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Eindhoven University of Technology

MASTER

Micro- and mesomixing in a spinning disc reactor : an investigation on themechanisms influencing the micromixing times

Manzano Martínez, A.N.

Award date:2016

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Micro- and Mesomixing in

a Spinning Disc Reactor

An investigation on the mechanisms

influencing the micromixing times

A.N. (Arturo) Manzano Martınez

Department of Chemical Engineering and ChemistryChemical Reactor Engineering Group

GRADUATION COMMITTEE:Graduation Supervisor: Dr.ir. J. van der Schaaf

Internal committee member: PDEng. V. KrzeljInternal committee member: Dr. M.F. Neira d’AngeloExternal committee member: Dr.ir. J.T. Padding

Master Thesis

Eindhoven, May 2016

“I am an old man now,

and when I die and go to heaven

there are two matters

on which I hope for enlightenment.

One is quantum electrodynamics,

and the other is the turbulent motion of fluids.

And about the former I am rather optimistic.”

- Horace Lamb -

Micro- and Mesomixing in a Spinning Disc Reactor iii

Abstract

The two known mesomixing mechanisms influencing the micromixing times were investigated inorder to explain the deviation between the theoretical times for micromixing and the experiment-ally obtained in a rotor-stator Spinning Disc Reactor. By using the Villermaux-Dushman method,consisting on two competitive parallel reactions (the iodide/iodate reaction system with neutral-ization), a segregation index XS is obtained. This value depends on the set of concentration ofreagents, but the micromixing time does not, meaning that regardless of the concentration of theacid the micromixing times must be the same in perfect mixing conditions for specific operationalconditions. The equivalent micromixing times are obtained from the segregation indexes by theEngulfment model. This model was adapted to include the effects of the mesomixing mechanism ofinertial-convective disintegration of eddies by varying the ratio of characteristic times M = τS/τE .A Turbulent Jet model was also developed to evaluate a theoretical segregation index as a functionof the turbulent diffusivity coefficient DT .

The results show influence of the mechanism of Turbulent Dispersion for low energy dissipationrates, and strong effects of the inertial-convective disintegration of eddies as the energy dissipationrates increased.

Micro- and Mesomixing in a Spinning Disc Reactor v

Preface

This Thesis is a summary of the nine-month-trip of the author, A. Manzano, to the land of theTheory of Turbulence. Although the route has had many complications, in the end the destinationwas reached. Unfortunately, this report will not be able to show the feelings of frustration or joyas a function of time (or any other variable), but will rather stick to the gotten achievements andfindings.

Thanks to Professor John van der Schaaf for giving me such an interesting, yet abstract anddifficult topic. I really enjoyed the challenge.

Micro- and Mesomixing in a Spinning Disc Reactor vii

Contents

Contents ix

List of Figures xi

List of Tables xiii

1 Introduction 1

2 Literature Review 3

2.1 Rotor-stator Spinning Disc Reactor . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Importance of Mixing in Fast Chemical Reactions . . . . . . . . . . . . . . . . . . . 4

2.2.1 Competitive-consecutive reactions . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.2 Competitive parallel reactions . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.2 Models of Turbulent Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3.3 Important concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.4 Mixing Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.1 Macromixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4.2 Mesomixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.3 Micromixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 The Engulfment Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5.1 Self-Engulfment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.6 Mesomixing Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6.1 Turbulent Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6.2 Inertial-Convective Mesomixing . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.7 Characterization of Micromixing: The Villermaux-Dushman method . . . . . . . . 18

2.7.1 Segregation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Experimental Design 21

3.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Reactants and Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Calibration Curve - Determination of the Extinction Coefficient . . . . . . . . . . . 23

3.4 Data Acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Mixing Models 25

4.1 E-Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Self-Engulfment Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3 Turbulent Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4 Inertial-Convective Mesomixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Micro- and Mesomixing in a Spinning Disc Reactor ix

CONTENTS

5 Results and Discussions 31

5.1 Energy dissipation rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5.2.1 Comparison between Micromixing times - Engulfment Model . . . . . . . . 325.2.2 Comparison between Micromixing times - Self-engulfment . . . . . . . . . . 335.2.3 Comparison between Segregation indexes - Turbulent Dispersion . . . . . . 345.2.4 Comparison between Micromixing times - Inertial-Convective

mesomixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 Characteristic times comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Conclusions and Recommendations 43

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Bibliography 49

Acknowledgments 53

Appendix 55

A Calibration Curves 55

B Curve Fitting 57

x Micro- and Mesomixing in a Spinning Disc Reactor

List of Figures

2.1 Schematic representation of the rotor-stator Spinning Disc Reactor . . . . . . . . . 42.2 Turbulence energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3 Normalized spectrum of the kinetic energy of turbulence . . . . . . . . . . . . . . . 102.4 Schematic representation of the energy cascade . . . . . . . . . . . . . . . . . . . . 112.5 Schematic representation of the EDD model . . . . . . . . . . . . . . . . . . . . . . 142.6 Mesomixing - Radial distribution of fresh feed forming a plume . . . . . . . . . . . 152.7 Inertial-convective mesomixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.8 Structure of the plume for the two mechanisms of mesomixing . . . . . . . . . . . . 172.9 Structure of partially segregated islands - Inertial-convective mesomixing . . . . . . 18

3.1 Schematic representation of experimental setup . . . . . . . . . . . . . . . . . . . . 223.2 Segregation Index with varying rotational speeds, for different acid concentration . 24

4.1 Variation of Segregation Index versus mixing times - Engulfment Model . . . . . . 264.2 Variation of Segregation Index versus mixing times - Engulfment Model considering

self-engulfment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.3 Comparison of Segregation Index versus mixing times for the E-model with and

without self-engulfment probability . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.4 Examples of the Turbulent Dispersion model . . . . . . . . . . . . . . . . . . . . . 284.5 Variation of Segregation Index versus mixing times - Engulfment Model considering

inertial-convective mesomixing effects . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.1 Micromixing time as a function of energy dissipation rate, using E-model . . . . . 335.2 Micromixing time as a function of energy dissipation rate, using E-model and con-

sidering self-engulfment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.3 Segregation index as a function of turbulent diffusivity coefficient . . . . . . . . . . 365.4 Micromixing time as a function of energy dissipation rate, using E-model and con-

sidering inertial-convective mesomixing with M = 1 . . . . . . . . . . . . . . . . . . 375.5 Micromixing time as a function of energy dissipation rate, using E-model and con-



sidering inertial-convective mesomixing with M = 10 . . . . . . . . . . . . . . . . . 395.8 Comparison of the characteristic times for the two mechanisms of mesomixing . . . 405.9 Comparison of the characteristic times for the two mechanisms of mesomixing and

the experimentally obtained mixing times . . . . . . . . . . . . . . . . . . . . . . . 41

A.1 Calibration curve for the inlet flowcell . . . . . . . . . . . . . . . . . . . . . . . . . 55A.2 Calibration curve for the outlet flowcell . . . . . . . . . . . . . . . . . . . . . . . . 56

B.1 Fitting curves for the Engulfment model . . . . . . . . . . . . . . . . . . . . . . . . 58B.2 Fitting curves for the Engulfment model, considering self-engulfment . . . . . . . . 59

Micro- and Mesomixing in a Spinning Disc Reactor xi

LIST OF FIGURES

B.3 Fitting curves for the Engulfment model, considering inertial-convective mesomix-ing, for Acid concentration = 0.02 . . . . . . . . . . . . . . . . . . . . . . . . . . . 60



xii Micro- and Mesomixing in a Spinning Disc Reactor

List of Tables

3.1 Set of concentrations used for the experiments . . . . . . . . . . . . . . . . . . . . . 22

5.1 Measured energy dissipation rates for the operational rotation speeds . . . . . . . . 32

Micro- and Mesomixing in a Spinning Disc Reactor xiii

Chapter 1

Introduction

In general terms of industrial process engineering, the treatment of an heterogeneous systemto make it homogeneous can be taken as a definition of the process of mixing. It is used in

almost all processes in the chemical industry, in order to reduce the gradient of temperature orconcentration along the system, or to disperse immiscible materials. However, these are basicallyphysical examples of the applications of mixing, and it might be more difficult to picture thechemical part of mixing and its impact. Since chemical reactions occur in a molecular scale, theydepend directly on the quality of the “micromixing”, which is the term given for the molecular-scalemixing of the reagents.

The importance of mixing when complex chemical reactions take place is a topic of extremeconcern, for example, in the fine chemicals or pharmaceutical industry, where high purity productsare required, and side products should be avoided. In order to avoid that situation, the mixingof the components must be as fast as the reaction time, which is determined by the kinetics. Insimple words, for fast reactions when the reagents cannot “find” each other to react and form adesired product, then they will react with other available materials and form undesired ones.

At Eindhoven University of Technology, investigation of a novel type of multiphase reactorhas been developed for the last years in the group of Chemical Reactor Engineering. This newreactor is the rotor-stator Spinning Disc Reactor, and previous works have shown high rates ofmass transfer and heat transfer[26]. It also exhibits a high degree of turbulence, and since thenature of turbulence itself promotes mixing in flowing fluids, this reactor could be expected toenhance the performance of fast reactions.

Research has not only been focused on mass and heat transfer but also on the process of mixingconsidering times and efficiency, the results lead towards low mixing times, in the range of fewmilliseconds. However, these times are higher compared to the theoretical values correspondingto pure micromixing models, meaning that mixing in a higher scale influences the system. Still,this scale is smaller than the “macromixing” time, which is the timescale for fresh material to mixwith the whole contents of the reactor. Hence, the term “mesomixing” refers to this timescalebetween the micromixing and the macromixing. In the literature, two mechanisms for mesomixinghave been identified: the turbulent dispersion, and the inertial-convective disintegration of eddies.Both mechanisms will be described in the following Chapter.

Based on the principle that the mixing time should not be affected by the concentration ofthe limiting reagent in perfect mixing conditions, this study has the purpose of determine howmesomixing affects the mixing efficiency in a rotor-stator Spinning Disc Reactor, identify themechanism responsible of the phenomenon, and attempt to include it in the current model formicromixing to predict more accurate mixing times.

Micro- and Mesomixing in a Spinning Disc Reactor 1

Chapter 2

Literature Review

The theory behind this project is relatively extensive. An attempt to introduce the conceptsthat led to the present research takes place in this chapter. It is not the intention to make

an exhaustive summary or compilation of the available literature, but rather set a background tosustain the base of the mixing models.

2.1 Rotor-stator Spinning Disc Reactor

Since the 1960’s, there has been research works on Spinning Disk Reactors (SDR), consistingon a rotating disc within a vessel, inducing a flow towards the disc. Due to the high rates ofmass transfer and heat transfer associated to the liquid film in the spinning disc, this type ofreactor is also seen as one of the foundations of the trend in Chemical Engineering called ProcessIntensification (PI), which aims to lower production scales in order to reduce time and costs offabrication, stocks, waste, energy costs, investment in equipment, and safety risks. Hence, itsunderstanding for applications in the chemical industry is relevant.

The rotor-stator Spinning Disc Reactor has a different configuration. It can be described as arotating disc (the rotor) within a fixed cylindrical housing (the stator), and the distance betweenthe rotor and the stator being only a few millimeters, in combination with the high rotationalspeeds of up to 2000 RPM, generates a large velocity gradient in that gap. This signifies highshear forces being applied into the liquid, resulting in a high degree of turbulence in the reactorfor most of the operational conditions [26]. Because it is a multiphase reactor, depending on thedesired application different configurations can be selected. For instance, the liquid is fed to thereactor through the top of the stator, close to the rotating axis, but a second liquid or a gas canbe injected either from below of the stator, or also from the top of the stator as a co-feeding.

Figure 2.1 shows a schematic representation of the concept of the rotor-stator Spinning DiscReactor. For the research performed by Meeuwse, it was of great interest to investigate themultiphase possibilities of the reactor, thus the scheme shows injection of gas and the generationof bubbles.


CHAPTER 2. LITERATURE REVIEW

Figure 2.1: Schematic representation of the rotor-stator Spinning Disc Reactor, reproduced from[26].

2.2 Importance of Mixing in Fast Chemical Reactions

All chemical reactions occur at a characteristic reaction rate, at a given temperature and a con-centration of the reactants. If the mixing time of the reagents fails to achieve a faster rate thanthe one for reaction, a significant delay may take place. In industrial applications this leads tolarger equipment compared to well mixed systems, in order to obtain the desired conversion ofreagents.

For some single reactions, e.g. an acid-base neutralization which is irreversible and with a rateconstant easily around 108 s−1, the reaction time is very short compared to the time needed formixing the reagents, allowing them to be described as instantaneous (relative to the mixing).

The mixing process will also affect multiple reactions systems, retarding those reactions thatare very fast compared to the mixing rates. But more important is to be aware that the yield ofthe products very often depend on the mixing efficiency. This leads to two approaches in whichthe understanding of mixing are important:

Industrial The objective is to improve the yield of a product by both chemical and physicalmeans. This implies taking into account temperature, stoichiometry, concentrations, catalyst, aswell as the reactor type, physical properties, how to supply reagents, agitation intensity, etc. Thereason is that it affects directly the raw material consumption, the purity of the product, andreduces the refining costs and waste.

Research Here the intention is to be able to describe how a product distribution can berelated to the mixing process. Definition of concepts, development of principles, experimentation,modeling, verification, all are tasks aiming to acquire sufficient knowledge to predict product yieldsand improve them.

The retardation of fast single chemical reactions due to mixing can be handled using basicconcepts of Chemical Reactor Engineering, and it can be said that it mainly affects the efficiencyof the equipment. On the other hand for complex reactions, experimental evidence has historicallyshown that measured product distribution and yields can be contradictory to the predictions based

4 Micro- and Mesomixing in a Spinning Disc Reactor


on the reaction kinetics, remarking the relevance of mixing processes to reduce the economical,environmental and safety issues regarding the formation of undesired products.

When speaking of complex or multiple reactions whose yield depend on mixing, they can beclassified into two main stoichiometric types, but what they have in common is that one or moreof the reagents is added separately from the rest, so they “compete” to produce different products,desired and undesired.

2.2.1 Competitive-consecutive reactions

These reactions commonly follow the scheme:

A + Bk1

R (2.1)

R + Bk2

S (2.2)

For this scheme, reagents A and B react to produce R, which also reacts further with B to formS. Typically this reactions have k1 ≫ k2 which means 2.1 is faster than 2.2. When S is theintended product, these two equations may be summed and with excess of B the reaction willproceed without problems. But most of the time the objective is to produce the intermediate R. Itis possible to predict the theoretical yield if the kinetics are known. However, in reality it is verycommon to see much smaller yields than the predicted by the kinetics, and this can be attributedto the mixing effect already described in this section.

Some examples of reactions where the yield of S was expected to be small due to k1 ≫ k2, butthe experimentally obtained yields were considerable and not according to the kinetics, are listed:

❼ Diamines and various isocyanates

❼ Diamines and various acylating agents

❼ Dibenzyl and nitronium ion

❼ Durene and nitronium ion

❼ Tyrosine and iodine

❼ Alkyl benzenes and nitronium ion

❼ Various diazo couplings

❼ Bromine and 1,3,5-trimethoxybenzene

❼ EDTA complex with precipitation and neutralization

2.2.2 Competitive parallel reactions

In this scheme, it is considered that an added reagent can react with various species present inthe system. If the reaction order of this component varies between the different reactions, one ofthe products will be favored due to selectivity advantage. This can be represented by:

A + Bk1

R (2.3)

C + Bk2

S (2.4)



In an ideal scenario, the concentrations of R and S would depend only in the ratio of k1 and k2

when they exhibit second-order kinetics,k1k2

≈

[R]

[S]. That situation would involve conditions free

of mixing limitation. But then again, many systems present mixing-dependency in their productdistribution. Some examples are:

❼ Benzene and toluene with nitronium ion

❼ Cobalt (III) complexes with chromium (II) ion

❼ Alkaline ester hydrolysis and neutralization

❼ Alkaline ester hydrolysis and precipitation

❼ Diazo coupling with decomposition of reagent

❼ Iodate / iodine reaction with neutralization

❼ Acetal hydrolysis with neutralization

2.3 Turbulence

Turbulence is present on all macroscopic scales, from the interior of the cells, to astrophysicalphenomena. Physically, it consists of eddies of different sizes. Although its occurrence in flowingfluids has always been there, it is an unsolved phenomenon in terms of mathematical physics.Since the 19th and 20th centuries, many physicists and engineers have studied the problem ofturbulence, but it is still not possible to understand completely in detail how it actually occurs,and even less being able to predict turbulent behavior.

2.3.1 Features

Despite the attempts of many scientists to find a proper, universally accepted definition for Tur-bulence, this has not been agreed. However, many features of turbulence have been identifiedthroughout history. Some of them, perhaps the most important ones to describe the features ofturbulence, were listed by Tennekes and Lumley[32] and discussed by Ba ldyga and Bourne[4]:

❼ Irregularity. Is what makes it impossible to find a deterministic approach to turbulence.Turbulence is also unpredictable, probably the most notorious difference from the laminarflow.

❼ Diffusivity. It is a result in its increased mixing property. The irregular motions presentin turbulence enhance mixing and transport of species, momentum and energy, resulting inmuch faster rates than for molecular diffusion.

❼ Continuum. Turbulence is treated as a continuum phenomenon, and it is governed by thecontinuity equations for momentum, mass and species balances.

❼ Large Reynolds numbers. Experiments by Osborne Reynolds in 1883 demonstrated thedifference between two patterns of flow, and in 1885 he introduced the idea of the Reynoldsnumber as a function of velocity, pipe diameter, and viscosity. He determined that for highReynolds number, the flow cannot be laminar, it is unstable and chaotic.

❼ Dissipation of kinetic energy. Due to the work of viscous deformation, the kinetic energy ofthe turbulence is transformed into internal energy of the fluid.

❼ Three-dimensional. Velocity fluctuations occur in all directions. There is only one knownmechanism for the above mentioned dissipation of energy, called vortex stretching. Withoutgoing into detail, this feature makes the chaotic and turbulent flow to be locally isotropic ina statistical sense.



2.3.2 Models of Turbulent Flow

In general, the theory of turbulence is equivalent to the theory of Navier-Stokes equations forhigh Reynolds numbers. But this is a matter of perspective. The typical approach to problemsof turbulence for chemical engineers, involves solving the averaged Navier-Stokes equations usingsemi-empirical or phenomenological models. On the other hand, for scientists the main problemis to obtain statistical solutions of the Navier-Stokes equations in order to predict the motions inturbulent flow. Hence, depending on the problem application and the desired results, differentmethods have been developed.

The description and comparison of the methods to approach problems of turbulence can befound in many publications [16, 21, 4], thus there is no need to elaborate in detail. A classification,only for reference, is:

1. Semi-empirical turbulence models

(a) Boundary layer flows

(b) Boundary-free shear flows

2. Classical statistical methods

(a) Models based on the eddy viscosity concept

(b) Models based on the stress transport equations

(c) Spectral dynamics of isotropic turbulence

3. Modern statistical methods

(a) Navier-Stokes equation in Fourier space

(b) The problem of closure revisited

(c) The quasi-normality hypothesis

(d) Direct interaction approximation

(e) Renormalization methods

(f) The Edwards-Fokker-Plank (EFP) theory

4. Deterministic methods

(a) Direct numerical simulations (DNS)

(b) Large eddy simulations (LES)

5. Chaos and turbulence

(a) Dissipative dynamical systems and strange attractors

(b) Strange attractors and the origin of turbulence

6. Multifractional approach

The above mentioned methods were developed to provide sufficient tools, namely empirical, math-ematical, statistical or deterministic, to find (or approximate) solutions to the problems of turbu-lence. However, they have naturally different limitations. The deterministic methods aiming tocompute numerical solutions to the Navier-Stokes equations require extremely high computationalcapacities, that not even the available supercomputers can supply. Statistical methods can onlyprovide approximate solutions to the Navier-Stokes equations, derived from different assumptionsand conditions. But such discussions and analysis are beyond this study.



2.3.3 Important concepts

Complete description of the models for turbulent flows will not be presented in this study, whichis not trying to find a solution to turbulence, but rather apply the concepts behind the theoryof turbulence to understand the behavior in a rotor-stator Spinning Disc Reactor. However, theyprovide important considerations that should be taken into account to develop a mixing model.Some of them are summarized.

Scales of Turbulence

Probably the biggest problem for the deterministic methods is that turbulent flows have a widerange of length and time scales. In general, for the continuum approach, there are four:

❼ The large scale. It has a characteristic length L, which is basically the size of the geometryof the problem. At this scale, the term U is typically used for the characteristic velocity.Considering the kinematic viscosity of the fluid, ν, the Reynolds number is given by:

Re =UL

ν(2.5)

❼ The integral scale, l, which is in the same order of magnitude than L. This scale containsthe largest size range of eddies. They have a characteristic velocity, that is in the sameorder of the turbulence intensity u, which is nothing more than the root mean square of thefluctuating contribution for instantaneous velocity:

u = u + u′ (2.6)

u =√

u′2 (2.7)

For homogeneous turbulence, where the fluctuations in the three dimensions are equal, theturbulence intensity can also be defined as a function of the turbulent kinetic energy:

u =

(

2k

3

)1/2

(2.8)

Considering that the energy of the eddies is dissipated at a certain rate, ε, this length scaleis derived:

l ∼k3/2

ε(2.9)

Analogously to 2.5, the Reynolds number associated with these large eddies is called theTurbulence Reynolds number, ReL, defined:

ReL =k1/2l

ν=

k2

εν(2.10)

❼ The Taylor microscale. Is the length scale used to characterize the turbulent flow. Itfalls between the large eddies (integral scale) and the small eddies (Kolmogorov scale). Itcannot be calculated easily, since it requires several correlation functions, expanding them ina Taylor series, and solving for some terms. According to literature [22], a valid expressionto obtain the Taylor microscale for isotropic turbulence is:

λ =

√

15νu

ε≈

√

10νk

ε(2.11)

By using this length and the velocity, the Taylor microscale Reynolds number can be calcu-lated:

Reλ =uλ

ν(2.12)



❼ The Kolmogorov microscale. Also known as the dissipation scale, is the smallest of theturbulence scales. A.N. Kolmogorov derived them[18, 20] by the assumption that at thisscale is where the energy is dissipated into heat, and the viscosity dominates. He definedthe length scale, time scale, and velocity scale, respectively, as:

η =

(

ν3

ε

)1/4

(2.13)

τη =

(

ν

ε

)1/2

(2.14)

uη = (νε)1/4 (2.15)

To summarize, the length scales are:

L ≫ l ≫ λ ≫ η

Energy Spectrum

Kolmogorov published a series of papers in 1941 [18, 20, 19], also referred as the K41 theory. There,the 2/3 law is provided, which leads to the prediction of a k−5/3 decay rate in what it is now knownas the inertial subrange of the energy spectrum, among many other important derivations.

Eddies are packed with kinetic energy, and depending on their size, the amount of energy theycarry varies. The Kolmogorov hypothesis states that the energy is transferred from large eddies tosmall eddies, until they reach the scale in which the energy is dissipated into heat. The frequenciesof small eddies present in a flow are much larger than for large eddies, and they correspond tothe scales presented in 2.3.3. It is possible to represent the turbulent kinetic energy as a functionof the wavenumber, as shown in Fig. 2.2, and when normalized on logarithmic scale, as shown inFig. 2.3, the (-5/3) slope predicted from the K41 theory becomes evident.

Figure 2.2: Turbulence energy spectrum, reproduced from [4].



Figure 2.3: Normalized spectrum of the kinetic energy of turbulence, reproduced from [25].

Energy cascade

Now that the different scales have been described, and the fact that the eddies contain energy hasbeen established, a brief explanation of the energy transfer will take place.

The bigger the eddies, the more unstable they are, which causes them to break up into smallereddies. When that happens, the energy contained in the large eddies is transferred to the smallereddies. These eddies will follow the process, breaking up into even smaller eddies, transferring theenergy until the process reaches the smallest scale, in which the energy is dissipated into heat.

This dissipation takes place at the end of the sequence, when the Kolmogorov microscale isreached, and the rate of dissipation, ε, is therefore determined by the first process in the sequence,namely the energy transferred from the largest eddies.

A schematic representation of this energy cascade is shown in Fig. 2.4, showing differentmethods to model the fine-scale properties of turbulence. More information about the limitationsand advantages of each model can be found in [4].



Figure 2.4: Schematic representation of the energy cascade by different methods: (a) KolmogorovK41 theory; (b) β-model; (c) Multifractional models. Reproduced from [4].

2.4 Mixing Scales

The diffusivity of turbulence has already been described as one of the main features of turbulence,and defined as the ability to mix. For non-reacting systems, the transport of species due to themomentum transfer is orders of magnitude faster than is done by molecular diffusion. The analysisof reactive materials is, however, more complex. Chemical reactions happen at a molecular scale,and the prediction of the reaction rate depends on many factors, such as the local instantaneousconcentrations. Solving simultaneously a set of partial differential equations is required to calculatethese concentrations, including balances for species, mass and energy. Thus, the whole analysis ofturbulent mixing including reaction might be convoluted.

In this complex process, it is possible to distinguish and describe three stages: macromixing,mesomixing, and micromixing[3].

2.4.1 Macromixing

This term refers to the process of mixing on the scale of the equipment. It is related to thelarge scale processes responsible of the large scale transport, such as the mean concentration ofspecies, temperature, residence time distribution, etc. The importance of the macromixing in thewhole process is that it determines the conditions for the mixing in smaller scales (micro- andmesomixing). Also called bulk blending, it is possible to identify the macromixing with the meanvelocity convection.



2.4.2 Mesomixing

Chemical reactions occur typically in just a few centimeters in length, what is called the reactionzone. Transport of reagents in the neighborhood of this zone happens due to turbulence, andexchange between fresh material and its surroundings takes place. Fast chemical reactions oftenare localized near the feeding of reagents, and as the material is introduced a plume is formed,which is of a bigger scale relative to the micromixing (Kolmogorov microscale) but also of a finescale relative to the system. Ergo, the term mesomixing is appropriate to describe this scale.

Two aspects of mesomixing can determine the environment for micromixing:

❼ The process of turbulent diffusion can be used to describe the spatial evolution of the plumeof fresh material at the feed point.

❼ The inertial-convective process of disintegration of large eddies.

2.4.3 Micromixing

Micromixing is the last stage of the turbulent mixing. When the turbulent energy cascade reachesa scale that is known as the viscous-convective subrange, a point somewhere between the limitof the inertial-convective range and the beginning of the Kolmogorov microscale, deformation ofthe fluid occurs. This is followed by molecular diffusion in the viscous-diffusive subrange, which issituated in the spectrum somewhere around the Kolmogorov microscale but larger than the scalewhere laminar deformation takes place. For liquids, micromixing is identified with mixing in theviscous-convective and the viscous-diffusice subranges.

2.5 The Engulfment Model

Micro- and Mesomixing can be modeled with help of mechanistic models that have been proposedto estimate micromixing times. Some of them are:

❼ The interaction by exchange with the mean (IEM) model

❼ The droplet erosion and diffusion model

❼ The engulfment deformation diffusion (EDD) model

❼ The engulfment model (E-model)

❼ The incorporation model

The E-model is a simplification of the EDD model proposed by Ba ldyga and Bourne in 1984.Figure 2.5 shows the discrete EDD model, from the disintegration of large eddies in the inertial-convective subrange (a) and (b), going through the finer-scale where laminar deformation occurswithin the viscous-convective subrange (c), and finishing in the small region where the reactionzone grows due to the action of vorticity (d). Later on, they demonstrated[2] that the effectsof molecular diffusion are negligible compared with the viscous-convective stretching that causesthe incorporation of fluid into the reaction zone. They consider the mechanism of engulfment ofenvironment into the vortex of the eddie to be the dominate mechanism in turbulent mixing.

The simplification of the model considers the growing of a small volume of concentrated reactantbeing added into a large volume of a second reactant. This allows to consider the reaction zoneto be equal to the region where the aggregate is present. Considering an environment rich in A,and an aggregate of B, the volume of the reaction zone grows according to engulfment:

dVm

dt= EVm; Vm(0) = VB0 (2.16)



Leading to a mass balance of a substance α in the reaction zone:

dVm〈cα〉

dt= EVm〈cα〉 + rαVm (2.17)

where 〈cα〉 represents the concentration of α in the environment of the growing eddy, and rα isthe production (or consumption) rate of α.

Considering that eddies are very small, the concentration of α in the growing reaction zone canbe assumed to be uniform. Substituting 2.16 into 2.17, an evolution for concentration is expressedby:

dcαdt

= E(〈cα〉 − cα) + rα (2.18)

The EDD model describes the engulfment parameter E as the responsible for the growing of theeddy. This is also valid for the E-model, thus:

V (t) = V (0) · eEt (2.19)

E ≃ 0.058

[

ε

ν

]1/2

(2.20)

In equation 2.19, V (0) represents the initial volume of the eddy. It grows proportional to theelongation of the vortex, as a result of the engulfment mechanism. The whole derivation of theEngulfment parameter E is described in[4], in Chapter 8.

Since E has units of s−1, and the model is based on the assumption that the viscous-convectiveprocess (engulfment) controls mixing at the smallest scales, it can be used to define a characteristictime for micromixing:

τE = 1/E (2.21)

2.5.1 Self-Engulfment

When reactant B is injected in an A-rich environment in very small volumes, the volume fractionXB is very small. However, when XB is relatively large, the model might overestimate the reactionrate because some self-engulfment between eddies of B will take place. This reduces the rate atwhich the reaction zone would grow if XB tends to zero. The term P , probability of engulfment,is introduced. Equations 2.16, 2.17 and 2.18 are expressed by:

dVm

dt= EPVm (2.22)

dVmcαdt

= EPVm〈cα〉 + rαVm (2.23)

dcαdt

= EP (〈cα〉 − cα) + rα (2.24)

Considering the case when the initial volume fraction XB0 ≪ 1, and it is growing close to unity,the probability of self-engulfment is estimated as the fraction of B. Hence, the probability ofengulfment fresh material from the surroundings:

P = 1 −XB (2.25)

By defining XB = VB/(VA + VB) and considering Vm = VB :

dXB

dt= EXB(1 −XB); XB(0) = XB0 (2.26)

dcαdt

= E(1 −XB)(〈cα〉 − cα) + rα (2.27)

And for constant E, solution for eq. 2.26 is:

XB =XB0e

Et

1 −XB0(1 − eEt)(2.28)



Figure 2.5: Schematic representation of the EDD model. (a,b) represent the deformations withininertial subrange in a large and a small scale, respectively; (c) shows the laminar deformations inthe viscous subrange; and (d) represents the action of vorticity of fluid elements with an initialthickness in the order of η. Reproduced from [4].

2.6 Mesomixing Coupling

Chemical reactions occur in a molecular scale, and micromixing directly affects this process forthe reasons previously described by the E-model (2.5). However, additional inhomogeneity causedby mesomixing processes has an indirect effect on the chemical reactions, which can be actuallyvery strong and become controlling.

In section 2.4.2 mesomixing is defined as the process in a bigger scale than micromixing, butsmaller than macromixing. Two mechanisms have been identified and analyzed for mixing in themesoscale [3, 5]. They are described in the following subsections.

2.6.1 Turbulent Dispersion

This mechanism is described as the development of a plume of fresh material, α, that is fed intothe system. The inlet is through a pipe of internal diameter 2a at a volumetric flow Q. Thisprocess is well represented by Fig. 2.6.



When the scale of mesomixing, LD, is smaller than the scale of the system, LX , but largerthan the feed pipe radius a, the feed point can be considered as a point source:

a ≪ LD ≪ LX (2.29)

Being u the local velocity of the fluid in the surroundings, the characteristic length for disper-sion can be estimated by:

LD =

(

Q

u

)1/2

(2.30)

Considering homogeneous turbulence with constant local velocity and constant turbulent diffus-ivity (DT ), an expression for the distribution of volume fraction of α-rich feed, X0

α is given by:

X0α(r, x) =

Qα

4πDTxexp

[

−ur2

4DTx

]

(2.31)

Equation 2.31 can be used to determine the environment for the inertial-convective mixing andmicromixing. Using t for the time since α left the pipe, then:

x = ut (2.32)

Substituting the distance x in equation 2.31 for t, employing equation 2.32:

X0α(r, t) =

Qα

4πDT utexp

[

−r2

4DT t

]

(2.33)

Now, a characteristic time for turbulent dispersion, τD, is identified:

τD =Qα

uDT(2.34)

Figure 2.6: Steady-state radial distribution of fresh feed at any position x downstream from thefeed point. Reproduced from [4].

2.6.2 Inertial-Convective Mesomixing

This mechanism is related to the disintegration of a large spot of reactant by turbulence. In amore perceptibly way, it consists on small-scale velocity fluctuations within the spot of reactantwhen is mixed by turbulence, generating also small-scale concentration fluctuations.

The disintegration of a large spot to a relevant scale for micromixing is represented in Fig 2.7from time t0 to t1. Then the action of micromixing takes place from t1 to t2, to be followed byanother disintegration. These two processes are simultaneous in reality, just for a representationthey were taken as steps.



Figure 2.7: Representation of the inertial-convective mesomixing. Reproduced from [4].

Back to the plume considered to represent the turbulent dispersion in subsection 2.6.1, Figure2.8 shows its difference in structure for both mechanisms. In (a) only turbulent dispersion isconsidered, meaning the plume consists of unmixed elements. For (b), disintegration of the eddiesand micromixing occur, resulting in the presented mixture structure.

In order to model the inertial-convective mesomixing, it is interpreted as the formation of smalleddies within big eddies[4, 5]. Considering a partially segregated fluid like shown in Fig. 2.7, itsstructure is represented by “islands” of fluid B with a volume fraction Xu and a composition Xe

B ,enclosed in a “sea” absent of B. Figure 2.9 represents this description.

Three regions are highlighted:

a) Between islands, there is absence of B.

b) In the islands, B is present on a coarse scale

c) Phenomenological points within the islands, where B is mixed on a molecular scale

In the whole structure, the volume fraction of B is:

XBM = XuXeB (2.35)

For a macroscopically well-mixed system, XBM has the same value at any point. But the structuresof the eddies change with time. Xu increases as eddies break up, and at the same time Xe

B

decreases. From [5]:dXe

B

dt=

−(XeB −XBM )

τS(2.36)

dXu

dt=

Xu(1 −Xu)

τS(2.37)

The solution for eq. 2.37 when the initial volume fraction of B is X0 and for constant τS comesto:

Xu =X0

X0 + (1 −X0)exp

(

−t

τS

) (2.38)

Analogous to eq. 2.25, the rate of growth of B-rich zones is affected by self-engulfment, and byintroducing XBu as the volume fraction of micromixed B-rich fluid in the islands:

P = 1 −XBu (2.39)

This “self-engulfment” moderates the rate of growth of the Volume of B:

dVB

dt= EVB(1 −XBu) (2.40)



Figure 2.8: (a) Plume of unmixed elements, only considering dispersive mesomixing; (b) Structureof the plume when disintegration of eddies is considered. Reproduced from [4].

Because XBu =VB

Vu=

VB

V

V

Vu=

XB

Xu, the micromixing due to engulfment becomes:

dXB

dt= EXB

(

1 −XB

Xu

)

(2.41)

Finally, the concentration of α is obtained by:

dcαdt

= E

(

1 −XB

Xu

)

(〈cα〉 − cα) + rα (2.42)

For this mechanism, a characteristic τS time can also be identified:

τS = A

(

Λ2C

ε

)1/3

(2.43)

ΛC is the characteristic scale for the inertial-convective subrange, and it takes a value in the sameorder of magnitude of the Taylor microscale, λ, described in Section 2.3.3. For fully developedturbulence in liquids, usually A = 2.



Figure 2.9: Structure of partially segregated islands: (a) B absent; (b) B present on coarse scale;(c) B mixed on molecular scale. Reproduced from [4].

2.7 Characterization of Micromixing: The

Villermaux-Dushman method

In 1995 [12], Fournier, Falk and Villermaux proposed a new system of parallel competing reactionsthat allows the study of micromixing in chemical reactors. It consists on a neutralization reaction2.44 and the Dushman reaction 2.45, following the scheme presented in Section 2.2.2. After themethod was presented, some papers were published aiming to obtain the kinetics of the system,determine the best set of concentrations, operating conditions, model the system, and determinemicromixing times and mixing efficiency [11, 14, 15, 6].

H2BO –3 + H+

⇋ H3BO3 (2.44)

IO –3 + 5 I– + 6 H+ → 3 I2 + 3 H2O (2.45)

The neutralization reaction (eq. 2.44) happens to be instantaneous relative to the micromixingtimes, while the Dushman reaction (eq. 2.45) is very fast, in the range of the micromixing process,but slower than the neutralization. The acid is added to the reactor in stoichiometric defect,becoming the limiting reactant. For ideal mixing, the acid would only be consumed by the firstreaction, but in poor mixing conditions the acid will consume the borate ions and the excess willreact to form iodine. The iodine formed is a measure of the segregation state of the fluid.

The system contains a premixed solution of iodide and iodate ions in a H2BO –3 /H3BO3 buffer,

so the iodine formed will react with the iodide to yield triiodide ions, in a quasi-instantaneousequilibrium:

I2 + I– ⇋ I –3 (2.46)

This reaction has an equilibrium constant KB that depends on the Temperature of the system T(in Kelvin), given by:

KB =[I−3 ]

[I2][I−](2.47)

log10 KB =555

T+ 7.355 − 2.575 log10 T (2.48)

And a reaction rate [27, 29, 30] given by:

r3 = k3[I−][I2] − k′3[I−3 ] (2.49)



The concentration of triiodide ions are measured by UV/Vis spectrophotometry at a wavelength of353 nm, being one of the most important advantages of this method. Applying the Beer-Lambertlaw for a resulting absorbance A, using the extinction coefficient for triiodide at 353nm ǫ and theoptical path lOPL withing the measurement cell:

[I−3 ] =A

ǫlOPL(2.50)

The reaction rate for the neutralization reaction is so fast that it cannot be determined, but itis believed[10] that for bimolecular reactions in aqueous solutions involving H+ and OH− ions,it is in the range of 1011 l mol−1s−1. For the Dushman reaction, the kinetics have been studiedsince 1888 and are still unclear. An empirical model, expressed as a fifth order rate that does notdepend on the temperature, but rather on the ionic strength [12], can be used:

r2 = k2[IO−

3 ][I−]2[H+]2 (2.51)

where k2 is calculated by:

µ =1

2

∑

i

Ciz2i (2.52)

µ < 0.166, log10(k2) = 9.281 − 3.664µ1/2 (2.53)

µ > 0.166, log10(k2) = 8.383 − 1.511µ1/2 + 0.237µ (2.54)

In equation 2.51 Ci denotes the concentration of species i, and zi the charge of that ion.

2.7.1 Segregation Index

The micromixing quality can be represented by the segregation index, XS . Defining Y as the ratioof acid consumed by the Dushman reaction divided by the total acid injected, and YST as thevalue of Y for infinitely slow mixing, the segregation index is estimated by:

XS =Y

YST(2.55)

Y =2(nI2 + n

I−

3

)

nH

+

0

=2Vtotal([I2] + [I−3 ])

Vinjection[H+]0(2.56)

YST =6[IO−

3 ]0

6[IO−

3 ]0 + [H2BO−

3 ]0(2.57)


Chapter 3

Experimental Design

Characterization of the micromixing time is possible by the Villermaux-Dushman methoddescribed in section 2.7. This chapter shows the characteristics of the experimental setup,

the materials used to perform the experiments, a brief summary of the calibration of the spectro-photometer, and finally the data collected from the experiments.

3.1 Experimental Setup

The rotor-stator spinning disc reactor consists of a stainless steel rotating disc, with a radius of0.135 m and thickness of 4 ·10−3 m, between a stainless steel top stator, and a bottom stator madeof Poly(methyl methacrylate) (PMMA). The stator, having an inside radius of 0.145 m, createsa radial gap of 1 · 10−2 m. Between the rotor and the bottom stator, the axial gap is 4 · 10−3

m, and between the rotor and the top stator it is 8 · 10−3 m. Thus, the reactor has a volume of7.62 · 10−4 m3.

A schematic representation of the reactor is shown in Fig. 3.1. The bulk solution is fedinto the reactor at the top, near the axis, through “Inlet 1”, pumped at a constant flowrate of7.5 · 10−6 m3s−1 with a pump regulated by a Bronkhorst Cori-Flow➋ mass flow controller. Theacid is injected through the “Inlet 2”, in the bottom stator, by a pipe with inner diameter of 0.5mm located at a radial position of 0.130 m, near the rim of the disc. A KNAUER Smartline Pump1000 is used to inject the acid at a constant flowrate of 1.5 · 10−7 m3s−1.

The rotor has a maximum rotational speed of 209 rad s−1 (2000 RPM), provided by a motorSEW-Eurodrive CFM71M. The energy transferred to the liquid will dissipate into heat and increasethe temperature significantly (up to 10 ◦C for rotation speeds of 180 rad s−1). Since the equilibriumconstant KB is highly sensitive to temperature as stated in equation 2.48, the outlet of the reactoris recirculated into the fed vessel, which consists of a double walled storage tank of 5 liters, cooledusing a Lauda ECO RE 630. The decision to recirculate it was made to prevent excessive wasteof bulk solution.

Real-time measurements of triiodide concentration ([I−3 ]) are possible using in-line UV-Visflowcells with optical length of 5 · 10−3 m. A Dual channel Avaspec-2048-2-USB2 spectrophoto-meter with an Avalight DH-S light source is used to measure the concentration of triiodide at353 nm. Measurements were recorded every 0.1 seconds, with an integration time for the UV-Visspectroscope of 1.1 · 10−3 s. Because it is a recirculating system, one flowcell is installed at theinlet of the reactor, and another at the outlet, so the difference in concentrations is equivalentto the production of triiodide. Due to the high sensibility of the optical fiber used to connectthe flowcells to the lamp and to the spectrophotometer, they were fixed using cable ties to avoidmovement. This will prevent disturbances in the measurements.

To calculate the absorbance, equation 3.1 is used. Here, N represents the number of countsmeasured for the sample. N0 is the number of counts measured for the dark reading, that is, withthe light source turned off. And Nref is the number of counts measured for the reference sample,


CHAPTER 3. EXPERIMENTAL DESIGN

demi water.

A = − logN −N0

Nref −N0

(3.1)

Figure 3.1: Schematic representation of experimental setup, adapted from [26].

3.2 Reactants and Preparation

The Villermaux-Dushman method for characterization of micromixing times requires two solutions:a bulk solution containing iodide, iodate and borate ions, and an acid. It has been stated [14]that technical grade products can be used, since the purity of the chemicals has no influence onthe results.

Preparation of 5 liter bulk solution was required for each run of experiments with the con-centrations shown in Table 3.1. All the solutions were prepared using demi water. First a twoliter buffer solution was prepared by dissolving Sodium Hydroxide and Boric Acid, in order todissociate the last one to form borate ions. Then, two separate solutions of one liter each wereadded, one containing dissolved Potassium Iodide, and the other one dissolved Potassium Iodate.It is important to prepare them separately and add them to the basic solution to avoid formationof iodine. Finally, water was added to complete the 5 liters.

Although sulfuric acid has been historically used, the second dissociation depends on manyfactors that might make the data to be unreliable. Thus, strong monoprotic (pKa = −9.24)perchloric acid was used. Technical grade acid was carefully diluted with water to obtain threedifferent concentrations, as shown in Table 3.1.

Bulk Solution

Sodium Hydroxide 0.006 MBoric Acid 0.012 MPotassium Iodate 0.0024 MPotassium Iodide 0.012 M

Acid

Perchloric Acid 0.02 M0.04 M0.08 M

Table 3.1: Set of concentrations used for the experiments


CHAPTER 3. EXPERIMENTAL DESIGN

3.3 Calibration Curve - Determination of the Extinction

Coefficient

As mentioned in Section 2.7, triiodide can be monitored by UV-Vis spectrophotometry due to theaffinity of this ion towards absorption of light at a wavelength of 353 nm. Equation 2.50 relatesthe concentration of triiodide to the absorbance values of the equipment. From the description ofthe setup in Section 3.1, the optical path lenght, lOPL, is known to be 5 · 10−3 m.

The extinction coefficient can be found in literature [14], ranging from 2300 to 2700 m2mol−1.It is therefore preferable to calculate it experimentally, as it seems to depend on the conditions ofthe flowcell. By preparing different solutions with known triiodide concentrations and reading theabsorbance from the UV-Vis, it is possible to obtain the extinction coefficient from eq. 2.50.

A 1 liter stock solution was prepared by dissolving 2g of Potassium Iodide in demi water, andadding 1.27g of Iodine. The solution was stirred for 6 hours, and then titrated using standardizedsodium thiosulfate with starch as an indicator to obtain the exact concentration of Iodine dissolved[28]. Several diluted solutions were prepared using the stock solution.

Measurements of these solutions resulted in two different calibration curves for each of theflowcells, as shown in Figures A.1 and A.2. Thus, the extinction coefficient for the inlet and outletflowcells are ǫinlet = 2406 m2mol−1 and ǫoutlet = 2361 m2mol−1, respectively. The differencebetween them can be attributed to the wear.

3.4 Data Acquisition

In a micromixing controlled regime where only the engulfment mechanism is relevant, the acidconcentration does affect the segregation index but should not influence the micromixing time.Based on that principle, experiments were performed varying the acid concentration, as shown inTable 3.1, in order to define which mechanism is controlling the micromixing time.

The Villermaux-Dushman reactions were carried out in the rotor-stator SDR, three times forevery acid concentration, varying the rotational speed from 11 to 188 rad s−1, in order to have abetter average of all the measurements.

Segregation indexes were obtained in accordance to previous studies [13, 33, 34], in the or-der of magnitude of 10−2, as shown in Figure 3.2. However, it is remarkable that due to thelower concentration of the acid used for these experiments in comparison to previous studies, theyet unexplained trend of an increasing segregation index with increasing rotation speed is morenotorious. The possible reasons will be discussed later in Chapter 5.


Figure 3.2: Effect of calculated Segregation Index, XS , with varying rotational speeds, ω, fordifferent acid concentration

Chapter 4

Mixing Models

Modeling the system has been done in the past and relatively good approximations of themicromixing times have been obtained [34], in the order of magnitude of 10−3s. The existing

model consists in solving all the mass balances, expressed as differential equations, that are involvedin the Villermaux-Dushman system. Additionally, the differential equation for the growing eddyvolume engulfing liquid from the surroundings is simultaneously solved. Thus, the system to besolved can be explicitly described as:

∂[H+]e∂t

=[H+]b − [H+]e

tm− r1 − 6r2 (4.1)

∂[H2BO−

3 ]e∂t

=[H2BO−

3 ]b − [H2BO−

3 ]etm

− r1 (4.2)

∂[H3BO3]e∂t

=[H3BO3]b − [H3BO3]e

tm+ r1 (4.3)

∂[I−]e∂t

=[I−]b − [I−]e

tm− 5r2 − r3 (4.4)

∂[IO−

3 ]e∂t

=[IO−

3 ]b − [IO−

3 ]etm

− r2 (4.5)

∂[I2]e∂t

=[I2]b − [I2]e

tm+ 3r2 − r3 (4.6)

∂[I−3 ]e∂t

=[I−3 ]b − [I−3 ]e

tm+ r3 (4.7)

∂Ve

∂t=

Ve

tm(4.8)

For these equations, the subscript e stands for eddy, while b is for the bulk liquid. Rewritingequation 2.56 and 2.57 for this model using the mentioned subscripts:

Y =2Ve([I2]e + [I−3 ]e)

Ve,0[H+]0(4.9)

YST =6[IO−

3 ]b

6[IO−

3 ]b + [H2BO−

3 ]b(4.10)

Bulk concentrations are kept constant and equal to the values shown in Table 3.1. The acid injectedonly contains H+ ions, hence all values for concentrations are zero except for the hydrogen ion.

The initial volume of the eddy has been arbitrary defined as Ve,0 = Qacid/1000. However, itdoes not have any influence in the model, as long as it takes a value different than zero. With allthe initial conditions specified, the model is executed using MATLAB [24] ode15s solver, and theintegration stops when the acid is consumed, i.e. when the pH reaches a value of 8.


CHAPTER 4. MIXING MODELS

From the experiments, it was determined that the average temperature of the reactor was20.2 ◦C. The Dushman reaction is independent of the temperature, and so is the neutralizationreaction. However, the equilibrium reaction for triiodide does have dependency on the temperat-ure. Thus, it was needed to extrapolate the values found in literature [27, 29, 30] using Arrheniusequation. The values taken for the model are k3 = 2.05 · 109 l mol−1s−1 and k′3 = 2.33 · 106 s−1.

4.1 E-Model

Figure 4.1 shows the segregation index as a function of mixing time for a range of acid concen-trations. The model includes the operational conditions described in Section 3.1, as well as theconsiderations mentioned in the beginning of this Chapter.

Figure 4.1: Variation of the Segregation Index versus mixing time, for the operational conditionsdescribed in Section 3.1, considering the Engulfment Model.

4.2 Self-Engulfment Probability

It is possible to easily implement the probability of self-engulfment described in Section 2.5.1 inthe model, by simply adding the term (1 −XB) to the mass balances so they look like Eq. 2.27.Including the differential equation for XB is not necessary, since algebraic Equation 2.28 shows

the solution for constant E. For the initial volume fraction, XB0 =Qacid

Qacid + Qbulkcan be used.

The model is represented in Figure 4.2. A comparison between the e-model and the e-modelconsidering self-engulfment is presented in Figure 4.3. The solid lines represent the e-model, whilethe dotted lines are for the model with self-engulfment probability. At a certain mixing time, ahigher value for the segregation index is expected when considering self-engulfment. This agreeswith the theory: when the eddy cannot engulf more bulk solution because it is self-engulfing, theneutralization reaction has already depleted the borate ions and the acid can only react to formiodine, until fresh bulk is engulfed.



Figure 4.2: Variation of the Segregation Index versus mixing time, for the operational conditionsdescribed in Section 3.1, considering self-engulfment in the Engulfment Model.

Figure 4.3: Variation of the Segregation Index versus mixing time, for the operational conditionsdescribed in Section 3.1, considering self-engulfment probability. The solid line (—) represents theengulfment model, while the dashed line (- - -) is for the model considering self-engulfment.



4.3 Turbulent Dispersion

A model for turbulent dispersion was adapted from a simple model of a Turbulent Jet, solvingthe mass balances for two concentric cylinders, the outer one representing the bulk flow and theinner one the acid injection. The model considers the radius of the outer cylinder to be halfthe distance between the rotor and the bottom stator (0.5h1), and creates a mesh along thisdistance. The radius of the inner cylinder is localized in the mesh at a distance proportionalto the ratio Qacid/Qbulk, assuming the flow instantaneously accelerates to reach the velocity ofthe bulk flow. The model solves an initial-boundary value problem for parabolic-elliptic partialdifferential equations using the pdepe function from Matlab [24].

The output generates a movie of the concentration profile over time by consecutively plottingthe concentration profile along the radial distance for a defined space of time, and in the endit calculates the segregation index when the acid is completely depleted. Figure 4.4 shows theoutput of the model for the concentrations of acid used in the experiments, considering a diffusioncoefficient D = 1 · 10−9 m2 s−1.

(a) Acid concentration = 0.02, t0 (b) Acid concentration = 0.02, t1 (c) Acid concentration = 0.02, t2

(d) Acid concentration = 0.04, t0 (e) Acid concentration = 0.04, t1 (f) Acid concentration = 0.04, t2

(g) Acid concentration = 0.08, t0 (h) Acid concentration = 0.08, t1 (i) Acid concentration = 0.08, t2

Figure 4.4: Examples of the output movie generated by the model, for the three different Acidconcentrations shown in Table 3.1, including a diffusion coefficient D = 1 · 10−9 m2 s−1 at threedifferent times: t0 = 0s, t1 ≈ 1s, t2 ≈ 10s.



4.4 Inertial-Convective Mesomixing

The effect of the inertial-convective mesomixing can also be implemented in the existing model by

adding the term

(

1−XB

Xu

)

to the mass balances, as explained in Section 2.6.2, Equation 2.42. It

is required to include the differential equation 2.41 and the algebraic equation 2.38, with an initial

value XB0 =Qacid

Qacid + Qbulk.

The ratio of time constants M = τS/τE can be taken as a measure of the effect of the inertial-convective mesomixing. By varying this ratio, it is possible to plot the segregation index as afunction of the mixing time for various scenarios, and measure the effect when this mechanismdominates the process.

(a) Acid concentration = 0.02 (b) Acid concentration = 0.04

(c) Acid concentration = 0.08

Figure 4.5: Variation of the Segregation Index versus mixing time, for the operational conditionsdescribed in Section 3.1, considering inertial-convective mesomixing effects for the three differentconcentrations of acid shown in Table 3.1.


Chapter 5

Results and Discussions

Heretofore has been presented an introduction to the theory of turbulence and some import-ant concepts to consider, passing through a brief description of the equipment and how the

experimental data were collected, and describing four mixing models that can be used to estimatethe corresponding mixing times to the segregation indexes obtained experimentally. However, alink between the experimental results for the segregation index and the suitable model is stillrequired.

From Section 2.5, the characteristic time for micromixing due to engulfment mechanism, provento be dominant at the microscale, can be rewrite:

τE = E−1 = 17.3

√

ν

ε(5.1)

This means that for a constant viscosity, the micromixing time is purely a function of the localaveraged energy dissipation rate. This energy comes from the power input, transferred from theequipment to the liquid.

In order to compare the experimental results with the theoretical ones, first it is needed torelate the obtained segregation indexes to the energy transfer from the high rotational speeds fromthe reactor. Later on, a series of parameters, correlations and assumptions will be described togive basis to the taken approach, and finally the comparison can take place.

5.1 Energy dissipation rate

According to the literature and previous research[7, 31, 8, 9], the flow regime that prevails inthe rotor-stator Spinning Disc Reactor is the torsional Couette flow regime. The local energydissipation rate per unit mass that follows the shear stress profile for that regime [13] is:

ε(r) = 0.0118(ωr)11/4ν1/4h−5/41 (5.2)

As mentioned before, the reactor transfers the energy from the rotor to the system. This powerinput, averaged for the whole reactor, is calculated by:

ε =

∫ 2π

0

∫ RD

0

∫ h1

0ρε(r)rdzdrdθ +

∫ 2π

0

∫ RD

0

∫ h2

0ρε(r)rdzdrdθ

ρVR(5.3)

It is possible to calculate the average energy dissipation rate from the experiments, so in order toobtain the local energy dissipation rate, the next correlation can be used:

φ =ε

ε(5.4)

For the configuration given in Section 3.1, φ = 5.33.


CHAPTER 5. RESULTS AND DISCUSSIONS

Also mentioned in the description of the setup, the motor responsible for the spinning of therotor is a SEW-Eurodrive CFM71M with a rated speed of 3000 rpm and 400V. From the technicalsheet of the motor, a standstill torque M0 = 6.5Nm and a standstill current I0 = 4.3A are taken.For each rotational speed used during the experiments, an averaged output current was measured.Thus the torque can be calculated:

τ =IM0

I0(5.5)

Considering that the net power input to the system is equal to the product of the torque and therotational speed, using Equation 5.5:

Pnet = ωτ =ωIM0

I0(5.6)

According to previous studies [13], the power loss due to internal current loss was fitted in a 4th

degree polynomial equation, which is used to calculate the effective net power input:

P0 = 2 · 10−9ω4 − 9.8 · 10−7ω3 + 3.7 · 10−4ω2 + 0.2278ω + 0.9621 (5.7)

Peff = Pnet − P0 (5.8)

Finally, the average energy dissipation rate in the reactor can be calculated by:

ε =Peff

ρVR(5.9)

The operational rotational speeds of the reactor during the experiments ranged from 100 −1800 RPM (10.5 − 188rad s−1). Thus, the equivalent average energy dissipation rates and thelocal dissipation rate are calculated and shown in Table 5.1 after measuring the output current Iin amperes.

RotationalSpeed [RPM] ε [Wkg−1] ε [Wkg−1]

100 0.8743 4.6568300 8.2678 44.037500 29.018 154.56600 56.412 300.47900 167.29 891.081000 207.69 1106.21200 356.69 1899.81500 647.31 3447.81800 1099.2 5855.1

Table 5.1: Measured energy dissipation rates for the operational rotation speeds, for a φ = 5.33.

5.2 Results

5.2.1 Comparison between Micromixing times - Engulfment Model

From the experiments performed, the segregation indexes were measured as a function of therotational speed and varying the acid concentration of the acid. These were shown in Section3.4, in Figure 3.2. When only engulfment controls the mixing process, the segregation index as afunction of the mixing time was modeled in Section 4.1 and shown in Figure 4.1.



Now, in order to relate the experimental data XS to the mixing times according to the model,a curve fitting for each acid concentration was done. They only considered, however, the range ofsegregation index relevant to the experiments (0.001 < XS < 0.1). Figure B.1 shows these curves.Thus, the experimental mixing times from the measured segregation indexes:

tm,Cacid=0.02 =

(

XS

0.87342

)1.0740

(5.10)

tm,Cacid=0.04 =

(

XS

2.6484

)1.1158

(5.11)

tm,Cacid=0.08 =

(

XS

6.0864

)1.1411

(5.12)

The comparison between the theoretical micromixing time and the corresponding mixing time forthe experiments are shown in Figure 5.1

Figure 5.1: Micromixing time as a function of energy dissipation rate, using E-model.

5.2.2 Comparison between Micromixing times - Self-engulfment

The same procedure from the previous Section can be implemented for the model in which selfengulfment is considered. Figure B.2 shows the fitted curves for the three different acid concen-trations. And the mixing times can be then calculated using:

tm,Cacid=0.02 =

(

XS

0.96823

)1.0787

(5.13)

tm,Cacid=0.04 =

(

XS

3.1597

)1.1255

(5.14)

tm,Cacid=0.08 =

(

XS

13.9041

)1.0682

(5.15)



Figure 5.2 shows an improvement in matching the mixing times from the experimental segregationindexes to the theoretical value of the micromixing time, but only for the values of energy dissip-ation rate lower than 2 · 102. It is also worth to mention that the deviation is bigger for the loweracid concentrations. This suggest that the micromixing time is being influenced by mesomixingeffects strongly in the region with high energy dissipation rate.

Figure 5.2: Micromixing time as a function of energy dissipation rate, using E-model and consid-ering self-engulfment.

5.2.3 Comparison between Segregation indexes - Turbulent Dispersion

In Section 4.3 was explained that a model for a Turbulent Jet was adapted to simulate theVillermaux-Dushman system. The resulting model requires the Turbulent diffusivity coefficientDT as an input in the flux equation, in order to substitute the mass diffusion coefficient D.This value was arbitrarily modified in order to converge to segregation indexes similar to theexperimentally obtained. Thus, for the range 0.001 < XS < 0.1, the simulations suggested valuesfor the turbulent diffusivity in the range 1 · 10−8 < DT < 1 · 10−4. From the K41 theory, theturbulent diffusivity coefficient is calculated by:

DT = 0.1k2

ε(5.16)

Although the energy dissipation rate can be measured, the value for the turbulent kinetic energy,k, cannot. In the theory of turbulence, some correlations can be found to estimate this value,at least in the correct order of magnitude. However, by doing that the geometry of the systembecomes relevant.

First, the average velocity in the reactor is related to the rotational speed. From [23], theaverage speed in a Taylor-Couette flow is calculated as 0.3 times the speed from the movingsurface. Thus, for this configuration where the injection of acid is located at the radial positionr = 0.130 m:

u = 0.3U = 0.3ωr (5.17)



The integral scale l contains the largest size range of eddies, and is in the order of magnitude ofthe large scale L. For this setup the large scale is equal to the distance between the rotor and thestator (L = h1 = 0.003 m). Another way to estimate this scale is [1]:

l ∼u

ε(5.18)

And since l must be in the same order of magnitude than L, a factor of 1/50 is used. This factormight seem very large, but in CFD is common to find values between 1/6 - 60 to relate lengthscales [1]. The turbulence Reynolds number is also calculated:

ReL =

(

l

η

)4/3

(5.19)

Allowing to estimate the intensity of turbulence, u, which leads to the turbulent kinetic energy:

u =ReLν

l(5.20)

k =3u2

2(5.21)

Even though these are merely estimations, they have been proven to give the correct orders ofmagnitude, becoming good correlations for these analysis. From the experiments, now it is possibleto relate the coefficient of turbulent diffusivity to the rotational speed, but this is only valid forthis configuration of the reactor. Modifying the gap between rotor and stator would require arecalculation of the above mentioned parameters, and of course perform the experiments again.

Figure 5.3 shows the comparison between the experimental and the theoretical segregationindexes as a function of the turbulent diffusivity coefficient. It is shown that for the experimentallyobtained values of XS , the coefficient DT differs for two orders of magnitude to the values obtainedfrom the model. According to literature[4, 17], DT often acquires (for liquids) orders of 10−6

to 10−3 m2s−1, meaning that the proposed estimation is relatively good and the mechanism ofturbulent dispersion is not dominant for high energy dissipation rates.

However, for the range of low energy dissipation rates, the results suggest that the turbulentdispersion should not be discarded as the dominant mechanism in the micromixing process. Theestimated values for DT corresponding to the rotational speeds of 100 to 600 RPM would matchto the experimental data if they were one order of magnitude lower, and unless an exact valuecould be measured the effect is still latent.

5.2.4 Comparison between Micromixing times - Inertial-Convective

mesomixing

The last of the mechanisms investigated, the inertial-convective mesomixing, was also treatedfollowing the same procedure than sections 5.2.1 and 5.2.2. Considering that the mechanism ischaracterized by M = τS/τE , the decision was to compare the corresponding micromixing timesfor different values of M , in order to obtain results closer to the theoretical micromixing times.

Figures B.3, B.4 and B.5 show the fitted curves resulting from modeling the segregation indexesas a function of the mixing time, considering the different intensities of the effect of the inertial-convective mesomixing. From the plots, the equivalent equations to calculate mixing time from



Figure 5.3: Segregation index as a function of turbulent dispersion coefficient.

the experimental segregation indexes are:

tm,Cacid=0.02,M=1 =

(

XS

1.1961

)1.1193

(5.22)

tm,Cacid=0.02,M=2 =

(

XS

1.4682

)1.1697

(5.23)

tm,Cacid=0.02,M=5 =

(

XS

3.1087

)1.1416

(5.24)

tm,Cacid=0.02,M=10 =

(

XS

6.4953

)1.1090

(5.25)

tm,Cacid=0.04,M=1 =

(

XS

3.1655

)1.1723

(5.26)

tm,Cacid=0.04,M=2 =

(

XS

6.0703

)1.1091

(5.27)

tm,Cacid=0.04,M=5 =

(

XS

13.7529

)1.0912

(5.28)


(

XS

16.6919

)1.1657

(5.29)



tm,Cacid=0.08,M=1 =

(

XS

6.6091

)1.2047

(5.30)

tm,Cacid=0.08,M=2 =

(

XS

13.8021

)1.1318

(5.31)

tm,Cacid=0.08,M=5 =

(

XS

31.857

)1.1150

(5.32)


(

XS

34.2898

)1.2020

(5.33)

In Figures 5.4, 5.5, 5.6 and 5.7, the comparisons between the theoretical micromixing times andthe corresponding mixing times for the experiments are shown.

Figure 5.4: Micromixing time as a function of energy dissipation rate, using E-model and consid-ering inertial-convective mesomixing with M = 1.

Equation 2.43 can be used to calculate the characteristic time for this mechanism. Eventhough ΛC cannot be easily determined, a valid approximation [5] is ΛC ≈ 0.272l, being l therange of the integral scale. Although computing the values for τS is possible, allowing to usemore concrete values for M in the model, still the values will not match exactly the experimentalresults. However, this can give an explanation for the trend of an increasing segregation indexwhen increasing rotational speed.



5.3 Characteristic times comparison

The characteristic times for the mesomixing mechanisms are described by equations 2.34 and2.43, and in this Chapter, estimations and correlations for all the involved parameters were given.Hence, it is possible to estimate these characteristic times and compare them to the micromixingtime.

Figure 5.8: Comparison of the characteristic times for the two mechanisms of mesomixing

These values are not exact but the trend can indicate that for this particular configurationof the reactor, the mechanism of turbulent dispersion might become significant at low energydissipation rates, while the inertial-convective disintegration of eddies can dominate the mixingprocess at high energy dissipation rates.

The estimated values are in accordance to the results analyzed in section 5.2, as it can be seenin Figure 5.3 that for low values of turbulent diffusivity DT (equivalent to low energy dissipationrate), the estimated segregation index is in the same order of magnitude than the experimentallyobtained. Additionally, it was explained in Section 2.3.3 that the scales of turbulence grow whenincreasing the energy dissipation rates and the turbulent kinetic energy. That would result on anincreasing integral scale l, which will directly increase the characteristic scale of inertial-convectivesubrange ΛC , leading to a great effect of mesomixing due to inertial-convective mechanism. Fromthe figures presented in section 5.2.4, an improvement is observed when increasing the ratio M athigh energy dissipation rates.

Finally, it is possible to compare these estimated values to the ones obtained experimentally inorder to observe the effects of mesomixing. Converting the segregation indexes to the equivalentmicromixing times using the model with the self-engulfment consideration, and the model ofinertial-convective mesomixing with a value of M = 5, and plotting them against the rotationalspeed, is shown in Figure 5.9.

For rotational speeds below 300 RPM, the mechanism dominating the micromixing is theTurbulent Dispersion, but for rotational speeds above 600 RPM the value of M will graduallyincrease and the Inertial-convective disintegration of eddies becomes more important. This canexplain the trend seen in Figure 3.2 of an increase in the segregation index for faster rotationalspeeds.


(a) self-engulfment

(b) Inertial-convective M = 5

Figure 5.9: Comparison of the characteristic times for the two mechanisms of mesomixing, andthe experimentally obtained mixing times.

Chapter 6

Conclusions and

Recommendations

6.1 Conclusions

The rotor-stator Spinning Disc Reactor is a novel type of reactor with a high potential toperform fast reactions due to the intense degree of turbulence and the high rates of mass

transfer and heat transfer. In the industry, specially in the fine chemicals and pharmaceutical,some complex chemical reactions can compete for a reagent, and in those scenarios the selectivityand yield of the desired products depend on the mixing quality of their components. Chemicalreactions occur in a molecular level, and the mixing at that scale is called the micromixing. Hence,the investigation of the micromixing in the rotor-stator SDR becomes important in order to predictits efficiency for industrial applications where very fast complex chemical reactions are involved.

This project arises from the experimental evidence of previous studies, where the micromixingtimes were estimated by the Villermaux-Dushman method and they resulted in higher values thanthe theoretically expected, meaning a mechanism on a higher scale called mesomixing has influenceon the system. Based on the premise that the mixing time is not affected by the concentrationof the limiting reagent in the micromixing regime, experiments were performed varying the acidconcentration of the Villermaux-Dushman system in order to identify the effects of mesomixing,and evaluate the mechanism responsible of this event.

Four models were adapted and developed in order to achieve this objective: an existing mi-cromixing model was corrected and modified to vary acid concentrations, then the effects of self-engulfment were included in that model, later the mechanism of inertial-convective disintegrationof eddies was also included, and last a Turbulent Jet model to investigate the turbulent dispersionwas completed.

The experimental data was compared to the predicted micromixing times, and by using cor-relations found in literature, it was concluded that for this particular geometry and operationalconditions the mechanism of Turbulent dipsersion has influence in the mixing times for low energydissipation rates, while for the high energy dissipation rates the mechanism of inertial-convectivedisintegration of eddies becomes dominant.

6.2 Recommendations

Mesomixing is intimately related to the geometry of the system, thus for further research, exper-iments should be performed for other configurations of the rotor-stator SDR, varying the dimen-sions, as well as the operational conditions. It is also recommended to vary the viscosity of thereagents. By definition, only the engulfment mechanism is directly influenced by the viscosity,which means that the micromixing time would increase, but the mesomixing effects should not.


Nomenclature

ν kinematic viscosity

ε average power dissipation rate in the reactor

u local average velocity

ǫ molar extinction coefficient

η Kolmogorov length scale

λ taylor scale

ΛC integral concentration scale

〈cα〉 concentration of α in the surroundings near reaction zone

µ ionic strength

ω rotational speed

φ proportionality constant relating average and local power dissipation rates

ρ specific gravity or density

τ torque

τη Kolmogorob time scale

τD characteristic time for turbulent dispersion

τE characteristic time for micromixing by engulfment

τS time scale for dissipation of segregation in inertial-convective subrange

A constant in equation 2.43

M0 standstill torque

u turbulence intensity, rms of velocity fluctuation

ε local energy dissipation rate per unit mass

A absorbance

a internal radius of the pipe

cα concentration of substance α

Ci concentration of species i

D diffusion coefficient


CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS

DT turbulent diffusivity

E engulfment rate coefficient

h1 axial gap between the rotor and the bottom stator

h2 axial gap between the rotor and the top stator

HD disc thickness

I measured current

I0 standstill current

k turbulent kinetic energy per unit mass

KB equilibrium constant

ki kinetic constant for reaction i

L scale of large eddies

l integral scale

LD characteristic length of dispersion

lOPL optical path length

LX scale of the system

M ratio of characteristic times for inertial-convective mesomixing and micromixing

N number of counts measured by the spectrophotometer

N0 number of counts measured for the dark reading

ni mole number of species i

Nref number of counts for the reference sample

P probability of engulfment

P0 power loss

Peff effective power input

Pnet net power input

pKa acid dissociation constant at logarithmic scale

Q volumetric flow

Qacid volumetric flow of the acid

Qbulk volumetric flow of the bulk solution

r radial coordinate transverse to streamline, radial position

rα rate of formation of substance α by reaction

RD rotor radius

ri reaction rate for reaction i

Re Reynolds number


CHAPTER 6. CONCLUSIONS AND RECOMMENDATIONS

Reλ Taylor microscale Reynolds number

ReL turbulence Reynolds number

T temperature

t time

tm micromixing time

U characteristic velocity of the large scale

u local instantaneous velocity

u′ fluctuating contribution for instantaneous velocity

uη Kolmogorov velocity scale

V volume of the reaction zone

VA volume of A-rich fluid in the reaction zone

VB0 initial volume of aggregate B in the reaction zone

VB volume of B-rich fluid in the reaction zone

Vinjection volume injected

Vm volume of the reaction zone

VR volume of the reactor

Vtotal total volume of the system

w radial gap between between rotor and stator

x coordinate measured along streamline

X0α volume fraction of α-rich fluid due to turbulent dispersion

XeB volume fraction of liquid B in large eddies

X0 volume fraction of B in feed or initial volume fraction B

XB0 initial volume fraction of B-rich fluid in the reaction zone

XBM average volume fraction of liquid B

XBu volume fraction of micromixed fluid in large eddies

XB volume fraction of B-rich fluid in the reaction zone

XS segregation index

Xu volume fraction of fluid containing B

Y selectivity of iodine

YST value of Y for infinitely slow mixing

zi charge number of ion i


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Acknowledgments

Foremost I would like to express my gratitude to dr.ir. John van der Schaaf, for giving me theopportunity to work on this amazing project. I still remember the meeting in which I asked

him for a challenging topic for my graduation project. I must admit that for the past few monthsI asked myself one question several times: “why did I do that?”. But he believed in me and healways helped me find the way to achieve what we were looking for. Thank you.

Besides my supervisor, I would also like to thank PDEng. Vladan Krzelj, dr. Fernanda Neirad’Angelo, and dr.ir. Johan Padding. Thank you all for agreeing to take part of the graduationcommittee.

I want to thank the staff of the SCR group for the support, but specials thanks to ing. PeterLipman, not only for all the assistance related to the experimental setup but also for the nice andpleasant conversations.

Also, I would like to thank my colleagues from the graduation room, the “vrieskist”. Al-though it was usually a very quite room, we also had fun building the window and talking aboutmiscellaneous topics during these nine months.

Furthermore, I would also like to express my gratitude to CONACYT for their financial sup-port.

Many friends were there in my best and worst moments during these months. I want to startthanking Jan Willem Oortwijn (the best intro-dad!) for helping me integrate in the Dutch cultureand becoming a true friend. And of course, my mexican friends Diana, Kevin, Manuel, Raul,Roberto (Benito), Gabie, and Rodrigo. You all have become like a family here. I also want tothank the only two persons that read my thesis and helped me with the corrections: my friendsBarbara Poblano, and Emilio Bajonero.

Finally, I want to thank my family. My parents Arturo and Ruby, who have always been verysupportive, thank you, because of you I am the person who I hope makes you proud. I would liketo thank Abi Raque, the best grandmother in the world. And the person I miss the most, mybrother Kevin.


Appendix A

Calibration Curves

The calibration of the spectrophotometer was done by preparing a stock solution as mentionedin Section 3.3, and diluting it to several solutions with known concentration of iodine. Then thecounts were measured for each of the solutions, and the resulting Absorbance (following equation2.50) was plotted versus the corresponding concentration of triiodide in order to obtained the realextinction coefficient. These plots were fitted using the tool cftool from Matlab [24], and are shownhere:

Figure A.1: Calibration curve for the inlet flowcell


APPENDIX A. CALIBRATION CURVES

Figure A.2: Calibration curve for the outlet flowcell


Appendix B

Curve Fitting

Here are shown the plots for curve fitting, mentioned in Chapter 5, that led to the equations thatallow to relate the experimental data for segregation index XS to the micromixing times accordingto each model.

The curve fitting was done using the cftool in Matlab [24] for each model, for each concentrationof acid, and for the case of the inertial-convective mechanism, for different values of M .

In order to achieve better results, the curve fitting only considered values of segregation indexin the range of the experimentally obtained. Than means in the range of 0.001 < XS < 0.1.Additionally, a power law was selected for the fitting, although the linear curve fitting was chosenin previous studies [13].


APPENDIX B. CURVE FITTING



Figure B.1: Fitting curves for the segregation index as a function of the mixing time, for thedifferent acid concentrations, E-model.





Figure B.2: Fitting curves for the segregation index as a function of the mixing time, for thedifferent acid concentrations, E-model considering self-engulfment.



(a) M = 1 (b) M = 2

(c) M = 5 (d) M = 10

Figure B.3: Fitting curves for the segregation index as a function of the mixing time, for acid con-centration = 0.02, E-model considering inertial-convective mesomixing effects, for various valuesof M.



(a) M = 1 (b) M = 2

(c) M = 5 (d) M = 10



micro- and mesomixing in a spinning disc reactor - …micro- and mesomixing in a spinning disc...

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