# Scaling of Droplet Breakup in High-Pressure Homogenizer Orifices. Part I: Comparison of Velocity Profiles in Scaled Coaxial Orifices

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Droplet Breakup Scaling Theory

#### 2.2. Materials

#### 2.2.1. Original Scale System

^{®}, Carl Roth, Karlsruhe, Germany) that prevents coalescence of droplets. For this mixture, Newtonian flow behavior was determined with a rotational rheometer (Anton Paar Physica MCR 301, Graz, Austria) at a temperature of 20 °C in the shear rate range of 0.1–100 s

^{−1}. A dynamic viscosity of 0.00425 Pa∙s was measured. As the disperse phase fraction was below 1 wt % for all experiments, it was expected that Newtonian flow behavior was also present at higher shear rates during the process [25,26]. The density of the continuous phase was determined with the density determination set DIS 11 (DCAT11, dataphysics, Filderstadt, Germany) to be 1145.3 kg/m

^{3}at 20 °C.

^{®}(IOI Oleo GmbH, Witten, Germany) and Miglyol 840

^{®}(IOI Oleo GmbH, Witten, Germany) were mixed in ratio of 41:59. Added to these was 0.012 wt % of the fluorescence color Nile red (9-(diethyl-amino)benzo[a]phenoxazin-5(5H)-one, Sigma-Aldrich Chemie GmbH, St. Louis, MO, USA), which was dissolved in the oil mixture and stirred overnight. Any undissolved Nile red crystals were removed by filtering the next day. Subsequently, the dynamic viscosity of the disperse phase was measured with a rotational rheometer (Anton Paar Physica MCR 301, Graz, Austria) at a temperature of 20 °C to 0.0149 Pa∙s with Newtonian flow behavior. A Wilhelmy plate (DCAT11, dataphysics, Filderstadt, Germany) was used to measure the interfacial tension between the continuous (for droplet visualization) and the disperse phase. For interfacial tension, a value of 4.316 mN/m was determined after a measuring time of 2 h at a temperature of 20 °C, while a density of 928.33 kg/m

^{3}was measured for the disperse phase with the density determination set DIS 11 (DCAT11, dataphysics, Filderstadt, Germany) at 20 °C.

#### 2.2.2. Five-Fold Scaled System

^{®}, Carl Roth, Karlsruhe, Germany). The density was determined with the density determination set DIS 11 (DCAT11, dataphysics, Filderstadt, Germany) to 1148.55 kg/m

^{3}at 20 °C.

^{®}(IOI Oleo GmbH, Witten, Germany) with 0.012 wt % Nile red (9-(diethyl-amino)benzo[a]phenoxazin-5(5H)-one, Sigma-Aldrich Chemie GmbH, St. Louis, MO, USA) was used as the disperse phase of the 5-fold scaled system. The Nile red was dissolved using the same procedure as in the original scale system. The dynamic viscosity of the disperse phase was 0.02947 Pa∙s with Newtonian flow behavior, which again was measured with a rotational rheometer (Anton Paar Physica MCR 301, Graz, Austria) at a temperature of 20 °C. Parallel to the originally scaled system, a Wilhelmy plate (DCAT11, dataphysics, Filderstadt, Germany) was used to measure the interfacial tension between the continuous (for droplet visualization) and the disperse phase, which was determined to 3.986 mN/m after a measuring time of 2 h at a temperature of 20 °C. The density of the disperse phase was 920 kg/m

^{3}according to the supplier’s datasheet.

#### 2.2.3. 50-Fold Scaled System

^{3}at 20 °C was determined with the density determination set DIS 11 (DCAT11, dataphysics, Filderstadt, Germany).

^{®}AK 100 (Wacker Chemie AG, Stuttgart, Germany) was used as disperse phase, which had a density of 960 kg/m

^{3}according to the supplier’s datasheet. Its dynamic viscosity was again measured with a rotational rheometer (Anton Paar Physica MCR 301, Graz, Austria) at a temperature of 20 °C to 0.1066 Pa∙s.

#### 2.3. Experimental Setup

#### 2.4. µ-PIV Measurements

^{2}, which corresponds to a spatial resolution of 1.5 µm/px. Compared to the 5-fold scaled setup, the visual field of the original scale setup has an area of about 12 × 12 mm

^{2}, which corresponds to a spatial resolution of 6 µm/px. A minimum of 2000 double pictures was taken in each measurement run for statistical convergence.

^{−2}mm (Märzhäuser Wetzlar GmbH & Co. KG, Wetzlar, Germany) to measure the velocity map farther downstream from the orifice exit. The orifice itself was moved a distance of $5\xb7d$ between two measurement runs to ensure a sufficient overlay of the measurement area. The velocity maps of the sections were interpolated on a new grid and, wherever overlapping, averaged with MATLAB 2019b (Mathworks, Nantucket, MA, USA).

#### 2.5. PIV Measurements

## 3. Results

#### 3.1. Matching of Orifice Dimensions and Material System

#### 3.2. Comparison of the Flow Pattern in the Orifices

#### 3.2.1. Comparison of the Normalized Velocity and the Normalized Velocity Fluctuations

#### 3.2.2. Comparison of the Normalized Velocity on the Center Axis

#### 3.2.3. Influence of Confinement on the Normalized Velocity on the Center Axis

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Experimental setup of the original scale system ($\mathrm{\Psi}=1$) with pressurized gas cylinder (

**a**), pressure vessel (

**b**), optically accessible orifice (

**c**) and needle valve (

**d**). The level of the pressure vessel (L) is measured constantly to avoid a complete emptying. The values of the pressure sensors (PR) and the scale (WR) are recorded during the experiment.

**Figure 2.**Experimental setup of the 5-fold scaled system ($\mathrm{\Psi}=5$) with pressurized gas cylinder (

**a**), pressure vessel (

**b**), droplet generator (

**d**), optically accessible orifice (

**e**) and needle valve (

**f**). The positive displacement pump (

**c**) was not used when performing velocity measurements. The values of the pressure sensors (PR), scale volume flow sensor (FR) and temperature sensor (TR) were recorded during the experiment. The pressure in the vessel (PI) and the level in the vessel (LI) are indicated at the test rig. The µ-PIV system was used to measure the velocity profile in the optically accessible orifice.

**Figure 3.**Experimental setup of the 50-fold scaled system ($\mathrm{\Psi}=50$) with the optically accessible inlet, outlet channel and orifice. The continuous phase is circulated by a centrifugal pump. The droplet generator was not used and disconnected during velocity measurements. The temperature (TI), the volume flow (FI) and the pressure (PI) are indicated at several places in the test rig. The motor of the pump (M) is controlled according to the pressure signal of the orifice pressure drop sensor (PIC).

**Figure 4.**Geometry of the original and the 5-fold scaled orifice: (

**a**) geometry of the 50-fold scaled orifice and (

**b**) image of the original scale orifice with demounted acrylic glass cover plate (flow from bottom to top). The protrude bar on the acrylic glass plate is fitted in the inlet channel (

**c**); image of the 50-fold scaled orifice (flow from left to right) (

**d**). The related dimensions can be found in Table 2.

**Figure 5.**(

**a**) Normalized velocity fields of all three scales at Re = 2000; (

**b**) normalized velocity fluctuation of all three scales at Re = 2000; (

**c**) normalized velocity fields of all three scales at Re = 5700; (

**d**) normalized velocity fluctuation of all three scales at Re = 5700. Due to the limited working distance of the microscope, it was not possible to conduct measurements within the 5-fold scaled system for any diameter ratios ${D}_{\mathrm{exit}}/d$ larger than 10.

**Figure 6.**(

**a**) Normalized velocity on the center axis at a Reynolds number Re = 5700 at the outlet of an orifice $\left(\mathrm{\Psi}=1\right)$ and of a 50-fold scaled orifice $\left(\mathrm{\Psi}=50\right)$ with an outlet channel diameter ratio of $\raisebox{1ex}{${D}_{\mathrm{exit}}$}\!\left/ \!\raisebox{-1ex}{$d$}\right.=20$. (

**b**) Normalized velocity on the center axis at a Reynolds number Re = 2000 at the outlet of an orifice $\left(\mathrm{\Psi}=1\right)$ and of a 50-fold scaled orifice $\left(\mathrm{\Psi}=50\right)$ with an outlet channel diameter ratio of $\raisebox{1ex}{${D}_{\mathrm{exit}}$}\!\left/ \!\raisebox{-1ex}{$d$}\right.=20$.

**Figure 7.**(

**a**) Normalized velocity on the center axis at a Reynolds number Re = 5700 at the outlet of an orifice $\left(\mathrm{\Psi}=1\right)$, of a 5-fold scaled orifice $\left(\mathrm{\Psi}=5\right)$ and of a 50-fold scaled orifice $\left(\mathrm{\Psi}=50\right)$ with an outlet channel diameter ratio of $\raisebox{1ex}{${D}_{\mathrm{exit}}$}\!\left/ \!\raisebox{-1ex}{$d$}\right.=10$; (

**b**) normalized velocity on the center axis at a Reynolds number Re = 2000 at the outlet of an orifice $\left(\mathrm{\Psi}=1\right)$, of a 5-fold scaled orifice $\left(\mathrm{\Psi}=5\right)$ and of a 50-fold scaled orifice $\left(\mathrm{\Psi}=50\right)$ with an outlet channel diameter ratio of $\raisebox{1ex}{${D}_{\mathrm{exit}}$}\!\left/ \!\raisebox{-1ex}{$d$}\right.=10$.

**Figure 8.**(

**a**) Normalized velocity on the center axis of the original scale orifice $\left(\mathrm{\Psi}=1\right)$ with altered outlet channel diameter ratio at a Reynolds number Re = 5700; (

**b**) normalized velocity on the center axis of the original scale orifice $\left(\mathrm{\Psi}=1\right)$ with altered outlet channel diameter ratio at a Reynolds number Re = 2000.

**Figure 9.**(

**a**) Normalized velocity on the center axis of a 50-fold scaled orifice $\left(\mathrm{\Psi}=50\right)$ with altered outlet channel diameter ratio at a Reynolds number Re = 2000; (

**b**) normalized velocity on the center axis of a 50-fold scaled orifice $\left(\mathrm{\Psi}=50\right)$ with altered outlet channel diameter ratio at a Reynolds number Re = 5700.

Scale | $\mathbf{\Psi}=1$ | $\mathbf{\Psi}=5$ | $\mathbf{\Psi}=50$ |
---|---|---|---|

${\rho}_{\mathrm{c}}/\mathrm{kg}\xb7{\mathrm{m}}^{-3}$ | 1145.3 | 1148.55 | 1145.4 |

${\rho}_{\mathrm{d}}/\mathrm{kg}\xb7{\mathrm{m}}^{-3}$ | 928.33 | 920 | 960 |

$\gamma /\mathrm{mN}\xb7{\mathrm{m}}^{-1}$ | 4.3104 | 3.986 | 20.074 |

${\eta}_{\mathrm{c}}/\mathrm{Pa}\xb7\mathrm{s}$ | 0.00425 | 0.00942 | 0.0314 |

${\eta}_{\mathrm{d}}/\mathrm{Pa}\xb7\mathrm{s}$ | 0.0149 | 0.02947 | 0.1066 |

Scale | $\mathbf{\Psi}=1$ | $\mathbf{\Psi}=5$ | $\mathbf{\Psi}=50$ |
---|---|---|---|

$d/\mathrm{mm}$ | 0.2 | 1 | 10 |

$l/\mathrm{mm}$ | 0.4 | 2 | 20 |

${D}_{\mathrm{in}}/\mathrm{mm}$ | 2 | 10 | 100 |

${D}_{\mathrm{out}}/\mathrm{mm}$ | 2/3/4 | 10 | 50/100/150/200 |

$R/\mathrm{mm}$ | - | - | 20 |

$\alpha $ | 60 | 60 | - |

${d}_{\mathrm{c}}/\mathrm{mm}$ | 0/0.5/1 | 0 | - |

**Table 3.**Resulting scaling factors, material properties and dimensions of the $\mathrm{\Psi}=5$ system.

d | D | l | ${\mathit{\rho}}_{c}$ | ${\mathit{\rho}}_{d}$ | $\mathit{\gamma}$ | ${\mathit{\eta}}_{c}$ | ${\mathit{\eta}}_{d}$ | |
---|---|---|---|---|---|---|---|---|

Target scaling factor | 5 | 5 | 5 | 1 | 1 | 1 | 2.361 | 2.361 |

Target value | 1 mm | 10 mm | 2 mm | 1145.3 kg/m^{3} | 928.33 kg/m^{3} | 4.3104 mN/m | 0.00950 Pa∙s | 0.03331 Pa∙s |

Actual value | 1 mm | 10 mm | 2 mm | 1148.55 kg/m^{3} | 920 kg/m^{3} | 3.986 mN/m | 0.00942 Pa∙s | 0.02947 Pa∙s |

Actual scaling factor | 5 | 5 | 5 | 1.0029 | 0.991 | 0.9247 | 2.22 | 1.978 |

**Table 4.**Resulting scaling factors, material properties and dimensions of the $\mathrm{\Psi}=50$ system.

d | D | l | ${\mathit{\rho}}_{c}$ | ${\mathit{\rho}}_{d}$ | $\mathit{\gamma}$ | ${\mathit{\eta}}_{c}$ | ${\mathit{\eta}}_{d}$ | |
---|---|---|---|---|---|---|---|---|

Target scaling factor | 50 | 50 | 50 | 1 | 1 | 1 | 7.071 | 7.071 |

Target value | 10 mm | 100 mm | 20 mm | 1145.3 kg/m^{3} | 928.33 kg/m^{3} | 4.3104 mN/m | 0.03005 Pa∙s | 0.10536 Pa∙s |

Actual value | 10 mm | 100 mm | 20 mm | 1145.4 kg/m^{3} | 960 kg/m^{3} | 20.074 mN/m | 0.0314 Pa∙s | 0.1066 Pa∙s |

Actual scaling factor | 50 | 50 | 50 | 1.00 | 1.03 | 4.6571 | 7.39 | 7.154 |

**Table 5.**Resulting dimensionless numbers of the droplet breakup for all three scales based on a Reynolds number Re = 2000 in the disruption unit and droplets with a diameter ${x}_{0}$ of 40, 200 and 2000 µm, respectively.

Scale | $\mathbf{\Psi}=1$ | $\mathbf{\Psi}=5$ | $\mathbf{\Psi}=50$ |
---|---|---|---|

${x}_{0}/\mu \mathrm{m}$ | 40 | 200 | 2000 |

$Re$ | 2000 | 2000 | 2000 |

$We$ | $146\times {10}^{2}$ | $155\times {10}^{2}$ | $343\times {10}^{1}$ |

$\kappa $ | 0.811 | 0.801 | 0.838 |

$\lambda $ | 3.51 | 3.13 | 3.39 |

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**MDPI and ACS Style**

Preiss, F.J.; Mutsch, B.; Kähler, C.J.; Karbstein, H.P.
Scaling of Droplet Breakup in High-Pressure Homogenizer Orifices. Part I: Comparison of Velocity Profiles in Scaled Coaxial Orifices. *ChemEngineering* **2021**, *5*, 7.
https://doi.org/10.3390/chemengineering5010007

**AMA Style**

Preiss FJ, Mutsch B, Kähler CJ, Karbstein HP.
Scaling of Droplet Breakup in High-Pressure Homogenizer Orifices. Part I: Comparison of Velocity Profiles in Scaled Coaxial Orifices. *ChemEngineering*. 2021; 5(1):7.
https://doi.org/10.3390/chemengineering5010007

**Chicago/Turabian Style**

Preiss, Felix Johannes, Benedikt Mutsch, Christian J. Kähler, and Heike Petra Karbstein.
2021. "Scaling of Droplet Breakup in High-Pressure Homogenizer Orifices. Part I: Comparison of Velocity Profiles in Scaled Coaxial Orifices" *ChemEngineering* 5, no. 1: 7.
https://doi.org/10.3390/chemengineering5010007