Experimental investigations of turbulent fragmenting stresses in a rotor-stator mixer. Part 2. Probability distributions of instantaneous stresses
Graphical abstract
Introduction
Rotor-stator mixers (RSMs) (also referred to as high-shear mixers) are commonly used for emulsification and mixing in chemical engineering processing. Although significant advances have been made during the last decade, the drop breakup process and its relation to RSM hydrodynamics are still relatively poorly understood (Atiemo-Obeng and Calabrese, 2004, Atiemo-Obeng and Calabrese, 2016). This can be seen most clearly when comparing it to other emulsification processes, such as the high-pressure homogenizer, where a large number of breakup visualization studies and experimental hydrodynamic investigations are now available, which has led to a substantial increase in the general understanding (see Bisten and Schuchmann, 2016 for a recent review). By analogy, we suggest that in-depth experimental characterization of RSM hydrodynamics and comparisons to drop breakup visualization could provide new insights into RSM emulsification as well.
The RSM gives rise to a highly turbulent flow (Mortensen et al., 2011) and drop breakup visualizations suggest a turbulent mechanism of drop breakup (Ashar et al., submitted for publication). Traditionally, theoretical attempts to predict or correlate drop diameters resulting from turbulent emulsification to design and operating conditions have been based on a stress analysis, comparing the average turbulent disruptive stress, 〈σ〉, to the stabilizing stress, σstab. The ratio between the average fragmenting stress and the stabilizing stress defines a dimensionless number
Stabilization occurs due to Laplace pressure and viscous resistance (Calabrese et al., 1986, Davies, 1985, Hinze, 1955, Vankova et al., 2007),Eq. (1) has traditionally, either by itself or in combination with other dimensionless numbers, been used to model under which conditions drops break and for interpreting emulsification experiments (e.g. Boxall et al., 2012, Calabrese et al., 1986, Gupta et al., 2016, Hinze, 1955). This approach to modeling stable drop diameters is often referred to as the Kolmogorov-Hinze theory.
The traditional approach has proven useful as a first approximation. However, it has often given unsatisfactory results for predictions, especially for RSMs (Håkansson et al., 2016, Hall et al., 2013). Moreover, the Kolmogorov-Hinze theory relies on two assumptions that can be questioned. First, it assumes that the three-dimensional turbulent flow can be characterized by a single mean-efficient or maximum stress level, often estimated from the mean effective or maximum dissipation rate of turbulent kinetic energy (TKE). However, the turbulent flow in emulsification equipment is inhomogeneous (Håkansson et al., 2011, Kresta and Wood, 1993, Mortensen et al., 2011, Utomo et al., 2009, Xu et al., 2014), which must be taken into consideration when discussing drop breakup.
Secondly, the turbulent stress at each location in the flow is not constant. Instantaneous stresses do not equal the average stress (as assumed in the Kolmogorov-Hinze theory) but vary stochastically over time. This complication to the Kolmogorov-Hinze theory was noted by Kolmogorov (1949) in his original discussion of drop breakup, and he continued investigating the effect in later studies (Kolmogorov, 1962). These stochastic fluctuations have been extensively studied in fluid mechanics literature and are often referred to as “intermittency” (Pope, 2000, Sreenivasan, 2004). However, with a few notable exceptions (Baldyga and Bourne, 1992, Baldyga and Bourne, 1995, Baldyga and Podgorska, 1998, Jasinska et al., 2015), it has rarely been discussed in relation to emulsification technology.
An improved understanding of the turbulent drop fragmentation in RSM requires hydrodynamic characterization of both spatial and temporal variations. Spatially resolved investigations of RSM fluid flow using particle image velocimetry (PIV) was first reported by Kevala et al. (2005) and later by Mortensen et al. (2011). In the first part of this study (Håkansson et al., 2017), we have extended these investigations by estimating the local fragmenting stress and compared it to breakup visualizations in the same RSM geometry. Fig. 1 summarizes the results of the first part of these investigations by showing the average stress in the RSM region compared to the drop breakup positions of Ashar et al. (submitted for publication) at two different rotor positions. It was concluded that the average turbulent stress predicts the most likely position of breakup but is unable to explain other drop breakup positions. In part 1, it was hypothesized that the stochastic distribution of disruptive stresses could help explain why breakup sometimes occurs at locations where the average stress is low.
There are, to the best of our knowledge, no previously reported studies on the statistical distribution of local turbulent stresses for RSMs, nor for any other emulsification process. However, two theoretical models describing the stochastic nature of disruptive stresses have been suggested and applied to emulsification. Drop fragmentation rate models often assume an exponential decaying probability density function for the kinetic energy at each turbulent eddy length-scale (e.g. Angelidou et al., 1979, Andersson and Andersson, 2006, Coulaloglou and Tavlarides, 1977, Hagsaether et al., 2002, Lou and Svendsen, 1996, Narsimhan et al., 1979). This corresponds to a specific assumption about the stress distribution that can be tested experimentally.
Another suggestion for how to incorporate the stress distribution comes from the multifractal emulsification theory by Baldyga and Bourne (1995). This framework has been used to propose both fragmentation rate models and steady-state drop correlations (Baldyga and Podgorska, 1998). From a theoretical perspective the approach is promising, but it has not been extensively applied by other research groups.
Neither of these models for the stochastic nature of disruptive stresses has been compared directly with measured instantaneous stresses for emulsification processing conditions. Experimental investigations are therefore much in need, both to understand RSM emulsification better and to validate the theoretically proposed models.
The long-term objective of this research project is to extend the scientifically based knowledge of drop fragmentation in RSMs. In light of the lack of experimental data and suggestion from previous investigation on the importance of stress distributions, the specific objectives of this study are threefold:
- i.
Quantify the probability distribution of local instantaneous stresses in the effective region of an RSM.
- ii.
Compare the experimental data to the previously suggested models for the statistical distribution of instantaneous stresses used in emulsification processing, and conclude on their suitability for describing emulsification in RSMs.
- iii.
Discuss implications of findings for the emulsification efficiency in RSMs.
Section snippets
The distribution of fragmenting stresses
In the turbulent inertial regime of drop breakup, the average disruptive stress is proportional to the average intermediary length-scale TKE, 〈kd〉, (see discussion in Håkansson et al., 2017),where kd is obtained by integrating the power spectrum from the smallest turbulent length-scales, η, to the limiting eddy length-scale, ld:
Eq. (4) is often simplified by a series of assumption (i.e. isotropic, homogeneous turbulence and ld within the inertial subrange, see
Tank, geometry and experimental setup
Experiments were conducted in a specially designed tank (Fig. 2) allowing for the optical access needed for PIV experiments (Mortensen et al., 2011) and breakup visualizations (Ashar et al., submitted for publication). A batch RSM (see Fig. 2A), consisting of a single six-bladed rotor with a diameter (D) of 0.188 m and a single stator screen with 26 slots (w = 6 mm, H = 30 mm) was mounted in the tank. The rotor-stator gap clearance (δ in Fig. 2B) was set to 0.5 mm. The geometry was chosen to resemble
Statistical convergence of moments and probabilities
In this study, two measures are used to characterize the probability distribution of disruptive stresses at each point in the measurement field: first, the pointwise statistical moments of the stress distribution, and secondly, the probability of having a stress exceeding the stabilizing stress, σstab.
Fig. 3A shows the convergence of the first three moments (mean, standard deviation and skewness) of the stress distribution in a representative point in the field (point III in Fig. 1, obtained at
Conclusions
This study provides the first experimental quantification of the local probability distribution of instantaneous turbulent stresses in an RSM. It is concluded that the stress distribution is approximately lognormal. However, the data indicate a higher probability for rare high intensity stresses than the lognormal model. Moreover, the relative distribution of the stress distribution (i.e. normalized with the average stress) appears to increase with the rotor tip-speed and varies systematically
Acknowledgment
Financial support from the Knowledge Foundation (KK-stiftelsen) grant number 20150023 and Tetra Pak Processing Systems is gratefully acknowledged.
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