# Transferring Bubble Breakage Models Tailored for Euler-Euler Approaches to Euler-Lagrange Simulations

^{*}

## Abstract

**:**

_{crit}) number served as a threshold value for the occurrence of bubble breakage events. We

_{crit}depended on the bubble daughter size distribution (DSD) and a set minimum time between two consecutive bubble breakage events. The commercial frameworks Ansys Fluent and M-Star were applied for EE and EL simulations, respectively. The latter enabled the implementation of LES, i.e., the use of a turbulence model with non-time averaged entities. By properly choosing We

_{crit}, it was possible to successfully transfer two commonly applied bubble breakage models from EE to EL. Based on the mechanism of bubble breakage, We

_{crit}values of 7 and 11 were determined, respectively. Optimum We

_{crit}were identified as fitting the shape of DSDs, as this turned out to be a key criterion for reaching optimum prediction quality. Optimum We

_{crit}values hold true for commonly applied operational conditions in aerated bioreactors, considering water as the matrix.

## 1. Introduction

_{32}is of particular importance. To predict the dissolved gas concentration gradients within the reactor, the local d

_{32}should be determined as accurately as possible. Often, Euler-Euler approaches are applied considering both the gas and the aqueous phase as a continuum, thereby integrating bubble breakage in population balance models (PBMs). In this context, Reynolds-averaged Navier-Stokes (RANS) turbulence models are commonly applied to finite volume (FV) solvers. Many existing breakage models have been calibrated under these conditions [1,2,3].

_{crit}, i.e., a critical ratio of fluid inertia versus surface tension, as a pivotal threshold value for starting bubble breakage. This dimensional criterion for bubble breakup, as used in EL contexts, has been documented in several recent investigations [15,17,21,22,23].

## 2. Materials and Methods

^{−1}at the ring sparger. To provoke and enhance bubble breakage, the liquid was steadily injected at the center of the bottom and simultaneously removed from the top of the reactor, mimicking an external fluid loop. Both the inlet and outlet tube had a diameter of 24 mm. The volume flow rate varied to expose bubbles to different turbulence intensities. The bubble diameter at the inlet remained at 16 mm. Fluid viscosity, density and surface tension were assumed to be similar to the properties of water. According to [24], the specific energy dissipation rate ${\epsilon}_{pump}$ caused by the pump was calculated directly from the pump volume flow rate $\dot{v}$ in Equation (1):

_{l}and the liquid density $\rho $, the liquid velocity at the inlet w was considered. The latter was defined as the pump volume flow in relation to the inlet area A in Equation (2):

#### 2.1. Bubble Column Simulation in the Euler-Euler Approach

^{−4}for all residuals. Once the threshold was achieved, steady-state conditions were assumed. The BSD was evaluated only in the upper third of the column, as it was assumed that the bubble breakage process was completed at this position. A summary of all settings is listed in Table 1.

#### 2.2. Bubble Column Simulation in the Euler-Lagrange Approach

_{i}. Each point is individually defined by the probability density function ${f}_{i}\left(x,t\right)$, representing fictitious parcels as a proxy for fluid movement. The LB algorithm consists of two steps: collision and streaming. During the streaming step, the parcels move with a constant velocity from node to node, each providing updating information after the respective time step Δt. The model presumes that the fluid relaxes locally to the equilibrium distribution function ${f}_{i}^{eq}\left(x,t\right)$ over a characteristic timescale $\tau $, which depends on the molecular viscosity of the fluid. In the standard LBM equation, the collision step is described on the right-hand side in Equation (3) [26,27].

^{−6}and 1.6 × 10

^{−5}s kept lattice density deviation below 2% (usually significantly lower). A summary of all settings is listed in Table 2.

#### 2.3. Bubble Breakage Models for the Euler-Euler Approach

#### 2.4. Bubble Breakage Model Description for the Euler-Lagrange Approach

_{crit}. Since $\rho $ and $\sigma $ are assumed to be constant during the simulation, $\epsilon $ and d are the crucial parameters for breakage. These influencing variables are also decisive in the original models. A critical We number needs to be found in such a way that it is able to reproduce the bubble breakup rates of the considered models. For bubble breakage, the condition of Equation (12) applies:

#### 2.5. Daughter Size Distribution for Bubble Breakage Models

#### 2.6. Bubble Breakage Duration

#### 2.7. Bubble Size Distribution Evaluation

_{crit}, the minimum time between two bubble breakages and a suitable DSD. The related parameters must be found individually for each scenario and are only valid for the given setting. In our study, the quality of the fitted solution was evaluated based on multiple criteria. The Sauter diameter d

_{32}weights the size classes and is influenced by the shape of the overall distribution, as shown in Equation (17).

## 3. Results and Discussion

#### 3.1. Influence of the Turbulence Model

#### 3.2. Describing Daughter Size Distribution (DSD) for the Euler-Lagrange Approach

^{−3}to 0.446 W m

^{−3}, depending on the pump energy input. Therefore, constant $\epsilon $ values were set for Luo et al. as 0.08 W m

^{−3}and for Lehr et al. as 0.1 W m

^{−3}. These $\epsilon $ settings enabled the creation of the characteristic distributions, as depicted in Figure 4 and Figure 5. Regarding the approach of [4], there is still a non-zero minimum at ${f}_{V}$= 0.5. Equal breakup, however, becomes more likely with rising d

_{parent}. As expected, the distribution of Lehr et al. transforms from a bell, via M to U-shape, with increasing d

_{parent}. For a d

_{parent}of 4 mm or smaller, similar-sized daughter bubbles are most likely.

#### 3.3. Optimal Critical We Number and Setting of ${t}_{Break}$

_{crit}numbers for EL were identified as 7 and 11 for the Luo et al. [4] and the Lehr et al. [1] models, respectively. As indicated in Figure 7 and Figure 8, they represent minima for the best estimates of summed Sauter diameter deviations between the EE and EL approaches. Note that the sensitivity on the critical We number at the optimum is more pronounced for the Luo approach than for the Lehr model. In the investigated setup, the Sauter diameter of [4] is smaller than that of [1] for all BSDs, e.g., 3.8-fold smaller at the energy input of 202 W m

^{−3}. This finding is in accordance with the studies of [45,46].

_{crit}numbers managed well to predict the original bubble distribution by the models of Luo et al. and Lehr et al. with the EL approach. Only small sums of Sauter diameter deviation of 3.06 and 2.07 mm were found, which is qualified as a fairly good estimate. Furthermore, previous studies identified critical We numbers of 6, 12 and 15 [16,47,48] for bubble breakage events, which underline the soundness of this study’s results. As We

_{crit}values intrinsically depend on the settings for DSD, turbulence model, and ${t}_{Break}$, the observed well accordance with independent studies may be taken as an additional hint for qualifying the appropriateness of these settings. Often, the critical We number of [49] is cited with We

_{crit}= 2.3. However, this value reflects breakage of a single bubble in water, which excludes the superimposing bubble-to-bubble effects that are created in the experimental set-up of this study mirroring real aeration conditions. An overview of all the fitted parameters is listed in Table 3.

^{−3}is installed. At 44 W m

^{−3}, it is underestimated by 0.91 mm. By trend, increasing energy inputs of 103 and 202 W m

^{−3}, lead to overestimated Sauter diameters with deviations of 0.83 and 1.38 mm. By analogy, Figure 10 depicts similar findings for the model of [1]. Again, only small deviations are observed, not indicating a clear trend. However, a vague hypothesis may be formulated: EE bubble simulations might be biased by the configuration of discrete bubble size classes. The latter might lead to classes of ‘lumped’ small bubbles that are less accurately describing real bubble distributions than the individual EL approaches.

#### 3.4. Comparison of Bubble Size Distribution between the Euler-Euler and Euler-Lagrange Approaches

_{pump}values. The gray bars represent the results of the new model simulated in EL. The new models are able to accurately capture partial population breakage. In the case of the Lehr et al. [1] model, the portion of bubbles that undergo breakage is smaller compared to the Luo et al. model, as seen in Figure 11 and Figure 12. This effect could be maintained by using the individual levels of We

_{crit}. The Luo et al. model predominantly generates bubbles around 1 mm in diameter due to the U-shaped DSD used, while the M-shaped DSD of the Lehr et al. model mainly produces bubbles of classes 4 and 7 mm. At ε

_{pump}of 13 W m

^{−3}, a fraction of bubbles kept the initial diameter of 16 mm in the Luo et al. model, as seen in Figure 11. With an energy input of 44 W m

^{−3}, no bubbles are left, showing an initial diameter of 16 mm. In both simulations, bubbles with a dominant diameter of approximately 4 mm are predicted. After the energy input was increased from 103 to 202 W m

^{−3}, distributions changed only slightly. Increasingly, more energy is needed to cause further breakage events. Notably, the specification of discrete bins prevents the occurrence of bubbles smaller than 1 mm in the EE approach. Although this theoretical barrier does not exist for EL, bubble breakage also ends at 1 mm because the energy demand for further breakage becomes too large. In the case of the model of Lehr et al., the portion of bubbles that undergo a breakage is smaller compared to the model of Luo et al., as seen in Figure 11 and Figure 12. The fraction of bubbles that have not broken once is almost two times higher in the case of the EE approach. The DSD compensates for this effect by forming smaller bubbles, resulting in a similar Sauter diameter. The influence of the M-shape can be seen at ${\epsilon}_{pump}$ of 44 W m

^{−3}, since very small bubbles around the size of 1 mm are not formed at all. As the energy input increases to 103 and 202 W m

^{−3}, the increase in the bubble breakage rate is minor compared to ${\epsilon}_{pump}$ values of 13 and 44 W m

^{−3}.

## 4. Conclusions

_{crit}number, a DSD and the minimum time between two consecutive bubble breakages. Two of the most widely used bubble breakage models have been successfully adapted to the new model description using EE solutions in Ansys Fluent as a template. The presented BSD was consistent with respect to both the Sauter diameter and arithmetic diameter. Furthermore, it reproduced the distribution shape of the bubbles well. The models were validated for a wide range of bubble diameters and a broad turbulence spectrum as encountered in high-volume multiphase reactors. The breakage models have no unknown parameters since all constants were determined in the original version. As such, they are applicable to both EL simulation and the higher sophisticated LES turbulence model. Given the rising importance of LES simulation, the framework of this study may enable the proper consideration of bubble breakage events in LES, which is a prerequisite for multiphase simulations in bioreactors. We

_{crit}represents an additional degree of freedom that acts as a fitting parameter and is adjustable to a wide range of datasets. Therefore, experimental datasets may well serve as input for realistic bubble breakage predictions in the LES context.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | Area of inlet | m^{2} |

$\alpha $ | gas hold-up | - |

$\beta $ | model constant | - |

${C}_{s}$ | Smagorinsky coefficient | - |

d_{32} | Sauter bubble diameter | m |

d | bubble diameter, | m |

d_{parent} | diameter of parent bubble | m |

$\epsilon $ | turbulent kinetic dissipation rate | W m^{−3} |

${\epsilon}_{pump}$ | specific energy dissipation rate | W m^{−3} |

${f}_{V}$ | bubble volume fraction ratio | - |

${f}_{V,min}$: | minimal bubble volume fraction ratio | - |

$\Delta {f}_{V}$: | bubble volume fraction increment | - |

$f\left(x,t\right)$ | probability density function | - |

${f}^{eq}\left(x,t\right)$ | equilibrium distribution function | - |

k | turbulent kinetic energy | m^{2} s^{−2} |

$\lambda $ | eddy length scale | m |

${P}_{B}$ | breakage probability | - |

$\rho $ | density | kg m^{−3} |

$\overline{S}$ | filtered strain rate tensor | s^{−1} |

$\sigma $ | surface tension | N m^{−1} |

t | time | s |

${t}_{Break}$ | minimum time between two consecutive breakages | s |

$\tau $ | relaxation time | s |

u | velocity | m s^{−1} |

$\dot{v}$ | pump volume flow | m^{3} s^{−1} |

V | bubble volume | m^{3} |

V_{l} | column volume | m^{3} |

${\nu}_{t}$ | sub-grid eddy viscosity | m^{2} s^{−1} |

w | velocity inlet | m s^{−1} |

${\mathsf{\Omega}}_{B}$ | breakage frequency | s^{−1} |

${\omega}_{B}$ | hitting eddy frequency | s^{−1} |

x | position | m |

${\Delta}_{x}$ | grid size | - |

$\xi =\frac{\lambda}{d}$ | eddy/bubble size ratio | - |

## Abbreviations

BSD | bubble size distribution |

CFD | computer fluid dynamic |

EE | Euler-Euler approaches |

EL | Euler-Lagrange approaches |

FV | Finite volume |

DSD | daughter size distribution |

LBM | lattice Boltzmann method |

LES | large eddy simulation |

PBM | population balance model |

RANS | Reynolds-averaged Navier-Stoke equations |

RNG | renormalization group |

We | Weber number |

We_{crit} | critical Weber number |

## Appendix A

**Figure A1.**Sauter diameter of the bubble breakage model of Luo et al. obtained for 4 different energy input rates. The solution from the EE approach and the fitted solution with 3 different critical We numbers are shown.

**Figure A2.**Sauter diameter of the bubble breakage model of Lehr et al. obtained with 4 different energy input rates. The solution from the EE approach and the fitted solution with 3 different critical We numbers are shown.

## Appendix B

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**Figure 1.**Mesh for bubble column simulation in Ansys Fluent. (

**a**): View from the top (

**b**): Section view from the side.

**Figure 2.**Mesh independence study with 3 different mesh densities. The turbulent dissipation rate on a vertical line in the center of the column is compared. (

**a**): Ansys Fluent simulation (

**b**): M-Star simulation.

**Figure 3.**Flow field as contour plot shown as a section plane, where turbulent dissipation rate is depicted. Simulation of bubble column in Ansys Fluent. (

**a**): Ansys Fluent simulation (

**b**): M-Star simulation.

**Figure 4.**Prediction of daughter size distributions using the approach of Luo et al. for different parent bubble sizes with the set value ε = 0.08 W m

^{−3}.

**Figure 5.**Prediction of daughter size distributions using the approach of Lehr et al. for different parent bubble sizes with the set value ε = 0.1 W m

^{−3}.

**Figure 6.**Overview of fitting parameters for the EL breakage model. The function, as well as the method for the determination of the parameters, is illustrated. The left column depicts the bubble breakage implementation in EE and the right column shows how the implementation is transferred into an EL implementation. (

**Top left**): Different bubble sizes exist in a PBM solely as distribution quantities that shift into smaller classes in case of a breakage. (

**Top right**): A breakage of an individual bubble is executed only if the breakage criterion in the form of $W{e}_{crit}$ is exceeded. (

**Center left**): The shape of the DSD changes three-dimensionally with varying bubble diameters and turbulent dissipation energy. (

**Center right**): The shape of the DSD changes solely with varying bubble diameter and the influence of the turbulent dissipation energy is replaced by a constant. (

**Bottom left**): The integro-differential equation of the PBM calculates bubble breakage events per unit of time. (

**Bottom right**): After an executed bubble breakage, a subsequent division can take place at the earliest after the time ${t}_{break}$.

**Figure 7.**The summed deviation of the Sauter mean bubble diameter for different critical We numbers of the fitted model for Luo et al. recreation.

**Figure 8.**The summed deviation of the Sauter mean bubble diameter for different critical We numbers of the fitted model for Lehr et al. recreation.

**Figure 9.**Arithmetic diameter and Sauter diameter of the bubble breakage model given by Luo et al. The solution is compared with the EE approach and the fitted model in EL. Shown for pump volume flow 13, 44, 103 and 202 W m

^{−3}.

**Figure 10.**Arithmetic diameter and Sauter diameter of the bubble breakage model given by Lehr et al. The solution is compared with the EE approach and the fitted model in EL. Shown for pump volume flow 13, 44, 103 and 202 W m

^{−3}.

**Figure 11.**Comparison of the resulting bubble size distribution created with the Luo et al. [4] model in the EE approach (black) and the fitted model for EL (gray). Shown for energy input of 13, 44, 103 and 202 W m

^{−3}.

**Figure 12.**Comparison of the resulting bubble size distribution created with the Lehr et al. [1] model in the EE approach (black) and the fitted model for EL (gray). Shown for energy input of 13, 44, 103 and 202 W m

^{−3}.

**Table 1.**Geometry, simulation set-up, boundary conditions and solver setting of bubble column simulation in Ansys Fluent.

Properties | Boundary Conditions | Units |
---|---|---|

Fluid inlet | Volume flow: 0.6, 0.9, 1.2, 1.5 | L s^{−1} |

Air inlet | Volume flow: 4 | L min^{−1} |

Outlet | Degassing | |

Wall | No-slip | |

Initial bubble size | 16 | mm |

Column diameter | 0.2 | m |

Column height | 1.3 | m |

Multiphase | Euler-Euler | |

Ring sparger diameter | Ring diameter × tube diameter (8.8 × 0.9) | cm |

Fluid inlet diameter | 24 | mm |

Population balance model | Discrete with 20 bins in range [1–16 mm] | |

Breakage model | Luo et al. and Lehr et al. | |

Turbulence | RNG k-$\epsilon $ model | |

Phase interactions | ||

Drag | [25] | |

Solution methods | ||

Pressure-velocity coupling | Phase coupled SIMPLE | |

Gradient | Least Squares Cell Based | |

Pressure | Body Force Weighted | |

Momentum | QUICK | |

Volume fraction | QUICK | |

Turbulent kinetic energy | Second Order Upwind | |

Turbulent dissipation rate | Second Order Upwind | |

Phase-2 Bin | Second Order Upwind | |

Transient Formulation | Bounded Second Order Implicit | |

Time step size | 0.01 | s |

Total mesh size | 862,750 |

**Table 2.**Geometry, simulation set-up, boundary conditions and solver setting of bubble column simulation in M-Star.

Properties | Boundary Conditions | Units |
---|---|---|

Fluid inlet | Volume flow: 0.6, 0.9, 1.2, 1.5 | L s^{−1} |

Air inlet | Volume flow: 4 | L min^{−1} |

Outlet | Free Surface | |

Wall | No-slip | |

Initial bubble size | 16 | mm |

Column diameter | 0.2 | m |

Column height | 1.3 | m |

Ring sparger diameter | Ring diameter × tube diameter (8.8 × 0.9) | cm |

Fluid inlet diameter | 24 | mm |

Multiphase | Euler-Lagrange | |

Breakage model | Modified Weber number | |

Turbulence | LES | |

Sub-grid model | [32] | |

Smagorinsky coefficient | 0.1 | |

Phase interactions | ||

Drag | [25] | |

Lift | [28] | |

Fluid-bubble coupling | Density | |

Velocity vector set | D3Q19 | |

Time step size | Between 3 × 10^{−6} s and 1.6 × 10^{−5} s | s |

Total mesh size | 24.1 M |

**Table 3.**Summary of all determined parameters to be able to implement the bubble breakage model of Luo et al. and Lehr et al. in LBM.

Luo et al. [4] | Lehr et al. [1] | ||
---|---|---|---|

Critical Weber number | $W{e}_{crit}$ | 7 | 11 |

Min. Daughter Volume, Fraction and Increment | ${f}_{V,min}$; $\Delta {f}_{V}$ | 0.01 | 0.01 |

Min. Time Between to Breakages | ${t}_{break}$ | 10 ms | 250 ms |

Daughter Size Distribution | $DSD\left(d,\epsilon \right)\underset{\epsilon constant}{\Rightarrow}DSD\left(d\right)$ | Original U-Shape | Original M-Shape |

Constant $\epsilon $ in DSD | 0.08 W m^{−3} | 0.1 W m^{−3} |

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## Share and Cite

**MDPI and ACS Style**

Mast, Y.; Takors, R. Transferring Bubble Breakage Models Tailored for Euler-Euler Approaches to Euler-Lagrange Simulations. *Processes* **2023**, *11*, 1018.
https://doi.org/10.3390/pr11041018

**AMA Style**

Mast Y, Takors R. Transferring Bubble Breakage Models Tailored for Euler-Euler Approaches to Euler-Lagrange Simulations. *Processes*. 2023; 11(4):1018.
https://doi.org/10.3390/pr11041018

**Chicago/Turabian Style**

Mast, Yannic, and Ralf Takors. 2023. "Transferring Bubble Breakage Models Tailored for Euler-Euler Approaches to Euler-Lagrange Simulations" *Processes* 11, no. 4: 1018.
https://doi.org/10.3390/pr11041018