Droplet Size Distribution in a Kenics Static Mixer: CFD Simulation and Experimental Investigation of Emulsions

The minimum achievable droplet sizes created by a simple in-line Kenics Static Mixer (KSM) under various flow rates and mixing time in oil in water (O/W) emulsion were investigated through turbulent flow system. First, a Computational Fluid Dynamics (CFD) method is utilized to predict final droplet sizes in different Reynolds number. Then, an experimental setup was used in order to validate CFD results. The droplet size was monitored using Dynamic Light Scattering (DLS) technique by means of a Malvern zetasizer machine. Breakup/coalescence of droplets under constant volume fractions of oil was studied when flow rate was varied from 36.7 to 85 ml/s. Results showed that droplet size distribution highly depends on flow rate and mixing time. Droplets break more easily and faster at higher flow rates. The results proved that the obtaining small enough droplets using static mixer in less than 40 minutes at the flow rates above 36.7 ml/s at moderate concentration of oil volume fraction.


Introduction
Over recent years a great deal of attention has been paid to the formation and stability of micro/nano scale emulsions and precise control of droplet size and size distribution [1]. Two phase liquid dispersion is one of the most complex processes among mixing operations. Agitating two immiscible liquids results in the dispersion of one phase in the other in the form of small droplets whose characteristics depend on the equipment and the operating conditions [2]. It is practically impossible to make stable dispersions of uniform droplet size distribution, because of the wide range of properties and flow conditions [3]. A large amount of work can be found in the literature concerning the prediction of drop size distributions in turbulent liquid-liquid dispersions in static mixers (SM). Most of them use the concept of a turbulent energy cascade to predict the maximum stable droplet diameter, referring to the Hinze-Kolmogorov theory [2][3][4][5][6][7][8].
Static mixers are introduced as an alternative device believed to have a significant industrial potential to produce stable emulsions [9]. While SMs are widely used in other agro and petrochemical processes, they have not been studied in depth for the generation of mini-emulsion droplets. It is clear that they can be economically practical, safely used and can be utilized on larger scales. However, in terms of dispersion systems, their role in droplet breakage is an area of ongoing research. In our previous work [10], we reported a successful experimental production of mini-emulsions which was produced by making the mixture circulate two immiscible liquids (oil and aqueous phases) through the pipe in which the SM was inserted.
In this investigation, a CFD code is used to calculate the flow in the KSM and results are validated by means of DLS measurements of a final droplets diameter in a specific time. In the first section, the numerical methods and governing equations are proposed and the model and simulation properties are described. In the second section, two cases in the same manner of experimental setup but different in order of flow rate are defined and material properties are described. Finally, the CFD results are compared with the experimental results to evaluate the results, then numerical results and the dynamic behavior of the KSM is discussed in more detail.

Theoretical Model-Numerical Methodology
Breakup of bubbles and droplets, has been the subject of investigation for several decades starting with the pioneering work by two researchers, Kolmogorov [11] and Hinze [12], who proposed a formula for the maximum drop size, independently. Thereafter, Luo base on spherical assumption of droplet shapes proposed a model for breakup of fluid drop, description of the stability of mono-dispersed colloids, Population Balance Equations (PBEs) have found diverse applications in areas involving particulate systems [13]. Recently, Solsvik proposed an algebra of the high-order least-squares method, which linked to the implementation issues of a problem describing the drop size distribution within a liquid-liquid emulsion [14]. The highaccuracy, low numerical diffusion of the least-squares method for these types of solution has been proved, regarding the published literatures [15][16][17][18].
In this work, a 3D CFD model of the two-phase flow in continuous KSM is developed. Based on an Eulerian-Eulerian two-fluid model, the high-order least-squares method (HOLS) is used to solve the PBE [19]. The PBE and CFD models are both solved by the non-commercial CFD code developed by our researcher team.
A general form of the population balance equation can be expressed as follows: Where, n(L;x,t) is the number density function with droplet diameter (L) as the internal coordinate, G(L)n(L;x,t) is the droplet flux due to molecular growth rate, B ag (L;x,t) and D ag (L;x,t) are the birth and death rate of droplets diameter (L) due to aggregation, respectively, and B br (L;x,t) and D br (L;x,t) are the birth and death rate of droplets diameter (L) due to breakage, respectively. In eqn 1, the first term on the left hand is the transient term, the second term is the convective term, and the terms on the right hand are the source term describing droplet growth, aggregation, and breakage dynamics, respectively.
Regarding to its actual properties, the simulated SM has an inner diameter of 25 mm, a height of 25 mm, and 10 standard static elements fabricated from polyacetal plastic, arranged alternatively at 90° ( Figure  2). In addition, Grid sensitivity was carried out initially, and the results indicated that a total amount of 325 K cells was adequate to conserve the mass of each phase in the dynamics model.
In order to obtain suitable mesh size in our CFD model at initial step, fluid velocity was varied from 0.11 to 10 m/s which provides Reynolds number from 3 K to 280 K and mesh size was adapted for minimal error based on numerical and experimental Re number. Significant differences was seen for course mesh size, however, after re-meshed the model with super fine mesh, (~10 -6 ) when looking at the overall flow characteristics, which are shown below, one can see that the differences are not too large and in general they agree well. Detailed information of mesh independency study and prediction errors is presented in Table 2.
It should be note that, although the physical geometries of SM is adapted with model ( Figure 1), there may exist some differences between results of our model and experiments due to small deviation of SM geometry and other assumptions.
The phase-coupled SIMPLE (Semi-Implicit Method for Pressure Linked Equations) algorithm was used to couple pressure and velocity [20]. A one stage calculation and two cases with different Reynolds numbers were implemented. The flow field was simulated with bulk velocity started from 36.7 ml/s, 60.6 ml/s and 62 ml/s in the first case and 36.7 ml/s up to 82 ml/s in the second case. Then results were compared with those obtained experimentally from DLS technique. The breakage and coalescence process was simulated by utilizing the energy and PBE model regarding droplet tracing technique until the average nano-size Sauter mean diameter (d 32 ) reached. It should be noted that since sufficient amount of literatures proved there is negligible ratio of breakage occurs in storage tank compared to those in SM, droplet breakage in storage can be disregarded [9,10,[21][22][23].

Experimental
To carry out the experimental studies two cases were considered for oil phases; methyl methacrylate (MMA) and sunflower oil. In both systems de-ionized water was used as continuous phase. General formulations of mini-emulsions are shown in Table 1.
In the second case, Sunflower oil as dispersed has a density of 902.4 kg/m 3 , Refractive index of 1.4646, and viscosity of 47.11 g/m.s, all measured at 25°C.Geometry and dimensions of SM were modeled using Solid Works 3D CAD software, and exported into commercial software GAMBIT 2.1 and an appropriate mesh is generated.
A schematic diagram of the experimental setup is provided in Figure 3. A circulator pump was made to function with variable electrical current to ensure a series of known flow-rates. The mixture of two immiscible fluids was pumped from a 2 liter capacity reservoir to the SM. The fluid flow unit consists of piping section (with inner diameter of 25 mm and total piping lengths of 1571 mm) preceded by an inlet section where two phases are co-axially introduced into the piping section without any pre-mixing process. However, immediately after entrance into the pipe they mixed due to the turbulence fluid flow system. As mentioned previously, the oil droplet size was measured by DLS after certain time achieving steady state condition. Furthermore, a feedback system is used to measure the flow rate.

Results and Discussion
At the initial phase of the validation process, the results of experimental emulsification using the KSMs according to first formulation were compared with those obtained from CFD model. These experimental results were previously published elsewhere [10]. Figure 4 shows droplet size as emulsification time. In this figure the points to the graph are experimentally captured for different flow rates, whereas the lines indicate calculated values using CFD code.
Regarding to the Figure 4, droplet diameter decreasing asymptotically with increasing homogenization time and smaller droplet obtained at higher flow rates. In other respects, an increase in mechanical energy can help overcome the limit imposed by interfacial tension, thereby inducing more breakage. One might theorize that at higher flow rates, more energy is input into the system allowing breaking up large droplets. By means of that, intensifies the distribution      The flow rate was set to 60.6 and 85 ml/s for experiments and varied from 36.7 to 85 ml/s for numerical studies. As it can be seen, at 60.6 ml/s flow rate, the experimental data shows larger droplets than numerical results. When flow rate is higher than 68 ml/s from the beginning of process for few minutes droplet diameter of emulsion is in good accordance with those of CFD results. However, in general there is a meaningful difference between experimental and CFD results. This is due to the fact of the problem of "lost" droplets, saying, droplet trajectories are trapped near a solid wall accentuate in lower flow rates [21].
The experimental results of 85 ml/s flow rates shows relatively lower difference of droplet size between numerical and experimental results in compared with those obtained for lower flow rates.
With increasing flow rate, the Non-linear relationship between the flow rate and the average droplets size appears even at first stage of emulsification.
In order to evaluate the validity of our CFD model results for a given homogenization time, droplets size of emulsion prepared within 40 minutes at 36.7 ml/s flow rate is compared with those obtained from numerical data in Figure 6. This figure clearly displays similar trends for numerical and experimental results.
It is possible to determine the frequency of coalescence and breakup for numerical results in Figure 6. This may help us to have an idea for experimentally coalescence and breakup of droplets. The brakeage of droplet has been studied extensively, the incorporation of two different breakage behavior that accounted for large droplets to break easier due to turbulent shear [24,25] and on the other hand, small droplets break due to collisions between droplets and turbulent eddies [26,27].
However, theoretical and experimental results are not ideally matched, but in order to gain an insight to the results of previous investigations it is worthy to discuss the frequency of coalescence and break up of droplets based on Figure 6 in three different group of smaller than 400 nm, between 400-800 nm and larger than 800 nm. Where, and decreases the average size. This leads to narrow distribution droplet size.
Considering Figure 4, discrepancy between the CFD and  ------------1332  1592   10 -2  408  391  418  382  427  371  ----------528       numerical results for the average frequency of droplets under 400 nm showed average coalescence of 1.243×E10 6 and 8.315×E10 2 of breakage per droplet. Whereas, these values are 6.218×E10 4 and 2.386×E10 4 for second group and also 9.624×E10 2 , 9.582×E10 8 for third group, respectively. These results only calculated during the time a droplet was a member of the groups. These comparison shows that smaller droplets tend to more coalescence and larger droplets should break up more frequently than smaller ones. These results are in agreement with those of other researchers. If the previously described breakage frequency is valid, then our experimental data supports the dependency of the breakage efficiency to the droplet size. Regarding the numerical data there is negligible breakage rates predicted for small droplet sizes.
It is reported that the coalescence of droplets, depends on the evolution of overall surface area and shape of drops [28,29] and/or on the diameter of drops [30] and/or on the volume of the droplet [30,31]. Now it is interesting to turn our attention to check whether or not these well-founded phenomena may satisfy with our CFD results. Since we use PBEs, the exact number of droplets is available for each of previously mentioned groups of droplets. It was determined that the number of droplets under 400 nm is only 2.89% of total number of droplets, while those between 400 to 800 nm are 39.24% and larger than 800 nm are 57.87%. Based on these results total surface area of droplets are 5.292×E10 9 nm 2 , 2.634×E10 14 nm 2 and 1.109×E10 14 nm 2 for droplets groups less than 400 nm, between 400-800 nm and larger than 800 nm, respectively.
Thus, considering the total surface area of droplets in compare with coalescence and breakage rate reveals a logical conformity with respect to previews judgment; this means that the coalescence rate strongly depends indirectly on the droplet size and with decreasing droplet sizes, harmonically increasing total droplets surface area, coalescence frequency increased in agreement. Figure 7 shows numerical results of oil droplet size distributions after 5, 10, 20, 30 and 40 minutes of homogenization for the 36.7 ml/s flow rate. One can see as the slope of the 5 min indicator curve increased dramatically after 900 nm droplet size, the ratio is express limitation of droplet breakage to the 900 nm.
It also reproduced the positive trend that the mean diameter decreased with increasing homogenization time. However, the numerical results show some difference in droplet diameter, especially for the lowest flow rates. Taken collectively, these results suggested that the functional dependencies of the mixing time and breakage rate was reasonable but that quantitative predictions with the base case model parameters may be difficult. Below also provided further numerical details of the full drop size distribution (See Table 3).

Conclusions
Droplet breakage using KSM has been simulated by means of CFD technique. In the preliminary validation stage, the simulation has captured the droplet changes successfully and reasonable difference between computed and measured results was shown. Fluid flow rate, mean droplet size and homogenization time were considered as important parameters. The CFD results were evaluated for two experimental systems with different oil phase. Droplet size was measured for these systems using Dynamic Light Scattering method. In theoretical model mesh size was adapted for our system using mesh dependency studies. Comparing theoretical results and experimental results may pursuit one that population balance equations can be suitable technique to simulate droplet creation at homogenization process. A more in depth study considering much more parameters is required to gain better understanding of homogenization process using SM.