Model-based optimization of the operation procedure of emulsification

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Model-based optimization of the

operation procedure of

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Model-based optimization of the

operation procedure of

emulsification

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

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

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

Martijn STORK

ingenieur in de bioprocestechnologie geboren te Ede

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

Samenstelling promotiecommissie:

Rector Magniﬁcus voorzitter

Prof. ir. O.H. Bosgra Technische Universiteit Delft, promotor Prof. dr.ir. P.M. van den Hof Technische Universiteit Delft

Prof. dr. ir. J. Grievink Technische Universiteit Delft Prof. dr. ir. G. van Straten Wageningen Universiteit

Prof. dr. ir. A.C.P.M. Backx Technische Universiteit Eindhoven Dr. ir. Z. Verwater-Lukszo Technische Universiteit Delft Dr. ir. J.A. Wieringa Unilever

ISBN-10 9090196862 ISBN-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 more eﬃcient”.

Copyright c 2005 by Martijn Stork

All rights reserved. No part of the material protected by this copyright notice may be 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.

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Voorwoord

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

Een groot aantal mensen hebben een belangrijke rol gespeeld bij mijn klim naar de top. Okko, jou wil ik met name bedanken voor de grote vrijheid die je mij hebt gegeven in het bepalen van de route naar de top! Jan, hartstikke bedankt voor de inhoudelijke discussies over druppels en je aanstekende enthousiasme. Ondanks dat je 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 eenvoudige boerenverstand moet hebben was het niet altijd even makkelijk om stand te houden in het Delftse theoretische geweld, maar al met al heb ik een super tijd met jullie gehad. 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 de hulp 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, relatief vlakke stuk naar de top bedanken. Elke, jij hebt het weliswaar niet zo zwaar gehad als de vrouw van Tolstoj (zij schreef al zijn manuscripten in het net), maar het corrigeren van mijn proefschrift op taal- en schrijﬀouten is toch ook niet bepaald een enerverende taak geweest. Heel erg bedankt daarvoor en voor je steun gedurende het promotietraject!

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

Martijn Stork

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Introduction

Emulsions are widely encountered in the food industry. As Becher (2001) has pointed out, the ﬁrst food we consume is an emulsion, namely breast milk. An other common food emulsion is mayonnaise. During 2002, 24200 ton of mayonnaise was sold worth 46.1 euro millions sales in the Netherlands1. A short history of food emulsions is listed in Table 1.1.

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

Food Date of introduction

Mammalian milk c. 2.4.107 B.C.E. Milk from domestic c. 8500 B.C.E. animals; butter and cheese

Sauces 15th-16th century

Ice cream c. 1740

Mayonnaise c. 1845

Margarine 1869

Applications of emulsions are also found in the cosmetic and the pharmaceutical industry. 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 is not (Chappat, 1994). The use of emulsions as blood substitute was suggested by the extraordinarily solubility of oxygen and carbon dioxide in certain perfluorchemi-cals. This fluorochemical-in-water emulsion could be used for transfusion in such

1_{Source: Food for Thought, 2003.}

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emergencies as a shortage of whole blood3, or in cases where the patient has a rare blood 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 is referred to Chappat (1994), Becher (2001) and F¨orster 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 motivamotiva-tion the problem statement is formulated; this is subject of Section 1.2. The approach followed to achieve the conﬁned problem statement is presented in Section 1.3. Finally, in Section 1.4 the outline of this thesis is described.

1.1 Background and motivation

First, some deﬁnitions and basic aspects of emulsions are described. This is subject of 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 several limitations 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 are discussed.

1.1.1 Basic aspects of emulsions

Emulsions are formed from two immiscible liquids: one constitutes the droplets which 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 hundred nanometers 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 and emulsions of aqueous droplets in an organic liquid as W/O4. Milk and mayonaisse are 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 be broken). In fact, only micro-emulsions (which are not the topic of this thesis) are thermodynamically stable dispersions of oil and water, which means that they form spontaneously and are stable indeﬁnitely. Most macro-emulsions (comprising most products) require the input of considerable amounts of energy for their production and can only be stable in a kinetic sense. A kinetic stable emulsion is an emulsion for which the inevitable process of separation has slowed to an extent that it is not of 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 emulsions are stored for longer periods of time and should remain stable for a period of 1-2 years (Chappat, 1994).

3_{Blood drawn from the body from which no constituent, such as plasma or platelets, has been}

removed.

4_{O/W- and W/O- emulsions are also referred to as water continuous and oil continuous}

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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 because surfactant molecules have both hydrophilic5 and lipophilic6 molecular groups), faci-litating the formation of emulsions by lowering the interfacial tension. They are also responsible 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 and the 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 group has an aﬃnity for oil.

The term stabilizer is used for macromolecules soluble in the continuous phase that are usually not surface-active. Stabilizers provide long term stability of emul-sions mainly by increasing the viscosity of the continuous phase or by producing yield 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 other with a thin ﬁlm between them (ﬂocculation), or may unite to a larger droplet (coa-lescence). This is illustrated in Figure 1.2.

5_{Having an aﬃnity for water; readily absorbing or dissolving in water.}

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Coalescence Flocculation

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 one larger droplet is formed.

Another main destabilizing phenomenon is creaming (or sedimentation); which is caused by the diﬀerent density of the oil and water phase. Once ﬂocculated or coalesced, the droplets sink faster to the bottom (or rise faster to the top) than droplets of the original size. This enhanced sedimentation leads to a concentrated emulsion in parts of the emulsion and subsequently, results in the breaking of the emulsion. 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 like when they coalesce (B→ C). This enhanced creaming shortens the time for breaking the 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 by appropriate additions of weighting agents to either phase.

• Reducing the droplet size.

• Increasing the viscosity of the continuous phase. This may be achieved by

the addition of high molecular weight polymers. These act to increase the viscosity of the continuous phase to such an extent that the creaming rate becomes negligible.

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In this and the previous section several basic aspects of emulsions were brieﬂy reviewed. 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 needed for the production of emulsions. The energy is needed to overcome the droplet’s Laplace pressure pL [Pa]. The Laplace pressure is deﬁned as the diﬀerence between

the 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 tension between oil and water. To breakup droplets into smaller ones, the droplets must be strongly deformed. Consequently, the stress needed to deform the droplet is higher for a smaller droplet. Since the stress is generally transmitted by the sur-rounding liquid via agitation, higher stresses require more vigorous agitation, hence more energy. There are many diﬀerent machines to make emulsions. Commonly encountered machines are:

• Vessels with high-speed stirrers: High-speed mixers may have a single blade,

multiple blades, or intermeshing blades. The eﬃciency of dispersion depends on the shear rate. The generated shear rates are typically orders of magnitude lower than in homogenizers and colloid mills. Therefore, high-speed stirrers are usually used to make uniform premixes that are fed into for example a high-pressure homogenizer or colloid mill.

• High-pressure homogenizers: Homogenizers produce emulsions by pumping the

mixture at high pressure (up to 80 MPa) through a narrow valve. The gap can be as small as 15µm. The mixture enters the valve at high pressure but low velocity. The sudden increase in velocity and decrease in pressure of the liquid as it passes through the small gap creates high-shear forces and cavitation due to vaporization.

• Colloid mills: The colloid mill consists of a stationary part, the stator and a

rotating part, the rotor. In the narrow gap (a gap width of 50µm is quite common) between these the intensity of the hydrodynamic forces acting on the droplets is very high, which causes breakage of the droplets.

For more information on equipment for the production of emulsions the reader is referred 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 circulation pipe. The vessel is equipped with a scraper stirrer: a device consisting of several blades 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 caused by the rotation of the colloid mill. The reservoir contains oil which is pumped into the vessel. The outlet valve is used to empty the vessel after production. Within this set-up there are two main variations:

• Conﬁguration I: In the majority of the production facilities the colloid mill

has a conical shape (set-up as shown in Figure 1.4). The rotating of the rotor and the conically shape causes a circulating ﬂow to the vessel. Hence, in this set-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 have

a conical shape and therefore it does not act as a pump. An extra pump is present to create the circulation ﬂow to the vessel. Note that in this set-up the shearing and pumping action are not coupled.

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

Scraper stirrer Stator Rotor Oil reservoir Cooling jacket Circulation pipe Colloid mill Vessel Stirrer blade Oil droplet Outlet valve Oil inlet pump

Figure 1.4: Equipment often used for the production of O/W-emulsions in the food industry. The equipment comprises three main parts: a stirred vessel, a colloid mill and 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 Value

Vessel volume l 500-1500 Gap width colloid mill mm 0.5

Length colloid mill cm 3 Entrance rotor diameter cm 9 Exit 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 added to the vessel. Then the stirrer and the rotor are switched on and the oil is pumped into the vessel. The inlet ﬂow rate and the stirrer and rotor speed have constant values in time. The values of these variables, the so called control variables, are product dependent and their values are often established based on experience (best practice). How this is done is explained later. After the oil addition the process is continued for a certain amount of time; the length of this time period is also often based on experience. After that the colloid mill is shut down and extra ingredients e.g. small onion parts, are added to the stirred vessel and mixed. The colloid mill is shut down in order to prevent these ingredients from being destroyed. Typical production times for O/W-emulsions in the food industry are in the order of 10-20 minutes.

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

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

a) The values of the inlet flow rate, the stirrer and rotor speed and the production time are always the same for a certain product. That means that every time that e.g. a certain mayonnaise type is produced the same operation procedure is applied. This does however not imply that the quality of this mayonnaise will be 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 mayonnaise quality will fluctuate from batch to batch. This is not desirable because in recent years there is an increased customer demand for consistent high product quality (see for example Harold and Ogunnaike, 2000; Verwater-Lukszo, 1995). It might even be that the product quality specifications are not met and then the product has to be classified as off-spec. From a cost point of view this is clearly undesirable.

b) For some new developed products a large experimental eﬀort is needed before it is possible to produce the product at industrial scale with a similar quality compared to the product that was produced in the kitchen. Hence, this could lead to a large time consuming eﬀort, implying possible high costs and 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 often the experimental eﬀort is stopped as soon as an operation procedure has been found 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 comparable product quality with a diﬀerent operation procedure that takes for example less time or energy.

In order to be able to enlarge the eﬃciency of the existing production processes and to shorten the time to market for new products - and therewith create an advantage over competition - it is necessary to overcome (some of) the limitations of 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 as function of the time and the control variables (i.e. the inlet flow rate, the stirrer and the rotor speed). The model should include the effect of equip-ment dimensions and the product composition (e.g. oil volume fraction, fluid viscosities), as they vary widely from factory to factory and from product to product. 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) as function of the time Mathematical

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 product composition.

it is for example possible to calculate the eﬀect of an increase of the rotor speed with 5 s−1 on the product quality as function of the time. Further, and even 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, product quality in e.g. minimal time. This could not only be a matter of adjusting e.g. the rotor speed to a certain fixed value for the whole process. In reality the 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 obtained in minimal time by rotating vigorously at the start of the process whereas the rotor speed is set to a lower value during the rest of the process. This way the process can be operated at the desired optimum. Further, it is expected that the time to introduce new products will be shorter, because it is now no longer necessary to establish the operation procedure experimentally for every new product. Hence, with this procedure it is not only possible to calculate how 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 quality in e.g. minimal time

Time Optimal rotor speed

Optimal stirrer speed Optimal inlet flow rate

Mathematical model

Optimization routine

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

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

Batch number Specifications

Product quality

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) it should be possible to minimize the number of products that are classiﬁed as oﬀ-spec and to manufacture products with consistent high product quality.

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As mentioned previously a model would be needed to describe the evolution of physical measurable variables, that determine the product quality, in time. Relating those physical measurable variables to the product quality is a very complex and a product dependent task (especially in the food and cosmetic industry) and does not ﬁt within the framework of this thesis. However, the droplet size distribution (DSD) and the emulsion viscosity aﬀect the product quality to a certain extent. Examples of these are:

• Thickness of e.g. ketchup and sauces correlates with the viscosity. As discussed

by Borwankar (1992), consumer perception of thickness of barbecue sauce is a 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 sauce is non-Newtonian (apparent viscosity depends on the shear rate). Therefore the viscosities relevant for the various sensory attributes are different since different shear rates are relevant (see Figure 1.8).

0 100 Shear rate [s -1_] Tasting Basting Pouring 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 the

viscosity at lower shear rates (F¨orster and Rybinski, 1998).

• The droplet size of pharmaceutical emulsions must be small (the largest droplet

should always be less than 5µm). The most important reason for this is that the droplets should not clog the blood vessels during their transport (Chappat, 1994).

• Destabilization phenomena were described previously and it was mentioned

that one of them, namely creaming, could be reduced by reducing the droplet size.

7_{To moisten (meat, for example) periodically with a liquid, such as melted butter or a sauce,}

especially while cooking.

8_{To hold fast or adhere to something, as by grasping, sticking, embracing, or entwining: clung}

to the rope to keep from falling; fabrics that cling to the body.

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• The color of an emulsion is aﬀected by its DSD. Table 1.3 shows the optical

characteristics for the range of droplet diameters encountered in emulsions. A theory to relate the color of emulsions to a.o. the droplet radius is presented in McClements (2002).

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

[µm] Pure white Exceeds 0.5 White to grey 0.1 to 0.3 Grey to translucent 0.01 to 0.14 Transparent 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 or color) they do not solely determine the product quality. Hence, the product quality of 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 statement can be formulated:

Investigate, based on a model, how the control variables should be chosen as a function 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 emulsion viscosity (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 most common configuration (Configuration I) the colloid mill acts as shearing device as 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 to establish which configuration enables the fastest production. To this end it will be investigated:

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

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

B) For the production of food emulsions it is desirable to produce emulsions with less oil while maintaining a comparable terminal DSD and emulsion viscosity (evaluated at a certain shear rate) as obtained originally. Normally this is established by the addition of additional stabilizers like e.g. starch. The aim of 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 of relevance 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 to produce 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 velopment of emulsions with a multi-modal DSD, could direct product de-velopment in new directions. It might for example be possible to develop multi-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 for the production of O/W-emulsions, ii) using a small scale version of the equipment offers the possibility to perform experiments at negligible costs and allows much more flexibility than when using a production facility, iii) emulsion products vary widely in their composition (e.g. different oil types, different surfactants, different ingredients), however all emulsion products contain water, oil and a surfactant, and iv) it is expected that the off-line optimization of the operation procedure will already yield valuable insight in how the process can be improved.

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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 dynamic optimization (or open loop optimal control). Literature regarding methods for sol-ving dynamic optimization problems as well as regarding applications of dynamic optimization is widely available. Good introductions to dynamic optimization are given by Bryson (1999) and Agrawal and Fabien (1999). Applications of dynamic optimization are encountered in many areas (for example chemical processes, fer-mentation processes and food processes). Two examples of applications of dynamic optimization problems are:

• The dynamic optimization of thermal processing. In its basic form, the

dy-namic optimization of thermal processing problems seeks to find the heating temperature (as a time-dependent profile) which maximizes the final nutrient retention 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-Vázquez and Mart´ınez, 1999; Kleis and Sachs, 2000).

• The dynamic optimization of a fed-batch reactor for ethanol production. This

dynamic 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 ethanol using 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 Jayaraman et 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 physical and chemical relationships. These models give a physical insight of the system and 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 the physical structure of the system: black-box models give an input/output re-lation of the process. These models are useful if a physical understanding of the system is absent or not relevant for the purpose of the model. Black-box models are identiﬁed on the basis of experimental data.

3. Knowledge about the process may be incomplete, which can result in models that use both white-box and black-box modelling strategies. Models that combine 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, physical properties, etc.. Rigorously speaking, pure ﬁrst-principles models are very rare, since

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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 and grey-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 in practice (real products, large scale equipment). Therefore a white-box model would be highly desirable (extrapolation properties, relatively easy to extend to other pro-duct compositions). However, given the complexity of emulsification processes, this was 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 of these fit parameters do depend on the specific equipment dimensions, therefore their values 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 these chapters will now be presented.

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

Chapter 3: The basic theory regarding droplet breakage in laminar flow was used for the modeling of the fed-batch emulsification process. Chapter 3 addresses the 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 emulsification process.

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

Chapter 5: A detailed description of the equipment, used ﬂuids and measuring instruments is given in this chapter. Also several preliminary experiments are described. The preliminary experiments were performed to establish e.g. the 0.95-conﬁdence intervals of the measured variables and the reproducibility of the process.

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

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performed to this end are discussed. One part of these experiment is used to determine the values of the ﬁt parameters. The other experiments were used for 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 the formulated sub-problems. The numerical strategy as used for the solution of the dynamic optimization problems is discussed in this chapter. Further, the results of the dynamic optimization problems are described. Based on this it is discussed how the control variables should be chosen as a function of time in order to produce emulsions with a certain, predeﬁned, terminal DSD and emulsion viscosity in minimal time.

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

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

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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 the modeling 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 (Sec(Sec-tion 2.3), (d) the number and the sizes of the daughter droplets (Section 2.4) and (e) breakup of droplets in concentrated emul-sions (Section 2.5). Phenomena like coalescence and droplet breakage in turbulent flow are not described because it is assumed, as discussed in Chapter 3, that these phenomena are negligible. For more information about droplet breakage and emul-sion formation the reader is referred to Stone (1994), Walstra (1993) and Walstra and Smulders (1998).

2.1 Breakage condition

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

Ω = ηc˙γr

σ , (2.1)

where ηc [Pa s] is the shear viscosity of the continuous phase and ˙γ [s−1] the shear

rate. 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 initially

spherical droplet is deformed into a stable ellipsoid. If Ω is greater than Ωcra stable

droplet shape does not exist and the droplet will be stretched continuously until it breaks. The critical capillary number Ωcr is a function of the ﬂow type and the

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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 [−]

Critical capillary number [−]

α=0 α=0.2 α=0.6 α=1

Figure 2.1: Critical capillary number for breakup of droplets in various types of two-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, generally speaking, that the liquid in the droplet, being more viscous than that around it, cannot flow as fast as the shear rate tries to cause deformation. The result is 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 marked effect on Ωcr and on its dependence on the viscosity ratio λ. This means that

it becomes more easy to breakup droplets, especially at high λ.

Note that Ωcr says nothing about the droplet sizes produced upon breakup: the

value of Ωcronly gives the maximum droplet size that can survive in a given ﬂow in

the absence of coalescence. The maximum stable droplet diameter dcr [m] is given

by dcr= 2σΩcr ˙γηc . (2.2)

2.2 Breakage mechanisms

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

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Necking

In this type of breakup, the two ends of the droplet form bulbous ends and a neck develops between them. The neck continuously thins until it breaks, leaving behind a few much smaller droplets (called satellite droplets) between two large droplets formed from the bulbous ends. This necking mechanism generally occurs during a sustained 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

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End-pinching

Relaxation of a moderately extended droplet under the influence of surface tension forces 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 beyond its maximum steady shape with a flow at the critical capillary number. Provided the droplet has been stretched sufficiently beyond its maximum steady shape, then after flow stoppage, the droplet first relaxes back towards a spherical shape, but subsequently 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 extended droplets. However when a droplet is suddenly subjected to a stress much greater than the critical stress for breakup (Ω >> Ωcr) the droplet is stretched aﬃnely

and becomes a highly extended thread. The extended droplet is unstable to minor disturbances and will eventually disintegrate into a number of large droplets with satellite 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 undeformed

droplet. Experimental results with regard to the breakup time in laminar ﬂow are published in Grace (1982), Elemans et al. (1993) and Wieringa et al. (1996). It is found that the average breakup time in simple shear ﬂow is given by

tb= 64 dηc

2σ(ηd/ηc)

0.3_, _(2.3)

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

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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) it is mentioned that no trend with Ω/Ωcrwas observed in the data. This is in contrast

with the results of Grace (1982) where it is concluded that the time needed for breakup decreases quite rapidly as the ratio Ω/Ωcrincreases. 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 with

regard to the breakup time in elongational flow are presented. The viscosity ratio as used for the experiments ranges from 10−3 to 102. It is concluded that, in the region of overlapping viscosity ratios, the breakup time as found in simple shear flow is 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 in Wieringa et al. (1996) and Grace (1982). In Wieringa et al. (1996) three diﬀerent values of Ω/Ωcr(2, 3 and 4) and λ (0.01, 0.31 and 1) were used for the experiments in

simple shear ﬂow. Based on this study the following relation is proposed to predict the number of daughter droplets ν [-] for λ = 1 and Ω/Ωcr≥ 2

ν =−140 + 80 Ω

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 results also 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 number of daughter droplets depends linearly on Ω/Ωcr. Experimental work to determine

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the number of daughter droplets in elongational ﬂow is lacking. This is also the case for 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 during the actual emulsification process is doubtful. In reality surfactants are present, the droplet is subject to continuously changing hydrodynamic conditions and droplet breakage 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 visvis-cosity throughout the model. 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 under simple shear flow. It was observed that the critical capillary number for breakup decreased 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 in single droplet experiments, did show breakup at elevated emulsions concentrations. All these effects were explained by means of a mean field model, which assumes simply that breakup of a droplet in a concentrated emulsion is determined by the average emulsion viscosity rather than the continuous phase viscosity. Mathemati-cally Ω = ηe˙γr σ (2.5) and λ = ηd ηe , (2.6)

where ηe [Pa s] is the emulsion viscosity. In the next chapter it is discussed how

the information described in this chapter, is used for the development of a dynamic model describing the DSD and emulsion viscosity as a function of the time and of the control variables for the fed-batch emulsiﬁcation process.

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Chapter 3

Dynamic modeling of

emulsification

In Chapter 1 it is mentioned that the DSD and the emulsion viscosity will be used as indicators for the product quality. A model capable of predicting those variables as a function of time is presented in this Chapter. Several models have been published in literature dealing with emulsiﬁcation in a stirred vessel equipped with a turbine or propeller stirrer. Most of these models predict some kind of a mean, e.g. the

d32 [m] (the volume/surface average or Sauter mean), in steady-state conditions as

function of typically the physical properties of the fluids (interfacial tension, density and viscosity), the volume fraction and the average power input per unit mass of fluids (see for example Arai et al., 1977; Kumar et al., 1991; Wichterle, 1995; Kumar et al., 1998; Zhou and Kresta, 1997). A limited number of models predict the evolution 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 turbulent flow conditions.

The emulsification process in colloid mills has not been widely studied. Wieringa et al. (1996) studied the emulsification of concentrated emulsions in a colloid mill with smooth rotor and stator surfaces under laminar flow conditions. Two models were compared. The first is based on a cascade of binary events, so that a large number of steps is needed to complete the breakup process for all droplets. The second model includes the capillary breakup process. This considerably reduces the number of breakup events needed to obtain a certain final droplet size and thereby the time scale of the process. Comparison of experimental and calculated mean droplet sizes showed the importance of the capillary mechanism.

To the authors knowledge no model is available in the literature that describes the DSD(t) for the system under study. Therefore a new model was developed. It is expected (as discussed in Section 3.1) that droplet breakage will primarily occur under laminar flow conditions. Droplet breakage under turbulent flow conditions differs 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

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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 model is 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 for the modeling of these functions information is needed about processes occurring at the 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 predicts the 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 the circulation 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 mode of flow (i.e. laminar or turbulent flow) during the fed-batch emulsification process is estimated (Section 3.1.1). Then, in Section 3.1.2, it is estimated in which parts of the equipment droplet breakage will primarily occur. Assumptions related to the surfactant are presented in Section 3.1.3 and in Section 3.1.4 assumptions related to the Reactor and the Flow rate model are presented. Finally, in Section 3.1.5, all assumptions are listed.

3.1.1 Mode of flow

The mode of flow affects the droplet breakage and the circulation flow rate. Therefore it is necessary to establish which mode of flow is acting during the process in the various parts of the equipment. As illustrated in Figure 1.4 the equipment consists of a stirred vessel in combination with a colloid mill and a circulation pipe. The vessel is equipped with a scraper stirrer: a device consisting of several blades that rotate at a small distance from the vessel wall. This is illustrated in Figure 3.6. The colloid mill consists of a stator and a rotor. The rotor and the stator surfaces of the colloid mill used are not smooth; both contain grooves. This is illustrated in Figure 3.1.

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

• The piping.

• The bulk ﬂow in the vessel.

• The boundary layer of the impeller. • The gap of the colloid mill.

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Rotor Stator b_g h_g h_cm Groove Gap

Figure 3.1: Picture (not on scale) of the grooves in the rotor and stator surfaces of the 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 the start of the production of e.g. mayonnaise the fluid consists of water, surfactant and several ingredients. The viscosity is low, typically around 3 mPa s. Due to the oil addition the viscosity increases and this could result in a change of the mode of flow from i.e. turbulent to laminar. In this section the values of the Reynolds numbers are calculated as function of the emulsion viscosity for the equipment parts listed previously. Based on these calculations the mode of flow in these equipment parts is estimated as function of the emulsion viscosity.

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

• The Reynolds number for transition from laminar to turbulent ﬂow in piping

is 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 ﬂuid

velocity in the piping, Dp [m] is the pipe diameter, ρ [kg m−3] is the ﬂuid

density and η [Pa s] is the ﬂuid viscosity. Note, that the emulsion viscosity is shear thinning (the viscosity decreases as the shear rate increases). Later it is explained how this is taken into account.

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

Reb=

NstDst2ρ

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where Nst [s−1] is the stirrer speed and Dst [m] is the stirrer diameter. The

Reynolds number for transition from laminar to turbulent ﬂow is around 1000 (van ’t Riet and Tramper, 1991).

• Relatively high shear rates are expected in the boundary layers around the

stirrer in the vessel. As a first approximation the scraper stirrer can be modeled as 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] is the distance along the blade. Taking v∞ = vtip, where vtip [m s−1] is the

impeller tip speed, enables the calculation of Rexalong the scraper stirrer. The

Reynolds number Rexfor transition from laminar to turbulent ﬂow over a ﬂat

plate is around 3.105(Schlichting, 1979), however the same author states that impeller rotation can considerably reduce the Reynolds number for transition.

• In Kataoka (1986) it is discussed how the modes of ﬂow can be characterized

for a system in which an uniform axial flow enters an annular space with the inner cylinder rotating and the outer at rest. In this system the modes of flow can be characterized as a function of two independent variables: the modified Reynolds number (the square root of the Taylor number)

Rem = πDN hcρ

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 mean

axial speed in the gap of the system. In Figure 3.2 it is shown how the mode of ﬂow depends schematically on Rem and Rez.

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

Reg= ρvgDh

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Rem Rez Turbulent flow Laminar flow Turbulent flow with vortices Laminar flow with 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 number

for transition from laminar to turbulent ﬂow is expected to have the same order of magnitude as in piping.

Estimation of Reynolds numbers

In Figure 3.3 and 3.4 the previously described Reynolds numbers are shown as function 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 the scraper stirrer blade width Lst, hence x = Lst/2. The density of the ﬂuid is taken as

1000 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 on the rotor speed. At the start of the process a typical value is 14 m3h−1; this value is used in the calculations. Note that using the maximum value for the circulation flow rate and for the stirrer and rotor speed will give “worst-case” estimates for the emulsion viscosity at which the possible transition from turbulent to laminar flow occurs.

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101 102 102

103 104 105

Emulsion viscosity [mPa s]

Reynolds number [−] Re b Re x Re p Re g

Figure 3.3: Reynolds numbers in the bulk ﬂow, the boundary layer, the piping and in 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 will be 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 Rebis smaller

than 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 is

smaller than 2100).

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

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

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101 102 100

101 102 103

Emulsion viscosity [mPa s]

Reynolds number [−]

Rem Rez

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 the continuous 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 be laminar in the colloid mill during most of the time. Based on this it is expected that droplet breakage in the colloid mill, during the time period of turbulent flow (grooves) and laminar flow with vortices (gap) is negligible compared to droplet breakage occurring during the time period of laminar flow. Therefore, and for the sake of simplicity, it is assumed that the flow is laminar in the colloid mill during the total process.

3.1.2 Breakage zones

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

1_{For emulsion products with a continuous phase viscosity larger than 60 mPa s it would be}

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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 of the blade the fluid approaches the blade and displays a stagnation line somewhere around half way along the blade. Thus, there is a plane hyperbolic flow on the blade. Similarly there is a boundary layer on the blade itself and a droplet present in this layer can experience strong shearing action leading to its breakage, provided the droplet diameter is smaller than the boundary layer thickness.

Hyperbolic flow Boundary layer Blade Rotation direction

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

Kumar et al. (1998) considered the following 3 breakage mechanisms controlling the 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 shear

ﬂow; 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 of each other. All these mechanisms have their corresponding dcr values, denoted by dt

cr [m], dscr [m] and decr [m] for droplet breakup in turbulent, simple shear and

elongational ﬂow ﬁelds, respectively. The observed value of dcr near the impeller

will be the smallest of the dcr values given by the 3 mechanisms. Besides these

mechanisms, high shear rates would be expected in the gap between the scraper stirrer blade and the vessel wall (see Figure 3.6). The corresponding dcr value is

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denoted by dg

cr [m]. Relations for obtaining rough estimates of the 4 critical droplet

diameters are described subsequently.

D_st/2 L_st x x=0 Centerline vessel 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 by diﬀerent 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)W e−0.6, (3.8)

where the Weber number W e [-] is given by

W e = ρcN 2 stD3st

σ . (3.9)

The value of coeﬃcient C1, as reported by Sprow (1967), is 0.125. Calabrese et

al. (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 best

ﬁt their experimental data for a turbine impeller. Equation 3.8 cannot be used when the dispersed phase is viscous or rheologically complex,

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

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in Lagisetty et al. (1986). The dt

cr for the model of Lagisetty et al. (1986) is given

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

dt_cr= Dst 1 + 4φ 8Red 3/4 1 + (1 + 4φ) 9/10_Re3/2 d 23/2W e6/5 , (3.10)

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

Red=

NstD2stρc ηd

. (3.11)

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

0)

dt_cr= 0.125Dst(1 + 4φ)1.2W e−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 boundary layer thickness δl[m], deﬁned as the distance from the impeller surface at which the

ﬂuid 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] is

given as (Schlichting, 1979) ˙γl= 0.332 v_∞ x Rex. (3.14)

Note, that the shear rate at the leading edge, x = 0, is not deﬁned. This is a result of 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 the average emulsion viscosity and taking v_∞= vtip, it follows that the critical droplet

diameter 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) modeled the flow towards and around the impeller blade by a uniform flow approaching a semi-infinite plate of the same width as the impeller blade. In Lamb (1945) a solution for the flow field around a semi-infinite flat plate kept perpendicular to the

Model-based optimization of the operation procedure of emulsification

Model-based optimization of the

operation procedure of

Model-based optimization of the

operation procedure of

emulsification

Voorwoord

Contents

Chapter 1

Introduction

1.1

Background and motivation

1.1.1

Basic aspects of emulsions

Oil

Water

A

B

C

D

1.1.2

Process equipment

1.1.3

Current operating procedure and limitations

1.1.4

Improving the operation procedure

1.2

Problem formulation

1.3

Approach and limitations

1.4

Outline of this thesis

Chapter 2

Theory of droplet breakup

2.1

Breakage condition

2.2

Breakage mechanisms

2.3

Breakup time

2.4

Number and sizes of daughter droplets

2.5

Breakup in concentrated emulsions

Chapter 3

Dynamic modeling of

emulsification

3.1

Assumptions

3.1.1

Mode of flow

3.1.2

Breakage zones