# Computational Fluid Dynamics (CFD)-Based Droplet Size Estimates in Emulsification Equipment

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Droplet Break-Up in 2D Linear Flows

_{C}RG/σ, where μ

_{C}denotes the viscosity of the continuous phase, R represents the radius of the initial spherical droplet, and σ is the interfacial tension. Physically this number can be interpreted as a measure of the ratio of the deforming hydrodynamic stress (order μ

_{C}G) and the counteracting Laplace pressure (order σ/R), or alternatively as the ratio of the droplet deformation timescale (order = μ

_{C}R/σ) and the process timescale (1/G). The other dimensionless number is the viscosity ratio μ

_{D}/μ

_{C}, henceforth denoted as λ. At sufficiently small values of Ca the droplet reaches a stable deformed shape, but for each value of the viscosity ratio a critical value Ca

_{C}exists above which no stable deformation is possible. Based on systematic experimental work in a four-roll mill, Bentley and Leal [14] have constructed the curves of this critical capillary number versus viscosity ratio for a series of 2D quasi-steady linear flows. De Bruijn [15] has complemented this work by single-droplet studies in simple shear flow, generated in a Taylor-Couette geometry as used also by Grace [13]. These data are reproduced in Figure 1, and analytical correlations for different values of α are provided by Equations (A1) and (A2) in Appendix A. The quality of the fit for the coefficients in Table A1 is illustrated in Figure A1 for α = 0 and α = 1.

_{C}the droplet typically breaks into two fragments, while for Ca >> Ca

_{C}it stretches into a long fluid thread that breaks into many fragments. For non-Newtonian continuous phases (e.g., non-dilute emulsions or continuous phases with thickeners) the continuous phase viscosity is usually replaced by the apparent emulsion viscosity at the prevailing shear rates (e.g., [9,12,17]).

_{C}R/σ) lies well below 0.1 s. Approximating a local flow as “quasi-steady linear” may then be a reasonable approximation in the context of the local droplet behavior. It is a pragmatic assumption, since approaches for droplet deformation in transient flows are more elaborate (e.g., [10,18]) and are not yet widely used in emulsification modelling.

_{32}) to be proportional to d

_{MAX}and write:

_{MAX}and d

_{32}and the prevailing value of Ca

_{C}. Additionally, it may empirically compensate to some extent for the differences between 2D and 3D flow, and for deviations from the quasi-steady flow condition. When detailed information about the flow type in the emulsification zone of the device is lacking, this can be a simple and useful approach for a given emulsion-device combination. However, it introduces a hidden λ dependence into the fit parameter A. Moreover it may be that the elongation-shear balance in the local flow changes with process conditions like rotor speed and throughput, which can have a substantial effect on the outcome according to Figure 1. In that case there is no single value for A that covers the whole range of the process conditions even for a given emulsion (fixed λ).

## 3. Droplet Break-Up in 3D Laminar Flows

_{MAX}. Hill [4] has suggested a mathematical procedure in which the local coordinate system is rotated such that the local velocity gradient tensor gets as close as possible to a 2D linear flow. In order to quantify the latter condition a merit function is defined as the sum of squares of all components L

_{ij}of the local velocity gradient tensor defined in Equation (1), except L

_{12}and L

_{21}. The procedure determines the coordinate system in which the merit function is minimal and transforms the local velocity gradient tensor to that base. We then have, by comparison to Equation (1), L

_{12}≈ G and α ≈ L

_{21}/L

_{12}.

^{T}represents “transposition”, i.e., the exchange of L

_{ij}and L

_{ji}. For the 2D linear flow of Equation (1) we then obtain:

_{D}= 0. While the name “invariant” suggests that the value is independent of the coordinate system, this is in fact only true for the invariants of the rate-of-strain tensor. We will get back to this point later. For the 2D linear flow of Equation (1) we can derive:

_{C}d

_{MAX}/σ = Ca

_{C}/G (or d

_{MAX}itself for given μ

_{C}and σ) across the computational grid for a given viscosity ratio. This requires minimal user coding and little addition to the overall computational effort of the single-phase CFD simulations. Extension to non-Newtonian continuous phases is straightforward, as mentioned in Section 2, provided a correlation is available to link the local value of μ

_{C}to the local value of the shear rate.

^{2}>> ω

^{2}) while a value close to −1 points to almost pure rotation (S

^{2}<< ω

^{2}). However, it is not obvious that the result always represents the “closest-resembling 2D flow” in the context of Figure 1. In order to get a better view on how well a given 3D local flow can be approximated by a 2D equivalent we propose to also calculate a dimensionless flow parameter that has been introduced by den Toonder et al. [20]:

_{N}varies between −2/√3 for biaxial elongation and 2/√3 for uniaxial elongation, being zero for 2D flow. In fact it is more convenient for our purposes to renormalize R

_{N}such that it varies between −1 and 1, similar to α:

_{N}close to zero then indicates that the 2D flow approximation should work, although we cannot suggest a limit for |R′

_{N}| as yet. It is worth noting that during the preparation of the present article we have become aware of a very recent paper by Nakayama et al. [21], in which the parameter R′

_{N}is proposed for characterization of flow patterns in complex mixers.

_{N}is an objective parameter, α and G are not. Objective parameters that incorporate both stretching and rotation have been developed ([22], and references therein), but using these in the present context would be complex and computationally elaborate. In principle the fact that α and G are not objective could lead to “jumps” in their values at the boundary between rotating and stationary mesh parts of a CFD simulation. However, as shown also in the next section, we have not yet observed such behavior in our simulations of rotor-stator devices.

## 4. Results for a Lab-Scale Toothed Rotor-Stator Mixer (Ultra-Turrax)

^{3}and a viscosity of 0.1 Pas. The simulations have been done with the commercial software FLUENT v.15. The CFD grid consists of three parts: the rotor, the stator, and a substantially larger cylindrical tank into which the mixing head was placed. The rotor was placed in a rotating part of the grid and the other two parts were placed in a non-rotating zone. The mass flow through the head was calculated by integrating the axial velocity over the six circular holes in the base plate of the rotor. The start design of the grid was inspired by the MSc thesis of Ko [23], in which a toothed rotor-stator with one rotor row and one stator row was simulated. His work demonstrated that the cells near the gap walls were more important than those near the center, hence, his use of grid refinement near the walls. Seeking a compromise between accuracy and simulation time, we tested additional variations of structured grids in simplified mill geometries. The grid of one section of the rotor and stator was locally refined (see the darker region in the IKA rotor grid in Figure 3) to check the influence of the calculation grid without facing extremely long calculation times. No significant differences in the local flow variables could be found when comparing the passing of the refined grid sections to the passing of the unrefined sections. A horizontal cross-section of our final grid choice is provided in Figure S1 of the the Supplementary Material.

^{−6}s. The results became periodical after about 4–5 rotor teeth had passed a given stator tooth. The unsteady calculation took another 2–3 days. An influence of the six circular holes in the base plate could not be found. For all flow variables second-order schemes have been applied. The gradients have been calculated with the least square cell-based method and the pressure-velocity coupling with the “Pressure Implicit with Splitting of Operator (PISO) ”scheme.

_{N}(Figure 6), α (Figure 7), Ca

_{C}/G (Figure 8) and u

_{RAD}Ca

_{C}/G (Figure 9) were calculated via custom field functions. This increased the overall computing time by 7%–8% compared to a calculation without these additional functions. The most complex parameter is R′

_{N}. When pictures of this parameter are calculated and exported for a sufficiently dense series of time steps to compose an animation, the duration of the simulation can increase up to 30% longer than a calculation without the custom field functions.

_{N}in these zones, as shown in Figure 6.

_{N}does not make a significant step change across these boundaries, we do not think that their visibility is related to the objectivity discussion at the end of Section 3. Rather we conclude that the velocity derivatives are more sensitive to numerical irregularities caused by the rather large aspect ratio of the grid cells adjacent to the boundaries than the velocity components themselves (Figure 4 and Figure 5). Interestingly, the R′

_{N}parameter predicts the flow in the circular holes in the rotor base plate to be rather 2D as well, although the relative magnitude of the velocity components does not readily suggests this. Above the base plates the flow is more 3D in the central part of the mixing head.

_{N}| is not small the α values are questionable. The same holds for the other plots that are based on the 2D flow approximation. According to Figure 1, the critical capillary number at a given viscosity ratio decreases with the increasing value of α. However, this does not necessarily imply that the zones with the highest α values in Figure 7 provide the lowest d

_{MAX}. The latter also depends on the corresponding local velocity gradient G. Figure 8 shows the group Ca

_{C}/G = μ

_{C}d

_{MAX}/2σ for a viscosity ratio of unity. Here we see that the smallest values for d

_{MAX}are, in fact, predicted for the outer shear gap, because the local value for G is much higher than in the elongational zones. Surprisingly, we also see small droplets being predicted for the flow through the holes in the base plate. For both low and high viscosity ratios the “emulsification zone” is expected to shift towards the elongational flows in the slots and near the tooth edges. This holds in particular for λ > 4, given the very steep increase in critical capillary numbers with increasing λ in that range for simple shear (Figure 1, α = 0). Indeed experimental data for a toothed rotor-stator reported by Weiss [25] (see also Figure S2 in the Supplementary Material) and for a slotted rotor-stator by Dicharry et al. [26] show that droplet break-up in such rotor-stator geometries remains possible up to high viscosity ratios, although the mean droplet size does increase gradually with λ at given process conditions. The latter is likely due to the fact that the mean diameter is an average of droplets that have followed different streamlines and did not all pass the zones of highest break-up effectivity, as also pointed out by Dicharry et al. [26]. As shown in our animation of “α iso-surfaces” in the Supplementary Material, the volume in which a given α value can be found decreases with increasing α for the Ultra-Turrax, and this will likely be true for toothed/slotted rotor-stators in general.

## 5. Discussion

_{N}(Equation (11)) has been introduced to characterize the 2D flow approximation, and for the Ultra-Turrax in Section 4 this seems to be valid in the zones that are expected to determine the final droplet size after multiple recirculations. It would be useful to also have an assessment for the validity of the assumption of quasi-steady flow compared to the droplet deformation timescale, for droplets that are already close to d

_{MAX}. This timescale is represented by the group Ca

_{C}/G = μ

_{C}d

_{MAX}/σ. One option could be to look at the distance that a droplet of size d

_{MAX}travels during its deformation. If this is small compared to the distance over which the local flow changes appreciably, the quasi-steady flow assumption should be locally valid. For a quick view we propose to characterize the travel distance by the product u

_{RAD}Ca

_{C}/G, since the radial velocity provides a measure of how quickly a droplet traverses a shear gap of a slot. We have calculated this parameter in Figure 9, and see that it is of the order of 0.1 mm or less in the relevant zones. This is small compared to the gap size and the slot dimensions, which may be taken as a rough and easy measure of the distance over which the local flow changes. Figure 9 then suggests that the assumption of quasi-steady flow may not be too bad for droplets close to d

_{MAX}.

_{MAX}along a streamline and assume that the smallest value will be the final one for droplets following that path. Doing this for a representative set of streamlines and combining the predicted values of d

_{MAX}with the flow rate along the streamlines, an average droplet size for the emulsion may be estimated. Monitoring the droplet deformation along the streamlines, as done by Feigl et al. [10], and would be more realistic, but would also be a step-change in computational effort.

## 6. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Table A1.**Coefficients of Equation (A2), taken from Hill [4].

α | a | b | c | d | e |
---|---|---|---|---|---|

0 | 0.935 | 0.557 | −2.150 × 10^{−5} | −1.901 10^{−3} | −0.754 |

0.2 | 0.327 | 0.473 | −9.177 × 10^{−3} | −0.248 | −1.369 |

0.4 | 0.286 | 0.380 | −1.713 × 10^{−2} | −0.292 | −1.938 |

0.6 | 0.244 | 0.286 | −2.509 × 10^{−2} | −0.336 | −2.509 |

0.8 | 0.310 | 0.273 | −0.352 | −0.435 | −3.525 |

1 | 0.465 | 0.454 | −0.526 | −0.665 | −5.259 |

**Figure A1.**Comparison of the correlation (A2) and the experimental data of Figure 1 for α = 0 and α = 1.

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**Figure 1.**Critical capillary number CaC versus viscosity ratio λ for different values of the flow parameter α. Open circles are data for simple shear flow (α = 0), taken from de Bruijn [15]. The other data were taken from Bentley and Leal [14] and represent α values of 0.2 (solid circles), 0.4 (solid triangles), 0.6 (crosses), 0.8 (open squares), and 1 (solid diamonds).

**Figure 2.**Rotor (

**left**), stator (

**middle**), and assembly (

**right**). Important dimensions are given in Table 1. The clearance between the teeth and opposite base plate is small and somewhat dependent on the tightening of the screw. It was taken to be 0.1 mm in the simulations.

**Figure 4.**Axial velocity at 6000 rpm in different horizontal planes (see also Figure 3). The left picture (y = 0) lies in the middle of the base plate of the rotor, hence the white areas. The middle- and right planes are located at 1 and 4 mm above the left plane, so just above the upper surface of the base plate, and approximately half of the tooth height, respectively. Color scale is in m/s.

**Figure 5.**Radial velocity at 6000 rpm for the same planes as in Figure 4. Color scale is in m/s.

**Figure 7.**Dimensionless flow parameter α at 6000 rpm for the same planes as in Figure 4.

**Figure 8.**Parameter Ca

_{C}/G = μCd

_{MAX}/2σ (s) at 6000 rpm for the same planes as in Figure 4. The viscosity ratio is unity and the corresponding value of Ca

_{C}has been locally calculated from correlation (A2).

Dimension | Rotor | Stator |
---|---|---|

Outer diameter (mm) | 40 | 45 |

Outer diameter middle row (mm) | - | 35 |

Outer diameter inner row (mm) | 30 | 25 |

Radial tooth thickness (mm) | 2 | 2 |

Tooth height inner rows (mm) | 7 | 7 |

Tooth height outer stator row | - | 15 |

Tangential slot width (mm) | 2 | 2 |

Number of slots | 20 | 18 |

Entrance hole diameter (mm) | 7 | - |

Gap between adjacent rows (mm) | 0.5 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Janssen, J.; Mayer, R.
Computational Fluid Dynamics (CFD)-Based Droplet Size Estimates in Emulsification Equipment. *Processes* **2016**, *4*, 50.
https://doi.org/10.3390/pr4040050

**AMA Style**

Janssen J, Mayer R.
Computational Fluid Dynamics (CFD)-Based Droplet Size Estimates in Emulsification Equipment. *Processes*. 2016; 4(4):50.
https://doi.org/10.3390/pr4040050

**Chicago/Turabian Style**

Janssen, Jo, and Roy Mayer.
2016. "Computational Fluid Dynamics (CFD)-Based Droplet Size Estimates in Emulsification Equipment" *Processes* 4, no. 4: 50.
https://doi.org/10.3390/pr4040050