# An Impact of the Discrete Representation of the Bubble Size Distribution Function on the Flow Structure in a Bubble Column Reactor

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Basic Assumptions of the Model

#### 2.2. Description of Polydispersity of Bubbly Medium

#### 2.3. Governing Equations

**V**is the velocity vector, p is the pressure,

**E**is the unity tensor, S is the source term responsible for the interphase momentum transport, D is the source term responsible for the bubble dispersion due to turbulence and $\mathit{\tau}$ is the strain rate tensor with its components expressed as:

#### 2.4. Interphase Momentum Exchange

#### 2.4.1. Buoyancy Force

#### 2.4.2. Drag Force

#### 2.4.3. Lift Force

#### 2.4.4. Virtual Mass Force

#### 2.4.5. Wall Lubrication Force

#### 2.5. Turbulence

#### 2.5.1. Turbulence Generation and Dissipation by Bubbles

#### 2.5.2. Bubble Path Dispersion

#### 2.6. Numerical Method

## 3. Problem Statement

#### 3.1. Simulation Domain and Conditions

#### 3.2. A Discrete Approximation of the BSD Function

**R1**,

**R2**and

**R3**. The entire interval of bubble size variation is replaced by the range $[{d}_{min},\phantom{\rule{0.277778em}{0ex}}{d}_{max}]$, where ${d}_{min}=exp(\mu -3\sigma ),\phantom{\rule{0.277778em}{0ex}}{d}_{max}=exp(\mu +3\sigma )$, which corresponds to 99.7% of all bubbles. The

**R1**distribution approximation is uniform with respect to the variable ${d}_{b}$ in the range $[{d}_{min},\phantom{\rule{0.277778em}{0ex}}{d}_{max}]$; the entire range is divided into N classes, each class having the same width with respect to ${d}_{b}$. The

**R2**distribution approximation is uniform with respect to the variable $ln\left({d}_{b}\right)$; the range $[ln\left({d}_{min}\right),\phantom{\rule{0.277778em}{0ex}}ln\left({d}_{max}\right)]$ is divided into N classes, the width of each class with respect to $ln\left({d}_{b}\right)$ is the same. The

**R3**approximation is a hybrid

**R2–R1**: the first $N/2$ classes are uniformly distributed with respect to the variable $ln\left({d}_{b}\right)$ in the range $[ln\left({d}_{min}\right),\phantom{\rule{0.277778em}{0ex}}\mu ]$; the subsequent $N/2$ classes are uniformly distributed with respect to the variable ${d}_{b}$ in the range $[exp\left(\mu \right),\phantom{\rule{0.277778em}{0ex}}{d}_{max}]$.

**R1**is used for validation purposes. A set of different numbers of classes is used with N = [1…10], covering a range from a monodisperse approach to a detailed polydisperse one. The validation is performed for the characteristic bubble size M = 0.5 mm.

**R1**), which corresponds to monodisperse bubbles, (N = 4,

**R3**), which corresponds to a compact economic distribution, (N = 7,

**R1**) and (N = 7,

**R2**), which allow a qualitative description of the polydisperse medium, and (N = 10,

**R3**), which provides a detailed description of the polydisperse bubble flow.

**R1**) does not describe the features of the distribution of small bubbles, providing for this domain only one class for bubbles from 0 to 0.25 mm. The distribution (N = 7,

**R2**) resolves small bubbles at the expense of details for large bubbles. The distribution (N = 4,

**R3**), despite some roughness, singles out a class for small and medium bubbles. The distribution (N = 10,

**R3**) is the most detailed, describing both small and large bubbles well over the entire range.

## 4. Results and Discussion

#### 4.1. Algorithm Verification

#### 4.2. An Impact of Characteristic Bubble Size and Parameters of Approximation of the BSD Function on the Flow Structure

**R1**), (N = 4,

**R3**), (N = 7,

**R1**) and (N = 10,

**R3**).

**R3**) has fewer bubble classes than (N = 7,

**R1**), it gives a solution close to (N = 10,

**R3**) for the volume fraction and the interfacial area, in contrast to (N = 7,

**R1**). A comparison with the experimental data [22] for ${d}_{b}$ = 0.5 mm shows that the option (N = 4,

**R3**) is preferable.

**R3**) as the most detailed representation of the BSD function. The comparison was carried out using the normalized variance of the obtained solution and the “reference” solution in the entire domain for a set of characteristic values of the polydisperse flow, such as the velocities of the carrier and dispersed phases, the volume fraction of the bubbles, and the interfacial area. The normalized variance was estimated according to the formula:

**R1**), 2 stands for (N = 4,

**R3**), 3 stands for (N = 7,

**R1**) and 4 stands for (N = 7,

**R2**) (Figure 8, Figure 9, Figure 10 and Figure 11).

**R1**), then through distributions (N = 4,

**R3**), (N = 7,

**R1**) and ending with (N = 7,

**R2**).

**R3**), a solution can be obtained that is close to the variant (N = 7,

**R2**) and has a smaller difference from the “reference” one than (N = 7,

**R1**). This can be explained by the fact that, in the case of the distribution (N = 7,

**R1**), small bubbles corresponding to the left side of the BSD function are not resolved at all in the simulation. However, their influence is important since they have the ability to accumulate in stagnation or recirculation zones, significantly influencing the flow parameters and changing the flow structure. Although the distribution (N = 4,

**R3**) is coarser than (N = 7,

**R1**), it still describes the behavior of small bubbles.

## 5. Conclusions

**R2**) provides a better solution, in terms of normalized variances, than that with the uniform bins with respect to ${d}_{b}$ (

**R1**), due to the correct resolution of the domain of small bubbles (with ${d}_{b}$ < M).

**R3**has been proposed which is uniform with respect to $ln\left({d}_{b}\right)$ for bubbles with sizes smaller than M and uniform with respect to ${d}_{b}$ for sizes larger than M. Such an approximation is potentially preferable with substantially large bubble sizes up to (${d}_{b}$ > 6 mm).

**R3**) with four classes of bubbles provides better results than the original

**R1**approximation with seven classes and is comparable to the

**R2**approximation with seven classes of bubbles. The obtained results were compared with the results for the

**R3**approximation with 10 classes of bubbles as well as with the experimental data, and demonstrated qualitative and quantitative agreement. This makes the proposed approach promising for flow pattern prediction and robust due to low computational resource requirements.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BCR | Bubble column reactor |

BSD | Bubble size distribution |

MUSIG | Multiple size group |

TVD | Total variation diminishing |

SIMPLE | Semi-implicit method for pressure-linked equations |

## References

- Yang, X.; Liu, Z.; Manhaeghe, D.; Yang, Y.; Hogie, J.; Demeestere, K.; van Hulle, S.W.H. Intensified ozonation in packed bubble columns for water treatment: Focus on mass transfer and humic acids removal. Chemosphere
**2021**, 283, 131217. [Google Scholar] [CrossRef] [PubMed] - Inkeri, E.; Tynjälä, T. Modeling of CO
_{2}Capture with Water Bubble Column Reactor. Energies**2020**, 13, 5793. [Google Scholar] [CrossRef] - Perez, B.J.L.; Jimenez, J.A.M.; Bhardwaj, R.; Goetheer, E.; van Sint Annaland, M.; Gallucci, F. Methane pyrolysis in a molten gallium bubble column reactor for sustainable hydrogen production: Proof of concept & techno-economic assessment. Int. J. Hydrogen Energy
**2021**, 46, 4917–4935. [Google Scholar] [CrossRef] - Vik, C.B.; Solsvik, J.; Hillestad, M.; Jakobsen, H.A. Interfacial mass transfer limitations of the Fischer-Tropsch synthesis operated in a slurry bubble column reactor at industrial conditions. Chem. Eng. Sci.
**2018**, 192, 1138–1156. [Google Scholar] [CrossRef] - Kantarci, N.; Borak, F.; Ulgen, K.O. Bubble column reactors. Process Biochem.
**2005**, 40, 2263–2283. [Google Scholar] [CrossRef] - Lau, Y.M.; Thiruvalluvan Sujatha, K.; Gaeini, M.; Deen, N.G.; Kuipers, J.A.M. Experimental study of the bubble size distribution in a pseudo-2D bubble column. Chem. Eng. Sci.
**2013**, 98, 203–211. [Google Scholar] [CrossRef] - Besagni, G.; Inzoli, F. Prediction of Bubble Size Distributions in Large-Scale Bubble Columns Using a Population Balance Model. Computations
**2019**, 7, 17. [Google Scholar] [CrossRef] [Green Version] - Pakhomov, M.A.; Terekhov, V.I. Simulation of the turbulent structure of a flow and heat transfer in an ascending polydisperse bubble flow. Tech. Phys.
**2015**, 60, 1268–1276. [Google Scholar] [CrossRef] - Yao, W.; Morel, C. Volumetric interfacial area prediction in upward bubbly two-phase flow. Int. J. Heat Mass Transf.
**2004**, 47, 307–328. [Google Scholar] [CrossRef] - Wang, T.; Wang, J.; Jin, Y. Population Balance Model for Gas-Liquid Flows: Influence of Bubble Coalescence and Breakup Models. Ind. Eng. Chem. Res.
**2005**, 44, 7540–7549. [Google Scholar] [CrossRef] - Lo, S. Application of Population Balance to CFD Modelling of Bubbly Flow via the MUSIG Model; AEA Technology: Rotherham, UK, 1996. [Google Scholar]
- Krepper, E.; Lucas, D.; Frank, T.; Prasser, H.-M.; Zwart, P.J. The inhomogeneous MUSIG model for the simulation of polydispersed flows. Nucl. Eng. Des.
**2008**, 238, 1690–1702. [Google Scholar] [CrossRef] - Ziegenhein, T.; Rzehak, R.; Lucas, D. Transient simulation for large scale flow in bubble columns. Chem. Eng. Sci.
**2015**, 122, 1–13. [Google Scholar] [CrossRef] - Olmos, E.; Gentric, C.; Vial, C.; Wild, G.; Midoux, N. Numerical simulation of multiphase flow in bubble column reactors. Influence of bubble coalescence and break-up. Chem. Eng. Sci.
**2001**, 56, 6359–6365. [Google Scholar] [CrossRef] - Fard, M.G.; Stiriba, Y.; Gourich, B.; Vial, C.; Grau, F.X. Euler–Euler large eddy simulations of the gas–liquid flow in a cylindrical bubble column. Nucl. Eng. Des.
**2020**, 369, 110823. [Google Scholar] [CrossRef] - Zhang, X.B.; Yan, W.C.; Luo, Z.H. Numerical simulation of local bubble size distribution in bubble columns operated at heterogeneous regime. Chem. Eng. Sci.
**2021**, 231, 116266. [Google Scholar] [CrossRef] - Wu, Y.; Liu, Z.; Li, B.; Gan, Y. Numerical simulation of multi-size bubbly flow in a continuous casting mold using population balance model. Powder Technol.
**2022**, 396, 224–240. [Google Scholar] [CrossRef] - Wu, Y.; Liu, Z.; Li, B.; Xiao, L.; Gan, Y. Numerical simulation of multi-size bubbly flow in a continuous casting mold using an inhomogeneous multiple size group model. Powder Technol.
**2022**, 402, 117368. [Google Scholar] [CrossRef] - Bhole, M.R.; Joshi, J.B.; Ramkrishna, D. CFD simulation of bubble columns incorporating population balance modeling. Chem. Eng. Sci.
**2008**, 63, 2267–2282. [Google Scholar] [CrossRef] - Chernyshev, A.S.; Schmidt, A.A. Investigation of the evolution of the bubble size distribution in the ascending and descending flows. J. Phys. Conf. Ser.
**2017**, 899, 032009. [Google Scholar] [CrossRef] [Green Version] - Lain, S.; Broder, D.; Sommerfeld, M. Experimental and numerical studies of the hydrodynamics in a bubble column. Chem. Eng. Sci.
**1999**, 54, 4913–4920. [Google Scholar] [CrossRef] - Lain, S.; Broder, D.; Sommerfeld, M.; Goz, M.F. Modelling hydrodynamics and turbulence in a bubble column using the Euler–Lagrange procedure. Int. J. Multiph. Flow
**2002**, 28, 1381–1407. [Google Scholar] [CrossRef] - Roghair, I.; Lau, Y.M.; Deen, N.G.; Slagter, H.M.; Baltussen, M.W.; van Sint Annaland, M.; Kuipers, J.A.M. On the drag force of bubbles in bubble swarms at intermediate and high Reynolds numbers. Chem. Eng. Sci.
**2011**, 66, 3204–3211. [Google Scholar] [CrossRef] - Auton, T.R. The lift force on a spherical body in a rotational flow. J. Fluid Mech.
**1987**, 183, 199–218. [Google Scholar] [CrossRef] - Tomiyama, A.; Tamai, H.; Zun, I.; Hosokawa, S. Transverse migration of single bubbles in simple shear flows. Chem. Eng. Sci.
**2002**, 57, 1849–1858. [Google Scholar] [CrossRef] - Menter, F.R.; Kuntz, M.; Langtry, R. Ten Year of Industrial Experience with the SST Turbulence Model. In Proceedings of the 4th International Symposium on Turbulence, Heat and Mass Transfer, Antalya, Turkey, 12–17 October 2003; pp. 625–632. [Google Scholar]
- Troshko, A.A.; Hassan, Y.A. A two-equation turbulence model of turbulent bubbly flows. Int. J. Multiph. Flow
**2001**, 27, 1965–2000. [Google Scholar] [CrossRef] - Sato, Y.; Sekoguchi, R. Liquid velocity distribution in two-phase bubble flow. Int. J. Multiph. Flow
**1975**, 2, 79–95. [Google Scholar] [CrossRef] - Sokolichin, A.; Eigenberger, G.; Lapin, A. Simulation of Buoyancy Driven Bubbly Flow: Established Simplifications and Open Questions. AIChE J.
**2004**, 50, 24–45. [Google Scholar] [CrossRef] - Moraga, F.J.; Larreteguy, A.E.; Drew, D.A.; Lahey, R.T., Jr. Assessment of turbulent dispersion models for bubbly flows in the low Stokes number limit. Int. J. Multiph. Flow
**2003**, 29, 655–673. [Google Scholar] [CrossRef] - Versteeg, H.K.; Malalasekera, W. An Introduction to Computational Fluid Dynamics. The Finite Volume Method, 2nd ed.; Pearson Education Limited: Harlow, UK, 2007; pp. 134–211. [Google Scholar]
- Chernyshev, A.S.; Schmidt, A.A. Numerical modeling of gravity-driven bubble flows with account of polydispersion. J. Phys. Conf. Ser.
**2016**, 754, 032005. [Google Scholar] [CrossRef] [Green Version] - Chernyshev, A.S.; Schmidt, A.A. Numerical simulation of gravity-driven mono- and polydisperse bubbly flows in a three-dimensional column in the framework of the Euler–Euler approach. J. Phys. Conf. Ser.
**2020**, 1697, 012236. [Google Scholar] [CrossRef] - Chen, P.; Sanyal, J.; Dudukovic, M.P. Numerical simulation of bubble columns flows: Effect of different breakup and coalescence closures. Chem. Eng. Sci.
**2005**, 60, 1085–1101. [Google Scholar] [CrossRef] - Allio, A.; Buffo, A.; Marchisio, D.; Savoldi, L. Two-fluid modelling for poly-disperse bubbly flows in vertical pipes: Analysis of the impact of geometrical parameters and heat transfer. Nucl. Eng. Technol.
**2022**, in press. [Google Scholar] [CrossRef]

**Figure 2.**BSD function and piecewise approximations. The characteristic bubble size is ${d}_{b}$ = 0.25 mm. (

**a**) N = 7,

**R1**; (

**b**) N = 7,

**R2**; (

**c**) N = 4,

**R3**; (

**d**) N = 10,

**R3**.

**Figure 3.**Distributions of flow parameters as a function of the number of bubble classes. The characteristic bubble size ${d}_{b}$ = 0.25 mm; (

**a**) the bubble volume fraction; (

**b**) the interfacial area density; (

**c**) the bubble velocity; (

**d**) the liquid velocity.

**Figure 4.**Radial profiles of the bubble volume fraction in the reference cross-section (see Figure 1) per different characteristic bubble sizes: (

**a**) M = 0.25 mm; (

**b**) M = 0.5 mm; (

**c**) M = 1.0 mm.

**Figure 5.**Radial profiles of the interfacial area density in the reference cross-section (see Figure 1) for different characteristic bubble sizes: (

**a**) M = 0.25 mm; (

**b**) M = 0.5 mm; (

**c**) M = 1.0 mm.

**Figure 6.**Radial profiles of bubble velocity in the reference cross-section (see Figure 1) for different characteristic bubble sizes: (

**a**) M = 0.25 mm; (

**b**) M = 0.5 mm; (

**c**) M = 1.0 mm.

**Figure 7.**Radial profiles of liquid velocity in the reference cross-section (see Figure 1) for different characteristic bubble sizes: (

**a**) M = 0.25 mm; (

**b**) M = 0.5 mm; (

**c**) M = 1.0 mm.

**Figure 8.**Distribution of a weighted variance ${\sigma}_{norm}$ for the bubble volume fraction, ${\alpha}_{b}$.

**Figure 9.**Distribution of a weighted variance ${\sigma}_{norm}$ for the interfacial area density, ${S}_{int}$.

**Figure 10.**Distribution of a weighted variance ${\sigma}_{norm}$ for the bubble velocity, ${V}_{b}$.

**Figure 11.**Distribution of a weighted variance ${\sigma}_{norm}$ for the liquid velocity, ${V}_{l}$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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**

Chernyshev, A.; Schmidt, A.; Chernysheva, V.
An Impact of the Discrete Representation of the Bubble Size Distribution Function on the Flow Structure in a Bubble Column Reactor. *Water* **2023**, *15*, 778.
https://doi.org/10.3390/w15040778

**AMA Style**

Chernyshev A, Schmidt A, Chernysheva V.
An Impact of the Discrete Representation of the Bubble Size Distribution Function on the Flow Structure in a Bubble Column Reactor. *Water*. 2023; 15(4):778.
https://doi.org/10.3390/w15040778

**Chicago/Turabian Style**

Chernyshev, Alexander, Alexander Schmidt, and Veronica Chernysheva.
2023. "An Impact of the Discrete Representation of the Bubble Size Distribution Function on the Flow Structure in a Bubble Column Reactor" *Water* 15, no. 4: 778.
https://doi.org/10.3390/w15040778