Modelling and parametric estimation of simulated moving bed chromatographic processes (SMB)

UNIVERSITE LIBRE DE BRUXELLES

Modelling and parametric estimation of simulated moving bed chromatographic

processes (SMB)

Thèse présentée en vue de l’obtention du grade de docteur en sciences appliquées par Valérie Grosfils

Promoteur : Prof. M. Kinnaert

Co-promoteur : Prof. A. Vande Wouwer avril 2009

2d edition

ACKNOWLEDGEMENTS

This thesis is the result of a seven-year work. The first four years were financially supported by the Walloon Region within the framework of the MOVIDA project. I spent the last years as teaching assistant. I would like to thank all the people who helped me to complete my work.

First of all, my greatest thanks are due to my promoter, Professor Michel Kinnaert, who initiated me in research. He always found time in his full time table to guide me with relevant ideas and advices.

Many thanks are due to my co-promoter, Professor Alain Vande Wouwer for his interest in my work and the idea of new directions he proposed.

I would like to express my gratitude to Professor Raymond Hanus for his interesting suggestions concerning my work and for allowing me to continue my thesis as a teaching assistant.

My great thanks are also for Professor Véronique Halloin for her pertinent comments during the board for the thesis follow-up.

Moreover, I gratefully acknowledge Professor Achim Kienle and his coworkers of the Max Planck Institute of Magdeburg, especially Henning Schramm, for their contributions to the experimental work.

I also would like to acknowledge Professor Philippe Bogaerts and Professor Jean- Pierre Corriou for their participation in the thesis jury.

I would like to express my gratitude to the Fondation Van Buuren for its financial support which helps me after the end of the MOVIDA project and before my nomination as teaching assistant.

I would like to thank Michel Hamende, Emile Cavoy, Sophie Vanlaethem and their co-workers of UCB Pharma for their interest in this work.

Many thanks to

all the people who joined the project Movida, especially Caroline Levrie.

my department colleagues, especially, Andrée Delhaye, Pascale Lathouwers, Laurent Catoire, Serge Torfs, Lhoussain El Bahir, Joseph Yame, Xavier Hulhoven, Thomas Delwiche, Angelo Buttafuoco, Jonathan Verspecht, Manuel Ricardo Galvez Carrillo, Cristina Retamal, Laurent Rakoto, Samuel Vagman, Valérie Decoux, and Mohamed El Aydam, for their kindness and the good atmosphere they contributed to install in the lab.

Antoni Severino and Guy De Weireld, from the FPMs, for their nice collaboration

Special thanks to my friends and my wonderful family for their love and their encouraging support.

David, Lucie, Simon, my little sunshine’s, thank you for your patience and your unconditional love.

CONTENTS

CHAPTER 1 : INTRODUCTION ... 1

PART 1: GENERAL CONCEPTS

CHAPTER 2 : AN INTRODUCTION TO CHROMATOGRAPHY AND SMB PROCESSES... 7

2.1.INTRODUCTION 7

2.2.BATCH CHROMATOGRAPHY 7

2.3.SMB PROCESS: GENERAL INTRODUCTION 9

2.3.1. Description... 9 2.3.2. Operating parameters ... 13

2.4.DESCRIPTION OF THE STUDIED SMB PLANT 14

2.5.MODIFICATIONS OF THE SMB PROCESS 16

CHAPTER 3 : MODELLING OF CHROMATOGRAPHIC PROCESSES . 17

3.1.INTRODUCTION 17

3.2.EQUILIBRIUM ISOTHERM 17

3.3.MODELLING OF A COLUMN AND INTRODUCTION TO THE WAVE THEORY 19 3.3.1. Column model ... 19 3.3.2. Wave theory... 25

3.4.MODELLING OF SMB PROCESSES 30

3.4.1. Counter-current movement ... 30 3.4.2. Connections between columns in a SMB process... 34 3.4.3. Column model in a SMB process ... 38

3.5.MODEL PARAMETERS AND OPERATING CONDITIONS 40

3.6.NUMERICAL SIMULATION 44

3.6.1. Introduction... 44 3.6.2. Approximation of the spatial derivatives... 44 3.6.3. Numerical Integration... 46

3.7.CONCLUSIONS OF CHAPTER 3 47

APPENDIX 3.1 EXPLANATION OF THE SHAPE OF THE SMB PROFILES 48 APPENDIX 3.2 EQUATIONS OF A SMB MODEL WITH 8 COLUMNS 55 APPENDIX 3.3 TRIANGLE THEORY WITH LANGMUIR ISOTHERMS 63 APPENDIX 3.4 APPROXIMATIONS OF SAPTIAL DERIVATIVES AT THE EXTREMITIES

65

APPENDIX 3.5 SIMULATION PARAMETERS 67

PART 2: CONTRIBUTIONS TO THE MODELLING OF SMB PROCESSES

CHAPTER 4 : A SIMPLIFIED MODELLING APPROACH TO SMB

PROCESSES... 71

4.1.INTRODUCTION 71 4.2.PRINCIPLE OF THE TRANSLATED TMB MODEL 72 4.3. ASSUMPTIONS 73 4.4.WAVE VELOCITY 73 4.5.TRANSLATION 74 4.6.TIME DELAY AND SMOOTHING OF THE CONCENTRATION CURVES 77 4.7.RESULTS, LIMITATIONS AND DISCUSSIONS 80 4.8.CONCLUSION 87 APPENDIX 4.1.:PARAMETERS AND OPERATING CONDITIONS 88 CHAPTER 5 : EXTRA-COLUMN DEAD VOLUME MODELLING... 89

5.1.INTRODUCTION 89 5.2.GENERAL EQUATION OF MASS BALANCE IN THE DEAD VOLUME 91 5.3.DEAD VOLUME IN THE CIRCULATING LOOP 91 5.4.DEAD VOLUME OF THE INPUT AND OUTPUT LINES 92 5.5.BOUNDARY CONDITIONS 93 5.6.MEASUREMENT EQUATIONS 93 5.7.NUMERICAL SIMULATION 94 5.8.VALIDATION WITH EXPERIMENTAL PROFILES 94 5.9.CONCLUSIONS 100

PART 3: CONTRIBUTIONS TO PARAMETER ESTIMATION IN SMB PROCESSES

6.1.INTRODUCTION 105 6.2.MODELLING AND UNKNOWN PARAMETERS 107 6.2.1. SMB modelling... 108

6.2.2. Batch Model ... 116

6.3.DIRECT METHODS 119 6.3.1. Dead volume ... 119

6.3.2. Porosity ... 119

6.3.3. Isotherm parameters ... 120

6.3.4. Diffusion coefficient of the ED model ... 121

6.3.5. Diffusion coefficient of the LDF model and mass transfer coefficients of the LDF model and of the kinetic model ... 121

6.3.6. Calibration coefficients... 122

6.4.INVERSE METHOD OR IDENTIFICATION - GENERAL PRINCIPLES 122 6.4.1. Notations ... 123

6.4.2. Optimization criterion ... 124

6.4.3. Parameter constraints... 124

6.4.4. Numerical Optimization Methods ... 125

6.4.5. Identifiability and experiment design ... 125

6.4.6. Confidence interval ... 127

6.4.7. Confidence envelope ... 139

APPENDIX 6.1 LEAST SQUARE ESTIMATOR WITHOUT ERROR ON

ζ ˆ

7.1.INTRODUCTION 151 7.2.PROBLEM STATEMENT AND IDENTIFICATION APPROACH 152 7.2.1. Batch experiments and definition of data sets... 152

7.2.2. Identification procedure... 153

7.2.3. Parameter constraints... 154

7.2.4. Cost function ... 154

7.2.5. Initial estimates ... 155

7.3.EXPERIMENT DESIGN AND IDENTIFIABILITY 156 7.3.1. Generation of the fictitious data... 156

7.3.2. Sensitivity analysis ... 156

7.3.3. Local identifiability study from fictitious measurements... 171

7.4.BASIN OF ATTRACTION FOR ERROR-FREE DATA 171 7.5.BASIN OF ATTRACTION FOR NOISY DATA 177 7.5.1. Basin of attraction... 177

7.5.2. Confidence Interval... 180

7.5.3. Confidence envelope ... 182

7.6.CONCLUSION 185 7.7.SUMMARY OF THE IDENTIFICATION PROCEDURE 186 Model ... 186

Identified parameters ... 186

Data ... 186

Cost function ... 186

Optimization algorithm ... 187

Identification procedure... 187

APPENDIX 7.1 FICTITIOUS MEASUREMENTS 188

APPENDIX 7.2 RESULTS OF THE SENSITIVITY ANALYSIS 195

CHAPTER 8 : VALIDATION WITH EXPERIMENTAL DATA... 203

8.1.INTRODUCTION 203

8.2.THE STUDIED SEPARATION 203

8.3.STATEMENT OF THE IDENTIFICATION PROBLEM 205

8.4.FIRST ESTIMATION OF MODEL PARAMETERS 206

8.5.BATCH IDENTIFICATION 210

8.5.1. Results, validation, comparison ... 210 8.5.2. Confidence interval ... 217 8.5.3. Conclusions... 218

8.6.VALIDATION WITH SMB EXPERIMENTS 219

8.7.CONFIDENCE ENVELOPE 224

8.8.REMARK: IDENTIFICATION FROM SMB EXPERIMENTS 228

8.9.CONCLUSIONS 228

APPENDIX 8.1 DEAD VOLUMES 230

APPENDIX 8.2 COVARIANCE MATRIX Q^ζe 233

APPENDIX 8.3 MODEL PARAMETERS AND SIMULATION PARAMETERS 234 CHAPTER 9 : CONCLUSIONS ... 237

BIBLIOGRAPHY 243

NOTATION

A: less retained component B: more retained component

b_{i :} isotherm parameter : ratio of the rate constants of adsorption and desorption for component i

c : concentration in the liquid phase cF:: injected concentration

Sl

c : injected concentration of data set S _l C : l weighting factor used in cost function J_cl;

)) t ( y ( max / )) t ( y ( max

C _S^mes

t mes

t S^{max l}

l=

d: index corresponding to dead volume V_d D : Gram determinant

D_{app :} apparent diffusion coefficient (ED model)

D_d: diffusion coefficient in a dead volume

DL: diffusion coefficient (LDF and general rate model) g_θ: sensitivity w.r.t. θ

H: Henry coefficient

h: height equivalent to a theoretical plate i : component A, B

j: zone

J: cost function

J_cl: “classical” cost function (Eq. 7-1) J_ln: cost function defined by (Eq. 7-2)

J_min: smallest cost function obtained with the multi start strategy: ^r_min

r

J min with r = 1, …, 2ⁿ

ki: mass transfer coefficient of component i (kinetic model)

k_i^rel: relative mass transfer coefficient of component i (kinetic model):

m i rel

i k /v

k =

k_F,i: mass transfer coefficient of component i (LDF model) L: column length

m: column number

m_δ: number of the column placed before detector δ M : l number of measurements in data set S _l

M_T: total number of measurements

∑

=

= ^p

n

1

T M

M

l l

n: number of parameters

n_p: number of data sets used in identification n_s: number of switching periods already performed N: number of components

N_i: the number of theoretical plates of component i Ni: the number of theoretical plates of component i N_c: number of columns

N_D: number of grid points in a dead volume NG: number of grid points per column

N_p: number of positions in the SMB plant (defined in section 5.1) N_z: zone number

p_trans: position of a concentration in the translated TMB profile p_TMB: position of a concentration in the TMB profile

PR: productivity Q: flow rate

Q_j: j = I,.., IV, flow rate in zone j Q_m m = 1, .., N_C, flow rate in column m QR: raffinate flow rate

Q_E: extract flow rate Q_F: feed flow rate Q_D desorbent flow rate

Q_S countercurrent solid flow rate Q^ζe Covariance matrix of ζ^ˆ_e

q : concentration in the solid phase q^eq: adsorbed equilibrium concentration

q_S: column saturation capacity (isotherm parameter) R : l weighting factor in cost function J_ln:

))) t ( y (ln(

max / ))) t ( y (ln(

max

R ^mes_S

t mes

Smax

t l

l =

S: column geometric cross-sectional area S l l=1,...,n_p: data set l;

{

}

Sl= _l l = l− l < l ₊

Smax: data set used in identification with _{,i S,i}

Smax c

c ≥ l (l=1,...,n_p),

t: time

t_D: time delay

t_Din: time delay of the injection due to dead volume before the unit t₀: the column dead time

t_p: injection duration

t_tr a constant characterizing the rise time of the pulse

t_R0i: the retention time of component i at analytical concentration t_S: the time elapsed since the last switch

u: inlet concentration profile

UV(i): calibration coefficient of component i

UV1, UV2, UV3, UV4: refers to one of the 4 UV detectors of the MPI plant v: fluid velocity

v_cc : fluid velocity in the TMB model v_S: solid velocity

V: column volume

V_d: dead volume

V_din: dead volume before the SMB unit

V_d,m^col,b: dead volume located before column m (moving) V_d,m^col,af: dead volume located after column m (moving)

V_d,p^port,b: dead volume located before a column at position p (fixed) V_d,p^port,af: dead volume located after a column at position p (fixed) V_inj: injected volume (in batch experiments)

w: weigthing factor in cost function W: adsorbent weight

W_i,1/2 the width halfway up the peak for component i.

y: model output y^mes: measurement vector z: axial coordinate

Greek alphabet

δ : refers to the δ^th UV detector

∆t : switching period ε: porosity

Γ: time span of experiment θ: parameter vector

θˆ : estimated parameter vector θ^*: transformed parameter vector

θ~

: estimation error on θˆ : ~ ˆ _tr θ

− θ

= θ

θf: parameter vector of one of the fictitious cases defined in Table A.7.1 ˆmin

θ : parameter value corresponding to J_min (with the cost function J_cl)

ln

ˆmin

θ parameter value corresponding to J_min (with the cost function J_ln) θtr: “true” value of θ

θsup, θinf: bound for parameter θ (constraints)

ζ^: vector containing a-priori known parameters (and not identified) (batch)

[

]

=

ζ _{δ δ}

) k

~ (

ζe Estimation error on ζe :~ (k) ˆ (k) (k)

etr e

e =ζ −ζ

ζ

ζe : part of vector ζ which may be corrupted by some errors;

T inj ,

B , F , A , m F

e, =[Q c c UV_δ(A) UV_δ(B) V ]

ζ _{l l}

l ev

ζ ζev =

[

]

ζwe: part of vector ζ known without error; ζ_we =[V_din ε]^T

SMBδ

ζ : vector containing a-priori known parameters (and not identified) (SMB with UV detector δ)

[

]

SMB

B ( UV ) A ( UV c

c V ...

Q Q Q

Q _{δ δ}

δ

ε ε

= ζ

Conventions for the vector operations

[

]

V= _T : vector of dimension MT x 1;

V(k): k^th element of vector V

V(i:j): vector composed of elements of vector V:

[

]

) j : i (

V = + −

CHAPTER 1 : Introduction

Chromatographic separation processes are based on the differential adsorption of the components of a mixture. In the conventional batch liquid chromatography, the separation is achieved by injecting a pulse of the solute mixture. The components move with different velocities through the column in function of their affinity for the solid phase. Hence, as the low retained component exits the column earlier than the more retained one, the separation is achieved. The simulated moving bed (SMB) process is a separation process made of a series of chromatographic columns in which a counter-current movement of the liquid and solid phases is allowed by periodically switching the inlet and outlet valves in the direction of the liquid flow.

This counter-current configuration offers significant advantages like the increase of the productivity and a decrease of the solvent consumption.

This process has been used for large scale production in the petrochemical and sugar industry since the 1950s. Since the 1990’s, the interest of the fine chemical, pharmaceutical and biotechnical industries for the SMB process has grown. This attractive technology is now used to separate chiral molecules, amino-acids, enzymes, and sugars, to purify proteins or to produce insulin. In these fields, the SMB process is viewed as a powerful purification process which offers high yield and high purity at reasonable production rate even for difficult separations. SMB plants are exploited at all production scales from laboratory, with small columns with a diameter of a few millimeters for separation of some grams per day, to production plants, with columns of one meter diameter for purification of hundreds of tons per year.

However, on the one hand, the transfer of the SMB technology to the separation of fine chemicals is not immediate. Indeed, the conditions (characteristics of the phases, interactions, etc.) are very different. Moreover, product purity is also subject

to tight constraints imposed by the pharmaceutical and food regulatory organisations. On the other hand, the optimal operating conditions, which, by definition, are achieved if the required purities are obtained with the highest possible productivity and the smallest possible solvent consumption, are not easy to determine. Most of the methods use a process model and are, thus, subject to modelling errors. Besides, the optimal conditions of the SMB operations are not robust to the changes of temperature, of the feed composition, or of the feed flow rate. Hence, most of the SMB units work at robust but suboptimal conditions. In this way, they satisfy to the specifications most of the time despite the disturbances.

Indeed, in general, there is no closed loop control. Hence, the main issues are the selection of optimal operating conditions and process control to achieve disturbance rejection, problems which require the development of a model of the process. The aim of this work is the study and the development of SMB process models and the estimation of model parameters.

The SMB models consist of mass balance equations in the liquid and in solid phases for the components to separate. Several SMB process models have been developed in the literature. They essentially differ in the description of mass transfer resistance and diffusion and in the representation of the counter-current movement of the solid phase. The first task of this work is the analysis of the existing models to compare their ability to reproduce SMB concentration profiles and their complexity in terms of the number of parameters to estimate.

In general, large discrepancies are observed between the experimental profiles and the simulated ones. Two weaknesses are mentioned to explain this problem. First, the model parameters are often roughly estimated from few experiments or modified heuristically to minimize the difference between both profiles. Secondly, the introduction of the dead volumes in the SMB models is not often considered. Hence, there is obviously a need for a systematic estimation procedure of the parameters of SMB process models and for an effective modelling of the dead volumes.

Besides, models with a low computational load, easy to use in optimization, monitoring and control, do not show the cyclic behaviour of the process.

Hence, based on the results of this preliminary study, the rest of the work aims at finding solutions to the weak points of the existing models.

Two kinds of models, missing until now, are developed:

- a simplified model showing the cyclic behaviour of the process for use in optimization, monitoring and control, tasks which require a model with a low computational load as they are performed on-line.

- a very precise model of the plant, including the dead volumes, to reproduce with details the working of the SMB. This kind of model should be useful for example, to understand with details the influence of each disturbance on the process. Such a model may also be used to generate fictitious data to develop and test the methods to optimize, control and monitor the SMB processes.

Moreover, a systematic estimation procedure of the parameters of a SMB process model is needed. Typically, in the literature, all these parameters are determined from batch experiments, performed on analytical columns or on SMB columns.

Most of the methods described in the literature suffer from a number of drawbacks (assumptions not verified, large amount of products needed, etc) and the results of the use of parameters estimated from batch experiments in SMB models are, in general, not conclusive. Besides, no systematic method for the estimation of the errors on the estimated parameters is reported. The aim of the third part of the work is to develop an identification method for determining, with good accuracy, from batch experiments, the isotherm parameters as well as the mass transfer coefficients and/or the diffusion coefficients for use in a SMB model. Distinctive features of the present study are the following:

- The chromatographic model is chosen within a class of three classical models thanks to a systematic comparison of the identifiability and of the computational load of the three model types.

- Parameter identifiability is thoroughly investigated, including parameter sensitivity, experiment design and the influence of local minima on the optimization problem.

- Confidence intervals are provided for each of the estimated parameters and confidence envelopes are computed for the simulated elution peaks.

The developed approach will be validated with experimental data.

In order to present the above described themes, this work is divided into three parts.

The 1^st part is a review of the literature concerning the chromatography, the SMB process and its modelling. The second part presents new modelling approaches: a simplified model and the modelling of the extra dead-volumes. Finally, the third part is dedicated to parameter estimation.

In particular, the first section begins with Chapter 2 which gives an introduction to the concept of chromatography and a description of the SMB process in general, and of the experimental plant in particular. Then, Chapter 3 provides a review of the literature on the modelling of chromatographic columns and SMB processes. A short introduction to the optimization of the operating conditions is also included. Besides an introduction to the methods used for numerical simulation of SMB models is presented.

The second part of the work is devoted to the presentation of our contributions in the modelling of SMB processes. Chapter 4 presents a simplified modelling approach, whereas Chapter 5 proposes a new strategy to model the dead volumes in a SMB process.

Finally, the third part of this thesis is dedicated to the development of the parameter estimation procedure. Chapter 6 gives some theoretical backgrounds about the direct and inverse method used in the following chapters. Chapter 7 presents the development of the identification procedure and the verification of its effectiveness on fictitious data generated from a batch model with known parameters. Finally, the

validation on SMB measurements is presented in Chapter 8.

PART I:

GENERAL CONCEPTS

CHAPTER 2 :

An introduction to chromatography and SMB processes

2.1. Introduction

Chromatographic separation is based on the differential adsorption of the components of a mixture on an adsorbent. In the case of liquid chromatography considered in this work, the components to separate are in the liquid mobile phase and the adsorbent is a packed bed of solid particles. Originally, chromatography is a discontinuous process (batch). However, several continuous processes like the SMB process have been developed. In this chapter, the principles of the batch chromatographic process and of the SMB process are first introduced. Then the SMB unit of the Max Planck Institute of Magdeburg, where experiments were performed, is described.

Note that, in this study, only binary mixtures are considered. The more retained component will be labelled B and the less retained component A.

2.2. Batch chromatography

Batch chromatography is performed on a column filled with porous solid. Figure 2-1 shows a chromatographic column at 6 stages of the separation process. A small volume of the mixture with components A and B to separate is introduced at time t = 0. The component A (in blue) has less affinity for the solid phase and migrates faster. The component B (in pink) is more adsorbed by the stationary phase and is withdrawn at the end of the column after component A. In Figure 2-2, the evolution of the concentration peaks along the column is given.

Figure 2-1: Schematic representation of a batch chromatographic separation

Figure 2-2: Concentration profiles in a batch chromatographic column at time t1, t2 and t4 defined in Figure 2-1; simulation parameters given in Appendix 3.4

This batch process is relatively simple to implement and to use. However it suffers from the following drawbacks: high solvent consumption, large dilution of products and low productivity. Continuous counter-current chromatographic processes, as illustrated in Figure 2-3, should alleviate these problems. They especially increase the exchange capabilities.

However, a real counter-current movement is difficult to realise in practice. Hence, simulated moving bed chromatographic processes (SMB), where there is no real solid movement but a simulated counter-current, have been developed. The process is described in details in the following section.

c_A,Smax = c_B,Smax

Mélange

t=t₁ t=t₂ t=t₃ t=t₄ t=t₅

0 z (m) 0.1

t1

t₂

t₄

c (vol%)

Figure 2-3: Schematic representation of a real counter-current chromatographic column (Lehoucq, 1999)

2.3. SMB process: general introduction 2.3.1. Description

This process is constituted of chromatographic columns connected in series. The counter-current movement is usually achieved by periodically switching the inlet and outlet ports in the direction of the liquid flow, as shown in Figure 2-4. Note that, in some small-scale processes, the input and output ports are fixed and the counter- current movement is obtained by rotating the columns in the counter-current direction of the fluid flow (Figure 2-5). In both cases, during the operation, the columns successively occupy different relative positions with respect to the position of the input and output ports. The role of each input and output port is explained in the next paragraph.

The SMB processes considered in this study are constituted of 4 zones delimited by the inlet and outlet ports. To understand the role of each zone, an equivalent counter- current representation of a SMB process for the separation of components A and B is given in Figure 2-6. The mixture to separate is fed between zone II and III. The raffinate which mostly consists of the less adsorbed component is withdrawn between zone III and IV, and the extract which mostly contains the more retained component is obtained between zone I and II. After being regenerated in section IV, the liquid phase is recycled to section I together with fresh solvent which is introduced between zone I and IV. This solvent is called desorbent because it helps to regenerate the solid phase.

Note that, in the following, the start-up of the plant coincides with the beginning of the injection of a continuous feed flow in the process filled with solvent.

B A

Fluid phase

A + B (Feed)

Solid phase

Figure 2-4: Schematic view of the simulated moving bed chromatographic process with switching of the inlet and outlet ports (Lehoucq, 1999)

Figure 2-5: Schematic view of the simulated moving bed chromatographic process using a rotative valve (column rotation)

n^th switching

period

Solvent Raffinate Extract A

B

Solvent Feed

A+B

Feed A+B Raffinate

A

Direction of the fluid flow Extract

B

(n+1)^th switching

period

Solvent

Extract B

Raffinate A

Fluid flow

Feed A+B Switching

I

I I

I I I I V

D e s o r b e n t D E x t r a c t E

F e e d F R a f f i n a t e R

s o l i d l i q u i d

A

B

A B

S + A S + A + B

S + B S

D D + B

D + A D + A + B

D + B

D + A + B D + A

D

B B

A A

Figure 2-6: Equivalent counter-current representation of a SMB process for the separation of components A and B (Haag et al., 2001)

In zone III, just after the feed injection, component B is preferentially adsorbed. The solid entering in this zone contains a small amount of component A which is desorbed. At the end of zone III, the liquid consists essentially of component A which is partially withdrawn as raffinate. The solid entering zone IV has been regenerated in zone I. Hence component A and the small amount of B are adsorbed.

The solid at the beginning of zone I mostly contains component B. In contact with the regenerated fluid coming from zone IV, B is desorbed and withdrawn as extract at the end of zone I. The solid entering zone II has been in contact with the feed and contains both components. Given their affinity for the solid phase, component A is desorbed and component B is adsorbed. In summary, in zone II and III, the separation is performed. In zone I, the adsorbent is regenerated and zone IV regenerates the solvent.

Because of the periodical switching of the ports, the SMB process is a periodic process. Figure 2-7 gives spatial concentration profiles. By convention, these profiles are represented with the desorbent input at the origin of the spatial axis.

Profiles are here given at steady state just after the switch, at 50% of a switching period, just before the next switch and just after the next switching. The concentration profiles move along the z-axis in function of the time elapsed since the last switch. At each switching time, the profiles jump back one column behind.

As seen in Figure 2-9 (a) and (b), the extract and raffinate concentrations are periodical signals. This period is equal to the switching period.

The following section is dedicated to the definition of the operating conditions.

Figure 2-7: SMB spatial concentration profiles with t, the time elapsed since the start-up of the plant, N, an integer, ∆∆∆∆t, the switching period, and δδδδ, a period of time with δδδδ << ∆∆∆∆t; _ _ component B, __ component A; simulation parameters given in Appendix 3.4.

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

t = N∆t + δ 0.7

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

t = N∆t+0,5∆t 0.7

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 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

z (m) Desorbant

Extract B

Feed

A + B Raffinate A Zone II

Zone I Zone III Zone IV

t = (N+1) ∆t + δ t = (N+ 1) ∆t - δ

2.3.2.Operating parameters

Besides the process characteristics like the column length, the column diameter, the number of columns per region, the bed particle size and distribution, the working conditions are defined by the choice of the switching time, the feed concentration, and 4 of the 8 flow rates. There are 4 internal flow rates (one per zone), Q_j,

IV ,..., I

j= , and 4 external flow rates (the feed flow rate, Q_F, the raffinate flow rate, Q_R, the extract flow rate, Q_E, and the desorbent flow rate, Q_D). In a classic SMB process, all these flow rates are constant during the operation. They are linked by the mass balance equations:

R E F

D Q Q Q

Q + = +

D IV

I Q Q

Q = +

E I

II

Q Q

Q = −

F II

III

Q Q

Q = +

R III

IV

Q Q

Q = −

2-1

The operating conditions are usually determined in order to - reduce solvent consumption;

- increase the productivity, PR, which is defined as: Q c^{F F}

PR= W with c_F, the feed concentration and W, the adsorbent weight;

- obtain the desired purities in the extract and in the raffinate.

Figure 2-9: a) extract concentration profile in function of time

60000 7000 8000

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

time (s)

CB (vol%)

60000 6500 7000 7500 8000

0.1 0.2 0.3 0.4 0.5 0.6 0.7

time (s)

CA (vol%)

Figure 2-9: b) raffinate concentration profile in function of time

Note that constraints have to be taken into account like the maximum acceptable pressure drop and the solubility limits of the products.

A lot of methods have been developed in the literature to determine the required operating conditions (Migliorini et al., 1998 ; Klatt et al., 2002; Biressi et al., 2000;

Jupke et al., 2002; Beste et al., 2000 ; Azedo et al., 1999; Pais et al., 1998^a ; Kleinert et al., 2002; Schramm et al, 2003^a; Abel et al., 2004 and 2005; Garcia et al. 2006).

One well-known method, the triangle theory (Mazzoti et al., 1997) is presented in section 3.5.

2.4. Description of the studied SMB plant

Figure 2-10: Pictures of the preparative SMB unit (CSEP C912, Knauer, Berlin, Germany) located at the Max Planck Institute of Magdeburg (Germany)

In this study, experiments were conducted in the Max-Planck-Institut Dynamik Komplexer Technisher Systeme in Magdeburg (Germany) on a preparative SMB unit (CSEP C912, Knauer, Berlin, Germany). Figure 2-10 shows pictures of the SMB plant and Figure 2-11 gives a schematic representation of this unit. In this process, the counter-current movement of the solid and of the liquid phases is achieved by switching the columns, thanks to a multi-function valve. This valve consists of a rotor and a stator with 24 ports each. The ports are connected to each other by continuous channels. Hence, in Figure 2-11, all the devices inside the inner circle move during the switching, whereas, the rest is fixed. Note that this SMB

plant is built for up to 12 columns but only 8 columns are introduced in the process used at MPI. Hence, as described in (Knauer, 2000), the free ports are connected by short capillaries and the valve switches alternatively one and two times successively during a full cycle (which is equal to 8 switching periods).

The process is equipped with two inlet pumps, one on the feed flow (P4), and another on desorbent flow (P3). Two other pumps are located in the circulating stream (P1 and P2).

Besides, this SMB process is also equipped with four UV detectors, two in the circulating stream (UV3 and UV4) (which move with the columns) and two on the product outlets (UV1 and UV2).

Figure 2-11: Schematic representation of the Knauer CSEP C912 unit (Max Planck Institute, Magdeburg, Germany) with 8 columns

Desorbent

Extract

Raffinate

UV 3

UV 4

Fluid flow Switching

UV 1

UV 2

P4

P1 P3

P2

Zone I

Zone II

Zone III Zone IV

Feed

2.5. Modifications of the SMB process

Note that several variants of the SMB process have been developed to improve the productivity. Some use a modulation of the feed concentration like the Modicon (Schramm et al, 2003^b and 2003^c). Others use variable internal and external flow rates like the Powerfeed (Zhang et al., 2003) or variations in the feed flow rate (Zang et al., 2002). In the Varicol (Ludemann-Hombourger et al., 2000; Ludemann- Hombourger et al., 2002), asynchronous switching is performed. Asynchronous switching of the different ports may also be used to compensate dead volumes (Hotier et al., 1996). As this study focuses on the classical SMB process, these variants are not described in details here.

CHAPTER 3 :

Modelling of chromatographic processes

3.1. Introduction

Chromatographic models consist of the mass balance equations of the components in the solid and liquid phases. In this section, the equilibrium isotherms which characterize the distribution of the solute between both phases are first introduced.

Then modelling of batch experiments is described. SMB modelling follows. The chapter ends with a description of the model parameters and a short introduction to the numerical simulation of these models.

3.2. Equilibrium isotherm

The adsorption isotherms characterize the equilibrium between the adsorbed concentration and the concentrations in the liquid phase at constant temperature.

At very low concentration, the adsorption is easy and one molecule has no influence on the other. The isotherms are then approximated by a linear function:

q_i^eq = H_i c_i 3-1

where i = A, B, H_i and q_i^eq are respectively the Henry coefficient and the adsorbed equilibrium concentration of component i. c_i is the concentration of component i in the fluid phase.

At higher concentration, as the capacity of the adsorbent is limited and many molecules are in competition, the adsorption is more difficult. Hence, the isotherms

are non-linear and are a function of the concentrations of the two components in the liquid phase. Figure 3-1 shows an example of non-linear competitive isotherms.

Many multicomponent non-linear isotherm equations have been described (Quinones et al., 1998; Guiochon, 1994). As it will be explained in Chapter 8, the Langmuir isotherm has been chosen in this study. This widely used model relies on the assumption that the stationary phase is composed of a fixed number of adsorption sites of equal energy, one molecule being adsorbed per adsorption site until monolayer coverage is achieved. The isotherm equation for component i is the following:

+ ∑

∑ =

= +

=

= i A,Bi i i i B

, A i i i

i i Si eq

i 1 b c

c H c

b 1

c b

q q ^3-2

q_Si is the column saturation capacity of component i, b_i, the ratio of the rate constants of adsorption and desorption and H_i, the Henry coefficient. Note that at low concentration, the term 









 ∑

=A,B

i bici is negligible and the isotherms become linear.

Figure 3-1: Competitive multi-components isotherms (c = c_A = c_B); __ less adsorbed component, _ _ more adsorbed component (Langmuir isotherms)

0 1 2 3 4 5 6 7 8 9 10

0 1 2 3 4 5 6 7

c (vol%)

qeq (vol%)

3.3. Modelling of a column and introduction to the wave theory

In this section, the modelling of a batch chromatographic column is presented and the wave theory is introduced. The latter is useful to understand a lot of phenomena in chromatography like the movement of the concentration profiles.

3.3.1. Column model

Hereafter several batch chromatographic models are presented. They differ in the assumptions used to build the mass balance equations (Guiochon et al., 1994;

Guiochon, 2002). As the aim of this study is the modelling and the parameter estimation of SMB processes, the column models described here will be compared further, in section 3.4.3, when they will be introduced in a SMB model. The modelling of the injected concentration profile is also described.

The following assumptions are used to derive the model equations:

- The column is assumed to be radially homogeneous;

- Compressibility of the mobile phase is negligible;

- Isothermal operating conditions are considered;

- Only components A and B are adsorbed.

3.3.1.1. Equilibrium stage model

In the equilibrium stage model, the bed elements are represented by a cascade of N mixing elements where equilibrium is achieved. Each element is called a theoretical plate.

Figure 3-2: Theoretical plates, p^th plate plate p+1 plate p-1

plate p p

ci h

1 p

ci⁺

The differential mass balance equation in the bulk mobile phase states that the difference between the amount of component i that enters plate p and the amount of component i that exits plate p is equal to the amount accumulated in plate p.

The input flow of component i is: εSvc_i^p+1

The output flow of component i is given by: εSvc^p_i

And the accumulation of component i is written: ) dt )dq 1 dt ( ( dc hS

p i p

i + −ε ε

with B , A

i= ; p =1, .., N;

P

ci , the concentration in the liquid phase of component i in the p^th plate;

t, the time;

v, the velocity of the fluid flow, ε, the porosity;

h, the height equivalent to a theoretical plate (HETP):

N h= L ; L, the column length;

S: the column geometric cross-sectional area.

Hence, the following equations describe the mass balances of the p^th plate in the fluid phase:

dt ) 1 dq dt (dc h vc vc

p i p

p i i 1 p

i ε

ε + − +

+ = 3-3

As equilibrium is achieved in plate p, the mass balance equation in the solid phase is described by:

peq i p

i q

q = ^3-4

with q_i^peq, the adsorbed concentration of component i in equilibrium with )

c , (c_{A,p B,p} .

At the origin, this model has been developed following an approach used in chemical engineering which transforms the distributed parameter system into a system with mixing elements in order to avoid the partial differential equation formalism. However, this model implies that the diffusion phenomenon is identical for both components which is not realistic (Dunnebier et al., 1998).

3.3.1.2. Linear Driving Force model (LDF) (Guiochon, 1994)

In this model, a continuous flow of the liquid phase is taken into account. The axial dispersion is considered in the fluid phase and the intraparticle mass transfer rate is modelled by a linear driving force.

The derivation of the LDF model is explained hereafter. The models described in the following sections have been built similarly.

The differential mass balance for component i in the bulk mobile phase states that the difference between the amount of component i that enters the slice of column with thickness ∆z during time ∆t and the amount of component i which exits the slice is equal to the amount accumulated in the slice (Figure 3-3).

Figure 3-3 : Chromatographic column m

The input flow of component i is :

t , z i Li

i )

z D c vc (

S ∂

− ∂

ε 3-5

The output flow is :

t , z z i Li

i )

z D c vc ( S

∆

∂ +

− ∂

ε 3-6

The accumulation term of component i in the slice of volume S∆z is:

t , zmean i

i )

t ) q 1 t ( ( c z

S ∂

ε ∂

−

∂ + ε∂

∆ ^3-7

where q_i is the concentration of component i in the solid phase and z, the axial coordinate. D_L,i is the axial dispersion coefficient for component i. S is the cross- section area.

If ∆z tends to zero, the following equation is obtained:

z

ε

z z+∆z

z v c z D c t 1 q t

c _i

2 i 2 Li i i

∂

− ∂

∂

= ∂

∂

∂ ε

ε + −

∂

∂ 3-8

The mass balance in the solid phase is written as:

) q q ( t k q

i eq i i ,

i = F −

∂

∂ 3-9

with k_F,i, the mass transfer coefficient.

3.3.1.3. Equilibrium dispersive model (ED) (Guiochon, 1994)

In the equilibrium-dispersive model, the effects of mass transfer resistance and of diffusion are lumped in an apparent diffusion coefficient D_app. Hence, equilibrium is assumed between the solid and the fluid phase. The mass balance equation in the fluid phase is written as:

z v c z D c t 1 q t

c _i

2 i 2 i , app i i

∂

− ∂

∂

= ∂

∂

∂ ε

ε + −

∂

∂ 3-10

where q_i =q^eq_i and

i i

i ,

app 2N

Lv 2

v

D = h = ^3-11

with h_i, the length equivalent to a theoretical plate and N_i, the number of theoretical plates.

In the case of a linear isotherm, the following relation has been obtained between the apparent diffusion coefficient of the ED model and the diffusion and mass transfer coefficients of the LDF model:

(

)

',i 0 i , L i ,

app v

k 1 k 1 D k D

+ +

= 3-12

with ^'_0,i 1 H_i

k ε

ε

= − ^.

H_i is the Henry coefficient (see Eq.3-1).

According to (Guiochon, 1994), this model is correct if the mass transfer is only controlled by diffusion and if the mass transfer is very fast.

3.3.1.4. Kinetic model

All the non-ideal effects are lumped into a mass transfer coefficient.

z v c t 1 q t

c_{i i i}

∂

− ∂

∂ =

∂ ε

ε + −

∂

∂ _3-13

) q q ( t k q

i eq i i

i = −

∂

∂ _3-14

k_i, the lumped mass transfer coefficient, is a linear function of the fluid velocity.

Hence, a relative mass transfer coefficient which is independent of the velocity is defined as: k^rel_i =k_i/v^.

3.3.1.5. General rate model

This model takes all the possible contributions to the mass transfer kinetics into account (Guiochon, 1994). As there are different ways to describe these effects, there are many versions of this model. In the SMB community, the axial diffusion, the mass transfer resistance and the pore diffusion are usually introduced in the equations. The solid phase is assumed to be constituted of porous, uniform and spherical particles with porosity εp and local equilibrium is considered within the pores. The concentration of component i in the liquid phase is the solution of the following mass balance equation:

rp r i , p i p

i , ext ext

ext i

2 2 i i , L

i (c c )

R 1 k

z 3 v c z D c t c

=

ε − ε

− −

∂

− ∂

∂

= ∂

∂

∂ 3-15

Inside the pores,

















∂ + ∂

∂

= ∂

∂ ε ∂

−

∂ + ε ∂

r c r 2 r D c t ) q 1 t (

c _p,i

2 i , 2 p i , i p p i

p,

p 3-16

r 0 c

0 r i p,

∂ =

∂

=

3-17

with r, the radial coordinate inside the pore, εext, the external porosity, εp, the internal porosity of the particles, r_p, the particle radius, k_ext, the external mass transfer coefficient, D_p, the diffusion coefficient inside the pores, c_p, the concentration in the stagnant fluid phase in the particle pores.

This is the most rigorous model (Guiochon, 2002). It is convenient for all kinds of isotherms.

3.3.1.6. Ideal Model (Guiochon, 1994)

The ideal model considers that the column has an infinite efficiency. It means that there is no axial dispersion and that the mass transfer between the mobile and the stationary phase is assumed to be instantaneous. The mass balance equation in the fluid phase is the following:

z v c t 1 q t

c_{i i i}

∂

− ∂

∂ =

∂ ε

ε + −

∂

∂ 3-18

with q_i =q^eq_i ^.

This model gives a first estimation of the concentration profiles but cannot accurately reproduce the elution peaks in case of low-efficiency columns. Indeed, in this case, the contributions of the mass transfer kinetics and axial dispersion become significant.

3.3.1.7. Modelling of the injection

In this section, the inlet concentration profiles are described for several classical batch experiments.

Elution peak

Elution peaks result from the injection of a small volume of the solution containing the components to separate. The ideal shape of the inlet concentration profile should be a rectangle but dispersion phenomena affect significantly the profile. In this study, the inlet concentration profile of component i is described as follows:

)) t / ) t t ( exp(

1 ( c )) t / t exp(

1 ( c ) t ( u

else

)) t / t exp(

1 ( c ) t ( u

t t if

tr p i

, F tr i

, F i

tr i

, F i

p

−

−

−

−

−

−

=

−

−

=

<

3-19

with c_F,i, the injected concentration of component i, t, the time, t_p, the injection duration and t_tr, a constant characterizing the rise time of the pulse (Figure 3-4).

Note that this injection profile is close to the one proposed in (Felinger et al., 2003) but it has less parameters.

Figure 3-4: Shape of the inlet concentration profile u; simulation parameters given in Appendix 3.4.

Concentration step

Other batch experiments consist in the injection of abrupt step changes of different concentrations. The injected front should be abrupt but dispersion phenomena affect significantly the profile. Hence the inlet concentration profile of component i is described as follows

)) t / t exp(

1 ( c ) t (

u_i = _F,i − − _tr 3-20

After this complete presentation of batch modelling, the wave theory which has been developed from batch models is presented. This theory notably helps to explain the movement of the peaks in the column.

3.3.2.Wave theory

The wave theory, presented in this section, has been built from the equation of the batch ideal model and helps to understand the behaviour of the concentration profiles in a chromatographic process. The wave theory is also used in Chapter 4 to build a new modelling approach.

Each elution peak resulting from the injection of a rectangular pulse in a column initially free of solute is constituted of two waves (Hellferich and Carr, 1993), one corresponding to the adsorption front, the other to the desorption front.

In a single component system, introducing

t c c q t q

∂

∂

∂

= ∂

∂

∂ into (3-18) gives:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0 0.05 0.1

time (s) u

(vol%)

z 0 v c t c

z =

∂ + ∂

∂

∂ 3-21

with v_z, the concentration velocity equal to :

c q 1 1

v_z v

∂

∂ ε

ε + −

= 3-22

In the case of ideal chromatography, the phases are in local equilibrium, hence, q can be expressed as a unique function of c and the partial derivative becomes a total derivative:

dc 1 dq 1 v_z v

ε ε + −

= 3-23

Hence v_z, the velocity of concentration c, is a function of the slope of the isotherm at concentration c.

Therefore, in the case of an adsorption characterized by a linear isotherm, the velocity is not a function of the concentration. As shown in Figure 3-5 (a) and (b), the shape of the wave does not change during the displacement.

However, in the case of a non-linear isotherm, the slope of the isotherm decreases when the concentration increases. Hence, the velocity is higher for larger concentration. A front of the form represented in Figure 3-5 (d) is then spreading.

Whereas, a front of the form shown in Figure 3-5 (c) tends to sharpen until it becomes a concentration discontinuity to avoid a physically impossible coexistence of different concentrations. It is called a shock. For non-linear isotherms, like the Langmuir isotherms, the final shock pattern is obtained rapidly. The velocity at which the shock moves is given by (Helfferich et al., 1993):

i i i

c 1 q 1 v v

∆

∆ ε

ε + −

∆ =

3-24 where ∆q_i and ∆c_iare the concentration differences between the downstream and upstream sides of the wave.

In conclusion, if the non-linearity increases, the symmetry of the elution peak decreases.