allow for a good quantitativedescription of the elution profiles, and that a non-ideal behavior inthe adsorbed phase has to be accounted for. This is the case whenapplying the RAST, which indeed achieves a good quantitative des-cription of the three selected profiles. One can conclude that at lowfeed concentrations, the shape of the elution profiles is mainly deter-mined by interactions in the liquid phase, affecting the liquid phaseactivities, while interaction in the adsorbed phase is low and has anegligible effect on the elution profiles. At high concentrations, inte-raction in the adsorbed phase becomes dominant, and a non-idealbehavior in the adsorbed phase has to be considered in order to des-cribe the experimental behavior.Simulations based on the RAST for all relevant feed conditions areoverlaid with the experimental profiles in Figures 4.13 to 4.17. A


very good agreement between simulated and experimental profi-les can be observed for the broad range of tested conditions, andeven close to the solubility limit as in BT14 and BT17. The qualityof the description deteriorates slightly when the fraction of metha-nol in the eluent decreases (see BT18 and BT19). At low methanolfractions, adsorption isotherms with respect to the liquid phase con-centrations are very steep (compare Figures 4.8 and 4.9). Hence, theimpact of an interaction in the adsorbed phase on the elution beha-vior is dominant, and minor inaccuracies in the description of suchinteraction have a strong effect on the shape of the elution profiles.Finally, a few profiles with high feed concentrations exhibit a stepin the TBP and PNT profiles before desorption, which evolves to apeak as feed concentrations approach the solubility limit. This phe-nomenon is particularly evident in the conditions BT16 and BT17,but is also indicated by BT14 and BT19. Similar peaks have beenobserved in single component adsorption profiles of TBP at highfeed concentrations (not shown here for the sake of brevity). It issuspected that this step is due to an interaction of methanol withthe adsorbent and hence with TBP and PNT in the adsorbed phase,which is neglected in our model. Even a minor interaction of met-hanol can slightly alter the composition of the eluent, which has alarge impact on the adsorption behavior of TBP and PNT, as ob-served in section 4.6. Accounting for an interaction of methanol isexpected to further enhance the quality of the model, however itconsiderably increases the complexity of the model and the numberof parameters to be determined, and it requires an accurate mea-surement of the (excess) adsorption of methanol. Batch adsorptionexperiments of methanol performed during this study indicated aslight adsorption of methanol, which however was too low to bequantified accurately.


0 2 4 6 8 10

t [min]

0

5

10

15

20

ci [

gL

-1]

exp1

exp2

sim

PNT

TBPBT1

0 2 4 6 8 10

t [min]

0

10

20

30

40

ci [

gL

-1]

exp1

exp2

sim

PNT

TBPBT2

0 2 4 6 8 10

t [min]

0

10

20

30

40

50

ci [

gL

-1]

exp1

exp2

sim

PNT

TBPBT3

0 2 4 6 8 10

t [min]

0

20

40

60

80

ci [

gL

-1]

exp1

exp2

sim

PNT

TBP

BT4

0 2 4 6 8 10

t [min]

0

20

40

60

80

ci [

gL

-1]

exp1

exp2

sim

PNT

TBP

BT5

0 2 4 6 8 10

t [min]

0

20

40

60

80

100

ci [

gL

-1]

exp1

exp2

sim

PNT

TBP

BT6

Figure 4.13: Binary breakthrough experiments BT1 to BT6 (weight fractionof eluent in the solvent r = 0.5742, ratio of TBP:PNT in the feed∼3:1). Experimental data points of PNT (red) and TBP (blue)are illustrated by filled triangles and circles (replicates), andare connected by a dashed line to guide the eye. Continuouslines correspond to the simulated profiles based on the realadsorbed solution theory (RAST) for PNT (red) and TBP (blue).Note that experimental results of experiments BT1 to BT13 havebeen reported previously.25,26


0 2 4 6 8 10

t [min]

0

0.5

1

1.5

2

2.5

3

3.5

ci [

gL

-1]

exp1

exp2

sim

PNT

TBP

BT7

0 2 4 6 8 10

t [min]

0

2

4

6

8

10

12

ci [

gL

-1]

exp1

exp2

sim

PNT

TBP

BT8

0 2 4 6 8 10

t [min]

0

5

10

15

20

ci [

gL

-1]

exp1

exp2

simPNT

TBP

BT9

0 2 4 6 8 10

t [min]

0

5

10

15

20

ci [

gL

-1]

exp1

exp2

simPNT

TBP

BT10

Figure 4.14: As in Figure 4.13, but showing binary breakthrough experi-ments BT7 to BT10 (weight fraction of methanol in the eluentr = 0.5742, ratio of TBP:PNT in the feed ∼1:3).


0 2 4 6 8 10

t [min]

0

2

4

6

8

10

12

ci [

gL

-1]

exp1

exp2

simPNTTBP

BT11

0 2 4 6 8 10

t [min]

0

5

10

15

20

ci [

gL

-1]

exp1

exp2

simPNTTBP

BT12

0 2 4 6 8 10

t [min]

0

5

10

15

20

25

ci [

gL

-1]

exp1

exp2

simPNTTBP

BT13

0 2 4 6 8 10

t [min]

0

50

100

150

ci [

gL

-1]

exp1

sim

PNT

TBP

BT14

Figure 4.15: As in Figure 4.13, but showing binary breakthrough experi-ments BT11 to BT14. The weight fraction of methanol in theeluent is r = 0.5742. Experiments BT11 to BT13 feature a ratioof TBP:PNT ∼1:1 in the feed, experiment BT14 was carried outat a feed composition located very close to the solubility limit,with a ratio of TBP:PNT ∼3:1.


0 5 10 15

t [min]

0

2

4

6

8

10

12

ci [

gL

-1]

exp1

exp2

sim

PNTTBP

BT15

0 5 10 15

t [min]

0

10

20

30

40

ci [

gL

-1]

exp1

exp2

sim

PNT

TBP

BT16

0 5 10 15

t [min]

0

20

40

60

80

100

ci [

gL

-1]

exp

sim

PNT

TBP

BT17

Figure 4.16: As in Figure 4.13, but showing binary breakthrough experi-ments BT15 to BT17. The weight fraction of methanol in theeluent is r = 0.50. The feed composition of BT17 is located veryclose to the solubility limit.

0 5 10 15 20

t [min]

0

5

10

15

ci [

gL

-1]

exp

sim

PNT

TBP

BT18

0 5 10 15 20

t [min]

0

5

10

15

20

ci [

gL

-1]

exp

sim

PNT

TBP

BT19

Figure 4.17: As in Figure 4.13, but showing binary breakthrough experi-ments BT18 and BT19. The weight fraction of methanol in theeluent is r = 0.43.

4.8 two-phase flow behavior 91

4.8 two-phase flow behavior

With reference to section 3.3.3, the hydrodynamic behavior of a sy-stem with multiple convective phases in a porous medium is ma-croscopically described by fractional flows fj, which are commonlya function of the phase viscosities ηj and the relative permeabilitieskr,j (compare equation 3.17). In this section, we derive relationshipsdescribing the phase viscosities ηj, as well as the relative permeabi-lities kr,j, of the ternary system PNT - methanol - water, in the ab-sence of TBP. These relationships, considered in equation 3.17, allowthe calculation of fractional flows fj, depending on phase composi-tions and saturations.

4.8.1 Phase viscosities

In the investigated system, the viscosity of the PNT-rich phase wasassumed to be equal to the viscosity of PNT at 23

C, ηP = 1.17mPas, which was interpolated linearly from the data reported for20C and 25

C.76 The viscosity of the PNT-lean phase was assumedto correspond to the viscosity of the water-methanol mixture at therelevant eluent composition at 23

C. Viscosities of water-methanolmixtures were estimated as described in the literature,77 with purecomponent relationships between viscosity and temperature fittedto data for water78 and for methanol79 (for the estimation at 296 Ksee Figure 4.18).

4.8.2 Relative permeabilities

Relative permeabilities for the system PNT - methanol - water withthe adsorbent Zorbax 300SB-C18 are investigated by displacing twoliquid phases in thermodynamic equilibrium (i.e. the two ends ofthe same tieline, located on the binodal curve). Since the two dis-placed phases have the same liquid phase activities ai of all com-ponents i, the adsorbed phase concentration nP, depending on aP,remains constant. As a consequence, no ad- or desorption, and nochanges in porosity ε occur throughout the displacement. Hence,all effects observed in the resulting elution profiles can be attribu-


0 0.2 0.4 0.6 0.8 1

zM

[mol:mol]

0.4

0.6

0.8

1

1.2

1.4

1.6

ηm

ix [

mP

a·s]

Figure 4.18: Viscosity estimation for the binary system methanol-water at296 K, according to the estimation procedure described in Ref.[77]. Pure component data at different temperatures taken fromRef. [78, 79].

ted to the hydrodynamic behavior of the two convective phases. Fordetails on the experimental procedure and the data evaluation seesection 4.3.7. The viscosities of the selected phases in equlibriumwere estimated as ηR = 1.17 mPas and ηL = 1.42 mPas for the PNT-rich and -lean phase, respectively (estimation procedure outlinedabove).The experimental pressure and flow profiles over the entire dura-

tion of the experiment (equilibration and two imbibition - drainagecycles) are presented in Figure 4.19. Furthermore, pressure profilesand the evolution of the average saturation of the PNT-rich phaseSR

av over time are overlaid for both cycles in Figure 4.20 (imbibition)and in Figure 4.21 (drainage).Pressure profiles during the secondary drainage cycles (SD(i) and

SD(ii)) are reproducible and slowly approach a stable pressure levelafter 15 to 20 min, indicating that the irreducible saturation Si ofthe wetting, PNT-rich phase is reached (Figure 4.21b). In contrast,pressure profiles during primary and secondary imbibition differconsiderably, suggesting a considerable hysteresis between the dis-placement behavior during primary and secondary cycles (Figure


0 20 40 60 80 100

t [min]

0

200

400

600

800

1000

FP

[g

L-1

]

0

40

80

120

160

200

∆p

[b

ar]

EQ SD(i)PI SD

(ii)SI

Figure 4.19: Flow (blue) and pressure (orange) profile over the entire dura-tion of the displacement experiment, displacing two phases inthermodynamic equilibrium. Different steps are indicated byvertical black lines and the respective abbreviation: EQ: equi-libration, PI: primary imbibition, SI: secondary imbibition, SD:secondary drainage ((i) and (ii) denote the first and the secondsecondary drainage cycle).

4.20b). Furthermore, stable pressure levels are neither reached du-ring primary nor during secondary imbibition, which indicates thatthe displacement (until reaching the residual saturation Sr of thenon-wetting, PNT-lean phase) was not completed.An instantaneous decrease in pressure drop can be noted at thestart of the imbibition steps, and is ascribed to the opening of theinjection loop: the HPLC pump adjusts only gradually to the addi-tional resistance in the system caused by the injection loop, whichresults in a temporary decrease in flow rate and thus in pressuredrop. Hence, this pressure drop is a systematic error and does notresult from the investigated hydrodynamic behavior. Neither thepressure drop nor the associated decrease in flowrate is accountedfor in the model.Experimentally determined flow profiles (Figure 4.19) exhibit a grea-ter scattering during imbibition than during drainage. The loweraccuracy of the data points during imbibition, determined by HPLCanalysis, is due to the additional manual dilution of HPLC samples,


0 2 4 6 8 10 12

t [min]

0

0.2

0.4

0.6

0.8

1

Sa

v

R [

-]

PI

SI

(a)

0 5 10

t [min]

20

40

60

80

100

∆p

[b

ar]

PI

SI

(b)

Figure 4.20: Profiles during primary and secondary imbibition cycles. (a)average saturation SR

av, determined by equation 4.30. (b) Pres-sure drop ∆p

0 5 10 15 20 25

t [min]

0

0.2

0.4

0.6

0.8

1

Sa

v

R [

-]

SD(i)

SD(ii)

(a)

0 5 10 15 20 25

t [min]

40

60

80

100

120

140

∆p

[b

ar]

SD(i)

SD(ii)

(b)

Figure 4.21: Profiles during secondary drainage cycles. (a) average satura-tion SR

av, determined by equation 4.30. (b) Pressure drop ∆p

which is necessary due to the high concentrations of PNT in the elu-ate fractions during imbibition. Imbibition profiles exhibit a shocktransition reaching directly to high values of FP. A subsequent wavetransition is not recognizable, since - in the case that a wave tran-sition exists - it would be very flat and also because the data isscattered during imbibition. In turn, during drainage, the wave partis more distinct, and spans a greater range of FP values.The discussed issues in the flow profiles (data scattering, low data

accuracy and short wave part during imbibition) are reflected in


0 0.2 0.4 0.6 0.8 1

SR

[-]

0

0.2

0.4

0.6

0.8

1k

r

X [

-]

exp (drainage)

exp (imbibition)

model

kr

R

(a)

kr

L

0 0.2 0.4 0.6 0.8 1

SR

[-]

0

0.2

0.4

0.6

0.8

1

f X

[-]

(b)

f L

f R

Figure 4.22: (a) Relative permeability data determined from the drainage(circles) and imbibition (crosses) data. Darker blue and redcircles were determined from experimental data of the firstdrainage cycle (SD(i)), lighter symbols were obtained by dataanalysis of the second drainage cycle (SD(ii)). Model descrip-tions based on equations 3.18 and 3.19 are illustrated by con-tinuous lines. (b) Fractional flow curves based on the realivepermability functions presented in (a) and assuming ηR = 1.17mPas and ηL = 1.42 mPas

the average saturation profiles obtained through equation 4.30 andillustrated in Figures 4.20a and 4.21a. Saturation profiles during im-bibition quickly reach a rather constant level close to SR

av = 1, anddue to the scattering and the great span of the shock transition awave part is not recognizable. From the profiles, a residual satura-tion of the non-wetting phase Sr ≈ 0 was inferred.During drainage, a wave transition can be clearly observed, startingat an average saturation SR

av = 0.5 down to the plateau value, whichfor the two cycles ranges between SR

av = 0.3− 0.4. From the plateauvalues, an irreducible saturation of Si ≈ 0.32 was estimated.From the saturation and pressure profiles presented, relative per-meability data was estimated as described in section 4.3.7. The re-sulting relative permeability over saturation data is presented inFigure 4.22a. Data estimated from the drainage curves are presen-ted as open circles, with the maximum relative permeability kL

r,maxat SR = 0.32 (Si = 0.32) determined from the pressure plateau rea-ched during both secondary drainage steps, and applying equation


Table 4.5: Parameters of the relative permeability functions for the experi-mental system

Parameters kLr,max kR

r,max Sr Si mL mR

Values 0.94 0.77 0.00 0.32 2.00 3.00

3.15 with fL = 1 (i.e., uL = u). Due to the multiple experimen-tal issues during imbibition (very short wave part, combined withdata scattering, as well as the discussed systematic error in the pres-sure profile), no reasonable relative permeability data was expectedapplying the method described in section 4.3.7 to the imbibitiondata. A rough estimate was only obtained for the maximum rela-tive permeability kR

r,max at SR = 1 (Sr = 0), considering the finalpressure drop during the secondary imbibition cycle in equation3.15 with fR = 1. This estimated value is indicated as a cross in Fi-gure 4.22. Finally, the parameters mR/L were adjusted in such a wayas to achieve a reasonable description of the experimental data, i.e.mR = 3 and mL = 2. The estimated parameter values, required inequations 3.18 and 3.19, are summarized in Table 4.5.We want to underline that the estimated relative permeability function,due to the multiple experimental issues and various underlyingphysical assumptions, only provides a rough description of the ac-tual hydrodynamic behavior. Hysteretic effects, both between pri-mary and secondary cycles and between drainage and imbibitionsteps, were neglected in the description. Furthermore, a very likelydependence of the hydrodynamic behavior on the interfacial ten-sion between the two phases (i.e. a dependence on the location ofthe tie-line with respect to the plait point) has not been taken intoaccount. In order to consider the latter effect, similar displacementexperiments as discussed in this section, but along different tie-lines,would have been necessary. Considering the lack of accuracy andthe considerable effort required by this type of experiments, it wasdecided to refrain from further experimental investigation whilekeeping the limitations of the established description in mind.Finally, from the relative permeability equations and the estimationof phase viscosities discussed in the previous section, one can infer

4.9 conclusions 97

a relationship of the fractional flows fj depending on Sj. This relati-onship is illustrated in Figure 4.22b for phase viscosities ηR = 1.17mPas and ηL = 1.42 mPas.

4.9 conclusions

The experimental results and theoretical relationships presented inthis chapter characterize the relevant physical properties of the in-vestigated system composed of phenetole (PNT), 4-tert-butylphenol(TBP), methanol and water with the adsorbent Zorbax 300SB-C18.The findings also provide first insights on the implications of multi-ple fluid phases in liquid chromatographic systems in general.

Fluid phase equilibria The investigated quaternary system exhi-bits a rather complex thermodynamic behavior, featuring solid-liquid,liquid-liquid and solid-liquid-liquid equilibria, which are describedwith high accuracy by a fitted UNIQUAC equation. Applied for thedescription of the chromatographic process (by implementation intothe material balance equations, see chapter 3), this model allows thecalculation of activities and phase compositions of the fluid phasesin thermodynamic equilibrium.

Adsorption equilibria The single-component (PNT or TBP) andbinary (PNT and TBP) adsorption behavior was investigated expe-rimentally by Frontal Analysis. Operating conditions were constrai-ned to the single-phase region (a single fluid phase), in order toexclude hydrodynamic effects which at this stage could not be eva-luated. Hence, any phenomena observed in the measured elutionprofiles could clearly be attributed to the adsorption behavior.The adsorption behavior was described by rigorously applying ther-modynamic concepts. Single-component isotherms were determi-ned as a function of the liquid phase activities, which were calcu-lated by the previously established UNIQUAC model. Binary inte-raction in the adsorbed phase was described by the RAST, based onthe single-component isotherms and a Redlich-Kister equation ac-counting for non-idealities in the adsorbed phase. The accuracy ofthe model, concerning the description of single-component adsorp-tion, and in particular of binary adsorption, is remarkable, conside-


ring that the model is based on various thermodynamic relations-hips and isotherm equations, which were determined through dif-ferent and independent series of experiments, and which introducea certain error in the model description. The range of conditionsinvestigated and described by the model is unprecedented, compri-sing solute concentrations reaching until the solubility limit as wellas a broad span of eluent compositions. Neglecting the interactionof the modifier methanol with the adsorbent is a shortcoming ofthe model. Accounting for such interaction might result in a slightimprovement of the description, at the cost of a huge increase inmodel complexity. All in all, the quantitative agreement betweenmodel and experiments confirms the underlying thermodynamicconcepts of an activity-based description and the adsorbed solutiontheory, and it highlights the general validity of such concepts, ena-bling their applicability to a broad range of operating conditions.While the investigation was constrained to operating conditions inthe single-phase region (for reasons explained above), it is expectedthat such model, based on liquid phase activities, can also be ap-plied in the presence of multiple fluid phases in thermodynamicequilibrium, i.e. featuring identical liquid phase activities.Given the general validity of the model for different operating condi-tions, such thermodynamically consistent model does not only con-stitute a tool for fundamental research about physical mechanisms,but it can also be exploited for process design. In this context, wewould like to point out that the experimental effort to establish theadsorption model, which was very high in this work to demonstratethe accuracy of the underlying assumptions, can be considerably re-duced by applying a predictive thermodynamic model (such as theUNIFAC model) for the liquid phase, and by decreasing the amountof measurements to assess single-component and binary adsorptionbehavior. In principle, few measurements are necessary to obtain afirst estimate of the adsorption behavior on the basis of activities,which can then be extrapolated to design and assess the perfor-mance of chromatographic processes upon an alteration of opera-ting conditions. Apart from the broad applicability in terms of fluidphase compositions demonstrated in this study, such model couldalso describe the adsorption behavior over varying temperatures. Inthe context of gas adsorption, an application to high pressure sys-

4.9 conclusions 99

tems might be interesting, where fugacities instead of partial pres-sures would be required to accurately describe the thermodynamicproperties of the gas phase.

Hydrodynamic behavior Hydrodynamic properties were investi-gated in the absence of TBP, for the ternary system PNT - methanol- water, by performing imbibition and drainage experiments withone selected pair of phases in thermodynamic equilibrium (featu-ring identical liquid phase activities). Since liquid phase activitiesdid not change during the displacement, and adsorption was foundto depend on the liquid phase activities, no ad- or desorption wasexpected to occur during these experiments. Hence, any observedeffects could be ascribed to the two-phase flow behavior.The PNT-rich phase was found to be the wetting phase, while thePNT-lean phase was non-wetting. Accordingly, the irreducible fractionof the PNT-rich phase upon displacement by the PNT-lean phasewas very high (Si ≈ 0.32). In turn, no residues of the PNT-leanphase upon displacement by the PNT-rich phase could be observed(Sr ≈ 0).The experimental relative permeability data of the liquid chroma-tographic system could be described accurately by an empiricalBrooks-Corey correlation, which is commonly applied in the contextof natural reservoirs, where it was found to describe a high numberof multiphase systems with good accuracy. The established modelneglects a dependence of the two-phase flow behavior on the inter-facial tension between the two phases. This is considered as a majorshortcoming when approaching the plait point, where the composi-tions of the two phases approach each other, the interfacial tensionapproaches 0, and hence the two phases should approach identicalinterstitial velocities (reaching single-phase flow properties at theplait point).Finally, it is worth pointing out that in this work, only the displa-cement of two immiscible phases was investigated, while a spon-taneous phase split due to an enrichment and supersaturation ofcomponents within the column was not considered. Such sponta-neous phase split can only be achieved in the presence of multipleadsorbing, interacting components, for example when consideringthe entire quaternary system (with TBP present).


The investigation of a spontaneous phase split within the columnis very complex for various reasons: First, the thermodynamic pro-perties of the quaternary system are more diverse and complex thanthose of the ternary system, featuring different types of phase equili-bria and imposing considerable difficulties concerning well-control-led and accurate dynamic column experiments. In addition, withTBP being a solid under the experimental conditions, its presencein the chromatographic system implies the risk of the formation of asolid phase, which might damage the adsorbent. Finally, a spontane-ous phase split is prone to kinetic limitations, which have not beenaccounted for in this work. For the above reasons, the investigationof a phase split is considered as time-intensive and challenging, butconstitutes an important and interesting future task.

5S O L U T I O N A N D E VA L U AT I O N O FC H R O M AT O G R A P H I C M O D E L S

5.1 introduction

This chapter is focused on the solution, evaluation and compari-son of the two models introduced in chapter 3 (equilibrium theoryand lumped kinetic model). While a numerical solution to the lum-ped kinetic model has already been provided in section 3.2.2, thesolution of the equilibrium theory model based on the method ofcharacteristics is described in detail in this chapter, which also pro-vides a deep insight into the impact of different thermodynamicand fluiddynamic effects on the shape of elution profiles.Equilibrium theory models are commonly used in liquid chromato-graphy to understand and to describe the behavior of experimentalsystems24,26,28,74,80–82 and to design chromatographic processes.83,84

While all these models account for adsorption, but do not considermultiple convective phases, equilibrium theory models in the con-text of different applications in natural reservoirs describe two- ormultiphase flow through a porous medium, but most often neglectadsorption effects.4,69,85,86 To the best of our knowledge, there ex-ists only one contribution accounting for both multiphase flow andadsorption effects in the context of enhanced coalbed methane reco-very and CO2 sequestration.10,18 Focused on the mathematical deri-vation of model solutions, such study assumes simple relationshipsto describe the thermodynamic equilibria between gas and liquidphase (constant distribution coefficients), and between convectiveand solid phases (Langmuir isotherm as a function of the partialpressures in the gas phase).Accounting for the physically realistic thermodynamic and hydro-dynamic properties established in the previous chapter for the sy-

Results presented in this section have been reported in: Ortner, F.; Mazzotti, M. Two-phase flow in liquid chromatography, Part 2: Modeling Ind. Eng. Chem. Res. 2018,57(9), 3292-3307

102 solution and evaluation of chromatographic models

stem phenetole (PNT), methanol and water with the adsorbent Zor-bax 300SB-C18, and considering both adsorption and two-phaseflow, the equilibrium theory model investigated in this chapter con-stitutes a combination and extension of equilibrium theory modelsused in liquid chromatography and in applications to natural reser-voirs.The solution of the equilibrium theory model and lumped kineticmodel under consideration of the established algebraic equationsenables the direct comparison of advantages and disadvantages ofthe two models under physically realistic conditions, and the pre-diction of the elution behavior of the investigated experimental sy-stem in the presence of adsorption and two fluid phases.The chapter is structured as follows: In section 5.2, we summarizethe physical relationships, which were established for the systemPNT - methanol - water with the adsorbent Zorbax 300SB-C18 in theprevious chapter, and which are implemented in the component ma-terial balances (equations 3.8). The model is solved by applying themethod of characteristics, assuming fluid phases with equal inters-titial velocities (section 5.3) and with different interstitial velocities(section 5.4). For both cases, the general derivation of the solution ispresented, properties of the characteristics in the hodograph planeare discussed, and the elution behavior is illustrated for exemplaryinitial and feed conditions. Subsequently, the lumped kinetic model(compare section 3.2.2) is solved based on a finite volume discre-tization scheme, and resulting elution profiles are compared to theequilibrium theory simulations (section 5.5). Conclusions are drawnin section 5.6.

5.2 model system

We consider the system PNT - methanol - water (in this chapter re-ferred to as components 1, 2 and 3, respectively) with the adsorbentZorbax 300SB-C18 at 296.15 K and with a total porosity of the co-lumn εref = 0.615, which was characterized in the previous chapter.Accordingly, the UNIQUAC model established for this system, withparameters provided in Table 4.2 (column “Parameters (ternary)”),is used to determine liquid phase activities for any ternary compo-sition, and, for an immiscible composition, to calculate the phase

5.3 solution of the et model assuming equal velocities 103

split (phase saturations Sj) and the compositions wij of the two li-quid phases in thermodynamic equilibrium. The binodal curve ofthe ternary system PNT - methanol - water is calculated by perfor-ming equilibrium calculations (see Appendix A.3.1) for 5,000 dif-ferent compositions between the w2 = 0 axis and the plait point,which in turn is determined based on the partial derivatives of theGibbs free energy of mixing.87 To avoid computationally expensiveequilibrium calculations and thus to increase the simulation speed,the 5,000 calculated tie-lines were used as a look-up table for solvingboth the equilibrium theory model and the lumped kinetic model.The implementation of the look-up table is model-specific and is ex-plained in detail in Appendix B.1.Single-component adsorption of PNT (n1) is described by the anti-Langmuir isotherm (see equation 4.41), being a function of the li-quid phase activities of PNT, while methanol and water are assu-med to be inert components (n2 = n3 = 0). Hydrodynamic pro-perties are described by the fractional flow functions established insection 4.8. Both the isotherm equation for PNT and the fractionalflow equation can be solved explicitly, hence implementation intothe equilibrium theory (ET) model and the lumped kinetic model isstraightforward.

5.3 solution of the et model assuming equal veloci-ties

In a first step, we analyze the dynamic behavior of a chromato-graphic system where the two liquid phases move at the same in-terstitial velocity, i.e. when assuming fj = Sj and thus Fi = Ci in themodel equations above. We first provide the general solution of theequilibrium theory model for equal velocities, using the method ofcharacteristics.24 Then we apply the general solution to the modelsystem of interest here. We calculate the network of characteristicsin the hodograph plane, i.e. the (C2,C1) plane, which provides ima-ges of the solutions independent of initial and feed states, and weexplore characteristic features and physical implications of the solu-tions. Finally, the chromatographic behavior is illustrated by elutionprofiles for specific initial and feed states.


5.3.1 General Solution

Under the assumption that fj = Sj and thus Fi = Ci, and keepingin mind that n2 = n3 = 0, equation 3.8 for components 1 and 2 canbe recast as:(

ε+ (1− εref)n11

(1−

C1ρ1

))∂C1∂τ

+

((1− εref)n12

(1−

C1ρ1

))∂C2∂τ

+∂C1∂ξ

= 0 (5.1a)(ε−

C2ρ1

(1− εref)n12

)∂C2∂τ

−

(C2ρ1

(1− εref)n11

)∂C1∂τ

+∂C2∂ξ

= 0 (5.1b)

Here, C1 and C2 are the unknown variables, and n1j denotes∂n1/∂Cj (j = 1, 2). Note that, assuming an isotherm depending onthe liquid phase activity a1, and not directly on the liquid phaseconcentration C1, the dependence of n1 on Cj (j = 1, 2) is nottrivial. The isotherm equation itself gives n1 as a function of a1,which is a function of the mole fractions of the three components,i.e. a1 = a1(Z1,Z2,Z3), through the UNIQUAC equation; molefractions Zi in turn depend on the liquid phase concentrations, i.e.Zi = Zi(C1,C2,C3), as specified by equation B.1 in the Appendix.Finally, based on the assumption of volume additivity, the numberof unknown variables is reduced to 2, by expressing C3 as a functionof C1 and C2 using equation 3.9 (for the details of such calculationsee Appendix B.1).Applying the method of characteristics,24 one can derive equationsfor the characteristics, i.e. the images of the solutions, in the ho-dograph (C2,C1)-plane and in the physical (ξ, τ)-plane, based onwhich it is then possible to determine elution profiles for specific ini-tial and feed conditions. The procedure of deriving these analyticalsolutions has been explained in great detail in the literature.24,26,28,74,82

For the sake of brevity, we keep explanations concerning the so-lution procedure to a minimum, and focus on the specific resultsobtained with this model.In the case of hyperbolic systems of two equations, such as that gi-


ven by equations 5.1, there are two sets of characteristics, Γ1 and Γ2,with slopes ζ1,2 = dC2/dC1, in the hodograph plane given by:

Γ1 : ζ1 = −n11n12

(5.2)

Γ2 : ζ2 = −C2

ρ1 −C1, (5.3)

The Γi characteristics, resulting from the solution of the ODEs 5.2and 5.3, describe the change in composition (C1, C2) that occurswhen connecting an initial state to a feed state. Equation 5.2 can bewritten as:

∂n1∂C1

dC1 +∂n1∂C2

dC2 = 0 = dn1, (5.4)

which allows to conclude that the adsorbed phase concentration n1remains constant along the Γ1 characteristics. Substituting equation3.9 both in finite and in differential form into equation 5.3 yields thefollowing differential equation in C2 and C3

dC3dC2

=C3C2

. (5.5)

Such equation holds along Γ2, and implies that the ratio of the con-centrations of the two inert components (in the following called sol-vent ratio) remains constant along the Γ2 characteristics.Each composition on the solution path mapping onto Γ1 (Γ2) charac-teristics moves through the column at a constant propagation velo-city λ1 (λ2). The reciprocals of the propagation velocities, σj = λ−1j ,determine the slopes of the characteristics in the physical plane(ξ, τ); for the investigated system of equations 3.8 the slopes are:

σ1 = ε (5.6)

σ2 = ε+ (1− εref)n11

(1−

C1ρ1

)−(1− εref)n12

C2ρ1

(5.7)

Since both n11 and n12 can be positive or negative, it is difficult tomake a general statement about whether the value of σ2 is larger


than ε or not; however, it should be pointed out that this is alwaysthe case for our system (thus, σ2 > σ1).We consider a Riemann problem with an initial (downstream) stateA (CA

i for ξ > 0 and τ = 0) and a feed (upstream) state B (CBi for

ξ = 0 and τ > 0). The solution in the physical plane (ξ, τ) consists ofthree states, namely the initial state A, an intermediate state I, andthe feed state B. These three states are connected by two transitions,each of which maps onto a Γj (j = 1, 2) characteristic. Depending onthe directional derivative of σj along the Γj characteristic, the tran-sition can be a contact discontinuity (directional derivative equalszero), a simple wave (positive directional derivative moving fromdownstream states D to upstream states U), a shock (negative di-rectional derivative moving upstream), or a semi-shock (directionalderivative changing sign).Since σ1 remains constant along Γ1, the directional derivative is 0

and consequently a transition mapping on a Γ1 characteristic is al-ways a contact discontinuity. In turn, the directional derivative ofσ2 along Γ2 can be positive or negative, allowing for different typesof transition. If σ2 increases when moving from downstream sta-tes D to upstream states U, the transition is a simple wave, in theopposite case, it is a self-sharpening shock, i.e. a “weak” solutionobtained from the integral form of the material balance and with aslope in the physical plane σ2 given by:

σ2 =[εC1] + [(1− εref)n1]

[C1]=

[εC2]

[C2]. (5.8)

The sign [·] denotes a jump of the enclosed quantity across the dis-continuity. Note that the conditions given in Equation 5.8 for compo-nents 1 and 2, which both have to be fulfilled for a shock transition,also provide the possible states which, assuming a certain down-stream state D (a certain upstream state U), can be connected to D(U) through a shock. These states are located on a shock path Σ2,which is tangent to the Γ2 characteristic in D (U). Since in our caseΓ2 characteristics are straight lines, a shock path Σ2 coincides witha characteristic Γ2. A further requirement for a shock connecting adownstream state D to an upstream state U is that σU

2 < σ2 < σD2 .23

If the directional derivative of σ2 changes in sign along the Γ2 cha-racteristic connecting the intermediate to the feed state, the transi-


tion will be a semi-shock (shock-wave or wave-shock). A detailedexplanation of the necessary conditions and the behavior of semi-shock transitions is provided in the literature.23,24,26

The two transitions in the solution of the Riemann problem mapon two different types of characteristics Γ1 and Γ2, which intersectat the intermediate state I and connect I to initial state A and feedstate B. In a physically meaningful solution, downstream states haveto propagate through the column faster than upstream states. Thus,with σ1 < σ2, hence λ1 > λ2, the intermediate state has to be con-nected to the initial (downstream) state A by a Γ1 characteristic, andto the feed (upstream) state B by a Γ2 characteristic.A physical interpretation of the derived mathematical solution isas follows. Along a Γ1 characteristic, the adsorbed phase concentra-tion n1 remains constant. Since neither adsorption nor desorptionoccur along such a characteristic, all states along it move at the samepropagation velocity, namely the interstitial velocity ε−1. In turn, al-ong a Γ2 characteristic, the concentration ratio of inert components2 and 3 remains constant. Since neither inert component interactswith the adsorbent, and all convective phases move with the sameinterstitial velocity, these components propagate at the same velo-city. Therefore, unless the concentration ratio of inert components isdifferent in the initial and in the feed state (i.e. these two states arenot located on the same Γ2 characteristic), there is no driving forcefor the ratio C2/C3 to change. As the concentration and activity ofcomponent 1 vary along these characteristics, adsorption or desorp-tion occur and propagation velocities vary as well, leading to theformation of waves, shocks or semi-shocks.

5.3.2 Hodograph plane

In the following, the equations derived above are applied to deter-mine characteristics in the hodograph plane for the model systemintroduced in section 5.2, while still assuming equal velocities ofthe convective phases (fj = Sj). Γ1 and Γ2 characteristics in the hodo-graph plane are plotted in Figure 5.1a and b, respectively. Γ1 charac-teristics were calculated by solving the ordinary differential equa-tion 5.2 with a built-in Matlab routine (ode113). In turn, the calcula-


tion of Γ2 characteristics is trivial, as they feature a constant solventratio C2/C3 and thus are straight lines in the ternary diagram.

With n1 being constant along blue Γ1 characteristics (see equa-tion 5.2), based on equation 4.41 Γ1 characteristics constitute pathsof constant a1. The qualitative behavior of the Γ1 characteristics isillustrated in Figure 5.2, where the miscible region is enlarged forbetter visibility. All Γ1 characteristics connect a point on the w2 = 0

axis with a point on the w3 = 0 axis. In the soluble region, these cha-racteristics are arc-shaped, while in the immiscible region they mapon the tie-lines. Characteristics emanating from a point S locatedon the w2 = 0 axis and close to the w3 = 1 corner remain entirelyin the miscible region. Increasing the mass fraction of component 1

(wS1) of this starting point, a Γ1 characteristic, located entirely in the

miscible region and tangent to the plait point, is attained. A furt-her increase of wS

1 results in Γ1 characteristics passing through theimmiscible region along tie-lines, and extending into the miscibleregion to reach the w2 = 0 and w3 = 0 axes. Once wS

1 reaches thesolubility limit of component 1 in component 3, the correspondingΓ1 characteristic maps onto the tieline located on the w2 = 0 axis.Finally, if S is located in the miscible region on the component 1 richside, the corresponding Γ1 characteristic again connects directly tothe w3 = 0 axis, without passing through the immiscible region. Itshould be kept in mind that along Γ1 characteristics, σ1 remains con-stant, thus the directional derivative equals 0 and transitions alongthese paths are always contact discontinuities.As illustrated in Figure 5.1b, Γ2 characteristics are straight lines al-ong which the solvent ratio C2/C3 remains constant (i.e., connectinga point on the w1 = 0 axis to the w1 = 1 corner). Red arrows in Fi-gure 5.1 indicate the direction in which σ2 increases, red squaresdenote locations at which the directional derivative of σ2 equalszero and changes sign. As discussed above, the non-monotonic be-havior of σ2 allows the formation of wave, shock and semi-shocktransitions. Note that at the intersection of Γ2 characteristics withthe binodal curve, σ2 is discontinuous, so as concentration plateauscan form at these locations in the corresponding elution profiles,although the solution path remains on the same characteristic.


20

40

60

80

20 40 60 80

20

40

60

80

comp 2

comp 3 comp 1

Γ1

(a)

20

40

60

80

20 40 60 80

20

40

60

80

(b)

Γ2

comp 2

comp 1comp 3

Figure 5.1: Characteristics in the Hodograph plane (mass fractions) for theternary system with one adsorbing component introduced insection 5.2, assuming equal interstitial velocities of all convectivephases. (a) Γ1 characteristics, (b) Γ2 characteristics. The blackcurve corresponds to the binodal curve. Red arrows indicate thedirection in which σ2 increases along a Γ2 characteristic. Redsquares denote conditions at which the directional derivative ofσ2 changes from positive to negative.


20

40

60

80

20 40 60 80

20

40

60

80

comp 1

comp 2

comp 3

Figure 5.2: Illustration of the qualitative behavior of Γ1 characteristics in thehodograph plane.

5.3.3 Solution of exemplary cases

The behavior of the exemplary system shall now be illustrated fortwo different cases. As the equilibrium theory solution under thecurrent assumption of equal interstitial velocities of the convectivephases is rather simple, the examples that follow can be easily ex-ploited to analyze solutions for different initial and feed states (alsofor those located in the immiscible region).With reference to Figure 5.3, the initial state A is located on thew1 = 0 axis, i.e. a mixture of the inert components 2 and 3. Let usconsider two feed states, B1 and B2, which are located in the so-luble region on the w3 = 0 axis, i.e. two different binary mixturesof components 1 and 2. In both cases we consider a complete chro-matographic cycle, consisting of the adsorption step displacing theinitial state A by the feed state Bi, and of the desorption step elutingthe feed state by the initial state. Accordingly, for the derivation ofthe solution of the desorption step, the initial state and the feed stateare swapped. The corresponding paths in the hodograph plane areshown in Figure 5.3, whereas elution profiles are illustrated in Fi-gure 5.4 with adsorption profiles on the left and desorption profileson the right. Black thin lines in the concentration profiles show thesolution obtained by solving numerically the lumped kinetic model


of equations 3.10 and 3.11 (see the discussion in section 5.5 below).

20

40

60

80

20 40 60 80

20

40

60

80

comp 1comp 3

comp 2

A

B2

Ia

B1I

d

Figure 5.3: Paths in the hodograph plane for exemplary cases with an ini-tial state A and two different feed states in the soluble region.Paths describe the solution for the entire chromatographic cycle,i.e. adsorption and desorption. Feed state B1: all paths are loca-ted in the soluble region; feed state B2: paths lead through theimmiscible region during desorption. States Ia and Id denoteintermediate states formed during adsorption and desorption,respectively. Note that blue and red paths correspond to Γ1 andΓ2 characteristics, as they are illustrated in Figure 5.1a and b,respectively.

As described in the general solution above, all paths connectinginitial to feed states emanate from the initial state along a Γ1 cha-racteristic until reaching an intermediate state I, which is then con-nected to the feed state by a Γ2 characteristic. As a consequence, thefirst transition connecting the initial state to I is always a contactdiscontinuity propagating with the interstitial velocity, i.e. appea-ring at the column outlet at the void time. The second transition inmost cases (and in our two examples) is a shock, although undercertain conditions it can also be a wave or a semi-shock.Whenever an initial state located on the w1 = 0 axis is connected

to a feed state located on the w3 = 0 axis, the intermediate state ofthe adsorption step Ia is located in the w2 = 1 corner. The physicalreason for this is that the inert component 3 (absent in the feed state)propagates at the interstitial velocity, and is thus eluted within the


0 0.5 1

τ [-]

0

100

200

300

400

500

600

700

800

900

Ci [

gL

-1]

0 1 2 3 4

τ [-]

0

100

200

300

400

500

600

700

800

900

ET

det

comp 2

comp 1

(a) Ia

Id

1A B

1B

1A

0 0.5 1

τ [-]

0

100

200

300

400

500

600

700

800

900

Ci [

gL

-1]

0 1 2 3 4

τ [-]

0

100

200

300

400

500

600

700

800

900

ET

det

(b) Ia

Id

2

comp 2

comp 1

A B2

B2

A

Figure 5.4: Concentration profiles of the chromatographic cycles connectingthe initial state A to the feed state (a) B1 and (b) B2. Left: ad-sorption, right: desorption. Colored thicker lines correspond toequilibrium theory solutions, thin black lines to solutions of thelumped kinetic model solved with a number of discretizationsteps N = 1000. The corresponding images of the solution in thehodograph plane are shown in Figure 5.3.

5.4 solution of the et model assuming different velocities 113

void time. In turn, the adsorbing component 1 (absent in the initialstate) propagates more slowly and arrives at the column outlet la-ter, hence at the intermediate state only component 2 is present. Ingeneral, for all initial states located on the w1 = 0 axis connectedto any feed state in the miscible region, the image of the adsorptionstep belongs entirely to the miscible region.The desorption step for initial and feed states located in the misci-ble region can pass through the two-phase region, as seen for theexample with feed state B2, where the two-phase region is crossedalong a Γ1 characteristic. The intermediate states of the desorptionsteps Id of the two exemplary cases are both located in the misci-ble region and on the same Γ2 characteristic (determined by stateA), but differ due to the different Γ1 characteristics to be considered(determined by state Bi). The resulting elution profiles (see Figure5.4, right column) are qualitatively similar, with both intermediatestates located in the miscible region, connected to states Bi througha contact discontinuity, and to A through a shock. A potential cros-sing of the two-phase region (as it is the case for example 2) occursduring the contact discontinuity. We want to point out that an analo-gue solution can be derived for most initial and feed states locatedin the soluble region. An intermediate state located in the two-phaseregion can occur under specific and rare initial and feed states, suchas in the case that during desorption, the component 1 lean phase ofthe tie-line involved in the solution path has a solvent ratio C2/C3,which is lower than the solvent ratio of state A. This is only the casefor feed states B located very close to the w1 = 1 corner, and is notconsidered any further in this work.

5.4 solution of the et model assuming different velo-cities

In this section, we derive the equilibrium theory solution, conside-ring the more realistic case of different interstitial velocities of theconvective phases, i.e. fj = fj(Sj) 6= Sj and Fi 6= Ci. The solutiondescribed in this section applies only within the immiscible region,while in the single-phase region the solution derived in section 5.3remains valid. We proceed as in the previous section, by first provi-


ding the general solution for the equilibrium theory model, beforeapplying it to the model system. Here the fractional flow functionintroduced in equations 3.17 to 3.19 is utilized. Again, we investi-gate the hodograph plane, as well as elution profiles derived forspecific initial and feed conditions. Note that in the main text, wefocus on the most important mathematical and physical attributesof the equilibrium theory solution. A detailed explanation of theconstruction of solutions is provided in Appendix B.3.

5.4.1 General solution

Solving the equilibrium theory model with different interstitial velo-cities based on the dependent variables C1 and C2 is possible, butresults in rather complex expressions for the characteristics in thehodograph and the physical plane. Instead, we perform the variabletransformation suggested by Ref. [4], by defining two new variables,namely the saturation of the phase rich in component 1, SR, and atie-line indicator η, uniquely defining the tie-line on which a state islocated. Here, we choose η to be the logarithm of the mass fractionof component 1 in the lean phase η = ln(wL

1). The original systemof equations 3.8 in the unknowns C1 and C2 is transformed into thefollowing system in the unknowns SR and η:

Ai∂η

∂ξ+Bi

∂η

∂τ+Ci

∂SR

∂ξ+Di

∂SR

∂τ= 0, i = 1, 2 (5.9)

with

Ai =fR dcR

i

dη+ (1− fR)

dcLi

dη

+(cRi − c

Li

) ∂fR∂η

, i = 1, 2 (5.10a)

B1 =ε

(SR dcR

1

dη+(1− SR

) dcL1

dη

)

+ (1− εref)

(1−

C1ρ1

)dn1dη

(5.10b)

B2 =ε

(SR dcR

2

dη+(1− SR

) dcL2

dη

)−C2ρ1

(1− εref)dn1dη

(5.10c)


Ci =(cRi − c

Li

) ∂fR∂SR , i = 1, 2 (5.10d)

Di =(cRi − c

Li

)ε, i = 1, 2 (5.10e)

Note that, with n1 being a function of the liquid phase activity ofcomponent 1 (a1) and ε being defined as in equation 3.5, both niand ε remain constant along the same tie-line, and thus ∂n1/∂SR =

∂ε/∂SR = 0. This variable transformation results in simpler expres-sions for the characteristics, allowing to draw interesting physicalconclusions, as seen below.We now define two sets of characteristics Γt and Γnt in the new hodo-graph plane with coordinates SR and η, with slopes ψt/nt = dη/dSR.These characteristics are the image of the solution in the hodographplane. Substituting ψ in equations 5.9 yields two equations in theunknown SR, which must be identical. This is the case if and only ifψ is a solution of the following quadratic equation:24

(A1B2 −A2B1)ψ2 + (B2C1 −B1C2 +A1D2 −A2D1)ψ

+C1D2 −C2D1 = 0(5.11)

Since C1D2 = C2D1, the following two expressions are the twoexact solutions of the last equation, which in fact define the twofamilies of characteristics:

Γt : ψt = 0 (5.12)

Γnt : ψnt =B2C1 −B1C2 +A1D2 −A2D1

A2B1 −A1B2, (5.13)

From equation 5.12 and the definition of η, it is clear that Γt charac-teristics map on the tielines. We therefore refer to Γt characteristicsas tieline characteristics and to Γnt characteristics as non-tieline cha-racteristics (hence the label used as subscript).The propagation velocities in the physical plane λx = dξ/dτ (x =

t, nt), associated to states on the corresponding characteristics Γx,are derived by the method of characteristics as:

λt =1

σt=C1D1

=1

ε

∂fR

∂SR (5.14)

λnt =1

σnt=A1ψnt +C1B1ψnt +D1

. (5.15)


The directional derivatives of both λt and λnt can be negative or po-sitive, thus no conclusions about the type of transition can be drawna priori. In the case where the directional derivative of λx increasesalong a Γx characteristic connecting a downstream state to an up-stream state, a “weak” solution is required, and the propagationvelocity of the corresponding shock transition can be derived fromthe integral material balance as

1

λ= σ =

[εC1 + (1− εref)n1]

[F1]=

[εC2]

[F2]. (5.16)

The Rankine-Hugoniot condition given by equation 5.16 also deter-mines the shock paths Σx in the hodograph plane, since all conditi-ons being connected to a specified initial condition through a shockare uniquely identified by the two expressions of σ. It is worth re-calling the existence of a further condition to be fulfilled by a shocktransition connecting a downstream state D to an upstream state U,namely that σU

x < σ < σDx (see section 5.3.1).

As discussed below, states in the immiscible region are most oftenconnected to states in the miscible region through shocks. Sinceequation 5.16 is derived from the integral component material ba-lances, it remains valid also for these shocks crossing the binodal,where for the single phase state it is obvious that Ci = Fi = ci.

5.4.2 Hodograph plane

We now apply the general solution to the model system to inves-tigate the behavior of the characteristics in the hodograph plane,and of the propagation velocities associated to the correspondingliquid phase compositions. Figure 5.5a presents the Γt and Γnt cha-racteristics in blue and red, respectively, in the (ln(wL

1),SR)-plane.

As expected, Γt characteristics map on the tie-lines, and are thusstraight horizontal (blue) lines in Figure 5.5a. Each Γt characteristicis tangent to two Γnt characteristics; the points of tangency wherethey meet are singular points where the discriminant of the quadra-tic equation 5.11 is zero, hence ψt = ψnt and λt = λnt. These pointsare called watershed points (or, in some contributions, equal eigen-value points). For the discussed system, there is an infinite numberof watershed points, which are all located in the physically relevant


domain of (η,SR) values.Let us now consider the propagation velocities λt and λnt along the

characteristics, which are illustrated for two selected tieline charac-teristics in Figure 5.6a, and for four selected non-tieline characteris-tics in Figure 5.6b. The selected characteristics are the highest andthe lowest tieline characteristic in Figure 5.5a, as well as the twopairs of non-tieline characteristics with watershed points located onthe highest and lowest tieline characteristic. Considering λt and λntalong their corresponding characteristics (Γt and Γnt, respectively),it can be observed that the propagation velocities do not exhibita strictly monotonic behavior. The velocity λt exhibits a maximumat a saturation SR ranging between 0.6 and 0.8, and decreases to0 when approaching 0 and 1. The slopes of the propagation velo-cities λnt along non-tieline characteristics exhibit multiple changesin sign. With directional derivatives of λx along Γx characteristicschanging in sign, complex semi-shock transitions, which possiblyexhibit multiple wave and shock parts, can be formed. Points wherethe directional derivatives are zero (and change sign) are indicatedin Figure 5.5a as open circles and squares.With reference to Figure 5.6, there are two points along each tielinecharacteristic (one point along each non-tieline characteristic) whereλt = λnt, which are indicated in Figure 5.6 as open black circles, andcorrespond to the watershed points. The states associated to thesepoints are special because their propagation velocity in the physi-cal plane is the same whether they occur within a transition whoseimage is a Γt or a Γnt characteristic. Therefore, such points can bereached through a path along a Γt (Γnt) characteristic and departedin the direction of a Γnt (Γt) characteristic without its propagationvelocities changing. As a consequence, and contrary to what hap-pens at all other points, a change of path through a watershed pointcan occur without any intermediate constant state emerging in thesolution.A final observation concerning Figure 5.6 is that, unlike in the caseof equal interstitial velocities treated in section 5.3, none of the twopropagation velocities λt and λnt is always larger than the other. Fo-cusing on the behavior along tieline characteristics (Figure 5.6a), itcan be noted that between the two watershed points λt > λnt. Inturn, beyond the watershed points towards SR → 0 or SR → 1, the


0 0.2 0.4 0.6 0.8 1

SR

[-]

-8

-6

-4

-2

0

ln(x

1L)

[-]

Γt

Γnt

(a)

0 0.2 0.4 0.6 0.8 1

SR

[-]

-8

-6

-4

-2

0

ln(x

1L)

[-]

Γ1

Γ2

(b)

Figure 5.5: Characteristics in the hodograph plane, constructed based onthe transformed variables SR and η = ln(wL

1) and assumingthe hydrodynamic behavior described by equations 3.17 to 3.19.(a) Tieline characteristics (Γt, blue) and non-tieline characteris-tics (Γnt, red), as defined in equations 5.12 and 5.13. (b) Γ1 andΓ2 characteristics, constructed by combining parts of Γt and Γntsuch that λ1 > λ2 always. Circles (in the colour of the characte-ristics) indicate a change in the directional derivative of σx alonga Γx characteristic (x = t, nt, 1, 2) from negative to positive, squa-res reflect a change from positive to negative.


0 0.2 0.4 0.6 0.8 1

SR

[-]

0

2

4

6

8

λ [

-]

λt

λnt

(a)

Γt characteristics

0 0.2 0.4 0.6 0.8 1

SR

[-]

0

1

2

3

4

5

6

7

λ [

-]

λt

λnt

(b)

Γnt

characteristics

Figure 5.6: Propagation velocities λt (blue) and λnt (red) (a) along tielinecharacteristics Γt, (b) along non-tieline characteristics Γnt. Con-tinuous and dashed lines in (a) correspond to the highest andlowest tieline (highest and lowest ln(wL

1)) illustrated in Figure5.5. In (b), continuous and dashed lines correspond to the twopairs of non-tieline characteristics with watershed points locatedon the highest and lowest tieline (two watershed points per tie-line). Open black circles indicate watershed points. Note that thenon-tieline characteristic with a watershed point at SR = 0.94 fe-atures a very narrow range of SR, such that only the watershedpoint and no curve is visible.


propagation velocities λnt are larger than λt. This behavior complica-tes the derivation of solution paths and elution profiles for specificinitial and feed states: As already discussed for the case of equalinterstitial velocities (section 5.3.1), in a physically meaningful solu-tion, downstream states have to propagate faster than upstream sta-tes. Hence, a physically meaningful intermediate state has to be rea-ched from upstream states with a propagation velocity higher thanthe propagation velocity with which it is connected to downstreamstates. For the solution discussed in section 5.3.1, with σ1 < σ2hence λ1 > λ2, this consideration leads to the simple rule that thesolution path would always map on a sequence of Γ1 characteristic(connecting a downstream state A to the intermediate state I) andΓ2 characteristic (connecting the intermediate state I to an upstreamstate B), intersecting at I. In the case of different interstitial veloci-ties discussed in this section, with no defined order of λt and λnt,the derivation of solution paths is therefore more complex.In order to simplify the derivation of solution paths and elution pro-files for specific initial and feed states, and following the suggestionof Helfferich,69 we define two new sets of paths, namely “fast” pathsand “slow” paths, with propagation velocities along the “fast” pathsbeing always larger (or equal) to the propagation velocities alongthe “slow” paths. With reference to the discussed behavior of Γt andΓnt characteristics and their propagation velocities, this correspondsto reconnecting segments of Γt and Γnt characteristics at the waters-hed points in a different manner. The resulting “fast” and “slow”paths are presented in Figure 5.5b. With the propagation velocitiesof “fast” paths being larger than those of “slow” paths, it followsthat solution paths always map on a sequence of “fast” and “slow”path, intersecting at the intermediate state I, and connecting I to thedownstream state A and the upstream state B, respectively. We haveobserved that, solving equation 3.8 with respect to the original vari-ables C1 and C2 (which is more complicated from a mathematicaland a computational point of view), the calculated characteristicsmap on the “fast” and “slow” paths, such that for the remainderof this work we call these paths Γ1 and Γ2 characteristics, with thecorresponding propagation velocities λ1 and λ2.The (ln(wL

1),SR)-plane presented in Figure 5.5 only shows the im-

miscible region. In order to assess the location of the characteris-


20

40

60

80

20 40 60 80

20

40

60

80

comp 2

comp 3 comp 1

Γ1

Γ2

Figure 5.7: Mapping of Γ1 and Γ2 characteristics (compare Figure 5.5b) inthe ternary diagram.

tics with respect to the fluid phase composition, and to take thesoluble region into account, Γ1 and Γ2 characteristics determined inFigure 5.5b are mapped into the ternary diagram in Figure 5.7. Inthis diagram, it can be noted that all Γ2 characteristics, and a fewΓ1 characteristics, intersect with the binodal curve. Γ1 characteris-tics which do not intersect with the binodal can only be accessedfrom the miscible region through shocks fulfilling equation 5.16. Inturn, characteristics in the immiscible region intersecting with thebinodal are continued in the soluble region along the Γ1 and Γ2characteristics derived in section 5.3. However, it should be notedthat the slopes of Γi characteristics, as well as the correspondingpropagation velocities λi, are discontinuous at the binodal curve.As a consequence, crossing the binodal curve along a characteristiceither occurs through a shock transition, or it produces a plateau inthe concentration profiles, which does not correspond to a classicintermediate state, as the solution path remains on the same cha-racteristic. For further explanations on the crossing of the binodalcurve, see Appendix B.3.It is worth making one last comment on the impact of adsorption


on the shape of the characteristics in the hodograph plane. With theS-shaped fractional flow functions (see Figure 4.22b), there is onepoint on every tieline at which fX = SX (apart from SX = 0 or 1), andthus Fi = Ci. At this point, both phases travel with the same intersti-tial velocity. Connecting all these points over all tielines, one obtainsthe “equivelocity” curve, which intersects all tie-lines and ends inthe plait point. In the absence of adsorption effects, it can be shownthat both the equivelocity curve, as well as the binodal curve arenon-tieline characteristics.4 In contrast, in our case, accounting foradsorption effects and for the resulting volume (porosity) changes,it can be shown that neither the equivelocity curve nor the binodalcurve constitute non-tieline paths (proof is provided in AppendixB.2 and in Ref. [10]).

5.4.3 Solution of exemplary cases

The results discussed in the previous section emphasize the com-plexity and diversity of possible solutions for different initial andfeed states, featuring (i) different possibilities of crossing the bino-dal, (ii) characteristics with peculiar shape, and (iii) directional deri-vatives of propagation velocities changing sign along these characte-ristics, possibly several times, thus producing complex semi-shocktransitions. We want to provide a few exemplary elution profilesand their mapping in the hodograph plane, but, for the sake of bre-vity, we only highlight the most relevant mathematical and physicalaspects. A detailed explanation is provided in Appendix B.3.With reference to Figure 5.8, we consider 5 exemplary cases, all ofthem sharing the same initial state A (mixture of inert components2 and 3), but featuring different feed states, namely B1 to B5. Thefirst three feed states are located in the miscible region, more preci-sely they are binary mixtures of adsorbing component 1 and inertcomponent 2. Feed states B4 and B5 are located in the immiscibleregion. Paths in the hodograph plane, concentration and flow pro-files (entire chromatographic cycles) for examples 1 to 3 (misciblefeed states) are provided in Figures 5.9 to 5.11, those for examples4 and 5 in Figures 5.12 and 5.13, respectively. Note that, apart fromthe equilibrium theory solutions, concentration and flow profiles of


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comp 2

comp 3 comp 1

B5

B4

B2

B1

B3

A

Figure 5.8: Initial and feed conditions (mole fractions) of the five differentexemplary cases solved by equilibrium theory. All cases consti-tute displacement cycles with the same initial state A (mixtureof inert components 2 and 3) and different feed states B1 to B5(B1 to B3 being mixtures of adsorbing component 1 and inertcomponent 2, B4 and B5 being located in the immiscible region).Connecting lines are for illustrative purposes only.

Figures 5.9 to 5.13 also exhibit thin black lines, corresponding tosimulations based on the lumped kinetic model, which will be dis-cussed in detail in section 5.5.First of all, it is worth noting that, with the assumption of diffe-rent interstitial velocities of the convective phases, Fi differs fromCi and thus concentration and flow profiles differ. While concentra-tion profiles describe the overall composition of the phase mixtureat a certain position in the column (here at the column end whereξ = 1), flow profiles describe the composition of the overall flux (inthis case again at ξ = 1).

Let us first focus on the cases with feed states in the miscibleregion. Comparing Figures 5.9, 5.10 and 5.11, it can be noted that,although changes in the feed composition are rather small, the pathsin the hodograph planes differ considerably in the three cases. Yet,all cases exhibit one unique solution, which in the three examples


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0 5 10 15

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-1]

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det

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Figure 5.9: Solutions for exemplary case 1 with initial state A (located onthe c1 = 0 axis) and feed state B1 (located in the w1 = 1 cor-ner). Top: Paths in the hodograph plane resulting from the equi-librium theory derivation (mass fractions). Center: concentrationprofiles. Bottom: flow profiles. Thick, colored lines in the centerand bottom graph provide the solution of the equilibrium the-ory model, thin black lines correspond to numerical simulationswith a number of discretization steps N = 200.


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-1]

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detcomp 1

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Figure 5.10: As in Figure 5.9, but with a feed state B2 (located on the c3 = 0

axis).


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comp 3 comp 1

comp 2

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B3

Id

Ia

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det

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Id

A

comp 2

comp 1

B3

0 5 10 15

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Fi [

gL

-1]

ET

det

comp 2

comp 1

Figure 5.11: As in Figure 5.9, but with a feed state B3 (located on the c3 = 0

axis).


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B4

Ia

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gL

-1]

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det

comp 1

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Figure 5.12: As in Figure 5.9, but with a feed state B4 (located in the immis-cible region).


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comp 2

comp 3 comp 1

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Ia

Pa

B5

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τ [-]

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700

Ci [

gL

-1]

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det

comp 1

comp 2

A AId

BPa

Ia

0 10 20 30 40

τ [-]

0

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Fi [

gL

-1]

ET

det

comp 1

comp 2

Figure 5.13: As in Figure 5.9, but with a feed state B5 (located in the immis-cible region).


cross the immiscible region, during adsorption and desorption incases 1 and 2, and during desorption in case 3.Concerning the adsorption step, one possible solution path entirelylocated in the soluble region is always available for initial states lo-cated on the w1 = 0 axis, connected to feed states located on thew3 = 0 axis. This path is the same solution that would be obtainedusing the method described in section 5.3.3, with an intermediatestate located in the w2 = 1 corner and being reached through acontact discontinuity propagating at the interstitial velocity. Undercertain conditions, a second path, crossing the immiscible region,should be considered, where the initial state is connected to a statein the immiscible region through a shock (or semi-shock). This se-cond solution is the physically valid solution, if the shock (semi-shock) emanates from the initial state with a propagation velocityfaster than the interstitial velocity (ε−1), and enables a further va-lid solution path connecting to the feed state. In examples 1 and 2,such transitions propagating faster than the interstitial velocity exist,hence we observe adsorption steps leading through the immiscibleregion. On the contrary, for example 3, the solution through the mis-cible region is valid.Two remarks are worth making. Due to the different interstitial velo-cities of the convective phases, states are able to propagate fasterthan the interstitial velocity, and can thus appear at the column out-let before the void time (which is impossible when a single-phaseflow is present, as it is commonly the case). This property can alsobe concluded from Figure 5.6a, as propagation velocities of bothtie-line and non-tieline characteristics feature values well above theinterstitial velocity. Secondly, certain states in the miscible regioncan be connected through two possible paths, one leading entirelythrough the miscible region, and one crossing the immiscible region.Neither of these two paths leads to an apparent physical inconsis-tency, but the physically correct path is only one, namely that featu-ring the faster transition emanating from the initial state.In contrast to the fast first transitions of solutions crossing the im-miscible region, the second transitions in some cases propagate veryslowly. A particularly striking example of this behavior consists ofthe desorption profiles of examples 1 to 3, which exhibit a longtailing, as compared to rather short intermediate states when assu-


ming equal velocities of the convective phases (see examples 1 and2 in section 5.3.3). The mathematical reason for this can again befound in the propagation velocities λt along tie-line characteristics(compare Figure 5.6a), which approach 0 when approaching the bi-nodal (i.e. as SR → 0 or SR → 1). Physically, this is explained by thefact that liquid phases at a low phase saturation (< SX

lim) are trappedin the column and theoretically do not propagate.Considering the desorption step in more detail, there are miscibleinitial and feed conditions which do not offer a possible connectionthrough the miscible region, as already observed in section 5.3.3. Inall the 3 cases considered here, Γ1 characteristics emanating from Biin the miscible region intersect with the binodal curve and generatea solution through the miscible region. In turn, when increasing thecontent of component 2 in the feed state B to above 15 % (wt.), apath through the miscible region is the physically valid solution, asit was described in the case of example 1 in section 5.3.3.While desorption paths of examples 2 and 3 in the hodograph planelook similar, and different from the one of example 1, the elutionprofiles of examples 1 and 2 partly overlap, but differ from profile3. The reason for this overlap in the last transition (semi-shocks forexamples 1 and 2), is that major parts of these transitions map onthe same path in the hodograph plane (and thus the wave-shocktransition is identical). In contrast, the intermediate state Id of pro-file 3 is located closer to A, and is therefore directly connected to Athrough a shock, instead of a semi-shock.This qualitative difference in desorption profiles can again be obser-ved in examples 4 and 5 (Figures 5.12 and 5.13). If the intermediatestate Id is located further away from state A (example 4), it is con-nected to A through a semi-shock (wave-shock). In turn, with anintermediate state Id located closer to state A (example 5), the twostates are connected through a shock. In addition, in the adsorptionstep of case 5, a plateau (Pa) can be noticed in the elution profiles,which does not correspond to an intermediate state. It is formedwhen the solution path in the hodograph plane reaches the binodal(where propagation velocities exhibit a discontinuity), since the do-wnstream states in the soluble region propagate considerably fasterthan the subsequent shock, connecting state Pa to the feed state B5.In all other examples considered here, the binodal is crossed directly

5.5 solution of the lumped kinetic model 131

through a (semi-)shock.The latter two examples 4 and 5 were added mainly for the purposeof comparing equilibrium theory solutions and numerical simulati-ons, and for the sake of brevity no further discussion of the observedbehavior and its physical implications is provided for these cases.

5.5 solution of the lumped kinetic model

Finally, the lumped kinetic model of equations 3.10 and 3.11 is dis-cussed as a possible alternative to the equilibrium theory model.The numerical solution of the model based on a semi-discrete finitevolume scheme was discussed in section 3.2.2. At every time stepof the numerical ODE solver, it was checked whether the compo-sition in each discretized cell was located in the miscible or in theimmiscible region, whose boundary was described by ten thousandpoints on the binodal curve calculated through equilibrium calcu-lations. Also for this purpose, a built-in Matlab routine (inpolygon)was used. While for compositions in the miscible region, thermo-dynamic properties (activities, adsorbed phase concentrations) werecalculated analytically, the same properties (as well as phase compo-sitions) for states in the immiscible region were determined throughlinear interpolation from a look-up table, to avoid computationallyexpensive equilibrium calculations. The interpolation procedure isdescribed in detail in Appendix B.1. It is obvious that modificati-ons to the two-phase flow behavior (e.g., equal/ different intersti-tial velocities) can be easily made by changing the fractional flowfunction.The resulting simulations for the same initial and feed conditions asthose considered in sections 5.3 and 5.4 are directly overlaid withthe equilibrium theory solutions in Figures 5.4 and 5.9 to 5.13 (nu-merical simulations are illustrated by thin, black lines). All numeri-cally calculated profiles feature negligible errors in the componentmaterial balances, namely smaller than 0.01%. After comparing nu-merical simulations and equilibrium theory solutions, three diffe-rent types of situations are observed, where the comparison highlig-hts (1) quantitative agreement between the two solutions, (2) minordifferences, with solutions of the lumped kinetic model convergingto the equilibrium theory solutions as the number of discretization


steps increases, and (3) major differences, with solutions of the lum-ped kinetic model not converging with increasing numbers of dis-cretization steps.Most of the investigated conditions, accounting both for equal andfor different velocities of the convective phases, belong to the firstcategory, exhibiting a good agreement between equilibrium theoryand numerical solutions, as expected. In a few cases, some differen-ces (category 2) are noticeable, such as in case 2 in Figure 5.4, wherethe second shock transition appears later in the numerical simula-tion. In this case, the numerical simulation approaches the equili-brium theory solution with an increasing number of discretizationsteps. We suspect that this difference is due to a numerical error re-lated to the peculiar adsorption properties along the solution path:the adsorbed amount remains constant throughout the entire firsttransition, but changes strongly over a comparatively small span ofliquid phase concentrations during the second transition, so as verysmall errors in the level of the intermediate state have strong effectson the elution time of this second transition. It seems like the nume-rical solution “captures the correct speeds to the wrong shocks”, asit was observed previously in the literature [88]. It should be poin-ted out that also in the presence of numerical errors, the errors inthe component material balances are negligible (as stated above). Inthe desorption of case 2 (see Figure 5.4), for example, the error in thematerial balance due to the later second transition is counterbalan-ced by a slightly earlier first transition (compared to the equilibriumtheory solution).Finally, for cases 2 and 3 accounting for different velocities of theconvective phases (see Figures 5.10 and 5.11), major differences (ca-tegory 3) are observed. While for case 2, these differences are local,and a general agreement between the two profiles (calculated withthe equilibrium theory model and the lumped kinetic model) is stillachieved, for case 3 the numerical solution fails completely. In bothcases, the numerical solution does not converge upon increasing thenumber of discretization steps, but numerical issues deteriorate andthe deviation with respect to the equilibrium theory solution beco-mes more pronounced.We suspect that the system is prone to numerical problems especi-ally close to the plait point, towards which the length of tie-lines

5.6 conclusions 133

decreases and thermodynamic properties change quickly, which isin line with similar issues reported in a thorough and elucidatingnumerical study of two-phase flow by Mallison et al. [88]. The ci-ted contribution also reports similar numerical artifacts (oscillations,overshooting) as the ones observed in Figure 5.10. A change in thetime stepping procedure (e.g., using ode45 instead of ode113), aswell as a test of different flux limiters (such as a 3rd-order WENOscheme [89] instead of the VanLeer flux limiter) did not achieve anoticeable improvement of the numerical results. We have also triedto introduce a dependence of the fractional flow function on thetie-line location, so as the two-phase flow approaches single-phaseflow properties (fj = Sj) towards the plait point as mentioned insection 3.3.3, which did not resolve the issue. Furthermore, it is no-teworthy that for cases 4 and 5, with very similar desorption pathsof the equilibrium theory solution as cases 2 and 3, respectively,the lumped kinetic model achieves correct results. Apparently, nu-merical issues are avoided for feed conditions located closer to theinitial conditions, in the immiscible region, but solving these issuesbeyond the attempts made and described above is beyond the scopeof this work.

5.6 conclusions

We have solved and evaluated an equilibrium theory model and alumped kinetic model, accounting for adsorption and convection ofmultiple phases through a porous medium. The equilibrium theorymodel assumes thermodynamic equilibrium between convective pha-ses, and between convective and adsorbed phases, neglecting disper-sive effects. While the lumped kinetic model is able to account forcertain kinetic limitations, such as dispersive effects or mass trans-fer resistances between the liquid phases and the adsorbed phase(all lumped in one term), the assumption of thermodynamic equi-librium between the convective phases is maintained. The modelswere set up to study two-phase flow effects in the context of liquidchromatography (where commonly only one convective phase andadsorption is accounted for). However, it is also of great interest forapplications in natural reservoirs, such as enhanced coalbed met-hane recovery or CO2 sequestration (where adsorption effects have


most often been neglected).The equilibrium theory model was solved using the method of cha-racteristics, considering both equal and different velocities of theconvective phases. Especially in the case of different velocities, thederivation of solutions becomes very complex and different initialand feed conditions offer a great diversity of qualitatively differentsolutions. However, these solutions also offer a deep understandingof the physical behavior implied by the combination of adsorptionand two-phase flow.In contrast, numerical simulations offer a great flexibility towardsadaptations of the model, such as the implementation of differentadsorption isotherms or fractional flow functions. Solutions to spe-cific initial and feed conditions can be obtained in a more genericmanner; however, due to the necessary determination of phase splitsand compositions, simulations are computationally expensive andtime consuming. In addition, the current study has revealed ma-jor numerical issues of the lumped kinetic model for specific initialand feed conditions, especially when accounting for different velo-cities of the convective phases and for distant initial and feed states,which lead to solution paths passing close to the plait point.

6E X P E R I M E N TA L VA L I D AT I O N

6.1 introduction

In the previous chapters, physical properties in the presence oftwo fluid phases were characterized experimentally and describedtheoretically, and model equations, accounting for adsorption andmulti-phase flow, were derived and solved. In this chapter, we wantto validate this model, as well as its underlying assumptions andphysical relationships. For this purpose, dynamic column experi-ments with different (and immiscible) initial and feed conditionsare performed with the ternary system PNT - methanol - water andthe adsorbent Zorbax 300SB-C18 (in the absence of TBP), which wasalready characterized in chapter 4 and was also considered in thecontext of the model solution in chapter 5. Model validation shallbe achieved by comparison of the obtained experimental data withmodeling results.The chapter is structured as follows: In section 6.2, we describe theexperimental procedure applied for the validation experiments. Theexperimental results and their comparison to model simulations arepresented in section 6.3. Finally, conclusions (section 6.4) are drawnfrom the experimental behavior and the quality of the model des-cription.

6.2 experimental

6.2.1 Material and basic methods

Validation experiments were performed with the system phenetole,methanol and water with the adsorbent Zorbax 300SB-C18. The che-mical substances and experimental equipment employed for the pre-

Results presented in this section have been reported in: Ortner, F.; Mazzotti, M. Two-phase flow in liquid chromatography, Part 1: Experimental investigation and theore-tical description Ind. Eng. Chem. Res. 2018, 57(9), 3274-91

136 experimental validation

paration of solutions and for chromatographic experiments are thesame as the ones discussed in section 4.3.

6.2.2 Dynamic column experiments involving adsorption and two-phaseflow

Displacement experiments with partially miscible initial (water-me-thanol mixtures) and feed (PNT-methanol mixtures) states, whichwere not in thermodynamic equilibrium, were carried out to va-lidate the model assumptions and the implemented relationships.For this purpose, the (50x4.6 mm) Zorbax 300SB-C18 column wasequilibrated with the initial state, and 4 mL of the feed state wereinjected through the 5 mL injection loop. When closing the injectionloop, the feed state in the column was re-displaced by the initialstate. With this procedure, the entire chromatographic cycle, i.e. theadsorption and the desorption step, was considered. As in the bi-nary breakthrough and imbibition/drainage experiments (see secti-ons 4.3.5 and 4.3.7), an additional pump was connected after the UVdetector to automatically dilute the eluate with methanol at a ratioeluate : diluent 1.2 : 10 (v:v). The diluted eluate was gathered infractions and the concentrations of PNT in the fractions were ana-lyzed offline by HPLC to obtain the flow profile FP. Fractions withhigh PNT concentrations were further diluted manually at a ratiosample : methanol 1:50 (v:v) before analysis. For details concerningthe HPLC analysis, see section 4.3.1.

6.3 comparison of model predictions and experimen-tal data

The validation of the model is performed with the ternary systemPNT - methanol - water and the adsorbent Zorbax 300SB-C18. Ac-cordingly, the UNIQUAC parameters on the right of Table 4.2, theanti-Langmuir isotherm for PNT (equation 4.41), and the fractionalflow functions determined in section 4.8 were considered in the mo-del equations presented in section 3.2.1. The model was solved byapplying the method of characteristics (see chapter 5), and concen-tration and flow profiles were derived for different initial and feed

6.3 comparison of model predictions and experimental data 137

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80

20 40 60 80

20

40

60

80

PNT

MeOH

H2O

A2

B4

A1

B2

B1

B3

Figure 6.1: Initial (Ai) and feed (Bi) states (mass fractions), investigated inthe validation experiments. Combinations of initial and feed sta-tes are connected by a continuous line.

sates. The simulated profiles were then compared to experimentalflow profiles obtained at the same initial and feed conditions. Itshould be stressed that in this context, the model was used in afully predictive manner (i.e. no adjustment of neither model para-meters nor implemented relationships).We investigate 8 different conditions, arising from the combination

of two different initial states (A1, A2, both mixtures of methanol andwater) with four different feed states (B1, B2, B3 and B4, all mixturesof PNT and methanol), which are visualized in the ternary phase di-agram shown in Figure 6.1. The resulting concentration (predicted)and flow (experimental and predicted) profiles are presented in Fi-gure 6.2 for all conditions with initial state A1 and in Figure 6.3 forall conditions with initial state A2. Note that the combination of ini-tial state A1 and feed states B1 and B3 corresponds to the cases 1 to3 discussed theoretically in section 5.4.3.The initial and feed states of the validation experiments are not in

thermodynamic equilibrium, and they are all located in the solubleregion, far from the binodal curve. As a consequence, the liquidphase activities of PNT aP of initial and feed states are not identical,resulting in ad- and desorption, which also brings along a changein porosity. Apart from conditions with feed state B4, all other con-ditions lead to the occurrence of states within the immiscible region,


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t [min]

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FP [

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A1 - B

4

Figure 6.2: Concentration (predicted, left column) and flow (experimentaland predicted, right column) profiles of PNT with initial stateA1 and feed state (a) B1, (b) B2, (c) B3, (d) B4. Entire chromato-graphic cycles, i.e. adsorption and desorption steps (separatedby a dashed vertical line), are considered. Data points in darkand light blue indicate experimental replicates. Red, green andblack lines present model predictions based on the different as-sumptions discussed in the text.


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gL

-1]

A2 - B

2

0 2 4 6 8

t [min]

0

200

400

600

800

1000

CP [

gL

-1]

(c)A

2 - B

3

0 2 4 6 8

t [min]

0

200

400

600

800

1000

FP [

gL

-1]

A2 - B

3

0 2 4 6 8

t [min]

0

100

200

300

400

500

600

700

CP [

gL

-1]

(d)A

2 - B

4

0 2 4 6 8

t [min]

0

100

200

300

400

500

600

700

FP [

gL

-1]

A2 - B

4

Figure 6.3: As in Figure 6.2, but for initial state A2.


thus resulting in two-phase flow. Owing to the fact that initial andfeed states are not in thermodynamic equilibrium and are not loca-ted on the binodal curve, no phase becomes permanently trappedwithin the column, as it was the case in displacement experimentsperformed in section 4.8. It is possible that phases are trapped tem-porarily, but are then slowly consumed by a continuous adjustmentof the thermodynamic equilibrium between the liquid phases.Let us first comment on the predicted profiles presented in Figu-res 6.2 and 6.3, which are based on three different models. Thefirst (red lines, designated as “ET”) is the model discussed in thiswork, accounting for both adsorption and different velocities of themultiple convective phases, and implementing all the relationshipsestablished previously. The other two models, designated as “ET(ads)” and “ET (hydro)”, are simplifications of the first model: Thesimplification “ET (ads)” (black lines) accounts for adsorption ef-fects (via the established adsorption isotherm), but assumes equalinterstitial velocities for all convective phases, i.e. it neglects hydro-dynamic effects (fj = Sj and Fi = Ci). This model represents theequilibrium theory model as it is commonly used to describe chro-matographic processes without liquid-liquid phase separation. Thesecond simplification “ET (hydro)” (green lines) accounts for diffe-rent velocities (via the established relative permeability functions),but it neglects adsorption effects (i.e. nP = 0). This type of modelis commonly used to describe two-phase flow in porous natural re-servoirs, where interactions with the rock matrix are ignored.4 Thesolution by the method of characteristics for model “ET (hydro)” isderived by Ref. [4]. For the models “ET (ads)” and “ET”, a detailedderivation of the solutions (specifically for the conditions with ini-tial state A1 and feed states B1 to B3) is provided in chapter 5. Solu-tions for conditions with initial state A2 and/or feed state B4 can bederived likewise, as outlined in sections 5.3 and 5.4.When assuming equal velocities (model “ET (ads)”), concentration(Ci) and flow (Fi) profiles are identical, whereas they differ consi-derably in the case of different velocities (models “ET (hydro)” and“ET”). While the overall liquid concentration Ci corresponds to theamount of component i per overall void volume at a certain positionin the column, the overall fractional flow Fi describes the amount ofcomponent i moving per overall volume of flow (over all the fluid


phases). Hence, the concentration profiles calculated at location L inFigures 6.2 and 6.3, left column, describe the concentration of PNTin the void space at the end of the column, whereas the flow profilesat position L (right column in Figures 6.2 and 6.3) correspond to theconcentration of PNT in the (single or two-phase) eluate. Ci and Fi,and hence concentration and flow profiles, are only identical in thecase of a single-phase flow, or if multiple fluid phases move withidentical interstitial velocities.Since conditions with feed state B4 map on paths in the hodographplane which do not pass through the immiscible region, there is notwo-phase flow occurring in the corresponding solution. As a conse-quence, the two models “ET (ads)” and “ET” yield identical elutionprofiles. In contrast, the model “ET (hydro)”, neglecting adsorptioneffects, predicts a simple plug flow displacement of the initial stateby the feed state, occurring at the dead time both during the displa-cement and during the re-displacement step.At all other conditions (feed states B1 to B3), predictions of all threemodels differ. However, similar elution profiles are produced withthe two models accounting for different velocities, while profilesbased on the model assuming equal velocities exhibit considerabledifferences. This indicates that, at least under the established ad-sorption and flow conditions, hydrodynamic effects dominate overadsorption effects in the immiscible region. Discrepancies are moredistinct during desorption than during adsorption. When displa-cing a PNT-lean phase (non-wetting) by a PNT-rich phase (wetting)during adsorption, no phase becomes temporarily trapped whenconsidering different velocities, since the residual saturation Sr = 0.In contrast, during desorption, i.e. displacing a PNT-rich (wetting)phase by a PNT-lean (non-wetting) phase, the ratio of wetting phasetemporarily trapped is very high, since Si = 0.32. This portion oftrapped phase is removed very slowly by constant adjustment of thethermodynamic equilibria between the liquid phases; this explainsthe long “tailing” in the desorption steps when accounting for diffe-rent velocities. In turn, when assuming equal velocities, both liquidphases always remain convective, and the PNT-rich phase can thusbe eluted much faster.In a next step, let us compare the model predictions with the expe-rimental profiles. We have no possibilities to experimentally deter-


mine compositions at a location inside the column (concentrationprofiles Ci at this location), but we can only determine the composi-tion of the eluate at the column outlet (flow profiles Fi at position L).Here, we focused on the determination of the flow profiles of PNT.Experimental replicates of several conditions show a good reprodu-cibility (see overlaid elution profiles in darker and lighter blue inFigures 6.2 and 6.3).The models accounting for adsorption (“ET” and “ET (ads)”) accu-rately describe the conditions with feed state B4, thus providing agood confidence in the established relationship describing the ad-sorption behavior (no two-phase flow under these conditions). Con-ditions with feed states B1 and B2 are described quantitatively whenaccounting for both adsorption and different velocities of the con-vective phases. The last part of the desorption profile (wave-shocktransition) is identical for both feed states with the same initial stateA1 or A2, since it maps on the same path in the hodograph plane(compare section 5.4).Elution profiles for conditions with feed state B3 clearly differ fromthe profiles for feed states B1 and B2: the last part of the desorp-tion profile is considerably shorter and does not overlap with theelution profiles for the other two feed states, both experimentallyand in simulations using the “ET” model. According to the equili-brium theory solution, this last part corresponds to an intermediatestate (plateau), followed by a shock transition (which differs fromthe wave-shock transition for feed states B1 and B2). However, theseconditions can only be described qualitatively by the “ET” model.In fact, the experimental profiles exhibit a shorter desorption pro-file, located between the predictions assuming different velocities,and the one assuming equal velocities. This quantitative mismatchis due to the established relative permeability function, neglectingan impact of the interfacial tension between the liquid phases onthe hydrodynamic behavior. The solution path for conditions withfeed state B3 in the hodograph plane passes through the immiscibleregion, very close to the plait point (compare Figure 5.11), wherethe two convective phases have similar compositions (low interfa-cial tension) and the flow behavior should thus approach the singlephase flow behavior (equal velocities). This is not taken into accountby the established relative permeability function. The experimental

6.4 conclusions 143

profile is therefore located between the predictions assuming diffe-rent velocities (but independent of the interfacial tension), and theprediction assuming equal velocities.From the comparison of the three model predictions with the experi-mental profiles, one can conclude that, in the immiscible region andfar from the plait point, the impact of the hydrodynamic two-phaseflow behavior on the elution profiles dominates over the impact ofthe adsorption behavior. Approaching the plait point, properties ofthe two convective phases become similar, the interfacial tensionbetween the phases decreases, and an equal velocity behavior is ap-proached. At the same time, the impact of the adsorption effects onthe elution profile increases. In the miscible region, with only oneconvective phase, hydrodynamic effects are absent and the adsorp-tion behavior is the dominating factor.

6.4 conclusions

This chapter concludes the development of a model description oftwo-phase flow in liquid chromatography, providing an experimen-tal validation of the model equations and solution. An equilibriumtheory model accounting for both adsorption and two fluid phaseswith different velocities was solved (see chapter 5) and applied herein a fully predictive way to simulate concentration and flow profi-les for specific initial and feed conditions. The chosen initial andfeed conditions were not in thermodynamic equilibrium, and thusresulted in adsorption or desorption during the elution, and six outof eight conditions led to two-phase flow. The resulting simulatedprofiles were compared to profiles obtained from dynamic columnexperiments at the same initial and feed conditions, and a quantita-tive agreement was achieved for a majority of the conditions.The good agreement between simulation results and experimentaldata confirms the validity of the mathematical model developed, interms of both physical assumptions and simplifications made andof specific mathematical relationships utilized. This demonstrates agood understanding of both the experimental system and the im-plications of multiple convective phases in liquid chromatography.In the following, we want to summarize the most important (expe-rimental and theoretical) findings, and evaluate them against the


background of possible applications in the context of liquid chro-matography.

• The comparative analysis of experimental results and modelpredictions clearly demonstrates a transition from a dominanceof hydrodynamic effects in the immiscible region to an adsorp-tion dominated behavior in the miscible region (where onlyone convective phase is present). The dominance of hydrody-namic effects in the immiscible region is particularly strong(indicated by fractional flows differing considerably from sa-turations, i.e. where fj 6= Sj), if one convective phase is consi-derably more wetting than the other, and when the interfacialtension between the convective phases is large. In the consi-dered system, this is the case for conditions in the immiscibleregion located far from the plait point. As the plait point isapproached during a dynamic column experiment, the con-vective phases feature similar properties (e.g. similar composi-tions, small interfacial tension between the convective phases),hence the dominance of hydrodynamic effects decreases.

• Strong hydrodynamic effects, i.e. big differences in the veloci-ties of the convective phases, correspond to components beingstrongly retained in the column and being eluted very slo-wly, as it is the case for the conditions A1-B1 and A1-B2 insection 6.3. This effect is in general undesired in liquid chro-matography. It can thus be concluded that strong hydrodyn-amic effects have a negative impact on the chromatographicseparation.

• Operation under two-phase flow conditions hence involves atrade-off between high productivities resulting from high li-quid phase concentrations (which are no longer limited to themiscible region), and between the discussed two-phase flowbehavior, resulting in slow elution and possibly high dilutionof the target product. Accepting a two-phase flow can be thepreferred option if the increase in productivity due to higherfeed concentrations outweighs the disadvantages correlated tothe two-phase flow behavior. This might be the case if hydro-dynamic effects are low to moderate (i.e. low interfacial ten-sion between the convective phases), or if the solubility of the

6.4 conclusions 145

different components is extremely low, implying severe con-straints to productivity under single-phase conditions.

• The dynamic column experiments considered in this chaptercreate a two-phase flow by injecting a feed state, which is im-miscible with the initial state. Another interesting phenome-non to be studied is a spontaneous phase split within the co-lumn, due to a considerable enrichment of one or more compo-nents, as it was observed in a previous study.25,26 In principle,the presented model could also account for the latter pheno-menon, with the underlying assumption that the phase splitis not kinetically hindered. As phase separations are proneto kinetic limitations, the physical behavior and mathematicaldescription of this phase split requires further investigation.

• The current validation is limited to a ternary system withonly one adsorbing component (which does not feature theformation of a spontaneous phase split, as discussed above).Since chromatographic processes commonly aim at the sepa-ration of different adsorbing components, it is crucial to ex-tend this study to binary and multicomponent systems. Theequilibrium theory model established in this work is able toaccount for a random number of components. Further, the es-tablished relationships describing physical properties, such asisotherms based on liquid phase activities, or fractional flowfunctions based on Brooks-Corey correlations, form a solid ba-sis, which however needs to be extended to account for a hig-her number of components. In chapter 4, we have established athermodynamic model describing fluid phase equilibria for aquaternary system, as well as a thermodynamically consistentdescription of binary adsorption. In a similar manner, fluid-dynamic relationships (i.e. fractional flow functions) need tobe extended, considering that a higher number of componentsmight also feature more than two fluid phases in equilibrium.Finally, the derivation of an equilibrium theory solution beco-mes more difficult and elaborate, since the system of PDEs tosolve increases linearly with the number of components. Mo-reover, the creation of a look-up table for phase equilibria toapply in the model solution increases in complexity, with mul-


tiple regions of different types of phase-equilibria forming (Nc-1)-dimensional bodies (Nc being the number of components).

7E VA L U AT I O N O F T W O - P H A S E F L O W I N AC H R O M AT O G R A P H I C R E A C T O R

7.1 introduction

Reactive chromatography, a hybrid process coupling chemical re-action and chromatographic separation, has received great attentionin the past decades (for an overview see Refs. [90, 91]). This processis particularly advantageous in the context of equilibrium limitedreversible reactions, as the selective removal of reaction productsfrom the reaction site allows to overcome the conversion limit impo-sed by chemical equilibrium.However, it can occur that products and/or reactants are only par-tially miscible.92 In this case, enrichment of the partially immis-cible components due to ongoing reaction within the chromato-graphic reactor can result in a liquid-liquid phase split - and con-sequently in two convective phases. A possible example of such asystem consists of esterification reactions, which constitute the re-action class considered most frequently in the context of chromato-graphic reactors.93–98 In these reversible, equilibrium limited reacti-ons, an alcohol (D) and an acid (C) react to form water (B) and anester (A) according to C + D −−−− A + B. In the presence of a poly-meric ion exchange resin acting as catalyst and selective adsorbent,the two products water and ester are commonly the strongest andthe weakest adsorbing components, respectively, while the two re-actants adsorb with intermediate strength.93,96,99 As a consequence,the two products are simultaneously separated and removed fromthe reaction site.In the case of long-chained, hydrophobic esters and reactants, mis-cibility of water and ester, and possibly of water and one or bothreactants, can be limited. As a consequence, the enrichment of wa-

Results presented in this section have been reported in: Ortner, F.; Mazzotti, M. The-oretical evaluation of two-phase flow in a chromatographic reactor Ind. Eng. Chem.Res. 2018, 57(16), 5639-52

148 evaluation of two-phase flow in a chromatographic reactor

ter due to ongoing reaction in the column can cause a phase splitand possibly two-phase flow.The occurrence of multiple convective phases in the context of li-quid chromatography has not been discussed in the literature untilrecently.25 Due to a lack of understanding, a common approachis to adapt process conditions to prevent a phase split, which forsome reactive chromatographic processes severely limits the appli-cable concentration ranges and the process performance, hence itsattractiveness.98 Yet it was shown experimentally that exceeding so-lubility limits in the exemplary case of methylformate hydrolysis ina packed-bed reactor (operated at steady state without chromato-graphic separation) can significantly increase the yield.100

In this chapter, we build on the deep physical understanding oftwo-phase flow in non-reactive chromatography attained in the pre-vious chapters, and we investigate theoretically the physical impli-cations of a phase split and subsequent two-phase flow in a chro-matographic reactor, as well as its consequences for process perfor-mance. We evaluate an exemplary system consisting of n-hexanol(D) and acetic acid (C), reacting to water (B) and n-hexyl-acetate (A).The evaluation is based on an equilibrium theory model, assumingthermodynamic equilibrium between all (convective and adsorbed)phases, as well as reaction equilibrium. Furthermore, one of the re-actants (in esterification reactions commonly the alcohol),93,97,98 isassumed to be present in excess (solvent), while all other compo-nents are diluted in the solvent. A simple analytical solution forsuch equilibrium theory model with a single convective phase hasbeen presented previously.101–103 This model shall now be extendedand solved, when accounting for the presence of multiple convectivephases (in the spirit of the previous chapters 3 to 6).The chapter is structured as follows: In section 7.2, we explain theunderlying model assumptions and derive the model equations. Then,the exemplary esterification system is introduced, characterizingcrucial system properties, such as thermodynamic equilibria, as wellas reaction and two-phase flow behavior (section 7.3). The equi-librium theory model is used to derive elution profiles for speci-fic initial and feed conditions, located in the single-phase regionand hence preserving a single-phase flow, or located in the two-phase region and thus resulting in phase split and two-phase flow

7.2 equilibrium theory model 149

(section 7.4). Qualitative properties of the elution profiles are dis-cussed (section 7.4.1), and the process performance in the presenceof single-phase and two-phase flow is evaluated (section 7.4.2). Forthe sake of brevity and readability, the underlying mathematicalderivation of the equilibrium theory solution, which is novel forthe applied model and provides a deeper mathematical and physi-cal understanding, is explained in Appendix C.1. Conclusions aredrawn in section 7.5.

7.2 equilibrium theory model

7.2.1 Derivation of model equations

We set up the following conservation laws, accounting for adsorp-tion, multi-phase flow and chemical reaction:

∂Ci∂τ

+ ν∂ni∂τ

+∂Fi∂ξ

= κiR, i = 1...3. (7.1)

Note that, in the context of the mathematical model, componentsA to D are renamed to components 1 to 4. The independent di-mensionless time and space variables are designated as ξ = x/Lcand τ = tu/εLc, where Lc is the column length and u is the su-perficial velocity. It is worth pointing out that in this chapter, thedefinition of τ differs slightly from the one in the previous chapters3 to 6 (defined in equation 3.7). Furthermore, in contrast to previouschapters where concentrations were given in mass per unit volume,all the concentrations in this chapter, such as the liquid and adsor-bed phase concentrations ci and ni, overall liquid concentrations Ciand overall fractional flows Fi are defined in mole per unit volumeof adsorbed or liquid phase. Since molar concentrations are usedconsistently throughout this chapter, and in order to clearly diffe-rentiate between concentrations and the transformed variables Ci,we refrain from indicating these molar concentrations by an over-bar (as it was done in chapter 4). The phase ratio ν is defined as(1 − ε)/ε, with ε denoting the total porosity. The adsorbed phaseconcentration of component i (in mole per unit volume of adsor-bent) is denoted as ni, while R is the overall specific reaction rate ofthe equilibrium reaction (over all convective phases) and κi is the


stoichiometric coefficient of the reaction, equaling 1 for components1 and 2 and -1 for component 3. Finally, variables Ci and Fi describethe overall liquid concentration and overall fractional flow of com-ponent i over all convective phases.It should be pointed out that only conservation laws for the firstthree components are considered, as the fourth component can bedetermined through the stoichiometric correlation

4∑i=1

Zi = 1, (7.2)

where Zi is the mole fraction of component i over all convectivephases. For the interconversion between Zi and Ci, see AppendixC.2. In this work, the fourth component (in our case the alcohol)is assumed to be present in excess, as a solvent. The overall liquidconcentrations Ci of the other three components (one reactant andthe two products) are assumed to be rather low, such that volume-tric changes due to adsorption, reaction or mixing are neglected. Asa consequence, constant volumes of adsorbed and convective pha-ses (constant porosity ε) and a constant overall flow rate (constantsuperficial velocity u) are assumed in equations 7.1. For convectivephases in particular, we assume specific volumes to equal the speci-fic volume of the bulk component (alcohol); details concerning thisassumption are discussed in Appendix C.2. In addition, we neglectheat effects upon adsorption or reaction and consider a thermod-stated chromatographic setup, hence the process is assumed isot-hermal (in agreement with various literature studies of differentreactive chromatographic systems92,93,104–106).The overall liquid concentrations Ci and overall fractional flows Fiare defined (analogue to equations 3.2) as:

Ci =

NP∑j=1

zijρmixj Sj =

NP∑j=1

cijSj, Fi =

NP∑j=1

zijρmixj fj =

NP∑j=1

cijfj (7.3)

Here, NP corresponds to the number of convective phases, zij andcij denote the molar fractions and concentrations of component i inphase j, respectively, and ρmix

j is the molar density of phase j.It is further assumed that all convective phases are in thermodyna-mic equilibrium (i.e., activities ai of each component are identical in

7.2 equilibrium theory model 151

all convective phases), and that convective phases and the adsorbedphase are in equilibrium (which is described by an adsorption isot-herm, being a function of the liquid phase activities, as discussed insection 7.3).Under the additional assumption of chemical equilibrium, the equi-librium relationship, defined through the equilibrium constant, pro-vides another constraint, namely:

K =a1a2a3a4

(7.4)

With this constraint, the degrees of freedom (number of unknownvariables), can be further reduced from 3 to 2. This means that, ina quaternary system, the concentrations of only 2 components uni-quely define the system at chemical and phase equilibrium.Since conservation equations 7.1 are linear in the reaction rate R, thislinear term can be eliminated by summation, as described in detailin Refs. [101–103, 107]. Doing so, the system of conservation equa-tions is reduced to two homogeneous partial differential equations:

∂Ci∂τ

+ ν∂ni∂τ

+∂Fi∂ξ

= 0, i = 1, 2. (7.5)

For the assumed type of reaction, xi = xi + x3, with x = n, c,C, F.Accordingly, one can also define Ci and Fi as:

Ci =

NP∑j=1

cijSj, Fi =

NP∑j=1

cijfj. (7.6)

Note that the reduction to the variables C1 and C2 (both ni = ni(C)

and Fi = Fi(C)) only provides a sufficient, unique description of thesystem under the assumption of chemical and phase equilibrium(where degrees of freedom are reduced to 2).Equations 7.5 constitute a system of first order, homogeneous par-tial differential equations, which can be solved analytically with themethod of characteristics,23,24 as described in greater detail in Ap-pendix C.1 (and analogous to the solution presented in chapter 5).


7.2.2 Critical assessment of the model assumptions

We would like to comment on the model assumptions outlinedabove, in particular regarding their physical relevance. As statedabove, one of the reactants (the alcohol) is assumed to be presentin excess, such that overall liquid concentrations of the other threecomponents are comparatively low. We would like to point out thatthis assumption does not entail thermodynamic ideality: Operatingat elevated concentrations close to or even above the solubility limitclearly requires the consideration of thermodynamic nonideality ofthe convective phases, and of a non-linear adsorption equilibrium(based on liquid phase activities). The main purpose of assumingone component to be present in excess is to allow neglecting volu-metric changes due to adsorption, mixing or reaction. It is obviousthat the assumption of a constant specific volume of the convectivephases (equaling the specific volume of the excess component), be-ars a certain error, which, as assessed in greater detail in AppendixC.2, is not expected to exceed 4 vol % for the system and conditionsconsidered in this work. We are hence convinced that, under theconditions investigated, the effects of volumetric changes are minoras compared to the impact of adsorption and hydrodynamic proper-ties on the elution behavior. It is also worth noting that relaxing theassumption of negligible volumetric changes results in a considera-ble complication of the model equations and its solution, since inthat case a variable porosity ε and/or superficial velocity u wouldhave to be considered.The second major assumption of the equilibrium theory model isthat of negligible kinetic limitations of any kind, entailing thermo-dynamic equilibria between all convective and adsorbed phases, aswell as chemical equilibrium of the reversible reaction at any timeand position in the column. As we could observe in chapter 6, thedisplacement of partially miscible states of a different chromato-graphic system, involving adsorption and two-phase flow, couldbe quantitatively described by a similar equilibrium theory modelas the one suggested in this chapter (also neglecting kinetic limi-tations). In general, we expect thermodynamic equilibria betweenthe different convective and adsorbed phases to be attained fast inchromatographic systems, due to the large interfacial areas between

7.3 hexyl-acetate system 153

the phases. In contrast, a spontaneous phase split, i.e. the creationof a new phase within the column, as well as reversible chemicalreactions, are prone to kinetic limitations. These aspects clearly re-quire an in-depth examination for the system under investigation,to assess whether the assumption of thermodynamic and chemicalequilibrium is close enough to reality.

7.3 hexyl-acetate system

We consider an exemplary model system, namely the esterificationof n-hexanol (component D/4) and acetic acid (C/3) to water (B/2)and n-hexyl-acetate (A/1), with the strongly acidic ion-exchange re-sin Amberlyst CSP2 as catalyst and adsorbent, operating at 298.15 K.The system has not been characterized experimentally by us in thescope of this study, but realistic model and parameter assumptionsare made based on literature data.

7.3.1 Thermodynamic equilibria

The thermodynamic behavior of convective (liquid) phases is esti-mated using the fully predictive modified UNIFAC model.75 Phaseequilibrium calculations were performed as outlined in AppendixA.3.1. Three of the four ternary diagrams, resulting when one ofthe four components is absent (zi = 0), are presented in Figure7.1. Blue tie-lines, connected through binodal curves, indicate theliquid-liquid phase regions, while white regions correspond to mis-cible regions of a single liquid phase. The fourth ternary diagramfor the system of n-hexanol, n-hexyl-acetate and acetic acid is trivial,since this ternary system is completely miscible. To further illustratethe behavior of the quaternary system, the simulated tie-lines in thequaternary diagram are presented in Figure 7.2.While n-hexanol, n-hexyl-acetate and water are all completely mis-cible with acetic acid, a considerable miscibility gap exists for bothn-hexanol and n-hexyl-acetate with water. As acetic acid is addedto these mixtures, the miscibility increases.In the equilibrium theory model presented in section 7.2, thermody-namic equilibrium (as predicted using the modified UNIFAC mo-


del) between the convective phases is assumed at every positionand time in the column, so as all species feature identical liquidphase activities in all convective phases.


20

40

60

80

20 40 60 80

20

40

60

80

H2O

AcOH

HexOH

20

40

60

80

20 40 60 80

20

40

60

80

AcOH

H2OHexOAc

20

40

60

80

20 40 60 80

20

40

60

80

H2O

HexOH

HexOAc

Figure 7.1: Ternary diagrams (mole fractions) of the model system (re-sulting from one of the four components being absent), pre-dicted by the modified UNIFAC model. Liquid-liquid equili-bria are indicated by blue tie-lines. The fourth ternary systemresulting when H2O is absent is completely miscible.


Furthermore, thermodynamic equilibrium is assumed between theconvective phase(s) and the adsorbed phase, which is describedthrough an adsorption isotherm, being a function of the liquid phaseactivities. As observed in chapter 4, the adsorption behavior of mul-ticomponent systems in the nonideal range (which is certainly thecase here, dealing with compositions close to and above the solubi-lity limit), can often be described with high accuracy as a functionof the liquid phase activities. In accordance with the literature,99 weuse a competitive Langmuir isotherm for this system:

ni =qsati Kiai

1+3∑k=1

Kkak

, (7.7)

where qsati and Ki are the saturation capacity and adsorption equili-

brium constant of component i, respectively. Adsorption of compo-nent 4 is neglected (qsat

4 = K4 = 0) because of the assumption thatit behaves as a solvent, while parameters for components 1 to 3 areestimated from Ref. [99] and reported in Table 7.1.

Table 7.1: Isotherm parameters for the exemplary system, estimated fromRef. [99].

component qsati [molL−1] Ki [-]

1 1.83 0.50

2 14.65 8.10

3 4.40 1.70

7.3.2 Reaction equilibrium

The chemical equilibrium between the esterification reaction and itsreverse ester hydrolysis reaction is defined through its equilibriumconstant K, see equation 7.4. From Ref. [99], K is estimated to beequal to 22.07 at 298.15 K. As discussed in section 7.2, consideringchemical equilibrium for the quaternary system at a specified tem-perature, the degrees of freedom determining possible compositions


8060402080

20

H2O

2040

80

40

80

40

60

40

20

6060

60

20

20

40

60

80

AcOH

80

HexOHHexOAc

Figure 7.2: Thermodynamic behavior of the quaternary model system, pre-dicted by the modified UNIFAC model. Light blue lines are tie-lines, connecting liquid phases in equilibrium, dark blue pointsindicate plait points.

reduce to two. Thus, compositions in the quaternary system in che-mical equilibrium are uniquely defined through two variables only,which may be z1 and z2, but they can also be chosen to be C1 andC2, as in the equilibrium theory equations 7.5 (for an interconver-sion between the variables see Appendix C.2).Concerning the two-phase region, not all tie-lines fulfill the condi-tion for reaction equilibrium (equation 7.4). Tie-lines fulfilling suchcondition are all located on the same surface, which is indicatedthrough exemplary tie-lines in Figure 7.3a. Converting the compo-sition z of the phases in equilibrium (ends of the tie-lines) to thevariables c1 and c2, the surface of tie-lines fulfilling chemical equi-librium can be mapped onto the (C2, C1) plane (compare Figure7.3b). It can be noted that on the horizonal axis (C2 = 0), C1 cor-responds to C1, the concentration of n-hexyl-acetate, while on thevertical axis (C1 = 0), C2 corresponds to C2, the concentration ofwater. The range of values spanned by the vertical axis is conside-rably larger than that spanned by the horizontal axis, which is dueto the fact that water (component 2) has a lower molecular weight,hence a higher molecular density, than n-hexyl-acetate (component1).


8060

4060

4020

80

20

40

60

20

HexAc

80

20

40

H2O

60

80

60

20

HexOH80 60 40 20

AcOH

80

40

(a)

0 2 4 6 8 10 12

C1 [mol L−1]

0

10

20

30

40

50

C2[m

olL−1]

(b)

Figure 7.3: Liquid phases in thermodynamic and reaction equilibrium, in-dicated by red tie-lines in the quaternary system in (a). In (b),these tie-lines are mapped on the (C2, C1) plane.


7.3.3 Two-phase flow behavior

As in the previous chapters, we define a relationship between thevolumetric fractional flows fj and the phase saturations Sj. For thispurpose, with the fractional flows fj being a function of the relativepermeabilities krj and the phase viscosities ηj (compare equation3.17), it is necessary to establish a relationship between the relativepermeabilities krj and the phase saturations Sj. Such relationshipis provided by the Brooks-Corey correlation, see equation 3.18 withequation 3.19.In this work, lacking experimental data for the exemplary system,we assume the most standard form of this correlation, with mX = 2,Si = Sr = 0 and kXr,max = 1 (X = R, L). Here, the superscripts Rand L denote the water-rich and the water-lean phase, respectively.Furthermore, for equation 3.17, we assume equal viscosities ηj forboth phases. The resulting relationship fR = fR(SR) is illustrated inFigure 7.4. The typical S-shape of the relationship accounts for thecommon flow behavior that at low volume fractions SX (X =R,L),the interstitial velocity of phase X is very slow (slower than that ofthe other phase Y). In turn, at high volume fractions SX (X =R,L),the interstitial velocity of phase X is higher than that of the otherphase Y.

0 0.2 0.4 0.6 0.8 1

SR [-]

0

0.2

0.4

0.6

0.8

1

fR[-]

Figure 7.4: Fractional flow function assumed for the esterification system(blue continuous line), and relationship obtained when assu-ming equal interstitial velocities (black dashed line).


7.4 elution properties and process performance

In this section we present and discuss elution profiles for differentinitial and feed states, resulting from the solution of the equilibriumtheory model (section 7.2) using the method of characteristics. Im-portant physical aspects of the elution profiles are discussed, andtheir implications on the process performance are assessed. For theinterested reader, appendix C.1 offers a detailed explanation of theequilibrium theory solution, which is novel for the model establis-hed in section 7.2 and provides a thorough mathematical and phy-sical insight on the chromatographic behavior.

7.4.1 Elution profiles for single- and two-phase flow conditions

In the following, we consider entire chromatographic cycles, con-sisting of an adsorption step, where an initial state A is displacedby a feed state B until the entire chromatographic column is equili-brated with state B, and a desorption step, where B is redisplacedby A until the column is reequilibrated with A. The column is ini-tially equilibrated with pure n-hexanol, such that the initial stateA is defined as CA

1 = CA2 = 0. At time τ = 0, a certain composi-

tion of acetic acid and n-hexanol (solution with concentration C3of acetic acid) is fed to the column at a constant superficial velocityu. The feed composition instantaneously achieves chemical equili-brium through reaction to n-hexyl-acetate and water, to fulfill equa-tion 7.4. Since equimolar amounts of n-hexyl-acetate and water areformed during the reaction, and none of the components was initi-ally present in the mixture, it can be concluded that the feed flowinto the column is characterized by FB

1 = FB2 = C3. In the case where

the feed flow at chemical equilibrium consists of a single phase, thefeed state within the column corresponds to CB

1 = CB2 = FB

1 = FB2.

In contrast, if the feed flow consists of two convective phases, theycan move with different velocities, so as one phase accumulates inthe column, while the other propagates faster. Then, the concentrati-ons CB

1 and CB2 of the corresponding feed state in the column differ

(CB1 6= CB

2), and can be derived from FB1, FB

2 on the basis of equati-ons 7.6 with the inverted fractional flow function (equations 3.17 to3.19).

7.4 elution properties and process performance 161

We consider four different feed states Bi (i = 1− 4), with FBi corre-

0 1 2 3 4

C1 [mol L−1]

0

2

4

6

8

10

C2[m

olL−1]

P3

A

B1

Id4 B4

B3

B2

Id3

Id2

Id1 Ia3 Ia4Ia2Ia1

Figure 7.5: Solution paths (blue) in the (C2, C1) plane for examples i = 1 -4 with initial state A and feed states Bi. Entire chromatographiccycles (i.e. adsorption and desorption steps) are considered. In-termediate states are denoted as Ix, with x = a, d for the adsorp-tion and the desorption step, respectively. The binodal curve isplotted as a continuous black curve, the dashed black line deno-tes states with C2 = C1.

sponding to 1.0, 1.5, 2.0 and 3.0 mol/L. The feed states Bi and theinitial state A are plotted in the (C2, C1) plane in Figure 7.5. Thisplane has already been considered in Figure 7.3b, and Figure 7.5constitutes a zoom to the lower left corner of Figure 7.3b, assuringrelatively low concentrations of components 1 to 3 and hence an ex-cess of component 4 (as assumed when developing the equilibriumtheory model). The continuous black line in Figure 7.5 constitutesthe binodal, separating the single phase from the two-phase region.Accordingly, feed states B1 and B2 are located in the single phaseregion, while feed states B3 and B4 are located in the two-phase re-gion and lead to a phase-split and two-phase flow within the chro-matographic column. Blue lines constitute the mapping of the equi-librium theory solution for the specific initial and feed states in the(C2, C1) plane, i.e. the evolution of C1 and C2 throughout the chro-


matographic cycles, for the mathematical derivation see AppendixC.1. In addition, grey lines and symbols belong to the general so-lution of the equilibrium theory model, independent of initial andfeed states, and are explained in detail in Appendix C.1. The dashedblack line in Figure 7.5 denotes the conditions C2 = C1, and onlythe first two feed states B1 and B2 in the single phase region arelocated on that line, for reasons explained above.Elution profiles (concentration Ci and flow Fi profiles) are presen-ted in Figure 7.6 (adsorption steps, i.e. displacement of A by Bi)and in Figure 7.7 (desorption steps, i.e. displacement of Bi by A).Concentration profiles represent the concentration of a componentover all liquid phases at a specific location of the column (here atthe end of the column, ξ = 1). Meanwhile, flow profiles describe theoverall amount of a component moving per volume of flow over allliquid phases (overall fractional flow), in this case again at ξ = 1.Experimentally, it is the overall fractional flows of the different com-ponents that can be easily measured by gathering the eluate in fracti-ons and analyzing these (see chapter 6).With reference to Figures 7.6 and 7.7 it can be noted that, where-

ver the solution path in Figure 7.5 is located in the miscible region,concentration and flow profiles are identical. This is the case for theentire profiles of the first two examples (located entirely in the mis-cible region), while it is only partly the case for examples 3 and 4.It can also be observed that, in any of the adsorption profiles (Fi-gure 7.6), there is a period (intermediate state Ia) where C1 6= 0 andC2 = 0 (and accordingly F1 6= 0 and F2 = 0). In Figure 7.5, this cor-responds to a part of the solution path, namely that between A andIa, mapping on the horizontal axis. From the definition of Ci andFi, one can deduce that during this period, only component 1, i.e.the main product n-hexyl-acetate, is present in the bulk componentn-hexanol, while components 2 and 3 (water and acetic acid) are ab-sent. Likewise, in any of the desorption profiles (Figure 7.7), thereis a period where C1 = 0 and C2 6= 0, which corresponds to partsof the solution paths in Figure 7.5 (between Id and A) mapping onthe vertical axis. During this period, only component 2 (water) ispresent in the solvent and components 1 and 3 are absent. We canhence conclude that a purification of both products can be achievedfor any of the four feed states, regardless whether a phase split and


A-B1

0 2 4 6 8 10 12

τ [-]

0

1

2

3

4

Ci[m

olL−1]

C2C1

B1Ia1A

0 2 4 6 8 10 12

τ [-]

0

1

2

3

4

Fi[m

olL−1]

F2F1

A Ia1 B1

A-B2

0 2 4 6 8 10 12

τ [-]

0

1

2

3

4

Ci[m

olL−1]

C1 C2

B2A Ia2

0 2 4 6 8 10 12

τ [-]

0

1

2

3

4

Fi[m

olL−1]

F2F1

A Ia2 B2

A-B3

0 2 4 6 8 10 12

τ [-]

0

2

4

6

8

10

Ci[m

olL−1]

C1

C2

A Ia3 P3 B3

0 2 4 6 8 10 12

τ [-]

0

1

2

3

4

Fi[m

olL−1]

F2

F1

A Ia3 B3P3

A-B4

0 2 4 6 8 10 12

τ [-]

0

2

4

6

8

10

Ci[m

olL−1]

C1

C2

B4Ia4A

0 2 4 6 8 10 12

τ [-]

0

1

2

3

4

Fi[m

olL−1]

F1F2

A Ia4 B4

Figure 7.6: Adsorption profiles for examples i = 1 - 4 with initial state Aand feed states Bi. Left column: concentration profiles, Rightcolumn: flow profiles


A-B1

0 2 4 6 8 10 12

τ [-]

0

1

2

3

4

Ci[m

olL−1]

C1C2

Id1B1 →A

0 2 4 6 8 10 12

τ [-]

0

1

2

3

4

Fi[m

olL−1]

F2

F1

B1 Id1 →A

A-B2

0 2 4 6 8 10 12

τ [-]

0

1

2

3

4

Ci[m

olL−1]

C1

C2

Id2B2 →A

0 2 4 6 8 10 12

τ [-]

0

1

2

3

4

Fi[m

olL−1]

F2

F1

Id2B2 →A

A-B3

0 2 4 6 8 10 12

τ [-]

0

2

4

6

8

10

Ci[m

olL−1]

C1

C2

B3 Id3 →A

0 2 4 6 8 10 12

τ [-]

0

1

2

3

4

Fi[m

olL−1]

F2F1

B3 Id3 →A

A-B4

0 2 4 6 8 10 12

τ [-]

0

2

4

6

8

10

Ci[m

olL−1]

C1

C2

Id4B4 →A

0 2 4 6 8 10 12

τ [-]

0

1

2

3

4

Fi[m

olL−1]

F2F1

B4 Id4 →A

Figure 7.7: Desorption profiles for examples i = 1 - 4 with initial state Aand feed states Bi. Left column: concentration profiles, Rightcolumn: flow profiles


two-phase flow occurs or not.Let us have a closer look at the adsorption profiles reported in Fi-gure 7.6. As we explain in greater detail in Appendix C.1, an initialstate A and a feed state Bi are commonly connected by two transi-tions and one intermediate state I. This is indeed the case for exam-ples 1, 2 and 4, whereas example 3 exhibits an additional plateau ata state P3. With reference to Figure 7.5, it can be noted that P3 is atthe intersection of the solution path with the binodal curve, hence itcorresponds to the solubility limit. From Figure 7.4, we have conclu-ded that phases with a low saturation S (here the water-rich phase)propagate with a very low interstitial velocity. Accordingly, states inthe two-phase region located close to the binodal curve feature a lowpropagation velocity, which is smaller than the propagation velocityof a state in the single-phase region close to the binodal curve. Cros-sing the binodal, i.e. connecting downstream single-phase states toupstream two-phase states close to the binodal curve, hence leads tothe creation of a plateau at the solubility limit, as observed for exam-ple 3. We have further deduced from Figure 7.4 that the interstitialvelocity of a phase increases with increasing saturation S. Therefore,states located in the two-phase region at a greater distance from thebinodal feature higher propagation velocities, which exceed the pro-pagation velocities of states in the miscible region. Connecting suchtwo-phase, upstream states to single-phase, downstream states nolonger results in the formation of a plateau at the solubility limit, asobserved for example 4.To conclude our discussion of the adsorption profiles, we focus onthe intermediate states Ia. The values of C1 or F1 of these interme-diate states increase with increasing feed concentrations, i.e. fromexample 1 to example 4. Meanwhile, the duration of these statesdecreases from example 1 to 2, stagnates from example 2 to 3 anddecreases again slightly from example 3 to 4. It is a well-known fact(see the relevant literature23,24,28 and Appendix C.1), that systemswith a curved downward isotherm, as the Langmuir isotherm assu-med here, feature a shock transition when a downstream state withlower concentration is connected to an upstream state with higherconcentration, and that the propagation velocity of such shock in-creases with increasing concentration of the upstream state. Shockswith a higher propagation velocity are expected to appear at the


column outlet at lower retention times. The retention time of thefirst shock transition in the adsorption profiles, being already closeto the void time, only decreases slightly from example 1 to 4. Theretention time of the second shock transition decreases considerablyfrom example 1 to 2. Since the second transition in example 3 con-nects the intermediate state to the state P3 instead of the feed stateB3, this transition spans a similar concentration range as the secondtransition in example 2, connecting Ia

2 to B2. The second transiti-ons of examples 2 and 3 thus feature similar retention times. Thisis the case for all feed states located in the two-phase region andsufficiently close to the binodal to create a second plateau in theadsorption profiles at the solubility limit. Since feed state B4 is loca-ted in the two-phase region at a greater distance from the binodaland enables a direct shock transition from the intermediate state Ia

4

to the feed state B4, this transition spans a greater range of concen-trations, hence it appears at a lower retention time than the secondtransition in example 3.Finally, with reference to Figure 7.7, we observe that the desorptionprofiles feature more complex transitions than the adsorption profi-les, as explained in detail in Appendix C.1. It is however importantto note that the last tailing (wave transition) occurs entirely in thesingle-phase region (along the vertical axis in Figure 7.5), and isidentical for all four examples. Hence, the same time is required inall four examples to completely regenerate the column, i.e. to reachinitial state A with C1 = C2 = 0. The physical reason for this beha-vior is that water, featuring a very high Henry constant H2, exhibitsa strong affinity to the adsorbent, so as in all four examples low con-centrations of water are desorbed very slowly during the mentionedwave transition. In this system, the desorption of small concentrati-ons of water requires more time than the removal of low fractionsof a water-rich phase forming during examples 3 and 4.

7.4.2 Process Evaluation

As discussed in the previous section, pure products n-hexyl-acetateand water (diluted in the solvent) can be collected during the inter-mediate states of the adsorption and desorption steps, regardlesswhether the feed state is located in the miscible or in the immiscible


region. Concentration levels of the corresponding pure product atthese intermediate states increase with increasing concentrations ofthe feed state. The duration of the intermediate state during adsorp-tion, featuring pure n-hexyl-acetate, remains virtually constant forimmiscible feed states with a certain proximity to the binodal (com-pare examples 2 and 3), and is only reduced for immiscible feedstates located at a greater distance from the binodal (see example4). At the same time, complete regeneration (corresponding to thecomplete removal of the strongest adsorbing component water), re-quires the same time in all four examples.Based on the elution behavior, let us now evaluate the productivity

0 0.5 1 1.5 2 2.5 3

FB1 [mol L−1]

0

5

10

15

P[m

olL−1]

single phase two phases

Figure 7.8: Cyclic productivity P as a function of the overall fractional feedflow FB

1. Since the overall process time until complete regene-ration (complete elution of component 2) is constant, P can becorrelated with the productivity (per cycle time and column vo-lume) of the batch process. The dashed line indicates the solubi-lity limit (assuming FB

1 = FB2).

of the process. We assume an identical time for the chromatographiccycle (which is legitimate, considering the fact that regeneration re-quires the same time in all cases), and an operation of the processon the same column. The area under the flow profile in the presenceof pure n-hexyl-acetate (area under state Ia) corresponds to the cy-


clic productivity P, i.e. to the amount of n-hexyl-acetate purifiedper chromatographic cycle and column void volume (not the totalcolumn volume, due to the definition of τ):

P = ∆τFI1 (7.8)

where FI1 denotes the overall fractional flow of n-hexyl-acetate at the

column outlet during state Ia, and ∆τ denotes the duration (dimensi-onless) of this state. Under the assumption of a constant duration ofthe chromatographic cycle, P can be directly related to the producti-vity in amount of purified product per time and column volume,and it is plotted as a function of the overall fractional feed flow FB

1

in Figure 7.8.The dashed line in Figure 7.8 indicates the solubility limit; values ofFB1 on the left of that line form one stable liquid phase (correspon-

ding to a single-phase flow), whereas overall fractional feed flowsexceeding this limit result in liquid-liquid phase split and two-phaseflow. For feed states located in the miscible region, the productivityapproaches a plateau value with increasing FB

1. This can be explai-ned by the trade-off between increasing concentration levels of Ia

(which has a positive effect on productivity), and the decreasingduration of state Ia (negative effect on productivity). When crossingthe binodal, the productivity increases almost linearly, since the con-centration level of Ia increases, while the duration of Ia remains vir-tually constant. At a certain value of FB

1 in the immiscible region,the profile exhibits another kink, and subsequently flattens, againapproaching a plateau. At overall fractional feed flows beyond thiskink, Ia is directly connected to the feed state B through a shocktransition, without the formation of a plateau P at the binodal. Thepropagation velocity of this shock increases with increasing concen-trations of B. Accordingly, the duration of the intermediate state Ia

in the elution profiles decreases with increasing values of FB1 and

limits the increase in productivity.To conclude, the effect of a spontaneous phase split and subsequenttwo-phase flow can considerably enhance the performance of thebatch process. Physically, this can be explained by the fact that thewater-rich phase propagates slowly at low saturations SR, thus en-hancing the retention of water, on top of its strong interaction withthe adsorbent. The water-rich phase accumulates to a certain extent

7.5 conclusion 169

within the column, and allows for a longer duration of the elutionof component 1 pure.Due to the high Henry constant of water, low concentrations ofwater exhibit a very low propagation velocity, and thus completedesorption occurs more slowly than the removal of the water-richphase (through continuous adjustment of thermodynamic equili-bria). For this reason, complete regeneration of the adsorbent re-quires the same amount of time, regardless of the location of thefeed state (single-phase or two-phase region). It should be pointedout that this is not necessarily the case, and that the removal of thesecond phase can be the time limiting factor over desorption (asobserved theoretically and experimentally in chapters 5 and 6), es-pecially in the case of lower Henry constants, of upwards curvedadsorption isotherms, or of trapped fluid phases (SR

lim > 0).

7.5 conclusion

We have assessed the separation behavior and performance of achromatographic batch reactor in the presence of a phase split andtwo-phase flow. An equilibrium theory model was derived, accoun-ting for adsorption, two-phase flow, and a reversible reaction inchemical equilibrium. This model and its solution is based on thechromatographic model accounting for multiphase flow, which wasderived, solved and validated in the previous chapters 3 to 6. In ad-dition, previous work in the field of chromatographic reactors101–103

serves as a foundation for this chapter. Elution profiles were de-rived for feed states located in the miscible and in the two-phaseregion for an exemplary esterification system (with main product n-hexyl-acetate and side product water). It can be concluded that pureproducts (diluted in eluent n-hexanol) are achieved for miscible orimmiscible feed states. Furthermore, the two-phase flow behaviorexhibits an advantageous effect on the separation behavior, enhan-cing the amount of purified main product per chromatographic cy-cle.The equilibrium theory model applied in this study is based onseveral strong assumptions, in particular neglecting kinetic limitati-ons concerning mass transfer, chemical reaction, and liquid-liquidphase split. It certainly has to be assessed for the specific applica-


tion, whether these assumptions are consistent enough with reality.Further, it is clear that the batch process discussed in this workwould not be considered for an industrial application, where moreefficient simulated moving bed reactors (SMBRs) are preferred. Ho-wever, the equilibrium theory solution for a single batch columnprovides a deep understanding of the physical implications of two-phase flow in a chromatographic reactor, which are expected to re-main valid also for continuous reactive chromatography. In parti-cular, it was found that the formation of a second fluid phase richin the strongest adsorbing component (and with low saturation Sj)can increase the separation efficiency hence the process producti-vity, but the increased retention of the strongest adsorbing compo-nent also bears the risk of making the column regeneration moredifficult. We are convinced that these features (discussed in detailin section 7.4) would affect the performance of the SMBR process ina similar manner.This chapter, together with the previous chapters, clearly shows thattwo-phase flow in chromatography can be well described and un-derstood, and that its effects can be tolerable for some applicationsand advantageous for others.

8C O N C L U S I O N S A N D O U T L O O K

8.1 summary and conclusion

This thesis started with a theoretical investigation of singular shocksin liquid chromatography, a topic which had already been exploredexperimentally and theoretically by Mazzotti and co-workers8,25,27

prior to this PhD project. A rigorous theoretical study in chapter 2

presented different mathematical approaches to derive conditionsand properties of singular shocks in liquid chromatography. Theseapproaches achieved identical results and hence provided additio-nal proof of the theoretical existence of singular shocks in the con-text of liquid chromatography. Meanwhile, apparent experimentalevidence27 had been refuted upon a more in-depth analysis,25,26

and instead of spikes of theoretically infinite concentration, a liquid-liquid phase split and two-phase flow had been observed. This fin-ding spurred the interest in the description and understanding oftwo-phase flow and its implications in liquid chromatography, whichformed the core of this thesis.

In chapter 3, we derived chromatographic models based on com-ponent material balances, which account for both adsorption andfor two-phase flow with different velocities of the convective phases.Kinetic effects were either neglected (equilibrium theory model) orlumped in a simplistic manner (lumped kinetic model), while al-ways assuming fluid phases in thermodynamic equilibrium. Thecomponent material balances describe the generic behavior of achromatographic system, but they have to be combined with alge-braic equations, which describe the physical properties of a specificsystem under investigation.

Such algebraic relationships were established in chapter 4, wherethe system PNT, TBP, methanol and water with the adsorbent Zor-bax 300SB-C18 was characterized experimentally and described the-

172 conclusions and outlook

oretically. The quaternary system exhibited diverse and complexthermodynamic properties, featuring solid-liquid, liquid-liquid, andsolid-liquid-liquid phase equilibria, which were described by a fit-ted UNIQUAC model. The adsorption behavior of PNT and TBPwas described by rigorously applying thermodynamic concepts, withsingle-component isotherms which are a function of the liquid phaseactivities, and the adsorbed solution theory being applied to ac-count for binary adsorption. This thermodynamically consistent mo-del was able to quantitatively describe the experimental behaviorover a remarkably broad range of solute concentrations (reaching upto the solubility limit) and of eluent compositions (ratio methanol:water). With its broad applicability in terms of operating conditions(in this work concerning eluent compositions and solute concen-trations, but also a temperature or pressure dependence could beconsidered), such isotherm model based on liquid phase activitiesmight also be considered for a robust process design and optimiza-tion.Finally, the fluiddynamic properties of the system in the absence ofTBP (i.e. of the ternary system PNT, methanol and water with thesame adsorbent) were investigated through a displacement of two li-quid phases in thermodynamic equilibrium. The experimental datacould be accurately described by an empirical Brooks-Corey corre-lation, which has previously been used in the context of multiplefluid phases in natural reservoirs, where it was capable of describinga high number of different systems. The PNT-rich phase was obser-ved to be the wetting phase with a high irreducible saturation, whilethe PNT-lean phase was observed to be non-wetting. A dependenceof the relative permeabilities on the interfacial tension between thefluid phases was neglected, which constitutes a shortcoming whenapproaching the plait point, where the interfacial tension should ap-proach zero and the fluiddynamic behavior is expected to approacha single-phase flow behavior (i.e., equal interstitial velocities of thefluid phases).

The equilibrium theory model and the lumped kinetic model, com-bined with the established algebraic equations, were solved by ap-plying the method of characteristics or a finite volume discretizationscheme, respectively, to simulate elution profiles for the ternary sy-

8.1 summary and conclusion 173

stem PNT, methanol and water (chapter 5). It was found that, whilethe equilibrium theory solution becomes rather complex in the pre-sence of multiple convective phases and adsorption, it is computa-tionally inexpensive, accurate, and allows a deep understanding ofthe observed chromatographic behavior. In turn, the lumped kineticmodel offers a generic solution scheme, which can be applied to abroad range of operating conditions and is flexible towards changesin the underlying algebraic equations. However, the offered numeri-cal solution is computationally expensive and time-consuming, andexhibits numerical issues, in particular at conditions close to theplait point.

In chapter 6, the equilibrium theory solution, as well as model as-sumptions and implemented physical relationships, were validatedby dynamic column experiments in the presence of ad- and desorp-tion and of two liquid phases. It should be pointed out that in thiscontext, the equilibrium theory model was used in a predictive man-ner, without any further adjustments of parameters. A quantitativedescription of the experimental behavior was achieved for a majo-rity of the investigated conditions. Notable differences between ex-perimental data and simulations were only observed in the vicinityof the plait point, which was ascribed to the mentioned shortco-ming of the fractional flow functions, neglecting a dependence onthe interfacial tension between the fluid phases. Simulations withdifferent variations of the established model, neglecting adsorptionor assuming equal interstitial velocities of the convective phases, re-vealed that in the presence of two convective phases, and far fromthe plait point (i.e. at high interfacial tensions), the impact of fluid-dynamic effects on the elution profiles is dominant over the impactof adsorption effects. The dominance shifts when approaching theplait point, and in the presence of a single convective phase, the cru-cial impact factor are adsorption effects. High irreducible or residualsaturations (i.e., high fractions of trapped phases) can, additionallyto a high affinity to the adsorbent, result in a strong retention andvery slow elution of some solutes, which is evaluated as a rathernegative effect in terms of the performance of a chromatographicprocess.


Finally, the established equilibrium theory model was extended toaccount for equilibrium reactions within the chromatographic co-lumn, and is applied to evaluate the impact of two-phase flow onthe performance of a chromatographic reactor (chapter 7). This the-oretical evaluation revealed that, for the separation of the two re-action products, the separation efficiency can be enhanced if a se-cond liquid phase enriched in the stronger adsorbing componentforms. Since this phase is likely to be more wetting (being mainlycomposed of components with a strong affinity to the adsorbent)and to be present at a rather low saturation, it is also likely to flowat a lower velocity, which increases the retention of the strongeradsorbing component. As a consequence, it is possible to obtain ahigher amount of purified, weaker adsorbing component per chro-matographic cycle, which thus enhances the process productivity.

The idea for this PhD project was initiated by the experimental ob-servation of two-phase flow in a liquid chromatographic system; aphenomenon which at the time had rarely been reported in the li-terature and had neither been investigated nor understood. At theend of the thesis, a good understanding of this phenomenon and itsphysical implications has been developed, a chromatographic mo-del has been established and validated, and this model has alsobeen used to evaluate the impact of two-phase flow on the perfor-mance of chromatographic processes. While effects on process per-formance depend on the type and physical properties of the chro-matographic process, and can range from unfavorable to tolerableto advantageous, the chromatographic behavior in the presence oftwo liquid phases was experienced to be reproducible, and couldbe well described and understood by the established model. Furt-her investigation of this phenomenon, and especially of its role indifferent chromatographic processes, is highly recommended.

8.2 outlook

While the presented work provides answers to a number of issues,such as the physical implications of two-phase flow in liquid chro-matography, their theoretical description, and a first evaluation ofprocess performance in the presence of multiple fluid phases, it

8.2 outlook 175

opens just as many new questions and exciting challenges to tackle.Some future research topics are outlined below.

Adsorption under non-ideal thermodynamic conditions Forthe reversed phase chromatographic system characterized in chap-ter 4, it was found that single-component and binary adsorptioncould be described over a broad range of operating conditions (so-lute concentrations and eluent compositions) as a function of liquidphase activities. Setting up such isotherms offers an accurate and ro-bust description, which, being thermodynamically consistent, mightalso be extrapolated to a limited extent to unexplored operating con-ditions. Hence, it might be conveniently applied to design and eva-luate processes at different operating conditions. Due to the discus-sed benefits expected from an adsorption model based on activities,the applicability of such model should be investigated in more de-tail, for a higher number of different components, different chroma-tographic modes (e.g. reversed phase, ion exchange, hydrophobicinteraction) and different operating variables (e.g. eluent compositi-ons, temperature, pressure). In addition, the use of such model asa design and evaluation tool, and its validity upon extrapolation,should be assessed.

Model extension and validation While the understanding of ad-sorption and fluiddynamic properties in the presence of two-phaseflow is advanced at the end of this thesis, some properties, opera-ting conditions and phenomena were neglected or excluded for thesake of simplicity, but deserve further consideration.First of all, kinetic limitations, such as mass transfer resistances ordispersive effects, were entirely neglected in the equilibrium theorymodel, and were treated in a very simplistic manner in the lumpedkinetic model. The assumption of negligible kinetic limitations is theonly key assumption of the equilibrium theory model which wasnot investigated through an independent experimental campaign inchapter 4. Kinetic limitations would result in band broadening inthe experimental elution profiles.38,108 With experimental profilesin section 6.3 exhibiting very sharp transitions, and being in quanti-tative agreement with the established equilibrium theory model, weconclude that kinetic effects play a negligible role for the investiga-


ted system. This finding is non-generic, and kinetic effects shouldindeed be considered when characterizing different systems.Secondly, the model validation and a considerable part of the cha-racterization of physical properties (fluiddynamic properties) wasperformed with a ternary system, featuring a single adsorbing com-ponent only. Since chromatographic processes commonly involvemore than one adsorbing component, it is crucial to extend the in-vestigation in this respect. This extension entails a complication ofthe physical relationships, as it was observed when moving fromsingle-component isotherms to a description of binary adsorptionin sections 4.6 and 4.7. It also renders the model solution more dif-ficult and elaborate, since such model in the presence of a highernumber of components is based on a larger system of PDEs, andfeatures solutions with multiple transitions and intermediate sta-tes. In addition, the creation of look-up tables, implemented in themodel for the fast determination of fluid phase equilibria, becomesmore complex, since multi-component systems can feature regionsof different phase equilbria with multi-dimensional shapes.Arising from the previous two issues (i.e. the disregard of kineticlimitations and of multi-component systems), a third shortcomingof this study concerns the examination of a spontaneous phase split.For the creation of a two-phase flow, two immiscible phases weredisplaced in this thesis. In turn, a spontaneous phase split in thecolumn requires the formation of supersaturation due to an enri-chment of one or multiple components. This can only be achievedin the presence of two or more adsorbing and interacting compo-nents, or in the case of a reaction with partially miscible productsand/or reactants. Furthermore, phase splits are particularly proneto kinetic limitations, i.e. a certain degree of supersaturation is requi-red for a phase to pass from a metastable to an unstable state andto separate into two or more phases. Nevertheless, a spontaneousphase split is expected to precede a multi-phase flow in a majorityof chromatographic applications. The in-depth investigation of thisphenomenon is hence a crucial, but at the same time an ambitiousfuture aim, which combines all the challenges arising in the contextof multi-component systems and kinetic limitations.

Short-cut characterization and description The model esta-

8.2 outlook 177

blished in this work provides a highly accurate, quantitative des-cription of the investigated system and hence confirms the under-lying assumptions and physical relationships. However, it shouldbe pointed out that, in order to achieve such description, the inde-pendent and accurate characterization of thermodynamic and fluid-dynamic properties required a considerable experimental effort. Inorder to make chromatographic processes involving a two-phaseflow attractive for industrial applications, it is necessary to consi-derably reduce the effort of system characterization. This can beachieved by using predictive models (such as the UNIFAC equationto describe thermodynamic equilibria of fluid phases), and by a ca-reful analysis of the absolutely necessary phenomena which have tobe accounted for in order to achieve a reliable model description. Asan example, it might be considered to establish benchmarks, deno-ting conditions at which fluiddynamic effects or adsorption effectsare dominant, or at which both effects have to be taken into account.

Evaluation of process performance A first evaluation of the im-pact of phase split and two-phase flow on the performance of chro-matographic processes is provided in chapter 7, where a chroma-tographic reactor (considering an esterification reaction) was inves-tigated theoretically. While the presented study provides a valua-ble evaluation and first insights on potential advantages of a two-phase flow in chromatographic processes, it is clear that this workonly constitutes the starting point. A more thorough evaluation ofchromatographic processes in the presence of multiple fluid phases,involving an experimental investigation and characterization, hasto follow. The two most arguable assumptions made in chapter 7,which have to be assessed in more detail, are the ones of negligiblekinetic limitations during the phase split, and of chemical (reaction)equilibrium. It is also obvious that the batch process evaluated inchapter 7 is not of interest for an industrial application, since con-tinuous chromatographic processes (simulated moving beds and si-mulated moving bed reactors) achieve higher productivities, pro-duct purities, and recoveries. Hence, the evaluation of process per-formance in the presence of phase split and two-phase flow has tobe extended to continuous operating modes.

9A D D E N D U M : I N T E R C O N V E R S I O N O FC A R B O H Y D R AT E S I N S I N G L E - P H A S EC H R O M AT O G R A P H Y

9.1 introduction

Due to the molecular similarity of different sugars, their purifica-tion is somewhat demanding. Both analytical and preparative se-paration is commonly achieved by chromatographic processes onpolystyrene-divenylbenzene resins.109,110 The interaction involvescomplexation between the analyte and the fixed metal cations onthe resin as well as size exclusion mechanisms. These are affectedby the type of cation bound to the resin, the degree of cross-linkingas well as the size of the particles.111–113

The ligand-ligand interaction between sugar and metal cation de-pends also on the structural form of the sugar.111,114 Different isome-ric and anomeric forms therefore result in different residence timesin the column. If the interconversion between the different formsis slow, which is the case at low temperatures, they can be separa-ted completely. On the contrary, at high temperature the intercon-version speeds up and the peaks are distorted until they coalesce,which occurs when the reaction equilibrium is reached fast enoughto prevent separation between the different forms.114,115

One of the first to discuss equilibrium reactions in chromatography,as a cause for distorted peak shapes and multiple peaks in elu-tion profiles, were Keller et al.116 The interplay between the re-action kinetics of sugar interconversion and chromatographic sepa-ration was described in a qualitative manner by Goulding.117 Dif-ferent models were developed to quantitatively describe the ob-servations. Apart from models based on component mass balan-ces,[115] which are most commonly applied in the context of chro-

Results presented in this section have been reported in: Ortner, F.; Wiemeyer, H.;Mazzotti, M. Interconversion and chromatographic separation of carbohydrate stere-oisomers on polystyrene-divinylbenzene resins J. Chromatogr. A 2017, 1517, 54-65

180 addendum :interconversion of carbohydrates

matography, also stochastic models for first order reactions werereported by Giddings118 and Keller et. al116. The developed mo-dels and the gained physical understanding were later employedto determine reaction kinetic parameters. While Carta et al.115 usedan inverse fitting method based on a numerical mass balance mo-del to obtain apparent reaction rate constants, Trapp derived ana-lytical equations for peak and plateau heights from the stochasticmodel.119–121 An even simpler method for the determination of ap-parent reaction rate constants,111,114 which will also be applied inthis contribution, is based on a simple analysis of the change ofpeak areas over time.In most contributions, the values obtained for the apparent reactionrate constants in the chromatographic column were either acceptedas they were or compared with literature values for reaction rateconstants or activation energies in aqueous solution.114,115 Rarely,an attempt was made to investigate the effect of the sorbent on thereaction rate and to describe possible differences between values inliquid and adsorbed phase.111

In the spirit of the work of Carta et al., this contribution presentsa quantitative analysis of the interplay of adsorption and mutaro-tation of Fructose and Glucose. The resin CK08EC from Mitsubishi(8% cross-linking, Ca-form, particle size 9µm) used for this studyhas similar chemical properties as the one used by Carta et al., butsmaller particle size. Differences can also be noted in the approachof fitting adsorption and reaction kinetic parameters: While Cartaet al. apply an inverse fitting procedure (of all parameters, separa-tely for each experimental condition), the current contribution ina first step performs a parameter estimation using the simple andfast fitting method based on the analysis of peak areas, similar tothe approaches proposed by Baker et al.111 and Nishikawa et al.114.This method provides also a deeper understanding of the effect ofmutarotation on chromatographic elution profiles. In a second step,the estimated values for reaction kinetic parameters are providedas an inital guess for an inverse fit to a series of experimental re-sults (at different flow rates and temperatures), using a numericalmass balance model. With the enhanced parameters, a very goodagreement between simulations with the numerical model and ex-perimental elution profiles is achieved. Accuracy of the estimated


parameters is confirmed further through a comparison with litera-ture data. Differences between adsorbed and liquid phase reactionrate constants were quantified through a comparison with kineticdata for aqueous solutions from the literature, which also allows todraw conclusions on the effect of the sorbent on interconversion.This paper is structured as follows. Section 9.2 aims at providingthe reader with the background on mutarotation, the applied chro-matographic model and the peak area method used for an estima-tion of reaction kinetic parameters. In section 9.3 the experimentalprocedure is described. Section 9.4 presents the experimental andmodeling results. Experimental profiles are presented and discus-sed in a qualitative way in section 9.4.1. Through a simple analysisof shape, elution times and peak areas of the experimental elutionprofiles, relevant model parameters (apparent dispersion coefficientin section 9.4.2, adsorption parameters in section 9.4.3 and reactionkinetic parameters in sections 9.4.4.1 and 9.4.4.2) are estimated. Tofurther enhance the accuracy of the description of reaction kinetics,estimated parameters of section 9.4.4.2 are used as an initial guessfor an inverse fitting (results see section 9.4.4.3), applying the nume-rical model introduced in section 9.2.2. Estimated model parametersare verified through a comparison with literature values (sections9.4.3 and 9.4.4.4). Further confirmation of the model parameters isprovided in section 9.4.5 through a comparison of experimental andsimulated elution profiles. Additionally, reaction rate constants forthe adsorbed phase are calculated and the effect of the resin on thekinetics is discussed (section 9.4.6). Finally, conclusions are drawnin section 9.5.

9.2 theoretical background

9.2.1 Mutarotation and adsorption of Glucose and Fructose

The term mutarotation refers to the interconversion between diffe-rent structural forms of a sugar molecule. The rate of such intercon-version has an effect on the peak shapes in chromatography.For Glucose, there are two different anomers in solution, α- and β-Glucose, whose structures differ in the orientation of the hydroxylgroup on the C-1 carbon. Equilibrium between the two forms can be


assumed to be temperature independent within the range conside-red in this work, with a ratio between the prevalent β-form and theα-form of Neq

β /Neqα = 1.72 in water;111,122 note that Neq

i is the massor mole amount of species i at equilibrium. The fraction of a thirdopen-chain form present at equilibrium is negligible (below 1%).Fructose forms four different structures, five(Furanose)- and six(Py-ranose)-membered ring forms, which form each two different ano-mers. As for Glucose, the presence of a fifth open-chain form is neg-ligible. Proportions of the different forms at equilibrium in water asa function of temperature are shown in Figure 9.1. It can be notedthat the β-Pyranose form is predominant, but its fraction at equi-librium decreases with increasing temperature, while the fractionof the two Furanose anomers increases. Due to its low occurrence,the α-Pyranose form will be neglected in this work. We further in-troduce a relationship, linearly depending on temperature, for theequilibrium amount of β-Pyranose with respect to the sum of theamounts of the two Furanose forms, which was fitted to the litera-ture data in the experimentally relevant range (0 to 31

C):

NeqβPyr

NeqαFur +N

eqβFur

= 0.0083 T − 2.0926 (9.1)

with T being the temperature in Kelvin.In chromatography, using ion-exchange PS-DVB resins, the differentGlucose and Fructose forms can be separated due to different affi-nities with respect to the cationic ligands. In the case of Glucose,the α-form is more retained than the β-form, while for Fructose,the β-Pyranose form is more retained, and the two Furanose formsco-elute. In the following, we will refer to the less retained (first elu-ting) form as form 1 and denote the more retained form as 2. Notethat for Fructose, due to the discussed co-elution, the two Furanoseforms are considered as one and the same form 1.


280 300 320 340 360

T [K]

0

10

20

30

40

50

60

70

80

90

co

mp

os

itio

n [

%]

α-pyranose

α-furanose

β-furanose

β-pyranose

Figure 9.1: Equilibrium composition of Fructose over temperature usingdata from Shallenberger123 (in light shades of red and blue) andAngyal124 (in dark shades) for aqueous solutions. Diamondsand circles indicate Pyranose and Furanose forms, respectively,blue and red symbols denote α- and β-forms.

9.2.2 Mass balance model

The differential mass balance in a chromatographic column for astructural form i, which is at adsorption equilibrium but undergoesa chemical reaction (equilibrium dispersive model22) reads

ε∂ci∂t

+ (1− ε)∂ni∂t

+ u∂ci∂z

=∂

∂zεDa

∂ci∂z

+ Ri; i = 1, 2 (9.2)

with u being the superficial velocity, z the axial coordinate alongthe column and t the time. The apparent dispersion coefficient isdesignated as Da. Note that the same apparent dispersion coeffi-cient is assumed for all structural forms of one sugar, but differentcoefficients are assumed for Glucose and Fructose. A detailed dis-cussion of the apparent dispersion coefficient is reported in section9.4.2. The last term, Ri, represents the net rate of change in overallconcentration (in liquid and adsorbed phase together) of the form i

due to the chemical reaction.The thermodynamic equilibrium between adsorbed and liquid phase


is described by a linear adsorption isotherm, accounting for the tem-perature dependence according to Van’t Hoff’s law

∂ lnHi∂T

=∆Hadsi

RT2(9.3)

Therefore, the adsorbed phase concentration of species i can be ex-pressed as

ni = Hici = H0i exp

(−∆Hads

i

RT

)ci, (9.4)

where the Henry constant is evaluated as a function of temperatureT and depends on the pre-exponential factor H0i and on the heat ofadsorption ∆Hads

i .The reaction occurs in the liquid as well as in the adsorbed phase,but the reaction rates in the two phases may differ. Consideringthe forward and backward reaction in both phases, the net overallreaction rate can be written as

Ri = ε(kmj cj − k

mi ci

)+ (1− ε)

(ksjnj − k

sini

)(9.5)

Where ki (i = 1, 2) is the reaction rate constant associated withcomponent i, which interconverts to the other form j, either in themobile phase, superscript m, or in the stationary phase, superscripts.Substituting Equation 9.4 into Equation 9.5, yields:

Ri =(εkmj + (1− ε)ks

jHj

)cj − (εkm

i + (1− ε)ksiHi) ci (9.6)

Possible differences in the reaction rates in the mobile and in the ad-sorbed phase cannot be distinguished experimentally, where onlythe combined reaction rate can be determined. We therefore intro-duce the apparent reaction rate constant kapp

i for component iwhichis defined as

kappi = εkm

i + (1− ε)ksiHi (9.7)

In the following, we want to prove that, taking into account theestablished adsorption equilibrium, the ratio kapp

1 /kapp2 is identical


to the ratio km1 /k

m2 in the mobile phase. To this aim, we consider the

reaction equilibria in the liquid phase and the solid phase, where

ceq1 k

m1 = c

eq2 k

m2 ; (9.8a)

neq1 k

s1 = n

eq2 k

s2. (9.8b)

The superscript eq indicates a concentration at reaction equilibrium.Note that, due to Equation 9.8, the ratio of rate constants km

2 /km1

equals the ratio Neq1 /N

eq2 , which corresponds to the equilibrium

composition in the liquid phase as discussed in section 9.2.1. Furt-hermore, considering the adsorption equilibrium defined in Equa-tion 9.4, one obtains:

ks1

ks2

=n

eq2

neq1

=H2H1

km1

km2

(9.9)

Using equation 9.9 in equation 9.7 for form 1 yields:

kapp1 = εkm

1 + (1− ε)H1H2k

m1

H1km2ks2

=km1km2

(εkm2 + (1− ε)H2k

s2

)=km1km2k

app2 (9.10)

Substituting Equation 9.4 and 9.7 into equation 9.2 and transfor-ming it into dimensionless form yields:

(ε+ (1− ε)Hi)∂φi∂τ

+∂φi∂ξ

=

∂

∂ξ

(ε

Pe

∂φi∂ξ

)+tr

ρi

(k

appj ρjφj − k

appi ρiφi

),

(9.11)

with the following dimensionless variables

φi =ciρi

(9.12a)

γi =niρi

(9.12b)

τ =t

L/uF =t

tr(9.12c)


ξ =z

L(9.12d)

µ =u

uF =u

L/tr(9.12e)

Pe =uFL

Da. (9.12f)

In these equations, L is the length of the column, tr the residencetime, ρi the density of the species i and uF the superficial feed velo-city. As we assume constant velocity in the column u = const. = uF,thus µ = 1.The system of partial differential equations (Equation 9.11) was sol-ved by employing a semi-discrete finite volume discretization schemewith a van Leer flux limiter.125 The spatial discretization results in asystem of ODEs with respect to time, which was solved by a built-inMatlab ODE-solver (ode113). Pulse injections were handled as a Rie-mann problem with the following initial and boundary conditions:

φi(0; ξ) = 0 for 0 6 ξ 6 1 (9.13a)

φi(τ; 0) =

φFi for 0 6 τ 6 τinj

0 for τinj < τ(9.13b)

9.2.3 Fitting procedure for rate constants: peak area method

In this contribution, a simple, computationally inexpensive fittingprocedure for the rate constants kapp

i is applied. This method is ba-sed on a thorough analysis of experimental elution profiles, relatingthe impact of reaction kinetics to the evolution of peak areas overtime.A typical elution profile is illustrated in Figure 9.2. Such profile can

be subdivided into three parts, namely the two peaks and the inter-mediate mixed zone, which connects the two peaks. The peaks canbe associated with the amount of unreacted moleculesNu

i of speciesi (species 1 and 2 corresponding to the green and red area in Figure9.2, respectively). Each of the two structural forms i moves with acharacteristic propagation velocity λi through the column, which isdetermined by its retention behavior. Consequently, the unreacted


13 14 15 16 17 18

t [min]

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

c1+

c2 [

gL

-1]

N r

N u

1

N u

2

Figure 9.2: Illustration of the fitting procedure based on the areas of thepeaks corresponding to the amount of unreacted molecules ofboth structural forms (Nu

1 and Nu2) relative to the total area in

the chromatogram Nu1 +N

u2 +N

r.

molecules elute at a characteristic retention time tRi (diffusion andmass transfer effects result in a Gaussian shape around tRi ). The in-termediate area between the peaks can be ascribed to the amount ofmoleculesNr having reacted (once or multiple times) from one formto the other. The retention time of reacted molecules is an averageof the characteristic retention times of the two anomers, weighedby the relative time spent as one or the other of the two structuralforms while propagating along the column.116,120,121 It is obviousthat the total amount of moleculesNtot (i.e. injected amount), equalsthe sum Nu

1 +Nu2 +N

r, and corresponds to the total area under theelution profile.With increasing residence time, the amount of unreacted species de-creases exponentially, according to the following first order decayfunction, which is obviously associated to the first order kinetics ofequations 9.5 to 9.7:120,121

ln

(Nui

N0i

)= −k

appi t, (9.14)

where N0i is the initially injected amount of species i, which can beobtained knowing the overall injected amount Ntot and the equili-


brium composition of species 1 and 2, taken from the literature anddiscussed in section 9.2.1.The amount of unreacted form i, Nu

i , at different residence times tRicorresponds to the peak area of the unreacted form obtained frompeak injections at different flow rates (but at constant temperature).A linear regression between ln

(Nui /N

0i

)and tRi , applying Equation

9.14, yields values for kappi .

To further reduce the number of fitted parameters, we recall thatk

app1 /k

app2 = km

1 /km2 = N

eq2 /N

eq1 (Equation 9.10). Assuming equili-

brium compositions as discussed in section 9.2.1, one can directlyrelate kapp

1 to kapp2 , and thus use experimental data obtained for

both species 1 and 2 to fit the reaction rate constant of one speciesonly.

9.3 experimental procedure

The experiments were carried out on a modular HPLC unit (Agi-lent Technologies 1200 Series) equipped with a quarternary pistonpump, an auto injector and a column compartment. The fluid phaseis heated up (or cooled down) before entering the column by flo-wing through a heat exchanger. The temperature (set in the HPLCcontrol software Agilent ChemStation A.02.10) was varied between0 C and 80 C. These extreme temperatures are unlikely to be rea-ched at the column, thus the actual temperature of the column wasmeasured with a thermocouple placed directly in the middle of thecolumn, onto the column wall. In the following, we differentiatebetween "set temperatures" (between 0 and 80 C) and "measuredtemperatures" (between 11 and 72 C). For detection, an RI-detector(Agilent Technologies, G1362A) was used.The experiments were perfomed on a column (7.8 × 300mm) pac-ked with a PS-DVB resin (CK08EC from Mitsubishi, 8% cross-linking,Ca-form, particle size 9µm). The porostiy of 24.7% was determinedby injection of dilute Calcium Chloride (Merck, Calciumchloride di-hydrate) aqueous solution with a concentration of 0.1mmol L−1.Calcium Chloride is excluded from the resin due to the Donnan

9.4 results 189

repulsion effect.126 The bed porosity can thus be determined fromthe retention time, tRCaCl2

, of the injected pulse of 10µL as:

ε =u tRCaCl2L

(9.15)

Note that throughout this work, all presented retention times andelution profiles have been corrected by the extracolumn dead time,measured by pulse-injections of a Glucose solution through a column-bypass.For pulse-injection experiments of sugars, pure water was used aseluent. Solutions were prepared using an analytical balance (Mett-ler Toledo AX205 DeltaRange) and Millipore water. Either pure D-Fructose (Sigma Aldrich, purity > 99%) or pure D-Glucose (SigmaAldrich, purity > 99.5%) in solution (2 g L−1) was used and pulseinjections of 10µL injection volume were performed. The elutionprofiles were detected by the RI detector and converted into con-centration profiles assuming a linear relationship between detectorsignal and liquid phase concentration.

9.4 results

9.4.1 Experimental campaign

In order to determine the reaction kinetic and adsorption equili-brium parameters required in the chromatographic model (see section9.2.2), pulse injection experiments at different flow rates between0.1 and 1.4mLmin-1 and at constant temperature were performed.The flow rates were increased until the pressure limit of the column(60 bar) was approached at the specified temperature. To investigatethe temperature dependence of reaction rate and Henry constants,experimental series were performed at different temperatures. Tem-perature ranges were chosen in such a way that peak distortion (dueto an interplay of chemical reaction and adsorption) was clearly vi-sible. Experiments were performed at measured temperatures of 21,38, 54 and 72 C for Glucose, and of 11, 17 and 21 C for Fructose.The obtained elution profiles (over elution time) are presented in

Figures 9.3 and 9.4 for Glucose and Fructose, respectively. They il-lustrate the effect of the flow rate and temperature on the extent


10 15 20 25

t [min]

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

c1+

c2 [

gL

-1]

(a)

0.6 0.5 0.30.4

21°C

10 15 20 25

t [min]

0

0.01

0.02

0.03

0.04

0.05

0.06

c1+

c2 [

gL

-1]

(b)

0.6 0.5 0.4 0.3

38°C

5 10 15 20 25

t [min]

0

0.01

0.02

0.03

0.04

0.05

c1+

c2 [

gL

-1]

0.3

0.40.5

0.60.7

0.80.9

(c)

54°C

4 6 8 10 12

t [min]

0

0.01

0.02

0.03

0.04

0.05

0.06

c1+

c2 [

gL

-1] 1.4 0.8

(d)

0.61.01.2

72°C

Figure 9.3: Experimental (solid lines) and simulated elution profiles (das-hed lines) for Glucose at different temperatures and flowrates.Flowrates are indicated next to the profiles in mLmin-1. Parame-ters used in the simulations correspond to the ones estimatedin section 9.4.2 (apparent dispersion coefficient), 9.4.3 (adsorp-tion parameters) and 9.4.4.3 (reaction kinetic parameters obtai-ned through inverse fitting). (a): 21 C (b): 38 C (c): 54 C (d):72 C

9.4 results 191

20 40 60 80 100 120 140

t [min]

0

0.005

0.01

0.015

0.02

c1+

c2 [

gL

-1]

0.1

(a)

0.4 0.3 0.2

12°C

10 20 30 40 50 60 70

t [min]

0

0.005

0.01

0.015

c1+

c2 [

gL

-1]

0.5

(b)

0.20.30.4

17°C

10 15 20 25 30 35 40

t [min]

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

c1+

c2 [

gL

-1]

0.6 0.5 0.4 0.3

(c)

21°C

Figure 9.4: Experimental (solid lines) and simulated elution profiles (das-hed lines) for Fructose at different temperatures and flowrates.Flowrates are indicated next to the profiles in mLmin-1. Parame-ters used in the simulations correspond to the ones estimatedin section 9.4.2 (apparent dispersion coefficient), 9.4.3 (adsorp-tion parameters) and 9.4.4.3 (reaction kinetic parameters obtai-ned through inverse fitting). (a): 12 C (b): 17 C (c): 21 C


of reaction. The trends are the same for both sugars but the inter-conversion is much faster in the case of Fructose, resulting in moredistorted peaks even at rather low temperature. In the case of Glu-cose, the predominant species is less retained, resulting in a largerfirst peak, while for Fructose, the less abundant Furanose forms areeluted earlier.The effect of changing the flow rate at constant temperature is illus-trated in Figure 9.5. At constant temperature, the retention volumeof the unreacted forms, corresponding to the position of the peakmaxima in Figure 9.5b, remains constant (within a certain margin ofexperimental error, which is very low, as can be deduced from thezoomed elution profiles in Figure 9.5b). In turn, the retention time(see Figure 9.5a) is inversely proportional to the flow rate. With de-creasing flow rate, the retention time and thus the time available forthe interconversion increases, resulting in an increase of the numberof molecules that have interconverted at least once while travellingalong the column. The peak maxima in the chromatograms decre-ase and the level of the intermediate plateau rises.The impact of temperature on the elution profiles can be clearly ob-served in Figures 9.3 and 9.4. Varying the temperature affects theHenry constants (see Equation 9.4), leading to changing peak positi-ons at constant flow rate. Additionally, reaction rates increase withtemperature (discussed in section 9.4.4.2), thus interconversion beco-mes more dominant at higher temperatures than chromatographicseparation; as a consequence one observes peak coalescence.

9.4.2 Apparent dispersion coefficient

The apparent dispersion coefficient Da can be estimated throughthe number of theoretical plates NT or the plate height HT:22

HT = LNT

(9.16)

Da = vHT2 , (9.17)

where v = u/ε is the interstitial velocity. The number of theoreticalplates NT, is typically determined through pulse injections, from

9.4 results 193

10 15 20 25

t [min]

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

c1+

c2 [

gL

-1]

(a)

0.5 0.4 0.30.6

5 5.5 6 6.5 7 7.5

V [mL]

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

c1+

c2 [

gL

-1]

(b)

0.3

0.6

Figure 9.5: Elution profiles resulting from Glucose pulse injections perfor-med at 38 C with varying flowrate. Profiles are plotted over(a) elution time and (b) elution volume. Flowrates are indicatednext to the profiles in mLmin-1.


the peak width at half height w1/2 and the retention time tR of theeluting peak, as:

NT = 5.545

(tR

w1/2

)2(9.18)

In this work, peaks corresponding to the unreacted, predominantspecies (form 1 in the case of Glucose and form 2 in the case ofFructose) were used. For an accurate determination of NT, the effectof the area corresponding to interconverted molecules on w1/2 ofthe peak corresponding to the unreacted form has to be negligible.Accordingly, all peaks of Glucose form 1 at 21 C, as well as thepeaks of Fructose form 2 at the three highest flowrates at 12 C, andat the highest flowrate at 17 C were used. It was further assumedthat different structural forms of the same sugar feature the sameapparent dispersion coefficient Da.The determined values of HT and of Da as a function of v for bothGlucose and Fructose are plotted in Figure 9.6a and b, respectively.It can be noted that HT and Da values are different for Glucoseand Fructose, and that both HT and Da increase with v. Linear re-gressions to the HT data for each Glucose and Fructose (see Figure9.6a) provide an accurate description for both HT and the lumpeddispersion coefficient Da. With this description, we account for thedependence of Da on v, independently for the two sugars, whileneglecting a possible impact of temperature.

9.4.3 Henry constants

The Henry constants of the unreacted forms 1 and 2 of Fructose andGlucose were obtained from the positions of the peak maxima cor-responding to the unreacted forms. The adsorption behavior wasassumed to be linear, considering the low concentration range rele-vant in this contribution, as well as adsorption data reported in theliterature.110,113 From the retention times tRi the Henry constantsof the forms i = 1, 2 were determined using the following, well-established equation:

Hi =1

ν

(utRiεL

− 1

)(9.19)

9.4 results 195

2 4 6 8 10

v [ms-1

] ×10-4

0

0.5

1

1.5

2

HT [

m]

×10-4

HG

T=1.718x10

-5+0.0537v

(a)

HF

T=-4.322x10

-5+0.3409v

Fructose

Glucose

2 4 6 8 10

v [ms-1

] ×10-4

0

1

2

3

4

5

6

7

Da [

m2s

-1]

×10-8(b)

Fructose

Glucose

Figure 9.6: (a) Van Deemter curves and (b) apparent dispersion coefficientDa over interstitial velocity v for Fructose (red) and Glucose(blue). Relationships of HT (fitted) and Da (based on the corre-lation for HT) as a function of v are indicated as dashed blacklines.


Not all measurements of the experimental campaign reported abovewere considered for the calculations. The plateau correspondingto the interconverted molecules shifts the peak maxima (associa-ted with the unreacted forms) if the extent of the reaction is toolarge. Therefore, only measurements with a low connecting bridgebetween the peak maxima were considered, thus excluding thoseat low flow rates and at high temperatures. No data was obtainedfor the experimental series performed with Glucose at 72 C. TheHenry constants at the different temperatures were determined asthe average of the values obtained at different flow rates (2 to 4 va-lues per temperature and species were available). The values for thetwo sugars are summarized in Table 9.1.It can be noted that Fructose, featuring higher Henry constants, is

Table 9.1: Henry constants for Glucose and Fructose at different temperatu-res.

H1 [-] H2 [-]

Glucose 21 C 0.219 ± 0.002 0.308 ± 0.001

38 C 0.223 ± 0.001 0.300 ± 0.002

54 C 0.231 ± 0.001 0.292 ± 0.002

Fructose 11 C 0.394 ± 0.000 0.880 ± 0.000

17 C 0.376 ± 0.000 0.796 ± 0.003

21 C 0.362 ± 0.000 0.721 ± 0.000

more retained than Glucose, and that the selectivity between thetwo structural forms 1 and 2 (ratio of the Henry constants) is higherfor Fructose.The temperature dependence is expressed using the Van’t Hoff equa-tion 9.3. The pre-exponential factor H0i and the heat of adsorption∆Hads

i were determined by linear regression of the measured ln(Hi)vs. 1/T (see Fig. 9.7), their values are reported in Table 9.2.Glucose form 2 and both Fructose forms exhibit a negative heat

of adsorption ∆Hadsi , hence an expected decrease in Henry constant

9.4 results 197

2.8 3 3.2 3.4 3.6

1/T [K] ×10-3

-1.6

-1.5

-1.4

-1.3

-1.2

-1.1

-1

-0.9

-0.8

-0.7

-0.6

ln(H

i) [-

](a)

H1

H2

Hlit

1

Hlit

2

Glucose

3.25 3.3 3.35 3.4 3.45 3.5 3.55 3.6

1/T [K] ×10-3

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

ln(H

i) [-

]

(b)

H2

H1

Hlit

1

Hlit

2

Fructose

Figure 9.7: Van’t Hoff Plot for both structural forms of Glucose (a) andFructose (b). The fit obtained from the measurements (cirles)is shown in blue (species 1) and red (species 2) lines and therelationship from Carta et al.115 in black lines.


Table 9.2: Fitted Van’t Hoff parameters for the structural forms 1 and 2 ofboth sugars.

H0i [-] ∆Hadsi [Jmol−1]

Glucose 1 0.370 1.29× 103

2 0.179 −1.33× 103

Fructose 1 0.0315 −5.98× 103

2 0.00264 −1.38× 104

(and thus in affinity to the adsorbent) with increasing temperature.In contrast, Glucose form 1 exhibits an unexpected opposite trend.However, the temperature effect for Glucose form 1 is rather small,corresponding to a change in elution volume of the peak maximumof only around 2% of the column void volume for a change in tem-perature of 20 K.The Van’t Hoff relationships obtained by Carta et al.115 (black solidlines in Figure 9.7) differ from those obtained in this work, as Henryconstants for all species (of both sugars) are higher and those of spe-cies 1 (both sugars) do not depend on temperature, i.e. ∆Hads

1 = 0.Higher values of the Henry constants can be explained consideringthe fact that the resin used by Carta et al. featured a lower cross-linking degree, thus allowing for better accessibility of the adsorp-tion sites, and resulting in a higher affinitiy for the sugars. Despitethe (quantitative and qualitative) difference in heat of adsorption∆Hads

i , both the ratio H01/H02 and the difference in heat of adsorp-

tion ∆(∆Hads) = ∆Hads2 −∆Hads

1 are very similar, thus indicating asimilar selectivity H1/H2. The latter can be ascribed to the chemicalsimilarity of the two resins (both in Ca-form), such that interactionsbetween sugars and sorbent can be expected to be similar.The similarities between this work and that of Carta et al.115 indicatea substantial consistency, especially when considering the differenttypes of adsorbents and different fitting procedures utilised.

9.4 results 199

9.4.4 Reaction rate parameters

9.4.4.1 Estimation of reaction rate constants by the peak area method

Apparent reaction rate constants as defined in Equation 9.7, des-cribing the overall interconversion in the liquid and the adsorbedphase combined, were fitted according to the procedure describedin section 9.2.3. To this aim, the logarithmic ratios ln

(Nui /N

tot) andln(N0i /N

tot) were determined from the elution profiles. The ini-tial amount of species i, N0i , was obtained by multiplying the totalamount Ntot (injected amount, corresponding to the total area un-der the elution profile) by 1/(1+Neq

j /Neqi ), where the ratioNeq

j /Neqi

is estimated from literature values as discussed in section 9.2.1. Theamount of unreacted species i, Nu

i , was estimated by mirroring thefront or back half of the peak (for front and back peaks, respecti-vely), at an axis through the peak maximum, and calculating thepeak’s area.The values of ln

(Nui /N

tot) obtained in this way are plotted againstthe retention times of the two species in Figure 9.8. The reactionrate constants at the different temperatures were obtained from theslope of the regression line and are reported in Table 9.3. Note thatall symbols of the same type (circles, squares, triangles) were usedto fit one reaction rate constant kapp

1 (T); then kapp2 (T) was calculated

using Equation 9.10. The intercept with the vertical axis was fixedto be equal to ln

(N0i /N

tot) = 1/(1+Neqj /N

eqi ). The filled symbols

were not considered for the fit for reasons that are discussed in thefollowing.The accuracy of values of ln

(Nui /N

tot), and thus of the appliedfitting procedure, decreases with increasing reaction rates. At hightemperatures (where reaction rates are high), and especially in thecase of smaller peaks corresponding to the less abundant species,large zones of interconverted molecules affect the peak shapes andthe position of peak maxima. This compromises the accurate deter-mination of the logarithmic ratios ln

(Nui /N

tot). Due to the lack ofreliability, the Glucose data of the less abundant form 2 at a tempe-rature of 54 C, as well as all the data of the less abundant form 1

of Fructose, was excluded from the fit (values are marked as filledsymbols in Figure 9.8). Elution profiles of Glucose at a tempera-


0 500 1000 1500

ti

R [s]

-2.5

-2

-1.5

-1

-0.5

0

ln(N

iu/N

tot )

[-]

54°C

21°C

38°C

21°C

54°C

38°C

species 2

species 1

(a)Glucose

0 1000 2000 3000 4000 5000 6000 7000 8000

ti

R [s]

-8

-7

-6

-5

-4

-3

-2

-1

0

ln(N

iu/N

tot )

[-]

11°C

17°C

21°C

11°C17°C21°C

species 1

(b)Fructose

species 2

Figure 9.8: The logarithmic ratios ln(Nui /N

tot) vs. retention time tRi for Glu-cose (a) and Fructose (b). Blue symbols and lines correspond tothe reaction of species 1 and red ones to species 2. Filled sym-bols were not considered for the fit of the reaction rate constant.For Glucose, triangles are values determined at a temperature of54, squares at 38 and circles at 21 C. For Fructose, triangles arethe values determined at 21, squares at 17 and circles at 12 C.

9.4 results 201

ture of 72 C, as well as at the lowest two flowrates at 54 C werecompletely excluded from the analysis, as peaks corresponding tounreacted forms were no longer visible.

Table 9.3: Apparent reaction rate constants for Glucose and Fructose at dif-ferent temperatures

kapp1 [s−1] (fitted) k

app2 [s−1] (calculated)

Glucose 21 C 6.91× 10−5 1.19× 10−4

38 C 3.10× 10−4 5.34× 10−4

54 C 9.67× 10−4 1.66× 10−3

Fructose 11 C 7.89× 10−4 2.06× 10−4

17 C 1.11× 10−3 3.39× 10−4

21 C 2.26× 10−3 7.74× 10−4

9.4.4.2 Temperature dependence of rate constants

The temperature dependence of the apparent rate constants wasinvestigated assuming an Arrhenius relationship:127

ln(k

appi

)= ln

(k0i

)−EAiR

(1

T

)(9.20)

A linear regression of ln(k

appi

)vs. 1/T provides the two parameters

EAi , being the activation energy, and the pre-exponential factor k0iof species i.The Arrhenius plots for both sugars are shown in Figure 9.9 (datafrom the peak area method as open circles, and linear regressions ofthe data as dashed lines), resulting parameters are summarized inTable 9.4 (column “Peak Area Method”). As expected, all slopes arenegative as the reaction speeds up with an increase in temperature.For Glucose, activation energies for both forward and backward re-action are identical, which follows from the fact that the equilibriumconstant was assumed to be independent of temperature.


2.8 3 3.2 3.4 3.6 3.8

1/T [K] ×10-3

-13

-12

-11

-10

-9

-8

-7

-6

-5

-4

ln (

kia

pp)

[s-1

]

(a)

k2

app

k1

app

k1

lit

k2

lit

Glucose

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

1/T [K] ×10-3

-11

-10

-9

-8

-7

-6

-5

-4

-3

ln (

kia

pp)

[s-1

]

(b)

k2

app

k1

app

k1

lit

k2

lit

Fructose

Figure 9.9: Arrhenius plot for the interconversion of structural forms 1

(blue) and 2 (red) for Glucose (a) and Fructose (b). Experimentaldata obtained by the peak area method are given as open circles,while the Arrhenius relationship obtained by linear regressionof this data is shown as dashed lines. Arrhenius relationshipsobtained from the inverse fitting procedure are plotted as solidlines. Solid black lines indicate the fit by Carta et al.115

9.4 results 203

Table 9.4: Arrhenius parameters for the apparent reaction rate constants ofboth forms 1 and 2 of the two sugars.

Peak Area Method Inverse Fitting

k0i [s−1] EAi [Jmol−1] k0i [s−1] EAi [Jmol−1]

Glucose 1 3.10× 107 6.56× 104 3.19× 107 6.66× 104

2 5.32× 107 6.56× 104 5.19× 107 6.68× 104

Fructose 1 3.61× 1010 7.46× 104 3.45× 1010 7.51× 104

2 3.21× 1013 9.39× 104 3.53× 1013 9.38× 104

9.4.4.3 Inverse fitting and validation of model parameters

As discussed in section 9.4.4.1, the applied fitting method based onpeak areas is simple and computationally inexpensive, but bears theshortcoming of low accuracy at a high extent of interconversion (i.e.at high temperatures and/or low flowrates). In addition, as only ex-perimental profiles with clearly visible peaks of unreacted speciescan be used in the fitting, the range of conditions for parameterdetermination is very limited. With the aim of improving accuracyand reliability of the reaction kinetic parameters k0i and EAi , we haveperformed an inverse fitting, providing the parameters estimated insection 9.4.4.2 as initial guess.For the inverse fitting, elution profiles were simulated using the mo-del presented in section 9.2.2, and assuming the same conditionsas those of the pulse injection experiments. Dispersion and adsorp-tion parameters estimated in sections 9.4.2 and 9.4.3 were used forthe simulations. Reaction kinetic parameters k0i and EAi were thenadjusted iteratively to minimize the error between simulated andexperimental elution profiles. To reduce the computational effort,only two conditions (highest and lowest flowrate) at each tempera-ture were considered in the fit, resulting in eight operating condi-tions for Glucose and in six for Fructose. Error minimization wasachieved with a Matlab built-in genetic algorithm, constraining theparameter space to ±10% (Glucose) and ±20% (Fructose) around


the initial parameter values of k0i and EAi , and providing the valuesestimated by the peak area method as initial guesses. One shouldnote that, by fitting reaction kinetic parameters of species 1 and 2

independently, the ratios kapp1 /k

app2 , and with this the equilibrium

composition Neq2 /N

eq1 in the liquid phase (compare Equation 9.10)

are no longer fixed.Parameter values resulting from the inverse fitting procedure arereported in Table 9.4, and the Arrhenius relationships are furtherpresented as solid blue and red lines in Figure 9.9. Rather smallchanges of the parameter values between the initial guesses (peakarea estimation) and the values obtained by inverse fitting can benoted, with a maximum change of around 3.5% of the parametervalues for Glucose and of around 10% for Fructose. From the illus-tration in Figure 9.9, it is obvious that the new relationships (solidlines) deviate from the data points (open circles), which were obtai-ned with the peak area method and thus are prone to the inaccura-cies discussed in that context. In contrast to the previous Arrheniusrelationships (dashed lines), which were directly obtained by linearregression through the data points, the new relationships are basedon the fitting of the entire elution profiles, hence they are not relatedto those specific data points.

9.4.4.4 Comparison of reaction kinetic parameters with literature data

In comparison with the results of Carta et al.,115 presented as solidblack lines in Figure 9.9, it can be noted that the slopes (correspon-ding to activation energies), as well as the absolute values in the ex-perimentally investigated range are of the same order of magnitude,as expected due to the use of chemically similar resins (Ca-form).The data of Glucose was also compared to the results of Baker etal.,111 who estimated the reaction rate constant at 1.5 C in the pre-sence of a resin of similar type (HPX-87C column, 8% cross-linking,Ca-form, particle size 9µm), with the peak area method. Valuesof 1.1× 10−5 s−1 and 1.6× 10−5 s−1 for species 1 and 2, respecti-vely, were reported. Calculating rate constants at 1.5 C with the Ar-rhenius relationship and with parameter values estimated from thepeak area method (see Table 9.3) yielded values of 1.02× 10−5 s−1for species 1 and 1.76× 10−5 s−1 for species 2. Lower values of

9.4 results 205

0.60× 10−5 s−1 (species 1) and 1.07× 10−5 s−1 (species 2) were obtai-ned when using parameter values from the inverse fitting. Theserates are in good agreement with the predictions of Baker et al., es-pecially considering that the temperature of 1.5 C lies outside theexperimental temperature range of this contribution.

9.4.5 Comparison of Simulated and Experimental Profiles

Elution profiles (for all conditions investigated experimentally) weresimulated both based on parameters obtained from the peak areamethod in section 9.4.4.2 (see Appendix, Figures D.1 and D.2) andwith the parameters obtained from the inverse fitting (see Figures9.3 and 9.4), using the numerical model introduced in section 9.2.2.A considerable improvement in the description of experimental pro-files from the initial (peak area method) to the final parameter va-lues can be noted for both Glucose and Fructose. The accurate des-cription of experimental profiles for both sugars does not only con-firm the reaction kinetic parameters fitted in section 9.4.4.3, whichmainly determine size and shape of the interconversion zone, butalso the dispersion coefficients and adsorption parameters estima-ted in sections 9.4.2 and 9.4.3, which affect widths and positions ofpeaks corresponding to the unreacted forms.While the entire set of experimental Glucose profiles can be des-cribed quantitatively, some noticeable differences between experi-ments and simulations can be observed for Fructose (especially con-cerning the shape of the interconverted zone, and the relative peakheights of unreacted species 1 and 2). In this context, one shouldkeep in mind the considerable simplifications made for the Fructosesystem for modeling purposes, i.e. merging two structural forms(Furanose forms) into one single species 1, and entirely neglectinganother form (α-Pyranose). The simplifications of the Fructose sy-stem can likely lead to the observed mismatch between experimentsand simulations. On the other hand, considering the system of fourstructural forms without simplifications would complicate the mo-del, considerably increase the number of parameters to be deter-mined (interconversion between and adsorption of 4 species), andpose a major experimental challenge, namely that of to determining


the large number of parameters for species which can hardly bedetected.

9.4.6 Reaction rates in the stationary phase

Since rate constants in the mobile phase (i.e. in aqueous solution) areavailable in the literature, the rate constants in the stationary phasecan be estimated from the apparent rate constants using Equation9.7. Comparison of the rates in mobile and adsorbed phase allows todetermine catalytic or inhibiting effects of the sorbent. Note that inthis section, we only consider the Glucose system, since the physicalrelevance of model parameters determined for the Fructose systemis limited, considering the discussed model simplifications.Reaction rate constants for both Glucose anomers in water are gi-ven by LeBarc for a range of temperatures between 30 and 45 C,128

whilst Ballash provides the sum of the rate constants of both Glu-cose anomers between 15 and 35 C.122 From these sums, values ofthe rate constants of forms 1 and 2 in water were obtained throughthe ratio of the rate constants km

2 /km1 , assuming a constant value of

1.72 as discussed in section 9.2.1.From the literature values of reaction rate constants in the mobilephase and from apparent reaction rate constants kapp

i calculatedwith the Arrhenius equation (parameters obtained through inversefitting), the reaction rate constants in the stationary phase ks

i wereestimated using Equation 9.7. The results are summarized in Table9.5 and Figure 9.10.The reaction rate constants for the stationary phase are of the same

order of magnitude of, or (at high temperatures) slightly larger thanthe ones in solution. This indicates a minor catalytic effect of the re-sin. In earlier contributions, it was observed that the complexationof sugar molecules with cations has a catalytic effect on the intercon-version reaction both in aqueous solution129,130 or when attached toa cation-exchange resin.111 However, in both cases (aqueous solu-tion or presence of an adsorbent), the effect of Ca2+ ions, which isinvestigated in this contribution, was found to be insignificant orless significant than that of other ions, such as Cu2+ or Pb2+.111,130

Similar values of rate constants in the stationary and the mobile

9.4 results 207

Table 9.5: Reaction rate constants for Glucose anomers 1 and 2 in adsorbedand mobile phase as well as apparent reaction rate constants (cal-culated with the Arrhenius relationship using parameters obtai-ned by inverse fitting). Reaction rate constants in the mobilephase are taken from the literature. Literature sources [122] and[128] are given in superscripts.

T kappi (Eq. 9.20) km

i (Lit.) ksi (calculated)

[C] ×104 [s−1] ×104 [s−1] ×104 [s−1]

species 1 15 0.27 0.56122 0.79

24.7 0.66 1.36122 1.96

25 0.68 1.45122 1.93

30 1.06 2.57128 2.53

35 1.62 2.84128 5.47

35.2 1.65 3.64122 4.46

40 2.46 3.45128 9.46

45 3.67 4.63128 14.8

species 2 15 0.41 0.97122 0.72

24.7 1.01 2.33122 1.89

25 1.04 2.49122 1.94

30 1.62 4.00128 2.76

35 2.49 4.28128 6.32

35.2 2.53 6.26122 4.35

40 3.77 5.15128 11.1

45 5.65 6.73128 17.9


3.1 3.2 3.3 3.4 3.5

1/T [K] ×10-3

-11

-10.5

-10

-9.5

-9

-8.5

-8

-7.5

-7

-6.5

-6

ln(k

i) [s

-1]

ki

m

ki

s

ki

app

species 1

species 2

Figure 9.10: Reaction rate constants in the mobile phase (filled circles, va-lues in light shades of blue and red are taken from Ref. [128]and darker shades from Ref. [122]), adsorbed phase (diamonds,obtained through calculation) and apparent reaction rate con-stants between 15 and 45 C for both Glucose anomers. Valuesfor species 1 and 2 are indicated in blue and red, respectively.

phase determined in the current work are thus in line with the fin-dings in literature.

9.5 conclusion

In this work, we investigated the interplay of mutarotation and chro-matographic separation of sugars both qualitatively and quantita-tively. Adsorption and reaction kinetic parameters were estimatedthrough a simple fitting method, based on elution times of peakmaxima and on the evolution of peak area ratios over residencetime in the chromatographic column. Values for reaction kinetic pa-rameters were further refined by an inverse fitting procedure, withparameter values resulting from the peak area method as an initialguess, and using a numerical mass balance model. The enhanced pa-rameters yielded a very good agreement between simulations withthe chromatographic model and experimental profiles. Parametervalues were further verified through a comparison with literaturedata. From the determined apparent reaction rate constants and

9.5 conclusion 209

using literature data for rate constants in aqueous solution, reactionrate constants in the adsorbed phase could be estimated and inter-preted.The high accuracy of the estimated model parameters for the Glu-cose system allows for a reliable prediction of the interplay of in-terconversion and chromatographic separation over a rather broadrange of operating conditions. As an example, the temperature ofpeak coalescence of the two anomers can be determined accuratelyfor specific flowrates (roughly ranging between 50 and 60

C fromlow to high flowrates). While adsorption parameters are specific forthe utilised sorbent, reaction kinetics should be generally applicableto Ca-form PS-DVB resins, since no significant effect of the sorbenton the reaction kinetics was observed.Chromatographic separation processes are well established in thesugar industry, and are commonly performed at elevated tempera-tures, where the mutarotation is fast and the components elute asa mixture of the different forms. However, the effect of mutarota-tion on elution profiles in the context of sugar separations has beendiscussed rarely,111,115,117 and has not been investigated in the re-cent years. Besides providing a quantitative and in depth analysisof adsorption and reaction properties for these systems, this paperintends also to raise awareness of the existence and the impact ofmutarotation on the chromatographic separation performance, as itcan result in distorted profiles, especially at low to moderate tempe-ratures.

B I B L I O G R A P H Y

1. Brooks, R.; Corey, A., Hydraulic properties of porous media. Hydrol.Pap. 1964, 3, Civ. Eng. Dep., Colo. State Univ., Fort Collins.

2. Corey, A.; Brooks, R., Properties of Porous Media Affecting FluidFlow. J. Irrig. Drain. Div. 1966, 4855, 61–88.

3. Hubbert, M., Darcy’s Law and the Field Equations of the Flow ofUnderground Fluids. J. Pet. Technol. 1956, 207, 222–239.

4. Orr, F. M., Theory of Gas Injection Processes. 1st. Holte, Denmark:Tie-Line Publications, 2007.

5. Orr, F. M.; Heller, J. P.; Taber, J. J., Carbon Dioxide Flooding forEnhanced Oil Recovery : Promise and Problems. AOCS AnnualMeeting. Toronto, 1982.

6. Pope, G. A., The Application of Fractional Flow Theory to EnhancedOil Recovery. Soc. Pet. Eng. J. 1980, 20(03), 191–205.

7. Thomas, S, Enhanced oil recovery-an overview. Oil Gas Sci. Technol.2008, 63(1), 9–19.

8. Mazzotti, M., Nonclassical composition fronts in nonlinearchromatography: Delta-shock. Ind. Eng. Chem. Res. 2009, 48,7733–7752.

9. Lu, X.-C.; Li, F.-C.; Watson, A. T., Adsorption measurements inDevonian shales. Fuel 1995, 74(4), 599–603.

10. Seto, C., Analytical Theory For Two-Phase, Multicomponent Flow inPorous Media with Adsorption. PhD Dissertation, Stanford Univ.2007.

11. White, C. M.; Strazisar, B. R.; Granite, E. J.; Hoffman, J. S.;Pennline, H. W., Separation and Capture of CO_2 from LargeStationary Sources and Sequestration in Geological Formations —Coalbeds and Deep Saline Aquifers. J. Air Waste Manage. Assoc. 2003,53(6), 645–715.

12. Blunt, M. J., Flow in porous media—pore-network models andmultiphase flow. Curr. Opin. Colloid Interface Sci. 2001, 6(3), 197–207.

13. Whitaker, S., Flow in Porous Media II: The Governing Equations forImmiscible , Two-Phase Flow. Transp. Porous Media 1986, 1, 105–125.

212 bibliography

14. Corey, A. T., The interrelation between oil and gas relativepermeabilities. Prod. Mon. 1954, 19, 38–41.

15. Buckley, S.; Leverett, M., Mechanism of fluid displacement in sands.Trans. AIME 1942, 146(1337), 107–116.

16. Lane, H. S.; Lancaster, D. E.; Watson, A. T., Characterizing the roleof desorption in gas production from devonian shales. EnergySources 1991, 13(3), 337–359.

17. Somasundaran, P.; Zhang, L., Adsorption of surfactants on mineralsfor wettability control in improved oil recovery processes. J. Pet. Sci.Eng. 2006, 52(1-4), 198–212.

18. Seto, C. J.; Jessen, K.; Orr, F. M., A Multicomponent ,Two-Phase-Flow Model for CO 2 Storage and EnhancedCoalbed-Methane Recovery. SPE J. 2009, (March), 30–40.

19. Joss, L.; Pini, R., Digital Adsorption: 3D Imaging of Gas AdsorptionIsotherms by X-ray Computed Tomography. J. Phys. Chem. C 2017,121, 26903–15.

20. Pini, R., Multidimensional quantitative imaging of gas adsorption innanoporous solids. Langmuir 2014, 30(37), 10984–10989.

21. Pini, R., Interpretation of net and excess adsorption isotherms inmicroporous adsorbents. Microporous Mesoporous Mater. 2014, 187,40–52.

22. Guiochon, G.; Felinger, A.; Shirazi, G., Fundamentals of Preparativeand Non-Linear Chromatography. 2nd. Academic Press, 2006.

23. Rhee, H.-K.; Aris, R.; Amundson, N. R., First-Order PartialDifferential Equations - 1. Theory and Application of Single Equations.Englewood Cliffs, N. J.: Prentice-Hall, 1986.

24. Rhee, H.-K.; Aris, R.; Amundson, N. R., First-Order PartialDifferential Equations - 2. Theory and Application of Hyperbolic Systemsof Quasilinear Equations. Englewood Cliffs, N. J.: Prentice-Hall, 1989.

25. Jermann, S.; Ortner, F.; Rajendran, A.; Mazzotti, M., Absence ofexperimental evidence of a delta-shock in the system phenetole and4-tert-butylphenol on Zorbax 300SB-C18. J. Chromatogr. A 2015, 1425,116–128.

26. Ortner, F.; Jermann, S.; Joss, L.; Mazzotti, M., Equilibrium TheoryAnalysis of a Binary Chromatographic System Subject to a MixedGeneralized Bi-Langmuir Isotherm. Ind. Eng. Chem. Res. 2015, 54(45),11420–11437.

bibliography 213

27. Mazzotti, M.; Tarafder, A.; Cornel, J.; Gritti, F.; Guiochon, G.,Experimental evidence of a delta-shock in nonlinearchromatography. J. Chromatogr. A 2010, 1217(13), 2002–2012.

28. Mazzotti, M., Local Equilibrium Theory for the BinaryChromatography of Species Subject to a Generalized LangmuirIsotherm. Ind. Eng. Chem. Res. 2006, 45, 5332–5350.

29. Sun, M., Delta shock waves for the chromatography equations asself-similar viscosity limits. Q. Appl. Math. 2011, 69(3), 425–443.

30. Cheng, H.; Yang, H., Delta shock waves in chromatographyequations. J. Math. Anal. Appl. 2011, 380(2), 475–485.

31. Li, X.; Shen, C., Viscous regularization of delta shock wave solutionfor a simplified chromatography system. Abstr. Appl. Anal. 2013,2013(ID 893465).

32. Keyfitz, B. L.; Kranzer, H., Spaced of Weighted Measures forConservation Laws with Singular Shock Solutions. J. DifferentialEquations 1995, 118, 420–451.

33. Tsikkou, C., Singular shocks in a chromatography model. J. Math.Anal. Appl. 2016, 439(2), 766–797.

34. Oberguggenberger, M., Multiplication of distributions and applicationsto partial differential equations. London: Longmean Scientific &Technical, 1992.

35. Nedeljkov, M., Delta and singular delta locus for one-dimensionalsystems of conservation laws. Math. Methods Appl. Sci. 2004, 27(8),931–955.

36. Krejic, N.; Krunic, T.; Nedeljkov, M., Numerical verification of deltashock waves for pressureless gas dynamics. J. Math. Anal. Appl. 2008,345(1), 243–257.

37. Estrada, R; Kanwal, R. P., Taylor Expansions for Distributions. Math.Methods Appl. Sci. 1993, 16(4), 297–304.

38. Gritti, F.; Guiochon, G., Mass transfer kinetics, band broadening andcolumn efficiency. J. Chromatogr. A 2012, 1221, 2–40.

39. Golshan-Shirazi, S.; Guiochon, G., Modeling of preparative liquidchromatography. J. Chromatogr. A 1994, 658, 149–171.

40. Javeed, S.; Qamar, S.; Seidel-Morgenstern, A.; Warnecke, G., Efficientand accurate numerical simulation of nonlinear chromatographicprocesses. Comput. Chem. Eng. 2011, 35(11), 2294–2305.

214 bibliography

41. Prausnitz, J. M.; Lichtenthaler, R. N.; Azevedo, E. G. de, Molecularthermodynamics of fluid-phase equilibria. 2nd. Prentice-Hall, 1986.

42. Wilson, G. M., Vapor-Liquid Equilibrium. XI. A New Expression forthe Excess Free Energy of Mixing. J. Am. Chem. Soc. 1964, 86(2),127–130.

43. Renon, H.; Prausnitz, J. M., Local compositions in thermodynamicexcess functions for liquid mixtures. AIChE J. 1968, 14(1), 135–144.

44. Abrams, D. S.; Prausnitz, J. M., Statistical thermodynamics of liquidmixtures: a new expression for the excess Gibbs energy of partly orcompletly miscibles systems. AIChE J. 1975, 21(1), 116–128.

45. Anderson, T. F.; Prausnitz, J. M., Application of the UNIQUACequation to calculation of multicomponent phase equilibria. 1.Vapor-liquid equilibria. Ind. Eng. Chem. Process Des. Dev. 1978, 17(4),552–561.

46. Anderson, T. F.; Prausnitz, J. M., Application of the UNIQUACequation to calculation of multicomponent phase equilibria. 2.Liquid-liquid equilibria. Ind. Eng. Chem. Process Des. Dev. 1978, 17(4),561–567.

47. Fredenslund, A.; Jones, R. L.; Prausnitz, J. M., Group-contributionestimation of activity coefficients in nonideal liquid mixtures. AIChEJ. 1975, 21(6), 1086–1099.

48. Larsen, B. L.; Rasmussen, P.; Fredenslund, A., A modified UNIFACgroup-contribution model for prediction of phase equilibria andheats of mixing. Ind. Eng. Chem. Res. 1987, 26(3), 2274–2286.

49. Michelsen, M. L., The Isothermal Flash Problem, Part 1: Stability.Fluid Phase Equilib. 1982, 9, 1–19.

50. Michelsen, M. L., The Isothermal Flash Problem. Part 2: Phase-SplitCalculation. Fluid Phase Equilib. 1982, 9, 21–40.

51. Heinonen, J.; Rubiera Landa, H. O.; Sainio, T.;Seidel-Morgenstern, A., Use of Adsorbed Solution theory to modelcompetitive and co-operative sorption on elastic ion exchange resins.Sep. Purif. Technol. 2012, 95, 235–247.

52. Nowak, J.; Poplewska, I.; Antos, D.; Seidel-Morgenstern, A.,Adsorption behaviour of sugars versus their activity in single andmulticomponent liquid solutions. J. Chromatogr. A 2009, 1216(50),8697–704.

53. Myers, A. L.; Prausnitz, J. M., Thermodynamics of mixed-gasadsorption. AIChE J. 1965, 11(1), 121–127.

bibliography 215

54. Radke, C. J.; Prausnitz, J. M., Thermodynamics of Multi-SoluteAdsorption From Dilute Aqueous Solutions. AIChE J. 1972, 18(4),761–768.

55. Antos, D.; Piatkowski, W., Thermodynamics of MulticomponentLiquid Chromatography. Chem. Ing. Tech. 2016, 88(11), 1586–1597.

56. Pinder, G.; Gray, W., Essentials of Multiphase Flow and Transport inPorous Media. New Jersey: John Wiley & Sons, 2008.

57. Sahimi, M., Flow and Transport in Porous Media and Fractured Rock.2nd ed. Weinheim: Wiley-VCH, 2011.

58. Whitaker, S., Flow in Porous Media I: A Theoretical Derivation ofDarcy’s Law. Transp. Porous Media 1986, 1, 3–25.

59. Talu, O.; Zwiebel, I., Multicomponent adsorption equilibria ofnonideal mixtures. AIChE J. 1986, 32(8), 1263–1276.

60. Hefti, M.; Marx, D.; Joss, L.; Mazzotti, M., Microporous andMesoporous Materials Adsorption equilibrium of binary mixturesof carbon dioxide and nitrogen on zeolites ZSM-5 and 13X.Microporous Mesoporous Mater. 2015, 215, 215–228.

61. Gritti, F.; Guiochon, G., New thermodynamically consistentcompetitive adsorption isotherm in RPLC. J. Colloid Interface Sci.2003, 264(1), 43–59.

62. Tarafder, A.; Mazzotti, M., A method for deriving explicit binaryisotherms obeying the ideal adsorbed solution theory. Chem. Eng.Technol. 2012, 35(1), 102–108.

63. O’Brien, J. A.; Myers, A. L., Rapid Calculations of MulticomponentAdsorption Equilibria from Pure Isotherm Data. Ind. Eng. Chem.Process Des. Dev. 1985, 24(4), 1188–1191.

64. O’Brien, J. A.; Myers, A. L., A Comprehensive Technique forEquilibrium Calculations in Adsorbed Mixtures: The GeneralizedFastIAS Method. Ind. Eng. Chem. Res. 1988, 27(11), 2085–2092.

65. Do, D. D., Adsorption Analysis: Equilibria and Kinetics. Series onChemical Engineering, Volume 2. London: Imperial College Press,1998.

66. Rubiera Landa, H. O.; Flockerzi, D.; Seidel-Morgenstern, A., AMethod for Efficiently Solving the IAST Equations with anApplication to Adsorber Dynamics. AIChE J. 2013, 59(4), 1263–1277.

67. Brunauer, S.; Emmett, P. H.; Teller, E., Adsorption of Gases inMultimolecular Layers. J. Am. Chem. Soc. 1938, 60(2), 309–319.

216 bibliography

68. Welge, H. J., A simplified method for computing oil recovery by gasor water drive. Petroleum Transactions, AIME 1952, 195, 92–98.

69. Helfferich, F. G., Theory of Multicomponent, MultiphaseDisplacement in Porous Media. Soc. Pet. Eng. J. 1981, 51–62.

70. Toth, J.; Bodi, T.; Szucs, P.; Civan, F., Convenient formulae fordetermination of relative permeability from unsteady-state fluiddisplacements in core plugs. J. Pet. Sci. Eng. 2002, 36, 33–44.

71. Perry, R. H.; Green, D., Perry’s Chemical Engineers’ Handbook. 6th ed.Singapore: McGraw-Hill, 1984.

72. Ta, H. D.; Seidel-Morgenstern, A., Impact of Liquid-Phase VolumeChanges on Estimating Reaction Rate Parameters: TheHomogeneously Catalyzed Hydrolysis of Methyl Formate. Ind. Eng.Chem. Res. 2017, 56(45), 13086–13095.

73. Ulrici, A.; Cocchi, M.; Foca, G.; Manfredini, M.; Manzini, D.;Marchetti, A.; Sighinolfi, S.; Tassi, L., Study of the dependence ontemperature and composition of the volumic properties ofethane-1,2-diol + 2-methoxyethanol + 1,2-dimethoxyethane + watersolvent system and graphical representation in the quaternarydomain. J. Solution Chem. 2006, 35(2), 139–159.

74. Ortner, F.; Joss, L.; Mazzotti, M., Equilibrium theory analysis ofliquid chromatography with non-constant velocity. J. Chromatogr. A2014, 1373, 131–140.

75. Gmehling, J.; Li, J.; Schiller, M., A modified UNIFAC model. 2.Present parameter matrix and results for different thermodynamicproperties. Ind. Eng. Chem. Res. 1993, 32(3), 178–193.

76. Ottani, S.; Comelli, F.; Bologna, I; Ciamician, C. G.; Bologna, I,Densities, Viscosities, and Excess Molar Enthalpies of PropyleneCarbonate + Anisole or + Phenetole at (293.15 , 303.15 , and 313.15)K. J. Chem. Eng. Data 2001, 46, 125–129.

77. Teja, A. S.; Rice, P, Generalized corresponding states method for theviscosities of liquid mixtures. Ind. Eng. Chem. Fundam. 1981, 20,77–81.

78. Haynes, W. M.; Lide, D. R.; Bruno, T. J., CRC Handbook of Chemistryand Physics. 95th ed. Productivity Pr Inc, 2014.

79. Mikhail, S.; Kimel, W., Densities and Viscosities of Methanol-WaterMixtures. J. Chem. Eng. Data 1961, 6(4), 533–537.

bibliography 217

80. Helfferich, F. G., Multicomponent Ion Exchange in Fixed Beds:Generalized Equilibrium Theorv for Systems with ConstantSeparation Factors. Ind. Eng. Chem. Fundam. 1967, 6(3), 362–364.

81. Zhang, W.; Shan, Y.; Seidel-Morgenstern, A., Breakthrough curvesand elution profiles of single solutes in case of adsorption isothermswith two inflection points. J. Chromatogr. A 2006, 1107, 216–225.

82. Mazzotti, M.; Rajendran, A., Equilibrium theory-based analysis ofnonlinear waves in separation processes. Annu. Rev. Chem. Biomol.Eng. 2013, 4(February), 119–41.

83. Migliorini, C.; Mazzotti, M.; Morbidelli, M., Continuouschromatographic separation through simulated moving beds underlinear and nonlinear conditions. J. Chromatogr. A 1998, 827, 161–173.

84. Mazzotti, M., Equilibrium theory based design of simulated movingbed processes for a generalized Langmuir isotherm. J. Chromatogr. A2006, 1126, 311–322.

85. Zhu, J., Multicomponent Multiphase Flow in Porous Media withTemperature Variation or Adsorption. PhD Dissertation, StanfordUniv. 2003.

86. Seto, C. J.; Orr, F. M., Analytical solutions for multicomponent,two-phase flow in porous media with double contact discontinuities.Transp. Porous Media 2009, 78(2), 161–183.

87. Marcilla, A; Serrano, M. D.; Reyes-Labarta, J. A.; Olaya, M. M.,Checking Liquid - Liquid Plait Point Conditions and TheirApplication in Ternary Systems. Ind. Eng. Chem. Res. 2012, 51,5098–5102.

88. Mallison, B. T.; Gerritsen, M. G.; Jessen, K; Orr, F. M., High-OrderUpwind Schemes for Two-Phase, Multicomponent Flow. SPE J. 2005,10(3), 297–311.

89. Lieres, E. von; Andersson, J., A fast and accurate solver for thegeneral rate model of column liquid chromatography. Comput.Chem. Eng. 2010, 34(8), 1180–1191.

90. Sundmacher, K.; Kienle, A.; Seidel-Morgenstern, A., IntegratedChemical Processes - Synthesis, Operation, Analysis and Control.Weinheim: Wiley-VCH, 2005.

91. Rodrigues, A. E.; Pereira, C. S.; Santos, J. C., ChromatographicReactors. Chem. Eng. Technol. 2012, 35(7), 1171–1183.

218 bibliography

92. Constantino, D. S.; Pereira, C. S.; Faria, R. P.; Loureiro, J. M.;Rodrigues, A. E., Simulated moving bed reactor for butyl acrylatesynthesis: From pilot to industrial scale. Chem. Eng. Process.: ProcessIntensification 2015, 97, 153–168.

93. Mazzotti, M.; Kruglov, A.; Neri, B.; Gelosa, D.; Morbidelli, M., AContinuous Chromatographic Reactor: SMBR. Chem. Eng. Sci. 1996,51(10), 1827–1836.

94. Mazzotti, M.; Neri, B.; Gelosa, D.; Kruglov, A.; Morbidelli, M.,Kinetics of Liquid-Phase Esterification Catalyzed by Acidic Resins.Ind. Eng. Chem. Res. 1997, 36(1), 3–10.

95. Lode, F.; Houmard, M.; Migliorini, C.; Mazzotti, M.; Morbidelli, M.,Continuous reactive chromatography. Chem. Eng. Sci 2001, 56(2),269–291.

96. Agrawal, G.; Oh, J.; Sreedhar, B.; Tie, S.; Donaldson, M. E.;Frank, T. C.; Schultz, A. K.; Bommarius, A. S.; Kawajiri, Y.,Optimization of reactive simulated moving bed systems withmodulation of feed concentration for production of glycol etherester. J. Chromatogr. A 2014, 1360, 196–208.

97. Kulkarni, A. A.; Zeyer, K. P.; Jacobs, T.; Kaspereit, M.; Kienle, A.,Feasibility studies and dynamics of catalytic liquid phaseesterification reactions in a micro plant. Chem. Eng. J. 2008, 135,270–275.

98. Mitra Ray, N.; Ray, A. K., Modelling, simulation, and experimentalstudy of a simulated moving bed reactor for the synthesis ofbiodiesel. Can. J. Chem. Eng. 2016, 94(5), 913–923.

99. Schmitt, M.; Hasse, H., Chemical equilibrium and reaction kineticsof heterogeneously catalyzed n-hexyl acetate esterification. Ind. Eng.Chem. Res. 2006, 45(12), 4123–4132.

100. Reymond, H.; Vitas, S.; Vernuccio, S.; Von Rohr, P. R., ReactionProcess of Resin-Catalyzed Methyl Formate Hydrolysis in BiphasicContinuous Flow. Ind. Eng. Chem. Res. 2017, 56(6), 1439–1449.

101. Grüner, S.; Kienle, A., Equilibrium theory and nonlinear waves forreactive distillation columns and chromatographic reactors. Chem.Eng. Sci. 2004, 59(4), 901–918.

102. Vu, T. D.; Seidel-Morgenstern, A.; Grüner, S.; Kienle, A., Analysis ofEster Hydrolysis Reactions in a Chromatographic Reactor UsingEquilibrium Theory and a Rate Model. Ind. Eng. Chem. Res. 2005, 44,9565–9574.

bibliography 219

103. Grüner, S.; Mangold, M.; Kienle, A., Dynamics of reactionseparation processes in the limit of chemical equilibrium. AIChE J.2006, 52(3), 1010–1026.

104. Pereira, C. S. M.; Zabka, M.; Silva, V. M.T. M.; Rodrigues, A. E., Anovel process for the ethyl lactate synthesis in a simulated movingbed reactor (SMBR). Chem. Eng. Sci 2009, 64(14), 3301–3310.

105. Minceva, M.; Gomes, P. S.; Meshko, V.; Rodrigues, A. E., Simulatedmoving bed reactor for isomerization and separation of p-xylene.Chem. Eng. J. 2008, 140(1-3), 305–323.

106. Graça, N. S.; Pais, L. S.; Silva, V. M.T. M.; Rodrigues, A. E., Analysisof the synthesis of 1,1-dibutoxyethane in a simulated moving-bedadsorptive reactor. Chem. Eng. Process.: Process Intensification 2011,50(11-12), 1214–1225.

107. Barbosa, D.; Doherty, M. F., A New Set of Composition Variables forthe Representation of Reactive-Phase Diagrams. Proc. R. Soc. A 1987,413(1845), 459–464.

108. Jessen, K.; Stenby, E. H.; Orr, F. M., Interplay of Phase Behavior andNumerical Dispersion in Finite-Difference CompositionalSimulation. SPE J. 2004, 9(2), 193–201.

109. Tiihonen, J.; Sainio, T.; Kärki, A.; Paatero, E., Co-eluent effect inpartition chromatography. Rhamnose–xylose separation with strongand weak cation-exchangers in aqueous ethanol. J. Chromatogr. A2002, 982(1), 69–84.

110. Luz, D. A.; Rodrigues, A. K. O.; Silva, F. R. C.; Torres, A. E. B.;Cavalcante, C. L.; Brito, E. S.; Azevedo, D. C. S., Adsorptiveseparation of fructose and glucose from an agroindustrial waste ofcashew industry. Bioresour. Technol. 2008, 99(7), 2455–2465.

111. Baker, J.; Himmel, M., Separation of Sugar Anomers by AqueousChromatography on Calcium- and Lead-Form Ion-ExchangeColumns. J. Chromatogr. 1986, 357, 161–181.

112. Huck, C. W.; Huber, C. G.; Bonn, G. K., HPLC of Carbohydrateswith Cation- and Anion-Exchange Silica and Resin-Based StationaryPhases. Carbohydrate Analysis by Modern Chromatography andElectrophoresis. Ed. by El Rassi, Z. Journal of ChromatographyLibrary, Elsevier Science, 2002. Chap. 5, 165–205.

113. Vente, J. a.; Bosch, H.; De Haan, A. B.; Bussmann, P. J. T.,Comparison Of Sorption Isotherms Of Mono- and DisaccharidesRelevant To Oligosaccharide Separations For Na, K, And Ca LoadedCation Exchange Resins. Chem. Eng. Commun. 2005, 192(1), 23–33.

220 bibliography

114. Nishikawa, T.; Suzuki, S.; Kubo, H.; Ohtani, H., On-columnisomerization of sugars during high-performance liquidchromatography: Analysis of the elution profile. J. Chromatogr. A1996, 720(1-2), 167–172.

115. Carta, G.; Mahajan, A.; Cohen, L.; Byers, C., Chromatography ofreversibly reacting mixtures: mutarotation effects in sugarseparations. Chem. Eng. Sci. 1992, 47(7).

116. Keller, R.; Giddings, C., Multiple Zones and Spots inChromatography. J. Chromatogr. 1960, 3, 205–220.

117. Goulding, R., Liquid Chromatography of Sugars and RelatedPolyhydric Alcohols on Cation Exchangers. J. Chromatogr. 1975, 103,229–239.

118. Giddings, J. C., Stochastic Considerations on ChromatographicDispersion. J. Chem. Phys. 1957, 26(1), 169.

119. Trapp, O.; Schoetz, G.; Schurig, V., Determination ofenantiomerization barriers by dynamic and stopped-flowchromatographic methods. Chirality 2001, 13(8), 403–414.

120. Trapp, O., Unified equation for access to rate constants of first-orderreactions in dynamic and on-column reaction chromatography.Anal. Chem. 2006, 78(1), 189–198.

121. Trapp, O., Fast and Precise Access to Enantiomerization RateConstants in Dynamic Chromatography. Chirality 2006, 18, 489–497.

122. Ballash, N.; Robertson, E., The Mutarotation of Glucose inDimethylsulfoxide and Water Mixtures. Can. J. Chem. 1973, 51,556–564.

123. Shallenberger, R. S.; Lee, C. Y.; Acree, T. E.; Barnard, J.;Lindley, M. G., Specific rotation of alpha-D- andbeta-D-fructofuranose. Carbohydr. Res. 1977, 58, 209–211.

124. Angyal, S. J., The Composition of Reducing Sugars in Solution. Adv.Carbohydr. Chem. Biochem. 1984, 42.

125. Van Leer, B., Towards the Ultimate Conservative Difference Scheme.II. Monotonicity and conservation Combined in a Second-OrderScheme. J. Comput. Phys 1974, 14, 361–370.

126. Jung, S.; Ehlert, S.; Pattky, M.; Tallarek, U., Determination of theinterparticle void volume in packed beds via intraparticle Donnanexclusion. J. Chromatogr. A 2010, 1217(5), 696–704.

127. Arrhenius, S. A., On the reaction velocity of the inversion of canesugar by acids. Z. Phys. Chem 1889, 4, 226–248.

bibliography 221

128. Le Barc, N.; Grossel, J. M.; Looten, P.; Mathlouthi, M., Kinetic studyof the mutarotation of D-glucose in concentrated aqueous solutionby gas-liquid chromatography. Food Chem. 2001, 74(1), 119–124.

129. O’Connor, C. J.; Odell, A. L.; Bailey, A. A., Mutarotation ofD(+)-GIucose. I evidence of electrophilic attack by hydratedmonomeric and dimeric Copper(II) species. Aust. J. Chem. 1982,35(5), 951–960.

130. Bobbio, P. A.; Bobbio, F. O.; Rodrigues, L. K.; Reyes, F. G. R.,Mutarotation I - Effect of Cuˆ2+ and Caˆ2+ on the Mutarotationof Glucose. An. Acad. Bras. Cienc. 1978, 50, 225.

131. Bard, Y., Nonlinear Parameter Estimation. New York: Academic Press,1974.

132. Ochsenbein, D. R.; Schorsch, S.; Vetter, T.; Mazzotti, M.; Morari, M.,Growth Rate Estimation of β l -Glutamic Acid from OnlineMeasurements of Multidimensional Particle Size Distributions andConcentration. Ind. Eng. Chem. Res. 2014, 53(22), 9136–9148.

133. De Albuquerque, I.; Mazzotti, M., Crystallization Process DesignUsing Thermodynamics To Avoid Oiling Out in a Mixture ofVanillin and Water. Cryt. Growth Des. 2014, 14, 5617–5625.

134. Sorensen, J.; Magnussen, T.; Rasmussen, P.; Fredenslung, A.,Liquid-Liquid Equilibrium Data: Their Retrieval, Correlation andPrediction. Fluid Phase Equilib. 1979, 3, 47–82.

135. Voskov, D.; Tchelepi, H., Compositional space parametrization formiscible displacement simulation. Transp. Porous Media 2008, 75(1),111–128.

136. Voskov, D.; Tchelepi, H., Compositional space parameterization:Theory and application for immiscible displacements. Soc. Pet. Eng.J. 2009, 14(3), 431–440.

137. Voskov, D.; Tchelepi, H., Compositional space parameterization:Multicontact miscible displacements and extension to multiplephases. Soc. Pet. Eng. J. 2009, 14(3), 441–449.

AS U P P L E M E N TA RY M AT E R I A L F O R C H A P T E R 4

a.1 uncertainty quantification of isotherm and rast

parameters

Standard deviations of the single component isotherm parameters,as well as the parameters of the Redlich-Kister equation and thespreading pressure dependence required by the Real Adsorbed So-lution theory, were quantified based on the theory of the MaximumLikelihood Estimate (MLE).131,132 It is assumed that measured vari-ables are independent and deterministic, with normally distributederrors of zero mean and given variance. These errors should furtherbe uncorrelated between the different measurements, and be inde-pendent for the different measured variables. Under these assump-tions the objective function φ according to the MLE theory shouldbe formulated as follows:

φ(p) =1

Nt

Nv∑i=1

ln

Nt∑j=1

(yij − yij(p)

)2 (A.1)

Here, p is the parameter vector, and yij and yij(p) correspond tothe experimental values and the predictions, respectively. In the con-text of the fitting of isotherm parameters, yij would correspond tothe adsorbed phase concentrations of component i (PNT or TBP),while for the fitting of the RAST parameters, yij − yij(p) wouldcorrespond to the overall error between the experimental and thecalculated concentration profile, normalized by the number of indi-vidual, measured data points within such profile. Nt is the numberof observations, which in both cases would correspond to the num-ber of experiments, while Nv is the number of independent outputs,which would equal 2 in both cases, corresponding to the values forthe two components PNT and TBP. We want to point out that noneof the objective functions (equations 4.40 and 4.44) used in this workare in the form of the function φ (equation A.1), but the applid ob-

224 supplementary material for chapter 4

jective functions clearly provided the better fit.Nevertheless, we use the MLE theory for the estimation of standarddeviations of the fitted parameters. For this purpose, we approxi-mate in a first step the diagonal measurement error covariance ma-trix V with the elements

Vii ≈ σ2 =1

Nt

Nt∑j=1

(yij − yij(p)

)2 . (A.2)

Applying, additionally to the multiple assumptions stated at thebeginning of this section, a linearization of the nonlinear systemequations (isotherm and chromatographic model equations), the pa-rameter covariance matrix Vp can be estimated as in equation A.3from V and the output sensitivity Wj = ∇pyj, with yj being thevector of variable values at observation j.

Vp ≈(WTV−1W

)−1. (A.3)

The matrices W and V are defined as:

W =

W1

W2...

WNt

, V =

V 0 · · · 0

0 V · · · 0...

. . .

0 0 · · · V

(A.4)

The standard deviation σpi equals V12p,ii.

It should be pointed out that, due to the various assumptions madeat the beginning of the section, the linearization applied for the esti-mation of Vp, and the fact that the MLE was not used for the para-meter estimation in the first place, the standard deviations given intable 4.2 are rough estimates. These values are meant to provide anidea about the quality of the parameter fit, rather than presenting aquantitatively reliable uncertainty estimation.

a.2 solution of the buckley-leverett equation

In the following, the solution of the Buckley-Leverett equation (equa-tion 4.24) by the method of characteristics shall be explained. This

A.2 solution of the buckley-leverett equation 225

derivation has been multiply presented and discussed in the con-text of two-phase flow,4,15,68,69 however, with this work being ratherfocused on chromatography, we would like to summarize the mainaspects of the solution. For the sake of convenience, we use the di-mensionless variables defined in equation 3.7, and transform theBuckley-Leverett equation accordingly:

ε∂S1∂τ

+∂f1∂ξ

= 0. (A.5)

We assume an S-shaped fractional flow function as illustrated inFigure A.1, which is commonly used for the description of two-phase flow behavior in a porous medium.1 At the limiting saturati-ons Slim

i indicated in Figure A.1, phase i becomes hydraulically dis-connected, and thus the corresponding flux fi approaches 0. Furt-hermore, with the implied S-shaped fractional flow behavior, thederivative f ′i(Si) approaches 0 when approaching the limiting satu-ration.Applying the method of characteristics23,24 to equation A.5, one canderive the characteristic propagation velocity λ of a specific satura-tion S1:

λ =dξdτ

=1

ε

df1dS1

=1

εf ′1(S1) (A.6)

With ni =const. and thus ε =const. for a displacement of phases inthermodynamic equilibrium, and with f ′1(S1) being independent oftime and space, the propagation velocity of a specific saturation S1remains constant along the column. As a consequence, characteris-tics in the physical plane, illustrating the movement of a state alongtime and space, are straight lines, with a slope σ = 1/λ. Characteris-tics in the physical plane for the displacement of phase 2 by phase 1

(in the following called displacement a) and for the displacement ofphase 1 by phase 2 (in the following called displacement b) are illus-trated as blue lines on the left of Figures A.2 and A.3, respectively.In the case where the propagation velocity (and thus f ′1(S1)) decre-ases monotonically from a downstream state to an upstream state,the movement of each state enclosed between the two states is defi-ned by its characteristic velocity. The characteristics in the physicalplane diverge and thus correspond to a diverging wave connecting


the downstream to the upstream state.In contrast, if f ′1(S1) increases monotonically from a downstream toan upstream state, upstream states tend to move faster than down-stream states, which creates a multivalued, unphysical situation. Aweak solution is required, which directly connects the downstreamstate to the upstream state through a shock.23 Through the integralform of the Buckley-Leverett equation, one can determine the pro-pagation velocity λ of this shock transition:

λ =[f1]

ε[S1](A.7)

where [·] denotes the jump in the quantity enclosed across the dis-continuity. Again, the propagation velocity is independent of timeand space, resulting in a linear propagation path of the shock in thephysical plane with a slope σ = 1/λ, which in Figures A.2 and A.3is illustrated as red dashed lines.In the following, we will explain the derivation of a solution for

the Buckley-Leverett equation for the exemplary case of a primarydisplacement a, i. e. starting with a column completely equilibratedwith phase 2 (S01 = S1(ξ, 0) = 0, 0 6 ξ 6 1) and feeding pure phase1 (SF

1 = S1(0, τ) = 1, τ > 0). Starting from the initial state, f ′1(S1) = 0and thus λ = 0 until reaching Slim

1 . However, continuing further al-ong the fractional flow curve (S1 > Slim

1 ) an upwards curvature canbe observed, implying an increase in f ′1(S1) and thus in λ. Accor-dingly, upstream states tend to move faster than downstream states,which creates an unphysical situation and leads to the formation ofa shock transition. Moving further along the fractional flow curvetowards the feed state SF

1 = 1, an inflection point, where the curva-ture switches from upward to downward, is reached. Beyond thisinflection point, the f ′1(S1) and λ decreases, and indicates the for-mation of a diverging wave. The change of transition from a shockto a wave occurs at the saturation Sa,p

1 , which propagates with thesame characteristic velocity as the preceding shock, connecting S01 toSa,p1 . Mathematically, this means that σ(Sa,p

1 ) = σ(S01,Sa,p1 ), which

graphically corresponds to the grey line in Figure A.1 connecting S01to Sa,p

1 being tangent to the fractional flow curve in Sa,p1 . The shock

path and wave characteristics in the physical plane correspondingto the primary displacement a are illustrated on the left of Figure

A.2 solution of the buckley-leverett equation 227

0 0.2 0.4 0.6 0.8 1

S1 [-]

0

0.2

0.4

0.6

0.8

1

f 1 [

-]

S1

a,s

S1

a,p

S1

lim 1-S2

lim

S1

b,p

S1

b,s

Figure A.1: Exemplary fractional flow curve (blue line), being a function ofthe phase saturation and exhibiting a typical S-shape. Slim

i de-note the limiting (irreducible or residual) saturation of phase i,where the phase becomes hydraulically disconnected. Grey li-nes connect two states which in the concentration profile areconnected through a shock, they span from an initial state(S1(ξ, 0)=0 or 1 for primary displacement and Si(ξ, 0) = Slim

i

for secondary displacement) to a state Sx,y1 , where x = a,b

denotes the displacement a from phase 2 to phase 1 or displa-cement b from phase 1 to phase 2; and where y = p, s denoteswhether the displacement is primary or secondary.

A.2. The resulting elution profile, exhibiting a semi-shock (transi-tion which starts as a shock and changes into a wave), is providedon the right of Figure A.2. Approaching a saturation of S1 = 1−Slim

2 ,f ′1(S1) and thus the propagation velocity approaches zero. As a con-sequence, this saturation is approached in the elution profile, but isnever reached.An equivalent reasoning can be applied to derive elution profiles

for the secondary displacement a, where the initial state is charac-terized by a certain amount of phase 1 being trapped (hydraulicallydisconnected), such that the initial saturation S01 = Slim

1 . Also theelution profiles for primary and secondary displacement b, withinitial states S01 = 1 and S01 = 1− Slim

2 , respectively, and a feed stateSF1 = 0, can be derived in an analogue way. Implying an S-shaped


0 0.5 1

ξ [-]

0

0.5

1

1.5

2

τ [-

]

0 0.5 1

S1 [-]

0

0.5

1

1.5

2

τ [-

]

S1

lim1-S

2

lim

secondary

primary

S1

b,s

S1

b,p

Figure A.2: Solution of the Buckley-Leverett equation for displacement a(displacement of phase 2 by phase 1). Left: Physical plane withcharacteristics for the primary displacement a. Continuous bluelines represent simple wave characteristics, red dashed lines il-lustrate shock paths. Right: Elution profile. Purple profile corre-sponds to the primary displacement, grey profile to the secon-dary displacement.

fractional flow function, all four cases (primary and secondary dis-placements a and b) result in a semi-shock (shock-wave) transitionin the elution profile. States which are connected by the shock transi-tion are connected by a gray line for each of the four cases in FigureA.1. Superscripts a,b indicate displacement case a or b, respectively,and superscripts p, s denote a primary or secondary displacement.Solutions for the primary displacements a and b are illustrated inFigures A.2 and A.3. Furthermore, the graphs on the right of FiguresA.2 and A.3 include the elution profiles of the respective secondarydisplacement cases, illustrated as grey lines. In all four cases, the

A.3 phase equilibria 229

0 0.5 1

ξ [-]

0

0.5

1

1.5

2τ

[-]

0 0.5 1

S1 [-]

0

0.5

1

1.5

2

τ [-

]

S1

lim1-S

2

lim

secondary

primaryS1

b,p

S1

b,s

Figure A.3: Solution of the Buckley-Leverett equation for displacement b(displacement of phase 1 by phase 2). Left: Physical plane withcharacteristics for the primary displacement b. Continuous bluelines represent simple wave characteristics, red dashed lines il-lustrate shock paths. Right: Elution profile. Purple profile corre-sponds to the primary displacement, grey profile to the secon-dary displacement.

limiting saturation Slimi of the displaced phase i is approached as

τ→∞, but it is not reached since f ′1 → 0 as Si → Slimi .

a.3 phase equilibria

a.3.1 Calculation of phase equilibria

The structure of the thermodynamic code to calculate phase equi-libria is outlined in Figure A.4. In a first step, the stability of theinvestigated mixture, which initially was assumed to be a liquid so-lution, was assessed. The solution is stable if its Gibbs energy is at


the global minimum. This is the case if the tangent plane criterion49

(abbreviated as TPC in Figure A.4) is fulfilled, and if the fugacityof the pure solid (TBP) is greater than the fugacity of the solute insolution, i.e.41,133

zTγT 6 exp[∆HmT (TmT )

RT

(T

TmT− 1

)]. (A.8)

Note that subscript T indicates a property of TBP. The melting ent-halpy ∆HmT and melting temperature TmT of TBP were determinedby differential scanning calorimetry as ∆HmT = 17032.95 ± 575.83Jmol−1 and TmT = 372.56± 0.13 K. Violation of the tangent planecriterion indicates the coexistence of two or multiple stable liquidphases, whereas violation of equation A.8 implies the formationof a solid phase (in equilibrium with one or multiple liquid pha-ses). According to the outcome of the stability checks, phase equi-libria were calculated based on isofugacity and material balanceconditions,41,133 which are specified in table A.1. An iterative solu-tion procedure for the calculation of liquid-liquid equilibria50 wasapplied and extended to the calculation of solid-liquid-liquid equi-libria, as outlined in table A.1. Stationary points resulting from theLL stability check (tangent plane criterion) were used as initial gues-ses for the LL and SLL phase split calculations. The resulting phasesin equilibrium were each checked for thermodynamic stability, andonly solutions with stable phases were accepted. Since only LL, SL,and SLL phase equilibria were encountered experimentally, no furt-her phase equilibrium options were considered in the code.It should be pointed out that for the investigated quaternary system,only one solid phase, composed of pure TBP of one crystal struc-ture, was assumed. This assumption is supported by XRD spectraof the solid phase resulting from SL and SLL equilibrium experi-ments, which are in good agreement with the XRD spectrum deter-mined from pure TBP. A selection of XRD spectra, overlaid with theoriginal spectrum of TBP, is provided in Figure A.5.

a.3.2 Fitting of UNIQUAC parameters

The 12 interaction parameters (2 for each pair of components in thequaternary system) of the UNIQUAC model were fitted to the ex-


Stability check:

TPC

Stability check:

TPC

Stability check:

TPC, sol. fug.

Calculate LL

phase split

Calculate SL

phase split

Calculate SLL

phase split

Result: LL

equilibrium

Result: SL

equilibrium

Result: stable

liquid phase

Result: SLL

equilibrium

stable unstable

LL unstable stable SL unstable

stable unstable

stable

Output:

Warning

unstable

stable unstable

Stability check:

solid fugacity

Stability check:

TPC, sol. fug.

Output:

Warning

Figure A.4: Structure of the thermodynamic code


TableA

.1:Phase

equilibriumconditions

a

Liquid-

LiquidSolid

-Liquid

Solid-

Liquid-

Liquid

Conditions

Isofugacityz

Iiγ

Ii=z

IIi γIIi ;∀i

zIiγ

Ii=z

IIi γIIi ;∀i

zTγ

T=

exp [∆HmT

(TmT

)

RT

(TTmT

−1 )]

zITγ

IT=z

IITγ

IIT=

exp [∆HmT

(TmT

)

RT

(TTmT

−1 )]

MaterialBalances

ztoti

=Sz

Ii+

(1−S)z

IIi ;∀i

zLi=Sz

Ii+

(1−S)z

IIi ;∀i

ztoti

=(1

−s)zi ;∀i\

Tz

toti

=(1

−s)z

Li ;∀i\

T

ztotT

=(1

−s)z

T+s

ztotT

=(1

−s)z

Li+s

Stoichiometry

N∑i=1

zXi

=1

N∑i=1

zXi

=1

N∑i=1

zXi

=1

aM

olarfractions

ofcom

ponentsi

areexpressed

aszXi

,w

herethe

superscriptsX

=tot;I;I

I;L,

denotethe

overallm

olefraction

overall

phasesin

equilibrium,the

mole

fractionin

thefirst

andthe

secondliquid

phasein

thermodynam

icequilibrium

,andthe

overallmole

fractionover

allliquidphases

inequilibrium

,respectively.

Sdenotes

them

olefraction

ofthe

firstliquid

phasew

ithrespect

tothe

overallm

olenum

berof

allliquid

phases,w

hiles

denotesthe

mole

fractionof

thesolid

phasew

ithrespect

tothe

overallmole

number

ofallphases

inequilibrium

.


Tabl

eA

.2:I

mpl

emen

tati

onof

phas

eeq

uilib

rium

cond

itio

ns

Liqu

id-

Liqu

idSo

lid-

Liqu

idSo

lid-

Liqu

id-

Liqu

id

Impl

emen

tati

on

Inpu

tz

tot

zto

tz

tot

Unk

now

nva

riab

les

zI ,S

sz

I ,S,s

Solu

tion

Iter

atio

n[5

0]

Num

eric

alro

ot-fi

ndin

g:It

erat

ion

(ext

ensi

onof

[50

])

Init

ialg

uess

zI ,z

IIIs

ofug

acit

yco

ndit

ion→z

TIn

itia

lgue

ssz

I ,zII

γI i=

f(z

I ,T),γ

II i=

f(z

II,T

)s=z

tot

T−z

T1−z

Tγ

I i=

f(z

I ,T),γ

II i=

f(z

II,T

)

Ki=γ

II iγ

I i=z

I iz

II iKi=γ

II iγ

I i=z

I iz

II i∑ N i=

1

(Ki−1)z

toti

S(Ki−1)+1

! =0→S(K

)∑ N i=

1

(Ki−1)z

L iS(Ki−1)+1

! =0→S(K

)

Mi=S(Ki−1)+1

Mi=S(Ki−1)+1=z

L iz

II i

zI i=z

totiKi

Mi

,zII i=z

totiMi

zII T=

exp[ ∆

Hm T

(Tm T

)

RT

( T Tm T

−1

)]γ

II T

Iter

ate

...s=z

tot

T−z

II TM

T1−z

II TM

T

zL i=z

toti1−s

;∀i\

T

zL T=z

tot

T−s

1−s

zI i=z

L iKi

Mi

,zII i=

zL i

Mi

Iter

ate

...


perimental data in the ternary systems, which are obtained in theabsence of one of the four components. Objective functions of thefitting are based on relative errors of the simulated weight fractionswith respect to the measured weight fractions, weighted by the num-ber of repetitions Nrep,j performed of the experimental condition j.For LL and SLL equilibria, these weighted relative errors were squa-red and summed up over all components Nc, both liquid phases Iand II, all repetitions Nrep,j, and finally all considered experimentalconditions Nexp to obtain a sum of squared errors φLL

φLL =

Nexp∑j=1

Nrep,j∑k=1

II∑p=I

Nc∑i=1

(1

Nrep,j

(wp,modi,k,j −w

p,expi,k,j

wp,expi,k,j

))2. (A.9)

For SL equilibria, weighted errors were only determined for TBP,being the component to form the solid phase once the fugacity ofthe initial mixture exceeds the fugacity of the solid (for details seeA.3.1). The errors were squared and summed up over the repetitionsand the considered experimental conditions to obtain φSL

φSL =

Nexp∑j=1

Nrep,j∑k=1

(1

Nrep,j

(wmod

T,k,j −wexpT,k,j

wexpT,k,j

))2(A.10)

Phase equilibria were calculated based on isofugacity and materialbalance conditions as outlined in A.3.1, using the UNIQUAC mo-del with an assumed set of interaction parameters. The UNIQUACinteraction parameters were then fitted in two steps: In a first step,10 of the 12 parameters (all except the 2 parameters describing theinteraction of PNT and TBP) were fitted to experimental LL and SLLconditions of the ternary systems PNT-methanol-water (PMW) andTBP-methanol-water (TMW), which corresponds to conditions 1 to14 in tables A.5 and A.6. The following objective function φ1 wasminimized:

φ1 =

(φLL

Nexp

)PMW

+

(φLL

Nexp

)TMW

. (A.11)

with the number of (S)LL conditions Nexp equaling 9 in the PMWsystem and 5 in the TMW system. With this formulation of the


objective function, the binary interaction parameters of PNT (TBP)with each water and methanol were determined by the data in thePMW (TMW) system, whereas for the determination of the binaryinteraction parameters of methanol and water the contributions ofthe two ternary systems PMW and TMW were weighted equally. Inthe TMW system, only LL and SLL data was selected for the fit-ting, since this data is located at the ratios of water and methanolwhich are of most relevance for the remaining work. For the 6 inte-raction parameters of the couples PNT - methanol, PNT - water andmethanol - water, the UNIQUAC parameters of the ternary system(compare Table 4.2) were used as initial guesses, while for the other4 interaction parameters, initial guesses were estimated from a pre-liminary assessment of suitable parameter values.The remaining 2 interaction parameters for the couple PNT - TBPwere fitted to the SL conditions in the system PNT - TBP - metha-nol (PTM), and to the water-lean phase of the SLL condition in thesystem PNT - TBP - water (PTW), corresponding to conditions 19 to22 in tables A.7 and A.8. Data of the water-rich phase of condition22 was not considered in the fit, in order to obtain a more accuratedescription in the regions relevant for subsequent applications (adetailed discussion is given below). The objective function φ2 of thesecond fit can be written as

φ2 = (φSL)PTM + (φLL)PTW . (A.12)

Again, initial guesses were obtained from a preliminary assessment.Both objective functions φ1 and φ2 were minimized using a built-inMatlab routine (fmincon). Note that the 6 UNIQUAC parameters ofthe ternary system PNT - methanol - water, determined in absenceof TBP (see Table 4.2) were fitted with a similar procedure, howeverin a single fitting step, using an objective function analogue to thatof equation A.9.It is to be said that the described two-step procedure for the fittingof the 12 parameter UNIQUAC model was not the first-choice fit-ting approach, but that many approaches were tested to achieve asatisfactory description of the phase equilibrium behavior. The cho-sen procedure features several unconventional aspects, which shallbe discussed in the following.The objective function was chosen to be based on weighted relative


errors in the weight fractions of phases in equilibrium. An alterna-tive approach would be to minimize differences in activities of pha-ses in equilibrium, which is computationally less expensive (since itdoes not require phase equilibrium calculations) and was indeed ap-plied in the preliminary assessments to find suitable initial guesses.However, equal activities of phases do not necessarily guarantee aglobal minimum of the Gibbs free energy, and thus phase stability.This shortcoming was noticeable in the fitting result, such that thecomputationally more intense fit based on weight fractions was cho-sen. Relative errors were considered to ensure that small and largeweight fractions were weighted equally. A good evaluation of fittingprocedures based on activities or phase compositions, consideringrelative or absolute errors, is provided in Ref. [134].Secondly, it should be noted that the degrees of freedom to be elimi-nated by the isofugacity conditions is 1, N+ 1 and N+ 2 for SL, LLand SLL equilibria, respectively (compare the number of unknownvariables in table A.1), while all other variables can be determinedthrough material balance conditions. Nevertheless, it was chosen tofit parameters to 2N data points (2Nweight fractions) per LL or SLLexperiment in order to weight all components and phases equally,instead of adjusting the number of data points per experiment tothe degrees of freedom.The two-step fitting procedure and its objective functions, appliedas described above, turned out to provide the most accurate des-cription of the relevant regions, i.e. of the system PNT - methanol -water, of the system TBP - methanol - water, particularly in the re-gion where the ratio methanol:water ranges between 40:60 and 60:40

(w:w), and of the interaction between TBP and PNT. Data close tothe water-rich corner was excluded from the fit, since this regionis not relevant for the further work, and inclusion of this data ledto a considerable deterioration of the description of relevant regi-ons. Accordingly, considerable differences between calculated andmeasured phase compositions are observed in the water-rich cor-ner, particularly concerning the solubility of TBP in water (compareexperimental data and model calculations of conditions 1, 15 and 22

in tables A.5 to A.8).We refrained from estimating standard deviations or uncertainty in-tervals of the fitted UNIQUAC parameters based on the MLE theory


(see A.1), which is very far from the applied, unconventional two-step fitting procedure and hence is not expected to provide realisticestimates.

a.3.3 Additional experimental data

10 15 20 25 30 35

2θ

0

0.2

0.4

0.6

0.8

1

inte

nsity [

a.

u.]

original

TW

(a)

10 15 20 25 30 35

2θ

0

0.5

1

1.5

2

inte

nsity [

a.

u.]

original

TM

(b)

10 15 20 25 30 35

2θ

0

1

2

3

4

inte

nsity [

a.

u.]

original

TP

(c)

10 15 20 25 30 35

2θ

0

0.5

1

1.5

inte

nsity [

a.

u.]

original

TMP30

(d)

Figure A.5: XRD spectra of the solid phase resulting from selected phaseequilibrium experiments, compared to the XRD spectrum me-asured for pure TBP. Selected SL equilibrium experiments: (a)TBP - water, (b) TBP - methanol, (c) TBP - PNT, (d) TBP witha mixture methanol:PNT 30:70 (w:w). All spectra were normali-zed with respect to the peak height at 2θ = 16.53.


Table A.3: Original quaternary compositions of phase equilibrium experi-ments. Replicates indicated by “r”.

Exp P T M W

1 50.03 0.00 0.00 49.97

1r 49.93 0.00 0.00 50.07

2 48.98 0.00 7.99 43.03

2r 48.97 0.00 8.00 43.03

3 47.94 0.00 15.99 36.07

3r 47.94 0.00 15.99 36.07

3rr 47.90 0.00 15.97 36.13

4 46.95 0.00 23.98 29.07

4r 46.91 0.00 23.96 29.13

5 47.07 0.00 27.95 24.98

6 47.51 0.00 31.02 21.47

6r 47.54 0.00 31.00 21.46

7 46.02 0.00 37.97 16.01

7r 46.00 0.00 38.00 16.00

8 46.04 0.00 43.95 10.01

8r 46.09 0.00 43.90 10.01

9 61.71 0.00 33.83 4.46

10 0.00 60.06 19.56 20.38

10r 0.00 42.16 21.92 35.92

10rr 0.00 47.13 21.21 31.66

10rrr 0.00 45.27 22.06 32.67

11 0.00 30.99 32.90 36.11

12 0.00 30.04 33.93 36.03

12r 0.00 30.00 33.93 36.08

13 0.00 28.96 35.01 36.02

13r 0.00 29.04 35.03 35.93

14 0.00 28.89 35.91 35.20

14r 0.00 29.01 35.97 35.02

15 0.00 23.46 0.00 76.54

15r 0.00 23.17 0.00 76.83

16 0.00 81.77 18.23 0.00

16r 0.00 83.07 16.93 0.00

17 0.00 40.59 17.77 41.64

17r 0.00 40.44 17.80 41.76


Table A.4: Original quaternary compositions of phase equilibrium experi-ments - continuation. Replicates indicated by “r”.

Exp P T M W

18 0.00 79.98 13.96 6.06

18r 0.00 80.10 13.87 6.03

19 19.66 71.49 8.86 0.00

19r 20.88 70.03 9.09 0.00

20 5.41 81.97 12.62 0.00

20r 5.51 81.66 12.83 0.00

21 56.84 43.16 0.00 0.00

21r 56.82 43.18 0.00 0.00

22 33.66 41.98 0.00 24.36

22r 34.06 41.23 0.00 24.71

24 26.59 15.34 30.04 28.02

25 10.42 22.25 36.50 30.83

26 42.32 7.45 30.66 19.57

27 8.82 50.93 10.01 30.25


Table A.5: Phase equilibria in the system PNT - methanol - water: Experi-mental and Calculated

Expphase I phase II

P M W P M W

LL 1 0.08 0.00 99.92 99.87 0.00 0.13

1r 0.12 0.00 99.88 99.86 0.00 0.14

1 model 0.03 0.00 99.97 99.87 0.00 0.13

LL 2 0.18 15.05 84.77 99.43 0.42 0.16

2r 0.16 14.83 85.02 99.49 0.36 0.15

2 model 0.11 15.37 84.52 99.47 0.37 0.16

LL 3 0.25 29.73 70.01 98.35 1.45 0.21

3r 0.25 29.96 69.79 99.01 0.80 0.20

3rr 0.26 29.00 70.74 98.79 1.01 0.20

3 model 0.35 30.11 69.55 98.96 0.86 0.19

LL 4 0.90 44.07 55.03 98.20 1.57 0.23

4r 0.88 44.14 54.98 98.19 1.58 0.23

4 model 1.07 44.05 54.88 98.19 1.57 0.23

LL 5 1.69 50.73 47.58 97.47 2.26 0.26

5 model 1.93 51.01 47.06 97.59 2.14 0.27

LL 6 3.03 56.10 40.86 97.08 2.62 0.30

6r 2.92 56.63 40.45 96.93 2.77 0.30

6 model 3.15 56.38 40.47 96.89 2.80 0.31

LL 7 8.02 63.78 28.20 95.03 4.53 0.44

7r 7.83 64.07 28.10 95.24 4.33 0.43

7 model 7.86 63.94 28.21 94.69 4.86 0.45

LL 8 22.12 63.03 14.85 89.48 9.65 0.87

8r 21.68 63.28 15.04 89.82 9.33 0.85

8 model 21.41 63.31 15.29 88.75 10.39 0.86

LL 9 41.09 50.91 7.99 78.04 19.99 1.97

9 model 40.42 51.66 7.93 78.25 19.98 1.76


Table A.6: Phase equilibria in the system TBP - methanol - water: Experi-mental and Calculated

Expphase I phase II

T M W T M W

SLL 10 63.29 21.82 14.89 1.33 32.49 66.18

10r 64.72 20.84 14.44 1.44 32.53 66.03

10rr 63.85 21.48 14.68 1.50 33.30 65.20

10rrr 62.79 22.15 15.07 1.83 33.52 64.65

10 model 59.06 24.08 16.85 1.42 32.40 66.18

LL 11 52.03 27.92 20.05 2.91 38.44 58.64

11 model 49.05 29.90 21.05 3.46 37.47 59.07

LL 12 48.67 29.55 21.78 3.71 39.21 57.07

12r 49.49 28.98 21.53 3.64 39.11 57.24

12 model 46.93 31.06 22.01 4.06 38.34 57.61

LL 13 45.52 30.72 23.76 5.18 39.93 54.88

13r 45.33 30.69 23.98 5.10 39.79 55.11

13 model 44.62 32.29 23.09 4.79 55.99 39.22

LL 14 39.06 33.08 27.86 8.35 40.04 51.62

14r 38.38 33.25 28.37 8.41 40.18 51.42

14 model 42.18 33.55 24.27 5.66 40.05 54.30

SL 15 6.14e-02 0.00 99.94

15r 6.25e-02 0.00 99.94

15 model 5.88e-18 0.00 100.00

SL 16 75.69 24.31 0.00

16r 75.47 24.53 0.00

16 model 74.07 25.93 0.00

SL 17 0.82 29.66 69.52

17r 0.73 29.66 69.61

17 model 0.76 29.68 69.56

SL 18 69.26 21.43 9.31

18r 68.80 21.74 9.46

18 model 65.21 24.26 10.54


Table A.7: Phase equilibria in the system PNT - TBP - methanol: Experimen-tal and Calculated

Expphase I

P T M

SL 19 26.55 61.49 11.96

19r 27.03 61.21 11.77

19 model 29.21 57.63 13.16

SL 20 8.27 72.45 19.28

20r 8.24 72.58 19.18

20 model 9.28 69.09 21.63

SL 21 74.16 25.84 0.00

21r 74.15 25.85 0.00

21 model 75.77 24.23 0.00

Table A.8: Phase equilibria in the system PNT - TBP - water: Experimentaland Calculated

Expphase I phase II

P T W P T W

SLL 22 59.19 39.00 1.81 0.04 6.38e-02 99.90

22r 59.15 39.04 1.81 0.04 6.53e-02 99.90

22 model 59.78 37.88 2.34 0.02 9.12e-18 99.98

Table A.9: Phase equilibria in the quaternary system PNT - TBP - methanol- water: Experimental and Calculated

Expphase I phase II

P T M W P T M W

LL 24 1.10 1.62 44.99 52.29 53.64 29.90 12.73 3.73

24 mod. 1.39 3.17 45.22 50.22 1.82 27.52 14.85 5.80

LL 25 1.17 3.97 45.14 49.71 20.51 42.01 25.35 12.13

25 mod. 1.29 6.66 44.92 47.14 19.56 37.85 28.08 14.51

LL 26 3.25 1.99 54.44 40.33 78.83 12.61 7.03 1.54

26 mod. 4.27 5.15 53.31 37.27 80.34 9.74 8.03 1.88

SLL 27 28.90 60.35 6.01 4.74 0.04 0.24 19.10 80.62

27 mod. 27.54 56.97 8.82 6.68 0.07 0.07 20.34 79.51

BS U P P L E M E N TA RY M AT E R I A L F O R C H A P T E R 5

b.1 thermodynamic properties in chromatographic co-des

An important problem in the solution of the equilibrium theory andthe numerical code is the determination of thermodynamic proper-ties such as activities, phase compositions, but also adsorbed phaseconcentrations (being a function of the liquid phase activities), aswell as their derivatives with respect to the unknown variables. Thisissue is solved individually for the soluble and the two-phase region,as well as for the equilibrium theory and the numerical code.

b.1.1 Soluble region

In the soluble region, no equilibrium calculations have to be perfor-med, but activities and their derivatives with respect to mole fracti-ons zi can be calculated analytically from the UNIQUAC equations.In turn, the mole fractions zi can be expressed as a function of theliquid phase concentrations:

zi =

ciMi

Nc∑k=1

ckMk

; i = 1...Nc (B.1)

As a consequence, also the analytical calculation of adsorbed phaseconcentrations through the isotherm equation is trivial, while wewould quickly like to outline the analytical determination of theirderivatives nij with respect to the concentration cj (only requiredfor the equilibrium theory code). For this, we define the matrices N ,N , A, Z and C:

Nij =∂ni∂Cj

, Nij =∂ni∂aj

, Aij =∂ai∂zj

, Zij =∂zi∂cj

, Cij =∂ci∂Cj


(B.2)

Note that, while ci corresponds to the liquid phase concentrationof any of the Nc components (i = 1...Nc), Cj with j = 1...Nc − 1denote the unknown variables, since component Nc is a functionof the other components, given through equation 3.9 (note furtherthat Cj = cj in the soluble region). Accordingly, the dimensions ofmatrices N , A, Z are Nc×Nc; those of N and C are Nc× (Nc− 1).It is a straightforward consequence that:

N = NAZC. (B.3)

The matrices N , A, Z and C can be determined analytically by dif-ferentiating the isotherm equation, the UNIQUAC equation, equa-tion B.1 relating zi to cj and the expression relating ci to Cj (assu-ming additivity of volumes), respectively. Specifically for the inves-tigated ternary system with one adsorbing component, matrix N is:

N =dn1da1

=H1

(1−K1a1)2

(B.4)

Differentiating equation B.1 yields Z:

Zii =

cmMiMm

+ cnMiMn(

3∑k=1

ckMk

)2 ; n 6= m 6= i; n,m, i = 1...3 (B.5a)

Zij = −

ciMiMj(3∑k=1

ckMk

)2 ; i 6= j; i, j = 1...3 (B.5b)

Finally, matrix C is defined as:

C =

(1 0 −ρ3/ρ1

0 1 −ρ3/ρ2

)(B.6)

The matrix A, involving the partial derivatives of the UNIQUACequation with respect to zi (i = 1...3) is not provided for the sake ofbrevity.

B.1 thermodynamic properties in chromatographic codes 245

b.1.2 Two-phase region in the numerical code

As described in section 5.2, a look-up table was implemented fromthe 5000 tie-lines determined through equilibrium calculations andspanning the entire two-phase region. The purpose of this look-uptable is the reduction of computational time by avoiding computati-onally expensive equilibrium calculations during the solution of themodel equations (as is also suggested in the literature135–137). Thefirst step is therefore to determine the tie-line which the state C1,C2 is located on. For this purpose, the normal vector from the stateto each tie-line in the look-up table, as well as its length (distanceof the state to the tie-lines) is determined. The closest (lowest dis-tance) of the 5000 tielines above and below the state are selected,and phase compositions, activities, and adsorbed phase concentra-tions are interpolated linearly (relating one distance to the sum ofthe two distances) from the properties of each tieline. Finally, thesaturation SR is calculated using:

SR =C1 − c

L1

cR1 − c

L1

, (B.7)

from which also the fractional flows fj and the overall fractionalflows Fi can be determined, using equations 3.17, 3.18 and 3.2. Notethat, due to the implementation of the formal mass transfer term inthe lumped kinetic model, no derivatives of adsorbed phase concen-trations and activities are required.

b.1.3 Two-phase region in the equilibrium theory code

For the solution of the equilibrium theory model in the two-phaseregion, a variable transformation to a tie-line indicator (here ln(xL

1))and the phase saturation SR is suggested (see section 5.4). With thetie-line indicator, the tie-line, which the state is located on, is alreadyknown. For the 5000 established tie-lines, the properties phase com-position, activities and adsorbed phase concentrations were calcula-ted. Subsequently, spline interpolations were performed to connectthe values of these properties as a function of the tie-line indicator.Differentiating these splines also provide the derivatives of the men-tioned properties with respect to the tie-line indicator. Thus, in the


equilibrium theory code, properties of the states are easily determi-ned through evaluating the established spline interpolations at thetie-line indicator value of the state.In the case of the simplified equilibrium theory model, assumingequal interstitial velocities and hence fR = SR, Γ1 and Γ2 characte-ristics map on tie-lines and on lines of constant solvent ratio, re-spectively, and thus do not have to be calculated by solving an ODE.With λ1 = ε−1, the propagation velocities along tie-lines are con-stant and equal to the velocity of the corresponding states on thebinodal curve (calculated as explained in Appendix B.1.1). Since Γ2characteristics in the case of the simplified equilibrium theory mo-del correspond to non-tieline characteristics Γnt, the expression ofλnt (equation 5.15), simplified for the case fR = SR, is used to calcu-late the propagation velocities along Γ2 characteristics.

b.2 analysis of the binodal and equivelocity curve

In this section, we will provide proof that the binodal and the equi-velocity curve only constitute non-tieline characteristics in the casethat n1 = const. or n1 = 0, but not in the presence of ad- anddesorption. For this purpose, we transform Equations 5.9, conside-ring ψ = dη/dSR and λ = dξ/dτ:[(

fRdcR1

dη+(1− fR

) dcL1

dη+(cR1 − c

L1

) ∂fR∂η

)

− λ

(ε

(SR dcR

1

dη+(1− SR

) dcL1

dη

)+ (1− εref)

(1−

C1ρ1

)dn1dη

)

+1

ψ

(cR1 − c

L1

)( ∂fR∂SR − λε

)]∂η

∂ξ= 0 (B.8a)[(

fRdcR2

dη+(1− fR

) dcL2

dη+(cR2 − c

L2

) ∂fR∂η

)

− λ

(ε

(SR dcR

2

dη+(1− SR

) dcL2

dη

)− (1− εref)

C2ρ1

dn1dη

)

+1

ψ

(cR2 − c

L2

)( ∂fR∂SR − λε

)]∂η

∂ξ= 0 (B.8b)

B.2 analysis of the binodal and equivelocity curve 247

b.2.1 Binodal curve

We consider the component 1 rich side of the binodal curve. Notethat an analogous proof is possible for the component 1 lean side.For the component 1 rich side, and keeping in mind the S-shapedpower law relationship between fR and SR, the following conditionshold:

SR = 1; fR = 1;∂fR

∂SR = 0;∂fR

∂η= 0 (B.9)

Considering these conditions in the expressions for λnt (see equa-tion 5.15) and for ψnt (see equation 5.13), the following simplifiedconditions can be obtained:

λnt =

dcR1

dη −cR1−c

L1

cR2−c

L2

dcR2

dη

εdcR1

dη − εcR1−c

L1

cR2−c

L2

dcR2

dη + (1− εref)

(1− C1

ρ1+cR1−c

L1

cR2−c

L2

C2ρ1

)dn1dη

(B.10)

ψnt =a

b(B.11)

with

a =ε(cR2 − c

L2

) dcR1

dη+ ε

(cR1 − c

L1

) dcR2

dη(B.12a)

b =

(ε

dcR2

dη− (1− εref)

C2ρ1

dn1dη

)dcR1

dη

−

(ε

dcR1

dη+ (1− εref)

(1−

C1ρ1

)dn1dη

)dcR2

dη(B.12b)

For n1 = const., λnt can be further simplified to λnt = 1/ε, and thedenominator of equation B.11 equals 0, such that 1/ψnt = 0.Considering the simplified expressions for λnt and ψnt, as well as


the expressions in equation B.9, in equations B.8, the componentmaterial balances are fulfilled in the case that n1 = const.. In turn,no further simplification of equations B.10, B.11 and B.8 is possiblein the presence of ad- or desorption. Therefore, the binodal curveonly constitutes a non-tieline characteristic Γnt in the absence of ad-sorption.

b.2.2 Equivelocity curve

The equivelocity curve connects those points within the immiscibleregion (and not located on the binodal curve), for which fR = SR.In the case that the viscosities µR and µL are constant and thus in-dependent of η, and that relative permeabilities kX

r are independentof η, fR is a function of SR only and both fR and SR are constantalong the equivelocity curve. This is not the case in our assumpti-ons, where µL changes with the phase composition and thus fR is afunction of SR and η. In that case, one can determine the change ofSR along the equivelocity curve, applying equation 3.17:

fR = SR =kRr (S

R)

kRr (S

R) + kLr(S

R)µR(η)µL(η)

(B.13)

In the following, we start with the assumption that the equivelocitycurve constitutes a non-tieline characteristic, and evaluate the condi-tions under which this assumption holds true and no conflicts arise.Resolving equation B.13 with respect to SR and differentiating theresulting expression g(η) with respect to η, yields an expression forψnt (assuming the equivelocity curve constitutes a non-tieline cha-racteristic).

SR = fR = g(η);dfR

dη=

dSR

dη=

1

ψnt= g ′(η) (B.14)

In general, the change in fR along a non-tieline characteristic can bedescribed as:

dfR

ds=∂fR

∂SRdSR

ds+∂fR

∂η

dηds

(B.15)

where s is the characteristic parameter describing the directionalmovement along the Γnt characteristic. With g(η) being a function of

B.3 et solution assuming different velocities (continued) 249

η only, this characteristic parameter s can also be chosen to equal η.Combining equations B.14 and B.15 yields:

g ′(η) =1

ψnt=

∂fR

∂η |SR

1− ∂fR

∂SR |η(B.16)

Analogue to the considerations with respect to the binodal curve(section B.2.1), we will now consider the condition fR = SR in equa-tions 5.15, and 5.13, to obtain simplified expressions for λnt and ψnt,respectively. These simplified expressions should, in the case thatthe equivelocity curve is a non-tieline curve, fulfill equations B.8.Replacing fR by SR in equations 5.15 only yields a significant simpli-fication in the case that n1 = const., for which λnt = 1/ε. In a similarmanner, the expression for ψnt (equations 5.13) only simplifies forn1 = const., yielding equation B.16. This provides a first indicationthat for n1 = const., the equivelocity curve indeed constitutes a non-tieline characteristic.Considering the resulting expressions for λnt and ψnt in equationsB.8, one can find that the component material balances are only ful-filled if n1 = const., i.e. the equivelocity curve only constitutes a Γntcharacteristic in the absence of adsorption.

b.3 equilibrium theory solution assuming different velo-cities (continued)

In section 5.4, we have outlined the solution of the equilibrium the-ory model accounting for different velocities, introducing Γnt andΓt characteristics in the hodograph plane, as well as Γ1 and Γ2 cha-racteristics. The latter allow a simpler derivation of solution paths,following a fixed sequence along a Γ1 characteristic (connecting do-wnstream state A to the intermediate state I) and a Γ2 characteristic(connecting intermediate state I to upstream state B). Here, we tho-roughly discuss some aspects of the derivation, which were omittedin section 5.4 for the sake of brevity. In particular, we show how sta-tes in the immiscible region are connected to states in the miscibleregion. Subsequently, we derive in detail the solutions to the threeexamples with miscible feed states B1 to B3, discussed in section5.4.3.


b.3.1 Connection of states in the miscible region to states in the immisci-ble region

Connecting the miscible region with the immiscible region, we haveto take into account the characteristics and their behavior derivedfor both regions, see section 5.3 for the miscible region and section5.4 for the immiscible region. Furthermore, one should recall thatfor a physically meaningful solution, upstream states always travelat a smaller propagation velocity than downstream states.

b.3.1.1 Crossing the binodal curve

From Figure 5.7 in section 5.4.2, it can be concluded that all Γ2 cha-racteristics (mapping on tieline characteristics close to the binodal),but only a few Γ1 characteristics (mapping on non-tieline charac-teristics close to the binodal) intersect with the the binodal curve.Hence, a major part of the Γ1 characteristics can only be accessedfrom the miscible region through a shock transition fulfilling theRankine-Hugoniot condition (equation 5.16). Characteristics inter-secting with the binodal are continued in the miscible region by thecorresponding Γ1 and Γ2 characteristics, which are illustrated in Fi-gure 5.1. At the binodal, the propagation velocities λi along the Γicharacteristics exhibit a discontinuity. This leaves two different op-tions for crossing the binodal: If λU

i (i.e., the propagation velocityat the binodal, approached from the upstream side) is smaller thanλDi (the propagation velocity at the binodal, approached from the

downstream side), the path along characteristic i can be continuedbeyond the binodal, and in the corresponding concentration profilea plateau is formed at the composition of the intersection with thebinodal. This concentration plateau does not correspond to a clas-sical intermediate state, since it does not mark the intersection ofdifferent types of characteristics, but the solution path is continuedalong the same characteristic i. If λU

i > λDi , the binodal has to be

crossed via a shock transition.Let us first consider the behavior of the propagation velocities λ2.

For the further discussion, we refer to Figure B.1, which presentspropagation velocities λi (i = 1, 2) along two selected tie-lines. It isobvious that λ2 approaches 0 as the binodal curve is approached


0 0.2 0.4 0.6 0.8 1

SR

[-]

0

2

4

6

8

λ [

-]

λ1

λ2

Figure B.1: Propagation velocities λ1 (blue) and λ2 (red) along selected tie-lines characteristics. For line and marker styles see Figure 5.6.

from the immiscible region (i.e., as SR → 0 or SR → 1). In the mis-cible region, propagation velocities λ2 are always > 0. Thus, alonga Γ2 characteristic, if downstream states are located in the immisci-ble region and upstream states in the miscible region, the binodal iscrossed through a shock. In contrast, if upstream states are locatedin the immiscible region and downstream states in the miscible re-gion, in theory a concentration plateau, mapping on the intersectionof the solution path with the binodal, can be formed. However, refer-ring again to Figure B.1, it can be noticed that λ2 exhibits a strong in-crease from the binodal towards the center of the immiscible region.Thus, upstream states in the immiscible region can only be accessedfrom the binodal through a shock. If the propagation velocity of thisshock is faster than the propagation velocity of downstream stateslocated in the miscible region, the binodal would again be crossedby a shock.Secondly, we consider the propagation velocities λ1, and their be-havior along Γ1 characteristics. Most of the Γ1 characteristics in theimmiscible region do not intersect with the binodal curve (compareFigure B.1), hence conditions located on these characteristics canonly be accessed from the miscible region through shocks. Since inthe miscible region, λ1 = 1/ε corresponds to the interstitial velocity,


the shocks crossing the binodal have to propagate faster than the in-terstitial velocity, if immiscible states are located upstream, and theyhave to be slower than the interstitial velocity, if immiscible statesare located downstream.Concerning the few Γ1 characteristics intersecting with the binodalcurve, it was observed that the propagation velocity λ1 = 1/ε alongΓ1 characteristics approaching the binodal from the miscible regionis greater than the propagation velocity along Γ1 characteristics ap-proaching the binodal from the immiscible region. As in the case ofthe Γ2 characteristics, if downstream states are located in the immis-cible region and upstream states in the miscible region, the binodalis crossed through a shock. In the opposite case, the formation of aconcentration plateau is possible. A strong increase of λ1 (compareFigure B.1) from the binodal towards the center of the immiscibleregion can result in a shock transition, which crosses the binodal,in the case that the propagation velocity of the shock is higher thanλ1 = 1/ε in the miscible region.Since for most conditions (among them the conditions consideredin this contribution), the binodal is crossed by a shock transition, wewill in the following focus on these cases. The propagation velocityof such shocks λ, and the Rankine-Hugoniot condition which hasto be fulfilled by conditions connected through a shock, is given byequation 5.16 in section 5.4.1, where Fi = Ci = ci for the singlephase state. We also want to point out that in our case, such shockcrossing the binodal does not map on the extension of a tieline intothe miscible region, as it is the case in the absence of adsorption(n1 = 0), or for an isotherm depending linearly on liquid phase con-centrations [4, 10]. Proof that mappings of these shock transitionsdeviate from tie-line extensions can be derived by a straightforwardtransformation of equation 5.16, in an analog manner to the proofprovided in Ref. [10].

b.3.1.2 Continuation in the miscible region

In the following, we want to discuss how shocks crossing a binodalare continued in the miscible region. In the absence of adsorptioneffects, shock transitions crossing the binodal always map on tielineextensions in the miscible region.4,69 This is not the case for our sy-


stem, as already indicated by Γ1 and Γ2 characteristics not mappingon tieline extensions in the miscible region (see section 5.3), and isshown mathematically in Ref. [10]. Instead, in order to determineshock paths connecting the miscible to the immiscible region, onehas to consider the characteristics derived for the miscible region insection 5.3.In the following, we will consider a Riemann problem with mis-cible initial and feed states A and B, respectively, which are con-nected through a solution path mapping partly in the immiscibleregion, and with an intermediate state I located in the immiscibleregion. All the exemplary cases considered in the following sectioncorrespond to such a Riemann problem. As a consequence of theassumed conditions, a miscible downstream state A is connected toan immiscible upstream state I through a Γ1 characteristic, and animmiscible downstream state I is connected to a miscible upstreamstate B through a Γ2 characteristic.Let us start from the initial (downstream) state A located in themiscible region. The path connecting this initial state to any otherupstream state in principle emanates along the Γ1 characteristic pas-sing through the initial state, which always results in a transitionbeing a contact discontinuity with constant propagation velocityλ1 = 1/ε (compare section 5.3). The only way that this initial state isconnected through a different transition than a contact discontinuityis a shock, reaching a state in the immiscible region, which has tofulfill the Rankine-Hugoniot condition (equation 5.16, section 5.4.1)and to feature a propagation velocity λ > λ1 = 1/ε, in order to fulfillthe self-sharpening condition of shock transitions. We would liketo emphasize that, starting from the considered initial state, thereis only two possible types of transitions, namely a contact discon-tinuity or a (semi-)shock, with the properties discussed above.Secondly, we consider the connection of an immiscible downstreamstate to the miscible (upstream) feed state B. This feed state is rea-ched through a Γ2 characteristic, along which the directional deri-vative of λ2 can change sign in the miscible region, as observed insection 5.3. As a consequence, the shock crossing the binodal mightalso constitute a semi-shock, continuing as a simple wave in themiscible region.


0 0.2 0.4 0.6 0.8 1

SR

[-]

-8

-6

-4

-2

0

ln(x

1L)

[-]

B

A

B

A

Figure B.2: Location of states S in the immiscible regions, connectedthrough shocks to initial and feed states A and B, respectively,in the miscible region. Points of maximum propagation velocityλt,max along tieline characteristics are indicated by blue circlesand connected through a dashed line. In the case that the transi-tion is to be continued from S within the immiscible region, theshock connecting A to S has to span over the point of maximumpropagation velocity λt,max of the respective tieline path. In turn,when S is reached through preceding (downstream) transitions,the shock connecting S to B does not span over the point ofmaximum propagation velocity.

b.3.1.3 Continuation in the immiscible region

Still considering the Riemann problem discussed in the previoussection, we now focus on the continuation of a shock crossing the bi-nodal in the immiscible region. The aspects discussed in this sectionare illustrated in Figure B.2, which shows the characteristics in theimmiscible region, plotted in the ln(xL

1) − SR plane in grey, and the

connection of miscible initial and feed states A and B to differentregions in this plane.If a path connecting a single phase state to a state in the immiscible

region should be continued within the immiscible region, there aretwo possible scenarios. Either, the single phase state is directly con-


nected through a shock to an intermediate state I located in the im-miscible region. The second option is a semi-shock transition, wherethe single phase state is connected through a shock to a state S inthe immiscible region with λ = λS

i , at which the transition changesfrom a shock to a wave.If the single phase state is located downstream (initial state A), λ1should decrease along the further path emanating from S towardsthe intermediate state, in order to enable the formation of a waveconnecting S to upstream immiscible states. Recalling the behaviorof λ1 along a tieline (illustrated in Figure B.1), it is clear that, star-ting from the miscible region (component 1 lean or rich side), theshock transition has to connect to a state S located beyond the pointwith the maximum propagation velocity λmax

1 . These points are indi-cated as blue circles in Figure B.2. Arrows in Figure B.2 indicate theconnection of initial states A through a shock to possible immisciblestates beyond the points of maximum propagation velocity.If the single phase state is located upstream (feed state B), the pro-pagation velocities λ2 emanating from state S downstream shouldincrease, for S to be connected to immiscible states located down-stream through a wave. A significant change in the values of λ2 canbe observed particularly along the part of Γ2 characteristics map-ping on tielines (compare also Figures 5.6a and b), and a decreasein λ2 can be evidenced towards the binodal. Accordingly, with re-spect to the state of maximum propagation velocity (λmax

1 ) on therespective tieline, the state S should be located in the direction to-wards the feed state B, i. e. the shock connecting S to B should notspan over the point of maximum propagation velocity (as illustratedby arrows in Figure B.2).

b.3.2 Case studies

In this section, we want to present a detailed construction of thesolutions to the exemplary cases presented in section 5.4, based onRiemann problems with a miscible initial state A in the component1 lean region (specifically a mixture of component 2 and 3 only) andwith miscible feed states Bi (i = 1− 3) in the component 1 rich re-gion (specifically a mixture of component 1 and 2 only). We considerthe entire chromatographic cycle, i.e. the displacement of A by Bi


(adsorption), as well as the re-displacement of Bi by A (desorption).

b.3.2.1 Displacement A-B1

Figure B.3 presents the derived adsorption and desorption path inthe hodograph plane; i.e. in the (ln(xL

1),SR) - plane in subfigure (a)

and in ternary diagram in subfigure (b). Red (ΣA) and blue (ΣB)continuous lines and in Figure B.3a indicate all conditions whichfulfill the Rankine-Hugoniot condition (Equation 5.16) when con-nected to the initial state A or the feed state B1, respectively. PointsSXi (X=A,B; i = 1, 2) denote switch points with λ = λS

i , where theshock transitions connecting X to SX

i can be continued by a wavetransition (semi-shock) within the immiscible region. Black continu-ous and dashed lines in both subfigures (a) and (b) illustrate theparts of the solution path, where the transition is a wave or a (semi-)shock, respectively. Note that the miscible region is not visible inFigure B.3a. Therefore, the first and last shock transition of each theadsorption and the desorption step, connecting the miscible state Xto a state in the immiscible region, can only be illustrated in FigureB.3b.

AdsorptionLet us first consider the adsorption step, displacing initial state A

by feed state B1. The corresponding path in the hodograph plane(Figure B.3) passes through SA

1 − Ka − Ia, and the resulting concen-tration and flow profile is provided in Figure B.4. Note that, withfR 6= SR, the flow profiles are not identical with the concentrationprofiles (compare section 5.4).Starting from the initial state A, in principle we have to consider theΓ1 characteristic mapping on the x1 = 0 axis, which suggests a con-tact discontinuity with propagation velocity 1/ε. However, statesin the immiscible region move with a higher propagation velocityand thus produce a shock transition into the immiscible region (seesection B.3.1).In order to continue the path within the immiscible region, the ini-tial state connects to the point SA

1through a (semi-)shock. With

λAS = λS1 and λ1 decreasing when moving from SA

1along the Γ1


0 0.2 0.4 0.6 0.8 1

SR

[-]

-8

-6

-4

-2

0

ln(x

1L)

[-]

Kd

Ka

S1

A

S2

AΣ

A

ΣB

ΣB

ΣA

S2

B S1

B

(a)

Ia

Id

20

40

60

80

20 40 60 80

20

40

60

80

comp 2

comp 3 comp 1Ia

S1

A

Ka

S2

A

Kd

Id

B1

A

(b)

Figure B.3: Paths in the hodograph plane for the displacement cycle A (ini-tial state) to B1 (feed state). (a) SR − ln(xL

1) plane. (b) ternary di-agram (mass fractions). Continuous red and blue lines indicateall conditions which fulfill the Rankine-Hugoniot condition forstate A and B, respectively. Continuous and dashed black linesindicate wave and shock paths, respectively. Points SX

i denotethe conditions which are connected to state X (A or B) by ashock and at which λ = λS

i . States Ix (x = a,d) denote the in-termediate states during the displacement (adsorption) and re-displacement (desorption). States Kx indicate the conditions atwhich the transition (semi-shock) changes from shock to waveor vice-versa.


0 1 2 3 4 5

τ [-]

0

200

400

600

800

1000

Ci [

gL

-1]

comp 1

Ia(a) S1

A KaA

comp 2

B1

0 1 2 3 4 5

τ [-]

0

200

400

600

800

1000

Fi [

gL

-1]

comp 1

(b)

comp 2

B1

IaA

Figure B.4: (a) Concentration and (b) flow profile for the adsorption step,i.e. the displacement of state A by state B1.


0 2 4 6 8 10 12

τ [-]

0

200

400

600

800

1000

Ci [

gL

-1]

0.2 0.3 0.40

500

1000

S2

A

Id

Kd

A

comp 2

comp 1

(a) IdB1

0 2 4 6 8 10 12

τ [-]

0

200

400

600

800

1000

Fi [

gL

-1]

(b)

comp 2

comp 1

B1Id A

Figure B.5: (a) Concentration and (b) flow profile for the desorption step,i.e. the displacement of state B1 by state A.


characteristic, the type of transition changes from shock to wave atSA

1. Reaching the watershed point, the directional derivative of λ1

changes from negative to positive, and thus the transition changesback from wave to shock. This second switch occurs at Ka, whereλK1 = λKI. The second shock part connects Ka to the intermediate

state Ia, the corresponding shock path deviates only slightly fromthe Γ1 characteristic. All in all, the initial state A is connected to theintermediate state Ia through a shock-wave-shock, with shock andwave parts switching at SA

1and Ka.

The intermediate state Ia is located at a slightly higher SR than thepoint SB

1, and thus λI

2 < λS2. As a consequence, the intermediate

state cannot be connected to the point SB1

through a wave, and isinstead directly connected to the feed state through a shock, withλI2 < λ

IB < λB1

2 .

DesorptionThe desorption step, i.e. displacing (initial) state B1 by (feed) state A,results in the path passing through Id −Kd − SA

2. The corresponding

concentration and flow profile is given in Figure B.5.As in the adsorption case, states in the immiscible region travel ata propagation velocity greater than λB

1 = 1/ε, and thus produce ashock connecting B to the immiscible region, which directly reachesthe intermediate state Id with λB1

1 < λBI < λIt. This intermediate

state is determined through the two Rankine-Hugoniot conditionscorresponding to the shocks B1-Id and Id-Kd.From the intermediate state Id, a Γ2 characteristic emanates with adirectional derivative of λ2 changing from positive to negative. Ac-cordingly, the second transition starts as a shock. At the state Kd,with λIK = λK

2 , the transition switches from a shock to a wave. AtSA

2, with λS

2 = λSA, the transition changes from a simple wave toa shock, connecting to the feed state A. One can conclude that thesecond transition, connecting Id to A, is a shock-wave-shock withshock and wave parts switching at Kd and at SA

2.


Paths in the hodograph plane for the adsorption and the desorp-tion step of the displacement cycle A-B2 are provided in Figure B.6.


0 0.2 0.4 0.6 0.8 1

SR

[-]

-8

-6

-4

-2

0

ln(x

1L)

[-]

ΣB

S2

B

S2

A

KaK

d

S1

AIa

Id

ΣB

ΣA

ΣA

(a)

S1

B

20

40

60

80

20 40 60 80

20

40

60

80

comp 2

comp 3 comp 1

Ia

Ka

S1

A

S2

BS

2

A

Id

A

B2

Kd

(b)


1) plane. (b) ternary di-agram (mass fractions). For notation and line styles, see FigureB.3


0 0.5 1 1.5 2

τ [-]

0

200

400

600

800

1000

Ci [

gL

-1]

S1

BKaIaA B2

(a)

comp 1

comp 2

0 0.5 1 1.5 2

τ [-]

0

200

400

600

800

1000

Fi [

gL

-1]

B2

Ia

A(b)

comp 1

comp 2

Figure B.7: (a) Concentration and (b) flow profile for the adsorption step,i.e. the displacement of state A by state B2.


0 5 10

τ [-]

0

200

400

600

800

1000

Ci [

gL

-1]

0.2 0.40

500

1000

S2

A

IdKd

A

comp 1

comp 2

S2

B

(a) B2

Id

0 5 10

τ [-]

0

200

400

600

800

1000

Fi [

gL

-1]

A

comp 1

comp 2

B2

Id(b)



With the initial state A being identical for all the considered cases,the shock paths ΣA (red continuous lines in Figure B.6a), as wellas points SA

i are identical for all cases. In contrast, shock paths ΣBconnecting to state Bi approach the plait point (higher ln(xL

1)) withincreasing methanol content in Bi.

AdsorptionAs in the previous example, states in the miscible region propagatefaster than the interstitial velocity 1/ε, and provoke a shock, con-necting the initial state A directly to the intermediate state Ia, withλA1 < λ

AI < λI1. The intermediate state Ia is determined through the

two Rankine-Hugoniot conditions corresponding to the shock tran-sitions A-Ia and Ia-Ka.Since λ2 increases along the respective Γ2 characteristic from Ia toKa, the second transition starts as a shock. At the watershed point,the directional derivative of λ2 changes from positive to negative.Accordingly, the transition changes from shock to wave at Ka andback to a shock at SB

1. The second transition, connecting Ia to thefeed state B2, is thus a shock-wave-shock with λI

2 < λIK = λK

2 andλS2 = λSB < λB

2.

DesorptionThe first transition connecting the initial state B2 to the intermedi-ate state Id is a shock-wave-shock, with changes between shock andwave parts in SB

2 and in Kd.Since Id is located at higher SR than SA

2 and thus λ2 is decreasingfrom Id to SA

2 , the intermediate state is connected to the feed stateA via a wave-shock, switching from wave to shock at SA

2 .It should be noted that, although the sequence of the paths in the ho-dograph plane and the types of transitions differ considerably fromthe ones derived for the desorption step B1-A, the correspondingelution (concentration and flow) profiles coincide in a major part,namely in the last wave-shock transition. In the hodograph plane,this overlap corresponds to the part of the solution path reachingtowards SA

2 and mapping on a tieline, which is identical for bothcases.



With B3 featuring a higher methanol content than B1 and B2, the cor-responding shock paths ΣB (blue lines) are closer to the plait point,see Figure B.9a. As can be deduced from Figures B.9a and b, onlythe desorption step passes through the immiscible region, while theadsorption step entirely proceeds through the miscible region.

AdsorptionBoth the initial state A and the feed state B can be connected to validpoints in the immiscible region, SA

1 and SB1 with λA

1 < λAS = λ

SA1

t and

λSB

1

t = λSB < λB2, respectively. However, none of the states located on

Σ ′A, enabling a connection to A through a shock, can be connectedthrough a physically meaningful path to any of the states locatedon ΣB, allowing a shock transition to the feed state B. Conditionson the second shock path Σ ′′A (red line containing SA

2 ) do not fulfillthe necessary shock condition that λ > λA

1 . Since conditions on theΣ ′A path do not allow a continuation of the path within the immis-cible region, and conditions on the Σ ′′A path do not present a validconnection to the initial state A, the only physically valid solutionleads through the miscible region, as explained in detail in section5.3 (therefore not illustrated here).

DesorptionThe desorption path is similar to the one derived for the case B2-A.However, the intermediate state Id is located at lower SR than thepoint SA

2 , and therefore λI2 < λS

2. As a consequence, Id is directlyconnected to the feed state A through a shock instead of the (shock-)wave-shock transitions exhibited in the cases B1-A and B2-A. Ac-cordingly, elution profiles for the case B3-A (see Figure B.10) do notoverlap with the previously considered cases B1-A and B2-A.


0 0.2 0.4 0.6 0.8 1

SR

[-]

-8

-6

-4

-2

0

ln(x

1L)

[-]

ΣB K

d

S2

AId

S2

B

ΣB

S1

A

S1

B

(a)

Σ'A

Σ''A

20

40

60

80

20 40 60 80

20

40

60

80

comp 2

comp 3 comp 1

Ia

Kd

S2

BId

A

B3

(b)


1) plane. (b) ternary di-agram (mass fractions). For notation and line styles, see FigureB.3


0 5 10

τ [-]

0

200

400

600

800

1000

Ci [

gL

-1]

0.2 0.40

500

1000

comp 1

comp 2

Kd

S2

B

Id

Id

A(a) B3

0 5 10

τ [-]

0

200

400

600

800

1000

Fi [

gL

-1]

comp 1

comp 2

Id AB

3(b)


CS U P P L E M E N TA RY M AT E R I A L F O R C H A P T E R 7

c.1 solution of the equilibrium theory model

In this section, we derive the solution of the equilibrium theory mo-del introduced in section 7.2 by the method of characteristics. Insections C.1.1 and C.1.2, the general solution of the model is pro-vided separately for the miscible and the immiscible region. Theresulting equations are then applied to the chromatographic systemunder investigation, and the equilibrium theory solution for this sy-stem is illustrated, independently of initial and feed states, in thehodograph plane in section C.1.3. Finally, the elution profiles forspecific initial and feed states presented in section 7.4.1 are derivedand explained mathematically in section C.1.4.

c.1.1 Solution in the miscible region

In the miscible region, with only one convective phase, the over-all fractional flows are identical to the overall liquid concentrati-ons (which equals the single liquid phase concentration), and thusFi = Ci = ci. In this case, equations 7.5 correspond to the modeldiscussed in Refs. [101–103]. With its mathematical structure beingidentical to the classical binary chromatographic model, the mathe-matical solution procedure has been discussed in great detail in theliterature.23,24,28,82 Furthermore, the general solution procedure ba-sed on the method of characteristics, yielding characteristics Γj inthe hodograph plane, propagation velocities λj = dξ/dτ, as well astheir reciprocals σj, has been outlined previously, in section 5.3.1.Here, we just provide the most important results, without explai-ning the mathematical solution procedure in detail.


There are two types of characteristics Γ1 and Γ2 in the hodograph(C1, C2) plane, defined through their slopes ζ as:

Γj : ζ3−j =dC1dC2

=θj − n22n21

; j = 1, 2. (C.1)

The variables nij are defined as ∂ni/∂cj, whereas θj are the eigen-values of the matrixN = [nij]. We assign the condition that θ1 < θ2.It is worth noting that the relationship of ni in terms of cj, and thusthe determination of the partial derivatives nij, is not trivial. Withadsorbed phase concentrations ni being described by the isothermsprovided in equation 7.7, the variables ni are function of the liquidphase activities aj of all four components. In turn, the liquid phaseactivities ai are function of the mole fractions zi of the first threecomponents (the mole fraction z4 can be determined from the otherthree mole fractions). These three mole fractions can be expressedin terms of the three liquid phase concentrations ci, which can thenbe converted into the variables ci = Ci. The determination of N isexplained in detail in Ref. [103], important equations are summari-zed in Appendix C.3.The Γi characteristics, resulting from the solution of the ODEs C.1,describe the change in composition (C1,C2) that occurs when con-necting an initial state to a feed state. Each composition located ona Γi characteristic travels along the column at a specific propagationvelocity λi. The inverse of the propagation velocity, σi = λ−1i , corre-sponds to the slope of the characteristics in the physical (ξ, τ) plane,which is defined as

σj = 1+ νθj (C.2)

With the definition θ1 < θ2, it can be concluded that σ1 < σ2 henceλ1 > λ2.Let us now consider a Riemann problem with an initial (down-stream) state A and a feed (upstream) state B. The solution in thephysical (ξ, τ) plane consists of three states, namely the initial stateA, the intermediate state I, and the feed state B, connected by twotransitions, each of which maps either on a Γ1 or a Γ2 characteristic.The type of transition depends on the directional derivative of σialong the Γi characteristic; if σi decreases from an upstream state

C.1 solution of the equilibrium theory model 271

U to a downstream state D (i.e. the propagation velocity λi increa-ses), then the transition is a simple wave. In the other case, when σiincreases from U to D, a simple wave transition would cause a mul-tivalued, unphysical solution. The physically meaningful solutionis then a discontinuous shock transition, which directly connects Uto D, and with a slope σ obtained from the integral form of theconservation laws, i.e.:

σ = 1+ νθ, (C.3)

with

θ =[n1]

[C1]=

[n2]

[C2]. (C.4)

Here, [·] denotes the jump in the quantity enclosed across the discon-tinuity. Equation C.4 is the Rankine-Hugoniot condition, definingthe possible states which, assuming a certain downstream state D(a certain upstream state U), can be connected to D (U) through ashock. These states are located on a shock path Σi, which is tangentto the characteristic Γi in D (U). A further condition of the shocktransition is that σU

i < σ < σDi .

The two transitions connecting A to B map on two different typesof characteristics Γ1 and Γ2, which intersect at the intermediate stateI. For a physically meaningful solution, downstream states have topropagate faster than upstream states. With σ1 < σ2 (λ1 > λ2), thesame state is reached faster through a Γ1 than through a Γ2 charac-teristic, such that intermediate states should always be connectedto downstream states through a Γ1 characteristic, and to upstreamstates through a Γ2 characteristic. As a consequence, the physicallycorrect path connecting the initial, downstream state A to the feed,upstream state B corresponds to the sequence A - Γ1 - I - Γ2 - B. Inthe case of shock transitions, Γi characteristics in this sequence arereplaced by Σi paths, which can deviate slightly from the correspon-ding characteristics.

c.1.2 Solution in the immiscible region

In the case of two convective phases, equations 7.5 are mathemati-cally very similar to the model discussed and solved in section 5.4.1,


with minor differences: In the current model, both modified compo-nents 1 and 2 are adsorbing (n2 6= 0), the porosity is assumed tobe constant, and the fractional flow fR is a function of SR only (notof a tie-line indicator η). Furthermore, recall that the variable τ isdefined here in a slightly different way as in the previous chapters(see section 7.2). Although the model could also be solved basedon the unknowns C1 and C2, it is mathematically simpler and pro-vides more physical insight to perform a variable transformationto the new unknowns η and SR. While SR is the phase saturationof the water rich phase, η denotes a tie-line indicator, which uni-quely identifies the tie-line. In this context, we define η as cL

1, i.e.the concentration of the modified component 1 in the water leanphase. Since the derivation is analogous to the derivation providedin section 5.4.1, only the main results are summarized here.The variable transformation results in a system of homogeneousPDEs in η and SR of the form given by equation 5.9. The parametersAi to Di (i = 1, 2) are in this chapter defined as

Ai = fR dcR

i

dη+ (1− fR)

dcLi

dη(C.5a)

Bi = SR dcR

i

dη+ (1− SR)

dcLi

dη+ ν

dnidη

(C.5b)

Ci =dfR

dSR

(cRi − c

Li

)(C.5c)

Di =(cRi − c

Li

)(C.5d)

and hence exhibit minor deviations from the corresponding expe-rissions in section 5.4.1. Note that, with cXi and ni depending onη only, and with fR depending on SR only, derivatives in equationsC.5 are normal (not partial) derivatives. The necessary values forcXi and ni are determined from spline interpolations fitted to the60 predetermined tie-lines (illustrated in Figure 7.3) as a functionof the tie-line indicator η. Derivatives of these spline functions withrespect to η also provide dcXi /dη and dni/dη.The two sets of characteristics Γx (x = t, nt) in the hodograph planeare defined by equations 5.12 and 5.13, but considering the newexpressions for Ai to Di (equation C.5). From equation 5.12, it isclear that Γt characteristics map on tie-lines, while the second set of


characteristics defined through equation 5.13 are non-tieline charac-teristics.Also the definitions of σt and σnt are identical to the ones in section5.4.1 in terms of the parameters Ai to Di (compare equations 5.14

and 5.15), yielding slightly altered expressions when consideringthe new expressions for Ai to Di:

σt =

(dfR

dSR

)−1

(C.6)

σnt =B1ψnt +D1A1ψnt +C1

(C.7)

The slopes σ of the shock paths in the physical plane, derived fromthe integral material balance, read:

σ =[C1] + ν[n1]

[F1]=

[C2] + ν[n2]

[F2]. (C.8)

As in the miscible region, the Rankine-Hugoniot condition given byequation C.8 also determines the shock paths Σx, as it identifies allstates U (D), which can be connected to a downstream state D (up-stream state U) through a shock, while satisfying the integral formof the material balances. Equation C.8 is also valid for shock transiti-ons crossing the binodal, i.e. connecting a miscible to an immisciblestate. In this case, Fi can be replaced by ci for the miscible state.In principle, tieline and non-tieline characteristics can be calculated

for the entire two-phase region, as shown in Figure C.1a. The cha-racteristics in the (C1, C2) plane are constructed by solving the or-dinary differential equations 5.12 and 5.13, and by transforming theresulting values for η and SR to C1 and C2. We would like to emp-hasize that characteristics in the hodograph plane exhibit the samequalitative properties as those in section 5.4. Furthermore, propaga-tion velocities λx (x =t,nt) along different tie-line characteristics areillustrated in Figure C.1b. With equation C.6 being a function of SR

only (and not of η), profiles of λt are identical along all tie-lines. Thisis not the case for the profiles of λnt. Neither of the two propagationvelocities λx is always greater than the other, as it is the case in themiscible region (λ1 > λ2). It is also worth noting that λt → 0 as thebinodal is approached (SR → 0 or SR → 1). The physical interpreta-tion of this behavior is that phases at a low saturation SX propagate


very slowly (low fX, compare Figure 7.4), thus approaching a fracti-onal flow of 0 at the limiting saturation SXlim, which according to ourassumption equals 0.Considering the entire immiscible region violates our assumptionof low overall liquid concentrations (and thus negligible volumetricchanges due to adsorption, reaction, or nonideal mixing). Hence, weconstrain the applicable region to concentrations C1 < 4 molL−1

and C2 < 10 molL−1, corresponding to the grey box in Figure C.1a.In this region (reaching a maximum saturation SR ≈ 0.2), one candeduce from Figure C.1b that λnt > λt. Following the terminology ofthe miscible region, with λ1 > λ2, we rename the Γnt and Γt charac-teristics to Γ1 and Γ2 characteristics, respectively, in the applicableregion. It is worth noting that, with the same reasoning as appliedfor the miscible region (section C.1.1), one can deduce the correctsolution, connecting an initial (downstream) state A to a feed (up-stream) state B, to map on a sequence A-Γ1-I-Γ2-B (or, in the case ofshock transitions, Σi paths instead of Γi characteristics). The stateI is the intermediate state determined by the intersection of the Γ1and the Γ2 characteristic, passing through A and B, respectively.


0 2 4 6 8 10 12

C1 [mol L−1]

0

10

20

30

40

50C

2[m

olL−1]

Γnt

Γt

(a)

0 0.2 0.4 0.6 0.8 1

SR [-]

0

0.5

1

1.5

2

λ[-]

λnt

λt

(b)

Figure C.1: (a) Γt (red) and Γnt (blue) in the entire immiscible region, map-ped in the (C2, C1) plane. (b) Propagation velocities λt (red) andλnt (blue) along the tie-line characteristics.


c.1.3 Illustration of the solution in the hodograph plane

The characteristics in the applicable region of the hodograph planefor the miscible and the immiscible region are plotted in Figure C.2.The miscible and the immiscible region are separated by a blackcontinuous line, corresponding to the binodal curve. Γ1 and Γ2 cha-racteristics are illustrated in blue and red, respectively. Furthermore,arrows in the color of the respective characteristic indicate the di-rection in which σi increases. Following characteristics in this di-rection when connecting a downstream to an upstream state, wavetransitions are physically possible, while in the opposite direction,shock transitions are required (note that shock paths Σi can deviateslightly from Γi characteristics). Blue filled circles indicate states atwhich the directional derivative of σ1 switches from negative to po-sitive. Crossing these states can create semi-shock transitions, whichare a combination of shock and wave transitions (for a detailed dis-cussion of semi-shocks see the relevant literature23,24,26,81).It can be noted that all Γ2 characteristics, but only a few Γ1 charac-

teristics in the immiscible region, intersect with the binodal. Thesecharacteristics are continued by the corresponding Γ2 and Γ1 charac-teristics in the miscible region. However, slopes of the Γi characte-ristics, as well as propagation velocities λi, or their reciprocals σi,exhibit a discontinuity at the binodal. If the jump in σi is positivewhen crossing the binodal from a downstream to an upstream state,the upstream limit at the binodal propagates more slowly than thedownstream limit. As a consequence a plateau in the concentrationprofile at the intersection of Γi with the binodal forms. Note that thisplateau does not correspond to a classical intermediate state, sincethe solution path at this plateau continues along the same type ofcharacteristic. In the opposite case, where the jump in σi at the bino-dal is negative when connecting a downstream to an upstream state,the binodal has to be crossed through a shock (since the upstreamlimit would propagate faster than the downstream limit, thus crea-ting a multivalued, non-physical situation). As the binodal is onlycrossed by few Γ1 characteristics, we neglect these cases and focuson the crossing of the binodal along Γ2 characteristics.According to Figure C.1b, λ2 → 0, and thus σ2 → ∞, when ap-proaching the binodal from the immiscible region. In the miscible


0 1 2 3 4

C1 [mol L−1]

0

2

4

6

8

10

C2[m

olL−1]

Γ1

Γ2

S2

S1

Figure C.2: Γ1 (blue) and Γ2 (red) characteristics of the model system inthe hodograph plane. The black line denotes the binodal (se-parating miscible and two-phase region). Arrows indicate thedirection in which the slope of the physical plane σi increa-ses. Blue filled circes denote the positions at which the directio-nal derivative of σ1 chances from negative to positive. The redempty circles on the veritical axis indicate two states S1 and S2which are connected through a contact discontinuity along a Γ2characteristic (σS1

2 = σ2 = σS2

2 ).

region, σ2 never approaches ∞. Thus, for any Γ2 characteristic, thelimiting value of σ2 in the immiscible region is greater than that inthe miscible region. As a consequence, downstream miscible statesconnected to upstream immiscible states through a Γ2 characteristiccreate a plateau at the intersection with the binodal, while upstreammiscible states connected to downstream immiscible states yield ashock crossing the binodal.First, let us consider the scenario of a downstream miscible state Dconnected to an upstream immiscible state U through a Γ2 charac-teristic, i.e. the scenario which in principle creates a plateau at theintersection with the binodal with composition P. According to thedecrease in σ2 along Γ2, indicated by the arrows in Figure C.2, Dshould be connected to P through a shock with slope σDP, and P


should further be connected to U through a shock with slope σPU.This is the case for states U in the two-phase region located relativelyclose to the binodal, where σDP < σPU, i.e. where the downstreamshock propagates faster than the upstream shock. In the oppositecase when σDP > σPU, which occurs for states U located at a greaterdistance from the binodal, the physically valid solution is a directshock transition connecting D to U with a slope σDU which travelsfaster than the shock connecting D to P.Second, we examine the case of a downstream immiscible state Dconnected to an upstream miscible state U through a Γ2 characte-ristic. In this case, increasing slopes σ2 along Γ2 (compare arrowsin Figure C.2) indicate that the connection of D to the binodal, andof the binodal to U, consist of simple wave transitions. However,the binodal itself has to be crossed by a shock. The physically cor-rect solution of such cases is the formation of semi-shock transiti-ons. As an illustrative example, we consider the Γ2 characteristicmapping on the vertical axis. The two empty circles in Figure C.2indicate two states S1 and S2, which are connected across the bi-nodal by a contact discontinuity, fulfilling the necessary conditionσS1

2 = σS1S2 = σS2

2 . Any downstream state D located above S1, isconnected to S1 through a simple wave, and every upstream state Ulocated below S2 is reached from S2 by a simple wave. In turn, do-wnstream and upstream states located between S1 and S2 are acces-sed through a shock which crosses the binodal. As a consequence,there exists a spectrum of different possible transitions, reachingfrom (1) a wave - contact discontinuity - wave for states D and Ulocated beyond S1 and S2, (2) a semi-shock (wave-shock or shock-wave) if one of the two states is located between and the other onebeyond S1 and S2, to (3) a shock if both states D and U are locatedbetween S1 and S2.

c.1.4 Derivation of elution profiles

In the following, we derive mathematically the solutions (entirechromatographic cycles, i.e. adsorption and desorption steps) forthe inital state A and feed states Bi (i = 1− 4) discussed in section7.4.1. We will refer to the solution paths in the hodograph planepresented in Figure 7.5, as well as to the adsorption and desorption


profiles in Figures 7.6 and 7.7, respectively.Let us first examine the solution paths in the hodograph plane (Fi-gure 7.5). The solutions for the adsorption and desorption steps ofall four exemplary cases always map on a sequence of Γ1 characteris-tic (emanating from the initial state) and Γ2 characteristics (passingthrough the feed state), which intersect at an intermediate state I. Su-perscripts a and d denote the affiliation of a state to the adsorptionor the desorption step, respectively. Due to the location of state A inthe origin, solution paths of the adsorption steps involve the Γ1 cha-racteristic located on the horizontal axis, such that all intermediatestates Ia are located on this axis. With the same reasoning, solutionpaths of the desorption step involve the Γ2 characteristic located onthe vertical axis, such that all intermediate states Id are located onthe vertical axis. As discussed in section 7.4.1, the location of partsof the solution paths (connecting A and the intermediate states Ia

or Id) on the horizontal and the vertical axis corresponds to regionsof pure component 1 (n-hexyl-acetate) during adsorption and purecomponent 2 (water) during desorption in the elution profiles (com-pare Figures 7.6 and 7.7) for any of the four examples. As a conse-quence, pure products are achieved in all four examples, regardlesswhether a phase-split and two-phase flow occurs (examples 3 and4) or not (examples 1 and 2). In addition, with the distance of inter-mediate states Ia

i and Idi from the origin increasing from example 1

to example 4 (compare Figure 7.5), concentration levels of the inter-mediate states in the corresponding elution profiles increase.With regard to the adsorption steps, all transitions are shock transi-tions, as one can deduce from the directional derivatives of σi alongΓi characteristics indicated qualitatively by arrows in Figure C.2. Forexamples 1 and 2, located entirely in the single-phase regime, thesolution consists of the usual sequence of A - transition - Ia - transi-tion - B. In the case of example 3, where the binodal is crossed alonga Γ2 characteristic, an additional concentration plateau P3 forms atthe solubility limit, as explained in the previous section C.1.3. Withthe feed state B4 being located in the two-phase region at a greaterdistance from the binodal, this state features a higher propagationvelocity, and is directly connected to Ia

4 through a shock, withoutthe formation of an additional plateau. Also this behavior was dis-cussed in section C.1.3.


Considering the desorption profiles in Figure 7.7, a variety of dif-ferent transitions, especially concerning the connection of Id to A,can be observed. However, it is important to note that the last partof these transitions, reaching state A, is always a simple wave map-ping on the same Γ2 characteristic (vertical axis). Accordingly, thislast part of the elution profiles is identical for all four cases, suchthat complete desorption (state A) is reached at the same time. Thedetailed explanation for this behavior is as follows: In the miscibleregion, intermediate states on the vertical axis (such as Id

1 and Id2)

are connected to A through a simple wave. As in the adsorptioncase, the transition becomes more complex for intermediate stateslocated in the immiscible region (examples 3 and 4), since the bino-dal has to be crossed to reach A. The intermediate state Id

3 is locatedbelow the state S1 defined in section C.1.3, whereas the intermedi-ate state Id

4 is located above this state. As a consequence (see sectionC.1.3), Id

3 is connected to A through a semi-shock (shock-wave), withthe shock part crossing the binodal and changing into a wave at astate located above S2 (see Figure C.2). From Id

4, state A is reachedthrough a wave - contact discontinuity - wave (see the discussion insection C.1.3). Despite the more complex transition in cases 3 and 4,state A is still reached through a simple wave, overlapping with thesolutions of cases 1 and 2.

c.2 interconversion of mole fractions and concentra-tions

For the sake of simplicity, and to avoid volume changes due to mix-ing and reaction, we assume a constant (mass) density ρmix,w = 814

g/m3 for the convective phase(s), which equals the component den-sity of the solvent n-hexanol at 298.15 K. Assuming a constant mo-lar density ρmix is far from reality, since the different componentsof the exemplary system, and therefore also different mixtures, dif-fer strongly in molar density. We are aware that the assumption ofρmix,w = 814 g/m3 for the water-rich phase (which is almost purewater, as indicated in Figure 7.3a) bears a considerable error (ap-prox. 20% based on the real density value). However, since the over-all fraction of water (over all convective phases) is rather small, alsothe volume fraction of the water-rich phase is rather low (SR < 0.2),

C.3 calculation of the jacobian 281

such that the impact of this error on the overall volume is tolerable.The molar density of a mixture can be determined from its massdensity through:

ρmix =

14∑i=1

ziMi

ρmix,w, (C.9)

where Mi is the molecular weight of component i. Accordingly, themolar concentration ci can be calculated from zi using:

ci = ziρmix =

zi4∑i=1

ziMi

ρmix,w. (C.10)

And accordingly, for ci:

ci =

zi + z34∑i=1

ziMi

ρmix,w, i = 1, 2. (C.11)

Thus, ci and ci can be determined explicitly from z. In turn, thecomposition z at chemical equilibrium can be determined from c1and c2 only implicitly, through solving the four equations 7.2, 7.4and C.11 simultaneously, where equation 7.4 involves a calculationof liquid phase activities by the underlying thermodynamic (UNI-FAC) model. It is worth pointing out that equations C.9 to C.11 arealso valid for overall mole fractions Zi and concentrations Ci (Ci),characterizing the composition of the mixture over all convectivephases.

c.3 calculation of the jacobian

For the calculation of the solution in the single phase region, theJacobian N = ∂n/∂c is required. Applying the chain rule to the Ja-


cobian (similar to the determination of the Jacobian N in AppendixB.1.1) yields:

N =∂n

∂c=∂n

∂a

∂a

∂z

∂z

∂c

∂c

∂c(C.12)

Note that a involves the activities of all four components, whilez and c are property vectors of the first three components only(the fourth component is dependent on the other 3 components).The vectors n and c are vectors with two entries only (two trans-formed variables determining the composition in chemical equili-brium). Hence, the matrices ∂n/∂a, ∂a/∂z, ∂z/∂c and ∂c/∂c havethe dimensions 2x4, 4x3, 3x3 and 3x2, respectively. The first par-tial derivative ∂n/∂a is simply obtained by summing the iso-thermequations 7.7 to obtain n, and differentiating the respective equati-ons analytically with respect to a. The derivative ∂a/∂z can be cal-culated by differentiating analytically the modified UNIFAC equati-ons. For the third factor, the following expression of z with respectto c is differentiated:

zi =ciM4

c1(M4 −M1) + c2(M4 −M2) + c3(M4 −M3) + ρmix,w (C.13)

Finally, the last derivative can be determined by Ref. [103]:

∂c

∂c= −

(∂G

∂c

)−1(∂G

∂c

)(C.14)

with G defined as

= G =

c1 − (c1 + c3)

c2 − (c2 + c3)

Ka3a4 − a1a2

. (C.15)

It is worth noting that the first two entries of G are functions of cand c, while the last entry is a function of c only. Evaluating thepartial derivatives of this last entry with respect to c again involvesthe partial derivatives ∂a/∂z and ∂z/∂c.

DS U P P L E M E N TA RY M AT E R I A L F O R C H A P T E R 9

d.1 profiles based on parameters obtained by the peak

area method

10 15 20 25

t [min]

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

c1+

c2 [

gL

-1]

(a)

0.6 0.5 0.4 0.3

21°C

10 15 20 25

t [min]

0

0.01

0.02

0.03

0.04

0.05

0.06

c1+

c2 [

gL

-1]

(b)

0.30.40.50.6

38°C

5 10 15 20 25

t [min]

0

0.01

0.02

0.03

0.04

0.05

c1+

c2 [

gL

-1]

0.9 0.80.7

0.6

0.5 0.4

0.3

(c)

54°C

4 6 8 10 12

t [min]

0

0.01

0.02

0.03

0.04

0.05

0.06

c1+

c2 [

gL

-1]

0.8 0.6

(d)

1.01.21.4

72°C

Figure D.1: Experimental (solid lines) and simulated elution profiles (das-hed lines) for Glucose at different temperatures and flowrates.Flowrates are indicated next to the profiles inmLmin-1. Parame-ters used in the simulations correspond to the ones estimatedin section 9.4.2 (apparent dispersion coefficient), 9.4.3 (adsorp-tion parameters) and 9.4.4 (reaction kinetic parameters obtainedthrough the peak area method). (a): 21 C (b): 38 C (c): 54 C (d):72 C (extrapolated model parameters)


20 40 60 80 100 120 140

t [min]

0

0.005

0.01

0.015

0.02

0.025

c1+

c2 [

gL

-1]

(a)

0.10.20.30.4

12°C

10 20 30 40 50 60 70

t [min]

0

0.005

0.01

0.015

c1+

c2 [

gL

-1]

(b)

0.20.5 0.4 0.3

17°C

10 15 20 25 30 35 40

t [min]

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

c1+

c2 [

gL

-1]

(c)

0.4 0.30.6 0.5

21°C

Figure D.2: Experimental (solid lines) and simulated elution profiles (das-hed lines) for Fructose at different temperatures and flowrates.Flowrates are indicated next to the profiles inmLmin-1. Parame-ters used in the simulations correspond to the ones estimatedin section 9.4.2 (apparent dispersion coefficient), 9.4.3 (adsorp-tion parameters) and 9.4.4 (reaction kinetic parameters obtainedthrough the peak area method). (a): 12 C (b): 17 C (c): 21 C

A C R O N Y M S

AST adsorbed solution theory

BET Brunauer-Emmett-Teller (isotherm)

BT breakthrough

DAD diode array detector

ET equilibrium theory

HPLC high performance liquid chromatography

IAST ideal adsorbed solution theory

LL liquid-liquid

MLE maximum likelihood estimate

NRTL non-random two-liquid

ODE ordinary differential equation

PD primary drainage

PDAE partial differential algebraic equation

PDE partial differential equation

PI primary imbibition

PNT phenetole

PS-DVB polystyrene-divinylbenzene

RAST real adsorbed solution theory

SD secondary drainage

SI secondary imbibition

SL solid-liquid

SLL solid-liquid-liquid

SMB(R) simulated moving bed (reactor)

TBP 4-tert-butylphenol

UNIFAC UNIQUAC functional-group activity coefficients

UNIQUAC universal quasi-chemical activity coefficients

286 acronyms

UV ultaviolet

WENO weighted essetially non-oscillatory

XRD X-ray diffraction

two-phase flow in liquid chromatography - experimental and theoretical investigation

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