University of Groningen

Collected tales on mass transfer in liquidsBollen, Arnoud Maurits

IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.

Document VersionPublisher's PDF, also known as Version of record

Publication date:1999

Link to publication in University of Groningen/UMCG research database

Citation for published version (APA):Bollen, A. M. (1999). Collected tales on mass transfer in liquids. Groningen: s.n.

CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).

Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.

Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.

Download date: 19-04-2020

RIJKSUNIVERSITEIT GRONINGEN

Collected tales on mass transfer in liquids

Proefschrift

ter verkrijging van het doctoraat in deWiskunde en Natuurwetenschappenaan de Rijksuniversiteit Groningen

op gezag van deRector Magnificus, dr. D.F.J. Bosscher,

in het openbaar te verdedigen opvrijdag 26 november 1999

om 14.15 uur

door

Arnoud Maurits Bollengeboren op 22 juli 1966

te Groningen

Promotor: prof. ir. J.A. Wesselingh

ISBN: 90-367-1157-6

‘Zo jong nog, en nu reeds niets gepresteerd.’ (W. Pauli)

Beoordelingscommissie:prof. dr. H.W. Hoogstratenprof. dr. R. Krishnaprof. E.L. Cussler

ACKNOWLEDGEMENTS

i

ACKNOWLEDGEMENTS

Now that the work is done, I am somewhat disappointed to discover that this thesis has becomea collection of rather loosely connected chapters instead of a sound, solid scientific epos with adistinct beginning and a distinct end. Well, it’s too late now, and I just hope that those whohave the courage to plough their way through it will feel, afterwards, that it was worth theeffort.

I would like to thank everybody who has directly contributed to the realisation of this thesis,especially those who did more than they were obliged to on account of their function. If theshoe fits, wear it. First of all, I should mention my promotor, Prof. Hans Wesselingh forcreating the most pleasant atmosphere in his group in which there was always room for a joke,a difference of opinions or an exchange of views on all sorts of subjects. Also, he managed topick the most agreeable persons as my collegues, even without consulting me first. I will alsokeep good memories of Trudi, who has never hesitated to forward her view on things, or to dowhat she felt was necessary, even if I sometimes felt that it was not. I greatly appreciated thework lunches and dinners over at their place.

Working in a pleasant atmosphere adds to the zest for work. Some persons who were notscientifically involved, have, by way of just being there, made important but invisiblecontributions to this thesis. First and foremost, I would like to say special thanks to allmembers of the ‘9-uur koffie’ (= nine o’clock coffee) bunch. The students whom I had thehonour to supervise were another source of pleasure: Bart Venneker, Jan Wegenaar, TeunDuijnstee, Martin de Boer and Lamberto Eldering. Many were the moments I enjoyed workingwith them. Many also were the moments I did not. They would then be asking either for myattention or for things of a more concrete nature, while I was trying to concentrate on somevery interesting and at that time probably very important matter. I may not always have beenvery accesible, for which I apologise. I am also sorry that I have not been able to include any ofthe work of Martin de Boer and Teun Duijnstee in this thesis. Finally, the students whopopulated room 5118.0210 (‘de waterzaal’), and most of whom were pleasant company, cannotbe left unmentioned, but they are too numerous to list here.

My brother Guido also made a very important contribution to this book by way of lending me acomputer on which many of the calculations were done, and also some of the text processing.Other people I should mention are my room mates Atze-Jan van der Goot and Cor Visser, thetechnicians Jan ‘Is-het-voor-thuis?’ Bolhuis and Marcel de Vries, and the librarians, whosomehow manage to stay friendly and helpful under circumstances that definitely would driveme crazy.

This research has been financially supported by the Council for Chemical Sciences of theNetherlands Organization for Scientific Research (CW-NWO). DSM-Research initiated andcontributed to the investigations on liquid-liquid extraction. In this respect, I would especiallylike to thank Ton Simons of DSM Research for his stimulating and enthousiastic co-operation.

SUMMARY

iii

SUMMARY

Diffusion in liquids determines the rate of mass transfer in many processes, both natural andindustrial. In liquid-liquid extraction for example. This thesis culminates in a mass transfermodel using the Maxwell-Stefan equations to describe liquid-liquid batch extraction. Thisprocess necessarily involves strongly non-ideal liquid mixtures, and diffusion in such mixturesis the thread of this thesis. After the first two introductory chapters (a general and a technicalone), a few matters concerning the Taylor dispersion method for the determination of diffusioncoefficients are discussed in chapters 3 and 4. Then, in chapter 5, the behaviour of diffusivitiesin demixing mixtures is investigated. In chapter 6 the model for liquid-liquid batch extractionis developed, which is subsequently applied in chapter 7 to systems in which emulsification isknown to occur during extraction.

A problem in multicomponent diffusion is obtaining the diffusion coefficients. One of theinstruments that allow routine measurements in liquids is the Taylor capillary. This capillaryusually is some 20 m long, and therefore it is usually coiled. In chapter 3 the effects of coilingon the accuracy of the measurements is investigated theoretically. In a few extreme cases, itturns out that small coil diameters can introduce large errors. In those cases it is necessary toreduce the flow drastically.

The Taylor dispersion method requires very sensitive and accurate composition measurementof the eluent flow. For a ternary mixture, it is commonly thought that two components have tobe monitored simultaneously, which makes the method more difficult to use. Yet, in chapter 3it is theoretically proven that it is possible to obtain multicomponent diffusivities bymonitoring only one composition. The price that must be paid is that at least two experimentsare needed for one set of ternary diffusivities.

The theory of the Taylor method assumes that the diffusivities and the total concentration ofthe mixture do not depend on composition. In reality such mixtures are rare. In chapter 4numerical simulations show that effects of composition variations are usually small, and thatquite large concentration differences between the tracer pulse and the eluent liquid can beapplied.

Liquid-liquid extraction usually involves diffusion in mixtures with compositions near ademixing boundary. There are indications that the diffusivities might behave differently in suchmixtures. These are based on diffusion measurements near consolute points in binary mixtures.Such consolute points are, in a way, analogues for the spinodal curve. To see whetherdiffusivities really misbehave, the data are re-examined in chapter 5. It turns out to be difficultto obtain accurate thermodynamic data of the mixtures near the consolute point. Nevertheless,no evidence is found of serious anomalies in the behaviour of the diffusion coefficients nearthe consolute point, but unambiguous proof that they behave normally cannot be given either.This means that there is no reason not to use the usual mass transfer equations for describingliquid-liquid (batch) extraction.

SUMMARY

iv

In chapter 6 a model is developed for a stirred ternary batch extractor with a flat liquid-liquidinterface. Important parts of the model are the thermodynamic equilibria used and the two masstransfer ‘films’ on either side of the interface. This type of model turns out to be very sensitiveto thermodynamic and mass transfer parameters. Numerical stability and accuracy are also aproblem. A good description of experiments from literature could only be obtained afteroptimising the model parameters.

During the extraction in certain mixtures, such as water-caprolactam-benzene, an emulsion canform in one of the phases. A possible explanation is that the multicomponent mass transferdrives the system into the liquid-liquid demixing zone. The extraction model of chapter 6 isused in chapter 7 to see whether this might be the case. The model can yield composition pathspassing through the demixing zone, but only for rather improbable values of the modelparameters. This strongly indicates that other processes must play a role in the formation of theemulsion.

A good model of liquid-liquid extraction will undoubtedly need a proper multicomponentmodel of the mass transfer in the extractor. This thesis shows how such models can be set upand used. It also shows many problems related to obtaining model parameters and doing modelcalculations. Finally, it contains indications that also other parts of liquid-liquid extractiontheory such as thermodynamics, the behaviour of interfaces and the two-phase flow inequipment need to be improved.

CONTENTS

v

ACKNOWLEDGEMENTS i

SUMMARY iii

1. INTRODUCTION 1

1.1 Liquid-liquid extraction fundamentals 11.1.1 The ternary composition diagram 11.1.2 The binodal curve 21.1.3 The spinodal curve 31.1.4 Final equilibrium 31.1.5 Composition paths 5

1.2 A microscopic view on extraction 51.2.1 Mass transfer across the interface 51.2.2 The role of diffusion in extraction 6

1.3 Practical extraction methods 71.3.1 Purity and efficiency 71.3.2 Mixer-settlers 71.3.3 Extraction columns 9

1.4 Structure of this thesis 9

2. A THEORETICAL FRAMEWORK FOR MASS TRANSFER 11

2.1 Notational matters 112.2 Defining systems 12

2.2.1 Components 122.2.2 Thermodynamic conditions 132.2.3 Composition 132.2.4 Frames of reference 14

2.3 Two mass transfer models 172.3.1 Fick’s transfer model 182.3.2 The Maxwell-Stefan transfer model 202.3.3 Choose your weapon 21Notation 22References 23

3. TAYLOR DISPERSION THEORY 25

3.1 The Taylor dispersion experiment 253.1.1 Theory of the dispersive process 253.1.2 The capillary: coiled or straight? 27

3.2 Retrieval of diffusivities from experimental data 333.3 Conclusions 38

Notation 39References 40

CONTENTS

vi

4. NUMERICAL SIMULATION OF TAYLOR DISPERSION 43

4.1 Derivation of the equations 444.2 Composition-independent parameters 46

4.2.1 Discretisation of the differential equation 474.2.2 Solving the discretised equation 49

4.3 Composition-dependent parameters 514.3.1 Discretisation and solution of the equations 524.3.2 Example: methanol / tetrachloromethane 52

4.4 Results 534.5 Conclusions 57

Appendix A. The convection-diffusion equation at the capillary axis 57Appendix B. Expressions for the coefficients for composition-independent

parameters 58Appendix C. Cubic spline interpolation 59Appendix D. The amount of tracer in the tube section 61Appendix E. Expressions for the coefficients for composition-dependent

parameters 62Notation 64References 65

5. DIFFUSION NEAR CONSOLUTE POINTS 67

5.1 The world according to Cussler 675.2 Determination of Γ from VLE data 70

5.2.1 The ideal gas GE method 705.2.2 The non-ideal gas GE method 725.2.3 The numerical integration method 73

5.3 The system triethylamine-water 755.4 The system nitrobenzene-n-hexane 805.5 Conclusions 82

Notation 82References 83

6. MODELLING LIQUID-LIQUID BATCH EXTRACTION 85

6.1 The physical system 856.2 Solving the flux equations 86

6.2.1 Difference approximation 87Linear mean composition 90Cubic mean composition 90Logarithmic mean composition 91

6.2.2 Numerical integration 916.3 Solving the dynamic equations 92

CONTENTS

vii

6.4 Composition dependence of relevant physical properties 926.4.1 Total concentration 936.4.2 Activities 936.4.3 Diffusivities 946.4.4 Film thicknesses 95

6.5 Results 966.5.1 Initial flux solutions 976.5.2 Dynamic model solutions 99

6.6 Comparison with the model of Krishna et al. 1036.7 Conclusions 104

Appendix A. Combined fit of the LLE and VLE data 105Appendix B. Multiple subfilm difference approximation with barrier layer 108Notation 109References 110

7. EMULSIFICATION IN LIQUID-LIQUID BATCH EXTRACTION 113

7.1 The physical system 1147.2 Solving the flux equations 1157.3 Solving the dynamic equations 1157.4 Composition-dependence of relevant physical properties 116

7.4.1 Total concentration 1167.4.2 Activities 1167.4.3 Diffusivities 1177.4.4 Film thicknesses 117

7.5 Batch extraction measurements 1177.6 Results 119

7.6.1 Initial flux solutions 1207.6.2 Dynamic model solutions 120

7.7 Conclusions 123Appendix A. Liquid-liquid equilibria of the systems used 124Appendix B. The binodal curve, the distribution coefficients and the plait point 126Notation 126References 127

SAMENVATTING (Dutch summary for laymen) 129

INTRODUCTION

1

1. INTRODUCTION

So, this thesis is on ‘mass transfer in liquids’. More precisely, it is on mass transfer in non-ideal liquid mixtures, with liquid-liquid extraction as the main subject. However, this does notmean that all of the thesis is dedicated to the description of this process, nor does this implythat every aspect of extraction is covered. The dual aim of this chapter isa) to provide a short introduction to the basics of liquid-liquid extraction, andb) to outline the structure of this thesis.

1.1 LIQUID-LIQUID EXTRACTION FUNDAMENTALS

In liquid-liquid extraction a solute is transferred from one liquid to another. Consider, forinstance, benzene ‘polluted’ with methanol. We could remove the methanol by contacting thebenzene with another liquid which has a higher affinity for methanol†. To be sure that onlymethanol is transferred and not benzene, the methanol-extracting liquid should not mix withbenzene. Water could be a suitable extract component, because we know that it mixes verywell with methanol, and rather poorly with benzene. Further, since benzene is nonpolar, andwater and methanol are polar, we expect that water has a higher affinity for methanol thanbenzene. So, it is clear that any extraction process involves at least three substances:• the solute (methanol) which is to be removed from• the raffinate component (benzene) by• the extract (water).Accordingly, the two immiscible phases are called the raffinate and the extract phase. In reallife, extraction processes usually involve more than three components, but to keep things assimple as possible, we will consider only ternary extractions. As we will see at the end of thisthesis, even this ‘simple’ case can be complicated enough. For a quantitative description of themutual miscibilities of the components, we need measurements or models. The results can bevisualised in a single graph, the ternary phase diagram.

1.1.1 The ternary composition diagram

An experienced eye can recognise the solubility characteristics of a ternary mixture from theternary phase diagram at one glance. The ternary diagram plots (sets of) ternary compositions.It is customary to define a mixture’s composition in terms of mass, mole or, less frequently,volume fractions. The fact that the sum of the fractions of all components always equals unityallows us to plot ternary compositions in two dimensions. If a mixture contains 10% water and70% methanol, it follows that benzene must make up for the remaining 20%. In chemicalengineering the fractions used are mostly mole fractions, usually denoted by x.

There are two ways of plotting compositions: the composition domain can either be anequilateral triangle, or the familiar Cartesian co-ordinate system. In the equilateral triangle, theco-ordinates of all components have the same scale, but the diagram is more difficult to plot † This no prerequisite for the extraction of the pollutant, but the higher the affinity of the extracting fluid,

the more efficient the extraction process will be.

Chapter 1

2

and to read. The Cartesian domain is easy to plot and to read, but distorts the co-ordinate of atleast one component. Otherwise, these two ways of plotting compositions are notfundamentally different. In this thesis, the Cartesian way of plotting is used.

1.1.2 The binodal curve

The phase diagram of a chemical system contains the set of all possible equilibriumcompositions. We know that benzene and water do not mix, and it is not likely that thepresence of a small amount of methanol will cancel this incompatibility. Varying methanolcontents will lead to different degrees of mixing of the immiscible species. If we want to knowhow the miscibility of water and benzene depends on the methanol content, we have to domeasurements. Liquid-liquid equilibrium measurements are usually carried out as follows.Measured amounts of the three species are mixed, so that the overall composition of themixture is known. If the mixture is unstable, that is, if the amounts of water and benzeneexceed their mutual solubilities, the mixture will split into two phases in equilibrium with eachother. Subsequently, the compositions of these two phases are analysed, and can then beplotted in a ternary diagram.

It is obviously impossible to measure all possible equilibrium compositions, because there areinfinitely many of them. Therefore, only a finite set of these compositions can be measured,and thermodynamic models are used to provide estimates for intermediate compositions. Thethermodynamic models most commonly used for this kind of work are the UNIQUAC and theNRTL models. These models enable us to calculate a continuous set of equilibriumcompositions, which, in the case of the example system forms a continuous curve, the binodalcurve. This curve only looks like a single continuous curve, but it consists of two branches,each of which represents a phase. They meet in what is called the plait point.

Once we have plotted this continuous curve, we can no longer see which composition of thebenzene phase is in equilibrium with which composition of the aqueous phase. To overcomethis minor difficulty, we also plot a few tie-lines, which connect equilibrium compositions ofboth phases. These tie-lines give an impression of how the components are distributed over thephases in equilibrium. In figure 1, the binodal curve of the system water-methanol-benzene isplotted along with a few tie-lines. The equilibrium compositions shown have been calculatedusing the UNIQUAC model.

Mixtures with an overall composition that lies within the area enclosed by the binodal curveand the water-benzene axis – the demixing zone – are thermodynamically unstable. That is, theGibbs free energy of the two phases in equilibrium is lower than that of the overall mixture.Mixtures within the demixing zone will usually, but not always, split. If they do, the phaseswill have compositions at the ends of a tie-line.

INTRODUCTION

3

xw

0.0 0.2 0.4 0.6 0.8 1.0

xm

0.0

0.2

0.4

0.6

0.8

1.0

Figure 1. The binodal curve (solid curve) and five tie lines (dashed lines) of the system water-methanol-benzene, calculated using the UNIQUAC model at 289.15 K (standard pressure). The spinodalcurve is also plotted (dotted curve, see the next section). The plait point is marked by a black square. Notethat, as a result of the unit total mole fraction condition, only the lower left triangle of the plot isaccessible.

1.1.3 The spinodal curve

There are areas within the demixing zone where a mixture must get over an energy barrierbefore it can demix. In the absence of external disturbances such a solution will remain mixed,and is called supersaturated or metastable. Given a thermodynamic model of the system, wecan calculate the set of compositions which form the boundary between the metastable and thetruly unstable compositions. This set of compositions, which form a curve within the demixingzone, is called the spinodal curve. It is shown in figure 1. At the plait point, the binodal and thespinodal curves always coincide.

Inside the spinodal curve mixtures will always spontaneously demix, while mixtures in thezones between the binodal and the spinodal curves can be metastable. It is impossible tomeasure spinodal compositions.

1.1.4 Final equilibrium

Assume that we have 4 mol of benzene contaminated with 1 mol of methanol. The methanolfraction of this raffinate phase is therefore 20 mol%. The extract phase initially consists of 10mol of pure water. When we put these two liquids phases in a flask, what will happen? Thedynamics of the extraction depend on whether we shake the flask, or whether we just put it ona shelf.

In the former case, the two phases break up into droplets. As a result, the contact area betweenthe two phases becomes very large, which greatly enhances the transfer of methanol from theraffinate to the extract phase. Also, both phases are well mixed, so the transport of methanol

Chapter 1

4

from the raffinate to the interface and from the interface into the extract phase is rapid. In short:the resistance against the transfer of methanol is decreased by shaking. If we stop shaking aftera while, the two phases will coalesce, and they will have attained their final equilibriumcompositions.

If the flask is left at rest, the interface is small, that is, if the flask you use for the experiment isof normal dimensions. Because the density of the extract phase is higher than that of theraffinate phase, the latter will ‘float’ on top of the former. Thus, the interfacial area will beequal to the cross-section of the flask. Further, the two phases will be more or less stagnant,although the composition changes at the interface will lead to density differences within thephases, which can induce viscous flow. Anyway, it is clear that the transport of methanol willbe much slower than in the previous case. You should not be surprised if you will have to waitfor days for the phases to attain equilibrium.

The dynamics of the two methods are different, but the result will eventually be the same: thesystem in the flask will reach equilibrium. The equilibrium compositions of both phases can bedetermined using the phase diagram. A very important composition in the determination of thefinal equilibrium is the overall composition of the mixture in the flask, which is invariant. Itcan be proved that at any time during the extraction, the line through the two pointsrepresenting the (average) composition of the two phases must also pass through this overallmixture composition point. So, to obtain the final equilibrium compositions, we must find thetie-line which passes through this point. Since it is unlikely that the tie-line we need is presentin the phase diagram, it will have to be calculated, or estimated from interpolation between twoother tie-lines. In figure 2, the initial and final compositions are shown.

xw

0.0 0.2 0.4 0.6 0.8 1.0

xm

0.0

0.2

0.4

0.6

0.8

1.0

Figure 2. Graphical construction of the final equilibrium compositions. White triangles: initialcompositions, white squares: final equilibrium compositions, black dot: overall composition. For furtherexplanation, see figure 1.

INTRODUCTION

5

After the extraction, it turns out that the methanol content in the benzene phase has fallen from20% to 1.2%, while the extract phase now contains 8.7% of methanol†. The total numbers ofmoles in both phases also have changed: 4.06 mol of raffinate remain and 10.94 mol of extract.

1.1.5 Composition paths

During the extraction, the (average) compositions of the two phases will ‘move’ from theinitial compositions towards the final equilibrium compositions. If we plot the transientcompositions of the phases in the phase diagram, they form curves called composition paths.The nature of these paths depends on the way in which the extraction is carried out, eventhough the start and end points are fixed. Consider, for example, the two extraction methodsdiscussed in the previous section. Apart from the fact that shaking the flask speeds up theextraction, the composition paths of the phases in the agitated flask will probably also have adifferent form than those of the flask which is left at rest. Usually, the composition paths do notcross the binodal, that is, they stay in within the part of the domain which represents non-demixing mixtures. Given the narrow gaps between the binodal and benzene-methanol andwater-methanol axes, it may seem that there is little room for variations, but that is just a matterof scale.

These differences between the composition paths are caused by the fact that shaking does notnecessarily accelerate the transfer of each of the substances in an equal way. This is a result ofthe complex way in which a system parameter, such as the level of agitation, influences theprocess of extraction. The system in the jar is highly non-linear, which results in unpredictableresponses to changes in the process parameters. It is the poor understanding of this behaviourwhich makes the design of an extraction column a difficult job, while the design of adistillation column is almost a routine matter nowadays.

1.2 A MICROSCOPIC VIEW ON EXTRACTION

In section 1.1.4 the distinction was made between transfer of the solute to (or from) theinterface and transfer across the interface. To have the extraction take place as rapid aspossible, the transport within the phases is enhanced by stirring or shaking, which is easy to do.Once the resistance in the bulks is largely eliminated, the transport across the interfacebecomes rate-determining. This resistance against mass transfer is much harder to reduce: themain method is to increase the interfacial area, but in practice this is possible only up to certainlimits. Because the transfer across the interface often determines the overall transfer ratesduring the extraction, knowledge of the mechanisms that govern this mode of transfer isessential to understand and control extraction.

1.2.1 Mass transfer across the interface

For the investigation of interfacial mass transfer it is necessary to remove the influences of thetransfer within the phases. This is achieved by eliminating the intra-phase transfer resistancescompletely. When both phases are well-mixed and the interface is relatively small, the masstransferred across the interface will be small compared to the mass present in the bulks, and the

† These numbers should be regarded with some reservation however, because of the large influence of the

thermodynamic model parameters; we do not know how reliable these are.

Chapter 1

6

phase compositions will change only slowly. Because of the mixing of the bulks, they will behomogeneous to a good approximation. Also, a well-defined interface is necessary to study thetransfer across the interface. An apparatus that enables us to do such experiments is the Lewis-cell, which is discussed in more detail in chapter 6. The situation at the interface in a Lewis-cell is often simplified to the schematic representation shown in figure 3.

Figure 3. A schematic representation of the situation at the interface.

In the bulks of the phases a level of turbulence is assumed that levels out all compositionvariations almost immediately. The phase boundary is conceived of as a sharp transitionbetween the two phases, and it is thought to be mechanically rigid to a degree that causesdamping of the turbulent eddies in the bulk near the interface. This leads to the presence ofstagnant fluid layers on both sides. The thicknesses of these films depend on such quantities asthe level of agitation in the phases and their viscosities. Usually, they are of the order of 10−5 min liquids, and 10−4 m in gases. Mass transfer in the films can only occur by diffusion. Thismodel of the interface is a rather crude simplification, but usually it works. There arealternative models, but these are less frequently used, and generally do not yield significantlybetter results.

1.2.2 The role of diffusion in extraction

According to the film model, the transfer between the two phases in a Lewis-cell is determinedby the interfacial area, the film thicknesses and the diffusion coefficients of the (three)substances. The area of the interface is known, and so the thicknesses and the diffusivitiesremain as the main parameters. The film thickness, which is a clearly defined length in the filmmodel, is only a vague concept in reality, and cannot be measured directly. Yet, as we knowhow diffusion and the film thicknesses interact, we can derive the film thicknesses from masstransfer measurements when the diffusion parameters are known. However, by the nature ofthe systems investigated in papers on this subject, the usefulness of the results is doubtful.

The diffusivities themselves are also a source of uncertainty. Although there is an extensiveamount of literature on this subject, the experiments described in it mainly concern diffusivitiesin binary mixtures. The chances of finding diffusivity data on an arbitrary ternary system ofsimple substances are very small. Fortunately, methods have been developed for theinterpolation of binary diffusivities to ternary compositions. Unfortunately, these methodswork best for ideal or moderately non-ideal mixtures, while here we are dealing with stronglynon-ideal mixtures. Since it is widely accepted that there is a link between the activities of thecomponents and their diffusivities, diffusivity interpolation may become inaccurate when thevariations in the activities are large. Apart from these interpolation problems, it is possible that

INTRODUCTION

7

there are more fundamental causes for deviant behaviour in multiphase systems. Especially inthe vicinity of the demixing zone it is not clear whether the diffusivities behave the same as inmiscible mixtures. There are indications that deviations from ‘normal’ behaviour can beexpected for compositions near the spinodal.

1.3 PRACTICAL EXTRACTION METHODS

Now that we have an idea of what goes on during extraction on a small scale, we turn to theindustrial scale. Extraction finds a fairly large application in chemical industry, where it ismostly used when distillation is not feasible. As in any industrial separation, an extractionprocess should not only meet requirements regarding the product purity, but it must also usethe available resources as efficiently as possible. In the next sections we will see that the needfor optimal efficiency leads to countercurrent columns, which are used for industrial extraction.

1.3.1 Purity and efficiency

If, in the example of section 1.1.4, the obtained purity of the benzene after the extraction is nothigh enough, then how do we proceed? Instead of carrying out the extraction in a flask, wecould have used a separation funnel. After shaking and settling of the phases (or waiting), thetwo phases are easily separated. To further reduce the methanol content of the benzene, anamount of fresh water can be added to the raffinate phase, and a new extraction is started. Ifagain 10 mol of water is used as the extract phase, the benzene would contain a mere 0.055%of methanol after the extraction, and repeating the procedure once again would reduce this to2.4·10−3%. To arrive at this low methanol level, we needed 30 mol of water. Had we used 5mol portions of water, the extraction would have been far more efficient: after four extractionsteps, we would have obtained a purity of 1.6·10−3% at the expense of only 20 mol of water.Generally, the extraction is most efficient (in terms of the amount of extract needed), when thesubsequent extract phases are infinitely small. There is one, in this case slight, disadvantage:the benzene loss also increases, because with each extraction step also a small amount ofbenzene is transferred to the aqueous phase. Because the solubility of benzene in water is verylow, these losses are small (about 0.5%).

1.3.2 Mixer-settlers

In our example, we had only a small amount of raffinate, and the extraction probably is bestcarried out following the above procedure. But what if we have a very large quantity or aconstant flow of raffinate? In that case, a continuous extraction process is needed. A simpleprocess is directly based on the step-by-step funnel method. We replace each funnel step by avessel in which the two phases are mixed, and a vessel where they are allowed to settle. Bothvessels can be fed with continuous streams, as is shown in figure 4. The combination of thetwo vessels is called a mixer-settler.

Chapter 1

8

Figure 4. Sketch of a mixer-settler.

Ideally, in the stream leaving the mixer the two finely dispersed phases are at equilibrium. Thepurity of the raffinate phase leaving the mixer-settler then only depends on thethermodynamics of the system and the flow ratio of the two phases. Apart from immiscibility,there are a few other properties that the mixture must have. First, the interfacial tensionbetween the two phases must be large enough to ensure quick coalescence in the settler.Second, there must be a density difference between the phases because phases with equaldensities are hard to separate too.

To obtain the desired raffinate purity, several mixer-settlers can be placed in series. Obviously,it makes no sense to feed a mixer-settler with the extract flow of the previous mixer-settler. Wecould instead feed each mixer with a fresh extract stream, or use the extract streams of settlersdownstream, which contain relatively little solute, to feed mixers further upstream. This willreduce the amount of extracting liquid needed. These two configurations are shown for fourmixer-settlers in figure 5.

Figure 5. Two possible configurations of four mixer-settlers in series.

INTRODUCTION

9

1.3.3 Extraction columns

The step from the second configuration to a countercurrent extraction column is only small.There are several considerations which make a vertical column a better proposition forcontinuous extraction than a series of mixer-settlers. First and foremost, there is the economicaspect: a single column is much cheaper to build than a train of mixer-settlers. Anotheradvantage is that we can let gravity do its work and transport the phases through the column ifthe light phase is fed to the bottom of it, and the heavy phase to its top. This requires that thereis a significant density difference between the two phases, but such a difference must also existto make a settler work.

Extraction columns are never just empty columns, because there are two problems related tothis particular design. In an extraction column, we always have a continuous phase and adispersed phase. The dispersed phase consists of droplets dispersed in the continuous phase.Because the droplets tend to coalesce, precautions must be taken to prevent this. The otherproblem is that in empty columns both phases seriously suffer from backmixing. There are afew widely used solutions to these problems.

Rotating disc contactors (RDC’s) and pulsed columns belong to the class of agitated columns.These types of columns are quite common. In an RDC mixer disks mounted on a vertical shaftin the column keep the droplets small. To reduce backmixing in an RDC, it iscompartmentalised by rings, which reduce the cross-section of the column between the mixerdisks. So, each disk mixes its own compartment, and an RDC can be regarded as a series ofnon-ideal mixers in series. In a pulsed column, the liquid in the column is agitated by pulsingit.

There are also non-agitated columns, such as sieve tray columns, which are analogous todistillation columns, and packed bed columns. The internals of these columns keep the dispersephase from coalescing in a passive way and simultaneously reduce the backmixing. Thesetypes of columns are not so common, because their performance is even less predictable thanthat of agitated columns. This is mainly caused by the flow patterns in these columns.

1.4 STRUCTURE OF THIS THESIS

This thesis contains 7 chapters. Here follows a brief overview of the contents of chapter 2through 7.

Chapter 2 is a kind of ‘technical’ introduction. It contains definitions and derivations ofequations needed for a clear understanding of the chapters to come.

Chapter 3 treats a method for measuring diffusivities in liquids, the Taylor dispersion method.As we have seen in section 1.2.2, diffusivities are essential parameters in extraction processes,and it is important to have an accurate and fast method for the determination of ternarydiffusivities. Although the Taylor dispersion technique is not new, and there is a considerableamount of literature on the subject, there are still several details that need clarification. Theway to determine the optimal experimental conditions is discussed, and a method to obtainmulticomponent diffusivities.

Chapter 1

10

Chapter 4 also concerns the Taylor dispersion method. Theoretically, the method can be usedonly for mixtures with constant diffusivities with respect to the composition. Such mixtures donot exist, and in this chapter the Taylor dispersion method is numerically simulated, and theinfluence of concentration dependent diffusivities is investigated.

In section 1.2.2 it was mentioned that anomalous diffusivity behaviour may be found near thespinodal curve. Such anomalies could be of influence on mass transfer in liquid-liquidextraction, since during extraction near-spinodal compositions occur all the time. In chapter 5,an attempt is made to see if there is a reason to believe that this is indeed the case.

In chapter 6, a model for liquid-liquid extraction in a Lewis-cell is developed. It is based on thefilm model of the phase interface discussed in section 1.2.1. What is new, is that it uses onlyinformation on the interfacial area, the initial state of the phases and physical properties of thepure substances and of the mixture. The model is verified against literature data.

During liquid-liquid extractions, a phenomenon called emulsification can occur: in one of thephases, a fine mist develops. This is undesirable, because it complicates the separation of thephases. In the final chapter, we try to find out whether the model developed in chapter 6 can beused to explain the formation of emulsions.

A THEORETICAL FRAMEWORK FOR MASS TRANSFER

11

2. A THEORETICAL FRAMEWORK FOR MASS TRANSFER

This introductory chapter explains the concepts and equations we will need for understandingthe subsequent chapters. All of the issues treated here have already been discussed by others,and the only reason for going over them once more is to make sure that a) everything is neatlyand unambiguously defined, and b) the reader is spared the trouble of having to lookeverything up. Well-informed readers may find this chapter a bit boring, but are neverthelessadvised to read it because the perspective and some of the definitions differ from what iscustomary in literature.

2.1 NOTATIONAL MATTERS

Before we engage on our mission, it is necessary to straighten out some notational matters,most of which concern the kinds of vectors that are used, and the operations applied to them.Basically, two kinds of vectors are used throughout this work:• Spatial vectors, which represent quantities having both a magnitude and a direction. These

vectors have, in principle, three elements, one for each co-ordinate in space. Vectors ofthis type are denoted by a bold-faced symbol. Examples of spatial vectors are the velocityof a body (denoted by u) or of a chemical species i (ui), or the flux of that substance (Ni).

• Component vectors, which contain data related to the components of a mixture. Forexample, a mixture contains certain masses of its component species, denoted by mi. Wecan order these data in a component vector, which has therefore as many elements as thereare species, at least, if it is to be complete. As a rule, the length of these vectors is relatedto the number of components n, but exactly how depends on their purpose. The vectors arerepresented as underlined symbols (m in this case).

A combined form is also possible. The fluxes of the individual components are ordered inspatial vectors, but these can, in turn, be ordered in component vectors. The result is acomponent vector with spatial vectors as its elements. The flux component vector is denoted byN.

In transfer phenomena gradients of composition-related extensive quantities frequently play animportant role, for they often represent the driving forces in the process. These gradients, canalso be ordered in a vector of the combined type. As an example, consider the gradients of theconcentration c, which are ordered in the vector ∇c. We adopt the following notationconvention:

∇ =∇∇F

HGG

I

KJJ ≡ ∇c

c

c c1

2

M

(1)

For many properties ordered in component vectors, both scalar and vectorial, it is meaningfulto define the total value as the sum of all elements. In the above examples, the sum of allcomponent masses is the total mass of the mixture, and the sum of all fluxes is the total flux.For example, consider the concentration vector c , then

Chapter 2

12

c c c cii

n

= =∑ ∑=

means1

(2)

where c stands for the total concentration. It is essential that ci is summed over all ncomponents. In many literature sources the total of an extensive property is denoted with thesubscript ‘t’, but this is not strictly necessary, because it is a scalar and therefore cannot bemistaken for the vector (c versus c). If we define a unit vector I whose elements are all equal to1, we could also write c = I · c instead of c = ∑ c. Further, no strict discrimination will be madebetween column matrices and vectors, and multiplication of a matrix by a vector is considereda legal operation as long as their sizes allow it.

2.2 DEFINING SYSTEMS

When speaking of mass transfer, we are confronted with the question: what is transferred andin what does this happen? Clearly, mass is transferred in space, but this is too general adescription. We are not concerned with the motion of cars on a highway, nor are we interestedin the motions of individual molecules or atoms. Our scales of interest are those of localtransport in (chemical) industrial equipment, roughly in the range between a micrometre and amillimetre. The physical surroundings in which the transfer takes place is usually called themedium. The gas, liquid, solid or, if you wish, plasma in which the transfer takes place is oftenalso referred to as system, mixture, solution or fluid. The nature of the medium depends on itscomponents, the thermodynamic conditions and its quantitative composition.

2.2.1 Components

In a mixture, we usually lump all entities with like properties together as a component.Obvious examples are chemical elements and compounds, but a group of similar compoundscan also be considered as one component. It is not always simple to discern the components ofa mixture. For example, how many components does an aqueous solution of NaCl contain?This depends on the circumstances. In simple bulk diffusion, without external forces, thesmallest possible number of components is two: water and NaCl, because the Na+ and Cl− ionscannot move independently to a significant extent without violating the electroneutralityprinciple. In the presence of an alternating electrical field, however, the two ionic species mustbe treated as separate components. Electroneutrality, which glued the ions together in the bulkdiffusion case, can now be violated on a small scale, but enough to be responsible for theconductivity of the electrolyte. In solutions containing different salts, electroneutrality can bepreserved in different ways, so that even for simple bulk diffusion, the ionic species must bedefined as independent components.

If microporous structures such as membranes and catalysts are involved, these solid matricescan also be considered as a component, although they themselves are not transferred. Moreprecisely: we are usually interested in transport relative to those structures. Nevertheless, theycan be treated as a component because the interaction between them and the other, mobile,components influences the transfer rates. We can even go one step further by defining virtualcomponents: in surface diffusion for example, the free sites at a surface may be thought of as aspecies with its own diffusional behaviour.

A THEORETICAL FRAMEWORK FOR MASS TRANSFER

13

It is hard to give an unambiguous definition of a component. Any physical body or chemicalsubstance that is of influence on a mass transfer process may be called a component. Often, thechoice which (combinations of) substances or bodies are called components depends oncircumstantial needs and limitations, but we must define how we choose them.

2.2.2 Thermodynamic conditions

The condition of a medium can be important. Conditions that we often encounter are:• isothermal, which means that at a given time its temperature is constant throughout its

entire extent,• isobaric, in which the pressure is constant,• homogeneous, which means that the total thermodynamic state is spatially constant, and

since the state of a thermodynamic phase is determined by its temperature, pressure andcomposition†, homogeneity implies that these quantities must be constant throughout themixture. Another implication is that the system consists of one phase only. Ahomogeneous mixture is not a very interesting one from our point of view, because on ourscales of interest nothing will change, and from now on such a medium will be calledstrictly homogeneous. The term homogeneous will be reserved for single-phase systemsboth isothermal and isobaric, but possibly with varying composition.

2.2.3 Composition

Each component contributes to the extensive properties of a mixture, for example to its mass,volume, number of moles or internal energy. The composition is defined by the extent that thecomponents contribute to such a property (so there are many different ways of definingcomposition). Mostly relative contributions of the species are used; this makes the compositionindependent of the total amount of mixture. These fractional contributions of the componentsto the property Q can be ordered in composition vectors, which we shall denote as xQ.Compositions of mixtures are most conveniently described by the mass, mole or volumefractions of the chemical species they contain. It is customary to denote the mass, mole andvolume fraction vectors by ω, x and ε respectively.

At this point it is useful to pay some attention to the ‘problem’ that there are different ways inwhich a composition-dependent quantity may be represented. If, in an n-component mixture,the mole fractions are chosen as the composition variables, they together form the n-vector x.The sum of any complete set of fractions equals unity: ∑ x = 1. Therefore, x is a dependentvector with one redundant element. If the kth element is discarded, an independent (n −1)-vector y remains:

yx i k

x k i nii

i

=≤ <≤ ≤ −

RST +

if

if

1

11

(3)

Component k is called the dependent component, while the others are said to be independent.We see that the most obvious choice for k is n, because if k ≠ n, the element numbers of y nolonger correspond to the substance numbers as they were chosen originally. This

† This is true only in the absence of other possible driving forces such as electric and magnetic fields and

interfacial tension differences. In this thesis these effects are assumed non-existent.

Chapter 2

14

correspondence is lost for elements with numbers k and higher. This brings about undesiredand unnecessary component bookkeeping. Further, we can always arrange the components insuch a way that the dependent one bears the number n. This does not imply that there isnothing to be said for k being a free number. In that case, we do not have to fiddle with thecomponent order if we want to change the dependent component. Anyway, the choice for k = nis so common that it is often stated only implicitly and therefore we will also adopt it. Thenotations x and y for the two different composition vectors will be used from now on.

The x, y-ambiguity has some consequences for the derivatives of composition dependentfunctions. Any composition-dependent function, Q say, can take two forms, one operating on x,Q (x), and another operating on y, Q ( y). Theoretically derived functions usually operate on x,while empirical functions mostly operate on y. The partial derivatives of these two functionsare related by:

∂

∂

∂

∂

∂

∂

Q y

y

Q x

x

Q x

xi n

i i n

e j b g b g l q= − ∈ −, ,...,1 1 (4)

It is assumed here that y is truly independent. The left derivative is called a constrainedderivative, while those on the right are called unconstrained. As an example, consider theequation defining the activity coefficient γi of component i in a mixture: ln γi = (∂ n gE / ∂ ni)T, P,where n stands for the number of moles, and gE = GE/RT, GE being the molar Gibbs excessenergy of mixing.

lnE

EE

EE

EE

γ∂

∂∂∂

∂∂

δ∂∂

δii i j

ij jj

n

jij j

j

nn g

ng n

g

ng

g

xx g

g

yy= = + = + − = + −

= =

−

∑ ∑d i d i1 1

1

(5)

The last step is not obvious, but follows from substitution of eq. (4). Again, this is nothingnew, but it is important to what is coming, and besides, in literature it is not always clearlyindicated which composition vector or which derivative is used.

2.2.4 Frames of reference

Consider a mixture of n components, in which we want to observe the flows or fluxes of acertain extensive quantity Q, for example mass, number of moles, volume or internal energy.We want to say something about flows, and therefore about velocities. Just as the concept ofposition, velocity has no independent meaning as such; it is a relative property. Therefore, aframe of reference must be fixed, which we may choose arbitrarily. But first, some things mustbe set straight about our conception of mixtures.

Usually, we consider a mixture as a continuum: its properties are continuous and more or lesswell-behaved functions in the space it occupies. However, at very small (read: molecular)scales this picture is not true: the smooth property functions break down into useless noise andlose their meaning. The continuum model is only applicable as long as the scales we work ondo not enter this molecular domain.

Now the difference between intensive and extensive properties becomes important. We cannotspeak of the magnitude of an intensive quantity of a certain amount of mixture withoutreferring to its average. For example, the temperature of an amount of water is its average

A THEORETICAL FRAMEWORK FOR MASS TRANSFER

15

temperature. This is not so for extensive quantities: an amount of mixture represents a certainamount of the quantity, no matter how this may be distributed over the mixture. Because anintensive property is not related to the amount of substance, it is, within certain limits,independent of the scale. Imagine a point P in the mixture and a spherical volume Vsurrounding it. If the volume is large, the property considered varies throughout the volume,but as we take the average we fail to recognise them. If the volume is gradually reduced,possible variations become less and less important and the magnitude of the property willconverge to a definite value, see figure 1. We define this value to be the magnitude of thequantity at P. But then, as we venture into the molecular domain, the amount of mixturecontained in the volume becomes so small that the quantity’s behaviour becomes unpredictableas a result of the disruptive effect that single atoms or molecules may have on the averagevalue. The transition from the continuum region to the molecular region is located at volumescontaining, say, 106 molecules, to be on the safe side. Under standard conditions thiscorresponds to volumes of approximately 10−24 m3 in liquids and 10−21 m3 in gases.

10 log (V / m3 )

-30 -20 -10 00

intensiveextensive

Intensivequantity

at P

quantityvalue

Figure 1. The graph of an arbitrary extensive quantity and the corresponding volume-averagedintensive quantity as functions of the spherical volume V around a point P. The behaviour of the extensivequantity is roughly exponential, because it is approximately inversely proportional to V.

Conclusion: intensive properties are suited for the continuous description of mixtures andtherefore for field theories. On the other hand, extensive quantities are suited for describingbodies, which have a certain extent. So, to use extensive quantities as continuous properties, wemust transform them into intensive ones by taking their average over an amount of mixture.There are several ways to do this: we can divide the properties by the mass of the sample (so-called specific quantities), by the number of moles (molar quantities) or by the volume of thesample, and we will then call them (quantity) densities.

We denote the contribution of component i to the Q-density of a mixture by ρiQ. The total Q-

density of the solution is the sum of the contributions of all components: ρQ = ∑i ρiQ. Assume

that, on average, each molecule of i contributes an amount of υiQ to the total Q-content of the

Chapter 2

16

mixture. The density ρiQ then equals the molecular concentration of i times υi

Q: ρiQ = xi c A υi

Q

= xi ~ρi

Q , where A is Avogadro’s constant. The component-specific density ~ρiQ is composition-

dependent, and is called the partial molar Q-density of component i.

ρ ρ ρ ρQ Q Q Q Qx x= = = ⋅∑ ∑ ~ (6)

For the three most commonly used choices for Q, the (partial molar) densities are listed in table1.

Table 1. The (partial molar) densities for the three most common definitions of Q. The symbol cstands for the total concentration, and ρ for the total (mass) density of the medium, Mi is the molar massof component i, ci (= c xi) its concentration, and ~vi its partial molar volume.

Now back to the main subject of this section, frames of reference. Suppose that the averagevelocity of the molecules of species i with respect to the arbitrary co-ordinate system is ui. Thisvelocity will simply be called the velocity of i. The displacement of i results in a flux of Q, Ni

Q

N uiQ

i iQ= ρ (7)

The total flux of Q is the sum of the fluxes caused by each of the species

N N uQ Q Q= = ⋅∑ ρ (8)

Usually, the reference frame is not arbitrary, but connected with the medium in some definiteway. The most common frames are the volume-fixed, the mole-fixed, and the mass-fixedframe. In these frames no total flux exists of the quantity with respect to which they are fixed.Another frame which is sometimes encountered is the ‘solvent’-fixed frame, which isstationary relative to one of the components in the mixture. These frames are members of awhole family of possible frames. In our definition a frame is determined by a referencequantity q and a reference vector r. This so-called (r, q)-frame has a velocity ur,q relative to theoriginal, arbitrary, frame. The four common frames suggest that this reference velocity must besuch that

r r q q⋅ ≡N 0, , (9)

N r q q, , is the q-flux vector in the new (r, q)-frame, which, according to equation (7), equals

N u uir q q

iq

ir q i n, , , , ,...,= − ∈ρ d i l q1 (10)

The combination of the above two equations gives

Q symbol ρQ ρiQ ~ρi

Q xiQ unit

number of moles n c ci c xi mol m−3

mass m ρ Mi ci Mi c ωi kg m−3

volume V 1 ~vi ci~vi c εi 1

A THEORETICAL FRAMEWORK FOR MASS TRANSFER

17

uu

r qi i

qi

i

n

q

r

r, =

⋅

⋅=∑ ρ

ρ1 (11)

For the first three frames mentioned above – volume-fixed, mole-fixed and mass-fixed –, thereference vectors are the unit vector I, whose elements are 1, and the reference quantities are,little surprisingly, the volume, the number of moles and the mass, respectively. For the‘solvent’-fixed reference frame, the reference vector elements are zero except for the elementthat corresponds to the solvent component, which is 1. The reference quantity may be anyproperty.

The fluxes N r q Q, , of a quantity Q in the (r, q)-frame are, for reasons outlined in section 2.3.1,called the diffusive fluxes. These fluxes are, of course, defined by

N u uir q Q

iQ

ir q i n, , , , , ...,= − ∈ρ d i l q1 (12)

It is sometimes important to know what the relation is between two diffusive fluxes of differentquantities in different frames. Let us assume that for any two quantities Q and q there exists alinear transformation [Q<q] such that

ρ ρQ qQ q= < (13)

It is then easy to prove that this transformation also applies to the fluxes of Q and q relative to acommon (r, p)-reference frame:

N Nr p Q r p qQ q, , , ,= < (14)

By extending the derivation of Toor [4], we find that two arbitrary fluxes are linked by therelation

N NNr q Q s p P P

s p P

PQ P

r

rr q P r, , , ,

, ,

= −⋅

⋅

FHG

IKJ

=∗

∗∗

< <ρρ

withT

(15)

The above definitions and expressions will be used in the next section for a concise discussionof models for mass transfer.

2.3 TWO MASS TRANSFER MODELS

Mass transfer comprises two different kinds of processes: diffusive and convective transfer.Together, these two modes of transfer yield the total rate of transfer:

N NQ r q Q r q Q= +, , , ,ΦΦ (16)

where the last term is the vector of convective fluxes. The diffusive fluxes have already beenintroduced in the previous section. The convective mode of transfer is relatively easilyunderstood: it is a consequence of translation of (parts of) the medium as a whole, and the fluxis simply the product of its Q-density ρi

Q and the reference velocity ur,q of the medium.

Chapter 2

18

ΦΦ r q Q r q Q, , ,= u ρ (17)

Although a proper description of this kind of transfer may be extremely complex and difficultfor some cases such as, for instance, turbulent flow, this is never caused by a defectiveunderstanding of the theoretical basis of the transfer process. More likely is it a result of thebad times one can have in computing the flows in the medium. Diffusive mass transfer usuallyinduces fluxes that are small in comparison with convective transfer. This often causesdiffusive transfer to be rate limiting in many ‘installations’, both natural and industrial. Thetwo modes of transfer often act simultaneously, mostly with convection as the long-distancetransfer mechanism and diffusion as its close-range counterpart. This thesis is not mainlyconcerned with the subject of convective mass transfer, but in many cases, the two phenomenaare connected.

2.3.1 Fick’s transfer model

Mass transfer in single-phase media is an important phenomenon often encountered ineveryday and engineering practice, and an adequate model for its description is thereforehighly desired. Ever since usable theories about diffusive mass transfer came into being, Fick’slaw of diffusion has prevailed, and there seemed to be no sensible contestants. An explanationof this dominance lies in the deceptive simplicity of Fick’s law, which in its most-used formstates that the diffusive mole flux of a chemical species in a homogeneous binary mixturerelative to the mole-fixed reference frame is proportional to its spatial concentration gradient:

N I n n D c n, , ,= − ∇ = 2 (18)

The proportionality factor D is called the diffusion coefficient or diffusivity. For several reasonsthe use of the concentration gradient does not seem most appropriate, and there has been somediscussion over the question whether it would not be wiser to restate equation (18) as

N I n n c D x n, , ,= − ∇ = 2 (19)

which is now regarded as the better formulation. The two notations are identical only if thetotal concentration c does not depend on position.

Depending on the frame of reference and the flux-quantity considered, Fick’s law can takemany forms. Apart from equation (19) forms in common use are (cf. Toor [4], Cussler [1]):

N

N

I V n I V n

I m m I m m

D c

Dn

, , , ,

, , , ,,

= − ∇= − ∇

=ρ ω

2 (20)

In literature diffusive mole and mass fluxes are often indicated by the symbols J and j,respectively. This notation we will not adopt. It can be proven that for ideal solutions thebinary diffusion coefficients in the above equations are equal. In fact, for homogeneous mediathere is a binary version of Fick’s equation for every combination of reference frame and flux-quantity which all have the same diffusion coefficient. If, namely, equation (19) is substitutedin equation (15), we get

A THEORETICAL FRAMEWORK FOR MASS TRANSFER

19

N r q Q q

Q

qD r

rn, , ,= − ⋅ ∇

⋅

FHG

IKJ

=ρρ

ρe j 2 (21)

where it has also been assumed that both [Q<n] and [q<n] are constant with respect tocomposition. As a result, [Q<q] is also constant (it equals [Q<n] [q<n]−1), and so we may write

N r q Q Q

q

qD r

rn, , ,= − ⋅ ∇

⋅

FHG

IKJ

=ρρ

ρe j 2 (22)

As was mentioned above, for most frames the reference vector is the unit vector I. This turnsthe last equation into a simple form, as could have been guessed from equation (20).

N I q Q Q qD x n, , ,= − ∇ =ρ 2 (23)

So, the diffusion coefficient is the same for all forms, and it satisfies the equation

D x x x∇ = −1 1 2 2 1u ub g (24)

The fluxes appear to be functions only of the composition gradient and the velocity differenceof the two components, which are both independent of the reference frame. This implies thatconvective transfer plays no role, which makes these fluxes diffusive by definition.

The use of Fick’s law is not limited to binary systems. There is also a multicomponent version,which reads:

N NI n n I n nc D x c D y, , , ,$= − ∇ = − ∇[ ] or [ ] (25)

(For the meaning of the vectors ∇x and ∇y, see page 11.) The elements of the non-singularmatrix [D] are related to those of [ $D ] by Dij = $D ij − $D in, where n is, as before, the number ofthe dependent component, which is usually taken to be the last component n. The dimensionsof the square matrices [D] and [ $D ] are n −1 and n, respectively. Analogous to the binaryversion, many variants of this multicomponent law exist, depending on the reference frame andthe flux quantity, but here the diffusion coefficients are in general not the same. Cullinan [2]proved that these diffusivity matrices are similar – a term which is used here in themathematical sense – and can therefore always be transformed into one another. For moreinformation on this topic, see references [2] and [4]. As a result, all variants of Fick’s modelrequire a minimum of (n −1)2 diffusion coefficients.

Resuming, we notice that the most important features of Fick’s mass transfer model are that• it requires (n −1)2 coefficients.• it separates diffusive and convective transfer.Although this may not be clear at this point, there are two limitations to Fick’s law:• it cannot incorporate other driving forces than chemical potential gradients,• non-ideality effects of the solution are incorporated in the diffusivities.

Chapter 2

20

2.3.2 The Maxwell-Stefan transfer model

An alternative to Fick’s model is presented by what is currently known as the Maxwell-Stefanor Stefan-Maxwell model, depending on which part of the world one happens to live in. It hasbeen named after its ‘inventors’, who, in the good scientific traditions, seem to have developedit each on their own. The model takes the spatial gradient of the potential of a substance as itsdriving force. Further, it tells us that the opposing forces are caused by the velocity differencesof the species. All this is expressed in the formula for an n-component mixture

∇µ =−

= − ∈≠ ≠

∑ ∑i jj i

j i

ijij j

j ij iR T x

Ðx i n

u uu uζ d i l q, ,...,1 (26)

where the ui denote the same molar average velocities as in section 2.2.4. The coefficients Ðij

and ζij (= RT /Ðij) are called the Maxwell-Stefan diffusion coefficients and friction coefficients,respectively. Here, the form where only the chemical potential acts as the motive force isgiven. More forces can be added without difficulties, and a discussion of the span ofapplications, as well as of the exact form the equations then take is given elsewhere [3, 5].With subsequent use of the equations (7) and (16) these Maxwell-Stefan equations can also bewritten as

x

R T

x x

c Ð

x x

c Ði ni i i j

nj i

n

ijj i

i jn

jn

ijj i

∇µ=

−=

−∈

≠ ≠∑ ∑

N N N Nr,q, r,q,i , ,...,1l q (27)

This is the most common form, but other flux quantities could have been chosen just as well.Due to the Onsager equations (Ðij = Ðji), the sum of the right sides of all n transport equationsvanishes, regardless of the fluxes N r,q,n, and so we get x·∇µ = 0, an expression which is alsoknown as the Gibbs-Duhem equation. In other words, one of the n transport equations can bediscarded without any loss of information. The remaining equations can also neatly be mouldedin a matrix form:

c y R T i ni i i j jr q n

j

n

∇µ = ∈ −=

∑β ,, ,$ , ,...,N

1

1 1l q (28)

The hat indicates that we are dealing with the complete flux vector. The elements of the(n −1×n)-matrix [ β] are given by

β βi ji

iji i

j

ijj i

x

Ð

x

Ði n j i, ,, ,..., ,= = − ∈ − ≠

≠∑ if 1 1l q (29)

The chemical potentials of the various substances are µi = µio + RT ln (xi γi), but i ≠ n and

therefore xi = yi. So, yi∇µi = RT (yi∇ln γi + ∇yi), and substitution of the expression for ∇ln γi

yields

c y yy

r q n

i j ii

jijΓ Γ∇ = = +β

∂ γ∂

δ$ , ,

,N whereln

(30)

Our goal is to find a solution for the n fluxes. Unfortunately, the matrix [ β], being non-square,has no inverse. Everything depends upon how the fluxes are interrelated, and thus upon thereference frame. The lack of a sufficient number of equations is made up for by an equation

A THEORETICAL FRAMEWORK FOR MASS TRANSFER

21

relating the fluxes called the bootstrap equation. For now, let us assume that it is of the linearform†

ss

s

r q n

nr q n j

njr q n

j k

⋅ = ⇒ = −≠∑$ , , , , , ,N 0 N N (31)

where s may be any scalar vector. With this expression we can rewrite eq. (30) as

c y B Bs

si j nr q n

i j i jj

ni nΓ ∇ = = − ∈ −N , ,

, , , , ,...,with β β 1 1l q (32)

Note that the flux vector now has only n −1 elements: Nnr,q,n has disappeared. This is the set of

equations we need to solve the fluxes. It is also often written in a flux-explicit form

N r q n c G y G B, , = − ∇ = − −with

1 Γ (33)

The contents of this section can be summarised by stating that the Maxwell-Stefan model• needs only n (n −1) / 2 coefficients,• does not necessarily separate convection and diffusion,• can easily be extended to account for driving forces other than chemical potential

gradients,• does not lump ideality and diffusive effects.

2.3.3 Choose your weapon

So, there we are, having two models at our disposal which perform the same task: describingmass transfer. Which of the two should we choose? It is clear that if other forces than chemicalpotential gradients are involved, the Maxwell-Stefan model is the only possible choice. If suchforces are absent the models must be essentially equivalent. Indeed, the coefficients of onemodel can be transmuted into those of the other model [3]. Therefore, if we choose a specificmodel, we do so on account of its form rather than its function. Undeniably, the Maxwell-Stefan model is the more versatile model, but in practice most diffusivity data available todayare Fick diffusivities, because these can be measured directly, see also chapter 3. To obtain theMaxwell-Stefan diffusion coefficients, we must be acquainted with the thermodynamics of thesystem in order to compute the Γ-matrix. Nevertheless, this weakness of the model is also itsstrength: the fact that it separates the diffusivities from the thermodynamic behaviour facilitatesthe estimation of the coefficients at compositions at which they are not known, especially innon-ideal mixtures [3, 5]. Moreover, the Maxwell-Stefan diffusivities often appear to have amore fundamental meaning than the Fick diffusivities. Anyway, our choice for either model isdetermined by the type of problem and by the nature of the available data.

† Bootstrap equations by no means need to be linear: in chapter 6 we will encounter a nonlinear equation,

and we will see that this gravely complicates the solution of the fluxes.

Chapter 2

22

NOTATION

Note: symbols for local use (which occur in a definite meaning in only one passage) are notlisted here.

[B] (n −1×n −1) coefficient matrix of the Maxwell-Stefan equations (32), (s m−2)c concentration, (mol m−3)D Fick diffusion coefficient, (m2 s−1)Ð Maxwell-Stefan diffusion coefficient, (m2 s−1)[G] Maxwell-Stefan ‘diffusivity’ matrix, equation (33), (m2 s−1)I unit component vector, Ii = 1 ∀ i ∈ {1,..., n}M molar mass, (kg mol−1)n number of components, (–)n number of moles, (mol)N Q Q-flux, ((Q) m−2 s−1)N r,q,Q diffusive Q-flux relative to (r, q)-frame, ((Q) m−2 s−1)R gas constant, (J mol−1 K−1)T absolute temperature, (K)Q<q transformation mapping a quantity q onto a quantity Q, ((Q) (q)−1)ur,q reference velocity of (r, q)-frame of reference, (m s−1)u (mole-based) average component velocity, (m s−1)~v partial molar volume, (m3 mol−1)x mole fraction, (–)xQ Q-fraction, (–)y mole fraction, (–)

Greek symbols[Γ] non-ideality matrix, (–)δ Kronecker deltaγ activity coefficientµ chemical potential, (J mol−1)ρQ Q-density, ((Q) m−3)~ρQ partial molar Q-density, ((Q) m−3)

ω mass fraction (–)ΦΦr,q,Q convective Q-flux relative to (r, q)-frame, ((Q) m−2 s−1)ζ friction coefficient, (kg mol−1 s−1)

Subscriptsi component number ij component number jij component pair i, jn last component, number nT, P at constant temperature and pressure

A THEORETICAL FRAMEWORK FOR MASS TRANSFER

23

Superscriptsm massn (number of) moles, or amounto pure componentV volumeT transpose of a matrix

REFERENCES

[1] Cussler E.L.‘Diffusion. Mass transfer in fluid systems.’Cambridge University Press, Cambridge, UK, 1984.

[2] Cullinan H.T.‘Analysis of the flux equations of multicomponent diffusion.’Ind. Eng. Chem. Fundam., 4(2), 1965, pp 133–139.

[3] Taylor R., Krishna R.‘Multicomponent mass transfer.’Wiley, New York, USA, 1993.

[4] Toor H.L.‘Reference frames in diffusion.’AIChE J., 8(4), 1962, pp 561.

[5] Wesselingh J.A., Krishna R.‘Mass transfer.’Ellis Horwood, Chicester, U.K., 1990.

TAYLOR DISPERSION THEORY

25

3. TAYLOR DISPERSION THEORY

In chapter 1 the important role of diffusion in liquid-liquid extraction has been pointed out. Fora quantitative description of a ternary extraction, we need the diffusion coefficients of thesystem. Measuring diffusion coefficients used to be, and still is, a rather time-consumingoccupation. There are two methods in common use: the diaphragm cell method [13] and theTaylor dispersion method. This chapter deals with the latter, which is named after G.I.Taylor,who, in the 1950’s, published a number of articles on the dispersion in fluid media flowingthrough a tube [15, 16]. Although it has taken a while before the principles he discussed wereactually applied, the method is now recognised as a fast and reliable way to measurediffusivities in homogeneous fluids.

This chapter consists of two parts. To start with, we look at the theory of the dispersiveprocess, but not in too great a detail, because this has been done by others. Instead, we will payspecial attention to the design of the dispersion tube. In the second part, we will see how toobtain multicomponent diffusivities from dispersion measurements.

3.1 THE TAYLOR DISPERSION EXPERIMENT

To decide what equipment exactly is needed, and how it must be designed, we must beacquainted with the theory behind the processes on which the measurements are based. Thuswe will briefly discuss this theory, and then some design rules.

3.1.1 Theory of the dispersive process

When a fluid flows through a tube, a velocity profile develops. This profile is not uniform overthe cross-section of the tube: at some radial positions, the fluid flows faster than at others. Thelaminar profile for Newtonian fluids has a parabolic velocity distribution. Matter present in thefluid moves along with it and is therefore dispersed along the tube axis. Apart from this purelyconvective dispersion, diffusion can also be of influence.

Suppose that somewhere in the tube we have a small ‘plug’ of fluid with a compositiondifferent from an otherwise similar bulk of fluid. The flow of the fluid causes this plug todisperse, but it also induces radial composition gradients. This leads to diffusive fluxes at thefront and back sides of the plug. These fluxes become important if the radial diffusive fluxesare roughly of the same order of magnitude as the convective axial fluxes. This is the casewhen either the axial velocity is very low, or when the radial distances are very small. Wheneither condition is satisfied, diffusion tends to keep the plug together, in contrast to what onemight intuitively expect. So, through the combined action of convection and diffusion, the plugwill leave the tube as a broadened, but still more or less compact plug. To achieve thissituation, Taylor dispersion experiments are usually carried out in tubes with small diameters,also called capillaries.

Chapter 3

26

Taylor considers the dispersion of a Dirac-pulse of a tracer fluid in a laminar flow. The tube iscircular with a diameter d, and the bulk fluid has a superficial velocity usup. The distributionafter t seconds is given by

∆ c z tn

d t

z u t

t11

2

2

2

4,b g d i

= −−F

HGG

I

KJJ

Esup

expπ π κ κ

(1)

Here, ∆c1 is the concentration of component 1 (the ‘tracer’†) minus its concentration in theeluent, n1

E is the excess number of tracer moles in the pulse, κ is the dispersion coefficient, andz is the down-stream distance from the initial location of the Dirac-pulse. The approximateexpression for κ given by Taylor [15] is

κ =u d

Dsup2 2

192(2)

where D is the molecular Fick diffusivity of the tracer with respect to the eluent. Aris [2] showedthat the exact solution is

κ = +u d

DDsup

2 2

192(3)

Later, Price [12] derived expressions for the concentration distributions for three componentsin a way analogous to Taylor, and which are therefore not exact.

∆

∆

c z tK

d t

z u t

t

K

d t

z u t

t

c z tK

d t

z u t

t

K

d t

z u t

t

11

21

2

1

22

2

2

2

23

21

2

1

42

2

2

2

2

4

2

4

2

4

2

4

,

,

b g d i d i

b g d i d i

= −−F

HGG

I

KJJ + −

−F

HGG

I

KJJ

= −−F

HGG

I

KJJ + −

−F

HGG

I

KJJ

π π κ κ π π κ κ

π π κ κ π π κ κ

exp exp

exp exp

sup sup

sup sup

(4)

with

κ κ κ κ κ

κ κ

κ κ

i i i

iu d

DD D

KD n D n

DK

D n D n

D

KD n D n

DK

D n D n

D

= = = + −

=− −

= −− −

=− −

= −− −

+0 0

2 2

12

1

1

2 2 2 1 1 2 2

2

2 2 1 1 1 2 2

3

1 1 2 2 2 1 1

4

1 1 1 2 2 1 1

1921* *

,*

, ,*

,

,*

, ,*

,

,

,

,

where tr disc

disc disc

disc disc

sup

E E E E

E E E E

b ge jd i d i

d i d i

(5)

† Strictly speaking, components 1 and 2 are not interchangeable except when the total number of excess

moles in the pulse is zero, that is, when the partial molar volumes are equal and constant. But this is aprerequisite for the derivation of eq. (1) anyway.

TAYLOR DISPERSION THEORY

27

So, by monitoring the concentration distribution we are able to determine the dispersioncoefficient(s).

3.1.2 The capillary: coiled or straight?

To obtain a concentration distribution at the end of the capillary which has spread enough toallow us to extract the value of the diffusivity with a reasonable accuracy, the dispersiveprocess must have had enough time to do its work. For several reasons outlined in this section,we cannot decrease the fluid flow at will, and therefore the capillary must have a definitelength, which turns out to be about 20 m. This makes it desirable to coil the capillary: thisconfiguration saves space, makes the capillary less vulnerable, shortens data communicationlines, and enables better temperature control. But can we do it without any repercussions?

Arguments for a straight capillary are of a theoretical nature: the solutions of the mathematicalequations governing the Taylor dispersion dynamics are simpler and require fewersimplifications in the case of a straight capillary. These solutions may therefore be assumed tobe more accurate, and also fitting these solutions to measured RTD curves favours simpleequations.

Usually, the best of two worlds is taken: the capillary is coiled but at the same time it isassumed to be mathematically straight. Under what circumstances are we allowed to do this?Taylor dispersion is usually carried out in a coiled capillary, the coil diameter δ being of theorder of several decimetres, about 4 dm† say. Although the measured diffusivities are probablyaccurate enough, literature does not provide a firm theoretical basis for neglecting the curvatureof the capillary. At least, it is impossible to satisfy all criteria for a Taylor dispersion capillaryfor all possible system parameters without falling out of the range of reasonable dimensionsand experimental conditions. These criteria are:

1. The flow within the capillary should be laminar, so

Re = <4

2000ΦV ρ

π ηd(6)

2. The volume of the injected sample Vinj must be very small compared with the capillary volumeVcap [13]. This gives the sample enough volume to disperse in, and allows us to ignore the initialshape of the pulse.

V

Vcap

inj

> 100 (7)

3. This criterion originates from the theory of Taylor dispersion [1, 2, 15, 16]. The cross-sectionaveraged second moment of the concentration distribution in the capillary, m2 , is given by

† Rutten [13] used a capillary with an inner diameter of 0.53 mm and a coil diameter of 0.40 m, for Snijder's

[14] capillary these measures were 0.56 mm and 0.1 m, and v.d. Ven [21] uses a capillary of similardimensions.

Chapter 3

28

m Du d

Dt

u d

D

D t

dii

i2

2 2 2 4

28

2

21

2192

8 14

= +FHG

IKJ − −

FHG

IKJ

LNMM

OQPP

−

=

∞

∑sup sup expαα

(8)

In this expression, α i represents the ith zero of the derivative of J0, the zeroth-order Besselfunction of the first kind. Although not strictly necessary, it is convenient to simplify the first termon the right-hand side: in most cases the diffusivity D will be much smaller than usup

2d 2/ 192 D,and when it is negligibly small, it can be omitted. However, under what conditions do we say thatone term is negligibly small compared to another term? Let us say that a positive number a isnegligible compared to another positive number b if a < ε b, ε being some small number between 0and 1, 0.01 for instance.

With the definition of the radial Péclet number Pe (= usup d / D) and with usup = 4 ΦV / π d 2, we find

Du d

D

u d

DPe+ = >sup sup if

2 2 2 22

192 192 1921

ε(9)

4. This condition also follows from equation (8). Note that the summation function is quiteimpractical for curve-fitting, and it would be convenient if it could be eliminated. First theexponential terms get in our way, but it turns out that all of them are usually much smaller than 1.Of all α i, α 1 has the smallest value: α1 ≈ 3.83, α 2 ≈ 7.01 etc. Hence, we only have to considerwhether the term containing this first root is negligible, because if it is, so are all the other terms.This means according to our negligibility definition:

Dt

d 2124

> −ln εα

(10)

Now, equation (8) reduces to

mu d t

D

u d

DA2

2 2 2 4

2968= −sup sup (11)

where A represents the sum Σ α i−8, which has a value of 2.1701...⋅10−5. The second term of this

expression can be ignored if

εε ε

u d t

D

u d

DA

Dt

d

Asup sup or2 2 2 4

2 2

2

96

8 768 167 10> > ≈

⋅ −.(12)

Both conditions (10) and (12) concern the same quantity D t /d 2, called the Fourier number Fo. Itturns out that whenever (12) is satisfied, so is (10), which is therefore a redundant criterion.

What is the meaning of t in eq. (12)? Strictly speaking, t is the time after injection of the sample. Arepresentative value of t is the mean residence time τ of the sample:

τπ

=L d 2

4 ΦV

(13)

TAYLOR DISPERSION THEORY

29

If this expression is inserted in (12) for t, then this equivalent to the statement that this criterionholds at t = τ, and it will be assumed that it still holds when the last traces of the sample leave thecapillary. The resulting inequality is

D L AπεΦV

>3072

(14)

5. This condition is a rather prosaic one, for it is imposed by the limitations of the HPLC-pumpused for pumping the fluid through the capillary. The maximum pressure drop ∆ Pmax such pumpsusually can generate is about 400 bar, and according to Poiseuille's law, the parameters must bechosen to satisfy the inequality

∆Φ

∆PL

dP= <

1284

Vmax

ηπ

(15)

6. The criteria discussed so far concern straight capillaries, and, apart from numbers 3 and 4, arenot likely to cause too many problems. The next criterion should take care that coiling does notlead to significant deviations of the measured diffusivities. Intuitively, one might argue that coilingthe capillary leads to a secondary flow perpendicular to the tube axis. The extra radial mixingdiminishes radial concentration gradients. Radial transfer in straight capillaries is a diffusiveprocess only, and therefore we would expect convection due to coiling to increase the radialtransport. However, it will be seen that coiling also influences the axial profile, and can evenreduce the dispersion coefficient.

Approximate solutions of the equations describing simultaneous convection and diffusion incurved pipes have been found by Erdogan and Chatwin [8], and Nunge, Lin and Gill [10]. In bothpapers power series expansions in the coil-capillary diameter ratio λ are used. In the first paper theauthors find the following expression for the dispersion coefficient κ, which is based on thevelocity profiles derived by Dean [6, 7]

κ λEC = + − +−Pe D a a a Sc a21 2

2 43

24Re d i (16)

Here the constants a1 to a4 have the following values

a a a a1 2 2 3 4

1

192

1

576 40

2569

15840

1109

43200= = = =; ; ; (17)

In the straight capillary case, that is, as λ → ∞, κEC → D Pe2 a1, which is the approximation foundby Taylor [15]. This indicates that the approximation is not accurate, since the limit should yieldthe Aris solution for straight tubes.

In the article of Nunge, Lin and Gill, equation (16) was extended to

κ κ λ κNLG = + −Pe D220

222d i (18)

with

Chapter 3

30

κ

κ κ

20 2 1

22 24

32

4 52

6 7 8 9 20

1= +

= − + + − + +

Pea

a a Sc a a a Sc a a a

r

Re Red i b g(19)

The constants a1,..., a4 have the same values as before, and

a a a a a5 6 7 8 9

1

41472

31

60

8499

4480

419

11520

1

4= = = = =; ; ; ; (20)

Figure 1 shows a plot of the deviation of κ NLG relative to the Aris solution,

∆ NLGNLG A

A

=−

=+

−κ κκ

λ κ2 222

21 1

Pe

Pe a(21)

We see that• The straight-tube limit of equation (18) is the Aris solution, as expected.• For high Re-numbers both dispersion coefficients κEC and κNLG tend to the same limit.• For normal liquids (Sc ≈ 103), an increase of the dispersion coefficient due to coiling is found

only at very low Re-numbers.• For higher Sc-values very large deviations may be found, even at moderate Re-numbers. To

keep the relative error smaller than 1% may require an extremely high λ.

Although both equations predict that the dispersion coefficient increases only for Re-, Sc- and λ-values within definite ranges, there are notable differences, and equation (18) is probably the betterapproximation. The criterion for coiling then becomes

ε λ κκ

220

22

1≥ (22)

Re (-)

0.0 2.5 5.0 7.5 10.0

λ2 ∆NLG (-)

-10

0

10

101001000

Sc (-)

Figure 1. The deviation of the dispersion coefficient according to Nunge, Lin and Gill for several Sc-numbers.

TAYLOR DISPERSION THEORY

31

7. A last criterion is that the sampling time must be long enough for accurate measurements. Wewill choose the minimum sampling time (tm) as 120 s. After having injected a Dirac-pulse in thefluid stream at the beginning of our capillary at t = 0, we stroll over to the other end of it and waitfor the pulse to come out. So, our position is fixed (z = L), and the remaining variable is the time t.The response of the system is given by equation (1), which is a Gauss function with respect to t.The ‘standard deviation’ σ of this function equals (2 κ t)½. Since the peak width interval is chosenarbitrarily, there is no need for excessively scrutinous consideration of which κ should be used, andhere it is taken to be the Aris approximation. The peak width of a response curve is usually definedas 4 σ [3], and the seventh criterion then becomes

tt

um = >4 2

120κ

sup

(23)

As stated above, the criteria 1, 2, and 5, which concern straight tubes and are not related to ε, areusually easily satisfied. The additional peak width criterion may not be met in all practical cases,but this is a problem we can live with. The most severe troubles can be caused by the criteria 3, 4and 6. For a given configuration these criteria can be written in the form ε > f (ΦV). In figure 2these three functions are plotted for a capillary with dimensions L = 20 m, d = 0.53 mm, andδ = 0.80 m. The relevant physical fluid parameters were taken as D = 5·10-9 m2 s−1, ρ = 1.5·103

kg m−3 and η = 2·10−4 Pa s. It is clear that the highest accuracy is obtained at a flow rate ofabout 1.5·10−9 m3 s−1.

109 ΦV (m3 s-1)

0.0 0.5 1.0 1.5 2.0

103 ε (-)

0.0

0.2

0.4

0.6

0.8

1.0

346

criterion #

bestaccuracy

Figure 2. The accuracy ε derived from the criteria 3, 4 and 6 as a function of the volume flow. Theequipment parameters are: L = 20 m, d = 0.53 mm, and δ = 0.80 m, and the physical fluid properties areD = 5·10−9 m2 s−1, ρ = 103 kg m−3 and η = 2⋅10−4 Pa s.

In table 1 the best obtainable accuracy of the Taylor dispersion method is listed for several sets ofphysical properties of the fluid. The best accuracy is obtained at a specific flow rate, which may,however, be too low for practical applications. Therefore, the accuracy is also computed for a moreconvenient flow rate of 2.5⋅10−9 m3 s−1 (= 0.15 ml min−1).

Chapter 3

32

Table 1. The best possible accuracy with the Taylor dispersion method for several sets of systemparameters, the corresponding flow rate, and the criteria by which this ε-minimum is determined.Capillary dimensions: L = 20 m, d = 0.53 mm, and δ = 0.80 m.

We see that, although the minimum accuracy is quite good in all cases, it is not always obtained atflow rates which allow quick measurements: τ is about 8 hours at a flow rate of 0.16⋅10−9 m3 s−1.But it appears that the accuracy deteriorates dramatically in some cases if we choose a moreconvenient flow. Thus, for mixtures with extreme properties, it is necessary to have a look at thecriteria before any measurements are carried out.

Extreme conditions as those of the last two rows of table 1 are not likely to occur, because lowviscosity is usually accompanied by high diffusivities, and only in the case of mixtures of largeentities such as macromolecules and low-viscous fluids, are problems to be expected.

With the above set of rules, it is no longer necessary to work with rules-of-thumb, as mostexperimenters seem to do. Rutten [13] does not explicitly derive his coiling criteria (λ > 100,Re < 50), although he claims them to be based on work done by Tijssen [17, 18, 20], Nunge et al.[10], and Erdogan and Chatwin [8]. Snijder [14] uses the results of Janssen [9] to justify his choiceof capillary dimensions. Janssen found – on the basis of numerical simulations – that ’for values ofDe2Sc < 100 there will be no significant difference with the straight tube’, whenever λ > 20. Whatis meant exactly with significant is not completely clear, but it probably refers to relative errors ofup to about 3%, which is much more than the minimum allowable deviation of 0.1% used here. Ifthe more restrictive condition De2Sc < 50 is used, the relative deviation shrinks to less than 0.5 %,which is in better accordance with the criterion given by Alizadeh et al. [1]: De2Sc < 20 andλ > 100.

Evidence supporting the notion that coiling effects cannot always be ignored was found by Chenand Blanchard [4, 5], who measured diffusivities of macromolecular substances (PS) in an organicsolvent (THF), and who found significant deviations from Taylor dispersion theory. Tijssen [19]showed that these deviations can probably be ascribed to the small curvature of the tube used in theexperiments.

physical fluid properties best accuracy ΦV = 2.5·10−9 m3 s−1

10−3 ρ(kg m−3)

103 η(Pa s)

109 D(m2 s−1)

104 ε(–)

109 ΦV

(m3⋅s−1)criteria 104 ε

(–)criterion

1.5 200 5 3.3 1.6 3, 4 5.3 40.6 200 5 3.3 1.6 3, 4 5.3 41.5 0.2 5 3.7 1.5 3, 6 30 60.6 0.2 5 3.3 1.6 3, 4 5.3 41.5 200 0.5 3.3 0.16 3, 4 53 40.6 200 0.5 3.3 0.16 3, 4 53 41.5 0.2 0.5 3.3 0.16 3, 4 3.0·103 60.6 0.2 0.5 3.3 0.16 3, 4 4.8·102 6

TAYLOR DISPERSION THEORY

33

3.2 RETRIEVAL OF DIFFUSIVITIES FROM EXPERIMENTAL DATA

We now know how to conduct an experiment, and what to expect from the data it produces.The final step is to extract diffusivities from the data. To this aim, eqs. (1) or (4) are fitted tothe experimental data. The following section treats the problems connected with this retrievalof the diffusivities from the experimental data.

The extraction of the diffusivities from set of measured composition profiles is not always as easilydone as it is said. Problems can arise due to unruliness of the model functions as well as of thedetection method. First, the functions. We wish to fit the functions (1) and (4) to signals measuredat the end of the capillary, that is, at z = L. We will use the functions in the mole fraction form: themole fraction is simpler to determine than the concentration, especially at the moment that thesolution is prepared. The equations then become

∆ y tb

tk

t

tb

n

c dk L1

2

13 2 2

12

21 2b g b g c h= −−F

HGIKJ

= =κ

τ

κ πexp with and

E

(24)

or

∆

∆

y tb

tk

t

t

b

tk

t

t

y tb

tk

t

t

b

tk

t

t

bK

c db

K

c db

K

c db

K

c d

11

1

2

1

2

2

2

2

23

1

2

1

4

2

2

2

11

3 2 2 22

3 2 2 33

3 2 2 44

3 2 2

1 1

1 1

2 2 2 2

b g b g b g

b g b g b g

= −−F

HGIKJ

+ −−F

HGIKJ

= −−F

HGIKJ

+ −−F

HGIKJ

= = = =

κ

τ

κ κ

τ

κ

κ

τ

κ κ

τ

κ

π π π π

exp exp

exp exp

, , ,

(25)

It will be shown below that there is little wrong with the first equation from a fitting point of view,but the similarity of the two component profiles in the second one will cause trouble. The right-hand members of these eqs. require the total concentration c, but because n E / c = y E Vinj

everything can be defined in terms of mole fractions. Vinj is a quantity that must be determinedanyway for the evaluation of the injected excess amounts of the independent components.

The detection of the concentration-time functions may give problems as well. A good measuringdevice is linear, and does not deform or delay the output signal. That is, if f ( t ) is the output of theapparatus, we have

f t S y t s r tb g b g b g= + + (26)

Ideally, the matrix [S ] of the detector is independent of y, and in even more ideal equipment, it isalso diagonal. In this section we will assume [S ] is constant for the composition ranges considered.The constant s is the offset of the detector, and it would be nice if it equals o but we shall seebelow that it is no problem if it is not. The function r (t) is the (white) noise function of themeasuring apparatus. The number of independent outputs, the dimension of f, is bound to amaximum of n − 1, for we cannot obtain more information from the n − 1 independent molefractions yi.

Chapter 3

34

Measuring equipment can exhibit baseline drift, and eq. (26) should also include a drift function. Ifthe drift is not too serious, it does not render an experiment useless. Any moderate, well-behavedfunction can be accurately described with a polynomial, and so, assuming the drift function is apolynomial, we get

f t S y t s r t a tjj

j

m

b g b g b g= + + +=∑

0

(27)

We can, of course, try to fit the baseline drift simultaneously with the other parameters. Because ofthe increased number of parameters, this slows down the fitting process, but above all it mayshrink the domain of convergence. This is the set of all combinations of initial parameterestimates for which the fitting routine† converges to a solution. Luckily, there is a simple andoften effective alternative, namely ‘manual’ pre-fit baseline correction. Given the functions (24)and (25), we can see that ∆ y (t) → 0 as | t | → ∞, so

f t s r t a t tjj

j

m

b g b g→ + + → ∞=∑

0

as (28)

We now make a selection of data points at both sides of the peak that clearly belong to the baselineof the curve. They represent measurements of the eluent composition. To these selected data we fita polynomial of a certain degree. (A degree of three will do perfectly in most cases.) Thispolynomial is then subtracted from all points in the data set. For moderate forms of baseline drift,this method gives us a neatly zeroed baseline. Apart from this, it also frees us of the constants s(offset) and [S ] ye

‡ (the eluent signal). These are incorporated in the constants a0, but this alsohappens when the baseline drift is fitted along with the signal, and that is why the offset of thedetector is not important. The resulting signal f ( t ), which can be used to fit the equations (24) and(25), is

f t S y t r tb g b g b g= +∆ (29)

This baseline correction method has one possible disadvantage. The drift function is only fitted to alimited selection of the data set. Any irregularities in the baseline in the part of the signal that doesnot belong to this selection are not ‘seen’ and remain present in the corrected data set. This is whyit would be better to fit the baseline along with the other parameters, had it not been for thedrawbacks mentioned above. Further, ‘manual’ baseline correction requires that the data containenough baseline points††, otherwise it is not possible to determine the correction polynomial withsufficient accuracy.

Do we need to know the linearity matrix [S ] to be able to fit eqs. (24) and (25)? We do not,although it may make things easier if we do. In the binary case, substitution of eq. (24) into (26)yields

† The Marquardt-Levenberg method [11] was used for all parameter optimisations.‡ As was mentioned at the beginning of this chapter, ∆c = c – ce. Since c is assumed constant, this is

equivalent to ∆y = y – ye. Hence: [S ] y(t) = [S ] ∆y(t) + [S ] ye.†† A good fit always requires a well-defined baseline, so normally this constraint should be fulfilled

anyway.

TAYLOR DISPERSION THEORY

35

f tS b

tk

t

tb g b g

= −−F

HGIKJ

1 1

21,

κ

τ

κexp (30)

So, in principle S1,1 b, D and τ can be determined from a single output data set. In this case practicefollows theory, and usually, the lumped parameter S1,1 b and the other two parameters can bedetermined from just one set of data with reasonable accuracy. Only if S1,1 is known, it is possibleto evaluate n1

E, but the injected excess amount of a substance is a less relevant parameter anyway.Besides, it can be determined beforehand while preparing the samples. If one really wants to fitn1

E, two experiments at different sample compositions are required, and eq. (24) must be fittedsimultaneously to the two resulting data sets.

In the ternary case things are more complicated, because of the similarity of ∆ y1 ( t ) and ∆ y2 ( t ).With three components, f can have two independent elements at most, but many measurementdevices have only one output (e.g. refractive index at one wave length, or conductivity). Theconstants S2,1 and S2,2 then are zero, and the output becomes

f tS b S b

tk

t

t

S b S b

tk

t

t11 1 1 1 2 3

1

2

1

1 1 2 1 2 4

2

2

2

1 1b g b g b g=

+−

−FHG

IKJ

++

−−F

HGIKJ

, , , ,

κ

τ

κ κ

τ

κexp exp (31)

The fit of a single data set gives values for (S1,1 b1 + S1,2 b3), (S1,1 b2 + S1,2 b4), κ1, κ2 and τ. Onlyunder certain conditions is one data set sufficient to fit all parameters. In most cases however, wewill need more than one set of experimental data. How many more? This depends on1. whether we know the excess number of moles n E for each experiment,2. whether the linearity [S ] is known3. the number of outputs of the detector.Suppose that we have a number of m experiments at constant eluent composition. Theparameters [D ] and [S ] are the same for all m data sets, while τ may differ, and n E certainlywill. The diffusivities are unknown by definition, whereas the mean residence time is a fitparameter for the sake of accuracy†. Table 2 contains an inventory of the number of unknownsfor all possible situations that can be encountered.

Table 2. The number of unknowns and equations for all possible sets of experimental conditions. Thenumbers in parentheses are the numbers of experiments needed for the determination of the parameters.The numbers in this table are valid for ternary dispersion only.

† Note that a varying τ implies that either L is not correct or usup deviates. Since L can be measured

accurately, it must be usup that potentially contains a significant error. Therefore, we should useusup = L / τ in eqs. (3) and (4), which makes κ a function of both D and τ.

detector # # unknowns #

calibrated outputs n E unknown n E known equationsYes 1 4 + 3 m (–) 4 + m (1) 2 + 3 mYes 2 4 + 3 m (1) 4 + m (½)* 2 + 5 mNo 1 6 + 3 m (–) 6 + m (2) 2 + 3 mNo 2 8 + 3 m (3) 8 + m (1½)* 2 + 5 m

* In the case of two outputs, half an experiment corresponds to one peak.

Chapter 3

36

The minimum number of unknowns is 4 + m: the four D’s and m τ’s. For an uncalibrateddetector this number must be increased by the number of unknown linearity factors, namely 2(single output) or 4 (dual output). The n E make for 2 m extra variables if they are not known.The simultaneous fit of the data set from a single-output detector yields a number of 2 + 3 mequations: namely values for κ1 and κ2, which are the same for all sets, and 3 m values for τand the composite parameters (S1,1 b1 + S1,2 b3) and (S1,1 b2 + S1,2 b4). Dual-output detectorssupply us with two peaks per experiment and therefore we have 5 m + 2 equations.

The table shows that with a single-output measurement apparatus it is impossible to find thevalues of all parameters unless we know n E. This is because every data set introduces as manynew unknowns as extra equations. With a dual-output device it is possible to fit all possibleparameters, which is why two outputs are to be preferred to one output. Thus, one peak sufficesto determine all 5 parameters with such a measuring device if the n E are known and if it iscalibrated. If it is not calibrated, we need three peaks. This may seem a bit strange, but thissituation is not essentially different from the single-output case with known n E, and we needtwo peaks from, say, the first output to determine all parameters except S2,1 and S2,2. For thedetermination of these two parameters, we need a third data set from the second output.

If, as has been said above, it is easy to determine the n E, then why would we like to fit them?This is because possible experimental errors in n E – or in [S ] if we choose to calibrate ourdetector – can then ‘move into’ the fitted parameters, of which the D’s are the ones that reallymatter. This is also the reason why the mean residence time of a sample is taken as a fitparameter, even though it can be determined experimentally with good accuracy. The fact that thisintroduces extra parameters does not weigh heavily, since τ usually can be fitted very accurately,because it is almost independent of other parameters. Moreover, it is possible to make good initialguesses, either from the flow rate and the capillary volume, or from the output data‡.

It must be realised that the numbers of experiments listed in table 2 are an absolute minimum.More data sets usually give better fits, and tend to increase the stability and enhance the domain ofconvergence of the fitting routine. As an example consider the case of a calibrated detector withone output. Suppose that we know n E, then according to the table only one experiment should do.To verify this, we use constructed data sets instead of measured ones so as to be sure about whatthe exact values of all parameters are. Let L = 19.0 m, d = 5.3·10−4 m, usup = 1.0·10−2 m s−1,c = 1.33·104 mol m−3, D1,1 = 3.0·10−9, D1,2 = 5.0·10−10, D2,1 = 1.3·10−9, D2,2 = 5.0·10−9 m2 s−1,S1,1 = 2.0, S1,2 = −1.0 V and s = o V. Artificial peaks were constructed with eq. (31) for 250 pointson the time interval [1.6, 2.2] ks (i.e., every 2.4 s). Noise was added according to

r t x t x trb g b gc h b g= −σ πcos ln2 2 (32)

which is said to give normally distributed random numbers with a variance of σr2 if x(t) is a

uniformly distributed random number on [0, 1⟩. The responses to the following samples werecalculated

‡ τ is located approximately at the (middle) extremum of the signal, but not exactly: if t0 is the position of

this extremum, then to a good approximation we have (for binary dispersion): τ – t0 ≈ κ / u2sup.

TAYLOR DISPERSION THEORY

37

I. (n E)T = (5.0 9.0)·10−7 mol, andII. (n E)T = (8.0 −2.0)·10−7 mol.The noiseless composition-time distributions of these two samples in the absence of baseline driftare shown in the figure 3.

t (ks)

1.6 1.8 2.0 2.2-6e-5

0e+0

6e-5

-6e-4

0e+0

6e-4

f (V)

II

I

Figure 3. The simulated ideal responses of the two samples. Notice that the response of pulse II has amaximum amplitude about 13 times larger than that of pulse I.

Although these distributions contain the same information about [D ], they are different when itcomes to parameter fitting. This is best illustrated by the actual fit results.

Table 3. One-peak fit results for both peaks I and II for varying σr. Calibrated detector, samplecompositions known. The diffusivities Dij are given in 109 m2 s−1, the estimated standard errors (ESE’s)∆Dij are given as a percentage of the fitted value of Dij . The τ’s, which are not shown, fitted very wellunder all circumstances – not only with respect to their magnitude, but also to their ESE –, a fact that hasalready been mentioned above. The enormous ESE’s are no mistakes!

The first thing that meets the eye, is the inaccuracy of the fits, but we will come to discuss that inthe next paragraph. What is of importance here, is that peak I is less sensitive to noise, that is, itretains the information about the diffusivities better than peak II if the noise gets worse. This isespecially so if it is realised that for a given σr, the signal-to-noise ratio is much smaller for peak Ithan for peak II. Not only do the fits of peak I give more accurate values of the Dij, but also are the

Sample # σr (V) D1,1 ± ∆D1,1 D1,2 ± ∆D1,2 D2,1 ± ∆D2,1 D2,2 ± ∆D2,2

10−10 2.91 ± 4% 0.494 ± 3% 0.932 ± 6·101% 5.02 ± 2%I 10−6 2.83 ± 6·104% 0.593 ± 2·104% 0.835 ± 8·105% 5.26 ± 3·104%

10−5 2.62 ± 3·105% 1.45 ± 2·105% 1.39 ± 2·106% 6.50 ± 1·105%

10−10 2.78 ± 2·101% 0.153 ± 7·102% 1.09 ± 3·101% 5.22 ± 9%II 10−6 2.77 ± 3·104% 0.208 ± 1·107% 1.28 ± 3·104% 6.15 ± 1·104%

10−5 2.77 ± 1·106% 0.485 ± 3·107% 7.80 ± 4·104% 31.5 ± 1·105%

Chapter 3

38

estimated standard errors (ESE’s) smaller on average. Moreover, it appears that the binarydispersion function (30) can also be fitted well to peak II, which indicates that the information onthe two superpositioned peaks is ‘weak’. Therefore, it is worthwhile to choose the samplecompositions such, that the recorder signal has a peak I appearance, but this may not always bepossible.

Of the diffusivities listed in the table, only those at σr-values of 10−10 V are usable. For the highernoise levels, the ESE’s are so big that the values of the diffusivities are statistically insignificant.That the fitted values of the Dij are not too far off, is a result of the ideal peaks: apart from the(almost) perfectly white noise there are no disturbing influences. For the example peaks, themaximum signal-to-noise ratios at a σr of 10−10 V are as high as 4·105 (peak I) and 5·106 (peak II).Since such levels can not be expected in practice, the conclusion must be that one peak simply isnot enough to determine the diffusivities with low errors. Fits to two peaks simultaneously, on theother hand, give perfect results with low ESE’s. This is illustrated in table 4. If the peaks are good,a fit of more than two peaks gives only slight improvement.

Table 4. Two-peak fit results for both peaks I and II for varying σr. Calibrated detector, samplecompositions known.

But there is more! From table 2, we see that we need two data sets to fit the diffusivities for anuncalibrated detector. With the result for the calibrated detector in mind, we might expect that wemay need more than two sets, but this is not so. The linearity [S ] can be fitted together with [D ]and the two τ’s without apparent loss of accuracy of the fitted value of [D ]. (In fact, it slightlyimproves them.) Thus we kill two birds with one stone: we avoid the work of calibration, and wekeep errors in the experimentally determined values of [S ] out of our other parameters.

3.3 CONCLUSIONS

The first conclusion that can be drawn from the foregoing is that for normal substances, no specialprecautions need to be taken for diffusivity measurements with the Taylor dispersion method. Fornormal capillary dimensions, the coil diameter can be relatively small, and flows can be chosenthat result in acceptable residence times of the pulses. However, in extreme cases such asmacromolecular substances and unusual combinations of density, viscosity and diffusivity, it maybe necessary to decrease the flow rate through the capillary to obtain good accuracy.

The second conclusion is that it is possible to measure multicomponent diffusivities with a detectorwhich can produce only one independent signal. As more is known about the samples and thedetector linearity, fewer measurements are needed to produce reliable results.

σr (V) D1,1 ± ∆D1,1 D1,2 ± ∆D1,2 D2,1 ± ∆D2,1 D2,2 ± ∆D2,2

10−10 3.00 ± 5·10−5% 0.500 ± 4·10−4% 1.30 ± 3·10−4% 5.00 ± 8·10−5%10−6 3.00 ± 4·10−1% 0.520 ± 4% 1.29 ± 3% 5.05 ± 8·10−1%10−5 2.96 ± 5% 0.700 ± 3·101% 1.17 ± 3·101% 5.45 ± 8%

TAYLOR DISPERSION THEORY

39

NOTATION

a1,..., a9 constants in coiled-capillary dispersion model, (–)A constant, Σ α i

−8 ≈ 2.1701⋅10−5, (–)b, b1,..., b4 fitting parameters, (m)c concentration, (mol m−3)D Fick diffusivity, (m2 s−1)d inner capillary diameter, (m)De Dean number, Re / λ½, (–)disc[D] discriminant of [D], tr 2 [D]− 4 | D |, (m4 s−2)f detector signal, (V)J0 zeroth order Bessel function of the first kindK1,..., K4 constants in Price’s formulas eqs. (4) and (5), (mol)L length of the capillary, (m)m2 second moment of the cross-section averaged concentration profile, (m2)n number of moles (mol)P pressure, (Pa)Pe radial Péclet number, usup d / D = Re Sc, (–)r noise function, (V)Re Reynolds number, ρ usup d / η, (–)Sc Schmidt number, η / ρ D, (–)S detector linearity, (V)s detector offset, (V)tr [D] trace of [D], ∑ i Dii, (m

2 s−1)t time, (s)u velocity, (m s−1)V volume, (m3)z axial capillary co-ordinate, (m)

Greek symbolsα i ith root of the derivative of J0

∆ difference, deviationδ coil diameter, (m)ε accuracy, (–)η fluid dynamic viscosity, (Pa s)κ dispersion coefficient, (m2 s−1)λ coil / capillary diameter ratio, δ / d, (–)ρ fluid mass density, (kg m−3)σ ‘standard deviation’ of the response peak, (m)σr standard deviation of the normally distributed noise function, (V)τ mean residence time, L / usup, (s)ΦV volumetric flow rate, (m3 s−1)

Chapter 3

40

SubscriptsA Ariscap capillaryEC Erdogan & Chatwini several countersinj injectedm measurementmax maximumNLG Nunge, Lin & Gillsup superficialV volume

SuperscriptsE excessT transpose

REFERENCES

[1] Alizadeh A., Nieto de Castro C.A., Wakeham W.A.’The theory of the Taylor dispersion technique for liquid diffusivity measurements.’Int. J. Thermophys., 1(3), 1980, pp 243–284.

[2] Aris R.’On the dispersion of a solute flowing through a tube.’Proc. Roy. Soc. A, 235, 1956, pp 69–77.

[3] Braam W.G.M.’Scheidingsmethoden: chromatografie.’1985, Wolters–Noordhoff, Groningen, NL.

[4] Chen H.R., Blanchard L.P.Can. J. Chem., 53, 1975, p 228.

[5] Chen H.R., Blanchard L.P.Can. J. Chem., 53, 1975, p 476.

[6] Dean W.R.’Note on the motion of fluid in a curved pipe.’Phil. Mag., 4, 1927, pp 208–223.

[7] Dean W.R.’The stream-line motion of a fluid in a curved pipe.’Phil. Mag., 5, 1928, pp 673–695.

[8] Erdogan M.E., Chatwin P.C.’The effects of curvature and buoyancy on the laminar dispersion of solute in a horizontaltube.’J. Fluid Mech., 29, pp 465–484.

[9] Janssen L.A.M.’Axial dispersion in laminar flow through coiled tubes’Chem. Eng. Sci., 31, 1976, pp 215–218.

[10] Nunge R.J., Lin T.-S., Gill W.N.’Laminar dispersion in curved tubes and channels.’J. Fluid Mech., 51, 1972, pp 363–383.

TAYLOR DISPERSION THEORY

41

[11] Press W.H. et. al.’Numerical Recipes.’1986, Cambridge University Press, Cambridge, UK.

[12] Price W.E.’Theory of the Taylor dispersion technique for three-component-system diffusionmeasurements.’J. Chem. Soc. Faraday Trans. 1, 84(7), 1988, pp 2431–2439.

[13] Rutten Ph.W.M.’Diffusion in liquids.’1992, Delft University Press, Delft, NL.

[14] Snijder E.D.’Metal hydrides as catalysts and hydrogen suppliers.’1992, Enschede.

[15] Taylor G.I.’Dispersion of soluble matter in solvent flowing slowly through a tube.’Proc. Roy. Soc. A, 219, 1953, pp 186–203.

[16] Taylor G.I.’Conditions under which dispersion of a solute in a stream of solvent can be used tomeasure molecular diffusion.’Proc. Roy. Soc. A, 225, 1954, pp 473–477.

[17] Tijssen R.’Effect of column-coiling on the dispersion of solutes in gas chromatography. Part 1:theory’Chromatographia, 3, 1970, pp 525–531.

[18] Tijssen R.’Effect of column-coiling on the dispersion of solutes in gas chromatography. Part 2:generalized theory’Chromatographia, 5, 1972, pp 286–295.

[19] Tijssen R.’On the dispersion of macromolecules in dilute solutions measured with the band-broadening technique.’Can. J. Chem. Eng., 55, 1977, pp 225–226.

[20] Tijssen R.’Axial dispersion and flow phenomena in helically coiled tubular reactors for flow analysisand chromotography’Anal. Chim. Acta, 114, 1980, pp 71–89.

[21] Ven-Lucassen I.M.J.J. van der, Kieviet F.G., Kerkhof P.J.A.M.‘Fast and convenient implementation of the Taylor dispersion method.’J. Chem. Eng. Data, 40, 1995, pp 407–411.

NUMERICAL SIMULATION OF TAYLOR DISPERSION

43

4. NUMERICAL SIMULATION OF TAYLOR DISPERSION

In chapter 3 the equations were given which allow the measurement of diffusivities usingTaylor dispersion. An important assumption made in the derivation of these equations is thatthe fluid mixture is ‘ideal’†, that is, its physical properties are independent of the compositionof the mixture. However, this is not generally true for real mixtures. In this chapter weinvestigate the effects of composition-dependence of certain physical mixture properties on theaccuracy of the measured coefficients. We do this by numerical simulation of the Taylordispersion phenomena, because the equations cannot be solved analytically. We will analysethe equations with the method of finite differences, which results in a computer program forcomputing a concentration profile.

We begin with the numerical solution of the ideal binary case. The results can be comparedwith the analytical answer. This allows us to determine the accuracy of the numerical method.It also enables us to establish the numerical parameter settings of the program for the secondcase, where non-ideal binaries are treated. This is what this chapter really is all about. Only thetwo composition effects that we expect to be most important are investigated, to keep theequations manageable. We will consider a varying diffusivity and total concentration for as faras they influence the diffusivity. That is, effects of concentration on viscosity or density are notaccounted for. So, we do not consider any influence on the fluid flow. In general, effects thatdo not disturb the cylindrical rotation symmetry are easily implemented, but as soon as thissymmetry breaks down, things become more complex.

What we want to do in this chapter is to compute a composition profile as it will leave thecapillary for a non-ideal mixture, compare this with equation (3.1), and see if there is asignificant discrepancy between the two. The practical significance of this chapter is that itmight help answer the question to what extent the composition difference between the sampleand the eluent influences the measured diffusivity. Usually, this difference is kept small, butthis can complicate detection of the concentration-time profile. Therefore, to facilitatedetection, we would like the two liquids to differ as much as possible without messing up themeasurement.

To compute the time-concentration function, a number of steps must be taken. The first stepsare equal for both cases, namely the derivation of the differential equation that must be solved.This equation belongs to the family of convection-diffusion equations, whose members arenotorious for their difficult numerical solvability. The equation is then discretised andotherwise adapted for numerical use. After that, the discussion diverges, because the solutionmethods for ideal and non-ideal mixtures differ. Finally, the results of both cases will beexamined.

†Here, the term ‘ideal’ is not used in the thermodynamic sense.

Chapter 4

44

4.1 DERIVATION OF THE EQUATIONS

The general non-steady mass transfer equation is:

− = ∇⋅ −∂∂

c

tRnN ‡ (1)

It is assumed here that there is no chemical reaction, and as a result the source terms R are zero.We choose a capillary-fixed frame, and the mole fluxes Nn with respect to this co-ordinatesystem can be expressed as the sum of a convective and a diffusive flux relative to a certain (s,q)-frame, see equations (2.7) and (2.12)

N N un s q n s q c= +, , , (2)

Equation (1) then transforms to

− = ∇⋅ + ⋅∇ + ∇⋅∂∂

c

tc cs q s q s q nu u N, , , , (3)

The product us, q·∇c is defined in eq. (6). The continuity equation for constant-density fluids isgiven by

∇⋅ =us q, 0 (4)

This equation holds irrespective of the reference frame, because it is just another way to say:“in is out”, and this is true for any velocity the fluid may have. Note that we neglect the effectof composition on the total concentration in eq. (4), and so on the local flow pattern, butremember that we decided to ignore the fluid-mechanical effects of this. According to thegeneralised Maxwell-Stefan theory, the diffusive fluxes in an n-species mixture are given byequation (2.33), and the differential equation becomes ((∇y ) i = ∇y i, see chapter 2, page 20).

− = ⋅∇ − ∇ ⋅ ∇∂∂

c

tc c G ys qu , e j (5)

In view of the cylindrical symmetry of the capillary, it is best to switch over to cylindrical co-ordinates (r, φ, z), and the two terms on the right go over to

us qrs q

s q

zs qc u

c

r

u

r

cu

c

z

c G yr r

r c Gr

yr

c G yz

c Gz

y

, ,,

,⋅∇ = + +

∇⋅ ∇ =FHG

IKJ +

FHG

IKJ +

FHG

IKJ

∂∂

∂∂ φ

∂∂

∂∂

∂∂

∂∂ φ

∂∂φ

∂∂

∂∂

φ

e j 1 12

(6)

where urs, q, uφ

s, q and uzs, q are the radial, tangential and axial components of us, q, respectively.

Now is the time to decide what frame of reference we wish to apply. From equation (2.11) andtable 2.1, it can be concluded that in the volume-fixed reference frame the total volume flux isequal to the reference velocity u I, V. The total volume flux is what is usually understood by thefluid velocity, and which we can compute as a function of the co-ordinates:

‡ Readers who have problems with the notation used should refer to chapter 2.

NUMERICAL SIMULATION OF TAYLOR DISPERSION

45

u u u u rar

I V I VzI V, , ,, ,= = = −0 0 2 1

2

φ sup c he j (7)

This is the velocity vector for fluids exhibiting Newtonian viscosity behaviour: the well-knownparabolic flow profile. The symbol a stands for the inner radius of the tubing, usup is thesuperficial fluid velocity. Furthermore, axial rotation symmetry is assumed, or ∂ /∂ φ = 0.Combining all these expressions, one gets

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

c

tu r

ac

z r rr c G

ry

zc G

zy= − +

FHG

IKJ +

FHG

IKJ2 1

12

sup c he j (8)

As is customary with the numerical solution of differential equations, the co-ordinates aretransformed so as to render them dimensionless. Here, we choose the transformations

Rr

a r a R= ⇒ =

∂∂

∂∂

1(9)

Zz u t

L z L Z=

−⇒ =sup ∂

∂∂

∂1

(10)

Tt

t t t T

u

L Z= ⇒ = −

mon mon

sup∂∂

∂∂

∂∂

1(11)

The transformation of z results in a co-ordinate system that moves along with the fluid’saverage velocity usup. For the numerical solution of the differential equation, it is important tomake a judicious choice of the parameters tmon and L. This will be explained further on. Theabove transformations convert equation (8) into

12 1

1 122 2t

c

T

u

LR

c

Z a R RR c G

Ry

L Zc G

Zy

mon

sup∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

= − +FHG

IKJ +

FHG

IKJd i (12)

For numerical accuracy, the co-ordinates are refined with the following transformations, whichallow the definition of non-uniformly spaced grids at a later stage. This feature enables us tohave a smaller mesh size at positions where the composition gradients are large, which(hopefully) enhances the accuracy of the discretisation. These grid transformations are

R RR R

ρ α ρ α ρ ρ ρ α ρ α∂

∂ ρ∂

∂ρρ ρ ρ ρb g d i b g d i b g= − + ∈ ′ = − + =′

1 0 1 2 112 , , ; ; (13)

Z ZZ Z

ζα ζ α ζ ζ

α ζ α ζ ζζ

α ζ α

α ζ α∂

∂ ζ∂

∂ζζ ζ

ζ ζ

ζ ζ

ζ ζ

b g d id i b g d i

d i b g=− + ∈

− + ∈ −′ =

− +

− +=

′

1 0 1

1 1 0

2 1

2 1

12

2

, ,

, ,; ; (14)

T TT T

τ τ τ τ α τ∂

∂ τ∂

∂ τα

τατ τb g b g b g= ∈ ′ = =

′−, ,0 1

11; ; (15)

The values of the refinement parameters α are subject to some restrictions. Since therefinement transformations are necessarily monotonously increasing functions, α τ must be

Chapter 4

46

positive and α ρ and α ζ must lie within the range [0, 2]. Furthermore, α τ is arbitrarily chosen tobe a natural number, and the ranges of attainable values for α ρ and α ζ are further restricted bythe conditions that the reciprocal of R ′ must be defined on the entire ρ-interval [0, 1] and Z ′ onthe ζ-interval [−1, 1]. In contrast, the derivative T ′ need not be non-zero at τ = 0: ∂ /∂T is notused at this time level, which corresponds to the initial condition of the convection-diffusionproblem, and which is assumed to be known. Thus, α τ∈N+ and α ρ , α ζ∈⟨0, 2⟩.

Now that the ζ and τ domains have been established, it can be inferred from equations (9)through (11), and (13) through (15) that tmon is the time during which the dispersive process ismonitored, and that L is half of the length of the tube section considered. In principle, themagnitude of tmon and L can be chosen freely, but in practice the value of L turns out to besubject to certain restrictions (see page 49). The discontinuous definition of Z has its raisond’être in the fact that in the axial direction the major composition gradients are, by virtue ofequation (3.1), expected around the middle of the Z-interval. The same is true for the radialgradients around the middle of the R-interval, but these gradients are expected to be relativelysmall anyway [3], and the parameter α ρ is therefore assigned the value 1, which means that thegrid remains equidistant in the radial direction.

With the grid refinement functions and the fact that the concentration vector c is defined as theproduct c y , the differential equation becomes

∂

∂ τ

τ

ζρ

∂∂ζ

τ

ρ ρ∂

∂ρ

ρ

ρ∂

∂ρ

τ

ζ∂

∂ζ ζ∂

∂ζ

c y t T u

L ZR c y

t T

a R R

R

Rc G y

t T

L Z Zc G y

=′

′−

+′

′ ′

FHG

IKJ

+′

′ ′

FHG

IKJ

mon sup

mon

mon

b gb g b ge jb g

b g b gb gb g

b gb g b g

2 1

1

2

2

2

(16)

This is our basic equation for both ideal and non-ideal equations, which we will discuss in thenext two sections. We will restrict ourselves to two-component systems, because they aresimpler and because no special effects are to be expected in a multicomponent case. The binaryform of the above equation is obtained by replacing [G] by the Fick diffusivity D, and y by y,where it is unimportant which mole fraction is meant, because it makes no difference.Notwithstanding this equivalence of the components, the key component whose behaviour ismonitored is arbitrarily called the ‘tracer’. In the next sections the symbol y is used withsubscripts, which, however, refer to grid point numbers rather than components.

4.2 COMPOSITION-INDEPENDENT PARAMETERS

It is clear that the differential equation is greatly simplified if c and D are independent of thecomposition of the fluid:

∂∂ τ

τ ρ ζ∂∂ζ

τ ρ∂

∂ρ

ρ

ρ∂∂ρ

τ ζ∂

∂ζ ζ∂∂ζ

yP

yQ

R

R

yS

Z

y= +

′

FHG

IKJ +

′

FHG

IKJ, , , ,b g b g b g

b g b g b g1

(17)

where

NUMERICAL SIMULATION OF TAYLOR DISPERSION

47

Pt T u

L ZRτ ρ ζ

τ

ζρ, ,b g b g

b g b ge j=′

′−mon sup 2 1

2(18)

Qt T D

a R Rτ ρ

τ

ρ ρ,b g b g

b g b g=′

′mon2

(19)

St T D

L Zτ ζ

τ

ζ,b g b g

b g=′

′mon

2(20)

However, equations (19) and, as a result, also (17) are not valid at the tube axis, because R(ρ)is zero there. In appendix A it is shown that at these positions the two equations become

∂∂ τ

τ ζ∂∂ζ

τ∂

∂ρ∂∂ρ

τ ζ∂

∂ζ ζ∂∂ζ

yP

yQ

R

yS

Z

y= +

′

FHG

IKJ +

′

FHG

IKJ, , , ,0 0

1

0

1b g b g b g b g b g (21)

and

Qt T D

a Rτ

τ,0

2

02b g b gb g=′

′mon (22)

4.2.1 Discretisation of the differential equation

We are now ready to discretise these equations, and to this aim the (τ, ρ, ζ)-domain is coveredwith a regular three-dimensional (m +1)×( p +1)×(2 q +1)-grid. The mesh sizes of the grid are∆τ (= 1/m), ∆ρ (= 1/p), and ∆ζ (= 1/q), and the corresponding grid point numbers are i∈{0,..., m}, j ∈{0,..., p} and k ∈{− q,..., q}. Grid points are denoted τ i, ρ j and ζ k, whichdesignate the grid positions i ∆τ, j ∆ρ and k ∆ζ, respectively. On this grid the derivative of ywith respect to τ on the left-hand side of equations (17) and (21) is discretised by means of thebackward formula

∂∂ τ

y y y yO

i j k

i j k i j k i j kFHG

IKJ =

− ++− −

, ,

, , , , , ,3 4

21 2 2

∆τ∆τd i (23)

while for the diffusive terms on the right-hand sides the following discretisations are used forboth the axial and the radial derivatives.

∂∂ρ

ρ∂∂ρ

gy

g y g g y g yi j k

j i j k j j i j k j i j kb g d iFHG

IKJ = − + +− − − + + +

, ,

, , , , , ,

12 1 1∆ρ ½ ½ ½ ½ (24)

∂∂ζ ζ

∂∂ζ

1 1 1 1 1 12 1 1′

FHG

IKJ =

′−

′+

′

FHG

IKJ +

′

LNMM

OQPP−

−− + +

+Z

y

Zy

Z Zy

Zy

i j k ki j k

k ki j k

ki j kb g

, ,

, , , , , ,∆ζ ½ ½ ½ ½

(25)

Chapter 4

48

The errors of both schemes are O(∆ρ2) and O(∆ζ2) respectively. The function g (ρ) equalsR(ρ)/R ′(ρ) for equation (17) and 1/R ′(ρ) for equation (21). The notations g j ± ½ and Z ′k ± ½ areshorthand for g (ρ j ± ½ ) and Z ′(ζ k ± ½ ).

The discretisation of the convective term of the two different forms of the differential equationis a little more difficult, since it is known that if a central discretisation scheme is used,numerical instability is likely to prevent us from finding a solution in case convectiondominates diffusion. Instead, a so-called upwind scheme must be implemented, but a first-order approach will confront us with another notorious problem, namely numerical diffusion.Therefore, a second-order upwind method is chosen.

Implementation of an upwind scheme requires that we know the direction of the ‘wind’. Thewind in our transformed and refined co-ordinate system is the fluid velocity uζ

u u R uu

L Zζ ζ ζ ζ= − =

′, ,sup supsupwith1 2 2d i b g (26)

with a zero at

ρ

α α α

αα

α

ρ ρ ρ

ρρ

ρ

t

if

if

=

− + + −

−∈

=

RS||

T||

2 2 2 1

2 10 2 1

1

22 1

d id i l q, \

(27)

For the axial velocity we find 0 ≤ uζ ≤ uζ, sup whenever 0 ≤ ρ ≤ ρt, and − uζ, sup ≤ u ζ < 0 ifρ t < ρ ≤ 1. Thus there exists a grid line number, jt say, where the discretisation of ∂ y/∂ ζ in theconvective term switches from backward to forward. The value of jt is found by truncation ofρ t /∆ρ. The grid line jt belongs to that part of the domain where u ζ is non-negative. This resultsin the following second-order discretisation of the convective term

∂∂ζ

y y y yO j j

y y yO j j p

i j k

i j k i j k i j k

i j k i j k i j k

FHG

IKJ =

− ++ ∈

=− + −

+ ∈ +

− −

+ +

, ,

, , , , , ,

, , , , , ,

,...,

,...,

3 4

20

3 4

21

1 2 2

1 2 2

∆ζ∆ζ

∆ζ∆ζ

d i l q

d i l q

if

if

t

t

(28)

However, at the upwind axial edges of the domain a first-order upwind scheme is used to keepthe number of unknowns in balance with the number of equations available. So,

∂∂ζ

y y yO k q j j

y yO k q j j p

i j k

i j k i j k

i j k i j k

FHG

IKJ =

−+ = − ∈

=−

+ = ∈ +

−

+

, ,

, , , ,

, , , ,

,...,

,...,

1

1

0

1

∆ζ∆ζ

∆ζ∆ζ

b g l q

b g l q

if and

if and

t

t

(29)

If all the discretisation formulas are combined with the differential equations (17) and (21), anequation emerges of the form

NUMERICAL SIMULATION OF TAYLOR DISPERSION

49

I A y B y C y E y

F y G y H yi j k i j k i j k i j k i j k i j k i j k i j k i j k

i j k i j k i j k i j k i j k i j k

, , , , , , , , , , , , , , , , , ,

, , , , , , , , , , , ,

= + + ++ + +

− − +

+ − +

2 1 1

2 1 1

(30)

Now it is time to concern ourselves with the boundary conditions of the problem to match thenumber of unknowns with the number of equations. The spatial domain has four boundaries,one at the capillary axis, one at its wall and two at the inlet and outlet sides. The latterboundaries present the toughest problem. We cannot be sure about the properties of thecomposition profiles at these positions, but we might assume that, provided L is chosen largeenough, we have

∂∂

2

2 0y

zz L z L= = = −at and (31)

This is the least restrictive choice we can make. With the analytical solution of the Taylordispersion equation for ‘ideal’ mixtures in mind, eq. (3.1), we can infer that these conditionswill only be satisfied if the compositions at the boundaries will (almost) be equal to the eluentcomposition. This means that the tracer peak is (almost) completely contained in the capillarysegment of length 2 L. After applying equations (10), (14) and (25) we finally arrive at

1 1 1 11 1′

−′

+′

FHG

IKJ +

′= − =

−−

− + ++Z

yZ Z

yZ

y k q k qk

i j kk k

i j kk

i j k½ ½ ½ ½

at and, , , , , , (32)

The radial boundary conditions at r = 0 and r = a are dictated by considerations of symmetry inthe former, and mass conservation in the latter case. Thus, at both positions the radial flux mustvanish, or, put in mathematical terms,

∂∂

y

rr r a= = =0 0at and (33)

In their discretised form these conditions read

y y y yi k i k i p k i p k, , , , , , , ,− − += =1 1 1 1and (34)

At last, the coefficients can be determined, and the formulas for doing so are listed in appendixB.

4.2.2 Solving the discretised equation

Now that we know how to compute the coefficients, yet another question arises. If theboundary conditions are to be valid, L must be large enough. However, how large is ‘largeenough’? The problem lies in the spreading of the peak. From Taylor dispersion practice weknow that under usual measurement conditions the peak spreads much faster than in the case ofstationary diffusion. In the former case, the binary dispersion coefficient κ id for constant-density, constant-diffusivity mixtures is given by eq. (3.3)

κ id

sup= +

u a

DD

d i2

48(35)

Chapter 4

50

while in the latter case it is equal to the molecular diffusivity D. This rapid spreadingcomplicates the choice of the length L of the tube section. On the one hand we would like tokeep it as small as possible for an accurate discretisation. On the other hand, the section musthave a certain minimum length to contain the entire peak, as a result of the conditions (31).This problem can be solved in different ways:1. Keep the tube section length small by taking tmon small. This option is not feasible since

the results obtained with it cannot be verified experimentally.2. Take L long enough to contain a ‘full-grown’ peak and take, to maintain reasonable

accuracy, a large number of radial grid lines q. This will not work either for it requireslarge amounts of computer memory and takes excessive run times.

3. Start with modest values of both L and q, and whenever the chosen tube section lengthturns out to become too short during the computations, stretch it a bit, keeping q constant.This is option chosen here. The axial domain stretch factor is denoted by ε.

The problem with this option is that if we do not want to overstretch the domain – which leadsto loss of information and accuracy – the grid positions in the old domain generally will notcoincide with those of the new (that is, stretched) domain. Therefore, we need a way to makeaccurate approximations of y at non-grid positions. This is achieved by cubic splineinterpolation of the y-values of the unstretched domain along each axial grid line, see appendixC. The spline ends are chosen to have zero second derivatives since this is in agreement withboundary conditions (31). This interpolation is then used to evaluate the profile for each of theaxial grid lines in the new domain. Logically, some of the grid points in the new domain liebeyond the edges of the old one. The tracer concentration at these positions, however, may beset equal to the tracer level of the eluent, because the tracer peak is assumed to be containedcompletely by the old tube segment.

The actual computation of the tracer profile proceeds as follows. Firstly, the initial tracerprofile, y (0, ρ, ζ), is defined: y0, j, k must be known for all j and k. This profile is taken to be ablock pulse:

yy k w

y k wj k0, , =>≤

RS|T|e

p

if

if(36)

with ye and yp the eluent and pulse compositions. Note that the initial peak is 2 w +1 grid pointswide. This profile then, allows us to compute the coefficients for the N = ( p +1)(2 q +1) linearequations (30) of the time level ∆τ (i = 1). Next, we have to solve this set of equations, whichcan be represented by the matrix equation [A] y = b. The vector y contains the tracer content ofthe fluid at all grid positions, and in the vector b all the coefficients I i, j, k are stored. The N×Ncoefficient matrix [A] holds all the coefficients A i, j, k to H i, j, k. It is banded and sparse becauseequation (30) indicates that the tracer level at a certain grid point is related only to its value at afew of the neighbouring points. Thus, a special algorithm† for the solution of such bandedsystems can be used, which saves a lot of storage capacity and computation time.

† We have used the routines F01LBF and F04LDF of the NAG-library (mark 16, double precision), a

library of FORTRAN-routines from the Numerical Algorithm Group at Oxford, UK.

NUMERICAL SIMULATION OF TAYLOR DISPERSION

51

The solution of the equations yields the tracer content at each of the N spatial gridpoints, withwhich we can evaluate the coefficients of the equations at the next time level, and we can thensolve the set of equations for this time level. We repeat this process until we arrive at the finaltime level, and the tracer profile which is thus computed is the profile we seek.

The last thing left to do after the final profile has been computed, is to extract the axialconcentration profile as we ‘see’ it, which is the radial average of the spatial profile. Thisaverage is again determined by a spline interpolation of the y-profile, this time along the radialgrid lines. The first derivative of the spline ends is set zero, in accordance with boundaryconditions (33). Subsequently, the resulting cubic equations are analytically integrated,yielding an axial sequence of mean concentrations that compose a profile suited forcomparison with measured profiles. For a more detailed discussion, see appendix D.

As a final check, the total tracer amount can be computed by integration of the axial tracerprofile in much the same way this profile itself was determined, albeit that the spline ends aremodified to match the boundary conditions at the tube section ends. Obviously, the total traceramount should remain constant throughout the computations. The method for computing theamount of tracer is also outlined in appendix D.

4.3 COMPOSITION-DEPENDENT PARAMETERS

Generally, both the concentration c and the Fick diffusivity D depend on the composition of themixture. Differential equation (16) cannot be simplified and therefore c and D must be keptinside the differentials. The equations (17) through (22) now become

∂∂ τ

τ ρ ζ∂∂ζ

τ ρ∂

∂ρ

ρ

ρ∂∂ρ

τ ζ∂

∂ζ ζ∂∂ζ

c yP

c yQ c D

R

R

yS

c D

Z

y= +

′

FHG

IKJ +

′

FHG

IKJ, , , ,b g b g b g

b g b g b g (37)

where ρ ≠ 0, and with

Pt T u

L ZR Q

t T

a R R

St T

L Z

τ ρ ζτ

ζρ τ ρ

τ

ρ ρ

τ ζτ

ζ

, , ,

,

b g b gb g b ge j b g b g

b g b gb g b g

b g

=′

′− =

′

′

=′

′

mon sup mon

mon

2 12

2

2

(38)

At the tube axis, we have

∂∂ τ

τ ζ∂∂ζ

τ∂

∂ρ∂∂ρ

τ ζ∂

∂ζ ζ∂∂ζ

c yP

c yQ

c D

R

yS

c D

Z

y= +

′

FHG

IKJ +

′

FHG

IKJ, , , ,0 0

0b g b g b g b g b g (39)

and

Qt T

a Rτ

τ,0

2

02b g b gb g=′

′mon (40)

Chapter 4

52

4.3.1 Discretisation and solution of the equations

Discretising this whole lot proceeds much along the same lines as in the previous case, and ityields, just as in section 4.2.1, an equation of the form of (30). The coefficients for thisequation are given in appendix E.

The solution of the final profile by stepping the time ladder is similar to the way the previouscase was treated: we just have to solve the set of N linear equations. The big difference with thecomposition-independent case is how we take each time step. The problem is that in order tocompute the values of these coefficients, we must have the solution at our disposal they aresupposed to help us find. Our way out of this vicious circle is iteration: we assume that thetracer profile at the next time level is identical to the current profile, and with this informationwe compute the coefficients. Then the equations are solved, yielding a new tracer profile.Unfortunately, this new profile cannot be taken as the profile for the next iterative step becausethis leads to instable oscillations. Therefore, a weighted average of the new and the previousprofiles is taken as the profile after each iteration: y = δ ynew + (1− δ) yprevious. The parameter δ iscalled the damping or relaxation factor. With the new profile, which we hope to be a betterapproximation to that of the next time level, the coefficients are computed again, and so on.We repeat this process until the relative difference between two subsequent profiles becomessmaller than a certain pre-set limit (less than 1‰ for all grid nodes). It is clear that this way ofgetting from one time level to the next is much more time consuming than the single-stepalgorithm that was used in the composition-independent case.

4.3.2 Example: methanol / tetrachloromethane

To connect the computational results to the real world, we choose a real binary liquid system:methanol/tetrachloromethane. This is a sort of worst-case scenario, since both the diffusivity Das well as the total concentration c vary exceptionally strong with composition. For a numberof mixture compositions diffusivities at 25 °C have been listed by Anderson and Babb [1].These data can be described fairly accurately by the equation

Dk yi

i

i

=−

=∑

10 9

0

4 (41)

The mole fraction y is understood to be that of methanol. The constants k i (in s m−2) are:k 0 = 0.3787, k 1 = 16.35, k 2 = − 47.56, k 3 = 48.04, and k 4 = −16.78. According to Rutten [2] thedensity data of Timmermans [4] fit a third-degree polynomial ρ = ∑ k i y

i. Note that in thissection ρ stands for the mixture’s density, not the transformed and refined radial co-ordinate.The coefficients k i (in kg m−3) are: k 0 = 1.5865, k 1 = − 0.4413, k 2 = 0.2540, and k 3 = − 0.6086.The total concentration of the liquid is related to the molar masses of both species byc = ρ / ((M2 − M1) y + M2), which can be refitted to a third-degree polynomial

c k yii

i

==∑103

0

3

(42)

NUMERICAL SIMULATION OF TAYLOR DISPERSION

53

The coefficients (in mol m−3) have the values: k 0 = 10.18, k 1 = 7.661, k 2 = −2.666, andk 3 = 9.191. Both the diffusivity and the concentration of a methanol/tetra mixture are shown infigure 1.

yMeOH (-)

0.0 0.5 1.0

109 D( m2 s-1 )

0

1

2

3

Dc10-4 c

( mol m3 )

Figure 1. The diffusivity D and the total concentration c as functions of the mole fraction methanol.The circles represent the measured data of Andersen and Babb [1].

4.4 RESULTS

In this section the numerical results will be presented of both the ideal fluid and the non-idealfluid program. As was mentioned above, the first case merely serves to test the reliability of themethod, and to determine the influence of certain numerical parameters on the solution. Thus,the first program enables us to minimise the number of runs of the second program, each run ofwhich takes about 15 times as long as a run of the first program†. The results of the non-idealfluid program constitute the really interesting part of this section, in which we hope to be ableto infer the effect of the composition difference between the injected sample and the eluent.

The program for the composition-independent case was run 16 times with different parametersettings. Some of the parameters were the same for all runs, namely: a = 2.65·10− 4 m,D = 1.0·10−9 m2 s−1, ye = 0.0, yp = 1.0, and usup = 1.0·10−2 m s−1. The standard settings of theremaining geometrical and numerical parameters were: L = 0.1 m, tmon = 1.0·103 s, w = 1,m = 1000, p = 50, q = 35, α ρ = 1.0, α ζ = 0.1, α τ = 2, δ = 0.5, ε = 1.5. For each run except runnumber 5, one or more of these parameters is altered, to see what the effect on the quality ofthe corresponding solution is. This quality is monitored as follows: the computed profile iscompared with an adapted form of the peak equation (3.1)

† The computations were carried out on a HP 9000/735 workstation. The time needed to complete a run of

the non-ideal fluid program was 2 to 4 weeks, depending on the number of ‘jobs’ run. Usually, we hadonly 15% to 25% of the processor’s attention.

Chapter 4

54

∆ yn

a t

L Z s

t= −

−LNMM

OQPP

E

mon mon

exp2 42

2

π π κ κb g

(43)

Here nE denotes, as it did in chapter 3, the excess number of tracer moles in the sample, and κis the Aris dispersion coefficient, equation (35). It turns out that the computed profile slightlyshifts away from the central position Z = 0, and this axial shift is represented by s. Forcomparison of the computed profile and equation (43) a curve fitting program (Tablecurve 1.0,Jandel scientific) is used. It supplies us with the four main quality indicators:• The correlation coefficient of the fit. It is often denoted r2, but to avoid confusion with the

spatial co-ordinate r, we will not use this symbol. In the ideal case, namely a perfect fit, itsmagnitude is 1.

• The relative difference ∆D between the fitted and the programmed diffusivities, Dfit and D:∆D = (Dfit − D) / D.

• The axial shift s of the peak.• The excess tracer amount nE, which should be conserved during the computations.For all runs, these quality indicators are listed in table 1 together with the values of thoseprogram parameters that differ from the default values listed above.

Here follow the most important conclusions that can be drawn from these data.• The axial shift is almost independent of the parameter settings, and is therefore nothing to

worry about. The origin of this shift is not clear.• The fitted diffusivity corresponds closely to the programmed value. For tmon-values of 100

s and more, Dfit is about 2% lower than the true D.• The correlation coefficient is always very close to the ideal value of 1, except for small

tmon, but this is in agreement with the fact that equation (43) is valid for all tmon only if theinitial pulse is a Dirac-pulse, which cannot possibly be implemented.

• The excess tracer amount nE in the sample is conserved quite well in the computations. Infact, better than may be deduced from the table. The initial profile represents a blockpulse, and the initial tracer amount is computed with the spline method discussed earlieron page 51. A drawback of this method is that it cannot cope with sharp edges, and theinitial tracer amount therefore tends to be overestimated. In the first few time steps, as theprofile becomes smoother, this amount falls rapidly to a lower value, and is then virtuallyconstant during the rest of the computations. It is therefore remarkable that, in many runs,the fitted amount corresponds better with the initial than the final amount. Only for smallinitial peak widths, when the tracer content is greatly overestimated, is this not so. Thecause of this phenomenon is unclear, but it is probably not very important anyway: as longas the tracer leakage out of the tube section is negligible, the exact amount does not mattermuch.

• In the ranges presented, the numerical parameter settings are of no great influence on thecomputed results. Time grid refinement contributes little to the accuracy, but run number 9indicates that axial grid refinement is worthwhile.

NUMERICAL SIMULATION OF TAYLOR DISPERSION

55

Table 1. Results of the composition-independent parameter program. The subscript numbers in thecolumn ‘correl. coeff.’ (correlation coefficient) denote the number of subsequent nines: 0.9389 means0.99989.

On the basis of these results, the parameters of new runs were chosen: L = 5.0·10−2 m,tmon = 5.0·102 s, w = 1, m = 500, p = 60, q = 40, α ζ = 0.1, α τ = 1, ε = 1.25. The other parameterswere not changed: a = 2.65·10−4 m, ye = 0.0, yp = 1.0, usup = 1.0·10−2 m s−1, α ρ = 1.0, δ = 0.5.Note that the diffusivity is not a parameter for the non-ideal fluid program, because it isdetermined by the composition of the mixture.

To determine the influence of the composition difference between the eluent and the sample,three more runs were carried out, see table 2. The purpose of runs 17 and 18 is to see whetherthe results of run 19 can be trusted. The former run serves to clarify if the ∆D of approximately−2% is also found with the new parameter settings, while run number 18 should tell us if bothprograms yield the same results when the non-ideal program is fed with constant D and cfunctions.

run # parameter correl. ∆D s nE (10−10 mol)settings coeff. (%) (cm) initial final fit

1 tmon=10.0, m=10 0.9105 +20.1 −0.9 2.590 2.539 2.581

2 tmon=50.0, m=50 0.9265 −2.23 −1.1 2.590 2.539 2.573

3 tmon=100.0, m=100 0.9289 −2.15 −1.1 2.590 2.537 2.560

4 tmon=500.0, m=500 0.9382 −1.96 −1.1 2.590 2.537 2.548

5 default settings 0.9389 −2.17 −1.1 2.590 2.537 2.547

6 tmon=2000.0, m=2000 0.9408 −2.59 −1.2 2.590 2.535 2.548

7 ατ =1 0.9389 −2.12 −1.1 2.590 2.537 2.548

8 ατ =1, w=5 0.9389 −2.13 −1.1 16.71 16.64 16.71

9 αζ=1.0, ατ =1 0.9378 −3.41 −1.0 18.91 18.91 19.00

10 tmon=500.0, m=500,L=0.05, w=5

0.9378 −1.94 −1.1 11.28 11.24 11.29

11 tmon=500.0, m=500,L=0.05, ατ =3, w=5

0.9379 −2.03 −1.1 8.357 8.317 8.356

12 tmon=500.0, m=500,L=0.05, w=5, ε=1.25

0.9378 −1.80 −1.1 8.357 8.322 8.356

13 tmon=500.0,L=0.05, w=5

0.9379 −2.03 −1.1 8.357 8.317 8.356

14 p=60 0.9389 −2.12 −1.1 2.590 2.537 2.547

15 q=40 0.9389 −1.55 −1.1 2.189 2.149 2.156

16 p=60, q=40 0.9389 −1.51 −1.1 2.189 2.149 2.156

Chapter 4

56

Table 2. Data of the three additional runs. The axial shifts s were −5.4, − 4.5 and − 4.4 mmrespectively. The value of ∆D for run number 19 is relative to the diffusivity of methanol in pure tetra (atinfinite dilution).

The cause of the difference of four decades between the tracer amounts from run number 17and those listed in table 1 on one side and the other two runs on the other is that in the ideal-fluid case the concentration was of no influence on the solution, and it was set to an arbitrary 1mol m−3. The true total concentration of the fluid is of the order of 1.0·104 mol m−3 for the lasttwo runs (see also figure 1), which explains the difference. The fact that the correlationcoefficient of run 19 is slightly lower than those of the other two additional runs indicates thatthe variant concentration and diffusivity lead to a small deformation of the peak. This peak isshown in figure 2 together with the best fit of equation (43). Since the tracer fractions of theeluent and the sample were set at 0 and 1 respectively, these liquids consist of pure tetra andmethanol. Thus the simulated Taylor-dispersion experiment should yield the diffusivity ofmethanol in tetra at infinite dilution, which, according to equation (41) equals 2.64·10−9 m2 s−1.The value of Dfit should not differ too much from this value.

Position in tube segment ( m )

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5-2

0

2

4

6

8

10

12

Tracerconcentration

(mol m-3 )

Figure 2. The computed profile for the case where a sample of pure methanol is injected into an eluentstream of pure tetra. The circles represent the radially averaged concentrations, and the line the best fit offunction (43). Note the high data density of the computed profile around the segment centre, which is aresult of the axial grid refinement.

run # program 109 D c correl. ∆D nE (10−9 mol)case (m2 s−1) (mol m−3) coeff. (%) initial final fit

17 ideal 2.451 1 0.9407 −1.25 0.1095 0.1075 0.1078

18 non-ideal 2.541 1.0·104 0.9411 −1.16 1095 1075 107819 non-ideal eq. (41) eq. (42) 0.9389 −8.98 2668 2620 2635

NUMERICAL SIMULATION OF TAYLOR DISPERSION

57

The obvious conclusion is that the difference between the eluent and sample compositionswould in this severe case lead to a significant underestimation of the diffusivity. From table 2this under-estimation can be estimated at approximately 8%. This is a considerable error, but assaid before, it is one of the worst examples possible. Had we chosen to inject a sample of puremethanol in tetra, the deviation would certainly be less pronounced, because at thiscomposition D is a weaker function of y.

4.5 CONCLUSIONS

The main conclusion of this chapter therefore is that the Taylor-dispersion technique is notcompletely insensitive to the sample/eluent composition difference, but this difference can bechosen larger than is often held advisable. Especially in cases where the concentration and thediffusivity are well-behaved functions of the composition, the risks are limited.

APPENDIX A. THE CONVECTION-DIFFUSION EQUATION AT THE CAPILLARY AXIS

The general form of the convection-diffusion equation in axisymmetric geometries withexclusively axial flow is given by

∂∂

∂∂

∂

∂∂

∂c

tu

c

z r

r N

r

N

zzI V r

I V n

zI V n

= − − −,

, , , ,1 d i(A.1)

The subscripts z and r signify the axial and radial components of a spatial vector. Clearly, theright-hand side of this equation becomes indefinite at the axis of symmetry (r = 0). The form ittakes at those positions can be deduced from the mass conservation equation for a cylindricalvolume element centred around the axis.

∂∂

∂∂

∂∂

c

tu

c

z rN

N

zzI V

rI V n z

I V n

= − − −, , ,, ,2

(A.2)

Since at the axis both equations must hold, and because we have

1r

r N

r

N

r

N

r

rI V n

rI V n

rI V n∂

∂∂

∂

, , , , , ,d i= + (A.3)

the following identity applies to the axis of symmetry

∂∂

N

r

N

rrI V n

rI V n, , , ,

= (A.4)

and therefore differential equation (A.1) becomes

∂∂

∂∂

∂∂

∂∂

c

tu

c

z

N

r

N

zzI V r

I V nzI V n

= − − −,, , , ,

2 (A.5)

Chapter 4

58

APPENDIX B. EXPRESSIONS FOR THE COEFFICIENTS FOR COMPOSITION-INDEPENDENTPARAMETERS

This appendix and appendix E are not meant to impress the reader, but to allow him to quicklyreproduce the results of the computations, provided he would want to.

The value of I is given by

Iy y

i j ki j k i j k

, ,, , , ,=

− +− −4

21 2

∆τfor all grid point numbers i, j and k. The values of the coefficients A through H depend uponthese numbers. For the computation of the coefficients G and H three different groups can bediscerned:1. j = 0:

G HQ

R Ri k i ki

, , , ,,

0 00

201 1

= =′

+′

FHG

IKJ− +∆ρ ½ ½

2. j ∈ {1,..., p− 1}:

GQ R

RH

Q R

Ri j ki j

ji j k

i j

j, ,

,, ,

,=′

FHG

IKJ =

′FHG

IKJ− +∆ρ ∆ρ2 2

½ ½

3. j = p:

GQ R

R

R

RHi p k

i p

p pi p k, ,

,, ,=

′FHG

IKJ +

′FHG

IKJ

LNMM

OQPP =

− +∆ρ2 0½ ½

For the coefficients A to F there are eight classes: four in that part of the domain where uζ ≥ 0and four in the rest of the domain. Below first the coefficients in the former classes are listed,which have in common that j ∈ {0,..., jt} and Fi, j, k = 0.1. k = − q:

A B EP Z

Z

C E G H

i j q i j q i j qi j q q

q

i j q i j q i j q i j q

, , , , , ,, ,

, , , , , , , ,

− − −− − −

− +

− − − −

= = =′

′

= − − − −

0 0

32

∆ξ

∆τ

½

½

2. k = − q +1:

A BP Z

Z

S

ZE

S

Z

C B E G H

i j q i j qi j q q

q

i q

qi j q

i q

q

i j q i j q i j q i j q i j q

, , , ,, , ,

, ,,

, , , , , , , , , ,

− + − +− + − −

− +

− +

− +− +

− +

− +

− + − + − + − + − +

= = − −′

′

FHG

IKJ +

′=

′

= − − − − −

1 11 1

2 11

21½

1 1 1 1 1

02

3

32

∆ξ ∆ξ ∆ξ

∆τ

½

½ ½

3. k ∈ {− q + 2,...,q − 1}:

AP

BP S

ZE

S

Z

C A B E G H

i j ki j k

i j ki j k i k

ki j k

i k

k

i j k i j k i j k i j k i j k i j k

, ,, ,

, ,, , ,

, ,,

, , , , , , , , , , , ,

= = − +′

=′

= − − − − − −

− +2

2

3

2

2 2∆ξ ∆ξ ∆ξ ∆ξ

∆τ

½ ½

4. k = q:

NUMERICAL SIMULATION OF TAYLOR DISPERSION

59

AP

BP

C A B G H Ei j qi j q

i j qi j q

i j q i j q i j q i j q i j q i j q, ,, ,

, ,, ,

, , , , , , , , , , , ,= = − = − − − − − =2

2 3

20

∆ξ ∆ξ ∆τ

At grid positions with a negative uζ, that is, with values of j in { jt+1,..., p}, the coefficients B toF also form four classes. For all classes we have A i, j, k = 0.5. k = − q:

B EP

FP

C E F G H

i j q i j qi j q

i j qi j q

i j q i j q i j q i j q i j q

, , , ,, ,

, ,, ,

, , , , , , , , , ,

− −−

−−

− − − − −

= = = −

= − − − − −

02

2

3

2

∆ξ ∆ξ

∆τ6. k ∈ {− q + 1,...,q − 2}:

BS

ZE

P S

ZF

P

C B E F G H

i j ki k

ki j k

i j k i k

ki j k

i j k

i j k i j k i j k i j k i j k i j k

, ,,

, ,, , ,

, ,, ,

, , , , , , , , , , , ,

=′

= +′

= −

= − − − − − −

− +∆ξ ∆ξ ∆ξ ∆ξ

∆τ

2 2

2

2

3

2

½ ½

7. k = q − 1:

BS

ZE

P Z

Z

S

ZF

C B E G H

i j qi q

qi j q

i j q q

q

i q

qi j q

i j q i j q i j q i j q i j q

, ,,

, ,, , ,

, ,

, , , , , , , , , ,

−−

−−

− +

−

−

−−

− − − − −

=′

= −′

′

FHG

IKJ +

′=

= − − − − −

11

21½

11 1

2 1

1 1 1 1 1

23 0

3

2

∆ξ ∆ξ ∆ξ

∆τ

½

½ ½

8. k = q:

BP Z

ZC B G H E Fi j q

i j q q

qi j q i j q i j q i j q i j q i j q, ,

, ,, , , , , , , , , , , ,= −

′

′= − − − − = =+

−∆ξ ∆τ½

½

32

0 0

APPENDIX C. CUBIC SPLINE INTERPOLATION

Preliminary remark: the notation used in this appendix bears no relation to that of the rest of thischapter, it is all just mathematics.

Given n points (x k, yk), where k = 1,..., n. The cubic spline function f (x) exists of n −1 third-degreepolynomials fi (x), each of which is valid on the ith interval [x i, x i+1]:

f x a x a x a x a i ni i i i ib g = + + + = −33

22

1 0 1 1, , , , , ,..., (C.1)

The evaluation of the spline function therefore requires that the 4 n − 4 coefficients are known.The functions fi are subject to the 4 n − 6 conditions

f x y i ni i ib g = = −, ,...,1 1 (C.2)

f x y i ni i i+ += = −1 1 1 1b g , ,..., (C.3)

′ = ′ = −+ + +f x f x i ni i i i1 1 1 1 2b g b g , ,..., (C.4)

′′ = ′′ = −+ + +f x f x i ni i i i1 1 1 1 2b g b g , ,..., (C.5)

Chapter 4

60

As a result of this limited number of equations, two more must be given for a completedefinition of f. However, these extra equations need not be specified until after the derivationof the functions fi. Since f is a third-degree polynomial everywhere on the interval [x1, x n], itfollows from condition (C.5) that its second derivative is a continuous piecewise linearfunction. Let Mi denote f ″(x i). We find that f ″ can be expressed as

′′ = ′′ = − + −++f x f x

M

hx x

M

hx xi

i

ii

i

iib g b g b g b g1

1 (C.6)

where h i is the length of the ith interval: h i = x i+1 − x i. Integration of this equation yields

′′ = ′ + = − − + −z ++f x x f x k

M

hx x

M

hx xi i i

i

ii

i

iib g b g b g b gd 1 1

2 1 2

2 2(C.7)

with ki1 an (as yet arbitrary) constant. Again integrating this expression gives

′ = + + = − + −z ++f x x f x k x k

M

hx x

M

hx xi i i i

i

ii

i

iib g b g b g b gd 1 2 1

3 1 3

6 6(C.8)

Substituting equations (C.2) and (C.3) gives

kh

M Mh

y y

kh

M x M xh

y x y x

ii

i ii

i i

ii

i i i ii

i i i i

1 1 1

2 1 1 1 1

6

1

61

= − − −

= − + −

+ +

+ + + +

b g b g

b g b g(C.9)

Now that the constants k i 1 and k i 2 are known, equations (C.7) and (C.8) go over to

′ = − − + − − − + −++

+ +f xM

hx x

M

hx x

hM M

hy yi

i

ii

i

ii

ii i

ii ib g b g b g b g b g

2 2 6

11

2 1 2

1 1 (C.10)

f xM

hx x

M

hx x

y

h

h Mx x

y

h

h Mx x

ii

ii

i

ii

i

i

i ii

i

i

i ii

b g b g b g

b g b g

= − + − +

−FHG

IKJ − + −

FHG

IKJ −

++

++ +

6 6

6 6

1

3 1 3

11 1

(C.11)

It is worth noting that f depends only on the co-ordinates of the n points in the data set and itssecond derivative at these points. So far, equations (C.4) have not been used yet. Theirsubstitution establishes relations between the M i -values:

Mh

Mh h

Mh y y

h

y y

hi ni

ii

i ii

i i i

i

i i

i6 3 61 21

12

1 2 1

1

1++

+ =−

−−

= −++

++ + +

+

+ , ,..., (C.12)

As was mentioned above, the evaluation of the M i requires two more equations. These extraconditions can be chosen arbitrarily. The most frequently used conditions are:• M1 = M n = 0 (natural spline).• M1 = M2, M n − 1 = M n (second degree end segments).• M1 = λ M2, λ M n − 1 = M n .

NUMERICAL SIMULATION OF TAYLOR DISPERSION

61

• M1 = M n , f ′(x1) = f ′(x n) (periodic spline).It is also possible to assign fixed values to any two out of the four variables M1, M n , f ′(x1) andf ′(x n). As an example, consider a spline with fixed tangents at the ends of the interval:f1′(x1) = p, and fn′−1(x n) = q. Substitution of these equations in (C.10) result in

Mh

Mh y y

hp M

hM

hq

y y

hnn

nn n n

n1

12

1 2 1

11

1 1 1

13 6 6 3+ =

−− + = +

−−

− − −

−

and (C.13)

Together with the n − 2 equations (C.12), these two relations constitute a tridiagonal set oflinear equations from which all M i can be solved†. With the M i the coefficients of equation(C.1) can be computed, as they follow from equation (C.11):

ah

M M ah

M x M x

ah

M h x M x h y y

ah

Mx h y x

Mx h y x

ii

i i ii

i i i i

ii

i i i i i i i i

ii

ii i i i

ii i i i

3 1 2 1 1

12

12

12 2

1

0 12 2

11 2 2

1

1

6

1

2

1 1

63 3

1

6 6

, ,

,

,

= − = −

= − + − + −FHG

IKJ

= − +LNM

OQP − − +L

NMOQP

FHG

IKJ

+ + +

+ + +

+ ++

+

b g b g

d i d i

d i d i

(C.14)

APPENDIX D. THE AMOUNT OF TRACER IN THE TUBE SECTION

The total amount of tracer, n (in moles), in the tube section equals

n c y r r z u tr

a

z u t L

L

= −=− =−zz 2

0

π d d sup

sup

d i (D.1)

However, instead of y (r, z) we have a discretised form of y (R, Z). Substitution of thetransformation functions Z and R yields

n a L c y R R ZRZ

===−zz2 2

0

1

1

1

π d d (D.2)

Since 2 π a2 L is the tube section volume, it follows that

⟨ ⟩ ===−zzc c y R R Z

RZ

tr d d0

1

1

1

(D.3)

If the total concentration of the mixture is independent of its composition, that is, if c ≠ c( y),this equation is equivalent to:

⟨ ⟩ ===−zzc c y R R Z

RZ

tr d d0

1

1

1

(D.4)

† For solving the tridiagonal set of equations, the F01LEF and F04LEF routines of the NAG-library were

used (see footnote on page 50).

Chapter 4

62

Because of the discrete character of the y-profile, these integrals cannot be computed directly.Instead, for the given time level τ i, first the inner integral (with R as the running variable) is‘solved’ along each grid line k ∈{− q,..., q}. This is achieved by cubic spline interpolation ofeither (c y) i, j, k or of y i, j, k with j running from 0 through p. Thus, we obtain a family of 2 q + 1splines Fi, k (R) with coefficients a 0, i, k through a 3, i, k (see eq. (C.1)), whose auxiliary equationsare chosen F ′i, k (0) = F ′i, k (1) = 0, so as to comply with boundary conditions (33). So,

c y R R F R Rs

a RR Z k

i k jR

R

j

p

s i k j

s

s R

R

j

p

j

j

j

j

d d= = =

−

−==

−z z∑ ∑∑FHG

IKJ = =

LNM

OQP

+ +

0

1

0

1

22

5

0

11 11

∆ξ

, , ,d i d i (D.5)

These 2 q + 1 values are then used to construct another spline Gi (Z) − coefficients b 0, i, k to b 3, i, k

−, with Gi″(−1) = Gi″(1) = 0 in accordance with eq. (31). This spline is then integrated:

⟨ ⟩ = =LNM

OQP

+ +z∑ ∑∑=−

−

−==−

−

c G Zs

b Zi kZ

Z

k q

q

s i k

s

s Z

Z

j q

q

k

k

k

k

tr db g d i1 11

11

41 1, (D.6)

The total tracer amount in the tube section simply equals ⟨ctr⟩ times the volume of the section.

APPENDIX E. EXPRESSIONS FOR THE COEFFICIENTS FOR COMPOSITION-DEPENDENTPARAMETERS

The value of I is given by

Ic x c x

i j ki j k i j k

, ,, , , ,=

− +− −

4

21 2b g b g

∆τfor all grid point numbers i, j and k. The values of the coefficients A through H do depend uponthese numbers. For the computation of the coefficients G and H three different groups can bediscerned. (Tip: do not try this at home!)1. j = 0:

G HQ c D

R

c D

Ri k i ki

i k i k, , , ,

,

, , ,0 0

0

20= =′

FHG

IKJ +

′FHG

IKJ

LNMM

OQPP− +∆ρ ½, ½

2. j ∈ {1,..., p − 1}:

GQ R c D

RH

Q R c D

Ri j ki j

i j ki j k

i j

i j k, ,

,

, ,, ,

,

, ,

=′

FHG

IKJ =

′FHG

IKJ− +∆ρ ∆ρ2 2

½ ½

3. j = p:

GQ R c D

R

R c D

RHi p k

i p

i p k i p ki p k, ,

,

, , , ,, ,=

′FHG

IKJ +

′FHG

IKJ

LNMM

OQPP =

− +∆ρ2 0½ ½

For the coefficients A to F there are eight classes: four in that part of the domain where uζ ≥ 0and four in the rest of the domain. Below first the coefficients in the former classes are listed,which have in common that j ∈ {0,..., jt} and Fi, j, k = 0.1. k = − q:

NUMERICAL SIMULATION OF TAYLOR DISPERSION

63

A B CP

c c E G H

cE

P Z

Zc

S

Zc D c D

i j q i j q i j qi j q

i j q i j q i j q i j q i j q

i j qi j q

i j q q

qi j q

i q

qi j q i j q

, , , , , ,, ,

, , , , , , , , , ,

, ,, ,

, ,, ,

,

, , , ,

...− − −−

− − − − − −

−−

− − −

− +− −

−

− +− + − −

= = = − − − − +

− =′

′+

′−

0 0

3

2

1

1 2

∆ξ

∆τ ∆ξ ∆ξ

d i

b g b g½

½ ½½ ½

2. k = − q +1:

A BP

cZ

Zc

S c D

Z

ES c D

ZC

Pc c

i j q i j qi j q

i j qq

qi j q

i q

i j q

i j qi q

i j qi j q

i j qi j q i j

, , , ,, ,

, , , ,,

, ,

, ,,

, ,, ,

, ,, , ,

− + − +− +

−− −

− +− −

− +

− +

− +− +

− +− +

− +− +

= = − + +′

′

FHG

IKJ

LNMM

OQPP +

′FHG

IKJ

=′

FHG

IKJ = −

1 11

11

2

11

21½

11

1

02

4 1

23 4

∆ξ ∆ξ

∆ξ ∆ξ

½

½ ½

, , ,

, , , , , , , ,, ,

...− − −

− + − + − + − +− +

+ +

− − − − −

q i j q

i j q i j q i j q i j qi j q

c

B E G Hc

1

1 1 1 113

2

d i

∆τ3. k ∈ {− q +2,..., q −1}:

AP

c BP

cS c D

ZE

S c D

Zi j ki j k

i j k i j ki j k

i j ki k

i j ki j k

i k

i j k, ,

, ,, , , ,

, ,, ,

,

, ,, ,

,

, ,

= = − +′

FHG

IKJ =

′FHG

IKJ− −

− +2

22 1 2 2∆ξ ∆ξ ∆ξ ∆ξ½ ½

CP

c c c A B E G Hc

i j ki j k

i j k i j k i j k i j k i j k i j k i j k i j ki j k

, ,, ,

, , , , , , , , , , , , , , , ,, ,= − + − − − − − −− −2

3 43

21 2∆ξ ∆τd i

4. k = q:

AP

c BP

cS

Zc D c Di j q

i j qi j q i j q

i j qi j q

i q

qi j q i j q, ,

, ,, , , ,

, ,, ,

,

, , , ,= = − +

′−− −

−− +2

22 1 2∆ξ ∆ξ ∆ξ ½

½ ½b g b g

CP

c c c A B G Hc

E

i j qi j q

i j q i j q i j q i j q i j q i j q i j qi j q

i j q

, ,, ,

, , , , , , , , , , , , , ,, ,

, ,

= − + − − − − −

=

− −23 4

3

2

0

1 2∆ξ ∆τd i

At grid positions with a negative uζ, that is, with values of j in { jt +1,..., p}, the coefficients Bto F also form four classes. A i, j, k = 0 for all classes.5. k = − q:

EP

cS

Zc D c D F

Pci j q

i j qi j q

i q

qi j q i j q i j q

i j qi j q, ,

, ,, ,

,

, , , , , ,, ,

, ,−−

− +−

− +− + − − −

−− += +

′− = −

2

21 2 2∆ξ ∆ξ ∆ξ½½ ½

b g b g

CP

c c c E F G Hc

B

i j qi j q

i j q i j q i j q i j q i j q i j q i j qi j q

i j q

, ,, ,

, , , , , , , , , , , , , ,, ,

, ,

−−

− − + − + − − − −−

−

= − + − − − − − −

=2

3 43

2

0

1 2∆ξ ∆τd i

6. k ∈ {− q +1,..., q −2}:

BS c D

ZE

Pc

S c D

ZF

Pci j k

i k

i j ki j k

i j ki j k

i k

i j ki j k

i j ki j k, ,

,

, ,, ,

, ,, ,

,

, ,, ,

, ,, ,=

′FHG

IKJ = +

′FHG

IKJ = −

−+

++∆ξ ∆ξ ∆ξ ∆ξ2 1 2 2

2

2½ ½

CP

c c c B E F G Hc

i j ki j k

i j k i j k i j k i j k i j k i j k i j k i j ki j k

, ,, ,

, , , , , , , , , , , , , , , ,, ,= − + − − − − − − −+ +2

3 43

21 2∆ξ ∆τd i

7. k = q − 1:

Chapter 4

64

BS c D

ZC

Pc c c

B E G Hc

F

EP

cZ

Z

i j qi q

i j qi j q

i j qi j q i j q i j q

i j q i j q i j q i j qi j q

i j q

i j qi j q

i j qq

q

, ,,

, ,, ,

, ,, , , , , ,

, , , , , , , ,, ,

, ,

, ,, ,

, ,

...−−

−−

−− +

− − − −−

−

−− +

=′

FHG

IKJ = − + − +

− − − − − =

= − +′

′

11

21½

11

1 1

1 1 1 11

1

11

23 4

3

20

24 1

∆ξ ∆ξ

∆τ

∆ξ

d i

½

−+

−

−

FHG

IKJ

LNMM

OQPP +

′FHG

IKJ½ ½

cS c D

Zi j qi q

i j q, ,

,

, ,1

1

2∆ξ

8. k = q:

BP Z

Zc

S

Zc D cD E Fi j q

i j q q

qi n k

i q

qi j q i j q i j q i j q, ,

, ,, ,

,

, , , , , , , ,= −′

′+

′− = =+

−+

−− +∆ξ ∆ξ

½

½ ½½ ½1 2 0 0b g b g

CP

c c B G Hc

i j qi j q

i j q i j q i j q i j q i j qi j q

, ,, ,

, , , , , , , , , ,, ,= − − − − −+∆ξ ∆τ1

3

2d i

NOTATION

[A] coefficient matrixa inner tube or capillary radius, (m)b vector of constantsc molar concentration, (mol m3)i time grid position counter, (–)j radial grid position counter, (–)k axial grid position counter, (–)k0,..., k4 polynomial coefficientsL length of capillary section, (m)m number of time grid subintervals, (–)N number of spatial grid points, or number of equationsP dummy factor in equations (17), (21), (37) and (39)p number of radial grid subintervals, (–)Q dummy factor in equations (17), (21), (37) and (39)q half the number of axial grid subintervals, (–)R transformed radial co-ordinate, (–)r radial co-ordinate, (m)S dummy factor in equations (17), (21), (37) and (39)s axial shift of the computed profile, (m)T transformed time co-ordinate, (–)t time, (s)u velocity, (m s−1)w initial peak width parameter, (–)y mole fraction, (–)y variable vector containing the mole fractions at all spatial grid points, (–)Z transformed axial co-ordinate, (–)z axial co-ordinate, (m)

NUMERICAL SIMULATION OF TAYLOR DISPERSION

65

Greek symbolsα grid refinement parameter, (–)∆ difference, deviationδ relaxation factor, (–)ε stretch factor, (–)φ tangential co-ordinate, (–)κ dispersion coefficient, (m2 s−1)ρ refined radial co-ordinate, (–)τ refined time co-ordinate, (–)ζ refined axial co-ordinate, (–)

Subscriptse eluentfit best fit to equation (43)i time node number iid ‘ideal’ solutionj radial node number jk axial node number kmon monitoredp sample pulser radial co-ordinateρ refined radial co-ordinatesup superficialt forward/backward convection transitiontr tracerτ refined time co-ordinateφ tangential co-ordinatez axial co-ordinateζ refined axial co-ordinate

SuperscriptsE excessI,V volume-fixed reference frames,q (s, q)-reference frame

REFERENCES

[1] Anderson D.K., Babb A.L.‘Mutual diffusion in non-ideal liquid mixtures. IV. Methanol-carbon tetrachloride anddilute ethanol-carbon tetrachloride solutions.’J. Phys. Chem., 67, pp. 1362-1363.

[2] Rutten Ph. W.M.‘Diffusion in liquids’Delft University Press, Delft, the Netherlands, 1992.

[3] Taylor G.I.’Dispersion of soluble matter in solvent flowing slowly through a tube.’Proc. Roy. Soc. A, 219, 1953, pp 186–203.

Chapter 4

66

[4] Timmermans J‘The physico-chemical constants of binary systems in concentrated solutions.’, 1-3Interscience Publishers, New York., USA, 1959.

DIFFUSION NEAR CONSOLUTE POINTS

67

5. DIFFUSION NEAR CONSOLUTE POINTS

This chapter can be considered as a prelude to the next two chapters, which are on ternaryliquid-liquid extraction. As has already been mentioned in chapter 1, there are doubts whetherthe Maxwell-Stefan theory holds near the spinodal curve. In this chapter, we consider diffusionin mixtures near their consolute or critical point. At such a point the mixture is on the verge ofdemixing. The consolute point is a binary analogue of the spinodal curve, and is thereforeespecially suited for the examination of possible anomalous diffusivity behaviour. Cussler [5,6] has questioned the validity of the Maxwell-Stefan theory near consolute points byexamining existing data on diffusivities. Here we will re-examine all data and the reasoning ofCussler to see whether we should doubt the Maxwell-Stefan equations.

As we have seen in chapter 2, we can take apart the Fick diffusivity D into a ‘mechanical’ part,represented by the Maxwell-Stefan diffusivity Ð, and a thermodynamic part, Γ:

D Ða

yy

y= = = +Γ Γ, with

ln

ln

ln∂∂

∂ γ∂

1

11

1

1

1 (1)

The factor Γ is called the non-ideality or thermodynamic correction factor. The Maxwell-Stefan diffusion coefficient, which has been freed of thermodynamic influences, expresses onlymechanical interactions, and these are much less susceptible to changes in pressure,temperature or composition than the thermodynamic functions. Hence, the diffusivity Ð is amore ‘relaxed’ function than the Fick diffusivity D, although this is not to say that Ð isindependent of these variables. Nevertheless, this is an advantage of the use of Ð over D.Cussler, however, argues that eq. (1) may not be valid in all cases.

Let us not beat about the bush: this is an attempt to rehabilitate these equations in the sense thatwe will try to show that they cannot be falsified on the basis of Cussler’s arguments. To beginwith, we will let Cussler have his say – although we will sometimes interrupt him –, afterwhich we will try to give alternative explanations for the phenomena that gave rise to hisdoubts in the first place.

5.1 THE WORLD ACCORDING TO CUSSLER

Cussler substantiates his statement by referring to articles by Haase and Siry [10], andClaesson and Sundelof [3]. Haase and Siry report measured diffusion coefficients in thesystems triethylamine(1)-water(2) and nitrobenzene(1)-n-hexane(2). The former system has alower consolute point at approximately T# = 18.3 °C and x1# = 0.0874 [10], while the othersystem has an upper consolute point at about T# = 19.8 °C and x1# = 0.42. Claesson andSundelof only deal with the system nitrobenzene-n-hexane.

Cussler bases his arguments exclusively on this latter system, presumably because he has noaccurate thermodynamic model for the other mixture. The upper consolute temperature of thesystem nitrobenzene(1)-n-hexane(2) is reported to be T# = 19.7 °C [10], 19.9 °C [3], or 20.2 °C[1]. In all three articles, the consolute composition is given as x1# = 0.42. For particular reasons,

Chapter 5

68

Claesson and Sundelof measured their diffusivities at x1 = 0.5. The measured diffusivities asreported in both references are shown in figure 1.

T ( K )

290 300 310

D (m2/s)

0

1e-10

2e-10

3e-10

Claesson & SundelofHaase & Siry

Figure 1. The Fick diffusion coefficients in the system nitrobenzene-n-hexane measured by Claessonand Sundelof, and Haase and Siry. The lines are fits of arbitrary polynomials, which just happen to lookall right. Strictly, the fit to the data of Claesson and Sundelof ought to intersect the horizontal axis at atemperature lower than T#, because they were not measured at x1#.

Apparently, as the temperature approaches T#, the diffusivity drops to zero. This is inaccordance with equation (1) because Γ → 0 as T → T# and x1 → x1#, whereas Ð may beexpected to retain its approximately constant, nonzero value. So, there should be nothing toworry about, but it is the trend of the measured diffusivities that bothers Cussler. He claimsthat the chemical potential of the system nitrobenzene-n-hexane “is found by experiment to fitthe equation”

µ µ1 1 1 22= + +o lnR T x z x (2)

In other words, the system considered is a so-called regular solution. At the consolute point, wehave x1# = 0.5 and z = 2 R T#. Thus, he obtains the following expression for Γ near the consolutepoint:

Γ = −1 4 1 2x xT

T# (3)

Since, he continues, the (linear) temperature variation of this factor near the consolute point ismuch greater than that of Ð, the Fick diffusivity should exhibit linear behaviour, and he noticesthat this is not the case. Ergo, exit equation (1).

Is it truly that bad? There seem to be a few weak spots in Cussler’s reasoning. First of all, acloser look at figure [5].1 or [6].7.3-1 reveals that he bases his arguments on just one set ofmeasurements, namely those of Claesson and Sundelof on the mixture nitrobenzene-n-hexane.Secondly, the assumption that the system behaves as a regular solution seems a bit awkward,

DIFFUSION NEAR CONSOLUTE POINTS

69

because x1# was experimentally determined to be 0.42, and not 0.5. It may be because of thisregular solution assumption that Cussler only considered the data of Claesson and Sundelof,which were not measured at the consolute composition of the mixture, but at x1 = 0.5. Ormaybe he chose to use these data because they represent the worst case: the non-linearity of thedata of Haase and Siry is not at all that dramatic. Finally, equation (3) is not strictly linear, butin view of the value of T#, the curvature at T = T# is so small (− 8 x1 x2 / T#

2) that we may indeedexpect near-linear behaviour.

First we will, following Cussler, assume that the mixture is a regular solution. We can thencalculate the Maxwell-Stefan diffusivities with equation (3) by rewriting (1) as

ÐD

=Γ

(4)

Ideally, these diffusivities should be only weak functions of temperature. If Γ is calculatedusing eq. (3), we obtain the Maxwell-Stefan diffusion coefficients shown in figure 2.

T ( K )

290 300 310

Ð ( m2/s )

0.0

5.0e-9

1.0e-8

1.5e-8

2.0e-8

Claesson & SundelofHaase & Siry

T#

Figure 2. The Maxwell-Stefan diffusion coefficients computed using the regular solutionthermodynamic model.

We see that, as far as the diffusivities at the true consolute compositions are concerned, Cusslershould have no objections against equation (1). The variation in the other diffusivities,however, is indeed larger than expected. What can be the cause of this? Above, we alreadymentioned that Cussler’s choice for regular solution thermodynamics is not very convincing.He admits that “the regular solution model is inexact near the critical point”, but he brushesaway this inconvenience with a reference to the work of Stanley [18]. In table 15.2 of his bookStanley states that the experimental consolute point exponent ν of the system agrees with thetheoretical value of 2/3. This statement is based upon the experiments of Chen and Polonsky,who write: “The data are not precise enough to test (T − T#)

−ν-type dependence.” Moreover,even if ν has the right value – as is indicated by Wu et. al. [21] – this certainly does not provethat the mixture is a regular solution, because Stanley’s table shows that this should then also

Chapter 5

70

be the case for the very asymmetric system 2-methyl-propanoic acid-water (x1# ≈ 0.12,T# ≈ 25.5 °C) [17]. Finally, there are others who doubt the correctness of the use of the regularsolution model for the system regarded. Thomaes [19] for example, says that it is a very roughapproximation, and that it “is obviously not valid near the critical solution point”.

The trouble is that it is difficult to obtain Γ with any great accuracy, especially for mixturesnear their consolute point. Yet we will try to extract it from vapour-liquid equilibrium dataavailable in literature. In the next section, the methods used for obtaining the thermodynamicfunctions from VLE data will be discussed.

5.2 DETERMINATION OF ΓΓ FROM VLE DATA

Much work has already been done in this field, and this has lead to well-established methodsfor extracting the thermodynamic functions from such P-x data. As was said above, it isdifficult to obtain accurate Γ values from such data for systems near their consolute point,which holds especially for the systems triethylamine-water. Still, we will give it a try, and threemethods for VLE data reduction will pass in review:1. the ideal gas GE method .2. the non-ideal gas GE method.3. the numerical integration method.

Before going into details, however, we consider a few properties that the Γ-function shouldhave:• For pure substances, Γ is always equal to 1.• For temperatures at which no demixing occurs, Γ is positive for all compositions.• At the consolute temperature, Γ is zero at the consolute composition, but only there.• For temperatures at which demixing occurs, Γ is negative at the consolute composition.

The two zeros at the composition axis are the spinodal compositions of the mixture.

5.2.1 The ideal gas GE method

Let us assume that• the vapour phase is ideal γi

g = 1 ∀ i.†

• the liquid phase molar volume is negligible compared to that of the vapour phase (noPoynting correction).

Then we can express the total vapour pressure P of the mixture as a function of the vapourpressures of the pure components Pi

o, the mixture composition and the (liquid phase) activitycoefficients γi. The expression reads

P P x P x= +1 1 1 2 2 2o oγ γ (5)

Given P (x1) and the pure component vapour pressures, we seem to have one degree of freedomin this equation. However, there is an extra constraint imposed on the activity coefficients, theGibbs-Duhem equation:

† The superscript ‘g’ refers to vapour phase quantitities. Absence of this superscipt automatically

indicates, when meaningful, a liquid phase quantity.

DIFFUSION NEAR CONSOLUTE POINTS

71

xH

R TT

V

R TPi i

i

d ln d dE E

γ∑ = − +2 (6)

We will only consider isothermal P-x1 data, whence dT = 0, and due to the above assumptions,we can also neglect the liquid-phase excess volume work term. A way to incorporate theGibbs-Duhem equation into eq. (5) is to take the molar Gibbs excess energy of the mixture, GE,as the key quantity. To get rid of the continuously reappearing factor RT, we use the reducedmolar Gibbs excess energy gE = GE / RT. Now let a prime denote the constrained derivative of afunction with respect to the mole fraction of component 1, then eq. (2.5) can be written as

ln lnE E E Eγ γ1 2 2 1= + ′ = − ′g x g g x g, (7)

These equations automatically satisfy the Gibbs-Duhem equation, which is exactly why gE issuch a handy tool to use in combination with eq. (5). Together with eq. (1) we find

Γ = 1 + x1 x2 gE″. So, all we need to do is to find an expression for gE(x1).

There are two types of such expressions: the so-called local composition models (e.g. Wilson,NRTL, UNIQUAC, etc.), and empirical equations. All these expressions enable us to computegE as a function of x1 and some (vector of ) adjustable parameters p. Our task is to find p suchthat the total vapour pressure P (as fixed by the function gE ) best fits the experimental data.However, as the local composition models have a limited number of parameters, they are ofteninadequate for describing strongly non-ideal systems. Alternatively, we can also simply choosea function gE(x1, p) which we think will give a satisfactory fit. Three frequently used functionsare the Redlich-Kister expansion of the Margules function, the (expanded) Van Laar equationand the SSF (Sum of Symmetric Functions, [12]) equation. All three functions – at least, forbinary mixtures – have the form gE = v (x1) s (x1, p), where v has the properties v (0) = v (1) = 0,and v > 0 ∀ x1 ∈ ⟨0, 1⟩. The function s is given by either of the following alternatives:

s x p x

s xp x

s xp

xp

x p

k

k

k

l

k

k

k

l

l k

kk

k

l

1 1

1

1

1

1

1

1

1

2

11

21

2

2 1

1

2 1

1

, ,

, ,

, ,

p

p

p

b g b g

b gb g

b gb gb g

= − −

=−

=

+ −FHG

IKJ

−

=

−

=

+

=

∑

∑

∑

Redlich Kister expansion

Expanded Van Laar

SSFdiv

div

(8)

The reasons for the first two expressions containing polynomials in 2 x1 − 1 are presumablymostly historic, since polynomials in x1 would do equally well. The degree l of thesepolynomials is usually taken ≤ 6, because higher degrees seldom yield usable results: theyoften lead to thermodynamic functions with too many extrema to be credible. This is especiallyso for those functions based on higher-order derivatives of gE. The number of parameters of theSSF equation is l too, but here l must be even, and is usually taken 4 or 6. The function v is thesame for all three functions. Usually, v = x1 x2, but as we will see in section 5.3, this functioncannot deal with very sharp bends in the P(x1) curve near the domain edges. This problem

Chapter 5

72

might be obviated by choosing another function. It turns out that functions with a point ofinflection on [0, 1] are not suited as this affects the credibility of the gE-function. Finally, thefunction

v p x p xl l= − −+ +1 11 1 2 2exp expb g b g (9)

was chosen as an alternative, since it has independently adjustable tangents at x1 = 0 and x1 = 1,and has no points of inflection.

Note that, strictly, we only need P-x1 data. Whenever vapour composition measurements arealso available, this information is usually merely used to check the consistency of the fitted gE-function. In most low-pressure cases the method outlined in this section works quite well formost of the systems encountered in ‘everyday life’, but there are a few remarks to be made.

Firstly, the choice for the gE-function is mostly empirical: it lacks a theoretical background asopposed to the local composition models. This is not a point of great importance though, sincethe theoretical foundations on which these models are built are not too sound either. The onlydrawback is that the function parameters have scant physical meaning. On the other hand: weare free to adjust the functions at will without interfering with any deep ideas behind them.Secondly, we assumed above that the pure component vapour pressures Pi

o are known. Inpractice, these quantities will be calculated with, for example, the Antoine equation.Sometimes it is better, however, to regard the Pi

o – or just one of them – as adjustableparameters too:• when the pure vapour pressures are part of the P-x1 data, it would be strange to impose the

calculated values on the fitted P(x1) function.• when the Antoine equation is inaccurate, it leads to erroneous values of the Pi

o in question.This has repercussions on the fit for the entire x1-domain. Compare, for instance theDECHEMA data sheets [9], part 1, page 539 and part 1b, page 347.

The simple GE-based method, as has already been stated above, is not always sufficient, as wewill see in sections 5.3 and 5.4. Maybe the two assumptions at the beginning of this section areto be blamed for this. Their removal leads to the following method.

5.2.2 The non-ideal gas GE method

It is known that in certain mixtures the gas phase may not be ideal, even at atmosphericpressure. In this case we cannot use eq. (5) to calculate P(x1), but we must use the equations(see, for instance, Raal and Mühlbauer [14])

ln ln

ln ln

g

o

o o g

g

o

o o g

γδ

γδ

11

1 1

11 1 1 2

2

12

22

2 2

22 2 2 1

2

12

= +− − +

= +− − +

x P

x P

B V P P P x

R T

x P

x P

B V P P P x

R T

d id i d i

d id i d i(10)

Again, the ln γi follow from eq. (7). It is clear that we need additional information to solvethese equations. First of all, for the Poynting correction – the incorporation of which hardlytakes an extra effort – we need the pure component molar liquid volumes Vi

o (assumed constant

DIFFUSION NEAR CONSOLUTE POINTS

73

over the pressure range considered). Also, for the non-ideality of the vapour phase, we need thesecond virial coefficients of the pure substances, B11 and B22, and that of the mixture, B12.Further, δ12 = 2 B12 − B11 − B22. Whether it is necessary to incorporate these parameters can onlybe established afterwards.

A disadvantage of the non-ideal method is that it defines P implicitly and that it is thereforemuch more laborious from a computational perspective. Also, the pure component virialcoefficients B11, B22 and especially B12 are often not known at low pressures. So we must resortto methods of estimating these quantities, with all the risks thereof.

A pleasantly concise overview of the methods for estimating the various quantities is given byReid, Prausnitz and Poling [15]. They recommend using the method of Tsonopoulos. Therequired properties of the pure components are given in the table below.

Table 1. Pure substance properties required for the estimation of the various quantities used inequations (10). * Estimated using eq. (2-3.4) from [15], ** source: [20]

There is one problem, however: in view of its large dipole moment it is well possible thatnitrobenzene is non-ideal in the vapour phase. Tsonopoulos’ equations for estimating thesecond virial coefficients do not seem to incorporate this effect because the data sets uponwhich these equations are founded contain only one nitrocompound: nitromethane. The same istrue for the tertiary amines. We have not been able to find better data for our systems in openliterature. Anyway, the effects of gas-phase non-ideality appear to be small, and they do notsolve the problems we shall discuss in sections 5.3 and 5.4.

5.2.3 The numerical integration method

When the first two methods do not work, then what should we do? Apparently, it is sometimesdifficult to find equations that correctly describe gE over the whole composition range. But weare not necessarily interested in that, we may only want to know the course of this function inthe neighbourhood of a certain point. In that case, a local fit of the P-x1 data might besufficient. A method which only needs a local description of these data is the numericalintegration method. (The term ‘local’ promises more than it can fulfil, see below.) Under theassumptions stated at the beginning of section 5.2.1 the so-called Duhem-Margules differentialequation holds:

d

d

g g g

g

x

P

x x

x x P1 1 2

1 1

=−d i (11)

The combination of eqs. (5) and (6) then leads to the following expression for the (constrained)derivative of P with respect to x1:

Substance Tc [16](K)

Pc [16](MPa)

Vc [16](cm3 mol−1)

ω [15](–)

µ [15](D)

Tb [16](K)

V o [9](cm3 mol−1)

nitrobenzene 712.0 3.50 357.5 0.370* 4.2g** 481.45 102.73n-hexane 507.9 3.02 368.4 0.299 0.0 341.93 131.61triethylamine 535.2 3.04 394.5 0.320 0.9 362.35 139.96water 647.3 22.11 56.5 0.344 1.8 373.15 18.03

Chapter 5

74

′ = −P P P1 1 2 2o oγ γd iΓ (12)

If we assume that we know P(x1), and therefore also its inverse x1(P), we have

d

d

gg

g g

g

x

Pf P x

x x

x x P P1

11 2

1 1

= =−

,d i b gd i (13)

We have two reference points for this equation, at which we know both x1 – and therefore alsoP – and x1

g: at x1 = 0 and x1 = 1 we have x1g = x1. Numerical integration could be started at these

compositions if it was not for f being indeterminate at those compositions. Formerly, thisproblem used to be solved by estimating x1

g at a point close to x1 = 0 or x1 = 1 using Raoult’slaw. This is not necessary, however, because we can write

d

d

d

d

g g g g

g

gx

x

x

PP

x x

x x P xP h x x1

1

1 1 2

1 1 1

1 1= ′ =−

′ =d i b g d i, (14)

The choice of x1 as the variable instead of P seems more natural. Besides, P(x1) is a one-to-onerelation, whereas x1(P) may not be. Substitution of eq. (12), x1

g = P1o x1 γ1 / P, and

x2g = P2

o x2 γ2 / P gives

hP P

P= γ γ1 2

1 22

o o

Γ (15)

which is well defined at the edges of the x1-interval [0,1]:

hP

Ph

P

Px x1 10 11

21 2

2

1=

∞=

∞= =γ γo

o

o

o, (16)

The activity coefficients at infinite dilution γi∞ follow immediately from the P(x1) function in

combination with eq. (12):

γ γ11

0 2 22

1 1

1 11 1

∞=

∞== ′ + = − ′

PP P

PP Px xo

oo

o, (17)

So, eq. (14) can be easily integrated from x1 = 0 to x1 = 1, provided that there is no azeotrope.At azeotropic compositions x1

g = x1, (x1g x2

g ) ≠ 0 and dx1g / dx1 ≠ 0, whence P′ = 0. In general,

at the azeotropic composition Γaz ≠ 0, and therefore, according to eq. (12), P1o γ1,az = P2

o γ2,az.Consequently, eq. (15) goes over to

hP

P

x

xazaz

o

azaz

azg

azaz az=

FHG

IKJ =

FHG

IKJ =

γ 1 1

2

1

1

2

, ,

,

Γ Γ Γ (18)

Of course, Γaz depends directly on the properties of P(x1): using eq. (14) we find

Γazaz

gaz

g

azg

azg

azg

az glim lim

1,az 1,az

1,az

= =′

−=

′′

′ −

FHGG

IKJJ→ →

=

x x x x

x x

hx x

P

P

x x

x x

P

P

x1 1

1

1 2

1 1

1 2

1 1

, , , , (19)

DIFFUSION NEAR CONSOLUTE POINTS

75

For the last step l’Hôpital’s indeterminate limit rule has been used. Since (x1g)′az = haz = Γaz, we

find that Γaz is the root of a square polynomial:

Γaz azg

azg az

az

= − +′′F

HGIKJ

LNMM

OQPP0 5 1 1 4 1 2

1 2

. , ,x xP

P(20)

The numerical integration itself can be carried out with any numerical differential equationsolver, such as the Runge-Kutta method.

5.3 THE SYSTEM TRIETHYLAMINE-WATER

As a test case we will look at the system triethylamine(1)-water(2) at 18.0 °C. This is a hardnut to crack: the temperature is close to the consolute temperature T#, and the P(x1) curve has asharp bend near x1 = 0. In figure 3, the fits of the UNIQUAC model and the three gE functionsto the data of Counsell [4] are shown. These data are the best available on this system. Chun,Clinkscales and Davison [2], as well as Kohler [11] also measured P-x1 data, but these are lessaccurate, and in the case of Kohler’s data, too sparse and therefore too widely spaced to beuseful here.

Counsell himself, but also Dudley and Tyrell [7], and Haase and Siry too, have derivedactivities from the same data, so why do we do it all over again? The former two do not give adetailed description of their methods, and so the reliability of their activity data is uncertain.One gets the impression that these methods involve the use of pen, paper and ruler. Further, inview of the Γ values derived from these activities (see figure 5), they are probably ofquestionable accuracy, although this is not to say that we hold our own method in an overlyesteem. Anyway, we will see if we can improve on these earlier results. The results of Haaseand Siry are doubtful for another reason: they claim that they have obtained them fromKohler’s data using the ideal gas GE method applied to a three-parameter Redlich-Kisterexpansion. The problem is, however, that their tabulated values do not correspond to thosepredicted by their GE fit.

First, we consider the GE-based methods as described above, and see what the UNIQUAC, theRedlich-Kister, the Van Laar and the SSF functions can do for us. Only for the UNIQUACmodel the Pi

o are calculated using the Antoine equation with constants from the DECHEMAdata sheet ([9], part 1b, p. 347). If the Pi

o are treated as parameters too, the optimisation routinecomes up with fitted values for these pressures with improbable deviations from the Antoine-predicted values. For the three empirical equations (8), it almost makes no difference whetherthe Pi

o are treated as adjustable parameters or not: both methods lead to similar P(x1) functions.For these equations the Pi

o are fitted along with the six (l = 6) other model parameters.

Chapter 5

76

x1 (-)

0.0 0.2 0.4 0.6 0.8 1.0

P (Pa)

2000

4000

6000

8000

SSF (67.9)Van Laar (67.8)R.-K. (99.5)UNIQUAC

0.0 0.1

6000

8000

Figure 3. Fit of the UNIQUAC model and the three gE

-based functions for the system TEA-water at18.0 °C (Counsell, circles). The graph in the lower right corner is a magnification of the upper left part ofthe main graph. The numbers between parentheses are the root mean square deviation (RMSD) values inPa.

Only the Van Laar and the SSF fits are possibly satisfactory. The usefulness of a fit depends onwhat one wants to do with it, and on the accuracy that is required. Maybe that for certain aimsthese two fits would be good enough. For us, the ultimate test for consistency is the positionand the depth of the minimum of Γ. We should have Γ(x1#, T#) = Γ′(x1#, T#) = 0. For lowerconsolute points Γ > 0 ∀ x1 ∈ [0, 1] if T < T#. Unfortunately, none of the fits fulfils thiscondition, as can be seen in figure 4. This plot also clearly demonstrates the concept of the‘credibility’ of a fit. The Γ corresponding to the Redlich-Kister equation and, to a lesser extent,the Van Laar equation are obviously not credible. So, the SSF function is the only one which isboth accurate and credible.

Application of eq. (9) leads to smaller RMSD values, especially for the Redlich-Kister (64.8Pa) and the Van Laar (66.0 Pa) functions. The credibility of the Van Laar equation remainsrather poor, but that of the Redlich-Kister expansion clearly improves, beit that it still has twomaxima. The improvements for the SSF functions are slight (67.8 Pa). Still, the problemremains that the minimum of Γ is negative for all three functions. As we already mentioned insection 5.2.2, the non-ideal gas method does not improve this situation, and all we can do isapply the numerical integration method.

DIFFUSION NEAR CONSOLUTE POINTS

77

x1 (-)

0.0 0.2 0.4 0.6 0.8 1.0

Γ (-)

-0.5

0.0

0.5

1.0

SSF Van Laar R.-K. UNIQUAC

Figure 4. The thermodynamic correction factors corresponding to the fits of figure 3. Note that allmodels except UNIQUAC predict the position of the (leftmost) minimum correctly, viz. at approximatelyx1#.

The question is: what should the P(x1) function look like? We could use a spline interpolationto connect the experimental P-x1 data. Then we do not have to worry about the relative errorsof the experimental points: all of them are equally important. Unfortunately, a cubic spline alsoexhibits an overshoot at the sharp bend in the data, so we must use a square spline. The bestfree condition seems to be to set the spline derivative at x1 = 0 at about 1.9515·105 Pa, becausethis minimises the oscillatory character of the interpolation. However, we still obtain a Γfunction in which every separate subinterval of the spline can be seen, a result of thediscontinuous second derivative of the square spline (see figure 5). Also, a point of inflectionshows up near x1 = 0, resulting in a maximum of Γ there.

The remaining option is to use a piecewise fit of the P-x1 data. The most obvious approach is tosplit up the function in a part left and a part right of the sharp bend. The problem lies in the firstpart: the relative positions of the first four points – remember that we are still discussingCounsell’s data at 18.0 °C – force the fit function to have a point of inflection, but this leads toΓ values greater than 1 near x1 = 0. If we choose to ignore this fact, and try to describe thesefour points with a concave function, we must fudge the weights of the data, although there isno reason to believe that these weights should not be equal. Further, the error in the pressuremeasurements given by Counsell is so small – ± 6.7 Pa – that for the second and third datumany function without a point of inflection is bound to have a residual far greater than this error.This problem may be soothed by the notion that the error in composition of the liquid phase,about which Counsell makes no statement, has a very strong effect due to the steepness ofP(x1) at this interval. However, Counsell’s measurements on the vapour phase compositionsupport the use of a sigmoidal function. As we will see below, all these considerations appearto have little influence on the position of the minimum of Γ.

Chapter 5

78

Still, it turns out to be quite difficult to find a satisfactory fit for the data left of the bend. Manyfunctions have been considered, and for all it appeared to be impossible to link them to the fitof the right-hand part of the data in a satisfactory manner, and simultaneously suppressovershoot near the bend. Therefore, we split up the data into three parts instead of two, whichconsist of points 1–4, 4–13 and 13–28 respectively. To begin with, a function was chosen thatgave a satisfactory fit with the data of the middle section. Of this function we determine thevalue, the first and the second derivative in both points number 4 and 13. Then the data in theleft and right sections were fitted to suited functions subject to three constraints, namelycontinuity of the value, the tangent and the curvature with the function of the middle section.The result is a function that looks natural, because the eye is not sensitive to discontinuities ofderivatives higher than order two. In figure 5, Γ as derived from this P(x1) function is shown.As was mentioned above, the thermodynamic correction factor can also be derived fromCounsell’s thesis, which lists both 10log γ1 and 10log γ2 as a function of x1. We distil Γ fromthese data by fitting a 10th degree polynomial to 10log γ1, the derivative of which is thensubstituted in eq. (1). The 10log γ2 data were discarded because we are mainly interested insolutions with low amine contents. Given the course of the 10log γ2-function for suchcompositions, the uncertainty in Γ derived from this function is far bigger than that in the10log γ1-based Γ. Moreover, Counsell points out that it is impossible to check the consistency ofhis two 10log γi functions, and therefore no extra information can be obtained by comparison ofthe two.

0.0 0.1

Γ (-)

0.0

0.5

1.0

1.5

x1 (-)

0.1 0.3 0.5 0.7

CounsellSpline interpolationPiecewise fit (concave)Piecewise fit (sigmoidal)

Figure 5. Γ at 18 °C according to Counsell and as obtained by numerical integration of several fits. Aswe are only interested in the minimum, integration is not carried out beyond the azeotrope. Note the twodifferent scale intervals of the horizontal axis.

Interesting features of this graph are:• The minimum is shallow for all alternatives• The minimum of the spline is located at too large amine contents, and is very slightly

negative (x1 = 7.82·10−2, Γ = −4.73·10−4).

DIFFUSION NEAR CONSOLUTE POINTS

79

• The minimum of the concave fit is located at roughly the right composition, and is positive(x1 = 6.82·10−2, Γ = 2.36·10−3).

• The location of the minimum of the sigmoidal fit is not too far off either, and is Γ positivetoo (x1 = 9.97·10−2, Γ = 2.36·10−3).

• The location of the minimum as predicted by Counsell’s 10log γ1 data lies at x1 = 7.62·10−2,but Γ is negative there: −5.51·10−2.

So, let us assume that at x1# we have Γ = 2.36·10−3. The VLE at 10 °C and 4 °C can be treatedin a similar way, but in this case the sigmoidal function for the four leftmost points is the onlycredible option. At these two temperatures, the Γ values at x1# turn out to be 5.15·10−2 and1.00·10−1, respectively.

Now, how do we check whether these data are in concordance with the diffusivity data ofHaase and Siry? It is insufficient to assume that Ð is a constant over the temperature rangeconsidered. A well-known approximation for the binary Maxwell-Stefan diffusivity of aninfinite dilution of substance i in ‘pure’ j is the Stokes-Einstein equation Ðij = k T / n π η j ri, kbeing the Boltzman constant. Apart from the viscosity of j, we also need to know the molecularor Van der Waals radius of i. This radius was determined at r1 = 3.112·10−10 m [8]. As theviscosity of TEA-water mixtures rapidly rises in the vicinity of x1 = 0, the viscosity is not takento be that of pure water, but that of a mixture with x1 = x1#. It is derived from interpolations ofdata from Kohler and Dudley and Tyrell. For diffusion in water the number n may varybetween, say, 2 (very small molecules) and 8 (large and irregularly shaped species) [8].However, because we are not dealing with an infinitely dilute system, we do not know quitewhat to expect, but it is clear that n should lie, say, somewhere between 1 and 10. To see howwell the combination of the Stokes-Einstein equation and eq. (1) correlates D to Γ, we use n tominimise the value of

D TT

T

k

n rmm m

mm

b g b g−FHG

IKJ∑ Γ

η π 1

2

(21)

where the index m ranges over all experimentally determined Γ’s. Not only those determinedfrom Counsell’s data are considered, but also those tabulated by Haase and Siry, although wedo not know how they arrived at these values. Because the set of D’s by Haase and Siry doesnot contain all D(Tm), the Fick diffusivities are represented by a fitted function. The best valueof n is 1.20 for our ‘own’ Γ’s, and 1.47 for those of Haase and Siry.

The results may not be brilliant, but probably the most negative way to put it is that eq. (1)cannot be rejected on the basis of the diffusivity data of Haase and Siry on the system TEA-water. More positive minds might find that figure 6 hints at its correctness.

Chapter 5

80

T ( K )273 283 293

D (m2 s-1)

0

1e-11

2e-11

3e-11

4e-11

5e-11

This workHaase and Siry

Γ:

Figure 6. Correlation between the experimentally determined D (Haase and Siry, line), and eq. (1) incombination with the Stokes-Einstein equation (circles and squares).

5.4 THE SYSTEM NITROBENZENE-N-HEXANE

Obtaining the thermodynamic functions of this system from the VLE measurements by Neckeland Volk [13] is not such an arduous task as it was with the previous system. The P-x1 behaverather well, and application of the ideal gas GE method as described in section 5.2.1 suffices.The SSF function with l = 6 and free Pi

o yields the best results, although the Redlich-Kister andVan Laar methods do pretty well too. Yet, for the temperature closest to T#, 21 °C, Γ is slightlynegative at x1#, −6.7·10−3, which cannot be true. It is possible that measurements atcompositions that do not lie in the direct vicinity of the consolute composition affect the fitthere. In an attempt to obviate this problem, fits were carried out with bigger weights for themeasurements near the consolute composition. However, although positive Γ values could beobtained that way, the minimum shifted away from the experimentally determined consolutepoint. The difficulty lies of course in the fact that the vapour pressure of the mixture is virtuallyconstant over a wide range of compositions, and that it changes rapidly at high nitrobenzenecontents. This is difficult to describe for the simple functions of eq. (8). There is no method forobtaining Γ that will not lead to a negative minimum, because the SSF fit almost describes thedata perfectly already. The causes can probably be ascribed completely to an insufficientnumber of experimental data, which maybe are too inaccurate as well.

Anyway, to see whether eq. (1) conflicts with the experimental data, we proceed as in theprevious section. Again, we will try to minimise D − Γ Ð, but eq. (21) is useless here, becauseof the negative minimum of Γ at T = 21 °C. Therefore, we assume that the trend of thecalculated Γ(T ) is correct, but that the offset is wrong. Also, in the current situation, theStokes-Einstein form of Ð cannot be used, because x1# is too far removed from infinite dilution.So, the expression to be minimised is

DIFFUSION NEAR CONSOLUTE POINTS

81

D TT

T

k

rmm m

mm

b g b gb g−

+FHG

IKJ∗∑

Γ ∆Γ

η π

2

(22)

Here, r* is a characteristic length, which must have a value somewhere near the mean Van derWaals radius of the two components, and ∆Γ is the Γ bias. To use this equation, we need themixture viscosity at x1# at 21 °C, 25 °C and 35 °C. Neckel and Volk measured the compositiondependence of the viscosity only at the former two temperatures, and so we must findalternative ways to obtain the viscosity at 35 °C. To this end, we use the viscosity estimationmethod of Teja and Rice† in combination with an estimation equation for pure liquids. Bothestimation methods are described in the book by Reid, Prausnitz and Poling [15]. It appearsthat this method predicts the measured viscosities very well. This encourages the use of theextrapolation to 35 °C. The quantities T, Γ and η are tabulated below.

Table 2. The magnitudes of the quantities derived from the experiments of Neckel and Volk.

Optimisation of eq. (22) yields the values 1.75·10−2 and 4.31·10−10 m for the Γ bias and thecharacteristic length. The Van der Waals radii of nitrobenzene and n-hexane are 2.96·10−10 and3.00·10−10 m, respectively. So, r* is about 1.3 times the mean of these two radii, which does notseem to be an unreasonable value. The Γ bias is bigger than we would like, however, but thereis little we can do about that. The result of the optimisation is shown in figure 7. If anythingsensible can be said about these data, it is that the system nitrobenzene-n-hexane does not seemto support the claims of Cussler either.

† The interaction parameter ψij of this model was given the value 1.

T(K)

Γ(–)

η(mPa s)

294.15 −6.65·10−3 0.608298.15 3.15·10−3 0.579308.15 3.01·10−2 0.515

Chapter 5

82

T ( K )

290 300 310

D (m2/s)

0

1e-10

2e-10

3e-10

Figure 7. Correlation between the experimentally determined D (Haase and Siry, line), and eq. (1) incombination with the Stokes-Einstein-like diffusivity (circles).

5.5 CONCLUSIONS

As we expected, it is diffucult to obtain accurate thermodynamic data from the measurementsof the system near the consolute point. Especially the system TEA-water proves to be verydifficult. Due to this inaccuracy, it is impossible to establish once and for all whether eq. (1) iscorrect. Yet, the measurements certainly do not seem to constitute evidence that this equationdoes not hold near consolute points. It looks as if it is a draw, beit, it seems, with advantage forthe adherents of eq. (1).

NOTATION

a activity, (–)B second virial coefficient, (m3 mol−1)D Fick diffusivity, (m2 s−1)Ð Maxwell-Stefan diffusivity, (m2 s−1)G molar Gibbs energy, (J mol−1)g reduced molar Gibbs energy, (–)H molar enthalpy, (J mol−1)n number of moles, (–)n Stokes-Einstein factor, (–)P pressure, (Pa)p parameter vector (which is not a spatial vector)R universal gas constant, (J mol− 1 K−1)r Van der Waals radius, (m)T temperature, (K) or (ºC)V molar volume, (m3 mol−1)x dependent mole fraction, (–)y independent mole fraction, (–)

DIFFUSION NEAR CONSOLUTE POINTS

83

Greek symbolsΓ thermodynamic correction factor, (–)γ activity coefficient, (–)η viscosity, (Pa s)µ chemical potential, (J mol−1) or dipole moment, (D)ω acentric factor, (–)

Subscripts1, i, j component indicesaz at azeotropec at critical point# at consolute point

Superscripts′ constrained derivative with respect to the mole fraction of component 1″ constrained second derivative with respect to the mole fraction of component 1E excess quantityg vapour phaseo pure component

REFERENCES

[1] Chen S.H., Polonsky N.‘Observation of anomalous damping and dispersion of hypersound in a binary liquidmixture near the solution critical point.’Phys. Rev. Lett., 20, 1968, pp 909–911.

[2] Chun, K.W., Clinkscales T.C., Davison R.R.‘Vapor-liqiuid equilibrium of triethylamine-water and methyldiethylamine-water.’J. Chem. Eng. Data, 16(4), 1971, pp 443-446.

[3] Claesson S., Sundelof L.-O.‘Diffusion libre au voisinage de la température critique de miscibilité.’J. Chim. Physique, 54, 1957, pp 914–919.

[4] Counsell J.F.Ph.D. ThesisBristol, UK, 1959.

[5] Cussler E.L.‘Cluster diffusion in liquids.’AIChE J., 26(1), 1980, pp 43–50.

[6] Cussler E.L.‘Diffusion. Mass transfer in fluid systems.’Cambridge University Press, 1984, Cambridge, UK.

[7] Dudley G.J., Tyrell H.J.V.‘Transport processes in binary and ternary mixtures containing water, triethylamine andurea. Part 2.’J. Chem. Soc. Farad. Trans. I, 69, 1973, pp 2200-2208.

Chapter 5

84

[8] Edward J.T.‘Molecular volumes and the Stokes-Einstein equation.’J. Chem. Education., 47 (4), 1970, pp. 261-269.

[9] Gmehling J., Onken U.‘Chemistry Data Series.’, Volume IDECHEMA, Frankfurt am Main, Germany, 1977.

[10] Haase R., Siry M.‘Diffusion im kritischen Entmischungsgebiet binärer flüssiger Systeme.’Z. phys. Chem., NF 57, 1968, pp 56–73.

[11] Kohler F.‘Zur Thermodynamik des Systems Wasser-Triäthylamin.’Mh. Chem., 82, 1951, pp 913–925.

[12] Malanowski S., Anderko A.‘Modelling phase equilibria. Thermodynamic background and practical tools.’Wiley, New York, USA, 1992.

[13] Neckel A., Volk H.‘Zur Thermodynamik des Systems n-Hexan–Nitrobenzol.’Mh. Chem., 95, 1954, pp. 822-841.

[14] Raal J.D., Mühlbauer A.L.‘Phase equilibria. Measurement and computation.’Taylor and Francis, Washington D.C., USA, 1998.

[15] Reid C.R., Prausnitz J.M., Poling B.E.‘The properties of gasses and liquids.’, 4th ednMcGraw-Hill, New York, USA, 1988.

[16] Simmrock K.H., Janowsky R., Ohnsorge A.‘Chemistry Data Series.’, Volume IIDECHEMA, Frankfurt am Main, Germany, 1977.

[17] Sørensen J.M., Arlt W.‘Chemistry Data Series.’, Volume VDECHEMA, Frankfurt am Main, Germany, 1980.

[18] Stanley H.E.‘Introduction to phase transitions and critical phenomena.’Oxford University Press, Oxford, UK, 1971.

[19] Thomaes J.‘Thermal diffusion near the critical solution point.’J. Chem. Phys., 25, 1956, pp 32–33.

[20] Weast R.C., Astle M.J., Beyer W.H. (eds).‘CRC Handbook of chemistry and physics.’CRC Press, Inc., Boca Raton, USA, 1983.

[21] Wu G., Feibig M., Leipertz A.‘Messung des binären Diffusionskoeffizienten in einem Entmischungssystem mit Hilfeder Photonen-Korrelationsspektroskopie .’Wärme- u. Stoffüb., 22, 1988, pp 365–371.

MODELLING LIQUID-LIQUID BATCH EXTRACTION

85

6. MODELLING LIQUID-LIQUID BATCH EXTRACTION

In this chapter we will investigate a two-film multicomponent diffusion model of a Lewis cell.Ideally, this model only requires information on certain physical properties of (mixtures of) thechemical components present, and on the conditions under which the extraction is carried out.

The model consists of two parts. The first part (section 6.2) concerns the computation of thefluxes through the phase interface, while in the second part (section 6.3) these fluxes are usedfor the calculation of the temporal composition behaviour of the phases. In the last section ofthis chapter the model will be validated against LL extraction measurements on the systemglycerol-water-acetone by C.Y.Low [11] (section 6.5). Also, the model is compared with analternative model proposed by Krishna et al. [8] (section 6.6). First of all however, we willbriefly discuss the process of batch extraction in a Lewis cell, so as to make sure that we knowwhat we are talking about.

6.1 THE PHYSICAL SYSTEM

The model outlined in this chapter is applicable to all ternary systems of type 1†. In liquid-liquid extraction, each component usually has a specific function. It is either• the solute, which is transferred from the raffinate to the extract phase,• the raffinate component, which constitutes the main part of the raffinate phase, or• the extract component, also called the solvent, which is the main component of the extract

phase.In the case of the system glycerol-water-acetone, water is obviously the solute. It is not quiteclear however, what the functions of glycerol and acetone are, because no-one in his right mindwould consider extracting water from acetone with glycerol or vice versa. At least, not as aserious separation process. Based on the distribution coefficient of water however, it seemsmost obvious to assume that acetone is the raffinate and glycerol the extract component (seefigure A2). The distinction between the extract and raffinate components becomes clearer inthe next chapter, where the extraction of an industrially relevant mixture is investigated.

The raffinate and extract components are only partly miscible. It is customary that in a ternaryphase diagram of type 1 systems, the two-phase region is bounded by the horizontal axis. Thisaxis must therefore represent either the raffinate or extract component, while the vertical axismarks the content of the solute. The components will be ordered and numbered accordingly:1. glycerol (independent),2. water (independent),3. acetone (dependent).For notational reasons, the phases will also be numbered. Phase number 1 is the acetone-richphase, and phase 2 therefore is the glycerol-rich phase.

† The classification of multiphase liquid systems as defined in the DECHEMA series [15] is used.

Chapter 6

86

Now that we have defined the chemical system we will have a look at the batch extractionapparatus, a Lewis cell (see figure 1a).

Figure 1. a) Side view of a Lewis cell for batch extraction. b) Schematic representation of the two-phase region near the interface, rotated over 90°. The horizontal axis represents the spatial co-ordinate zperpendicular to the phase interface, and the vertical axis the composition of the liquid, x. The symbolsδ(1) and δ(2) denote the film thicknesses.

The two immiscible phases are in contact through their interface. Each phase is mixed by astirrer, and baffles are put in to prevent the interface from being deformed by a vortex. Thedesign of Lewis cells found in literature often includes all sorts of bells an whistles to improvethe flow patterns in both phases as well as the stability of the phase interface. We will assumethat under these conditions the film model applies at the interface. Figure 1b shows a schematicrepresentation of the two-phase system near the interface. With this picture in mind, we can setup the equations that will eventually yield solutions for the fluxes through the phase interface.

6.2 SOLVING THE FLUX EQUATIONS

To start with, we give away that the solution of the fluxes did not proceed without difficulties.Below, a number of solution methods is given in order of increasing complexity. Naturally, wewould not have given the more intricate ones if the rougher methods would have worked. Thesections to follow can be seen as a summary of the trials we have made. Not that it is complete:the deadwood (viz. those efforts that failed and that could not be used to devise more advancedmethods) has been cut away so as to save paper.

The central question of this section is how to evaluate the fluxes between the two immisciblephases in a ternary mixture for given phase compositions. The film model is chosen as thebasis for the description of the mass transfer across the interface. The intra-film mass transfer isdescribed with the Maxwell-Stefan transport equations:

MODELLING LIQUID-LIQUID BATCH EXTRACTION

87

x ac

x x

Ðii i

i jn

j in

ijj i

n

∇ =−

=≠

∑ln1

1 2N N

, , (1)

From now on, the strict notation introduced in chapter 2 is abandoned: in this chapter a flux isunderstood to be a mole flux. To facilitate the mathematics, it is assumed that the problem isone-dimensional. Still, the equations are too complex to solve analytically, and we have toresort to numerical methods. The two options open to us are• the use of a difference approximation of the transport equation, or• numerical integration.The first method is faster by far and it also turns out to be more stable from a numericalperspective. But it is also less accurate, and the question is how serious this is in the case of thestrongly non-ideal systems we are dealing with. Both methods are discussed below.

6.2.1 Difference approximation

The aim of using a difference approximation is to simplify the differential equations toalgebraic equations by eliminating the ∇-operator. There are two sensible options for thedifference approximation of ∇ln ai. If we let δ( p) be the thickness of the film in phase p, it caneither be taken as ∆ln ai

( p) / δ( p) or as ∆ ai( p) /( a i

( p) δ( p) ). The bar over the symbol ai – or overany other symbol – indicates that the quantity considered is to be evaluated at a compositionthat is representative for the film. Usually this is some sort of mean composition, and itsdefinition, which is somewhat arbitrary, is considered later on. Meanwhile, we write thetransport equations in the form

τδ

ip

ip

p

p ijp

j i

n

dc

f i pb g b gb gb g

b g= − = = =≠

∑ 0 1 2 1 2, , , , (2)

The driving force terms di( p) depend on which difference approximation is used for ∇ln ai

( p).The di

( p) that correspond to the two different approximations are

d x aip

ip

ipb g b g b g= ∆ ln (3)

and

d xa

a

ai

pi

p ip

ip

ip

ip

b g b gb g

b gb g

b g= =∆ ∆

γ(4)

Equation (3) has the advantage that it is easier to combine with the UNIQUAC or NRTLmodel, which give an explicit expression for ln ai rather than for ai itself. We will not use thisequation however, because the resulting equations are numerically less stable than thosederived with equation (4). Besides, (3) is said to be a less accurate approximation anyway. Thedifference of ai over the film in phase p equals:

∆ a a aip

ip

ipb g b g b g= −+ −, , (5)

The subscripts ‘+’ and ‘−’ stand for the upper and lower film boundaries, respectively. Thefriction terms fij

( p) are given by

Chapter 6

88

fx N x N

Ðf fij

p ip

jp

jp

ip

ijp ij

pji

pb gb g b g b g b g

b gb g b g=

−⇒ = − (6)

From these equations we can see that the full set of variables consists of• the bulk compositions x−

(1) and x+(2),

• the interface compositions x+(1) and x−

(2),• the fluxes Ni

( p) of all components in both films,• the film thicknesses δ(1) and δ(2),• the total concentrations c( p),• the activities of all components for all bulk and interface compositions,• the activities ai in both phases, and• the diffusivities Ðij for both phases.In general, the total concentration, the activities, the diffusivities and even the film thicknessesdepend on composition. To reduce the number of variables, we need mathematical expressionsfor these dependencies, which will be specified in section 6.4. With these relations the last fiveitems can be eliminated from the above enumeration of variables, because they can then bederived from the compositions in the film. As a result, the concentration and the film thicknessin eq. (2), as well as the diffusivities in eq. (6) will be evaluated at the representative filmcomposition.

We are now ready to identify the process parameters and variables. In this section the bulkcompositions x−

(1) and x+(2) are assumed to be known and so they are parameters. The fluxes,

whose values we seek to obtain, are variables. Furthermore, the transport equations depend inmany ways upon the interface compositions x+

(1) and x−(2) – mostly through the mean film

compositions – and therefore these are variables too.

So, we have 12 variables: 3 interface mole fractions and 3 fluxes for each of the two phases. Tofind numerical values for all of these means that we must have an equal number of equations.These equations are:• 2 mass transfer equations for the film in phase 1,• 2 mass transfer equations for the film in phase 2,• 2 interface mole fraction summations: ∑ x+

(1) = ∑ x−(2) = 1,

• 3 flux continuity equations at the interface.So we need three more equations. We will assume that the compositions at both sides of theinterface are at equilibrium, and then the three additional equations are• 3 interface equilibrium equations εi, one for each component.These three are the bootstrap equations which tie the transport equations to the system in thecell†. They can be defined in two ways. In the first place on the basis of distributioncoefficients:

ε i i i im x x i n= − = ∈− +, , , ,...,1 2 0 1b g b g l q (7)

† As has already been mentioned in chapter 2, this is a case where the bootstrap equations are not linear.

In fact, they do not even explicitly involve the fluxes at all.

MODELLING LIQUID-LIQUID BATCH EXTRACTION

89

The distribution coefficients mi can be fitted as functions of one of the four variables yi,±( p)

from binodal data. The second definition of the equilibrium equations is the thermodynamicdefinition of phase equilibrium:

ε i i ia a i n= − = ∈+ −ln ln, , , ,...,1 2 0 1b g b g l q (8)

The first method is more robust, but it is also less precise. The solution of the flux equationswith eq. (7) can be used as a starting guess for the solution of the flux equations with eq. (8).

In any case, we could now solve the above equations but it is better to reduce the number ofvariables beforehand. Especially the relations between the mole fractions, and the fluxcontinuity equations are easily substituted into the other equations, which reduces the numberof variables to 7. The remaining equations are

τε

ip

i

i p

i

b g = == =

0 1 2

0 1 2 3

, , ,

, , ,(9)

The set of (independent) variables of these equations is

y y y y N N N11

21

12

22

1 2 3, , , ,, , , , , ,+ + − −b g b g b g b g (10)

It may seem that in the transport equations, through the fi 3( p), we have introduced two

unknowns too many. However, x3( p) is not so much a variable in these equations but merely a

symbol denoting 1 − x1( p) − x2

( p). What the whole problem now boils down to, is to solve theset of equations (9), which we will do with the Newton-Raphson method.

In view of the crudeness of this method for solving the flux equations, we should not besurprised when it turns out to be inaccurate for the strongly nonideal fluid systems considered,especially if the phases are far from equilibrium. A way to mitigate this ‘inconvenience’ is tosubdivide the films into a number of subfilms or ‘slices’. This requires a slight modification ofthe transport equations (2). If ξ( p) denotes the number of slices in film p, and r is the currentslice number, we have

τγ

δξi r

p i rp

i rp

rp

rp i j r

p

j ii j r

p i rp

j j rp

i

i j rp

pa

cf f

x N x N

Ðr,

,

,

, ,, ,

,

, , ,...,b gb g

b gb gb g

b g b gb g b g

b gb g{ }= − =

−∈

≠∑

∆1 (11)

The overbars now refer to quantities at slice-representative compositions. Similarly, thedifference ∆ a equals a r + − a r −. The only quantity whose value is not immediately obvious isthe slice thickness δr

( p). The simplest way to evaluate it is to calculate the total film thickness atthe overall mean film composition, and let δr

( p) be one-ξ( p)th part of that. On the other hand:now that we know more about the composition profiles in the films than just the bulk andinterface mole fractions, we are able to make a better guess about the mean compositions. Wecould define them to be the average of the mean slice compositions:

x xp

p rp

r

p

b gb g

b gb g

==

∑1

1ξ

ξ

(12)

Chapter 6

90

These compositions can then be used to calculate the overall film thicknesses, which in turndetermine the slice thicknesses.

There are 2 (ξ(1) + ξ(2)) + 3 variables, namely

y y y y y y y y N N N1 11

2 11

1

1

2

11 1

22 12

1

2

2

21 2 31 1 2 2, , , , , , , ,

, , ..., , , , ,..., , , , ,b g b g b g b g b g b g b g b gb g b g b g b gξ ξ ξ ξ

(13)

These variables are balanced by the equations

τ τ τ τ τ τ τ τ ε ε εξ ξ ξ ξ1 1

12 11

1

1

2

11 12

2 12

1

2

2

21 2 31 1 2 2, , , , , , , ,

, ,..., , , , ,..., , , , ,b g b g b g b g b g b g b g b gb g b g b g b g (14)

It is clear that the solution of this multi-slice method takes more time than that of the single-slice method, because of the increased number of variables.

At this point it becomes necessary to straighten out how the representative film compositionsare defined. As was mentioned above, this composition is usually chosen as the meancomposition. The mean of the composition over a film which extends from z− to z+ is

yz z

y z zz

z

=−+ − −

+z1 b gd (15)

The distance z+ − z− is the film thickness δ. The problem is, that we do not know much aboutthe composition profiles y(z), and we will have to make certain suppositions about them to beable to evaluate their mean. Three possible choices will be discussed below: the linear mean,the cubic mean, and the logarithmic mean.

6.2.1.1 Linear mean composition

The first and simplest possibility is to assume them to be linear, in which instance their meanbecomes equal to the arithmetic mean:

y y y= +− +

12 e j (16)

This expression is pleasantly simple, because it does not depend on δ. We expect this approachto become increasingly inaccurate as the composition profiles become more strongly curved.Such curved profiles arise mainly in the case of large fluxes, or in other words: if the twophases are far removed from equilibrium. Therefore, it may be important to have a moresophisticated method for such cases.

6.2.1.2 Cubic mean composition

Given the composition and the fluxes anywhere in the film, we can use the differential form ofthe Maxwell-Stefan equations (20) to evaluate the spatial composition gradient y′. Thus, at anyof the (sub)film boundaries we do not only know the composition, but also the compositiongradient. Given the thickness of a (sub)film, there is one cubic polynomialy (z) = a z3 + b z2 + c z + d which satisfies all four conditions y ( z−) = y−, y ( z+) = y+, y′( z−) = y′−and y′ ( z+) = y′+. The mean of this polynomial over the (sub)film is given by

MODELLING LIQUID-LIQUID BATCH EXTRACTION

91

y y y y y= + + ′ − ′− + − +

12

16 δ e je j (17)

This expression does depend on the film thickness. This is not a fundamental problem, but itmay force us to include the film thicknesses in the set of variables, and certainly doescomplicate the (analytical) evaluation of the Jacobian of the transport equations (11) a greatdeal.

6.2.1.3 Logarithmic mean composition

Another approach that could be used in cases where the linear mean is not good enough is thelogarithmic mean

y y yi i i= − +, , (18)

Although the trials from section 6.5.1 seem to suggest that this method gives better results thanthe linear mean, it is completely unclear why this should, in general, be so. Therefore, we willnot pursue this option any further.

6.2.2 Numerical integration

The flux equations can also be solved by (numerical) integration of the transport equations. Tothis end, we use their general formulation used in eq. (2.33). The gradient of y is given by

′ = −y

cB n1 1Γ N (19)

In the current situation there are only three components and there is only one spatial co-ordinate that matters. Hence, this equation can be simplified to

′ = −

′ = − +=

−∑ ∑∑ ∑

y f f c

y f f cf

x N x N

Ðj j

j jij

i j j i

i j

1 2 2 1 1 2 2

2 2 1 1 1 1 2

Γ Γ

Γ Γ, ,

, ,

/

/

ΓΓ

ΓΓwith (20)

These relations express y′ as a function of y and N only, because, as in the previous sections,we assume that all other quantities involved (γi, c and Ðij) are functions of y. Thus, for givenfluxes, and with the (known) bulk composition as the starting point, we can find thecomposition profile in a film by numerical integration. The length of the interval over whichthe equations are integrated, the film thickness δ, is assumed to depend on the compositionprofiles. These profiles in turn depend on the integration interval length. This circulardependence makes it necessary to treat the film thicknesses as variables. This problem hasalready been mentioned in the discussion of the cubic mean composition.

Another difficulty is that we cannot directly use a simple non-linear equation solver to evaluatethe variables, because the transport equations form sets of coupled differential equations. Infact, what are the unknowns, and what equations do we have to solve them from? Clearly, asthe interface compositions follow from the integration of the Maxwell-Stefan equations, theycan no longer be variables, which means that the set of remaining variables consists of 5elements: the fluxes and the δ’s. Thus, the differential equations can be seen as a function

Chapter 6

92

relating these variables to the interface compositions, and the solution is the set of fluxes andthicknesses for which these compositions are such that• equilibrium reigns at the interface (εi = 0, see eq. (8)), and• the variables δ( p) and the thicknesses computed using eqs. (27) and (28) are the same:

δ( p) − δ( η ( p) ) = 0.Again, there are several ways to compute the film thickness with this latter equation, dependingon how the mean composition is defined. Once the function that maps the five variables ontothe above five conditions is established, a conventional root-finding routine can be used to findthe solution.

Numerical integration, with its higher accuracy as compared to the difference approximationmethod, is more time consuming and less stable. The solution of the variables found with eithermethod can be used to simulate the dynamic processes in the Lewis batch extraction cell.

6.3 SOLVING THE DYNAMIC EQUATIONS

During the extraction process, mass is exchanged between the two phases, and so thecomposition as well as the volume of both phases change in time. These changes are governedby the differential equations

d

d

n

tN S i pi

pp

i

b gb g= − = =1 1 2 3 1 2, , , , , (21)

The symbol S stands for the area of the phase interface. For the computation of the timedependence of the amounts of the three components in the two phases these three differentialequations must be solved simultaneously. It is assumed that the initial amounts of thecomponents in both phases are known. Equations (21) can be solved with the Euler method,which is rather primitive, but apart from its simplicity it has, in this case, another advantage. Itis much easier, namely, to restart the calculations in case the program crashes.

The transient compositions of the two liquids is computed as follows. We know the initialamounts of each substance in both phases. With these amounts we can calculate the bulk molefractions, and subsequently solve the flux equations. Next, a time step ∆t is defined. Providedthat this time interval is short enough, the amounts of the components transferred during thetime interval ∆t (almost) equal ∆ ni

( p) = (−1) p Ni S ∆t. The mole numbers at t = ∆t thus becomeni

( p) (t = ∆t) = ni( p) (t = 0) + ∆ ni

( p). Because we now know the amounts at the new time level, thesame operation is repeated over and over again until a situation is reached which meets certainpredefined stop conditions. The accuracy of the Euler method can be tuned by varying ∆t. Itappears that for the dynamic equations satisfactory accuracy levels can be obtained with ∆tvalues that result in acceptable computation times. Before we can actually solve the dynamicequations, however, we must say something about the composition dependence of thequantities we need for solving the flux equations.

6.4 COMPOSITION DEPENDENCE OF RELEVANT PHYSICAL PROPERTIES

To solve the flux equations, expressions must be defined that link the total concentration, theactivities and the diffusivities to the composition of the mixture. These expressions depend on

MODELLING LIQUID-LIQUID BATCH EXTRACTION

93

the chemical system and temperature regarded: glycerol(1)-water(2)-acetone(3) at 298.15 K. Inthis section these expressions are specified.

6.4.1 Total concentration

A lack of experimental data on this subject forces us to use a simple model here. We willassume that the partial molar volumes are composition invariant, and therefore equal the molarvolumes vi

o of the pure substances. The molar volume v then becomes

v x vi ii

n

==∑ o

1

(22)

The molar volumes vio can be evaluated from the mass densities and the molar masses of the

pure substances, which are listed in table 3 at page 95. The total concentration of the mixture,being the reciprocal of its molar volume, can then easily be evaluated.

6.4.2 Activities

For the thermodynamics of the system we need a thermodynamic model such as UNIFAC,UNIQUAC or NRTL. The UNIFAC model has the advantage that no experimental data areneeded, but is therefore also the least accurate, so we must use either the UNIQUAC or NRTLmodel. The UNIQUAC and NRTL interaction parameters of the system glycerol(1)-water(2)-acetone(3) are listed in table 1. The volume parameters ( r in UNIQUAC jargon) for the threecomponents are 3.5857, 0.9200 and 2.5735, respectively. The area parameters (often called q)are, in the same order: 3.0600, 1.4000 and 2.3360. These parameters are fixed and component-specific. For a description of the UNIQUAC and NRTL model equations, see reference [15].

Table 1. UNIQUAC and NRTL interaction parameters of the ternary system (Krishna et al. [9]).

We must keep in mind, however, that these parameters result from parameter optimisationsagainst LLE data only. Hence, they are essentially valid only locally, that is, near the binodalcurve. They do not necessarily correctly predict the activities for compositions that do not lie inthe direct vicinity of the demixing zone. For our computations we not only need the binodalactivities at the phase interface, but also at other, non-binodal compositions and the question is:how accurate are the above parameters for the description of such compositions? When eitherof the above models model is compared with VLE data available for both miscible binarymixtures (see figure A1), we see that the answer is: not very accurate. This may adverselyaffect the results of the batch extraction model.

UNIQUAC NRTL (αij = 0.2)

component Aij Aji Aij Aji

i j (K) (K)

1 2 30.029 −403.23 −385.51 −453.181 3 162.61 191.62 258.79 735.362 3 146.20 −23.520 624.75 −198.33

Chapter 6

94

To test this, we need a sufficiently accurate model that describes both the LLE data and theVLE data of the two miscible binaries. Preferably, this new model must predict the LLE data atleast as well as the UNIQUAC model does. Refitting the UNIQUAC model therefore is not anoption. What model should we use then? An obvious candidate is the multicomponentextension of the Margules equation: gE = ∑ i ∑ j g

Ei, j, with j ∈ {2,..., n} and 0 < i < j. The binary

terms gEi, j equal x i x j ∑ k B k, i, j (x i − x j) k, where k ranges from 0 to any non-negative integer l.

Its abundance of freely adjustable parameters, the possibility to set the number of parametersdifferently for each (i, j)-binary, as well as its mathematical simplicity make the Margulesequation very attractive. It turns out that for a fit that performs better than the LLE fit ofKrishna et al. we need a four-parameter extension for the glycerol-acetone mixture.Unfortunately, we find that a satisfactory fit can only be obtained with sets of parameters forwhich gE

1,3 has too many zeros to be of any use.

Maybe the SSF equation, famed for its capability of dealing with non-ideal systems, and whichgave the best results in sections 5.3 and 5.4, can do the job. The form of the multicomponentSSF equation is essentially the same as that of the Margules equation: gE = ∑ i ∑ j g

Ei, j. The gE

i, j-terms are given by

g x x x xB

x b xi n j i ni j i j i j

k i j

i k i j jk

l

,, ,

, ,

, ,..., , ,...,E = ++

= − = +=

∑d id i

2

21

1 1 1 (23)

This equation differs slightly from the form given by Rogalski and Malanowski [13]†. Thereason is that this form requires less floating point operations, but it does not reveal that allb k, i, j must be positive. Further, to ensure invariance under component permutation, we musthave b k, i, j = 1/ b k, j, i and B k, i, j = B k, j, i / (b k, j, i)

2. Taking l = 2 it is possible to find a satisfactorycombined VLE / LLE fit with this equation. The corresponding parameters are tabulated below.A more detailed description of how these parameters were obtained is given in appendix A.

Table 2. Parameter values of the multicomponent SSF model for the combined LLE and VLE data.The number of digits may be on the high side, but there was no time to test the sensitivity of the model toround-off.

6.4.3 Diffusivities

For the estimation of the three diffusivities at arbitrary compositions, we have two methods atour disposal: one by Wesselingh and Krishna [16] † Eq. (23), which uses 2l parameters, also differs from the binary SSF equation, eq. (5.8), which has l

parameters. The B’s are ‘fundamentally’ different from the b’s, and eq. (23) is therefore preferred to eq.(5.8).

i, j = 1, 2 i, j = 1, 3 i, j = 2, 3

B1, i, j −0.2745099 2.998848 0.4675026b1, i, j 1.469884 0.7747245 0.5631358B2, i, j −0.1128089 −0.5751752 1.650551b2, i, j 0.3320776 0.5369349 1.708807

MODELLING LIQUID-LIQUID BATCH EXTRACTION

95

Ð Ð Ði j i jx x x

i jx x x

jj i

ii j

= → + −→ + −1 1 2

1 1 2e j d id i d i/ /(24)

and another by Kooijman and Taylor [7]

Ð Ð Ð Ð Ði j i jx x

i jx x

i kx

j kx x

k i j

nj

ji

ik k

k= → → → →

≠∏1 1 1 1

12e j d i d i

,

(25)

where the superscript xk → 1 serves to indicate the diffusivity at infinite dilution in a mixture of‘pure’ k. These n ( n − 1) diffusivities at infinite dilution can be estimated with an adapted formof the Stokes-Einstein equation, which was derived by Rutten [14], who found that thisequation generally gives better results than the original version:

Ðk T

r ri jx

j i j

j → =1

24 π ηo(26)

The symbol k designates the Boltzmann constant, η jo is the dynamic viscosity of pure j, and ri

is some characteristic measure of the molecular size of i, which is taken to be the Van derWaals radius [3]. The viscosities and the radii for the pure components are listed in the tablebelow.

Table 3. Molar masses, mass densities, viscosities and Van der Waals radii of the pure components at298.15 K [4, 6, 10].

6.4.4 Film thicknesses

We have no reliable relations for the values of the film thickness in both phases, nor do wehave experimental data to derive them from. We could choose to use them as ‘fitting’parameters to make the model outcome match the experimental results. But we also haveanother, rather crude, option: we can assume that the film thickness of a phase is proportionalto a certain power b of its viscosity. For dilute aqueous phases, δ is of the order of 10−5 m andthis thickness therefore corresponds to a 1 mPa s bulk viscosity. Hence, δ is related to theviscosity by

δ η ηb g = −103 5b b (27)

Unfortunately, there is no unambiguous answer to the question as to what value b should have.The relations derived in the extensive literature on the subject of mass transfer in stirred cellsare often contradictory, use Fickian diffusivities and mainly concern binary systems. Therefore,some scientific flexibility is required when we use the relation of Bulicka and Procházka [2],which is recognised as one of the more accurate. It states that for the Sherwood number inphase p we have Sh( p) ~ ψ (Sc( p) )0.5 (Re( p) )0.75. Because ψ depends on the viscosity ratio of both

component M(10−3 kg mol−1)

ρ(103 kg m−3)

ηo

(mPa s)r

(10−10 m)

glycerol 92.10 1.260 934.00000 2.73water 18.02 0.997 0.900 1.70

acetone 58.08 0.784 0.306 2.49

Chapter 6

96

phases, it could be of some influence only in the beginning of the transfer process. If weneglect this factor, we find that Sh ~ D0.5 η− 0.25. Since Sh = k d /D, and because we defined k asÐ /δ, we write δ ~ Ð − 0.5 η0.25. This proportionality also incorporates Ð because of its viscositydependence: it can be shown that the Maxwell-Stefan diffusivities as given in section 6.4.3 areinversely proportional to the viscosity. This is true only for binary mixtures, but then, theSherwood relation is also based on (pseudo-) binary diffusion. Anyway, we find that the valueof the exponent b is ¾.

Still, the exact magnitude of the film thickness remains unknown. Apart from the fact that wedo not know what value of η to use (bulk or some kind of average), it is also unclear how thisquantity is related to the composition of the mixture. To clear up this latter problem, theviscosity of a mixture is calculated by interpolation of the pure component viscosities by

ln ln oη η= ∑ xi ii

(28)

Eq. (27) cannot be expected to be very good. Firstly, for the above-mentioned reason that it isnot al all clear what viscosity should be used. Secondly, because the thickness of a film is alsolikely to depend on the level of agitation of the two phases.

6.5 RESULTS

In the introduction to this chapter, we already mentioned that we will validate the modelagainst the measurements by C.Y. Low, who investigated the process of liquid-liquid batchextraction of the system glycerol(1)-water(2)-acetone(3) extensively. The interfacial area S ofthe Lewis cell he used can be deduced from an illustration in Low’s thesis: it must have beenabout 2.835·10−2 m2. For our purposes the experiment Low indicates as ‘run 1’ suffices,because it describes the transient behaviour of a system which is initially far removed fromequilibrium. Therefore, this experiment represents the toughest test of our model. Themeasured compositions of this experiment are given in the table below.

How reliable are these measurements? Let yov be the overall composition of the contents of theentire cell. The reliability can be assessed by using the fact that at any time during theextraction the line between the two phase points in the composition diagram should passthrough the point yov. Because Low determined the initial compositions by weighing, weassume them to be more accurate than the other compositions, which were determined by othermeans. For each of Low’s measurements we can calculate the tangent of the line through yov

that minimises the distance between this line and the measured compositions. The smaller thisdistance, the more reliable the measurement (probably) is, but how do we define it?

The distance could be defined as the sum of the geometrical distances between the points andthe line. In this case the best line turns out to be the one through yov and the most distant of thetwo phase compositions. It seems a bit awkward, however, to saddle the composition of onephase with the experimental errors of both points. Instead, we will use a definition of thedistance that is related to the quantity that will be used to measure the performance of thedynamic model later on. We have a set of l (= 6) measurements yj = (y1, j

(1), y2, j(1), y1, j

(2), y2, j(2))T.

The quantity χ2, which is used to indicate the deviation between these data and thecorresponding model values yj

*, is defined as

MODELLING LIQUID-LIQUID BATCH EXTRACTION

97

χσ

2

2

11

1

1

=−F

HGIKJ

∗

==

−

=∑∑∑

y yi jp

i jp

i jp

pi

n

j

l, ,

,

b g b g

b gΠ

(29)

The integer Π (= 2) denotes the number of phases, and the σi, j( p) are the standard errors in the

measured mole fractions. We will assume that the errors are equal, and for convenience setthem all equal to unity. The measurements can be adjusted in such a way that they obey massconservation and simultaneously minimise χ2 between the measured and the adjusted data.This minimum is found to be χ2 = 1.172·10−3. This is the lowest obtainable deviation for ourmass transfer model, because mass conservation is one of its basic assumptions.

Table 4. Transient compositions of Low’s run number 1. Initial total mole numbers in phases 1 and 2were 49.47 and 46.25, respectively, whence the overall mole fractions yov,i are 0.4106 and 0.1916. Theleft numbers in each column are the data reported by Low, the right numbers are the compositions that

satisfy the mass conservation condition and simultaneously minimise χ2.

The fact that the differences between the measured and adjusted values significantly exceed thestandard errors in the measurements given by Krishna et al. [8], indicates that the measuredcompositions somehow suffer from systematic errors. This is also the reason that we do not usethese reported errors in the χ2 function.

In section 6.5.2 we will see how the model performs in describing these measurements. Butbefore that, we will go into the solution of the flux equations for the situation in the cell at t = 0s, which appears to be far from simple.

6.5.1 Initial flux solutions

First of all, the question arises as to what combination of methods and physical propertiesshould be used. All in all, the number of possible combinations is rather large. In addition tothe various flux solution methods, which are listed below for convenience, we can choosebetween the three thermodynamic models discussed in section 6.4.2, the two diffusivityinterpolation eqs. (24) and (25) from section 6.4.3, and finally, we could also consider differentfilm thickness exponents b in eq. (27). For the time being, these options are set as follows:UNIQUAC thermodynamics, eq. (24) for diffusivity interpolation and b = ½†.

† The use of this value of b has its origin in ‘historical’ circumstances.

Time Phase 1 Phase 2(ks) y1

(1) y2(1) y1

(2) y2(2)

0.0 0.0000 / 0.0000 0.2302 / 0.2302 0.8498 / 0.8498 0.1502 / 0.1502 1.8 0.0124 / 0.0124 0.2005 / 0.1982 0.7824 / 0.7824 0.1877 / 0.1853 3.6 0.0201 / 0.0204 0.1798 / 0.1733 0.7471 / 0.7474 0.2145 / 0.2072 5.4 0.0295 / 0.0301 0.1502 / 0.1454 0.7164 / 0.7201 0.2348 / 0.2290 9.0 0.0310 / 0.0333 0.1311 / 0.1191 0.6749 / 0.6780 0.2588 / 0.242914.4 0.0447 / 0.0488 0.0881 / 0.0752 0.6313 / 0.6375 0.2838 / 0.2645

Chapter 6

98

I. Difference approximation methoda. Linear mean composition

1. Without subfilms2. With subfilms

i Whole film composition average based thicknessii Subfilm composition average based thickness

b. Cubic mean composition1. Without subfilms2. With subfilms

i Whole film composition average-based thicknessii Subfilm composition average-based thickness

c. Logarithmic mean composition1. Without subfilms2. With subfilms

i Whole film composition average-based thicknessii (Subfilm composition average-based thickness)

II. Numerical integrationi Whole film composition average-based thicknessii Subfilm composition average-based thickness

Let us start at the beginning: method (Ia1). The solutions of this method corresponding to thesituation at t = 0 s, are too inaccurate to be of any use other than to calculate starting values forthe other methods. It has one advantage though: it seldom fails. The next method (Ia2i), incontrast, confronts us with a problem or two.

The most important difficulty is what we will call the dustbin effect: the interdependence of thecomplete set of transport equations (see chapter 2, page 20) forces us to discard one of them.For exact solutions, this is all right of course, but in the case of numerical approximations, itmay lead to serious problems. By discretising the equations, we introduce errors in thesolutions of the independent components, that is, those substances whose transport equationswe did not discard. These errors are automatically dumped in the solution for the flux of thedependent component, which therefore acts, so to say, as a dustbin: all the garbage ends up init. This does not necessarily lead to catastrophes, but when the exact flux of the dependentcomponent is small compared to those of the other components, the magnitude of the error mayexceed that of the flux itself, thus resulting in a solution of the wrong sign. It is thereforeadvisable to reorder the species in such a way that the component with the smallest flux is oneof the independent ones.‡ Under the current circumstances the acetone flux turns out to be verysmall compared to that of the other two components. Hence we cannot maintain the componentorder we used thus far. For programming reasons we would like to keep glycerol as the firstcomponent and therefore we change the order to glycerol-acetone-water.

Now that the dustbin effect can no longer cause the acetone flux to become negative, we cantest the various methods. Below, a summary is given of the results obtained using UNIQUACthermodynamics, eq. (24) diffusivity interpolation and a film thickness exponent b = ½.

‡ This is still no guarantee that its calculated flux is always in the right direction.

MODELLING LIQUID-LIQUID BATCH EXTRACTION

99

The numerical integration method (IIi), which is, by the way, not significantly sensitive to thedustbin effect, serves as the standard for the difference approximation trials. It shows that theacetone flux is very small as compared to the other two fluxes, but that it is definitely positive.The values of the five variables N and δ( p) are −3.016·10−2, 1.387·10−3, 6.263·10−8 mol m−2 s−1,7.379·10−6 and 5.682·10−5 m, respectively. As could be expected, the film in phase 2 is muchthicker than the one in phase 1. For the difference approximation method we find that• the fluxes depend on the component order,• the finer the discretisation, the better the results, and the smaller the dependence on the

component order,• the solution is far more sensitive to the discretisation in phase 2 than in phase 1,• the use of the cubic mean composition instead of the linear mean yields a significant

(relative) improvement of the acetone flux, but a slight deterioration of the other twofluxes,

• the use of the mean subfilm compositions to calculate the overall film thicknesses may,depending on the way in which it is done, cause them to increase or to decrease or mayeven lead to convergence failure,

• the use of a thin layer adjacent to the bulk in phase 2 in which the friction force terms areevaluated at the bulk compositions is not an effective remedy against negative acetonefluxes, see appendix B.

Based on these trials we choose method (Ia2i) with 5 subfilms in phase 1, and 20 in phase 2(from now on called 5/20 discretisation) to simulate the dynamic extraction process. Thecomponent order is glycerol-acetone-water.

The application of the NRTL model yields results very similar to those calculated usingUNIQUAC model, and the NRTL model will therefore no longer be considered. It also appearsthat the use of the SSF model, the diffusivity interpolation formula of Kooijman and Taylor, orof b = ¾, invariably leads to convergence failure. Closer analysis of these latter attempts showthat a significant portion – up to 60% – of the film in phase 2 is inactive, that is, thecompositions do not change there. This may indicate that the film thickness is seriouslyoverestimated by eq. (27), but it may also have other causes. In the next section we will seethat this does not mean that we have to write off these model options. In any case, it is clearthat the difference approximation method can be used to yield sufficiently accurate results,which saves us a lot of computation time in simulating the dynamic process.

6.5.2 Dynamic model solutions

In section 6.3 it has been explained that given the solution of the fluxes, we are able to simulatethe temporal composition paths of these phases. For example, when the fluxes are solved usinga 5/20 discretisation, the UNIQUAC model, eq. (24) and b = ½, and when the time step ∆t isset to 10.0 s†, we find the composition trajectories shown in figure 2.

† Comparison with a run with a time step of 1.0 s showed that 10.0 s is small enough. Moreover, since the

Euler method will turn out to be inadequate for the system regarded in the next chapter, it will bereplaced there by the RK4 method. A retrospective check again confirmed that 10.0 s suffices.

Chapter 6

100

Phase 1

y1

0.00 0.01 0.02 0.03 0.04 0.050.00

0.05

0.10

0.15

0.20

0.25Phase 2

y1

0.6 0.7 0.8 0.9

0.15

0.20

0.25

0.30y2

Figure 2. Plain model composition paths (solid lines). Difference approximation method (Ia2i) with5/20 discretisation, UNIQUAC, eq. (24), b = ½ and ∆t = 10.0 s. The solid black circles represent Low’sdata, the open circles the adjusted data. The triangles are the model results corresponding to Low’smeasurements, and the squares mark the final equilibrium compositions. The dotted line is the binodalcomputed using UNIQUAC, the dashed one the composition space boundary. The solid line along thebinodal is the interface composition path, which, for phase 2, lies beyond the graph boundaries. Aremarkable feature of this figure is that the measured composition of phase 1 at 14.4 ks lies within thedemixing zone. That is, the calculated demixing zone, but a comparison of the LLE data from Krishna etal. [9] shows that this point is located approximately on the binodal curve. The adjusted point, however,most likely is situated within the true two-phase region. In the next chapter, we will show that this is notan impossible situation.

It is clear that the model in this plain form does not suffice. (In view of the assumptions onwhich the model is based this is not really surprising, on the contrary.) Not only are the fluxratios wrong, which results in wrong directions of the paths, but the fluxes of water andacetone are also too small. The χ2 value is a poor 6.846·10−2. What do you do when realityrefuses to agree with your model? You take to ‘parameter optimisation’, the emergency exit forall modellers who have run out of basic principles. Clearly, the film thicknesses and thediffusivities are the main weaknesses of our model. Let us assume that the relations given insection 6.4 predict the right trends, but maybe not the right magnitudes. The ‘true’ Ð*

ij andδ*( p) are related to those predicted by equations (24) and (27) by Ð*

ij = αij Ðij andδ*( p) = β( p) δ( p). So, there are five adjustable parameters whose optimisation yields α12 = 2.157,α13 = 22.32, α23 = 0.1647, β(1) = 6.757, and β(2) = 0.2971 as well as a χ2 of 1.495·10−3. Note thatthese results are obtained by fitting the model to the original data set of Low, not to theadjusted set, although for the parameters it would not make much difference if we did. Thecomposition trajectories corresponding to the set of optimised parameters are shown in figure3.

MODELLING LIQUID-LIQUID BATCH EXTRACTION

101

Phase 1

y1

0.00 0.01 0.02 0.03 0.04 0.050.00

0.05

0.10

0.15

0.20

0.25Phase 2

y1

0.6 0.7 0.8 0.9

0.15

0.20

0.25

0.30y2

Figure 3. Fitted composition paths. Method (Ia2i), 5/20 discretisation, UNIQUAC, eq. (24), b = ½ and∆t = 10.0 s. For further explanation, see the caption of figure 2.

Now that looks better! Before we get over-enthousiastic, though, there are a few things that wehave to give account of. Firstly, the β-parameters do not accord with their expected values,because they tell us that the films in phase 1 and 2 have a thickness of about 45 and 21 µm,respectively. So, what happened to the viscosity dependence? Further, α13 and α23 are quiteextreme as well. Maybe we are willing to accept the latter factor, for it is known that both theestimation of the diffusivities at infinite dilution as well as the interpolation methods are not atall reliable, but a factor of over 20? On the other hand, extreme parameter values can also pointto insensitivity of the model to that particular parameter. Here, this is indeed the case, as can bededuced from the estimated standard errors (ESE’s) of the fitted parameters.

Although it is not possible to be exact about the magnitude of the ESE’s because of theuncertainty about the experimental errors, we can at least evaluate their relative magnitudesfrom the Hessian matrix of χ2 with respect to the parameters. The inverse of this matrix iscalled the covariance matrix, whose diagonal elements are a measure of the ESE’s. Theirvalues relative to that of α13 are ∆α12 = 9.67·10−2, ∆α13 = 1, ∆α23 = 7.38·10−3, ∆β(1) = 3.03·10−1,and ∆β(2) = 1.33·10−2. Clearly, the model is least sensitive to the values of the parameters α13

and β(1), while α23 cannot be changed much without causing a significant increase of χ2.Nevertheless, even though the model cannot be rejected as none of the diffusivities and filmthicknesses exceed the limit of the entirely incredible, the parameter values give rise to thesuspicion that there is still something wrong.

Luckily, it may be possible to devise some improvements, because the combinations of modeloptions that did not work without parameters in the previous section, might yield better resultsthan the one tested above. As an example, let us see what the SSF model can do. The result isshown in figure 4.

Chapter 6

102

Phase 1

y1

0.00 0.01 0.02 0.03 0.04 0.050.00

0.05

0.10

0.15

0.20

0.25Phase 2

y1

0.6 0.7 0.8 0.9

0.15

0.20

0.25

0.30y2

Figure 4. Fitted composition paths. Method (Ia2i), 5/20 discretisation, SSF, eq. (24), b = ½ and∆t = 10.0 s. The binodal curve and the final equilibrium compositions are now based on the SSF model.For further explanation, see the caption of figure 2.

The first thing that meets the eye is the different position of the binodal curve in the graphbelonging to phase 1. The corresponding shift of the final equilibrium composition makes thebend at the end of the bulk composition path in the previous figure disappear, which allows acloser approach of the rightmost points. This effect is probably the main cause of the fact thatχ2 is somewhat smaller than that of the above fit: 1.453·10−3. The optimised parameters are2.325, 9.386, 0.9014, 6.204, and 0.2488. Apart from β(2), the parameters now have moremoderate values, which proves that it may be worthwhile trying other combinations as well.Because the composition paths differ little from the ones shown in the figures above, we willconfine ourselves to tabulating the values of the parameters and χ2.

Table 5. A few more fit results.

Ia2iUNIQUAC

eq. (24)b = ¾

Ia2ii*

SSFeq. (24)b = ½

Ib2ii*

SSFeq. (24)b = ½

Ia2ii* SSF

eq. (25) b = ½

α12 02.145 1.921 1.930 00.4951α13 22.89 6.703 6.698 01.314α23 00.1662 0.4953 0.4946 01.159β(1) 08.031 5.384 5.423 28.17β(2) 00.1158 0.1690 0.1682 00.0526810+3·χ2 1.512 1.424 1.424 1.406

*δ

ξδ η η η

ξrp

p

p p p p

p rp

r

x x xb gb g

b g b g b g b gb g

b ge j e j= = = ∑1 1, ,

MODELLING LIQUID-LIQUID BATCH EXTRACTION

103

The conclusions that can be drawn from this table are:• the value b = ½ is too big rather than too small, at least, according to the model,• the thermodynamic model is of considerable influence on the parameter values, but not on

the magnitude of χ2,• the application of the cubic mean does not give significantly better results than the linear

mean,• the interpolation scheme of Kooijman and Taylor yields beautiful α-values, but disastrous

β’s.Especially the second point is interesting, because it shows that the model can hardly berejected on the basis of the parameter values alone.

6.6 COMPARISON WITH THE MODEL OF KRISHNA ET AL.

At this point, it has become almost inevitable to compare the model predictions with the articleof Krishna et al. [8]. They showed that the extraction paths can be described quantitativelyusing a method that is far less laborious than the one outlined in this chapter. The mainobjective of their study was to show it is impossible to describe liquid-liquid (batch) extractionwithout taking account of the coupled diffusion mechanism. To this end they only needed toprove that the Fick diffusivity matrix contained significant off-diagonal elements, and indeed itdid. The calculation of the diffusivity matrix required two important assumptions: (a) thevolumetric mass transfer coefficient matrices [K] = c S [D] / n δ† of both phases are constantduring the process, and (b) the interface immediately attains its final equilibrium state.

Do these assumptions agree with our model results? From the figures above it is immediatelyclear that the latter assumption does not. The validity of the assumption that [K] is constant iseasily verified. At any instant during the process, we are able to calculate the Maxwell-Stefandiffusivities as well as the concentration and the film thickness. The matrix [D] (which must beevaluated relative to the mole-fixed frame) follows from substitution of eq. (2.31) with s = I,into eq. (2.32). Comparison of eq. (2.33) with (2.25) shows that [D] = − [B]−1 [Γ], where [Γ] isthe familiar thermodynamic factor matrix, and with

− =+

+ −+

−+ +

+

L

N

MMMM

O

Q

PPPPB

x

Ð

x x

Ð

x

Ð

x

Ðx

Ð

x

Ð

x

Ð

x x

Ð

2

12

1 3

13

1

12

1

13

2

12

2

23

1

12

2 3

23

(30)

The transfer coefficient matrices thus calculated are:

K K

K K

1 1

2 2

0 350 0 00497

0 00624 0 401

0 651 0187

0 420 0 545

0 471 0 0819

0 267 0 462

0131 0 00455

0 0585 0 256

0 0823 0 00908

0 0323 0144

0 0988 0 00455

0 0413 0181

b g b g

b g b g

=−LNM

OQP

LNM

OQP =

LNM

OQP

=−L

NMOQP

LNM

OQP =

LNM

OQP

. .

. .

. .

. .,

. .

. .

. .

. .

. .

. .,

. .

. .

L

L

† Here, n stands for the total number of moles in the cell.

Chapter 6

104

The model settings used to obtain these results are: method (Ia2ii), SSF, eq. (24), b = ½ and thecorresponding parameters listed in table 5. The above values are, from left to right, thecoefficients at t = 0, at t = 14.4 ks and the time-average coefficients. All of them are evaluatedat the mean film compositions as defined at the bottom of table 5. It is clear that the transfercoefficients are far from constant, and they also differ quite a lot from the values found byKrishna et al. A reason to have a closer look at their paper, which leads to the followingcomments.• It is not clear where the reported final equilibrium compositions come from; at least, we

have not been able to find them Low’s thesis.• The model does not, in spite of all the sincere words about mass conservation at the

beginning of the article, reveal the intrinsic inconsistencies in Low’s measurements. Thephases have become two completely separated entities.

• The authors state that an equilibration experiment takes about 6 hours. The compositionpaths they present to prove the correctness of their equations do not, however, correspondto this time span. In fact, the path of phase 2 has then only just reached the sharp bend. Thetime required to allow this phase to approach its final equilibrium state within one mole%(|| y − yeq|| < 0.01) is over 24 hours.

• Finally, the model presented is a bit of a self-fulfilling prophecy, because everythingdepends on the path one has in mind. The method works as follows: first, consider themeasured compositions and draw a path that looks all right. Whether it is the correct path isimmaterial (within certain limits of course). Determine the tangents to the compositiontrajectory at the initial and final compositions. This roughly fixes the form of the path,because the model predicts that the tangent between these two points will gradually changefrom the initial to the final value. The form of the path is not enough, though, because ittells us nothing about the transfer rates. These rates can be determined by linear regressionof the logarithm of the difference of the interface and the measured composition versustime, but after the composition differences have been multiplied by the modal matrix of[K ]. This modal matrix, and that is the trick, also depends upon the chosen tangents. Theeffect is that, for example, the tangent of phase 2 at the final composition point can beincreased by a factor 2.3 (Krishna et al. determined it to be 1/2.3) without any ill effects. Infact, the sum of square deviations of the linear regression used for the determination of the‘fast’ eigenvalue decreases more than 10%.

These comments do not affect the conclusions of the article in question, namely that coupleddiffusion plays an important role in liquid-liquid mass transfer. It is clear, however, that inview of the above, we do not have to be too disturbed by the discrepancies between our modeland the results obtained by Krishna and co-workers.

6.7 CONCLUSIONS

The most important conclusion of this chapter is that the plain model is not accurate enough tobe of use in a practical sense: one glance at figure 2 suffices to prove this. Whether this is aresult of fundamental flaws in the model or of a defective knowledge of the physical propertiesinvolved, is not clear. But then, it was clear beforehand that the development of this model ismore or less an academic exercise. In the first place, the examination and modelling of batchextraction in a Lewis cell is an academic occupation by definition, and, secondly, Krishna et al.have shown that the whole process can be predicted by methods that are a lot less laboriousthan the ones discussed in this chapter. Then why did we do it? Because, as we stated in the

MODELLING LIQUID-LIQUID BATCH EXTRACTION

105

first paragraph of this chapter, it is the intention to use the model in the next chapter to showthat coupled diffusion can, in theory, lead to the formation of dispersions in extractionprocesses. Also, the model enabled us to check some of the assumptions made by Krishna etal., which were seen not to hold.

The optimised model is able to describe the measurements of C.Y. Low reasonably well, butthere are a few points on which it may be criticised. In the first place, the number ofparameters, five, is more than one would like, because it affects the credibility of the model.Secondly, the parameter values depend strongly on the model settings: the differenceapproximation method, the thermodynamic model, the diffusivity interpolation method and thefilm thickness exponent. For some combinations the parameters, which ideally should equalunity, are quite extreme, for others they fall within the (subjective) boundaries of what iscredible. Whichever combination of settings we choose, however, they all share oneremarkable property: the film thickness in the phase with the highest viscosity is predicted notto be thicker than that in the low-viscous phase. This is in contradiction to what we intuitivelyexpect, and we have no explanation for this phenomenon. It may have something to do with theenormous viscosity differences between the phases. This difference also has its influence onthe diffusivities at infinite dilution: the diffusivities in ‘pure’ glycerol are very low as comparedto those in the other pure liquids. Such great differences are liable not to improve the accuracyof any model. A suggestion to anyone who is planning to investigate liquid-liquid extraction isto compare it to measurements on a system where such differences are absent.

APPENDIX A. COMBINED FIT OF THE LLE AND VLE DATA

In this appendix, we take a closer look at how the SSF parameters for the system glycerol-water-acetone (table 2) were obtained. Assume that we have a set of lLLE measurementsyj = (y1, j

(1), y2, j(1), y1, j

(2), y2, j(2))T. Usually, the parameters of the thermodynamic model are

adjusted so as to minimise the optimisation objective function χ2

χσ

2

2

11

1

1

=−F

HGIKJ

∗

==

−

=∑∑∑

y yi jp

i jp

i jp

pi

n

j

l, ,

,

b g b g

b gΠLLE

(A1)

This equation is virtually identical to the objective function of the mass transfer model, eq. (29). The difference lies in the model values yj

*. In the case of the mass transfer model, the modeloutcome was linked to a particular measurement by the measurement time. Here it is notimmediately clear which model values correspond to the measured ones. Let us thinkpositively, and assume that there are no systematic errors in the measurements, and thattherefore the yj

* are the model compositions nearest to the measured ones. If all experimentalerrors σi, j

( p) equal unity, χ2 is the same as the function Fx which is used in the DECHEMAseries [15]†. However, in doing so we deprive ourselves of the possibility to weight the datadifferently.

The RMSD (Root Mean Square Deviation) between the data and the model is defined as

† Part 2, section 7: ‘Parameter Estimation from Ternary LLE Data’, eq. (2).

Chapter 6

106

RMSDl n

y yi jp

i jp

pi

n

j

l

=−

−RS|T|

UV|W|∗

==

−

=∑∑∑1

1

2

11

1

1

0 5

LLE

LLE

b g e jb g b gΠ

Π

, ,

.

(A2)

In the case of unit errors, we have RMSD = (χ2 / lLLE (n −1) Π )½.

The calculation of the RMSD is rather complicated, because it requires the points yj* on the

binodal closest to the measured compositions. Rigorous computation of these points takes toomuch time, and therefore they are estimated. To that end, we compute a set of more or lessevenly spaced binodal compositions††, for all of which the distance to every measurement canbe calculated. The smallest distance between a measurement and the binodal curve isdetermined by means of interpolation. The accuracy of these estimates depends upon thespacing between the computed binodal compositions. This spacing is quantified by the length|| ∆ y ||, which is the maximum distance on either branch of the binodal between two calculatedequilibrium compositions, and which therefore determines the ‘density’ of these points. Thesmaller || ∆ y ||, the more steps are required to compute the binodal, and the more time it takes,of course. If, however, || ∆ y || is too big, the calculation of sharply curved binodals becomesdifficult. The influence of || ∆ y || on the RMSD is tabulated below.

Table A1. The computed RMSD’s for both the UNIQUAC and NRTL model. The values correspond tonew fits to the experiments of Krishna et al. [9]

Krishna et al. give RMSD’s of 1.013·10−2 and 1.014·10−2, respectively. This is strange, sinceour method when fed with their set of parameters yields values of 1.070·10−2 and 1.079·10−2.The source of the differences between these values is not clear. Anyway, on account of theabove table, we decide that a norm of 1.0·10−3 is sufficiently accurate for a goodapproximation.

Now for the real work: the simultaneous fit of the LLE and binary VLE data. The addition ofthe latter data requires an adaption of the objective function (A1):

χσ σ σ

2

2

11

1

1

12 12

12

2

1

23 23

23

2

1

12 23

=−F

HGIKJ

+−F

HGIKJ +

−FHG

IKJ

∗

==

−

=

∗

=

∗

=∑∑∑ ∑ ∑

y y P P P Pi jp

i jp

i jp

pi

n

j

lj j

jj

lj j

jj

l, ,

,

, ,

,

, ,

,

b g b g

b gΠLLE

(A3)

The integers l12 and l23 are the numbers of vapour pressure measurements of the systemsglycerol-water and water-acetone. The model vapour pressures P*

12 of the system glycerol-water and P*

23 of the system water-acetone are calculated using the ideal gas GE method treatedin chapter 5. The pure component vapour pressures – at 298.15 K, of course – were not

†† For the computation of the binodal curve, see refs. [1] and [12].

|| ∆ y || (–) RMSDUNIQUAC (–) RMSDNRTL (–)

1.0·10−2 1.038·10−2 1.035·10−2

5.0·10−3 1.039·10−2 1.038·10−2

1.0·10−3 1.039·10−2 1.040·10−2

5.0·10−4 1.039·10−2 1.041·10−2

MODELLING LIQUID-LIQUID BATCH EXTRACTION

107

regarded as fit parameters, but were set at 0.0, 3.158·103, and 3.079·104 Pa for glycerol, waterand acetone, respectively. The vapour pressure of pure glycerol could not be found inliterature, but from extrapolations it is clear that it must be very low. The σ’s in the objectivefunction were maintained especially for the purpose of assigning different weights to thedifferent data sets. The difficulty is namely that the LLE data and the VLE data differ severalorders of magnitude. Setting the errors of all data sets equal will most likely lead to a bad fit ofthe LLE set, because it is then ‘underweighted’.

At this point we could go on a while about all kinds of practical and theoretical considerationson this and other subjects, and about the considerable adrenaline levels that were sometimesreached during the quest for an acceptable result, but that is probably not in the interest of thereader. Remains to say that, in the end, all three data sets were simultaneously put through aMarquard-Levenberg fit against the SSF equation (23) with four parameters per componentpair. The variance of the LLE data was fixed at 5.0·10−5 absolute, and the errors of the VLEdata set at 5% of the measured value. Not that we believe that these are the true errors(especially the latter are much smaller in reality), but they just happen to lead to reasonableresults. The numerical parameter values are given in table 2. A graphical representation of theresults is given in the figure below.

water / acetone

yacetone

0.0 0.2 0.4 0.6 0.8 1.0

γ

0

2

4

6

8

VLE fit UNIQUAC SSF

glycerol / water

ywater

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

Figure A1. Comparison of the ‘true’ and UNIQUAC-predicted activity coefficients of the two non-demixing binaries of the ternary system glycerol-water-acetone. Glycerol-water: Kirgentsev andLukyanov ([5], part 1b, p. 186), water-acetone: Wang, Tang and Hu, ([5], part 1b, p. 149). The ‘true’activities are found by fitting a four-parameter binary SSF equation (eq. (5.8)) to the VLE data, whichgives accurate results.

These results may not seem ideal, and given the amount of work needed to arrive at them, wecannot but concur. Yet, the resulting model is a better description of the ternary system as awhole than the UNIQUAC model using the parameter set of Krishna et al. The LLE-basedRMSD has decreased notably from 1.070·10−2 (or 1.013·10−2 if you wish) to 9.369·10−3. Do notlet yourself be fooled by the appearance of the left-hand graph of figure A2, also have a closelook at the distribution ratio. Moreover, the difference between the model and the

Chapter 6

108

measurements for the glycerol-rich phase at low water contents is not likely to interfere withthe extraction calculations anyway as the interface compositions usually lie further up thebinodal curve. Also, the activities at infinite dilution of acetone in water and of glycerol inwater do not look overly trustworthy, but then, such water contents are never reached duringthe extraction process either.

y1

0.0 0.2 0.4 0.6 0.8

y2

0.0

0.1

0.2

0.3

0.4

0.5UNIQUAC SSFExperiment

y2(1)

0.0 0.1 0.2 0.3 0.4

m2

0

2

4

6

8

Figure A2. Left: comparison of the measured binodal curve and its appearance as predicted by theUNIQUAC and the SSF model. Squares: plait points, SSF (black) & UNIQUAC (white). Right: thedistribution coefficients of water.

APPENDIX B. MULTIPLE SUBFILM DIFFERENCE APPROXIMATION WITH BARRIER LAYER

During the test trials it appeared that even if acetone is not the dependent component its fluxcan still be negative. The dustbin effect (see page 98) cannot be held responsible in this case.Then how can this be? When a difference approximation as described in section 6.2.1 is used,both the driving force terms and the friction terms are evaluated at the mean (sub)filmcomposition. For acetone this means that these terms are evaluated at non-zero acetone molefractions, even though the acetone content itself is zero in the bulk. This possibly causes theproblems. The differential form of driving force term di (see eqs. (1) and (2)) divided by δequals

dx

a

z

x

z

x

x

x

z

x

zxi

ii i i

i

i

j

j

j

ni

iδ γ∂ γ∂

= = + → →=

∑d ln

d

d

d

d

d

d

d1

0b g (B.1)

A look at figure 1b reveals that the mole fraction gradient of acetone must be negative in thefilm near z = δ(2). The difference approximation used throughout this chapter also always yieldsa negative value. Therefore, the difference approximation of the driving force term cannot leadto fundamentally wrong results. Thus, the problem must lie in the friction terms. At z = δ(2), thesum of all friction terms in the transport equation of acetone equals

MODELLING LIQUID-LIQUID BATCH EXTRACTION

109

f Nx

Ðijj i

n

ij

ijj i

n

≠ ≠∑ ∑= − (B.2)

This, together with eqs. (B.1) and (2), conclusively proves that the acetone flux is positivewhenever its gradient is negative and its content zero. (Not a very surprising discovery). In thecase of the difference approximation of the friction terms (eq. (6)) the fact that the meanacetone fraction is nonzero can cause the acetone flux to be negative, no matter how manysubfilms may be defined.

To solve this problem, it seemed worth trying to define a thin barrier layer adjacent to the bulk,where the friction terms are defined on the basis of the bulk compositions instead of on themean compositions. The idea is that the layer acts as a sort of membrane for acetone, whoseflux cannot become negative, provided it is chosen as one of the independent components. Ifthe layer is thin enough, its contribution to the friction of the entire film is small, and thesolution for the components water and glycerol should not be influenced by its presence a greatdeal. As we have concluded above, the driving force terms can safely be defined on the basis ofthe mean barrier compositions, although it will not make much difference if for these terms thebulk composition is used as well.

Although the layer improves the fluxes compared to the non-layer method, it does not preventthe fluxes from becoming negative. This implies that this phenomenon is also partly caused bythe behaviour of the other components. Moreover, for small numbers of subfilms in phase 2,the barrier layer method may lead to convergence problems. For sufficiently finediscretisations in this film, it yields results similar to the non-layer method. All in all thereappears to be little incentive to use this method.

NOTATION

A UNIQUAC or NRTL model interaction parameters, (K)a activity, (–)Ð Maxwell-Stefan diffusivity, (m s−2)d driving force term, (–)c molar concentration, (mol m−4)f friction force term, (mol m−4)[K ] mass transfer coefficient matrix, (m s−1)k mass transfer coefficient (m s−1) or Boltzman constant (J K−1)m distribution coefficient, (–)N flux, see chapter 2S interfacial area, (m2)t time, (s)x mole fraction, (–)y mole fraction, (–)z spatial co-ordinate, (m)

Chapter 6

110

Greek symbolsα diffusivity fit multipliers, (–)β film thickness fit multipliers, (–)[Γ] non-ideality thermodynamic matrix, (–)δ film thickness, (m)ε equilibrium equation residual, (–)η dynamic viscosity, (Pa s)τ transport equation residual, (–)ξ number of subfilms

Subscripts− lower (sub)film boundary+ upper (sub)film boundary± either at lower or upper (sub)film boundaryi, r component i in subfilm number rij, r component pair i-j in subfilm number rov overall

Superscripts( p) phase number p (= 1 or 2)T transpose

An overbar indicates that the quantity considered is to be evaluated at the mean (sub)filmcomposition.

REFERENCES

[1] Bollen A.M., Wesselingh J.A.‘The behaviour of the binodal and spinodal curves for near-binary compositions.’Fluid Phase Equilibria, 149, 1998, pp. 17–25.

[2] Bulicka J., Prochàzka J.‘Mass transfer between two turbulent phases.’Chem. Eng. Sci., 31, 1976, pp. 137–146.

[3] Edward J.T.‘Molecular volumes and the Stokes–Einstein equation.’J. Chem. Education., 47 (4), 1970, pp. 261–269.

[4] Foerst W. (editor)‘Ullmanns Encyklopädie der technischen Chemie.’Urban und Schwarzenberg, München–Berlin, Germany, 1951 (third edition).

[5] Gmehling J., Onken U.‘Chemistry Data Series.’, Volume IDECHEMA, Frankfurt am Main, Germany, 1977.

[6] Janssen L.P.B.M., Warmoeskerken M.M.C.G.‘Transport phenomena data companion.’Delft University Press, Delft, the Netherlands, 1991.

MODELLING LIQUID-LIQUID BATCH EXTRACTION

111

[7] Kooijman H.A., Taylor R.‘Estimation of diffusion coefficients in multicomponent liquid systems.’Ind. Eng. Chem. Res., 30, 1991, pp. 1217–1222.

[8] Krishna R., Low C.Y., Newsham D.M.T., Olivera–Fuentes C.G., Standart G.L.‘Ternary mass transfer in liquid–liquid extraction.’Chem. Eng. Sci., 40(6), 1985, pp 893–903.

[9] Krishna R., Low C.Y., Newsham D.M.T., Olivera–Fuentes C.G., Paybarah A.‘Liquid–liquid equilibrium in the system glycerol–water–acetone at 25 °C.’Fluid Phase Equilibria, 45, 1989, pp 115–120.

[10] Kroschwitz J.I., Howe-Grant M. (editors)‘Kirk–Othmer encyclopedia of chemical technology.’Wiley, New York, USA, 1991 (fourth edition).

[11] Low C.Y.‘Isothermal liquid–liquid equilibria and mass transfer in ternary extraction.’PhD thesis, UMIST, Manchester, UK, 1979.

[12] Novák J.P., Matouš J., Pick J.‘Liquid–liquid equilibria.’Elsevier, Amsterdam, NL, 1987.

[13] Rogalski M., Malanowski S.‘A new equation for correlation of vapour-liquid equilibrium data of strongly non-idealmixtures.’Fluid Phase Equilibria, 1, 1977, pp 137-152.

[14] Rutten Ph. W.M.‘Diffusion in liquids.’Delft University Press, Delft, the Netherlands, 1992.

[15] Sørensen J.M., Arlt W.‘Chemistry Data Series.’, Volume VDechema, Frankfurt am Main, Germany, 1980.

[16] Wesselingh J.A., Krishna R.‘Mass transfer.’Ellis Horwood, Chicester, U.K., 1990.

EMULSIFICATION IN LIQUID-LIQUID BATCH EXTRACTION

113

7. EMULSIFICATION IN LIQUID-LIQUID BATCH EXTRACTION

This chapter is a sequel to the previous one†. It serves to investigate whether mass transfer canbe the main mechanism behind the formation of dispersions in liquid-liquid extraction. Thetheory of the previous chapter only needs a few minor additions to be applicable here.Therefore, the structure of these two chapters is almost the same, and we will only indicatewhat these additions are and to which section of chapter 6 they apply.

The idea that emulsification might be a mass transfer driven process stems from problems inthe industrial extraction of caprolactam from water. Caprolactam is the main feedstock for themanufacture of nylon-6,6, and there are several processes for its industrial manufacturing. Inone of these processes, which is used by DSM in the Netherlands, the stream leaving thereactor is an aqueous solution of caprolactam. Caprolactam is extracted from this stream withbenzene in a rotating disk contactor (RDC). In the top of the extraction column, where thereactor stream is fed, a very fine dispersion forms in the benzene phase, which hinders theseparation of the two phases. This, in turn, negatively influences the purity of the product.

In the past, three mechanisms for emulsification have been proposed:1. break-up of the interface due to interfacial turbulence caused by mass transfer,2. mechanical break-up as a result of low interfacial tension,3. ‘chemical’ break-up due to supersaturation.The question is: to what extent is each of the above effects responsible for the emulsification?The first effect has been investigated by Aranow and Witten [1, 2], but it is hard to judge itsinfluence, because the criteria for break-up of the interface depend on quantities which aredifficult to determine. The second option also offers an explanation, since the interfacialtension in this section of the column is low. It is likely however, that mass transfer plays a role,and it is on this option that we will focus.

The idea is that at high caprolactam concentrations, the large chemical potential difference ofcaprolactam between the two phases results in a high flux of this substance. Water will bedragged along, perhaps to such an extent that demixing will occur, resulting in an aqueousdispersion in the benzene phase. The idea is not new: Rushak and Miller [7] showed that undercertain circumstances spontaneous emulsification occurs if two semi-infinite immiscible phasesof homogeneous composition are contacted. The main flaw of their dynamic penetration-type

† In fact, this chapter is also the predecessor of chapter 6. Originally, the entire liquid-liquid batch

extraction model was treated in this chapter, whence chapter 6 did not exist. The current chapter wassent in its original form to a few experts on the subject of LL extraction. As a result of their comments,we decided to test the model on the extraction of glycerol-acetone-water, to show that it could deal with‘simple’, non-emulsifying extractions as well. At the time we expected this to be a piece of cake: all wewould need to do was to feed the model the pure component properties, a thermodynamic model, andthe initial compositions. In reality, however, all kinds of trouble turned up, and it took considerableefforts to square all problems away. By the time everything worked more or less properly, the wholeexercise had gotten so extensive that we decided to make it a separate chapter. As a result, theemulsifcation chapter could be slimmed down considerably.

Chapter 7

114

model is the assumption of pseudo-binary Fickian diffusion under circumstances where this isnot likely, as was shown in chapter 6. Moreover, their solution applies only to systems withconstant diffusivities and density. Apart from these theoretical matters there is a practical point:in the RDC both phases can hardly be assumed stagnant.

Jackson [4] derived a steady-state solution for mass transfer between phases of constantcomposition, which are separated by a stagnant zone, the ‘diffusion tube’. He used theMaxwell-Stefan equations even though he states that „there is scant evidence that the Stefan-Maxwell equations give an accurate description of diffusion in condensed phases”. Ananalytical solution is possible only under simplifying conditions. With this solution, the authorsuggests that emulsification may occur in systems where the two phases are not ‘on oppositesides’ of the plait point. He proposes a multi-interface mechanism, although there does notseem to be any experimental evidence to back up this hypothesis.

Neither article gives qualitative or quantitative solutions for our current problem. The rest ofthis chapter concerns the development of a model to describe the batch extraction of ternarysystems, that does provide both a theoretical explanation and quantitative results.

7.1 THE PHYSICAL SYSTEM

In the previous chapter, where water was the solute, it was difficult to say whether glycerol wasthe raffinate and acetone the extract component or vice versa. Here, there are no problems inassigning these functions to the substances. It is clear that caprolactam is the solute, water isthe raffinate and benzene is the extract component. The components are numbered as follows:1. water,2. caprolactam,3. benzene.Phase number 1 is, as before, the phase rich in component 3, and is therefore the extract phase,whence phase number 2 is the raffinate phase. Appendix A contains the phase diagram of thissystem.

The situation at the interface is essentially the same as depicted in figure 1b of the previouschapter. However, when in one of the phases (or in both, for that matter) a finely dispersedextra phase is present, it must be extended to account for this situation. For now, we assumethat the two film layers are free of dispersed phases, so that a dispersion is containedcompletely in the bulk of a phase. As the bulks are well-mixed, so are the dispersed phases.Further, because the dispersion/bulk interface area is much bigger than that of the bulk/bulkinterface, we neglect the mass transfer resistance between the continuous part and the dispersedpart of a phase. As a result the bulk and the dispersed phase are at instantaneous equilibrium.Figure 1 depicts this situation.

EMULSIFICATION IN LIQUID-LIQUID BATCH EXTRACTION

115

Figure 1. The phases in the Lewis batch extraction cell in case of the formation of dispersions. Thedispersed and the continuous parts of the phases are at equilibrium, which is visualised by the whitearrows. In the situation shown here, only phase 1 exhibits a phase split, and therefore the composition ofthe dispersed part of phase 2 is not displayed.

Note that if the extract phase is to become oversaturated, the water flux must be negative andthat water must therefore diffuse against its gradient. This is, as has been clearly demonstratedin the preceding chapter, not uncommon in extraction processes. More surprisingly, it is notsome small water flux we are talking about. From the nature of the binodal curve (figure A1)we can see that this flux must be roughly the same as the molar caprolactam flux. For thismoment however, these are mere speculations.

7.2 SOLVING THE FLUX EQUATIONS

The solution of the flux equations requires no adaptations relative to the corresponding sectionin chapter 6.

7.3 SOLVING THE DYNAMIC EQUATIONS

Again, the dynamic process is described by the differential equations (6.21). However, in thecurrent situation, we must continually be aware of the possibility that, during the process, thecomposition of a phase enters the two-phase region. It is assumed that if this happens, thephase splits up in a disperse and a continuous part. This is of course not necessarily true: aslong as the composition path of a phase does not cross the spinodal curve, we cannot be surethat demixing occurs. The model may need some modification on this point. For the time beinghowever, the demixing zone is taken to be the entire area enclosed by the binodal curve.

The phase split is determined by the equilibrium equations for all components and the massbalance for the total of the phase considered. The mass balance requires that the overallcomposition y* and the compositions y′ and y″ of the continuous and the disperse parts of thephase lie on the same line in the phase diagram:

Chapter 7

116

ln ln′ − ′′= =

− ′ ′′− ′ + − ′ ′ − ′′ =∗ ∗

a a i

y y y y y y y y

i i 0 1 2 3

01 1 2 2 2 2 1 1

, , ,

d ib g d ib g (1)

After these equations are solved with y′ and y″ as the variables, the resulting composition of thecontinuous part then plays the role of the bulk composition (see figure 1). This is why theformation of the dispersion affects mass transfer: it influences the bulk composition of a phaseand therefore also the outcome the flux equations.

Further, the Euler method turns out to be inadequate in the current situation, because the fluxeschange too rapidly in the initial stages of the extraction process. Due to the small time steprequired to ensure sufficient accuracy in this case, the total computation time of a batchextraction would take too long. Therefore, we are forced to implement an adaptive step sizemethod. We employ the four-term Runge-Kutta method. The results of this model arepresented in section 7.6, while the next section deals with measurements that were conductedto test the model.

7.4 COMPOSITION-DEPENDENCE OF RELEVANT PHYSICAL PROPERTIES

Not much news here either, except for the physical constants of the components.Unfortunately, we could not use the original chemical system water-caprolactam-benzene,because benzene was considered too toxic. We replaced it by toluene, in the hope that thiswould exhibit the same behaviour of forming dispersions. We were not disappointed in thishope. The physical constants of this new system are therefore also included in this section.

7.4.1 Total concentration

See the corresponding section in chapter 6.

7.4.2 Activities

Here, the replacement of benzene with toluene faces us with the problem that for this systemno thermodynamic data were available. We decided to measure equilibrium data ourselves, andto derive UNIQUAC parameters from these data. The measured equilibrium data are includedin appendix A. The interaction parameters are listed in table 1. Since some of the simulationsalso involve the original industrial system, the parameters for the components water-caprolactam-benzene are given in the same table as well.

Table 1. UNIQUAC interaction parameters of the ternary systems water(1)-caprolactam(2)-toluene(3)(left) and water(1)-caprolactam(2)-benzene(3) (right) at a temperature of 25 °C. The parameters of theformer system were derived from our own data.

component Aij Aji Aij Aji

i j (K) (K)

1 2 678.56 −385.82 −127.56 −163.081 3 405.88 697.30 334.20 657.572 3 328.89 −107.72 −112.71 – 39.534

EMULSIFICATION IN LIQUID-LIQUID BATCH EXTRACTION

117

The volume parameters for water, caprolactam, toluene and benzene are 0.9200, 4.6106,3.9228 and 3.1878, respectively. The area parameters are, in the same order: 1.4000, 3.7240,2.9680 and 2.4000.

7.4.3 Diffusivities

Again, we use the two interpolation methods of Wesselingh and Krishna [9] and of Kooijmanand Taylor [6]. In the determination of the diffusivities at infinite dilution, we find that theproblem with our ternary systems is that caprolactam in its pure state is a solid, and it istherefore a bit troublesome to determine its viscosity. We cheat our way around this byextrapolating the viscosity data of liquid water-caprolactam mixtures to solid caprolactam withthe logarithmic equation (6.28) and assume that the ηi

o are given by the Andrade equation

ln lnoηi ii

i iEA

B

TC T D T i= + + + (2)

The parameters A to E for water and caprolactam are then found to be: −52.27, 3.665·103,5.786, −5.846·10−29 and 10 for water, and −240.07, 1.6545·104, 32.159, − 4.7493·10−28 and,again, 10 for caprolactam. The viscosities and the radii for the pure components are listed inthe table below.

Table 2. Molar masses, mass densities, viscosities and Van der Waals radii of the pure components[3, 5, 8].

7.4.4 Film thicknesses

In chapter 6, we saw that eq. (6.27) is one of the most doubtful elements of the transfer model.However, as it is unlikely that the nonoptimised model is going to turn out to be sufficient here,we will simply maintain the film thickness equation and hope for the parameter optimisation tobrush away the errors this equation introduces.

7.5 BATCH EXTRACTION MEASUREMENTS

The experiments discussed in this section were carried out in a Lewis batch extraction cell asshown in figure 1a of chapter 6. The dimensions of the cell and its internals are listed in table3.

component M(10−3 kg mol−1)

ρ(103 kg m−3)

ηo

(mPa s)r

(10−10 m)

water 18.02 0.9971 0.900 1.80caprolactam 113.180 1.0200 259.00000 3.03

toluene 92.13 0.8647 0.560 2.86benzene 78.12 0.8790 0.649 2.68

Chapter 7

118

Table 3. The main dimensions of the experimental equipment. The stirrers were flat disks.

An experiment was carried out as follows: first the heavy raffinate phase, a mixture of waterand caprolactam, was poured into the cell. Then the lighter extract phase was brought upon theraffinate phase without disturbing the interface. This was achieved by pouring the extract phasethrough a co-axial tube along the stirrer axis on top of the upper stirrer, which was positionedon top of the raffinate phase (see figure 2). Once the two phases lay quiescently on top of eachother, the stirrers were positioned in the middle of the phases, after which the stirrers were setin motion (at 85 rpm). This marked the start of the experiment. Additional measurementsshowed that under these conditions the phases were well mixed on a time scale of the order ofone second.

During the experiments samples were taken simultaneously from both phases at more or lessregular intervals, the compositions of which were determined with GC analysis. The sampleswere representative for the phases as a whole: when a dispersion was present, a sampleconsisted of both the continuous and the disperse part of that phase. Acetone was used as aninternal standard, which had the pleasant side effect that two-phase samples were homogenisedbefore analysis. A more detailed description of the methods and materials used for analysis canbe found in appendix A.

Figure 2. Experimental knack for applying the extract phase without disturbing the raffinate phase.

A total of six experiments were conducted. The initial extract phase in all experiments waspure toluene, while the raffinate phase was a mixture of water and caprolactam, with 20, 30,40, 50, 60 and 70 mass % caprolactam. Dispersions in the extract phase were observed only atcaprolactam fractions of 50% and higher, which accords with our theory as described in the

dimension magnitude

cell height, (cm) 12.0cell diameter, (cm) 9.8baffle height, (cm) 12.0baffle width, (mm) 5.2number of baffles, (1) 4 stirrer diameter, (cm) 6.0stirrer thickness, (mm) 1.6stirrer distance, (cm) 4.8

EMULSIFICATION IN LIQUID-LIQUID BATCH EXTRACTION

119

introduction of this chapter. From the beginning of these experiments the formation of thedispersion could be seen, first only at the interface like a kind of smoke, and then after a whilethrough the entire extract phase. When the experiments were about 15 to 25 minutes underway, the dispersion became so dense that the upper stirrer could no longer be discerned, eventhough its distance from the glass cell wall was only about 2 cm. Then, some 30 to 40 minuteslater, the dispersion became perceptibly less dense and 90 minutes after the beginning of theexperiment it was gone completely.

7.6 RESULTS

In this section the results of the model are compared with the experiment in which the initialraffinate phase was the 30 / 70 mass % water-caprolactam mixture. Under these conditions, theeffects leading to the dispersion are most pronounced. The initial amounts of the threecomponents were 8.214 moles of water, 3.047 moles of caprolactam and 4.667 moles oftoluene. The measured transient compositions are listed in the table below. These compositionsare the average of two duplicate measurements taken of the same sample. Closer examinationof the table shows that datum number 8 (logged at 7.20 ks) cannot possibly be correct (also seefigure 3), and it is therefore rejected. The differences between the duplicates appear to be moreor less time invariant for all four yi

( p). The averages of these differences, with exception ofthose belonging to datum number 8 of course, are therefore assumed representative for therelative standard deviations of the four independent mole fractions. The (relative) σi, j

( p) (seeeq. 6.29) are therefore equal for all j ∈ {0,..., 9} \ {8}, are: σ1, j

(1) = 4.46, σ2, j(1) ≡ 1.0,

σ1, j(2) = 2.59, and σ2, j

(2) = 2.50.

As in the previous chapter, the remaining measurements are checked for their consistency withrespect to the total mass conservation condition. The adjusted compositions, which satisfy thiscondition and simultaneously minimise χ2 relative to the measurements, are also listed in thetable. The sum of these minimum χ2 values is 1.101·10−4, which is, for reasons explained in theprevious chapter, the lowest value the model can obtain.

Table 4. Transient compositions of the 30/70 run. Initial total mole numbers in phases 1 and 2 were4.667 and 11.26, respectively, whence the overall mole fractions yov,i are 0.5157 and 0.1913. The leftnumbers in each column are the measured fractions, the right numbers are the compositions that satisfy

the mass conservation condition and simultaneously minimise χ2.

Point Time Phase 1 Phase 2# (ks) y1

(1) y2(1) y1

(2) y2(2)

0 0.00 0.0000 / 0.0000 0.0000 / 0.0000 0.7294 / 0.7294 0.2706 / 0.27061 1.02 0.0391 / 0.0388 0.0222 / 0.0232 0.7330 / 0.7284 0.2533 / 0.26622 1.80 0.0406 / 0.0404 0.0364 / 0.0373 0.7303 / 0.7266 0.2484 / 0.25963 2.70 0.0665 / 0.0662 0.0560 / 0.0570 0.7310 / 0.7270 0.2413 / 0.25444 3.60 0.0784 / 0.0783 0.0791 / 0.0794 0.7270 / 0.7261 0.2415 / 0.24515 4.50 0.0819 / 0.0819 0.0955 / 0.0957 0.7310 / 0.7305 0.2366 / 0.23866 5.40 0.0872 / 0.0870 0.1087 / 0.1096 0.7436 / 0.7417 0.2239 / 0.23437 6.30 0.0720 / 0.0719 0.1199 / 0.1200 0.7405 / 0.7402 0.2254 / 0.22738 7.20 0.1122 / 0.1122 0.1144 / 0.1147 0.7360 / 0.7353 0.2296 / 0.23309 9.00 0.0558 / 0.0558 0.1307 / 0.1310 0.7482 / 0.7477 0.2179 / 0.2217

Chapter 7

120

7.6.1 Initial flux solutions

The various methods to evaluate the fluxes have been considered in the corresponding sectionof chapter 6. In the current situation, no nasty surprises like the dustbin effect turned up. Eventhe solutions of the equations without subdivisions are not complete nonsense. Nevertheless, itis better to see how the ratio between the computational effort required and the accuracy of thesolution can be optimised. In other words: how many subdivisions do we need and how mustthese intervals be distributed over the two films? To find this out, we use the solution of thenumerical integration method (IIi) (see page 98) as a yardstick: N and δ( p) are −7.219·10−2,−9.408·10−2 and 3.991·10−3 mol m−2 s−1, and 8.541·10−6 and 1.751·10−5 m, respectively. Thecomputed profiles reveal that, in contrast to the extraction of the system glycerol-water-acetone, both films are more or less equally important. Therefore, the number of subintervals isset the same for both phases, and it appears that 12/12 discretisation is accurate enough. Withthe use of difference approximation method (Ia2i), the magnitudes of the above five variablesare found to be –7.249·10−2, −9.424·10−2 and 3.981·10−3 mol m−2 s−1, and 8.536·10−6 and1.751·10−5 m.

7.6.2 Dynamic model solutions

Just as in chapter 6, we start with the plain model, the results of which are shown in figure 3.Most remarkably, the model predicts that no dispersion is formed: an oversaturated phasewould have betrayed itself by crossing the binodal. From the left graph it is also clear that themodel predicts higher transfer rates in phase 1 than found experimentally, which means that themass transfer resistance is somehow underestimated. Further, the figure enables us to see whatthe effect of adjusting the experimental compositions is. Although on average the distancebetween the measured and the adjusted points is less than 1 mol%, for some of the points it isconsiderably more. Anyway, these are the measurements we will have to make do with, and theuncertainty about the measured data cannot conceal that the model as it is does not suffice.This is also reflected by the magnitude of χ2: 1.027·10−2.

Given the results in the previous chapter, it would have been unrealistic to hope for the plainmodel to come up with a ‘perfect’ set of trajectories. Yet, the above results are not what weexpect. In the toluene phase, the initial interface fractions of water and caprolactam are smalland approximately equal, and the bulk fractions are zero. A one-step difference approximationof eq. (6.1) gives di ≈ 2 xi , f13 ≈ x 3 N1 / Ð13, and f23 ≈ x 3 N2 / Ð23, while f12 is found to be smallcompared to these two friction terms. This implies that N2 / N1 ≈ Ð23 / Ð13. When thediffusivities are calculated using eq. (6.24), the expected flux ratio, and therefore the tangent ofthe composition path at x1

(1) = 0 in the left graph, equals 0.35, instead of the model value of 1.3.Apparently, the difference is caused by factors which lie outside the toluene phase.

EMULSIFICATION IN LIQUID-LIQUID BATCH EXTRACTION

121

Phase 1

y1

0.00 0.03 0.06 0.09 0.12

y2

0.00

0.03

0.06

0.09

0.12

Phase 2

y1

0.70 0.72 0.74 0.76

0.22

0.24

0.26

0.28

0

1

2

34

5

67

8

91

2

3

4

5

6

78

9

0

Figure 3. Plain model composition paths (solid lines). Difference approximation method (Ia2i) (seepage 98) with 12/12 discretisation, UNIQUAC, eq. (6.24) and b = ½. The solid black circles represent themeasured data, the open circles the adjusted data. In those cases where the adjusted and measured pointslie so far apart that it can cause confusion, they are joined by a dotted line. The numbers near the pointsare the measurement numbers (see table 4). The triangles are the model results corresponding to theexperiments, and the squares mark the final equilibrium compositions. The dotted curve is the binodalcomputed using UNIQUAC, the dashed one the composition space boundary. The solid lines along thebinodal are the interface composition paths.

Anyway, to improve the model results, we use the same trick as in the preceding chapter: thediffusivities and the film thicknesses are furnished with premultiplication factors, which aresubsequently used as fitting parameters. The quality of a fit is again quantified by eq. (6.29),but now χ2 is not the only quantity that matters, because it does not take into account the timeduring which the dispersion exists, te, also called the emulsification time. Yet, this time spanought to be considered in the comparison of the model with the experimental data, because twomodels with marginally different χ2-values could have notably different te. However, because itis clear that the extraction model is still too crude to be truly predictive, it is probably not worththe effort to account for te in the optimisation objective function. Instead, we carry out theoptimisation with eq. (6.29) as the objective function, and will see about te afterwards.

The results of the optimised model are shown in figure 4. There is no emulsification†, and χ2 isstill far too big (1.999·10−3) compared to the lowest obtainable value (1.101·10−4). Theparameters corresponding to these profiles are: α12 = 0.1951, α13 = 0.2778, α23 = 15.31,β(1) = 1.808 and β(2) = 1.472.

† It is not impossible to find demixing: by fiddling the relative weights of the measurements, a trajectory

can be fitted that does enter the demixing zone. However, this does require rather extreme values of theparameters.

Chapter 7

122

Phase 1

y1

0.00 0.03 0.06 0.09 0.12

y2

0.00

0.03

0.06

0.09

0.12

Phase 2

y1

0.70 0.72 0.74 0.76

0.22

0.24

0.26

0.28

Figure 4. Composition paths (solid lines) of the five-parameter optimised model. Differenceapproximation method (Ia2i) with 12/12 discretisation, UNIQUAC, eq. (6.24) and b = ½. For furtherexplanation, see the caption of figure 3.

This disappointing result may be caused by the fact that the measurements are too inaccuratefor this kind of work. Their poor quality affects the model in two ways, firstly through thethermodynamic description of the system, and secondly through the composition trajectories.The first effect is probably the worst, because it cannot be corrected by the five fittingparameters. An indication that the thermodynamic description of the system is indeed bad isgiven by the composition trajectory in phase 2, which suggests that the equilibriumcomposition in that phase is wrong. The measurements 7 and 9 in phase one, however, alsorouse suspicions: it is not likely that, after more than 90 minutes, the system has enough energyleft to struggle out of the grasp of the equilibrium composition so far as these measurementssuggest. Yet, it can also be that the binodal is steeper in reality than computed, but in that case,the measured trajectory cuts much deeper into the demixing zone.

What to do? The most obvious approach is to see what the influence of the thermodynamicmodel is. It could either be that the distribution coefficients are wrong, or the form of thebinodal, or both. The distribution coefficients are most easily adjusted. Departing from thebinodal as it follows from the LLE fit, one of the distribution coefficients represents a degreeof freedom, and we use m2 for this purpose. In appendix B it is shown that a distribution cannotbe defined arbitrarily, but that it must satisfy certain conditions. Unfortunately, theseconditions do not allow us complete control over the nature of the m2 function. Even so, anumber of adjustments of this distribution coefficient have been tried: higher coefficientfunctions than the one shown in appendix A, lower ones, functions where the ascending partnear y1

(2) = 0 is largely eliminated, and a coefficient function which leads to a equilibriumcomposition in phase 2 which seems to fit the measured trajectory better. None of these weresatisfactory, and as an example, the fit with the latter adjustment is shown in figure 5.

EMULSIFICATION IN LIQUID-LIQUID BATCH EXTRACTION

123

Phase 1

y1

0.00 0.03 0.06 0.09 0.12

y2

0.00

0.03

0.06

0.09

0.12

Phase 2

y1

0.70 0.72 0.74 0.76

0.22

0.24

0.26

0.28

Figure 5. Composition paths (solid curves) of the optimised model for the system with an adjusteddistribution coefficient function which yields a more probable final equilibrium composition in phase 2.Difference approximation method (Ia2i) with 12/12 discretisation, UNIQUAC, eq. (6.24) and b = ½. Forfurther explanation, see the caption of figure 3.

In this specific case, we see that in shifting the equilibrium composition of phase two down thebinodal, we are forced by the overall composition point to shift that in the other phase up.Although we cannot be sure, it seems that this new equilibrium point, (0.1225, 0.1499), is nowlocated too far up the binodal, and that we have moved the problem from one phase to theother. There are, however, two more general problems with most of the thermodynamicdefinitions of the system, including the original one. First, the optimisation leads to bigparameter values for α23. Also, it is difficult or even impossible to optimise the model using thediffusivity interpolation scheme of Kooijman and Taylor, eq (6.25). That is, the parametersalways ended up in regions where no solution of the flux equations could be found, eventhough the optimisation was carried out carefully, using the Marquardt-Levenberg algorithm incombination with a homotopy-like method. Therefore, it seems unlikely that a feasibleoptimum does exist, but this cannot be ruled out completely.

In view of all these difficulties, we believe that the cumulative effects of the experimentalinaccuracies makes the measurements unusable for quantitative analysis of the emulsificationprocess. Retrospectively, it would probably have been better to analyse the liquid samples by acombination of two simple measurement techniques, like, for instance, refractive index anddensity, although in that case it would have been difficult to analyse the phase in which thedispersion occurs.

7.7 CONCLUSIONS

The proposed model fails to describe the measured composition trajectories. This can becaused by the poor quality of the measurements, by the assumptions on which the model isbased, or by inaccurate component property data. Probably, the failure is a combination ofthese three factors. The measurements are inaccurate to a degree that makes them unsuitable

Chapter 7

124

for our purposes. Nevertheless, it appears to be very difficult, if not impossible, to find acredible set of model parameters that yields even a qualitatively correct description of theemulsification process. This may be due to flaws in the part of the model that describes thetransfer across the interface, or in the assumptions concerning the processes taking place in thesupersaturated phase. The possible flaws in the extraction model have already been discussedin section 6.7. Especially the fact that the enormous viscosity differences between the puresubstances may seriously affect the model outcome, also applies to the systems consideredhere. But this probably only affects the model outcome on a quantitative level, and the causesof the qualitative failure must be sought in the emulsification mechanism.

As to these emulsification-related assumptions: it is well possible that the dispersion is formedby a combination of the three mechanisms listed at the first page of this chapter. The first twoeffects have been neglected completely in this chapter’s model. Yet for the system water /caprolactam / benzene, it is known that the interfacial tension differences are very small, and itis likely that this is true also for the toluene-containing system. This may be of greatimportance, not only because it allows easy break-up of the interface, but also because ithinders coalescence of the dispersion once it is formed. It is not clear to what extent each of thethree mechanisms is responsible for the formation of the dispersion, but as Rushak and Miller[7] have shown, absence of the purely mechanical break-up of the interface caused by stirringdoes not prohibit emulsification. Therefore, since the model suggests that emulsification isexclusively a diffusion-induced phenomenon, the mechanism of interfacial break-up due tomass transfer driven interfacial turbulence probably accounts for at least a part of the formationof the dispersion.

APPENDIX A. LIQUID-LIQUID EQUILIBRIA OF THE SYSTEMS USED

The absence of data on the liquid-liquid system which we used for our batch extraction forcedus to determine the binodal compositions experimentally. To this aim, known (weighed)amounts of the three components were thoroughly mixed with an ultrasonic bath and kept at aconstant temperature of 25 °C for several hours. Then from each of the two phases formedsamples were taken, which were analysed using packed bed gas chromatography. The glasscolumn used had an internal diameter of 4.0 mm and a length of about 20 cm and was packedwith Porapack Q. Helium was used as the carrier gas at a flow rate of 20 ml min−1. The injectedsample volume was 0.1 µl. For detection thermal conductivity was applied. The same analysismethod was used for the determination of the compositions of the batch extractionexperiments.

The compositions thus found were then used to find the best set of UNIQUAC interactionparameters. Originally, this was done by DSM Research, for at the time we did not yet have theproper software ourselves. However, this set of parameters leads to anomalies for the water-rich phase at low caprolactam contents: the spinodal intersects the binodal there, and thiscauses strange behaviour of the distribution coefficient of caprolactam between the two phases.By the time chapter 6 was finished, we were also able to compute the parameters ourselves,and it appeared that the data were indeed difficult to fit. After a few ‘creative’ measures, weobtained the set of UNIQUAC parameters listed in table 1. In the figure below, the experimentsare compared with the UNIQUAC results using the fitted set of interaction parameters. Clearly,

EMULSIFICATION IN LIQUID-LIQUID BATCH EXTRACTION

125

the model fits the experimental data quite well, but the sharp ascent of the distributioncoefficients at low caprolactam contents, may not exist in reality.

y1

0.0 0.2 0.4 0.6 0.8 1.0

y2

0.0

0.1

0.2

0.3

y2(1)

0.0 0.1 0.2 0.3

m2

0

1

2

3

Figure A1. The UNIQUAC fit for the system water-caprolactam-toluene. Left: the binodalcompositions, both experimental (circles) and computed using UNIQUAC (solid line) The squarerepresents the plait point, the dashed line is the spinodal curve. Right: the distribution coefficient ofcaprolactam. The sharp ascent at low caprolactam contents may not reflect reality, but is hard to avoid.

In the next figure the phase diagram of the system with toluene is compared with the systemwith benzene. Notice that the distribution coefficient of caprolactam in the benzene-containingsystem is more favourable for the extraction.

y1

0.0 0.2 0.4 0.6 0.8 1.0

y2

0.0

0.1

0.2

0.3

Figure A2. The phase diagrams of the two type 1 systems water-caprolactam-toluene and water-caprolactam-benzene at 25 °C. Constructed with the UNIQUAC model, parameters from table 1. Asbefore, the squares are the plait points.

Chapter 7

126

APPENDIX B. THE BINODAL CURVE, THE DISTRIBUTION COEFFICIENTS AND THE PLAIT POINT

Let y2 = f ( y1) be a functional description of the (form of the) binodal curve, and assume that ithas only one maximum on the part of the y1-domain for which it is positive. Let (y1, m, y2, m)designate the position of this maximum. Further, assume that we know the distributioncoefficient of the second component as a function of y2

(1): y2(2)

/ y2(1) = m2 ( y2

(1)). Intuitively, onefeels that there must be some relation between these two functions and the position of the plaitpoint. In this appendix, we will see what this relation is.

To start with, we have dy2(1)

/ dy1(1) = f ′( y1

(1)), and since y2(2) = y2

(1) m2, we also havedy2

(2) / dy2

(1) = m2 + y2(1) m′2. Therefore, dy2

(2) / dy1

(1) = (m2 + y1(1) m′2) f ′. Now suppose that m2 > 1

everywhere on [0, y2, p⟩†, where y2, p is the plait point mole fraction of component 2. We will

show that this implies y1, p ≤ y1, m. The fact that, in spite of the nature of the distributioncoefficient, the plait point may coincide with the maximum of f, may seem illogical, yet wewill show that it is true.

First, we assume that y1, p < y1, m. Let y↓1(1) be the lower bound of the y1

(1)-domain, then for somey1

(1) ∈ ⟨ y1↓(1), y1, p⟩, y2

(2) passes through a maximum, namely y2(2) = y2, m Thus, dy2

(2) / dy1

(1)

which, as we have seen, equals (m2 + y1(1) m′2) f ′, must have a zero on ⟨ y1↓

(1), y1, p⟩ whichrepresents this maximum. Hence, either f ′ must vanish, or (m2 + y2

(1) m′2). The former option isnot possible, because it represents the maximum of the binodal curve, which, by assumption, isnot in the domain of y1

(1). Therefore, only the second option remains: (m2 + y2(1) m′2) = 0 must

hold somewhere on ⟨ y↓1(1), y1, p⟩ in order to have y1, p < y1, m. If this condition is not satisfied, we

must have y1, p ≥ y1, m. However, since m2 > 1 on [0, y2, p⟩ we also have y1, p ≤ y1, m and thereforewe find y1, p = y1, m. This solution corresponds to the root of f ′.

As an example, consider the linear distribution coefficient m2 = m2,0 + y2(1) (1 − m2,0) / y2, p, where

m2,0 > 1 is the distribution coefficient at y2(1) = 0. The condition (m2 + y2

(1) m′2) = 0 holds aty2

(1) = ½ m2,0 y2, p / (m2,0 − 1), and because 0 < y2(1) < y2, p, we must have m2,0 > 2. But this is not

enough, since y2(1) must be in equilibrium with y2, m, and so we find that m2,0 must also satisfy

the equality m2,0 = 2 ( y2, m / y2, p) [1 ± (1 − ( y2, p / y2, m))½]. If these conditions do not hold, the plaitpoint must be located at the maximum of f.

The same argumentation can be applied to distribution coefficients smaller than unity simplyby swapping components 1 and 3. Moreover, this knowledge can be used to alter thedistribution function of a system at will without having to change the form of the binodal or theposition of the plait point.

NOTATION

No new notation was used in this chapter. For notation, see page 109.

† m2 ( y2, p) ≡ 1

EMULSIFICATION IN LIQUID-LIQUID BATCH EXTRACTION

127

REFERENCES

[1] Aranow R.H., Witten L.‘Effect of diffusion on interfacial Taylor instability.’Phys. Fluid., 6(4), 1963, pp. 535-542.

[2] Aranow R.H., Witten L.‘Diffusion-induced interfacial instability.’Phys. Fluid., 10(6), 1967, pp. 1194-1199.

[3] Edward J.T.‘Molecular volumes and the Stokes-Einstein equation.’J. Chem. Education., 47 (4), 1970, pp. 261-269.

[4] Jackson R.‘Diffusion in ternary mixtures with and without phase boundaries.’Ind. Eng. Chem. Fundam., 16(2), 1976, pp. 304-306.

[5] Janssen L.P.B.M., Warmoeskerken M.M.C.G.‘Transport phenomena data companion.’Delft University Press, Delft, the Netherlands, 1991.

[6] Kooijman H.A., Taylor R.‘Estimation of diffusion coefficients in multicomponent liquid systems.’Ind. Eng. Chem. Res., 30, 1991, pp. 1217-1222.

[7] Rushak, K.J., Miller C.A.‘Spontaneous emulsification in ternary systems with mass transfer.’Ind. Eng. Chem. Fundam., 11(4), 1972, pp. 534-540.

[8] Weast R.C., Astle M.J., Beyer W.H. (eds).‘CRC Handbook of chemistry and physics.’CRC Press, Inc., Boca Raton, USA, 1983.

[9] Wesselingh J.A., Krishna R.‘Mass transfer.’Ellis Horwood, Chichester, U.K., 1990.

SAMENVATTING

129

SAMENVATTING

Volgens de titel is dit is een proefschrift over stoftransport in vloeistoffen. Eigenlijk gaat hetvoornamelijk over diffusief stoftransport in niet-ideale vloeistoffen. De niet tot chemicusopgeleide lezer weet misschien niet wat ‘diffusief stoftransport’ is en wat ‘niet-idealevloeistoffen’ zijn. Omdat dit twee centrale begrippen zijn in dit proefschrift, zullen we daareerst maar eens dieper op in gaan. Daarna komt de inhoud van dit proefschrift aan bod.

DIFFUSIE

Om te begrijpen wat diffusie is, doen we het volgende (gedachten)experiment. We nemen eenkopje water en op de bodem daarvan – waar anders? – leggen we een suikerklontje. Om tevoorkomen dat het water verdampt, dekken we het kopje af met een schoteltje. Vervolgensdoen we een hele tijd niks, dat wil zeggen, niet met het kopje plus inhoud. We kunnenbijvoorbeeld rustig met vakantie gaan. Als we terugkomen, blijkt dat het suikerklontje isopgelost en dat de suiker homogeen verdeeld is over het water, dat dus overal in het kopje evenzoet is. Dat lijkt misschien vanzelfsprekend, maar dat is het niet. Waarom zou de suiker van deplaats waar het suikerklontje ligt, bewegen naar andere delen van het water in het kopje? Datkomt doordat de moleculen in vloeistoffen en gassen voortdurend in beweging zijn.

Door die beweging botst een suikermolecuul in de vloeistof voortdurend op watermoleculen enandere suikermoleculen. Na elke botsing heeft het een andere, willekeurige richting. Neem nueen denkbeeldig vlak met aan de ene kant een hogere suikerconcentratie dan aan de anderekant. De kans dat individuele suikermoleculen die zich dicht bij dit vlak bevinden naar deandere kant zullen gaan, is aan beide zijden van het vlak even groot. Doordat zich echter aan dekant met de hoge concentratie meer suikermoleculen bevinden, zullen er ook meer moleculenvan deze kant de oversteek maken. Er bewegen zich dus netto meer moleculen van de hogenaar de lage concentratie dan andersom. Dit leidt er toe dat suiker van plaatsen met een hogesuikerconcentratie migreert naar delen van de vloeistof met een lage concentratie. Bovendienblijkt, dat dit nettoverschil groter is naarmate het concentratieverschil aan beide zijden van hetvlak toeneemt.

De snelheid waarmee stoffen diffunderen hangt af van een aantal factoren. Ten eerste van detemperatuur, want die bepaalt de heftigheid waarmee moleculen bewegen: hoe warmer, hoesneller en dus wilder. Doordat de moleculen in vloeistoffen nogal dicht opeen zitten, botsen zezeer vaak, terwijl de moleculen in gassen, die veel minder op een kluitje zitten, een stukjekunnen ‘vliegen’ voordat ze een ander deeltje tegenkomen. Dit verklaart het feit dat diffusie ingassen veel sneller verloopt dan in vloeistoffen en ook dat diffusie in gassen veel sterkerafhangt van de druk dan diffusie in vloeistoffen. Immers, in gassen is het aantal deeltjes pervolume-eenheid sterk afhankelijk van de druk, terwijl dat bij vloeistoffen nauwelijks het gevalis. Als je bij een gas de druk flink opvoert, neemt het volume duidelijk af, zodat hetzelfdeaantal gasdeeltjes het met een kleiner volume moet doen. Bij vloeistoffen daarentegen kun jede druk opvoeren wat je wilt, een beduidende volumeverkleining zit er niet in. Tenslotte hangtdiffusie af van de samenstelling van de vloeistof c.q. het gas. Belangrijke grootheden zijn: de

SAMENVATTING

130

grootte en de massa van de moleculen, de viscositeit en de mate van idealiteit. In het algemeengeldt dat grote moleculen langzamer diffunderen dan kleine en dat zware moleculen trager zijndan lichte. De viscositeit van een gas of een vloeistof is een maat voor de mobiliteit van dedeeltjes. In hoogvisceuze vloeistoffen ondervinden de moleculen tijdens hun bewegingen veelwrijving en dat maakt dat ze trager bewegen, hetgeen de diffusie vertraagt. De discussie overde effecten als gevolg van (niet-)idealiteit op diffusie sluit mooi aan bij het tweede moeilijkebegrip, ‘niet-ideale vloeistoffen’ en daarom behandelen we ze in de volgende paragraaf.

IDEALITEIT VAN VLOEISTOFFEN

Niet-idealiteit is een gevolg van krachten die de deeltjes op elkaar uitoefenen, deintramoleculaire krachten. Er zijn grote verschillen tussen de effecten van deze krachten invloeistoffen en gassen. Hierboven is al gezegd, dat de afstand tussen de moleculen in gassenveel groter is dan in vloeistoffen. Doordat de krachten tussen moleculen alleen op kleineafstanden werkzaam zijn, is hun rol in gassen bij lage drukken beperkt. In gassen blijft deinvloed van de samenstelling op diffusie dan ook voornamelijk beperkt tot de afmeting en demassa van de moleculen. In vloeistoffen echter, zijn de intramoleculaire krachten van grootbelang, want de gemiddelde afstand tussen de moleculen is vele malen kleiner dan in gassen.

We hebben nu de hele tijd over gassen en vloeistoffen gesproken, zonder dat daarbij desamenstelling ter sprake kwam. Het was niet duidelijk of we het nu over mengsels of overzuivere stoffen hadden en dat was ook niet nodig. Het begrip idealiteit heeft echter alleenbetekenis in verband met mengsels. Als we twee verschillende vloeistoffen bij elkaar doen, danzullen ze de neiging hebben te mengen tot een homogeen mengsel, ook als we niet roeren. Hoedat op moleculaire schaal in zijn werk gaat, is al ter sprake gekomen. Deze neiging tot mengenis een illustratie van de tweede hoofdwet van de thermodynamica, die vreemd genoeg bij veelmensen bekend lijkt te zijn. Meestal wordt deze wet dan als volgt verwoord: “de natuur streefteen zo hoog mogelijke graad van chaos na”. In ons geval vertegenwoordigt het homogeengemengde systeem een hogere graad van chaos dan de twee afzonderlijke vloeistoffen en zezullen dus willen mengen.

Dat wil echter niet zeggen dat dat ook altijd gebeurt! Doe maar eens olie en water bij elkaar. Jekunt wachten tot je 0.1 kg† weegt, maar mengen: ho maar. Dat er geen menging optreedt, is eengevolg van de intramoleculaire krachten. Oliemoleculen en watermoleculen houden niet vanelkaar. Ze stoten elkaar af en wel zodanig dat zelfs de dwang van de chaos ze er niet toe kanbrengen te mengen. In dat geval spreken we van een sterk niet-ideaal mengsel. Er zijn ookmengsels van stoffen die graag in elkaars omgeving verkeren, bijvoorbeeld omdat ze erg opelkaar lijken. Deze stoffen mengen in alle verhoudingen met elkaar en vormen (bijna-)idealemengsels. Tussenvormen zijn er natuurlijk ook: stoffen die wel in alle verhoudingen met elkaarmengen, maar toch niet dol op elkaar zijn. De antipathie is dan echter niet zo groot, dat die dechaosdwang kan overwinnen. Dat zijn gewoon niet-ideale mengsels. Zuivere stoffen zijn perdefinitie ideaal. Overigens zijn gasmengsels nooit dusdanig niet-ideaal dat ze ontmengen,daarvoor is de invloed van de intramoleculaire krachten te gering.

Het is eenvoudig in te zien dat niet-idealiteit van mengsels invloed heeft op diffusie. Kijk maarnaar het mengsel van olie en water. Je kunt ze bij elkaar voegen, maar toch zal er nauwelijks † Vroeger noemden we dit een ons, maar dat mag al lang niet meer.

SAMENVATTING

131

diffusie optreden. Nauwelijks? Ja, nauwelijks, want er lost altijd wel een klein beetje water opin de olie en andersom. Diffusie in niet-ideale mengsels is lastiger te beschrijven en tevoorspellen dan in ideale mengsels.

INHOUD VAN DIT PROEFSCHRIFT

Het belangrijkste deel van dit proefschrift gaat over de beschrijving van dediffusieverschijnselen die zich afspelen in het scheidingsproces vloeistof-vloeistofextractie.(Een korte beschrijving van dit proces volgt op pagina 134.) De rest is gewijd aan zaken dieaan dit hoofdonderwerp gerelateerd zijn, namelijk: het meten van diffusiesnelheden, en diffusiein ontmengende vloeistofmengsels.

Diffusiecoëfficiënten meten

In deze samenvatting is tot nu toe slechts in kwalitatieve termen over diffusie gesproken.Echter, we willen de snelheid waarmee diffusie plaatsvindt kwantificeren, zodat we ermeekunnen rekenen. Dit doen we door getallen (parameters) te introduceren die aangeven hoe hoogde diffusiesnelheid is. We noemen deze parameters diffusiecoëfficiënten. Deze diffusie-coëfficiënten zijn, zoals we al gezien hebben, afhankelijk van de temperatuur, de druk en deaard van het mengsel waarin diffusie optreedt. In dit proefschrift gaat het uitsluitend overdiffusie in vloeistoffen van een constante temperatuur en dus kunnen we de invloed van dedruk en de temperatuur verwaarlozen. Dat betekent dat we alleen nog maar dediffusiecoëfficiënten hoeven te weten van de gebruikte mengsels bij de juiste temperatuur.Doordat er zoveel verschillende chemische stoffen zijn en het aantal mogelijke mengselspraktisch oneindig is, mag je van geluk spreken als de diffusiecoëfficiënten die je nodig hebt inde literatuur te vinden zijn. In veel gevallen zullen we ze moeten meten.

Het meten van diffusiecoëfficiënten is geen sinecure. Diffusie is namelijk een traag proces endaardoor krijgen verstorende invloeden alle tijd de meting te bederven. Bovendien zijn demeeste meetmethoden nogal tijdrovend. Een relatief eenvoudige en snelle manier omdiffusiecoëfficiënten in vloeistoffen te meten, is de Taylor(dispersie)methode. Stel, we willende diffusiecoëfficiënten bepalen in een mengsel van een bepaalde samenstelling. Bij deTaylormethode wordt dit mengsel (het eluens) door een capillair gepompt, dat is een lange,zeer dunne buis. Gebruikelijke afmetingen zijn een lengte van 20 meter en een diameter vaneen halve millimeter(!). Op een gegeven moment wordt aan het begin van de capillair in deeluensstroom een tracerpuls geïnjecteerd. Dit is een kleine hoeveelheid mengsel waarvan deconcentraties afwijken van die van het eluens. Door het verschil in concentraties tussen de pulsen het eluens kan er diffusie optreden. De tracerpuls wordt meegevoerd door de eluensstroomen wordt gedurende zijn reis door de capillair vervormd door stroming en diffusie. Op eengegeven moment komt de puls er bij het einde van de capillair uit en daar meten we dan hoe desamenstelling van de vloeistofstroom varieert met de tijd. Uit deze gegevens kan dan dediffusiecoëfficiënt bepaald worden, tenminste, als het om een mengsel van slechts twee stoffengaat. In het geval van mengsels met meer stoffen is de zaak wat ingewikkelder.

Meestal wordt verondersteld dat er voor het meten van diffusiecoëfficiënten in mengsels vanmeer dan twee stoffen ook meer dan één concentratieafhankelijke eigenschap van de vloeistofgemeten moet worden. Dan zijn namelijk van meer stoffen de concentratievariaties bekend.Echter, dit stuit op een aantal praktische bezwaren. Zo moet de meetapparatuur zeer gevoelig

SAMENVATTING

132

zijn, want het gaat om minieme concentratievariaties. Dat betekent dat er niet zo veelmeetapparaten bestaan die geschikt zijn en de apparaten die wel aan de eisen voldoen, zijnnogal prijzig. Verder vervormt elk meetapparaat het geïnjecteerde vloeistofvolume een beetje,zodat een volgend meetapparaat een andere concentratievariatie meet, hetgeen weer leidt totfouten in de berekende diffusiecoëfficiënten.

Uit dit proefschrift blijkt echter dat het ook anders kan, door bij één mengselsamenstellingmetingen te doen bij een aantal injecties van verschillende samenstellingen. Dat betekentuiteraard dat de tijd die nodig is om de diffusiecoëfficiënten te bepalen toeneemt. DeTaylormethode is echter goed te automatiseren, dus dat hoeft geen al te groot probleem te zijn.Zo worden dus twee vliegen in één klap geslagen: de aanschaf van dure apparatuur is onnodigen de nauwkeurigheid neemt toe.

Een tweede probleem van de Taylormethode is dat de capillair om praktische redenen meestalwordt opgerold tot een spiraal die wat beter hanteerbaar is. Echter, de vergelijkingen waarmeede diffusiecoëfficiënten uit het samenstellingssignaal berekend worden, zijn gebaseerd op deveronderstelling dat de capillair recht is. Er kunnen daardoor afwijkingen in dediffusiecoëfficiënten ontstaan en het is daarom zaak te onderzoeken onder welke voorwaardenhet oprollen is toegestaan. Het blijkt dat er voor de meeste mengsels geen problemen teverwachten zijn. Voor de bepaling van de diffusiecoëfficiënten van zeer grote en zwaremoleculen kan het echter nodig zijn de stromingssnelheid door de buis drastisch te verlagen.Deze maatregel is ook noodzakelijk in gevallen waarbij de vloeistof een bepaalde, zeldzame,combinatie van eigenschappen heeft.

Tenslotte kan er bij de Taylormethode nog op een andere wijze een fout in de berekendediffusiecoëfficiënten sluipen. Om de diffusiecoëfficiënten te kunnen berekenen, hebben we dusverondersteld dat de capillair recht is. Er wordt echter ook van uit gegaan dat dediffusiecoëfficiënten en de totale concentratie niet afhangen van de samenstelling van hetmengsel. (De totale concentratie is het aantal moleculen per eenheidvolume.) In echte mengselsis dat nooit het geval. Nu hoeft dit geen probleem te zijn, want als we het verschil in desamenstelling tussen de tracerpuls en de hoofdstroom klein kiezen, dan zal de veronderstellingbij zeer goede benadering kloppen. Immers, kleine samenstellingsverschillen betekenen ookkleine variaties in diffusiecoëfficiënten en totale concentratie. Toch is dit geen goedeoplossing, want tijdens de tocht door de capillair nemen de oorspronkelijkeconcentratieverschillen sterk af, zodat die dan aan het eind van het capillair nauwelijks nog temeten zijn.

Het is dus raadzaam de oorspronkelijke concentratieverschillen zo groot mogelijk te nemen. Indit proefschrift is gekeken wat de effecten hiervan kunnen zijn op de nauwkeurigheid van deberekende diffusiecoëfficiënten. Om de zaken zo eenvoudig mogelijk te houden, is gekekennaar mengsels die uit twee stoffen bestaan. Helaas is het zelfs voor zulke mengsels nietmogelijk de fysische processen die plaatsvinden tijdens een Taylordispersiemeting metanalytische wiskunde door te rekenen. Daarom hebben we numerieke simulatie gebruikt: hetop de computer ‘nadoen’ van de processen in de capillair. Deze berekeningen tonen aan dat hetmogelijk is dat er een fout ontstaat in de berekende diffusiecoëfficiënt als deconcentratieverschillen tussen de puls en de hoofdstroom groot zijn. Voor de meeste mengselsis die fout klein. Er zijn echter mengsels waarvan de diffusiecoëfficiënt en de totale

SAMENVATTING

133

concentratie zeer sterk variëren met de samenstelling van het mengsel. In die zeldzamegevallen kan er een belangrijke afwijking ontstaan.

Diffusie in ontmengende vloeistoffen

In de paragraaf ‘Idealiteit van vloeistoffen’ is al iets gezegd over diffusie in niet-mengendevloeistoffen. De mate waarin stoffen mengbaar zijn is vaak een kwestie van temperatuur. In hetalgemeen geldt, dat hoe hoger de temperatuur, hoe beter stoffen in elkaar oplossen. Immers, alsde temperatuur stijgt, zullen de moleculen sneller bewegen en de afstotende krachten zullensteeds sterker moeten zijn om de moleculen uit elkaar te houden. Er zijn uitzonderingen opdeze regel, maar de oorzaken daarvan zijn te complex om hier uit de doeken te doen. We zullenhet hier verder alleen over de ‘normale’ mengsels hebben.

Door de temperatuurafhankelijkheid van de oplosbaarheid kan het gebeuren dat een mengselvan twee stoffen dat in eerste instantie homogeen is, plotseling begint te ontmengen als we detemperatuur verlagen. Na ontmenging houden we twee vloeistoffen over, die fasen genoemdworden. De samenstelling van deze fasen ligt vast, en hangt alleen af van de temperatuur.Eigenlijk ook van de druk, maar we hebben hierboven al gezien dat vloeistoffen niet zo onderde indruk zijn van drukveranderingen, en deze afhankelijkheid is dus slechts zwak. Bij detemperatuur waarbij de ontmenging begint, zijn de twee fasen (bijna) identiek vansamenstelling, maar naarmate de temperatuur verder verlaagd wordt komen ze verder en verderuit elkaar te liggen. De temperatuur waarbij de ontmenging begint en de bijbehorendesamenstelling van de fasen heet het kritisch mengpunt. Dit punt is voor ons interessant, omdater mechanismen bestaan die diffusie in zulke vloeistoffen beïnvloeden.

In dit proefschrift gebruiken we voor de beschrijving van diffusief stoftransport de zogenaamdeMaxwell-Stefanvergelijkingen. Op grond van deze vergelijkingen verwachten we dat dediffusiecoëfficiënten op een bepaalde wijze afhangen van de idealiteit van een mengsel en vande temperatuur. In het verleden zijn er diffusiecoëfficiënten gemeten in mengsels bij hetkritisch mengpunt, en er zijn aanwijzingen gevonden dat de diffusiecoëfficiënten zich daaranders gedragen dan de Maxwell-Stefanvergelijkingen voorspellen. Het is dus zaak eens tekijken of wij ons hier zorgen over moeten maken. Eén ding staat vast: in het kritisch mengpuntis de diffusiecoëfficiënt nul. Immers, er vindt geen diffusie plaats, hoewel de twee fasen tochvan samenstelling verschillen. Het onderzoek spitst zich toe op de manier waarop dediffusiecoëfficiënt naar nul gaat als we de temperatuur verlagen tot het kritisch mengpunt.

De idealiteit van het mengsel in de buurt van het kritisch mengpunt speelt een centrale rol. Hetis duidelijk dat een mengsel daar sterk niet-ideaal is. De mate van niet-idealiteit kan bepaaldworden uit het verband tussen de dampdruk en de samenstelling van het mengsel. Deze niet-idealiteit moet nauwkeurig bepaald worden om vast te stellen of diffusiecoëfficiënten wel ofniet in overeenstemming zijn met de Maxwell-Stefanvergelijkingen. Het bleek helaas nietmogelijk dit te doen met een nauwkeurigheid die ons toestaat definitief te bewijzen dat devergelijkingen ook bij het kritisch mengpunt opgaan. Aan de andere kant is het evenminmogelijk het tegendeel te bewijzen. Een analyse van de resultaten lijkt er echter eerder op tewijzen dat de Maxwell-Stefanvergelijkingen gewoon geldig zijn bij het kritisch mengpunt. Methet oog op het volgende onderwerp is dit een belangrijk resultaat. Het gaat dan namelijk overvloeistof-vloeistofextractie, en daar is diffusie in ontmengende vloeistoffen aan de orde van dedag.

SAMENVATTING

134

Vloeistof-vloeistofextractie

Vloeistof-vloeistofextractie is een scheidingsproces dat gebruik maakt van de onmengbaarheidvan vloeistoffen. Stel, we hebben benzeen verontreinigd met methanol. De methanol kanverwijderd worden door het mengsel in contact te brengen met een andere vloeistof die nietmengt met benzeen, maar die wel de methanol opneemt. Het beste is het, als de methanol bijvoorkeur in deze vloeistof gaat zitten. Water is een uitermate geschikte kandidaat: het mengtzeer slecht met benzeen, en als de methanolconcentratie laag is, gaat er 10 keer zo veelmethanol in het water zitten als in de benzeen.

Om de extractie uit te voeren, is het het eenvoudigst het benzeenmengsel samen met een portiewater in een fles te doen, en dan goed te schudden. Waarom moeten we schudden? Wel, dathoeft niet per se, maar als we het niet doen, dan gaat het erg lang duren voordat de extractieklaar is. Zeker als het om wat grotere hoeveelheden vloeistof gaat. Als we de vloeistoffen in defles doen, dan zal de benzeenfase op het water gaan drijven, doordat het lichter is. Er wordt eenduidelijk zichtbaar, horizontaal grensvlak tussen de twee vloeistoffen gevormd. Alle methanoldie uit de benzeenfase naar het water gaat moet natuurlijk door dat grensvlak. Als we nietschudden, moet alle methanol uit de benzeenfase naar het grensvlak diffunderen alvorens hetnaar het water kan worden overgedragen. We weten inmiddels, dat diffusie een traag proces is,en daardoor zal de extractie lang duren. Een extra vertragende factor is, dat het grensvlak maarklein is. Het zou dus al helpen, als we een fles met een grote diameter nemen. Als we gaanschudden, dan mengen we de vloeistoffen, waardoor de methanol niet alleen door diffusie naarhet grensvlak wordt getransporteerd, maar ook door stroming. Bovendien zullen devloeistoffen verdeeld worden in kleine druppeltjes, zodat het grensvlak tussen de fasen veelgroter wordt. Deze twee factoren versnellen het extractieproces vele malen.

In dit proefschrift proberen we de stofoverdrachtprocessen te beschrijven die zich tijdensextractie aan het grensvlak afspelen. Als we deze processen willen onderzoeken, is het niet zohandig een fles te nemen en die goed te schudden. De toestand die dan in de fles ontstaat isnamelijk nogal chaotisch en we weten niet goed hoe we die moeten beschrijven. Voor ons ishet handiger de extractie uit te voeren in een opstelling die leidt tot een duidelijk gedefinieerdetoestand van de fasen en het grensvlak. In de linkerfiguur is een dergelijke opstelling, die weeen Lewis-cel noemen, geschetst. De rechterfiguur geeft de situatie weer aan het grensvlakzoals we die ons voorstellen.

SAMENVATTING

135

Figuur 1. a) Een schematische schets van een zijaanzicht van een Lewis-cel. Het is eigenlijk niet meerdan een glazen pot met een ronde doorsnede, voorzien van twee roerders en keerschotten. Dekeerschotten moeten voorkomen dat er een draaikolk ontstaat die het grensvlak verstoort. b) Weergavevan de situatie aan het grensvlak, over 90° gedraaid.

Door elk van de twee fasen te roeren, uiteraard zonder het grensvlak te verstoren, zijn beidefasen volledig gemengd en dus homogeen. Dit is in figuur 1b weergegeven door de constantesamenstelling in de bulk van beide fasen. De (turbulente) stromingen die door het roerenontstaan, worden gedempt aan het grensvlak. Zo vormt zich aan beide zijden van het grensvlakeen dun laagje stilstaande vloeistof waarin geen stroming meer is. Stoftransport door dezezogenaamde filmlagen kan daardoor uitsluitend door diffusie gebeuren. Hiervoor zijnconcentratieverschillen nodig, en in de filmlagen verandert de samenstelling van beide fasenfors, zoals te zien is in figuur 1b. De dikte van deze filmlagen hangt af van een aantal factoren,waarvan de viscositeit en de mate van turbulentie van de fasen de belangrijkste zijn. Dezetoestand in de Lewis-cel trachten we te beschrijven met een wiskundig model. Voor dit modelis het noodzakelijk te weten hoe de dikte van de filmlagen, de diffusiecoëfficiënten en deidealiteit van het mengsel afhangen van de samenstelling en andere factoren. Helaas zijn dezeafhankelijkheden vaak niet goed bekend, en moeten we er schattingen voor gebruiken. Debetrouwbaarheid van deze schattingen is echter twijfelachtig.

Het model vergelijken we vervolgens met gegevens uit de literatuur over extractiemetingen ineen Lewis-cel. Het blijkt dat het model deze metingen niet goed beschrijft. Waarschijnlijk is diteen gevolg van de bovengenoemde onbetrouwbaarheid van de schattingen van de dikte van defilmlagen, de diffusiecoëfficiënten en de idealiteit van het mengsel. Om dit probleem teondervangen hebben we een aantal van deze grootheden vermenigvuldigd met eensjoemelfactor. In feite kunnen we, door die factoren te veranderen, de waarde van dezegrootheden aanpassen. Vervolgens hebben we, door de sjoemelfactoren goed te kiezen,geprobeerd het model met de metingen in overeenstemming te brengen. Dat bleek inderdaadmogelijk te zijn, zonder dat de sjoemelfactoren onwaarschijnlijke waarden kregen. Het lijkt erop dat het model een beschrijving van het extractieproces geeft die redelijk dicht bij dewerkelijkheid ligt. Zekerheid over de juistheid van het model kan natuurlijk pas gegevenworden als de waarden van grootheden die nu nog geschat zijn, vervangen worden doornauwkeurig gemeten waarden.

SAMENVATTING

136

Tenslotte kijken we of het aldus ontwikkelde extractiemodel in staat is een tamelijk zeldzaamfenomeen te beschrijven: de spontane vorming van emulsies tijdens de extractie. Het kangebeuren, dat in een van beide fasen een ‘mist’ ontstaat van zeer fijne, met het blote oog niet teonderscheiden druppeltjes. Als ons model in staat zou zijn dit soort gedrag te voorspellen, danzou dat aangeven dat deze emulsievorming zuiver een gevolg is van diffusie. Immers, dat is hetenige stofoverdrachtmechanisme dat het model gebruikt.

Hoe zou de emulsie door pure diffusie gevormd kunnen worden? Een extractie waarbij eenemulsie gevormd wordt, is die van caprolactam (een grondstof voor nylon) uit water metbenzeen. Caprolactam gaat heel graag uit de waterfase naar de benzeenfase, doordat hetdaarmee een idealere oplossing vormt dan met water. Het diffundeert dus snel van de ene naarde andere fase. Het idee is nu, dat het daarbij water ‘meesleurt’ naar de benzeenfase door deonderlinge wrijving tussen de moleculen. Doordat water echter slecht oplost in benzeen, zouhet kunnen dat er te veel water in de benzeen terechtkomt, en dat deze fase dan gaatontmengen. Helaas blijkt het model grote moeite te hebben met dit scenario. Weliswaar is hetmogelijk ervoor te zorgen dat het model emulsievorming voorspelt door de sjoemelfactorenaan te passen, maar deze hebben dan extreme waarden. Het is moeilijk te geloven dat deschattingen van de bijbehorende grootheden zo ver bezijden de werkelijkheid zouden zijn. Hetis waarschijnlijker dat er andere mechanismen zijn die een rol spelen bij het ontstaan vanemulsies.

Al met al geeft het onderzoek, beschreven in dit proefschrift, meer inzicht in een aantal zakenomtrent diffusie in niet-ideale vloeistoffen. Het is echter duidelijk dat er nog veel moetgebeuren op dit gebied.