Modelling the effect of particle size distribution on Expanded Bed Adsorption processes

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 eect of particle size distribution on Expanded Bed Adsorption processes.

Markus Wistrand

Sammanfattning

Proteinrening är en vanlig syssla i läkemedelsindustrin, t.ex. för att rena fram den aktiva substansen för ett läkemedel. Det är viktigt att under reningsprocessen minimera såväl tidsåtgång som förluster. Expanded Bed Adsorption (EBA) är en relativt ny teknik med potential 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 skräpprodukter passera genom en kolonn med ett absorberande media ("solid fas"). Skräpprodukterna går rakt igenom medan proteinerna binder specikt 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 solida fasen av partiklar tillhörandes en viss storleksfördelning. I detta examensarbete

intresserade jag mig särskilt för hur olika storleksfördelningar påverkar reningsprocessen och studerade detta genom att sätta upp en teoretisk modell för EBA och utifrån den skriva ett simuleringsprogram.

Slutsatsen av projektet är att solid fas bör ha en stor andel små partiklar för att binda maximalt av det protein som man önskar rena fram. Förhoppningsvis ger dessa resultat nya insikter i hur media för EBA bör se ut och det är också möjligt att genomföra nya studier med hjälp av programmet.

Examensarbete 20 p. i Molekylär bioteknikprogrammet

Uppsala universitet, januari 2002

Contents

1 Introduction 3

1.1 What has been done on EBA . . . . 4

1.1.1 Prior experiments on EBA . . . . 4

1.1.2 Prior modelling of EBA . . . . 6

1.2 Objectives . . . . 6

2 Modelling EBA 7 2.1 Model part one: Bed expansion (hydrodynamics) . . . . 7

2.2 Mechanisms of mass transfer . . . . 8

2.2.1 Model parameters . . . . 9

2.2.2 Dependence on bed height and radius . . . 10

2.2.3 The liquid phase . . . 10

2.2.4 The solid phase . . . 11

2.2.5 Intraparticle mass transfer . . . 11

2.3 Non-dimensional units . . . 13

2.4 Models on non-dimensional form . . . 13

2.5 The homogeneous model on non-dimensional form . . . 14

2.6 The porous model on non-dimensional form . . . 14

2.7 Validation of the models . . . 14

3 Simulations 16 3.1 Model parameters . . . 16

3.2 Particle size distributions . . . 18

3.3 Measurement of column utilization and productivity . . . 18

3.4 Parametric sensitivity analysis . . . 19

3.5 Simulations . . . 20

4 Results and discussion 21 4.1 Concentration proles in liquid and solid phase . . . 21

4.1.1 Eect of intraparticle mass transfer . . . 22

4.2 Eect of particle size distribution . . . 23

4.3 Parametric sensitivity analysis . . . 24

4.3.1 Capacity and productivity . . . 24

4.4 Concluding remarks . . . 33

5 Acknowledgements 34

6 Nomenclature 37

A Parameters 39

A.1 Axial dispersion coecient as a function of voidage . . . 39

A.2 Overall Bodenstein number . . . 39

B Numerical techniques 40

B.1 Liquid phase . . . 40 B.2 Solid phase . . . 41 B.3 Homogeneous particle . . . 41

C Proles for all simulations performed 42

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 process consuming both time and money. Techniques able to interfere and create shortcuts in long purication schemes are therefore of great value. Expanded Bed Adsorption (EBA) is a relatively new technique aiming to reduce the number of steps needed in purication, thereby saving both time and money as well as reducing product loss.

Traditional purication 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 point are the cells which are lysed to produce crude feed-stocks. These feed-stocks are a mishmash of cells, cell debris, proteins and impurities of all kinds. Applying the feed-stock directly to a packed bed chromatography column would cause column clogging.

Traditional purication schemes instead starts by steps aiming for the removal of cell debris, such as centrifugation and microltration. These procedures clarify and concentrate the feed-stock to make application to a chromatographic column possible. However, centrifugation and micro-

ltration have drawbacks which are essentially manifested in product losses due to degradation and spilling, loss of time and high costs. Being able to apply the crude feed-stock to a column for direct adsorption would thus be desirable. As is depicted in Fig. 1.1, EBA is a one-unit process aiming to clarify and concentrate the feed-stock as well as to obtain a rst purication.

Figure 1.1:

The link between fermentation and purication and where EBA enters the purication scheme.

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 expanded instead 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 the particles is exactly matched by the gravitational force, leading to an equilibrium. For a given adsorbent-feed system, the equilibrium is dependent on the ow in the column: a high ow gives a high degree of expansion. However, an upper limit on ow rate always exists as too high a ow would carry particles out of the column.

The voidage is the ratio of liquid phase volume to column volume (²

L

or voidage). Packed bed

chromatography typically operates with a voidage that is independent of column height and in

the region ²

L

=0.3-0.4 [1]. In EBA, normal values of overall voidage are in the region²

L

=0.7-0.8,

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 reduces clogging 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 is both to stabilize the expanded bed and to make media production easier and cheaper. Assuming all particles are of the same density (a simplication), 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 a consensus that theory as well as empirical studies support a distribution of particles inside the column, with respect to size. Consequently, due to the particle size distribution (PSD), as well as a possible density distribution, a voidage gradient is formed in the uidized bed, with the lowest 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 a base matrix that should be a both chemically and mechanically stable support for the ligands coupled to it. These ligands are chosen to suit the specic requirements in each application, i.e.

the binding characteristics (selectivity) needed for the purication of a certain protein. In EBA, the density of the adsorbent particles is higher than in normal chromatography to stabilize the bed and allow for a high ow rate. The higher density is achieved by placing an inert core in the middle of the adsorbent particle or from small pieces of quartz scattered in the particles.

Any EBA process is aected by two types of parameters. The hydrodynamic parameters govern the degree of expansion and mixing inside the column. These are uid velocity, mixing in liquid phase (axial dispersion), mixing in solid phase (solid dispersion), density of the adsorbent and liquid phase viscosity. Other parameters govern the mass transfer of solute inside the column.

These are diusion coecient inside the adsorbing material and lm mass transfer coecient.

Changing the particle size and the particle size distribution will aect both the hydrodynamics and 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 liquid

phase which reduces separation performance [1], [2], [3]. Increased knowledge of the mechanisms

behind dispersion in both solid and liquid phase has led to better products. It is thought that a distribution of particle radii adds to the stability by forming a bed where particles are classied by 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, Willoughby et al. [5] have veried experimentally that beds are indeed classied by size, although a certain degree of mixing of particles is present. Results pointing in the same direction, although showing less degree of classication, 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 along the bed. The results support a gradient of bed voidage inside the column and shows how axial dispersion decreases with increasing porosity, i.e. the highest dispersion is found at the bottom of the column.

Karau et al. [4] have examined the eect of particle size distribution on axial dispersion. They worked with three PSDs: 120-160 µm, 120-300 µm and 250-300 µm. They reported that size distribution aect 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 to for a narrow one. This underlines the need to view EBA as a technique dierent from traditional packed bed chromatography. While, in a packed bed, particles of dierent sizes are more or less completely mixed, they are much more classied 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 the particles is most often considered to occur in two distinct steps called external and internal mass transfer. The rst step accounts for a solute transport through an imaginary stagnant lm/layer around the uidized particle, whereas the second step takes care of mass transfer phenomena occurring inside the particle, such as intraparticle diusion and adsorption. The rate of external mass 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 as well as ligand size and its distribution. The rate of internal transport is characteristic for each media.

The internal resistance and the lm resistance together characterize mass transfer to and inside the solid phase. Depending on system parameters such as particle size, solute size and liquid velocity the importance of each of the two steps varies [8], [9].

Large particles have long diusion paths which adds to the internal resistance. It can generally be said that, for large diameter particles, internal resistance will be more important than lm resistance, 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 limiting step, and that lm resistance and axial dispersion can be neglected. However, as pointed out by Karau et al., this is not the case in EBA. They have measured [4] the quote between the two resistances, and come to the conclusion that for particles ranging in diameter over the interval 120-300 µm, the lm resistance goes from being equally important for small particles to become several times lower in value for large particles. Film resistance also decreases in importance when the linear uid velocity increases. To conclude, they remark that to optimize EBA, small particles with a relatively high density should be used to allow for a high linear velocity without extreme expansion, low lm resistance and yet short diusion paths inside particles.

The dynamic capacity is an often used measure of column eciency and can be stated as the

degree of average saturation of the adsorbent material in the column. In general, the more

dispersed the concentration front is, the lower is the dynamic capacity. Bruce and Chase [6] have

measured the dynamic capacity and reported higher capacity in the top region of the column

than in the bottom region. This was assigned to the following characteristics at the top of the

column, compared to the bottom: low axial dispersion, low linear velocity (as the voidage is

high, the velocity per void is low) and smaller diameter particles giving short diusion paths.

1.1.2 Prior modelling of EBA

In modelling Expanded Bed Adsorption, people have mainly focused on either the dynamics of the 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 dierent mass transfer models, in both case assuming mono-sized particles of the same density. Their two models describe dierent types of mass transfer mechanisms, applicable to two dierent types of adsorbent material. In the homogeneous model, particles are thought of as gel-like in which the solute diuses driven by the solid phase concentration gradient. In the porous model, the solute diuses in the pores of the particles while all the time being in equilibrium with the adsorbed concentration of solute on the matrix. The simulations were for both models in good agreement with experimental data.

The authors also did a parametric sensitivity analysis using simulations, trying to assess the impact of hydrodynamic parameters (velocity and axial dispersion) and mass-transfer parameters (particle radius, lm mass transfer and solid diusion) for dierent degrees of expansion. This was done by changing one parameter at the time and comparing how the breakthrough time changed. It was found that velocity and particle size were the most important parameters while axial dispersion, lm mass transfer and intraparticle diusion were of less importance.

A model of how, for a given particle size distribution, the voidage varies along the bed height in a uidized bed, has been developed by Al-Dibouni and Garside [12]. They presented one model assuming mixing of particles in the column and one assuming perfect classication of particles with respect to size. Both were compared to experimental results and both showed good agreement, with the mixing model proving to be the best one. However, it needs experimental input, and the authors reasoned that for most practical situations using the classication model should be good enough.

Thelen et al. [13] have developed a distributed parameter model, which is capable of predicting the dynamic response of bed height to step changes in the uidization velocity. The model captured both the convective transport of solid phase due to the liquid phase ow rate, and a proposed erratic (dispersive) transport caused by hydrodynamic eects. Particles were considered mono-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 compared favourably with experiments.

1.2 Objectives

The primary objective of this work is to study the eect of the particle size distribution on

Expanded Bed Adsorption process. This is done by constructing a model which combines the

hydrodynamics involved in bed expansion assuming a perfectly classied bed, with a model

describing mass transport in an EBA process. In Wright and Glasser's work [11] referred to

above, no attention was paid to the distribution of particles in the column due to dierent

sizes, and the gradient in voidage and interstitial velocity along the bed that follows. This is

a simplication of reality and we show here how dierent particle size distributions aect the

performance of the EBA system by performing computer simulations. For each distribution, a

parametric sensitivity analysis is performed changing both hydrodynamic properties and mass-

transfer properties. Results are reported in terms ofutilization (U) dened as dynamic capacity

divided by maximum capacity and productivity (P) dened as utilization per hour process. We

also provide concentration proles of both liquid and solid phase along the bed and within

adsorbent particles at dierent heights. Together these may give valuable insight in the dynamics

of EBA.

2 Modelling EBA

In this chapter, we will go through the dierent processes involved in modelling EBA. Basically the model consists of two separate parts. The rst part contains the model for bed expansion and assigns a voidage and a particle size to each segment of the bed. The second part deals with the 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. Once expanded, 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 classied bed. The basic assumption made is that each particle is axially stationary when the bed expansion has 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 mixing in the column due to circular movements is neglected.

Before explaining the model, it should be claried how an individual particle's terminal velocity is predicted. For low particle Reynolds number, Re

p

<0.2, this should be done by Stoke's law, while for higher particle Reynolds number a correcting term should be added [3]. Re

p

is given by

Re

_p

= ρ

_L

ud µ²

_L

where u is the supercial velocity, d is the particle diameter, ρ

L

is the liquid phase density, ²

L

is the bed voidage and µ is the liquid phase viscosity. The terminal velocity (u

t

) is then for the respective regions of Re

p

written as

u

_t

= d

²

g(ρ

_s

− ρ

_L

)

18µ Re

_p

< 0.2 (2.1)

u

_t

= d

²

g(ρ

_s

− ρ

_L

)

18µ(1 + 0.15Re

^0.687_p

) Re

_p

> 0.2 (2.2) where g is the acceleration of gravity and ρ

s

is the particle density.

It can be veried that for Re

p

= 0.2 , the two expressions do not match (a 5% error). Using the equations in their respective region therefore results in a discontinuity inu

t

. To avoid this, we have chosen to use only Eq. (2.2) for all calculations. The error this infers decreases rapidly from the 5% error, when Re

p

approaches 0.

However, Eq. (2.2) does not account for the eects of other particles in the system. This is corrected for through the Richardson-Zaki equation which relates terminal velocity to bed voidage (²

L

) and supercial velocity (u).

u = u

_t

(d)²

ⁿ_L

(2.3)

where the value of n for all values of Re

p

is given by [12].

(5.1 − n)/(n − 2.7) = 0.1Re

^0.9_p

(2.4)

For a distribution containing suciently large particles, a layer of the bed at the bottom of the column might stay sedimented. The necessary condition for expansion is that Eq. (2.3) gives a higher voidage than the value for the sedimented bed, i.e.

µ u u

_t

¶

_1/n

≥ ²

_sed

(2.5)

In a perfectly classied bed, particles with diameter betweend and (d − dd) will be found in a segment of the column between z and (z + dz). The volume gel in a segment of the column (dV ) can be written

dV = (1 − ²

_L

)Adz (2.6)

where A is the column area. The area can be expressed in terms of the total volume of gel in the column (V

gel

), the sedimented bed height (H

sed

) and the voidage of the sedimented bed (²

sed

).

A = V

_gel

H

_sed

(1 − ²

_sed

) (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 R

_∞

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 = H

_sed

f (d) (2.9)

and for the expanded part of the column dz

dd = H

_sed

(1 − ²

_sed

) (1 − (

_u^u0

t(d)

)

^1/n

) f (d) (2.10) These equations are integrated to get column height as a function of particle radius. Once this is done, the local voidage is calculated as a function of column height from Eq. (2.3) and the expansion by simply dividing H with H

sed

.

In case some of the particles are small enough, the voidage at higher segments of the bed will come close to 1 and the denominator in Eq. (2.10) will approach zero. It is then not possible to perform the integration. This is solved by cutting the distribution (excluding the smallest particles), and integrate Eq. (2.10) with the cut distribution and the cut settled bed height that follows. In an experiment this would mean that particles that are light enough to leave the column 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 by Eq. (2.3). Inserting the condition f(d) = 1, and Eq. (2.3) into Eq. (2.10), yields.

H

H

_sed

= 1 − ²

_sed

1 − ²

_L

(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 particles

described by lm resistance and diusion, respectively. We will in this section rst shortly explain

the model parameters and then give the model equations.

2.2.1 Model parameters

Convective transport: Convective transport of solute in liquid phase is driven by the liquid phase ow from below, which forces the solute through the column.

Dispersive transport: Dispersive transport is the result of inhomogeneities in liquid phase concentration 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 static expanded bed. While the bed is expanded, it is assumed to be classied by size. Due to tur- bulence and to the fact that particles are not saturated simultaneously, particles may however slightly move. For example should a particle that for some reason is more saturated (and there- fore heavier) than its neighbours, slightly settle. This will aect not only the mass transfer to the particles, but also the bed voidage distribution. In modelling the solid dispersion we follow the method outlined by Wright [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 given by a linear driving force and is dependent on the concentration dierence between liquid phase and the particle surface.

Intraparticle diusion: The internal resistance to mass transfer is modelled by a diusion process. It has, however, been shown by several authors that mass transfer in dierent types of chromatography 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 diusion model and the pore diusion model.

In the homogeneous model, the medium is seen as homogeneous and gel-like. The solute is adsorbed at the particle surface and diuses towards the center in a process governed by a driv- ing force due to the gradient in adsorbed solute concentration, and a diusion coecient (D

h

).

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 diusion inside the particle pores, with a driving force expressed in terms of the gradient in pore concentration, and a characteristic dif- fusion coecient (D

p

). Pore concentration is all the time in equilibrium with the adsorbed solute concentration. Wright et al. have shown that the porous model well describes mass uptake by Streamline media [11], [15].

Adsorption: In this work the equilibrium between adsorbed concentration and the liquid phase concentration in close vicinity of the adsorbing surface, is assumed instantaneous. The adsorption process, 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 = q

_max

C

_f

K

_s

+ C

_f

(2.12)

where q is the adsorbed concentration, C

f

is the liquid phase concentration at the particle surface, K

s

is the dissociation constant characteristic for each adsorption process andq

max

is the maximum capacity of the adsorbent. Assumptions behind applying a Langmuir isotherm are [10]:

1. Molecules are absorbed at a xed number of well-dened localized sites.

2. Each site can hold one adsorbate molecule.

3. All sites are energetically equivalent.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

50 100 150

c (mg/ml)

q (mg/ml)

Figure 2.1:

A favourable Langmuir isotherm. qmax= 150mg/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 low liquid concentration), with parameters q

max

= 150 mg/ml and K

s

= 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 change with 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. The following parameters are, however, position dependent vectors: D

ax

(z) is the axial dispersion coecient, D

s

(z) is the solid dispersion coecient, ²

L

(z) is the voidage, ²

s

(z) is the fraction solid phase to column volume (²

s

(z) = 1 − ²

_L

(z)) , k

f

(z) is the lm mass transfer coecient and R

p

(z) is the particle radius.

Further, the variables (which the model is solved for) are both dependent on position and time and should correctly be written: C(z, t) is the liquid phase concentration, C

p

(z, r, t) is the pore concentration, q

s

(z, r, t) is the adsorbed concentration, q(z, r, t) is the solid phase concentration, and q

⁰

(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 written as follows:

Accumulation = mass in

_{bulkf low}

− mass out

_{bulkf low}

+ mass in

_dispersion

− mass out

_dispersion

− 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 conditions

seems to be the most commonly applied ([9], [14], [19]), and assume no axial dispersion in the

sections immediately before and after the column. Initial condition is an empty column and the

model equations are written as follows:

∂C

∂t = − ∂

∂z µ

−D

_ax

∂C

∂z + u

²

_L

C

¶

− 3k

_f

(1 − ²

_L

)(C − C

_f

)

²

_L

R

_p

(2.13)

I.C: at t = 0, C(z, 0) = 0 for 0 ≤ z ≤ H (2.14)

B.C 1: at z = 0, C − D

_ax

²

_L

u

∂C

∂z = C

₀

, t>0 (2.15)

B.C 2: at z = H, ∂C

∂z = 0 , t>0 (2.16)

where C is the liquid phase concentration,C

0

is the feed concentration, D

ax

is the axial dispersion coecient, H is the expanded bed height, z indicates bed height, u is the supercial velocity, k

f

is the lm mass transfer coecient R

p

is the particle radius at the actual segment andC

f

is the liquid phase concentration close to the particle surface. The latter is given from the Langmuir isotherm (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, written as:

Accumulation = mass in

_dispersion

− mass out

_dispersion

+ mass uptake by adsorbent

The boundary conditions used reect the fact that there is no solid phase outside the column and the initial condition that, when the process starts, no adsorption has yet occured. The equations governing solid phase are then as follows:

∂q

⁰

∂t = D

_s

∂

²

q

⁰

∂z

²

+ 3k

_f

(C − C

_f

)

R (2.17)

I.C: at t = 0, q

⁰

(z, 0) = 0 for 0 ≤ z ≤ H (2.18)

B.C 1: at z = 0, ∂q

⁰

∂z = 0 , t>0 (2.19)

B.C 2: at z = H, ∂q

⁰

∂z = 0 , t>0 (2.20)

where D

s

is the solid phase dispersion coecient andq

⁰

is the average solid phase concentration.

2.2.5 Intraparticle mass transfer

Diusion in homogeneous media Mass transfer inside the adsorbent media is for the ho- mogeneous model governed entirely by diusion, and the equation is written as:

∂q

∂t = D

_h

1 r

²

∂

∂r µ

r

²

∂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 = r

c

, ∂q

∂r = 0 , t>0 (2.23)

B.C 2: at r = R

p

, ∂q

∂r = R

_p

D

_s

3D

_h

∂

²

q

⁰

∂z

²

+ k

_f

D

_h

(C − C

_f

) , t>0 (2.24) where r indicates radius, r

c

is the radius of the inert core at the center, q is the concentration inside the particle, R

p

is the particle radius and D

h

is the intraparticle diusion coecient.

Initially adsorbent contains no solute which is described by the initial condition. The rst boundary condition describes the fact that there is a zero ux to the inert core placed at the center of the particle. The second boundary condition relates the uptake at the particle boundary to the average solid phase concentration (q'). Noting that the ratio volume to area isR

p

/3 , the following relation holds at the particle surface:

∂q

∂r = R

_p

3D

_h

∂q

⁰

∂t (2.25)

Inserting

^∂q_∂t⁰

from Eq. (2.17) gives the second boundary condition and relates the ow across the boundary both to solid dispersion and to lm resistance.

Diusion in porous media Modelling of the porous media involves one more variable com- pared to the homogeneous case, as we have both pore concentration (C

p

) and adsorbed con- centration (q

s

). However, for the case of instantaneous equilibrium, it is possible to reduce computations 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]

²

_p

∂C

_p

∂t + (1 − ²

_p

) ∂q

_s

∂t = ²

_p

D

_p

1 r

²

∂

∂r µ

r

²

∂C

_p

∂r

¶

(2.26) where ²

p

is the particle porosity and D

p

is the pore diusion coecient. The chain rule gives:

∂q

_s

∂t = ∂q

_s

∂C

_p

∂C

_p

∂t (2.27)

The Langmuir isotherm describes the instantaneous equilibrium between pore concentration (C

p

) and adsorbed concentration in the particles (q

s

).

q

_s

= q

_max

C

_p

K

_s

+ C

_p

(2.28)

Dierentiating the isotherm with respect to C

p

gives

∂q

_s

∂C

_p

= K

_s

q

_max

(K

_s

+ C

_p

)

²

(2.29)

Inserting Eq. (2.27) and Eq. (2.29) into Eq. (2.26), the time derivative of the pore concentration can be expressed as

∂C

_p

∂t = α 1 r

²

D

_p

∂

∂r µ

r

²

∂C

_p

∂r

¶

(2.30) where

α = ²

_p

²

_p

+ (1 − ²

_p

)

_∂C^∂qs_p

(2.31)

The initial and boundary conditions used are the same as for the homogeneous model: no

adsorbed solute when the process starts (I.C), a zero ux to the inert core at the center of the

particle (B.C 1) and the ux across the surface of a particle is related to the change in average particle concentration (B.C 2).

I.C: at t = 0, C

p

(z, r, 0) = 0 for 0 ≤ z ≤ H (2.32)

B.C 1: at r = 0, ∂C

_p

∂r = 0 , t>0 (2.33)

B.C 2: at r = R

p

, ∂C

_p

∂r = R

_p

D

_s

3D

_p

∂

²

q

⁰

∂z

²

+ k

_f

D

_p

(C − C

_p

) , t>0 (2.34) 2.3 Non-dimensional units

If changes in parameter values occur on very dierent scales, it might be hard to assess whether observed 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 and easier to follow. All equations were therefore set on dimensionless form using the dimensionless groups listed in Tab. 2.1. C

0

is the feed concentration and, D is the intraparticle diusion coecient (D

h

or D

p

) and R is the reference radius. The latter is set to the radius of the smallest particle in the column as concentration changes the fastest in this particle.

Table 2.1:

Non-dimensional units.

β = _C^C

0 Dimensionless liquid phase concentration Q⁰= _q^q0

max Dimensionless average solid phase concentration Q = _q^q

max Dimensionless solid phase concentration βp=^C_C^p

0 Dimensionless pore concentration η = _R^r Dimensionless radius

ξ = _H^z Dimensionless bed height Θ = _R^D2t Dimensionless time P e = _²^uH

LDax Peclet number Bi = ^kf_D^R Biot number Θr= ^HD²_uR2^L Mass transfer unit

2.4 Models on non-dimensional form Liquid phase:

∂β

∂Θ = 1

P eΘ

_r

∂

²

β

∂ξ

²

− 1 Θ

_r

∂β

∂ξ µ

1 − ²

_L

Hu

∂D

_ax

∂ξ

¶

+ βu H

∂²

⁻¹_L

∂ξ − 3Bi ²

_s

R

²

_L

R

_p

(β − β

_f

) (2.35)

I.C: at Θ = 0, β(ξ, 0) = 0 for 0 ≤ ξ ≤ 1 (2.36)

B.C 1: at ξ = 0, β − 1 P e

∂β

∂ξ = 1, Θ > 0 (2.37)

B.C 2: at ξ = 1, ∂β

∂ξ = 0 , Θ > 0 (2.38)

Solid phase:

∂Q

⁰

∂Θ = D

_s

R

²

DH

²

∂

²

Q

⁰

∂ξ

²

+ 3Bi C

₀

q

_max

(β − β

_f

) (2.39)

I.C: at Θ = 0, Q

⁰

(ξ, 0) = 0 for 0 ≤ ξ ≤ 1 (2.40)

B.C 1: at ξ = 0, ∂Q

⁰

∂ξ = 0 , Θ > 0 (2.41)

B.C 2: at ξ = 1, ∂Q

⁰

∂ξ = 0 , Θ > 0 (2.42)

2.5 The homogeneous model on non-dimensional form The homogeneous particle:

∂Q

∂Θ = 1 η

²

∂

∂η (η

²

∂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 = r

_c

R

_p

, ∂Q

∂η = 0 , Θ > 0 (2.45)

B.C 2: at η = R

_p

R , ∂Q

∂η = R

_p

RD

_s

3H

²

∂

²

Q

⁰

∂ξ

²

+ Bi C

₀

q

_max

(β − β

_f

) , Θ > 0 (2.46) 2.6 The porous model on non-dimensional form

Porous particle

∂β

_p

∂θ = α 1 η

²

∂

∂η (η

²

∂β

_p

∂η ) (2.47)

where

α = ²

_p

²

_p

+ (1 − ²

_p

)

_∂C^∂qs_p

(2.48)

with initial and boundary conditions

I.C: at Θ = 0, β

p

(ξ, η, 0) = 0 for 0 ≤ ξ ≤ 1 (2.49)

B.C 1: at η = r

_c

R

_p

, ∂C

_p

∂η = 0 , Θ > 0 (2.50)

B.C 2: at η = R

_p

R , ∂β

_p

∂η = R

_p

RD

_s

q

_max

3H

²

C

₀

∂

²

β

_p

∂ξ

²

+ Bi

²

_p

(β − β

_p

) , Θ > 0 (2.51) 2.7 Validation of the models

In order to conrm that model equations and the numerical scheme were correctly used, the model predictions were compared to either analytical or numerical solutions for some specic cases 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 innite volume. Amount of mass present in the sphere at any time is

0 0.2 0.4 0.6 0.8 1 0

0.1 0.2 0.3 0.4 0.5

Error %

r/R

0 0.2 0.4 0.6 0.8 1

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

C/Cfinal

r/R

0 0.2 0.4 0.6 0.8 1

0 0.1 0.2 0.3 0.4 0.5

Error %

r/R

0 0.2 0.4 0.6 0.8 1

0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

C/Cfinal

r/R Homogenous

Porous

Figure 2.2:

Comparison of the models with the analytical solution for a homogeneous sphere in a batch with constant concentration, assuming lm resistance. Percentage error and concentration prole inside the sphere 30 minutes after the empty sphere was placed in the batch. First row: The homogeneous model.

Second row: The porous model, ²p= 1.

assumed negligible compared to the amount solute in the batch, and does not aect batch con- centration. There is no adsorption inside the sphere or at its surface, only diusion. The surface condition is

D ∂C

∂r = α(C

₀

− C

_f

)

where α is a constant, C

f

is the concentration at the surface of the sphere andC

0

is the concentra- tion required to maintain equilibrium with the surrounding atmosphere. This surface condition is the same as the one describing lm resistance withα = k

f

.

The implementations of the two mass transfer models were veried against this solution by

"cutting lose" the particles from the two overall models and setting parameters to the following values: C

0

= 2 mg/ml, ²

L

= 0.99999 , k

f

= 5 ∗ 10

⁻⁴

cm/s, and in addition for the porous model

²

_p

= 1 . A linear isotherm with K

s

=1 was substituted for the Langmuir isotherm.

C(R

_p

) = K

_s

C

_f

(2.52)

Simulations were performed with 15 discretization points in the particle. Both models were in good 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 lm resistance 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 ofC

0

. The model was veried by setting model parameters to: ²

L

= 0.5 , C

0

= 2 mg/ml and, for the porous model, ²

p

= 1 . In addition, the lm coecient was set to a high value (k

f

= 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 15 discretization points, the model was in good agreement with the analytical solution (Fig. 2.3).

3) Packed bed with non-adsorbing particles

To test the equations governing the column as well as the boundary conditions for the column,

0 0.2 0.4 0.6 0.8 1 0.2

0.25 0.3 0.35 0.4 0.45 0.5

Error %

r/R

0 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 1

0.2 0.25 0.3 0.35 0.4 0.45 0.5

Error %

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 homogeneous sphere in a well-stirred tank. Percentage error and concentration prole inside the sphere 30 minutes after the empty sphere was placed in the batch. First row: The homogeneous model. Second row: The porous model, ²p= 1.

the model was compared to simulations performed by Seidel-Morgenstern [19]. He has presented breakthrough curves for dierent Peclet numbers using the same boundary conditions as those used in this work. Using the author's parameters and setting absorbent capacity toq

max

= 0 (non-adsorbing particles), we repeated his simulation. The results tted Seidel-Morgenstern's results exactly (Fig. 2.4).

In addition, the model predictions were compared with the analytical solution derived by Ku£era [21] for an innite 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 at a given time. These moments can also be calculated from the breakthrough curves of our simu- lations, which was done. As Ku£era considers an innite column he do not have any boundaries.

We excluded the eect of boundaries by taking the dierence of two simulations with dierent column length. For example: The rst moment was calculated at 40 cm for a simulation with a 40 cm column, and at 20 cm for a simulation with a 20 cm column. The dierence between these moments were compared to the analytical solution for the rst moment at 20 cm. The rst moments were identical. Higher moments were not tested.

However, including the boundaries, the simulation did not match the analytical solution. As simulations matched the solution of Seidel-Morgenstern, the implementation of the boundaries is done correctly. The conclusion is that Danckwerts' boundary conditions, although they include axial 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 along

the bed as interstitial velocity, voidage and particle size changes. For those, correlations obtained

from literature have been used.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Θ

C/C0 Pe=0.01

Pe=1

Pe=20

Figure 2.4:

Packed bed with non-adsorbing particles. Comparison of model simulations with breakthrough curves in [19].

Axial dispersion in EBA was investigated by Bruce and Chase [6]. They reported values of the Bodenstein number as well as the voidage at dierent column heights. We have used their results to nd a correlation between axial dispersion and voidage. At each cross section of the column, a value is assigned to D

ax

from the voidage at this level (see further in section A.1).

D

_ax

= 0.2991 − 0.5101²

_L

+ 0.2163²

²_L

(3.1) The equation governing the liquid phase gives rise to terms involving the dierentiation ofD

ax

and

_²¹_L

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 express these terms by nite dierences of the values for D

ax

and

_²¹_L

.

For the solid dispersion coecient, D

s

, we use the correlation Wright ascribes to Van Der Meer et al. [11]. It relates solid dispersion to the supercial velocity only, and solid dispersion is thus assumed to be constant along the bed. The factor 0.04 in the correlation is not dimensionless.

D

_s

= 0.04u

^1.8

m/s (3.2)

The lm mass transfer coecient was for each section of the bed calculated from the correlation for the Sherwood number in a packed bed recommended in Chemical Engineer's Handbook [22].

It is dened as

k

_f

= D

_m

Sh/d (3.3)

where the Sherwood number is given as:

Sh = 1.09

²

_L

Re

^0.33_p

Sc

^0.33

D

_m

is the molecular diusion coecient of the solute,Sh is the Sherwood number, d the diameter of the adsorbent particle, Re

p

the particle Reynolds number and Sc the Schmidt number. The

lm mass transfer coecient increases with solute diusivity and uid velocity and decreases

with particle size [23].

0.0050 0.01 0.015 50

100 150 200

r (cm)

f(r)

0.0050 0.01 0.015

50 100 150 200

r (cm)

f(r)

0.0050 0.01 0.015

100 200 300 400

r (cm)

f(r)

0.0050 0.01 0.015

2 4 6 8 10x 10⁴

r (cm)

f(r)

1 2

3 4

Figure 3.1:

The four particle size distributions (PSDs) considered. 1: pyramidal, 2: biased to small particles, 3: biased to large particles and 4: mono-sized. -(blue) Number basis, - -(red) volume basis.

3.2 Particle size distributions

Studying the eect of particle size distribution on the EBA process is the main objective of this work. In order to evaluate this eect, the following four particle size distributions (PSDs) were chosen: 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 dierent on a volume basis than it might appear looking at the radii, as volume depend cubically on the radius. The particles were for all PSDs distributed in the same interval, i.e. the smallest and the largest particles were of the same size for all PSDs.

In all simulations, a settled bed voidage of ²

sed

= 0.36 is assumed. All beds therefore contain the same volume gel and, as particle sizes are dierent, the expanded beds will for the four cases not have the same expansion, for a given supercial velocity.

3.3 Measurement of column utilization and productivity

The breakthrough curve is a plot of how solute concentration at the column outlet varies with time. In practice, the process is often stopped when outlet concentration reaches 1-10% of the feed concentration, to avoid product losses. The dynamic capacity is a measure of how much of the protein loaded onto the column that has been adsorbed. It is most often dened as the amount solute bound to solid phase per volume settled bed at a certain breakthrough concentration. However, from an experimental point of view, all protein resident in the column may be of interest. This amount which could be called an "apparent" dynamic capacity, will, at the moment of breakthrough, include also the amount of solute in liquid phase. Working with high values of maximum capacity as we do, makes the dierence between the two capacities small and we will only consider the amount solute in solid phase. The dynamic capacity is written as

Q

_d

= 1 (1 − ²

_sed

)H

_sed

Z

_H

0

²

_s

q 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 direct comparisons between bed of dierent maximum capacity.

U = Q

_d

/q

_max

(3.5)

A high value of the dynamic capacity can be reached by working with a uid velocity close to the

minimum velocity necessary to get bed expansion. This gives proteins plenty of time to diuse