Emulsification in turbulent flow: 1. Mean and maximum drop diameters in inertial and viscous regimes
Graphical abstract
Transition from inertial to viscous regime of turbulent emulsification is induced by increasing the viscosity of the continuous phase and/or the oil volume fraction. Much smaller drops are obtained in the viscous regime of emulsification.
Introduction
The emulsification process can be considered as consisting of two opposite “elementary reactions”: drop breakup leading to formation of several smaller drops from a larger one, and drop–drop coalescence leading to formation of a larger drop from two smaller drops. In the general case, the evolution of the drop-size distribution during emulsification is governed by the competition of these two opposite processes [1], [2], [3], [4], [5], [6]. At high surfactant concentrations, the contribution of the drop–drop coalescence is negligible and the process of drop breakup determines the evolution of the drop-size distribution in the formed emulsions. After a sufficiently long emulsification time, a “steady-state” is reached, which is characterized by a relatively slow change of the drop-size distribution in the formed emulsions. In the current study, we consider the mean and the maximum drop sizes (see below for precise definitions), obtained as a result of the drop-breakup process in turbulent flow, in the steady-state stage of the emulsification process. The complementary study of the kinetics of drop-breakup in the same systems, which requires more elaborate analysis of the drop-breakup process, is presented in two subsequent papers [7], [8]. For discussion of the kinetic aspects of reaching the steady-state period in the actual emulsification experiments, see Refs. [9], [10], [11] and Section 4 in Ref. [7].
The classical studies of the emulsification process in turbulent flow, performed by Kolmogorov [12] and Hinze [13], showed that two different regimes of emulsification should be distinguished, which are termed “turbulent inertial” and “turbulent viscous” regimes, respectively (see Fig. 1). In the turbulent inertial regime, the drops are larger in diameter than the smallest turbulent eddies in the continuous phase, whereas in the turbulent viscous regime the drop diameter is smaller than the size of the smallest eddies. In the turbulent inertial regime, the maximum diameter of the stable drops (those able to resist the disruptive forces of the flow) is determined by the balance between the fluctuations in the hydrodynamic pressure of the continuous phase (which act on drop surface and induce drop deformation), and the drop capillary pressure, which opposes the drop deformation [12], [13]. In contrast, in the turbulent viscous regime, the maximum diameter of the stable drops is determined by the balance between the viscous stress acting from the continuous phase on the drop surface and the drop capillary pressure. The transition between these two regimes of emulsification depends on the size of the smallest eddies in the turbulent flow, (determined mainly by the rate of energy dissipation, ε, and the viscosity of the aqueous phase, ) and on the maximum drop size of the formed emulsion—see Section 2 for the respective equations.
Theoretical expressions relating the maximum diameter of the stable drops with the rate of energy dissipation, ε (which characterizes the intensity of the turbulent flow in Kolmogorov's theory), and with the interfacial tension of the drops, σ, were derived for these two regimes of emulsification, see Eqs. (4), (5), (6) below [12], [13]. The expression for the inertial turbulent regime was verified experimentally by several investigators [14], [15] for oil drops with viscosity close to that of the continuous aqueous phase, , and at relatively low oil volume fraction, .
The theory of emulsification of more viscous drops, , in the inertial regime, was further developed by Davies [16], Lagisetty et al. [17] and Calabrese et al. [18], [19], [20], [21]—see Section 2 below. Large set of experimental results for the effects of drop viscosity and interfacial tension on the maximum drop diameter was presented in the papers by Calabrese et al. [18], [19], [20], [21], and a good description by the theoretical expressions was observed.
The studies on the oil emulsification in turbulent viscous regime are scarce [22], [23], [24]. In this regime, the drops should be smaller than the size of the turbulent eddies, which means that higher rate of energy dissipation is required to achieve this regime, at fixed viscosity of the aqueous phase and drop size [22], [23], [24]. On the other hand, different dependences of the maximum drop size on the various governing parameters were derived for these two regimes of emulsification (Eqs. (4), (6) below), which predict that smaller droplets could be formed in the turbulent viscous regime, as compared to the turbulent inertial regime. To the best of our knowledge, this option has not been explored systematically so far. Therefore, one of the major aims of our study is to compare the mean and the maximum drop sizes, after emulsification in these two regimes.
In our previous papers [25], [26], [27] we studied the effects of several factors on the mean drop size during emulsification in the inertial turbulent regime, by using the so-called “narrow-gap homogenizer”—see Section 3.2 below for its description and mode of operation. Among the other results, we showed that an equation proposed by Davies [16] describes relatively well the maximum diameter of the stable drops in emulsions, prepared under different hydrodynamic conditions and interfacial tensions, and for oils with viscosity varied between 3 and 100 mPa s [27]. The current study complements our previous work in several aspects. First, we performed experiments with more viscous oils (up to 10,000 mPa s) to check whether the conclusions from the previous studies are still applicable for such viscous oils. Second, the experimental results for the mean drop size and polydispersity of the formed emulsions are compared to theoretical expressions and experimental results of other authors [16], [17], [18], [19], [20], [21]. Third, we demonstrate that the emulsification of the viscous oils is much more efficient (smaller drops are formed) when the emulsification is performed at higher viscosity of the aqueous phase and/or at high oil volume fraction. The latter result is explained by analyzing the conditions for transition from the inertial turbulent regime to the viscous turbulent regime of emulsification.
The paper is organized as follows: Section 2 summarizes the theoretical expressions known from the literature for the maximum drop size and for the drop-size distribution in emulsions, prepared in the two turbulent regimes of emulsification. Section 3 describes the used materials and experimental methods. Section 4 presents the experimental results from the characterization of the hydrodynamic conditions during emulsification. In Section 5, the results from the emulsification experiments in the inertial turbulent regime are presented and compared to theoretical expressions and experimental data by other authors. The results from the emulsification experiments in the viscous turbulent regime are presented in Section 6. Section 7 summarizes the conclusions.
Section snippets
Theoretical background
Drops placed in turbulent continuous phase could break upon the action of viscous or inertial stress acting on the drop surface. Which of these stresses dominates depends on the ratio of the drop size and the size of the smallest turbulent eddies in the flow, see Fig. 1. The size of the smallest eddies, , is given by the so-called “Kolmogorov scale,” defined as [12] where is the viscosity and is the mass density of the continuous phase, while ε is the rate of energy
Materials
Several surface-active emulsifiers were used to ensure a wide range of oil–water interfacial tensions: nonionic surfactant polyoxyethylene-20 hexadecyl ether (Brij 58, product of Sigma); anionic surfactant sodium dodecyl sulfate (SDS, product of Acros); mixture of the amphoteric surfactant cocoamidopropyl betaine (betaine, product of Goldschmidt Chemical Corporation) with the anionic surfactant sodium dodecyl-polyoxyethylene-3 sulfate (SDP3S, product of Stepan Company) in a molar ratio 3:2; and
Determination of the rate of energy dissipation
For comparison of the experimental data for the maximum and mean drop diameters with the theoretical predictions, Eqs. (4), (5), (6), (7), (8), one should know the rate of energy dissipation, ε, the interfacial tension, σ, and the viscosities of the dispersed and continuous phases, and . The values of and were determined as described in Section 3.4. The values of σ are discussed in Section 5.1. In a previous study [27], the values of ε were determined by numerical simulations of the
Inertial turbulent regime of emulsification
In the current section we compare our experimental results for the maximum drop diameter, , with the respective theoretical expressions for the inertial turbulent regime of emulsification, see Eqs. (7), (8). Particular attention is paid to the values of the numerical constants, , entering these expressions to clarify which set of values ensures satisfactorily description of the experimental data.
Emulsification in turbulent viscous regime
According to Kolmogorov theory [12], the increase of the viscosity of the aqueous phase, , leads to increase of the size of the smallest turbulent eddies, (see Eq. (1)). As a result, the emulsified drops could become smaller than and the emulsification would occur in the viscous regime (see Eq. (6) and Fig. 1). The effect of the emulsification regime (inertial turbulent or viscous turbulent) on the drop size is explored in the current section.
Conclusions
Systematic series of emulsification experiments is performed to quantify the effects of several factors on the mean drop size, , the maximum drop size, , and the emulsion polydispersity after emulsification in turbulent flow. These factors include: oil viscosity, , viscosity of the continuous phase, , interfacial tension, σ, oil volume fraction, Φ, and rate of energy dissipation in the turbulent flow, ε. The results clarify that the emulsification could be performed in the two
Acknowledgements
The authors of Ref. [40] are gratefully acknowledged for allowing us to use the data for dynamic interfacial tension prior to their publication. The valuable experimental help by D. Sidzhakova and the help in drop-size determination by M. Paraskova and E. Kostova (all from the Sofia University) are deeply appreciated. This study was supported by BASF Aktiengesellschaft, Ludwigshafen, Germany.
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