Modeling and numerical simulation of preparative chromatography for industrial applications

UPTEC X09 023

Examensarbete 30 hp Juli 2009

Modeling and numerical simulation of preparative chromatography

for industrial applications

Martin Enmark

Molecular Biotechnology Programme

Uppsala University School of Engineering

UPTEC X 09 023 Date of issue 2009-07

Author

Martin Enmark

Title (English)

Modeling and numerical simulation of preparative chromatography for industrial applications

Title (Swedish) Abstract

A new implementation of a numerical solver of a PDE describing a non-linear chromatographic process was developed and evaluated. Solutions of the algorithm was compared to those of available methods and experimental data.

Results indicate more accurate solutions of this PDE at low column efficiencies, typically found at production processes at Astra Zeneca. Possible implications of algorithm are more realistic solutions to model and therefore a more accurate basis for modeling and optimization of industrial separation processes.

Keywords

Chromatography, modeling, simulation, process Supervisors

Dr. Robert Arnell, Astra Zeneca AB

Scientific reviewer

Patrik Forssén, SweCrown AB

Project name Sponsors

Language

English ^Security

ISSN 1401-2138 Classification

Supplementary bibliographical information

Pages

43

Biology Education Centre Biomedical Center Husargatan 3 Uppsala

Box 592 S-75124 Uppsala Tel +46 (0)18 4710000 Fax +46 (0)18 555217

Modeling and numerical simulation of preparative chromatography for industrial applications

Martin Enmark

Sammanfattning

Preparativ v¨atskekromatografi (HPLC) ¨ar en v¨alanv¨and separationsmetod inom biotekniken. Den anv¨ands bl.a. p˚a Astra Zeneca f¨or att i stor skala rena fram specifika enantiomerer fr˚an race- mat. Ingen separationsprocess ¨ar den andra lik och betingelser s˚asom eluent och l¨ampliga sta- tion¨arfaser m˚aste hittas f¨or varje produktion. I dessa produktioner vill man best¨amma optimala experimentella betingelser, d.v.s. optimera f¨or utbyte, kostnad och t.ex. ˚atg˚ang av milj¨ofarliga l¨osningsmedel.

I syfte att modellera och optimera kan man antingen i) genomf¨ora ett empiriskt ”trial-and- error” f¨orfarande och med erfaranhet variera experimentella betingelser ii) variera experimentella betingelser genom att variera motsvarande parametrar i numerisk modell.

En befintlig algoritm till en specifik modell av preparativ kromatografi finns implementerad och har under detta examensarbete f¨or f¨orsta g˚angen anv¨ants med framg˚ang. Dock ¨ar l¨osningarna erh˚allna med denna algoritm ifr˚agasatta d˚a man simulerar vid vissa betingelser.

Detta examensarbete syftar till att implementera en ny algoritm med h¨ogre nogrannhet f¨or att kunna f¨ors¨akra sig om att de erh˚allna l¨osningarna alltid konvergerar mot de f¨or modellen verkliga dito. Ett annat syfte ¨ar att generellt p˚apeka m¨ojligheterna med numeriska simuleringar framf¨or andra metoder.

Examensarbete 30hp

Civilingenj¨orsprogrammet i Molekyl¨ar Bioteknik

Uppsala Universtet juli 2009

UPPSALA UNIVERSITY

Process Research and Development Astra Zeneca AB S¨ odert¨ alje Department of Physical and Analytical Chemistry, Surface Biotechnology

Master of Molecular Biotechnology programme Degree project 30 credits

July 4, 2009

Modeling and numerical simulation of preparative chromatography for industrial applications

Martin Enmark Degree project 30 credits

Master of Molecular Biotechnology programme Uppsala University

Supervisor: Dr. Robert Arnell Astra Zeneca

Contents

Abbreviations 1

1 Introduction 2

1.1 Purpose of Study . . . . 4

2 Chromatographic theory 5 2.1 The Chromatogram . . . . 5

2.2 The Chromatographic Process . . . . 6

2.3 Adsorption Isotherms . . . . 7

3 Industrial & Economical Perspective 9 3.1 Concepts and Definitions . . . . 9

3.2 Design of a Separation Process at Astra Zeneca . . . 12

4 Modeling 13 4.1 The Equilibrium Dispersive Model . . . 14

4.1.1 Numerical Solutions . . . 15

5 Determination of Adsorption Isotherms 17 5.1 The Inverse Method . . . 17

6 Methods and Materials 19 6.1 Implementation of Algorithm . . . 19

6.1.1 Accuracy and Validity . . . 21

6.1.2 Stability . . . 21

6.1.3 Performance . . . 22

6.2 Synthetic Algorithm Comparisons . . . 22

6.2.1 Linear Conditions . . . 22

6.2.2 Nonlinear Conditions . . . 22

6.3 Separation of Methyl Mandelate on a Chiral Protein Column . . . 24

6.3.1 Determination of Adsorption Isotherm Parameters . . . 24

6.4 Substances in Production at Astra Zeneca . . . 25

6.4.1 N1 . . . 25

6.4.2 N2 . . . 25

6.4.3 N3 . . . 26

6.4.4 Determination of Adsorption Isotherm Parameters . . . 26

6.5 Methyl-mandelate on a Chiral Cellulose Column . . . 27

6.5.1 Determination of Adsorption Isotherm Parameters . . . 27

6.6 Synthetic Determination of Isotherm Parameters . . . 28

7 Results 29 7.1 Algorithm Implementation . . . 29

7.1.1 Validity, Stability and Performance . . . 29

7.1.2 Linear Comparison . . . 30

7.1.3 Nonlinear Comparisons . . . 32

7.1.4 Synthetic Determination of Isotherm Parameters . . . 33

7.2 Methyl Mandelate on AGP Column . . . 34

7.3 Methyl Mandelate on Chiral Cellulose Column . . . 35

7.4 N1 . . . 36

7.5 N2 . . . 37

7.6 N3 . . . 38

8 Conclusions and Discussion 41

9 Acknowledgements 42

References 43

Abbreviations

[a

₁

a

₂

. . . a

_n

] Parameter a of component n D

_a

Axial dispersion constant ED-model Equilibrium-Dispersive model



t

Total porosity

F Volumetric phase ratio

HPLC High Performance Liquid Chromatography

N Number of theoretical plates / plate number / column efficiency OCFE Orthogonal Collocation on Finite Elements

PFD Precise Finite Difference RFD Rouchon Finite Difference t

₀

Column hold up time [s]

t

_r

Retention time [s]

t

_sys

Pre-column injection time [s]

u Linear flow rate [m/s]

V

_vol

Volumetric flow rate [l/s]

1 Introduction

Liquid chromatography (LC) is a conceptually straight-forward chemical separation method, yet many of its possible modes of application can be considerably complex [Schmidt-Traub, 2005].

A classical definition of the technique is:

”A separation process that is achieved by the distribution of the substances to be separated between two phases, a stationary phase and a mobile phase.

Those solutes distributed preferentially in the mobile phase, will move more rapidly through the system than those distributed preferentially in the stationary phase. Thus, the solutes will elute in order of their increasing distribution coefficients with respect to the stationary phase.” [Cazes and Scott, 2002]

The methods of LC was initially published in 1903 by the Russian botanist M.S Tswett. In his ground breaking article he describes separation of α- and β-carotenes (solutes) by using inulin (plant fibres) as stationary phase (adsorbent) and a ligroin (a petroleum distillation product) as eluent (mobile phase) [Guiochon et al., 2006].

The works made by Tswett were per definition in the realm of preparative chromatography, where the focus is collection of solutes rather than characterization of them as is the case in analytical chromatography. Nevertheless, chromatography matured as an analytical technique. The initial development and realization of Twsett’s technique went rather slow and it wasn’t until the 1940

’re-discovery’ rapid development in techniques and theory took place. By the late 1960’s, pumped flow High Performance LC (HPLC) using column packed with small particles, coupled to UV detection became available, essentially the basics of todays technology. [Cox, 2005]

The notion of different adsorption-desorption equilibrium as the ”driving force” behind separation has been known for at least a century, yet a general theory enabling predictable modeling has not been found. However, many applicable models have been found and are today employed with great success [Guiochon et al., 2006]. All kinds of applications of chromatography can benefit from insights drawn from simulations using these models.

Common for the mass balance equations of these mathematical models of chromatography is the lack of analytical solutions. This has necessitated the development of numerical solutions.

This project will focus on the applications of non-linear

¹

chromatography where the purpose is to collect eluted components.

In an industrial viewpoint, modeling and numerical simulation of production processes should be a very interesting approach compared a traditional, empirical approach in order to better undstand the process. Because the experimental space that can be spanned by synthetic experiments vastly expands that of the empirical ”trial-and-error” approach, in order to seek to optimize factors such as purity, productivity, eluent consumption, production rate etc., there is a great economical po- tential.

Numerical solutions of any mathematical model, particularly based on partial differential equa- tions should when applied converge to the true, analytical solution and hence accurately describe the physical meaning of the model as long as it is valid.

The work as measured by CPU-time needed to converge to this true solution may grow exponen- tially and as CPU-capacity is finite, one must also consider the calculation time, but at the same time bearing in mind that CPU-time is cheap compared to that of manual labour.

1Terms such as ”non-linear”, ”overloaded” and ”preparative” will be used interchangeably throughout this paper

In the end, from an industrial viewpoint, the total time from development of a process to the execution/delivery is the most important, regardless of which method is applied.

Earlier work with a particular numerical solution of a model of non-linear chromatography have been successful. However, recently there have been some discoveries when the algorithm seem to diverge from the true solution. This project will investigate solutions with this algorithm, imple- ment another, more precise numerical approximation and evaluate its solutions. Both synthetic, small lab scale experiments performed at the university and large scale Astra Zeneca production case studies will be investigated.

Figure 1: A crude schematic set-up of a typical LC separation. A sample container of two com-

ponents that are to be separated is connected via a valve to an eluent container. The column

outlet is connected to an UV-detector. As indicated, the modeling in this project concerns the

thermodynamic adsorption process occuring inside the column.

1.1 Purpose of Study

• Study and understand the Equilibrium-Dispersive (ED) model of chromatography. Briefly study analytical solutions but focus on numerical solutions of non-linear chromatography.

• Study the current implementation of simulation of 2-components only, obeying the two- component competitive Bi-Langmuir model in a separation process governed by the ED- model.

• Expand and rewrite the entire current implementation to adapt for simulations of the N-component problem obeying the n-component competitive Langmuir and Bi-Langmuir model. Construct a MATLAB text-based user-interface and and Fortran implementation of the actual PDE solver.

• If programming is successful, try to port the proprietary MATLAB-MEX(-FORTRAN) im- plementation to an free software solution of Octave-Fortran.

• Implement the algorithm as a solver in the current implementation of the inverse method algorithm for determination of adsorption isotherm parameters.

• Use the inverse solver to determine parameters from i) performed experiments at UU and AZ ii) synthetic data generated by both methods.

• Participate in a real production at Astra Zeneca, using the modeling approach in small scale columns then scaling up and performing the actual separation.

• Obtain experimental data from earlier Astra Zeneca productions or potential productions and use the new algorithm in concordance with the inverse solver algorithm to compare the optimized adsorption isotherm parameters. Furthermore, try to exemplify and quantify the meaning of any difference in the obtained parameters. A concrete application would be the accuracy in determining the cut point in a large-scale production process.

• Design/replicate and perform at least one practical experiment to i) gain insight and ex- perience chromatography to alleviate any possible unnecessary future puzzlements when performing simulations ii) determine isotherm parameters for validation of the implemented algorithm.

• Benchmark the implementation to validate accuracy and stability. Perform synthetic studies comparing the earlier implementation of numerical solver of the ED-model. Try to draw con- clusions about when obtained solutions diverge and the potential consequences for modeling

• Obtain reference solutions to the PDE by method considered most accurate, the orthogonal collocation on finite element (OCFE) method.

• For the special interest of Astra Zeneca PR&D, try to i) motivate the use of simulations

in the first place, ii) motivate the use of the inverse solver for determination of adsorption

isotherm parameters iii) attract the attention and questioning the solutions obtained with

RFD at low column efficiency.

2 Chromatographic theory

2.1 The Chromatogram

The outcome of every chromatographic operation can be visualized by the chromatogram. At linear conditions, when the adsorbed amount of solute is proportional to the amount of solute in mobile phase, we observe the linear chromatogram. This means that an injection of k components will result in k Gaussian peaks, each with a standard deviation depending on the theoretical column efficiency for the particular solute. Another term for column efficiency is number of theoretical

Figure 2: A chromatogram of a two-component separation at non-overloaded linear conditions.

The time Component at t

0

isn’t retained in the column and is used to determine the total column porosity 

t

.

plates, N.

N, or the variance per column length is determined by the van Deemter equation for a particular column and stationary phase [Harris, 2003]:

N = L

H ≈ L

A +

^B_u

+ Cu (2.1)

Where H is the height of a theoretical plate, L the column length, A the variance contribution of multiple paths, B the contribution of longitudinal diffusion, C the contribution from the finite equilibration time for a solute between the mobile and stationary phase and u the linear flow rate.

N can be experimentally determined from the following relation.

N = 5.545 t

²_r

w

²_1/2

(2.2)

Where t

_r

is the retention time of a Gaussian peak, w

_1/2

the width of half peak maximum of the same Gaussian peak (See Fig. 2).

A factor which affects N is the column total porosity 

t

. It’s defined as the ratio between the total volume in which solutes can traverse the column to the geometrical volume of the column.

As 

t

increases, so does N. Because decreasing N increases the variance of the Gaussian peak (See Fig. 2) it simultaneously decreases the chromatographic resolution R

s

. This is a measure of how well two peaks are separated.

Measures such as retention times, efficiency and resolution are well defined for the linear analytical chromatogram but diffuse for the non-linear realm of chromatography (see Fig. 3). Here, the adsorbed amount of solute is no longer proportional to the amount of solute in mobile phase.

The appearance of the peaks is still characterized by the column efficiency, but there is no defined resolution or retention times [Schmidt-Traub, 2005]. The most important contribution to the band profiles here are the thermodynamic effects of the adsorption isotherm, discussed in more detail in section 2.3.

Figure 3: A chromatogram of a two-component separation at overloaded non-linear conditions.

The retention time are described for the fronts of the eluted solutes.

2.2 The Chromatographic Process

The chromatographic process for a typical liquid phase system is at the fundamental level a simple

process of solute adsorption and desorption. A general discussion is applicable for almost all scales

of chromatography.

A column is packed with a suitable stationary phase. This is often beads with different chemical composition. It could be inorganic such as active carbon, zeolites, silica or alumina or organic such as cross-linked organic polymers (cellulose, peptides or proteins). [Schmidt-Traub, 2005].

Depending on the sought column efficiency N, column dimensions, flow rate etc, the diameter of the beads are chosen appropriately. Based on the nature of adsorbent, the solubility of the solutes an appropriate mobile phase is chosen. It can be a simple composition of organic or inorganic solvents or a more complex composition.

A typical separation process

²

begins with the column being equilibrated under constant temper- ature with solvent also kept at constant temperature. The outlet of the column is continuously monitored typically for stable UV-absorbance.

Considering batch-chromatography only, an injection of solutes dissolved in mobile phase is per- formed, either by an injector or by an pump. After the solutes have percolated through the column they are detected by the outlet, typically but not limited to UV or MS.

Figure 4: The principle of adsorption chromatography. A mixture of components that are to be separated are injected into a column and are continuously separated along their way by basis of the equilibrium between their affinity for the stationary phase and mobile phase respectively. Here, the circular red component has higher affinity for the adsorbent and will be retained longer.

2.3 Adsorption Isotherms

To quantify the phenomena of adsorption equilibrium inside the column, hence determine the ap- pearance of both the linear and non-linear chromatogram we must define the adsorption isotherms.

They describe the adsorption-desorption equilibrium between the solute and the stationary phase and have the general form:

q = f(C)

Where q and C is the concentrations of a solute in the stationary and mobile phase respectively.

As will be explained later, knowledge of q is fundamental regardless of which model one chooses to describe the actual chromatographic process [Guiochon et al., 2006].

The adsorption isotherm can have a varying level of complexity. If considering more than one component it can describe a process which assumes i) no adsorbate-adsorbate interactions and

2Every process considered in this project

only one adsorbate-adsorbent interaction ii) no adsorbate-adsorbate interactions but with more than one adsorbate-adsorbent interaction iii) adsorbate-adsorbate interactions exists but only one adsorbent-adsorbent interaction iv) adsorbate-adsorbent interactions with many adsorbate-sorbent interactions. In this project I will predominantly use isotherms of type i and ii.

Lets begin with breaking down the most simple case of them all, an adsorption isotherm of type i of a single component injected onto a column:

q = f(C) = αC 1 ± βC At linear chromatography we observe:

C→0

lim αC

1 ± βC = αC

At non-linear chromatography we will observe either a convex (1 + βC) or concave (1 − βC) type adsorption isotherm. The convex variant is the very common Langmuir adsorption isotherm.

Lets expand the previous discussion to two component, which doesn’t interact with each other but compete to adsorb to the sorbent, that is still type i.

q

₁

= f(C

₁

, C

₂

) = α

₁

C

₁

1 ± β

₁

C

₁

± β

₂

C

₂

q

2

= f(C

2

, C

1

) = α

2

C

2

1 ± β

1

C

1

± β

2

C

2

The generalization into the k component competitive Langmuir adsorption isotherm as used in this work.

q

_i

= a

_i

C

_i

1 + P

^k_j=1

b

j

C

j

(2.3) A typical adsorption behaviour often observed when separating chiral substances on chiral adsor- bents is that we have two interaction sites, one corresponding to the chiral interaction, unique for each enantiomer, and another corresponding to the non-chiral interaction, common of both enan- tiomers. The simplest adsorption isotherm describing this kind of behaviour is the Bi-Langmuir isotherm which simply describes the sum of the two interactions with Langmuir terms. A great example of a case when this model is appropriate is the separation of the β-blocker propranolol [Fornstedt et al., 1999]

q

_i

= a

_1,i

C

_i

1 + P

ⁿ_j=1

b

1,j

C

j

+ a

_2,i

C

_i

1 + P

ⁿ_j=1

b

2,j

C

j

(2.4)

A more complex model which takes into account solute-solute interactions and multiple adsorbent- sorbent interactions is quadratic or statistical isotherm model

³

, here described for the binary case:

q

1

= b

₁

C

₁

+ b

₃

C

₁

C

₂

+ 2b

₄

C

₁²

1 + b

1

C

1

+ b

2

C

2

+ b

3

C

1

C

2

+ b

4

C

₁²

+ b

5

C

₂²

(2.5) q

₂

= b

2

C

2

+ b

3

C

1

C

2

+ 2b

5

C

₂²

1 + b

₁

C

₁

+ b

₂

C

₂

+ b

₃

C

₁

C

₂

+ b

₄

C

₁²

+ b

₅

C

₂²

(2.6)

3A ratio of two polynomials which is more of strict mathematical nature than physical, since coefficents lack physical meaning.

3 Industrial & Economical Perspective

One could ask one self which is the main driving force behind the study of the fundamentals of non-linear chromatography. Curiosity most of you might argue, minimizing the cost of production by optimizing the separation process variables others might hastily interject [Guiochon et al., 2006].

Regardlessly, it is a fact that for example the Food and Drug Administration (FDA) has put pressure on for example the pharmaceutical industry about using process control not based on empirical observation (trial-and-error) but instead knowledge about the physio-chemical nature of the processes [Guiochon et al., 2006].

The scope of this section is to introduce concepts drawn from the chromatographic process, the chromatogram and briefly the economics to be able to communicate with any one involved in managing, designing or executing a chromatographic process. The section will only consider sequential column injections i.e. batch chromatography.

3.1 Concepts and Definitions

To simplify the further discussions, lets consider a case example of a separation process. The aim is to separate two enantiomers (1 & 2) with only one constraint being to recover component 1 with 100 % purity.

A ”semi-optimized” process have already been developed and the resulting chromatogram is de- scribed in Fig. 5.

Having introduced the basic ”time stamps” of the chromatogram (Fig. 2) we can introduce another important measure, the cycle time (∆t

cycle

). This is often the minimal time between subsequent injections onto the column. It is typically dynamically adjusted to account for any process variations and column regeneration time by for example using an UV-detection threshold.

The total amount of injected product (1) per cycle is easily calculated:

n

1

= V

inj

C

₁⁰

[g] (3.1)

Where C

₁⁰

is the injection concentration. The collected amount product A

₁

per cycle then deter- mines the yield, Y

₁

, of the production:

Y

₁

=

A1

z }| {

V

_vol

tC,2,1

Z

tC,1,1

C

₁

(t)dt

n

1

(3.2)

Where V

vol

is the volumetric flow, t

C,1,1

the time when enantiomer 1 starts eluting from the column and t

C,2,1

when the second enantiomer starts eluting. In this particular example, t

C,2,1

coincides with the cut point, t

_cut

which divides the two fractions.

The purity of the product, P u

₁

is determined by the cut point and is defined as:

P u

1

= A

1

A

1

+ V

vol tcut

Z

tC,1,1

C

2

(t)dt

| {z }

A1∩B1

(3.3)

Where C

₂

(t) is the response curve of component 2. In our example A

₁

∩ B

₁

= 0. The production

Time [min]

C on ce nt ra ti on [g /l ]

C

1

(t) C

2

(t)

t

C,1,2

t

C,2,2

t

cycle

A

1

t

C,1,1

A2 ∩ B1

A

2

B

1

t

cut

t

C,2,1

0 1 2 3 4 5 6 7 8

0 1 2 3 4 5 6 7 8

Figure 5: A fictional semi-optimized process chromatogram describing the separation of two enan- tiomers. Each is described by its response curve C

₁

and C

₂

which have been derived by simulations.

The peaks are defined by the time they start eluting t

_C,n,1

and the time they have eluted t

_C,n,2

. The cut point, t

_cut

separates the fraction containing product, A

₁

from waste, A

₂

and B

₁

.

rate P r of component 1 and the specific production SP r can then be defined:

P r

1

= n

₁

Y

₁

∆t

cycle

[g/min]

SP r

₁

= V

_vol

P r

1

[l/g]

(3.4)

Having made these definitions, its possible to quantify the search for the optimal separation pro- cess. The basic problem is to determine the individual band profiles C

₁

and C

₂

as functions of the decision variables.

These variables are the parameters available to change during an optimization process, typically but not necessarily or limited to the column dimensions and saturation capacity, total porosity, feed concentration and injection volume and flow rate. A very common constraint is the purity of the collected product.

How the determination of C

1

and C

2

is carried out varies. For narrowing down the scope, I’ll

continue using the fictional enantiomer production. The robust albeit time consuming empirical

method consists of first performing experiments in a small scale, finding possible experimental

t

cut

[min]

%

Purity Yield

3.5 4 4.5 5 5.5 6 6.5

0 10 20 30 40 50 60 70 80 90 100

Figure 6: A graph of purity and yield as functions of the cut-point t

_cut

for the example chro- matogram in figure 5.

conditions which includes stationary and mobile phase. Then, the process is scaled up, experi- mental injections are performed and evaluated. To investigate the momentanous composition at the outlet at different time points, fractions are collected and analyzed. After fine-tuning of of the separation and ensuring reproducible injections, the separation process can commence. Continu- ous samples can be taken from the product fraction ensuring a stable production.

In this project I will briefly investigate the numerical approach to process optimization. Section 4 will introduce chromatographic modeling, here it suffice to state that C

1

(t) and C

2

(t) are solutions of mathematical models of the chromatographic separation process.

Still, using this numerical approach, possible experimental conditions must be empirically found.

Once they have been found, a couple of suitable injections of scalable injection concentrations should be made. From these experimental profiles, the inverse method of finding a possible ad- sorption isotherm and its parameters is employed.

Now it’s possible to model the scale up the separation by changing column dimensions and effi- ciency and evaluating the results. The outcome from these simulations can serve as a guideline for where cut-points should be set. In the actual production, the cut point needs to be dynamically adjusted depending on the process.

After finding the parameters, the experimental space which is set up is only limited to the current

stationary and the maximal injection concentration. All other decision variables such as injection

concentrations and volumes and flow rate can be varied and the outcome determined by simula- tions.

Each member of the set of all possible outcomes will determine the value of the objective function used in the optimization process. Commonly used objective functions are the production rate and specific production (Eq. 3.4). Another useful function is the productivity, which states the amount of product produced per amount stationary phase per day. [Cox, 2005]

P t

₁

= n

₁

Y

₁

m

_stattcycle²⁴

[g/g/day] (3.5)

Where m

_stat

is the amount of stationary phase used in production. It may be obvious from a technical viewpoint which of these objective functions should be maximized or minimized, but uncertain if the total cost of the production is minimized when doing so. In fact, technical and economical optima seldom coincides [Guiochon et al., 2006].

A unit price of an arbitrary purification can be defined:

P rice = P rC

P r

₁

= F iC + OpC + F eC

P r

₁

[$/g/day] (3.6)

Where P rC is the total production cost, P r

₁

the production rate, F iC fixed costs such as facilities, equipment and labour cost. OpC is the operating cost, which includes solvent, solvent recovery, waste management, energy consumption etc. F eC is the feed cost which is inversely proportional to the yield. The higher the yield, the less lost feed, the less the feed cost. The task of minimizing the price, a multidimensional nonlinear optimization problem will not be further discussed here.

However, the general insight is that one must balance the two objective functions of production rate and specific production, considering constraints such as yield which can lower the F eC and other requirements such as purity.

3.2 Design of a Separation Process at Astra Zeneca

The Chromatography Team at the division for PRD constantly receives substances, often phar- maceutical intermediates which needs to be purified to for example continued synthesis.

Detailed knowledge of molecular structure of product and major contaminants are given, allowing research engineers to design a separation process.

Once suitable separation conditions have been found, the process is readily scaled up to production scale. Here, test injections are made and fractions carefully collected during the elution. These fractions are analyzed for example purity and yield. Based on the findings, a cut point is set to separate product from waste. This cutpoint is set dynamically by using a certain UV threshold.

The growing product and waste fractions are also continuously monitored to ensure a stable pro- duction.

Whenever a process is completed, it is always tried to recycle used solvents.

The most recent development is to use numerical determination of adsorption isotherm parameters

from small scale experiments and then simulating the large scale production process for rapid

determination of fraction cut-points. This approach was successfully employed in production N1,

see section 6.4.1.

4 Modeling

Accurately describing the relation between the sum of all physical processes, retaining and dispers- ing a given component throughout its migration through the column and the chromatogram, is crucial for several reasons. There is the the industrial, academical and ”personal understanding”

perspective [Schmidt-Traub, 2005].

If chromatography is used in a larger process, a purely empirical approach in finding the optimal controllable process parameters can be both time consuming, maybe impossible and also expen- sive. A model which takes into account some or all of the process parameters and which can be analytically or numerically solved can render many experiments unnecessary. The process of modeling in it self and its applications can lead to an increase in the understanding of the chro- matographic process and also serve as a pedagogical tool for students and people who wish to gain a better understanding of chromatography.

Many models of chromatography have been derived with varying complexity and assumptions made. [Guiochon et al., 2006; Schmidt-Traub, 2005]. They can be classified in four families of models: The linear and ideal, the linear and non ideal, the nonlinear and ideal and the nonlinear and non ideal models [Guiochon and Lin, 2003]. The model implemented in this study is a non- linear and non ideal model.

Each of the models generally take into account phenomena such as convection, dispersion, different type of diffusion and adsorption equilibrium or adsorption kinetics.

The basic equation of any

⁴

model is the differential mass balance equation.

∂C

∂t + F ∂q

∂t + u ∂C

∂x = D

_L

∂

²

C

∂x

²

(4.1)

This equation describes the difference between the influx and outflux of a compound considering an infinitesimally thin slice dx during an infinitesimally short time dt. C is the mobile phase concentration of the compound, q is the stationary phase concentration of the compound, F is the phase ratio, D

L

the axial dispersion coefficient.

The second fundamental part of any model is the adsorption behavior of a component, which is described by the adsorption isotherm q. As described earlier, it suffices to state:

q = f(C) (4.2)

To state a solution to 4.1, initial and boundary conditions must be considered. In this project and typically in most elution chromatography [Guiochon et al., 2006] the mobile phase doesn’t contain any of the components one wish to separate, hence the following initial condition:

C(x, 0) = q(x, 0) = 0 (4.3)

The boundary conditions most often assumed in simulations is the rectangular pulse injection. For an injection time of t

_inj

and a pulse concentration C

₀

the left, and the unrelated right boundary condition can be described:

C(0, t) = 0 t < 0 C(0, t) = C

₀

0 < t ≤ t

_inj

C(0, t) = 0 t

inj

< t

 ∂C

∂x



x=L

= 0

(4.4)

4This equation assumes fast mass mass transfer and simple kinetics. Other more complex mass balance equations exists, for example the mass balance equation of the pore model [Guiochon and Lin, 2003]

4.1 The Equilibrium Dispersive Model

The Equilibrium-Dispersive (ED) model is a generally accepted and frequently used model of HPLC. [Guiochon et al., 2006; Guiochon and Lin, 2003; Schmidt-Traub, 2005; Arnell, 2006]. De- spite its many simplifications of the chromatographic process, it has been successfully employed in modeling the separation of many substances.



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

∂C

_i

(x, t)

∂t + F ∂q

_i

(x, t)

∂t + u ∂C

_i

(x, t)

∂x = D

_a,i

∂

²

C

_i

(x, t)

∂x

²

0 ≤ x ≤ L, t ≥ 0, i = 1, . . . n,

C

_i

(x, 0) = C

_0,i

,

∂C

i

(L, 0)

∂x = 0, C

i

(0, t) = φ

i

(t)

(4.5)

F = V

s

V

₀

= 1 − 

t



_t

D

a,i

= Lu 2N

_i

(4.6)

Equation 4.5 is written for component i. For the n-component problem, there will be n equations 4.5 which needs to be solved simultaneously.

C

_i

and q

_i

are the mobile and stationary phase concentration respectively. F is the phase ratio, that is the quotient between the stationary phase volume and mobile phase volume.

D

_a,i

is the apparent dispersion constant which is dependent on the length L, volumetric flow of the mobile phase, u and the column efficiency N of a component in the column.

φ

_i

(t) is the left boundary condition, that is a concentration profile of the injected sample(s). (See Fig. 7 on page 15 for details)

The assumptions of the model are numerous. Below is a list of major assumptions [Guiochon et al., 2006].

• Since the only independent spatial variable is related to the axial position x of the column, the column is assumed to be radially homogeneous, that is, the concentration is independent of the distance from the column axis. This is a good approximation which also can be valid for carefully packed large scale columns up to 80 cm in diameter.

• Often, and in this project, the mass balance equation of the mobile phase is neglected. This approximation is obviously only valid if the mobile phase doesn’t adsorb to the stationary phase.

• Effects such as mass transfer kinetics, the finite time kinetics of the equilibrium between the sorbent and adsorbent, axial and eddy diffusion is lumped into one coefficient D

_a

. This coefficient is assumed being constant. This is a good approximation only if the viscosity of the samples are near that of the mobile phase which means that the diffusion doesn’t depend on the concentration.

• There exists no thermal effects during the adsorption and desorption or heating of the column

due to friction and adsorption/desorption. Also, parameters such as temperature, mobile

phase flow rate, viscosity, column pressure are assumed to be constant. (It’s easy to model

non-constant flow rate)

t [min]

φ( t) [m M ]

Experimental profile Rectangular/linear profile Rectangular profile

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

0 20 40 60 80 100 120

Figure 7: Three different injection profiles which in decreasing order of accuracy describe the left boundary condition of Eq. 4.5: Experimental, rectangular, rectangular/linear. The experimental injection profile recorded was recorded when performing experiments described in section 6.5

4.1.1 Numerical Solutions

The only analytical solutions to Eq 4.5 which exists are limited to when either case i-ii holds true.

i

 n = 1 q = aC

ii







n = 1

D

a

= 0 q = f(C)

This fact has necessitated the development of numerical methods to solve the general form of Eq.

4.5.

Existing methods range from using different kinds of finite difference approximations to finite element approximations (Orthogonal Collocation of Finite Elements, OCFE). [Guiochon, 2002].

OCFE is considered to be the most accurate method currently available but requires considerable computational time [Kaczmarski, 2007].

A particular variant of the finite difference method is the ”Rouchon forward-backward scheme”

[Rouchon et al., 1987]. This scheme makes some clever simplifications, drastically reducing com- putational time and has been used in many applications as solver of Eq. 4.5 [Samuelsson, 2008;

Arnell, 2006; Forss´en, 2005].

In this project, whenever the Rouchon-algorithm (RFD-method) is invoked, it uses the implemen- tation by P. Forss´en et al [Forss´en et al., 2006]. The following space and time discretization and update formula is used:



 

 

 

 

 

 

 

 

 

 

 

 

 

 



 

 

 

 

 

 

 

 

 

 

 

 

 

 



C

_i,jⁿ⁺¹

− C

_i,jⁿ

∆t + F q

ⁿ⁺¹_i,j

− q

_i,jⁿ

∆t + u C

_i,j+1

− C

_i,jⁿ⁺¹

∆x = 0

C

_i,j+1ⁿ⁺¹

= C

_i,jⁿ⁺¹

− ∆x

u∆t C

_i,jⁿ⁺¹

− C

_i,jⁿ

+ F q

ⁿ⁺¹_i,j

− q

ⁿ_i,j

i = 1, . . . , N

_components

, j = 0, 1, . . . , N

_spacesteps

, n = 0, 1, . . . , N

_timesteps

C

_i,j⁰

= C

0,i

C

_i,j⁰

= C

0,i

C

_i,0ⁿ

= φ

i

(t), 0 ≤ n ≤ N

injectionsteps

+ 1

∆x = L

N

_average

∆t = 2L

N

avg

u

^linear_avg

(4.7)

It may seem as this discretization is a solution to the homogeneous form of Eq. 4.5, but by the particular choices of ∆x and ∆t, the numerical error elegantly estimates the physical effects lumped in D

a

. This estimation is only exact whenever q is linear and only one component is simulated, or the different components have the same efficiency N and/or migration rate.

Even though this seldom is the case in preparative chromatography, the Rouchon algorithm has

many successful applications. Its accuracy has been questioned, both in the simulations of two

components [Guiochon, 2002] and more recently, in simulating both single, binary and ternary

separations, described by different adsorption isotherms and also concluding the worst accuracy

when simulating at low column efficiency [Kaczmarski, 2007].

5 Determination of Adsorption Isotherms

Since the behavior of any chromatographic separation as modeled by Eq. 4.5 is governed by the adsorption isotherm, q = f(C), any use of this model in the purpose of designing/and or optimiz- ing a preparative chromatographic process requires the accurate determination of its parameters [Felinger et al., 2003].

As was described in section 2.3, there are a multitude of different adsorption isotherms, many of which have been derived on physical basis and some of strict mathematical nature as the quadratic isotherm, coinciding with the Pad´e approximation [Guiochon and Lin, 2003]. While the latter may approximate the former, the physical meaning is lost and no further insight into the adsorption process can be drawn from such an approximation.

Many experimental methods of determining adsorption isotherm parameters can only be applied to the case of single component separation. Due to the competitive nature of adsorption equilib- rium, this data can’t be extrapolated to the case of multicomponent separation.

Examples of popular methods not dissected in this project are frontal analysis (FA), elution by characteristic point (ECP), FA by ECP, (FACP), injection on plateau method (PM) and the per- tubation peak method (PP) [Samuelsson, 2008].

These methods have in common that one first determine the value of q, then determine f(C) such that q = f(C) using nonlinear regression, choosing a suitable model. Ideally, one should perform this regression against all known models, then afterwards evaluate which model has the better fit.

Currently, there is no way of knowing beforehand, given a particular solute and adsorbent, which level of complexity is required, that is, which model can be used to fit experimental data [Forss´en et al., 2006]. The previously mentioned methods will give the most accurate determination of parameters.

5.1 The Inverse Method

The inverse method uses a work flow that is, as expected, opposite to that of the earlier mentioned methods [Felinger et al., 2003]. The general approach is to perform a number (the more the better) of experiments and record the chromatograms.

The next step is to determine which differential mass balance model to use (e.g. Eq 4.5), which type of adsorption isotherm is applicable and then guess the start values of the isotherm. In theory, this is a very straight-forward approach but in reality there are complications. Because the recorded detector response is the sum of the response from all the components present at the detector but the simulated responses are per component, there can be problems doing a proper detector calibration if the response isn’t linear. This can be solved by using other detection methods than UV [Guiochon et al., 2006] and is of no problem when simulating the separation of enantiomers which have identical and linear UV-response [Forss´en et al., 2006]. If separating enantiomers, one can take two approaches: one is to create a calibration curve by injecting a series of know concentrations into the detector or either injecting different amounts of enantiomers into the column, the integrating the total area of the chromatogram, hence doing the actual detector calibration.

The next step is to load the experimental chromatograms, superimpose the simulated chro- matograms and calculate the degree of overlap in a nonlinear least-square procedure.

Then an optimization routine is started where parameters first are varied or perturbed according

to a specific algorithm, for example a gradient based [Forss´en et al., 2006] or based on a genetic

algorithm [Zhang et al., 2008]. The change in degree of overlap is noted and depending on the

direction and its magnitude, the parameters will change accordingly. The optimization routine

will end when some convergence criteria is reached or the algorithm can’t adjust the parameters to give a better degree of fit. As with the experimental approach mentioned earlier, one should try to loop through different adsorption isotherm models and evaluate which has the best fit.

The large number of experiments and high consumption of solvent and samples needed when per- forming the classical methods of isotherm determination can be drastically reduced by the use of the inverse method [Forss´en et al., 2006].

Figure 8: A flowchart of the inverse method of determining adsorption isotherm parameters, as used in this project. Adapted and modified from [Forss´en et al., 2006]

In this work, the optimization routine imple- mentation of Patrik Forss´en is used [Forss´en et al., 2006].

This algorithm uses an objective function (Eq.

5.1) which is minimized by calculating its Jaco- bian with respect to the particular adsorption isotherm parameters. The actual optimization is a gradient based algorithm, lsqnonlin im- plemented in MATLAB.

min

Np

X

k=1 Nc

X

i=1 Ns,k

X

n=1

C

_iⁿ

(~a, k) − ¯ C

_i

(n, k)

²

(5.1)

The objective function as used in this project.

N

p

is the number of experimental chro- matograms, N

c

number of components, N

s,k

the number of time samples from the k

^th

ex-

perimental profile, C

_iⁿ

(~a, k) the simulated nth

time sample of the ith component for the kth

experimental condition, using isotherm param-

eters ~a. ¯ C

_i

(n, k) is the experimental nth time

sample of the ith component and kth chro-

matogram.

6 Methods and Materials

6.1 Implementation of Algorithm

The problem is to find a numerical approximation to Eq. 4.5, reprinted below.



 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

∂C

i

(x, t)

∂t + F ∂q

i

(x, t)

∂t + u ∂C

i

(x, t)

∂x = D

a,i

∂

²

C

i

(x, t)

∂x

²

0 ≤ x ≤ L, t ≥ 0, i = 1, . . . n,

C

_i

(x, 0) = C

_0,i

,

∂C

i

(L, 0)

∂x = 0, C

i

(0, t) = φ

i

(t)

The first step is to approximate the partial derivates using ordinary finite difference quotients.

This can be made in many ways [Guiochon and Lin, 2003] but in this project the following approximations have been used.

 ∂C

i

∂t



n j

≈ C

_i,jⁿ⁺¹

− C

_i,jⁿ

∆t + O(∆t)

 ∂C

_i

∂x



n j

≈ C

_i,j+1ⁿ

− C

_i,j−1ⁿ

2∆x + O(∆x

²

)

 ∂

²

C

_i

∂x

²



n j

≈ C

_i,j+1ⁿ

− 2C

_i,jⁿ

+ C

_i,j−1ⁿ

∆x

²

+ O(∆x

²

)

(6.1)

Since q = f(C) the chain rule can be applied to the partial derivative

^∂q_∂tⁱ

. For k components we have:

∂q

_i

∂t = X

^k

z=1

∂q

_i

∂C

z

∂C

_z

∂t (6.2)

By substituting Eq. 6.2 and 6.1 into Eq. 4.5 we have:

C

_i,jⁿ⁺¹

− C

_i,jⁿ

∆t + F X

^k

z=1

∂q

i

∂C

_z

C

_z,jⁿ⁺¹

− C

_z,jⁿ

∆t + u C

_i,j+1ⁿ

− C

_i,j−1ⁿ

2∆x = D

ai

C

_i,j+1ⁿ

− 2C

_i,jⁿ

+ C

_i,j−1ⁿ

∆x

²

(6.3)

An explicit statement of Eq. 6.3 for two components:



1 + F ∂q

₁

∂C

1

 C

_1,jⁿ⁺¹

− C

_1,jⁿ

∆t + F ∂q

₁

∂C

2

C

_2,jⁿ⁺¹

− C

_2,jⁿ

∆t

!

+ u C

_1,j+1ⁿ

− C

_1,j−1ⁿ

2∆x = D

_a1

C

_1,j+1ⁿ

− 2C

_1,jⁿ

+ C

_1,j−1ⁿ

∆x

²



1 + F ∂q

2

∂C

₂

 C

_2,jⁿ⁺¹

− C

_2,jⁿ

∆t + F ∂q

2

∂C

₁

C

_1,jⁿ⁺¹

− C

_1,jⁿ

∆t

!

+ u C

_2,j+1ⁿ

− C

_2,j−1ⁿ

2∆x = D

_a2

C

_2,j+1ⁿ

− 2C

_2,jⁿ

+ C

_2,j−1ⁿ

∆x

²

(6.4)

The generalization into k components leads to an equation system of k equations with k unknowns on the form:















a

11C_1,jⁿ⁺¹−Cⁿ_1,j

∆t

a

12C_2,jⁿ⁺¹−C_2,jⁿ

∆t

· · · a

1kC_k,jⁿ⁺¹−C_k,jⁿ

∆t

| c

1

a

21C_1,jⁿ⁺¹−Cⁿ1,j

∆t

a

22C_2,jⁿ⁺¹−C2,jⁿ

∆t

· · · a

2kC_k,jⁿ⁺¹−C_k,jⁿ

∆t

| c

2

... ... ... ... ...

a

k1C_1,jⁿ⁺¹−C_1,jⁿ

∆t

a

k2C_2,jⁿ⁺¹−C_2,jⁿ

∆t

· · · a

kkCⁿ⁺¹_k,j −C_k,jⁿ

∆t

| c

k















(6.5)

Where the coefficients are calculated as follows:

a

_αβ

=

( 1 + F

_∂C^∂qα_β

α = β

F

_∂C^∂qα_β

α 6= β c

_κ

= D

_aκ

C

_κ,j+1ⁿ

− 2C

_κ,jⁿ

+ C

_κ,j−1ⁿ

∆x

²

− u C

_κ,j+1ⁿ

− C

_κ,j−1ⁿ

2∆x (6.6)

An update formula for determining the unknown variables C

_κ,jⁿ⁺¹

can be explicitly stated:









 C

_1,jⁿ⁺¹

C

_2,jⁿ⁺¹

C

_κ,jⁿ⁺¹

...











=∆t

⁻¹





















a

₁₁

a

₁₂

· · · a

_1k

a

₂₁

a

₂₂

· · · a

_2k

... ... ... ...

a

_k1

a

_k2

· · · a

_kk











−1









 c

₁

c

₂

c ...

_κ









 +









 c

ⁿ_1,j

c

ⁿ_2,j

c

ⁿ_κ,j

...



















 (6.7)

This update formula needs to be evaluated at every time step n for each room step j. Additionally

∂qα

∂Cβ

needs to updated the same number of times.

In the implementation, no matrix inversion operation is performed but the linear system of equa- tions of Eq. 6.5 is solved by using the standard solver DGESV as implemented in the LAPACK v3.2 library [Anderson et al., 1999]

The stability criteria for a certain finite difference implementation of lesser accuracy [Guiochon and Lin, 2003] has been derived. It basically states that the number of room steps in which the column needs to be discretized should be equal to the number of theoretical plates. This is an intuitive condition considering the definition of a theoretical plate.

The second criteria on the time step is less intuitive, but it obviously depends on magnitude of the derivative of the adsorption isotherm and also the lenght of a room step. No effort has been made to the complex task of deriving similar critiera for this particular implementation. In this project, the room step is chosen to fulfill the earlier derived criteria but the second will be empirically determined and dynamically adjusted to reach stability. The definition of stability in this case is conserved mass.

The column is discretized as follows.

M =max (N

i

)

Number of space steps

dx = L

M

Length of a space step

dt =λ dx

max 

L~ tR



Length of a time step

Where ~ t

_R

is a vector containing all the retention times of the components as derived using the current adsorption isotherm (See Sec. 2.3) at infinite dilution. dt is therefore chosen as a fraction λ of the time it takes for the fastest component to traverse a space step.

Figure 9: A flowchart of the algorithm used in this project

Eq. 6.3 is then iteratively solved from t = 0 to t = t

end

, from j = 0 to j =

_dx^M

At the left boundary, the current concentration of sample is equal to the concentration of the injec- tion stream φ(t) (See Fig. 7). If φ(t) is an discretized injection profile or of type ”rectangular/linear” the value of C(t, 0, k) = φ(t, k) is used. Oth- erwise it’s approximated by Danckwerts left boundary condition[Guiochon et al., 2006]. At j =

^M_dx

the update for- mula is modified to the condition C

_κ,jⁿ

= C

_κ,j+1ⁿ

The actual algorithm was first implemented solely in MATLAB as proof of concept.

The final version uses a MATLAB inter- face to compiled Fortran code. The user supplies simulation parameters in the MAT- LAB interface but the iterative solution of 6.3 is performed in the compiled Fortran code.

The algorithm was named the ”Precise Finite Difference” method, PFD, after briefly evau- lating its results compared to the ”Rouchon Finite Difference” method.

6.1.1 Accuracy and Validity

To determine reasonable accuracy and va- lidity of the algorithm, it was decided to compare simulations performed in the im- plemented algorithm versus the earlier im- plemented RFD-method. Such a comparison can only be made for a special case. Since the only case when Eq. 4.7 converges to the real solution of Eq. 4.5 is when q is linear and only one component is simulated, whereas Eq. 6.3 in theory will converge independently of the number of components and q.

6.1.2 Stability

To investigate the stability of the algorithm as a function of the column discretization a large

synthetic grid of space and time discretization values were spanned and simulated for two compo-

nent separation, using fixed plate numbers. The following simulation parameters were used. All

parameters but the number of theoretical plates were taken from actual experiment performed

(See section 6.3).

Table 1: Table of parameters used in the accuracy and validation testing of the implemented algorithm

PFD RFD

Adsorption isotherm Langmuir

Parameters [a b] [2 0]

Theoretical plates 200. . . 2000

φ (t) Rectangular

Sample konc. [mM] 1. . . 1000

Eluent konc. [mM] 0

Injection volume [µl] 0. . . 1000 Column dimensions (WxD) [cm] 25x0.46

Flow [ml/min], t

0

[min] 1,2.82

Table 2: Table of parameters for stability testing

Adsorption isotherm Bi-Langmuir PFD

Parameters 1 [a

1

b

1

a

2

b

2

] 3.065 146.612 4.859 3595.987 Parameters 2 [a

₁

b

₁

a

₂

b

₂

] 3.281 318.333 7.020 4489.679

Theoretical plates [400 400]

φ (t) Experimental profile for 50ul injection

Sample konc. [mM] [1 1]

Eluent konc. [mM] [0 0]

Injection volume [µl] 50 µl

Column dimensions (WxD) [cm] 10x0.4

Flow [ml/min], t

0

[min] 1,1.3

6.1.3 Performance

To investigate the performance of the algorithm as a function of the column discretization, the calculation time was measured using the same grid as was used in the stability tests. All simulations were performed using MATLAB 7.4.0.336 (R2007a) and compiled Fortran code via G95 0.92 on Ubuntu 8.10, kernel 2.6.27-14 running on an IBM ThinkPad Z61p, Intel Core 2 Duo T7200 2.0GHz, 2GB RAM.

6.2 Synthetic Algorithm Comparisons

6.2.1 Linear Conditions

To briefly investigate the performance of the algorithm at linear conditions, a couple of experiments were performed. These were aimed at simulating analytical injections of racemic methyl mandelate on a chiral AGP column. See section 6.3 and 6.3.1 for more details on how these simulations were motivated and designed.

6.2.2 Nonlinear Conditions

To investigate the solutions to Eq. 4.5 obtained by the implemented algorithm (PFD) and RFD, a

thorough synthetic and experimental study was set up. Since most previous studies [Kaczmarski,

2007; Guiochon et al., 2006] have shown that solutions obtained using a variety of numerical

Table 3: Parameters used for the linear study comparing solutions of PFD and RFD

PFD RFD

Adsorption isotherm Bi-Langmuir

Parameters 1 [a

1

b

1

a

2

b

2

] 3.065 146.612 4.859 3595.987 Parameters 2 [a

₁

b

₁

a

₂

b

₂

] 3.281 318.333 7.020 4489.679

Theoretical plates [3393 2862] [3128]

φ (t) Rectangular/linear

Sample konc. [µM] [40 40]

Eluent konc. [µM] [0 0]

Injection volume [µl] 7

Column dimensions (WxD) [cm] 10x0.4

Flow [ml/min], t

0

[min] 0.7,1.3

schemes markedly differ at low column efficiency, focus was directed to artificially varying the column efficiency.

Because solutions obtained using the OCFE method are considered to always converge to the real solution, it was imperative to include these solutions as well. Since no working implementation of the OCFE method has been implemented at the lab, all solutions that will be presented are thanks to professor Krzysztof Kaczmarski. We don’t have access to his source code but have ensured that simulations are performed using identical parameters and injection profiles.

In total, 72 simulations were performed using each algorithm. Due to unforeseen problems with loss of mass using the RFD algorithm and experimental injection profiles, all simulations had to be repeated using a modified interpolated injection profile which assures conserved mass. For the comparisons at linear and non-linear conditions, see sections 6.3 and 6.3.1 for how the simulations were designed.

Table 4: Parameters used for the non-linear study

PFD RFD OCFE

Adsorption isotherm Bi-Langmuir

Parameters 1 [a

1

b

1

a

2

b

2

] 3.065 146.612 4.859 3595.987 Parameters 2 [a

₁

b

₁

a

₂

b

₂

] 3.281 318.333 7.020 4489.679

Theoretical plates [3393 2862]. . . [113 95] 3127. . . 104 [3393 2862]. . . [189 159]

φ (t) Interpolated experimental 50 µl

Sample konc. [µM] See section 6.3, table 5

Eluent konc. [µM] [0 0]

Injection volume [µl] 25,50,75,100

Column dimensions [cm] 10x0.4

Flow [ml/min], t

0

[min] 0.7,1.3

6.3 Separation of Methyl Mandelate on a Chiral Protein Column

A 10x0.4 cm chiral AGP column (ChromTech, H¨agersten, Sweden) consisting of 5µm silica beads with immobilized α

1

-glycoprotein as chiral selector was was installed in a Hewlett-Packard, Agi- lent 1100 system.

Eluent was prepared by dissolving appropriate amounts of acetic acid and sodium acetate, creating an 75 mM acetate buffer at pH 5, ionic strength 50 mM. Additionally, methanol was added to a concentration of 1.4 % (v/v). The buffer was filtered through a 45 µm filter before use.

The column was equilibrated with eluent at a constant flow at 0.7 ml/min and kept at 20

^◦

C during all experiments. UV-detection was made simultaneously at 6 different wavelengths but all further data processing used data obtained at 254 nm where the detector response was found to be linear.

A 2mM stock solution of (+/-)-methyl mandelate was prepared by dissolving appropriate amounts of (+/-)-methyl mandelate (Sigma-Aldrich, Stockholm, Sweden) directly in the mobile phase.

A calibration curve of absorbance vs concentration was constructed by overriding the automatic injector and manually injecting 0.5 ml of one enantiomer by a different valve bypassing the col- umn and reaching response plateaus.

⁵

. To validate the calibration, some replicates of the known amounts of the other enantiomer were also injected.

A linear regression of the data was performed using polyfit as implemented in MATLAB R2007a (The MathWorks, Natick, USA). Experimental chromatograms were exported and processed using this calibration polynomial.

To measure the injection profile of the system, the capillary tube connecting the injector and column was connected directly into the UV-detector whereupon a triplicate of 50 µl injections of 0.5 mM (+)-methyl mandelate were made.

Initial analytical injections to determine column porosity and efficiency of the two enantiomers were made in triplicate by injecting 7 µl 40 µM racemate and single injections of each enantiomer.

A number of overloaded injections were made after experimenting with suitable injection amounts and different ratios between the enantiomers. The experiments are listed in Table 5.

No void-volume marker was used.

6.3.1 Determination of Adsorption Isotherm Parameters

After performing the actual experiments all experimental chromatograms were processed and con- verted from UV-response vs time to concentration vs time. The porosity of the column was determined, as was the efficiency of the both components by using equation 2.2.

The measured injection profile at 50 µl was also converted and interpolated by a piecewise polyno- mial. The polynomial was then normalized to the known concentration for which it was recorded.

Thus, its possible to use this polynomial and interpolate an arbitrary injection concentration. The optimization employs the RFD-algorithm. The experimental profiles and parameters used for the inverse solver are listed in Table 6.3.

5The absorbance reading at the plateau corresponds to the response at the particular injection concentration