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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
iv
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
vi
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
xii
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
xviii
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
xx
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
6 2.1 ENZYME KINETICS: ω-TRANSAMINASES
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
CHAPTER 2 LITERATURE STUDY 7
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)
8 2.1 ENZYME KINETICS: ω-TRANSAMINASES
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)
CHAPTER 2 LITERATURE STUDY 9
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).
10 2.1 ENZYME KINETICS: ω-TRANSAMINASES
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:
CHAPTER 2 LITERATURE STUDY 11
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
12 2.1 ENZYME KINETICS: ω-TRANSAMINASES
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.
CHAPTER 2 LITERATURE STUDY 13
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).
CHAPTER 2 LITERATURE STUDY 15
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
CHAPTER 2 LITERATURE STUDY 17
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
CHAPTER 2 LITERATURE STUDY 19
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)
CHAPTER 2 LITERATURE STUDY 21
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).
26 3.2 COMPUTATIONAL FLUID DYNAMICS
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).
CHAPTER 3 MATERIALS AND METHODS 27
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)
28 3.2 COMPUTATIONAL FLUID DYNAMICS
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.
CHAPTER 3 MATERIALS AND METHODS 29
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.
accumulation︷︸︸︷dC
dt=
reaction︷︸︸︷r (3.13)
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.
accumulation︷︸︸︷dC
dt=
axial dispersion︷ ︸︸ ︷D · d
2C
dx2+
reaction︷︸︸︷r (3.14)
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
CHAPTER 3 MATERIALS AND METHODS 31
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).
CHAPTER 3 MATERIALS AND METHODS 33
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
38 4.2 COMPUTATIONAL FLUID DYNAMICS
(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
CHAPTER 4 RESULTS 39
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
40 4.2 COMPUTATIONAL FLUID DYNAMICS
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)
CHAPTER 4 RESULTS 41
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)
42 4.2 COMPUTATIONAL FLUID DYNAMICS
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
CHAPTER 4 RESULTS 43
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
44 4.2 COMPUTATIONAL FLUID DYNAMICS
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
CHAPTER 4 RESULTS 45
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
46 4.2 COMPUTATIONAL FLUID DYNAMICS
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
CHAPTER 4 RESULTS 47
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.
48 4.2 COMPUTATIONAL FLUID DYNAMICS
(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
CHAPTER 4 RESULTS 49
(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,
for substrates SA and SB.
(c) RTD test for a residence time equal to 30.9 minutes,
for substrates SA and SB.
(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
50 4.2 COMPUTATIONAL FLUID DYNAMICS
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
CHAPTER 4 RESULTS 51
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
52 4.3 SIMPLIFIED MODELS
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
CHAPTER 4 RESULTS 53
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
54 4.3 SIMPLIFIED MODELS
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.
CHAPTER 4 RESULTS 55
(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
model, and the TIS model with 2630 tanks for a resi-
dence time equal to 20.6 minutes.
(d) Results of the tracer test simulation for the CFD
model, and the TIS model with 3600 tanks for a resi-
dence time equal to 30.9 minutes.
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.
56 4.3 SIMPLIFIED MODELS
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
CHAPTER 4 RESULTS 57
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.
58 4.3 SIMPLIFIED MODELS
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