Extent and mechanism of coalescence in rotor-stator mixer food-emulsion emulsification

Accepted Manuscript Extent and mechanism of coalescence in rotor-stator mixer food-emulsion emulsification Andreas Håkansson, Måns Askaner, Fredrik Innings PII:

S0260-8774(15)30085-6

DOI:

10.1016/j.jfoodeng.2015.12.015

Reference:

JFOE 8429

To appear in:

Journal of Food Engineering

Received Date: 14 October 2015 Revised Date:

21 December 2015

Accepted Date: 21 December 2015

Please cite this article as: Håkansson, A., Askaner, M., Innings, F., Extent and mechanism of coalescence in rotor-stator mixer food-emulsion emulsification, Journal of Food Engineering (2016), doi: 10.1016/j.jfoodeng.2015.12.015. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Extent and mechanism of coalescence in rotor-stator mixer food-emulsion emulsification

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Andreas Håkansson*1, Måns Askaner1, Fredrik Innings2 *) Corresponding author: [email protected], +46 44 20 38 26

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1) Kristianstad University, Food and Meal Science, School of Education and Environment, SE-

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291 88 Kristianstad, Sweden.

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2) Tetra Pak Processing Systems, Lund, Ruben Rausings gata, SE-221 86 Lund, Sweden

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7 Abstract

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Food-emulsions often have high volume fractions of dispersed phase and are thus expected to

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show coalescence during emulsification, however, food-emulsion coalescence is difficult to

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measure in homogenizer equipment. This study experimentally estimates the rates of

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fragmentation and coalescence in a high viscosity and high volume fraction model emulsion

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subjected to pilot-scale rotor-stator mixing in order to quantify the relative effect of coalescence

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and discuss the mechanism of coalescence during batch processing of high-fat emulsion foods.

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Rate constants of both processes are estimated using a previously suggested method relying on

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parameter fitting from the dynamic evolution of the total number of emulsion drops (Hounslow

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and Ni, 2004). The results show substantial coalescence taking place. Scaling of rates with

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respect to rotor tip speed suggests coalescence and fragmentation controlled by a turbulent

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viscous mechanism.

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Keywords

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Coalescence; Emulsification; Food-emulsion; Fragmentation; Rotor-stator mixer.

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1. Introduction 1

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Emulsification can be described as a combination of drop fragmentation and coalescence

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(Håkansson et al., 2009; McClements, 2005; Walstra, 2005). Industrial processing of food

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emulsions (Santana et al., 2013) is designed to favor fragmentation, reduce coalescence and thus

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obtain small drops with narrow drop size distributions at minimal energy input and processing

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time. Studies on understanding emulsification therefore often focus on the effect of

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fragmentation. High intensity emulsification fragmentation is often classified in three broad

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mechanistic classes (Walstra, 2005): Turbulent inertial (TI) fragmentation brought about by

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interactions between drops and turbulent eddies smaller than the drop (Hinze, 1955), turbulent

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viscous (TV) fragmentation from shearing of drops by eddies larger than the drop (Hinze, 1955),

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and in case of laminar flow, a laminar viscous (LV) shear mechanism (Grace, 1982). For each

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mechanism, a basic scaling existsbetween resulting drop diameter (d) and emulsion

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characteristics, such as disperse and continuous phase viscosities (µD, µC) densities, (ρD, ρC),

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interfacial tension (σ) and the dissipation rate of turbulent kinetic energy (ε) (Hinze, 1955;

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Walstra, 2005; Zhang et al., 2012), −3 / 5

d ∝ ρC

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d ∝ µC−1 / 2 ρC−1/ 2ε −1/ 2σ

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(TI)

(1)

(TV)

(2)

or in the laminar case, the velocity gradient (G) (Grace 1982; Walstra, 2005)

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ε −2 / 5σ 3 / 5

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µ  d ∝ c D  µ C−1G −1σ  µC 

(LV).

(3)

where c in Eq. 3 is a concentration ratio dependent constant. Many food emulsions (e.g. mayonnaises, cake batters, creamy sauces and dressings) have high

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volume fraction of disperse phase and consequently high emulsion viscosity, and are often

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processed with rotor-stator mixers where the mean effective dissipation rate of turbulent kinetic

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& ) or tip speed (U) (Zhang et al., energy is assumed proportional to the cube of rotor frequency ( N

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2012)

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ε ∝ N& 3 ∝ U 3

(4a)

and laminar shear rate is proportional to rotor tip speed

G ∝ N& ∝ U

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(4b)

Combinations of Eqs. 1-3 with 4 suggest different scaling between measurables such as rotor

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frequency and drop diameters, and hence, comparisons between empirical and theoretical scaling

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have been used extensively to determine dominant mechanisms of fragmentation for emulsions in

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similar devices (Rueger and Calabrese, 2013a, 2013b; Tcholakova et al., 2011). Theoretically, d

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in Eqs. 1-3 should be interpreted as the maximum stable drop diameter (Hinze, 1955); however, it

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is often replaced by a mean drop diameter (e.g. volume or surface weighted average, d43 and d32

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respectively) in applications, since these are arguably proportional to each other (Rueger and

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Calabrese, 2013a), at least when disperse phase viscosity is low (c.f. Becker et al., 2013).

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This methodology for finding dominant regimes assumes that coalescence is sufficiently low

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not to influence the final drop size. However, experimental measurements with different methods

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show substantial coalescence during emulsification (Howarth, 1967; Lobo et al., 2002; Niknafs et

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al., 2011; Taisne et al., 1996), especially in high volume fraction systems (Mohan and

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Narsimhan, 1997; Niknafs et al., 2011). Emulsification of complex high volume fraction food

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emulsions such as mayonnaises, spreads and creamy sauces are therefore expected to be

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influenced by coalescence, but the extent and impact in industrial conditions is still largely

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unknown.

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Mechanistic understanding –for understanding and optimizing emulsification and equipment – thus requires measuring rates of the underlying processes of fragmentation and coalescence, 3

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rather than the combined result in terms of resulting drop size distributions. Several methods for

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estimating rates of fragmentation (e.g. Becker et al., 2014; Vankova et al., 2007) and coalescence

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(Howarth, 1967; Karbaschi et al., 2014; Lobo et al., 2002; Miller et al., 1963; Mohan and

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Narsimhan, 1997; Niknafs et al., 2011; Taisne et al., 1996) have been suggested; however, only

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the reflectivity technique (Howarth, 1967; Niknafs et al., 2011) and the moment evolution

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method (Hounslow and Ni, 2004) allow the determination of both processes using the same

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technique. Of these two, the latter has the advantage of allowing estimations from offline

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measurements in pilot- and production scale (c.f. Håkansson and Hounslow, 2013) and of

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allowing both processes to be quantified in one experiment.

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Several theoretical models for rates of coalescence and fragmentation kernels have been offered; comprehensive reviews on both fragmentation (Liao and Lucas, 2009) and coalescence

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kernels (Liao and Lucas, 2010) are available elsewhere. Whereas the number of proposed

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fragmentation rate models is large (Liao and Lucas, 2009) and growing (e.g. Becker et al., 2014;

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Maindarkar et al., 2015; Raikar et al., 2010), coalescence rate expressions of early origin (e.g.

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Delichatsios and Probstein, 1975; Saffman and Turner, 1956; von Smoluchowski, 1916) are still

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used extensively in literature.

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The objective of this study is to apply the moment evolution rate extraction method to a pilot

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scale rotor-stator emulsification system with high dispersed phase volume fraction and emulsion

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viscosity comparable to a complex emulsion food such as a mayonnaise or a creamy sauce (Pons

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et al., 1994; Singla et al., 2013) in order to, first, estimate the influence of coalescence on the

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emulsification process and, secondly, by investigating the scaling of fragmentation and

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coalescence rates discuss implications on dominant mechanism of coalescence and fragmentation

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in emulsification of high disperse phase volume fraction food emulsions.

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2. Theory and Calculations

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The moment evolution method (Hounslow and Ni, 2004; Håkansson and Hounslow, 2013)

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estimates the rates of fragmentation and coalescence by fitting the experimental evolution of

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moments of the drop size distribution to theoretical models. Assuming that the fragmentation rate

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(g) is of first order with regards to drop volume (v)

g (v) = g 0 ⋅ v ,

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(5a)

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that the coalescence rate (β) can be approximated by a sum kernel

β (v1 , v2 ) = β 0 ⋅ (v1 + v2 )

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(5b)

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with constants g0 and β0, and that each breakup gives rise to m drops on average, the per unit

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volume number of emulsion drops (N) is described by (Hounslow and Ni, 2004)

dN = − β 0ϕ D N + g 0 (m − 1)ϕ D , dt

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witch is solved by

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N ( 0) = N 0

N (t ) = N 0 exp(−tϕ D β 0 ) +

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g 0 (m − 1)

β0

(1 − exp(−tϕ D β 0 ))

(6)

(7)

where φD is the volume fraction of disperse phase. Whereas the simultaneous determination of

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coalescence and fragmentation from size distributions is generally ill-posed (Ramkrishna, 2000,

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pp. 222), the specific form of Eq. 7, with a time-scale depending only on coalescence rate, makes

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it suitable for determination of both rates (Hounslow and Ni, 2004).

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For a system without coalescence, the corresponding expressions are dN = g 0 (m − 1)ϕ D , dt

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N ( 0) = N 0

(8)

and

N (t ) = N 0 + g 0 (m − 1)ϕ D t .

(9) 5

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The number of drops at time t can be obtained by combining the volume fraction of disperse

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phase and the volumetric mean drop diameter (d43):

N (t ) =

6ϕ D πd 433 (t )

(10)

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Thus, rates of coalescence (β0) and fragmentation (m·g0) for a system can be obtained by fitting

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measured d43 over time to Eqs. 7 or 9 and 10. The relative fit to models with fragmentation only

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(Eq. 9) and fragmentation with coalescence (Eq. 7) could be used to determine if substantial

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coalescence occurs.

It should be noted that the method does not allow for independent estimates of fragmentation

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rate, g0, and the number of fragments per breakup, m. Following previous applications of this

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method (Håkansson and Hounslow, 2013; Hounslow and Ni, 2004), it is assumed that an average

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of four fragments are formed per fragmentation event (m = 4). There is no scientific consensus on

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the true number of fragments formed per breakup during emulsification under different

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conditions, and the experimental technique did not allow for verification of the assumption,

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however, m = 4 is of order of magnitude similar to previous studies (c.f. Liao & Lucas, 2009 and

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references therein). Moreover, as long as the number of fragment does not depend on the rotor tip

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speed or continuous phase viscosity, it will not influence scaling behavior of rates. Consequently,

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scaling instead of absolute rates are used for drawing conclusions on dominating mechanisms in

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this study.

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Due to the inhomogeneous distribution of velocity gradients and turbulence in the mixer

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(Mortensen et al. 2011; Utomo et al., 2008), inhomogeneous fragmentation and coalescence rates

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are expected. However, the moment evolution method is based on global average drop sizes and

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does not allow for investigations of local variations of rates. Thus, it does not provide information

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on where in the mixer coalescence and fragmentation takes place. 6

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Due to the inhomogeneity of the rates, care must be taken when interpreting the rates obtained

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from Eq. 7 or 9. Previous studies use one of two alternatives when interpreting rates from global

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data on inhomogeneous emulsification; they either (i) estimating a global mean rate over the

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whole mixer volume by setting the emulsification time (t) in Eq. 7 and 9 equal to the mixer

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processing time (tP) (c.f. Niknafs et al., 2011) or (ii) apply the models to the most likely region of

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fragmentation and coalescence and thus solve Eq. 7 and 9 with emulsification time equal to the

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proportion of time spent in this supposed region of emulsification (c.f. Mohan and Narsimhan,

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1997). Both approaches were tested in this study in order to see if the assumption influences the

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conclusions on dominant mechanism. The two approaches will be referred to as the global mean

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approach (GMA) (i) and the dissipation region approach (DRA) (ii).

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Experimental measurements by Mortensen et al. (2011) show substantial turbulence and velocity gradients in the rotor-stator region of the mixer and the jet formed downstream of the

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stator hole (Mortensen et al., 2011). Since both rates of fragmentation (Liao and Lucas, 2009) and

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coalescence (Liao and Lucas, 2010) increase strongly with dissipation rate of turbulent kinetic

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energy, the jet region downstream of the stator hole is a likely candidate when trying to identify

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the effective region of emulsification. Thus, for the DRA, emulsification time (t) in Eqs. 7 and 9

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is taken to be the time spent in the jet region of high shear and turbulence. If ideal mixing is

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assumed, the spatial probability distribution for any emulsion drop is uniform throughout the

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tank, and consequently, the fraction of time spent in the high shear and turbulent region equals

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the ration between the volume of this region and the volume of the tank. With Nslots rectangular

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slots with height h and length L, and a jet extending approximately10 h from the stator

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(Mortensen et al., 2011), the fraction of time under effective emulsification (t) to the processing

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time (tP) in DRA is thus

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t Vdiss (h ⋅ L) ⋅ N slot ⋅ 10h = = tP V V

(11)

where V is the total mixer volume.

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3.1 Experimental model emulsion

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Oil-in-water emulsions were formed with rapeseed oil (AAK Sweden AB, Karlshamn, Sweden)

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as the disperse phase and mixtures of aqueous sugar solution (Nordic Sugar A/S, Copenhagen,

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Denmark) and tap water as the continuous phase. The volume fraction of disperse phase was set

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to 52% (v/v) except in a low volume fraction control experiment performed at 0.9% (v/v).

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Polysorbate 80 (Tween80®, Sigma-Aldrich Sweden AB, Stockholm, Sweden) at an oil to

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Polysorbate mass ratio of 0.07 was used as emulsifier (for the high volume fraction experiments).

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A large excess of emulsifier (1.0 emulsifier to oil mass ratio) was used for the low volume

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fraction experiment. The equilibrium interfacial tension of Polysorbate 80 above the critical

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micelle concentration is approximately 6 mN/m (Tesch et al., 2002).

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The continuous phase consisted of aqueous sugar solutions at a solid concentration between 61

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and 65 % (w/w), corresponding to a continuous phase viscosity between 71 and 149 mPas

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(Sugartech, 2015). Concentrations were chosen to obtain continuous phase viscosity similar to

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the emulsion viscosity of high fat food emulsions such as mayonnaise or creamy sauces (c.f. Pons

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et al., 1994; Singla et al., 2013).

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3.2. Rotor-stator mixer and operation

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A Tetra Almix B120-25VA pilot-scale (V = 25 L) batch mixer was used throughout the study.

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The stator consisted of 60 slots (5x14 mm) and had a stator diameter (D) of 120 mm. During 8

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emulsification, the mixer was operated at tip speeds between 10 m/s and 30 m/s and with jacketed

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temperature control set to 20°C.

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Ingredients were added to the mixer and pre-emulsified for two minutes at a rotor-tip speed of 8 m/s in order to obtain a homogenous and stable pre-emulsion. After removing a sample of pre-

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emulsion, the mixer was run, stopped, sampled and run generating a series of emulsion with 20 s,

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40 s, 60 s, 300 s and 540 s of processing for each emulsion and rotor tip speed or continuous

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phase viscosity investigated. One emulsification experiment per rotor tip speed and continuous

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phase viscosity was performed. An additional replicate experiment at U = 20 m/s and µC = 149 m

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Pas was performed in order to assess reproducibility of obtained drop sizes and estimated rates of

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fragmentation and coalescence.

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3.3 Drop size measurements

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Drop sizes were measured within 3 hours of production using laser diffraction in a MasterSizer

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2000 (Malvern, Worcestershire, UK) with a sample stirring speed of 1200 rpm and sample

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obscuration (optical concentration) between 4 % and 12 % (chosen as low as possible to avoid

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multiple scattering while still ensuring high between-replicate reproducibility). The refractive

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index and absorbance of the disperse phase (rapeseed oil) was set at 1.473 and 0.0001

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respectively. The drop size distributions were calculated using the ‘general purpose spherical’

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light scattering model with calculation sensitivity set to ‘enhanced’.

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3.4 Rate extraction fitting and evaluation

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Fitting to the fragmentation and coalescence model was done using a Levenberg-Marquardt

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nonlinear least squares algorithm, used as implemented in MATLAB 2015a (MathWorks, Natick,

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MA). Since the two models (fragmentation-only, Eq. 9, and fragmentation with coalescence, Eq. 9

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7) have different number of parameters, the unadjusted goodness of fit (R2) is unsuitable for

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comparing the fit of the models, thus, the degree of freedom adjusted measure was used

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throughout the study to compare the fit between the two models (Levine et al., 2001),

(

)

p 1− R2 . n − p −1

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(12)

In Eq. 12, n is the number of observations per fit and p is the number of parameters (i.e. one for

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Eq. 9 and two for Eq. 7).

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The replicate emulsification experiment was used to estimate the reproducibility of the rate

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estimations. Data from each of the experiments were individually fitted to Eq. 7 and the relative

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between replicate standard deviation was used as a measure of overall uncertainty of the

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estimated rates.

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3.5 Estimated dissipation rate of turbulent kinetic energy

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The dissipation rate of turbulent kinetic energy of the turbulent jet (for the DRA) was estimated

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from

ε=f⋅

P ρ EVdiss

(13)

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where P is the measured power consumption of the mixer, ρE is the emulsion density and Vdiss is

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the dissipation volume (see Eq. 11). The fraction of energy dissipated in the dissipation volume, f,

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was set to 0.2 based on a previously reported CFD-simulated dissipation rate of turbulent kinetic

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energy distribution in a similar RSM geometry, where it was concluded that approximately 20 %

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of the energy was dissipated in the jet region (Utomo et al., 2008).

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4. Results and Discussion

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The results and discussion section is divided in five subsections. In Section 4.1, the proposed

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method is applied to a test cases showing that it is able to distinguish coalescing from non-

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coalescing conditions. Section 4.2-3 investigate the scaling of rates with respect to rotor tip speed

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and continuous phase viscosity respectively; this in order to discuss dominating mechanisms of

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fragmentation and coalescence. Section 4.4 discusses the relative overall influence of coalescence

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on the emulsification process and Section 4.5 discusses methodological strengths and weaknesses

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in comparison to alternative methods.

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4.1 Comparison of coalescing and non-coalescing conditions

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Systems with low volume fraction of oil and high emulsifier loads are expected to show low rates

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of coalescence. In order to test the ability of the method to separate coalescing from non-

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coalescing systems, it was applied to (A) a low volume fraction (φD = 0.9 %(v/v)) with a high

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emulsifier concentration (φE = 100 (w/w)) system, and (B) a high volume fraction (φD =52

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%(v/v)) with a low emulsifier concentration (φE = 7% (w/w)) system. Figure 1 (markers) shows

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drop size decrease with processing time for both systems. The low volume fraction (A) system

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results in smaller drops, indicating that fragmentation is faster and/or coalescence slower. Data

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from both experiments were fitted to the fragmentation with coalescence model (Eq. 7) and the

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fragmentation-only model (Eq. 9). The best fit of the models can be seen in Fig 1A-B for the two

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experiments respectively. For model A, the overall fit between measurements and fitted model is

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high (see Table 1), however, the drop sizes at intermediary times (20-60 s) are underestimated,

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illustrating that the assumption in Eq. 5a is not able to describe the details of the fragmentation

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throughout the entire process. Moreover, there is little improvement between the one-parameter

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fragmentation-only model and the two-parameter fragmentation with coalescence model for

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system A, indicating that coalescence is not substantially influencing the resulting drop size

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evolution for this system. The same conclusion can be drawn by comparing the adjusted

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goodness of fits of the two models (see Table 1).

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For system B, the fragmentation with coalescence model offers a substantial improvement over the fragmentation-only model (see Figure 1B), which could also be seen in the higher adjusted

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goodness of fit for this model in Table 1. Moreover, system B does not show the systematic

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deviation between modeled and measured drop sizes at intermediary times. This suggests that the

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assumptions of Eq. 5, although overly simplistic for a purely fragmenting system, is able to

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describe the drop size evolution rather well for a coalescing system, most likely due to the slower

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dynamics brought about by the countering effect of coalescence.

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In summary, the method is able to discriminate between a system with (B) and without (A) coalescence and describe the evolution of drop sizes well for a coalescing system. All subsequent experiments were performed on emulsion B (φD = 52 % (v/v), φE = 7 % (w/w). In all cases, the models including coalescence fit the data better than the fragmentation only

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model (see adjusted goodness of fits in Tables 2-3), indicating substantial levels of coalescence in

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all cases.

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Drop size decreases with increasing rotor tip speed, the average drop size after 540 s of

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processing at different tip speeds can be see Fig 2. The standard error of emulsification-

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experiment reproducibility was estimated using the replicate experiment and is illustrated in the

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form of error bars showing measured value plus/minus two standard errors in Fig. 2. The between

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replicate difference is small in comparison to the effect of rotor tip speed.

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The scaling between mixing intensity and drop size is often used in order to determine the

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dominant regime of emulsification. Loglog-regression of d43 to rotor tip speed (U) results in an 12

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exponent of -1.6 (R2 = 92%) which compared to the scaling of mechanisms TI (-1.2 from Eqs. 1

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and 4a), TV (-1.5 from Eqs. 2 and 4a) and LV (-1.0 from Eqs. 3 and 4b), suggests turbulent

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viscous fragmentation. The Kolmogorov length-scale, λ 1/ 4

  

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 µ3 λ =  C 3  ε ⋅ ρC

(14)

is large (λ > 70 µm at U between 10 and 30 m/s) in comparison to the drops (d43 < 1 µm at U

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between 10 and 30 m/s), and thus TV fragmentation is expected based on previous studies

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(Tcholakova et al., 2011; Walstra, 2005). However, the scaling of resulting drop diameters should

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not be over-interpreted if coalescence is present since the theoretical models (Eqs. 1-3) assume

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coalescence-free emulsification. Thus, the estimation of fragmentation and coalescence rate is

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expected to give more insight into the process than scaling of final drop diameters.

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Estimated rates of fragmentation and coalescence can be seen in Table 2, expressed both as global mean values over the entire mixer (GMA) and expressed relative to the jet high-dissipation

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region (DRA). The absolute values of fragmentation and coalescence rates are highly dependent

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on how large volumes they are averaged over (i.e. on the choice of GMA or DRA) since the rates

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have an approximately inverse relationship to emulsification time in Eqs. 7 and 9. Moreover, the

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fragmentation rate is dependent on the assumed number of fragments formed per breakup. Thus,

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the absolute rates should not be used in drawing mechanistic conclusions. However, the

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difference in estimated rates between replicate experiments is smaller than the difference across

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tip speeds for both approaches (see Table 2), suggesting that the relative scaling of rates could be

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used to discuss the effect of rotor tip speed.

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Figure 3 displays the scaling of fragmentation and coalescence rate with tip speed. Standard

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error from the replicate experiment was used to draw the error bars (showing measured value

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plus/minus two standard errors). Fitting power law regression models to the relative rates results

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in

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β 0 ∝ U 1.7 ∝ ε 0.57

(15a)

with R2 = 74 % and

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g 0 ∝ U 6.2 ∝ ε 2.1

(15b)

with R2 = 99 %. (Eq. 4a is used to transform the correlations from U to ε). The obtained scaling

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constants are identical for the two approaches for handling dissipation region (GMA and DRA),

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and for different assumptions of m between 2 and 40, showing that the scalings (as opposed to the

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absolute rates) are insensitive to these assumptions.

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For coalescence, the scaling with rotor speed in Eq. 15a is within the interval suggested by previous studies; 1.8 according to Madden and Damerell (1962), 0.9-3.2 according to Miller et al.

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(1963) and 1.3 -1.7 according to Howarth (1967). (The reported scaling constants of literature

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data have been adjusting for drop size as suggested by Howarth, 1967). The dependence on ε can

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be compared to suggestions from theoretical models, where different dependence on dissipation

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rate of turbulent kinetic energy is expected based on different assumptions on the drop-drop

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collision process, i.e. 0.33 for turbulent inertial (Delichatsios and Probstein, 1975) and 0.50 for

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turbulent viscous (Saffman and Turner, 1956) based collision models and 0.33 for laminar shear

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driven collisions (von Smoluchowski, 1916). The empirical scaling is best described by the

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turbulent viscous drop collision model. As previously mentioned, this is expected since the drops

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are small in comparisons to the Kolmogorov length-scale.

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Despite their sensitivity to modelling assumptions, it is interesting to compare absolute

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coalescence rates to the theoretical drop-drop collision rates suggested in literature. The ratio of

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coalescence to collision rate defines the coalescence efficiency, α, (c.f. Liao and Lucas, 2010). 14

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Assuming coalescence taking place in the dissipate jet region (i.e. using the DRA) and using the

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corresponding estimation of dissipation rate of turbulent kinetic energy (Eq. 13), the coalescence

325

efficiency can be estimated as the coalescence rate divided by the calculated collision rate

326

(Saffman and Turner, 1956). Figure 4 shows the estimated coalescence efficiency at varying tip

327

speeds (with error bars based on standard error between replicate coalescence rate experiments).

328

Figure 4 displays a coalescence efficiency of order magnitude 10-3-10-2, indicating that

329

approximately 0.1-1% of the collisions give rise to coalescence .This is qualitatively reasonable

330

compared to models (Liao and Lucas, 2010). Niknafs et al. (2011) estimate an order of

331

magnitudes lower efficiency for a similar system, however, based on global averaged, and thus

332

not comparable, rates.

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For the fragmentation scaling, the exponent in Eq. 15b is higher than previously measured for high-pressure homogenization in a low-volume fraction system (Håkansson and Hounslow,

335

2013), indicating a strong effect of energy input which could explain why the overall scaling of

336

mean drop size with tip speed is high (1.6) despite the substantial effect of coalescence and

337

increase thereof with increased energy input (Eq. 15a). The corresponding scaling in theoretical

338

models depends on drop size and empirical fitting parameters (see Liao and Lucas, 2009) which

339

makes direct theoretical comparison difficult. Furthermore, the assumptions made when deriving

340

the available fragmentation kernels are not fulfilled for the experiment (i.e. volume fraction of

341

dispersed phase is very high). Although, recent advances has provided promising kernels for high

342

viscosity cases (Becker et al., 2014; Raikar et al., 2010), theoretical fragmentation rate models

343

taking the turbulence modulation of high disperse phase volume fraction into account are still not

344

available.

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4.3 Effect of continuous phase viscosity 15

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Emulsion drop sizes obtained from emulsification at different continuous phase viscosities (after

348

540 s of processing) can be seen in Figure 5. Error bars based on estimated replicate

349

emulsification-experiment standard error has also been inserted. As seen in Fig. 5, the difference

350

in final drop diameter versus continuous phase viscosity is small in comparison to the

351

experimental uncertainty. Thus, the scaling of final drop size to continuous phase viscosity

352

cannot be used to compare dominating mechanism.

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Estimated rates of fragmentation and coalescence at varying continuous phase viscosities can

354

be seen in Table 3, and the rate relative to the lowest continuous phase viscosities is displayed in

355

Figure 6. Best fitting of exponential scaling between rate of fragmentation and coalescence are g 0 ∝ µC

0.44

(16)

with R2 = 99 %, and

358

β 0 ∝ µ C −0.014 with R2 < 1 %.

(17)

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For coalescence, the experimental uncertainty in the estimated rate is larger than the effect of

361

continuous phase viscosity (error bars are not shown in Fig. 6 in order to increase readability, but

362

are wider than the span). Thus the effect of continuous phase viscosity is not sufficiently large in

363

comparison to the experimental uncertainty to draw any mechanistic conclusions on coalescence

364

here. For fragmentation rate, on the other hand, the difference in estimated rates across

365

continuous phase viscosity is larger than the between replicate uncertainty (see Fig. 6), indicating

366

increasing fragmentation rates with increasing continuous phase viscosity. The same difficulties

367

as noted in Section 4.3 remain when trying to compare fragmentation rate scaling to theoretical

368

models. However, increasing rates of fragmentation with continuous phase viscosity is clearly

369

inconsistent with turbulent inertial fragmentation, since viscosity mainly act to suppress

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turbulence in this mechanism. A crude comparison could be made with the turbulent viscous

371

fragmentation model by Håkansson et al. (2009) which in the limit of a low critical Capillary

372

number predict an exponent of 1.5 – higher than the measured dependence but qualitatively in

373

agreement, predicting increased rates of fragmentation at increasing continuous phase viscosity.

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374 4.4 Relative influence of coalescence

376

Summarizing the findings above, substantial coalescence rates can be found for all the

377

investigated high volume fraction emulsions. For understanding the practical relevance of these

378

results, it would be interesting to translate it to an effect on the emulsification rate or required

379

emulsification time. The proposed method allows for an estimation of the size of the coalescence

380

effect on the rate of drop size reduction during processing by comparing the decrease in drop

381

diameter with coalescence (Eq. 7) and without coalescence (Eq. 9) using the estimated

382

fragmentation and coalescence rates (Tables 1-3). For system B in Table 1 the average rate of

383

drop diameter (d43) decrease (0-540 s of processing time) is 18 nm/s but would be 16% faster (i.e.

384

21 nm/s) without coalescence. Eliminating coalescence would consequently imply that a 0.5 µm

385

drop size reduction would take 24 s (=0.5 µm / 21 nm/s) instead of 28 s (=0.5 µm / 18 nm/s). This

386

decrease in the rate of drop reduction during processing could be used as a description of the

387

impact of having coalescence during emulsification.

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No systematic difference in this decrease rate can be seen across mixer intensities and

389

continuous phase viscosities (Tables 2-3); the presence of coalescence lowers the rate of drop

390

size reduction by between 9 and 17 %. Quenching coalescence thus offers a potential of

391

increasing the rate of emulsification – or reducing the processing time – to a corresponding

392

amount for the studied system.

17

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393

In interpreting the implications of these findings on complex high disperse phase volume fraction food-emulsification such as the production of mayonnaises, dressings and creamy sauces,

395

it should be noted that these food emulsions come in a large range of volume fractions (~30-

396

80%), and all the experiments of this study was carried out at 52 %. Furthermore, higher volume

397

fraction products (i.e. mayonnaise) also displays substantial shear thinning behavior which might

398

influence the mechanism of emulsification, and food-emulsion emulsifiers are often

399

macromolecular as compared to the non-ionic surfactant used in the model emulsion. Thus, the

400

relative effect of coalescence (expressed as the emulsification rate reduction) of a given product

401

needs to be determined separately. The determination of this measure for a range of different

402

food-emulsions, using the proposed methodology, and comparing it to surface properties is an

403

interesting continuation of the present study that could further increase understanding of the

404

effect of coalescence on food-emulsification.

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4.5 Methodological strengths and limitations

407

The coalescence and fragmentation rate quantification technique utilized in the present study

408

(Hounslow and Ni, 2004) is one of several different suggestions in emulsification literature. It has

409

the advantage of simultaneously estimating the scaling of both kernels from standard

410

emulsification data (e.g. volume weighted drop mean diameter), and offers a method to determine

411

if the emulsification is influenced by coalescence or entirely fragmentation dominated. However,

412

the method also suffers from limitations and draw-backs. First, both models for the time

413

evolution of the number of drops (Eqs. 7 and 9) are based on crude, however not unreasonable,

414

assumption; the turbulent viscous based collision model by Saffman and Turner (1956) is first

415

order with respect to drop volume but is not a sum kernel as assumed in Eq. 5b. Analysis of the

416

data from this study similar to those performed by Håkansson and Hounslow (2013) shows that

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18

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no other combination of first and zeroth degree scaling produce a better fitting with observed

418

drop sizes than Eq. 5. Still, the zeroth to first degree scaling with drop size remains an

419

assumption. The validity and reliability of the method also rely on how accurately the drop size at

420

different processing times can be measured. With high disperse phase volume fractions, there is

421

generally a substantial risk of coalescence during storage. No such size increase could be

422

observed during the three hours between emulsification and measurement, probably due to the

423

small drop sizes and high emulsion viscosity obtained.

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From the large number of suggested coalescence rate extraction methodologies (e.g. Howarth,

425

1967; Lobo et al., 2002; Miller et al. 1963; Niknafs et al., 2011; Taisne et al., 1996), it is apparent

426

that a variety of techniques is needed to study and quantify coalescence of different systems and

427

under different experimental conditions; neither the innovative reflectivity technique (Niknafs et

428

al., 2011) or the off-line step response method (Mohan and Narsimhan, 1997) are well suited for

429

large scale emulsifying equipment due to difficulties in positioning the probe, or in quickly

430

extracting samples. Similarly, fluorescent probe or other drop marking techniques (Lobo et al.,

431

2002; Taisne et al., 1996) are less suitable for large volume system due to high costs and

432

environmental impact of the required chemicals. However, due to the differences in methods and

433

assumptions made in the methods, systematic comparisons would be of high interest and could be

434

used in order to further validate the methods used in this and previous studies.

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436

5. Conclusions

437

The investigated high disperse phase volume fraction and high continuous phase viscous food-

438

emulsion model system experiences substantial levels of coalescence during emulsification:

439

* The presence of coalescence in this system reduces the rate of drop size decrease with

440

approximately 15%. 19

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* Coalescence rate increases with rotor frequency and the scaling is consistent with a turbulent

442

viscous mechanism.

443

* Fragmentation rate increases with increasing rotor tip speed and continuous phase viscosity.

444

The scaling of fragmentation rate with rotor frequency is consistent with a turbulent viscous

445

mechanism.

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446 Acknowledgments

448

This study was funded by the Knowledge Foundation (grant number 20150023) and Tetra Pak

449

Processing Systems.

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450

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20

Notation

453

Abbreviations

454

DRA

Dissipation region approach.

455

GMA

Global mean approach.

456

LV

Laminar viscous (regime of emulsification).

457

TI

Turbulent inertial (regime of emulsification).

458

TV

Turbulent viscous (regime of emulsification).

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459 Roman symbols

461

c

Constant in Eq. 13, -.

462

d

Drop diameter, m.

463

d32

Surface weighted average drop diameter, m.

464

d43

Volume weighted average drop diameter, m.

465

f

Proportion of energy dissipated in the jet region, -.

466

g

Fragmentation rate, s-1.

467

G

Shear rate, s-1.

468

h

Stator slot height, m.

469

L

Stator slot length, m.

470

m

471

n

472

N

473

N&

Rotor frequency, s-1.

474

Nslots

Number of stator slots, -.

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460

Average number of fragments formed per breakup event, -. Number of observations in Eq. 12, -.

Number of drops per unit volume of emulsion, m-3.

21

P

Mixer power input, W.

476

p

Number of fitting parameters in Eq. 12, -.

477

R2

Goodness of fit, -.

478

R2adj

Adjusted goodness of fit (defined in Eq. 12), -.

479

t

Emulsification time, s.

480

tP

Overall processing time in the mixer, s.

481

U

Rotor tip speed, m s-1.

482

v

Drop volume, m3.

483

V

Mixer liquid volume, m3.

484

Vdiss

Volume of a dissipation region, m3.

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485

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Greek symbols

487

α

Coalescence efficiency, -.

488

β

Coalescence rate, m3 s-1.

489

ε

Dissipation rate of turbulent kinetic energy, m2 s-3.

490

λ

Kolmogorov length-scale, m.

491

µC

Continuous phase viscosity, Pa s.

492

µD

Disperse phase viscosity, Pa s.

493

ρC

494

ρE

495

σ

496

φD

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486

Continuous phase density, kg m-3.

Emulsion density, kg m-3. Interfacial tension, N m-1.

Volume fraction of disperse phase, -.

497

22

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References

499

Becker, P.J., Puel, F., Chevalier, Y., & Sheibat-Othman, N. (2013). Monitoring silicone oil

500

droplets during emulsification in stirred vessel: effect of dispersed phase concentration and

501

viscosity. The Canadian Journal of Chemical Engineering, 92(2), 296-306.

RI PT

498

502

Becker, P.J., Puel, F., Jakobsen, H.A., & Sheibat-Othman, N. (2014). Development of and

504

improved breakage kernel for high dispersed viscosity phase emulsification. Chemical

505

Engineering Science, 109, 326-338.

M AN U

506

SC

503

507

Delichatsios, M.A., & Probstein, R.F. (1975). Coagulation in turbulent flow: theory and

508

experiment. Journal of Colloid and Interface Science, 51(3), 394-405.

509

Grace, H.P. (1982). Dispersion phenomena in high viscosity immiscible fluid systems and

511

application of static mixers as dispersion devices in such systems. Chemical Engineering

512

Communications, 14, 225-277.

TE D

510

EP

513

Håkansson, A., & Hounslow, M. J. (2013). Simultaneous determination of fragmentation and

515

coalescence rates during pilot-scale high-pressure homogenization. Journal of Food Engineering,

516

116(1), 7-13.

517

AC C

514

518

Håkansson, A., Trägårdh, T., & Bergenståhl, B. (2009). Dynamic simulation of emulsion

519

formation in a high pressure homogenizer. Chemical Engineering Science, 64 (12), 2915–2925.

520

23

ACCEPTED MANUSCRIPT

521

Hinze, J., (1955). Fundamentals of the hydrodynamic mechanism of splitting in dispersion

522

processes. AIChE Journal, 1(3), 289–295.

523 Hounslow, M.J., & Ni, X. (2004). Population balance modelling of droplet coalescence and

525

break-up in an oscillatory baffled reactor. Chemical Engineering Science, 59(4), 819-828.

RI PT

524

526

Howarth, W. (1967). Measurements of coalescence frequencies in an agitated tank. AIChE

528

Journal, 13(5), 1007-1013.

SC

527

M AN U

529

Karbaschi, M., Orr, R., Bastani, D., Javadi, A., Lofti, M., & Miller, R. (2014). A novel technique

531

to semi-quantitatively study the stability of emulsions and kinetics of coalescence under different

532

dynamic conditions. Colloids and Surfaces A: Physiochemical and Engineering Aspects, 460,

533

327-332.

534

TE D

530

Liao, Y., & Lucas, D. (2009). A literature review of theoretical models for drop and bubble

536

breakup in turbulent dispersions. Chemical Engineering Science, 64, 3389-3406.

537

EP

535

Levine, D.M., Ramsey, P.P., & Smidt, R.K. (2001). Applied statistics for engineers and

539

scientists. Prentice Hall, Upper Saddle River.

540

AC C

538

541

Liao, Y., & Lucas, D. (2010). A literature review on mechanism and models for the coalescence

542

process of fluid particles. Chemical Engineering Science, 65, 2851-2864.

543

24

ACCEPTED MANUSCRIPT

544

Lobo, L., Svereika, A., & Nair, M. (2002). Coalescence during emulsification. 1. Method

545

development. Journal of Colloid and Interface Science, 253(2), 409-418.

546 Madden, A.J., & Damerell, G.L. (1962). Coalescence frequencies in agitated liquid-liquid

548

systems. AIChE Journal, 8(2), 233-239.

RI PT

547

549

Maindarkar, S.N., Hoogland, H., & Henson, M.A. (2015). Predicting the combined effects of oil

551

and surfactant concentrations on the drop size distributions of homogenized emulsions. Colloids

552

and Surfaces A: Physiochemical and Engineering Aspects, 467, 18-30.

M AN U

SC

550

553 554

McClements, D.J. (2005). Food Emulsions, Principles, Practices, and Techniques. CRC Press,

555

Boca Raton.

TE D

556

Miller, R.S., Ralph, J.L., Curl, R.L., & Towell, L. (1963). Dispersed phase mixing: II.

558

Measurements in Organic Dispersed Systems. AIChE Journal, 9(2), 196-202.

559

EP

557

Mohan, S., & Narsimhan, G. (1997). Coalescence of protein-stabilized emulsions in a high-

561

pressure homogenizer. Journal of Colloid and Interface Science, 192(1), 1-15.

562

AC C

560

563

Mortensen, H.H., Calabrese, R.V., Innings, F., & Rosendahl, L. (2011). Characteristics of a batch

564

rotor-stator mixer performance elucidated by shaft torque and angle resolved PIV measurements.

565

Canadian Journal of Chemical Engineering, 89(5), 1076-1095.

566

25

ACCEPTED MANUSCRIPT

567

Niknafs, N. Spyropoulos, F., & Norton, I.T. (2011). Development of a new reflectance technique

568

to investigate the mechanism of emulsification. Journal of Food Engineering, 104(4), 603-611.

569 Pons, M., Galotto, M.J., & Subirats, S. (1994). Comparison of the steady rheological

571

characterization of normal and light mayonnaises. Food Hydrocolloids, 8(3-4), 389-400.

RI PT

570

572

Raikar, N.B., Bhatia, S.R., Malone, M.F., McClements, D.J., Almeida-Rivera, C., Bongers, P., &

574

Henson, M.A. (2010). Prediction of emulsion drop size distributions with population balance

575

equation models of multiple drop breakage. Colloids and Surfaces A: Physiochemical and

576

Engineering Aspects, 361(1-3), 96-108.

577 578

M AN U

SC

573

Ramkrishna, D. (2000). Population balances. Academic Press, London.

TE D

579

Rueger, P.E., & Calabrese, R.V. (2013a). Dispersion of water into oil in rotor-stator mixer. Part

581

1: Drop breakup in dilute systems. Chemical Engineering Research and Design, 91(11), 2122-

582

2133.

583

EP

580

Rueger, P.E., & Calabrese, R.V. (2013b). Dispersion of water into oil in rotor-stator mixer. Part

585

2: Effect of phase fraction. Chemical Engineering Research and Design, 91(2), 2134-2141.

586

AC C

584

587

Saffman, P.G., & Turner, J.S. (1956). On the collision of drops in turbulent clouds. Journal of

588

Fluid Mechanics, 1(1), 16-30.

589

26

ACCEPTED MANUSCRIPT

590

Santana, R.C., Perrechil, F.A., & Cunha, R.I. (2013). High- and low-energy emulsification for

591

food applications: A focus on process parameters. Food Engineering Reviews, 5(2), 107-122.

592 Singla, N., Verma, P., Ghoshal, G., & Basu, S. (2013). Steady state and time dependent

594

rheological behavior of mayonnaise (egg and eggless). International Food Research Journal,

595

20(4), 2009-2016.

RI PT

593

SC

596 Sugartech (2015). Material Properties. The Sugar Engineers.

598

http://www.sugartech.co.za/matlprop (accessed 2015-11-19).

M AN U

597

599 600

Taisne, L., Walstra, P., & Cabane, B. (1996). Transfer of oil between emulsion droplets. Journal

601

of Colloid and Interface Science, 184(2), 378-390.

TE D

602

Tcholakova, S., Lesov, I., Golmanov, K., Denkov, N.D., Judat, S., Engel, R., & Danner, T.

604

(2011). Efficient emulsification of viscous oils at high drop volume fraction. Langmuir, 27(24),

605

14783-14796.

606

EP

603

Tesch, S., Gerhards, C., & Schubert, H. (2002). Stabilization of emulsions by OSA starches.

608

Journal of Food Engineering, 54(2), 167-174.

609

AC C

607

610

Utomo, A.T., Baker, M., & Pacek, A.W. (2008). Flow pattern, periodicity and energy dissipation

611

in a batch rotor-stator mixer. Chemical Engineering Research and Design, 86(12), 1397-1409.

612

27

ACCEPTED MANUSCRIPT

613

Walstra, P. (2005). Emulsions. In Fundamentals of Interface and Colloid Science (Ed: J.

614

Lyklema). Elsevier, Amsterdam.

615 Vankova, N., Tcholakova, S., Denkov, N.D., Vulchev, V.D., & Danner, T. (2007). Emulsification

617

in turbulent flow. 2. Breakage rate constants. Journal of Colloid and Interface Science, 313(2),

618

612-629.

RI PT

616

SC

619

von Smoluchowski, M. (1916). Zusammenfassende bearbeitungen. Physikalische Seitschrift, 18,

621

585-594.

M AN U

620

622

Zhang, J., Xu, S., & Li, W. (2012). High shear mixers: A review of typical application studies on

624

power draw, flow pattern, energy dissipation and transfer properties. Chemical Engineering and

625

Processing: Process Intensification, 57-58, 25-41.

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Figure captions Figure 1. Drop diameter reduction with processing time from experiments (o) and best fit of fragmentation-only (-) and fragmentation with coalescence (--) models. A) φD = 0.9%,(v/v), B) φD

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= 52% (v/v).

Figure 2. Final drop diameter (after 540 s of processing) versus rotor tip speed. Experimental data (o) and a power law regression model (-). (µC= 149 mPa s). Error bars show measured

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values plus/minus two standard errors as estimated from the replicate experiments.

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Figure 3. Experimentally estimated rates of fragmentation (o) and coalescence (◊) as compared to the lowest tip speed. Lines show power-law regressions. (µC= 149 mPa s) Error bars show measured values plus/minus two standard errors as estimated from the replicate experiments.

Figure 4. Estimated coalescence efficiency. Error bars show mean values plus/minus two

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standard errors as estimated from the replicate experiments.

Figure 5. Final drop size (after 540 s of processing) versus continuous phase viscosity.

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Experimental data (markers) and a power law regression model (solid line). (U = 20 m/s). Error bars show measured values plus/minus two standard errors as estimated from the replicate

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Figure 6. Rate of fragmentation (o) and coalescence (◊) as compared to the lowest continuous phase viscosity. (U = 20 m/s). Lines show least squares fitting of fragmentation (-) and coalescence (--). Error bars show measured values plus/minus two standard errors as estimated from the replicate experiments. (Error bars for coalescence rate are large and has been omitted to increase readability)

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Table 1. Estimated rates of fragmentation and coalescence (expressed as the rates for a drop with diameter d*= 1 µm) and degrees of freedom adjusted goodness of fit. Values given both as

jet region (DRA). Emulsion

g(d*) [s-1]

φD

β(d*,d*) [m3/s]

[% (v/v)] DRA

GMA

DRA

A

0.9 %

8.4 109

1.0 1012

-

-

B

52 %

2.4 108

2.8 1010

2.3 10-16

2.7 10-14

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GMA) Global mean approach, DRA) Dissipation region approach, see Section 2.

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R2adj

R2adj

(Frag)

(Frag+Coal)

[Eq. 9]

[Eq. 7]

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GMA

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effective mean values over the entire mixer (GMA) and calculated based on the high turbulence

97.1%

98.3%

55.7%

99.8%

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Table 2 .Estimated fragmentation and coalescence rates at varying rotor tip speed. (µC= 149

U [m/s]

g(d*) [s-1]

GMA

β(d*,d*) [m3/s]

DRA

GMA

R2adj

R2adj

Emulsification

(Frag)

(Frag+Coal)

rate reduction by

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mPa s)

DRA

coalescence1

2.9 107

3.5 109

7.6 10-17

9.0 10-15

93.7 %

99.9%

18%

15

6.0 108

7.2 1010

7.4 10-17

8.8 10-15

94.0 %

99.7%

9.4%

20

2.4 109

2.8 1011

2.3 10-16

2.7 10-14

55.7 %

99.8%

16%

20

2.3 109

2.7 1011

1.4 10-16

1.7 10-14

76.6 %

99.9%

17%

23

5.5 109

6.6 1011

3.0 10-16

3.6 10-14

37.2 %

100%

16%

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d*= 1 µm. GMA) Global mean approach, DRA) Dissipation region approach, see Section 2. 1 ) Measured as the difference in rate of drop size reduction with and without coalescence present during the first 20 s of processing, see Section 4.4.

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Table 3. Estimated fragmentation and coalescence rates at varying continuous phase viscosity. (U = 20 m/s) g(d*) [s-1]

β(d*,d*) [m3/s]

[mPa s] GMA

DRA

GMA

R2adj

R2adj

Emulsification rate

(Frag)

(Frag+Coal)

reduction by

DRA

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µC

coalescence1

8.3 108

9.9 1010

1.0 10-16

1.2 10-14

88.3 %

99.5 %

101

1.4 109

1.7 1011

1.4 10-16

1.7 10-14

76.8 %

99.9 %

149

2.4 109

2.8 1011

2.3 10-16

2.7 10-14

55.7 %

99.8 %

6.7% 16% 16%

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d*= 1 µm. GMA) Global mean approach, DRA) Dissipation region approach, see Section 2. 1 ) Measured as the difference in rate of drop size reduction with and without coalescence present during the first 20 s of processing, see Section 4.4.

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Highlights

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Coalescence and fragmentation rates for a 25 L rotor-stator mixer was measured. Coalescence rate scaling suggests a turbulent viscous collision driven process. Fragmentation scaling is in agreement with a turbulent viscous mechanism. Coalescence decreases the rate of drop size reduction by 9-17 percent.

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• • • •