# Modeling the Separation of Microorganisms in Bioprocesses by Flotation

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## Abstract

**:**

## 1. Introduction

## 2. Modeling Approaches

#### 2.1. Aggregation Kernel

#### 2.1.1. Frequency of Encounters

#### 2.1.2. Encounter Efficiency

#### 2.2. Spatial Models

#### 2.2.1. Two-Zone Model

#### 2.2.2. One-Way Coupling

^{®}Microfloat

^{®}, see Figure 1a) for a two-phase system consisting of the liquid and gaseous phase has been performed with Ansys Fluent. The gaseous phase was described by bubbles with a diameter of 40 µm and a density of an air bubble with a maximum loading of microorganisms (18.28 kg/m

^{3}). Those estimates were based on experimental data for the flotation of yeasts [29]. For the liquid phase, water properties were applied. The central inlet was a mass flow inlet, the free surface a degassing boundary and one bottom opening (side) a pressure outlet. The mesh consisted of hexahedral cells (approximately 0.5 Mio.elements, which Buffo [46] (p. 129) found to be a sufficient mesh density for mesh independent results for a column with bubbly flow. Furthermore, Hecht et al. [47] observed agreement with experimental data using a similar mesh density for a column with bubbly flow). Most of the tank was expected to be laminar, while the inlet region had a Reynolds number in the turbulent regime. Therefore, different turbulence models (k-omega SST, k-epsilon) and a model without turbulence (laminar) were evaluated, but no effect on gas hold-up or flow field was observed. For drag on the bubbles, the commonly-used model by Tomiyama et al. [48] for slightly contaminated interfaces was used. Even though using a different drag model is expected to change the results of the CFD simulation slightly, this will affect all investigated models in this study in the same manner. Thus, for the main aim of model comparison, no significant changes are to be expected.

#### 2.3. Heteroaggregation Modeling

#### 2.3.1. Averaged

#### 2.3.2. Not Averaged

#### 2.3.3. Poly.Cells

#### 2.3.4. Poly.Bubbles

#### 2.3.5. Clustering

#### 2.3.6. Clustering Averaged

## 3. Numerical Methods

## 4. Results and Discussion

^{®}Microfloat

^{®}Rundzelle). To compare different simulations, a set of standard parameters (Table 2) was introduced. The parameters have been used for the simulations, unless other parameter settings are mentioned. The turbulence dissipation rate and the shear rate were chosen in a range as observed at the inlet of the investigated flotation tank. The residence time was chosen arbitrarily, whereas the other standard parameters were set to reasonable values.

#### 4.1. Comparison of the Different Aggregation Models

#### 4.1.1. Influence of the Cell Size Distribution

#### 4.1.2. Mechanisms Leading to the Formation of Clusters

#### 4.1.3. Scope of Aggregation Models

#### 4.2. One-Way Coupling

^{®}Microfloat

^{®}) with a flow rate of 75 L/h. The geometry and the distribution of the gas volume fraction of the tank are illustrated in Figure 1a. From the inlet to the top of the tank, the gas volume fraction was quite high in a narrow area, which was the main stream of the tank. At the top of the tank, most of the bubbles remained in the foam. The other areas of the tank did not have such a high gas volume fraction. Close to the outlet, $\Phi $ was zero. Exemplarily, the heteroaggregation on one streamline of the DAF tank was investigated in detail.

#### 4.2.1. Gas Volume Fraction

#### 4.2.2. Aggregation Mechanisms

#### 4.2.3. Comparison of Different Aggregation Models

## 5. Conclusions

^{®}Microfloat

^{®}Rundzelle).

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Model Averaged-Algebraic Solution for Two-Zone Model

#### Appendix A.1. Aggregation Equation

#### Appendix A.2. Collision Efficiency Due to Surface Coverage

#### Appendix A.3. Non-Dimensionalization

#### Appendix A.4. Analytical Solution

## Appendix B. Encounter Efficiency Due to Surface Coverage for Clustering

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**Figure 1.**Process conditions of the investigated flotation tank. (

**a**) Gas volume fraction and velocity directions of fluid; (

**b**) evolution of the gas volume fraction of a single streamline.

**Figure 2.**Illustration of the different aggregation models, where grey spheres represent bubbles and dark spheres correspond to microorganisms: (

**a**) Averaged: several monodisperse cells aggregate on monodisperse bubbles; loading of bubbles is averaged; (

**b**) Not Averaged: several monodisperse cells aggregate on monodisperse bubbles; loading of bubbles is distributed; (

**c**) Poly.Cells: several polydisperse cells aggregate on monodisperse bubbles; (

**d**) Poly.Bubbles: several monodisperse cells aggregate on polydisperse bubbles; (

**e**) Clustering Averaged: formation of aggregates with multiple monodisperse bubbles and monodisperse cells; bubble and cell number per cluster are averaged; (

**f**) Clustering: formation of aggregates with multiple monodisperse bubbles and monodisperse cells; bubble and cell number per cluster are distributed.

**Figure 3.**Comparison between the different aggregation models (Poly.Cells: ${\sigma}_{rel}\left({d}_{c}\right)=0.25$; Poly.Bubbles: ${\sigma}_{rel}\left({d}_{b}\right)=0.25$). (

**a**) Separation efficiency depending on the aggregation number at ${\Pi}_{1}=0.099$; (

**b**) separation efficiency depending on maximal surface coverage at ${\Pi}_{3}=0.971$.

**Figure 4.**Number density distribution of unbounded cells for the model Poly.Cells at different simulation times; standard parameter are used, and ${\sigma}_{rel}=0.25$.

**Figure 5.**Investigation on the effect of the variation of turbulent eddy dissipation and shear rate for constant ${\Pi}_{1}$ and ${\Pi}_{3}$ on cluster formation. (

**a**) Separation efficiency; (

**b**) number density distribution of clusters for the two extreme cases.

**Figure 6.**Aggregation mechanisms on a single streamline and resulting separation efficiencies for different aggregation models for this streamline. (

**a**) Importance of aggregation mechanisms on a single streamline; (

**b**) separation efficiencies for different aggregation models.

Encounter Frequency | Due to: | Eq. |

${K}_{S,C}=\frac{1}{6}\xb7{({D}_{1}+{D}_{2})}^{3}\xb7\sqrt{A}$ $A=2\xb7\left(\right)open="["\; close="]">{\left(\right)}^{\frac{\partial u}{\partial x}}2+{\left(\right)}^{\frac{\partial w}{\partial z}}2$ $+{\left(\right)}^{\frac{\partial u}{\partial y}}2+{\left(\right)}^{\frac{\partial v}{\partial z}}2$ | Laminar shear | (4) |

${K}_{G,C}=\frac{\pi}{4}\xb7{({D}_{1}+{D}_{2})}^{2}\xb7|{v}_{1}-{v}_{2}|$ | Sedimentation | (5) |

${K}_{T,C}=\frac{1.3}{8}\xb7{({D}_{1}+{D}_{2})}^{3}\xb7\sqrt{\frac{\u03f5}{\nu}}$ | Turbulence | (6) |

Encounter Efficiency Due to Surface Coverage | For Model: | Eq. |

${P}_{A,S}\left(l\right)=1-l,\phantom{\rule{2.em}{0ex}}l\in [0,1]$ | Averaged | (7) |

${P}_{A,S}\left(j\right)=1-\frac{j}{{j}_{max}},\phantom{\rule{2.em}{0ex}}j\in [0,{j}_{max}]$ | Not Averaged | (8) |

${P}_{A,S}\left(l\right)=1-l,\phantom{\rule{2.em}{0ex}}l\in [0,1]$ | Poly.Cells | (9) |

${P}_{A,S}(j,{d}_{b})=1-\frac{j}{{j}_{max\left({d}_{b}\right)}},$ $j\in [1,{j}_{max\left({d}_{b}\right)}]\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}{d}_{b}\in [{d}_{{b}_{min}},{d}_{{b}_{max}}]$ | Poly.Bubbles | (10) |

${P}_{A,S}(i,j,m,l)={R}_{p}(i,j)\xb7{R}_{b}(m,l)+{R}_{p}(m,l)\xb7{R}_{b}(i,j)$ | Clustering | (11) |

${P}_{A,S}(i,j)=2\xb7{R}_{p}(i,j)\xb7{R}_{b}(i,j),\phantom{\rule{2.em}{0ex}}\mathrm{for}\mathrm{two}\mathrm{clusters}$ | Clustering Averaged | (12) |

${P}_{A,S}(i,j)={R}_{b}(i,j),\phantom{\rule{2.em}{0ex}}\mathrm{for}\mathrm{a}\mathrm{cluster}\mathrm{and}\mathrm{a}\mathrm{cell}$ | (13) | |

${P}_{C}=\left(\right)open="("\; close=")">\frac{3}{2}+\frac{4}{15}\xb7{\left(\right)}^{\frac{{D}_{2}\xb7U}{\nu}}0.72\xb7{\left(\right)}^{\frac{{D}_{1}}{{D}_{2}}}2$ | Hydrodynamic encounter efficiency | (14) |

Parameter | Abbreviation | Value | Unit |
---|---|---|---|

Residence time | ${t}_{residence}$ | 10 | s |

Gas volume fraction | $\Phi $ | 0.03 | - |

Bubble diameter | ${d}_{b}$ | 40 | $\mathsf{\mu}$m |

Cell diameter | ${d}_{c}$ | 5 | $\mathsf{\mu}$m |

Cell concentration of feed | ${c}_{feed}$ | 10 | $\mathrm{g}\xb7{\mathrm{L}}^{-1}$ |

Ratio recycle flow to total flow | $\frac{{\dot{V}}_{Recycle}}{{\dot{V}}_{Total}}$ | $\frac{5}{6}$ | - |

Viscosity fluid | $\mu $ | $0.89$ | mPa · s |

Turbulent dissipation rate | $\u03f5$ | $0.0025$ | ${\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-3}$ |

Shear rate | $\dot{\gamma}$ | $\sqrt{125}$ | s${}^{-1}$ |

Properties | Averaged | Not Averaged | Poly.Cells | Poly.Bubbles | Clustering | Clustering Averaged |
---|---|---|---|---|---|---|

Cells polydisperse | No | No | Yes | No | No | No |

Bubbles polydisperse | No | No | No | Yes | No | No |

Loading averaged | Yes | No | No | No | No | Yes |

Clustering considered | No | No | No | No | Yes | Yes |

Simulation time | Very low | Low | Medium | Medium | Very high | Low |

Complexity | Very low | Low | Medium | Medium | High | Low |

Standard deviation of bubble diameter | Small | Small | Small | High | Small | Small |

Standard deviation of cell diameter | Small | Small | High | Small | Small | Small |

Influence of clustering | Negligible | Negligible | Negligible | Negligible | Important | Important |

Cells have to be smaller than bubbles | Yes | Yes | Yes | Yes | No | No |

Difficulty of direct coupling to CFD simulations | Low | High | High | High | Very high | Medium |

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## Share and Cite

**MDPI and ACS Style**

Schmideder, S.; Kirse, C.; Hofinger, J.; Rollié, S.; Briesen, H.
Modeling the Separation of Microorganisms in Bioprocesses by Flotation. *Processes* **2018**, *6*, 184.
https://doi.org/10.3390/pr6100184

**AMA Style**

Schmideder S, Kirse C, Hofinger J, Rollié S, Briesen H.
Modeling the Separation of Microorganisms in Bioprocesses by Flotation. *Processes*. 2018; 6(10):184.
https://doi.org/10.3390/pr6100184

**Chicago/Turabian Style**

Schmideder, Stefan, Christoph Kirse, Julia Hofinger, Sascha Rollié, and Heiko Briesen.
2018. "Modeling the Separation of Microorganisms in Bioprocesses by Flotation" *Processes* 6, no. 10: 184.
https://doi.org/10.3390/pr6100184