# Experimental Investigation and Modelling of the Droplet Size in a DN300 Stirred Vessel at High Disperse Phase Content Using a Telecentric Shadowgraphic Probe

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## Featured Application

**The presented image-based telecentric probe is unique in its functionality and can be applied to determine particulate properties of various multiphase flows.**

## Abstract

## 1. Introduction

#### 1.1. Motivation and State of the Art

#### 1.2. Optical Multimode Online Probe

#### 1.3. Theory and Modelling

_{max}and a correlation coefficient C

_{1}. Based on this theory, Hinze [46] derived a Weber number $W{e}_{\left(d\right)}$ for the maximum stable droplet diameter:

_{kin}is the turbulent kinetic energy and E

_{σ}is the surface energy, with the density of the continuous phase ${\rho}_{d}$, the mean relative velocity $\overline{u}$, the drop diameter d and the interfacial tension σ

_{d,c}. If the turbulent kinetic energy and the surface energy are equal, the droplet is stable. According to Hinze a droplet remains stable at a Weber number $W{e}_{\left(d\right)}$ ≤ 1, otherwise it breaks up into one or more smaller droplets [46]. Substituting Equation (1) in Equation (2) gives:

_{max}is:

_{2}. In stirred tanks, $\overline{\epsilon}$ is independent of the liquid properties according to the theory of isotropic turbulence:

_{50}or the Sauter mean diameter d

_{32}are linearly related to the maximum stable diameter via a correlation coefficient ${C}_{4}$ [49]:

_{4}reflects the impeller design, while C

_{5}is for coalescence. The exponent for the Weber number in Equation (6) is valid in systems without coalescence, and some authors found C

_{9}as a function of the phase fraction φ [14,17]. Laso et al. [51] developed another interesting approach, which covers high disperse phase fractions with the general form as:

_{6}= 0.118, C

_{7}= 0.27, C

_{8}= −0.056 and C

_{9}= −0.4. Noteworthy is that it takes the phase fraction as a nonlinear function and the viscosities of both phases into account.

## 2. Materials and Methods

#### 2.1. Substances

_{2}SO

_{4}. Thereby, the continuous phase is not altered in its physical properties as given in Table 1. The standardization of the continuous water phase results in a defined electric conductivity of $8800$ µS/cm, which is necessary to guarantee measurement repeatability damping the influence of hardly detectable impurities on coalescence [48,52]. The density of the water/Na

_{2}SO

_{4}solution and the oil are measured with a density meter (DMA55, Stabinger Messtechnik GmbH, Graz, Austria). The kinematic viscosity of the oil was measured with a viscometer (Viscoboy 2, LAUDA Scientific GmbH, Lauda-Königshofen, Germany), the value for water was taken from literature. The interfacial tension of oil in water was determined with a contact angle microscope (DataPhysics Instrument GmbH, Filderstadt, Germany) using the pendant drop method.

#### 2.2. Experimental Setup

_{0}= 400 mm. The tank is filled to H = 290 mm height to ensure a ratio of H/T = 1. The tank is surrounded by another square tank, which is filled with water and ensures a constant temperature in the inner tank by means of thermostat T1 and pump P1. A six-blade Rushton impeller with a diameter of D = 98 mm (D/T ≈ 0.3) is positioned at an off-bottom-height C = T/2 = 145 mm and driven by a stirring unit (Heidolph RZR 2052 control). The impeller has the identical dimensions as in the work of Montante et al. [53,54], a disc diameter of 73.5 mm, a blade height of 19.6 mm, and a blade width of 24.5 mm. The tank is equipped with four radial equally distributed baffles. They have a width of B = T/10 = 29 mm, a thickness of 3 mm and cover the entire height H

_{0}of the tank.

_{1}. The measuring point is arranged radially at 87 mm from the center of the impeller shaft with an offset of 45° between two baffles. The probe is fitted with a Basler Aca 1300-60 gm camera and 1× telecentric lens. The chosen resolution is 1024 pixel × 1024 pixel, which equals 5.4 mm × 5.4 mm.

_{2}near the liquid surface was additionally investigated experimentally up to 10 vol.-% and simulations are reported by Rave et al. [21].

#### 2.3. Experimental Procedure

_{d}is the volume of the disperse phase and ${V}_{c}$ the volume of the continuous phase. Each experiment with a specific impeller speed N and phase fraction φ was carried out twice using the following routine.

_{2}SO

_{4}was dissolved in the water and added to the tank and then the paraffin oil was added. The filling level $H$ was checked manually with a scale. Then, the impeller was turned on and a period of 40 min was used to ensure that a steady-state droplet size was established. The temperature of the emulsion was checked using a thermometer. After that, the droplet measurement is started taking 2.5 min to acquire the 1500 pictures. After each experiment, the chemicals were disposed and new ones were used for the next experiment in order to preclude a change in the physical properties due to long-term accumulation of impurities. Hence, for each impeller speed and phase fraction two droplet size measurements are available resulting from two separate experiments. One exception is the experiment for N = 400 rpm and φ = 5 vol.-%, which was measured only once without a repetition.

#### 2.4. Data Analysis and Modelling

_{32}, is sufficient. It is calculated from the second moment, M

_{2,0}and third moment, M

_{3,0}of the number distribution as follows:

^{®}Curve Fitting Tool (Matlab

^{®}version 9.6.0.1072779; R2019a and Curve Fitting Toolbox 3.5.9).

## 3. Results and Discussion

#### 3.1. Influence of the Impeller Speed

_{3}, since the fine dispersion has a negligible contribution to the volume distribution as depicted in Figure 5b. The volume distributions are nearly ${q}_{3,(<100\mathsf{\mu}\mathrm{m})}$ ≈ 0 µm

^{−1}for particle classes smaller than 100 µm and are normally distributed in shape, which indicates a coalescing system [3]. As expected, the droplet size in the number as well as in the volume distribution decreases with increasing impeller speed. In addition, the distribution becomes narrower with increasing energy input and thus increasing impeller speed. This trend is observed for all experiments. DSD data of all q

_{0}and q

_{3}is given in the Supplementary material. For a simpler visualization of multiple experiments, Figure 6 depicts the Sauter mean diameters calculated according to Equation (13). The Sauter mean diameter decreases with increasing impeller speed. The qualitative trend of the Sauter mean diameter at different impeller speeds is similar for all phase fractions, with the trend shifting towards a larger droplet diameter for higher phase fractions.

#### 3.2. Influence of the Phase Fraction

#### 3.3. Modelling of the Sauter Mean Diameter

_{8}of the viscosity in Equation (9) was taken from the data of Laso et al. [51], because the influence of the viscosity is not studied in this work.

^{2}> 0.95. The model of Laso et al. [51] represents the experimental data slightly better (higher value of R

^{2}). The deviation from the model to the experimental data is less than 12% absolute and 4.0% as mean deviation for Laso et al. [51] and less than 22% absolute and 5.7% as mean deviation for Doulah [50]. The errors are randomly distributed with the approach of Laso et al. [51], over all data. The standard approach by Doulah [50] has the highest errors for φ = 5 vol.-%, which is a result of the linear dependence of the phase fraction in the model. The value of the Weber exponent is for both models around C

_{9}≈ −0.65, which is close to the theoretical one of −0.6 being obtained if coalescence is neglected [47]. The results with a Weber exponent C

_{9}= −0.6 are shown in the Supplementary Material in Tables S1 and S2 and in Figure S1.

## 4. Conclusions

## Supplementary Materials

_{9}= −0.6; Table S1: Coefficients for the fitted models with the theoretical Weber exponent C

_{9}= −0.6; Table S2: Relative and absolute deviations for the fitted models with the theoretical Weber exponent C

_{9}= −0.6; Tables S3–S48: Measured DSD’s and Sauter mean diameters of all experiments.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Kraume, M. Transportvorgänge in der Verfahrenstechnik; Springer: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Kresta, S.M.; Etchells, A.W.; Dickey, D.S.; Atiemo-Obeng, V.A. (Eds.) Advances in Industrial Mixing: A Companion to the Handbook of Industrial Mixing; Wiley: Hoboken, NJ, USA, 2016. [Google Scholar]
- Zerfa, M.; Brooks, B.W. Prediction of vinyl chloride drop sizes in stabilised liquid-liquid agitated dispersion. Chem. Eng. Sci.
**1996**, 51, 3223–3233. [Google Scholar] [CrossRef] - Cull, S.G.; Lovick, J.W.; Lye, G.J.; Angeli, P. Scale-down studies on the hydrodynamics of two-liquid phase biocatalytic reactors. Bioprocess Biosyst. Eng.
**2002**, 25, 143–153. [Google Scholar] [PubMed] - Angle, C.W.; Hamza, H.A. Predicting the sizes of toluene-diluted heavy oil emulsions in turbulent flow Part 2: Hinze–Kolmogorov based model adapted for increased oil fractions and energy dissipation in a stirred tank. Chem. Eng. Sci.
**2006**, 61, 7325–7335. [Google Scholar] [CrossRef] - Abidin, M.I.I.Z.; Raman, A.A.A.; Nor, M.I.M. Review on Measurement Techniques for Drop Size Distribution in a Stirred Vessel. Ind. Eng. Chem. Res.
**2013**, 52, 16085–16094. [Google Scholar] [CrossRef] - Kumar, S.; Ganvir, V.; Satyanand, C.; Kumar, R.; Gandhi, K.S. Alternative mechanisms of drop breakup in stirred vessels. Chem. Eng. Sci.
**1998**, 53, 3269–3280. [Google Scholar] [CrossRef] - EL-Hamouz, A.; Cooke, M.; Kowalski, A.; Sharratt, P. Dispersion of silicone oil in water surfactant solution: Effect of impeller speed, oil viscosity and addition point on drop size distribution. Chem. Eng. Processing
**2009**, 48, 633–642. [Google Scholar] [CrossRef] - Maaß, S.; Wollny, S.; Voigt, A.; Kraume, M. Experimental comparison of measurement techniques for drop size distributions in liquid/liquid dispersions. Exp. Fluids
**2011**, 50, 259–269. [Google Scholar] [CrossRef] - Lovick, J.; Mouza, A.A.; Paras, S.V.; Lye, G.J.; Angeli, P. Drop size distribution in highly concentrated liquid-liquid dispersions using a light back scattering method. J. Chem. Technol. Biotechnol.
**2005**, 80, 545–552. [Google Scholar] [CrossRef] - Barrett, P.; Glennon, B. In-line FBRM Monitoring of Particle Size in Dilute Agitated Suspensions. Part. Part. Syst. Charact.
**1999**, 16, 207–211. [Google Scholar] [CrossRef] - Barrett, P.; Glennon, B. Characterizing the Metastable Zone Width and Solubility Curve Using Lasentec FBRM and PVM. Chem. Eng. Res. Des.
**2002**, 80, 799–805. [Google Scholar] [CrossRef] [Green Version] - Lichti, M.; Bart, H.-J. Particle Measurement Techniques in Fluid Process Engineering. Chembioeng. Rev.
**2018**, 5, 79–89. [Google Scholar] [CrossRef] - Amokrane, A.; Maaß, S.; Lamadie, F.; Puel, F.; Charton, S. On droplets size distribution in a pulsed column. Part I: In-situ measurements and corresponding CFD–PBE simulations. Int. J. Chem. Eng.
**2016**, 296, 366–376. [Google Scholar] [CrossRef] - Schlüter, M. Lokale Messverfahren für Mehrphasenströmungen. Chem. Ing. Tech.
**2011**, 83, 992–1004. [Google Scholar] [CrossRef] - Godfrey, J.C.; Grilc, V. (Eds.) Drop Size and Drop Size Distributions for Liquid-Liquid Dispersions in Agitated Tanks of Square Cross Section. In Proceedings of the 2nd European Conference on Mixing, Cambridge, UK, 22–27 June 1977; BHRA Fluid Engineering: Cambridge, UK, 1977; pp. 1–20. [Google Scholar]
- Desnoyer, C.; Masbernat, O.; Gourdon, C. Experimental study of drop size distributions at high phase ratio in liquid–liquid dispersions. Chem. Eng. Sci.
**2003**, 58, 1353–1363. [Google Scholar] [CrossRef] - Qi, L.; Meng, X.; Zhang, R.; Liu, H.; Xu, C.; Liu, Z.; Klusener, P.A.A. Droplet size distribution and droplet size correlation of chloroaluminate ionic liquid–heptane dispersion in a stirred vessel. Int. J. Chem. Eng.
**2015**, 268, 116–124. [Google Scholar] [CrossRef] - Kraume, M.; Gäbler, A.; Schulze, K. Influence of Physical Properties on Drop Size Distribution of Stirred Liquid-Liquid Dispersions. Chem. Eng. Technol.
**2004**, 27, 330–334. [Google Scholar] [CrossRef] - Gäbler, A.; Wegener, M.; Paschedag, A.R.; Kraume, M. The effect of pH on experimental and simulation results of transient drop size distributions in stirred liquid–liquid dispersions. Chem. Eng. Sci.
**2006**, 61, 3018–3024. [Google Scholar] [CrossRef] - Rave, K.; Hermes, M.; Wirz, D.; Hundshagen, M.; Friebel, A.; Harbou, E.V.; Bart, H.-J.; Skoda, R. Experiments and fully transient coupled CFD-PBM 3D flow simulations of disperse liquid-liquid flow in a baffled stirred tank. Chem. Eng. Sci.
**2022**, 120, 117518. [Google Scholar] [CrossRef] - Mickler, M.; Bart, H.-J. Optical Multimode Online Probe: Erfassung und Analyse von Partikelkollektiven. Chem. Ing. Tech.
**2013**, 85, 901–906. [Google Scholar] [CrossRef] - Schuhmann, R.; Thöniß, T. Telezentrische Systeme fuer die optische Mess- und Prueftechnik. Technol. Mess
**1998**, 65, 131–136. [Google Scholar] [CrossRef] - Wirz, D.; Bart, H.-J. Advances in particle size analysis with transmitted light techniques. Bulg. Chem. Commun.
**2020**, 52, 554–560. [Google Scholar] - Lichti, M. Optische Erfassung von Partikelmerkmalen: Entwicklung einer Durchlichtmesstechnik für Apparate der Fluidverfahrenstechnik. Ph. D. Thesis, TU Kaiserslautern, Kaiserslautern, Germany, 2018. [Google Scholar]
- Lichti, M.; Roth, C.; Bart, H.-J. Vorrichtung für Bildaufnahmen eines Messvolumens in Einem Behälter. Patent DE102015103497A1, 15 September 2016. [Google Scholar]
- Lichti, M.; Cheng, X.; Stephani, H.; Bart, H.-J. Online Detection of Ellipsoidal Bubbles by an Innovative Optical Approach. Chem. Eng. Technol.
**2019**, 42, 506–511. [Google Scholar] [CrossRef] - Wirz, D.; Hofmann, M.; Lorenz, H.; Bart, H.-J.; Seidel-Morgenstern, A.; Temmel, E. A Novel Shadowgraphic Inline Measurement Technique for Image-Based Crystal Size Distribution Analysis. Crystals
**2020**, 10, 740. [Google Scholar] [CrossRef] - Steinhoff, J.; Bart, H.-J. Settling Behavior and CFD Simulation of a Gravity Separator. In Extraction 2018; The Minerals, Metals & Material Series; Davis, B.R., Moats, M.S., Wang, S., Gregurek, D., Kapusta, J., Battle, T.P., Schlesinger, M.E., Flores, G.R.A., Jak, E., Goodall, G., et al., Eds.; Springer International Publishing: Cham, Switzerland, 2018; pp. 1997–2007. [Google Scholar]
- Steinhoff, J.; Charlafti, E.; Reinecke, L.; Kraume, M.; Bart, H.-J. Investigation and development of gravity separators with a standardized experimental setup. Can. J. Chem. Eng.
**2020**, 98, 384–393. [Google Scholar] [CrossRef] - Schmitt, P.; Hlawitschka, M.W.; Bart, H.-J. Centrifugal pumps as extractors. Chem. Ing. Tech.
**2020**, 262, 12215. [Google Scholar] [CrossRef] [Green Version] - Lichti, M.; Bart, H.-J. Bubble size distributions with a shadowgraphic optical probe. Flow Meas. Instrum.
**2018**, 60, 164–170. [Google Scholar] [CrossRef] - Jasch, K.; Schulz, J.; Bart, H.-J.; Scholl, S. Droplet Entrainment Analysis in a Flash Evaporator with an Image-Based Measurement Technique. Chem. Ing. Tech.
**2021**, 93, 1071–1079. [Google Scholar] [CrossRef] - Schulz, J.; Usslar, M.; Bart, H.-J. Impact of weir design on entrained liquid in tray columns. AIChE J.
**2021**, 67, A483. [Google Scholar] [CrossRef] - Schulz, J.; Bart, H.-J. Analysis of entrained liquid by use of optical measurement technology. Chem. Eng. Res. Des.
**2019**, 147, 624–633. [Google Scholar] [CrossRef] - Schulz, J. Local Image-Based and Conventional Integral Entrainment Analysis; Shaker Verlag: Düren, Germany, 2021. [Google Scholar]
- Lichti, M.; Schulz, J.; Bart, H.-J. Quantification of Entrainment Using an Optical Inline Probe. Chem. Ing. Tech.
**2019**, 91, 429–434. [Google Scholar] [CrossRef] - Hough, P.V.C. Method and Means for Recognizing Complex Patterns. U.S. Patent US3069654A, 18 December 1962. [Google Scholar]
- Yuen, H.K.; Princen, J.; Illingworth, J.; Kittler, J. Comparative study of Hough Transform methods for circle finding. Image Vis. Comput.
**1990**, 8, 71–77. [Google Scholar] [CrossRef] [Green Version] - Illingworth, J.; Kittler, J. A survey of the hough transform. Comput. Graph. Image Processing
**1988**, 44, 87–116. [Google Scholar] [CrossRef] - Mickler, M.; Didas, S.; Jaradat, M.; Attarakih, M.; Bart, H.-J. Tropfenschwarmanalytik mittels Bildverarbeitung zur Simulation von Extraktionskolonnen mit Populationsbilanzen. Chem. Ing. Tech.
**2011**, 83, 227–236. [Google Scholar] [CrossRef] - LeCun, Y.; Bengio, Y.; Hinton, G. Deep learning. Nature
**2015**, 521, 436–444. [Google Scholar] [CrossRef] [PubMed] - Schäfer, J.; Schmitt, P.; Hlawitschka, M.W.; Bart, H.-J. Measuring Particle Size Distributions in Multiphase Flows Using a Convolutional Neural Network. Chem. Ing. Tech.
**2019**, 83, 992. [Google Scholar] [CrossRef] [Green Version] - Steinhoff, J.; Charlafti, E.; Leleu, D.; Reinecke, L.; Becker, K.; Kalem, M.; Sixt, M.; Franken, H.; Braß, M.; Borchardt, D.; et al. ERICAA, Energie- und Ressourceneinsparung durch Innovative und CFD-Basierte Auslegung von Flüssig/Flüssig-Schwerkraftabscheidern, Abschlussbericht; Wiley: Hoboken, NJ, USA, 2019. [Google Scholar]
- Kolmogorov, A. The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds’ Numbers. Dokl. Akad. Nauk SSSR
**1941**, 30, 301–305. [Google Scholar] - Hinze, J.O. Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes. AIChE J.
**1955**, 1, 289–295. [Google Scholar] [CrossRef] - Shinnar, R.; Church, J.M. Statistical theories of turbulence in predicting particle size in agitated dispersions. Ind. Eng. Chem.
**1960**, 52, 253–256. [Google Scholar] [CrossRef] - Gebauer, F. Fundamentals of Binary Droplet Coalescence in Liquid–Liquid Systems; Verlag Dr. Hut GmbH: München, Germany, 2018. [Google Scholar]
- Sprow, F.B. Distribution of drop sizes produced in turbulent liquid—liquid dispersion. Chem. Eng. Sci.
**1967**, 22, 435–442. [Google Scholar] [CrossRef] - Doulah, M.S. An effect of hold-up on drop sizes in liquid-liquid dispersions. Ind. Eng. Chem.
**1975**, 14, 137–138. [Google Scholar] [CrossRef] - Laso, M.; Steiner, L.; Hartland, S. Dynamic simulation of agitated liquid—liquid dispersions—II. Experimental determination of breakage and coalescence rates in a stirred tank. Chem. Eng. Sci.
**1987**, 42, 2437–2445. [Google Scholar] [CrossRef] - Villwock, J.; Gebauer, F.; Kamp, J.; Bart, H.-J.; Kraume, M. Systematic Analysis of Single Droplet Coalescence. Chem. Eng. Technol.
**2014**, 37, 1103–1111. [Google Scholar] [CrossRef] - Montante, G.; Lee, K.C.; Brucato, A.; Yianneskis, M. Numerical simulations of the dependency of flow pattern on impeller clearance in stirred vessels. Chem. Eng. Sci.
**2001**, 56, 3751–3770. [Google Scholar] [CrossRef] - Montante, G.; Brucato, A.; Lee, K.C.; Yianneskis, M. An experimental study of double-to-single-loop transition in stirred vessels. Can. J. Chem. Eng.
**1999**, 77, 649–659. [Google Scholar] [CrossRef] - Rave, K.; Lehmenkühler, M.; Wirz, D.; Bart, H.-J.; Skoda, R. 3D flow simulation of a baffled stirred tank for an assessment of geometry simplifications and a scale-adaptive turbulence model. Chem. Eng. Sci.
**2021**, 231, 116262. [Google Scholar] [CrossRef] - Zlokarnik, M. Stirring: Theory and Practice; Wiley-VCH: Weinheim, Germany; Chichester, UK, 2010. [Google Scholar]

**Figure 1.**Scheme of a OMOP in a DN100 extraction column. The invasive tubes are mounted eccentric and do not interfere with the impeller shaft.

**Figure 3.**Experimental setup of the DN300 stirred vessel and the measurement position MP

_{1}of the OMOP probe; (

**a**) side view and (

**b**) top view of the vessel.

**Figure 4.**CNN analysis at different phase fractions at N = 325 rpm. (

**a**) 10 vol.-%, (

**b**) 30 vol.-% and (

**c**) 50 vol.-%. The green circles represent the detected droplet circumference.

**Figure 5.**(

**a**) Number distribution at different rpm at φ = 40 vol.-% phase fraction; (

**b**) volume distribution at different rpm at φ = 40 vol.-% phase fraction. The lines are a guide to the eye.

**Figure 6.**Experimental Sauter mean diameter (experiment 1: blue solid symbols, repetition experiment 2: red hollow symbols) at different rpm and phase fractions (dotted lines are a guide to the eye).

Substance | Density ρ [kg/m ^{3}] | Kinematic Viscosity ν [mm ^{2}/s] | Interfacial Tension σ _{d,c} [mN/m] |
---|---|---|---|

Water + Na_{2}SO_{4} (50 mmol/L) | 1000 | 1.0 | - |

Paraffin oil FC 2006 | 825 | 13.1 | 53 |

Coefficients | Confidence Intervals | ||||||
---|---|---|---|---|---|---|---|

Values Equation (8) | C_{4} | C_{5} | C_{9} | R^{2} | C_{4} | C_{5} | C_{9} |

0.1601 | 1.9190 | −0.6456 | 0.9503 | 0.1113 to 0.2089 | 1.6180 to 2.2200 | −0.6948 to −0.5964 | |

Values Equation (9) | C_{6} | C_{7} | C_{9} | R^{2} | C_{6} | C_{7} | C_{9} |

0.4283 | 0.2933 | −0.6475 | 0.9719 | 0.3306 to 0.5260 | 0.2693 to 0.3173 | −0.6845 to −0.6105 |

Calculated Errors | ||||
---|---|---|---|---|

Maximum Relative Error | Mean Relative Error | Maximum Absolute Error | Mean Absolute Error | |

$\mathsf{\Delta}{\mathit{d}}_{32}^{\mathit{m}\mathit{a}\mathit{x},\mathit{\%}}\phantom{\rule{0ex}{0ex}}\mathbf{in}\%$ | $\mathsf{\Delta}{\overline{\mathit{d}}}_{32}^{\mathit{m}\mathit{e}\mathit{a}\mathit{n},\mathit{\%}}\phantom{\rule{0ex}{0ex}}\mathbf{in}\%$ | $\mathsf{\Delta}{\mathit{d}}_{32}^{\mathit{m}\mathit{a}\mathit{x},\mathsf{\mu}\mathbf{m}}\phantom{\rule{0ex}{0ex}}\mathbf{in}\mu \mathbf{m}$ | $\mathsf{\Delta}{\overline{\mathit{d}}}_{32}^{\mathit{m}\mathit{e}\mathit{a}\mathit{n},\mathsf{\mu}\mathbf{m}}\phantom{\rule{0ex}{0ex}}\mathbf{in}\mu \mathbf{m}$ | |

Values Equation (8) | 21.9 | 5.7 | 63 | 24 |

Values Equation (9) | 11.4 | 4.0 | 50 | 18 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wirz, D.; Friebel, A.; Rave, K.; Hermes, M.; Skoda, R.; von Harbou, E.; Bart, H.-J.
Experimental Investigation and Modelling of the Droplet Size in a DN300 Stirred Vessel at High Disperse Phase Content Using a Telecentric Shadowgraphic Probe. *Appl. Sci.* **2022**, *12*, 4069.
https://doi.org/10.3390/app12084069

**AMA Style**

Wirz D, Friebel A, Rave K, Hermes M, Skoda R, von Harbou E, Bart H-J.
Experimental Investigation and Modelling of the Droplet Size in a DN300 Stirred Vessel at High Disperse Phase Content Using a Telecentric Shadowgraphic Probe. *Applied Sciences*. 2022; 12(8):4069.
https://doi.org/10.3390/app12084069

**Chicago/Turabian Style**

Wirz, Dominic, Anne Friebel, Kevin Rave, Mario Hermes, Romuald Skoda, Erik von Harbou, and Hans-Jörg Bart.
2022. "Experimental Investigation and Modelling of the Droplet Size in a DN300 Stirred Vessel at High Disperse Phase Content Using a Telecentric Shadowgraphic Probe" *Applied Sciences* 12, no. 8: 4069.
https://doi.org/10.3390/app12084069