Model-based optimization of theoperation procedure of

emulsification

Model-based optimization of theoperation procedure of

emulsification

PROEFSCHRIFT

ter verkrijging van de graad van doctoraan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof. dr. ir. J.T. Fokkemavoorzitter van het College voor Promoties,

in het openbaar te verdedigen op maandag 7 november 2005 om 13.00door

Martijn STORK

ingenieur in de bioprocestechnologiegeboren te Ede

Dit proefschrift is goedgekeurd door de promotor:Prof. ir. O.H. Bosgra

Samenstelling promotiecommissie:

Rector Magnificus voorzitterProf. ir. O.H. Bosgra Technische Universiteit Delft, promotorProf. dr.ir. P.M. van den Hof Technische Universiteit DelftProf. dr. ir. J. Grievink Technische Universiteit DelftProf. dr. ir. G. van Straten Wageningen UniversiteitProf. dr. ir. A.C.P.M. Backx Technische Universiteit EindhovenDr. ir. Z. Verwater-Lukszo Technische Universiteit DelftDr. ir. J.A. Wieringa Unilever

ISBN-10 9090196862ISBN-13 9789090196862

Keywords: emulsification, modeling, optimization

This work was supported with a grant from the Dutch Programme EET (Econ-omy, Ecology and Technology). Project title: “Batch processes - cleaner and moreefficient”.

Copyright c© 2005 by Martijn Stork

All rights reserved. No part of the material protected by this copyright notice maybe reproduced or utilized in any form or by any means, electronic or mechanical, in-cluding photocopying, recording or by any information storage and retrieval system,without written permission from the publisher: Martijn Stork.

Printed in Germany

Voorwoord

Een groot verschil tussen klimmen en een promotieonderzoek is, dat het bij klimmenduidelijk is waar de top ligt. Bij een promotieonderzoek verleg je de ligging van detop voortdurend, waardoor je de top nooit bereikt.

Een groot aantal mensen hebben een belangrijke rol gespeeld bij mijn klim naarde top. Okko, jou wil ik met name bedanken voor de grote vrijheid die je mij hebtgegeven in het bepalen van de route naar de top! Jan, hartstikke bedankt voor deinhoudelijke discussies over druppels en je aanstekende enthousiasme. Ondanks datje nu bij de “vijand” werkt, hoop ik dat we contact zullen blijven houden. Mannen,bedankt voor de fantastische tijd in Delluf! Voor iemand die het van zijn eenvoudigeboerenverstand moet hebben was het niet altijd even makkelijk om stand te houdenin het Delftse theoretische geweld, maar al met al heb ik een super tijd met julliegehad. Enkele hoogtepunten:

• Hilarische discussies tijdens de lunch: beklimming van de Wageningse Berg,riekdarten, kontflossen, definitie van humor, meneer Peer etc.

• Bierwerpen en het feit dat ik daar duidelijk niet voor in de wieg ben gelegd.

• Hardlopen en de koniningsgehaktballen van Rob. Zonder dat was de MILP-aanpak er nooit gekomen.

• Middagjes strand, Beestenmarkt, stappen in Delluf etc.

Lex en Nienke, jullie wil ik hartelijk bedanken voor de vele praktische tips en dehulp om de experimentele opstellingen aan de praat te krijgen en te houden. Super!Sjoerd en Debby, jullie wil ik met name voor de steun gedurende het laatste, relatiefvlakke stuk naar de top bedanken. Elke, jij hebt het weliswaar niet zo zwaar gehadals de vrouw van Tolstoj (zij schreef al zijn manuscripten in het net), maar hetcorrigeren van mijn proefschrift op taal- en schrijffouten is toch ook niet bepaaldeen enerverende taak geweest. Heel erg bedankt daarvoor en voor je steun gedurendehet promotietraject!

Terugkijkend waren het schitterende jaren. De top heb ik weliswaar niet gehaald,maar ik ben tevreden met de bereikte hoogte!

Martijn StorkArlen, 14 augustus 2005.

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Contents

Voorwoord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

1 Introduction 11.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Basic aspects of emulsions . . . . . . . . . . . . . . . . . . . . 21.1.2 Process equipment . . . . . . . . . . . . . . . . . . . . . . . . 51.1.3 Current operating procedure and limitations . . . . . . . . . 71.1.4 Improving the operation procedure . . . . . . . . . . . . . . . 9

1.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Approach and limitations . . . . . . . . . . . . . . . . . . . . . . . . 141.4 Outline of this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Theory of droplet breakup 172.1 Breakage condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Breakage mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . 182.3 Breakup time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Number and sizes of daughter droplets . . . . . . . . . . . . . . . . . 212.5 Breakup in concentrated emulsions . . . . . . . . . . . . . . . . . . . 22

3 Dynamic modeling of emulsification 233.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.1 Mode of flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1.2 Breakage zones . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1.3 The surfactant . . . . . . . . . . . . . . . . . . . . . . . . . . 363.1.4 Other assumptions . . . . . . . . . . . . . . . . . . . . . . . . 373.1.5 List of assumptions . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Reactor model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.2.1 Vessel compartment . . . . . . . . . . . . . . . . . . . . . . . 403.2.2 Colloid mill gap compartment . . . . . . . . . . . . . . . . . . 413.2.3 Colloid mill groove compartment . . . . . . . . . . . . . . . . 413.2.4 Piping compartment . . . . . . . . . . . . . . . . . . . . . . . 413.2.5 Simplification of the PBEs . . . . . . . . . . . . . . . . . . . . 42

3.3 Droplet models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.1 Modeling of S(v, t) . . . . . . . . . . . . . . . . . . . . . . . . 433.3.2 Modeling of ν(w, t) . . . . . . . . . . . . . . . . . . . . . . . . 44

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3.3.3 Modeling of P ′(v|w, t) . . . . . . . . . . . . . . . . . . . . . . 463.4 Viscosity model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.5 Flow rate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.5.1 Modeling of the pumping capacity of the colloid mill . . . . . 523.5.2 Modeling of the pressure drop over the colloid mill . . . . . . 533.5.3 Modeling of the pressure drop over the piping . . . . . . . . . 543.5.4 Flow rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 Numerical solution of the model 594.1 Discretization of a general PBE . . . . . . . . . . . . . . . . . . . . . 594.2 Model discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5 Experimental set-up 655.1 Lab-scale equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2 Measuring instruments . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.2.1 Viscosity measurements . . . . . . . . . . . . . . . . . . . . . 665.2.2 Flow rate measurements . . . . . . . . . . . . . . . . . . . . . 675.2.3 DSD measurements . . . . . . . . . . . . . . . . . . . . . . . 67

5.3 Fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.4 Preliminary experiments . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.4.1 Confidence intervals of the measurements . . . . . . . . . . . 705.4.2 Reproducibility of the process . . . . . . . . . . . . . . . . . . 735.4.3 Sample stability . . . . . . . . . . . . . . . . . . . . . . . . . 75

6 Model validation and parameter estimation 776.1 Flow rate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.1.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.1.2 Adaptation of the Flow rate model . . . . . . . . . . . . . . . 796.1.3 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . 806.1.4 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.2 Viscosity model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.3 Reactor model and Droplet models . . . . . . . . . . . . . . . . . . . 88

6.3.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.3.2 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . 896.3.3 Adaptation of the Droplet models . . . . . . . . . . . . . . . 926.3.4 Parameter estimation . . . . . . . . . . . . . . . . . . . . . . 946.3.5 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . 101

7 Optimization of the operation procedure 1037.1 General formulation of the dynamic optimization problems . . . . . 1047.2 Solving dynamic optimization problems . . . . . . . . . . . . . . . . 1067.3 Initial guess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.3.1 MLD System . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.3.2 End-point constraints and the objective function . . . . . . . 111

7.4 Optimization studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 1127.4.1 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 112

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7.4.2 Optimization problem A . . . . . . . . . . . . . . . . . . . . . 1147.4.3 Optimization problem B . . . . . . . . . . . . . . . . . . . . . 1217.4.4 Optimization problem C . . . . . . . . . . . . . . . . . . . . . 1337.4.5 Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . 137

7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

8 Conclusions and recommendations 1438.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.1.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1448.1.2 Operation procedure . . . . . . . . . . . . . . . . . . . . . . . 1458.1.3 Overall conclusion . . . . . . . . . . . . . . . . . . . . . . . . 147

8.2 Recommendations for future research . . . . . . . . . . . . . . . . . . 148

Bibliography 148

Glossary of symbols 155

A Viscosity measurements 163A.1 Viscosity of water/glucose syrup mixtures . . . . . . . . . . . . . . . 163A.2 Viscosity of water/surfactant mixtures . . . . . . . . . . . . . . . . . 164

B Micelle volume fraction 167

C Branch-and-bound method - basic operation 169

D Results sensitivity analysis 171

Summary 181

Samenvatting 183

Curriculum Vitae 185

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Chapter 1

Introduction

Emulsions are widely encountered in the food industry. As Becher (2001) has pointedout, the first food we consume is an emulsion, namely breast milk. An other commonfood emulsion is mayonnaise. During 2002, 24200 ton of mayonnaise was sold worth46.1 euro millions sales in the Netherlands1. A short history of food emulsions islisted in Table 1.1.

Table 1.1: A short history of food emulsions (Becher, 2001).

Food Date of introductionMammalian milk c. 2.4.107 B.C.E.Milk from domestic c. 8500 B.C.E.animals; butter and cheeseSauces 15th-16th centuryIce cream c. 1740Mayonnaise c. 1845Margarine 1869

Applications of emulsions are also found in the cosmetic and the pharmaceuticalindustry. Skin creams and lotions are examples of cosmetic emulsions. Pharmaceu-tical emulsions can be used e.g. as carriers of drugs or as blood substitute. Experi-ments have shown that transportation of water soluble drugs by water-in-oil (W/O)emulsions2 is much more efficient than injecting these drugs as aqueous solutions.In fact, an aqueous solution is digested in the stomach whereas a fatty emulsion isnot (Chappat, 1994). The use of emulsions as blood substitute was suggested bythe extraordinarily solubility of oxygen and carbon dioxide in certain perfluorchemi-cals. This fluorochemical-in-water emulsion could be used for transfusion in such

1Source: Food for Thought, 2003.2In Section 1.1.1 water-in-oil (W/O) and oil-in-water (O/W) emulsions are described.

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emergencies as a shortage of whole blood3, or in cases where the patient has a rareblood type that cannot be matched in available supplies or donors (Becher, 2001).Emulsions are used in many other applications and for an overview the reader isreferred to Chappat (1994), Becher (2001) and Forster and Rybinski (1998).

This chapter is organized as follows. In Section 1.1 the background and motiva-tion are described. Based on the background and motivation the problem statementis formulated; this is subject of Section 1.2. The approach followed to achieve theconfined problem statement is presented in Section 1.3. Finally, in Section 1.4 theoutline of this thesis is described.

1.1 Background and motivation

First, some definitions and basic aspects of emulsions are described. This is subjectof Section 1.1.1. Then, in Section 1.1.2 equipment as commonly used for the produc-tion of emulsions is briefly reviewed. The current operating procedure and severallimitations of the current operation procedure are described in Section 1.1.3. Finally,in Section 1.1.4 requirements for the improvement of the operation procedure arediscussed.

1.1.1 Basic aspects of emulsions

Emulsions are formed from two immiscible liquids: one constitutes the dropletswhich are dispersed in the other liquid which is referred to as the continuous phase (Is-raelachvili, 1994). The droplets of the dispersed phase are between a few hundrednanometers and a few tens of micrometers in size. Emulsions of droplets of an or-ganic liquid (an “oil”) in an aqueous liquid are indicated by the symbol O/W andemulsions of aqueous droplets in an organic liquid as W/O4. Milk and mayonaisseare examples of O/W-emulsions and margarine is an example of a W/O-emulsion.

Emulsions made by agitation of the pure immiscible liquid phases are very un-stable and separate rapidly into the two liquid phases (the emulsion is said to bebroken). In fact, only micro-emulsions (which are not the topic of this thesis) arethermodynamically stable dispersions of oil and water, which means that they formspontaneously and are stable indefinitely. Most macro-emulsions (comprising mostproducts) require the input of considerable amounts of energy for their productionand can only be stable in a kinetic sense. A kinetic stable emulsion is an emulsionfor which the inevitable process of separation has slowed to an extent that it is notof importance during the time period in which the emulsion is handled. For mayon-naise and sauces this is typically in the order of 6 months; pharmaceutical emulsionsare stored for longer periods of time and should remain stable for a period of 1-2years (Chappat, 1994).

3Blood drawn from the body from which no constituent, such as plasma or platelets, has beenremoved.

4O/W- and W/O- emulsions are also referred to as water continuous and oil continuous emul-sions respectively.

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Surfactants and/or stabilizers could be added for the stabilization of the emul-sion. Surfactants absorb at the interface between oil and water (this is becausesurfactant molecules have both hydrophilic5 and lipophilic6 molecular groups), faci-litating the formation of emulsions by lowering the interfacial tension. They are alsoresponsible for short- and long-term stability by preventing coalescence of droplets.Figure 1.1 shows a schematic representation of both O/W- and W/O-emulsions andthe action of surfactants.

Water Oil

Oil

Water

Surfactant

W/O-emulsion O/W-emulsion

Hydrophilic group

Lipophilic group

Figure 1.1: Schematic representation of both O/W- and W/O-emulsions. The hy-drophilic group of the surfactant has an affinity for water and the lipophilic grouphas an affinity for oil.

The term stabilizer is used for macromolecules soluble in the continuous phasethat are usually not surface-active. Stabilizers provide long term stability of emul-sions mainly by increasing the viscosity of the continuous phase or by producingyield strength. The main destabilization phenomena are described subsequently.

Destabilization phenomena Emulsion droplets (the dispersed phase) are in per-petual motion in an emulsion and collide with each other frequently. After the col-lision the droplets may separate again (stable emulsion), may stick to each otherwith a thin film between them (flocculation), or may unite to a larger droplet (coa-lescence). This is illustrated in Figure 1.2.

5Having an affinity for water; readily absorbing or dissolving in water.6Having an affinity for, tending to combine with, or capable of dissolving in lipids.

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CoalescenceFlocculation

Figure 1.2: Flocculation leaves the two droplets intact but aggregated to each other.When coalescence occurs the thin liquid film between the droplets bursts and onelarger droplet is formed.

Another main destabilizing phenomenon is creaming (or sedimentation); whichis caused by the different density of the oil and water phase. Once flocculated orcoalesced, the droplets sink faster to the bottom (or rise faster to the top) thandroplets of the original size. This enhanced sedimentation leads to a concentratedemulsion in parts of the emulsion and subsequently, results in the breaking of theemulsion. This process is graphically illustrated in Figure 1.3.

Oil

Water

A B C D

Figure 1.3: When droplets flocculate (A → B) the creaming rate is increased likewhen they coalesce (B → C). This enhanced creaming shortens the time for breakingthe emulsion (C → D).

Creaming can be reduced by:

• Reducing the density difference between the dispersed and continuous phase.This may be achieved either by selection of density-matched bulk phases or byappropriate additions of weighting agents to either phase.

• Reducing the droplet size.

• Increasing the viscosity of the continuous phase. This may be achieved bythe addition of high molecular weight polymers. These act to increase theviscosity of the continuous phase to such an extent that the creaming ratebecomes negligible.

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In this and the previous section several basic aspects of emulsions were brieflyreviewed. For more detailed information the reader is referred to Friberg and Larsson(1997), Binks (1998) and Morrison and Ross (2002).

1.1.2 Process equipment

Besides oil, water, surfactants and possible additional ingredients, energy is neededfor the production of emulsions. The energy is needed to overcome the droplet’sLaplace pressure pL [Pa]. The Laplace pressure is defined as the difference betweenthe pressure inside and outside the droplet, given by (for a spherical drop)

pL =2σ

r, (1.1)

where r [m] is the initial droplet radius and σ [N m−1] is the interfacial tensionbetween oil and water. To breakup droplets into smaller ones, the droplets mustbe strongly deformed. Consequently, the stress needed to deform the droplet ishigher for a smaller droplet. Since the stress is generally transmitted by the sur-rounding liquid via agitation, higher stresses require more vigorous agitation, hencemore energy. There are many different machines to make emulsions. Commonlyencountered machines are:

• Vessels with high-speed stirrers: High-speed mixers may have a single blade,multiple blades, or intermeshing blades. The efficiency of dispersion dependson the shear rate. The generated shear rates are typically orders of magnitudelower than in homogenizers and colloid mills. Therefore, high-speed stirrersare usually used to make uniform premixes that are fed into for example ahigh-pressure homogenizer or colloid mill.

• High-pressure homogenizers: Homogenizers produce emulsions by pumping themixture at high pressure (up to 80 MPa) through a narrow valve. The gap canbe as small as 15 µm. The mixture enters the valve at high pressure but lowvelocity. The sudden increase in velocity and decrease in pressure of the liquidas it passes through the small gap creates high-shear forces and cavitation dueto vaporization.

• Colloid mills: The colloid mill consists of a stationary part, the stator and arotating part, the rotor. In the narrow gap (a gap width of 50 µm is quitecommon) between these the intensity of the hydrodynamic forces acting onthe droplets is very high, which causes breakage of the droplets.

For more information on equipment for the production of emulsions the reader isreferred to Walstra and Smulders (1998), Becher (2001) and Morrison and Ross(2002).

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The equipment often used for the production of O/W-emulsions in the food in-dustry consists of a stirred vessel in combination with a colloid mill and a circulationpipe. The vessel is equipped with a scraper stirrer: a device consisting of severalblades that rotate at a low speed (typically 0.5 s−1) at a small distance from the ves-sel wall. A cooljacket suppresses the heating of the liquid, which is primarily causedby the rotation of the colloid mill. The reservoir contains oil which is pumped intothe vessel. The outlet valve is used to empty the vessel after production. Withinthis set-up there are two main variations:

• Configuration I: In the majority of the production facilities the colloid millhas a conical shape (set-up as shown in Figure 1.4). The rotating of the rotorand the conically shape causes a circulating flow to the vessel. Hence, in thisset-up the colloid mill acts like a shearing device as well as a pump.

• Configuration II: In other production facilities the colloid mill does not havea conical shape and therefore it does not act as a pump. An extra pump ispresent to create the circulation flow to the vessel. Note that in this set-upthe shearing and pumping action are not coupled.

A schematic picture of the equipment (Configuration I) is shown in Figure 1.4.

Scraperstirrer

Stator

Rotor

Oilreservoir

Cooling jacket

Circulation pipe

Colloid mill

Vessel

Stirrer blade

Oil droplet

Outlet valve

Oil inletpump

Figure 1.4: Equipment often used for the production of O/W-emulsions in the foodindustry. The equipment comprises three main parts: a stirred vessel, a colloid milland a circulation pipe.

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In order to give some feeling for the equipment dimensions of a production facility,typical values are listed in Table 1.2.

Table 1.2: Typical dimensions of a production facility.Equipment part Unit ValueVessel volume l 500-1500

Gap width colloid mill mm 0.5Length colloid mill cm 3

Entrance rotor diameter cm 9Exit rotor diameter cm 12

Pipe length m 4

1.1.3 Current operating procedure and limitations

The process is usually operated fed-batch wise. In the food industry, first the water,the surfactant, usually egg-yolk, and the ingredients, e.g. sugar and salt are addedto the vessel. Then the stirrer and the rotor are switched on and the oil is pumpedinto the vessel. The inlet flow rate and the stirrer and rotor speed have constantvalues in time. The values of these variables, the so called control variables, areproduct dependent and their values are often established based on experience (bestpractice). How this is done is explained later. After the oil addition the process iscontinued for a certain amount of time; the length of this time period is also oftenbased on experience. After that the colloid mill is shut down and extra ingredientse.g. small onion parts, are added to the stirred vessel and mixed. The colloid millis shut down in order to prevent these ingredients from being destroyed. Typicalproduction times for O/W-emulsions in the food industry are in the order of 10-20minutes.

Previously it was mentioned that the operation procedure is often establishedbased on best practice. When developing a new product, e.g. a certain type of lowfat dressing, kitchen trials are performed first in order to produce new prototypeproducts on small scale. During these kitchen trials, it is determined how muchsugar should be added, which aromas should be used, what the concentration ofthe various ingredients should be, etc. Hence, here the product composition isestablished. At this stage 1 kg of product is typically produced per batch. Severalprototypes products are then selected and with these products trials are performedto establish how the operation procedure should be chosen to be able to produce thedesired product quality at pilot plant scale. At this stage typically 30 kg is producedper prototype product. Finally, with one or two selected prototypes, industrial trialsare performed in order to determine how the operation procedure should be chosento get the desired product quality at industrial scale.

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The here described operation procedure and the way to establish this operationprocedure have several limitations. Three key limitations are mentioned below.

a) The values of the inlet flow rate, the stirrer and rotor speed and the productiontime are always the same for a certain product. That means that every timethat e.g. a certain mayonnaise type is produced the same operation procedureis applied. This does however not imply that the quality of this mayonnaisewill be the same from batch to batch. On the contrary, due to variationsin e.g. the surfactant quality, the oil quality or in the oil temperature themayonnaise quality will fluctuate from batch to batch. This is not desirablebecause in recent years there is an increased customer demand for consistenthigh product quality (see for example Harold and Ogunnaike, 2000; Verwater-Lukszo, 1995). It might even be that the product quality specifications are notmet and then the product has to be classified as off-spec. From a cost pointof view this is clearly undesirable.

b) For some new developed products a large experimental effort is needed beforeit is possible to produce the product at industrial scale with a similar qualitycompared to the product that was produced in the kitchen. Hence, this couldlead to a large time consuming effort, implying possible high costs and the riskthat competitors might launch a similar product earlier.

c) Due to time pressure and lack of resources it is most of the time not in-vestigated how the process could be operated at its optimum. Quite oftenthe experimental effort is stopped as soon as an operation procedure has beenfound enabling the production of the product with the desired product quality.However, it might be that it is possible to produce the same or a comparableproduct quality with a different operation procedure that takes for exampleless time or energy.

In order to be able to enlarge the efficiency of the existing production processesand to shorten the time to market for new products - and therewith create anadvantage over competition - it is necessary to overcome (some of) the limitationsof the current operation procedure.

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1.1.4 Improving the operation procedure

It is expected that the previously mentioned limitations can be largely reduced by:

i) A mathematical model describing the relation between the product quality asfunction of the time and the control variables (i.e. the inlet flow rate, thestirrer and the rotor speed). The model should include the effect of equip-ment dimensions and the product composition (e.g. oil volume fraction, fluidviscosities), as they vary widely from factory to factory and from product toproduct. This is illustrated schematically in Figure 1.5. With such a model

Control variables(i.e. inlet flow rate and the stirrer and rotor

speed)

Equipment dimensions(e.g. gap diameter, vessel volume)

Product composition(e.g. oil volume fraction, viscosity)

Product quality (e.g.viscosity, flavour) asfunction of the timeMathematical

model

Figure 1.5: The mathematical model describes how the evolution of the product qua-lity is affected by the control variables, the equipment dimensions and the productcomposition.

it is for example possible to calculate the effect of an increase of the rotorspeed with 5 s−1 on the product quality as function of the time. Further, andeven more important, it can also be used to calculate how the control vari-ables should be chosen in order to produce a certain product quality in e.g.minimum time. This is discussed under ii).

ii) An off-line optimization routine is needed to calculate, based on the model,how the process could be operated to reach a certain, predefined, productquality in e.g. minimal time. This could not only be a matter of adjustinge.g. the rotor speed to a certain fixed value for the whole process. In realitythe most beneficial way could be to vary the rotor speed during the process.It might for example be the case, that the desired product quality is obtainedin minimal time by rotating vigorously at the start of the process whereas therotor speed is set to a lower value during the rest of the process. This waythe process can be operated at the desired optimum. Further, it is expectedthat the time to introduce new products will be shorter, because it is now nolonger necessary to establish the operation procedure experimentally for everynew product. Hence, with this procedure it is not only possible to calculatehow the process should be operated to produce a certain mayonnaise quality:it even provides the operation procedure for obtaining this mayonnaise in e.g.minimal time. This is illustrated in Figure 1.6.

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Equipment dimensions

Product composition

Desired product qualityin e.g. minimal time

Time

Optimal rotor speed

Optimal stirrer speed

Optimal inlet flow rate

Mathematicalmodel

Optimizationroutine

Figure 1.6: The optimization routine enables to calculate, based on the model, howthe control variables should be chosen as a function of the time in order to reach thedesired product quality in e.g. minimal time.

iii) It has already been mentioned that e.g. the surfactant and or the oil qualitycould vary from batch to batch. That might result in variations of the productquality. Because of this a feed-back controller is needed to ensure that thepredefined, terminal product quality is reached in face of unavoidable andpersistent variations in external conditions and ingredient characteristics (e.g.variations in the surfactant quality). With the controller in place it shouldbe possible to minimize the number of products that are classified as off-specand to manufacture products with consistent high product quality. This isillustrated in Figure 1.7.

Batch number

Specifications

Pro

duct

qua

lity

Off-spec

Without controller With controller

5 10 15 20 25 30 35 40

Figure 1.7: With the controller in place (in this example from Batch number 25) itshould be possible to minimize the number of products that are classified as off-specand to manufacture products with consistent high product quality.

10

As mentioned previously a model would be needed to describe the evolution ofphysical measurable variables, that determine the product quality, in time. Relatingthose physical measurable variables to the product quality is a very complex and aproduct dependent task (especially in the food and cosmetic industry) and does notfit within the framework of this thesis. However, the droplet size distribution (DSD)and the emulsion viscosity affect the product quality to a certain extent. Examplesof these are:

• Thickness of e.g. ketchup and sauces correlates with the viscosity. As discussedby Borwankar (1992), consumer perception of thickness of barbecue sauce isa combination of perception of viscosities from several different sensory at-tributes: how the sauce pours out of the bottle; perception during basting7;its cling8; and, finally its mouthfeel. The rheological9 behavior of the sauceis non-Newtonian (apparent viscosity depends on the shear rate). Thereforethe viscosities relevant for the various sensory attributes are different sincedifferent shear rates are relevant (see Figure 1.8).

0 100

Shear rate [s -1]

Tasting

BastingPouring

Visual

Figure 1.8: Shear rates operating in sensory perception of barbecue sauce (Bor-wankar, 1992).

• The consistency of skin creams preferred by customers correlates with theviscosity at lower shear rates (Forster and Rybinski, 1998).

• The droplet size of pharmaceutical emulsions must be small (the largest dropletshould always be less than 5 µm). The most important reason for this is thatthe droplets should not clog the blood vessels during their transport (Chappat,1994).

• Destabilization phenomena were described previously and it was mentionedthat one of them, namely creaming, could be reduced by reducing the dropletsize.

7To moisten (meat, for example) periodically with a liquid, such as melted butter or a sauce,especially while cooking.

8To hold fast or adhere to something, as by grasping, sticking, embracing, or entwining: clungto the rope to keep from falling; fabrics that cling to the body.

9Rheology is defined as the study of deformation and flow of matter.

11

• The color of an emulsion is affected by its DSD. Table 1.3 shows the opticalcharacteristics for the range of droplet diameters encountered in emulsions. Atheory to relate the color of emulsions to a.o. the droplet radius is presentedin McClements (2002).

Table 1.3: Optical characteristics of emulsions (Becher, 1983).

Appearance Droplet diameter[µm]

Pure white Exceeds 0.5White to grey 0.1 to 0.3Grey to translucent 0.01 to 0.14Transparent Less than 0.01

(micro-emulsions)

It must be emphasized, that although the DSD and the emulsion viscosity (eva-luated at a certain shear rate) do affect certain quality attributes (e.g. thickness orcolor) they do not solely determine the product quality. Hence, the product qualityof products with the same DSD and emulsion viscosity might be quite different.

1.2 Problem formulation

With this motivation and background in the mind, the following problem statementcan be formulated:

Investigate, based on a model, how the control variables should be chosen as afunction of the time in order to produce emulsions (for a given oil volume frac-tion) with a certain, predefined, terminal droplet size distribution and/or emulsionviscosity (evaluated at a certain shear rate) in minimal time.

Related to this problem the following sub-problems were defined:

A) Two equipment configurations were presented in Section 1.1.2. In the mostcommon configuration (Configuration I) the colloid mill acts as shearing deviceas well as a pump. It only acts like a shearing device in the Configuration II;the circulation flow is due to a pump. The aim of this sub-problem is toestablish which configuration enables the fastest production. To this end itwill be investigated:

– How the control variables of Configuration I (i.e. the rotor speed andthe inlet flow rate) should be chosen as a function of the time to reach acertain, predefined, terminal DSD, volume fraction and emulsion viscosityin minimal time.

12

– How the control variables of Configuration II (i.e. the rotor speed, theinlet flow rate and the circulation flow rate) should be chosen as a functionof the time to reach the same terminal DSD, volume fraction and emulsionviscosity in minimal time.

B) For the production of food emulsions it is desirable to produce emulsions withless oil while maintaining a comparable terminal DSD and emulsion viscosity(evaluated at a certain shear rate) as obtained originally. Normally this isestablished by the addition of additional stabilizers like e.g. starch. The aimof this sub-problem is to investigate for both configurations:

– If this can be established by adapting only the operation procedure.Hence, the addition of e.g. starch will not be considered. This is ofrelevance for the industry from a cost point of view and because of the”low carb” trend in mainly the USA.

– How the control inputs should be chosen as a function of the time toproduce such an emulsion in minimal time.

C) To establish if and how emulsions (for a given volume fraction) with a multi-modal DSD can be produced. Applications of these emulsion types are cur-rently not known to the author. However, giving insight in the possible de-velopment of emulsions with a multi-modal DSD, could direct product de-velopment in new directions. It might for example be possible to developmulti-modal DSD emulsions with a range of mouth feelings.

Given the available time the research is confined to:

• A small scale version (7 l) of the equipment, for the production of O/W-emulsions, as shown in Figure 1.4.

• A model emulsion consisting of water, oil and a surfactant.

• The off-line optimization of the operation procedure of the emulsification pro-cess.

The main reasons for these choices are: i) this equipment type is often used forthe production of O/W-emulsions, ii) using a small scale version of the equipmentoffers the possibility to perform experiments at negligible costs and allows muchmore flexibility than when using a production facility, iii) emulsion products varywidely in their composition (e.g. different oil types, different surfactants, differentingredients), however all emulsion products contain water, oil and a surfactant, andiv) it is expected that the off-line optimization of the operation procedure will alreadyyield valuable insight in how the process can be improved.

13

1.3 Approach and limitations

The confined problem statement has not been studied in the literature to the aut-hor’s knowledge. However, the defined problem belongs to the domain of dynamicoptimization (or open loop optimal control). Literature regarding methods for sol-ving dynamic optimization problems as well as regarding applications of dynamicoptimization is widely available. Good introductions to dynamic optimization aregiven by Bryson (1999) and Agrawal and Fabien (1999). Applications of dynamicoptimization are encountered in many areas (for example chemical processes, fer-mentation processes and food processes). Two examples of applications of dynamicoptimization problems are:

• The dynamic optimization of thermal processing. In its basic form, the dy-namic optimization of thermal processing problems seeks to find the heatingtemperature (as a time-dependent profile) which maximizes the final nutrientretention of a food subject to a constraint on the microbiological lethality.Many authors have studied this problem (see for example Terajima and Non-aka, 1996; Chalabi et al., 1999; Alvarez-Vazquez and Martınez, 1999; Kleis andSachs, 2000).

• The dynamic optimization of a fed-batch reactor for ethanol production. Thisdynamic optimization problem considers the optimization of a fed-batch re-actor involving the production of ethanol by Saccharomyces cerevisiae. The(free terminal time) optimal control problem is to maximize the yield of ethanolusing the feed rate as the control variable. This problem is studied by for exam-ple Chen and Hwang (1990); Luus (1993); Banga et al. (1997) and Jayaramanet al. (2001).

One of the essential steps in solving dynamic optimization problems is the de-velopment of the process model. Models are usually divided in three types:

1. White-box (or first-principles) models, are derived from well known physicaland chemical relationships. These models give a physical insight of the systemand can even be built when the system is not yet constructed.

2. Black-box (or date-driven) models do not use any structure that reflects thephysical structure of the system: black-box models give an input/output re-lation of the process. These models are useful if a physical understanding ofthe system is absent or not relevant for the purpose of the model. Black-boxmodels are identified on the basis of experimental data.

3. Knowledge about the process may be incomplete, which can result in modelsthat use both white-box and black-box modelling strategies. Models thatcombine both approaches are called grey-box (or hybrid) models.

First-principles models are usually composed of macroscopic and/or microscopic ba-lances for energy, mass and momentum, plus other relationships for kinetics, physicalproperties, etc.. Rigorously speaking, pure first-principles models are very rare, since

14

there is almost always some sort of empirical relationship (e.g. for physical proper-ties) present. For more detailed information regarding white-box, black-box andgrey-box models the reader is referred to Henson and Seborg (1997), Ljung (1999)and Sohlberg (1998).

The ultimate goal is to optimize the operation procedure of emulsification inpractice (real products, large scale equipment). Therefore a white-box model wouldbe highly desirable (extrapolation properties, relatively easy to extend to other pro-duct compositions). However, given the complexity of emulsification processes, thiswas not considered feasible and it was decided to develop a grey-box model instead.This model can be used for equipment with various dimensions. The model com-prises several fit parameters with no clear physical interpretation. The values ofthese fit parameters do depend on the specific equipment dimensions, therefore theirvalues have to be determined for each equipment dimension separately.

1.4 Outline of this thesis

The remainder of this thesis consists of seven chapters. A brief overview of thesechapters will now be presented.

Chapter 2: In this chapter basic theory regarding droplet breakage in laminar flowis briefly reviewed. The information is confined to those subjects that arerelevant for the modeling of emulsification. Information is presented aboutthe following items: (a) the breakage condition, (b) the different breakagemechanisms, (c) the breakup time, (d) the number and the sizes of the daughterdroplets and (e) breakup of droplets in concentrated emulsions.

Chapter 3: The basic theory regarding droplet breakage in laminar flow was usedfor the modeling of the fed-batch emulsification process. Chapter 3 addressesthe development of a dynamic model describing the DSD and emulsion visco-sity as a function of the time and of the control variables for the emulsificationprocess.

Chapter 4: The model consists of a coupled set of nonlinear integro-differentialequations. In order to achieve the confined problem statement, a time domainsolution is needed. It is very unlikely that analytical solutions of the modelexist. Therefore a numerical method was used to arrive at a time domainsolution. The numerical method used is described in Chapter 4.

Chapter 5: A detailed description of the equipment, used fluids and measuringinstruments is given in this chapter. Also several preliminary experiments aredescribed. The preliminary experiments were performed to establish e.g. the0.95-confidence intervals of the measured variables and the reproducibility ofthe process.

Chapter 6: The model comprises several fit parameters. The parameter estima-tion and the model validation are described in Chapter 6. The experiments

15

performed to this end are discussed. One part of these experiment is used todetermine the values of the fit parameters. The other experiments were usedfor the model validation (comparing simulations with new measurement data,independent of the data used for the parameter estimation).

Chapter 7: Several dynamic optimization problems were formulated to study theformulated sub-problems. The numerical strategy as used for the solution ofthe dynamic optimization problems is discussed in this chapter. Further, theresults of the dynamic optimization problems are described. Based on this itis discussed how the control variables should be chosen as a function of timein order to produce emulsions with a certain, predefined, terminal DSD andemulsion viscosity in minimal time.

Chapter 8: The conclusions and the recommendations for future research are givenin Chapter 8.

Appendices: The final part presents the appendices in which essential backgroundinformation is gathered.

16

Chapter 2

Theory of droplet breakup

In this chapter basic theory about droplet breakage in laminar flow is briefly re-viewed. The information is confined to those subjects that are relevant for themodeling of emulsification. Information is presented about the following items: (a)the breakage condition (Section 2.1), (b) the different breakage mechanisms (Sec-tion 2.2), (c) the breakup time (Section 2.3), (d) the number and the sizes of thedaughter droplets (Section 2.4) and (e) breakup of droplets in concentrated emul-sions (Section 2.5). Phenomena like coalescence and droplet breakage in turbulentflow are not described because it is assumed, as discussed in Chapter 3, that thesephenomena are negligible. For more information about droplet breakage and emul-sion formation the reader is referred to Stone (1994), Walstra (1993) and Walstraand Smulders (1998).

2.1 Breakage condition

The breakup of a single droplet in steady two-dimensional shear flow in the absenceof surfactant has been studied widely. All linear (laminar) two-dimensional flowscan be represented by the variable α [-]. In particular, α = 0 for simple shear flowand α = 1 for plane hyperbolic (elongational) flow (no rotation present). It has beenshown that deformation of a droplet primarily depends on the ratio of the externalstress over the Laplace pressure, expressed in a dimensionless capillary number Ω [-]given by

Ω =ηcγr

σ, (2.1)

where ηc [Pa s] is the shear viscosity of the continuous phase and γ [s−1] the shearrate. The deformation of the droplet, which may be expressed in various ways,increases with increasing Ω. If Ω is less than a critical value Ωcr [-] the initiallyspherical droplet is deformed into a stable ellipsoid. If Ω is greater than Ωcr a stabledroplet shape does not exist and the droplet will be stretched continuously until itbreaks. The critical capillary number Ωcr is a function of the flow type and theviscosity ratio λ [-] being defined as λ ≡ ηd/ηc, where ηd [Pa s] is the shear viscosity

17

of the dispersed phase. Typical curves for Ωcr are presented in Figure 2.1 (Stone,1994).

10−4

10−3

10−2

10−1

100

101

102

0

1

2

3

4

5

6

7

8

9

10

Viscosity ratio [−]

Crit

ical

cap

illar

y nu

mbe

r [−

] α=0 α=0.2α=0.6α=1

Figure 2.1: Critical capillary number for breakup of droplets in various types oftwo-dimensional laminar flow.

Important items to be noted from Figure 2.1 are:

1. At λ > 4 breakup does not occur in simple shear flow. The reason is, generallyspeaking, that the liquid in the droplet, being more viscous than that aroundit, cannot flow as fast as the shear rate tries to cause deformation. The resultis that the droplet as a whole starts to rotate without being further deformed.

2. A fairly small elongational component in the flow pattern (α > 0) has a markedeffect on Ωcr and on its dependence on the viscosity ratio λ. This means thatit becomes more easy to breakup droplets, especially at high λ.

Note that Ωcr says nothing about the droplet sizes produced upon breakup: thevalue of Ωcr only gives the maximum droplet size that can survive in a given flow inthe absence of coalescence. The maximum stable droplet diameter dcr [m] is givenby

dcr =2σΩcr

γηc. (2.2)

2.2 Breakage mechanisms

When a droplet breaks it does so by one of the following four mechanisms (Ottinoet al., 2000): (1) necking, (2) tip streaming, (3) end-pinching and (4) capillaryinstabilities. These four mechanisms are briefly discussed here.

18

Necking

In this type of breakup, the two ends of the droplet form bulbous ends and a neckdevelops between them. The neck continuously thins until it breaks, leaving behinda few much smaller droplets (called satellite droplets) between two large dropletsformed from the bulbous ends. This necking mechanism generally occurs during asustained flow where Ω is relatively close to Ωcr.

Necking mechanism

Figure 2.2: Schematic representation of the necking mechanism.

Tip streaming

In this breakup type, small droplets break off from the tips of moderately extended,pointed droplets. Tip streaming is caused by the presence of surfactants (De Bruin,1993).

Tipstreaming mechanism

Figure 2.3: Schematic representation of the tip streaming mechanism.

19

End-pinching

Relaxation of a moderately extended droplet under the influence of surface tensionforces when the shear rate is low, may lead to breakup by the end-pinching mecha-nism. For example, this type of breakup occurs if a droplet is deformed beyondits maximum steady shape with a flow at the critical capillary number. Providedthe droplet has been stretched sufficiently beyond its maximum steady shape, thenafter flow stoppage, the droplet first relaxes back towards a spherical shape, butsubsequently fragments, forming a number of smaller droplets.

End-pinching mechanism

Figure 2.4: Schematic representation of the end-pinching mechanism.

Capillary instabilities

The three breakup mechanisms previously discussed occur for moderately extendeddroplets. However when a droplet is suddenly subjected to a stress much greaterthan the critical stress for breakup (Ω >> Ωcr) the droplet is stretched affinelyand becomes a highly extended thread. The extended droplet is unstable to minordisturbances and will eventually disintegrate into a number of large droplets withsatellite droplets in between. A schematic representation is shown in Figure 2.5.

2.3 Breakup time

The breakup time tb [s] is the time that is needed to breakup an initially undeformeddroplet. Experimental results with regard to the breakup time in laminar flow arepublished in Grace (1982), Elemans et al. (1993) and Wieringa et al. (1996). It isfound that the average breakup time in simple shear flow is given by

tb = 64dηc

2σ(ηd/ηc)0.3, (2.3)

where d [m] is the initial droplet diameter. The experiments cover viscosity ratiosfrom 10−4 to 4, interfacial tensions from 1 to 25 mN m−1 and continuous phase

20

Capillary instability mechanism

Figure 2.5: Schematic representation of the capillary instability mechanism.

viscosities from 1 to 280 mPa s. In Wieringa et al. (1996) and Elemans et al. (1993) itis mentioned that no trend with Ω/Ωcr was observed in the data. This is in contrastwith the results of Grace (1982) where it is concluded that the time needed forbreakup decreases quite rapidly as the ratio Ω/Ωcr increases. A possible explanation,according to Elemans et al. (1993), is that Grace (1982) might have observed end-pinching which yields a much smaller value for tb. In Grace (1982) also results withregard to the breakup time in elongational flow are presented. The viscosity ratioas used for the experiments ranges from 10−3 to 102. It is concluded that, in theregion of overlapping viscosity ratios, the breakup time as found in simple shear flowis approximately equal to the breakup time as found in elongational flow.

2.4 Number and sizes of daughter droplets

Experimental work to establish the number of daughter droplets is presented inWieringa et al. (1996) and Grace (1982). In Wieringa et al. (1996) three differentvalues of Ω/Ωcr (2, 3 and 4) and λ (0.01, 0.31 and 1) were used for the experiments insimple shear flow. Based on this study the following relation is proposed to predictthe number of daughter droplets ν [-] for λ = 1 and Ω/Ωcr ≥ 2

ν = −140 + 80Ω

Ωcr. (2.4)

The results of Wieringa et al. (1996) show that the number of daughter droplets,for the same value of Ω/Ωcr, increases as λ increases. Results presented in Grace(1982), also in simple shear flow, cover a much wider range of Ω/Ωcr (from 1-100)and 4 different values of λ: 1.78.10−4, 1.79.10−3, 1.69.10−2 and 0.01. Their resultsalso show that the number of daughter droplets increases as Ω/Ωcr is increased.However the dependency on λ is not clear nor does it become clear that the numberof daughter droplets depends linearly on Ω/Ωcr. Experimental work to determine

21

the number of daughter droplets in elongational flow is lacking. This is also the casefor experiments to establish the sizes of the daughter droplets.

2.5 Breakup in concentrated emulsions

The results presented so far deal with the breakage of a single droplet without sur-factant in a well defined flow field. Whether these results are also valid duringthe actual emulsification process is doubtful. In reality surfactants are present, thedroplet is subject to continuously changing hydrodynamic conditions and dropletbreakage occurs in the presence of a large population of droplets of different sizes.This affects the breakage process. Wieringa et al. (1996) and Janssen and Meijer(1995) incorporated the influence of the surrounding droplets by replacing the vis-cosity of the continuous phase for the apparent emulsion viscosity throughout themodel. This idea is further tested by Jansen et al. (2001). They present an experi-mental study on the conditions for droplet breakup in concentrated emulsions undersimple shear flow. It was observed that the critical capillary number for breakupdecreased by more than an order of magnitude for the most concentrated emulsions.Moreover, droplets with viscosity ratio λ > 4, which are known not to break insingle droplet experiments, did show breakup at elevated emulsions concentrations.All these effects were explained by means of a mean field model, which assumessimply that breakup of a droplet in a concentrated emulsion is determined by theaverage emulsion viscosity rather than the continuous phase viscosity. Mathemati-cally

Ω =ηeγr

σ(2.5)

andλ =

ηd

ηe, (2.6)

where ηe [Pa s] is the emulsion viscosity. In the next chapter it is discussed howthe information described in this chapter, is used for the development of a dynamicmodel describing the DSD and emulsion viscosity as a function of the time and ofthe control variables for the fed-batch emulsification process.

22

Chapter 3

Dynamic modeling ofemulsification

In Chapter 1 it is mentioned that the DSD and the emulsion viscosity will be used asindicators for the product quality. A model capable of predicting those variables asa function of time is presented in this Chapter. Several models have been publishedin literature dealing with emulsification in a stirred vessel equipped with a turbineor propeller stirrer. Most of these models predict some kind of a mean, e.g. thed32 [m] (the volume/surface average or Sauter mean), in steady-state conditions asfunction of typically the physical properties of the fluids (interfacial tension, densityand viscosity), the volume fraction and the average power input per unit mass offluids (see for example Arai et al., 1977; Kumar et al., 1991; Wichterle, 1995; Kumaret al., 1998; Zhou and Kresta, 1997). A limited number of models predict theevolution of the DSD in time (see for example Coulaloglou and Tavlarides, 1977;Tsouris and Tavlarides, 1994; Chen et al., 1998) in a stirred vessel under turbulentflow conditions.

The emulsification process in colloid mills has not been widely studied. Wieringaet al. (1996) studied the emulsification of concentrated emulsions in a colloid millwith smooth rotor and stator surfaces under laminar flow conditions. Two modelswere compared. The first is based on a cascade of binary events, so that a largenumber of steps is needed to complete the breakup process for all droplets. Thesecond model includes the capillary breakup process. This considerably reduces thenumber of breakup events needed to obtain a certain final droplet size and therebythe time scale of the process. Comparison of experimental and calculated meandroplet sizes showed the importance of the capillary mechanism.

To the authors knowledge no model is available in the literature that describesthe DSD(t) for the system under study. Therefore a new model was developed. Itis expected (as discussed in Section 3.1) that droplet breakage will primarily occurunder laminar flow conditions. Droplet breakage under turbulent flow conditionsdiffers from breakage under laminar flow conditions. Therefore, the models descri-bing the DSD(t) in a stirred vessel under turbulent flow conditions were of limited

23

value for the model derivation. On the other hand, the results of Wieringa et al.(1996) were quite useful and parts of their work are incorporated in the model.

The outline of this chapter is as follows. First, in Section 3.1, the assumptions un-derlying the model are discussed. Then, in Section 3.2, the so-called Reactor modelis derived. It is a compartment model and for each compartment a population ba-lance equation (PBE) is derived. PBEs contain so-called breakage functions and forthe modeling of these functions information is needed about processes occurring atthe droplet level (i.e. the breakage condition and the number of daughter droplets).This is described in the so-called Droplet model and is presented in Section 3.3.Then, in Section 3.4 the Viscosity model is presented. The Viscosity model predictsthe emulsion viscosity as function of a.o. the volume fraction and the shear rate.Finally, in Section 3.5 the Flow rate model is described. This model predicts thecirculation flow rate as function of a.o. the rotor speed and the emulsion viscosity.

3.1 Assumptions

In this section the assumptions underlying the model are discussed. First, the modeof flow (i.e. laminar or turbulent flow) during the fed-batch emulsification processis estimated (Section 3.1.1). Then, in Section 3.1.2, it is estimated in which partsof the equipment droplet breakage will primarily occur. Assumptions related to thesurfactant are presented in Section 3.1.3 and in Section 3.1.4 assumptions relatedto the Reactor and the Flow rate model are presented. Finally, in Section 3.1.5, allassumptions are listed.

3.1.1 Mode of flow

The mode of flow affects the droplet breakage and the circulation flow rate. Thereforeit is necessary to establish which mode of flow is acting during the process in thevarious parts of the equipment. As illustrated in Figure 1.4 the equipment consistsof a stirred vessel in combination with a colloid mill and a circulation pipe. Thevessel is equipped with a scraper stirrer: a device consisting of several blades thatrotate at a small distance from the vessel wall. This is illustrated in Figure 3.6. Thecolloid mill consists of a stator and a rotor. The rotor and the stator surfaces of thecolloid mill used are not smooth; both contain grooves. This is illustrated in Figure3.1.

In this section the mode of flow during the production process will be estimatedin:

• The piping.

• The bulk flow in the vessel.

• The boundary layer of the impeller.

• The gap of the colloid mill.

• The grooves of the colloid mill.

24

RotorStator

bg

hg

hcm

Groove

Gap

Figure 3.1: Picture (not on scale) of the grooves in the rotor and stator surfaces ofthe colloid mill.

It is explicitly stated that the mode of flow will be estimated during the pro-duction process while the mode of flow might change during the process. At thestart of the production of e.g. mayonnaise the fluid consists of water, surfactant andseveral ingredients. The viscosity is low, typically around 3 mPa s. Due to the oiladdition the viscosity increases and this could result in a change of the mode of flowfrom i.e. turbulent to laminar. In this section the values of the Reynolds numbersare calculated as function of the emulsion viscosity for the equipment parts listedpreviously. Based on these calculations the mode of flow in these equipment partsis estimated as function of the emulsion viscosity.

A short overview of Reynolds numbers to characterize the mode of flow in thevarious parts of the equipment is given next.

• The Reynolds number for transition from laminar to turbulent flow in pipingis around 2100 (Boyle, 1986) with

Rep =vpDpρ

η, (3.1)

where Rep [-] is the Reynolds number in piping, vp [m s−1] is the mean fluidvelocity in the piping, Dp [m] is the pipe diameter, ρ [kg m−3] is the fluiddensity and η [Pa s] is the fluid viscosity. Note, that the emulsion viscosity isshear thinning (the viscosity decreases as the shear rate increases). Later it isexplained how this is taken into account.

• The Reynolds number for the bulk flow in a stirred vessel Reb [-] is given by

Reb =NstD

2stρ

η, (3.2)

25

where Nst [s−1] is the stirrer speed and Dst [m] is the stirrer diameter. TheReynolds number for transition from laminar to turbulent flow is around1000 (van ’t Riet and Tramper, 1991).

• Relatively high shear rates are expected in the boundary layers around thestirrer in the vessel. As a first approximation the scraper stirrer can be modeledas a stationary flat plate with fluid moving over it. The Reynolds number Rex

[-] for flow over a flat plate is given by (van ’t Riet and Tramper, 1991)

Rex =ρv∞x

η, (3.3)

in which v∞ [m s−1] is the free fluid velocity along the plate and x [m] isthe distance along the blade. Taking v∞ = vtip, where vtip [m s−1] is theimpeller tip speed, enables the calculation of Rex along the scraper stirrer. TheReynolds number Rex for transition from laminar to turbulent flow over a flatplate is around 3.105 (Schlichting, 1979), however the same author states thatimpeller rotation can considerably reduce the Reynolds number for transition.

• In Kataoka (1986) it is discussed how the modes of flow can be characterizedfor a system in which an uniform axial flow enters an annular space with theinner cylinder rotating and the outer at rest. In this system the modes of flowcan be characterized as a function of two independent variables: the modifiedReynolds number (the square root of the Taylor number)

Rem =πDNhcρ

η

√hc

R(3.4)

and the axial Reynolds number

Rez =2vahcρ

η, (3.5)

where D [m] is the diameter of the inner cylinder, R [m] is its radius, N [s−1]is the revolution speed, hc [m] is the gap width and va [m s−1] is the meanaxial speed in the gap of the system. In Figure 3.2 it is shown how the modeof flow depends schematically on Rem and Rez.

As mentioned previously the rotor and the stator surfaces of the colloid millused are not smooth; both contain grooves. The flow field in such a system is highlycomplex. Because of the lack of information on the actual flow field in a colloid millwith grooves it is assumed that the mode of flow in the gap of the colloid mill canbe characterized with Equation 3.4 and 3.5. Further it is assumed that the mode offlow in the grooves can be characterized with the following Reynolds number

Reg =ρvgDh

η, (3.6)

26

Rem

Rez

Turbulent flow

Laminar flow

Turbulent flowwith vortices

Laminar flowwith vortices

50

1000

2000

100

Figure 3.2: Schematic of modes of flow in an annulus with axial flow (Kataoka,1986).

where vg [m s−1] is the mean axial speed in a groove of the colloid mill and Reg [-]is the Reynolds number in a groove of the colloid mill. The hydraulic diameter Dh

[m] is calculated as

Dh =4A

P=

2bghg

bg + hg, (3.7)

where A [m2] is the area of the wetted cross section, P [m] is the wetted perimeter,bg [m] is the groove width and hg [m] is the groove depth. The Reynolds numberfor transition from laminar to turbulent flow is expected to have the same order ofmagnitude as in piping.

Estimation of Reynolds numbers

In Figure 3.3 and 3.4 the previously described Reynolds numbers are shown asfunction of the emulsion viscosity. The equipment dimensions used in the calcu-lations are listed in Table 5.1. The value of x in Equation 3.3 is set to half thescraper stirrer blade width Lst, hence x = Lst/2. The density of the fluid is taken as1000 kg m−3. Further a stirrer and rotor speed of 0.5 and 50 s−1 (maximum values)are used respectively. The circulation flow rate depends on the fluid viscosity and onthe rotor speed. At the start of the process a typical value is 14 m3 h−1; this valueis used in the calculations. Note that using the maximum value for the circulationflow rate and for the stirrer and rotor speed will give “worst-case” estimates for theemulsion viscosity at which the possible transition from turbulent to laminar flowoccurs.

27

101

102

102

103

104

105

Emulsion viscosity [mPa s]

Rey

nold

s nu

mbe

r [−

]

Reb

Rex

Rep

Reg

Figure 3.3: Reynolds numbers in the bulk flow, the boundary layer, the piping andin a groove of the colloid mill as function of the emulsion viscosity.

Based on the results shown in Figure 3.3 and 3.4 it is expected that the flow willbe laminar for emulsion viscosities larger than approximately:

• 60 mPa s in the piping (laminar flow expected as Rep is smaller than 2100).

• 45 mPa s for the bulk flow in the vessel (laminar flow expected as Reb is smallerthan 1000).

• 1 mPa s in the boundary layer of the impeller (laminar flow expected as Rex

is smaller than 3.105).

• 10 mPa s in the gap of the colloid mill (Figure 3.2 and 3.4).

• 8 mPa s in the grooves of the colloid mill (laminar flow expected as Reg issmaller than 2100).

Emulsions are shear thinning as mentioned previously. Note that this implies thatthe emulsion viscosities listed are the viscosities at the shear rates as encountered inthe different equipment parts. The emulsion viscosity at the start of the productionof an O/W-emulsion will be at least 1 mPa s (viscosity of water).

Given the previously calculated emulsion viscosities it would be expected thatduring the fed-batch production of many emulsion products the mode of flow changesfrom turbulent to laminar in the vessel, the piping and in the grooves of the colloidmill and from laminar flow with vortices to laminar flow in the gap of the colloid

28

101

102

100

101

102

103

Emulsion viscosity [mPa s]

Rey

nold

s nu

mbe

r [−

]

RemRez

Figure 3.4: Modified and axial Reynolds number as function of the emulsion visco-sity.

mill1.

When this transition occurs depends on the oil addition rate, the viscosity of thecontinuous phase and possible effects of additional ingredients and/or the surfactant(e.g. egg-yolk forms a network in the emulsions affecting the viscosity considerably).However, it is estimated that, for a typical production process, the flow will belaminar in the colloid mill during most of the time. Based on this it is expectedthat droplet breakage in the colloid mill, during the time period of turbulent flow(grooves) and laminar flow with vortices (gap) is negligible compared to dropletbreakage occurring during the time period of laminar flow. Therefore, and for thesake of simplicity, it is assumed that the flow is laminar in the colloid mill duringthe total process.

3.1.2 Breakage zones

In this section it is estimated in which parts of the equipment droplet breakage willprimarily occur. The critical droplet diameters are estimated for the various partssubsequently.

1For emulsion products with a continuous phase viscosity larger than 60 mPa s it would beexpected that the flow remains laminar during the total process in all parts of the equipment.

29

Critical droplet diameter due to breakup in the vessel

It is generally accepted that, in a stirred vessel, droplet breakup occurs predomi-nantly in a small zone outside the edge of the moving impeller (Kumar et al., 1991).The flow pattern around the impeller has been reported by van ’t Riet and Smith(1973) for a Rushton turbine and is shown in Figure 3.5. Near the front face ofthe blade the fluid approaches the blade and displays a stagnation line somewherearound half way along the blade. Thus, there is a plane hyperbolic flow on theblade. Similarly there is a boundary layer on the blade itself and a droplet presentin this layer can experience strong shearing action leading to its breakage, providedthe droplet diameter is smaller than the boundary layer thickness.

Hyperbolicflow

Boundarylayer

Blade

Rotation direction

Figure 3.5: Flow field around a rotating blade (view from above). The arrowsindicate fluid velocities relative to the impeller blade.

Kumar et al. (1998) considered the following 3 breakage mechanisms controllingthe critical droplet diameter dcr in stirred vessels equipped with Rushton turbines:

• Droplet breakage in the turbulent flow field.

• Droplet breakage in the boundary layer on the impeller blade (simple shearflow; α = 0).

• Droplet breakage in the hyperbolic flow field in front of a rotating blade (elon-gational flow; α = 1).

All 3 mechanisms were assumed to operate simultaneously, but independent ofeach other. All these mechanisms have their corresponding dcr values, denoted bydt

cr [m], dscr [m] and de

cr [m] for droplet breakup in turbulent, simple shear andelongational flow fields, respectively. The observed value of dcr near the impellerwill be the smallest of the dcr values given by the 3 mechanisms. Besides thesemechanisms, high shear rates would be expected in the gap between the scraperstirrer blade and the vessel wall (see Figure 3.6). The corresponding dcr value is

30

denoted by dgcr [m]. Relations for obtaining rough estimates of the 4 critical droplet

diameters are described subsequently.

Dst/2

L st

xx=0

Centerlinevessel

Impeller blade

Vessel wall

h st

Figure 3.6: Schematic (not on scale) of a blade of the scraper stirrer.

Droplet breakup in the turbulent flow field Various expressions used bydifferent investigators to calculate dt

cr have been reviewed by Coulaloglou and Tav-larides (1976). In general the correlation is given by

dtcr = DstC1(1 + C2φ)We−0.6, (3.8)

where the Weber number We [-] is given by

We =ρcN

2stD

3st

σ. (3.9)

The value of coefficient C1, as reported by Sprow (1967), is 0.125. Calabrese etal. (1986) and Chen and Middleman (1967) determined C1 to be around 0.85.Coulaloglou and Tavlarides (1976) have estimated coefficient C2 to be 4.47 to bestfit their experimental data for a turbine impeller. Equation 3.8 cannot be used whenthe dispersed phase is viscous or rheologically complex,

The rate with which the droplet deforms has been ignored in the derivation ofEquation 3.8. For non viscous droplets this is acceptable. However, for viscous orrheologically complex droplets the equation no longer holds. A model of breakageof droplets accounting for the effect of rheology of the dispersed phase is described

31

in Lagisetty et al. (1986). The dtcr for the model of Lagisetty et al. (1986) is given

by the following explicit equation2 (Kumar et al., 1998)

dtcr = Dst

[1 + 4φ8Red

]3/4√

1 +(1 + 4φ)9/10Re

3/2d

23/2We6/5, (3.10)

where Red [-], the droplet Reynolds number, is given by

Red =NstD

2stρc

ηd. (3.11)

Equation 3.8 reduces to the following simpler equation for low viscosity drops (ηd →0)

dtcr = 0.125Dst(1 + 4φ)1.2We−0.6. (3.12)

Note, that the form of this equation matches quite well with Equation 3.8.

Droplet breakup in the simple shear flow field For laminar flow the boundarylayer thickness δl [m], defined as the distance from the impeller surface at which thefluid velocity reaches 99 % of the free fluid velocity, is (Schlichting, 1979)

δl ≈ 5√

ηx

ρv∞. (3.13)

The maximum shear rate (for Newtonian fluids) in this boundary layer γl [s−1] isgiven as (Schlichting, 1979)

γl = 0.332v∞x

√Rex. (3.14)

Note, that the shear rate at the leading edge, x = 0, is not defined. This is a resultof the failure of the boundary layer theory for very small values of x.

Using Equation 2.2, where the continuous phase viscosity is replaced for theaverage emulsion viscosity and taking v∞ = vtip, it follows that the critical dropletdiameter ds

max is given as

dsmax =

2σΩscrx

0.332vtipηe

√Rex

, (3.15)

where Ωscr [-] is the critical capillary number in simple shear flow in the boundary

layer on the scraper blade.

Droplet breakup in the elongational flow field Kumar et al. (1998) modeledthe flow towards and around the impeller blade by a uniform flow approachinga semi-infinite plate of the same width as the impeller blade. In Lamb (1945) asolution for the flow field around a semi-infinite flat plate kept perpendicular to the

2The equation presented in Lagisetty et al. (1986) has to be solved iteratively.

32

flow direction is given. Based on this solution Kumar et al. (1998) derived that, foran impeller with diameter Dst and blade width Dst/4, the solution can be writtenas

γe

6πNst= f(q1, q2), (3.16)

where q1 and q2 are the elliptical coordinates, γe [s−1] is the shear rate and f isa function depending on the location, and not on either the rotational speed orthe diameter of the impeller. In Kumar et al. (1998) it is shown that the highestpossible accelerating rates exist in the tip region of the impeller. The flow field inthis tip region is highly non-uniform. Kumar et al. (1998) approximated the dropletbreakage, in this non-uniform flow around the blade tip, by breakage of a droplet inan average uniform stationary hyperbolic flow. Thus,

γe

6πNst= f. (3.17)

Inserting Equation 3.17 in 2.2, where the continuous phase viscosity is replaced forthe average emulsion viscosity, it follows that the critical droplet diameter de

max canbe written as

decr =

2σΩecr

6πNstηef, (3.18)

where Ωecr [-] is the critical capillary number in plane hyperbolic flow (α = 1). The

constant f was estimated experimentally by Kumar et al. (1998). Its value waschosen such that Equation 3.18 predicted exactly one data point; a value of 7.8 isreported by Kumar et al. (1998). Kumar et al. (1998) performed experiments witha vessel equipped with a Rushton turbine stirrer. As f is independent of the stirrerspeed and impeller diameter, it seems reasonable to assume that the value of f hasthe same order of magnitude for a scraper stirrer. Using f = 7.8 gives

decr =

σΩecr

73.5Nstηe. (3.19)

Droplet breakup in the gap between the stirrer blade and the vessel wallThe shear rate in the gap between the stirrer blade and the vessel wall γst,g [s−1] is(linear velocity profile) approximated as

γst,g =πNstDst

hst, (3.20)

where hst [m] is the gap width. The critical droplet diameter dgcr is now given as

(using Equation 2.2)

dgcr =

2σhstΩgcr

πNstDstηe. (3.21)

where Ωgcr [-] is the critical capillary number in simple shear flow in the gap between

the scraper blade and the vessel wall.

33

Critical droplet diameter due to breakage in the colloid mill

High shear zones would be expected in the gap and in the grooves of the colloid mill.The corresponding critical droplet diameters are denoted by dcm1

cr [m] and dcm2cr [m]

respectively. Equations for the estimation of their values are described subsequently.

Droplet breakup in the colloid mill gap The shear rate originating from therotation γcm,rot [s−1] is approximated as (linear velocity profile)

γcm,rot =πNcmDr

hcm, (3.22)

where Ncm [s−1] is the rotation speed of the rotor, hcm [m] is the gap width and Dr

[m] is the mean rotor diameter. The mean rotor diameter is calculated as

Dr =Dr,i + Dr,o

2, (3.23)

where Dr,i [m] and Dr,o [m] are the diameter of the rotor (without grooves) at theentrance and the exit respectively. It follows that the shear rate in the gap due tothe rotation (at the rotation speed of 50 s−1) is equal to 34000 s−1. It is estimatedthat the shear rate originating from the throughput in the axial direction is smallerthan that from the rotation of the rotor. Therefore, only the shear rate according toEquation 3.22 is taken into account. The critical droplet diameter due to breakagein the gap of the colloid mill, dcm1

cr , is now given as (using Equation 2.2)

dcm1cr =

2σhcmΩcm1cr

πNcmDrηe

, (3.24)

where Ωcm1cr [-] is the critical capillary number in the gap of the colloid mill.

Droplet breakup in a colloid mill groove The maximum shear rate in a grooveγg [s−1] originating from the throughput in the axial direction is approximatedas (Boyle, 1986)3

γg =32Fg

πD3h

, (3.25)

where Fg [m3 s−1] is the flow rate through a single groove; it is estimated as

Fg =Vcm2Fcm,p

ngVcm, (3.26)

where Vcm2 [m3] is the emulsion volume in the grooves of the colloid mill, Vcm [m3]is the total emulsion volume in the colloid mill, ng [-] is the total number of grooves

3Approximating the maximum shear rate in a groove with Equation 3.22, where Dr and hcm

are replaced with Dr − hg and hcm + hg respectively, gives the same order of magnitude of thecritical droplet diameters as listed in Table 3.2.

34

and Fcm,p [m3 h−1] is the circulation flow rate. The emulsion volume in the groovesof the colloid mill is estimated as

Vcm2 = ngbghglcm, (3.27)

where lcm [m] is the length of the colloid mill. The emulsion volume in the gap ofthe colloid mill, Vcm1 [m3], is estimated as

Vcm1 =Ar,g

Arπ((Rr + hcm)2 − R

2

r)lcm, (3.28)

where Ar,g [m2] is the rotor area without grooves calculated as (πDr − 12ngbg), Ar

[m2] is the total rotor area calculated as (πDr) and Rr [m] is the mean rotor radius.The critical droplet diameter due to breakage in a groove of the colloid mill dcm2

cr

is now given as (using Equation 2.2 and 3.25)

dcm2cr =

σΩcm2cr πD3

h

16Fgηe, (3.29)

where Ωcm2cr [-] is the critical capillary number in a groove of the colloid mill.

Estimation of critical droplet diameters

Equations 3.10, 3.15, 3.19, 3.21, 3.24 and 3.29 are used for the calculation of the cri-tical droplet diameter due to breakage in the turbulent flow field (dt

cr), the boundarylayer on the scraper blade (ds

cr), the elongational flow field near the scraper stirrer(de

cr), the narrow gap between the vessel wall and the scraper blade (dgcr), the gap

of the colloid mill (dcm1cr ) and in a groove of the colloid mill (dcm2

cr ) respectively.The equipment dimensions and physical property values (typical values during

the experiments) used in the calculations are listed in Table 5.1 and 3.1 respectively.Typical values of the micelle volume fraction φm [-] and the droplet radius r [m]during the experiments described in Chapter 6 are 0.08 and 2 µm respectively; thesevalues are also used for the calculation of the emulsion viscosity.

Table 3.1: Physical properties used for the calculation of the critical droplet diame-ters.

Property Symbol Unit ValueFluid density ρc kg m−3 1000

Continuous phase viscosity ηc mPa s 3Dispersed phase viscosity ηd mPa s 50

Interfacial tension σ mN m−1 8Micelle radius am nm 26

Further a stirrer and rotor speed of 0.5 and 50 s−1 (maximum values) are usedrespectively. The circulation flow rate is set to 14 m3 h−1. The value of α is notknown exactly for the flow in the gap and in the grooves of the colloid mill. However,

35

it seems likely that a small elongational component will be present (this is discussedin more detail in Chapter 6). In the calculations α, for the flow in the colloid mill,is set to 0.2. Values of ds

cr are evaluated at x = 14Lst.

The calculated critical droplet diameters are listed in Table 3.2 for various oilvolume fractions. The emulsion viscosity, corresponding to a certain oil volumefraction φ, is calculated with the Viscosity model described in Section 3.4. The shearrates in the boundary layer, the elongational flow, the gap between the scraper bladeand the vessel wall, the gap of the colloid mil and in a groove of the colloid millare calculated with equation 3.14, 3.17, 3.20, 3.22 and 3.25 respectively. The listedemulsion viscosity ηe,∞ [Pa s] is the viscosity evaluated at a shear rate of 34000 s−1

(high shear emulsion viscosity).

Table 3.2: Critical droplet diameters as function of the oil volume fraction.φ ηe,∞ dt

cr dscr(

14Lst) de

cr dgcr dcm1

cr dcm2cr

[mPa s] [µm] [µm] [µm] [µm] [µm] [µm]0.2 5 1460 no breakup 3940 no breakup 25 5300.35 10 2022 no breakup 1991 no breakup 12 2420.47 20 2491 4822 981 869 5 1110.58 50 Laminar flow 932 437 146 2 45

At a volume fraction of 0.2 and 0.35 the viscosity ratio is larger than 4 in theboundary layer on the scraper blade and in the gap between the stirrer blade andthe vessel wall. For these regions simple shear flow has been assumed. Because ofthis, no breakup would be expected for these volume fractions. In Chapter 6 severalemulsification experiments are described. For all these experiments it was found thatthe diameter of the droplets is smaller than 50 µm already after 45 s of emulsification.It took approximately 20 min to reach a steady-state for all experiments. Based onthis and given the results in Table 3.2 it is expected that droplet breakage in thevessel and in the grooves of the colloid mill will be negligible in comparison to dropletbreakage in the gap of the colloid mill. Therefore it is assumed that droplet breakageoccurs primarily in the gap of the colloid mill.

3.1.3 The surfactant

The interfacial tension σ in the model is taken to be the equilibrium value of theinterfacial tension. In the actual physical process this might not be the case. Whenthe droplets are suddenly stretched, the interface concentration of surfactant willbe reduced, resulting in a higher interfacial tension. At a later stage, this canbe compensated by diffusive transport from the water phase towards the oil-waterinterface. In Walstra and Smulders (1998) the following estimate for the adsorptiontime scale tads [s], at the interface of a small droplet of radius r in a laminar flow atshear rate γ is given as

tads ≈ 3πΓrγmc

, (3.30)

36

where mc [mol m−3] is the surfactant concentration and Γ [mol m−2] is the surfaceexcess. Plateau values for the surface excess Γ∞ are generally about 6.10−6 mol m−2

(Walstra and Smulders, 1998). Taking r = 5 µm, γ = 34000 s−1 and mc = 100 mol m−3

(typical values during the experiments) gives a value for tads of 3.3.10−6 s. Thebreakup time is typically several orders larger. Thus it would be expected that theenlargement of the interface can be compensated for by interface-bulk surfactant ex-change before breakup occurs. The excessive amount of surfactant further ensuresthat the equilibrium value of the interfacial tension can be reached during the wholeemulsification process. Further it is assumed that, due to the excessive amount ofsurfactant, coalescence will be negligible. In addition to the above it is worth men-tioning that the breakup time is predicted by an equation (Equation 2.3) which isbased on droplet experiments without surfactant. In the model it is assumed thatthis relation is also valid when surfactant is present.

3.1.4 Other assumptions

For the derivation of the Reactor model and the Flow rate model it is necessary tomake a few additional assumptions. These are discussed next.

For the derivation of the Reactor model an assumption on the type of flow inthe various equipment parts should be made. Is the flow in e.g. the piping ideallymixed, is it a plug-flow or is it merely something in between? Detailed knowledgeregarding the flow pattern is lacking and in order to limit the model complexity andto keep the optimization problems computationally tractable it is proposed to modelthe fluid in the vessel, the colloid mill and in the piping as ideally mixed. In a latterstage, when comparing the simulations with measurement data, it might of coursebe necessary to adapt this.

The geometry of the colloid mill is rather complex, which complicates the deriva-tion of the Flow rate model considerably. Therefore it is proposed to model thecolloid mill as being cylindrical and to assume that the velocity profile in the gapis not affected by the presence of grooves in the rotor and stator surfaces. Theseassumptions simplify the derivation of the Flow rate model significantly.

Finally, it is assumed that temperature differences will be negligible. Given thepresence of the cooljacket this seems a reasonable assumption.

3.1.5 List of assumptions

The assumptions as previously discussed are listed below:

• Coalescence is negligible.

• Droplet breakup occurs primarily in the gap of the colloid mill.

• The flow in the colloid mill is laminar.

• The shear rate in the gap is uniform.

• Droplet breakup is governed by laminar flow. Droplet breakup due to mecha-nical and/or droplet-droplet interactions is negligible.

37

• The shear rate that breaks up the droplets in the colloid mill is entirely causedby the rotation of the rotor.

• The interfacial tension remains at its equilibrium value during the process.

• The breakup time for droplets with surfactants is given by Equation 2.3.

• The fluid in the vessel, the colloid mill and in the piping is ideally mixed.

• The colloid mill has a cylindrical geometry.

• The velocity profile in the gap of the colloid mill is not affected by the presenceof grooves in the rotor and stator surfaces.

• Temperature differences are negligible.

3.2 Reactor model

The Reactor model is a so-called compartment model. It consists of 4 compartments:one for the vessel (the Vessel compartment), one for the gap of the colloid mill (theColloid mill gap compartment), one for the grooves of the colloid mill (the Colloidmill groove compartment) and one for the piping (the Piping compartment). Foreach compartment a Population balance equation (PBE) is derived. For a generaltreatment of PBEs the reader is referred to Ramkrishna (2000). First, the formula-tion of a general PBE is discussed, then the PBEs for the several compartments arederived.

Population balance equation

Consider the flow system shown in Figure 3.7. Fin(t) [m3 s−1] and Fo(t) [m3 s−1]are the flow rate of the in- and outlet fluid flow respectively. V (t) [m3] is the fluidvolume in the vessel at time t [s]. It is described by

dV (t)dt

= Fin(t) (3.31)

The number-based concentration of droplets in the inlet flow, in the fluid inthe vessel and in the outlet flow as function of the droplet volume v [m−3] andthe time are denoted with nin(v, t) [m−3droplet volume m−3fluid volume], n(v, t)[m−3droplet volume m−3fluid volume] and no(v, t) [m−3droplet volume m−3fluidvolume] respectively. Note that the integral∫ v2

v1

n(v, t)dv (3.32)

gives the number of droplets, with a droplet volume between v1 [m3] and v2 [m3],per fluid volume at time t in the vessel.

It is assumed that the vessel is ideally mixed and therefore the system boundaryhas been chosen as shown in Figure 3.7. Assuming further that the droplets are

38

Systemboundary

n(v,t)

no(v,t)

nin(v,t)

Fin(t)

Fo(t)

V(t)

Vessel

Stirrer

Figure 3.7: Stirred vessel with an in- and outlet flow.

subject to breakage only the macroscopic PBE for droplets with volume v [m3] isgiven by

∂[V (t)n(v, t)]∂t

= Fin(t)nin(v, t) − Fo(t)n(v, t) + h(v, t), (3.33)

where the net generation (due to breakage) of droplets with volume v per fluid volumeat a certain time is given by h(v, t) [m−3 s−1]. This equation simply states that thenumber-based concentration of droplets with volume v in the vessel is affected bythe net transport (in - and outlet flow) and by the net generation (due to breakage)of droplets with volume v. The next discussion, on the modeling of h(v, t), closelyfollows the lines of Ramkrishna (2000). The function h(v, t) is expressed as thedifference between a source term h+(v, t) and a sink term h−(v, t), both due to thebreakage process. The specific breakage rate of droplets with volume v at time t,S(v, t) [s−1], represents the fraction of droplets with volume v breaking per unit timein a certain fluid volume. This gives

h−(v, t) = V (t)S(v, t)n(v, t), (3.34)

the number of droplets with volume v lost by breakage per unit time. The functionS(v, t) is often called the breakage frequency. Note that Equation 3.34 leads toexponential behaviour in time for n(v, t). As n(v, t) becomes smaller, breakagebecomes slower. It actually takes an infinite time for n(v, t) to become zero. In orderto characterize the source term h+(v, t) the following quantities are important:

1. ν(w, t) [-]: The number of droplets formed from the breakup of a single dropletwith volume w [m3] at time t.

2. P ′(v|w, t) [m−3]: This function (the probability density function) describes the

39

size(s) of the daughter droplets. The integral∫ v2

v1

P ′(v|w, t)dv (3.35)

gives the fraction of daughter droplets, with a droplet volume between v1 andv2, that is formed from the breakage of a single droplet with volume w at timet.

Collectively, the functions S(v, t), ν(w, t) and P ′(v|w, t) are referred to as the break-age functions. The source term for droplets with volume v originating from breakupof droplets with volume w is written as

h+(v, t) = V (t)∫ +∞

v

ν(w, t)S(w, t)P ′(v|w, t)n(w, t)dw, (3.36)

which reflects the production of droplets with volume v by breakage of larger droplets.The integrand on the right-hand side of Equation 3.36 is derived as follows. The num-ber of droplets with volume w that break per unit time is given by V (t)S(w, t)n(w, t)dw.This results in the formation of V (t)ν(w, t)S(w, t)n(w, t)dw new droplets of whicha fraction P ′(v|w, t)dw represents droplets with volume v. The net generation ofdroplets with volume v is given by h(v, t) ≡ h+(v, t) − h−(v, t) the right-hand sideof which is given by Equations 3.34 and 3.36. The PBE for droplets of size v in anideally stirred vessel with an in- and outlet flow and with breakage only is now givenas

∂[V (t)n(v, t)]∂t

= Fin(t)nin(v, t) − Fo(t)n(v, t)

+V (t)∫ +∞

v

ν(w, t)S(w, t)P ′(v|w, t)n(w, t)dw

−V (t)S(v, t)n(v, t). (3.37)

3.2.1 Vessel compartment

The evolution of the DSD in time in the Vessel compartment is only affected throughthe in- and outlet fluid flows. Two inlet fluid flows are present: one from the Pipingcompartment and one due to the filling of the vessel. Further there are two outletflows: one to the Colloid mill gap compartment and one to the Colloid mill groovecompartment. This gives the following PBE

∂[Vv(t)nv(v, t)]∂t

= Fin(t)(t)nin(v, t) + Fp,v(t)np(v, t) − Fv,cm1(t)nv(v, t)

−Fv,cm2(t)nv(v, t). (3.38)

Fp,v(t) [m3 s−1] is the flow rate of the emulsion from the Piping compartment tothe Vessel compartment The flow rate from the Vessel compartment to the Colloidmill gap compartment is given by Fv,cm1(t) [m3 s−1] and to the Colloid mill groove

40

compartment by Fv,cm2(t) [m3 s−1]. The number-based concentration of dropletswith volume v in the Vessel compartment and the Piping compartment are denotedwith nv(v, t) [m−3 m−3] and np(v, t) [m−3 m−3] respectively. The emulsion volumein the Vessel compartment at time t is given by Vv(t) [m3].

3.2.2 Colloid mill gap compartment

In this compartment the DSD(t) is not only affected through in- and outlet flowsbut also through the breakage of droplets. The inlet flow is coming from the Vesselcompartment. Further, there is one outlet flow. The PBE is now written as

∂[Vcm1ncm1(v, t)]∂t

= Fv,cm1(t)nv(v, t) − Fcm1,p(t)ncm1(v, t)

+Vcm1

∫ +∞

v

ν(w, t)S(w, t)P ′(v|w, t)ncm1(w, t)dw

−Vcm1S(v, t)ncm1(v, t), (3.39)

where Vcm1 [m3] is the constant emulsion volume in the Colloid mill gap com-partment. The flow rate of the emulsion from the Colloid mill gap compartment tothe Piping compartment is described by Fcm1,p(t) [m3 s−1]. The number-based con-centration of droplets with volume v in the Colloid gap mill compartment is givenby ncm1(v, t) [m−3 m−3].

3.2.3 Colloid mill groove compartment

The DSD(t) is only affected through in- and outlet flows. The inlet flow is comingfrom the Vessel compartment. There is one outlet flow. The PBE is written as

∂[Vcm2ncm2(v, t)]∂t

= Fv,cm2(t)nv(v, t) − Fcm2,p(t)ncm2(v, t), (3.40)

where Vcm2 [m3] is the emulsion volume in the Colloid mill groove compart-ment which is constant. The flow rate of the emulsion from the Colloid mill groovecompartment to the Piping compartment is described by Fcm2,p(t) [m3 s−1]. Thenumber-based concentration of droplets with volume v in the Colloid mill groovecompartment is given by ncm2(v, t) [m−3 m−3].

3.2.4 Piping compartment

The evolution of the DSD in time in the Piping compartment is affected by two inletflows and one outlet flow only. This gives the following PBE

∂[Vpnp(v, t)]∂t

= Fcm1,p(t)ncm1(v, t) + Fcm2,p(t)ncm2(v, t)

−Fp,v(t)np(v, t), (3.41)

41

where Vp [m3] is the constant emulsion volume in the Piping compartment. Apicture of the compartments is shown in Figure 3.8.

C.M. gap compartment

C.M. groovecompartment

Vesselcompartment

Fin Fp,v

Fcm,pFv,cm1

Fv,cm2

Pipingcompart-

ment

Fcm1,p

Fcm2,p

Figure 3.8: Picture of the Vessel, the Piping, the Colloid mill gap and the Colloidmill groove compartment.

3.2.5 Simplification of the PBEs

The PBEs as derived previously can be simplified by using the following relations

Fp,v(t) = Fcm1,p(t) + Fcm2,p(t) = Fcm,p(t), (3.42)

Fcm1,p(t) =Vcm1

VcmFcm,p(t) = Fv,cm1(t), (3.43)

Fcm2,p(t) =Vcm2

VcmFcm,p(t) = Fv,cm2(t), (3.44)

where Vcm [m3] is the emulsion volume in the Colloid mill. Inserting these relationsin the previously derived PBEs and using the chain rule gives the following relationsfor the 4 compartments

42

PBE Vessel compartment

∂[nv(v, t)]∂t

=Fin(t)Vv(t)

nin(v, t) − Fin(t)Vv(t)

nv(v, t)

+Fcm,p(t)

Vv(t)[np(v, t) − nv(v, t)]. (3.45)

PBE Colloid mill gap compartment

∂[ncm1(v, t)]∂t

=Fcm,p(t)

Vcm[nv(v, t) − ncm1(v, t)] − S(v, t)ncm1(v, t)

+∫ +∞

v

ν(w, t)S(w, t)P ′(v|w, t)ncm1(w, t)dw. (3.46)

PBE Colloid mill groove compartment

∂[ncm2(v, t)]∂t

=Fcm,p(t)

Vcm[nv(v, t) − ncm2(v, t)]. (3.47)

PBE Piping compartment

∂[np(v, t)]∂t

=Fcm,p(t)

Vp

[Vcm1

Vcmncm1(v, t) +

Vcm2

Vcmncm2(v, t) − np(v, t)

].(3.48)

3.3 Droplet models

In this section the modeling of the breakage functions is considered. First, themodeling of the breakage frequency S(v, t) is discussed, then the modeling of thenumber of daughter droplets ν(w, t) is presented and finally the modeling of theprobability density function P ′(v|w, t) is discussed.

3.3.1 Modeling of S(v, t)

The specific breakage rate S(v) is modeled as

S(v) = cS

tbif Ω ≥ Ωcr

0 if Ω < Ωcr, (3.49)

where cS [-] is a fit parameter which value has to be determined experimentally. Itsimply states that the breakage frequency is equal to 0 as Ω < Ωcr and is equalto cS divided by the breakup time as Ω ≥ Ωcr. Hence, it states that the breakagefrequency increases as the droplet diameter decreases. This relation is expected tohold for droplet breakage in an uniform shear field; this is discussed in Chapter 6.

43

The critical capillary number as function of λ and α (see Figure 2.1) is describedwell with the following correlation

Ωcr = 0.1 10−f1(λ)+√

f21 (λ)−f2(λ). (3.50)

The functions f1 [-] and f2 [-] are given by

f1(λ) =e(α) + b(α) log(λ)

c(α)(3.51)

and

f2(λ) =a(α) log2(λ) + 2d(α) log(λ) + 1

c(α). (3.52)

The values of a(α) [-], b(α) [-], c(α) [-], d(α) [-] and e(α) [-] are listed in Table 3.3.

Table 3.3: Values of a, b, c, d and e.α a b c d e0.0 0.935 0.557 −2.150.10−5 −1.901.10−3 -0.7540.2 0.327 0.473 −9.177.10−3 -0.248 -1.3690.4 0.286 0.380 −1.713.10−2 -0.292 -1.9380.6 0.244 0.286 −2.509.10−2 -0.336 -2.5090.8 0.310 0.273 -0.352 -0.435 -3.5251.0 0.465 0.454 -0.526 -0.665 -5.259

The capillary number Ω and the breakup time tb are predicted with Equation2.1 and 2.3 respectively. For the calculation of these variables as well as for the cal-culation of λ, the continuous phase viscosity ηc is replaced for the emulsion viscosityηe [Pa s] to incorporate the effect of surrounding droplets on the breakage.

3.3.2 Modeling of ν(w, t)

Experimental work to establish the number of daughter droplets is scarcely availablein literature. In Wieringa et al. (1996) and Grace (1982) experimental work to esta-blish the number of daughter droplets in simple shear flow is presented (see Section2.4). All experiments were performed with single droplets without surfactant. Thenumber of daughter droplets was measured for several values of λ and Ω/Ωcr. Theexperimental conditions are listed in Table 3.4.

Wieringa et al. (1996) found that the number of daughter droplets depends li-nearly on Ω/Ωcr. Their results also show that the number of daughter droplets, forthe same value of Ω/Ωcr, increases as λ increases. A mathematical relation descri-bing the number of daughter droplets as function of Ω/Ωcr and λ was not presented.The results of Grace (1982) do not show the linear dependence with Ω/Ωcr nor anydependency on λ. These discrepancies might be due to the different experimentalconditions.

44

Table 3.4: Experimental conditions.Author λ Ω/Ωcr

Wieringa et al. (1996) 0.01, 0.31, 1 2, 3, 4Grace (1982) 1.78.10−4, 1.79.10−3 1-100

1.69.10−2, 0.01

From the foregoing it can be concluded that even for breakage in simple shearflow it is not yet clear how the number of daughter droplets depends on Ω/Ωcr andλ. The situation during practical emulsification is even more complex: the flowpattern will probably contain an elongational component (due to the presence ofgrooves in the rotor and stator surfaces), the shear rate depends on the position inthe colloid mill (due to the grooves and the conical shape of the colloid mill) andsurfactant is present. Further, it is the case that the range of λ as used duringthe experiments does not cover the range as encountered in practice. During thefed-batch emulsification process oil is added during the process. Because of this theoil volume fraction and hence the emulsion viscosity increases (the modeling of theemulsion viscosity is discussed in Section 3.4). Therefore λ (λ = ηd/ηe) decreases.The evolution of λ as function of the time during the emulsification process dependson the emulsion product (the volume fraction and the continuous phase viscositydiffer from product to product). However, it is estimated that the decrease of λmight be quite considerable (a decrease from approximately 50 to 0.5 is expectedfor mayonnaise). The experiments described in Wieringa et al. (1996) and Grace(1982) cover only a small part of this range. Hence, much more research is neededto establish the number of daughter droplets for the various types of laminar flow asfunction of Ω/Ωcr and λ. This is not the objective of this research and the following(simple) equation is proposed to predict the number of daughter droplets in laminarflow for Ω/Ωcr ≥ 2

ν = 2 + c1ν + c2ν + c3ν

(Ω

Ωcr− 2)

, (3.53)

where c1ν [-], c2ν [-] and c3ν [-] are fit parameters. It has been reported in theliterature (see for example Wieringa et al., 1996) that 2 daughter droplets withseveral satellite droplets are formed at Ω/Ωcr = 1. Therefore for 1 ≤ Ω/Ωcr < 2 thefollowing relation is proposed

ν = 2 + c1ν + c2ν

(Ω

Ωcr− 1)

. (3.54)

Note, that the number of daughter droplets is equal to 2 + c1ν for Ω/Ωcr = 1 and ifΩ/Ωcr = 2 the number of daughter droplets is equal to 2 + c1ν + c2ν . Note further,that Equation 3.53 and 3.54 do not depend on λ explicitly.

45

3.3.3 Modeling of P ′(v|w, t)

Experimental work to establish the sizes of the daughter droplets seems to be lacking.Various functional forms of P ′(v|w, t) have been proposed in the literature for anumber of process units and materials. However, results regarding the equipmentand the material under study are not reported. The typical approach for determiningP ′(v|w, t) is as follows. First, functional forms with a number of fit parameters areassumed. Then, the DSD is measured as a function of time. Finally, parameteroptimization is used for the estimation of the values of the fit parameters. A reviewof the functional forms used for the description of P ′(v|w, t) is given subsequently.

Valentas et al. (1966) considered that for a particle breaking into ν smaller par-ticles, the probability density function conforms to the normal distribution

P ′(v|w) =1

σ√

2πexp

(− (v − v)2

2σ2

), (3.55)

with mean droplet volume v [m3] equal to w/ν. The standard deviation σ [m3] isset equal to v/3, this way the probability of resulting particles with v < 0 or v > wis equal to 0.4 %. The consideration of a normal distribution is intuitive, but can besupported (as the authors claim), by the central limit theorem, since the functionrepresents a combination of a large number of independent random effects, eachone contributing very little to the total distribution. This approach was adoptedby Coulaloglou and Tavlarides (1977) and also by Ribiero et al. (1995). Anotherprobability density function is used by Chatzi et al. (1989), concerning breakageof monomer droplets in suspension polymerization reactors. They propose that thebreakage event results in a number of daughter and satellite droplets. The function isobtained by the superposition of two normal distributions, each having a maximumsize corresponding to one of the two types of resulting droplets. Austin et al. (1976),for the case of grinding processes, propose4

P ′(v|w) =ζγ

3wν

( v

w

) γ3 −2

+(1 − ζ)β

3wν

( v

w

) β3 −2

. (3.56)

The fit parameters γ [-], β [-] and ζ [-] depend on the material being grounded andon the type of equipment. A method for deriving theoretical probability densityfunctions for multiple particle breakage is presented in Hill and Ng (1996). Theresulting breakage functions are discussed in more detail later.

In this thesis three functional forms are proposed for the modeling of P ′(v|w, t).This results in 3 Droplet models comprising several fit parameters. The estimationof the values of the fit parameters is described in Chapter 6; in the same chapterthe simulations of the 3 Droplet models are compared with experimental data. Thefollowing 3 Droplet models are proposed:

1. Droplet model A where the probability density function is modeled as a normaldistribution as proposed by Valentas et al. (1966) (P ′

A(v|w)).

4Austin et al. (1976) proposed a mass-based probability density distribution. The correspondingnumber-based distribution, as derived by Hill and Ng (1995), is shown here.

46

2. Droplet model B where two super positioned normal distributions as proposedby Chatzi et al. (1989) are used for the modeling of the probability densityfunction (P ′

B(v|w)).

3. Droplet model C where the probability density function as derived by Hill andNg (1996) is used (P ′

C(v|w)).

The functional forms of P ′A(v|w), P ′

B(v|w) and P ′C(v|w) are discussed subsequently.

Note that the variable t is omitted because the proposed probability density functionsare not a function of the time.

Probability density function P ′A(v|w)

Probability density function P ′A(v|w) is given by Equation 3.55. The mean volume v

and the standard deviation σ are calculated as w/ν and (cPAv)/3 respectively. cPA

[-] is a fit parameter affecting the width of the normal distribution; it is allowed tovary between 0 and 1.

Probability density function P ′B(v|w)

Probability density function P ′B(v|w) consists of 2 super positioned normal distri-

butions. The governing equations are given by

P ′B(v|w) = P ′

B1(v|w) + P ′B2(v|w), (3.57)

where P ′B1(v|w) is the probability density function of the daughter droplets and

P ′B2(v|w) is the probability density function of the satellite droplets. P ′

B1(v|w) isgiven as

P ′B1(v|w) =

ν1

νσ1

√2π

exp

(− (v − v1)2

2σ21

), (3.58)

where ν1 [-] is the number of daughter droplets. The mean volume of the daughterdroplets and the corresponding standard deviation are denoted with v1 [m3] and σ1

[m3] respectively. P ′B2(v|w) is calculated with the same equation. The only diffe-

rence is that ν1, σ1 and v1 are replaced for ν2, σ2 and v2 respectively. The standarddeviations σ1 and σ2 are calculated as (c1PB

v1)/3 and (c2PBv2)/3 respectively. The

fit parameter c1PBaffects the width of the normal distribution corresponding to the

daughter droplets whereas c2PBdoes the same for the normal distribution corres-

ponding to the satellite droplets. Both parameters are allowed to take any valuebetween 0 and 1.

For the calculation of P ′B(v|w) the ratio between v1 and v2 and between ν1 and

ν2 must be specified. These ratios are denoted with c3PB[-] and c4PB

[-] respectively.Mathematically

c3PB=

v1

v2, (3.59)

andc4PB

=ν1

ν2. (3.60)

47

Relations for v1 and v2 are easily derived assuming that all daughter droplets havethe same size and that this is also the case for the satellite droplets. Now thefollowing relation holds (conservation of mass)

ν1v1 + ν2v2 = w. (3.61)

Inserting the relation for c3PBgives the following relations for v1 and v2

v1 =w

ν1 + ν2/c3PB

(3.62)

andv2 =

w

ν1c3PB+ ν2

. (3.63)

The effect of the fit parameters c3PBand c4PB

on P ′B(v|w, t) is illustrated in

Figure 3.9 and 3.10. The default values of the parameters are: c1PB= c2PB

= 1,c3PB

= 5, c4PB= 1 and ν1 = 5.

0 10 20 30 40 50 600

1

2

3

4

5

6x 10

14

Droplet diameter [µm]

P’ B

(v|w

) [m

−3 ]

c3PB

=1 c

3PB=5

c3PB

=10

Figure 3.9: Probability density function B for different values of c3PB.

As c3PBis set to 1 (the mean droplet volume of the satellite droplets is equal to

the mean droplet volume of the daughter droplets) a single normal distribution isformed. Note that, for this case, P ′

B(v|w) is equal to P ′A(v|w) with ν = 5 + 5 and

cPA= 1. Increasing c3PB

(the mean droplet volume of the daughter droplets becomeslarger than the mean droplet volume of the satellite droplets) results in the formationof two peaks. The distance between the two peaks increases as c3PB

increases. Theheight of the peak corresponding to the satellite droplets increases because the meandroplet volume of the satellite droplets decreases. Changing the value of c4PB

does

48

0 10 20 30 40 50 600

1

2

3

4

5

6x 10

14

Droplet diameter [µm]

P’ B

(v|w

) [m

−3 ]

c4PB

=1 c

4PB=0.5

c4PB

=0.2

Figure 3.10: Probability density function B for different values of c4PB.

not affect the ratio between the mean droplet volume of the daughter and the satellitedroplets. Decreasing the value of c4PB

causes the formation of more satellite dropletsand the curve as a whole shifts to the left.

Probability density function P ′C(v|w)

In Hill and Ng (1996) a method is described for the derivation of theoretical pro-bability density functions for multiple particle breakage. The method will not bediscussed here; only the results are given. First the equations describing P ′

C(v|w)are given. Then P ′

C(v|w) is shown graphically as function of the droplet diameterfor various values of the fit parameter cPC

[-].If the number of daughter droplets is larger than 2 the probability density

function is given by

P ′C(v|w) =

AcPC,ν

νwcPC+ν−1

[vcPC (w − v)ν−2

ν−3∏i=0

[1

i + 1

]]

+AcPC

,ν

νwcPC+ν−1

[(w − v)cPC

+ν−2

(ν − 1

AcPC,ν−1

)], (3.64)

where AcPC,p is calculated with the following recursive relationship

ν

AcPC,ν

=(ν − 2)!cPC

!(cPC

+ ν − 1)!

ν−3∏i=0

[1

i + 1

]+

(ν − 1

AcPC,ν−1

)1

cPC+ ν − 1

. (3.65)

49

If the number of daughter drops is equal to 2 the probability density function iscalculated as

P ′C(v|w) =

cPC+ 1

2wcPC+1 [vcPC + (w − v)cPC ], (3.66)

where AcPC,2 is given as

AcPC,2 = cPC

+ 1. (3.67)

Figure 3.11 illustrates the effect of the fit parameter cPCon P ′

C(v|w) with ν = 3.In Figure 3.12 this is shown for ν = 10. For cPC

= 0 the probability densityfunction represents the uniform distribution function where it is equally likely toform a daughter droplet of any size less than that of the mother droplet. As cPC

isincreased from 0, the probability density function goes through an inflection pointand then forms a depression around the droplet diameter of 45 and 40 µm for ν = 3and ν = 10 respectively. That means, there are more small and large daughterdroplets, but fewer medium-sized daughter droplets.

0 10 20 30 40 50 600

2

4

6x 10

−5

Droplet diameter [µm]

P’ C

(v|w

) [m

−3 ]

cPc

=0c

Pc=4

cPc

=8

ν=3

Figure 3.11: Probability density function C for different values of cPCwith ν = 3.

3.4 Viscosity model

There is general agreement in the literature that the shape of the viscosity curves issimilar to those of suspensions of solid particles: a shear rate independent viscosityat low shear rate (if yield stress is absent), a shear thinning region, and a high shearplateau. Although many viscosity studies of all kinds of emulsions are reported inliterature, only a minor part of them actually deals with a search for a physicalinterpretation (see for example Princen and Kiss, 1989; Pal, 1997). Recently Jansen

50

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−4

Droplet diameter [µm]

P’ C

(v|w

) [m

−3 ]

cPc

=0c

Pc=4

cPc

=8

ν=10

Figure 3.12: Probability density function C for different values of cPCwith ν = 10.

et al. (2001) proposed a new scaling parameter for the viscosity of surfactant stabi-lized emulsions. It is suggested that the attractive force between emulsion dropletsis caused by the small surfactant micelles in the continuous phase of an emulsion.The new scaling parameter is referred to as the depletion flow number Fld [-] and itis defined as the ratio between the viscous energy and the depletion energy (Asakuraand Oosawa, 1958). The viscous energy is needed to separate the droplets whereasthe depletion energy opposes this separation. The depletion flow number is given by

Fld =4πηcγr2am

kTφm, (3.68)

where k [J K−1] is the Boltzmann constant, T [K] is the temperature, φm [-] themicelle volume fraction and am [m] is the micelle radius. In Jansen et al. (2001) it isshown that plots of the relative emulsion viscosity ηr = ηe/ηc [-] versus the depletionflow number result in the overlap of data for systems with different componentsand different surfactants but with similar volume fraction φ (the ratio between theoil and the total volume) and viscosity ratio. Finally it is shown that almost allviscosity data for surfactant stabilized, monodispersed, emulsions could be accuratelypredicted using the following empirical model for φ ≤ 0.6

ηr = ηr,∞(φ, λ) +ηr,0(φ, λ) − ηr,∞(φ, λ)

1 + 0.84Fl0.8d

. (3.69)

51

The high shear relative viscosity ηr,∞(φ, λ) [-] and the low shear relative viscosityηr,0(φ, λ) [-] are described with

ηr,∞(φ, λ) = exp

[(2.5λ + 1λ + 1

)φ

1 − (φ/1.15)

], (3.70)

ηr,0(φ, λ) = exp

[(10λ + 8λ + 1

)φ0.7

]. (3.71)

It should be noted that λ is calculated as ηd/ηc. It is not clear from Jansen et al.(2001) how the depletion flow number should be calculated in case of polydispersity(a range of droplet sizes).

In this thesis it is proposed to calculate it as follows

Fld =4πηcγam

kTφm

∫∞0

np(v, t)r2dr∫∞0

np(v, t)dr=

4πηcγam(r∗)2

kTφm, (3.72)

with

r∗ =

(∫∞0

np(v, t)r2dr∫∞0

np(v, t)dr

)0.5

. (3.73)

Note, that Equation 3.72 reduces to Equation 3.68 for an emulsion consisting ofdroplets all having the same size.

3.5 Flow rate model

The circulation flow is caused by the pumping action of the colloid mill. The pum-ping action is expressed in terms of a pressure difference, which is balanced by thedissipative pressure drop due to viscous forces in all parts of the system. In Sec-tion 3.5.1 the pressure difference created by the rotation of the rotor is modeled.The modeling of the dissipative pressure drop over the colloid mill and the piping isdiscussed in Section 3.5.2 and 3.5.3 respectively.

3.5.1 Modeling of the pumping capacity of the colloid mill

The radial pressure difference Pc [Pa] created by the rotation of the rotor in acolloid mill is the result of centrifugal forces. The pressure increases going outwardfrom the inlet. The pressure difference can be approximated by assuming a differentgeometry than the conical shape of the colloid mill and by assuming that the colloidmill has smooth surfaces. This is illustrated in Figure 3.13. Imagine a cylindricalgeometry and consider the pressure difference between the rotor and stator surfaces.The diameter of the rotor at the entrance is Dr,i [m]; the diameter of the statorat the exit is called Dst,o [m]. As the value of (Dst,o − Dr,i)/Dr,i decreases, theapproximation of the cylindrical geometry becomes better.

The fluid motion in the cylindrical geometry can be calculated analytically bysolving the Navier-Stokes equations, as in Bird et al. (1959). Rewriting their solution

52

Dr,i

Dst,o

h cm

lcm

Figure 3.13: Geometry of the colloid mill (solid lines) and the cylindrical geometry(broken lines) used for the analytical pressure calculation.

gives

Pc = cPc

ρe(NcmπD2r,iDst,o)2

(D2st,o − D2

r,i)2

[2ln

(Dr,i

Dst,o

)+

12

(Dst,o

Dr,i

)2

− 12

(Dr,i

Dst,o

)2]

,

(3.74)where ρe [kg m−3] is the emulsion density. The fit parameter cPc

[-] is introducedto account for the effect of the grooves on the pumping capacity.

3.5.2 Modeling of the pressure drop over the colloid mill

The pressure drop over the colloid mill can be approximated by assuming the samecylindrical geometry as used previously. The velocity distribution for a Newtonianfluid in the annulus is given as (Boyle, 1986)

u = − 14ηcm

∆Pcm

lcm

[R2

r − r2 +ln Rr(R2

r − (Rr + hcm)2)ln(Rr/(Rr + hcm))

− R2r

], (3.75)

where u [m s−1] is the velocity in the axial direction at radius r. The length of thecolloid mill is denoted with lcm [m]. ∆Pcm/lcm [Pa m−1] is the pressure gradient andRr [m] is the rotor radius. In the gap the shear rates are very high. This impliesthat most of the time the emulsion viscosity has reached the high shear plateau.Variations of the viscosity over the gap are therefore neglected and the emulsionviscosity in the gap of the colloid mill ηcm [Pa s] is calculated as ηr,∞ηc. The volumeflow rate F [m3 s−1] is given by F =

∫ a

bu2πrdr. Substitution for u, from Equation

3.75, performing the integration and solving for ∆Pcm gives

∆Pcm =−8lcmηcmF

π(R4

r − (Rr + hcm)4 − (R2r−(Rr+hcm)2)2

ln Rr/(Rr+hcm)

) . (3.76)

53

In reality the colloid mill has a conical shape and the rotor and stator surfaces containgrooves. In order to account for this it is proposed to replace Rr and Rr +hcm withthe fit parameters c1Pcm

[m] and c2Pcm[m] respectively.

3.5.3 Modeling of the pressure drop over the piping

The pressure drop over the piping originates from friction loss along the pipe lengthand minor losses caused by the presence of 2 elbows and 1 pipe exit. The minorlosses are not necessarily small, and the total minor loss may even be larger thanthe friction loss in a pipe system. The total pressure drop over the pipe ∆Pp [Pa] isgiven as (constant pipe diameter) (Kreider, 1985)

∆Pp = −ρv2p

2

fLp

Dp+

∑fittings

Kf

, (3.77)

where Lp [m] is the straight pipe length. The modeling of the friction factor5 f [-]and of the loss coefficient Kf [-] is discussed later. This equation holds for laminarand turbulent flow as well for Newtonian and non-Newtonian fluids.

For the modeling of the friction and minor losses knowledge about the viscosityfunction ηp as a function of the shear stress τ [Pa] is needed. In principle it would bepossible to use the viscosity model given in the previous section. However, insteadof using this rather complicated model, a so-called Power Law Model is used as anapproximation, to simplify the derivations considerably. The outline of this sectionis as follows. First, the approximation of the original model as a Power Law Modelis discussed. Then, the modeling of the friction loss is presented and finally it isshown how the minor losses are modeled.

Approximation of the viscosity model

The Power Law Model is simply the equation of a straight line on a log-log plot ofshear stress versus shear rate. Since most data can be adequately represented bysuch a relation over a limited range of shear rate or shear stress (which may coverup to two orders of magnitude, for some fluids), this is the most widely used modelfor non-Newtonian viscosity. The equation is (Darby, 1988)

τ = mγn, (3.78)

where n [-] is the flow index and m [Nsnm−2] is called the consistency index. Thecorresponding apparent viscosity ηa [Pa s] for this model is

ηa = mγn−1. (3.79)

5The friction factor used in this text is known as the Darcy-Weisbach factor. Other texts,particularly British ones, may use the Fanning friction factor which has a value of 1/4 of theDarcy-Weisbach factor.

54

Note, that if n = 1, the model reduces to that of Newtonian fluid, with m = η. Theviscosity decreases with increasing shear rate if n < 1 and is thus shear thinning. Ifn > 1, the viscosity increases with increasing shear rate, and is thus shear thickening.

Typically, the two parameters n and m are determined through curve fitting. Adifferent approach is followed here. The idea is to approximate the viscosity modelof Section 3.4 through the Power Law Model over a range of shear rates. It isestimated that at a shear rate of 14 m3 h−1 the maximum shear rate in the piping isabout 620 s−1. The minimal rotor speed is 5 s−1 and during experiments, with fluidviscosities up to 100 mPa s, at that speed a flow rate of 1 m3 h−1 was measured.This corresponds to a shear rate of approximately 40 s−1. Therefore, it was decidedto approximate the original viscosity model with the Power Law Model over a shearrange from 40 to 620 s−1. This approach gives the following analytical expressionfor n

n = 1 +log ηe(620) − log ηe(40)

log (620) − log (40), (3.80)

where ηe(620) and ηe(40) are the emulsion viscosities given by Equation 3.69 evalua-ted at shear rates of 620 and 40 s−1 respectively. The value of m is now calculatedas

m = 10[log ηe(620)−(n−1) log (620)]. (3.81)

A comparison between the emulsion viscosity as predicted with Equation 3.69 andwith Equation 3.79 (with n and m calculated with Equation 3.80 and 3.81 respecti-vely) is shown in Figure 3.14 for 1 ≤ γ ≤ 1000 s−1and φ = 0.5. The other parametervalues are: ηc = 3 mPa s, am = 26 nm, T = 297 K, φm = 0.03 and r = 2 µm.

100

101

102

103

101

102

Shear rate [s−1]

Em

ulsi

on v

isco

sity

[mP

a s]

Model Jansen Power Law model

Figure 3.14: Emulsion viscosity as calculated with the model of Jansen (Equation3.69) and with the Power Law approximation (Equation 3.79).

55

Differences between the emulsion viscosity as predicted with Equation 3.69 and3.79 are less than 10 % over a range of shear rate values from 10 − 1000 s−1; this isconsidered as acceptable for the intended model use.

Friction loss

For laminar flow of Power-Law fluids in a pipe it is well known that (Chhabra andRichardson, 1999)

f =64

ReMR, (3.82)

where the generalized Reynolds number ReMR [-] is defined as

ReMR =ρv2−n

p Dnp

8n−1m(

3n+14n

)n . (3.83)

The flow is laminar up to a value of ReMR of approximately 2100 (Chhabra andRichardson, 1999). The familiar Blasius expression for the velocity profile in turbu-lent flow is given by (Chhabra and Richardson, 1999)

f = 0.316Re−14 , (3.84)

where Re = ReMR with n = 1. Based on the Blasius expression together withmodifications based on experimental results, Irvine (1988) proposed the followingBlasius-like expression for power-law fluids

f = 4(

D(n)ReMR

)1/(3n+1)

, (3.85)

where

D(n) =2n+4

77n

(4n

3n + 1

)3n2

. (3.86)

Note that this expression does reduce to the Blasius expression for n = 1 and isexplicit in the friction factor, f . Note further that this equation does not take intoaccount the effect of the relative pipe roughness ε/Dp on the friction factor. For non-smooth pipes this might results in predicted values of f that are too low. Equation3.85 was stated to predict the values of the friction factor with an average of ±8%in the range of conditions: 0.35 ≤ n ≤ 0.89 and 2000 ≤ ReMR ≤ 50000.

Minor losses

The value of Kf,elbow [-] for the elbows is calculated with the following expres-sion (Kreider, 1985)

Kf,elbow =K1

Re+ K2

[1 +

2.54.10−2

Dp

], (3.87)

56

where K1 [-] and K2 [-] are constants. Little reliable information is available on thepressure drop for the flow of non-Newtonian fluids through pipe fittings. In Chhabraand Richardson (1999) it is mentioned that some work suggests that the shear-dependence of viscosity exerts little influence on such minor losses and thereforevalues for Newtonian fluids can be used. However, other work suggests that pressuredrops are much larger than for Newtonian fluids. Overall there appears to be littledefinitive information for the minor losses in various fittings. In this work valuesof K1 and K2 for Newtonian fluid are used (K1 = 800 and K2 = 0.4) and Re isreplaced for ReMR in Equation 3.87. For the pipe exit the loss coefficient Kf,e canbe found analytically. It is given as (Chhabra and Richardson, 1999)

Kf,e =3(3n + 1)2

(2n + 1)(5n + 3), (3.88)

with the assumption that the vessel radius is much larger than the pipe radius. If nwere equal to zero, the velocity would be uniform across the pipe cross section andequation 3.88 would give Kf,e = 1. This is in line with the value for turbulent flowwhen the velocity profile is assumed to be approximately flat. For laminar flow of aNewtonian fluid the value of Kf,e is 2. It is proposed to calculate the value of Kf,exit,during the transition from laminar to turbulent flow, as Kf,exit = 1/ReMR + 1.

Total pressure drop

The total pressure drop over the piping is now given as

∆Pp = −ρv2p

2

[fLp

Dp+ 2Kf,elbow + Kf,exit

], (3.89)

where f is calculated with Equation 3.82 and 3.85 for laminar and turbulent flowrespectively. Equation 3.87 is used for the calculation of Kf,elbow and Equation 3.88for the calculation of Kf,exit. Equation 3.89 is expected to be valid for laminar aswell turbulent flow (up to values of ReMR of 50000).

3.5.4 Flow rate

Combining Equation 3.74, 3.89 and 3.76 and solving for Fcm,p [m3 h−1] determinesthe flow rate through the colloid mill

∆Pc + ∆Pcm + ∆Pp = 0. (3.90)

Note that this equation can be solved iteratively only.

57

58

Chapter 4

Numerical solution of themodel

The model consists of a coupled set of nonlinear integro-differential equations. It isvery unlikely that analytical solutions of the model exist, hence to arrive at a timedomain solution numerical methods are needed. Several numerical techniques, e.g.method of moments, method of weighted residuals, method of discretization andMonte Carlo simulation techniques have been proposed to solve PBEs (for a reviewsee Ramkrishna (2000)). Computationally, however, the method of discretizationof continuous PBEs has emerged as an attractive alternative. A large variety ofdiscretization techniques have been proposed in the literature to solve PBEs. For anoverview the reader is referred to Kumar and Ramkrishna (1996). The discretizationmethod as developed by Kumar and Ramkrishna (1996) has been tested for a largenumber of cases and the numerical predictions for all cases studied (that involvedbreakage only) agree well with the analytical solutions. This method is used for thediscretization of the model equations.

The outline of this chapter is as follows. First, in Section 4.1, the discretizationmethod for a PBE with breakage only is explained. The discussion follows closelythe lines of Kumar and Ramkrishna (1996). Then, in Section 4.2, the method isapplied to the model.

4.1 Discretization of a general PBE

Consider the PBE for a population of droplets that undergo breakage in a batchvessel with a constant volume.

∂[n(v, t)]∂t

=∫ +∞

v

ν(w, t)S(w, t)P ′(v|w, t)n(w, t)dw − S(v, t)n(v, t). (4.1)

59

Integrating Equation 4.1 over a discrete size interval, say si [m3] to si+1 [m3], gives

d[Ni(t)]dt

=∫ si+1

si

∫ +∞

v

ν(w, t)S(w, t)P ′(v|w, t)n(w, t)dwdv

−∫ si+1

si

S(v)n(v, t)dv, (4.2)

whereNi(t) =

∫ si+1

si

n(v, t)dv, (4.3)

is the number of droplets with a droplet volume between si and si+1 per fluid volume.Equations 4.2 reflect a loss of autonomy1. A closed set of equations can however beobtained by representing the right-hand side of Equation 4.2 in terms of Ni(t) [m−3].The numerical technique as presented in Kumar and Ramkrishna (1996) divides theentire size range into small sections. The size range contained between two sizes si

and si+1 is called the i th section. It is assumed that the particle population in thissection is concentrated at a representative volume vi [m3] (a grid point). When anew droplet has its volume corresponding to a representative volume, all propertiesassociated with it are naturally preserved. However, when such a situation does notexist, the droplet is assigned to the adjoining representative volumes such that twoprechosen properties of interest are exactly preserved (for example the mass andthe number of daughter droplets). Thus, the formation of a droplet of volume v involume range vi, vi+1 is represented by assigning fractions a(v, vi) [-] and b(v, vi+1)[-] to particle populations at vi and vi+1, respectively. For the conservation of twogeneral properties f1(v) and f2(v), these fractions satisfy the following equations

a(v, vi)f1(vi) + b(v, vi+1)f1(vi+1) = f1(v) (4.4)

anda(v, vi)f2(vi) + b(v, vi+1)f2(vi+1) = f2(v). (4.5)

It is clear from these equations that for preservation of two properties, the particlepopulation at vi gets net droplets assigned to it for every new droplet that is born inthe size range vi−1, vi+1. Given the previous discussion Kumar and Ramkrishna(1996) proposed to modify the birth term due to droplet breakage, given by

RB =∫ si+1

si

∫ +∞

v

ν(w, t)S(w, t)P ′(v|w, t)n(w, t)dwdv (4.6)

to

RB =∫ vi+1

vi

a(v, vi)∫ +∞

v

ν(w, t)S(w, t)P ′(v|w, t)n(w, t)dwdv

+∫ vi

vi−1

b(v, vi)∫ +∞

v

ν(w, t)S(w, t)P ′(v|w, t)n(w, t)dwdv. (4.7)

1The term “autonomy” of a set of equations is used to express the property of being closed ina set of unknowns (dependent variables).

60

Since the particle population is assumed to be concentrated at representative sizes,v′

is, the number-based concentration of droplets n(v, t) can be expressed as

n(v, t) =M∑

k=1

Nk(t)δ(v − vk), (4.8)

where k [-] is an index variable and M [-] is the number of grid points. Substitutingfor n(v, t) from Equation 4.8 in Equation 4.7 gives

RB =∑k≥i

νk(t)Sk(t)Nk(t)∫ vi+1

vi

a(v, vi)P ′(v|vk, t)dv

+∑k≥i

νk(t)Sk(t)Nk(t)∫ vi

vi−1

b(v, vi)P ′(v|vk, t)dv, (4.9)

where Sk(t) = S(vk, t) and νk(t) = ν(vk, t). In the next discussion it is explainedhow Equation 4.9 can be further simplified for the situation in which the propertiesnumber and mass are conserved. Note however that the method of Kumar andRamkrishna (1996) is applicable for two general properties of interest. The propertiesnumber and mass are conserved by

a(v, vi)ν + b(v, vi+1)ν = ν (4.10)

anda(v, vi)νvi + b(v, vi+1)νvi+1 = vν, (4.11)

respectively. Hence, if ν droplets with volume v are formed in the size rangevi, vi+1, then the droplets are assigned to vi and vi+1 in such a way that thenet number and volume of droplets as formed at vi and vi+1 are equal to ν and vνrespectively. Solving Equation 4.10 and 4.11 for a(v, vi) and b(v, vi) gives

a(v, vi) =vi+1 − v

vi+1 − vi(4.12)

andb(v, vi) =

v − vi−1

vi − vi−1. (4.13)

Substitution of these equations in Equation 4.9 gives

RB =M∑

k=i

ni,kνk(t)Sk(t)Nk(t), (4.14)

where ni,k, being the contribution to the population at the i th representative sizedue to the breakage of a droplet with volume vk, is given as

ni,k =∫ vi+1

vi

vi+1 − v

vi+1 − viP ′(v|vk, t)dv +∫ vi

vi−1

v − vi−1

vi − vi−1P ′(v|vk, t)dv. (4.15)

61

The first and second integral term are zero for i = k and i = 1, respectively. Thediscrete version of the death term in Equation 4.2 is readily obtained by the substi-tution of Equation 4.8, giving

RD =∫ si+1

si

S(v, t)n(v, t)dv = Si(t)Ni(t). (4.16)

Substitution of Equations 4.9 and 4.16 in Equation 4.2 results in the discretizedversion of the PBE for pure breakup, and is given as

d[Ni(t)]dt

=M∑

k=i

ni,kνk(t)Sk(t)Nk(t) − Si(t)Ni(t). (4.17)

The discretization method is independent of the form of the breakage function andalso of the grid choice.

4.2 Model discretization

Application of the previously described discretizaton method to the model equations3.45, 3.46, 3.47 and 3.48 gives the following discretized model equations.

Vessel compartment

d[Ni,v(t)]dt

=Fin(t)Vv(t)

Ni,in(t) − Fin(t)Vv(t)

Ni,v(t)

+Fcm,p(t)

Vv(t)[Ni,p(t) − Ni,v(t)], (4.18)

where Ni,v(t) [m−3] is the number of droplets with a droplet volume vi per emulsionvolume in the Vessel compartment at time t and Ni,in(t) [m−3] is the number ofdroplets with a volume vi per emulsion volume in the inlet flow at time t.

Colloid mill gap compartment

d[Ni,cm1(t)]dt

=Fcm,p(t)

Vcm[Ni,v(t) − Ni,cm1(t)] − Si(t)Ni,cm1(t)

+M∑

k=i

ni,kνk(t)SkNk,cm1(t), (4.19)

where Ni,cm1(t) [m−3] is the number of droplets with volume vi per emulsion volumein the Colloid mill gap compartment at time t. The function ni,k is given by Equation4.15 and the integral terms appearing in this function are evaluated analytically forP ′

A(v|w) and P ′B(v|w) as given by Equation 3.55 and 3.57 respectively. The integral

terms are evaluated numerically for P ′C(v|w) as given by Equation 3.64.

62

Colloid mill groove compartment

d[Ni,cm2(t)]dt

=Fcm,p(t)

Vcm[Ni,v(t) − Ni,cm2(t)], (4.20)

where Ni,cm2(t) [m−3] is the number of droplets with a droplet volume vi per emul-sion volume in the Colloid mill groove compartment at time t.

Piping compartment

d[Ni,p(t)]dt

=Fp,cm(t)

Vp

[Vcm1

VcmNi,cm1(t) +

Vcm2

VcmNi,cm2(t) − Ni,p(t)

], (4.21)

where Ni,p(t) [m−3] is the number of droplets with a droplet volume vi per emulsionvolume in the Piping compartment at time t.

The discretized model equations (Equation 4.18, 4.19, 4.20 and 4.21) can besolved by using standard numerical solvers available in for example MATLAB.

63

64

Chapter 5

Experimental set-up

In this chapter the experimental set-up is presented. First, the lab-scale equipmentis described (see Section 5.1). Then, in Section 5.2, the used fluids are discussed.The measuring instruments are presented in Section 5.3 and finally, in Section 5.4,the preliminary experiments are discussed.

5.1 Lab-scale equipment

A schematic picture of the lab-scale equipment is shown in Figure 5.1. It consists ofa stirred vessel (7 l) in combination with a colloid mill and a circulation pipe.

Flowmeter

Scraperstirrer

Stator

Rotor

Oilreservoir

Cooling jacket

Sample point

Circulation pipe

Colloid mill

Vessel

Outlet valve

Oil inletvalve

Figure 5.1: Schematic drawing (not on scale) of the lab-scale equipment.

65

The colloid mill consists of a conical stator, in which a rotor rotates. Groovesare present on the rotor and stator surfaces. The rotation speed of the rotor couldbe controlled between 5 and 50 s−1 with a frequency converter. A cooling jacketsuppresses the heating of the liquid, which is primarily caused by the rotation of thecolloid mill. A sample point is mounted on the circulation pipe; its pipe diameteris chosen such that droplet breakage is negligible. The flow rate could be measuredwith a flow meter that is mounted in the middle of the circulation pipe.

Note, that the inlet flow rate could not be controlled in the lab scale equipment.Further, it was found, that when the process was operated fed-batch wise, thisresulted in extensive foaming. Therefore it was decided to perform the experimentsbatch-wise. This is described in more detail in 6.3.1. The equipment dimensions arelisted in Table 5.1.

Table 5.1: Equipment dimensions.Equipment part Symbol Unit Value

Maximum working volume vessel Vv,max l 7Stirrer diameter Dst cm 30

Width of scraper blade Lst cm 6.0Height of scraper blade hst,bl cm 8.0

Gap width between scraper hst mm 0.5blade and vessel wall

Gap width colloid mill hcm mm 0.50Length colloid mill lcm cm 2.7

Entrance rotor diameter Dr,i cm 9.1Exit rotor diameter Dr,o cm 12.4

Groove width bg cm 1.0Groove depth hg cm 1.0

Number of grooves ng - 24Pipe length Lp cm 85

Pipe diameter Dp cm 4.0

5.2 Measuring instruments

5.2.1 Viscosity measurements

Viscosity measurements were obtained with a Carri-Med Rheometer using a cone-and-plate geometry. The cone-and-plate rheometer consists essentially of a plateand a cone, the apex just not touching the plate. The sample of liquid is insertedinto the narrow gap between the cone and the plate, where it is retained by capillaryforces. The cone is almost a flat plate, in this case 178 degrees. The torque resultingfrom rotating the cone at constant angular speed is recorded. The design ensuresthat all portions of the sample are subjected to the same shear rate.

66

Measurements were performed with a cone diameter of 6 cm, an angle of 2 degreesand a gap of 50 µm. Each viscosity measurement took 10 min.

Two types of measurements were performed:

• The viscosity was measured as function of the shear rate. During these ex-periments the temperature was set to a value of 24 C and the shear rate wasincreased from 5 to 300 s−1.

• The viscosity was measured as function of the temperature. During these ex-periments the shear rate was set to 300 s−1 and the temperature was increasedfrom 20 to 30 C.

5.2.2 Flow rate measurements

Flow rate measurements were performed with an electromagnetic flowmeter; theSpeedmag Autozero 3000 from Endress+Hauser. The liquid flows in a circular cross-sectional tube. A magnetic field is created across the pipe, usually by coils excitedby an alternating current. The tube itself must be made from nonmagnetic materialso that the magnetic field can penetrate the tube. The liquid moves through themagnetic field thereby generating voltages. The voltages are measured betweenelectrodes, which are located in the wall of the tube. The voltage is proportionalto the mean fluid velocity in the tube in meters per second. For more details thereader is referred to Baker (2000).

5.2.3 DSD measurements

Droplet size distributions were measured using electrozone detection (a CoulterCounter was used as measuring equipment). Electrozone detection of suspendedparticles uses the change in electrical resistance when a single particle passes througha sensing zone (see Figure 5.2). The change in resistance as a particle passes throughthe sensing zone is proportional to the volume of the particle so that the techniquereports a number distribution of volumes. Up to a million particles can be countedin a short time, providing high precision in particle-size distribution. It is a popularand well established technique for sizes ranging from 0.6 to 1200 µm. Different sen-sors are used to optimize sensitivity for different size ranges. For more details aboutelectrozone detection the reader is referred to Morrison and Ross (2002).

Measurements were performed with a measuring tube with an orifice diameterof 100 µm. With this tube droplets with a diameter from 1.9 to 62.23 µm could bemeasured. This range comprises 64 measurement channels. The maximum dropletdiameter dc [µm] of each channel is given by

dc = 62.23(21/12.5

)U

. (5.1)

U [-] is calculated as

U = 64(

X

64− 1)

, (5.2)

67

where X [-] is the channel number; its value lies between 1 and 64. Each DSDmeasurement took 3 min.

Liquid flow

Measuring tube

Electrodes

Oil droplet

Figure 5.2: Schematic of the sensing zone of an electrozone detector.

5.3 Fluids

The continuous phase consists of tap water and the surfactant NEODOL 91-8.NEODOL 91-8 is a Shell trade mark and is a mixture of polyoxyethylene alcohols.The molecular formula is H − (CH2)ns

− O − (CH2 − CH2O)ms− H, where ns

has a value of 9, 10 or 11 and ms = 8 is the average of a Poisson distribution1.The molecular weight of NEODOL 91-8 is 512 g mol−1 and the density at 40 Cis 996 kg m−3. The equilibrium interfacial tension was measured as function of theNEODOL 91-8 concentration cs [mol m−3] using the Wilhelmy plate technique.The results are shown in Figure 5.3. Each data point is the average of two repeatedexperiments.

The measurement data show that the critical micelle concentration (CMC) isapproximately 10 mol m−3; the interfacial tension is then 8.2 mN m−1.

The dispersed phase consists of sunflower oil. The oil viscosity was measured asfunction of the shear rate twice. It was found (measurements are not shown) thatthe oil viscosity does not depend on the shear rate. Hence, the oil is Newtonian andthe mean value of the oil viscosity is 53 mPa s at a temperature of 24 C. The oilviscosity was also measured twice as a function of the temperature. The shear rate

1Source:http://www.shellchemicals.com/neodol.

68

10−1

100

101

102

0

2

4

6

8

10

12

14

16

18

20

Surfactant concentration [moles m−3]

Inte

rfac

ial t

ensi

on [m

N m

−1 ]

Figure 5.3: Equilibrium interfacial tension for the O/W/NEODOL 91-8 system.

was set to a value of 300 s−1 and the temperature was increased from 20 to 30 C.The results are depicted in Figure 5.4.

The results show that the oil viscosity depends strongly on the temperature: itdecreases from 62.5 to 42.5 mPa s as the temperature increases from 20 to 30 C.

5.4 Preliminary experiments

Preliminary experiments were performed to establish:

• The 0.95-confidence intervals of the measured variables (i.e. the DSD, theviscosity and the flow rate).

• The reproducibility of the process.

• The sample stability.

The first two preliminary experiments were performed to establish the order of mag-nitude of the random part of the experimental error. This information is necessaryto determine whether changes in the measured variables are due to different expe-rimental conditions (for example a different rotor speed), or that these changes canbe explained by the random part of the experimental error.

The reason for performing the preliminary experiment to establish the samplestability is explained subsequently. During the experiments, described in Chapter 6,samples were taken to measure the evolution of the DSD and the emulsion viscosity intime. The measurement of these variables takes, for 1 sample, approximately 20 min.

69

20 21 22 23 24 25 26 27 28 29 3040

45

50

55

60

65

Temperature [oC]

Oil

visc

osity

[mP

a s]

Figure 5.4: Oil viscosity as function of the temperature.

Since the time scale of the process is much smaller, it was not possible to performthese measurements immediately after the sampling. Instead, all measurementswere performed after an experiment. The experiments described in Chapter 6 tookapproximately 1 hour and typically 17 samples were taken. The DSD was measuredfor all samples; one measurement (including cleaning) took more or less 5 min. From10 samples the viscosity was measured; one measurement (including cleaning) tookapproximately 15 min. Thus, it took approximately 4 hours to measure the DSDand viscosity of all samples. The objective of the third preliminary experiment wasto establish if the DSD and/or the viscosity had been changed during this timeinterval.

The outline of this section is as follows. First, in Section 5.4.1, the confidenceintervals of the measurement variables are estimated. Then, in Section 5.4.2, ex-periments, performed to establish the reproducibility of the process, are described.Finally, in Section 5.4.3, the preliminary experiment, to establish if it is allowed tomeasure the DSD and the viscosity of the emulsion after an experiment, is discussed.

5.4.1 Confidence intervals of the measurements

First, it is explained how confidence intervals are calculated. Then, the confidenceintervals of the flow rate, viscosity and DSD measurements are calculated.

Calculation of confidence intervals

The random part of the experimental error causes scatter and reflects the qualityof the instrument design and construction. The random error may be calculated by

70

taking a series of repeated readings resulting in the value of the standard deviation ofa limited number of samples nr [-], and sometimes called the experimental standarddeviation

s =

√√√√ 1nr − 1

nr∑j=1

(qj − q)2, (5.3)

where q is the mean of nr measurements qj . If nr ≥ 30 the standard deviation σis taken as the experimental standard deviation s (Baker, 2000). If nr < 30 thestandard deviation is taken as σ = s

√nr/

√nr − 1. The confidence intervals are

calculated asq − t

σ√nr

< µ < q + tσ√nr

, (5.4)

where µ is the true value of x. The value of t [-] follows from the Student t distributionand depends on the significance level α [-] and the number of repeated readings.Equation 5.4 is for given σ a stochastic interval that contains the fixed (thoughunknown) value µ with probability of (1 − α). Such a stochastic interval is called a(1 − α)-confidence interval for µ.

Confidence intervals for the flow rate measurements

Flow rate measurements were repeated 10 times. This was done for different fluidviscosities and for different rotor speeds. Figure 5.5 shows the ratio (in %) betweenthe confidence intervals and the mean flow rate as function of the mean flow rate for5 different fluid viscosities.

0 5 10 15−5

−4

−3

−2

−1

0

1

2

3

4

5

Flow rate [m3 h−1]

Rat

io b

etw

een

conf

. int

. and

mea

n flo

w r

ate

[%]

ηc=100 mPa.s

ηc=45 mPa.s

ηc=27 mPa.s

ηc=19 mPa.s

ηc=12 mPa.s

Figure 5.5: Ratio between the 0.95-confidence interval and the mean flow rate asfunction of the mean flow rate for different fluid viscosities.

71

The results show that the ratio between the confidence interval and the meanflow rate depends slightly on the flow rate. Its value is:

• ±1.5% for flow rates between 0 and 2.5 m3h−1 and between 12.5 and 15 m3h−1.

• ±1% for flow rates between 2.5 and 12.5 m3h−1.

The measurement data do not show a dependency on the fluid viscosity.

Confidence intervals for the viscosity measurements

Viscosity measurements were repeated 4 times for 2 samples with different high shearemulsion viscosities ηe,∞. The ratio between the confidence interval and the meanemulsion viscosity as function of the shear rate is shown in Figure 5.6.

50 100 150 200 250 300−30

−20

−10

0

10

20

30

Shear rate [s−1]

Rat

io b

etw

een

conf

. int

. and

mea

n vi

scos

ity [%

]

High shear viscosity =13.1 mPa sHigh shear viscosity =15.2 mPa s

Figure 5.6: Ratio between the 0.95-confidence interval and the mean emulsion vis-cosity as function of the shear rate for two values of ηe,∞.

These results show, for both samples, that the measurement uncertainty:

• Decreases from ± 25 to ± 3 % as the shear rate is increased from 5 to 100 s−1.

• Remains more or less constant at a value of ± 3 % as the shear rate is increasedfrom 100 to 300 s−1.

72

Confidence intervals for the DSD measurements

DSD measurements were repeated 4 times for 2 different samples. The median andthe interquartile range (IQR) are used to characterize the DSD. The median is the50th percentile of the sample and it is a measure for the location of a DSD. Themedian is, in contrast with the mean, robust to outliers. A measure of spread ofa DSD is the IQR, being the difference between the 75th and the 25th percentileof the data. The IQR is also robust to outliers. The measured median values, themean value, the experimental standard deviation, the corresponding 0.95-confidenceintervals and the ratio between the confidence interval and the mean median arelisted in Table 5.2. In Table 5.3 this is shown for the IQR.

Table 5.2: Repeated DSD measurements: median values.Sample Median Mean of s Conf. Int. Ratio between

Median Conf. Int. andMean Median

[µm] [µm] [µm] [µm] [%]1 12.62; 12.96 12.54 0.32 ± 0.59 ± 4.70

12.31; 12.262 14.79; 14.58 14.49 0.24 ± 0.45 ± 3.11

14.26; 14.32

Table 5.3: Repeated DSD measurements: IQR values.Sample IQR Mean of s Conf. Int. Ratio between

IQR Conf. Int. andMean IQR

[µm] [µm] [µm] [µm] [%]1 5.83; 5.74 5.90 0.24 ± 0.45 ± 7.63

6.26; 5.772 6.06; 6.10 6.30 0.55 ± 0.55 ± 8.69

6.31; 6.71

5.4.2 Reproducibility of the process

The reproducibility of the process is determined by performing one experiment induplicate. Although this experiment should be repeated more than twice for anaccurate calculation of the confidence intervals, it does give a good qualitative insightin the reproducibility of the process. The experiment was performed as follows:

• An emulsion with a volume fraction of 0.5 was prepared at a stirrer and rotorspeed of 0.5 and 25 s−1 respectively.

73

• Samples were taken after 2, 6, 15 and 30 min. The DSD was measured for allthese samples. The viscosity was only measured for the last 2 samples.

The measured DSDs are shown in Figure 5.7. The values of the median and theIQR at the different time points are listed in Table 5.4 for the experiment carried outin duplicate. The results show that the mean value of the median and the IQR varieswith approximately ± 0.3 and ± 0.4 µm respectively. Note, that these variations aredue to measurement error (as discussed in the previous section) and to variations inthe experimental conditions (e.g. a slightly different volume fraction).

100

101

102

0

0.01

0.02

0.03

0.04

0.05

0.06

Droplet diameter [µm]

DS

DV

N [−

] afte

r 2

min

100

101

102

0

0.01

0.02

0.03

0.04

0.05

0.06

Droplet diameter [µm]

DS

DV

N [−

] afte

r 6

min

100

101

102

0

0.01

0.02

0.03

0.04

0.05

0.06

Droplet diameter [µm]

DS

DV

N [−

] afte

r 15

min

100

101

102

0

0.01

0.02

0.03

0.04

0.05

0.06

Droplet diameter [µm]

DS

DV

N [−

] afte

r 30

min

Figure 5.7: Measured DSD of duplicate experiment at various points in time.

Table 5.4: Results duplicate experiment.Time Median Median duplicate IQR IQR duplicatemin [µm] [µm] [µm] [µm]2 18.08 18.61 12.35 14.216 15.20 14.18 9.95 9.6715 14.01 13.50 8.68 9.1230 12.38 12.66 8.05 8.30

The ratio between ηe-ηe (ηe [Pa s] is the mean emulsion viscosity) and ηe asfunction of the shear rate for 2 different samples is shown in Figure 5.8.

74

50 100 150 200 250 300−30

−20

−10

0

10

20

30

Shear rate [s−1]

100(

η e −M

ean

η e)/M

ean

η e [%]

t=15 mint=30 min

Figure 5.8: Ratio between ηe-ηe and ηe as function of the shear rate for 2 differentsamples.

These results show the same trend as the results of Figure 5.6:

• The ratio between ηe-ηe and ηe decreases as the shear rate is increased from 5to 100 s−1.

• The ratio remains more or less constant as the shear rate is increased from 100to 300 s−1.

The absolute value of the ratio between ηe-ηe and ηe is approximately ±8% as theshear rate is increased from 100 to 300 s−1. This is slightly higher than found inSection 5.4.1; this is due to small variations in the experimental conditions of theduplicate experiment (e.g. slightly different volume fraction).

5.4.3 Sample stability

The objective of this preliminary experiment was to establish if it is allowed tomeasure the DSD and the viscosity of the samples after an experiment. This wasestablished as follows:

• An emulsion, with a volume fraction of 0.5, was prepared at a stirrer and rotorspeed of 0.5 and 50 s−1 respectively.

• After 1 and after 30 min a sample was taken. The DSD and the emulsionviscosity were measured immediately after sampling.

• These measurements were repeated for both samples after 1, 2, 4 and 24 hours.

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It was found that the differences between the median values measured at the differenttime points and the mean median value were less than 0.4 µm. For the IQR thesedifferences were less than 0.5 µm. The relative differences between the emulsionviscosities, measured as function of the shear rate at the different time points, andthe corresponding mean values were less than 3 % different for shear rates between100 to 300 s−1. The differences decreased from 20 to 3 % as the shear rate wasincreased from 5 to 100 s−1. The results of Section 5.4.1 show that these differencescan be explained by the measurement uncertainty. Therefore, it was concluded thatit was allowed to perform the measurements immediately after an experiment.

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Chapter 6

Model validation andparameter estimation

Before the model can be used for the optimization of the operation procedure, thevalues of the fit parameters in the model have to be determined. Several experimentswere carried out to this end. Other experiments were performed for the modelvalidation (comparing simulations with new measurement data, independent of thedata used for the parameter estimation).

With respect to the model validation it is worthwhile to keep the following philo-sophical ideas in mind. The universality of a hypothesis can never be proven, butconfidence in its universality can be improved by induction, i.e. experience. Onthe other hand, a hypothesis can be falsified : it takes just one example to prove itwrong. Although this is true for a hypothesis in logics or mathematics it is slightlymore complicated when experimental data are used. Taking into consideration theuncertainty of the used data, falsification is no longer absolute. In other words, asthe results of more and more simulations (over different operation conditions and/orproduct compositions) agree with the experimental data, the confidence in the modeluniversality is improved. On the other hand, the model is considered falsified if thehypothesis, that differences between the model and experimental data can be ex-plained by measurement uncertainty, is rejected with a certain probability. Note,that the concept of falsification is not coupled to the intended use of the model.Hence a model that is falsified for a certain operating range can still be useful. Forexample, suppose that the simulations are in disagreement with experimental datafor experiments at rotation speeds below 5 s−1 and that the simulations are in linewith experimental data for higher rotation speeds. Suppose further that the optimaltrajectory of the rotor speed lies between 5 and 50 s−1, then the model, althoughfalsified, is still useful.

Ideally, the experiments for the parameter estimation and for the model vali-dation should be designed such that, if the model would be falsified, it would bepossible to identify the wrong model relation(s) and/or assumption(s). For exam-ple, if it turns out that the model does not predict the evolution of the DSD in time

77

correctly then it would be desirable to know what underlying model equation(s) is(are) incorrect. Is e.g. the flow rate predicted incorrectly and if so which relationin the Flow rate model is then wrong? Or is the number of daughter droplets notpredicted correctly? Although this is desirable it is laborious and even questionablewhether it is possible to discriminate between the various effects occurring. As afirst step, the experiments are designed such that the fit parameters and the qualityof the Flow rate model, the Viscosity model and of the rest of the model (Dropletand Reactor model) could be established separately. This is described in Section6.1, 6.2 and 6.3 respectively.

6.1 Flow rate model

The Flow rate model comprises 3 fit parameters: cPc, c1Pcm

and c2Pcm(see Section

3.5.1 and 3.5.2). The experiments for the estimation of the values of these parametersare described in Section 6.1.1. In Section 6.1.2 it is explained that the Flow ratemodel is incorrect and a (simple) model adaptation is proposed. The estimationof the parameters of the adapted Flow rate model is subject of Section 6.1.3. Thevalidation of the Flow rate model is described in Section 6.1.4.

6.1.1 Experiments

Experiments were performed with mixtures of water and glucose syrup at variousrotation speeds. The volume fraction of glucose syrup was varied in order to adjustthe viscosity of the mixtures. Four water/glucose syrup mixtures were prepared:mixture 1 to 4. The volume fraction of glucose syrup decreases from mixture 1 to4. The viscosity of these mixtures was measured and the measurement data areshown in Appendix A.1. The data show that the viscosity decreases slowly as theshear rate is increased from 1 to approximately 50 s−1. At higher shear rates theviscosity remains constant. The viscosity values measured at 150 s−1 and the densityof the mixtures are listed in Table 6.1. The temperature increased slightly duringthe experiments (from 20 to 23 C) and this caused a decrease of the viscosity of themixtures. Because of this the viscosity values, as listed in Table 6.1, are not singlenumbers, but ranges. In the flow rate calculations mean viscosity values were used.The mixture densities differ, because the density of glucose syrup is higher than thatof water.

Table 6.1: Experiments.Mixture Viscosity range Density

[mPa s] [kg m−3]1 44-46 12202 26-29 12063 19-20 11854 12-13 1185

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The flow rate was measured for the various mixtures as function of the rotorspeed (rotor speed was varied between 5 and 45 s−1). Measurement data are shownin Figure 6.1. Although only one data point is shown for a given rotor speed for agiven mixture, it should be mentioned that this point is a mean value based on 10flow rate measurements.

0 5 10 15 20 25 30 35 40 450

5

10

15

Rotor speed [s−1]

Flo

w r

ate

[m3 h

−1 ]

Mixture 1Mixture 2Mixture 3Mixture 4

Figure 6.1: Measured flow rate as function of the rotor speed for various mixtures.

As the rotor speed is increased from 5 s−1 the flow rate increases. This increaseof the flow rate as function of the rotor speed is higher as the mixture viscosity islower. As the flow rate exceeds a value of approximately 12 m3 h−1 the flow ratebecomes less sensitive for a further increase of the rotor speed and gradually reachesa plateau value of approximately 14 m3 h−1.

6.1.2 Adaptation of the Flow rate model

The model structure of the Flow rate model is such that it can not describe that theflow rate reaches a plateau value. Based on the model it is expected that the flowrate increases either quadratically with the rotor speed (if the pressure drop over thecolloid mill is much larger than over the piping) or linearly with the rotor speed (ifthe pressure drop over the piping is much larger than over the colloid mill and if theminor losses are much larger than the frictional loss). Based on these experimentsit is not possible to establish which model relation(s) and/or assumption(s) is (are)incorrect.

79

However, it seems likely that Equation 3.74, predicting the pumping capacityof the colloid mill, does no longer hold. This equation is based on the followingassumptions:

• Laminar flow in the colloid mill.

• Velocity profile in the gap is not affected by the presence of grooves in therotor and stator surfaces.

The first assumption is made on the basis of estimates of the modified Reynoldsnumber Rem and the axial Reynolds number Rez (see Section 3.1.1). Although itcan not be guaranteed that the flow is laminar (the modes of flow as shown in Figure3.2 are determined for smooth rotor and stator surfaces), this assumption seemsmore reliable than the second one which is merely made for the sake of simplicity.Of course, this is rather speculative and more experimental work is needed to settlethis point.

For the purpose of the optimization of the operation procedure of the emulsifi-cation process in the equipment under study it is sufficient to have a model thatpredicts the flow rate as function of the rotor speed and the fluid viscosity. Whetherthis is a white- or black-box model is not relevant. Therefore it was decided to useEquation 3.90 for the prediction of the flow rate up to the point where the predictedflow rates are higher than 14 m3h−1. From that point on the predicted flow rate isset to a value of 14 m3h−1. A limitation of this approach is that the model does notgive insight in the cause(s) of the model mismatch. An other limitation (as discussedin Chapter 1) is that the model can not be used directly for the optimization of theoperation procedure of the emulsification process in equipment with different colloidmill dimensions. To that end, it would be necessary to perform some experimentsin that equipment and to determine the parameters again.

6.1.3 Parameter estimation

The fit parameters cPc, c1Pcm

and c2Pcmare determined by casting the estimation

problem as a nonlinear optimization problem. The (least-squares) objective J [%]is defined as

J(Fcm,p(i, j), Fcm,p(i, j); cPc, c1Pcm

, c2Pcm) = 100

Nexp∑i=1

Nd∑j=1

(Fcm,p(i, j) − Fcm,p(i, j)

Fcm,p(i, j)

)2

,

(6.1)where Fcm,p(i, j) and Fcm,p(i, j) denote the measured and simulated value of theflow rate respectively, taken at the jth sampling instant during experiment i. Nexp

[-] is the number of experiments and Nd [-] is the number of samples per experiment.For the parameter estimation the measurement data of mixture 1, 2 and 3 (up tovalues of 12 m3 h) were used. The fit parameters are constrained. The minimumvalue of c1Pcm

is (Dr/2− hg) and its maximum value is Dr/2 (see Figure 6.2). Thevalue of c2Pcm

lies between Dst/2 and (Dst/2 + hg).

80

Rotor

Stator

h g

Groove

Gap

Dst /2

Dr /2

Figure 6.2: Picture (not on scale) of the grooves in the rotor and stator surfaces ofthe colloid mill.

The mean rotor diameter Dr [m] and the mean stator diameter Dst [m] arecalculated as Dr = (Dr,i +Dr,o)/2 and Dst = Dr +hcm respectively. Dr,o [m] is therotor diameter at the exit. The parameter cPc

reflects how the pumping capacity ofthe colloid mill is affected by the presence of grooves in the rotor and stator surface.It is expected that cPc

is in the range of 0.5-20.The optimal parameter values are found by solving the following nonlinear opti-

mization problem

mincPc ,c1Pcm ,c2Pcm

J(Fcm,p(i, j), Fcm,p(i, j); cPc, c1Pcm

, c2Pcm) (6.2a)

subject to:

Dr/2 − hg ≤ c1Pcm≤ Dr/2, (6.2b)

Dst/2 ≤ c2Pcm≤ Dst/2 + hg, (6.2c)

0.5 ≤ cPc≤ 20. (6.2d)

The optimization problem was solved using a standard optimization routine (lsqnon-lin.m) available in MATLAB 5.3. The optimization parameter ‘TolFun’ was set to1.10−3 and default values were used for the other parameter values of the optimiza-tion routine. Numerical perturbation was used for the gradient calculation. Thecalculations were performed with a Pentium 4 (CPU 2.00 GHz and 0.99 GB ofRAM). The initial fit parameter values were set to 5.38 and 5.43 cm for c1Pcm

andc2Pcm

respectively; cPcwas set to 5.

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The optimum parameter values were found after 10 iterations (approximately 3min). The initial value of the objective function was 158259 and the (local) optimumvalue was 864. The corresponding optimal parameter values and the 95% confidenceintervals are listed in Table 6.2.

Table 6.2: Optimal parameter values and 95% confidence intervals for the Flow ratemodel.

Parameter Unit Optimal value Conf. Int.c1Pcm

cm 5.27 ± 32.3c2Pcm

cm 5.54 ± 31.4cPc

- 2.79 ± 0.48

The 95% confidence intervals of c1Pcmand c2Pcm

are very large, because differentcombinations of those parameters values can give the same optimal value of J . Themeasured and simulated flow rates for mixture 1, 2 and 3 are shown in Figure 6.3,6.4 and 6.5.

0 5 10 15 20 25 30 35 40 450

5

10

15

Rotor speed [s−1]

Flo

w r

ate

[m3 h

−1 ]

ExpSim

Figure 6.3: Simulated and measured flow rate of mixture 1 as function of the rotorspeed.

82

0 5 10 15 20 25 30 35 40 450

5

10

15

Rotor speed [s−1]

Flo

w r

ate

[m3 h

−1 ]

ExpSim

Figure 6.4: Simulated and measured flow rate of mixture 2 as function of the rotorspeed.

0 5 10 15 20 25 30 35 40 450

5

10

15

Rotor speed [s−1]

Flo

w r

ate

[m3 h

−1 ]

ExpSim

Figure 6.5: Simulated and measured flow rate of mixture 3 as function of the rotorspeed.

83

6.1.4 Model validation

The measurement data of mixture 4 were used for the validation of the Flow ratemodel. The simulated and measured flow rate of mixture 4 as function of the rotorspeed is shown in Figure 6.6.

0 5 10 15 20 25 30 35 40 450

5

10

15

Rotor speed [s−1]

Flo

w r

ate

[m3 h

−1 ]

ExpSim

Figure 6.6: Simulated and measured flow rate of mixture 4 as function of the rotorspeed.

The relative errors, calculated as 100(Fcm,p(i, j) − Fcm,p(i, j))/Fcm,p(i, j), areshown in Figure 6.7 for the 4 mixtures as function of the measured flow rate values.

The results of the parameter estimation and of the model validation show that:

• Both the simulations and the measurements show an increase of the flow rate asthe rotor speed increases. At some point the flow rate does no longer increaseand a plateau value is reached.

• The simulations as well as the measurements show that the increase rate ofthe flow rate as function of the rotor speed decreases as the fluid viscosityincreases.

• The relative errors are between approximately -10 and 10 % for flow ratesbetween 1 and 11 m3 h−1. Within this interval the relative errors are ran-domly divided, suggesting that there is no reason to reject the model structurewithin this interval. Note, that the relative errors are slightly larger than themeasurement uncertainty (Section 5.4.1). It is unclear why this the case.

• For flow rates between 11 and 14 m3 h−1 the relative errors show a clear trendfor all mixtures: the simulated flow rates are higher than the measured flow

84

0 2.5 5 7.5 10 12.5 15−30

−20

−10

0

10

20

30

Measured flow rate [m3 h−1]

Dev

iatio

n [%

]

Mixture 1Mixture 2Mixture 3Mixture 4

Figure 6.7: Relative errors between simulated and measured flow rates as functionof the measured flow rates for the various mixtures.

rates and the relative error decreases (from 15 to 0 %) as the flow rate increases.This indicates that the model structure is no longer correct. Given the size ofthe relative errors and the model purpose this is not considered as a problem.

Overall it can be concluded that the simulated flow rates are in reasonable agree-ment with the measured values. Differences between the simulated and the measuredflow rates are between approximately -10 and 10 % for rotor speeds between 5 and45 s−1 and for (Newtonian) fluids with viscosities between 12 and 50 mPa s. It isfurther worthwhile to note that the mode of flow in the piping changes from laminarto turbulent in this range of experimental conditions. With Equation 3.1 it is esti-mated that the mode of flow changes from laminar to turbulent flow in the pipingfor rotor speeds larger than 20, 15, 10 and 7 s−1 for mixture 1, 2, 3 and 4 respecti-vely. Hence, the simulated and measured flow rates are in reasonable agreement forlaminar and turbulent flow and for the transition region.

6.2 Viscosity model

The Viscosity model does not comprise fit parameters. Emulsions of different volumefractions were prepared for the model validation. Samples, with different DSDs,were taken during the emulsification process. The experimental conditions are listedin Table 6.3. The viscosity of the continuous phase differs slightly between theexperiments. This is due to different surfactant concentrations (see Appendix A.2).Based on the surfactant concentration and the total oil surface area (which increases

85

during the process due to the breakage of droplets into smaller ones) it is estimatedthat the micelle volume fraction decreases negligible during the experiments. Thisis explained in detail in Appendix B.

Table 6.3: Experimental conditions.Exp. φ Surfactant ηc φm

concentration[-] [mol m−3water] [mPa s] [-]

P1 0.50 110.3 1.51 0.051P2 0.60 162.8 1.80 0.077

The emulsion viscosity of the samples was measured as function of the shear rateat 24 C. Simulated and measured emulsion viscosities are shown in Figure 6.8 and6.9 for experiment P1 and P2 respectively. The viscosities are plotted as functionof the shear rate for 4 different values of r∗. The value of r∗ is calculated withEquation 3.73. The model calculations are performed with the following parametervalues: k = 13.8062.10−24 J K−1, am = 26 nm (Jansen et al., 2001) and T = 297 K.

101

102

101

102

Shear rate [s−1]

Em

ulsi

on v

isco

sity

[mP

a s]

r*=2.42 µm, Simr*=2.42 µm, Expr*=2.08 µm, Simr*=2.08 µm, Exp

101

102

101

102

Shear rate [s−1]

Em

ulsi

on v

isco

sity

[mP

a s]

r*=1.98 µm, Simr*=1.98 µm, Expr*=1.88 µm, Simr*=1.88 µm, Exp

Figure 6.8: Simulated (solid lines) and measured (dotted lines) emulsion viscositiesfor experiment P1 as function of the shear rate and for various values of r∗.

86

101

102

101

102

103

Shear rate [s−1]

Em

ulsi

on v

isco

sity

[mP

a s]

r*=2.07 µm, Simr*=2.07 µm, Expr*=1.67 µm, Simr*=1.67 µm, Exp

101

102

101

102

103

Shear rate [s−1]

Em

ulsi

on v

isco

sity

[mP

a s]

r*=1.44 µm, Simr*=1.44 µm, Expr*=1.33 µm, Simr*=1.33 µm, Exp

Figure 6.9: Simulated (solid lines) and measured (dotted lines) emulsion viscositiesfor experiment P2 as function of the shear rate and for various values of r∗.

The important items to be noted from Figure 6.8 and 6.9 are:

• The simulations as well as the measurements show a decrease of the emulsionviscosity as the shear rate is increased; the decrease rate decreases as the shearrate increases. The simulated decrease rate is in line with the measurementdata.

• The measured and simulated emulsion viscosities show an increase as the vo-lume fraction is increased.

• The simulations and the measurements show that the shear thinning behaviorincreases as r∗ decreases. The order of magnitude of the simulated effect of r∗

on the emulsion viscosity agrees quite well with the measured effect for bothexperiments.

• All simulated emulsion viscosities of experiment P1 are higher than the measu-red values. This is probably due to small deviations of the values of the experi-mental conditions from their nominal values. The simulated emulsion viscosityis especially sensitive for changes in the volume fraction: using a value thatis only 1 % lower than the nominal value (listed in Table 6.3) is sufficient toeliminate the bias.

87

• For experiment P2 the simulated emulsion viscosities are lower than the measu-red values for values of r∗ of 1.44 and 1.33 µm. Probably this is due to anoverestimation of these values of r∗. The reason for this is that the correspon-ding DSDs fall partly outside the measuring range (see Figure 6.14). Settingr∗ to 1.22 and 1.13 µm respectively for those 2 samples results already in agood fit.

Overall it can be concluded that the simulated emulsion viscosities are in rea-sonable agreement with the measured emulsion viscosities. Differences between thesimulated and the measured emulsion viscosity values of experiment P1 and P2 arebetween -25 and 25 % for shear rates between 5 and 100 s−1 and between -15 and10 % for shear rates between 100 and 300 s−1. These deviations are due to measu-rement uncertainty (see Chapter 5.4.1) and to the reasons as discussed previously.The results further suggest that it is justified to calculate the depletion flow numberwith Equation 3.72 in case of polydispersity.

6.3 Reactor model and Droplet models

The Droplet models contain several fit parameters. The Reactor model does notcontain fit parameters. Droplet model A and B as well as Droplet model C comprisethe following 5 fit parameters:

• The parameters c1ν , c2ν and c3ν appear in Equation 3.53 and 3.54; with theseequations the number of daughter droplets is calculated.

• Equation 3.49 comprises the parameter cS ; this equation is used for the calcu-lation of the breakage frequency.

• Equation 3.50, used for the calculation of the critical capillary number, com-prises the parameter α.

Besides these fit parameters, Droplet model A, B and C contain the followingmodel specific parameters for the calculation of P ′

A(v|w), P ′B(v|w) and P ′

C(v|w)respectively:

1. Droplet model A comprises the parameter cPA(Equation 3.55).

2. Droplet model B comprises the parameters c1PB, c2PB

, c3PBand c4PB

(Equa-tion 3.57).

3. Droplet model C comprises the parameter cPc(Equation 3.64).

Experiments for the estimation of the values of these parameters are describedin Section 6.3.1; the parameter estimation is discussed in Section 6.3.2. In Section6.3.3 it is explained that the simulated DSDs are not totally in agreement withthe measured DSDs and an adaptation of the Droplet models is proposed. Theestimation of the fit parameters of these adapted models is subject of Section 6.3.4.The validation of one of these models is described in Section 6.3.5.

88

6.3.1 Experiments

Four experiments were performed for the parameter estimation; the experimentalconditions are listed in Table 6.4. Emulsions with different volume fractions wereprepared at various rotor speeds. The rotation speed was kept constant during theprocess. The experiments were performed batch-wise and the process was operatedduring 1 hour. First, the vessel was filled completely with water, surfactant and oil.Then, the rotor was switched on and samples were taken at different points in time.The temperature of the samples was measured immediately after each sample wastaken. The DSD and the viscosity were measured after the experiments.

Table 6.4: Experimental conditions.Experiment φ Ncm ηc ηd

[-] [s−1] [mPa s] [mPa s]P1 0.50 50 1.7-1.4 66.0-46.0P2 0.60 50 1.9-1.5 60.0-36.0P3 0.49 25 1.7 64.0P4 0.51 35 1.7 64.0

A fifth experiment, experiment P5, is used for the model validation. The experi-ment is performed with a volume fraction of 0.51 and a rotor speed that varied as afunction of time. During the first 30 min the rotor speed is set to 25 s−1 after whichthe rotor speed is increased to 35 s−1 for half an hour. The rotor speed is then setto 50 s−1 during 10 min. After this period the rotor speed is decreased to 15 s−1 forhalf an hour after which it is increased again to 45 s−1.

Despite the cooling, the temperature increased from 18 to 28 C during expe-riment P1, from 21 to 34 C during experiment P2 and from 17 to 25 C duringexperiment P5. The temperature remained constant at 19 C during experimentP3 and P4. Due to the temperature increase the viscosity of the dispersed phase,the continuous phase and of the emulsion decreased during experiment P1 and P2.The minimum and maximum values of the dispersed and continuous phase viscosityare listed in Table 6.4 for P1-P4. The continuous and the dispersed phase viscosityof experiment P5, at 21.1 C (mean value), are 1.6 and 61.0 mPa s respectively. Inprinciple the temperature increase could also have affected the value of the interfa-cial tension. The temperature effect on the interfacial tension was not measured forthe O/W/NEODOL 91-8 system. However, the surface tension of water againstair decreases with 2 % if the temperature is increased from 20 to 30 C (Janssenand Warmoeskerken, 1987). It is therefore expected that the temperature increasedid not affect the interfacial tension.

6.3.2 Parameter estimation

The values of the fit parameters of the Droplet models are determined by castingthe estimation problem as a nonlinear optimization problem. The (least-squares)

89

objective function J is defined as

J(N∗k,p(i, j), N

∗k,p(i, j); θ) =

Nexp∑i=1

Nd∑j=1

M∑k=1

(N∗k,p(i, j) − N∗

k,p(i, j))2, (6.3)

where N∗k,p(i, j) and N∗

k,p(i, j) denote the measured and simulated value of the nor-malized (volume-based) DSD in the piping at vk, respectively, taken at the jth samp-ling instant during experiment i. θ represents the unknown parameter vector. Theoptimal parameter values are found by solving the following nonlinear optimizationproblem

minθ

J(N∗k,p(i, j), N

∗k,p(i, j); θ) (6.4a)

subject to:

θl ≤θ ≤ θu, (6.4b)

where θl represents the vector with minimum parameter values and θu represents thevector with the maximum parameter values. For solving this optimization problemthe same method as discussed in Section 6.1.3 was used. The model equations areintegrated 4 times, for both the gradient calculation and for the calculation of theobjective function. The initial conditions of the differential equations were set to theDSDs measured after 15 s for the 4 different experiments. Mean temperature values,obtained during the experiments, and corresponding continuous and dispersed phaseviscosities were used (see Table 6.4). Other parameter values used are listed in Table5.1. Emulsion viscosity values and flow rates were calculated with the Viscosity andFlow rate model respectively. The parameter optimizations were performed for all3 models and the optimizations were started from 3 different initial guesses for eachDroplet model. The optimization results are listed in Table 6.5.

Table 6.5: Results parameter optimization for Droplet model A, B, and C.Droplet No. of fit Initial No. of Calculation J0 J∗

model parameters guess iterations time[min]

A 6 A1 10 67 31360 8750A2 9 64 32206 10645A3 6 39 7656 7656

B 9 B1 11 103 17674 7135B2 11 112 20456 12538B3 8 73 12364 7254

C 6 C1 6 23 7075 7075C2 9 41 37781 14071C3 10 39 13476 7563

Depending on the initial guess the solver converges to different (local) optimumvalues. Note that the optimal parameter vector of A3 and C1 is the same as the

90

initial guess. The number of iterations varies between 6 and 11. Correspondingcalculation times are between 23 and 112 min. The lowest values of the optimalobjective function J∗ are found when the parameter optimizations are started fromA3, B1 and C1 for Droplet model A, B and C respectively. The correspondingvalues of J∗ are: 7656, 7135 and 7075 for Droplet model A, B and C respectively.This suggests that the simulated DSDs with Droplet model C correspond slightlybetter with the measured DSDs than when simulated with Droplet model A or B.However, the values of J∗ are local optimum values. Therefore it is possible that theglobal optimum of for example Droplet model B is lower than the global optimumof Droplet model A or C. The optimal parameter values θ∗ and the corresponding95 % confidence intervals are listed in Table 6.6 for A3, B1 and C1.

Table 6.6: Optimal parameter values and 95% confidence intervals for Droplet modelA, B and C.

θ θl θu A3, θ∗ and B1, θ∗ and C1, θ∗ andConf. Int. Conf. Int. Conf. Int.

α 0 1 0.2 ± 5.10−3 0.21 ± 1.3.10−3 0.2 ± 3.7.10−3

c1ν 0 100 20 ± 1.443 4.86 ± 0.92 10 ± 2.69c2ν 0 100 0 ± 2.114 0 ± 1.82 0 ± 5.91c3ν 0 100 5 ± 2.392 0 ± 1.64 2 ± 8.18cS 1.10−3 10 0.01± 4.8.10−3 0.012± 5.6.10−3 0.01± 5.6.10−2

cPA1.10−3 1 0.5 ± 0.274 - -

c1PB1.10−3 1 - 1 ± 0.35 -

c2PB1.10−3 1 - 0.26 ± 0.87 -

c3PB1 100 - 7.86 ± 1.84 -

c4PB1.10−3 100 - 0.54 ± 0.58 -

cPC0 50 - - 5 ± 0.83

The measured and simulated DSDs are shown in Figure 6.10 for experiment P1at various points in time. The simulations are performed with the optimal parametervalues obtained with the optimizations started from C1.

The figure clearly shows that the simulated evolution of the DSD as functionof time does not agree with the measured evolution. The measured DSD graduallyshifts to the left. The shape of the DSD remains more or less constant during theemulsification process. The model predicts a completely different behavior. TheDSD does not shift to the left gradually. Instead, the DSD to the right of the criti-cal droplet diameter (approximately 9 µm) decreases and to the left of the criticaldroplet diameter a peak, with a steep right edge, is formed with a fixed positionat approximately 7 µm. This dynamic behavior is representative for all performedsimulations and experiments. These differences can not be explained by deviationsof the values of the experimental conditions from their nominal values nor by measu-rement uncertainty. And although it can not be guaranteed1 that Droplet model

1In theory it is possible that the optimum parameter values obtained with the optimizationsstarted from A3, B1 and C1 are bad local optima.

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Figure 6.10: Simulated and measured evolution of the normalized (volume-based)DSD at various points in time [min] for experiment P1. Simulations are performedwith the optimal parameter values obtained with the optimizations started from C1.

A, B and C are not capable to simulate a shift to the left, this seems very unlikely.Therefore, it is concluded that the model structure of Droplet model A, B and C isnot correct. Hence, these models are considered falsified.

6.3.3 Adaptation of the Droplet models

In this section it is first discussed that the time-scale of the process is much longerthan expected. Then a possible explanation is given and finally, a model adaptationis proposed.

Expected time to reach a steady-state

The time needed to reach a steady-state is at least 15 min for the 4 experimentscarried out. The expected time needed to reach a steady-state is calculated basedon the experimental conditions of experiment P1 (the estimated time for the otherexperiments is more or less the same): the viscosity of the dispersed phase and ofthe emulsion is set to 56 mPa s (mean value of Table 6.4) and 14 mPa s respectively.The emulsion viscosity is calculated with the Viscosity model using a shear rateof 10000 s−1. The total fluid volume Vt was equal to 11.10−3 m3. Equipmentdimensions used in the calculations are listed in Table 5.1 and the value of theinterfacial tension is set to 8.2 mN m−1.

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The time needed to reach a steady-state is estimated as follows:

• With a circulation flow rate of 14 m3 h−1 the mean residence time in the gapof the colloid mill, given by Vcm/Fcm,p, is 17 ms. The fluid volume in the col-loid mill is given by Vcm = Vcm1 + Vcm2, where Vcm1 and Vcm2 are calculatedwith Equation 3.28 and 3.27 respectively. The breakup time of droplets witha diameter of 60 µm (maximum droplet size after approximately 30 s of emul-sification) is 5 ms for the given experimental conditions. Comparing the meanresidence time with the breakup time suggests that droplets with a diameterof 60 µm break at least once during a pass through the gap of the colloid millduring experiment P1.

• Droplets with a diameter below 10 µm are not expected to breakup duringexperiment P1. Assuming binary equally sized breakage, 8 breakage eventswould be needed to breakup a droplet with a diameter of 60 into droplets of9.5 µm (60 → 48 → 37 → 30 → 24 → 19 → 15 → 12 → 9.5 µm). Basedon this one would expect that 2 passes through the gap of the colloid mill areneeded to break a droplet with an (initial) diameter of 60 µm down to dropletswith a diameter below 10 µm. Note that assuming capillary breakage, whichwould be expected on basis of the results with smooth colloid mills of Wieringaet al. (1996), would result in even less passes.

• It is estimated that it takes approximately 65 s before all fluid has passed thegap of the colloid mill once during experiment P1. This value is calculated by(Vt/Fcm,p)(Vcm/Vcm1). Based on these estimates it is expected that a steady-state would be reached after 2.2 min for experiment P1.

Although these calculations are rather crude (assuming plug-flow, equal-sizedbreakage) they clearly show that the measured time to reach a steady-state is longerthan expected.

Possible explanation and model adaptation

Up till now it has been assumed that droplets break in the gap of the colloid millin laminar flow with a uniform shear rate. This is an extremely simplified viewof reality. A more realistic view could be the following. In practice the dropletsexperience a varying shear rate and flow type during the flow through the colloidmill. These variations are caused by the conical shape of the mill (the diameterand hence the shear rate increases from the inlet to the outlet of the mill) andthrough the presence of the grooves in the rotor and stator surfaces. With a rotorspeed of 50 s−1 and a mean residence time of 17 ms the rotor has made almost onerevolution during the pass of 1 droplet. It seems therefore likely that this droplethas experienced both the presence of the gap and of the grooves several times duringthe flow through the mill. The shear rate profile and flow type as experienced willdiffer from droplet to droplet (depending on the path followed through the mill).With this view it could be argued that the probability of droplet breakage increasesas the droplet diameter increases. Hence, droplets that are 10 times larger than the

93

critical droplet diameter have a higher breakage frequency than droplets that are2 times larger than the critical droplet diameter. Note that this is the opposite ofwhat would be calculated with Equation 3.49. It is proposed to replace this equationfor the black-box relation

S(v) =

c2S(d − dcr)c3S if Ω ≥ Ωcr

0 if Ω < Ωcr, (6.5)

where c2S [s−1] and c3S [-] are fit parameters. The Droplet models A, B and C withS given by Equation 6.5 are denoted as Droplet model D, E and F respectively. Oneof the limitations of using a black-box relation (see also Section 6.1.2) is that thevalues of the fit parameters will depend on the geometry of the colloid mill (e.g. thedepth of the grooves). Therefore it will not be possible to use this model to studythe effect of e.g. the groove depth on the evolution of the DSD in time. To that enda white-box model is needed.

The question of the effect of the colloid mill geometry on the evolution of theDSD in time (e.g. gap width, groove depth and/or width) is an interesting one,because the previous discussion suggests that in a colloid mill with smooth rotorand stator surfaces less passes are needed, to break droplets down below a certainvalue, than in a colloid mill with grooves! A benefit of the grooves is however that thepumping capacity and hence the refreshment frequency is enlarged. This is becausethe pressure drop over the colloid mill is lower as grooves are present. An otherbenefit could be that, due to the presence of the grooves, an elongational componentis introduced in the flow (this is discussed in more detail in the next section). Thismeans that it becomes more easy to breakup droplets especially at high viscosityratios (λ). Based on this it would be expected that the optimal colloid mill designwould be somewhere in between the current situation and the situation where thecolloid mill is equipped with smooth rotor and stator surfaces.

6.3.4 Parameter estimation

Parameter optimizations were performed to establish the values of the fit parametersof Droplet model D, E and F. The parameter optimizations were started from 3different initial guesses for each Droplet model and results are listed in Table 6.7.

The lowest values of the optimal objective function J∗ are found when the pa-rameter optimizations are started from D1, E2 and F3 for Droplet model D, E andF respectively. The corresponding values of J∗ are: 3743, 2749 and 2918 for Dropletmodel D, E and F respectively. These values are at least half as low as found withDroplet model A, B and C. Hence, the proposed model adaptation results in a bet-ter agreement between the measured and simulated DSDs. The values of Dropletmodel E and F are clearly lower than the value of Droplet model D and the valueof Droplet model E is slightly lower than that of Droplet model F. The optimalparameter values and the corresponding 95% confidence intervals are listed in Table6.8 for D1, E2 and F3.

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Table 6.7: Results parameter optimization with Droplet model D, E and F.Droplet No. of fit Initial No. of Calculation J0 J∗

model parameters guess iterations time[min]

D 7 D1 9 66 10464 3743D2 2 15 4002 4002D3 10 73 11308 4333

E 10 E1 24 269 5423 2758E2 11 128 3493 2749E3 7 69 18457 5681

F 7 F1 11 50 15211 2979F2 11 53 9770 3498F3 16 71 8142 2918

The optimal value of α is set to 0.19. This suggests that the flow contains (asmall) elongational component. The value of λ = ηd/ηe is 4.03, 1.28, 4.54 and 3.95for experiment P1, P2, P3 and P4 respectively. At λ > 4 breakup does not occurin simple shear flow. Therefore it is concluded that the flow contains elongationalcomponents as expected. This is probably due to the presence of grooves in therotor and stator surfaces because a colloid mill equipped with smooth rotor andstator surfaces gives fairly pure simple shear flow (Wieringa et al., 1996). Theoptimal values of c2ν and c3ν are 0 with the optimizations started from E2 and F3respectively. These values are also quite low for D1. Hence, the calculated numberof daughter droplets of Droplet model D, E and F is (almost) constant.

Table 6.8: Optimal parameter values and 95 % confidence intervals for Droplet modelD, E and F.

θ θl θu D1, θ∗ and E2, θ∗ and F3, θ∗ andConf. Int. Conf. Int. Conf. Int.

α 0 1 0.18 ± 3.10−3 0.19 ± 4.10−3 0.19 ± 3.4.10−3

c1ν 0 100 18.83± 1.18 6.32 ± 0.91 29.61 ± 10.77c2ν 0 100 0.87 ± 1.63 0 ± 0.80 0 ± 4.92c3ν 0 100 2.85 ± 1.61 0 ± 0.88 0 ± 7.06c2S 1.10−3 1.103 2.30 ± 0.48 2.31 ± 0.37 2.50 ± 0.51c3S -5 5 1.59 ± 0.11 1.87 ± 0.10 1.88 ± 0.11cPA

1.10−3 1 1 ± 0.29 - -c1PB

1.10−3 1 - 1 ± 0.31 -c2PB

1.10−3 1 - 0.44 ± 0.55 -c3PB

1 100 - 7.51 ± 1.06 -c4PB

1.10−3 100 - 0.28 ± 0.26 -cPC

0 50 - - 8.09 ± 1.03

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Single droplet breakage experiments in simple shear flow show an increase of thenumber of daughter droplets when the ratio between the capillary and the criticalcapillary number increases (see Chapter 2). Experimental work to determine thenumber of daughter droplets in elongational flow is lacking to the authors knowledge.Therefore it is unclear if it is reasonable to expect a constant number of daughterdroplets after breakage of a single droplet in elongational flow. Further experimentalwork would be needed to establish this.

Results D1 and E2 Measured and simulated DSDs obtained after 45 min areshown in Figure 6.11 and 6.12 for experiment P1-P4. These results are also repre-sentative for measured and simulated DSDs for other time points.

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Figure 6.11: Simulated (D1) and measured normalized (volume-based) DSD after45 min for experiment P1-P4.

The simulations are performed with the optimal parameter values obtained withthe optimizations started from D1 and E2 respectively. Results with the optimalparameter values obtained with the optimizations started from F3 are shown later.

These results show that the position of the simulated and measured DSDs after45 min are in reasonable agreement. The deviations between the measured and sim-ulated DSDs obtained with Droplet model E are clearly smaller than the deviationsas obtained with Droplet model D.

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Figure 6.12: Simulated (E2) and measured normalized (volume-based) DSD after 45min for experiment P1-P4.

The daughter droplets of Droplet model D break according to a single normaldistribution, which width is set to its maximum value (cPA

= 1). The measured andsimulated DSDs obtained with Droplet model E are in better agreement becauseof the formation of a second peak (c3PB

= 7.51); a broad one (c1PB= 1) for the

daughter droplets and a more narrow one (c2PB= 0.44) for the satellite droplets.

Although this results in the lowest objective value, the shape of the simulated DSDis clearly not in agreement with the measured DSD. Based on these results it isconcluded that the model structure of Droplet model D and E are not correct.

Results F3 Measured and simulated DSDs as obtained at various points in timeare shown in Figure 6.13, 6.14, 6.15 and 6.16. The simulations are performed with theoptimal parameter values as obtained with optimization F3. In industrial practicethe DSD is often characterized with the d43 [m] defined as

d43 =

∫∞0

d4n(v, t)dd∫∞0

d3n(v, t)dd(6.6)

The simulated and measured d43(t) for experiment P1-P4 is shown in Figure 6.17.

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Figure 6.13: Simulated (F3) and measured evolution of the normalized (volume-based) DSD at various points in time [min] for experiment P1.

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Figure 6.14: Simulated (F3) and measured evolution of the normalized (volume-based) DSD at various points in time [min] for experiment P2.

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Figure 6.15: Simulated (F3) and measured evolution of the normalized (volume-based) DSD at various points in time [min] for experiment P3.

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Figure 6.16: Simulated (F3) and measured evolution of the normalized (volume-based) DSD at various points in time [min] for experiment P4.

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Figure 6.17: Simulated (F3) and measured evolution of the d43 as function of timefor experiment P1-P4.

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6.3.5 Model validation

The measurement data of experiment P5 are used for the model validation. Figure6.18 shows the simulated (F3) and measured evolution of the DSD at various pointin time. The corresponding d43(t) is shown in Figure 6.19.

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Figure 6.18: Simulated (F3) and measured evolution of the normalized (volume-based) DSD at various points in time [min] for experiment P5.

The results of the parameter estimation and the model validation show that:

• The simulated as well as the measured DSDs shift to the left; this is a clearimprovement compared with Droplet model A, B and C.

• The simulated and measured DSDs shift further to the left (lower value of thed43) when the volume fraction and/or the rotor speed increases.

• Although the shape differs slightly, the position and the width of the simulatedDSDs are in reasonable agreement with the measured values.

• The decrease rate of the simulated d43 is in good agreement with that of themeasured d43: it decreases relatively fast during the first 15 min of emulsifi-cation after which it decreases much slower during the last 45 min. Relativedifferences between the measured and simulated d43 vary between -20 and 30 %for experiment P1-P5.

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0 20 40 60 80 100 120 1404

6

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12

14

16

18

20

Time [min]

d 43 [µ

m]

SimExp

Figure 6.19: Simulated (F3) and measured evolution of the d43 as function of thetime for Experiment P5.

• During experiment P5 the rotor speed is switched from 45 to 15 s−1 for halfan hour after 70 min. The measured (and simulated) d43 remains more or lessconstant during this time period. Hence, the assumption that coalescence isnegligible (Section 3.1.5), seems justified.

The slight differences in shape, as mentioned earlier, can not be explained com-pletely by measurement uncertainty nor by deviations of the values of the experi-mental conditions from their nominal values. Therefore, it is concluded that themodel structure of Droplet model F is not fully correct. More experiments would beneeded to establish which model relation(s) is (are) not fully correct. Despite theseslight differences in shape, it can be concluded that there is reasonable agreementbetween the simulated and measured DSDs. Given the overall performance of thetotal model (Flow rate model, Viscosity model, Droplet model F and the Reactormodel) it is expected that the model is accurate enough to render practical relevantoptimization results. This is the subject of the next chapter.

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Chapter 7

Optimization of theoperation procedure

In Chapter 1 the problem statement is formulated as follows: “Investigate, based on amodel, how the control variables should be chosen as a function of the time in order toproduce emulsions (for a given volume fraction) with a certain, predefined, terminalDSD and/or emulsion viscosity (evaluated at a certain shear rate) in minimal time.”Related to this problem statement the sub-problems A, B and C were defined; theyare briefly reviewed here.

A) Two equipment configurations are considered in this thesis: Configuration I,where the pumping and shearing action are coupled, and Configuration II,where these actions are not coupled. Sub-problem A aims at establishingwhich configuration enables the fastest production.

B) In Chapter 1 it is explained that it is desirable to produce emulsions with lessoil while maintaining a similar DSD and emulsion viscosity. It will be examinedif this can be established by adapting only the operation procedure. Hence, theaddition of additional stabilizers like e.g. starch will not be considered. It willalso be determined how the control inputs should be chosen as a function of thetime to produce such an emulsion in minimal time. This will be investigatedfor both equipment configurations.

C) Sub-problem C aims at establishing if emulsions (for a given volume fraction)with a multi-modal DSD can be produced. If this is the case, it will be exami-ned, for Configuration I and II, how the operation procedure should be chosento produce such an emulsion in minimal time.

In this chapter several dynamic optimization problems (A, B and C) are formulatedand solved in order to study these sub-problems. The results of the optimizationproblems A, B and C are used to study sub-problem A. The results of optimizationproblem B and C are used to study sub-problem B and C respectively.

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The outline of this chapter is as follows. First, in Section 7.1, a general mathe-matical formulation for the dynamic optimization problems A, B and C is given.A method for solving dynamic optimization problems is described in Section 7.2.With this method the optimizations are started from an initial guess of the opera-ting procedure. The choice of this initial guess affects the quality of the optimalsolution and a method for obtaining a good initial guess is described in Section 7.3.In Section 7.4 specific formulations and the solutions of the dynamic optimizationproblems A, B and C are described. In order to get an idea about the sensitivityof the optimization results for changes of several parameter values (e.g. values ofthe fit parameters, fluid properties), a sensitivity analysis is performed. This is alsodescribed in Section 7.4. Finally, in Section 7.5 the conclusions are given.

7.1 General formulation of the dynamic optimiza-tion problems

The goal of all 3 dynamic optimization problems is to choose the control inputs(the rotor speed Ncm(t), the inlet flow rate Fin(t) and, for Configuration II, thecirculation flow rate Fcm,p(t)) such that a certain predefined, terminal DSD, volumefraction and emulsion viscosity (evaluated at a certain shear rate) are reached inminimal time. Hence, the objective J is to minimize the final time tf [s] subject tothe following constraints:

1. The dynamic model describing the evolution of the DSD, the volume fractionand the emulsion viscosity as a function of the time. It is given by the followingdifferential algebraic equation

x = f(x, y, u), (7.1)0 = g(x, y, u), (7.2)

x(0) = x0, (7.3)

where x ∈ Rnx, y ∈ R

ny and u ∈ Rnu are respectively the state, algebraic

and the control variables. The vector x consists of the number of dropletswith volume vi per emulsion volume in the Vessel Ni,v(t), the Colloid mill gapNi,cm1(t), the Colloid mill groove compartment Ni,cm2(t), the Piping com-partment Ni,p(t) and the oil volume Voil(t) and its length is given by nx. Thevalue of nx [-] is 4 times the number of grid points M plus 1. The initialcondition x(0) is denoted with x0. The vector y consists of the emulsion vis-cosity evaluated at a certain shear rate ηe(γ, t), the oil volume fraction φ(t)and, for Configuration I, the circulation flow rate Fcm,p(t), hence ny [-] is 2and 3 for Configuration II and I respectively. The vector u comprises Ncm(t),Fin(t) and, for Configuration II, Fcm,p(t). Hence nu [-], the number of controlvariables, is 2 for Configuration I and 3 for Configuration II.

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2. A constraint to enforce that a certain predefined normalized (volume-based)droplet size distribution is reached at t = tf in the piping. Mathematically

N∗i,p,l ≤ N∗

i,p(tf ) ≤ N∗i,p,h. (7.4)

The normalized (volume-based) value of Ni,p(tf ), the number of droplets witha droplet volume vi per emulsion volume in the Piping compartment at timetf , is denoted by N∗

i,p(tf ) [m−3]. With this equation it is enforced that N∗i,p(tf )

lies between the lower bound N∗i,p,l [m−3] and the upper bound N∗

i,p,h [m−3].This constraint allows the formulation of an arbitrary target DSD.

3. In order to prevent phase-inversion (inversion of an O/W-emulsion to a W/O-emulsion or visa versa) the inlet flow rate is not allowed to exceed a certainvalue. This value is product dependent; during the optimization studies itis set to 0.09 m3 h−1. The values of the other control inputs are also con-strained between a lower bound ul and an upper bound uh. The correspondingconstraint is formulated as

ul ≤ u ≤ uh. (7.5)

The values of the upper and lower bounds of the control inputs used duringthe optimization studies are listed in Table 7.1. The maximum value of the

Table 7.1: Input constraints.u unit ul uh

Ncm s−1 0 50Fcm,p m3 h−1 0 14Fin m3 h−1 0 0.09tf min 0 60

final time is set to 60 min because longer production times are not expectedto be of practical relevance. The maximum values of Ncm and Fcm,p are basedon the equipment used.

4. A constraint to enforce that a certain predefined emulsion viscosity (evaluatedat a certain shear rate) is reached at the final time. The shear rate valueat which the emulsion viscosity is evaluated should ideally correspond to theshear rate that determines certain quality attributes. This is of course stronglyproduct dependent. However certain quality attributes (e.g. the consistencyof skin creams as described in Chapter 1) correlate with the viscosity at lowershear rates. For the optimization studies a low shear rate (10 s−1) was thereforechosen. The constraint to enforce that, the emulsion viscosity (evaluated at ashear rate of 10 s−1) at the final time ηe(10, tf ), lies between the lower boundηe,l [Pa s] and the upper bound ηe,h [Pa s] is formulated as

ηe,l ≤ ηe(10, tf ) ≤ ηe,h. (7.6)

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Note that constraints enforcing emulsion viscosities evaluated at other shearrates can easily be added.

5. The volume fraction differs from product to product (for dressings the volumefraction is approximately 40 % whereas it is more or less 74 % for mayonaisses).Therefore a constraint is formulated to enforce that the terminal oil volumefraction is equal to a desired value φref [-]. The constraint is given by

φ(tf ) = φref . (7.7)

The general formulation of the dynamic optimization problems A, B and C isnow given as

minu

J = tf (7.8a)

subject to:

x = f(x, y, u), (7.8b)0 = g(x, y, u), (7.8c)

x(0) = x0, (7.8d)φ(tf ) = φref , (7.8e)

ul ≤ u ≤ uh, (7.8f)ηe,l ≤ ηe(10, tf ) ≤ ηe,h, (7.8g)

N∗i,p,l ≤ N∗

i,p(tf ) ≤ N∗i,p,h. (7.8h)

7.2 Solving dynamic optimization problems

There is extensive literature available on numerical strategies for the solution of dy-namic optimization problems. An often used method is the so-called direct methodwhere the infinite dimensional dynamic optimization problem is approximated asa finite dimensional nonlinear program (NLP). Within the framework of the directmethod, there are two general strategies: the sequential method (see for exampleEdgar and Himmelblau, 1988; Vassiliadis et al., 1994a,b) and the simultaneous orcollocation method (see for example Biegler, 1984; Biegler et al., 2002). The se-quential method reduces the infinite dimensional problem to a finite dimensionalproblem through approximation of the control profiles u(t) by a finite family of ba-sis functions. Typically, piecewise constant approximation over equally spaced timeintervals is made for the inputs. This method is therefore called Control Vector Pa-rameterization (CVP) in the literature. In the collocation method both the controlsu and the states x are discretized using polynomials on finite elements. The bigadvantage of the simultaneous approach is that the objective function and the pro-cess model equations converge simultaneously (infeasible path method), while theprocess model equations are necessarily satisfied in every iteration in the sequentialapproach. However, the sequential approach is simpler in implementation and es-pecially for stiff systems, it may actually be an advantage instead of a disadvantage

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that the model equations are satisfied every iteration. For these reasons, the CVPmethod is used for the solution of the dynamic optimization problems in this thesis.

The NLP may be solved with either gradient based local approaches or gradient-free global approaches. Gradient based methods are of local nature, that is, theyconverge to local solutions, usually the nearest maximum (or minimum) in the neigh-borhood of the initial guess. Sequential Quadratic Programming (SQP) is at presentone of the most popular gradient based methods. For an overview of SQP the readeris referred to Nash and Sofer (1996). Global optimization methods aim to find theglobal (overall best) solution of multimodal problems. Stochastic strategies (i.e. ge-netic algorithms and simulated annealing) are popular global optimization methods.For an overview of these methods the reader is referred to Banga et al. (2003). Itshould be noted that stochastic methods do not guarantee global optimality andtypically a large number of iterations (several tens of thousands) are needed to solvethe optimization problem. One simulation of the emulsification model takes appro-ximately 30 s. Therefore, it was expected that stochastic methods would be of littleuse for our purposes and it was decided to use SQP for the solution of the NLPs.

As mentioned previously, gradient based methods are of local nature. The choiceof the initial guess is very important. It affects:

1. The quality of the (local) optimum.

2. Whether the optimizer is able to find a feasible solution at all.

3. The number of iterations (and hence the computation time) needed.

In Section 7.3 a method is described that enables the calculation of a good initialguess. The method approximates the original dynamic optimization problem asa Mixed Integer Linear Program (MILP). The MILP can be solved for its globaloptimum by using well-proven, standard optimization codes. The most popular andby far best known solution method for MILPs is the branch-and-bound method.The basic operation of the method is explained in Appendix C. In Section 7.4 it isshown that the MILP method yields good initial guesses for the SQP optimizations.Figure 7.1 shows a schematic representation of the various solution methods.

7.3 Initial guess

The method to approximate the original dynamic optimization problem as a MILPis derived as follows. First, model analysis shows that:

• The right-hand side of Equation 3.46 depends in a strong nonlinear fashion onthe rotor speed and the emulsion viscosity (via the capillary number). A verysmall increase of the rotor speed and/or the emulsion viscosity can already leadto the breakage of certain droplet sizes that would not break with a slightlylower value of the rotor speed and/or emulsion viscosity.

• Within small intervals of the capillary number (which is affected by the rotorspeed and the emulsion viscosity) the droplet sizes which are formed and brokenup remain unchanged. These intervals form the modes of the system.

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Infinite dimensionaldynamic optimization

problem

Finite dimensionalnonlinear program

(NLP)

Mixed IntegerLinear Program

(MILP)

Local optimum ofNLP

Simplificationsdescribed inSection 7.3

Global optimumof simplified

problem

Input parametrization(CVP-method)

SQP-method

Branch-and-bound method

Figure 7.1: Schematic representation of the various solution methods.

• A transition from one mode to another is triggered when the capillary numberpasses a certain critical value.

• The mathematical behavior of a mode is described by a given set of evolutionnonlinear differential equations. The nonlinearity is caused by:

– Products of the circulation flow rate and the states (e.g. Fcm,p(t)Ni,v(t)).

– Products of the inlet flow rate and the states.

– The nonlinear relation between the flow rate and the rotor speed and theemulsion viscosity (which depends in turn on the DSD via r∗). Note, thatthis is not the case for Configuration II where the circulation flow rate isa control input.

The model analysis suggests that the model can be reformulated as a state-transition network (Avraam et al., 1998) where nonlinear dynamics describe thebehavior in a mode and where transitions between different modes are modeled usinginteger decision variables. However, this would result in a Mixed Integer NonlinearProgram (MINLP) which is hard to solve in general.

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A further complicating factor is that the capillary number is affected by theemulsion viscosity. This variable can not be adapted explicitly but its dynamic be-havior is determined by the trajectory of the inlet flow rate. This implies that thetransition times are not known in advance, complicating the solution of the opti-mization problem considerably. In order to overcome these problems the followingsimplifications are proposed that render the model linear time invariant:

• The rotor speed within a mode is set to its maximum value. If for examplethe system is in mode s if the rotor speed lies between 10 and 12 s−1, then therotor speed is set to 12 s−1 for this mode. The maximum value is selected be-cause this way the exchange rate between the compartments and the breakagefrequency are maximized within that mode.

• The model calculations are performed as if the process is operated batch wise.The oil volume fraction is set to its desired terminal value. Due to this simplifi-cation the products of the inlet flow rate and the states vanish and the emulsionviscosity in the gap of the colloid mill (the viscosity used for the calculationof the capillary number, the critical capillary number, the breakage frequencyand the pressure drop over the colloid mill) remains approximately constantduring the process. The latter is the case because the oil volume fraction isconstant and the shear rate in the gap of the colloid mill is, even at low rotationspeed, quite high (3400 s−1 at 5 s−1).

• For the calculation of the emulsion viscosity in the piping (the viscosity usedfor the calculation of the pressure drop over the piping), r∗ is set to a valueof 2 µm. This simplification has a minor effect on the circulation flow rate,because the circulation flow rate is only slightly affected by the value of r∗ asshown in Section 7.4.

With these simplifications a piece-wise linear time-invariant dynamic system isobtained. A method to express such a system as a mixed logical dynamical (MLD)system (system that is described by linear dynamical equations subject to linearmixed-integer inequalities) and a framework for modeling and controlling MLD sys-tems is presented in Bemporad and Morari (1999). The basic idea is followed,however some modifications are made to facilitate the solution of the MILP. The ob-jective, to reach a certain end-point condition in minimum time, is enforced throughthe introduction of another set of integer decision variables. The mathematical for-mulation of the optimization problem as a MILP is discussed in Section 7.3.1 and7.3.2.

As mentioned earlier several simplifications are made to approximate the originaldynamic optimization problem as a MILP. As a consequence the solution of the MILPis not necessarily a feasible solution of the original optimization problem. However,in Section 7.4 it is shown that this method yields in general initial guesses that arein the neighborhood of (local) optimal solutions.

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7.3.1 MLD System

Define a set of periods (k=1,...,N) of fixed duration. The following binary variableis introduced to characterize the system behavior

δsk =

1 if the system is in mode s during period k0 otherwise . (7.9)

The system can only be in one mode at a certain point in time. This is expressedmathematically as

∑ns

s=1 δsk = 1, with ns [-] the number of modes of the system.

For each mode s the discrete time model xk+1 = Asxk holds, where the vectorxk is composed of Ni,v, Ni,cm1, Ni,cm2 and Ni,p. While the system is in mode s,the corresponding equations characterizing the behavior of the system must holdin that mode. In Bemporad and Morari (1999) an approach is suggested for this.The main idea is followed, however some modifications are made and these aremotivated later. Define zs

k ≡ xkδsk. With linear equality and inequality constraints

it is enforced that zsk = xk if the system is in mode s and that zs

k = 0 if this is notthe case. Mathematically

xk+1 =ns∑

s=1

(Aszsk), (7.10)

ns∑s=1

zsk = xk, (7.11)

zsk ≤ Pδs

k, (7.12)zs

k ≥ 0, (7.13)

where P is a weighting matrix. Hence, if δsk is 0 then zs

k is 0. Since δsk is 1 for only

one mode, zsk = xk for that mode. In order not to restrict the feasible region, P must

be chosen such that it is always possible to satisfy Equation 7.12. In Bemporad andMorari (1999) a slightly different formulation is used: zs

k ≡ (Asxk)δsk and

xk+1 =ns∑

s=1

zsk, (7.14)

zsk ≤ Asxk, (7.15)

zsk ≥ Asxk − P (1 − δs

k), (7.16)zs

k ≤ Pδsk, (7.17)

zsk ≥ 0. (7.18)

This results however in a larger integrality gap. The reader, who is not familiar withthe branch-and-bound method is referred to Appendix C, for the next discussion.A large integrality gap implies that no or hardly any bounding is likely to happenwhich makes the branch and bound process tend to complete enumeration. Thelarger integrality gap is caused because it is no longer enforced explicitly that thesum of zs

k is equal to Asxk. This is only the case if δsk = 1. However, during the

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solution of the relaxed MILP sub-problems, the binary variables are substituted bycontinuous variables with lower bound 0 and upper bound 1. Now it is no longerguaranteed that

∑ns

s=1 zsk = Asxk. Only an upper bound for the value of zs

k ispresent; it is equal to Asxk. Because of this, much more freedom is allowed inthe choice of zs

k. This results in higher objective values of the MILP sub-problemscompared to our formulation.

7.3.2 End-point constraints and the objective function

The formulation of the inequality end-point constraints and the objective functionJ will be handled next. The following binary variable is introduced to characterizethe end-point constraints

Yk =

1 if the inequality constraints are met during period k0 otherwise . (7.19)

A certain predefined terminal DSD is now enforced with the following inequalityconstraints

Ykxl − xk ≤ 0, (7.20)xk − xh − (1 − Yk)Q ≤ 0, (7.21)

where xl is the vector with the lower bounds of the state vector x and xh is the vectorwith its upper bounds. The weighting matrix Q must be chosen such that Equation7.21 can always be satisfied. Further, a certain predefined terminal emulsion viscosityhas to be enforced. The emulsion viscosity depends in a nonlinear fashion on the DSD(via r∗, see Equation 3.69 and 3.72), which would lead to a MINLP formulation.However, from Equation 3.73 it is clear that (r∗)2 depends linearly on the DSD.The emulsion viscosity is affected, for a given volume fraction and shear rate, byr∗ only. Hence, enforcing a certain predefined terminal emulsion viscosity is, fora given volume fraction, the same as enforcing a certain predefined value of (r∗)2.The following inequality constraints are now formulated to enforce that the value of(r∗)2 at tf is between its lower bound (r∗l )2 and its upper bound (r∗u)2

Yk(r∗l )2 − (r∗)2k ≤ 0, (7.22)

(r∗)2k − (r∗h)2 − (1 − Yk)R ≤ 0, (7.23)

where R is a weighting matrix; chosen such that Equation 7.23 can always be sa-tisfied. Hence, if xk or (r∗)2k are less than their lower bounds then Yk must be 0 tosatisfy constraint 7.20 and 7.22. If xk or (r∗)2k are larger than their upper boundthen Yk must be 0 to satisfy constraint 7.21 and 7.23. Yk can be either 0 or 1 if thebounds are met. Due to the formulation of the objective function J , Yk is set to 1.The objective function is written as

max J =N∑

k=1

Yk. (7.24)

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The binary variable Yk is set to 1 in the situation where the end-point inequalityconstraints are satisfied, because this way the number of periods where the end-point inequality constraints are satisfied, are maximized. Hence, the time needed tosatisfy these constraints is minimized. Note, that once the constraints are met, theycan be kept feasible by switching off the rotor speed.

7.4 Optimization studies

The formulation and the solution of the dynamic optimization problems A, B and Care described in this section. First, some implementation issues are discussed. Thenthe optimization studies are described.

7.4.1 Implementation

DSD of the inlet flow Up till now only batch emulsification simulations areperformed. Therefore the modeling of the number based droplet concentration forthe inlet fluid flow nin(v, t) [m−3 m−3] has not been discussed so far. The inletflow consists of oil only and does not contain oil droplets. Hence, strictly speakingthe inlet flow does not have a number based droplet concentration. However, formodeling purposes it is assumed that the oil breaks down instantaneously in a widerange of droplet sizes as soon as it enters the vessel. This is modeled as if the inletflow has a certain droplet size distribution nin(v, t). The product of the dropletvolume v and nin(v, t) is modeled with a log-normal distribution

vnin(v, t) =1

cdσ√

2πe−

(ln d−µ)2

2σ2 . (7.25)

Integrating vnin(v, t) from 0 to the maximum droplet volume should give a valueof 1 m3 droplet volume m−3 fluid volume, because the inlet fluid flow consists of oilonly. This implies that the constant c [m3 fluid volume m−2 droplet area] is givenby

c =∫ ∞

0

1dσ

√2π

e−(ln d−µ)2

2σ2 dv. (7.26)

The values of the mean value µ and the standard deviation σ are set to 5.7(log(300)) and 1.6 µm respectively. The corresponding DSD of the inlet flow isshown in Figure 7.3. In order to get an idea about the sensitivity of the optimizationresults for changes in the values of µ and σ a sensitivity analysis is performed. Thisis described in Section 7.4.5.

Grid A geometric grid is used for the discretization of the PBEs. A relatively finegrid is used in the region from 0.1 till 100 µm (the region of main interest) while acoarser grid is used from 100 till 1300 µm. This second region is mainly of interestfor the modeling of the inlet flow and is off less importance, because the incomingoil droplets breakup rapidly in droplets with a diameter below 100 µm.

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The grid points di [µm] are chosen as follows:

• dIi+1 = 1.2dI

i for i = 1, .., 38 where dI1 = 0.1 µm.

• dIIi+1 = 1.4dII

i for i = 1, .., 7 where dII1 = 1.2dI

39.

The discretized model consists of 188 states: 47 for the Vessel compartment(Ni,v(t) with i = 1, .., 47), 47 for the Colloid mill gap compartment (Ni,cm1(t) withi = 1, .., 47), 47 for the Colloid mill groove compartment (Ni,cm2(t) with i = 1, .., 47)and 47 for the Piping compartment (Ni,p(t) with i = 1, .., 47).

Input parametrization For the MILP method as well as for the CVP method thenumber of control intervals have to be selected. In principle it is desirable to use alarge number of control intervals, because this way the optimizer has “more freedom”to choose the optimal operation procedure. However, using control intervals witha length of e.g. 0.1 s is of little use, because the transient behavior of the DSD ismuch slower. Next to that, the optimization problems contain a large number ofoptimization parameters when using a large number of control intervals. With thisdiscussion in mind the number of control intervals was selected as follows:

• 15 control intervals with a time period of 1 min are used with the MILP methodfor Configuration I.

• 10 control intervals with a time period of 1.5 min are used with the MILPmethod for Configuration II.

• 10 control intervals are used with the CVP method for both configurations.

Note that the final time is fixed with the MILP method whereas it is an opti-mization variable with the CVP method. Therefore, the time period of 1 controlinterval, given by the ratio between the final time and the number of control inter-vals, is constant with the MILP method whereas it varies with the CVP method.

MILP method The system consists of 17 modes for all cases studied. Thus, theMILPs for Configuration I consist of 17 ∗ 15 = 255 binary variables and several tensof thousands of inequality constraints. The control input Fcm,p is incorporated in theoptimization problem of Configuration II by selecting 4 values of Fcm,p: 5, 8, 11 and14 m3 h−1. Only 4 values are selected in order to keep the problem computationallytractable. The MILPs for Configuration II consist of 17∗4∗10 = 680 binary variablesand several tens of thousands of inequality constraints. The MILPs are solved withGAMS/CPLEX 7.0.

CVP method The number of optimization parameters is equal to 2 ∗ 10+ 1 = 21(plus 1, because the final time tf is also an optimization parameter) and 3∗10+1 = 31for Configuration I and II respectively.

The optimization problem was solved using a SQP solver (fmincon.m) available inMATLAB. The optimization parameter ‘TolFun’ was set to 0.1 and default values

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were used for the other parameter values of the optimization routine. Numericalperturbation was used for the gradient calculation. The differential algebraic equa-tions are solved with a standard numerical solver available in MATLAB (ode15s.m).All calculations were performed with a Pentium 4 (CPU 2.00 GHz and 0.99 GB ofRAM).

The SQP optimizations were started from 4 initial guesses for Configuration Iand II. The initial guesses for the optimization studies related to Configuration Iare:

• uI1: A reference operation procedure which is described in Section 7.4.2.

• uI2: All control variables are set to their maximum values and the total ope-ration time is 10 min.

• uI3:

– A time-varying rotation speed: a rotor speed of 35 s−1 during the first 3control intervals, then it is set to 45 s−1 during the next 3 control intervalsand for the remaining time period a value of 50 s−1 is used.

– The inlet flow rate is set to 2.10−5 [m3 s−1] until the target volume fractionis reached after which it is switched off.

– The total operation time is set to 10 min.

• uI4: The solution of the corresponding MILP.

The initial guesses for the optimization studies related to Configuration II, uII1−uII4, are almost the same. The only difference is that an initial guess for thecirculation flow rate is specified: it is set to its maximum value for uII1 and uII2

and to a constant value of 8 m3 h−1 for uII3. uII4 is the solution of the correspondingMILP.

7.4.2 Optimization problem A

The results of optimization problem A are used, together with the results of opti-mization problems B and C, to establish which equipment configuration enables thefastest production. First a reference operation procedure is described. Based onthis reference operation procedure a target, terminal DSD and emulsion viscosityare defined. Then it is investigated, for Configuration I as well as for ConfigurationII, how these target DSD and emulsion viscosity can be produced as fast as possible.

Detailed information regarding the operation procedure for the manufacturingof emulsion products in industrial practice is not available in the literature to theauthors knowledge; this is probably due to reasons of confidentiality. Althoughthis detailed information is lacking, and despite the fact that the precise operationprocedure is product dependent, the general procedure is as follows: i) the continuousphase (water, surfactant and ingredients) is added to the vessel, ii) the stirrer andthe rotor are switched on and the oil is pumped into the vessel. The inlet flow rateand the stirrer and rotor speed have constant values in time; the values of these

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variables are product dependent, and iii) the process is operated for a fixed amountof time.

The following operation procedure is used as reference case (uI1):

• The rotor speed is set to 35 s−1.

• The inlet flow rate is fixed to 1.25.10−5 [m3 s−1] during 9.2 min; the oil volumefraction after this time period is 0.6.

• After the filling the process is continued during 6.1 min, hence the total ope-ration time is 15.3 min.

The viscosity of the continuous phase and of the dispersed phase is set to 15and 53 mPa s respectively and the volume of the continuous phase is 4.4.10−3 m3

during the simulations. The equipment dimensions used are listed in Table 5.1 andthe other parameter values were taken from Chapter 6.

10−1

100

101

102

103 0

5

10

15

20

0

0.02

0.04

0.06

0.08

0.1

0.12

Time [min]

Droplet diameter [µm]

DS

DV

N [−

]

Figure 7.2: Evolution of the normalized (volume-based) DSD of the Piping com-partment for the reference case.

The evolution of the normalized (volume-based) DSD is shown in Figure 7.2.During the first 9.2 min the oil volume increases, resulting in an increase of theemulsion viscosity. Because of this the capillary number increases. The effect onthe critical capillary number will be small (α = 0.19 and λ < 3.3). This impliesthat Ω/Ωcr will increase during the first 9.2 min, hence the critical droplet diameter

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decreases and the DSD shifts to the left. When all the oil is added to the vesselthe emulsion viscosity, in the gap of the colloid mill, stays more or less constantand the critical droplet diameter ceases to decrease. The final emulsion viscosityηe,ref (10, tf ) is 239 mPa s and the terminal normalized (volume-based) DSD of thePiping compartment is shown in Figure 7.3 together with the initial normalized(volume-based) DSD. The peak value of the normalized DSD corresponds to N∗

17,p

(its value is slightly higher than the value of N∗16,p).

10−1

100

101

102

103

104

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Droplet diameter [µm]

DS

DV

N [−

]

Terminal DSDInitial DSD N*

16,p N*

17,p

Figure 7.3: Terminal normalized (volume-based) DSD of the Piping compartmentobtained with the reference operation procedure. The initial normalized (volume-based) DSD is also shown.

For both equipment configurations it is investigated if it is possible to producea similar terminal DSD and emulsion viscosity in less time by adapting the opera-tion procedure. Instead of enforcing exactly the same terminal DSD and emulsionviscosity it is enforced that:

• The peak value of the target normalized DSD should correspond with N∗17,p.

• The value of the normalized DSD corresponding to droplet diameters largerthan 11 µm should be less than the value of N∗

27,p,ref .

• The terminal emulsion viscosity should have a value between 0.97ηe,ref (10, tf )and 1.03ηe,ref (10, tf ).

This way the optimizer has “more freedom” for the choice of the optimization pa-rameters.

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Optimization problem A is now formulated as

minu

J = tf (7.27a)

subject to:

x = f(x, y, u), (7.27b)0 = g(x, y, u), (7.27c)

x(0) = x0, (7.27d)φ(tf ) = 0.6, (7.27e)

ul ≤ u ≤ uh, (7.27f)0.97ηe,ref (10, tf ) ≤ ηe(10, tf ) ≤ 1.03ηe,ref (10, tf ), (7.27g)

0 ≤ N∗16,p(tf ) ≤ N∗

17,p(tf ), (7.27h)

0 ≤ N∗18,p(tf ) ≤ N∗

17,p(tf ), (7.27i)

0 ≤ N∗27,p(tf ) ≤ N∗

27,p,ref (tf ). (7.27j)

MILP method A local optimum of optimization problem A for Configuration Iwas found after 30 nodes (13 min). The objective value is 9. Fifteen control intervalswith a length of 1 min are used for the MILPs corresponding to Configuration I.Hence, after 7 min of emulsification the constraints are met. When the local optimumwas found the integrality gap (see Appendix C) was 4. This implies that the upperbound to the solution of the MILP is 13. After 10000 nodes (approximately 2 days)the integrality gap was still 3 and the optimization was aborted. Because of thisit can not be concluded that the found optimum is also the global optimum of theMILP.

It took 21 nodes (24 min) to find the global optimum of optimization problem Afor Configuration II; its value is 7. As mentioned earlier 10 control intervals with alength of 1.5 min are used for the MILPs corresponding to Configuration II. Hence,after 6 min of emulsification the constraints are met.

CVP method Results regarding the number of iterations, the calculation timeand the optimal objective value (the optimal operation time) are listed in Table 7.2.These results show that the number of iterations varies, depending on the initialcondition, between 3 and 50; the calculation time lies between 0.5 and 32 hours.No feasible solution is found when the optimizations are started from uII1. Note,that the solution of the MILPs are no feasible solutions for the SQP optimizations(otherwise the value of J∗ would have been 7.0 and 6.0 for uI4 and uII4 respectively).As mentioned earlier this is due to the rather rigorous simplifications that were madeto approximate the original dynamic optimization problem as a MILP. However, thenumber of iterations needed to solve the optimization problems indicate that thesolution of the MILPs yields good initial guesses for the SQP optimizations. Theoptimal operation time found does hardly depend on the initial guess; it is 9.3 and8.9 min for Configuration I and II respectively. These results show that the operation

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time can be decreased significantly with respect to the reference case: with -39.5 %and -41.8 % for Configuration I and II respectively.

Table 7.2: CVP results of optimization problem A.Configu- Initial No. of Calculation J0 J∗

ration guess iterations time[h] [min] [min]

I uI1 6 1.0 15.3 9.3uI2 3 0.5 10.0 9.3uI3 27 9.1 10.0 9.4uI4 4 0.6 7.0 9.3

II uII1 50 31.6 15.3 no feasiblesolution found

uII2 41 26.2 10.0 8.9uII3 47 32.0 10.0 8.9uII4 20 4.5 6.0 8.9

The terminal DSDs obtained with the reference operation procedure and withthe optimal operation procedures for Configuration I and Configuration II are shownin Figure 7.4. The evolution of the emulsion viscosity (evaluated at a shear rate of10 s−1) as a function of the time is shown in Figure 7.5 for the reference and optimaloperation procedures. The optimal filling procedure, for both configurations, is toadd the oil as fast as possible. With the maximum inlet flow rate of 2.5.10−5 [m3 s−1]it takes 4.2 min to reach the reference volume fraction of 0.6.

During the oil addition the emulsion viscosity increases rapidly from 15 mPa sto 230 mPa s. After the oil addition it continues to increase (although at a muchlower speed), this is because the DSD shifts further to the left. The lower bound onthe emulsion viscosity is satisfied after already 4.2 min when the optimal operationprocedures are applied. However, at that time point the constraint on N∗

27 is notsatisfied; there are still too many large droplets. This is not unexpected because theinlet flow contains droplets up to 1000 µm. In order to satisfy the constraint on N∗

27

the process has to be continued for approximately 5 min for both configurations.The optimal operation procedure for both configurations is to operate the colloid

mill at its maximum rotating speed. This solution is found independent of theinitial guess. Surprisingly, the optimal operation procedure for Configuration II isnot to circulate as fast as possible. This is only the case during the start of theprocess whereas it is lowered during the remainder of the process; this is explainedin more detail later. The optimal circulation flow rate for Configuration II, whenthe optimizations are started from uII2, is shown in Figure 7.6 as function of thetime. The optimal circulation flow rate obtained when the optimizations are startedfrom uII3 is qualitatively the same, however the exact trajectory differs slightly.In the same figure the reference circulation flow rate and the circulation flow ratecorresponding to a rotation speed of 50 s−1 (optimal trajectory for Configuration I)are shown.

118

10−1

100

101

102

0

0.05

0.1

0.15

Droplet diameter [µm]

DS

DV

N [−

]

Reference Optimal; Config. IOptimal; Config II

N*27,p

Figure 7.4: Terminal normalized (volume-based) DSDs of the Piping compartmentobtained with the reference operation procedure and with the optimal operationprocedures of optimization problem A for Configuration I and II.

0 2 4 6 8 10 12 14 160

50

100

150

200

250

300

350

Time [min]

Em

ulsi

on v

isco

sity

at 1

0 s−

1 [mP

a s]

Reference Optimal; Config. I Optimal; Config. IIConstraints

Figure 7.5: Emulsion viscosities (evaluated at a shear rate of 10 s−1) as functionof the time for the reference operation procedure and for the optimal operationprocedures of optimization problem A for Configuration I and II.

119

0 2 4 6 8 10 12 14 165

6

7

8

9

10

11

12

13

14

15

Time [min]

Circ

ulat

ion

flow

rat

e [m

3 h−

1 ]

Reference Optimal; Config. I Optimal; Config. II

Figure 7.6: Circulation flow rate as function of the time obtained with the referenceoperation procedure and with the optimal operation procedure of optimization prob-lem A for Configuration I. The optimal circulation flow rate of optimization problemA for Configuration II is also shown.

The reference circulation flow rate starts at a value of 13.5 m3 h−1 and decreasesto a value of 5.9 m3 s−1. This decrease is due to the increase of the emulsion viscositywhich is caused mainly by the oil addition. After 9.2 min all the oil is added and fromthat moment on the reference circulation flow rate remains more or less constant1.The decrease rate of the reference circulation rate flattens for a short period of time(between 3 and 4 min); this is due to the transition from turbulent to laminar flow.

The circulation flow rate obtained with the optimal operation procedure for Con-figuration I remains at its maximum value of 14 m3 h−1 during 3 min, then the vis-cosity (and the corresponding pressure drop over the colloid mill and the piping)becomes too large and the circulation flow rate starts to decrease. After approxima-tely 4.5 min all the oil is added; the circulation flow rate is then about 10.5 m3 h−1.Overall it can be seen that, during most of the operation time, the circulation flowrate obtained with the operation procedure of Configuration I is much higher thanobtained with the reference operation procedure. Because of this the mixing timedecreases; the significant reduction of the operation time is mainly due to this andto less extent to the increased breakage frequency (lower value of dcrit; see Equation6.5) due to the higher rotation speed. The following simulations with ConfigurationII were performed to quantify this: i) a simulation with all control inputs set totheir maximum values during 10 min, ii) a simulation with the rotor speed set to

1Actually it continues to decrease very slowly due to a further increase of the emulsion viscosityevaluated at low shear rates (caused by the decrease of r∗), but this effect is so small that it cannot be noted from Figure 7.6.

120

90% of its maximum value while the other control inputs remain at their maximumvalues and iii) a simulation with the circulation flow rate set to 90% of its maximumvalue while the other control inputs remain at their maximum values. The valuesof N∗

27,p and ηe(10) obtained after 10 min were determined. It was found that N∗27,p

and ηe(10) changed with +1.8 and -0.04% respectively when the rotor speed was setto 90% of its maximum value. The same variables changed with +30.4 and -5.4%when the circulation flow rate was set to 90% of its maximum value. Based onthese results it is concluded that the decreased operation time is mainly due to theincreased circulation flow rate.

The optimal circulation flow rate trajectory for Configuration II is qualitativelythe same as for Configuration I: during the first minutes it is set to its maximum valueafter which it is decreased. However, the circulation flow rate for Configuration II isthe same or higher than that of Configuration I during almost the total process. Thisresults in a further decrease of the operation time (-3.8 % compared to the operationtime of Configuration I). A bigger benefit would be expected if the circulation flowrate would be set to its maximum value during the total process. Next it is discussedwhy this solution is not feasible. If the circulation flow rate would be set to itsmaximum value the emulsion viscosity reaches its upper bound (246 mPa s) after8.25 min. The value of N∗

27,p(tf ) is then 3.7.10−3; in order to satisfy the constraint itmust be smaller than 3.4.10−3. Since all control variables are set to their maximumvalues, the only way to decrease the value of N∗

27,p(tf ) further is to increase theoperation time. However, the emulsion viscosity should be less than (246 mPa s).This is the reason why the optimizer increases the operation time while decreasingthe circulation flow rate.

It should be mentioned that the reductions in operation time do depend on theproduct composition. For products with lower emulsion viscosities (caused by a lowervolume fraction and/or continuous phase viscosity) it is expected that the reductionin operation time would be less, because for those products the circulation flow rateis expected to be higher. Similarly, for products with higher emulsion viscositieslarger benefits are expected.

7.4.3 Optimization problem B

In this section it is investigated if it is possible to produce an emulsion with lessoil without lowering the emulsion viscosity and while maintaining a similar DSD asobtained with the reference operation procedure. Three optimization problems areformulated to investigate this. The aim of these optimization problems is to esta-blish how the operation procedure must be chosen in order to produce the followingemulsions in minimal time:

B1: An emulsion with an oil volume fraction of 0.55, an emulsion viscosity of atleast 1.03ηe,ref (10, tf ) and a terminal DSD which is similar to the terminalDSD obtained with the reference operation procedure.

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B2: An emulsion with an oil volume fraction of 0.55. Instead of enforcing anemulsion viscosity of at least 1.03ηe,ref (10, tf ), now an even higher emulsionviscosity (1.25ηe,ref (10, tf )) is enforced. The DSD specifications are the sameas for B1.

B3: An emulsion with an oil volume fraction of 0.5; the other specifications are thesame as for B1.

Optimization problem B1

Optimization problem B1 is formulated as

minu

J = tf (7.28a)

subject to:

x = f(x, y, u), (7.28b)0 = g(x, y, u), (7.28c)

x(0) = x0, (7.28d)φ(tf ) = 0.55, (7.28e)

ul ≤ u ≤ uh, (7.28f)1.03ηe,ref (10, tf ) ≤ ηe(10, tf ) ≤ ∞, (7.28g)

0 ≤ N∗16,p(tf ) ≤ N∗

17,p(tf ), (7.28h)

0 ≤ N∗18,p(tf ) ≤ N∗

17,p(tf ), (7.28i)

0 ≤ N∗27,p(tf ) ≤ N∗

27,p,ref (tf ). (7.28j)

MILP method A local optimum of optimization problem B1 for Configuration Iwas found after 2042 nodes (19 hours). The objective value is 6 (integrality gap is1), hence after 10 min of emulsification the constraints are met. After 2600 nodes(24 hours) the integrality gap was still 1 and the optimization was aborted.

The global optimum of optimization problem B1 for Configuration II was foundafter 19 nodes (20 min); its value is 5. Hence, after 9 min of emulsification thedesired emulsion is produced.

CVP method Results regarding the number of iterations, the calculation timeand the optimal objective value are listed in Table 7.3. Note that the lowest numberof iterations is needed when the optimizations are started from the initial guessobtained with the MILP method (from uI1 no feasible solution is found).

The results show that it is possible to produce an emulsion with a volume frac-tion of 0.55 while maintaining a similar terminal DSD and without decreasing theemulsion viscosity. The time needed to produce such an emulsion is the same forboth configurations: 10.7 min.

The terminal DSDs obtained with the reference operation procedure and theoptimal operation procedures for Configuration I and Configuration II are shown in

122

Table 7.3: CVP results of optimization problem B1.Configu- Initial No. of Calculation J0 J∗

ration guess iterations time[h] [min] [min]

I uI1 7 1.7 15.3 no feasiblesolution found

uI2 25 5.3 10.0 10.7uI3 37 12.2 10.0 10.7uI4 12 2.1 10.0 10.7

II uII1 8 1.9 15.3 10.7uII2 4 0.7 10.0 10.7uII3 25 6.0 10.0 no feasible

solution founduII4 4 0.9 9.0 10.7

Figure 7.7. The terminal DSD obtained with the optimal operation procedure forConfiguration I and II is narrower than the one obtained with the reference operationprocedure.

Because of this narrower shape (with the same peak position) the correspondingvalue of r∗ is lower (1.11 µm) than obtained with the reference operation procedure(5.68 µm). A decrease of r∗ results in an increase of the shear thinning behaviorof the emulsion (see Equation 3.68 and 3.69). Due to this it is possible to producean emulsion with an oil volume fraction of 0.55 without decreasing the emulsionviscosity (evaluated at a shear rate of 10 s−1).

With respect to this it should be noted that the high shear emulsion viscosityηe,∞ of the emulsion with the volume fraction of 0.55 is of course lower than ofthe emulsion with the volume fraction of 0.6 (the reference case). This is the casebecause the high shear viscosity is determined by the oil volume fraction and theviscosity of the continuous phase and not by the DSD. However, as mentioned earlier,certain quality attributes correlate to the viscosity at lower shear rates. For thoseemulsion products the previously described results are valuable. The evolution ofthe emulsion viscosity obtained with the reference and optimal operation procedureof Configuration I and II is shown in Figure 7.8 as function of the time.

During the oil addition the emulsion viscosity increases rapidly. The emulsionviscosity obtained with the optimal operation procedure for Configuration I andII increases only rapidly till 150 mPa s because the oil volume fraction is 0.55 in-stead of 0.6 as for the reference case. After the oil addition it continues to increase(although at a much lower speed), this is because the DSD shifts further to theleft. After 10.7 min the emulsion viscosity reaches the minimal desired value of1.03ηe,ref (10, tf ). At that moment the other constraints are also met and the pro-cess is finished.

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10−1

100

101

102

0

0.05

0.1

0.15

Droplet diameter [µm]

DS

DV

N [−

]

Reference Optimal; Config. IOptimal; Config II

Figure 7.7: Terminal normalized (volume-based) DSDs of the Piping compartmentobtained with the reference operation procedure and the optimal operation proce-dures of optimization problem B1 for Configuration I and II.

0 2 4 6 8 10 12 14 160

50

100

150

200

250

300

350

Time [min]

Em

ulsi

on v

isco

sity

at 1

0 s−

1 [mP

a s]

Reference Optimal; Config. I Optimal; Config. IIConstraints

Figure 7.8: Emulsion viscosities (evaluated at a shear rate of 10 s−1) as functionof the time for the reference operation procedure and for the optimal operationprocedures of optimization problem B1 for Configuration I and II.

124

The optimal operation procedure is as follows:

• The oil is added as fast as possible. With the maximum inlet flow rate of2.5.10−5 m3 s−1 it takes 3.7 min to reach a volume fraction of 0.55.

• For both configurations the rotor speed is set to its maximum value most ofthe time. These periods are altered with periods at which the rotor speed isset to a lower value. The exact rotor speed trajectory depends on the initialguess. The optimal rotor speed trajectory is shown in Figure 7.9 for bothconfigurations.

• The circulation flow rate is (for Configuration II) set to its maximum valueduring the total process. This is shown in Figure 7.10 together with the circu-lation flow rate corresponding to the optimal rotor speed trajectory for Con-figuration I.

0 2 4 6 8 10 120

10

20

30

40

50

60

Time [min]

Rot

or s

peed

[s−

1 ]

Optimal; Config. I Optimal; Config. II

Figure 7.9: Optimal rotor speed of optimization problem B1 as function of time forConfiguration I and II.

Increasing the circulation flow rate and the rotor speed results in a decrease ofthe mixing time and in an increase of the breakage frequency. This is the reasonwhy the optimizer has the tendency to set these variables to their maximum values.However this is not the optimal operation procedure, because at the time point thatthe position of the peak of the DSD corresponds with that of the reference DSD theemulsion viscosity is 220 mPa s whereas a value of 246 mPa s is desired.

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0 2 4 6 8 10 1210

11

12

13

14

15

Time [min]

Circ

ulat

ion

flow

rat

e [m

3 h−

1 ]

Optimal; Config. I Optimal; Config. II

Figure 7.10: Circulation flow rate trajectory obtained with the optimal operationprocedure of optimization problem B1 for Configuration I. The optimal circulationflow rate of optimization problem B1 for Configuration II is also shown.

The only way to increase the emulsion viscosity further, while maintaining theposition of the peak of the DSD, is by decreasing the rotor speed and increasing theoperation time. During the periods of less vigorous rotating the droplet breakagecontinues, however the average droplet size of the newly formed droplets is largerthan when the rotor speed would be set to its maximum value. Therefore, the righttail of the DSD continues to shift to the left whereas the peak position of the DSDremains the same.

Optimization problem B2

In the previous section it is shown how the operation procedure should be chosen inorder to produce an emulsion with a volume fraction of 0.55 but without loweringthe emulsion viscosity. Now, it will be investigated whether it is possible to increasethe emulsion viscosity even further while maintaining a similar terminal DSD.

126

The mathematical formulation of optimization problem B2 is given as

minu

J = tf (7.29a)

subject to:

x = f(x, y, u), (7.29b)0 = g(x, y, u), (7.29c)

x(0) = x0, (7.29d)φ(tf ) = 0.55, (7.29e)

ul ≤ u ≤ uh, (7.29f)1.25ηe,ref (10, tf ) ≤ ηe(10, tf ) ≤ ∞, (7.29g)

0 ≤ N∗16,p(tf ) ≤ N∗

17,p(tf ), (7.29h)

0 ≤ N∗18,p(tf ) ≤ N∗

17,p(tf ), (7.29i)

0 ≤ N∗27,p(tf ) ≤ N∗

27,p,ref (tf ). (7.29j)

MILP method After 3533 nodes (26 hours) still no integer solution was foundfor Configuration I. At that point the upper bound was 11. This implies that,in principle, there might exist an integer solution with an objective value of 11.Of course it can also be the case that no integer feasible solution exists. A possiblereason for this could be that it is simply not possible to produce the desired emulsionin 15 min (15*1 min) or less. This idea was supported by the first CVP results(started from uI1 − uI3) indicating that 17 min are needed for the production of anemulsion with the desired specifications. Therefore the optimization was abortedand a new optimization was started. The new optimization problem comprises 20instead of 15 control intervals of 1 min. Hence, the maximum operation time is now20 min.

A local optimum of optimization problem B2 for Configuration I was found after6448 nodes (65 hours). The objective value is 4 (integrality gap is 7). As twentycontrol intervals with a length of 1 min are used for the MILPs corresponding toConfiguration I, it takes 17 min of emulsification in order to satisfy the constraints.

The global optimum of optimization problem B2 for Configuration II was foundafter 127 nodes (2 hours); its value is 4. Hence, after 10.5 min of emulsification thedesired emulsion is produced.

CVP method In Table 7.4 the optimization results of optimization problem B2are listed. Note that again the lowest number of iterations is needed when theoptimizations are started from the initial guess obtained with the MILP method.

These results show that it is possible to produce an emulsion with a volumefraction of 0.55 while maintaining a similar DSD and with an even higher emulsionviscosity as obtained with the reference case. The operation time of the productionof such an emulsion is much shorter with Configuration II than with ConfigurationI. With Configuration I it takes 16.6 min whereas only 11.6 min are needed with

127

Table 7.4: CVP results of optimization problem B2.Configu- Initial No. of Calculation J0 J∗

ration guess iterations time[h] [min] [min]

I uI1 19 6.7 15.3 17.1uI2 50 20.1 10.0 max. number of

iterations reacheduI3 15 3.2 10.0 no feasible

solution founduI4 20 8.4 17.0 16.6

II uII1 26 7.3 15.3 no feasiblesolution found

uII2 11 3.7 10.0 11.6uII3 18 5.3 10.0 11.6uII4 4 2.5 10.5 11.7

Configuration II: a reduction of the operation time with 30 %. At the end of thissection it is discussed why this is the case.

With respect to these results it is worthwhile to remember that the optimalsolutions found are local solutions. Hence, it might be that the global optimaloperation times are lower than the values listed in Table 7.4. This also implies thatthe benefit of using Configuration II instead of Configuration I might be different.

In Figure 7.11 the terminal DSDs obtained with: i) the reference operation pro-cedure, ii) the optimal operation procedure for Configuration II of optimizationproblem B1, and iii) the optimal operation procedure for Configuration II of op-timization problem B2 are shown. The terminal DSD obtained with the optimaloperation procedure for Configuration I of optimization problem B2 is very similarto the terminal DSD of Configuration II and is therefore not shown.

These results show that the terminal DSD obtained with the optimal operationprocedure of optimization problem B2 for Configuration II is slightly narrower thanthe terminal DSD obtained with the optimal operation procedure of optimizationproblem B1. Because of this, the value of r∗ is also slightly lower: the terminalvalue of r∗ is 0.84 µm instead of 1.11 µm. This lower value of r∗ results in aneven stronger shear thinning behavior of the emulsion. Due to this it is possible toproduce an emulsion with an oil volume fraction of 0.55 with an emulsion viscosityof 1.25ηe,ref (10, tf ) while maintaining a similar DSD (same peak position).

The optimal operation procedure (for both configurations) is again to fill as fastas possible. The circulation flow rate is set to its maximum value for ConfigurationII; this way the mixing time is minimized. The rotor speed is chosen such thatthe DSD does not shift too far to the left. Qualitatively this is in close agreementwith the trajectory obtained with optimization problem B1: periods of vigorousrotation are altered with periods of less vigorous rotation. However, the rotationspeed, during these periods of less vigorous rotation, is considerably lower than

128

10−1

100

101

102

0

0.05

0.1

0.15

Droplet diameter [µm]

DS

DV

N [−

]

Reference Optimal B1; Config. IIOptimal B2; Config. II

Figure 7.11: Terminal normalized (volume-based) DSD of the Piping compartmentobtained with the reference and the optimal operation procedures of optimizationproblem B1 and B2 for Configuration II.

those obtained with optimization problem B1. The optimal rotor speed trajectoryfor Configuration I and II are shown in Figure 7.12. In Figure 7.13 the circulationflow rate trajectory obtained with the rotor speed trajectory for Configuration I andthe optimal circulation flow rate for Configuration II are shown.

As with optimization problem B1 the periods of less vigorous rotation are nec-essary to ensure that the DSD does not shift too far to the left while, at the sametime, the right tail of the DSD continues to decrease (which is necessary in order toreduce the value of r∗ further). During these periods of less vigorous rotation thecirculation flow rate decreases considerably for Configuration I. With ConfigurationII the pumping and shearing action are not coupled and the circulation flow rate isset to its maximum value during the total process. This is the main reason2 whythe operation time, for the production of the desired emulsion, is much smaller forConfiguration II than for Configuration I.

2Overall, the rotor speed of Configuration II is larger than that of Configuration I. This impliesthat the breakage frequency is also larger. As explained earlier this effect is less important thanthe effect on the mixing time.

129

0 2 4 6 8 10 12 14 16 180

10

20

30

40

50

60

Time [min]

Rot

or s

peed

[s−

1 ]

Optimal; Config. I Optimal; Config. II

Figure 7.12: Optimal rotor speed as function of the time of optimization problemB2 for Configuration I and II.

0 2 4 6 8 10 12 14 16 180

5

10

15

Time [min]

Circ

ulat

ion

flow

rat

e [m

3 h−

1 ]

Optimal; Config. I Optimal; Config. II

Figure 7.13: Circulation flow rate trajectory obtained with the optimal operationprocedure of optimization problem B2 for Configuration I. The optimal circulationflow rate of optimization problem B2 for Configuration II is shown as well.

130

Optimization problem B3

In this section it will be studied whether it is possible to produce an emulsion with alower volume fraction (φ = 0.5) and with the same terminal emulsion viscosity andDSD specifications as defined for optimization problem B1. Mathematically

minu

J = tf (7.30a)

subject to:

x = f(x, y, u), (7.30b)0 = g(x, y, u), (7.30c)

x(0) = x0, (7.30d)φ(tf ) = 0.5, (7.30e)

ul ≤ u ≤ uh, (7.30f)1.03ηe,ref (10, tf ) ≤ ηe(10, tf ) ≤ ∞, (7.30g)

0 ≤ N∗16,p(tf ) ≤ N∗

17,p(tf ), (7.30h)

0 ≤ N∗18,p(tf ) ≤ N∗

17,p(tf ), (7.30i)

0 ≤ N∗27,p(tf ) ≤ N∗

27,p,ref (tf ). (7.30j)

MILP method A local optimum of optimization problem B3 for Configuration Iwas found after 103 nodes (40 min). The objective value is 6 (integrality gap is 4),hence after 10 min of emulsification the constraints are met. After 1526 nodes (6hours) the integrality gap is still 3 and the optimization was aborted.

It took 21 nodes (15 min) to find the global optimum of optimization problemB3 for Configuration II; its value is 4. Hence, after 10.5 min of emulsification thedesired emulsion is produced.

CVP method Results regarding the number of iterations, the calculation timeand the optimal objective value are listed in Table 7.5. These results show that itis possible to produce an emulsion with a volume fraction of 0.5 while maintaininga similar DSD and without reducing the emulsion viscosity as obtained with thereference operation procedure. The time needed to produce such an emulsion is11.3 min for both configurations.

The terminal DSDs obtained with the reference operation procedure and withthe optimal operation procedures for Configuration I and II are shown in Figure 7.14.The terminal DSDs obtained with the optimal operation procedure for ConfigurationI and II are exactly the same and considerably narrower than the terminal DSDobtained with the optimal operation procedure of optimization problem B1. Theterminal value of r∗ is 0.67 µm, whereas it is 1.11 µm for optimization problem B1.This results in a further increase of the shear thinning behavior of the emulsion.Due to this it is possible to produce an emulsion with an oil volume fraction of 0.5and an emulsion viscosity of 1.03ηe,ref (10, tf ) in both Configuration I and II.

131

Table 7.5: CVP results of optimization problem B3.Configu- Initial No. of Calculation J0 J∗

ration guess iterations time[h] [min] [min]

I uI1 12 2.0 15.3 11.3uI2 3 0.4 10.0 11.3uI3 3 0.4 10.0 11.3uI4 2 0.3 10.0 11.3

II uII1 6 1.3 15.3 11.4uII2 3 0.5 10.0 11.3uII3 14 4.4 10.0 11.3uII4 4 1.0 10.5 11.3

10−1

100

101

102

0

0.05

0.1

0.15

Droplet diameter [µm]

DS

DV

N [−

]

Reference Optimal; Config. IOptimal; Config II

Figure 7.14: Terminal normalized (volume-based) DSD of the Piping compartmentobtained with the reference and the optimal operation procedures of optimizationproblem B3 for Configuration I and II.

132

The optimal operation procedures for Configuration I and II are:

• To fill as fast as possible. With the maximum inlet flow rate of 2.5.10−5

m3 s−1 it takes 2.9 min to reach a volume fraction of 0.5.

• To rotate at the maximum speed of 50 s−1.

• To circulate at the maximum circulation flow rate of 14 m3 h−1 during thetotal process.

• To emulsify during 11.3 min.

The circulation flow rate, corresponding to a rotation speed of 50 s−1, of Con-figuration I remains at its maximum value of 14 m3 h−1 during the total process.Therefore, the simulations are exactly the same for Configuration I and II for thegiven optimal operation procedures. With a volume fraction of 0.6 the circulationflow rate started to decrease (at a rotor speed of 50 s−1) after 3 min. The reasonthat the circulation flow rate remains now at its maximum value is that the pressuredrop over the colloid mill is lower due to the lower emulsion viscosity in the colloidmill.

The optimal rotor speed trajectories of optimization problems B1 and B2 bothshow the following qualitative behavior: periods of vigorous rotating are alteredwith periods of less vigorous rotating. These periods of less vigorous rotating are nolonger present in the optimal rotor speed trajectory of optimization problem B3: itremains at its maximum value during the total process. The periods of less vigorousrotating were necessary to ensure that the DSD wouldn’t shift too far to the left.With the volume fraction of 0.5 the rotor speed can be kept at its maximum value,because the emulsion viscosity in the colloid mill is lower than with a volume fractionof 0.55. Because of this, the position of the peak of the DSD will not shift too farto the left.

7.4.4 Optimization problem C

The first objective of optimization problem C is to establish if the operation proce-dure can be chosen such that a multi-modal DSD is produced. The second objectiveis to determine how the operation procedure should be chosen to produce such aDSD in minimal time.

The constraints of optimization problem C are formulated rather loose in ordernot to constrain the feasible region too much. It is only enforced that: i) the terminalDSD should be at least bi-modal; other multi-modal DSDs are however also allowed,and ii) the number of droplets with a diameter larger than 85 µm should be “small”.To be more specific:

• The value of N∗20,p(tf ) should be at least 2.5.10−3 higher than the value of

N∗25,p(tf ).

• The value of N∗30,p(tf ) should be at least 2.5.10−3 higher than the value of

N∗25,p(tf ).

133

• The value of N∗38,p(tf ) (corresponding to droplets with a diameter of 85 µm)

should be smaller than or equal to 1.10−3.

Note that nor the position of the peaks nor their height have been specified. Becauseof this loose formulation the optimizer has more freedom for the choice of the inputtrajectories. The mathematical formulation of optimization problem C is now givenas

minu

J = tf (7.31a)

subject to:

x = f(x, y, u), (7.31b)0 = g(x, y, u), (7.31c)

x(0) = x0, (7.31d)φ(tf ) = 0.6, (7.31e)

ul ≤ u ≤ uh, (7.31f)

0 ≤ N∗25,p(tf ) ≤ N∗

20,p(tf ) + 2.5.10−3, (7.31g)

0 ≤ N∗25,p(tf ) ≤ N∗

30,p(tf ) + 2.5.10−3, (7.31h)

0 ≤ N∗38,p(tf ) ≤ 1.10−3. (7.31i)

MILP method After 12598 nodes (50 hours) still no integer feasible solution wasfound for optimization problem C for Configuration I. Although the upper boundwas still 10 the optimization was aborted.

It took 1257 nodes (15 hours) to find the global optimum of optimization problemC for Configuration II; its value is 6. Hence, after 7.5 min of emulsification anemulsion with a multi-modal DSD is obtained.

CVP method From none of the initial guesses a feasible solution was obtainedfor Configuration I: the solver crashed after a few iterations. Note, that this doesnot necessarily imply that it is physically impossible to produce the desired emulsion(for the given emulsion composition, equipment dimensions etc.) with ConfigurationI. Other possible explanations are:

• The SQP solver was not able to find a feasible solution because the initialguesses were “too far” from a local optimum. In other words: it can not beguaranteed that the NLP has no feasible solution.

• In order to arrive at an NLP the original infinite dimensional dynamic opti-mization problem was approximated: the control profiles u(t) were approxi-mated by piecewise equally spaced time intervals. Hence, even if the NLP doesnot have a feasible solution it can still be the case that the original optimizationproblem has a feasible solution.

Despite this it is most likely physically impossible to produce the desired emulsionwith Configuration I, as will be discussed later.

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For Configuration II a feasible solution was found when the SQP solver wasprovided with the initial guess obtained with the MILP method. It took 6 iterations(2.5 hours) to find this solution. No feasible solution was found from the other initialguesses: the solver crashed after a few iterations.

The evolution of the normalized (volume-based) DSD of the Piping compartmentobtained with the optimal operation procedure is shown in Figure 7.15. At the startof the process a peak with relatively small droplets at 5 µm is formed. After 2.4 min(3 control intervals) the value of N∗

20,p is 3.6.10−2 larger than that of N∗25,p. From

that moment on a second peak is formed. This peak consists of relatively largedroplets (the peak position is at 24 µm). It takes 3.9 min to ensure that the valueof N∗

30,p is at least 2.5.10−3 larger than the value of N∗25,p. At that time point

Equation 7.31g and 7.31h are satisfied. However, the constraint enforcing that N∗38,p

should be less than or equal to 1.10−3 (Equation 7.31i) is not satisfied yet. Thisis not unexpected because during the oil addition (which takes 4.2 min) large oildroplets enter the vessel. The process has to be operated for an other 4.1 min inorder to ensure that Equation 7.31i is also satisfied. The terminal DSD obtainedafter 7.89 min is shown in Figure 7.16.

10−1

100

101

102

103 0

2

4

6

8

0

0.02

0.04

0.06

0.08

0.1

Time [min]

Droplet diameter [µm]

DS

DV

N [−

]

Figure 7.15: Evolution of the normalized (volume-based) DSD of the Piping com-partment obtained with the optimal operation procedure of optimization problem Cfor Configuration II.

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10−1

100

101

102

103

0

0.02

0.04

0.06

0.08

0.1

Droplet diameter [µm]

DS

DV

N [−

]

N*38,p

N*30,p

N*25,p

N*20,p

Figure 7.16: Terminal normalized (volume-based) DSD of the Piping compartmentobtained with the optimal operation procedure of optimization problem C for Con-figuration II.

The optimal operation procedure for Configuration II to produce this bi-modalDSD as fast as possible is:

• To rotate vigorously during a short time period at the start of the process.During this time period the peak at 5 µm droplets is formed. After this shorttime period the process should be continued at a low rotation speed. Duringthis period the remaining large droplets are broken down and the second peakat 24 µm is produced. The rotor speed trajectory is shown in Figure 7.17.

• To circulate and to fill as fast as possible.

• To emulsify during 7.89 min.

Using the same rotation speed trajectory for Configuration I would result ina circulation flow rate of 4.10−3 m3 h−1when the vessel is filled completely. Thismeans that there is almost no circulation. Therefore, it seems likely that it is indeedphysically impossible to produce the desired DSD with Configuration I (for the givenemulsion properties, equipment dimensions etc.).

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0 1 2 3 4 5 6 7 80

10

20

30

40

50

60

Time [min]

Rot

or s

peed

[s−

1 ]

Figure 7.17: Optimal rotor speed as function of the time of optimization problem Cfor Configuration II.

7.4.5 Sensitivity analysis

In order to get an idea about the sensitivity of the optimization results to changes ofthe parameter values (e.g. values of the fit parameters, fluid properties, equipmentdimensions), a sensitivity analysis is performed. Parameter values usually containerrors or uncertainties. Therefore, information concerning the sensitivity of theoptimization results to changes of parameter values is important. The sensitivitiesare determined as follows for optimization problem A, B1, B2 and B3 for bothequipment configurations:

i) The terminal peak position, the value of N∗27,p(tf ) and the value of ηe(10, tf )

are determined for the base case,

ii) Each parameter is changed separately (one at a time) with +5 % from thenominal value and then the new terminal peak position, the value of N∗

27,p(tf )and of ηe(10, tf ) are calculated.

For optimization problem C the values of N∗20,p(tf )−N∗

25,p(tf )−2.5.10−3, N∗30,p(tf )−

N∗25,p(tf )−2.5.10−3 and N∗

38,p(tf ) are determined. The results are listed in AppendixD. The results show for optimization problem A, B1, B2 and B3 that:

• The effect of changes of the parameters on the peak position is relatively small.The terminal peak position of the base case corresponds with N∗

17,p(tf ) (dropletdiameter of 1.85 µm). Changing the parameter values has a modest effect onthe peak position: it varies between N∗

17,p(tf ) and N∗16,(tf ) (droplet diameter of

1.54 µm). The only exception is caused by c1Pcmfor all optimization problems

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with Configuration I. Changing this parameter with +5 % results in a peakposition at N∗

31,p(tf ) corresponding with a droplet diameter of 23.7 µm. Thereason for this is that, because of the smaller gap, the pressure drop over thecolloid mill is increased, resulting in a considerably lower circulation flow rate.Therefore, the DSD shifts much less further to the left than with the base case.With Configuration II this is not the case, because there the circulation flowrate is a control variable.

• The right tail of the terminal DSD of the base case is rather sensitive forchanges of the parameters. Deviations larger than 10 % are listed in Table 7.6and 7.7 for Configuration I and II respectively.

Table 7.6: Sensitivity of N∗27,p(tf ) for changes of the parameters for

Configuration I. Deviations, in percentage, are shown between brackets.Opt A Opt B1 Opt B2 Opt B3

c1Pcm(+1174) c1Pcm

(+7841) c1Pcm(+19007) c1Pcm

(+24701)φ (+69.9) φ (+76.0) φ (+175.3) φ (+45.0)

c2Pcm(−34.3) Dr,i (−33.3) c2Pcm

(−77.2) Dr,i (−41.2)c3S (−17.1) c3S (−26.4) Dr,o (−33.6) tf (−31.1)Dr,o (−16.8) tf (−18.3) Ncm(t) (−28.9) c3S (−30.7)tf (−16.8) hcm (−20.3) tf (−27.2) hcm (−24.7)

Ncm(t) (−12.4) c3S (−24.8)Dr,i (−12.2) hcm (−22.9)hcm (−11.8) ηc (+15.7)

lcm (+10.7)

Table 7.7: Sensitivity of N∗27,p(tf ) for changes of the parameters for

Configuration II. Deviations, in percentage, are shown between brackets.Opt A Opt B1 Opt B2 Opt B3

φ (+39.1) φ (+48.9) φ (+53.4) φ (+45.1)Dr,i (−20.5) Dr,i (−34.0) Dr,i (−37.2) Dr,i (−41.2)tf (−18.3) tf (−26.8) tf (−29.0) tf (−31.1)c3S (−17.8) c3S (−26.4) c3S (−28.2) c3S (−30.7)hcm (−11.8) hcm (−20.2) hcm (−22.2) hcm (24.7)

Fcm,p(t) (−15.7) ηd (+11.9) Fcm,p(t) (−19.9)

• The value of ηe(10, tf ) is less sensitive for changes in the parameters thanN∗

27,p(tf ). Deviations larger than 10 % are listed in Table 7.8 and 7.9 forConfiguration I and II respectively.

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Table 7.8: Sensitivity of ηe(10, tf ) for changes of the parameters forConfiguration I. Deviations, in percentage, are shown between brackets.

Opt A Opt B1 Opt B2 Opt B3c1Pcm

(+1174) c1Pcm(+7841) c1Pcm

(+19007) c1Pcm(+24701)

φ (+69.9) φ (+76.0) φ (+175.3) φ (+45.0)c2Pcm

(−34.3) Dr,i (−33.3) c2Pcm(−77.2) Dr,i (−41.2)

c3S (−17.1) c3S (−26.4) Dr,o (−33.6) tf (−31.1)Dr,o (−16.8) tf (−18.3) Ncm(t) (−28.9) c3S (−30.7)tf (−16.8) hcm (−20.3) tf (−27.2) hcm (−24.7)

Ncm(t) (−12.4) c3S (−24.8)Dr,i (−12.2) hcm (−22.9)hcm (−11.8) ηc (+15.7)

lcm (+10.7)

Table 7.9: Sensitivity of ηe(10, tf ) for changes of the parameters forConfiguration II. Deviations, in percentage, are shown between brackets.

Opt A Opt B1 Opt B2 Opt B3φ (+39.1) φ (+48.9) φ (+53.4) φ (+45.1)

Dr,i (−20.5) Dr,i (−34.0) Dr,i (−37.2) Dr,i (−41.2)tf (−18.3) tf (−26.8) tf (−29.0) tf (−31.1)c3S (−17.8) c3S (−26.4) c3S (−28.2) c3S (−30.7)hcm (−11.8) hcm (−20.2) hcm (−22.2) hcm (24.7)

Fcm,p(t) (−15.7) ηd (+11.9) Fcm,p(t) (−19.9)

The sensitivity depends on the optimization case and the configuration studied.However, in general it can be concluded that:

• The values of φ and tf should be adjusted carefully during the implementationof the optimal operation procedures of the optimization problems for Config-uration I as well as II. This is also the case for Ncm(t) for Configuration I andfor Fcm,p(t) for Configuration II. It is estimated that it should be possible toadjust the parameters tf and Ncm(t) within 0.5 %; φ can be adjusted within0.01 %3. Perturbing the parameter φ with +0.01 % and tf and Ncm(t) with+0.5 % gives deviations of at most -3.3 and +1.3 % for N∗

27,p(tf ) and ηe(10, tf )respectively; this is considered acceptable.

• The values of Dr,o, Dr,i and hcm should be measured accurately. The values ofthese parameters (listed in Table 5.1) are measured within 0.8 %. Perturbingthese parameters with +0.8 % gives deviations of at most −5.5 % and +2.1 %for N∗

27,p(tf ) and ηe(10, tf ) respectively; this is considered acceptable.

3In Section 6.2 it is mentioned that the bias between the simulated and the measured emulsionviscosity of experiment P1 is eliminated by using a value of φ that is 1 % lower than the nominalvalue (listed in Table 6.3). For this experiment (batch-emulsification) this is a reasonable value,because during the addition of the oil/water mixture to the vessel some fluid was spilled.

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• The values of the fit parameters c1Pcm, c2Pcm

should be estimated accurately;to a lesser extent this is also the case for c3S . The results listed in Table 6.8show that the value of c3S is estimated as 1.88 ± 0.11. Hence, this value isdetermined within 5.9 %. Given the results of the sensitivity analysis thisis relatively large. Therefore it is desirable to estimate this parameter valuemore accurate. The results listed in Table 6.2 show that the values of c1Pcm

and c2Pcmare estimated as 5.27 ± 32.2 and 5.54 ± 31.4 cm respectively. The

0.95-confidence intervals are extremely large because different combinations ofthese parameter values can give the same circulation flow rate; in the field ofSystem and Control Theory this is called an identification problem (Ljung,1999). Note that this does not imply that these parameters can be changedindependently with for example 30 cm. As a matter of fact performing theparameter estimation described in Section 6.1.3 with the parameter c2Pcm

setto 5.54 cm gives an estimated value of 5.28 ± 0.027 cm (0.5 %) for c1Pcm

.Perturbing c1Pcm

with +0.5 % gives deviations of at most +82.6 % and −29.1% for N∗

27,p(tf ) and ηe(10, tf ) respectively; this is not considered acceptable.Therefore it is desirable to estimate these parameters values more accurate.

• The results of optimization problem C are relative sensitive for changes of thevalue of the parameter σ of the inlet flow. Therefore it is desirable to estimatethis parameter value more accurate.

7.5 Conclusions

The results of the optimization studies show that (categorized per sub-problem A,B and C):

A) The results of the optimization problems A, B and C show that:

– The operation time with Configuration II was at most 30 % lower (opti-mization problem B2) than with Configuration I.

– For one optimization case (optimization problem C) the optimizer wasnot able to find an operation procedure enabling the production of thedesired emulsion with Configuration I. Such an operation procedure wasfound for Configuration II.

– For three other cases (optimization problem A, B1 and B3) the operationtime differed only slightly. The operation time was, for these cases, atmost 4 % less with Configuration II than with Configuration I.

Overall it can be concluded that Configuration II allows the production of awider range of emulsions and is always at least as fast as Configuration I.

It should be mentioned that the reductions in operation time do depend onthe product composition. For products with lower emulsion viscosities (causedby a lower volume fraction and/or continuous phase viscosity) it is expectedthat the reduction in operation time would be less, because for those products

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the circulation flow rate is expected to be higher. Similarly, for products withhigher emulsion viscosities, which will be the case for most food emulsions,larger benefits are expected. With respect to this it is worthwhile to rememberthat in the calculations it is assumed, that Configuration II is equipped witha “perfect” pump. In this pump shear rates are negligible (compared with theshear rate in the gap of the colloid mill) and the circulation flow rate can becontrolled, independently of the emulsion viscosity, between 0 and 14 m3 h−1.This implies that the obtained benefits with Configuration II can be less inpractice.

B) In the optimization cases (optimization problems B1, B2 and B3) studied theoil volume fraction was lowered from 0.6 to 0.5. Based on the results of thesecases it can be concluded that:

– It is possible to produce emulsions with less oil while maintaining a similarDSD (same position of the peak value) and value of ηe(10, tf ) by adaptingonly the operation procedure.

– This can be realized by narrowing the DSD, which causes an increaseof the shear thinning behavior of the emulsion with the lower volumefraction, and hence of ηe(10, tf ). This way ηe(10, tf ) of the emulsionwith the lower oil volume fraction can become at least the same as of theemulsion with the original oil volume fraction while maintaining a similarDSD. With respect to this it is worthwhile to mention that although it waspossible to narrow the DSD it was not possible to produce an emulsionconsisting of equally sized droplets.

– The operation procedure for the production of such an emulsion in mini-mal time is product dependent.

The results of these 3 optimization cases depend strongly on the Viscositymodel. As explained in Section 3.4 it is proposed to calculate the depletion flownumber in case of polydispersity with Equation 3.72. The results so far suggestthat that this is justified indeed. However, more experimental work is neededto gain more confidence in this model. Further, it should be emphasized thatthese results do not imply that it is always possible to produce emulsions withless oil while maintaining a similar DSD and emulsion viscosity. This dependsof course on e.g. the desired volume fraction: when the volume fraction waslowered to 0.4 no feasible solution was found.

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C) The results of optimization problem C show that:

– It is possible to produce a bi-modal DSD with Configuration II.

– Such a DSD can be produced by vigorous rotation during a short pe-riod of time at the start of the process. During this time period a peakwith relatively small droplets is formed. After this short time period theprocess must be continued at a low rotation speed. During this periodthe remaining large droplets are broken down and a second peak withrelatively large droplets is formed.

In order to verify if these results are also obtained in practice the calculated op-timal input trajectories should be implemented in the equipment used and the sim-ulated terminal DSD and emulsion viscosity should be compared with the measuredvalues.

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Chapter 8

Conclusions andrecommendations

In Chapter 1 the problem statement was formulated as follows:

Investigate, based on a model, how the control variables should be chosen as afunction of the time in order to produce emulsions (for a given volume fraction)with a certain, predefined, terminal DSD and/or emulsion viscosity (evaluated at acertain shear rate) in minimal time.

Related to this problem statement the sub-problems A, B and C were defined; theyare briefly reviewed here.

A) Two equipment configurations are considered in this thesis: Configuration I,where the pumping and shearing action are coupled, and Configuration II,where these actions are not coupled. Sub-problem A aims at establishingwhich configuration enables the fastest production.

B) It is desirable to produce emulsions with less oil while maintaining a similarDSD and emulsion viscosity. Sub-problem B aims at examining if this is possi-ble by adapting only the operation procedure and if so, how the control inputsshould be chosen as a function of the time to produce such an emulsion inminimal time.

C) Sub-problem C aims at establishing if and how emulsions (for a given volumefraction) with a multi-modal DSD can be produced in minimal time.

8.1 Conclusions

In order to solve the previously described sub-problems a model was developed andseveral optimization problems were formulated. The conclusions of this thesis arecategorized as follows. First in Section 8.1.1 conclusions regarding the modeling

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are given. Then in Section 8.1.2 the conclusions, answering the sub-problems, aredescribed. Finally, in Section 8.1.3, the overall conclusion is given.

8.1.1 Modeling

The model comprises 4 main parts: the Flow rate model, the Viscosity model, theDroplet model and the Reactor model. The conclusions are categorized accordingly.

• Flow rate model: The results of the simulations are qualitatively in goodagreement with the experimental results. Both the simulations and the mea-surements show:

– An increase of the flow rate as the rotor speed increases. At some pointthe flow rate does no longer increase and a plateau value is reached.

– That the increase rate of the flow rate as function of the rotor speeddecreases as the fluid viscosity increases.

Differences between the simulated and the measured flow rates are betweenapproximately -10 and 10 % for rotor speeds between 5 and 45 s−1 for (New-tonian) fluids with viscosities between 12 and 50 mPa s. The relative errorsare randomly divided for flow rates between 1 and 11 m3 h−1, suggesting thatthe model structure is correct within this interval. For flow rates between11 and 14 m3 h−1 the relative errors show a clear trend for all mixtures: thesimulated flow rates are higher than the measured flow rates and the relativeerrors decrease (from 15 to 0 %) as the flow rate increases. This indicates thatthe model structure is no longer correct. Given the size of the relative errorsand the model purposes this is not considered as a problem.

• Viscosity model: The results of the simulations are qualitatively in good agree-ment with the experimental results. Both the simulations and the measure-ments show:

– A decrease of the emulsion viscosity as the shear rate is increased; thedecrease rate decreases as the shear rate increases.

– That the shear thinning behavior increases as r∗ decreases.

Differences between the simulated and the measured emulsion viscosity valuesof experiment P1 and P2 are between -25 and 25 % for shear rates between 5and 100 s−1 and between -15 and 10 % for shear rates between 100 and 300 s−1.As explained in Section 6.2 these deviations can be explained by measurementuncertainty and by small deviations of the values of the experimental conditions(e.g. volume fraction and r∗) from their nominal values. The results furthersuggest that it is justified to calculate the depletion flow number with Equation3.72 in case of polydispersity.

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• Droplet and Reactor model: The results of the simulations are qualitativelyin reasonable agreement with the experimental results. Both the simulationsand the measurements show that:

– The DSD shifts towards smaller droplet sizes (to the left) during theprocess.

– The simulated and measured DSDs shift further to the left (lower valueof the d43) when the volume fraction and/or the rotor speed increases.

– Although the shape differs slightly, the position and the width of thesimulated DSDs are in reasonable agreement with the measured values.

– The decrease rate of the simulated d43 is in good agreement with thatof the measured d43: it decreases relatively fast during the first 15 minof emulsification after which it decreases much slower during the last 45min.

Relative differences between the measured and simulated d43 vary between-20 and 30 % for experiment P1-P5. The slight differences in shape can notbe explained completely by measurement uncertainty nor by deviations of thevalues of the experimental conditions from their nominal values. Therefore, itis concluded that the model structure of Droplet model F is not fully correct.Despite these slight differences in shape, it can be concluded that there isreasonable agreement between the simulated and measured DSDs.

Given the overall performance of the total model (Flow rate model, Viscositymodel, Droplet model F and the Reactor model) it is expected that the model qua-lity is sufficient to render practical relevant optimization results. It is expected thatthe calculated operation procedure, for reaching a certain, predefined, terminal DSDand/or emulsion viscosity in minimal time, will be qualitatively correct. However,quantitatively (e.g. exact values of the operation time and/or of the control vari-ables), differences are expected compared to the trajectories found when the modelwould be “perfect”.

8.1.2 Operation procedure

In this section conclusions related to the problem formulation are presented, cate-gorized per the sub-problems A, B and C.

A) The results of the optimization problems A, B and C show that:

– The operation time with Configuration II was at most 30 % lower (opti-mization problem B2) than with Configuration I.

– For one optimization case (optimization problem C) the optimizer wasnot able to find an operation procedure enabling the production of thedesired emulsion with Configuration I. Such an operation procedure wasfound for Configuration II.

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– For three other cases (optimization problem A, B1 and B3) the operationtime differed only slightly. The operation time was, for these cases, atmost 4 % less with Configuration II than with Configuration I.

Overall it can be concluded that Configuration II allows the production of awider range of emulsions and is always at least as fast as Configuration I.

It should be mentioned that the reductions in operation time do depend onthe product composition. For products with lower emulsion viscosities (causedby a lower volume fraction and/or continuous phase viscosity) it is expectedthat the reduction in operation time would be less, because for those productsthe circulation flow rate is expected to be higher. Similarly, for products withhigher emulsion viscosities, which will be the case for most food emulsions,larger benefits are expected. With respect to this it is worthwhile to rememberthat in the calculations it is assumed, that Configuration II is equipped witha “perfect” pump. In this pump shear rates are negligible (compared with theshear rate in the gap of the colloid mill) and the circulation flow rate can becontrolled, independently of the emulsion viscosity, between 0 and 14 m3 h−1.This implies that the obtained benefits with Configuration II can be less inpractice.

B) In the optimization cases (optimization problems B1, B2 and B3) studied theoil volume fraction was lowered from 0.6 to 0.5. Based on the results of thesecases it can be concluded that:

– It is possible to produce emulsions with less oil while maintaining a similarDSD (same position of the peak value) and value of ηe(10, tf ) by adaptingonly the operation procedure. Hence, the addition of e.g. starch is notconsidered.

– This can be realized by narrowing the DSD, which causes an increaseof the shear thinning behavior of the emulsion with the lower volumefraction, and hence of ηe(10, tf ). This way ηe(10, tf ) of the emulsionwith the lower oil volume fraction can become at least the same as of theemulsion with the original oil volume fraction while maintaining a similarDSD.

– The operation procedure for the production of such an emulsion in mini-mal time is product dependent.

The results of these 3 optimization cases depend strongly on the Viscositymodel. As explained in Section 3.4 it is proposed to calculate the depletion flownumber in case of polydispersity with Equation 3.72. The results so far suggestthat that this is justified indeed. However, more experimental work is neededto gain more confidence in this model. Further, it should be emphasized thatthese results do not imply that it is always possible to produce emulsions withless oil while maintaining a similar DSD and emulsion viscosity. This dependsof course on e.g. the desired volume fraction: when the volume fraction waslowered to 0.4 no feasible solution was found.

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C) The results of optimization problem C show that:

– It is possible to produce a bi-modal DSD with Configuration II.

– Such a DSD can be produced by vigorous rotation during a short pe-riod of time at the start of the process. During this time period a peakwith relatively small droplets is formed. After this short time period theprocess must be continued at a low rotation speed. During this periodthe remaining large droplets are broken down and a second peak withrelatively large droplets is formed.

The previously described results show that the way the control variables shouldbe chosen as a function of the time in order to produce emulsions with a certain,predefined, terminal DSD and/or emulsion viscosity in minimal time, does stronglydepend on the product composition and on the desired DSD and/or emulsion visco-sity.

8.1.3 Overall conclusion

In Chapter 1 it is mentioned that the current operating procedure has the followingkey limitations:

a) The values of the inlet flow rate, the stirrer and rotor speed and the produc-tion time are fixed for a certain product. That means that every time thate.g. a certain mayonnaise type is produced the same operation procedure isapplied. This does however not imply that the quality of this mayonnaise willbe the same from batch to batch. On the contrary, due to variations in e.g.the surfactant quality, the oil quality or in the oil temperature the mayonnaisequality will fluctuate from batch to batch. This is not desirable and it mighteven be that the product quality specifications are not met and then the pro-duct has to be classified as off-spec. From a cost point of view this is clearlyundesirable.

b) For some new developed products a large experimental effort is needed beforeit is possible to produce the product at industrial scale with a similar qualitycompared to the prototype product that was produced in the kitchen. Hence,this could lead to a large time consuming effort, implying possible high costsand the risk that competitors might launch a similar product earlier.

c) Due to time pressure and lack of resources it is most of the time not in-vestigated how the process could be operated at its optimum. Quite oftenthe experimental effort is stopped as soon as an operation procedure has beenfound enabling the production of the product with the desired product quality.However, it might be that it is possible to produce the same or a comparableproduct quality with a different operation procedure that takes for exampleless time.

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In the work reported a first step is set in order to overcome these limitations.A model describing the DSD and the emulsion viscosity as function of the timewas developed and several off-line optimization studies were performed. Detailedconclusions regarding those two items were given in Section 8.1.1 and Section 8.1.2respectively. Although the optimization studies have been performed for a modelemulsion, small scale equipment and are not yet experimentally validated, the re-sults of this work strongly suggest that it is possible to minimize the productiontimes (limitation c) and to shorten the product development times for new pro-ducts (limitation b). The developed model can also be used as a starting point forthe development of a feedback controller, which would enable the manufacturing ofproducts with consistent high product quality (limitation a). Therefore it is worth-while and recommended to expand this research work in the direction of industrialemulsions.

8.2 Recommendations for future research

The following recommendations are given for future research:

1. Implement the calculated optimal input trajectories in the equipment usedand compare the simulated terminal DSD and emulsion viscosity with themeasured values. If these are in agreement, the model has to be adapted forindustrial emulsions and equipment dimensions (recommendation 2). Based onthe results it can also be decided that the model has to be improved further.

2. Adapt the model for industrial emulsions and equipment dimensions and toperform optimization studies with this model. In Chapter 1 it is already des-cribed that a white-box model would be highly desirable, because such a modelcan be relatively easily adapted to other product compositions and equipmentdimensions. However, given the complexity of emulsification processes, thiswas not considered feasible and in this work it was decided to develop a grey-box model instead. Consequence of this is however that it is more laboriousto adapt the model to other product compositions and equipment dimensions.It would be at least necessary to perform experiments in the large scale equip-ment with an industrial emulsion in order to estimate the values of the fitparameters again.

3. Design of a feedback controller in order to be able to produce products ofconsistent high product quality and therewith reduce the number of productsthat are classified as off-spec.

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154

Glossary of symbols

a fraction [-]am micelle radius [m]A wetted cross section area [m2]Aoil oil surface area [m2]ADSD total surface area of measured DSD [m2]b fraction [-]bg groove width [m]c constant [m3 m−2]cPA

fit parameter [-]c1PB

fit parameter [-]c2PB

fit parameter [-]c3PB

fit parameter [-]c4PB

fit parameter [-]cPC

fit parameter [-]c1Pcm

fit parameter [m]c2Pcm

fit parameter [m]c∆PC

fit parameter [-]cS fit parameter [-]c2S fit parameter [s−1]c3S fit parameter [-]c1ν fit parameter [-]c2ν fit parameter [-]c3ν fit parameter [-]C1 fit parameter [-]C2 fit parameter [-]CMC critical micelle concentration [mol m−3]d initial droplet diameter [m]d43 mean diameter given by Equation 6.6 [m]dc maximum droplet diameter of a channel [m]dcr maximum stable droplet diameter [m]dcm1

cr maximum stable droplet diameter due to breakage [m]in the gap of the colloid mill

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dcm2cr maximum stable droplet diameter due to breakage [m]

in a groove of the colloid millde

cr maximum stable droplet diameter due to breakage [m]in elongational flow

dgcr maximum stable droplet diameter due to breakage [m]

in the gap between the stirrer blade and the vessel wallds

cr maximum stable droplet diameter due to breakage [m]in the boundary layer

dtcr maximum stable droplet diameter due to breakage [m]

in turbulent flowD inner cylinder diameter [m]Dh hydraulic diameter [m]Dr,i rotor diameter at the entrance [m]Dr,o rotor diameter at the exit [m]Dst,i stator diameter at the entrance [m]Dst,o stator diameter at the exit [m]Dp pipe diameter [m]Dr mean rotor diameter [m]Dst mean stator diameter [m]Dst stirrer diameter [m]f Darcy-Weisbach friction factor [-]f fit parameter [-]F flow rate [m3 s−1]Fcm,p circulation flow rate [m3 s−1]Fcm1,p flow rate Colloid mill gap to the Piping compartment [m3 s−1]Fcm2,p flow rate Colloid mill groove to the Piping compartment [m3 s−1]Fin inlet flow rate [m3 s−1]Fg flow rate in a groove of the colloid mill [m3 s−1]Fo outlet flow rate [m3 s−1]Fv,cm1 flow rate from the Vessel [m3 s−1]

to the Colloid mill gap compartmentFv,cm2 flow rate from the Vessel [m3 s−1]

to the Colloid mill groove compartmentFp,v flow rate from the Piping to the Vessel compartment [m3 s−1]Fcm,p simulated flow rate [m3 s−1]Fld depletion flow number [-]h net generation of droplets [m−3 s−1]h+ source term [m−3 s−1]h− sink term [m−3 s−1]hc gap width [m]hcm gap width between the rotor and the stator [m]hg groove depth [m]hst gap width between scraper blade and vessel wall [m]

156

hst,bl height of scraper blade [m]i index variable [-]j index variable [-]J objective functionk Boltzman constant [J K−1]k index variable [-]K1 constant [-]K2 constant [-]Kf loss coefficient [-]Kf,e loss coefficient exit [-]Kf,elbow loss coefficient elbow [-]Kf,exit loss coefficient exit [-]lcm rotor length [m]Lp straight pipe length [m]Lst width of scraper blade [m]m consistency index [Nsnm−2]mc surfactant concentration [mol m−3]M number of grid points [-]Ms molecular weight of the surfactant [kg mol−1]n flow index [-]n number-based concentration of droplets [m−3 m−3]

per fluid volumeni,k contribution to the population at the ith representative [-]

size due to the breakage of a droplet with volume vk

nv number-based concentration of droplets [m−3 m−3]in the Vessel compartment

ncm1 number-based concentration of droplets [m−3 m−3]in the Colloid mill gap compartment

ncm2 number-based concentration of droplets [m−3 m−3]in the Colloid mill groove compartment

ng number of grooves [-]nin number-based concentration of droplets [m−3 m−3]

in the inlet flownp number-based concentration of droplets [m−3 m−3]

in the Piping compartmentnr number of repeated readings [-]ns number of modes [-]nu number of control variables [-]ny number of algebraic variables [-]nx number of state variables [-]N revolution speed [s−1]Ncm rotor speed [s−1]

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Nexp number of experiments [-]Nd number of samples per experiment [-]Ni,cm1 number of droplets with volume vi per emulsion [m−3]

volume in the Colloid mill gap compartmentNi,cm2 number of droplets with volume vi per emulsion [m−3]

volume in the Colloid mill groove compartmentNi,in number of droplets with volume vi [m−3]

per inlet flow volumeNi,p number of droplets with volume vi per [m−3]

emulsion volume in the Piping compartmentN∗

i,p normalized (volume based) value of Ni,p [-]N∗

i,p,l lower bound of N∗i,p [-]

N∗i,p,u upper bound of N∗

i,p [-]Ni,v number of droplets with volume vi per [m−3]

emulsion volume in the Vessel compartmentN∗

k,p measured normalized (volume-based) value of Nk,p [-]N∗

k,p simulated normalized (volume-based) value of Nk,p [-]Nst stirrer speed [s−1]pL Laplace pressure [Pa]P wetted perimeter [m]P weighting matrixP ′ probability density function [m−3]P ′

A probability density function Droplet model A [m−3]P ′

B probability density function Droplet model B [m−3]P ′

C probability density function Droplet model C [m−3]q measurement valueq1 elliptical coordinateq2 elliptical coordinateq mean value of nr measurementsr initial droplet radius [m]r coordinate axis [m]r∗ droplet radius given by Equation 3.73 [m]R inner cylinder radius [m]R weighting matrixRB birth term [m−3 s−1]RD death term [m−3 s−1]Rr rotor radius [m]Reb Reynolds number for the bulk flow of the vessel [-]Reg Reynolds number in a groove of the colloid mill [-]Rex Reynolds number over a flat plate [-]Rem modified Reynolds number [-]ReMR generalized Reynolds number [-]

158

Rez axial Reynolds number [-]Rep Reynolds number in piping [-]s experimental standard deviations index variable [-]si droplet volume [m3]S breakage frequency [s−1]Sad amount of adsorbed surfactant [mol]Sex excess surfactant [mol]Sk breakage frequency of droplets with volume vk [s−1]t time [s]t variable of Student t distribution [-]tads adsorption time [s]tb breakup time [s]tf operation time [s]T temperature [K]u velocity in axial direction at radius r [m s−1]u control variablesul lower bound of the control variablesuh upper bound of the control variablesv droplet volume [m3]v mean droplet volume [m3]v1 droplet volume [m3]v2 droplet volume [m3]v1 mean droplet volume [m3]v2 mean droplet volume [m3]va mean axial speed [m s−1]vi representative volume [m3]vp mean fluid velocity in piping [m s−1]vtip impeller tip speed [m s−1]v∞ free fluid velocity [m s−1]V emulsion volume [m3]Vc continuous phase volume [m3]Vcm emulsion volume in the colloid mill [m3]Vcm1 emulsion volume in the [m3]

Colloid mill gap compartmentVcm2 emulsion volume in the [m3]

Colloid mill groove compartmentVoil oil volume [m3]Voil,DSD oil volume of the measured DSD [m3]Vp emulsion volume in the Piping compartment [m3]Vv emulsion volume in the Vessel compartment [m3]Vv,max maximum working volume of the vessel [m3]w volume mother droplet [m3]

159

We Weber number [-]x coordinate axis [m]x state variablesxl lower bound of state variablesxh upper bound of state variablesx0 initial conditionX channel number [-]y algebraic variablesY binary variable [-]z state variablesα parameter describing type of laminar flow [-]α significance level [-]β fit parameter [-]γ fit parameter [-]γ shear rate [s−1]γcm,rot shear rate originating from rotation in gap colloid mill [s−1]γe shear rate in elongational flow [s−1]γg shear rate in a groove of the colloid mill [s−1]γl shear rate in the boundary layer [s−1]δ binary variable [-]δl boundary layer thickness [m]∆Pc radial pressure difference in the colloid mill [Pa]∆Pcm pressure drop over the colloid mill [Pa]∆Pp pressure drop over the piping [Pa]ε pipe roughness [m]ζ fit parameter [-]η shear viscosity [Pa s]ηa apparent shear viscosity [Pa s]ηc shear viscosity of the continuous phase [Pa s]ηcm shear viscosity in the Colloid mill gap compartment [Pa s]ηd shear viscosity of the dispersed phase [Pa s]ηe shear viscosity of the emulsion [Pa s]ηe,l lower bound on shear emulsion viscosity [Pa s]ηe,h upper bound on shear emulsion viscosity [Pa s]ηe mean shear viscosity of the emulsion [Pa s]ηp shear viscosity in the Piping compartment [Pa s]ηr relative emulsion viscosity [-]ηr,∞ high shear relative emulsion viscosity [-]ηr,0 low shear relative emulsion viscosity [-]θ parameter vectorθl lower bound parameter vectorθu upper bound parameter vector

160

λ viscosity ratio [-]µ mean [m]ν number of droplets formed due to [-]

breakage of one dropletν1 number of daughter droplets [-]ν2 number of satellite droplets [-]νk number of droplets formed due to [-]

breakage of one droplet with volume vk

ρ fluid density [kg m−3]ρm micelle density [kg m−3]ρe emulsion density [kg m−3]σ interfacial tension [N m−1]σ standard deviationσ1 standard deviationσ2 standard deviationτ shear stress [Pa]φ oil volume fraction [-]φm micelle volume fraction [-]φref target terminal oil volume fraction [-]Γ surface excess [mol m−2]Γ∞ plateau value of surface excess [mol m−2]Ω capillary number [-]Ωcr critical capillary number [-]Ωcm1

cr critical capillary number in gap colloid mill [-]Ωcm2

cr critical capillary number in groove colloid mill [-]Ωe

cr critical capillary number in elongational flow [-]Ωg

cr critical capillary number in the gap [-]between the vessel wall and the stirrer blade

Ωscr critical capillary number in simple shear flow [-]

161

162

Appendix A

Viscosity measurements

A.1 Viscosity of water/glucose syrup mixtures

The viscosity measured as function of the shear rate is shown in Figure A.1 for 4water/glucose syrup mixtures. The measurements were performed at 24 C. Thecomposition of the mixtures is listed in Table 6.1. Figure A.2 shows the viscosity asfunction of the temperature for the 4 water/glucose syrup mixtures. The measure-ments were performed at a shear rate of 150 s−1.

101

102

5

10

15

20

25

30

35

40

45

50

55

60

Shear rate [s−1]

Vis

cosi

ty [m

Pa

s]

Mixture 1Mixture 2Mixture 3Mixture 4

Figure A.1: Viscosity of water/glucose syrup mixtures as a function of the shearrate at 24 C.

163

20 21 22 23 24 25 26 27 28 29 30

5

10

15

20

25

30

35

40

45

50

55

60

Temperature [oC]

Vis

cosi

ty [m

Pa

s]

Mixture 1Mixture 2Mixture 3Mixture 4

Figure A.2: Viscosity of water/glucose syrup mixtures as a function of the tempe-rature at 150 s−1.

The results show that:

• Until approximately 50 s−1 the viscosity decreases slowly as function of theshear rate. At higher shear rates the viscosity remains approximately constant.

• The viscosity (evaluated at a shear rate of 150 s−1 and at a temperature of24 C) of mixture 1, 2, 3, and 4 is 37, 24, 17 and 11 mPa s respectively.

• An increase of the temperature results in a decrease of the viscosity of the wa-ter/glucose mixtures. The decrease rate increases as the glucose syrup concen-tration increases. It decreases with approximately 0.5 mPa s C−1 for mixture4 and with approximately 1.75 mPa s C−1 for mixture 1.

A.2 Viscosity of water/surfactant mixtures

Figure A.3 shows the viscosity as function of the shear rate for the continuous phaseof mixture P1 and P2. The composition of both mixtures is listed in Table 6.3. Theshear rate is plotted from 30 s−1, because viscosity measurements at lower shear ratevalues are unreliable. Figure A.4 shows the viscosity as function of the temperaturefor both mixtures. Each measurement was performed twice and the mean values areshown in the 2 figures.

164

102

0

0.5

1

1.5

2

2.5

3

3.5

4

Shear rate [s−1]

Vis

cosi

ty [m

Pa

s]

P1P2

Figure A.3: Viscosity for 2 mixtures of water and NEODOL 91-8 as function ofthe shear rate (from 30 to 150 s−1) at 24 C.

20 25 30 350

0.5

1

1.5

2

2.5

3

Temperature [oC]

Vis

cosi

ty [m

Pa

s]

P1P2

Figure A.4: Viscosity for 2 mixtures of water and NEODOL 91-8 as a function ofthe temperature at 150 s−1.

165

These results show that:

• The viscosity does hardly depend on the shear rate.

• The viscosity (evaluated at a shear rate of 150 s−1 and at a temperature of24 C) of mixture P1 and P2 is 1.51 and 1.80 mPa s respectively. The viscosityof mixture P2 is slightly higher because of the higher surfactant concentration.

• An increase of the temperature results in a decrease of the viscosity of thewater/surfactant mixtures with approximately 0.04 mPa s C−1.

166

Appendix B

Micelle volume fraction

The micelle volume fraction, φm [-] follows from the micelle density and the excesssurfactant. The excess surfactant Sex [mol] is the amount of surfactant in excess ofthe CMC plus the amount of adsorbed surfactant. The calculations are performedas follows:

1. The amount of adsorbed NEODOL 91-8, Sad [mol] follows from the totaloil surface area Aoil [m2] and the plateau value of the surface excess Γ∞.Mathematically

Sad = AoilΓ∞. (B.1)

A typical value of Γ∞ is 1.7.10−6 [mol m−2] (Janssen et al., 1994); this valueis used in the calculations.

2. The DSD of only a small emulsion volume is measured. In order to obtain thetotal oil surface area the measured surface area is multiplied with the ratiobetween the total oil volume and the measured oil volume. Mathematically

Aoil = ADSDVoil

Voil,DSD, (B.2)

where ADSD [m2] is the surface area of the measured DSD, Voil [m3] is the oilvolume used during the experiments and Voil,DSD [m3] is the oil volume of themeasured DSD.

3. The excess surfactant Sex is given by

Sex = (mc − CMC)Vc − Sad, (B.3)

where mc [mol m−3] is the surfactant concentration in the continuous phaseand Vc [m3] is the continuous phase volume.

4. The micelle volume fraction is now calculated as follows

φm =SexMs

ρmVc, (B.4)

167

where Ms [kg mol−1] is the molecular weight of the surfactant and ρm [kgm−3]is the micelle density, which is taken as that of the surfactant density in thecalculations. The molecular weight and the density of NEODOL 91-8 are0.51 kg mol−1 and 996 kg m−3 respectively; the CMC is 10 mol m−3 (see Sec-tion 5.3).

The calculated values are listed in the Table B.1 for experiment P1 and P2. Thecomposition of both mixtures is listed in Table 6.3.

Table B.1: Calculation of the micelle volume fraction during experiment P1 and P2.Exp. Time mc Vc Aoil Sad φm

[min] [mol m−3] [m3] [m2] [mol] [-]P1 0.5 110.3 5.84.10−3 4245 0.007 0.051

1 5346 0.009 0.0516 6329 0.011 0.05130 7150 0.012 0.051

P2 0.5 162.8 4.80.10−3 6458 0.011 0.0771 8992 0.015 0.0774 9346 0.016 0.07730 14539 0.025 0.076

Due to the breakage of the oil droplets into smaller ones, the total oil surfacearea increased considerably during experiment P1 and P2. After 30 s the total oilsurface area is already 6458 m2 (almost 1 football field) for experiment P2 and after30 min this value has even doubled. However, the micelle volume fraction remainedapproximately constant during the experiments. This is due to the excessive amountof surfactant.

168

Appendix C

Branch-and-bound method -basic operation

The basic operation of the branch-and-bound (BB) method will be explained inthis section. For a more thorough investigation the reader is referred to Hillier andLieberman (1995).

Initialization The BB solution process initializes via the solution of a relaxationof the MILP. The relaxation is obtained by the substitution of continuous variableswith lower bound 0 and upper bound 1 for the binary variables. This relaxation is aLinear Program. Linear Programs can be solved efficiently for their global optimumusing e.g. the Simplex method or Interior Point optimization (Nash and Sofer, 1996).Depending on the solution of the initial relaxation the following can be concluded:

1. If the initial relaxation is infeasible, then the MILP will be infeasible as welland the solution process can be terminated.

2. If a fractional (i.e. integer-infeasible) solution results then there exists nointeger solution with a higher objective value1, hence the obtained maximumacts as a upper bound for the integer optimal solution (if one exists).

3. If the solution of the relaxed problem satisfies the integrality constraint thenit is also optimal for the MILP and the solution process can be terminated.

Branching, bounding and fathoming After initialization a first node of the BBdecision tree is created by selecting a single integer variable on which the branchingis started. Branching refers to the fixation of an integer variable to respectively 0 and1. The construction of the decision tree proceeds via branching on successive integervariables. For each branch a relaxation is solved, where the integrality constraintsof the remaining integer variables are removed.

1In case of the maximization of a certain objective J .

169

Depending on the solution of the relaxations the following conclusions may bedrawn:

• If the relaxation is infeasible, then there can not be a feasible integer solu-tion in the corresponding branch so this branch can be omitted form furtherconsideration. Coming to such a conclusion is referred to as fathoming.

• If the solution to the relaxation satisfies the integrality constraints, then alower bound to the solution of the MILP is found. The highest lower boundis denoted incumbent. If the lower bound is higher than the incumbent thenthe incumbent is updated accordingly (bounding). The distance between theincumbent and relaxations of a MILP is denoted the integrality gap.

• If the solution to the relaxation is fractional (i.e. does not satisfy the inte-grality constraints), then further investigation in the branch is required. Thecorresponding node is referred to as an open node. However, if a lower boundis present and the relaxed solution for the open node is lower (i.e. worse) thanthe lower bound, then it can be concluded that the optimum is not located inthe corresponding branch after which fathoming can take place. As soon as abetter incumbent is found all open nodes are checked for fathoming. When nofurther fathoming is possible new nodes are introduced by selecting the nextinteger variable that is branched upon.

Optimality test When there are no remaining nodes open the procedure can beterminated. The current incumbent is the global optimal solution to the originalMILP problem. If there is no incumbent then the MILP is infeasible.

Using the basic BB process as outlined above theoretically guarantees to find theglobal optimum. However, there is no guarantee on the solution speed, which maybe impractically slow even for problems of moderate size. Nevertheless, if an incum-bent is present, then the worst-case distance from the optimum is known (this isthe difference between the incumbent and the ‘best’ open node) so practically thesolution process can be terminated as soon as a solution is obtained for which theworst-case distance is acceptably small.

170

Appendix D

Results sensitivity analysis

171

Table D.1: Results sensitivity analysis optimization problem A for Configuration I.Each parameter is changed separately with +5% from the nominal value.

Perturbed Unit Nominal Peak N∗27,p(tf ) ηe(10, tf )

parameter value [-] [%] [%]Fit

parameterscPc

- 2.79 N∗17,p -5.9 +0.7

c1Pcmcm 5.27 N∗

31,p +1174 -34.9c2Pcm

cm 5.54 N∗17,p -34.3 +6.6

α - 0.19 N∗17,p -0.6 0

c1ν - 29.6 N∗17,p -1.6 0

c2S s−1 2.50 N∗17,p -4.1 0

c3S - 1.88 N∗17,p -17.1 0

cPc- 8.09 N∗

17,p +1.0 0Inlet flow

µ µm 5.70 N∗17,p +0.6 -0.3

σ µm 1.60 N∗17,p -6.1 +1.0

Equipmentdimensions

hcm mm 0.50 N∗17,p -11.8 +1.0

lcm cm 2.70 N∗17,p -0.5 -0.4

Dr,i cm 9.10 N∗17,p -12.2 +0.6

Dr,o cm 12.40 N∗17,p -16.8 +2.3

Experimentalconditions

φ - 0.60 N∗17,p +69.9 +29.9

ηc mPa s 15.00 N∗17,p +2.7 +2.7

ηd mPa s 53.00 N∗17,p +1.3 +1.4

σ mN m−1 8.15 N∗17,p +0.7 0

ControlvariablesNcm(t) N∗

17,p -12.4 +1.5Fin(t) N∗

17,p -4.6 +0.4tf N∗

17,p -16.8 +1.5

172

Table D.2: Results sensitivity analysis optimization problem A for Configuration II.Each parameter is changed separately with +5% from the nominal value.

Perturbed Unit Nominal Peak N∗27,p(tf ) ηe(10, tf )

parameter value [-] [%] [%]Fit

parameterscPc

- 2.79 N∗17,p 0 0

c1Pcmcm 5.27 N∗

17,p 0 0c2Pcm

cm 5.54 N∗17,p 0 0

α - 0.19 N∗17,p -0.6 0

c1ν - 29.6 N∗17,p -0.9 0

c2S s−1 2.50 N∗17,p -4.3 0

c3S - 1.88 N∗17,p -17.8 0

cPc- 8.09 N∗

17,p +1.1 0Inlet flow

µ µm 5.70 N∗17,p +0.7 -0.3

σ µm 1.60 N∗17,p -6.4 +1.0

Equipmentdimensions

hcm mm 0.50 N∗17,p -11.8 +1.1

lcm cm 2.70 N∗17,p -4.3 0

Dr,i cm 9.10 N∗17,p -20.5 +1.9

Dr,o cm 12.40 N∗17,p -0.4 0

Experimentalconditions

φ - 0.60 N∗17,p +39.1 +31.3

ηc mPa s 15.00 N∗17,p -0.4 +3.1

ηd mPa s 53.00 N∗17,p -0.2 +1.5

σ mN m−1 8.15 N∗17,p +0.7 0

ControlvariablesNcm(t) N∗

17,p -0.6 0Fcm,p(t) N∗

17,p -8.4 +1.1Fin(t) N∗

17,p -6.2 +0.6tf N∗

17,p -18.3 +1.9

173

Table D.3: Results sensitivity analysis optimization problem B1 for ConfigurationI. Each parameter is changed separately with +5% from the nominal value.

Perturbed Unit Nominal Peak N∗27,p(tf ) ηe(10, tf )

parameter value [-] [%] [%]Fit

parameterscPc

- 2.79 N∗17,p -0.1 0

c1Pcmcm 5.27 N∗

31,p +7841 -39.8c2Pcm

cm 5.54 N∗17,p -0.1 0

α - 0.19 N∗16,p -1.3 +0.2

c1ν - 29.6 N∗16,p -1.3 +0.1

c2S s−1 2.50 N∗16,p -6.7 +0.3

c3S - 1.88 N∗16,p -26.4 +0.8

cPc- 8.09 N∗

17,p +1.6 -0.1Inlet flow

µ µm 5.70 N∗17,p +1.0 -1.5

σ µm 1.60 N∗16,p -7.7 +4.5

Equipmentdimensions

hcm mm 0.50 N∗16,p -20.3 +8.3

lcm cm 2.70 N∗16,p -6.4 +0.2

Dr,i cm 9.10 N∗16,p -33.3 +15.0

Dr,o cm 12.40 N∗16,p -0.8 +0.1

Experimentalconditions

φ - 0.55 N∗16,p +76.0 +2.3

ηc mPa s 15.00 N∗16,p -0.6 +2.1

ηd mPa s 53.00 N∗16,p -0.3 +1.3

σ mN m−1 8.15 N∗17,p +1.4 -0.2

ControlvariablesNcm(t) N∗

16,p -3.7 +1.3Fin(t) N∗

16,p -5.4 +2.1tf N∗

16,p -26.2 +11.4

174

Table D.4: Results sensitivity analysis optimization problem B1 for ConfigurationII. Each parameter is changed separately with +5% from the nominal value.

Perturbed Unit Nominal Peak N∗27,p(tf ) ηe(10, tf )

parameter value [-] [%] [%]Fit

parameterscPc

- 2.79 N∗17,p 0 0

c1Pcmcm 5.27 N∗

17,p 0 0c2Pcm

cm 5.54 N∗17,p 0 0

α - 0.19 N∗16,p -1.3 +0.2

c1ν - 29.6 N∗16,p -1.2 +0.1

c2S s−1 2.50 N∗16,p -6.7 +0.3

c3S - 1.88 N∗16,p -26.4 +0.8

cPc- 8.09 N∗

17,p +1.6 -0.1Inlet flow

µ µm 5.70 N∗17,p +1.0 -1.5

σ µm 1.60 N∗16,p -7.7 +4.5

Equipmentdimensions

hcm mm 0.50 N∗16,p -20.2 +8.3

lcm cm 2.70 N∗16,p -6.7 +0.3

Dr,i cm 9.10 N∗16,p -34.0 +15.4

Dr,o cm 12.40 N∗16,p -0.8 +0.1

Experimentalconditions

φ - 0.55 N∗16,p +48.9 +8.8

ηc mPa s 15.00 N∗16,p -0.9 +2.2

ηd mPa s 53.00 N∗16,p -0.4 +1.3

σ mN m−1 8.15 N∗17,p +1.4 -0.2

ControlvariablesNcm(t) N∗

16,p -1.3 +0.2Fcm,p(t) N∗

16,p -15.7 +8.3Fin(t) N∗

16,p -6.2 +2.5tf N∗

16,p -26.8 +11.8

175

Table D.5: Results sensitivity analysis optimization problem B2 for ConfigurationI. Each parameter is changed separately with +5% from the nominal value.

Perturbed Unit Nominal Peak N∗27,p(tf ) ηe(10, tf )

parameter value [-] [%] [%]Fit

parameterscPc

- 2.79 N∗17,p -20.2 +9.6

c1Pcmcm 5.27 N∗

31,p +19007 -54.8c2Pcm

cm 5.54 N∗16,p -77.2 +56.0

α - 0.19 N∗16,p -1.8 +0.5

c1ν - 29.6 N∗16,p -2.2 +0.2

c2S s−1 2.50 N∗16,p -6.3 +0.4

c3S - 1.88 N∗16,p -24.8 +0.9

cPc- 8.09 N∗

17,p +1.5 -0.1Inlet flow

µ µm 5.70 N∗17,p +0.9 -1.6

σ µm 1.60 N∗16,p -7.4 +4.9

Equipmentdimensions

hcm mm 0.50 N∗17,p -22.9 +9.5

lcm cm 2.70 N∗16,p +10.7 -7.1

Dr,i cm 9.10 N∗16,p -5.3 -3.0

Dr,o cm 12.40 N∗16,p -33.6 +17.7

Experimentalconditions

φ - 0.55 N∗16,p +175.3 -15.4

ηc mPa s 15.00 N∗16,p +15.7 -5.2

ηd mPa s 53.00 N∗16,p +1.8 +0.5

σ mN m−1 8.15 N∗17,p +2.0 -0.5

ControlvariablesNcm(t) N∗

16,p -28.9 +15.1Fin(t) N∗

16,p -3.1 +1.2tf N∗

16,p -27.2 +11.9

176

Table D.6: Results sensitivity analysis optimization problem B2 for ConfigurationII. Each parameter is changed separately with +5% from the nominal value.

Perturbed Unit Nominal Peak N∗27,p(tf ) ηe(10, tf )

parameter value [-] [%] [%]Fit

parameterscPc

- 2.79 N∗17,p 0 0

c1Pcmcm 5.27 N∗

17,p 0 0c2Pcm

cm 5.54 N∗17,p 0 0

α - 0.19 N∗16,p -1.5 +0.4

c1ν - 29.6 N∗16,p -1.2 +0.1

c2S s−1 2.50 N∗16,p -7.3 +0.6

c3S - 1.88 N∗16,p -28.2 +1.3

cPc- 8.09 N∗

17,p +1.7 -0.1Inlet flow

µ µm 5.70 N∗17,p +1.0 -1.6

σ µm 1.60 N∗16,p -7.9 +4.7

Equipmentdimensions

hcm mm 0.50 N∗17,p -22.2 +9.2

lcm cm 2.70 N∗16,p -7.3 +0.6

Dr,i cm 9.10 N∗16,p -37.2 +17.3

Dr,o cm 12.40 N∗16,p -0.9 +0.2

Experimentalconditions

φ - 0.55 N∗16,p +53.4 +6.0

ηc mPa s 15.00 N∗16,p -1.0 +2.0

ηd mPa s 53.00 N∗16,p +11.9 -5.2

σ mN m−1 8.15 N∗17,p +1.6 -0.4

ControlvariablesNcm(t) N∗

16,p -1.5 +0.4Fcm,p(t) N∗

16,p -17.6 +9.2Fin(t) N∗

16,p -6.4 +2.6tf N∗

16,p -29.0 +12.9

177

Table D.7: Results sensitivity analysis optimization problem B3 for ConfigurationI. Each parameter is changed separately with +5% from the nominal value.

Perturbed Unit Nominal Peak N∗27,p(tf ) ηe(10, tf )

parameter value [-] [%] [%]Fit

parameterscPc

- 2.79 N∗17,p 0 0

c1Pcmcm 5.27 N∗

31,p +24701 -58.6c2Pcm

cm 5.54 N∗17,p 0 0

α - 0.19 N∗17,p -2.0 +0.9

c1ν - 29.6 N∗17,p -1.3 +0.3

c2S s−1 2.50 N∗17,p -8.0 +0.9

c3S - 1.88 N∗17,p -30.7 +1.9

cPc- 8.09 N∗

17,p +1.8 -0.2Inlet flow

µ µm 5.70 N∗17,p +1.1 -1.2

σ µm 1.60 N∗17,p -8.1 +3.8

Equipmentdimensions

hcm mm 0.50 N∗17,p -24.7 +6.9

lcm cm 2.70 N∗17,p -8.0 +0.9

Dr,i cm 9.10 N∗17,p -41.2 +14.1

Dr,o cm 12.40 N∗17,p -1.2 +0.5

Experimentalconditions

φ - 0.50 N∗16,p +45.0 +8.3

ηc mPa s 15.00 N∗17,p -1.5 +2.2

ηd mPa s 53.00 N∗17,p -0.6 +1.3

σ mN m−1 8.15 N∗17,p +2.2 -0.9

ControlvariablesNcm(t) N∗

17,p -5.1 +2.0Fin(t) N∗

17,p -5.6 +1.8tf N∗

17,p -31.1 +10.0

178

Table D.8: Results sensitivity analysis optimization problem B3 for ConfigurationII. Each parameter is changed separately with +5% from the nominal value.

Perturbed Unit Nominal Peak N∗27,p(tf ) ηe(10, tf )

parameter value [-] [%] [%]Fit

parameterscPc

- 2.79 N∗17,p 0 0

c1Pcmcm 5.27 N∗

17,p 0 0c2Pcm

cm 5.54 N∗17,p 0 0

α - 0.19 N∗17,p -2.0 +0.9

c1ν - 29.6 N∗17,p -1.3 +0.3

c2S s−1 2.50 N∗17,p -8.0 +0.9

c3S - 1.88 N∗17,p -30.7 +1.9

cPc- 8.09 N∗

17,p +1.8 -0.2Inlet flow

µ µm 5.70 N∗17,p +1.1 -1.2

σ µm 1.60 N∗17,p -8.1 +3.8

Equipmentdimensions

hcm mm 0.50 N∗17,p -24.7 +6.9

lcm cm 2.70 N∗17,p -8.0 +0.9

Dr,i cm 9.10 N∗17,p -41.2 +14.1

Dr,o cm 12.40 N∗17,p -1.2 +0.5

Experimentalconditions

φ - 0.50 N∗16,p +45.1 +8.2

ηc mPa s 15.00 N∗17,p -1.4 +2.2

ηd mPa s 53.00 N∗17,p -0.5 +1.3

σ mN m−1 8.15 N∗17,p +2.2 -0.9

ControlvariablesNcm(t) N∗

17,p -2.0 +0.9Fcm,p(t) N∗

17,p -19.9 +7.1Fin(t) N∗

17,p -5.6 +1.8tf N∗

17,p -31.1 +10.0

179

Table D.9: Results sensitivity analysis optimization problem C for Configuration II.Each parameter is changed separately with +5% from the nominal value.

Perturbed Unit Nominal (N∗20,p(tf ) (N∗

30,p(tf ) N∗38,p(tf )

−N∗25,p(tf ) −N∗

25,p(tf )−2.5.10−3) −2.5.10−3)

parameter value [%] [%] [%]Fit

parameterscPc

- 2.79 0 0 0c1Pcm

cm 5.27 0 0 0c2Pcm

cm 5.54 0 0 0α - 0.19 -41.8 +3.0 -0.3

c1ν - 29.6 -2.8 -1.5 0c2S s−1 2.50 +41.0 +2.2 -0.3c3S - 1.88 +146.8 +6.2 -1.8cPc

- 8.09 -25.5 -5.0 +0.1Inlet flow

µ µm 5.70 -15.7 +6.1 +2.9σ µm 1.60 +144.2 -32.9 -15.0

Equipmentdimensions

hcm mm 0.50 +97.2 -6.6 -14.5lcm cm 2.70 +41.0 +2.2 -0.3Dr,i cm 9.10 +81.3 +0.5 -23.9Dr,o cm 12.40 -24.8 +2.1 -0.2

Experimentalconditions

φ - 0.60 -456.5 +13.8 +67.4ηc mPa s 15.00 -15.4 +2.7 -0.2ηd mPa s 53.00 -27.7 +0.8 -0.1σ mN m−1 8.15 +36.9 -6.0 +0.3

ControlvariablesNcm(t) -41.8 +3.1 -0.3Fcm,p(t) +17.1 -3.3 -14.8Fin(t) +5.5 -8.2 -10.3

tf +66.2 -9.2 -24.3

180

Summary

Emulsions are widely encountered in the food and cosmetic industry. The firstfood we consume is an emulsion, namely breast milk. Other common emulsions aremayonnaise, dressings, skin creams and lotions.

Equipment often used for the production of oil-in-water emulsions in the food in-dustry consists of a stirred vessel in combination with a colloid mill and a circulationpipe. Within this set-up there are two main variations: i) Configuration I where thecolloid mill acts like a shearing device and at the same time as a pump. This con-figuration is used in the majority of the production facilities, and ii) ConfigurationII where the shearing and pumping action are not coupled.

The operation procedure for obtaining a certain predefined emulsion quality isoften established based on experience (best practice). This is most probably time-consuming (e.g. large experimental efforts for new developed products) and it is alsounclear if the process is operated at its optimum (e.g. in minimum time). An otherdrawback is that there is no feedback during the production process. Hence, it isnot possible to deal with disturbances acting on the process. A possible consequenceis that, at the end of the production process, the product quality specifications arenot met and the product has to be classified as off-spec.

In order to be able to enlarge the efficiency of the production processes and toshorten the time to market of new products - and therewith create an advantage overcompetition - it is necessary to overcome these limitations of the current operationprocedure. In the work reported a first step is set into this direction. A modeldescribing the droplet size distribution (DSD) and the emulsion viscosity as functionof the time was developed and several off-line optimization studies were performed.

The model comprises several fit parameters and experiments were performed inorder to estimate the values of these parameters. A number of additional experimentswere performed to compare the simulated results with the measurements (modelvalidation). The results of the parameter estimation and the model validation showthat the simulated results are qualitatively in good agreement with the measurementdata. Given the overall performance of the model it is expected that the modelquality is sufficient to render practical relevant optimization results.

Although the optimization studies have been performed for a model emulsion,small scale equipment and are not yet experimentally validated, the results of thiswork strongly suggest that it is indeed possible to minimize the production times andto shorten the product development times for new products. This overall conclusion

181

is based on the following observations:

• The optimization results show that it is beneficial to produce emulsions withConfiguration II:

– Configuration II allows the production of emulsions with a bi-modal DSD.No operation procedure was found for the production of such an emulsionin Configuration I.

– The production of emulsions in Configuration II is always at least as fastas in Configuration I.

• The followed approach allows to calculate:

– If an emulsion with a certain, predefined, DSD and emulsion viscosity canbe produced.

– How the process should be controlled in order to produce such an emul-sion.

– How the process should be controlled to produce this emulsion in minimaltime.

• The optimization results show that it is possible to produce emulsions with:

– A bi-modal DSD.

– Less oil while maintaining a similar DSD and value of the emulsion visco-sity (evaluated at a shear rate of 10 s−1) by adapting only the operationprocedure. Hence, the addition of extra stabilizers is not considered.

This offers possibilities for the production of a broader range of emulsion pro-ducts and could direct product development in a new direction.

Based on this, it is worthwhile and therefore recommended to expand this re-search work in the direction of industrial emulsions.

182

Samenvatting

Emulsies komen veel voor in de voedingsmiddelen- en cosmetische industrie. Heteerste voedsel dat we krijgen is een emulsie, namelijk moedermelk. Andere veelvoorkomende emulsies zijn mayonaisse, dressings, huid cremes en lotions.

Apparatuur die vaak voor de productie van olie-in-water emulsies wordt gebruiktin de voedingsmiddelenindustrie bestaat uit een geroerd vat in combinatie met eencolloıd molen en een circulatie pijp. De twee belangrijkste variaties binnen dezeopzet zijn: i) Configuratie I waar de colloıd molen de afschuifkrachten levert entegelijkertijd als pomp fungeert. Deze configuratie wordt in de meeste productielocaties gebruikt, en ii) Configuratie II waar de afschuifkrachten ontkoppeld zijn vande pompwerking.

De bedrijfsvoering voor het verkrijgen van een bepaalde emulsie kwaliteit wordtvaak vastgesteld op grond van ervaring (best practice). Dit is zeer waarschijnlijk nietalleen erg tijdrovend (vereist bijvoorbeeld veel experimenteel werk voor sommigenieuw ontwikkelde producten), maar het is ook onduidelijk of het proces optimaalbedreven wordt, bijvoorbeeld zo snel mogelijk. Een ander nadeel is dat er geenterugkoppeling is gedurende het productieproces. Hierdoor is het niet mogelijk omverstoringen te onderdrukken, waardoor het zou kunnen dat op het eind van hetproductieproces het product niet aan de specificaties voldoet.

Om de efficiency van de productie processen te vergroten en om de “time tomarket” van nieuwe producten te verkorten - en daardoor een voordeel ten opzichtevan de concurrentie te creeren - is het noodzakelijk om de genoemde nadelen vande huidige bedrijfsvoering te verhelpen. In dit proefschrift is een eerste stap in dezerichting gezet. Een model is ontwikkeld dat beschrijft hoe de druppelgrootteverdeling(DSD) en de emulsie viscositeit veranderen als functie van de tijd en er zijn meerdereoff-line optimalisatie studies uitgevoerd.

Het model bevat een aantal fit parameters; experimenten zijn uitgevoerd om dewaarden van deze fit parameters te schatten. Andere experimenten zijn uitgevoerdom de simulaties te vergelijken met de metingen (model validatie). De resultatenvan de parameterschatting en de model validatie laten zien dat de simulaties kwali-tatief goed overeenkomen met de metingen. Op grond van deze resultaten is het deverwachting dat de model kwaliteit voldoende is om praktisch relevante optimalisatieresultaten te verkrijgen.

Ofschoon de optimalisatie studies zijn uitgevoerd voor een model emulsie, eenkleinschalige opstelling en nog niet experimenteel getoetst zijn, suggereren de resul-

183

taten dat het inderdaad mogelijk is om de productie tijd te minimalizeren en omde product ontwikkeling te versnellen. Deze conclusie is gebaseerd op de volgendeobservaties:

• De optimimalisatie resultaten laten zien dat het gunstig is om emulsies inConfiguratie II te produceren:

– Het is mogelijk om emulsies met een bi-modale DSD in Configuratie II teproduceren. Het is niet gelukt om dit type emulsie in Configuratie I teproduceren.

– De productie van emulsies in Configuratie II is altijd minstens even snelals in Configuratie I.

• De gevolgde aanpak maakt het mogelijk om te berekenen:

– Of een emulsie met een bepaalde, vastgestelde, DSD en emulsie viscositeitgeproduceerd kan worden.

– Hoe het proces aangestuurd moet worden om dit type emulsie te produ-ceren.

– Hoe het proces aangestuurd moet worden om dit type emulsie in minimaletijd te produceren.

• De optimalisatie resultaten laten zien dat het mogelijk is om emulsies te pro-duceren met:

– Een bi-modale verdeling.

– Minder olie zonder dat de emulsie viscositeit (bij een shear rate van10 s−1) lager wordt en waarbij tegelijkertijd de eind DSD vergelijkbaaris. Dit is gerealiseerd door alleen de bedrijfsvoering te veranderen, datwil zeggen zonder toevoeging van extra stabilisatoren.

Dit biedt mogelijkheden voor de productie van een grotere verscheidenheidaan emulsie producten en dit kan de product ontwikkeling in nieuwe richtingenleiden.

Gezien het bovenstaande is het waardevol en wordt het dan ook aanbevolen omdit werk uit te breiden in de richting van industriele emulsies.

184

Curriculum Vitae

Martijn Stork was born on January 14, 1972 in Ede, The Netherlands.

1984–1990 Pre-University Education (VWO) at Heldring College, Zetten, TheNetherlands.

1990–1996 MSc. Bioprocess Engineering at Wageningen University with aspecialization in modeling and control. Title MSc. project: Vali-dation of a mathematical model describing the growth of Rhizopusoligosporus on solid media.

1996–1999 Project leader process technology at TNO Nutrition in Zeist, TheNetherlands.

1999–2004 Ph.D. Mechanical Engineering Systems and Control Group of DelftUniversity of Technology, The Netherlands. This work was sup-ported with a grant from the Dutch Programme EET (Economy,Ecology and Technology). Project title: “Batch processes - cleanerand more efficient”.

2004 – Project manager at Nestle Product Technology Center in Singen,Germany.

185