# Validation of Novel Lattice Boltzmann Large Eddy Simulations (LB LES) for Equipment Characterization in Biopharma

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Reactor Setup

^{3}, $\eta $ = 1.0016 mPas, $\sigma $ = 0.0728 N/m) at 20 °C. A baffled (3 baffles 120°) and an unbaffled system are examined. Stirring speed is set to n = 250 and 350 rpm. For better optical access to the stirred vessel, an octahedral water basin is installed around the stirred vessel in the experiment. In addition, the baffles are made of acrylic glass to eliminate shadowing by the baffles. Further details about the experiments will be published separately.

#### 2.2. Experimental Setup

#### 2.3. Numerical Simulations

^{−4}; however, for higher rotating speeds, only $\approx $10

^{−3}was reached. The power number is calculated according to the following:

## 3. Results and Discussion

#### 3.1. Steady State Simulations

^{4}). As shown in Figure 2A, a mesh of about 350,000 polyhedral cells is fine enough to determine power numbers $Po$ by torque, as the power number converges to a stable value of $\approx 8.1$. Consequently, the rotating frame volume study is performed with this mesh. Convergence for the power number $Po$ at a relatively coarse grid size is also shown in the literature [16]. The influence of the rotating reference volume size on the evolution of the power number is clearly visible in Figure 2B. If the chosen impeller diameter to the reference frame diameter (D-ratio) and the blade height to the height of the rotating reference frame (RRF) volume ratio (H-ratio) is below 1.5, the value for the power number is increasing sharply. Whereas an elevated D-ratio showed no impact until a ratio of 2, higher D-ratios resulted in a decrease in the power number. Increasing the H-ratio further up to a H-ratio of 8 led to a convergence of power number up to a mean value of$Po=8.4\pm 0.2$. A combined volume of the upper and lower impeller results in a power number $Po$= 7.2 $\pm $ 0.2 for increasing height of the combined RRF (data not shown). Deviations in the flow field prediction dependent on the rotating reference frame size have been reported in several studies [22,40,41], leading to the general conclusion to enlarge the dimension of the rotating part toward a region where the flow variables’ gradient is low to reduce numerical errors due to interpolation inaccuracies in transferring the results between both domains. Some authors suggest locating the interface between the rotating and stationary frame 0.5 impeller radius from the impeller tip, whereas others recommend a ratio of 1.5 [21,40]. For this study, a D-ratio between 1.5 and 2.0 is considered appropriate, as higher RRF diameters would place the interface of the RRF too close to the baffles, where too-high velocity gradients exist. Additionally, for the axial extension of the rotating zone, different propositions are made reaching from one to four impeller radius above and below the impeller [42]. Although in this study, the value for the combined RRF is closer to the experimental value of $Po$ = 7.5, marked as a dashed line in Figure 2, the converged power number $Po$ for the increased H-ratios of the two separate RRF is preferred, as huge instabilities within the prediction of turbulent parameters are noticed for the combined RRF. Considering that in typical cell culture bioreactors, many installations, such as probes or sampling ports, are installed that influence the flow structure, the size of the rotating domain has to be evaluated for each individual case. Especially since the influence of the rotating domain size on power number determination is as significant as the grid size, we recommend a RRF size study for reliable power number determination.

#### 3.2. Transient Simulations

#### 3.2.1. Grid Refinement Study

#### 3.2.2. Validation of Numerical Simulations by 4D PTV Data

^{−3}% of the total volume. At $n$ = 350 rpm, the volume fraction depicting maximal velocity magnitudes of up to ${V}_{mag}$ = 0.66 m/s for the simulation and ${V}_{mag}$ = 0.70 − 0.80 m/s for the experiment is less than 10

^{−3}%. Deviations of the simulation and experiment in higher velocities may be due to the finer time-step resolution of the simulation, thereby averaging higher occurring velocities within the impeller discharge stream compared to the tip speed. Consequently, using the Eulerian approach to generate time-averaged velocity profiles is unsuitable to detect velocities larger than the tip speed. However, as these elevated velocities are present within the impeller discharge stream, only transient profiles or Lagrangian particle trajectories are capable of resolving the apparent velocities, which is the advantage of the approach later discussed in this paper.

_{mag}= 2.734 mm/s is chosen for the histogram representation, corresponding to a subdivision into 256 velocity classes. It is remarkable that the maximum of the most frequently occurring velocity components of about ${V}_{mag}$ = 0.08 m/s can be observed, both in the experiment and in the simulation. The chosen representation method has the decisive advantage that, in particular, the transient velocity components are included in the comparison and are not filtered as in the representation of the time-averaged velocity vectors. This becomes clear by the fact that in both the experiment and simulation, particle velocities occur which are clearly above the stirrer tip velocity of ${V}_{mag}$ = 0.47 m/s.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviation

Latin | Greek | |||

$C$ | Off-bottom clearance, m | $\Delta C$ | Impeller spacing, m | |

$Co$ | Courant number | $\Delta t$ | Time step, s | |

$d$ | Impeller diameter, m | $\Delta x$ | Grid spacing, m | |

$D$ | Tank diameter, m | $\eta $ | Dynamic viscosity, Pa s | |

$f$ | Probability density function | $\rho $ | Density, kg m^{−3} | |

$g$ | Gravitational acceleration, m s^{−2} | $\mathsf{\Omega}$ | Collision operator | |

$H$ | Tank height, m | $\epsilon $ | Energy dissipation rate, m^{2} s^{−3} | |

$h$ | Surface height, m | $\sigma $ | Surface tension, N m^{−1} | |

$M$ | Torque, N m | |||

$n$ | Agitation rate, rpm | |||

$p$ | Pressure, Pa | |||

$P$ | Power, W m^{−3} | |||

$P$_{0} | Power number | |||

${P}_{t}$ | Power by torque, W | |||

${P}_{\epsilon}$ | Power by energy dissipation, W | |||

r | Radial distance, m | |||

Re | Reynolds number | |||

$s$ | Spatial resolution, m | |||

$t$ | Time, s | |||

$U$ | Three-dimensional velocity vector, m s^{−1} | |||

${V}_{tip}$ | Tip speed, m s^{−1} | |||

${V}_{mag}$ | Velocity magnitude, m s^{−1} | |||

$V$ | Volume, m^{3} |

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**Figure 1.**Geometry of the bioreactor. Shaft mount (

**a**), lid (

**b**), bearing box (

**c**), baffles (

**d**), shaft (

**e**), impeller (

**f**), vessel (

**g**).

**Figure 2.**Mesh study and variation of rotating reference frame volume. Evolution of power number by torque over number of grid cells for a fixed rotating reference frame volume (

**A**) and depending on the ratio of impeller diameter or height to diameter or height of the RRF volume respectively (

**B**) with the experimental value of $Po$ = 7.5, shown as dashed line.

**Figure 3.**Time and azimuthal averaged profiles of normalized velocity magnitude. Profiles are averaged at different radial positions within the impeller discharge stream of top and bottom impeller.

**Figure 4.**Evolution of power number $Po$ determined by torque ${P}_{t}$ and by integral predicted energy dissipation ${P}_{\epsilon}$ over number of grid points (

**A**) with the experimental value of $Po$ = 7.5 as dashed line (simulated average $Po=$ 7.5 $\pm $ 0.5) and $Po$as function of Re for the baffled system (

**B**).

**Figure 5.**Comparison of simulated (left of the stirrer shaft, mirrored) and experimental (right of the stirrer shaft) velocity field depicted as slice between two baffles of the baffled system at $n$ = 250 rpm (

**A**). Experimental (red) and simulated (blue) velocity distribution (3D) of the baffled system for $n$ = 250 rpm (crosses) $n$ = 350 rpm (circles) (

**B**).

**Figure 8.**Visualization of experimental (magenta) and simulated particle trajectories for the baffled system at $n$ = 250 rpm (

**A**). Velocity distribution (upper) and absolute displacement of filtered particle trajectories (

**B**).

**Figure 9.**Visualization of experimental (magenta) and simulated particle trajectories for the unbaffled system at 250 rpm (

**A**). Velocity distribution (upper) and absolute displacement of filtered particle trajectories (

**B**).

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**MDPI and ACS Style**

Kuschel, M.; Fitschen, J.; Hoffmann, M.; von Kameke, A.; Schlüter, M.; Wucherpfennig, T. Validation of Novel Lattice Boltzmann Large Eddy Simulations (LB LES) for Equipment Characterization in Biopharma. *Processes* **2021**, *9*, 950.
https://doi.org/10.3390/pr9060950

**AMA Style**

Kuschel M, Fitschen J, Hoffmann M, von Kameke A, Schlüter M, Wucherpfennig T. Validation of Novel Lattice Boltzmann Large Eddy Simulations (LB LES) for Equipment Characterization in Biopharma. *Processes*. 2021; 9(6):950.
https://doi.org/10.3390/pr9060950

**Chicago/Turabian Style**

Kuschel, Maike, Jürgen Fitschen, Marko Hoffmann, Alexandra von Kameke, Michael Schlüter, and Thomas Wucherpfennig. 2021. "Validation of Novel Lattice Boltzmann Large Eddy Simulations (LB LES) for Equipment Characterization in Biopharma" *Processes* 9, no. 6: 950.
https://doi.org/10.3390/pr9060950