Model-based analysis of the transaminase process · is focused on the model based analysis of the...

Faculteit Bio-ingenieurswetenschappen

Academiejaar 2013-2014

Model-based analysis of the transaminase

process

Daan Van Hauwermeiren

Promotor: Prof. dr. ir. Ingmar Nopens

Tutor: ir. Timothy Van Daele

Masterproef voorgedragen tot het behalen van de graad van

Master in de bio-ingenieurswetenschappen: Milieutechnologie

De auteur en promotor geven de toelating deze scriptie voor consultatie beschikbaar te stellen en delen

ervan te kopieren voor persoonlijk gebruik. Elk ander gebruik valt onder de beperkingen van het

auteursrecht, in het bijzonder met betrekking tot de verplichting uitdrukkelijk de bron te vermelden bij

het aanhalen van resultaten uit deze scriptie.

The author and promoter give the permission to use this thesis for consultation and to copy parts of it

for personal use. Every other use is subject to the copyright laws, more specifically the source must be

extensively specified when using results from this thesis.

Ghent, June 2014

The promoter, The tutor, The author,

Prof. dr. ir. Ingmar Nopens ir. Timothy Van Daele Daan Van Hauwermeiren

Dankwoord

After a year of hard labour, this thesis saw the break of light. Of course, this work is no solo-slim, and

I would like to thank the following people for their input in this work.

First, I would like to thank my tutor Timothy Van Daele for the mentoring me this whole year. Not

only did I learn a lot from him, he also happens to be a very nice bloke. Although there was a clear

language barrier, ’t stad versus the marginal triangle, communication was always smoothly, especially

regarding Monty Python references. Furthermore I would also like to apologise for my stubbornness, as

I can assume it has yielded you quite some worrying and frustration. I don’t apologise for the tomatoes

in your headphones, that wasn’t me.

Second, I would like to thank my promotor Ingmar Nopens for offering me the chance to work on this

subject, and giving me an opportunity to collaborate in the Biointense project for two more years to

come.

Next, the Biomath-Kermit Thesis Team is certainly worth mentioning. From the Simulation Lab, to our

holiday getaway room, and back to the New Simulation Lab, these partners in crime helped me through

some tough moments: thank you Sophie, Michael, Wouter, Anton, Stijn, Chaim, Ruben, Robin, and

Marlies.

Sophie, for showing me that meshing is fun and requires very little effort. Also thanks for the beautiful

comments on my outfit-matched socks. Michael, for enlightening me with the fact that the cake always

lies, and yields weird velocity profiles like the mushroom, and the turbulent flow in disguise. Also many

thanks for doing the quantum computing lecture with me, I was fun. Or it was not, I don’t know, the

sentiment is still in superposition until further measurement. Anton, for being a cool guy with the right

underflow, yet I cannot grasp the fact that you are still working with OpenSCHUIM. I am also very

grateful to almost have you converted to the church of Linux. Wouter (aka the Good Guy Technical

Hotline), for doing this magical stuff with Mathematica, which I will probably never understand. Stijn,

for showing me that is it socially acceptable to wear custom-made yellow Beats by Dr. Dre-headphones.

Seriously, they’re sooo pretty. Chaim, because I like to watch people work in the lab from the coffee

room, and thinking that I’m glad I didn’t have any labwork.

Special notification for the die-hards among the thesis team who have were here during the weekends

at the simulation lab. Otherwise, there was nobody here to join me for a match of table tennis.

Furthermore, I would like to thank all the people from Biomath, to have welcomed me in this research

unit with its lovely atmosphere. Thank you: Ingmar, Thomas, Wouter, Elena, Youri, Lieven, Ivaylo,

Ashish, Sverine, Niels, Timothy, Stijn, and Giacomo.

Stijn (fully accredited member of Team Python), for supporting me with all me Python needs, and

converting me to the MODERATifarianism. Also, I am delighted that you finally switched to Linux:

welcome to the dark side. Niels, I really have to cancel that holiday in Copenhagen this August. Wouter,

because a Von Karmann wake is a valid approximation of the beer brewing process. Lieven, for the

occasional philosophical talk, and facts of life. Ivaylo, I wish my grandmother made moonshine! Tine,

because pie is always the answer.

i

In a non professional context, I would like some friends who have supported me during this year. Jonas,

my Duvel Triple Hop is ready to consumed! Tom, I really hate your ferret, but you are a true friend.

Yacine, parce que ta mere fait le meilleur lasagne du monde. Also, do not succumb to the corporate

doctrine: do not shave your beard! Jens, because pape di poopi. Mugabe, because everyone wants a

picture with Lukaku. Haek, your enthusiasm fills the room. Wout, because you are an even bigger nerd

than me. Lore, life can be dreadfull. Marjolein, hummus and crackers are perfectly edible at 4 o’clock

in the morning.

Furthermore, I would like to thank the following institutions: Stack Overflow, thank you anonymous

programmer for solving nearly all my issues. Thank you Wikipedia, correction: Web of Science, for the

provided accurate knowledge base. Thank you Eraser1, for not erasing my simulations.

Thank you mystery man who makes the coffee in the morning, you’re the real MVP.

Last but certainly not least, thank you Jasmien for the support in these dark times. And for the oreos,

thank you very much for the oreos: you really know how to make me a happy man. Ich mag deinen

Stil, Fraulein.

ii

Summary

At present, bioprocesses designed to produce chemical precursor products by means of enzymatic produc-

tion still suffer from low productivity and low process intensity compared to more traditional chemical

processes. The widespread use of these bioprocesses is hampered by these shortcomings. This thesis

is focused on the model based analysis of the transaminase process in microreactors. The synthesis

with microreactors is a continuous process, which uses lab scale optimised processes in an industrial

context by use of parallel upscaling. The goal of this thesis was to build a model that can accurately

predict reactor outlet concentrations for a variety of reactor setups, and inlet conditions. The reaction

was analysed starting from a physical model (Computational Fluid Dynamics (CFD) combined with

enzyme kinetics), implemented in the opensource CFD library OpenFOAM. The mesh generation for

the CFD model was executed by means of a flexible Python scripting environment as a wrapper around

the geometry and mesh generation software Salome. A scenario analysis tool was written in Python as

a wrapper around the OpenFOAM C++ libraries to ease the output analysis of different simulations

on the same case. Next, this physical CFD model was simplified to a mixed flow model, an ideal plug

flow model, a Tanks-In-Series model, and a Compartmental Model. For the simplified models, only the

Tanks-In-Series model was able to accurately describe the hydraulic behaviour of the reactor. Regard-

ing the prediction of outlet concentrations, only the predictions of the ideal plug flow model were in

the same order of magnitude of the physical model. It was concluded that the ideal plug flow model

determines the upper limit of the conversion rate, whereas the flexible CFD model for investigating

different scenarios and their effect on the overall process efficiency proved very useful and powerful.

iii

Samenvatting

De huidige aanpak voor bioprocessen, ontworpen om chemische precursoren te maken met behulp van

enzymes, kent nog steeds een lage productiviteit en een lage proces intensiteit in vergelijking met

traditionele chemische processen. Het wijdverspreid gebruik van deze bioprocessen wordt gehinderd

door deze tekortkomingen. Het zwaartepunt van deze thesis ligt bij de modelgebaseerde analyse van het

transaminase proces in microreactoren. Microreactoren worden gebruikt voor een continue synthese van

deze producten, welke gebruik maken van op laboratorium schaal geoptimaliseerde processen, toegepast

in een industriele context door middel van parallelle opschaling. Het doel van de thesis was om een

model te bouwen de concentratie aan het einde van de microreactor accuraat kan voorspellen voor een

verscheidenheid van reactoropstellingen en procescondities. De reactie werd geanalyseerd door uit te

gaan van een fysisch model (numerieke stromingsmechanica (CFD) gecombineerd met enzymkinetiek),

geımplementeerd in de open source CFD bibliotheek OpenFOAM. Het creeren van de mesh voor het

CFD model is uitgevoerd aan de hand van een flexibele Python scripting omgeving die functioneert als

een wrapper rond de gebruikte software voor geometrie en mesh: Salome. Nadien werd een scenario

analyse tool aangemaakt in Python die meerdere simulaties kan lopen voor eenzelfde geometrie en

ook toelaat om deze achteraf onderling te vergelijken. Hierna werd dit CFD model vereenvoudigd

tot een ideaal gemengd doorstroom model, een propstroom model, een Tanks-In-Series model, en een

Compartimenteel Model. Het hydraulisch gedrag van de reactor kon enkel accuraat beschreven worden

met behulp van het Tanks-In-Series model. Echter, de voorspelling van de uitlaat concentraties kon

gebeuren met behulp van het propstroom model. Deze gaf resultaten in dezelfde grootteorde als het

fysisch model. Er wordt besloten dat het mogelijk is om met het ideale propstroom model de bovenlimiet

van de omzettingsgraad te bepalen, waarna de flexibele CFD omgeving zeer nuttig is om verschillende

scenarios te analyseren en hun effect op de algemene procesefficientie te onderzoeken.

v

Contents

Dankwoord i

Summary iii

Nederlandse samenvatting v

Contents viii

List of Symbols ix

List of Abbreviations xiii

List of Figures xiv

List of Tables xvii

1 Problem statement 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Objectives of this research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.4 Outline: The roadmap through this dissertation . . . . . . . . . . . . . . . . . . . . . . . 2

2 Literature study 3

2.1 Enzyme Kinetics: ω-transaminases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.1 Enzyme nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.2 Biochemical features of ω-transaminases . . . . . . . . . . . . . . . . . . . . . . . 3

2.1.3 Production of optically pure amines . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.4 Process challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.5 Non-aqueous solvents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 Microreactor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Basic concepts of micro-reaction technology . . . . . . . . . . . . . . . . . . . . . 14

2.2.2 Synthetic micro-reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.3 Navier-Stokes system in microreactors . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.4 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Materials and Methods 23

3.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Computational Fluid Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Discretisation of the Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . 23

vii

3.2.2 Pressure - velocity coupling: the PISO- and SIMPLE-loop . . . . . . . . . . . . 24

3.2.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2.4 Kinetic model of ω-TA reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.5 Solution procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3 Simplified models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.1 Mixed flow model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.2 Plug flow model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.3 Dispersion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.4 Tanks-In-Series model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3.5 Compartmental Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.1 Python . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.2 OpenFOAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4.3 Salome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.4.4 ParaView . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Results 35

4.1 Uncertainty on rate equation of the kinetic model . . . . . . . . . . . . . . . . . . . . . 36

4.2 Computational fluid dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.1 Flexible mesh generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.2.2 Python package: scenario analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.3 Mesh dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.2.4 Residence time distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.2.5 Enzyme kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.3 Simplified Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.1 Mixed flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3.2 Plug flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.3 Tanks-In-Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3.4 Compartmental Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Discussion and perspectives 59

5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Bibliography 63

A Error propagation 67

B Theoretical velocity profile 69

viii

List of Symbols

Ω(t) parcel of fluid for which no material enters or leaves

∇ Nabla or del operator

A coefficient matrix

aP central coefficient

αp pressure under-relaxation factor

αU velocity under-relaxation factor

A∗ estimation of the coefficient matrix

b vector with constants

C volumetric concentration

Co Courant number

D dispersion coefficient

D diffusion constant

DH hydraulic diameter

dA surface elements

∆G change in Gibbs free energy

∆t time step

∆x grid spacing

dV volume elements

f other body forces: gravitational and centrifugal

H(U) transport part

I summation of the quantity i over a control volume

i quantity (scalar, vector or tensor) of the fluid

K thermodynamic equilibrium constant

KAi inhibition parameter for solute A

KAM Michaelis Menten parameter for solute A

KBM Michaelis Menten parameter for solute B

KEQ chemical equilibrium constant

ix

Kfcat catalytic turnover coefficient of the forward reaction

KISi uncompetitive substrate inhibition parameter

Kn Knudsen number

KPM Michaelis Menten parameter for solute P

KPPi core inhibition parameter for solute PP

KPPM Michaelis Menten parameter for solute PP

KPPSi product inhibition constant

KPQM Michaelis Menten parameter for solute PQ

KPQSi product inhibition constant

KQi inhibition parameter for solute Q

KQM Michaelis Menten parameter for solute Q

Krcat catalytic turnover coefficient of the reverse reaction

KSAi core inhibition parameter for solute SA

KSAM Michaelis Menten parameter for solute SA

KSASi substrate inhibition constant

KSBi core inhibition parameter for solute SB

KSBM Michaelis Menten parameter for solute SB

KSBSi substrate inhibition constant

λ second viscosity

λIM intermolecular length for the fluid molecules

λM molecular mean free path length

Ma Mach number

µ kinematic viscosity

N number of tanks

n outward-pointing unit-normal

p kinematic pressure, ratio of pressure and a constant density

Pe Peclet number

pKa acid dissociation constant at logaritmic scale

pnew approximation of the pressure field, to be used in the next momentum predictor

pold current pressure field used in the momentum predictor

PP product P (1-phenylethylamine)

pp solution of the pressure equation

PQ product Q (acetone)

Q volumetric fluid flow

Qin inlet fluid flow

Qout outlet fluid flow

R universal gas constant

r volumetric source or sink term

Re Reynolds number

ρ density of the fluidum

x

S outward pointing area vector

SA substrate A (acetophenone)

SB substrate B (isopropylamine)

T total stress tensor

Θ temperature

Usound velocity of sound waves in the fluidum

U velocity of the fluidum

Uf velocity of the fluidum at the face f

Uavg average flow velocity

V Volume of the reactor

x vector with variables

xi

List of Abbreviations

α-TA α-transaminase

ω-TA ω-transaminase

APH acetophenone

AroAT aromatic-amino-acid transaminase

ASCII American Standard Code for Information Interchange

Asp aspartic acid, an amino acid

AspAT aspartate transaminase

BSD Berkeley Software Distribution, a UNIX-like operating system

CAD Computer Aided Design

CD Central Differencing

CFD Computational Fluid Dynamics

CFL Courant-Friedrichs-Lewy stability condition

CLEA cross-linked enzyme aggregate

CM Compartmental Model

CSTR Completely Stirred Tank Reactor

DKR Dynamic kinetic resolution

DRIE Deep Reactive Ion Etching

E-PLP the active form of an ω-TA enzyme

E-PMP pyridoxamine 5’-phosphate form of an enzyme

EOF electroosmotic flow

EPF electrophoretic flow

Glu glutamic acid, an amino acid

GPL GNU General Public License

GUI Graphical User Interface

IR infrared

IS(c)PR In situ (co-)product removal

LGPL GNU Lesser General Public License

xiii

LIGA LIthographie Galvanoformung Abformung

MBA 1-phenylethylamine

ODE ordinary differential equation

OpenFOAM Open-source Field Operation And Manipulation toolbox

PISO Pressure Implicit with Splitting of Operator

PLP pyridoxal 5’-phosphate, a derivate of vitamin B6

PMP pyridoxamine 5’-phosphate

PSF Python Software Foundation

QUICK Quadratic Upwind Interpolation for Convective Kinetics

RTD Residence Time Distribution

SIMPLE Semi-Implicit Method for Pressure Linked Equations

TA transaminase

TIS Tanks-In-Series model

TVD Total Variance Diminishing

UD Upwind Differencing

UV ultraviolet

xiv

List of Figures

2.1 Schematic representation of the Ping Pong Bi Bi reaction scheme (Cleland notation).

The following abbreviations are used: Enzyme (E), amino substrate (A), keto product

(P), keto substrate (B) and amino product (Q). The reaction rate constants are denoted

with the letter k, followed by the index number of the reaction. A positive number

stands for the forward reaction, a negative number for the reverse reaction (Modified

from: Biswanger (2008)). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 ω-transaminase (ω-TA) reaction pathway of a) oxidative deamination of an amino donor

(shown in a box) which converts active form of an ω-TA enzyme (E-PLP) to pyridoxam-

ine 5’-phosphate form of an enzyme (E-PMP) and b) reductive amination of an amino

acceptor (in a triangle) which accompanies regeneration of E-PLP (Malik et al., 2012). . 4

2.3 Illustration of the large (L) and small (S) binding pockets. In this case, the 3-dimensional

structure of the enzyme was altered to yield an enzyme with a different substrate speci-

ficity (Mathew and Yun, 2012). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Transaminase reaction systems. (1) During kinetic resolution, transaminase converts a

racemic amine into an optically pure enantiomer and ketone with a theoretical yield

of 50%. (2) During asymmetric synthesis, pro-chiral ketones are converted into optically

pure enantiomers at a theoretical yield of 100% in the presence of a favourable equilibrium.

(3) During the deracemization reaction, the racemic amine is converted into an optically

pure enantiomer with a theoretical yield 100% (Mathew and Yun, 2012). . . . . . . . . . 6

2.5 Dynamic Kinetic Resolution (DKR) and deracemation processes. (a) DKR by combining

an enantioselective transformation with an in situ racemation step. (b) Deracemation

of racemic α-amino acids by combining an enantioselective amine oxidase with a non-

selective chemical reducing agent (Turner, 2004). . . . . . . . . . . . . . . . . . . . . . . 7

2.6 King-Altman representation of the ω-TA reaction mechanism proposed by Al-Haque et al.

(2012). SA is the amine acceptor, SB is the amine donor, PP is the amino product and

PQ is the keto-co-product. Modified from: Al-Haque et al. (2012) . . . . . . . . . . . . 9

2.7 Excess of amine donor required in function of the equilibrium constant (K) for a certain

conversion efficiency (85%, 90% or 95%). Example: for increasing the total conversion of

90% to 95% at a K value of 10−2, the amount of excess donor required needs to be more

than doubled (Tufvesson et al., 2011). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.8 Concentration of co-product in solution in function of the equilibrium constant (K) for

a certain conversion efficiency (85%, 90% or 95%). For a constant conversion efficiency,

the required concentration of (co-)product in the solution is directly proportional to the

equilibrium constant, K. Lower values of K, require a lower (co-)product concentration

(Tufvesson et al., 2011). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.9 An example configuration of the microreactor used in this thesis (Micronit, 2014). . . . 15

2.10 Classification of flows from free molecular flow to continuous flow in function of the

Knudsen number (Li, 2006). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

xv

3.1 Effect of numerical dissipation and dispersion on wavelike solutions: (a) exact solution, (b)

numerical solution with strong dissipation, (c) numerical solution with strong dispersion

(Zikanov, 2010). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Representation of ideal plug flow (left) and mixed flow (right) behaviour (Levenspiel, 1972) 30

3.3 Representation of the velocity profile in a plug flow model (left), and in the dispersion

model (right) (Levenspiel, 1972) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4 Representation of the tanks-in-series model (Levenspiel, 1972) . . . . . . . . . . . . . . . 31

3.5 Visual representation of the tailing phenomenon: the form is similar to an asymmetrical,

skewed bell curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.1 Overview of the methodology followed within this thesis to go from a physical model to

a simplified model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.2 Experimental setup for the forward reaction, the reaction conditions are as follows: E0 =

1.8 g/L, CSA = 1.7mM, CSB = 1000mM, CPP = 0.5mM, 2mM PLP, 100mM phosphate

buffer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Experimental setup for the reverse reaction, the reaction conditions are as follows: E0 =

3.6 g/L, CSA = 0mM, CPP = 5mM, CPQ = 1000mM, 2mM PLP, 100mM phosphate buffer 38

4.4 Visualisation of the percentage error on the concentrations for the recreated simulations

of Al-Haque et al. (2012). The full lines denote the error on the concentration in function

of time, the dotted lines represent the time averaged error on the concentration profile. . 38

4.5 Details about the generated non-equidistant mesh: visualisation of the mesh (left), and

cell centres on the width of the reactor (right) . . . . . . . . . . . . . . . . . . . . . . . . 41

4.6 Theoretical velocity profile in the microreactor, combined with the experimental results

for a mesh with 12 and 30 cells on the width of the reactor. The boundary condition at

the reactor wall is the no-slip condition: the velocity is equal to zero at the reactor wall.

The form of the velocity profile is independent of inlet flow, hence the use of a normalised

velocity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.7 Results of the tracer test simulations on the small test case: concentration at the outlet

and relative cumulative mass in function of time, and number of cells on the width of the

reactor for substrate SA (left) and SB (right). The gray area visualises a 5% error band

around the baseline solution (100 cells along the width). . . . . . . . . . . . . . . . . . . 45

4.8 Results of the tracer test simulations on the small test case: cumulative mass percentage

in function of time, and number of cells on the width of the reactor for substrate SA (left)

and SB (right). The gray area visualises a 5% error band around the baseline solution

(100 cells along the width). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.9 Residence Time Distribution (RTD) test with a mesh with 100 cells on the width of the

reactor. The concentration shown in the figure is for the slow diffusion solute SA. Even

at this resolution, small oscillations in the concentration are observed . . . . . . . . . . . 47

4.10 Results of the tracer test simulations on the small test case: cumulative mass percentage

in function of time, and time step for substrate SA (left) and SB (right) for a mesh size

equal to 20 cells on the width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.11 Results for the tracer test simulations on the full scale reactor for a theoretical residence

time of 10.3 (figure 4.11a), 20.6 (figure 4.11b), and 30.09 minutes (figure 4.11c): molar

concentration of the solute SA and SB in function of time. Figure 4.11d is an illustration

of the extraction procedure for obtaining the discrete cell velocities used to create the

dotted lines representing the theoretical residence time for that velocity in the other three

figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

xvi

4.12 Normalised experimental concentrations, and numerical simulations for the determina-

tion of diffusion constants (Bodla et al., 2013). The three solutes exhibit a similar break-

through curve, with the moment of breakthrough at almost the same time. Yet, for ace-

tophenone (APH), the curve converges to a lower level then 1-phenylethylamine (MBA).

Two numerical fits are proposed with two different diffusion constants. The highest dif-

fusion constants predicts the moment of breakthrough the most accurately, which is the

characteristic one should focus on to estimate the diffusion constant. The lower final

concentrations suggests sorption or loss of mass within the reactor. . . . . . . . . . . . . 50

4.13 RTD tracer test for a residence time of 10.3 minutes, the original simulation of sub-

strate SB is plotted together with the simulation for substrate SA with the new diffusion

constant: DSA = 8.27 · 10−12 m2 s−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.14 Result of a tracer test with the Mixed flow model. At time 0, a pulse is added to

the reactor. One can see a sharp increase in solute concentration, which is afterwards

decreasing slowly with first order kinetics. . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.15 Steady state reaction of the enzyme kinetics. The dotted lines resemble the 99% level of

the steady state concentration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.16 Comparison of the RTD profile obtained with Computational Fluid Dynamics (CFD)

simulations and the profile obtained with the Tanks-In-Series model (TIS) model. In

figures 4.16a, 4.16c, and 4.16d, the optimised number of tanks is shown in comparison

with the CFD simulation for a residence time of 10.3, 20.6, and 30.9 minutes. Figure

4.16b shows the outcome of the tracer test for different number of tanks for a residence

time of 10.3 minutes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.17 Visual representation of the Compartmental Model . . . . . . . . . . . . . . . . . . . . . 56

4.18 Simulation results for the Compartmental Model (CM): in figure 4.18a, the effect of the

volume ratio is shown for a constant flow factor equal to 0.01. In figure 4.18b, the effect

of the flow factor on the RTD curve is visualised for a constant volume ratio equal to

0.5. Figure 4.18c visualises the CM simulations for a high volume ratio equal to 0.99 in

function of the flow factor. Figure 4.18d is a magnification from figure 4.18c at the top

of the RTD curve. All simulations are perform with the number of tanks in the largest

dimension equal to 1400, i.e. 2800 tanks in total. . . . . . . . . . . . . . . . . . . . . . . 57

xvii

List of Tables

2.1 Examples of additional equations needed to fully describe the fluid regime. . . . . . . . 20

2.2 Different Knudsen regimes for fluids. (Gad-el hak, 1999) . . . . . . . . . . . . . . . . . 21

4.1 Parameter values and confidence intervals for the Al-Haque kinetic model: equation 3.6

(Al-Haque et al., 2012). The equilibrium constant (KEQ) and its confidence interval is

taken from Tufvesson et al. (2012). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.2 Formulas for the calculation of the propagation of errors (Bevington and Robinson, 2002). 36

4.3 Output of the Open-source Field Operation And Manipulation toolbox (OpenFOAM)

checkMesh utility applied on the fullscale geometry. . . . . . . . . . . . . . . . . . . . . 39

4.4 Overview of the maximum and mean error on the simulated fluid velocity compared to

the analytical velocity, and CPU time needed for calculation for different number of cells

on the width of the reactor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.5 Diffusion constants for the solutes used in the Biointense project, modified from Bodla

et al. (2013). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.6 Simulation time for the different mesh sizes. . . . . . . . . . . . . . . . . . . . . . . . . 46

4.7 Simulation time and Courant number for the different time steps. The normalised sim-

ulation time is the simulation time of the case for that specific time step divided by the

simulation time of the base case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.8 Overview of the boundary conditions for the different scenarios set up for the CFD cal-

culations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.9 Results of the 2D CFD simulation for different scenarios and different residence times. 53

4.10 Results for the simplified models (mixed flow, ideal plug flow, and TIS), and the physical

model (CFD) for different residence times. . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.1 Comparison of the CPU time for the different models: CFD, mixed flow, ideal plug flow,

and TIS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

xix

CHAPTER 1Problem statement, research objectives

and outline

1.1 Introduction

At present, bioprocesses designed to produce chemical precursor products by means of enzymatic pro-

duction still suffer from low productivity and low process intensity in comparison with more traditional

chemical processes. The widespread use of these bioprocesses is hampered by these shortcomings. A

possible solution to enhance the efficiency of the overall process is to integrate the production and

separation, which are now being implemented sequentially without direct coupling of both processes.

This integration of processes allows to move from batch to continuous production, thereby reducing the

required reactor volumes. In practical applications, this can be achieved by combining a microreactor

(used for production) and a membrane (used for separation).

This thesis is part of the Biointense project. Biointense is a single stage knowledge based bio economy

(KBBE) collaborative project. It is EC-funded through the 7th Framework Cooperation Programme

that has the strategic objective of supporting research activities to gain or consolidate leadership in

key scientific and technology areas and to encourage international competitiveness whilst promoting

research that supports EU policies. The main objectives in Biointense are to increase biocatalyst pro-

ductivity and process intensity. This will result in economically feasible processes through integration

and intensification and should also shorten the development times by developing optimized tools and

protocols that can be widely applicable in industry.

1.2 Problem Statement

The hydrodynamics of microreactors can be described using CFD, which can accurately predict local

concentrations of the different substrates, end-products and enzymes. However, this is only true in the

absence of process kinetics. The presence of enzymes in the reactor will lead to conversion of substrate

to end-product and the incorporation of the membrane in the reactor will lead to the extraction of the

end-product. Therefore, the conversion kinetics need to be coupled with the CFD model in order to

get an appropriate model to meet the objective. This coupled model can be used to avoid potential

problems or efficiency losses in an early stage of reactor design, but also for process optimisation and

control.

1.3 Objectives of this research

The goal of this thesis is to couple the CFD model with different kinetic models. To counteract computa-

tional load, a model reduction of the coupled CFD - kinetic model will be investigated in order to obtain

2 1.4 OUTLINE: THE ROADMAP THROUGH THIS DISSERTATION

simplified models. This allows performing calculations more efficiently and at a lower computational

expense which is useful for exploring the system design and to test different control strategies.

1.4 Outline: The roadmap through this dissertation

First, in section 2, an overview of the literature is given regarding enzyme kinetics, microreactor tech-

nology, and fluid dynamics. Second, in section 3, this knowledge is applied in a modelling environment.

The section is focused on CFD, and simplified models. To conclude that section, an overview of the used

software is given. Next, in section 4, the results of the model based analysis are elucidated. Finally, in

section 5, the discussion and perspectives are given.

CHAPTER 2Literature study

2.1 Enzyme Kinetics: ω-transaminases

Currently, the production of optically pure amines is largely based upon the resolution of racemates, in

practice this is done by recrystallisation of diastereomeric salts or by enzyme-catalysed kinetic resolution

of racemic substrates using lipases and acylases. However, these resolution approaches are inherently

inefficient (maximum yield equal to 50%) and is therefore increasingly viewed as non-competitive from

an economic perspective.

By shifting towards enzyme processes it is theoretically possible to obtain optically pure amines with

both a process and an enantiomeric yield of 100%. These enantiomeric pure chiral amines are used as re-

solving agent, catalyst for asymmetric synthesis, chiral auxiliaries/bases and because of their pronounced

biological activity, they are used as intermediate for pharmaceuticals and agrochemicals. Therefore they

are of an increasing economical value and in the area of interest of both pharmaceutical and fine chemical

industry (Turner et al., 2011).

2.1.1 Enzyme nomenclature

Transaminases are enzymes that typically require pyridoxal 5’-phosphate, a derivate of vitamin B6

(PLP) as a prosthetic group for the catalytic reaction. The field of transaminases is subdivided into

four subgroups based on the sequence alignment of 51 transaminases. Subgroups I, III and IV (called

α-transaminases (α-TAs)) exclusively accept α-amino acids and α-keto acids as substrate pairs. The

enzymes in subgroup II are the so-called ω-transaminases (ω-TAs) and represent the transaminases that

can transfer the amino group from a carbon atom that does not carry a carboxyl group (Malik et al.,

2012).

2.1.2 Biochemical features of ω-transaminases

Complicated enzyme properties repeatedly cross the path of bio-catalytic process design. It is a ne-

cessity to overcome these limitations for clearing the path to genuine industrial process development.

A thorough understanding of the biochemistry of a given enzyme is the key factor to accomplish this

goal. In the following three paragraphs, a brief overview of the relevant biochemical properties will be

provided. This subsection is based on the review article of Malik et al. (2012) unless stated otherwise.

Reaction chemistry

The reactions under study are bisubstrate reactions. The Ping Pong Bi Bi reaction will be discussed,

as this is the reaction type typical for transaminases. A bisubstrate system where the first product (P)

is released before the second substrate (B) is bound is called Ping Pong. The Ping Pong mechanism

4 2.1 ENZYME KINETICS: ω-TRANSAMINASES

applied to ω-TA gives the following reaction steps: the amino substrate (A) binds to the enzyme, then

the keto product (P) is released before the second substrate, ketone (B), binds to the modified enzyme.

The last step is the release of the amino product (Q), in this step the enzyme goes back to its original

structure (Leskovac, 2003). Schematically the reaction is visualised in figure 2.1.

Figure 2.1: Schematic representation of the Ping Pong Bi Bi reaction scheme (Cleland

notation). The following abbreviations are used: Enzyme (E), amino substrate (A),

keto product (P), keto substrate (B) and amino product (Q). The reaction rate

constants are denoted with the letter k, followed by the index number of the reaction.

A positive number stands for the forward reaction, a negative number for the reverse

reaction (Modified from: Biswanger (2008)).

Following the global mechanistic description of the reaction in the paragraph above, a more profound

description regarding the chemical reaction is elucidated below. The ω-TA reaction pathway can be

divided in two half reactions: the oxidative deamination of an amine donor and the reductive amination

of an amino acceptor as shown schematically in figure 2.2. To have a working enzyme, PLP is added

in order to form a Schiff base with an active site lysine, this is called the PLP form of the enzyme

(E-PLP). In the first half reaction, the amino group from a donor (A) is transferred to E-PLP. The

result of this half reaction is a corresponding ketone (P) and the pyridoxamine 5’-phosphate form of

the enzyme (E-PMP). In the second half reaction, the amino group of PMP is relocated to an amino

acceptor substrate (B), causing the formation of an amino product (Q) and the regeneration of E-PLP.

Figure 2.2: ω-TA reaction pathway of a) oxidative deamination of an amino donor

(shown in a box) which converts E-PLP to E-PMP and b) reductive amination of

an amino acceptor (in a triangle) which accompanies regeneration of E-PLP (Malik

et al., 2012).

CHAPTER 2 LITERATURE STUDY 5

Substrate specificity and stereoselectivity

As stated in section 2.1.1, the substrate specificity of ω-TA is much broader then α-transaminase (α-

TA). A number of structurally diverse primary amines have been demonstrated to show amino donor

reactivities. Most of the reactive amino donors were found among arylalkyl amines rather than aliphatic

amines. Pertaining to the amine acceptor, high reactivities were observed with α-keto acids (typically

pyruvate) and aldehydes (typically propanal and benzaldehyde). The reactivity of ketones as amino

acceptor is significantly lower than pyruvate, which is one of the most serious hurdles in the asymmetric

synthesis of chiral amines.

Active site model

The natural occurring ω-TA identified in research are from different microbial origin, yet the substrate

specificity’s were found found to be remarkably similar. On the basis of the relationship between

substrate structure and reactivity, an active site model of ω-TA from Vibrio fluvialis JS17 was proposed

by Shin and Kim (2002). A two site binding model was proposed to explain the substrate specificity

as well as the stereoselectivity. This model consists of a large and a small pocket (figure 2.3). The

key determinants were found to be dual recognition of hydrophobic and carboxylate groups in the large

pocket and strong repulsion of the carboxylate group in the small pocket. The small pocket contains

the steric constraints which are vital for substrate recognition and disallows entry of a substituent

larger than an ethyl group. This dual recognition mode of ω-TA is reminiscent of the side chain

pockets of aspartate transaminase (AspAT) and aromatic-amino-acid transaminase (AroAT). The former

example also operates via dual substrate recognition, in other words, the enzyme is able to recognise

and selectively bind two amino acids (Aspartic acid and Glutamic acid) with different side chains. This

knowledge of the three-dimensional enzyme structure is crucial for obtaining an improved understanding

of the substrate specificity. The two site model is visualised in figure 2.3. In the former figure, an

illustration is provided of different steps in the alternation process of an enzyme. By altering the

enzyme structure, the recognition and repulsion of the large and small pocket is changed to yield a

different specificity.

Figure 2.3: Illustration of the large (L) and small (S) binding pockets. In this case,

the 3-dimensional structure of the enzyme was altered to yield an enzyme with a

different substrate specificity (Mathew and Yun, 2012).

2.1.3 Production of optically pure amines

The production of optically pure amines is subdivided in three methods: kinetic resolution, asymmetric

synthesis and deracemization. This section provides a brief explanation of these technique for the

production of optically pure amines. Each method has its advantages and disadvantages, possible ways

to circumvent these drawbacks are explained in the next section (section 2.1.4).

Kinetic resolution

In this method, racemic amines are converted into enantiomerically pure amines with a theoretical

yield of 50%. The production can be categorised into two classes: the first uses hydrolysed catalysed


Figure 2.4: Transaminase reaction systems. (1) During kinetic resolution, transam-

inase converts a racemic amine into an optically pure enantiomer and ketone with

a theoretical yield of 50%. (2) During asymmetric synthesis, pro-chiral ketones are

converted into optically pure enantiomers at a theoretical yield of 100% in the pres-

ence of a favourable equilibrium. (3) During the deracemization reaction, the racemic

amine is converted into an optically pure enantiomer with a theoretical yield 100%

(Mathew and Yun, 2012).

aminolysis in a non-aqueous medium, the second uses transaminases (TAs) in an aqueous medium. As

the scope of this thesis lies within the field of ω-TA, only the latter case will be discussed.

First, the unwanted enantiomer is removed by ω-TA by converting it to the corresponding ketone. In the

example given in figure 2.4, the desired amine is the (R)-enantiomer. An (S)-ω-TA is used to remove the

unwanted amine from the racemic mixture, however one obtains a mixture of the pure (R)-enantiomer

and the ketone (Mathew and Yun, 2012).

Applications which utilise ω-TA has increased since the mid 1990s. Its main disadvantage is product

and substrate inhibition by inhibitory ketones (Koszelewski et al., 2010).

Asymmetric synthesis

In the asymmetric synthesis technique an amino group is transferred to prochiral ketones yielding

enantiomerically pure amines. This process is of great interest because of the potential yield of 100%,

twice the theoretical yield of kinetic resolution (Koszelewski et al., 2010). Nevertheless similar to the

kinetic resolution, product inhibition and unfavourable thermodynamic equilibrium forestall large scale

industrial applications. For ideal asymmetric synthesis, the amine donor and acceptor should have high

reactivity. Analysis of protein-ligand docks can help in the search for ideal substrate pairs for this

matter (Mathew and Yun, 2012).

Deracemation

As the name suggests, in deracemization, a racemic mixture is converted into a single enantiomer with

100% theoretical yield. This can be realised by two approaches: Dynamic Kinetic Resolution (DKR)

and a two-step one-pot process.

The one-pot, two-step process works as follows: first the kinetic resolution of the racemic mixture is

performed using ω-TA. This is executed by a stereoselective amination using an opposite enantioselective

ω-TA. In this way the unwanted enantiomer is converted into its ketone analogon. However one is


interested in the (R) or (S) enantiomer, therefore the ketone analogon needs to be converted to the

enantiomer of interest.

Next to enzymes, metal catalysts are also used in DKR. This process is widely used for the production

of secondary alcohols. For amines, certain issues arise: amines have the tendency to bind metal ions

and therefore hamper the reaction. Higher chemoselectivity and intrinsic mild reaction conditions make

enzymes the ideal catalyst. The presence of two optically opposite active biocatalysts and the addition

of an external ketone to facilitate the amino group transfer between the substrates enhance the speed

of racemation (Mathew and Yun, 2012). The DKR procedure is visualised in figure 2.5

Figure 2.5: DKR and deracemation processes. (a) DKR by combining an

enantioselective transformation with an in situ racemation step. (b) Deracema-

tion of racemic α-amino acids by combining an enantioselective amine oxidase with

a non-selective chemical reducing agent (Turner, 2004).

2.1.4 Process challenges

In this subsection the process challenges of enzymatic production will be discussed with a special focus

on ω-TA. In the reaction step, three strategies are used to obtain the target chiral amine: kinetic

resolution of a racemic amine, deracemation or direct asymmetric synthesis. The challenges discussed

here are solely focused on direct asymmetric synthesis, as this is in line with the research objective on of

this thesis. Many complications encountered with transaminases are also common to other biocatalytic

processes. Consequently, many parallels can be drawn to other biocatalytic reactions (Tufvesson et al.,

2011).

Thermodynamic limitations of the reaction system

First, this section starts with the issues sprouting from a thermodynamic view. Knowledge about the

thermodynamics of the system will yield information on which solutions are economically feasible on an

industrial scale.

The transamination reaction is reversible and the maximum achievable conversion is thus determined by

the initial concentrations and the thermodynamic equilibrium constant (K) of the reaction. A general

reversible reaction is given in equation 2.1, its equilibrium constant is defined by equation 2.2 (the

brackets denote the activity of the compound in the reaction medium). K is in turn determined by the

change in Gibbs free energy (∆G) for the reaction. This change is equal to the difference in ∆G between

the products and the reactants and is equal to the negative product of the universal gas constant (R),

the temperature (Θ) and the natural logarithm of the equilibrium constant (Equation 2.3). The nature

of the reactants determines the value of ∆G.

αA+ βB... ρR+ σS... (2.1)


K =RρSσ...AαBβ ...

(2.2)

∆G = Gproducts −Greactants = −RΘ lnK (2.3)

For instance for the amine transfer from an amino acid to an alpha keto acid, the change in Gibbs free

energy is small, resulting in a K-value around unity. Considering the amino transfer from an amino

acid to acetophenone, the equilibrium is strongly shifted towards the products side, i.e. the K-value is

larger than unity. From these examples it can be deduced that different reactants results in different

equilibria and different approaches are required to cope with the thermodynamic limitations.

Enzymatic, or more general biocatalytic reactions are considered to obey mixed order reaction kinetics,

this means a reaction order between between zero and first. This type of kinetics is similar to the

Langmuir-Hinshelwood model, also denoted as the Michaelis and Menten kinetics (Al-Haque et al.,

2012).

This kinetic model by Michaelis and Menten is in fact the most simplified form of the quasi-steady-state

assumption, also called the pseudo-steady-state-hypothesis. The idea behind this assumption is that

the concentration of intermediate complexes and their relative ratios do not change on the time-scale of

product formation. In other words, the intermediate complexes are formed at the same rate at which

they are decomposed Briggs and Haldane (1925).

The general reaction scheme for a reaction with one ligand, one catalytic site, and one enzyme-substrate

complex obeying Michaelis and Menten kinetics is shown in equation 2.4. In this equation, E represents

the free enzyme, S for the substrate, P for the product, and ES for the enzyme-substrate complex.

This type of kinetics assumes that only the early components of the reaction are at equilibrium, this

assumption is called the quasi-equilibrium or rapid equilibrium assumption. This assumption is shown

in equation 2.4 by the double arrow between E and S, and ES (reversible reaction) and the single arrow

pointing from ES to E and P (irreversible reaction). From this it follows that the overall reaction rate

is limited by the breakdown of ES to E and P. The general procedure described above can be used to

obtain equilibrium and velocity equations for all rapid equilibrium systems, including those involving

multiple ligands (Segel, 1993).

E + SKS−−−− ES

kp−→ E + P (2.4)

Al-Haque et al. (2012) has used the King-Altman method for deriving the reaction scheme. The King-

Altman method is a quick, schematic method for deriving quasi-steady-state equations for complex

enzymatic reactions by making use of a set of geometric rules designed to simplify an algebraic procedure.

The requirement of the enzymatic reaction is that it should only consist of a series of reactions between

different forms of the enzyme and it is not applicable to non-enzymatic reactions, mixtures of enzymes,

and reactions that contain non-enzymatic steps. In practice, the rate equations are written down for the

n enzyme forms, for which n-1 equations are independent. Combining n-1 independent equations with

the mass balance for the different enzyme forms, a solvable system of n equations with n unknowns is

set up (Cornish-Bowden, 2004). The King-Altman representation of the ω-TA reaction mechanism is

shown in figure 2.6.

The general form of an equilibrium controlled bisubstrate reaction can be formulated as equation 2.5.

The general rate equation proposed by Al-Haque et al. (2012) is shown in equation 2.6. This equation

consists of seven parameters including the catalytic turnover of the reaction (Kfcat, K

rcat), the Michealis

Menten parameters (KAM , KB

M , KPM , KQ

M ), inhibition parameters (KAi , KQ

i ) that are derived from

the core mechanism, and uncompetitive substrate inhibition parameter (KISi) due to the formation of

nonproductive complexes. Al-Haque used a method described by Segel (1993) for deriving this general

rate equation. In this method it is assumed that intermediate enzyme complexes cannot be measured.

From this assumption, a simplification of the model is executed: the kinetic constants are grouped into

the seven parameters mentioned above.

A+B ↔ P +Q (2.5)


Figure 2.6: King-Altman representation of the ω-TA reaction mechanism proposed

by Al-Haque et al. (2012). SA is the amine acceptor, SB is the amine donor, PP is

the amino product and PQ is the keto-co-product. Modified from: Al-Haque et al.

(2012)

rQ = −rA =E0K

fcatK

rcat

([A] [B]− [P ][Q]

KEQ

)(1 + [A]

KISI

)( Kr

catKBM [A] +Kr

catKAM [B] +

KfcatK

QM

KEQ[P ] +

KfcatK

PM

KEQ[Q] +Kr

cat [A] [B](2.6)

+KfcatK

QM

KAi KEQ

[A] [P ] +Kfcat

KEQ[P ] [Q] +

KrcatK

AM

KQi

[B] [Q] )

However, Ishikawa et al. (1998) states that the pseudo-steady-state treatment of complicated reactions

mechanisms leads to equations and constants so complex that the basic kinetic properties of the mech-

anism may be obscured. Segel (1993) has mentioned that in the most general case of multi-substrate

kinetics it is not possible to group elementary rate constants into convenient constants (Michaelis con-

stants and inhibition constants). If the substrate saturation curve in the complex cases is not a rect-

angular hyperbola, it does not make sense to speak about a half-maximum velocity and the simple

Michaelis and Menten mechanism is not applicable.

Clearly, there is lack of a straightforward consensus in the literature about this type of enzyme kinetics.

In the Biointense-project, the kinetic model of Al-Haque et al. (2012) is used. As the research is still

ongoing, the model of Al-Haque et al. (2012) is considered valid until proven otherwise.

Equation 2.7 is the representation of the chemical equilibrium using the Haldane relationship. Evolution

has made sure that enzymes evolved to operate efficiently in a natural environment. The conditions

in the natural environment (low concentration) are in contrast with industrial applications, where high

substrate concentrations are used to ensure manageable costs and to ease the downstream processing.

It is essential that the effects of higher concentrations on the enzymes kinetics are well known. Under

these industrial conditions inhibition is observed. This inhibition has three main origins: substrate,

product or other components in the reaction medium (Al-Haque et al., 2012). Inhibition can be divided

into three groups: competitive inhibition, non-competitive inhibition, and uncompetitive inhibition. In

competitive inhibition, the binding of the inhibitor (mostly reversible) on the active site on the enzyme

prevents binding of the substrate and vice versa. In non-competitive inhibition, the activity of the

enzyme is reduced by the binding of the inhibitor. The affinity of the inhibitor for the enzyme is not

a function of the substrate binding on the enzyme. As a contrast to the former type of inhibition, in

uncompetitive inhibition, the inhibitor only binds the enzyme substrate complex (Segel, 1993).


KEQ =

(Kfcat

Krcat

)2KPMK

QM

KAMK

BM

=

(Kfcat

Krcat

)KPi K

QM

KAi K

BM

=

(Kfcat

Krcat

)KPMK

Qi

KAMK

Bi

=KPi K

Qi

KAi K

Bi

(2.7)

Next follows a concise description of a couple of strategies to alleviate these thermodynamic and in-

hibitory issues in ω-TA reactions:

• Addition of excess amine donor:

By far the easiest option to shift the equilibrium towards the product side is by addition of excess

amine donor. However, this strategy is only applicable for the cases where the equilibrium is

only slightly unfavorable. For low K-values, the amount of donor amine needed for high rates of

conversion exceeds the solubility of the amine donor. As consequence, adding an excess of amine

donor for process configurations with a K-value lower than 10−2 is not sufficient to reach the

desired conversion rate. From an economic viewpoint it can be seen that addition of excessive

amounts of amine donor leads to an overly expensive and not economically viable process. In

figure 2.7, the relation between the excess of donor needed and the equilibrium constant for a

certain conversion efficiency is represented.

Figure 2.7: Excess of amine donor required in function of the equilibrium constant

(K) for a certain conversion efficiency (85%, 90% or 95%). Example: for increasing

the total conversion of 90% to 95% at a K value of 10−2, the amount of excess donor

required needs to be more than doubled (Tufvesson et al., 2011).

• In situ (co-)product removal:

A second strategy is the removal of a (co-)product from the media during the reaction to shift the

reaction towards the product side. Similar to the addition of excess donor, the equilibrium constant

determines how low the concentration of (co-)product needs to be to achieve the target yields. For

In Situ (co-)Product Removal (IS(c)PR), the most commonly utilised physico-chemical properties

are charge, hydrophobicity, molecular size, volatility and solubility. There are limitations for

every separation strategy. Selectivity of the separation and the relative concentrations of the

reaction components are common limiting factors. For instance, ketones and amines have similar

distribution behaviour when using liquid-liquid extraction as separation technique. This can be

solved by adding another driving force, e.g. ionisation. Nevertheless ionisation has its own issues:


similarity between the pKa value of the product and the amine donor excludes ionisation as

a separation technique. Evaporation of a volatile product is a third option. In this case the

volatility of any (co-)solvent and the donor amine has to be taken into account. Finally, next

to increasing the yield, lowering the product concentration in the phase where the reaction takes

place reduces the product inhibition of the reaction which will in turn also increase the yield of

the overall process. In figure 2.8, the relation between concentration of the co-product and the

equilibrium constant to reach a certain efficiency is represented.

Figure 2.8: Concentration of co-product in solution in function of the equilibrium

constant (K) for a certain conversion efficiency (85%, 90% or 95%). For a constant

conversion efficiency, the required concentration of (co-)product in the solution is

directly proportional to the equilibrium constant, K. Lower values of K, require a

lower (co-)product concentration (Tufvesson et al., 2011).

• Auto-degradation of a product: This option is a very convenient one, but not widely applicable.

For example it can be used when the product has the ability to cyclise spontaneously, thus lowering

the product concentration and favouring the reaction in the direction of the product.

• Enzymatic cascade reactions: An approach which has gained a lot of interest last years is to

couple the transamination reaction to other enzymatic steps that convert the co-product back to

the original substrate or into a non-reactive species. Next to that, many other configurations are

possible: from cascade reactions to coupled parallel reactions, operated under a variety of reactor

configurations: from a stirred tank reactor to packed bed reactor to even membrane reactors.

These different types of process layout are not limited to ω-TA, but are applicable to a broad

spectrum of enzymes (Santacoloma et al., 2011). Regardless of which configuration is used, the

interactions and compatibility of each of the enzymes and their associated reagents need to be

considered. As a result of the complexity of this approach, enzyme cascade reactions are currently

only applied for research purposes.

• Whole-cell biocatalysis: Enzymatic cascade reactions have huge potential, however, the economi-

cal burden of using multiple enzymes (and co-factors) is significant. A suggested solution for this

limitation is to use a whole cell as the biocatalyst. This is very promising for bioconversions which


usually require a cofactor addition and/or regeneration, such as ω-TA. Different microorganisms

can be used: wild-type microorganisms expressing the desired enzyme naturally or recombinant

microorganisms. However, currently the number of available ω-TAs with a known gene sequence

is limited. Disadvantages for whole-cell biocatalysis in general are uncontrollable side reactions

leading to unwanted side products, and slower reaction rates due to trans membrane diffusion

limitations and higher metabolic burden. As a consequence the lower cost of using whole cells

(eliminating the enzyme purification step) has to be evaluated against the aforementioned short-

comings to select the most suitable catalyst form.

Limitations of the biocatalyst in the reaction system

For a broad range of substrate ketones, matching transamines can be found. Nevertheless the stability

and activity of the enzyme needs to be high enough for achieving a profitable process. The cost of the

biocatalyst is dependent on a number of factors including the efficiency of the fermentation protocol,

enzyme specific activity, the expression level and the form of the biocatalyst.

The enzymes under study often suffer from substrate and product inhibition. In particular cases, the

inhibitory effects are already severe at millimolar concentrations. This obstacle can be overcome by

using multiphasic reactions or by modification of the enzyme itself. A couple of solutions are elaborated

below.

• Improvement of the biocatalyst: The methodology to develop the enzymes matching process re-

quirements is based on random changes in the protein structure combined with addition of selec-

tive pressure to find the improved mutants and on understanding the relationship between protein

structure and its properties. To acquire the craved properties it is essential that the screening

takes place under the preferred reaction process conditions. Gradual adaptation of the enzyme

is common as it is challenging to screen for all the desired properties at once. Due to high cost

associated with enzymatic improvement, it should be coupled with process enhancement.

• Separation and recycling of biocatalyst: Fast and easy separation of the biocatalyst from the

reaction medium is a key factor for two reasons: first, it is needed to ensure product purity and

avoid problems with emulsions and foaming downstream of the reactor. Second, the separation is

essential if the reaction has to be stopped at a certain point. The most rudimentary method is

lowering the pH in the reactor which causes the enzyme to denature into an insoluble precipitate.

In that specific case, the downstream purification step consists of a simple filtration. Bear in mind

that this basic method is only economically feasible for high value products which compensate the

loss of enzyme.

• Immobilization: Immobilisation has a number of key advantages over free enzymes, including: easy

recovery and reuse of the enzyme, improved operational and storage stability and the possibility

for continuous operation and minimisation of the the protein contamination in the product. On

the other hand, enzyme immobilisation has some drawbacks: loss in activity due to introduction of

mass transfer limitation, and by loss of active enzyme due to steric hindrance. Next to these prac-

tical implications, immobilisation is an extra preparation step and thus it will increase the overall

production cost. However, this increased cost is compensated by the fact that immobilised enzyme

can by used for many reaction cycles. In practice, immobilisation for whole cell ω-transaminase

can only be done by entrapment in calcium alginate beads. Free ω-transaminase can be immo-

bilised by covalent linkage to solid support materials or by entrapment in sol-gel matrices. When

upscaling a reaction using immobilised biocatalyst, the resistance of the particles to mechanical

forces (e.g. shear stress) puts an upper limit on the flow speed and the amount of mixing in the

reactor. This issue of mechanical forces can be alleviated by using a packed bed reactor, but in

this configuration the pressure drop over the reactor can become limiting.


Solubility limitations and use of solvents

In aqueous biocatalytic processes, many of the substrates are characterised by a low solubility in water.

A low maximum in substrate concentration leads to a low volumetric productivity and high costs

associated with downstream processing (purification and recovery). Substrate availability limitations

are not only caused by low solubility, but also by slow dissolution rate, inhibition and toxicity. A

common solution to overcome substrate availability limitation is the controlled feeding of substrate into

the reactor medium. This approach also helps minimising imine dimer formation. Another solution to

increase productivity is by addition of a solvent. Water miscible co-solvents increase the solubility of

the substrate in the aqueous phase. Water immiscible solvents act as a reservoir for the substrate. It

is important to note that solvent addition decreases the stability of the biocatalyst and can potentially

cause problems in downstream processing. Furthermore is the usefulness of a water miscible solvent

limited as the increase in solubility is quite small.

2.1.5 Non-aqueous solvents

Enzymes are traditionally used in an aqueous environment, considering the original environment of

enzymes, this looks like a reasonable choice. But this approach has its limitations: many compounds are

insoluble in water and water is responsible for unwanted side reactions and the degradation of common

organic reagents. Furthermore the thermodynamic equilibrium of many reactions is unfavourable in

water and product recovery is difficult in this medium. In principle many of these drawbacks can

be overcome by switching to organic solvents. At first sight this seems impossible because of the

conventional idea that proteins are denatured in organic solvents. It is important noting that this

thought sprouts from examining enzymes in aqueous-organic mixtures, not in pure (organic) solvents.

It has now been shown that this assumption was wrong, in absence of water enzymes are very rigid

because water acts like a molecular lubricant. It is correct that the drive of the enzyme to unfold

(and thus denature) is greater in organic solvents, but the pliability necessary to proceed is lacking.

Even in anhydrous organic solvents, crystalline enzymes retain their native structures. In the following

paragraphs, several aspects and implications of the use of organic solvents in combination with enzymes

will be disclosed (Klibanov, 2001).

Enzymatic activity in organic solvents

The absence of water can lead to new enzymatic reactions not possible in aqueous environment. As

illustration, consider the hydrolysis of esters by lipases, esterases and proteases. This reaction uses water

as a catalyst. Due to the absence of the catalyst, this process cannot occur in anhydrous environments.

Addition of alternative nucleophiles leads to reactions which are normally suppressed in water (Klibanov,

1987). The usage of alcohols, amines and thiols in combination with former mentioned enzymes leads to

transesterification, aminolysis and thiotransesterification, respectively. Another example is the synthesis

of esters from acids and alcohols (reverse hydrolysis) which becomes more thermodynamically favourable.

However, in most of the cases, the catalytic activity of enzymes is inferior to the activity in water and

they experience mass-transfer limitations due to the insolubility of enzymes in organic solvents (Schmitke

et al., 1996). The upside is a greater tendency to strip tightly bound water from the enzyme, which

benefits catalytic activity (Zaks and Klibanov, 1988).

Enzymatic activity in an aqueous environment is greatly affected by pH. It has been found that enzymes

can exhibit a property called pH memory: their catalytic properties reflects the pH of the last aqueous

solution to which they were exposed. This phenomenon can be explained by the fact that the ionogenic

groups of the protein retain their last ionisation state when being dehydrated and subsequently placed

in organic solvents. This property can be used to maximise the activity in organic solvent by addition of

appropriate buffer pairs into the aqueous solution from which the enzymes are to be extracted (Klibanov,

1987). This memory effect is not only valid for pH, but also for ligand-induced memory effects. Theses

ligands, often competitive inhibitors, cause conformational changes in the active sites. After removal

14 2.2 MICROREACTOR

of these ligands, the imprints are retained in anhydrous media because of the enzymes rigidity in these

media. Since the structure of these ligand-imprinted enzymes is distinct from their original, so are their

catalytic properties (Klibanov, 1995).

Before bringing the enzymes in contact with the organic solvent, they are usually lyophilised. Freeze-

drying, also called lyophilisation, is a gentle dehydration process whereby aqueous solutions are frozen

solid and the ice is sublimised under vacuum conditions. Denaturation is an important issue regarding

enzymes. It is found that either crystalline or lyophilised enzymes are not suspicious to denaturation

in anhydrous solvents as this environment locks the enzyme molecule kinetically in its prior confor-

mation. Despite this fact, the lyophilisation step itself can cause significant denaturation (Griebenow

and Klibanov, 1995). This negative effect can be diminished by lyophilising in presence of structure

preserving lyoprotectants, such as sugars and polyethylene glycol (Dabulis and Klibanov, 1993), certain

inorganic salts (Khmelnitsky et al., 1994), substrate resembling ligands (Russel and Klibanov, 1988)

and crown esters (Broos et al., 1995).

Reduced structural flexibility is another cause of lowered enzymatic activity in organic solvents. As

mentioned above, in organic solvents, the enzyme lacks the conformational mobility which it possesses

in aqueous environment (Rupley and Careri, 1991). This is a result of the hydrogen bonds in water

and its large dielectric constant (Affleck et al., 1992). The enzymes can be loosened up by adding small

quantities of water, solvent capable of forming hydrogen bonds (e.g. glycerol and ethylene glycol) or

denaturing co-solvents (in quantities insufficient to cause full denaturation) resulting in an enzymatic

activity increase (Almarsson and Klibanov, 1996).

Stability of enzymes in organic solvents

The thermal instability of enzymes can be categorised in two types. The first is a time-dependent,

gradual irreversible loss of enzymatic activity on exposures to high temperatures. The second is a heat-

induced cooperative unfolding, which is usually reversible and instantaneous (Klibanov, 2001). In both

types water is the pivotal participant (Rupley and Careri, 1991). By switching to non-aqueous media, an

improved stability is documented for both types of thermal inactivation. It follows that the resistance to

thermal unfolding decreases as the water content of the lyophilised enzyme powder rises (Garza-Ramos

et al., 1990). Other research has obtained the knowledge that the thermostability is the same for enzyme

powders exposed to organic solvents, air and argon. Therefore it is deduced that a hydrophobic solvent

is essentially inert towards the enzyme (Volkin et al., 1991). Next to thermostability, enzymes become

far more stable against proteolysis (Zaks and Klibanov, 1988). This is due to the fact that both enzyme

and offending protease are insoluble in the organic solvent and thus cannot interact.

2.2 Microreactor

At present, the greater part of synthetic reactions are implemented with techniques that are in place for

decades. These conventional techniques lack the efficient upscaling from laboratory conditions to full

scale plant production. Micro-reaction technology has the potential to bypass this issue by replicating

unit processes (parallel upscaling) instead of scaling-up. This enables direct transfer of laboratory

optimised conditions to full production scale.

2.2.1 Basic concepts of micro-reaction technology

In the context of this thesis, micro-reactors are defined as a device containing micro-structured features

(with sub-millimeter precision), in which chemical or enzymatic reactions take place in a continuous

system (Watts and Wiles, 2007).


Figure 2.9: An example configuration of the microreactor used in this thesis (Mi-

cronit, 2014).

Fabrication

For micro-structured reactors, the choice of substrate for making microreactors is determined by the end

use and the fabrication technique. The key substrates are silicon, quartz, metals, polymers, ceramics and

glass. This micro-reactor substrate has to be evaluated for chemical compatibility (i.e. no unwanted

side reactions with reaction substrates, products, solvents or (bio-)catalysts), thermal and pressure

resistance, and ease of fabrication.

Several fabrication techniques are available, including laser ablation, Deep Reactive Ion Etching (DRIE),

LIthographie Galvanoformung Abformung (LIGA), photolithography, powder blasting, microlamination,

hot embossing and injection moulding.

The first two (laser ablation and DRIE) have been proven to produce microreactors of outstanding

quality regarding surface quality, definition and reproducibility. However, mass production via these

techniques is not used in practice due to the high costs affiliated with precision engineering and the

serial nature of the technique.

The last two techniques (hot embossing and injection molding) have the advantage that a master/tem-

plate approach can be used for the fabrication. This attribute provides a relatively inexpensive approach

to mass-produced microreactors of excellent quality (Watts and Wiles, 2007).

Fluid flow

In the macro world, the fluid flow is generally forced by applying external forces using mechanical

pumps. On the micro-scale, several non-mechanical techniques can be used to displace fluids. Both

types, mechanical and non-mechanical techniques, will be discussed in the paragraph below.

Mechanical pumps deliver fluids in discrete aliquots by displacement of a membrane. This pumping

mechanism is independent of the device material. A major disadvantage is that the flow is often pulsed

instead of smoothly continuous, although there exist techniques to smoothen the pulses. At research

level, there is a large demand for pumps with the ability to deliver stable bi-directional flow, external

displacement pumps can deliver these requirements and are therefore widespread in use. The main

weaknesses one should pay attention to in this approach are leak-free connections, realisation of a low

dead volume and uniform control strategies for dealing with multiple reagent inlets.

Non-mechanical pumps directly use the transfer of energy from which a steady and pulse-free flow is

produced. Following techniques are used in non-mechanical pumps: electrochemical displacement (bub-

ble formation), thermal expansion, microsphere deformation, and pumps utilising electrohydrodynamic,

capillary or evaporation forces. The prevailing advantages are that no moving parts are used, the tech-

niques are fairly simple and its ability to produce pulse-free flows, even at low flow rates. However for

electrokinetic flow, which is a combination of electroosmotic flow (EOF) and electrophoretic flow (EPF),

16 2.2 MICROREACTOR

the performance of the pump is directly coupled with the properties of the liquid (Watts and Wiles,

2007).

Control of reaction conditions

Mixing is traditionally achieved on macro-scale by large eddies generated by magnetic or mechanical

stirrers, allowing bulk diffusion to dominate. In microreactors high viscous forces prevent the induction

of turbulence, the governing flow is laminar. Therefore mixing is dominated by molecular diffusion. The

most popular approach to increase the amount of mixing is by increasing the contact area by lamination.

This idea can be executed by splitting the stream into thin laminae and subsequently bringing them

back together. At the point of conclusion, complete mixing is achieved in no less then 15 ms (Bessoth

et al., 1999). As a general rule of thumb: n laminae corresponds to n2 times faster mixing. From this, it

follows that reactions in miniaturised systems are, in theory, simply limited by their inherent reaction

kinetics, given that efficient mixing can be achieved within the microreactors.

Temperature control in full scale vessels is often limited and slow. Fluctuations in temperature are

thus difficult to counter. On the micro-scale, changes in temperature are observed almost immediately.

As the flow regime in the micro-scale reactors is laminar, diffusion theory can be used to make an

approximation of the time needed to enable thermal mixing across the micro channel. A decrease in

channel diameter results in an increase in the rate of thermal mixing and in an even higher surface to

volume ratio. This last fact results in rapid heat dissipation: for silicon channels practical applications

have shown heat dissipation up to 41.000 Wm−2K−1 and for glass channels up to 740 Wm−2K−1. In

practice, this fact is utilized to ensure process safety, e.g. the prevention of hot-spots and thermal

runaway in highly exothermic reactions (Watts and Wiles, 2007).

Process intensification

The preceding paragraphs have elucidated some profound theoretical advantages of microreactors. How-

ever, for each microreactor, only small quantities of desired product can be synthesised at once. The

current modus operandi in process engineering is based on the scale-up of lab-scale or bench-optimised

processes. Micro-reaction technology achieves high production volumes by replication of successful reac-

tion units. By keeping these laboratory-optimised conditions, time and costs can be saved as no difficult

scaling has to be performed. This approach eliminates changes in surface-volume ratio, which greatly

affect the thermal and mass transfer properties of the reaction. Moreover, using microreactors improves

process flexibility as reactors can be configured to fit multiple operations. Reactions which were previ-

ously unscalable to an industrial level, can now be carried out by using multiple microreactors. This is

particularly interesting for the fine chemical and pharmaceutical industry (Watts and Wiles, 2007).

2.2.2 Synthetic micro-reactions

Continuous-flow solution-phase reactions

Continuous-flow solution-phase reactions are generally performed on chip-type microreactors. The stan-

dard process is performed by injecting substrate and enzyme solutions into separate inlets. This type

of process mainly relies on rapid mass transfer of the different reactants. By applying this technique on

trypsin-catalysed or glycosidase-catalysed hydrolysis reactions, the reaction yields are greatly improved.

The improvement in reaction yield was about three to five times higher compared to the original (batch-

wise) yield (Miyazaki and Maeda, 2006).

Stopped-flow reaction

Rather than utilising continuous flow, microreactors can also be operated in stopped-flow mode. In this

mode, the reactants are temporarily immobilised in the microreactor for a certain time period utilising


a physical and/or chemical field over the reactor. The removal of this external field results in a stopped-

flow in the reactor. The result is an effective increase in residence time without physically altering the

microreactor itself. As an addition the stopped-flow can be locally heated (IR) or cooled (IR diode

laser), this photothermal stimulation can enhance the reaction speed (Miyazaki and Maeda, 2006).

Enzyme immobilization on beads or monoliths

The use of immobilised enzymes is preferable for two reasons: no need to recover the biocatalysts from

the product stream and it eases the downstream processing. In macro-scale reactors, enzymes can be

immobilised on beads or monoliths. This approach can be extrapolated to microreactors. With respect

to the beads, several materials have been used successfully for the creation of the beads. From classic

glass beads to polystyrene to agarose derivates and even magnetic beads. Monolithic immobilisation

in microreactors is often executed either by a porous polymer or by a silica derivate. The immobili-

sation method used is either physical adsorption or cross-linking. Generally speaking, preparation of

immobilised enzymes is more straightforward with a monolith or powdered material. Despite easier

preparation, it is disadvantageous in large-scale arrangements due to the susceptibility to increasing

pressure (Miyazaki and Maeda, 2006).

Enzyme immobilisation on microchannel surfaces

Immobilisation on the microreactor surface uses the large surface area as an advantage but without

the increased pressure observed in for example monoliths. An easy way to achieve immobilisation is by

physical means. For example the biotin-avidin system was frequently used in microreactors to immobilise

enzymes, yet it is limited to streptavidin-conjugated enzymes. Another approach is the formation of

nanostructures on a silica microchannel surface utilising a modified sol-gel technique (using a copolymer

of 3-aminopropylsilane and methylsilane). These nanostructures increase the surface area at the channel

wall and allow a tenfold increase in immobilisation capacity. The enzymes can be secured on the

nanostructures by covalent cross-linking (disulfide, amine-bond, His-tag or by a modifying succinate

spacer compared with a single-layer immobilisation). A particle-arrangement shows an even higher

increase in kinetics (1.5 times) and correlated surface area (≈ 1.5 times). Silica nanoparticles are

immobilised on the channel wall surface by slowly evaporating a particle suspension in a completely

filled microreactor. Enzyme immobilisation is achieved by first subjecting the silica particles to a 3-

aminopropyltriethoxysilane treatment and subsequently covalent cross-linking the enzymes with the

amino groups. Despite the promising results, the physical stability of the particle-arrangement still

has to be improved. The last option for immobilisation of enzymes on the surface of a microreactor

is polymer coating. Research has shown that alkaline phospatase, ureases and several other enzymes

incorporated in a poly(ethylene glycol)-hydrogel can be coated on the reactor wall by exposure to UV

light (Miyazaki and Maeda, 2006).

Enzyme immobilisation on membranes

Enzymes can also be immobilised on a membrane, creating a chemicofunctional membrane. For example:

a nylon-membrane can be formed at the interface of two solutions formed in a microchannel. Due to

the technical difficulty accompanied with the creation of the membrane and the instability of the nylon-

membrane in organic solvents, the applications are limited. A different technique used in batch wise

organic synthesis, Cross-Linked Enzyme Aggregate (CLEA), forms an enzyme-immobilising membrane

on the microchannel surface. The procedure is quite straightforward: the microreactor has to be loaded

with an enzyme solution and a mixture of glutaraldehyde and paraformaldehyde and a CLEA membrane

will be formed on the microchannel wall. In contrast with the nylon membrane, the CLEA membrane

shows good stability against organic solvents and can be used for prolonged times (>40 days) (Miyazaki

and Maeda, 2006).

18 2.3 FLUID DYNAMICS

Enzyme separation by multiphase flow

In batch reactor, the usage of immiscible fluids (liquid-liquid reactions) requires vigorous stirring in order

to increase the interfacial area between both phases, along with extended reaction times. Performing

phase transfer reaction in microreactors has the benefit that a large longitudinal interface is created of

5,000 up to 50,000 m2m−3. The reactions proceed more rapidly and efficiently compared to analogous

stirred reactions. This parallel flow can also be utilised to realise continuous purifications, by means of

liquid-liquid extractions. An alternative to this parallel flow is slug flow, where liquid or gas slugs are

generated in an immiscible continuous phase. This approach allows the interfacial surface area to be

further increased (Watts and Wiles, 2007).

On-line purification

In macro-scale processes, product purification generally is a three step process:: first an aqueous work-

up (removal of inorganic material), then column chromatography (removal of unreacted substrates and

unwanted side products), and ending with recrystallization of the desired product. This approach is

a batch process, it would not make sense to treat the continuous product stream from a microreactor

in a similar manner, as this eliminates the big advantages of microreactors (speed and automation).

Several techniques have made a successful transition from the macro to the micro scale, including µ-

dialysis, µ-filtration and liquid-liquid extraction. The latter is showing the most promise with respect

to universal applicability. A stable miniaturised two-phase flow has a high degree of phase separation

combined with efficient analyte extraction due to short diffusion lengths and high interfacial surface

area. Next to product purification, the recovery of catalysts from reactions mixtures is a tough nut to

crack especially with the small reaction volumes in microreactors. For metal catalysts, several studies

have reported the successful extraction and prevention of precipitation within the reactor by using ionic

liquids. An alternative approach is to incorporate support reagents, catalysts and scavengers into the

miniaturised device. This approach eliminates the requirement for off-line or in-line purification steps,

only evaporation of the solvent is needed (Watts and Wiles, 2007).

2.3 Fluid Dynamics

2.3.1 Introduction

Fluid dynamics is the analysis of systems involving fluid flow, heat transfer and associated phenomena

such as chemical reactions. Fluid dynamics is a mathematical model of the real world, with the conser-

vation laws as its foundational axioms. These axioms are derived from classical mechanics: conservation

of mass, energy (first law of thermodynamics) and linear momentum (second law of Newton). Next to

these axioms, the fluid is regarded as a continuum: the molecular structure of matter and molecular

notions may be ignored. Hence, the behaviour of the fluid is described in terms of macroscopic prop-

erties which can be seen as an average over suitably large number of molecules. In this study, the

Navier-Stokes system will be used to describe the fluid behaviour (Versteeg and Malalasekera, 2002).

2.3.2 Governing equations

Considering an inertial frame of reference, the general form of the Navier-Stokes equation is stated in

equation 2.8. This equation is derived by application of the second law of Newton (one of the axioms of

fluid dynamics) to fluid flow combined with the assumption that the stress in the fluid can be described

as a sum of a diffusing viscous and a pressure term. In equation 2.8, U represents the velocity vector, ρ

the volumetric density, p the pressure, T the deviatoric component of the total stress tensor (a tensor of

second order), ∇ the del operator, and f the other body forces (they often consist of only gravitational

forces, but in non-inertial coordinate systems, they are used for the forces associated with rotating


coordinates).

ρ

(∂U

∂t+ U · ∇U

)= −∇p+∇ ·T + f (2.8)

A simplified equation can be derived for an incompressible, Newtonian fluid. The latter assumption rules

out the occurrence of shock and sound waves. Using the two assumptions, the Navier-Stokes equation

for incompressible fluid with constant viscosity reads:

Inertia (per volume)︷︸︸︷ρ( ∂U

∂t︸︷︷︸Unsteady

acceleration

+ U · ∇U︸︷︷︸Convectiveacceleration

)=

Cauchy/total stress tensor︷︸︸︷−∇p︸︷︷︸

Pressuregradient

+ µ∇2U︸︷︷︸Viscosity

+

Otherbodyforces︷︸︸︷

f (2.9)

In equation 2.9, µ represents the dynamic viscosity of the fluid. The sole difference between equations

2.8 and 2.9 is the viscous stress term (Versteeg and Malalasekera, 2002). In equation 2.8, the effect of

stress in the fluid is given by two terms: ∇p and ∇ ·T. The former term is derived from the isotropic

part (normal stresses) of the Cauchy stress tensor and is called the pressure gradient. It is worth noting

that in the Navier-Stokes equations, the gradient of the pressure matters, not the pressure itself. The

latter term is the anisotropic part of the Cauchy stress tensor and describes viscous forces. By making

assumptions on the nature of these stresses, the two terms can be expressed in functions of other flow

variables, i.e. velocity and density (Batchelor, 1967). In Newtonian fluids, the viscous stresses are

proportional to the rates of deformation. In a three dimensional compressible flow, two constants arise

that describe this behaviour: the first is the (dynamic) viscosity (µ), which relates stresses to linear

deformation, and the second viscosity (λ), which relates stresses to volumetric deformation. Not much

is known about the second viscosity, as its effect is negligible in practice. For incompressible flows,

volumetric deformation is non-existent: the second viscosity disappears from the equation. Given that

Newtonian fluids are isotropic (uniformity of the fluid in all directions), the dynamic viscosity is simply a

constant and the viscous stress term simplifies to the product of the dynamic viscosity and the laplacian

of the velocity tensor (second term of the right hand side of equation 2.9) (Versteeg and Malalasekera,

2002).

These equations state that changes in momentum only depend on changes in external pressure and

internal viscous forces acting on the fluid. The Navier Stokes system is a set of differential equations

and can only be solved analytically in the simplest cases. In fact, the Navier-Stokes smoothness and

existence problem is one of the seven millennium prize problems (Clay Mathematics Institute, 2014):

Prove or give a counter-example of the following statement:

In three space dimensions and time, given an initial velocity field, there exists a vector

velocity and a scalar pressure field, which are both smooth and globally defined, that solve

the Navier-Stokes equations.

Otelbaev (2013) has proposed a solution for the smoothness and existence problem at the end of 2013.

Due to the fact that the article is written in Russian and a complete translation of the article is lacking

at the time of writing, the solution will not be discussed in this thesis.

The Navier-Stokes equations are an expression of the conservation of momentum. It is worth noting

that a solution of these equations is a velocity field, which is different from classical mechanics where

solutions are typical trajectories of all individual water molecules. The former is called an Eulerian

coordinates system, it represents the velocity of the fluid at a certain position and time. The latter is

called a Lagrangian coordinate system, the flow is described by the position of a certain fluid parcel at

a certain time. The relationship between these two coordinate systems is described by the Reynolds

transport theorem. Equation 2.10 represents the Reynolds transport theorem, with Ω(t) a parcel of fluid

for which no material enters or leaves, n is an outward-pointing unit-normal, U denotes the velocity, i is

a quantity (scalar, vector or tensor) of the fluid under study, dA and dV are respectively the surface and

volume elements. In words, the former equations state that the time rate of change of I, the summation

of the quantity i over a control volume, within a system (left hand side of the equation) equals the sum

20 2.3 FLUID DYNAMICS

of the time rate of change of I within the control volume (Ω(t)), and the net flux of I through the control

surface (Batchelor, 1967).

d

dt

∫Ω(t)

i dV =

∫Ω(t)

∂i

∂tdV +

∫∂Ω(t)

(U · n)i dA (2.10)

To fully describe the fluid regime, additional equations are required. Depending on the case, different ad-

ditional information may be required. This information may include conservation of mass, conservation

of energy, boundary equations (no-slip surface, inlet/outlet etc.), equations of Maxwell (magnetohydro-

dynamics) and equations of state (temperature, solutes etc.) (Versteeg and Malalasekera, 2002). Table

2.1 lists some examples of additional terms.

Table 2.1: Examples of additional equations needed to fully describe the fluid regime.

Equation Description

∂ρ∂t +∇ · (ρU) Mass continuity equation

∂(ρφi)∂t +∇ · (ρiU) = −p∇ ·U +∇ · (k∇T ) + Φ + Si Energy continuity equation

p = p (ρ, T ) Equation of state for pressure∂(ρφ)∂t +∇ · (ρφU) = ∇ · (Γ ∇φ) + Sφ Equation of state for variable φ

2.3.3 Navier-Stokes system in microreactors

The fluid regime in this type of microreactors is laminar due to the combined effect of small scale pipes

(low hydraulic diameter) and low fluid velocity. This fluid regime has a low Reynolds (Re, see equation

2.11) and Mach number (Ma, see equation 2.12), and is incompressible which is in accordance with

the initial assumptions of a incompressible, Newtonian fluid (Koo and Kleinstreuer, 2003). In equation

2.11, U is the fluid velocity, DH is the hydraulic diameter, ρ is the density is the fluid under study and

µ is its dynamic viscosity. In equation 2.12, Usound stands for the speed of sound waves in the fluid

under study.

Re =ρUDH

µ(2.11)

Ma =U

Usound(2.12)

Considering the computational part, the question arises whether the fluid in microreactors can still be

approximated as a continuum (Koo and Kleinstreuer, 2003). A method to quantify this is the Knudsen

number (Kn, see equation 2.13), it equals the ratio of the molecular free path (λM ) to a representative

physical scale length (L, e.g. hydraulic diameter, DH).

Kn =λML

=

√πγ

2

Ma

Re(2.13)

Depending on the value of the Knudsen number, several regimes can be distinguished, see table 2.2.

From free molecular flow at high Knudsen numbers, to Navier-Stokes and Euler regime at low Knudsen

numbers. For liquids, Kleinstreuer (2003) suggests a modification for this condition: the general Knudsen

number for liquids (equation 2.14).

Knl =λIML

(2.14)

The general Knudsen number for liquids equals the ratio of the intermolecular length for the fluid

molecules (λIM ) to a representative physical scale (L). Given that the intermolecular length for water

molecules is 3 ·10−10m, the Navier-Stokes equations with no-slip boundary condition hold up to microre-

actors with a hydraulic diameter as low as 0.3 µm (Kleinstreuer, 2003). Koo and Kleinstreuer (2003)


state that the surface roughness effects should only be taken into account for DH ≤ 10 µm and turbu-

lence effects only become important when the Reynolds number exceeds 1000. Koo and Kleinstreuer

(2003) also state that viscous dissipation effects are not negligible for conduits with DH ≤ 100 µm, but

since temperature modeling is not within the scope of this thesis, these effects are considered negligi-

ble. Sharp (2001) states that non-Newtonian fluid behaviour only occurs when there exists long chain

polymers or when fine particle suspensions are considered. Hence, it is proven that the assumption of a

continuous, incompressible, Newtonian fluid is valid in microreactors.

Table 2.2: Different Knudsen regimes for fluids. (Gad-el hak, 1999)

Fluid regime Condition

Euler equations (neglect molecular diffusion) Kn → 0 (Re → ∞)

Navier-Stokes equations with no-slip boundary conditions Kn ≤ 10−3

Navier-Stokes equations with slip boundary conditions 10−3 ≤ Kn ≤ 10−1

Transition regime 10−1 ≤ Kn ≤ 10

Free-molecule flow Kn > 10

2.3.4 Closing remarks

In contrast to the Navier-Stokes equations, the Boltzmann transport equation, an integral-differential

equation which characterises the dynamics and kinetics of the distribution of micro-scale particles, is

applicable over the entire domain of Kn. Next to the Navier-Stokes equations, it is also possible to

derive the continuity and the energy equation from the Boltzmann equation (Li, 2006). The Lattice

Boltzmann method has some distinct advantages over the more frequently used Navier-Stokes system:

mesh-free (complex and moving geometries are easier to implement), intrinsic linear scalability in parallel

computing and efficient inter-phase interaction handling for multiphase flow. The downsides of the

approach overwhelm the upsides: computational very expensive, issues regarding turbulence modelling

and boundary conditions, and the impossibility to run steady state simulations. Due to its maturity,

the Navier-Stokes system is still the most reliable approach for the simulation of fluid flow (Shengwei,

2011). Consequently, the Navier-Stokes system will be used in this thesis.

Figure 2.10: Classification of flows from free molecular flow to continuous flow in

function of the Knudsen number (Li, 2006).

CHAPTER 3Materials and Methods

3.1 Objective

After a brief summary of the literature available on the subjects of enzyme kinetics, fluid dynamics, and

microreactor technology in the previous chapter, this chapter focuses on the application of the knowledge

in a modelling environment. First, the methods to numerically solve the Navier-Stokes equations will be

discussed in the section computational fluid dynamics. In the same section, more details will be given

on the stability of the system, the kinetic model of ω-TA reactions, and the solution procedure. Next to

CFD, a number of simplified models will be discussed in the next part of this chapter. Finally a brief

overview of the used software is given.

3.2 Computational Fluid Dynamics

3.2.1 Discretisation of the Navier-Stokes equations

The Navier-Stokes equations (equation 2.9) are a system of continuous partial differential equations.

Solving this system numerically, requires a discretisation of the equation in both time and space. The

ideas given in this chapter are general applicable, but the equations and procedures mentioned are

specific for OpenFOAM, the software package used in this thesis. In OpenFOAM notation, the final

form of the discretised Navier-Stokes system, using the finite volume method, is given by equation 3.1.aPUP = H (U)−

∑f

S(p)f

∑f

S ·

[(1

aP

)f

(∇p)f

]=∑f

S ·(

H (U)

aP

)f

(3.1)

F = S ·Uf = S ·

[(H (U)

aP

)f

−(

1

aP

)f

(∇p)f

](3.2)

The fluxes on the cell faces are calculated using equation 3.2. Here, aP denotes the central coefficient

(coefficient on the diagonal in a linear system), S stands for an outward-pointing face area vector (for

attaining a projection perpendicular to the surface of the cell), and an H(U)H-term consisting of a

transport and a source part, derived from the integral form of the momentum equation. The first

equation in system 3.1 is derived from the momentum equation: the left hand side denotes the velocity

in the cell multiplied by the central coefficient. The first term on the right hand side is the effect of

the transport and source part of neighbouring cells on the cell, whereas the second term is the effect

of pressure interpolated to the cell faces. The first equation in system 3.1 is derived from the pressure

equation: it relates the sum of the pressure gradient over all the cell surfaces (left hand side) to the

24 3.2 COMPUTATIONAL FLUID DYNAMICS

transport and source term effects on the flow (right hand side). Equation 3.2 is a discretised form of

the continuity equation, i.e. the mass balance over the cell (Jasak, 1996).

3.2.2 Pressure - velocity coupling: the PISO- and SIMPLE-loop

Consider the discretised form of the Navier-Stokes equations written in the section above. These equa-

tions show linear dependence of the pressure on the velocity and vice-verse. A special treatment is re-

quired to cope with this inter-equation coupling. Two approaches can be distinguished: a simultaneous

and a segregated approach. Simultaneous algorithms solve the coupled equations simultaneously over

the whole domain. The computational and memory cost of such an algorithm is huge and will therefore

not be used. In contrast, the segregated approach solves the coupled equations sequencially. Pres-

sure Implicit with Splitting of Operator (PISO), Semi-Implicit Method for Pressure Linked Equations

(SIMPLE) and their derivatives are the most common used algorithms to cope with the inter-equation

coupling. In this thesis, the PISO algorithm will be used for transient problems, whereas the SIMPLE

algorithm will be used for steady-state problems. The PISO algorithm, proposed by Issa (1986), can be

described as follows:

• The algorithm starts solving the momentum equation first. At this stage, the exact pressure

gradient source term is not known. Instead, the pressure field from the previous time-step is used

to perform the calculations. This stage is called the momentum predictor. The outcome of this

step is an approximation of the new velocity field.

• The predicted velocities acquired in the previous step are used to assemble the H(U) operator

and the pressure equation can be formulated. Solving this equation results in the first estimate of

the new pressure field. This step is called the pressure solution.

• The second equation in the discretised Navier-Stokes system (Equation 3.1) gives a set of con-

servative fluxes consistent with the new pressure field. As a consequence of the new pressure

distribution, the velocity field should be corrected, since it was initiated using the pressure field

of the previous time-step. Velocity correction is done in an explicit manner, using equation 3.3.

This is the explicit velocity correction stage.

UP =

Transported influence ofcorrections of neighbouring velocities︷︸︸︷

H (U)

aP− 1

aP∇p︸︷︷︸

Correction due to the changein the pressure gradient

(3.3)

The fact that the velocity is corrected explicitly, means that the transported influence of corrections of

neighbouring velocities is neglected. The assumption here is that the overall velocity error comes from

the error in the pressure term. As this is not the case in reality, it is necessary to correct the H(U)

term, formulate a new pressure equation and repeat the procedure. From the disquisition above follows

that the PISO loop consists of an implicit velocity predictor followed by a series of pressure solutions

and explicit velocity corrections. The iteration continues until a pre-determined criterion or tolerance

is reached.

Next to the assumption above, there is a second issue regarding the PISO loop: H(U) coefficients are

dependent on the flux field. After each pressure solution step, new flux fields are available. Although it

is possible to calculate new H(U) coefficients, this is not done. The second assumption states that the

non-linear pressure-velocity coupling is less important than the pressure-velocity coupling, consistent

with the linearisation of the momentum equation. The H(U) coefficients are only changed in the

momentum predictor, i.e. once per iteration in the PISO loop (Jasak, 1996).

When addressing steady state problems with the SIMPLE algorithm, a few considerations can be made.

First, if the steady state problem is solved iteratively, therefore it is not necessary to fully resolve

CHAPTER 3 MATERIALS AND METHODS 25

the linear pressure-velocity coupling. This comes from the fact that the changes between consecutive

solutions are no longer small. Second, the non-linearity of the system becomes a more acute problem,

since the effective time step is much larger (Jasak, 1996). The SIMPLE algorithm by Patankar (1980)

is formulated to take advantage of these facts.

• By solving the momentum equation, an approximation of the velocity field is obtained. Next, the

pressure gradient term is determined using the pressure distribution from the previous iteration,

or from an initial guess at the first iteration. The velocity under-relaxation factor (αU ) is used to

make sure that the equation is implicitly under relaxed.

• The new pressure distribution is obtained by defining and solving the pressure equation.

• Equation 3.2 is used to calculate a new set of conservative fluxes. The new pressure field, calculated

in the previous step, includes the pressure error and the convection-diffusion error. For a better

approximation of the pressure field, it would be necessary to redo the calculation. As stated before,

the non-linear effects are of greater importance than the pressure-velocity coupling in steady state

simulations. It is sufficient to keep this pressure field approximation and recalculate the H(U)

coefficients with the new set of conservative face fluxes. The pressure solution is therefore under

relaxed in order to take into account the velocity part of the error (Equation 3.4). With pnew, the

approximation of the pressure field that will be used in the next momentum predictor, pold stands

for the current pressure field used in the momentum predictor, pp is the solution of the pressure

equation and αp is the pressure under-relaxation factor (0 < αp ≤ 1). The recommended values

for the under relaxation factors are 0.2 for pressure and 0.8 for momentum (Jasak, 1996).

pnew = pold + αp(pp − pold

)(3.4)

3.2.3 Stability

In general mathematics, the Courant-Friedrichs-Lewy stability condition (CFL) is a necessary (but not

sufficient) condition for convergence when solving partial differential equations numerically with the

finite difference and finite volume method (Jasak, 1996). This condition can be explained with an

example: the domain of interest is a discrete spatial grid with a wave moving across it. To calculate the

amplitude of the wave at discrete time steps of equal length, this time interval must be less than the time

for the wave to travel to adjacent grid points. If the space between two adjacent grid points is reduced,

then the upper limit for the time step also decreases. For a one-dimensional case: the CFL-condition

has the following form:

Co =Uf ∆t

∆x≤ Comax (3.5)

with Uf the fluid velocity at the face, ∆x is the grid spacing and ∆t the time step. In this case, the

dimensionless number Co is called the Courant number. Generally speaking, if the Courant number

is larger than unity, the effective diffusion becomes negative and the system becomes unstable. The

effective diffusion equals the sum of the numerical diffusion from the differentiation scheme (positive)

and the numerical diffusion from the temporal discretisation (negative) (Jasak, 1996). Pure implicit

schemes are an exception: depending on the exact configuration of equations and the discretisation,

the Courant number can be up to a few orders of magnitude above the unity before instability occurs.

This limitation on the Courant number is quite severe, especially when solving a steady state problem.

It can be shown that the size of the maximum value for numerical diffusion is closely related to the

Courant number. Numerical diffusion is a phenomenon wherein the simulated medium exhibits a higher

diffusivity than the true medium. The name comes from the nature of the error introduces through

discretisation, it is diffusive in nature, i.e. dependent on the gradient of variable (Courant et al., 1928).


Figure 3.1: Effect of numerical dissipation and dispersion on wavelike solutions: (a)

exact solution, (b) numerical solution with strong dissipation, (c) numerical solution

with strong dispersion (Zikanov, 2010).

Next to temporal discretisation, the spatial discretisation has to be considered. As mentioned above,

the Courant number is directly proportional to the space between two adjacent grid points. Second,

the spatial discretisation determines the level of detail of the solution. If the number of computational

points is not sufficient to describe the changes in a particular region, the shape of the solution will be

lost. Therefore a consideration has to be made between obtaining a detailed solution and attaining this

solution in a reasonable time frame. Adding more detail to the solution by generating a smaller spatial

mesh will decrease the maximum time-step, presuming that the Courant number is kept at a constant

level (Equation 3.5). This decrease in maximum time step leads to an increase of the number of needed

iterations and will thus increase the computational expense of the simulation. The spatial discretisation

can induce to two more errors in stability and quality of the solution: non-orthogonality and skewness

error. For a mesh of reasonable quality, i.e. (almost) orthogonal mesh edges and skewness close to

zero, the introduced error is expected to be smaller than the numerical diffusion from the convection

differencing scheme. Only on highly distorted meshes the influence of these terms becomes significant

(Jasak, 1996).

3.2.4 Kinetic model of ω-TA reactions

The King-Altman representation proposed by Al-Haque et al. (2012) is shown in figure 2.6. This

mechanism includes the product and substrate inhibition observed in experiments. One can observe

in figure 2.6, four nonproductive complexes of the mechanism: E-PLP-SA, E-PLP-SB, E-PLP-PQ and

E-PLP-PQ. These complexes are characterised in the reaction rate equation (Equation 3.6) by substrate

inhibition constants KSASi and KSB

Si in the forward direction, and KPPSi and KPQ

Si in the reverse direction.

The derived quasi-steady-state rate equation is shown in equation 3.6. Next to the substrate and product

inhibition constants, the latter equation is characterised by the Michealis Menten parameters (KSAM ,

KSBM , KPP

M , KPQM ), the catalytic turnover of the reaction (Kf

cat, Krcat), the substrate and product

inhibition constant (KSBi and KPP

i )and the chemical equilibrium constant (KEQ) (Al-Haque et al.,

2012).


r[PP ] = −r[SA] =[E0]Kf

catkrcat

([SB] [SA]− [PQ][PP ]

KEQ

)(KrcatK

SAM [SB]

(1 + [SB]

KSBSi

+ [PP ]

KPPSi

)+Kr

catKSBM [SA]

(1 + [SA]

KSASi

+ [PQ]

KPQSi

) (3.6)

+Kfcat

KPPM [SA]

KEQ

(1 +

[SA]

KSASi

+[PQ]

KPQSi

)+Kf

cat

KPQM [PP ]

KEQ

(1 +

[PP ]

KPPSi

+[SB]

KSBSi

)+Kr

cat [SB] [SA] +Kfcat

KPPM [SB] [PQ]

KEQKSBi

+Kfcat

[PQ] [PP ]

KEQ+Kr

cat

KSBM [SA] [PP ]

KPPi

)The Haldane relationship, used for the formulation of the chemical equilibrium is given in equation

2.7. As mentioned before in section 2.1.4, ω-TA reactions suffer severely from inhibition at industrial

process setup (Al-Haque et al., 2012). Equation 2.6 is an example of a kinetic relationship exhibiting

uncompetitive inhibition.

At this moment it is not yet known whether a unique set of parameter values can be found for this

model. Assuming that the data is informative and has a high signal-to-noise ratio, a lack of practical

identifiability of the model under consideration can have two possible reasons: the insensitivity of a

parameter and correlation between two or more parameters. If the data is not informative enough, ad-

ditional qualitative data collection is necessary. Through the lack of uniqueness, the physical meaning

of the parameter is lost as several sets of parameter values can describe the same system behaviour.

Consequently, the model structure as a whole has to be reviewed. The analysis to investigate this

uniqueness and sensitivity of the different parameters is called identifiability analysis (Walter and Pron-

zato, 1997). Considering the initial period of the reaction (5 minutes), equation 3.6 can be further split

up into equation 3.7 and 3.8 for reaction conditions where initially only substrate or product is present

in the reaction medium. When the product concentrations are close to zero, it can be assumed that

product inhibition is non-existing and the reverse reaction is not occurring. Following this assumption,

equation 3.6 can be simplified, which is shown in equation 3.7. The same assumptions can be drawn for

the backward reaction, shown in equation 3.8.

r[PP ] = −r[SA] =[E0]Kf

cat [SA] [SB]

KSAM [SB]

(1 + [SB]

KSBSi

)+KSB

M [SA](

1 + [SA]

KSASi

)+ [SB] [SA]

(3.7)

r[SA] = −r[PP ] =[E0]Kr

cat [PP ] [PQ]

KPPM [PQ]

(1 + [PQ]

KPQSi

)+KPQ

M [PP ](

1 + [PP ]

KPPSi

)+ [PP ] [PQ]

(3.8)

The decomposition of the full model into an initial rate model has the advantage that less parameters

need to be estimated simultaneously and thus will lead to an estimation of the parameter values with

potentially a lower uncertainty (Al-Haque et al., 2012). Although the goal of Al-Haque et al. (2012)

was to create a robust model for parameter estimation, the results presented in the paper indicate that

some of the estimated parameters are still heavily correlated. This is not necessarily a problem if these

parameters can be estimated independent of each other, however two heavily correlated parameters

were estimated based on the same experiment. At the time of writing, research is being carried out

at the Biomath research unit regarding overparameterisation, uniqueness of the estimated parameters

and optimal experimental design. Without fully mature results, in this thesis, the model of Al-Haque

is considered as valid.

The rate equations mentioned above are not coupled with the Navier-Stokes system. They have no

influence on the flow in the reactor as they are modelled as solutes. It follows that these equations

can be solved separately from the PISO or SIMPLE algorithm, which hugely simplifies the solution

procedure. In this thesis, the velocity profiles are considered steady state. Hence, from these velocity

profiles, the advection-diffusion-reaction equations can be calculated separately. The general form of

the advection-diffusion-reaction equation is stated in equation 3.9.

∂C

∂t= ∇ · (DC∇C)−∇ · (UC)− rC (3.9)


with C, the volumetric concentration of the solute, DC the diffusion constant of the solute in the fluid

under study, and rC the source or sink term. As an example, the result of this equation applied on the

substrate A (SA) is written down in equation 3.10.

∂ [SA]

∂t= ∇ · (DSA∇ [SA])−∇ · (U [SA])− r[SA] (3.10)

3.2.5 Solution procedure

Combining the procedures described in the subsection above, a solution procedure can be derived to

solve the Navier-Stokes system with additional coupled transport equations (in this thesis, a diffusion

model for solutes and reaction kinetics). In general: two types of solution procedures are distinguished:

transient and steady state solvers. When using a transient solver, all inter-equation couplings apart

from the pressure velocity system are lagged, i.e. the pressure and velocity field are calculated before

the other fields, disregarding the coupling with other equations. If it is necessary to ensure a close

coupling between all equations, e.g. the coupling of pressure and energy in combustion processes, the

additional transport equations are included in the PISO loop. If this is not the case, computational

load can be thrift by solving these additional equations after the PISO loop. To obtain a steady state

profile, an additional solver was created based on the SIMPLE-algorithm. The general solver algorithm

is described as:

1. Set up the initial conditions and boundary constraints for all fields.

2. Start the calculation for a new time step.

3. Using the available face fluxes, assemble and solve the momentum predictor.

4. Initiate the PISO or SIMPLE loop and iterate until the tolerance, specified in the initial conditions,

for the pressure-velocity system is reached. The new fields for the variables pressure and velocity

and a new set of conservative fluxes are available.

5. The new set of conservative fluxes will be used to calculate equation 3.6.

6. If the goal of the calculation is achieving a steady state profile for all the fields, equation 3.11 can

be used. On the calculated residuals, several checks can be perform to quantify the convergence

(e.g. maximum residual, volume weighted average residual, root mean squared residual etc.). The

outcome is compared with the initial set-point. If the convergence criterion is reached, break the

loop.

residualsfield k = | kprevious timestep − kcurrent timestep | (3.11)

7. If the goal is not a steady state profile, a simple check is performed: if this time step is not equal

to the final time step, go back to step 2.

As mentioned in 2.3, the flow in microreactors is laminar. Adding to that the assumption of a steady

state flow, the solution procedure can be simplified to greatly diminish the computational load. The

diminishing is accomplished by eliminating the need to perform the PISO or SIMPLE loop in each time

step. The new solution procedure then becomes:

1. Set up the initial conditions and boundary constraints for the fields pressure and velocity.

2. Using the available face fluxes, assemble and solve the momentum predictor.

3. Initiate the PISO or SIMPLE loop and iterate until the tolerance, specified in the initial conditions,

for the pressure-velocity system is reached. The new fields for the variables pressure and velocity

and a new set of conservative fluxes are available.


4. Calculate the residuals for both the pressure and the velocity field by calculating the absolute

difference between the values at this time step and the values at the previous time step. Do a

calculation on said residuals and compare that with an initially set tolerance. If the calculated

tolerance is below the set-point, go to the next step, if not, go back to step 2.

5. Start the calculation for a new time step.

6. Use the steady state pressure and velocity fields to calculate the transport equations and reaction

kinetics for the desired time steps. If desirable, calculating steady state concentration profiles is

possible, using the same methodology as step 4.

3.3 Simplified models

Below, three types of simplified models are described. Their main purpose is to simulate the process

under consideration at a significantly reduced computational load while keeping the same level of accu-

racy for the variables of interest. For attaining this computational load, several assumptions are made.

As a result, the simplified models are only valid in a limited operational parameter space. Yet, within

this range they are proven useful for fast simulation and process control (Levenspiel, 1972).

3.3.1 Mixed flow model

In the mixed flow model, the reactor is modelled as one continuously stirred tank reactor (Completely

Stirred Tank Reactor (CSTR)). This model assumes perfect mixing inside the reactor. The output

composition of the fluid flow is identical to the composition inside the reactor (Levenspiel, 1972). The

species in the reactor are modelled by using a material balance over the reactor.

accumulation︷︸︸︷dC

dt=

in︷︸︸︷Qin · Cin

V−

out︷︸︸︷Qout · C

V+

reaction︷︸︸︷r (3.12)

Here, C is the concentration of the solute, Qin and Qout are respectively the inlet and the outlet fluid

flow, V is the volume of the reactor, and r is the volumetric sink/source term.

3.3.2 Plug flow model

In the plug flow model, the fluid flow in the reactor is modelled as a series of infinitely thin coherent

plugs, moving in the axial direction, i.e. in the direction of the fluid velocity, of the reactor. In this

model, the fluid is perfectly mixed in the radial direction, i.e. perpendicular to the flow velocity vector,

but not in the axial direction. In the plug flow model, each infinitesimal volume or plug can by considered

as a CSTR (Levenspiel, 1972). A representation of the velocity profile can be found in figure 3.3. The

species in each plug can be modelled by using equation 3.13.


dt=


30 3.3 SIMPLIFIED MODELS

Figure 3.2: Representation of ideal plug flow (left) and mixed flow (right) behaviour

(Levenspiel, 1972)

3.3.3 Dispersion model

The fluid mechanics in conventional CFD modelling, i.e. solving the Navier-Stokes system can be

replaced by more straightforward models which have less computational load. According to Levenspiel

(1972), the dispersion model, or dispersed plug flow model, is a plug flow model with some degree of

back-mixing superimposed on top of it. The magnitude of back-mixing is independent of the position

within the vessel. The latter implies that there is no occurrence of stagnant pockets, nor gross bypassing,

nor short-circuiting of fluid in the vessel. With varying intensities of intermixing, the model predictions

range from ideal plug flow to dispersed flow, i.e. a Completely Stirred Tank Reactor (CSTR). Figure

3.3 shows a good representation of the dispersion model. The back-mixing is quantified using a axial

dispersion coefficient (D). The theory behind this is similar to Fick’s law of diffusion. The coefficient

can be determined by performing a tracer test using the mean time of passage and the spread of the

curve. For a dispersion coefficient approaching zero, the dispersion becomes negligible, and the model

becomes the plug flow model. On the other hand, if the dispersion coefficient approaches infinity, the

dispersion becomes large, and the model becomes the mixed flow model. The dispersion model can be

modelled by using equation 3.14.


dt=

axial dispersion︷︸︸︷D · d

2C

dx2+


Figure 3.3: Representation of the velocity profile in a plug flow model (left), and in

the dispersion model (right) (Levenspiel, 1972)

3.3.4 Tanks-In-Series model

In Tanks-In-Series model (TIS), the whole reactor is modelled as a series of CSTRs. The TIS model has

the advantage of being a simple model and can easily be extended to any arrangement of compartments

(Levenspiel, 1972). Next to that, the TIS model has been widely applied in the modelling of activated

sludge waste-water treatment plants. However, the simplicity is its main drawback. The fluid flow is only


modelled in one direction and back-mixing can only be introduced in the model by retaining the liquid

in the system for a longer time. Whereas back-mixing can somehow be introduced, recirculation fluxes

cannot be represented with TIS. The major limitation of systemic models, e.g. the dispersion model

and TIS, is that, when combined with a (bio)kinetic model, the degrees of freedom of the (bio)kinetic

model will be used to compensate for the flaws in the mixing model. This method of ”calibration” is bad

modelling practice and will severely reduce the predictive power of the model (Alvarado et al., 2012).

The number of mixed tanks is determined through analysis of the RTD curve and the Peclet number

(the ratio of convective transport to the molecular transport). The number of tanks are varied until the

simulated tracer test sufficiently approximates the experimental one according to a predefined tolerance

(Levenspiel, 1972). An approximation of the required number of tanks can be calculated using equation

3.15 (Alvarado et al., 2012).

2 · (N − 1) = Pe =Uavg ·∆xDsolute

(3.15)

with N , the number of equivolume fully mixed tanks, which are connected in series, along the major flow

dimension. Pe is the Peclet number, Uavg is the average flow velocity, ∆x is the characteristic length

of the reactor, and Dsolute is the diffusion constant of the solute used in the tracer test. The TIS model

can be implemented using a system of Ordinary Differential Equations (ODEs), describing the mass

balance between consecutive reactors, see equation 3.16. In equation 3.16, Q stands for the volumetric

fluid flow, V is the volume of each tank, Cin is the inlet concentration, and Ci is the concentration in

tank i.

dC1

dt = Q·(Cin−C1)V

dC2

dt = Q·(C1−C2)V

...

dCN

dt = Q·(CN−1−CN )V

(3.16)

Figure 3.4: Representation of the tanks-in-series model (Levenspiel, 1972)

3.3.5 Compartmental Model

A CM holds in between a TIS and a CFD model, it consists of a number of compartments in more than

one dimension which are interconnected by both a recirculation flow and a forward flow. CM can be

seen as an extension over TIS: an increased freedom in defining the compartments and the ability to

use more than one dimension. Similar to TIS, each compartment is considered as a fully mixed volume.

Generally speaking, the following steps need to be performed in order to set up the CM (Alvarado et al.,

2012):

1. Set up a CFD-model to predict steady-state fluid flow in the container of interest.

2. Determine the different zones and volume of said zones using the CFD predictions and tracer tests.

32 3.4 SOFTWARE

3. Determine the number of compartments per zone to approximate the mixing behaviour. Equation

3.15 can be used for an initial estimation.

4. Determine the exchange and convective fluxes in and between zones by using the turbulence

characteristics of the flow and a mass balance between the different zones.

It is worth noting that the CM model can only be used if the whole flow pattern is not significantly

influenced by varying inlet conditions (Levenspiel, 1972). The approach mentioned above will be used

in the thesis to simplify the CFD models. With CM, it is possible to simulate tailing (see figure 3.5),

which is often observed for slow diffusion solutes. In tailing, an initial bell shaped curve is skewed, and

a tail is formed: the RTD curve looses its symmetry.

Figure 3.5: Visual representation of the tailing phenomenon: the form is similar to

an asymmetrical, skewed bell curve.

3.4 Software

In this thesis, only open source software was used.

3.4.1 Python

Python is a high-level programming level designed in the nineties. The Python philosophy relies on

readability of the code and the ability to write algorithms in fewer lines than C or C++ code. Python

supports different programming paradigms: object-oriented, procedural, imperative and functional pro-

gramming (Python-Software-Foundation, 2014). The license of the Python releases are held by the

Python Software Foundation (PSF), this PSF-license is compatible with the GNU General Public Li-

cense (GPL) (Python Software Foundation, 2014). The license for Python libraries and packages pro-

vided by third party software developers can differ from the general Python license, and thus has to be

checked with the provider of the library/package. In this thesis, Python is mainly used as a scripting

language (e.g. automation scripts and scenario analysis) and as a bridge between different software

packages.

3.4.2 OpenFOAM

OpenFOAM is an open source collection of flow solvers and utilities for the calculation of numerical flow

problems. OpenFOAM is not a program as such, it is a collection of binary files which can be edited

or created by the user. As OpenFOAM does not have a Graphical User Interface (GUI), the binaries

are called in the command line. The programming language of OpenFOAM is C++, both for solvers

as well as settings and case specific options (e.g. viscosity of the fluid, inlet speed, linear matrix solvers

. . . ). OpenFOAM is distributed by the OpenFOAM Foundation and is freely available and open source,

licensed under the GPL (OpenFOAM-Foundation, 2014).


3.4.3 Salome

The open-source software package Salome is used for the generation of the geometry and the mesh. The

source code is written in Python, and it thus can easily be controlled by Python scripts. This feature

makes the generation of large meshes on calculation clusters an easier task. It is also possible to generate

the geometry and the mesh with OpenFOAM, yet for large and complex configurations, the use of a

Salome is preferred. Salome provides more possibilities in generating meshes, it has a GUI interface

and compatibility with Computer Aided Design (CAD) files is included in the program. Salome is

distributed as open-source software under the terms of the GNU Lesser General Public License (LGPL)

(Open Cascade, 2014).

3.4.4 ParaView

ParaView is an open-source visualisation tool for large data sets. In this thesis it is used for the

visualisation and interpretation of the simulated flow patterns. As with Salome, ParaView contains a

Python console, which enables straightforward visualisation in a standardised manner in such a way

that different cases can be easily compared. Paraview is released by the Sandia Corporation under the

Berkeley Software Distribution, a UNIX-like software license (BSD) license (Sandia Corporation and

Kitware Inc, 2014).

CHAPTER 4Results

And now for something completely different

— Monty Python

The goal of this thesis is to provide a flexible model to predict the product concentration at the outlet

of the reactor. The final model should be fast and reliable, therefore several steps are necessary to

accomplish this. At the one hand a kinetic model is needed to calculate the local reaction kinetics.

At the other hand one needs a CFD model to account for the spatial variations in the reactor. By

combining the kinetic model with the CFD, one gets a very accurate and reliable model. This kind of

models are very flexible, but also have a high computational footprint. To counter this computational

load, two simplified models are investigated to check whether further speed-up of the simulations is

possible: TIS and CM. The TIS model is the easiest model, but can only predict bell shaped curves in

an RTD test. With the CM on the other hand, it is possible to simulate tailing.

In this chapter the uncertainty on the kinetic model will be calculated first. In this way a trade-off can

be made between accuracy and calculation speed for the RTD calculations. Second, CFD study will be

performed to examine whether the obtained solution is independent of the mesh under consideration

(mesh independency check). Finally, the kinetic and CFD model can be combined and be used to

calibrate and validate the simplified models. A schematic overview of the materials in this chapter can

be found in figure 4.1.

Figure 4.1: Overview of the methodology followed within this thesis to go from a

physical model to a simplified model

36 4.1 UNCERTAINTY ON RATE EQUATION OF THE KINETIC MODEL

4.1 Uncertainty on rate equation of the kinetic model

As stated in chapter 3, the kinetic model from the work of Al-Haque et al. (2012) is used to describe the

enzymatic reactions (equation 3.6). In this work, Al-Haque mentioned the 95% confidence intervals on

the estimated parameters (table 4.1). However, the confidence interval on the rate equation, or on the

simulated concentrations is not given. To establish a baseline for the CFD simulations, an estimation

of the error on the kinetic model is calculated to quantify the maximum tolerated error on the CFD

models. Al-Haque et al. (2012) mentioned that in an initial estimation, parameters KSBSi and KPP

Si were

extremely large compared to the operating concentration of the reactants (7.2 · 104 and 1.1 · 104 mM

respectively). The significance of these terms could thus be considered negligible and were therefore

omitted from the kinetic model (equation 3.6).

Table 4.1: Parameter values and confidence intervals for the Al-Haque kinetic model:

equation 3.6 (Al-Haque et al., 2012). The equilibrium constant (KEQ) and its confi-

dence interval is taken from Tufvesson et al. (2012).∗ No information was given about the confidence interval of KSA

i and KPPi .

Parameter Parameter value 95% CI Unit

Kfcat 0.0078 +− 0.0005 min−1

Krcat 0.0013 +− 0.0070 min−1

KSAM 1.85 +− 4.78 mM−1

KSBM 101.28 +− 38.23 mM−1

KPPM 0.12 +− 0.01 mM−1

KPQM 148.99 +− 2.91 mM−1

KSASi 4.1500 +− 0.0003 mM−1

KPPSi 10.3800 +− 0.0003 mM−1

KSAi 0.09 - ∗ mM−1

KSBi 4281.00 +− 0.63 mM−1

KPPi 100000.0 - ∗ mM−1

KPQi 0.11 +− 0.01 mM−1

KEQ 0.033000 +− 0.003234 -

The variance on the rate equation is calculated by propagating the error from the individual parameters.

The rules used to propagate the error are listed in table 4.2.

Table 4.2: Formulas for the calculation of the propagation of errors (Bevington and

Robinson, 2002).

Function Variance

f=aA σ2f = a2σ2

A

f=aA+ bB σ2f = a2σ2

A + b2σ2B + 2ab covAB

f=AB σ2f ≈ f2

[(σA

A

)2+(σB

B

)2+ 2 covAB

AB

]f=A

B σ2f ≈ f2

[(σA

A

)2+(σB

B

)2 − 2 covAB

AB

]The complete analytic derivation of the error on the rate equation is given in appendix A. The analysis

is implemented in Python using the Python uncertainties-package (Lebigot, 2014). The results from

this package are equal to those from the analytic calculations. For further analysis, the Python package

is used to obtain more flexibility in the calculations.

To check the implementation of the rate equation and the error on this equation, the simulated concen-

trations mentioned in Al-Haque et al. (2012) are reproduced. However, it was not possible to reproduce

CHAPTER 4 RESULTS 37

the results reported in the article. One possible reason is non-matching units in the rate equation:

the enzyme concentration is given in mass concentration, where a molar concentration is expected to

yield the expected unit for the rate equation ( mmolL min ). It was not possible to retrieve the density of the

enzyme under study, so no conversion to molar concentration could be made. For the simulations, it

was assumed that molar concentrations were meant in the article. The discussion ahead will focus on

the full rate equation, as this is the equation which will be used to simulated the solute concentrations.

The comparison between Al-Haque and our simulations can be found in figures: figures 4.2a, 4.2b, 4.3a

and 4.3b. A more detailed view regarding the relative error is given in figures 4.4a and 4.4b.

Not only was it impossible to repeat the simulations of Al-Haque et al. (2012), the error on the simulated

concentrations is high for both the substrates and the products. Although Al-Haque states that the

model is fully validated, as the simulations approximate the experimental results in a reasonable way.

From the uncertainty analysis carried out in this work it is clear that the model calibration and validation

is not as good as stated in the article.

Figures 4.4a and 4.4b visualise the percent error on the solutes, see equation 4.1 for the method of

calculation. From the results, it can clearly be seen that, the error margins are large. This means that

the results obtained with this equation are uncertain, and a better parameter estimation and/or model

structure may be required to model the enzyme kinetics more accurately.

percent error =absolute error

nominal value· 100% (4.1)

The work on the rate equation is still ongoing in the Biointense project. Due to lack of fully mature

results, the rate equation of Al-Haque will be considered as accurate, and will be used in further

simulations. The OpenFOAM solvers created to solve the enzyme kinetics are created as customisable

as possible to ease the modification of the rate equation, when it would be necessary later on in the

Biointense project. For further calculations, a baseline error margin of 5% will be taken. This baseline

is chosen this low to ensure that the largest contribution to the errors in the simulations will be because

of the kinetic model.

(a) The descending curve is the substrate SA, the ris-

ing curve is the product PP. Experimental values (trian-

gles), and modelled concentrations (full line) from Al-

Haque et al. (2012)

(b) Attempt to recreate the results from Al-Haque et al.

(2012), with 95% confidence intervals on the concentra-

tions.

Figure 4.2: Experimental setup for the forward reaction, the reaction conditions are

as follows: E0 = 1.8 g/L, CSA = 1.7mM, CSB = 1000mM, CPP = 0.5mM, 2mM

PLP, 100mM phosphate buffer


(a) The descending curve is the product PP, the rising

curve is the substrate SA. Experimental values (trian-

gles), and modelled concentrations (full line) from Al-

Haque et al. (2012)

(b) Attempt to recreate the results from Al-Haque et al.

(2012), with 95% confidence intervals on the concentra-

tions.

Figure 4.3: Experimental setup for the reverse reaction, the reaction conditions are

as follows: E0 = 3.6 g/L, CSA = 0mM, CPP = 5mM, CPQ = 1000mM, 2mM PLP,

100mM phosphate buffer

(a) Visualisation of the relative error for the forward

reaction (figure 4.2b)

(b) Visualisation of the relative error for the reverse

reaction (figure 4.3b)

Figure 4.4: Visualisation of the percentage error on the concentrations for the recre-

ated simulations of Al-Haque et al. (2012). The full lines denote the error on the

concentration in function of time, the dotted lines represent the time averaged error

on the concentration profile.

4.2 Computational fluid dynamics

4.2.1 Flexible mesh generation

As stated in section 3.4, mesh generation by use of the Salome software package can be done by Python

scripting. This approach has several advantages over traditional GUI geometry and mesh generation.

First, it is repeatable: modifying a mesh after it is created is often a tedious task, as the links between the

different geometry and/or mesh objects are often hard coded. Changing the mesh implies going through


the whole point-and-click process again for each modification: an inefficient and time consuming effort.

Generating, viewing and analysing a modified mesh by means of a script is simply a task of running

that script through the Python interpreter of Salome.

Second, once the script has been made, no GUI is needed to generate the mesh. This fact opens the

possibility for automated analysis and automated geometry optimisation as no GUI operation is required

from the user. Geometry optimisation does not fall within the scope of this thesis. The Python script is

therefore only used to provide a flexible and repeatable method to generate the meshes used to calculate

the results discussed further in this chapter, but is a nice piece of work looking at the future needs.

Finally, meshing by means of a script makes collaboration and sharing of code between researchers an

easier task. However, one should bear in mind that the learning curve for non graphical mesh generation

is much steeper than the conventional graphical approach.

The entire reactor is made in function of the following parameters: width, height, length, number of

cells per 100 µm, equidistant or non equidistant mesh, 2D/3D mesh, and whether or not to export the

mesh. At the moment, these parameters are hard coded into the script. However by applying a small

change, the desired value for a certain parameter can by given when the script is called in the command

line. This option is only vital for geometry optimisation, therefore it is not yet included in the current

version of the script.

In order to decrease the computational load and to simplify the mesh generation procedure, a simpli-

fication of the mesh is performed. This simplification is based on the work of Plazl and Lakner (2010)

and Stojkovic et al. (2011). In both articles, a meandering microreactor, similar to the one used in this

thesis (figure 2.9), is simplified to a straight pipe with a rectangular cross section. The original width,

height, and channel length are preserved: 0.2 mm x 0.4 mm x 334.1 mm.

The mesh quality is evaluated with the OpenFOAM checkMesh utility. This utility does a thorough

check of the geometry and topology of the mesh: bounded volumes, connectivity between cells, no double

volumes . . . A selection of the most important criteria is given in table 4.3. The output states that

the boundaries of the mesh, and the inter-cell connectivity are valid. Further, the output confirms the

generation of a fully orthogonal mesh with flat faces, and a reasonable aspect ratio. The cell volumes

and face area magnitudes are both valid (no negative/zero volumes/areas). Finally, the checkMesh

utility confirms the reactor dimensions: the length equal to 0.3441 m, the width equal to 0.0002 m, and

a height equal to 0.0004 m. According to the checkMesh utility, the mesh generated by the Python

script is valid for use in OpenFOAM simulations.

Table 4.3: Output of the OpenFOAM checkMesh utility applied on the fullscale

geometry.

Criterion Value for the studied case

Number of hexahedral cells 1548450

Boundary definition OK

Face-face connectivity OK

Overall domain bounding box (0 -0.0001 0) (0.3441 0.0001 0.0004)

Maximum aspect ratio 2.29749 (OK)

Face area magnitudes OK

Cell volumes OK

Mesh non-orthogonality Max: 0 average: 0

Face flatness (1 = flat, 0 = butterfly) average = 1 min = 1

Mesh OK


4.2.2 Python package: scenario analysis

In the OpenFOAM library, the input and output files are written in American Standard Code for

Information Interchange (ASCII) typesetting. This fact allows the easy generation of scripts which

can be automated to change certain parameters of the simulation. The OpenFOAM community has

provided a collection of Python scripts (PyFoam), which act as a wrapper around OpenFOAM source

code. This Python package increases the versatility of OpenFOAM.

In this thesis, the scripts from the PyFoam Python package are used to create a scenario analysis

tool written in Python. This tool combines the PyFoam script with OpenFOAM binaries and bash

commands in a fully flexible and general way.

The standard procedure of CFD-analysis is to set up the case, change the desired parameters, run the

case and save/analyse the output. For each scenario this procedure has to be carried out, which is an

inefficient and time consuming effort. The scenario analysis tool allows to merely define each scenario,

and the script will execute them one at a time. In this thesis, the script was used to calculate the RTD

tracer tests for different inlet velocities, which allows to build a general CM that can predict different

inlet conditions. Furthermore, the script will be used to calculate enzyme kinetics for different reactor

configurations.

The current version of the scenario analysis tool is built for OpenFOAM version 2.2.x. The tool can

change any parameter, boundary condition, and can even be used to test the effect of different numerical

schemes. The scenarios can be run on single or multi node jobs, as desired by the user. Furthermore,

the user has the option to choose what output should be stored of each scenario: the choice of time steps

and calculated fields to be kept. After the simulation, an overview of the analysed scenarios is created,

and the output is stored in such a way that comparing and post-processing the different scenarios with

ParaView can be done without any additional modification by the user.

4.2.3 Mesh dependency

The goal of a mesh dependency test is to investigate the effect of the spatial discretisation on the

solution. For consequently smaller mesh sizes, the solution should converge to the actual solution. As

small mesh sizes have more cells and require smaller time steps to calculate, the computational load

rapidly increases for decreasing mesh size. Hence, a trade-off has to be made between solution accuracy

and computational expense.

To investigate the effect of the spatial discretisation on the solution, a small test case is built. This

small test case is, compared to the actual reactor, shortened lengthwise by a factor of approximately

1/35 (shortened length is equal to 0.01 m). The solution under consideration is a steady state velocity

and pressure profile. The solution is considered as converged if the residuals between two consecutive

iterations fall below 10−6. The sparse matrix solvers used in OpenFOAM are iterative, i.e. they are

based on reducing the equation residual over a succession of solutions. As an illustration, the general

form of a system of linear equations in matrix notation is denoted in equation 4.2, with x the vector

with variables, and b the vector of constants, to be solved for the coefficient matrix A. The residual is

defined as the difference between the right and the left hand side of equation 4.2 for an estimation of

the coefficient matrix: A∗, written down in equation 4.3. For the errors in the solution, the residuals

are used as a method of measurement, the smaller the residual, the more accurate the solution. The

residual is evaluated by substituting the current solution into the equation and taking the magnitude

of the difference between the left and right hand sides, after that the residual is normalised to make it

independent of the scale of the problem.

A · x = b (4.2)

residual = b−A∗ · x (4.3)


The generated mesh is one of the structured type consisting of regular hexahedrons with an aspect

ratio (the ratio of the largest to smallest side in the cell) as close as possible to unity. The mesh can

be generated with two degrees of freedom: the number of cells on the width of the reactor, and the

ratio of the largest to smallest cell. For the two-dimensional simulations, multiple meshes are generated

to quantify the mesh dependency. The number of cells on the width were varied between 8 and 50,

with steps of 2, and some larger cases of 60, 70 and 80 cells on the width were also simulated. A

non-equidistant mesh is used: the cells at the wall of the reactor are smaller than those in the middle

to be able to cope with the steep velocity gradient near the reactor wall. Good CFD practice states

that the mesh should decrease in size in regions where the gradient is large (Versteeg and Malalasekera,

2002). A cell ratio of 4 is chosen for these simulations: the cells at the reactor wall are sufficiently small

and the cells in the middle of the reactor not too large. This can be quantified by the aspect ratio which

is equal to 2.30 for a mesh with 30 cells along the width (see table 4.3). The cell ratio is not chosen

larger than this value because aspect ratios which differ a lot from the unity (both smaller and larger)

can lead to unstable solutions (Versteeg and Malalasekera, 2002). A detail of a mesh with 30 cells on

the width is visualised in figure 4.5a

(a) The cells (30 on the

width) are smaller towards

the walls of the microreac-

tor. Close to the wall, the

gradient of the solution is

high, and small cells are

needed to attain a solution

with a sufficient resolution

(b) Illustration of the non equidistant mesh cell centres. The cell centres

are plotted (x-axis) in function of the number of cells on the width of the

reactor (y-axis).

Figure 4.5: Details about the generated non-equidistant mesh: visualisation of the

mesh (left), and cell centres on the width of the reactor (right)


Figure 4.6: Theoretical velocity profile in the microreactor, combined with the ex-

perimental results for a mesh with 12 and 30 cells on the width of the reactor. The

boundary condition at the reactor wall is the no-slip condition: the velocity is equal

to zero at the reactor wall. The form of the velocity profile is independent of inlet

flow, hence the use of a normalised velocity.

The simulations discussed further in this chapter will be based on a theoretical residence time between

10.3 and 30.9 minutes on the full scale reactor for which a fixed mesh size of 30 cells on the width of the

reactor ( 15 cells0.001 m ) will be used. From the definition of the Courant number (equation 3.5), the fastest

velocity, or lowest residence time, will determine the stability of the system (lower limit for the time

step in transient calculations). A residence time of 10 minutes is attained with a uniform inlet velocity

equal to 5.57 · 10−4 m s−1, corresponding to a flow rate of 4.45 · 10−11 m3 s−1.

The difference between the different mesh sizes is studied as follows: raw cell data is extracted from

the solution by means of the OpenFOAM sample utility. The reactor is sampled perpendicular to the

flow, i.e. on the width of the reactor. The sampling line is taken at about 75% of the reactor length

so that the flow is fully developed, and outlet effects are avoided. The cell values are compared with

the theoretical solution, i.e. the parabolic velocity profile. The derivation of the theoretical velocity

profile is summarised in appendix B. The analytical velocity profile, together with a 5% error margin is

visualised in figure 4.6.

In table 4.4, the maximum and average percent error are compared to the analytical solution. From

this table, it can be concluded that the 5% error margin is already attained at 12 cells on the width of

the reactor. For all mesh sizes, the simulated flow approximates the theoretical flow very well: accurate

for 5 decimals. This means that the mass balance is correct for all mesh sizes. The ideal mesh size will

thus be determined by error on the velocity.

A mesh with 12 cells on the width is sufficient according to the chosen baseline error. However, when

using only 12 cells, the parabolic form of the curve is lost. In the middle of the reactor, the distance

between the two cell centres is large (figure 4.6). From this figure it can be concluded that a mesh with

30 cells on the width has a low error on the velocity, retains the theoretical form of the velocity curve,

and has a very acceptable computational expense. For further calculations, a mesh with 30 cells on the

width is considered as sufficiently accurate.

4.2.4 Residence time distribution

Prior to making simplified models, a number of things have to be considered. One of the most important

is the Residence Time Distribution (RTD). The RTD is a distribution which describes how much mass

of fluid or solute leaves the reactor in function of time. This kind of experiments can help to characterise


Table 4.4: Overview of the maximum and mean error on the simulated fluid velocity

compared to the analytical velocity, and CPU time needed for calculation for different

number of cells on the width of the reactor.

Nocells errormax(%) errormean (%) CPU time (s)

8 8.36 4.34 0.21

10 5.59 2.93 0.28

12 3.98 2.09 0.48

14 2.98 1.62 0.58

16 2.50 1.29 0.69

18 2.25 1.04 0.91

20 2.05 0.87 1.18

22 1.89 0.74 1.38

24 1.75 0.63 1.56

26 1.62 0.55 1.93

28 1.52 0.49 2.22

30 1.42 0.43 2.48

32 1.34 0.38 3.00

34 1.27 0.35 3.25

36 1.21 0.31 4.12

38 1.15 0.29 4.70

40 1.09 0.26 5.70

42 1.04 0.24 6.43

44 1.00 0.22 7.46

46 0.96 0.20 8.69

48 0.93 0.19 9.93

50 0.89 0.18 11.52

60 0.76 0.13 20.68

70 0.54 0.09 36.33

80 0.53 0.07 63.87

non-ideal mixing and flow behaviour in reactors (Levenspiel, 1972). The mass flux is obtained by adding

additional code to OpenFOAM by using the SWAK4FOAM libraries: the mass (kmol) of solute leaving

the reactor in function of time equals the sum over the outlet surface of the product of the fluid face

flux (obtained from the velocity tensor field) and the solute concentration (equation 4.4).

mass flux solute =

N∑cell i =1

(Ucell i · Scell i · Ccell i) (4.4)

In equation 4.4, Ucell i is the fluid velocity in cell i, Scell i is the outward-pointing surface area vector

of the outlet patch for cell i, and Ccell i is the solute concentration in cell i. In contrast to the mesh

dependency simulations which are based on steady state velocity and pressure profiles, this type of

simulations are time-dependent. For these transient simulations, the choice of the time step is crucial:

it needs to be small enough to guarantee a stable numerical solution, yet large enough to keep the

computational expense to an acceptable level. The time step is chosen to be the highest possible step

while still maintaining a Courant number lower than unity (equation 3.5).

The convection-diffusion equation was added to the OpenFOAM code to account for the behaviour of

solutes. Therefore one needs the diffusion constants of all the solutes under consideration. The diffusion

constants used in the Biointense project are based on the work of Bodla et al. (2013). New estimations

were executed based on previous work. These results are summarised in table 4.5. However, these


results are preliminary and need to be validated by further experimental work to be performed in the

Biointense project.

Table 4.5: Diffusion constants for the solutes used in the Biointense project, modified

from Bodla et al. (2013).

Solute Diffusion constant (m2s−1)

SA (acetophenone) 1.0·10−13

SB (isopropylamine) 9.1·10−10

PP (1-phenylethylamine) 6.9·10−10

PQ (acetone) 9.1·10−10

E (transaminase) 5.0·10−12

A small test case is built to investigate the effect of the mesh size on the concentration profiles. After this

mesh dependency check, the effect of the time step was investigated. This small test case is, compared

to the actual reactor, shortened lengthwise by a factor of approximately 1/70 (length = 0.005m). It is

assumed that errors in this small test case will persist in the full scale reactor and vice versa. Similar

to the mesh dependency, the fastest residence time is selected to obtain the lower limit of the time step.

The simulation time is 15 seconds, during the first second, a pulse with a concentration of 1 mol/L is

added to the reactor.

During initial simulations, it was observed that the slow diffusion solute (SA) lead to unstable solutions

when using the standard discretisation algorithms. The issues are related to the interpolation of the

divergent scheme: a cell Peclet number (see equation 4.5) exceeding the value of two leads to instabilities

with the standard choice of interpolation which is Central Differencing (Central Differencing (CD)).

Pe =ρ ·U ·∆xDsolute

(4.5)

In equation 4.5, Pe is the cell Peclet number, ρ is the density of the fluid, U is the fluid velocity, ∆x

is the grid spacing, and Dsolute is the diffusion constant of the solute. The scheme can be stabilised by

lowering the cell Peclet number by lowering the grid spacing, i.e. utilising a finer mesh. However, this

would require a refinement of over a hundred fold of the mesh due to the very low diffusion constant

of SA. This refinement would lead to a massive increase in computational load. Another option is to

use a different interpolation scheme like Upwind Differencing (Upwind Differencing (UD)), Quadratic

Upwind Interpolation for Convective Kinetics (Quadratic Upwind Interpolation for Convective Kinetics

(QUICK)), Total Variance Diminishing (Total Variance Diminishing (TVD)), SUPERBEE, Van Leer,

Van Albada or Min-Mod scheme. UD is a first order scheme, whereas the other six are higher order

schemes which use flux limiters (Versteeg and Malalasekera, 2002). It was found that only the QUICK

and UD scheme are stable on the chosen geometry. However, due to the nature of the QUICK scheme,

over- and undershoots of the solution occur, yielding negative concentrations in the reactor, which is

physically impossible. The UD scheme is chosen as the interpolation scheme for the divergent term of

the numerical scheme. However, due to the fact that UD is only a first order scheme, the accuracy will

be lower than higher order schemes. This a consideration that has been made: generating an acceptable

solution in a limited time frame while keeping the computational load to an acceptable level.

To check the results obtained in the mesh dependency test, the RTD simulations are executed at different

mesh sizes, using the same methodology as in the mesh dependency. A mesh with 100 cells on the width

of the reactor is chosen as the baseline to compare with the other mesh sizes. Similar to the mesh

dependency test, an error margin of 5% applied to the baseline (the results from the mesh with 100 cells

on the width) will be used to check the simulations.

The RTD simulations are performed on the solute with the lowest (SA) and highest (SB) diffusion

constant to ensure that the whole range of possible outcomes is in the design space. The following

variables will be compared: mass flux leaving the reactor, cumulative mass, and relative cumulative


mass. For substrate SA, the results are visualised in figures 4.8a, and 4.7a. For substrate SB, the results

are visualised in figures 4.8b, and 4.7b

The cumulative mass percentage of substrate SA is visualised in figure 4.8a. The calculation of the

cumulative mass percentage is given in equation 4.6. Except for a minor percentage between minutes 5

and 6, the cumulative mass percentage of all the mesh sizes lies within the error band of 5%.

cumulative mass percentage =

∑tt=0 (mass SA)∑tend

t=0 (mass SA)· 100% (4.6)

The mass flux of substrate SA leaving the reactor is visualised in the upper part of figure 4.7a. As can

be expected, the solution approaches the baseline for increasing mesh sizes. In the tail of the curve

oscillations are observed. They have their origin in spatial discretisation errors: due to the almost

non-diffusive nature of substrate SA, the concentration profile gives rise to the observed oscillations. In

fact, these oscillations are still present in the baseline mesh, but now with very small amplitude (see

figure 4.9). These errors have no effect on the total mass in the reactor, as can be seen in figure 4.8a:

the cumulative mass in the reactor does not significantly change by altering mesh sizes.

In the lower part of figure 4.7a, the relative cumulative mass of substrate SA is visualised. It is calculated

using equation 4.7. This figure combined with the upper part of the same figure shows that although

the oscillations are physically not correct, the effect on the cumulative mass flow is marginal.

(a) Concentration at the outlet (top) and relative cu-

mulative mass (bottom) in function of time and number

of cells on the width for substrate SA

(b) Concentration at the outlet (top) and relative cu-

mulative mass (bottom) in function of time and number

of cells on the width for substrate SB

Figure 4.7: Results of the tracer test simulations on the small test case: concentration

at the outlet and relative cumulative mass in function of time, and number of cells

on the width of the reactor for substrate SA (left) and SB (right). The gray area

visualises a 5% error band around the baseline solution (100 cells along the width).

relative cumulative mass percentage =

∑tt=0 (mass SA)∑tend

t=0 (mass SA) ·∑tend

t=0 (mass SA baseline)· 100% (4.7)

The cumulative mass percentage of substrate SB is visualised in figure 4.8a. The calculation of the

cumulative mass percentage is similar to the calculation for substrate SA and is given in equation 4.6.

Similar to the cumulative plot of SA, only a small part of the curve lies outside the error band of 5%:

which is around minute 6.

In the upper part figure 4.7b, the mass flux of SB leaving the reactor is visualised. From this plot,

it is clear that mesh refinement only has a minor effect on the simulated mass flux. Contrary to the

simulations for SA, no oscillations are observed. In the lower part of figure 4.7b, the relative cumulative


mass percentage of substrate SB is visualised. Combining these two figures, it is concluded that the

solution dependency of the mesh will be determined by substrate SA, as substrate SB has oscillation-free

profiles, even for lower mesh sizes. It is worth noting that the RTD curve of SA has a higher peak and

a longer tail. These two properties can be assigned to the low diffusivity: the pulse has a sharper front,

and the solute at the wall of the reactor diffuses less towards the centre where the velocity is larger

(longer tail).

(a) Cumulative mass percentage in function of time, and

number of cells on the width of the reactor for substrate

SA

(b) Cumulative mass percentage in function of time,

and number of cells on the width of the reactor for sub-

strate SB

Figure 4.8: Results of the tracer test simulations on the small test case: cumulative

mass percentage in function of time, and number of cells on the width of the reactor

for substrate SA (left) and SB (right). The gray area visualises a 5% error band

around the baseline solution (100 cells along the width).

Table 4.6: Simulation time for the different mesh sizes.

Nocells Simulation time (s) Simulation timenormalised

20 61.69 0.37

30 166.11 1.00

40 326.06 1.96

50 578.98 3.49

60 1763.59 10.62

70 2550.21 15.35

80 3332.42 20.06

90 4313.89 25.97

100 10122.70 60.94


Figure 4.9: RTD test with a mesh with 100 cells on the width of the reactor. The

concentration shown in the figure is for the slow diffusion solute SA. Even at this

resolution, small oscillations in the concentration are observed

The second variable that can be changed is the time step. The upper limit of this time step is determined

by the velocity and the mesh size, following equation 3.5, to ensure that the Courant number is lower

than unity. Running simulations with a time step lower than the maximum allowed time step increases

the likelihood of having a stable numerical scheme (as the Courant number is a necessary, but not a

sufficient condition), reduces the time step error, yet it increases the computational load at the same time.

Figures 4.8a, 4.7a, 4.8b, and 4.7b are obtained using the largest time step possible while maintaining a

stable numerical scheme.

The three mesh sizes of interest (20, 30, and 40 cells on the width of the reactor) are calculated using

different time steps. The results for a mesh size equal to 20 are visualised in figures 4.10a and 4.10b. As

can be seen from these figures, reducing the time step has no significant effect on the simulation. The

same result is found for mesh sizes equal to 30 and 40 cells on the width (not visualised). However, a

reduced time step leads to an increase in computational load, as can be seen in table 4.7. Therefore it is

concluded that in the following simulations, the highest time step that still satisfies the CFL condition

(Courant number lower than unity) will be used.


(a) Concentration at the outlet (top) and relative cu-

mulative mass (bottom) in function of time and time

step for substrate SA

(b) Concentration at the outlet (top) and relative cu-

mulative mass (bottom) in function of time and time

step for substrate SB

Figure 4.10: Results of the tracer test simulations on the small test case: cumulative

mass percentage in function of time, and time step for substrate SA (left) and SB

(right) for a mesh size equal to 20 cells on the width

Table 4.7: Simulation time and Courant number for the different time steps. The

normalised simulation time is the simulation time of the case for that specific time

step divided by the simulation time of the base case.

Nocells Time step (s) tsimulation (s) tsimulation, normalised (-) Comax (-)

0.0100 41.91 1. 0.96

0.0050 71.23 1.70 0.48

20 0.0020 147.24 3.51 0.19

0.0010 287.09 6.85 0.10

0.0005 540.54 12.90 0.05

0.0100 96.56 0.61 1.49

0.0050 157.41 1. 0.75

30 0.0020 352.04 2.24 0.30

0.0010 686.29 4.36 0.15

0.0005 1367.97 8.69 0.07

0.0100 195.62 0.60 1.85

0.0050 325.71 1. 0.92

40 0.0020 702.41 2.16 0.37

0.0010 1353.75 4.16 0.18

0.0005 2628.53 8.07 0.09


(a) RTD test for a residence time equal to 10.3 minutes,

for substrates SA and SB.

(b) RTD test for a residence time equal to 20.6 minutes,


(c) RTD test for a residence time equal to 30.9 minutes,


(d) Visualisation of the discrete velocities. At each cell,

ci, the velocity is extracted for the given residence time.

These velocities are used to create the dotted lines rep-

resenting the theoretical residence time for that velocity

in figures 4.11a, 4.11b, and 4.11c.

Figure 4.11: Results for the tracer test simulations on the full scale reactor for

a theoretical residence time of 10.3 (figure 4.11a), 20.6 (figure 4.11b), and 30.09

minutes (figure 4.11c): molar concentration of the solute SA and SB in function of

time. Figure 4.11d is an illustration of the extraction procedure for obtaining the

discrete cell velocities used to create the dotted lines representing the theoretical

residence time for that velocity in the other three figures.

After an initial determination of the optimal simulation parameters, the RTD tests are performed on

the full scale reactor. Theoretical residence times for the fluid of 10.3, 20.6, and 30.9 minutes are chosen

for the simulations. The diffusion constants of table 4.5 are used. The results of these tracer tests

are visualised in figures 4.11a, 4.11b, and 4.11c. The concentration profile of SB approaches a bell

curve for the three simulations. The profile of the slow diffusing solute SA, on the other hand, has

some anomalies. However, this unrealistic profile can be explained by the slow diffusion constant, and

the spatial discretisation. It was presumed that SA diffusion was so slow that lateral transition in the

reactor barely occurs. This presumption was confirmed by plotting the theoretical residence time for the

discrete velocity steps (dotted lines in the figures). These lines are created by extracting the velocity

from the reactor for each cell on the width of the reactor. Visually, this can be represented by the

discrete velocities for each cell in figure 4.11d. Combining this data with the length of the reactor, the


residence time can be calculated for solutes without diffusion. The peaks of the concentration profile

coincide almost perfectly with the theoretical residence time. The forward offset can be explained by

numerical diffusion, the rounding of the peaks by the limited occurrence of diffusion of the solute. Aside

from these two facts, the uniform inlet velocity can play its role in the creation of the profile for SA.

The solute SA clearly exhibits unrealistic behaviour. The issue has its origin in the diffusion constant

(ignoring all numerical errors). The difference in diffusion constant for SA and PP is almost four orders

of magnitude, for two substances which differ very little in molecular structure. Moreover, the diffusion

constant of SA is estimated to be lower than the diffusion constant of the much larger and bulkier

enzyme, which is in fact rather counterintuitive.

Figure 4.12: Normalised experimental concentrations, and numerical simulations for

the determination of diffusion constants (Bodla et al., 2013). The three solutes ex-

hibit a similar breakthrough curve, with the moment of breakthrough at almost the

same time. Yet, for acetophenone (APH), the curve converges to a lower level then

1-phenylethylamine (MBA). Two numerical fits are proposed with two different diffu-

sion constants. The highest diffusion constants predicts the moment of breakthrough

the most accurately, which is the characteristic one should focus on to estimate the

diffusion constant. The lower final concentrations suggests sorption or loss of mass

within the reactor.

The original work of Bodla et al. (2013) is examined. Figure 4.12 is extracted from said article: it

shows the experimental data combined with numerical simulations. All three have the same time of

breakthrough, yet acetophenone (APH) converges to a lower level. From this figure it is deduced that

the experimental setup may be suffering from faulty measurements, loss of mass or sorption within

the reactor, or a combination of these. Next to experimental setup, some questions arise about the

numerical fit. For the determination of the diffusion constant, the data points in the first half of the

data set are more important as they show the moment of breakthrough, which is a measure for the

diffusion constant. In this figure, the moment of breakthrough of all three solutes is very similar, thus

a diffusion constant in the same order of magnitude would be expected instead of a difference of three

orders of magnitude.

In literature, six other estimations of the diffusion constant of SA are found: Milozic et al. (2014), Li

and Carr (1997), GSI International (2014), and New Jersey department of Environmental Protection

(2014). These six lie in the same order of magnitude as the corresponding product PP, as would be

expected for compound with similar molecular structure. These diffusion constants also lie in the order

of magnitude of the fitted diffusion constant in figure 4.12 (black line). For further simulations, the

diffusion constant for solute SA is taken as the average of these six literature values: 8.27 · 10−12 m2

s−1.

A new tracer test is executed for SA with the new diffusion constant, the result is shown in figure 4.13.

The figure shows an almost identical profile for both solutes. For further analysis (TIS) it is assumed


that all solutes can be approximated sufficiently by the tracer test of solute SB. This assumption will

lead to slight loss in accuracy, but it does not outweigh the computational load of the tracer test:

calculation time equal to one week on 15 nodes for one tracer test.

Figure 4.13: RTD tracer test for a residence time of 10.3 minutes, the original

simulation of substrate SB is plotted together with the simulation for substrate SA

with the new diffusion constant: DSA = 8.27 · 10−12 m2 s−1

4.2.5 Enzyme kinetics

The rate equation (3.6) derived by Al-Haque et al. (2012) is implemented in an OpenFOAM solver. As

stated in 3.2.5, the rate equation is decoupled from the velocity solver: a steady state velocity profile is

calculated first, next the enzyme kinetics are calculated. Since the production setup is continuous with

constant inlet conditions, a steady state kinetic solver is made. A transient and a steady state solver are

tested on the a small test case (a case identical to the test case for tracer tests). Both solvers predict

the same output, the difference lies in the computational expense. Calculation time for the steady state

solver is a couple of seconds for a single node calculations, whereas the calculation with the transient

solver takes over 6 hours for a 30 node job. Further calculations are executed with the steady state

kinetic solver (both on the small test case).

In more detail, the steady state solver requires some attention towards stability and diagonal dominance

of the linear system. The temporal derivative has a beneficial influence on the diagonal dominance:

it increases the diagonal dominance. This diagonal dominance has great advantages regarding the

solution procedure of linear systems. For instance, the Jacobi and Gauss-Seidel methods for solving

linear systems converge if the matrix is strictly diagonally dominant. Generally speaking, diagonal

dominance is beneficial for the convergence of the linear system. However in steady state calculations,

this influence does not exist. In order to enhance the diagonal dominance, under relaxation of the

solution is introduced similar to the SIMPLE algorithm mentioned in section 3.2.2. Equation 3.4 is

used to under relax the solution between iterations.

Three scenarios are set up, the boundary conditions for the different scenarios are listed in table 4.8.

The concentrations are obtained from other partners in the Biointense project: these concentrations

are used for determining the kinetic parameters. In the first scenario, the input conditions for both


inlets are the same for all solutes. The second scenario is identical to first, except for the enzyme: it

is fixed on the reactor wall. The third scenario is a variation on the second scenario: the flow is split

into two parts, with each substrate entering the reactor from only one inlet. The scenarios are set up in

a way that the overall mass flow of substrate or enzyme in the reactor is equal for all three scenarios.

The enzyme fixed on the wall is modelled as enzyme present in the cells adjacent to the reactor wall.

The total volume of these cells is 1/33 of the total reactor volume. The concentration of the enzyme is

increased a 33-fold so that the assumption of the same mass of enzyme in the reactor is valid. The inlet

product concentration for all scenarios are equal to zero. These three scenarios are analysed for three

residence times: 10.3, 20.6, and 30.9 minutes.

Table 4.8: Overview of the boundary conditions for the different scenarios set up for

the CFD calculations.

Scenario Solute Inlet north (10−3 molL ) Inlet south (10−3 mol

L )

SA 10 10

1 SB 500 500

E 0.03795 0.03795

SA 10 10

2 SB 500 500

E fixed on wall fixed on wall

SA 20 0

3 SB 0 1000

E fixed on wall fixed on wall

The results for the three scenarios are listed in table 4.9. From this table it is a clear that the CFD

simulations predict conversion percentages that are very close to each other for each residence time.

This means that the reactor setup in terms of splitting the inlet or fixing the enzyme on the reactor wall

is not a dominating factor for the overall process conversion rate. The kinetics of the enzymatic reaction

dominate the process: a longer residence time yields a higher efficiency. From these simulations, fixing

the enzyme on the reactor wall is valid consideration for reactor setup. The fixing of the enzyme does

not affect in a loss of efficiency, assuming that this fixing does not effect the kinetics in a negative way.

Fixing the enzyme on the wall leads to a more cost efficient process, as the enzyme can be reused in the

reactor itself and does not have to be extracted from the product stream.

The results from the simulations above were performed in two dimensions. The three dimensional output

was obtained by assuming a uniform third dimension. Performing these calculations in three dimensions

raises the computational load: if the spatial resolution is kept the same ( 15 cells0.0001 m ), the 3D mesh has

92.9 ·106 cells, which corresponds to a 62-fold increase in number of cells. Due to the high computational

load, the three scenarios were only performed for the lowest residence time. The calculated fluxes, outlet

concentrations and conversion ratios of the 3D and the 2D model were below a relative difference of 1%.

As this deviation is below the previously set error base line of 5%, the simulations in 2D are considered

as a valid simplification of the reactor.

4.3 Simplified Models

4.3.1 Mixed flow

The mixed flow model does not predict the hydraulics of the microreactor accurately, as can be seen

in figure 4.14. Instead of a bell curve, the concentration profile shows the profile of a first order decay


Table 4.9: Results of the 2D CFD simulation for different scenarios and different

residence times.

Res time (min) Scenario PPflux, out(10−17molL s ) PPconc, out (10−6mol

L ) Conversion(10−3%)

1 7.23 1.62 16.22

10.3 2 7.23 1.62 16.22

3 7.20 1.62 16.17

1 7.21 3.24 32.37

20.6 2 7.21 3.24 32.37

3 7.20 3.23 32.32

1 7.19 4.84 48.44

30.9 2 7.19 4.84 48.43

3 7.18 4.84 48.39

process, as could be expected from equation 3.12. In comparison with the CFD model, the conversion

ratios obtained with the mixed flow model are systematically lower (see table 4.10).

Figure 4.14: Result of a tracer test with the Mixed flow model. At time 0, a pulse

is added to the reactor. One can see a sharp increase in solute concentration, which

is afterwards decreasing slowly with first order kinetics.

4.3.2 Plug flow

In practice, the ideal plug flow model can be modelled by executing the rate equation for the time equal

to the residence time of the fluid in the reactor. The profile of the tracer test modelled with plug flow,

retains its form as there is no exchange between the discrete plugs in the model. This plug flow model

is the most ideal approximation of the reactor kinetics. The outlet concentrations and conversion rates

(see table 4.10) are the upper limit of what is theoretically achievable with this type of reaction kinetics

as it neglects all mass transfer limitations. Compared with the CFD model, the ideal plug flow model

predicts slightly higher conversion rates. The model can be used to easily check how long it would take

to come to a steady state. This is visualised in figure 4.15. From this figure its is clear that there are

issues with the kinetic model: after approximately one year, 99% of the steady state concentration is


attained. From experimental work done in the Biointense project, steady state reaction is observed after

merely a couple of days.

Figure 4.15: Steady state reaction of the enzyme kinetics. The dotted lines resemble

the 99% level of the steady state concentration.

4.3.3 Tanks-In-Series

The mathematical representation of the TIS model is a system of ODEs. This system is implemented

in Python using the biointense Python package developed at Biomath. First a RTD tracer test is

performed to determine the number of tanks needed to properly simulate the hydraulic behaviour of the

reactor. For a residence time of 10.3, 20.6, and 30.9 minutes: 1400, 2630, and 3600 tanks are needed to

yield a good prediction of the RTD concentration profile. The results of the tracer tests are visualised in

figures 4.16a, 4.16b, 4.16c, and 4.16d. Figures 4.16a, 4.16c, and 4.16d show the results from the tracer

test from the CFD simulation and the TIS simulation with optimised number of tanks for each residence

time. Figure 4.16b shows a comparison of tracer tests with the TIS model using a different number of

tanks. A higher number of tanks results in a sharper peak, a lower number results in a more smeared

out concentration profile. The number of tanks can be related to the diffusion constant of the solute

used: a slower diffusion solute (lower diffusion constant) will require less tanks than a faster diffusing

solute (larger diffusion constant).

To simulate enzyme kinetics with TIS, equation 3.16 is extended with a rate term (equation 3.6) to

equation 4.8. The system consists of 4 ODEs for each tank multiplied with the number of tanks. Each

tank is modelled by four equations, one for each solute. The calculated time for the simulations is

equal to the mean residence time obtained with the tracer test. The reasoning is that the TIS model

should be able to predict alterations in the reactor inlet conditions and their effect in function of time.

For this prediction, a necessary approximation of the reactor hydraulics is essential. The results of the

simulations for the TIS model with reaction kinetics can be found in table 4.10. The simulated outlet

concentration and conversion rate from the TIS model is lower than that of the ideal plug flow model

and the CFD model. From the same table it can be concluded that nearly the same predictions are

attained for the simulations with different number of tanks.


(a) Results of the tracer test simulation for the CFD

model, and the TIS model with 1400 tanks for a resi-

dence time equal to 10.3 minutes.

(b) Results of the tracer test simulation for the CFD

model, and the TIS model in function of the number of

tanks for a residence time equal to 10.3 minutes.

(c) Results of the tracer test simulation for the CFD



(d) Results of the tracer test simulation for the CFD



Figure 4.16: Comparison of the RTD profile obtained with CFD simulations and the

profile obtained with the TIS model. In figures 4.16a, 4.16c, and 4.16d, the optimised

number of tanks is shown in comparison with the CFD simulation for a residence time

of 10.3, 20.6, and 30.9 minutes. Figure 4.16b shows the outcome of the tracer test

for different number of tanks for a residence time of 10.3 minutes.


dSA1

dt = Q·(SAin−SA1)V − reaction

dSB1

dt = Q·(SBin−SB1)V − reaction

dPP1

dt = Q·(PPin−PP1)V + reaction

dPQ1

dt = Q·(PQin−PQ1)V + reaction

dSA2

dt = Q·(SA1−SA2)V − reaction

dSB2

dt = Q·(SB1−SB2)V − reaction

dPP2

dt = Q·(PP1−PP2)V + reaction

dPQ2

dt = Q·(PQ1−PQ2)V + reaction

...

dSAN

dt = Q·(SAN−1−SAN )V − reaction

dSBN

dt = Q·(SBN−1−SBN )V − reaction

dPPN

dt = Q·(PPN−1−PPN )V + reaction

dPQN

dt = Q·(PQN−1−PQN )V + reaction

(4.8)

4.3.4 Compartmental Model

The Compartmental Model (CM) is set up using the geometry visualised in figure 4.17. The reasoning

behind this configuration is as follows: In the middle of the reactor the fast moving part of the fluid

flow occurs, whereas near the wall, the fluid is nearly stagnant (see figure 4.6). The axial fluid flow is

dominated by this fast moving part (the larger tank), whereas retention is attained by fluid exchange

with the smaller tank (representing the fluid near the reactor wall). In comparison with TIS, CM has

two additional degrees of freedom: the volume ratio of the large versus the small tank, and the exchange

flux between those tanks. In this setup, the large and the small tanks have the same volume throughout

the whole axial direction. The exchange flux is modelled as a certain percentage of the axial flux, the

flow factor. In the standard way, this exchange flux in determined by the turbulence properties in the

reactor (Alvarado et al., 2012). The flow profile in the microreactor is laminar, no turbulence properties

can be described. The flow factor was determined by trail and error.

Figure 4.17: Visual representation of the Compartmental Model

With CM , it is possible to simulate a tailing effect in a tracer test, as can be seen in figure 3.5. Increasing

the volume ratio, i.e. the big tank becomes larger in comparison with the small tank, leads to sharper

peaks (see figure 4.18a). Increasing the exchange flux between the small and the big tanks leads to

the same effect (see figure 4.18b). With CM, the only good fits for the tracer test were obtained for

very high volume ratios (see figures 4.18c and 4.18d). For very high volume ratios, the CM actually


becomes an approximation of a TIS model for this specific configuration. However, the computational

expense is much larger, as the system of ODEs contains double the amount of ODEs compared to the

TIS model. It is concluded that for this reactor setup, CM does not yield advantages over the TIS

model, and further investigation of this model will not be pursued in this thesis. However for other

reactor configurations, CM can yield much better results compared to TIS (Alvarado et al., 2012).

(a) RTD curve for the CM with a fixed flow factor equal

to 0.01, and a variable volume ratio.

(b) RTD curve for the CM with a fixed volume ratio

equal to 0.5, and a variable flow factor.

(c) RTD curve for the CM with a fixed volume ratio

equal to 0.99, and a variable flow factor.

(d) Magnification of figure 4.18c at the top of the RTD

curve.

Figure 4.18: Simulation results for the CM: in figure 4.18a, the effect of the volume

ratio is shown for a constant flow factor equal to 0.01. In figure 4.18b, the effect of

the flow factor on the RTD curve is visualised for a constant volume ratio equal to

0.5. Figure 4.18c visualises the CM simulations for a high volume ratio equal to 0.99

in function of the flow factor. Figure 4.18d is a magnification from figure 4.18c at

the top of the RTD curve. All simulations are perform with the number of tanks in

the largest dimension equal to 1400, i.e. 2800 tanks in total.


Table 4.10: Results for the simplified models (mixed flow, ideal plug flow, and TIS),

and the physical model (CFD) for different residence times.

Res time Model PPflux, out PPconc, out Conversion

(min) (10−17 molL s ) (10−6 mol

L ) (10−3%)

CFD scenario 1 7.23 1.62 16.22

CFD scenario 2 7.23 1.62 16.22

CFD scenario 3 7.20 1.62 16.17

Mixed flow 4.57 1.03 10.26

10.3 Ideal Plug Flow 7.24 1.63 16.25

TIS (100 tanks) 6.68 1.54 15.40

TIS (500 tanks) 6.95 1.56 15.61

TIS (1000 tanks) 6.91 1.55 15.50

TIS (1400 tanks) 6.91 1.55 15.52

CFD scenario 1 7.21 3.24 32.37

CFD scenario 2 7.21 3.24 32.37

CFD scenario 3 7.20 3.23 32.32

Mixed flow 4.57 2.05 20.53

20.6 Ideal Plug Flow 7.24 3.25 32.50

TIS (100 tanks) 6.89 3.09 30.92

TIS (500 tanks) 6.99 3.14 31.40

TIS (1000 tanks) 7.01 3.15 31.47

TIS (2630 tanks) 7.00 3.14 31.41

CFD scenario 1 7.19 4.84 48.44

CFD scenario 2 7.19 4.84 48.43

CFD scenario 3 7.18 4.84 48.39

Mixed flow 4.57 3.08 30.79

30.9 Ideal Plug Flow 7.24 4.87 48.73

TIS (100 tanks) 6.90 4.65 46.49

TIS (500 tanks) 7.01 4.72 47.22

TIS (1000 tanks) 7.04 4.74 47.40

TIS (2000 tanks) 7.04 4.74 47.42

TIS (3600 tanks) 7.03 4.73 47.34

CHAPTER 5Discussion and perspectives

Welcome to the show where everything is made up, and the points don’t matter

— Drew Carey (Whose Line Is It Anyway?)

5.1 Discussion

First, the diffusion constant used for substrate SA in the Biointense project has some issues. The

obtained estimation in the work of Bodla et al. (2013) is remarkably low. As stated in section 4.2.4,

several questions arise regarding the experimental setup, measurements, and the numerical fit. Further

research is needed to address these issues properly.

Second, regarding the kinetics for ω-TA reaction: three main problems arise. The first problem is the

parameter estimation: propagation of the error on the individual constants has revealed that the overall

error on the rate equation is remarkably large. The second problem is related to long term reaction times:

the current model predicts that 99% of the steady state condition will be attained after approximately

one year. In contrast, internal documents in the Biointense project, the steady state condition is observed

after a couple days in experimental work. The third and final issue lies with in fact that Al-Haque et al.

(2012) removed two parameters from the rate equation after an initial parameter estimation yielded that

the effect of these parameters is negligible. This modus operandi raises the question whether the model

structure used in that article is valid for modelling ω-TA reaction. Instead of removing parameters with

a negligible effect, the underlying causes should be addressed in future work.

Further, the models used to simulate enzyme reactions in microreactors can be reviewed. The most

complex model is the physical model (CFD), which has the highest flexibility in reactor configuration.

The most simple model is the ideal plug flow model, for which only one degree of freedom is available:

the reaction time. The mixed flow model is not a valid model for simulating enzyme reactions in

microreactors, as it cannot accurately predict fluid behaviour nor reaction kinetics. The TIS model

can accurately predict the hydraulic behaviour of the reactor, yet for enzyme kinetics, it consequently

predicts a conversion rate lower than the CFD or ideal plug flow for a broad range in number of tanks.

The proposed CM cannot accurately predict the hydraulic behaviour of the reactor: the model setup

which yields an acceptable fit is actually an approximation of the TIS model (high volume ratios). As

the fluid behaviour could not be described accurately, no kinetic simulations were performed. These

models can also be compared based on the computational expense: a summary of required CPU time

is given in table 5.1. As a quick note: these CPU times should not be interpreted as exact numbers:

depending on the load of the server the simulation time can be lower or higher. The CPU times should

be seen as an estimation of the order of magnitude for the duration of the calculation. From table

5.1 a first distinction can be made between steady state and transient calculations (tracer tests): these

tracer tests require remarkably more computational expense. The use of a 2D CFD model reduces

the computational expense remarkably, and retains the same level of accuracy as stated previously in

60 5.2 PERSPECTIVES

section 4.2.5. As a conclusion for the used models, the following methodology can be suggested: first,

the reactor hydraulics and mean residence time are calculated using the CFD model. Second, the ideal

plug flow model is used to determine the upper limit of the conversion in the reactor. Finally the

physical model (CFD) is used to explore different scenarios and their impact of the conversion rate. The

computational expense for the ideal plug flow model and the 2D CFD model is within an acceptable

range. It is concluded that the physical model gives the best results. With the combination of the tools

developed in this thesis (the flexible meshing environment and scenario analysis tool), the CFD model is

the ideal approach for simulating enzyme reactions in microreactors at a excellent accuracy and modest

computational expense. However, until these models are validated with experimental data, no certain

claims can be made for which model performs the best.

The results of the CFD simulation predict a nearly equal conversion rate for the three scenarios. The

scenario where the enzyme is fixed on the reactor wall does not perform significantly worse than the

scenario where the enzyme is present in the bulk. Fixing the enzyme on the wall eases post-processing

of the product stream, and retains the costly enzyme within the reactor. Fixing the enzyme on the wall

leads to less waste and a more efficient use of resources. However, it should be investigated whether

the fixing of the enzyme results in loss of activity, and how much enzyme can be fixed on the reactor

wall. Next, it is possible that the reactor needs to be replaced when the enzymatic activity becomes

too low. The downtime and replacement of the reactor should be included in the determination of the

cost effectiveness of this reactor setup compared to the enzyme in the bulk. Due to a lack of knowledge

on these issues, in this thesis it was assumed that the enzyme did not loose enzymatic activity, and the

amount of enzyme that can be fixed on the reactor surface is not limited.

Table 5.1: Comparison of the CPU time for the different models: CFD, mixed flow,

ideal plug flow, and TIS.

Model Calculation environment CPU time

CFD 2D velocity 30 nodes 0.5 hours

CFD 2D kinetics 30 nodes 0.5 hours

CFD 2D tracer test 20 nodes 5 days

CFD 3D velocity 40 nodes 20 hours

CFD 3D kinetics 40 nodes 20 hours

Mixed Flow 1 node seconds

Ideal Plug Flow 1 node seconds

TIS tracer test 1 node 1 hour

TIS kinetics 1 node 12 hours

CM tracer test 1 node 3 hours

5.2 Perspectives

Future plans for using the scenario analysis tool as an optimisation algorithm: scenarios are run in order

to optimise the output of a certain objective function defined by the user. This optimisation algorithm

can be focused solely on parameter optimisation, but flexible meshing by means of a Python scripts also

opens the possibility to evaluate a variety of microreactor configurations. The scenario analysis tool

needs to be made compatible with the latest version of OpenFOAM.

Future work can focus more in dept on the simplified models. For instance, the dispersion model, which

was not set up in this thesis, can be analysed for its performance, and compared to the ideal plug flow

model. Next, the TIS model can be improved by setting up the model with a dynamic number of tanks

CHAPTER 5 DISCUSSION AND PERSPECTIVES 61

to cope for multiple inlet velocities making it very flexible. To implement this practically, backfluxes

between tanks can be introduced. The magnitude of these backfluxes are a function of the inlet velocity.

It is worth noting that the CFD model performs very well, and new simplified models might not be

needed as everything can be simulated in a flexible way with the CFD model.

The reactor set up can be investigated: in this thesis, only a simple microreactor with a single fluid phase

was considered. Possibilities for future work include multiphase, membrane, or packed bed reactors. The

fluid flow in these reactors will be substantially different from the basic single phase microreactor. A

thorough analysis has to be made to find the model which can accurately describe the flow behaviour in

these reactor types. Problems can arise for the modelling of interfacial, recirculation or dead zones. It is

possible that simplified models which cannot describe the fluid behaviour for the basic reactor properly,

such as the CM, can be of great value for other reactor setups, certainly regarding computational expense

as for instance the mesh generation for a packed bed reactor will be a tedious task. It is a possibility

that the simplified models outperform the CFD model for complex reactor configurations.

All the developed models should be applied carefully until they are thoroughly validated with exper-

imental data. Future works must supply experimental data of high quality to compare the modelled

conversion rates to the experimental ones.

As adduced in the discussion above, the kinetics of ω-TA reactions are not yet fully understood. Future

work must perform an extra parameter estimation, or investigate the model structure as a whole to

improve the rate equation.

Future work is needed to determine the loss in activity and maximum amount of enzyme that can be

fixed on the reactor wall. If this data is available, a true comparison can be made between the scenarios,

and it can be determined whether microreactor technology is a valid option for running continuous

production lines.

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66

APPENDIX AError propagation

Full rate equation, equal to equation 3.6 without the non-relevant parameters KSBi and KPQ

i as stated

in section 4.1:

r[PP ] = −r[SA] =[E0]Kf

catKrcat

([SB] [SA]− [PQ][PP ]

KEQ

)(KrcatK

SAM [SB]

(1 + [PP ]

KPPSi

)+Kr

catKSBM [SA]

(1 + [SA]

KSASi

) (A.1)

+Kfcat

KPPM [PQ]

KEQ

(1 +

[SA]

KSASi

)+Kf

cat

KPQM [PP ]

KEQ

(1 +

[PP ]

KPPSi

)+Kr

cat [SB] [SA] +Kfcat

KPPM [SB] [PQ]

KEQKSBi

+Kfcat

[PQ] [PP ]

KEQ+Kr

cat

KSBM [SA] [PP ]

KPPi

)Splitting the full rate equation in a numerator and denominator part:(

σ(

NumeratorDenominator

)Numerator

Denominator

)2

=

(σ (Numerator)

Numerator

)2

+

(σ (Denominator)

Denominator

)2

(A.2)

Numerator:

Numerator = [E0]KfcatK

rcat

([SB] [SA]− [PQ] [PP ]

KEQ

)(A.3)

Calculation of the error on the numerator:

(σ (Numerator)

Numerator

)2

=

σ(E0K

fcatK

rcat

)E0K

fcatK

rcat

2

+

σ(

[SB] [SA]− [PQ][PP ]KEQ

)[SB] [SA]− [PQ][PP ]

KEQ

2

(A.4)

σ(E0K

fcatK

rcat

)E0K

fcatK

rcat

2

= [E0]2 ·

σ

(Kfcat

)Kfcat

2

+

(σ (Kr

cat)

Krcat

)2

(A.5)

σ(

[SB] [SA]− [PQ][PP ]KEQ

)[SB] [SA]− [PQ][PP ]

KEQ

2

= [PP ]2 · [PQ]

2 ·(σ (Keq)

Keq

)2

(A.6)

Final form of the error on the numerator:

(σ (Numerator)

Numerator

)2

= [E0]2 ·

σ

(Kfcat

)Kfcat

2

+

(σ (Kr

cat)

Krcat

)2

+ [PP ]2 · [PQ]

2 ·(σ (Keq)

Keq

)2

(A.7)

67

Splitting the denominator into sums:

Denominator =

8∑i=1

Ni (A.8)

σ2 (Denominator) =

8∑i=1

σ2 (Ni) (A.9)

Solving each term of the denominator:

σ2 (N1) = (N1)2 · [SB]

2 ·

(σ (Krcat)

Krcat

)2

+

(σ(KSAM

)KSAM

)2

+

[PP ] · σ(KPPSi )

KPPSi

1 + [PP ]

KPPSi

2 (A.10)

σ2 (N2) = (N2)2 · [SA]

2 ·

(σ (Krcat)

Krcat

)2

+

(σ(KSBM

)KSBM

)2

+

[SA] · σ(KSAM )

KSAM

1 + [SA]

KSAM

2 (A.11)

σ2 (N3) = (N3)2 · [PQ]

2 ·

σ

(Kfcat

)Kfcat

2

+

(σ(KPPM

)KPPM

)2

+

(σ (Keq)

Keq

)2

+

[SA] · σ(KSASi )

KSASi

1 + [SA]

KSASi

2

(A.12)

σ2 (N4) = (N4)2 · [PP ]

2 ·

σ

(Kfcat

)Kfcat

2

+

σ(KPQM

)KPQM

2

+

(σ (Keq)

Keq

)2

+

[PP ] · σ(KPPSi )

KPPSi

1 + [PP ]

KPPSi

2

(A.13)

σ2 (N5) = (N5)2 · [SA]

2 · [SB]2 ·(σ (Kr

cat)

Krcat

)2

(A.14)

σ2 (N6) = (N6)2 · [SB]

2 · [PQ]2 ·

σ

(Kfcat

)Kfcat

2

+

(σ(KPPM

)KPPM

)2

+

(σ (Keq)

Keq

)2

+

(σ(KSBi

)KSBi

)2

(A.15)

σ2 (N7) = (N7)2 · [PP ]

2 · [PQ]2 ·

σ

(Kfcat

)Kfcat

2

+

(σ (Keq)

Keq

)2

(A.16)

σ2 (N8) = (N8)2 · [SA]

2 · [PP ]2 ·

(σ (Krcat)

Krcat

)2

+

(σ(KSBM

)KSBM

)2

+

(σ(KPPi

)KPPi

)2 (A.17)

68

APPENDIX BTheoretical velocity profile

Koo and Kleinstreuer (2003) stated that the flow in microreactor is laminar due to the combined effect of

small scale pipes (low hydraulic diameter) and low fluid velocity. This fluid regime has a low Reynolds

(equation 2.11) and Mach number (equation 2.12). The mathematical representation of the velocity

profile in 2 dimensions is a parabolic function. To obtain a symmetric function, the origin is taken at

the x-value for the top of the parabola. Using this knowledge, the general form of the velocity profile

can be written in equation B.1, with U the velocity, y the coordinates along the width of the reactor, a

a negative real number, and c a real number.

U = a · y2 + c (B.1)

The flow is calculated by integrating the velocity along the width of the reactor. Using the fact that

the width of the reactor equals 2 · 10−4 m, and using y = 0 as a symmetry axis to ease the calculation,

the equation for the flow rate is derived in equation B.2 trough B.6.

Q =

∫ 10−4

−10−4

U · dy (B.2)

Q = 2 ·∫ 10−4

0

U · dy (B.3)

Q = 2 ·∫ 10−4

0

(a · y2 + c

)· dy (B.4)

Q = 2 ·[a

3· y3 + c · y

]10−4

0(B.5)

Q =2 · a

3· 10−12 + 2 · c · 10−4 (B.6)

The flow calculated in equation B.6 must be equal to the inflow in the reactor for ensuring a sound mass

balance. The inflow velocity is uniform along the width of the reactor. Using this information, equation

B.7 can be derived from equation B.6.

2 · 10−4 · vin =2 · a

3· 10−12 + 2 · c · 10−4 (B.7)

Due to the no-slip boundary condition, the flow at the reactor wall is equal to zero. Mathematically,

this fact means that(10−4; 0

)and

(−10−4; 0

)are points on the velocity curve. This leads to a system

of 2 equations with two unknown constants (a and c), written down in equation B.8. This system can

be solved for the unknown parameters. The solution is written down in equation B.9.vin = a3 · 108 + c

0 = a · 10−8 + c(B.8)

69

a = −32 · vin · 108

c = 32 · vin

(B.9)

The final equation for the theoretical velocity profile is written down in equation B.10.

U =−3

2· vin · 108 · y2 +

3

2· vin (B.10)

70

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