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UPTEC X 02 003 ISSN 1401-2138 JAN 2002
MARKUS WISTRAND
Modelling the effect of particle size distribution on Expanded Bed Adsorption processes
Master’s degree project
Molecular Biotechnology Programme Uppsala University School of Engineering
UPTEC X 02 003 Date of issue 2002-01 Author
Markus Wistrand Title (English)
Modelling the effect of particle size distribution on Expanded Bed Adsorption processes
Title (Swedish) Abstract In order to investigate the effect of particle size distribution (PSD) on Expanded Bed Adsorption (EBA) process, a model accounting for a gradient in particle size within the expanded bed as a function of process parameters, was constructed and solved. Two mechanisms for intraparticle mass transfer were considered: solid/homogeneous diffusion and pore diffusion. A parametric sensitivity analysis was also performed, based on column capacity utilisation and productivity. A high percentage of small particles in the PSD was found to increase column capacity utilisation, while productivity was almost unaffected. Changes in intraparticle diffusion coefficient, liquid phase viscosity, particle radius and linear velocity all had an important effect on utilisation and, to a lower degree, on productivity. The effects of changes in axial and solid dispersion coefficients were close to negligible.
Keywords Expanded bed adsorption; particle size distribution; modelling; hydrodynamics Supervisors
Dr. Karol Lacki Amersham Biosciences
Examiner Patrik Forssén
Department of Scientific Computing, Uppsala University
Project name Sponsors Amersham Biosciences
Language English
Security
ISSN 1401-2138
Classification
Supplementary bibliographical information Pages 38
Biology Education Centre Biomedical Center Husargatan 3 Uppsala Box 592 S-75124 Uppsala Tel +46 (0)18 4710000 Fax +46 (0)18 555217
Modelling the e�ect of particle size distribution onExpanded Bed Adsorption processes.
Markus Wistrand
Sammanfattning
Proteinrening är en vanlig syssla i läkemedelsindustrin, t.ex. för att rena fram den aktivasubstansen för ett läkemedel. Det är viktigt att under reningsprocessen minimera såvältidsåtgång som förluster. Expanded Bed Adsorption (EBA) är en relativt ny teknik medpotential att göra just detta genom att utföra �era steg i processen på en gång.Principen i EBA är att man låter en lösning med proteiner och en rad skräpprodukterpassera genom en kolonn med ett absorberande media ("solid fas"). Skräpprodukternagår rakt igenom medan proteinerna binder speci�kt till molekyler i solida fasen.Proteinerna kan sedan tas tillvara genom att villkoren förändras inne i kolonnen (t.ex.pH-värdet) så att bindningen släpper och proteinerna sköljs ut. I EBA består den solidafasen av partiklar tillhörandes en viss storleksfördelning. I detta examensarbeteintresserade jag mig särskilt för hur olika storleksfördelningar påverkar reningsprocessenoch studerade detta genom att sätta upp en teoretisk modell för EBA och utifrån denskriva ett simuleringsprogram.Slutsatsen av projektet är att solid fas bör ha en stor andel små partiklar för att bindamaximalt av det protein som man önskar rena fram. Förhoppningsvis ger dessa resultatnya insikter i hur media för EBA bör se ut och det är också möjligt att genomföra nyastudier med hjälp av programmet.
Examensarbete 20 p. i Molekylär bioteknikprogrammet
Uppsala universitet, januari 2002
Contents
1 Introduction 31.1 What has been done on EBA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Prior experiments on EBA . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.2 Prior modelling of EBA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Modelling EBA 72.1 Model part one: Bed expansion (hydrodynamics) . . . . . . . . . . . . . . . . . . 72.2 Mechanisms of mass transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.1 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Dependence on bed height and radius . . . . . . . . . . . . . . . . . . . . 102.2.3 The liquid phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.4 The solid phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.5 Intraparticle mass transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Non-dimensional units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Models on non-dimensional form . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5 The homogeneous model on non-dimensional form . . . . . . . . . . . . . . . . . 142.6 The porous model on non-dimensional form . . . . . . . . . . . . . . . . . . . . . 142.7 Validation of the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Simulations 163.1 Model parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Particle size distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Measurement of column utilization and productivity . . . . . . . . . . . . . . . . 183.4 Parametric sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Results and discussion 214.1 Concentration pro�les in liquid and solid phase . . . . . . . . . . . . . . . . . . . 21
4.1.1 E�ect of intraparticle mass transfer . . . . . . . . . . . . . . . . . . . . . . 224.2 E�ect of particle size distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Parametric sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3.1 Capacity and productivity . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5 Acknowledgements 34
6 Nomenclature 37
A Parameters 39A.1 Axial dispersion coe�cient as a function of voidage . . . . . . . . . . . . . . . . . 39
1
A.2 Overall Bodenstein number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
B Numerical techniques 40B.1 Liquid phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40B.2 Solid phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41B.3 Homogeneous particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
C Pro�les for all simulations performed 42
2
1 Introduction
Proteins of interest in the pharmaceutical industry are expressed in systems such as bacteria,yeast and mammalian cells. To purify the proteins from these sources is an every-day processconsuming both time and money. Techniques able to interfere and create shortcuts in longpuri�cation schemes are therefore of great value. Expanded Bed Adsorption (EBA) is a relativelynew technique aiming to reduce the number of steps needed in puri�cation, thereby saving bothtime and money as well as reducing product loss.Traditional puri�cation techniques involve several steps often including a number of chromato-graphic steps to be sure to reduce impurities and contaminants to a minimum. The starting pointare the cells which are lysed to produce crude feed-stocks. These feed-stocks are a mishmashof cells, cell debris, proteins and impurities of all kinds. Applying the feed-stock directly to apacked bed chromatography column would cause column clogging.Traditional puri�cation schemes instead starts by steps aiming for the removal of cell debris, suchas centrifugation and micro�ltration. These procedures clarify and concentrate the feed-stockto make application to a chromatographic column possible. However, centrifugation and micro-�ltration have drawbacks which are essentially manifested in product losses due to degradationand spilling, loss of time and high costs. Being able to apply the crude feed-stock to a column fordirect adsorption would thus be desirable. As is depicted in Fig. 1.1, EBA is a one-unit processaiming to clarify and concentrate the feed-stock as well as to obtain a �rst puri�cation.
Figure 1.1: The link between fermentation and puri�cation and where EBA enters the puri�cationscheme.
Expanded Bed Adsorption is a technique of its own using specially designed columns and media.The working principle is the same as in traditional chromatography but the bed is expandedinstead of being packed. Expansion is the result of letting a �ow enter the column from below,pushing adsorbent particles upwards. At a certain degree of expansion, the drag force on theparticles is exactly matched by the gravitational force, leading to an equilibrium. For a givenadsorbent-feed system, the equilibrium is dependent on the �ow in the column: a high �ow givesa high degree of expansion. However, an upper limit on �ow rate always exists as too high a �owwould carry particles out of the column.The voidage is the ratio of liquid phase volume to column volume (εL or voidage). Packed bedchromatography typically operates with a voidage that is independent of column height and inthe region εL=0.3-0.4 [1]. In EBA, normal values of overall voidage are in the regionεL=0.7-0.8,
3
Figure 1.2: Voidage gradient and density gradient in an expanded bed
equivalent to a bed expansion of 2-3 times the settled bed [1]. It is this high voidage that reducesclogging in EBA systems compared with a packed bed system.A characteristic in EBA related to the voidage is the distribution of solid phase inside the column.Adsorbent particles are neither mono-sized, nor do they all have the same density. The reason isboth to stabilize the expanded bed and to make media production easier and cheaper. Assumingall particles are of the same density (a simpli�cation), smaller particles are upon �uidization,found in the upper part of the EBA column while larger particles are found at the bottom.The degree of mixing of adsorbent material inside the column has been debated, but there is aconsensus that theory as well as empirical studies support a distribution of particles inside thecolumn, with respect to size. Consequently, due to the particle size distribution (PSD), as wellas a possible density distribution, a voidage gradient is formed in the �uidized bed, with thelowest voidage in the bottom of the column and the highest in the upper regions Fig. 1.2.The adsorbing media is at the very heart of all types of chromatography. It is made from abase matrix that should be a both chemically and mechanically stable support for the ligandscoupled to it. These ligands are chosen to suit the speci�c requirements in each application, i.e.the binding characteristics (selectivity) needed for the puri�cation of a certain protein. In EBA,the density of the adsorbent particles is higher than in normal chromatography to stabilize thebed and allow for a high �ow rate. The higher density is achieved by placing an inert core in themiddle of the adsorbent particle or from small pieces of quartz scattered in the particles.Any EBA process is a�ected by two types of parameters. The hydrodynamic parameters governthe degree of expansion and mixing inside the column. These are �uid velocity, mixing in liquidphase (axial dispersion), mixing in solid phase (solid dispersion), density of the adsorbent andliquid phase viscosity. Other parameters govern the mass transfer of solute inside the column.These are di�usion coe�cient inside the adsorbing material and �lm mass transfer coe�cient.Changing the particle size and the particle size distribution will a�ect both the hydrodynamicsand the mass transfer characteristics.
1.1 What has been done on EBA
1.1.1 Prior experiments on EBA
Early �uidized beds were unstable. Unstable beds give rise to increased axial mixing in liquidphase which reduces separation performance [1], [2], [3]. Increased knowledge of the mechanisms
4
behind dispersion in both solid and liquid phase has led to better products. It is thought that adistribution of particle radii adds to the stability by forming a bed where particles are classi�edby size, i.e. a bed where each region in the column is made up of particles of a certain size [4].By using special columns, where samples can be withdrawn along the column length, Willoughbyet al. [5] have veri�ed experimentally that beds are indeed classi�ed by size, although a certaindegree of mixing of particles is present. Results pointing in the same direction, although showingless degree of classi�cation, have been presented by Bruce and Chase [6]. In the same article,the authors present empirical data on bed voidage and axial dispersion in liquid phase alongthe bed. The results support a gradient of bed voidage inside the column and shows how axialdispersion decreases with increasing porosity, i.e. the highest dispersion is found at the bottomof the column.Karau et al. [4] have examined the e�ect of particle size distribution on axial dispersion. Theyworked with three PSDs: 120-160 µm, 120-300 µm and 250-300 µm. They reported that sizedistribution a�ect the degree of dispersion, with the lowest dispersion for the wide size distribu-tion. Interesting to note, this is at odds with results for a packed bed presented by Han et al. [7].They showed that, axial dispersion was higher for a wide particle size distribution, compared tofor a narrow one. This underlines the need to view EBA as a technique di�erent from traditionalpacked bed chromatography. While, in a packed bed, particles of di�erent sizes are more or lesscompletely mixed, they are much more classi�ed in an expanded bed.In addition to the dispersion due to axial mixing in solid and liquid phase, mass transfer char-acteristics adds to the dispersed concentration front inside the column. Mass transport to theparticles is most often considered to occur in two distinct steps called external and internal masstransfer. The �rst step accounts for a solute transport through an imaginary stagnant �lm/layeraround the �uidized particle, whereas the second step takes care of mass transfer phenomenaoccurring inside the particle, such as intraparticle di�usion and adsorption. The rate of externalmass transfer depends on �lm thickness which is a function of liquid phase viscosity and velocity.The intraparticle mass transfer depends on the ratio of pore size to solute size, pore topology aswell as ligand size and its distribution. The rate of internal transport is characteristic for eachmedia.The internal resistance and the �lm resistance together characterize mass transfer to and insidethe solid phase. Depending on system parameters such as particle size, solute size and liquidvelocity the importance of each of the two steps varies [8], [9].Large particles have long di�usion paths which adds to the internal resistance. It can generallybe said that, for large diameter particles, internal resistance will be more important than �lmresistance, while for smaller particles the relative importance of �lm resistance increases [4], [10].In packed bed processes, it is often assumed that the internal resistance is the rate limitingstep, and that �lm resistance and axial dispersion can be neglected. However, as pointed out byKarau et al., this is not the case in EBA. They have measured [4] the quote between the tworesistances, and come to the conclusion that for particles ranging in diameter over the interval120-300 µm, the �lm resistance goes from being equally important for small particles to becomeseveral times lower in value for large particles. Film resistance also decreases in importancewhen the linear �uid velocity increases. To conclude, they remark that to optimize EBA, smallparticles with a relatively high density should be used to allow for a high linear velocity withoutextreme expansion, low �lm resistance and yet short di�usion paths inside particles.The dynamic capacity is an often used measure of column e�ciency and can be stated as thedegree of average saturation of the adsorbent material in the column. In general, the moredispersed the concentration front is, the lower is the dynamic capacity. Bruce and Chase [6] havemeasured the dynamic capacity and reported higher capacity in the top region of the columnthan in the bottom region. This was assigned to the following characteristics at the top of thecolumn, compared to the bottom: low axial dispersion, low linear velocity (as the voidage ishigh, the velocity per void is low) and smaller diameter particles giving short di�usion paths.
5
1.1.2 Prior modelling of EBA
In modelling Expanded Bed Adsorption, people have mainly focused on either the dynamics ofthe expanded bed or the adsorption characteristics of the bed assuming mono-sized particles.Wright and Glasser [11] modelled mass transfer and hydrodynamics in EBA with two di�erentmass transfer models, in both case assuming mono-sized particles of the same density. Their twomodels describe di�erent types of mass transfer mechanisms, applicable to two di�erent types ofadsorbent material. In the homogeneous model, particles are thought of as gel-like in which thesolute di�uses driven by the solid phase concentration gradient. In the porous model, the solutedi�uses in the pores of the particles while all the time being in equilibrium with the adsorbedconcentration of solute on the matrix. The simulations were for both models in good agreementwith experimental data.The authors also did a parametric sensitivity analysis using simulations, trying to assess theimpact of hydrodynamic parameters (velocity and axial dispersion) and mass-transfer parameters(particle radius, �lm mass transfer and solid di�usion) for di�erent degrees of expansion. Thiswas done by changing one parameter at the time and comparing how the breakthrough timechanged. It was found that velocity and particle size were the most important parameters whileaxial dispersion, �lm mass transfer and intraparticle di�usion were of less importance.A model of how, for a given particle size distribution, the voidage varies along the bed heightin a �uidized bed, has been developed by Al-Dibouni and Garside [12]. They presented onemodel assuming mixing of particles in the column and one assuming perfect classi�cation ofparticles with respect to size. Both were compared to experimental results and both showed goodagreement, with the mixing model proving to be the best one. However, it needs experimentalinput, and the authors reasoned that for most practical situations using the classi�cation modelshould be good enough.Thelen et al. [13] have developed a distributed parameter model, which is capable of predictingthe dynamic response of bed height to step changes in the �uidization velocity. The modelcaptured both the convective transport of solid phase due to the liquid phase �ow rate, and aproposed erratic (dispersive) transport caused by hydrodynamic e�ects. Particles were consideredmono-sized and non-adsorbing and the density of solid and liquid phase were thus kept constant.The model predictions for both step decreases and step increases in �uidization velocity comparedfavourably with experiments.
1.2 Objectives
The primary objective of this work is to study the e�ect of the particle size distribution onExpanded Bed Adsorption process. This is done by constructing a model which combines thehydrodynamics involved in bed expansion assuming a perfectly classi�ed bed, with a modeldescribing mass transport in an EBA process. In Wright and Glasser's work [11] referred toabove, no attention was paid to the distribution of particles in the column due to di�erentsizes, and the gradient in voidage and interstitial velocity along the bed that follows. This isa simpli�cation of reality and we show here how di�erent particle size distributions a�ect theperformance of the EBA system by performing computer simulations. For each distribution, aparametric sensitivity analysis is performed changing both hydrodynamic properties and mass-transfer properties. Results are reported in terms ofutilization (U) de�ned as dynamic capacitydivided by maximum capacity and productivity (P) de�ned as utilization per hour process. Wealso provide concentration pro�les of both liquid and solid phase along the bed and withinadsorbent particles at di�erent heights. Together these may give valuable insight in the dynamicsof EBA.
6
2 Modelling EBA
In this chapter, we will go through the di�erent processes involved in modelling EBA. Basicallythe model consists of two separate parts. The �rst part contains the model for bed expansionand assigns a voidage and a particle size to each segment of the bed. The second part deals withthe modelling of mass transfer in liquid and solid phase, given the outcome from the �rst part.As a starting point, the following assumptions have been made:
∗ Adsorbent particles are spherical.∗ Particle radius is assumed to follow a given distribution.∗ Adsorption is instantaneous.∗ The thermodynamics of the adsorption process are described by a Langmuir isotherm.∗ All particles have the same density.∗ The change in mass of adsorbent material due to the uptake of solute is neglected. Onceexpanded, the bed is assumed static.
∗ Radial dispersion in the column is not considered.
2.1 Model part one: Bed expansion (hydrodynamics)
We followed the model outlined by Al-Dibouni and Garside [12] for the case of a perfect classi�edbed. The basic assumption made is that each particle is axially stationary when the bed expansionhas reached an equilibrium. This occurs when the liquid interstitial velocity equals the settling(terminal) velocity at each height segment in the bed. As particle density is assumed constant,the particle radius alone determines in which segment a particular particle is found. All mixingin the column due to circular movements is neglected.Before explaining the model, it should be clari�ed how an individual particle's terminal velocityis predicted. For low particle Reynolds number,Rep<0.2, this should be done by Stoke's law,while for higher particle Reynolds number a correcting term should be added [3]. Rep is givenby
Rep =ρLud
µεL
where u is the super�cial velocity, d is the particle diameter, ρL is the liquid phase density, εLis the bed voidage and µ is the liquid phase viscosity. The terminal velocity (ut) is then for therespective regions of Rep written as
ut =d2g(ρs − ρL)
18µRep < 0.2 (2.1)
ut =d2g(ρs − ρL)
18µ(1 + 0.15Re0.687p )
Rep > 0.2 (2.2)
where g is the acceleration of gravity and ρs is the particle density.It can be veri�ed that for Rep = 0.2, the two expressions do not match (a 5% error). Using theequations in their respective region therefore results in a discontinuity inut. To avoid this, wehave chosen to use only Eq. (2.2) for all calculations. The error this infers decreases rapidlyfrom the 5% error, when Rep approaches 0.However, Eq. (2.2) does not account for the e�ects of other particles in the system. Thisis corrected for through the Richardson-Zaki equation which relates terminal velocity to bedvoidage (εL) and super�cial velocity (u).
u = ut(d)εnL (2.3)
where the value of n for all values ofRep is given by [12].
(5.1− n)/(n− 2.7) = 0.1Re0.9p (2.4)
7
For a distribution containing su�ciently large particles, a layer of the bed at the bottom of thecolumn might stay sedimented. The necessary condition for expansion is that Eq. (2.3) gives ahigher voidage than the value for the sedimented bed, i.e.
(u
ut
)1/n
≥ εsed (2.5)
In a perfectly classi�ed bed, particles with diameter betweend and (d − dd) will be found in asegment of the column between z and (z +dz). The volume gel in a segment of the column (dV )can be written
dV = (1− εL)Adz (2.6)where A is the column area. The area can be expressed in terms of the total volume of gel in thecolumn (Vgel), the sedimented bed height (Hsed) and the voidage of the sedimented bed (εsed).
A =Vgel
Hsed(1− εsed)(2.7)
Furthermore, gel volume in the segment can be expressed as
dV = f(d)dd (2.8)
where f(d) is a volume based frequency function, normalized to satisfy∫∞0 f(d)dd = 1. Eqs.
(2.3, 2.6, 2.7 and 2.8) gives for the non-expanded part of the column
dz
dd= Hsedf(d) (2.9)
and for the expanded part of the column
dz
dd=
Hsed(1− εsed)(1− ( u0
ut(d))1/n)
f(d) (2.10)
These equations are integrated to get column height as a function of particle radius. Once thisis done, the local voidage is calculated as a function of column height from Eq. (2.3) and theexpansion by simply dividing H with Hsed.In case some of the particles are small enough, the voidage at higher segments of the bed willcome close to 1 and the denominator in Eq. (2.10) will approach zero. It is then not possibleto perform the integration. This is solved by cutting the distribution (excluding the smallestparticles), and integrate Eq. (2.10) with the cut distribution and the cut settled bed heightthat follows. In an experiment this would mean that particles that are light enough to leave thecolumn at the given �ow are eluted.In the case of mono-sized particles, voidage will be the same all over the column and is given byEq. (2.3). Inserting the condition f(d) = 1, and Eq. (2.3) into Eq. (2.10), yields.
H
Hsed=
1− εsed
1− εL(2.11)
Eq. (2.11) is a well-known expression for bed expansion using mono-sized particles.
2.2 Mechanisms of mass transfer
In modelling the EBA process, we formulate a model describing convective and dispersive trans-port in liquid phase, dispersion in solid phase and mass transport to and inside the EBA particlesdescribed by �lm resistance and di�usion, respectively. We will in this section �rst shortly explainthe model parameters and then give the model equations.
8
2.2.1 Model parameters
Convective transport: Convective transport of solute in liquid phase is driven by the liquidphase �ow from below, which forces the solute through the column.
Dispersive transport: Dispersive transport is the result of inhomogeneities in liquid phaseconcentration which causes local concentration gradients. The inhomogeneities come from back-mixing and dead volumes in the column as the �uid passes through the bed.
Solid dispersion: Solid dispersion introduces the possibility of having a not perfectly staticexpanded bed. While the bed is expanded, it is assumed to be classi�ed by size. Due to tur-bulence and to the fact that particles are not saturated simultaneously, particles may howeverslightly move. For example should a particle that for some reason is more saturated (and there-fore heavier) than its neighbours, slightly settle. This will a�ect not only the mass transfer tothe particles, but also the bed voidage distribution. In modelling the solid dispersion we followthe method outlined byWright [14] and do only account for the axial component of the dispersion.
Film mass transfer: The external resistance to mass transfer is modelled by a hypotheti-cal stagnant �lm/layer that is formed around the particle. The �ux of solute in the �lm is givenby a linear driving force and is dependent on the concentration di�erence between liquid phaseand the particle surface.
Intraparticle di�usion: The internal resistance to mass transfer is modelled by a di�usionprocess. It has, however, been shown by several authors that mass transfer in di�erent types ofchromatography media cannot be explained by the same model (see for example [16] and [15]).In this work, two models that are the most frequently used when modelling ion exchange chro-matography, were considered. These are the homogeneous di�usion model and the pore di�usionmodel.In the homogeneous model, the medium is seen as homogeneous and gel-like. The solute isadsorbed at the particle surface and di�uses towards the center in a process governed by a driv-ing force due to the gradient in adsorbed solute concentration, and a di�usion coe�cient (Dh).Wright et al. have shown that this model well describes mass uptake by S-HyperD LS media[11], [15] .In the porous model, intraparticle mass transfer occurs by di�usion inside the particle pores, witha driving force expressed in terms of the gradient in pore concentration, and a characteristic dif-fusion coe�cient (Dp). Pore concentration is all the time in equilibrium with the adsorbed soluteconcentration. Wright et al. have shown that the porous model well describes mass uptake byStreamline media [11], [15].
Adsorption: In this work the equilibrium between adsorbed concentration and the liquid phaseconcentration in close vicinity of the adsorbing surface, is assumed instantaneous. The adsorptionprocess, that occurs at the surface of the homogeneous particle and inside the porous particle,is assumed to follow a Langmuir isotherm (Eq. (2.12)).
q =qmaxCf
Ks + Cf(2.12)
where q is the adsorbed concentration, Cf is the liquid phase concentration at the particlesurface, Ks is the dissociation constant characteristic for each adsorption process andqmax isthe maximum capacity of the adsorbent. Assumptions behind applying a Langmuir isotherm are[10]:
1. Molecules are absorbed at a �xed number of well-de�ned localized sites.2. Each site can hold one adsorbate molecule.3. All sites are energetically equivalent.
9
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
50
100
150
c (mg/ml)
q (m
g/m
l)
Figure 2.1: A favourable Langmuir isotherm. qmax = 150 mg/ml, Ks = 0.01 mg/ml
4. There is no interaction between molecules adsorbed on neighbouring sites.
Fig. 2.1 shows an example of a favourable Langmuir isotherm (high adsorption already at lowliquid concentration), with parameters qmax = 150 mg/ml and Ks = 0.01 mg/ml.
2.2.2 Dependence on bed height and radius
The model equations given below are stated as if parameters and variables did neither changewith time, nor with position. Correctly stated, their dependence on bed height (z), radius (r)and time (t) should be indicated, but this is omitted to make equations easier to read. Thefollowing parameters are, however, position dependent vectors: Dax(z) is the axial dispersioncoe�cient, Ds(z) is the solid dispersion coe�cient, εL(z) is the voidage, εs(z) is the fraction solidphase to column volume (εs(z) = 1− εL(z)), kf (z) is the �lm mass transfer coe�cient andRp(z)is the particle radius.Further, the variables (which the model is solved for) are both dependent on position and timeand should correctly be written: C(z, t) is the liquid phase concentration,Cp(z, r, t) is the poreconcentration, qs(z, r, t) is the adsorbed concentration, q(z, r, t) is the solid phase concentration,and q′(z, t) is the average solid phase concentration.
2.2.3 The liquid phase
For a given segment between height z and z +dz in the liquid phase, the mass balance is writtenas follows:Accumulation = mass inbulkflow − mass outbulkflow + mass indispersion − mass outdispersion
− mass uptake by adsorbent
At the boundaries of the column (inlet and outlet) we used the boundary conditions that were�rst presented by Danckwerts [17] for a case with axial dispersion. These boundary conditionsseems to be the most commonly applied ([9], [14], [19]), and assume no axial dispersion in thesections immediately before and after the column. Initial condition is an empty column and themodel equations are written as follows:
10
∂C
∂t= − ∂
∂z
(−Dax
∂C
∂z+
u
εLC
)− 3kf (1− εL)(C − Cf )
εLRp(2.13)
I.C: at t = 0, C(z, 0) = 0 for 0 ≤ z ≤ H (2.14)
B.C 1: at z = 0, C − DaxεLu
∂C
∂z= C0 , t>0 (2.15)
B.C 2: at z = H,∂C
∂z= 0 , t>0 (2.16)
where C is the liquid phase concentration,C0 is the feed concentration,Dax is the axial dispersioncoe�cient, H is the expanded bed height, z indicates bed height, u is the super�cial velocity, kf
is the �lm mass transfer coe�cientRp is the particle radius at the actual segment andCf is theliquid phase concentration close to the particle surface. The latter is given from the Langmuirisotherm (eq. (2.12)).
2.2.4 The solid phase
The mass balance for the solid phase is for a given segment between heightz and z +dz, writtenas:Accumulation = mass indispersion − mass outdispersion + mass uptake by adsorbent
The boundary conditions used re�ect the fact that there is no solid phase outside the column andthe initial condition that, when the process starts, no adsorption has yet occured. The equationsgoverning solid phase are then as follows:
∂q′
∂t= Ds
∂2q′
∂z2+
3kf (C − Cf )R
(2.17)
I.C: at t = 0, q′(z, 0) = 0 for 0 ≤ z ≤ H (2.18)
B.C 1: at z = 0,∂q′
∂z= 0 , t>0 (2.19)
B.C 2: at z = H,∂q′
∂z= 0 , t>0 (2.20)
where Ds is the solid phase dispersion coe�cient andq′ is the average solid phase concentration.
2.2.5 Intraparticle mass transfer
Di�usion in homogeneous media Mass transfer inside the adsorbent media is for the ho-mogeneous model governed entirely by di�usion, and the equation is written as:
∂q
∂t= Dh
1r2
∂
∂r
(r2 ∂q
∂r
)(2.21)
I.C: at t = 0, q(z, r, 0) = 0 for 0 ≤ z ≤ H (2.22)
B.C 1: at r = rc,∂q
∂r= 0 , t>0 (2.23)
11
B.C 2: at r = Rp,∂q
∂r=
RpDs
3Dh
∂2q′
∂z2+
kf
Dh(C − Cf ) , t>0 (2.24)
where r indicates radius, rc is the radius of the inert core at the center, q is the concentrationinside the particle, Rp is the particle radius and Dh is the intraparticle di�usion coe�cient.Initially adsorbent contains no solute which is described by the initial condition. The �rstboundary condition describes the fact that there is a zero �ux to the inert core placed at thecenter of the particle. The second boundary condition relates the uptake at the particle boundaryto the average solid phase concentration (q'). Noting that the ratio volume to area isRp/3, thefollowing relation holds at the particle surface:
∂q
∂r=
Rp
3Dh
∂q′
∂t(2.25)
Inserting ∂q′∂t from Eq. (2.17) gives the second boundary condition and relates the �ow across
the boundary both to solid dispersion and to �lm resistance.
Di�usion in porous media Modelling of the porous media involves one more variable com-pared to the homogeneous case, as we have both pore concentration (Cp) and adsorbed con-centration (qs). However, for the case of instantaneous equilibrium, it is possible to reducecomputations by expressing one variable through the other, through the Langmuir isotherm (eq.(2.12)).The equation governing mass transport within a porous particle is given by [18]
εp∂Cp
∂t+ (1− εp)
∂qs
∂t= εpDp
1r2
∂
∂r
(r2 ∂Cp
∂r
)(2.26)
where εp is the particle porosity and Dp is the pore di�usion coe�cient. The chain rule gives:
∂qs
∂t=
∂qs
∂Cp
∂Cp
∂t(2.27)
The Langmuir isotherm describes the instantaneous equilibrium between pore concentration (Cp)and adsorbed concentration in the particles (qs).
qs =qmaxCp
Ks + Cp(2.28)
Di�erentiating the isotherm with respect toCp gives
∂qs
∂Cp=
Ksqmax
(Ks + Cp)2(2.29)
Inserting Eq. (2.27) and Eq. (2.29) into Eq. (2.26), the time derivative of the pore concentrationcan be expressed as
∂Cp
∂t= α
1r2
Dp∂
∂r
(r2 ∂Cp
∂r
)(2.30)
whereα =
εp
εp + (1− εp) ∂qs
∂Cp
(2.31)
The initial and boundary conditions used are the same as for the homogeneous model: noadsorbed solute when the process starts (I.C), a zero �ux to the inert core at the center of the
12
particle (B.C 1) and the �ux across the surface of a particle is related to the change in averageparticle concentration (B.C 2).
I.C: at t = 0, Cp(z, r, 0) = 0 for 0 ≤ z ≤ H (2.32)
B.C 1: at r = 0,∂Cp
∂r= 0 , t>0 (2.33)
B.C 2: at r = Rp,∂Cp
∂r=
RpDs
3Dp
∂2q′
∂z2+
kf
Dp(C − Cp) , t>0 (2.34)
2.3 Non-dimensional units
If changes in parameter values occur on very di�erent scales, it might be hard to assess whetherobserved changes are relevant or not. Scaling each parameter with respect to a reference value(e.g. feed concentration for the liquid phase concentration) makes changes non-dimensional andeasier to follow. All equations were therefore set on dimensionless form using the dimensionlessgroups listed in Tab. 2.1. C0 is the feed concentration and, D is the intraparticle di�usioncoe�cient (Dh or Dp) and R is the reference radius. The latter is set to the radius of thesmallest particle in the column as concentration changes the fastest in this particle.
Table 2.1: Non-dimensional units.β = C
C0Dimensionless liquid phase concentration
Q′ = q′
qmaxDimensionless average solid phase concentration
Q = qqmax
Dimensionless solid phase concentrationβp = Cp
C0Dimensionless pore concentration
η = rR Dimensionless radius
ξ = zH Dimensionless bed height
Θ = DR2 t Dimensionless time
Pe = uHεLDax
Peclet numberBi = kf R
D Biot numberΘr = HDεL
uR2 Mass transfer unit
2.4 Models on non-dimensional form
Liquid phase:
∂β
∂Θ=
1PeΘr
∂2β
∂ξ2− 1
Θr
∂β
∂ξ
(1− εL
Hu
∂Dax
∂ξ
)+
βu
H
∂ε−1L
∂ξ− 3Bi
εsR
εLRp(β − βf ) (2.35)
I.C: at Θ = 0, β(ξ, 0) = 0 for 0 ≤ ξ ≤ 1 (2.36)
B.C 1: at ξ = 0, β − 1Pe
∂β
∂ξ= 1, Θ > 0 (2.37)
B.C 2: at ξ = 1,∂β
∂ξ= 0 , Θ > 0 (2.38)
Solid phase:∂Q′
∂Θ=
DsR2
DH2
∂2Q′
∂ξ2+ 3Bi
C0
qmax(β − βf ) (2.39)
13
I.C: at Θ = 0, Q′(ξ, 0) = 0 for 0 ≤ ξ ≤ 1 (2.40)
B.C 1: at ξ = 0,∂Q′
∂ξ= 0 , Θ > 0 (2.41)
B.C 2: at ξ = 1,∂Q′
∂ξ= 0 , Θ > 0 (2.42)
2.5 The homogeneous model on non-dimensional form
The homogeneous particle:∂Q
∂Θ=
1η2
∂
∂η(η2 ∂Q
∂η) (2.43)
I.C: at Θ = 0, Q(ξ, η, 0) = 0 for 0 ≤ ξ ≤ 1 (2.44)
B.C 1: at η =rc
Rp,
∂Q
∂η= 0 , Θ > 0 (2.45)
B.C 2: at η =Rp
R,
∂Q
∂η=
RpRDs
3H2
∂2Q′
∂ξ2+ Bi
C0
qmax(β − βf ) , Θ > 0 (2.46)
2.6 The porous model on non-dimensional form
Porous particle∂βp
∂θ= α
1η2
∂
∂η(η2 ∂βp
∂η) (2.47)
whereα =
εp
εp + (1− εp) ∂qs
∂Cp
(2.48)
with initial and boundary conditions
I.C: at Θ = 0, βp(ξ, η, 0) = 0 for 0 ≤ ξ ≤ 1 (2.49)
B.C 1: at η =rc
Rp,
∂Cp
∂η= 0 , Θ > 0 (2.50)
B.C 2: at η =Rp
R,
∂βp
∂η=
RpRDsqmax
3H2C0
∂2βp
∂ξ2+
Bi
εp(β − βp) ,Θ > 0 (2.51)
2.7 Validation of the models
In order to con�rm that model equations and the numerical scheme were correctly used, themodel predictions were compared to either analytical or numerical solutions for some speci�ccases reported in literature.
1) Mass transfer to a homogeneous sphere in a batch assuming �lm resistance(Chap-ter 6.3.4, "Surface evaporation", Crank [20]).The analytical solution describes a homogeneous sphere, initially at uniform concentration, sub-merged in a liquid of in�nite volume. Amount of mass present in the sphere at any time is
14
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
Err
or %
r/R
0 0.2 0.4 0.6 0.8 10.65
0.7
0.75
0.8
0.85
0.9
0.95
1
C/C
final
r/R
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5E
rror
%
r/R
0 0.2 0.4 0.6 0.8 10.65
0.7
0.75
0.8
0.85
0.9
0.95
1
C/C
final
r/R
Homogenous
Porous
Figure 2.2: Comparison of the models with the analytical solution for a homogeneous sphere in a batchwith constant concentration, assuming �lm resistance. Percentage error and concentration pro�le insidethe sphere 30 minutes after the empty sphere was placed in the batch. First row: The homogeneous model.Second row: The porous model, εp = 1.
assumed negligible compared to the amount solute in the batch, and does not a�ect batch con-centration. There is no adsorption inside the sphere or at its surface, only di�usion. The surfacecondition is
D∂C
∂r= α(C0 − Cf )
where α is a constant,Cf is the concentration at the surface of the sphere andC0 is the concentra-tion required to maintain equilibrium with the surrounding atmosphere. This surface conditionis the same as the one describing �lm resistance withα = kf .The implementations of the two mass transfer models were veri�ed against this solution by"cutting lose" the particles from the two overall models and setting parameters to the followingvalues: C0 = 2 mg/ml, εL = 0.99999, kf = 5 ∗ 10−4 cm/s, and in addition for the porous modelεp = 1. A linear isotherm with Ks=1 was substituted for the Langmuir isotherm.
C(Rp) = KsCf (2.52)
Simulations were performed with 15 discretization points in the particle. Both models were ingood agreement with the analytical solution (Fig. 2.2).2) Mass transfer to a homogeneous sphere in a well stirred tank(Chapter 6.3.3, "Dif-fusion from a well-stirred solution of limited volume", Crank [20]).The analytical solution describes a homogeneous sphere in a batch of limited volume. No �lmresistance and no adsorption is considered and the sphere is initially free from solute. The con-centration in the batch is uniform with a starting value ofC0. The model was veri�ed by settingmodel parameters to: εL = 0.5, C0 = 2 mg/ml and, for the porous model, εp = 1 . In addition,the �lm coe�cient was set to a high value (kf = 10000 cm/s) to practically get rid of �lm resis-tance, and a linear isotherm (Eq. (2.52)) was substituted for the Langmuir isotherm. Using 15discretization points, the model was in good agreement with the analytical solution (Fig. 2.3).3) Packed bed with non-adsorbing particlesTo test the equations governing the column as well as the boundary conditions for the column,
15
0 0.2 0.4 0.6 0.8 10.2
0.25
0.3
0.35
0.4
0.45
0.5
Err
or %
r/R0 0.2 0.4 0.6 0.8 1
0.8
0.9
1
1.1
C/C
final
r/R
0 0.2 0.4 0.6 0.8 10.2
0.25
0.3
0.35
0.4
0.45
0.5E
rror
%
r/R
0 0.2 0.4 0.6 0.8 1
0.8
0.9
1
1.1
C/C
final
r/R
Homogenous
Porous
Figure 2.3: Comparison of the models with the analytical solution for mass transfer to a homogeneoussphere in a well-stirred tank. Percentage error and concentration pro�le inside the sphere 30 minutesafter the empty sphere was placed in the batch. First row: The homogeneous model. Second row: Theporous model, εp = 1.
the model was compared to simulations performed by Seidel-Morgenstern [19]. He has presentedbreakthrough curves for di�erent Peclet numbers using the same boundary conditions as thoseused in this work. Using the author's parameters and setting absorbent capacity toqmax = 0(non-adsorbing particles), we repeated his simulation. The results �tted Seidel-Morgenstern'sresults exactly (Fig. 2.4).In addition, the model predictions were compared with the analytical solution derived by Ku£era[21] for an in�nite column with a packed bed of non-adsorbing particles. At a given time (t = 0)a pulse is introduced into the column and Ku£era gives analytical solutions for the moments ata given time. These moments can also be calculated from the breakthrough curves of our simu-lations, which was done. As Ku£era considers an in�nite column he do not have any boundaries.We excluded the e�ect of boundaries by taking the di�erence of two simulations with di�erentcolumn length. For example: The �rst moment was calculated at 40 cm for a simulation witha 40 cm column, and at 20 cm for a simulation with a 20 cm column. The di�erence betweenthese moments were compared to the analytical solution for the �rst moment at 20 cm. The �rstmoments were identical. Higher moments were not tested.However, including the boundaries, the simulation did not match the analytical solution. Assimulations matched the solution of Seidel-Morgenstern, the implementation of the boundaries isdone correctly. The conclusion is that Danckwerts' boundary conditions, although they includeaxial dispersion, can not match a (hypothetic) column without boundaries.
3 Simulations
3.1 Model parameters
The model parameter values used were obtained from literature. Some parameters vary alongthe bed as interstitial velocity, voidage and particle size changes. For those, correlations obtainedfrom literature have been used.
16
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Θ
C/C
0 Pe=0.01
Pe=1
Pe=20
Figure 2.4: Packed bed with non-adsorbing particles. Comparison of model simulations with breakthroughcurves in [19].
Axial dispersion in EBA was investigated by Bruce and Chase [6]. They reported values of theBodenstein number as well as the voidage at di�erent column heights. We have used their resultsto �nd a correlation between axial dispersion and voidage. At each cross section of the column,a value is assigned to Dax from the voidage at this level (see further in section A.1).
Dax = 0.2991− 0.5101εL + 0.2163ε2L (3.1)
The equation governing the liquid phase gives rise to terms involving the di�erentiation ofDax
and 1εL
with respect to bed height (z). As these are not constant with height in our model(unlike a model assuming mono-sized particles), these terms can not be neglected. We expressthese terms by �nite di�erences of the values forDax and 1
εL.
For the solid dispersion coe�cient,Ds, we use the correlation Wright ascribes to Van Der Meeret al. [11]. It relates solid dispersion to the super�cial velocity only, and solid dispersion is thusassumed to be constant along the bed. The factor0.04 in the correlation is not dimensionless.
Ds = 0.04u1.8 m/s (3.2)
The �lm mass transfer coe�cient was for each section of the bed calculated from the correlationfor the Sherwood number in a packed bed recommended in Chemical Engineer's Handbook [22].It is de�ned as
kf = DmSh/d (3.3)where the Sherwood number is given as:
Sh =1.09εL
Re0.33p Sc0.33
Dm is the molecular di�usion coe�cient of the solute,Sh is the Sherwood number, d the diameterof the adsorbent particle, Rep the particle Reynolds number and Sc the Schmidt number. The�lm mass transfer coe�cient increases with solute di�usivity and �uid velocity and decreaseswith particle size [23].
17
0.005 0.01 0.0150
50
100
150
200
r (cm)
f(r)
0.005 0.01 0.0150
50
100
150
200
r (cm)
f(r)
0.005 0.01 0.0150
100
200
300
400
r (cm)
f(r)
0.005 0.01 0.0150
2
4
6
8
10x 10
4
r (cm)
f(r)
1 2
3 4
Figure 3.1: The four particle size distributions (PSDs) considered. 1: pyramidal, 2: biased to smallparticles, 3: biased to large particles and 4: mono-sized. -(blue) Number basis, - -(red) volume basis.
3.2 Particle size distributions
Studying the e�ect of particle size distribution on the EBA process is the main objective of thiswork. In order to evaluate this e�ect, the following four particle size distributions (PSDs) werechosen: 1) pyramidal , 2) biased to small particles, 3) biased to large particles and 4) mono-sized(Fig. 3.1). Important to note is that the distributions are much more di�erent on a volumebasis than it might appear looking at the radii, as volume depend cubically on the radius. Theparticles were for all PSDs distributed in the same interval, i.e. the smallest and the largestparticles were of the same size for all PSDs.In all simulations, a settled bed voidage of εsed = 0.36 is assumed. All beds therefore containthe same volume gel and, as particle sizes are di�erent, the expanded beds will for the four casesnot have the same expansion, for a given super�cial velocity.
3.3 Measurement of column utilization and productivity
The breakthrough curve is a plot of how solute concentration at the column outlet varies withtime. In practice, the process is often stopped when outlet concentration reaches 1-10% ofthe feed concentration, to avoid product losses. The dynamic capacity is a measure of howmuch of the protein loaded onto the column that has been adsorbed. It is most often de�nedas the amount solute bound to solid phase per volume settled bed at a certain breakthroughconcentration. However, from an experimental point of view, all protein resident in the columnmay be of interest. This amount which could be called an "apparent" dynamic capacity, will, atthe moment of breakthrough, include also the amount of solute in liquid phase. Working withhigh values of maximum capacity as we do, makes the di�erence between the two capacities smalland we will only consider the amount solute in solid phase. The dynamic capacity is written as
Qd =1
(1− εsed)Hsed
∫ H
0εsq dh (3.4)
Column capacity utilization (utilization, U) is a non-dimensional measure that relates the dy-namic capacity to the maximum capacity of the expanded bed. This measure allows a directcomparisons between bed of di�erent maximum capacity.
U = Qd/qmax (3.5)
A high value of the dynamic capacity can be reached by working with a �uid velocity close to theminimum velocity necessary to get bed expansion. This gives proteins plenty of time to di�use
18
into the adsorbent. However, there are disadvantages with such a procedure. Most obvious, highexpansion and consequently high �ow rate may be desired to avoid column clogging. Working atlow �ow is also a tedious procedure and cost a lot in terms of time, thus lowering the productivity.There is an obvious trade-o� between the time of the process and the dynamic capacity reachedin the process, and one ought to consider a productivity measure. We have chosen to de�neproductivity (P ) as the utilization of total column capacity per hour process time and boththe utilization and the productivity are reported at 1% and at 10% breakthrough. It may ofcourse be discussed whether our de�nition of productivity is a relevant measure as each run inthe laboratory need time to prepare in addition to the actual running time. Not consideringthe preparation time, turns fast processes with low dynamic capacity more favorable than theyreally are.
3.4 Parametric sensitivity analysis
A parametric sensitivity analysis of the model has been made to better understand how di�erentparameters a�ect the EBA process. In any chromatographic process, the following two timeconstants are important.
DR2 Governing the rate of mass transfer in the particlesuH Governing the residence time in the column
Keeping these two ratios constant, should for a packed bed give the same breakthrough curve.One could for example for a given process, double both column height and velocity, and thebreakthrough curve should still look the same and appear at the same time. For an expandedbed, things are more complicated as bed height and voidage is dependent on �uid velocityand particle size. Working at a high �ow rate, gives, for a given particle size, high expansionand may therefore not lead to short residence time. Ideally, in all chromatographic techniques,fast intraparticle di�usion is desirable to get fast saturation of the solid phase. The rate ofdi�usion is, however, �nite and usually slower compared to the di�usion in free solution. Inorder to optimize the performance, one could then focus on the denominator (R2) in the �rsttime constant and lower the value of the particle radius. Having very small particles does howevergive high expansion for a given �ow, leading to long process times. Lowering the �ow gives lowerexpansion, more mass transfer into the adsorbent, but once again have negative e�ects on overallprocess time. It can also lead to column clogging.Clearly, the process is more complex for EBA than for a packed bed. Only looking at the timeconstants above is misleading as hydrodynamic and mass transfer parameters are interrelated.A parametric sensitivity analysis may provide answers to how these interrelations are. We havesystematically changed operating parameters from a "base case" for both models and for allfour radius distributions. Parameters that have been investigated are particle size, intraparticledi�usion coe�cient, axial dispersion, solid dispersion, �uid phase viscosity and �uid velocity. Foreach series, expansion was kept constant by adjusting the solid phase density.Four series were made for each model, one for each PSD considered. The series are named withan H or P indicating the model, a number (1-4) indicating the PSD and a letter (A-K) indicatingthe case parameter values. A column with a settled bed height (Hsed), of 10 cm, a settled bedvoidage (εsed) of 0.36 and a maximum capacity (qmax) of 150 mg/ml solid, was modelled. Thefeed concentration (C0) was 2 mg/ml and the dissociation constant (Ks) was 0.01 mg/ml for allsimulations. All other parameters are given in Tab. 4.1. Parameters that have been changedwith respect to the respective base case, are in bold type.
19
Figure 3.2: Algorithm for the program.
3.5 Simulations
The column was divided into segments and simulations started with the hydrodynamics (bedexpansion) giving each segment of the column its characteristics, whereupon the mass transferof the EBA process was modelled. The program was written in Matlab and a scheme for thealgorithm is depicted in Fig. 3.2.Calculations of bed expansion starts from the size distribution of adsorbent particles de�nedon radius basis (Fig. 3.1), from which the volume distribution f(d) is calculated. Given asuper�cial velocity, u, bed height, z, as a function of particle diameter, d, is calculated bynumerical integration of Eq. (2.10). This relation is then inverted to get the particle size foreach segment of the column. Knowing the diameter, the voidage of the segment is calculatedfrom the Richardson-Zaki equation (Eq. (2.3)).The use of Eq. (2.2) to calculate the terminal velocity,ut, provides an extra di�culty as it needsthe local voidage, εL, as input. This is solved by computing bed expansion in an iterative schemeuntil the voidage pro�le is stabilized. (The �rst guess ofut is provided by Stoke's law withoutthe correcting term (Eq. (2.1)). Once diameter and voidage is known for each column segment,the axial dispersion coe�cient, Dax, and the �lm mass transfer coe�cient, kf , are calculatedfrom the respective correlation (Eqs. 3.1 and 3.3).The parameters calculated are used in the mass transfer model described by Eqs. (2.13 and2.17) and Eq. (2.21) for the homogeneous model or Eq. (2.26) for the porous model. Theseequations are coupled nonlinear partial di�erential equations. Finite di�erences are used todiscretize equations both in the column and in the particle, which gives a system ofNcol ∗Npart
ordinary di�erential equations. These are solved simultaneously using ode15s which is a pre-built
20
Matlab solver designed for sti� di�erential equations [24]. Tolerance levels were at least set toMatlab's default values, but lower tolerance levels did not change the results signi�cantly. Thenumber of discretization points chosen in this work were 50 in the column (Ncol = 50) and 15(Npart = 15) in the particle. The limit was set by computational time and computer memory.In the case of the porous model, which is the sti�est problem, a denser discretization of theparticle could perhaps have in�uence on the results, but for the homogeneous model, the numberof discretization points seems su�cient (see section 2.7).It is important to understand under what conditions each simulation has been performed. Fourseries of simulations were performed, one for each PSD. In the di�erent simulations, some modelparameters are constant and others will vary with height along the column as a result of thePSD. For example, all four "base cases" (one for each PSD) were modelled using the same su-per�cial velocity, u, but as the radii were di�erently distributed, expansion grades were di�erent.Further, in the parametric sensitivity analysis, parameters that are changed will a�ect dependentparameter values through the correlations.
4 Results and discussion
4.1 Concentration pro�les in liquid and solid phase
As mentioned in section 1.2 results from simulations include concentration pro�les inside thecolumn and how these changes with time. In this section we will discuss an example (Fig. 4.1)to clarify how results obtained for each case considered are plotted. Plots of the concentrationpro�les for all simulations performed are found in the appendix (section C). Each pro�le plot ismade up of eight subplots (A-H), containing:
A: The particle size distribution (PSD) used in the simulation.B: Column characteristics along the expanded bed (Rp, kf and εL)C: Concentration pro�les with time for a particle at the column inlet.D: Concentration pro�les with time in liquid phase.E: Concentration pro�les with time for a particle in the middle of the column.F: Concentration pro�les with time for average solid phase concentration.G: Concentration pro�les with time for a particle at the column outlet.H: Breakthrough curve.Subplot A shows the radius distribution used in the simulation. It is indicated where the radiushas been cut due to elution of the smallest particles. Column characteristics along the expandedbed are shown in B. The column is seen lying with the inlet to the left and the x-axis is normalizedwith respect to expanded bed height. The plot shows how the radius of the adsorbent, the voidageand the �lm mass transfer coe�cient varies along the bed. The radius and the �lm coe�cientare normalized with respect to their maximum values, which are given in the corresponding table(in this case Tab. 4.2).Subplots C-G depicts the evolution of concentration pro�les in the column and inside the par-ticles. There is a �xed interval of approximately six minutes between each pro�le (this intervalmay however change slightly between di�erent simulations). Subplot D shows what the liquidphase concentration pro�les look like inside the column, with column height normalized withrespect to the expanded bed height on the x-axis and concentration on the y-axis. Subplot Fshows the average solid phase concentration pro�les along the column and illustrates the degreeof saturation of adsorbent phase at each column level. It is interesting to see how, for somesimulations, a lower degree of media saturation is reached in the bottom layer of the column,than in higher layers. This is an e�ect of the di�erent particle sizes at di�erent heights of thebed, and the gradient in di�usion path length that follows.Another way to illustrate solid phase concentration is to show concentration pro�les inside theparticles at each column level. We have chosen to plot pro�les for the particles in the inlet, in
21
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0 H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure 4.1: homogeneous model: Base case parameters (H1A). A: - Radius distribution, - -Particles tothe left of this line were eluted upon expansion. B: Pro�les of parameters along the bed, divided by theirrespective maximum value. - -(red) Particle radius ,• εL , -(blue) �lm resistance. C, E, G: Developmentof concentration pro�les with time inside the particles. C is particle at the column inlet, E is particle inthe middle of the column and G at the column outlet. The respective particle radius and its frequencyis indicated in subplot A. D: Evolution of liquid phase concentration pro�les with time along the bed. F:Evolution of solid phase concentration pro�les with time along the bed. H: Breakthrough curve. The timeinterval between pro�les in each plot is approximately 6 min.
the middle and in the outlet segment of the column (subplots C, E and G). The subplots showan cross-section of the particle from the center to the surface, with radius normalized to particleradius on the x-axis. One can see that at the inlet, mass transfer to particles starts immediatelyupon simulation take o�, but saturation is reached relatively slowly. In contrary, at the outlet,it takes some time before particles start being saturated, but, once they do, it is a fast process.Subplot H, �nally, shows the breakthrough curve.
4.1.1 E�ect of intraparticle mass transfer
The homogeneous and the porous model describe mass transfer in di�erent types of media. Fig.4.2 present the concentration pro�les inside the particles from simulations with the two models.Parameters are for both cases set to the respective base values and the distribution used is thepyramidal one. The hydrodynamics, i.e. the expansion and the voidage gradient, are the samefor both cases. The �gure presents results from the homogeneous model in the left column andfrom the porous model in right column. It is also indicated in which column segment the particles
22
are located. For the porous model, we have chosen to plot the total concentration of proteininside the particles (adsorbed + dissolved in the liquid phase �lling the pores).While pro�les are smooth for the homogeneous model, they are steep for the porous model anda higher discretization would be needed to make pro�les look good. The di�erence betweenthe pro�les is no wonder and follows from the respective model equations. In the homogeneousmodel, the internal mass transfer is entirely governed by a homogeneous di�usion process andthe pro�les have the characteristic bow representative for a di�usion process. In the porousparticle, concentration gradients are much steeper. The steepness is a function of the Langmuirisotherm which in this work is clearly favourable. This pattern describes a situation where themass transfer is dependent on strong ligand-solute interaction. The solute di�uses in the poresand adsorbs already at low concentrations strongly to very the speci�c ligands.
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
0 0.2 0.4 0.6 0.8 10
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1
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ax
r/R
0 0.2 0.4 0.6 0.8 10
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1
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ax
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1
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ax
r/R
0 0.2 0.4 0.6 0.8 10
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1
q/q m
ax
r/R
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
Column inlet
Mid−column
Column outlet
Figure 4.2: Comparison of particle pro�les with time for the homogeneous model (case H1A, left column)and for the porous model (case P1A, right column). Particles position inside the column are indicatedfor each line. The time interval between pro�les is in each plot approximately 6 min.
4.2 E�ect of particle size distribution
To visualize the e�ect of particle size distribution on EBA performance is the main objective ofthis work. Fig. 4.3 shows, for each radius distribution, breakthrough curves for the base cases ofthe homogeneous model. First thing to note is that assuming mono-sized particles (as Wright andGlasser do [11]), seems to give results almost equal to those assuming a pyramidal distribution.This holds both for the breakthrough curve (Fig. 4.3), and for the column utilization (Fig. 4.4).The productivity is for all distributions strikingly similar, while column utilization is the highestwith small particles and the lowest with large ones.Breakthrough curves for the porous model are oscillating slightly (Fig. 4.5) and apparentlythe solution su�ers from instability. As it seems, this is due to the parameters chosen for thesimulations. Instabilities are not seen for a higher value of the di�usion coe�cient (Fig. C.69).Although not thoroughly explored, very di�erent values in the time constants governing theevolution in the column and in the particles respectively, could be the cause of the instabilities(for time constants, see section 3.4). A further analysis is however not possible within the time
23
0 20 40 60 80 100 120 1400
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (min)C
/C0
1
2
3
4
Figure 4.3: Homogeneous model, base case parameters. Breakthrough curves for the four radius dis-tributions. 1) Pyramidal. 2) Biased to small particles. 3) Biased to large particles. 4) Mono-sizedparticles.
1 100
0.2
0.4
0.6
0.8
1
% breakthrough
U
1 100
0.1
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0.3
0.4
0.5
0.6
0.7
% breakthrough
P (
s−1 )1
2
3 4
4 4 4 2 2 2 1 1 1
3 3 3
Figure 4.4: Column utilization and productivity at 1% and 10% breakthrough for the four PSDs (1-4).Homogeneous model, base case parameters.
frame of this project. Of course, a denser grid of discretization points would reduce instabilitiesbut time and computer power has not allowed for this. However, late in the project, a simulationwas performed for the porous model on a new, much better computer. Base case parametervalues were used and the discretization was doubled inside the particle (Npart = 30), whilekept constant in the column (Ncol = 50). The pattern of pro�les was basically the same, butoscillations were gone. This tells us that we can, for most cases, rely on our simulations, althougha higher discretization would be good.The choice of PSD had much the same e�ect for the porous model as for the homogeneous model(Fig. (4.6)). The pattern we see is the highest capacity for the PSD biased to small particles, thelowest for the PSD biased to high particles and no e�ect from size distribution on productivity.
4.3 Parametric sensitivity analysis
Tables (4.2-4.9) present the results for each simulation in terms of expansion, overall Bodensteinnumber (Bo), column utilization (dynamic capacity/maximum capacity) and time. How theoverall Bo was estimated is given in the appendix (section A.2). The case number makes it easyto �nd the corresponding plot in the appendix.
4.3.1 Capacity and productivity
Axial dispersion: Mixing in liquid phase has been much discussed ([6], [2], [1]) and it has beensuggested that high dispersion could lower the performance in EBA. We performed simulations,
24
0 50 100 150 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time (min)
C/C
0
1
2
4
3
Figure 4.5: Porous model, base case parameters. Breakthrough curves for the four radius distributions.1) Pyramidal. 2) Biased to small particles. 3) Biased to large particles. 4) Mono-sized particles.
1 100
0.1
0.2
0.3
0.4
0.5
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0.7
% breakthrough
U
1 100
0.1
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0.6
0.7
% breakthrough
P (
s−1 )
4 4 3
3
1
2
3
4
1
3 1 2
2 2
4
1
Figure 4.6: Column utilization and productivity at 1% and 10% breakthrough for the four PSDs (1-4).Porous model, base case parameters.
Table 4.1: Parameter values for each run. All values except the intraparticle di�usion coe�cients arethe same for the two models. Bold type indicate that the parameter value has been changed with respectto the base case.
u R Daxmax Ds ρs µ kfmax Dh Dp
Run (cm/s) (µm) (cm2/s) (cm2/s) (g/cm3) (Cp) (cm/s) (cm2/s) (cm2/s)
A 300 100 5.7e-2 0.0011 1.2 1.5 5.1e-4 1e-8 3.5e-7B 300 100 2.9e-2 0.0011 1.2 1.5 5.1e-4 1e-8 3.5e-7C 300 100 11.5e-2 0.0011 1.2 1.5 5.1e-4 1e-8 3.5e-7D 300 100 5.7e-2 0.00011 1.2 1.5 5.1e-4 1e-8 3.5e-7E 300 100 5.7e-2 0.012 1.2 1.5 5.1e-4 1e-8 3.5e-7F 300 100 5.7e-2 0.0011 1.2 1.5 5.1e-4 2e-9 7.0e-8G 300 100 5.7e-2 0.0011 1.2 1.5 5.1e-4 5e-8 1.75e-6H 450 100 6.0e-2 0.0011 1.2 1.0 5.8e-4 1e-8 3.5e-7I 225 100 5.6e-2 0.0011 1.2 2.0 4.6e-4 1e-8 3.5e-7J 500 50 5.7e-2 0.0011 2.33 1.5 9.6e-4 1e-8 3.5e-7K 500 100 5.8e-2 0.0011 1.33 1.5 6.0e-4 1e-8 3.5e-7
25
Table 4.2: Simulation results: Homogeneous model, pyramidal PSD.CaseH1 expansion Bo U1% U10% t1%(min) t10%(min)
A 2.79 91.6 0.62 0.84 61.5 85.5B 2.79 183 0.64 0.85 63.7 86.5C 2.79 46.1 0.59 0.84 58.3 84.2D 2.79 91.6 0.61 0.84 60.2 84.2E 2.79 91.6 0.69 0.88 73.3 96.1F 2.79 91.6 0.29 0.46 29.8 47.4G 2.79 91.6 0.70 0.91 30.7 51.9H 2.79 139 0.46 0.76 30.7 51.87I 2.79 68.7 0.70 0.88 92.2 118.5J 2.79 153 0.87 0.96 52.8 58.3K 2.79 155 0.41 0.74 25.0 45.47
Table 4.3: Simulation results: Homogeneous model, PSD biased to small particles.CaseH2 expansion Bo U1% U10% t1%(min) t10%(min)
A 4.25 209 0.74 0.90 73.0 91.3B 4.25 412 0.75 0.91 74.9 92.5C 4.25 105 0.72 0.90 71.1 90.3D 4.25 209 0.73 0.90 71.7 90.3E 4.25 209 0.78 0.92 85.2 102.2F 4.25 209 0.40 0.54 40.0 56.0G 4.25 209 0.80 0.95 79.1 95.7H 4.46 343 0.60 0.84 39.7 57.0I 4.26 156 0.81 0.93 106.3 125.2J 4.24 345 0.93 0.98 123.4 141.2K 4.27 350 0.56 0.81 33.3 50.0
Table 4.4: Simulation results: Homogeneous model, PSD biased to large particles.CaseH3 expansion Bo U1% U10% t1%(min) t10%(min)
A 2.29 62.3 0.48 0.78 48.0 78.5B 2.29 126 0.51 0.78 50.6 79.1C 2.29 31.5 0.44 0.77 45.6 77.2D 2.29 62.3 0.47 0.77 46.8 77.2E 2.29 62.3 0.58 0.83 58.9 86.8F 2.29 62.3 0.19 0.37 19.9 37.8G 2.29 62.3 0.60 0.88 59.6 88.4H 2.26 92.7 0.30 0.67 20.2 45.2I 2.26 47.5 0.58 0.83 93.7 127.8J 2.30 106 0.82 0.93 49.0 56.0K 2.28 105 0.25 0.63 15.1 38.8
Table 4.5: Simulation results: Homogeneous model, mono-sized particles.CaseH4 expansion Bo U1% U10% t1%(min) t10%(min)
A 2.96 98.0 0.60 0.82 60.5 83.6B 2.96 49.1 0.62 0.83 62.8 84.9C 2.96 196 0.57 0.81 57.3 82.3D 2.96 98.0 0.59 0.81 59.9 82.9E 2.96 98.0 0.66 0.86 65.6 87.4F 2.96 98.0 0.33 0.50 33.9 52.2G 2.96 98.0 0.65 0.87 65.6 88.7H 2.96 148 0.49 0.76 32.7 51.6I 2.96 73.7 0.66 0.85 88.7 115.6I 2.96 164 0.83 0.94 50.3 57.0K 2.96 166 0.44 0.74 26.9 45.5
26
Table 4.6: Simulation results: Porous model, pyramidal PSD.CaseP1 expansion Bo U1% U10% t1%(min) t10%(min)
A 2.79 91.6 0.24 0.41 24.2 42.4B 2.79 183 0.25 0.43 25.9 43.9C 2.79 46.1 0.21 0.39 21.6 40.4D 2.79 91.6 0.24 0.41 24.2 42.4E 2.79 91.6 0.23 0.40 23.4 41.8F 2.79 91.6 0.11 0.15 12.7 17.3G 2.79 91.6 0.56 0.83 55.6 83.6H 2.79 139 0.14 0.28 10.1 19.7I 2.79 68.7 0.32 0.53 43.2 72.1J 2.79 153 0.64 0.80 38.1 48.0K 2.79 155 0.12 0.26 8.0 16.3
Table 4.7: Simulation results: Porous model, PSD biased to small particles.CaseP2 expansion Bo U1% U10% t1%(min) t10%(min)
A 4.25 209 0.34 0.53 33.6 54.3B 4.25 412 0.35 0.55 35.2 55.7C 4.25 105 0.31 0.52 31.4 52.7D 4.25 209 0.34 0.53 33.6 54.1E 4.25 209 0.32 0.53 33.0 55.6F 4.25 209 0.18 0.22 17.1 21.1G 4.25 209 0.71 0.91 69.5 91.1H 4.46 343 0.22 0.38 15.4 26.1I 4.26 156 0.44 0.65 58.4 87.7J 4.24 345 0.78 0.90 46.0 53.8K 4.27 350 0.19 0.33 12.2 21.0
Table 4.8: Simulation results: Porous model, PSD biased to large particles.CaseP3 expansion Bo U1% U10% t1%(min) t10%(min)
A 2.29 62.3 0.16 0.30 16.0 30.6B 2.29 126 0.17 0.31 17.5 31.7C 2.29 31.5 0.13 0.28 13.5 28.8D 2.29 62.3 0.16 0.30 16.0 30.9E 2.29 62.3 0.15 0.27 15.7 28.3F 2.29 62.3 0.09 0.13 9.8 14.6G 2.29 62.3 0.38 0.75 37.8 75.6H 2.26 92.7 0.09 0.21 6.4 14.9I 2.31 47.5 0.22 0.39 29.1 53.2J 2.30 106 0.50 0.67 29.5 40.4K 2.28 105 0.08 0.19 5.1 12.5
Table 4.9: Simulation results: Porous model, mono-sized particles.CaseP4 expansion Bo U1% U10% t1%(min) t10%(min)
A 2.96 98.0 0.29 0.47 29.8 48.2B 2.96 196 0.31 0.48 31.7 49.3C 2.96 49.1 0.26 0.45 27.1 46.6D 2.96 98.0 0.29 0.48 29.9 49.2E 2.96 98.0 0.27 0.42 27.9 42.8F 2.96 98.0 0.12 0.18 13.8 20.2G 2.96 98.0 0.56 0.79 56.2 80.9H 2.96 148 0.18 0.34 12.8 24.2I 2.96 73.7 0.38 0.57 51.6 77.5J 2.96 164 0.66 0.77 39.7 46.9K 2.96 166 0.16 0.31 10.3 20.0
27
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1
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s−1 )
Dax
(cm2/s)
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U
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1
U10
−210
−10.4
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1
P (
s−1 )
Dax
(cm2/s)
10−2
10−1
0.4
0.5
0.6
0.7
0.8
0.9
1
U
1 2
3 4
Figure 4.7: Parametric sensitivity analysis: Axial dispersion, homogeneous model. Utilization (bold,blue), productivity (thin, red), (- -) at 1% breakthrough, (-) at 10% breakthrough.
10−2
10−1
0
0.1
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0.3
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P (
s−1 )
Dax
(cm2/s)
10−2
10−1
0
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U
10−2
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Dax
(cm2/s)
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U
10−2
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Dax
(cm2/s)
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0
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U
10−2
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P (
s−1 )
Dax
(cm2/s)
10−2
10−1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
U1
3
2
4
Figure 4.8: Parametric sensitivity analysis: Axial dispersion, porous model. Utilization (bold, blue),productivity (thin, red), (- -) at 1% breakthrough, (-) at 10% breakthrough.
28
10−4
10−3
10−2
0.4
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s−1 )
Ds (cm2/s)
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U
10−4
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Ds (cm2/s)
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U
10−4
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Ds (cm2/s)
10−4
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U
10−4
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1
P (
s−1 )
Ds (cm2/s)
10−4
10−3
10−2
0.4
0.5
0.6
0.7
0.8
0.9
1
U
1 2
4 3
Figure 4.9: Parametric sensitivity analysis: Solid dispersion, homogeneous model. Utilization (bold,blue), productivity (thin, red), (- -) at 1% breakthrough, (-) at 10% breakthrough.
changing the axial dispersion coe�cient from the base value (Dax, see Tab. 4.1) to 0.5Dax
and 2Dax. The corresponding overall Bodenstein numbers (Bo), are given in each result table(Tab. 4.2-4.9). The plate number (N) is related to Bo as Bo ' 2N , and the higher is theaxial dispersion, the lower is the value ofBo and N . A general rule of thumb, recommended byAmersham Biosciences [1], is that Bo>40 is su�cient to ensure that the adsorption performanceis not limited by liquid mixing. All our base cases meet this criterion. It is interesting to notehow the size distribution a�ects liquid mixing. The highest Bodenstein number (less dispersion)is obtained for the PSD biased to small particles.Simulations did, neither for the homogeneous nor for the porous model, point at a great in�uenceof axial dispersion on the performance, not even forBo<40. No important di�erence in impactcould be seen between the four particle size distributions. Productivity was almost constantwhile a slight decrease in capacity was seen with higher values of axial dispersion. This e�ectwas most important at early breakthrough, which seems logical as dispersion in liquid phasesmears out concentration pro�les. Results are presented in Fig. 4.7 for the homogeneous caseand Fig. 4.8 for the porous case.
Solid dispersion: Solid dispersion a�ects the stability of the adsorbent inside the column. Apriori, one may think solid dispersion should have the same kind of smearing e�ect on break-through as axial dispersion. We changed the solid dispersion coe�cient from the base valuewith a factor of 10 (0.1Ds and 10Ds). The e�ect was for the porous model Fig. 4.10 close tonegligible for both capacity and productivity. For the homogeneous model Fig. 4.9 there seemsto be a small increase in capacity at high levels ofDs, which is, however, completely o�set bylonger process time and thus not seen in the productivity. This conclusion is valid for any of thedistributions considered.However, results concerning solid dispersion should be considered with some scepticism as massbalance seems to break down at high Ds. This is probably due to two e�ects. 1) High valuesof solid dispersion makes Ds the dominant parameter in the system of equations, causing insta-bilities. We have seen that a denser discretization, if not closes the mass balance, so at leastdecreases the error. 2) The model equations might not correctly handle the �ux from liquid phasedue to simpli�cations made. Allowing for solid phase dispersion the way we do in Eq. (2.17),
29
10−4
10−3
10−2
0
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Ds (cm2/s)
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U
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U
10−4
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10−4
10−3
10−2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
U
1 2
3 4
Figure 4.10: Parametric sensitivity analysis: Solid dispersion, porous model. Utilization (bold, blue),productivity (thin, red), (- -) at 1% breakthrough, (-) at 10% breakthrough.
smears out average solid phase concentration (q′) in the expanded bed. This will a�ect masstransfer to particles through the boundary condition, but does not mean that particles changeposition in the column. The model could be said to account for solid dispersion in a somewhathalf-hearted (but possible) way and ought to be further investigated. We have investigated massbalance for all simulations, and the problem only occurs for cases with the homogeneous modeland a high solid dispersion coe�cient.
Intraparticle di�usion coe�cient: Values of the intraparticle di�usion coe�cients are notthe same in the porous and in the homogeneous model. This is natural as the models describetwo di�erent mass transfer mechanisms. In the parametric sensitivity analysis, we varied thedi�usion coe�cients by a factor 5, resulting in the series 0.2D, D and 5D. From the resultsobtained, it is seen that the impact of the di�usion coe�cient is important (Fig. 4.11 and Fig.4.12). Capacity increases considerably for both models when the di�usion coe�cient increasesand this is the more pronounced the more large particles the size distribution contains. Thelatter agrees with the results of Karau et al. [4] on the magnitude of �lm resistance compared tointernal resistance, presented in section 1.1. For large particles, the internal resistance to masstransfer is probably the rate limiting step, while for small particles, �lm resistance has also to betaken into account. Thus, changing the intraparticle di�usion coe�cient should have a greaterimpact on mass transfer in large particles, as our results shows.Productivity gains are not at all seen for the homogeneous model. In the porous model theyexist but are small. As it seems, the gain from higher di�usion rate is almost completely o�setby longer process time. The choice of PSD has no e�ect on productivity gains.
Viscosity: Simulations were made for three levels of liquid phase viscosity (µ = 1 Cp, µ = 1.5Cp and µ = 2 Cp). Changing viscosity a�ects the terminal velocity (see Eq. (2.2)) and to keepexpansion and voidage pro�le constant, the super�cial velocity was changed in order to keepεLconstant. Doing so holds under the assumption thatn in Eq. (2.3) does not change and as isshown in Tab. 4.2-4.9, the degree of expansion did not vary much between the simulations.For both models, increasing viscosity leads to increase in the dynamic capacity (Fig. 4.13 and
30
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U
10−8
0
0.2
0.4
0.6
0.8
1
P (
s−1 )
D (cm2/s)
10−8
0
0.2
0.4
0.6
0.8
1
U
10−8
0
0.2
0.4
0.6
0.8
1
P (
s−1 )
D (cm2/s)
10−8
0
0.2
0.4
0.6
0.8
1
U10
−80
0.2
0.4
0.6
0.8
1
P (
s−1 )
D (cm2/s)
10−8
0
0.2
0.4
0.6
0.8
1
U
1 2
3 4
Figure 4.11: Parametric sensitivity analysis: Intraparticle di�usion coe�cient, homogeneous model.Utilization (bold, blue), productivity (thin, red), (- -) at 1% breakthrough, (-) at 10% breakthrough.
10−7
10−6
0
0.2
0.4
0.6
0.8
1
P (
s−1 )
D (cm2/s)
10−7
10−6
0
0.2
0.4
0.6
0.8
1
U
10−7
10−6
0
0.2
0.4
0.6
0.8
1
P (
s−1 )
D (cm2/s)
10−7
10−6
0
0.2
0.4
0.6
0.8
1
U
10−7
10−6
0
0.2
0.4
0.6
0.8
1
P (
s−1 )
D (cm2/s)
10−7
10−6
0
0.2
0.4
0.6
0.8
1
U
10−7
10−6
0
0.2
0.4
0.6
0.8
1
P (
s−1 )
D (cm2/s)
10−7
10−6
0
0.2
0.4
0.6
0.8
1
U1 2
3 4
Figure 4.12: Parametric sensitivity analysis: Intraparticle di�usion coe�cient, porous model. Utilization(bold, blue), productivity (thin, red), (- -) at 1% breakthrough, (-) at 10% breakthrough.
31
0.01 0.015 0.020.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P (
s−1 )
µ (g*s−1*cm−1)
0.01 0.015 0.02
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
U
0.01 0.015 0.020.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P (
s−1 )
µ (g*s−1*cm−1)
0.01 0.015 0.02
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
U
0.01 0.015 0.020.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P (
s−1 )
µ (g*s−1*cm−1)
0.01 0.015 0.02
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
U
0.01 0.015 0.020.3
0.4
0.5
0.6
0.7
0.8
0.9
1
P (
s−1 )
µ (g*s−1*cm−1)
0.01 0.015 0.02
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
U
1 2
3 4
Figure 4.13: Parametric sensitivity analysis: Changing viscosity and velocity to keep expansion constant.Homogeneous model. Utilization (bold, blue), productivity (thin, red), (- -) at 1% breakthrough, (-) at 10%breakthrough.
0.01 0.015 0.020
0.2
0.4
0.6
0.8
1
P (
s−1 )
µ (g*s−1*cm−1)
0.01 0.015 0.02
0
0.2
0.4
0.6
0.8
1
U
0.01 0.015 0.020
0.2
0.4
0.6
0.8
1
P (
s−1 )
µ (g*s−1*cm−1)
0.01 0.015 0.02
0
0.2
0.4
0.6
0.8
1
U
0.01 0.015 0.020
0.2
0.4
0.6
0.8
1
P (
s−1 )
µ (g*s−1*cm−1)
0.01 0.015 0.02
0
0.2
0.4
0.6
0.8
1
U
0.01 0.015 0.020
0.2
0.4
0.6
0.8
1
P (
s−1 )
µ (g*s−1*cm−1)
0.01 0.015 0.02
0
0.2
0.4
0.6
0.8
1
U1 2
3 4
Figure 4.14: Parametric sensitivity analysis: Changing viscosity and velocity to keep expansion constant.Porous model. Utilization (bold, blue), productivity (thin, red), (- -) at 1% breakthrough, (-) at 10%breakthrough.
32
Fig. 4.14). However, the longer process time due to low �uid velocity decreases productivitysharply. The e�ect of PSD is negligible.
Radius: The median radius was changed from the value in the base case (r=100µm) to halfthis value (r=50µm). The size distributions were also changed by reducing the di�erence betweenthe largest and the smallest particle (rmax−rmin) by a factor two, but keeping the overall formconstant. Note that although particle size is changed, gel volume and maximum capacity is keptconstant. This study was performed with a higher �uid velocity than all other studies (u=500cm/s). As in the study of viscosity, we wanted to keep expansion constant and therefore changedthe solid density in accordance with Eq. (2.2). Once again, this holds under the assumption ofa constant n in Eq. (2.3).
40 60 80 100 1200.9
0.95
1
1.05
1.1
P (
s−1 )
R (µm)
40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
U40 60 80 100 120
0.9
0.95
1
1.05
1.1
P (
s−1 )
R (µm)
40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
U
40 60 80 100 1200.9
0.95
1
1.05
1.1
P (
s−1 )
R (µm)
40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
U
40 60 80 100 1200.9
0.95
1
1.05
1.1
P (
s−1 )
R (µm)
40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
U
1 2
3 4
Figure 4.15: Parametric sensitivity analysis: Changing particle radius at constant velocity. Changeddensity to keep expansion constant. Homogeneous model. Utilization (bold, blue), productivity (thin, red),(- -) at 1% breakthrough, (-) at 10% breakthrough.
For both models, dynamic capacity increases signi�cantly when radius is reduced (Fig. 4.15and Fig. 4.16). Reducing radius size also has a sharpening e�ect on the breakthrough curvewhich at least for the homogeneous model is manifested in a narrowing di�erence betweenU1%
and U10%. The productivity is for the homogeneous model slightly higher for a shorter radius.Interesting is also that the productivity at 1% breakthrough is clearly higher than at 10% at thishigh velocity. This has not been seen in the simulations at lower velocity and is probably due tothe fact that a higher velocity gives a more dispersed breakthrough, compared to the residencetime (see corresponding pro�les in appendix). Breakthrough will therefore appear early at lowconcentration, but much later at higher concentration. For the porous model, productivity ise�ected in the same direction as dynamic capacity, i.e. it increases when particle radius isreduced.As discussed in section 3.4, particle radius and solid di�usion rate make up one of the timeconstants governing chromatography (D
R2 ). It is therefore no wonder that decreasing the radiushas much the same impact as increasing the intraparticle di�usion coe�cient.
Velocity: A comparison between the two simulations at di�erent velocity but same expansionis also worth considering. Expansion was kept constant by changing the density of the solidphase. Fig. 4.17 and Fig. 4.18 show that, for both models and all size distributions, capacity
33
40 60 80 100 1200.9
0.95
1
1.05
1.1
P (
s−1 )
R (µm)
40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
U
40 60 80 100 1200.9
0.95
1
1.05
1.1
P (
s−1 )
R (µm)
40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
U
40 60 80 100 1200.9
0.95
1
1.05
1.1
P (
s−1 )
R (µm)
40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
U
40 60 80 100 1200.9
0.95
1
1.05
1.1
P (
s−1 )
R (µm)
40 60 80 100 120
0
0.2
0.4
0.6
0.8
1
U
1 2
3 4
Figure 4.16: Parametric sensitivity analysis: Changing particle radius at constant velocity. Changeddensity to keep expansion constant. Porous model. Utilization (bold, blue), productivity (thin, red), (- -)at 1% breakthrough, (-) at 10% breakthrough.
decreases when velocity is increased. In return productivity increases quite signi�cantly. In otherwords: with a high �uid velocity, adsorbent is less saturated at breakthrough but, as �uid velocityis high, breakthrough occurs early enough to nevertheless result in a higher productivity. Thiscan also be seen from the pro�les in the appendix. For a high velocity, solid phase concentrationis to a higher extent smeared out along the bed.
4.4 Concluding remarks
In order to better understand the importance of particle size distribution as well as of severaloperating parameters, we have in this work performed simulations of EBA. We found that thePSD chosen did have a major e�ect on column utilization, both for the porous and the homo-geneous model of particle mass transfer. The degree of utilization was higher the higher wasthe percentage of small particles in the PSD. However, the productivity was almost not at alla�ected by the choice of PSD.Changes in the intraparticle di�usion coe�cient, liquid phase viscosity, particle radius and linearvelocity all had an important e�ect on column utilization and, to a lower degree, on productivity.The e�ect of changes in the axial and solid dispersion coe�cients were close to negligible. Thesestatements hold for all particle size distributions considered in this work. However, the e�ectof changes in parameters was most important for the PSD biased to large particles, and leastimportant for the PSD biased to small particles.To optimize EBA performance, our results suggest that one should use a PSD biased to smallparticles to get short internal di�usion paths. The particles should be of high density to allowfor a high linear velocity without extreme expansion, thus increasing process productivity.This work has primarily focused on setting up a �rst version of a model for EBA, includingboth the e�ects of hydrodynamics and mass transfer. It can be recommended to con�rm modelpredictions by comparison with experimental data using experimental parameter values and realparticle size distributions.Further work can also focus on modelling the dynamics of the bed. In this work, the bedwas assumed static once expanded but in reality the bed settles during process as particles
34
turn heavier due to mass uptake. The degree of bed expansion as well as bed characteristicsat a certain height (particle radius, voidage, dispersion and �lm mass transfer coe�cient) aretherefore not constant over time. It should not be too di�cult a task to include these e�ects infuture version.
5 Acknowledgements
I would like to thank Amersham Biosciences in Uppsala for giving me the opportunity to carryout this Master's degree project. I hope the program and the result presented will be useful.In particular I would like to thank Dr. Karol Lacki, my supervisor, and Mattias Bryntesson, aPh.D student at the Royal Institute of Technology in Stockholm and Amersham Biosciences, fordirecting me, supporting me, teaching me and laughing at and with me.
35
200 300 400 500 6000
0.2
0.4
0.6
0.8
1
P (
s−1 )
u (cm/h)
200 300 400 500 600
0
0.2
0.4
0.6
0.8
1
U
200 300 400 500 6000
0.2
0.4
0.6
0.8
1
P (
s−1 )
u (cm/h)
200 300 400 500 600
0
0.2
0.4
0.6
0.8
1
U
200 300 400 500 6000
0.2
0.4
0.6
0.8
1
P (
s−1 )
u (cm/h)
200 300 400 500 600
0
0.2
0.4
0.6
0.8
1
U
200 300 400 500 6000
0.2
0.4
0.6
0.8
1
P (
s−1 )
u (cm/h)
200 300 400 500 600
0
0.2
0.4
0.6
0.8
1
U
1 2
3 4
Figure 4.17: Parametric sensitivity analysis: Changing particle radius at constant velocity. Solid phasedensity is also changed to keep expansion constant. Homogeneous model. Utilization (bold, blue), produc-tivity (thin, red), (- -) at 1% breakthrough, (-) at 10% breakthrough.
200 300 400 500 6000
0.2
0.4
0.6
0.8
1
P (
s−1 )
u (cm/h)
200 300 400 500 600
0
0.2
0.4
0.6
0.8
1
U
200 300 400 500 6000
0.2
0.4
0.6
0.8
1
P (
s−1 )
u (cm/h)
200 300 400 500 600
0
0.2
0.4
0.6
0.8
1
U
200 300 400 500 6000
0.2
0.4
0.6
0.8
1
P (
s−1 )
u (cm/h)
200 300 400 500 600
0
0.2
0.4
0.6
0.8
1
U
200 300 400 500 6000
0.2
0.4
0.6
0.8
1
P (
s−1 )
u (cm/h)
200 300 400 500 600
0
0.2
0.4
0.6
0.8
1
U1 2
3 4
Figure 4.18: Parametric sensitivity analysis: Changing particle radius at constant velocity. Solid phasedensity is also changed to keep expansion constant. Porous model. Utilization (bold, blue), productivity(thin, red), (- -) at 1% breakthrough, (-) at 10% breakthrough.
36
6 NomenclatureC: concentration in liquid phase, mg/mlliquid.q: solid phase concentration, mg/mlsolid.q′: average solid phase concentration,mg/mlsolid.Cp: concentration in particle pore, mg/mlliquid
Cf : equilibrium concentration at particle surface,mg/mlliquid
qs: adsorbed concentration, mg/mlsolid
C0: feed concentration, mg/mlliquid
qmax: maximum capacity of particles, mg/mlsolid
Ks: dissociation constant, mg/mlQ: dimensionless concentration in particle, q/qmax
εL: bed voidageεs: 1-εL
εp: fraction volumepore/volumeparticle
u: super�cial velocity, cm/sz: axial position in the column, cmR: reference radius, (radius of the smallest particle) cmRp: radius of particle, cmrc: radius of core in particle center, cmr: radial position in the particle, cmη: dimensionless radial position, r/RDax: axial dispersion coe�cient, cm2/sDh: intraparticle di�usion coe�cient (homogeneous model),cm2/sDp: pore di�usion coe�cient, cm2/sDs: solid dispersion coe�cient, cm2/sDm: molecular di�usion coe�cient, cm2/sHsed: sedimented bed height, cmH: expanded bed height, cmkf : �lm mass transfer coe�cient, cm/sPe: Peclet number, uH/(εLDax)Bi: Biot number, kfR/DBo: Bodenstein number, uH/Dax
Greek lettersτ : characteristic time for the particle mass transfer,D/R2
β: dimensionless concentration in liquid phase,C/C0
ξ: dimensionless axial position in column, z/Hη: dimensionless radial position, r/Rθ: dimensionless time, Dt/Rθr: dimensionless mass transfer unit,HεL/(uτ)
References[1] Pharmacia Biotech (1997). Expanded bed adsorption-principles and methods. Handbook.Uppsala,
Sweden: Pharmacia Biotech.
[2] Hjorth R., Kämpe S., Carlsson M. (1995). Analysis of some operating parameters of novel adsorbentfor recovery of proteins in expanded beds.Bioseparation, 5, 217-223.
[3] Coulson J.M., Richardson J. F., Backhurst J. R., Harker J. H. (1987).Chemical Engineering. Volumetwo. (3rd ed.). Pergamon Press, Headington Hill Hall, Oxford OX3 0BW, England.
[4] Karau A., Benken C., Thömmes J., Kula M. R. (1997). The In�uence of Particle Size Distributionand Operating Conditions on the Adsorption Performance in Fluidized Beds.Biotechnology andBioengineering, 55(1), 54-64.
[5] Willoughby N. A., Hjorth R., Titchener-Hooker N. J. (2000). Experimental Measurement of Par-ticle Size Distribution and Voidage in an Expanded Bed Adsorption System.Biotechnology andBioengineering, 69(6), 648-653.
[6] Bruce L. J., Chase H. A. (2000). Hydrodynamics and adsorption behaviour within an expanded bedadsorption column studied using in-bed sampling.Chemical Engineering Science, 56, 3149-3162.
37
[7] Han N-W., Bhakta J., Carbonell R. G. (1985). Longitudinal ans Lateral Dispersion in Packed Beds:E�ect of Column Length and Particle Size Distribution.AlChE Journal, 31(2), 277-288.
[8] Sofer G., Hagel L. (1997).Handbook of Process Chromatography: A Guide to Optimization Scale-up,and Validation. Academic Press.
[9] Subramanian G. (1998).Bioseparation and Bioprocessing. Volume 1: Biochromatography. MembraneSeparations, Modeling, Validation.Wiley-VCH.
[10] Ruthven D. M. (1984). Principles of Adsorption and Adsorption Processes. John Wiley & Sons:University of New Brunswick, Fredericton, USA
[11] Wright P. R., Glasser B. J. (2001). Modeling Mass Transfer and Hydrodynamics in Fluidized-BedAdsorption of Proteins. AlChE Journal, 47(2), 474-488.
[12] Al-Dibouni M. R., Garside J. (1979). Particle mixing and classi�cation in liquid �uidized beds.TransIChemE, 57, 94-103.
[13] Thelen T. V., Ramirez W. F. (1997). Bed height dynamics of expanded beds.Chemical EngineeringScience, 52, 3333-3344.
[14] Wright P. R., (2000). The E�ect of Mass Transfer and Hydrodynamics of Fluidized Bed Adsorptionof Proteins. Graduate School-New Brunswick, The State University of New Jersey, USA.
[15] Wright P. R., Muzzio F. M., Glasser B. J. (1998). Batch Uptake of Lysozyme: E�ect of SolutionViscosity and Mass Transfer on Adsorption.Biotechnology Progress, 56, 3149-3162.
[16] Weaver L. E., Carta G. (1996) Protein Adsorption on Cation Exchangers: Comparison of Macrop-orous and Gel-Composite Media.Biotechnology Progress, 12, 342-355.
[17] Danckwerts P. V. (1953). Continuos �ow systems: Distribution of Residence Times.Chemical En-gineering Science, 2(1), 1-18.
[18] Horstmann B. J., Chase H. A. (1989). Modelling the a�nity adsorption of immunoglobulin G toprotein A immobilised to agarose matrices.Chemical Engineering Research and Design, 67, 243-254.
[19] Seidel-Morgenstern A. (1991). Analysis of Boundary Conditions in the Axial Dispersion model byapplication of numerical Laplace inversion.Chemical Engineering Science, 46, 2567-2571.
[20] Crank J. (1975). The Mathematics of Di�usion. (2nd ed.). Oxford University Press.
[21] Ku£era E. (1965). Contribution to the theory of chromatography: linear non-equilibrium elutionchromatography. Journal of Chromatography, 19, 237-248.
[22] Perry R. P., Green. D. W. (1998). Perry's Chemical Engineer's Handbook (7th ed.). McGraw-Hillinternational editions.
[23] Carta G. (2001). Mass Transfer in Liquid Chromatography.Department of Chemical Engineering,University of Virginia, Charlottesville, Virginia, USA.
[24] The MathWorks (1998).Matlab: Using Matlab. (Version 5). Natick, Massachusetts, USA: The Math-Works, Inc.
[25] Tyn M. T., Gusek T. W. (1990). Prediction of Di�usion Coe�cients of Proteins.Biotechnology andBioengineering, 35, 327-338.
[26] Levenspiel O. (1999). Chemical Reaction Engineering. (3rd ed.). John Wiley & Sons: Departmentof Chemical Engineering, Oregon State University.
[27] Heath M. T. Scienti�c Computing: An Introductory Survey. McGraw-Hill, International Editions.
38
A Parameters
A.1 Axial dispersion coe�cient as a function of voidage
The axial dispersion coe�cient (Dax) was estimated from an article by Bruce and Chase [6].They present values of the Bodenstein number at di�erent voidage along an expanded bed.These were calculated from experiments with blue dextran as tracer, which hardly enters solidphase. Samples were taken at three positions along the 40 cm high bed (at 10, 25 and 40 cm)and the mean residence time (tm) and the variance (σm) at each position was calculated from theresidence time distributions. From these could the Bodenstein numbers (assessing the mixing inliquid phase in the intervals) and the voidage, be calculated. Tab. A.1 recapitulates their data.
Table A.1: Parameter values from Bruce and Chase [6] along an expanded bed.Column zone Bo Column zone εL Column zone Mean residence time (s)
0-10 cm 5.2 0-10 cm 0.50 0-10 cm 1120-25 cm 21 10-25 cm 0.69 10-25 cm 1930-40 cm 44 25-40 cm 0.86 25-40 cm 227
0-25 cm 2980-40 cm 532
We used Bruce and Chase's data to correlate Dax with εL. The Bodenstein number can beexpressed in two ways
Bo =uH
Dax(A.1)
Bo ' 2t2mσ2
m
(A.2)
The variance was calculated for the intervals (0-10 cm, 0-25 cm and 0-40 cm), using Eq. (A.2)and the values in Tab. A.1. As the variance is additive for independent variables, the variancesfor the remaining intervals (10-25 cm and 25-40 cm) could then be calculated and, using Eq.(A.2) again, the Bodenstein numbers were calculated for these same intervals. Finally, Eq. (A.1)gave the axial dispersion coe�cient for the three intervals along the bed for which we also knowthe voidages from the estimation by Bruce and Chase. Tab. A.2 summarizes the values.
Table A.2: Values derived from Bruce and Chase [6].Column zone ∆H Bo Dax εL
0-10 cm 10 cm 5.2 0.1015 0.5010-25 cm 15 cm 20.5 0.0386 0.6925-40 cm 15 cm 23.4 0.0338 0.86
To the values calculated, we added the axial dispersion coe�cient for the boundary caseεL = 1,which we set to the molecular di�usion coe�cient for lysozyme,Dm = 3.8 ∗ 10−7 cm/s2 [25].A second order polynomial function describing the relationship betweenDax and εL was thenfound by doing a least square �t in Matlab. This polynomial function was used in the programto estimate axial dispersion coe�cients for along the bed.
Dax = 0.2991− 0.5101εL + 0.2163ε2L
A.2 Overall Bodenstein number
Overall Bodenstein numbers were estimated from simulations under non-adsorbing conditions.Parameter values were set to keep solid phase almost free from solute(qmax = 1 ∗ 10−5 mg/mladsorbent, Dh = 1 ∗ 10−10cm2/s and Ks=10000 mg/ml). Setting qmax=0 was not possiblenumerically. A peak injection was simulated as a pulse with a width of 1 s. The �rst and second
39
moments were calculated from the breakthrough curve as
µk =
∫∞0 tkC(t)dd∫∞
0 Cdd(A.3)
The variance is calculated from the moments asσ2 = µ2 − µ2
1 (A.4)
Under the boundary conditions of a closed vessel, the variance is related to the Bodensteinnumber according to Eq. (A.5) [26].
σ2
µ1= 2
1Bo
− 21
Bo2(1− e−Bo) (A.5)
B Numerical techniques
All equations were implemented in non-dimensional form. A pre-de�ned Matlab solver for sti�ODE-problems (ode15s) was used to solve them. It integrates the system of di�erential equationsover a de�ned time interval, provided it is given 1) a vector of initial values and 2) the timederivative of each variable at each time point. The derivatives were approximated by the methodof �nite di�erences [27]. For a general case, for an interval [a,b] divided into n equally spacedsubintervals, each with a length of a, the derivatives are by this method written as
y′i(t) 'yi+1 − yi−1
2hand y′′i (t) ' yi+1 − 2yi + yi−1
h2(B.1)
where i=1,...,N are the spatial discretization points.
B.1 Liquid phase
Following the method of �nite di�erences for a column divided into N+1 discretization points(1,...,N+1), the derivative in Eq. (2.35) was expressed as
∂βi
∂Θ=
1PeΘr
βi+1 − 2βi + βi−1
(∆ξ)2− 1
Θr
βi+1 − βi−1
2∆ξ
(1− εL
Hu
Daxi+1 −Daxi−1
2∆ξ
)
+βu
H
ε−1i+1 − ε−1
i−1
2∆ξ− 3Bi
εsR
εLRp(β − βf ) (B.2)
where i=2,...,N. At the boundaries, Eq. (B.2) can not be directly applied as the concentration isnot de�ned at i=0 and i=N+2. The derivative ofDax and εL is at the boundaries calculated as
(∂y
∂z
)
1
' y2 − y1
∆zat i=1 and
(∂y
∂z
)
N+1
' yN+1 − yN
∆zat i=N+1
At ξ = 0, the �nite di�erence of the boundary condition (Eq. (2.37)) is:
β1 − β2 − β0
2∆ξ= 1 (B.3)
This is solved for β0, and substituted into Eq. (B.2) to get an expression for the derivative withi=1.At ξ = 1, the boundary condition (Eq. (2.38)) gives for i=N+1
βN+2 − βN
2∆ξ= 0 or βN+2 = βN (B.4)
Substituting this into Eq. (B.2) gives an expression for the derivative with i=N+1.
40
B.2 Solid phase
The technique applied to discretize the liquid phase was used also for the solid phase. Thederivative in Eq. (2.39) was thus approximated with
(∂Q′
∂Θ
)
i
=DsR
2
DH2
Q′i+1 − 2Q′
i + Q′i−1
2∆ξ+ 3Bi
C0
qmax(β − βf ) (B.5)
where i=2,...,N. At the inlet boundary whereξ = 0, the �nite di�erence is given from Eq. (2.41).
Q′2 −Q′
0
2∆ξ= 0 or Q′
0 = Q′2 (B.6)
This is substituted into Eq. (B.5) which is solved for i=1.At ξ = 1, the �nite di�erence is given from Eq. (2.42).
Q′N+2 −Q′
N
2∆ξ= 0 or Q′
N+2 = Q′N (B.7)
This is substituted into Eq. (B.5) which is solved for i=N+1.
B.3 Homogeneous particle
Also the particle was divided into equally spaced intervals (1,...,N+1) and the derivatives ex-pressed by the method of �nite di�erences.
(∂Q
∂Θ
)
i
=1η
Qi+1 −Qi−1
2∆η+
Qi+1 − 2Qi + Qi−1
(∆η)2(B.8)
where i=2,..., N. Where the media meets the kernel at the particle core (η = rrc), the boundary
condition described by Eq. (2.45) is discretized as
Q2 −Q0
2∆η= 0 or Q2 = Q0 (B.9)
This is substituted into Eq. (B.8) which is solved for i=1.The boundary condition at the particle surface (Eq. (2.46)) is the most complicated. It speci�esthe �rst derivative (∂Q
∂Θ) but not the second derivative (∂2Q∂Θ2 ). QN+2 is needed to express the
second derivative by �nite di�erences, but it does not exist and there is no "zero �ux" boundaryto help (as at the particle center). A Taylor expansion is used instead
QN−1 ' QN −∆η∂Q
∂η N
+12(∆η)2
∂2Q
∂Q2N
(B.10)
which can be simpli�ed to(
∂2Q
∂η2
)
N
' 2∆η
(QN−1 −QN
∆η+
∂Q
∂η N
) (B.11)
This is inserted into Eq. (B.8) and terms are rearranged.
∂Q
∂ΘN= 2
(1η
+1
∆η
)Q
η N
+2
∆η(QN−1 −QN ) (B.12)
Inserting Eq. (2.46) into Eq. (B.12), and using �nite di�erences, gives the time dependentderivative at the particle surface.
41
(∂Q
∂Θ
)
η=RpR
= 2(
1η
+1
∆η
)[Bi
C0
qmax(β − βf ) +
(RpRDs
3H2
)(Q′
N+1 − 2Q′N + Q′
N−1)]
+2
(∆η)2(QN−1 −QN ) (B.13)
The expression is slightly di�erent at the column inlet and outlet as the discretization of Q' isdi�erent here (see section B.2). This is left out here, as well as is the discretization scheme forthe porous particle, which is very much the same as for the homogeneous particle.
C Pro�les for all simulations performed
Pro�les from simulations with the homogeneous model are �rst presented, followed by the porousmodel. Each pro�le is labelled according to the scheme in section 3.4 and the parameter valuesused are given in Tab. 4.1.
42
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.1: H1A. See caption of Fig. 4.1.
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.2: H2A. See caption of Fig. 4.1
43
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.3: H3A. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.4: H4A. See caption of Fig. 4.1
44
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.5: H1B. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.6: H2B. See caption of Fig. 4.1
45
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.7: H3B. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.8: H4B. See caption of Fig. 4.1
46
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.9: H1C. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.10: H2C. See caption of Fig. 4.1
47
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.11: H3C. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.12: H4C. See caption of Fig. 4.1
48
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.13: H1D. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.14: H2D. See caption of Fig. 4.1
49
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.15: H3D. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.16: H4D. See caption of Fig. 4.1
50
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.17: H1E. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.18: H2E. See caption of Fig. 4.1
51
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.19: H3E. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.20: H4E. See caption of Fig. 4.1
52
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.21: H1F. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.22: H2F. See caption of Fig. 4.1
53
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 200 2500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.23: H3F. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.24: H4F. See caption of Fig. 4.1
54
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.25: H1G. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.26: H2G. See caption of Fig. 4.1
55
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.27: H3G. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.28: H4G. See caption of Fig. 4.1
56
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 20 40 60 80 1000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.29: H1H. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 20 40 60 80 1000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.30: H2H. See caption of Fig. 4.1
57
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 20 40 60 80 1000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.31: H3H. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 20 40 60 80 1000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.32: H4H. See caption of Fig. 4.1
58
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.33: H1I. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.34: H2I. See caption of Fig. 4.1
59
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.35: H3I. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.36: H4I. See caption of Fig. 4.1
60
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 20 40 60 800
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
20 30 40 50 60 70 800
100
200
300
400
r (µm)
f(r)
A
C
E
G
Figure C.37: H1J. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 20 40 60 800
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
z/H
q’/q
max F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
q/q m
ax
r/R
G
20 30 40 50 60 70 800
100
200
300
400
r (µm)
f(r)
A C
E
G
Figure C.38: H2J. See caption of Fig. 4.1
61
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 20 40 60 800
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
20 30 40 50 60 70 800
100
200
300
400
r (µm)
f(r)
A
C
E
G
Figure C.39: H3J. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 20 40 60 800
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10.4
0.6
0.8
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
30 40 50 60 700
5
10x 10
7
r (µm)
f(r)
A
Figure C.40: H4J. See caption of Fig. 4.1
62
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 20 40 60 80 1000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.41: H1K. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 20 40 60 800
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.42: H2K. See caption of Fig. 4.1
63
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 20 40 60 80 1000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.43: H3K. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 20 40 60 80 1000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.44: H4K. See caption of Fig. 4.1
64
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.45: P1A. See caption of Fig. 4.1.
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.46: P2A. See caption of Fig. 4.1
65
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.47: P3A. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.48: P4A. See caption of Fig. 4.1
66
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.49: P1B. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.50: P2B. See caption of Fig. 4.1
67
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.51: P3B. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.52: P4B. See caption of Fig. 4.1
68
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.53: P1C. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.54: P2C. See caption of Fig. 4.1
69
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.55: P3C. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.56: P4C. See caption of Fig. 4.1
70
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.57: P1D. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.58: P2D. See caption of Fig. 4.1
71
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.59: P3D. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.60: P4D. See caption of Fig. 4.1
72
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.61: P1E. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.62: P2E. See caption of Fig. 4.1
73
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.63: P3E. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.64: P4E. See caption of Fig. 4.1
74
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 200 2500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.65: P1F. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 200 2500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.66: P2F. See caption of Fig. 4.1
75
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 200 2500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.67: P3F. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 200 2500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.68: P3F. See caption of Fig. 4.1
76
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.69: P1G. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.70: P2G. See caption of Fig. 4.1
77
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.71: P3G. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.72: P4G. See caption of Fig. 4.1
78
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.73: P1H. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.74: P2H. See caption of Fig. 4.1
79
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.75: P3H. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.76: P4H. See caption of Fig. 4.1
80
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.77: P1I. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.78: P2I. See caption of Fig. 4.1
81
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 150 200 2500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.79: P3I. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 150 2000
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.80: P4I. See caption of Fig. 4.1
82
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 20 40 60 800
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
20 30 40 50 60 70 800
100
200
300
400
r (µm)
f(r)
A
C
E
G
Figure C.81: P1J. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 20 40 60 800
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
20 30 40 50 60 70 800
100
200
300
400
r (µm)
f(r)
A C
E
G
Figure C.82: P2J. See caption of Fig. 4.1
83
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 20 40 60 800
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
20 30 40 50 60 70 800
100
200
300
400
r (µm)
f(r)
A
C
E
G
Figure C.83: P3J. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 20 40 60 800
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
30 40 50 60 700
5
10x 10
7
r (µm)
f(r)
A
Figure C.84: P4J. See caption of Fig. 4.1
84
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
C
E
G
Figure C.85: P1K. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A C
E
G
Figure C.86: P2K. See caption of Fig. 4.1
85
0 0.2 0.4 0.6 0.8 10.2
0.4
0.6
0.8
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)C
/C0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
50
100
150
200
r (µm)
f(r)
A
CE
G
Figure C.87: P3K. See caption of Fig. 4.1
0 0.2 0.4 0.6 0.8 10.7
0.8
0.9
1
ε L, kf, R
z/H
B
0 50 100 1500
0.5
1
Time (min)
C/C
0
H
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
C/C
0
D
0 0.2 0.4 0.6 0.8 10
0.5
1
z/H
q’/q
max
F
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
C
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
G G
0 0.2 0.4 0.6 0.8 10
0.5
1
q/q m
ax
r/R
E
50 100 1500
5
10x 10
7
r (µm)
f(r)
A
Figure C.88: P4K. See caption of Fig. 4.1
86