Influence of Interfacial Force Models and Population Balance Models on the kLa Value in Stirred Bioreactors
Abstract
:1. Introduction
1.1. Oxygen Demand of Cells
1.2. Breakup and Coalescence
1.3. Population Balance Modeling
1.4. Computational Fluid Dynamics
2. Materials and Methods
2.1. CFD Simulations
2.2. Richardson Extrapolation
2.3. Statistical Analysis
2.4. Validation
3. Results and Discussion
3.1. Mesh Study
3.2. Euler–Euler Simulations
3.2.1. Influence of the Bubble Diameter
3.2.2. Influence of Drag Force
3.2.3. Influence of Lift Force
3.2.4. Influence of the Virtual Mass Force
3.2.5. Influence of Turbulent Dispersion Force
3.3. Population Balance Modelling
3.3.1. Influence of Initial Gas Bubble Size Distribution
3.3.2. Influence of the Coalescence Model
3.3.3. Influence of the Breakup Model
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
ANOVA | Analysis Of Variance |
CFD | Computational Fluid Dynamics |
CFL | Courant-Friedrichs-Lewy |
CPU | Central Processing Unit |
DSD | Daughter Size Distribution |
HPC | High Performance Computer |
MOM | Method Of Moments |
MRF | Moving Reference Frame |
NDF | Number Density Function |
OTR | Oxygen Transfer Rate |
OUR | Oxygen Uptake Rate |
PBE | Population Balance Equation |
PBM | Population Balance Modelling |
RANS | Reynolds-averaged Navier–Stokes |
VOF | Volume of Fluid |
Nomenclature
Latin symbols | ||
A | Area | [] |
a | Specific interfacial area | [m−1] |
B | Birth rate | [m−3 s−1] |
c | Constant value | [-] |
Drag force coefficient | [-] | |
Lift force coefficient | [-] | |
Dissolved oxygen concentration at the gas liquid interphase | [mol m−3] | |
Dissolved oxygen concentration in the liquid bulk | [mol m−3] | |
Specific heat capacity | [J kg−1 K−1] | |
Turbulent dispersion force coefficient | [-] | |
Coefficient of variation | [-] | |
Virtual mass force coefficient | [-] | |
Wall lubrication force coefficient | [-] | |
Biomass concentration | [cells L−1] | |
Sauter mean diameter | [] | |
D | Death rate | [m−3 s−1] |
D | Dimensions | [-] |
Bubble diameter | [] | |
E | Error | [-] |
F | Force | [] |
Sum of volume-related interfacial forces [ | [] | |
Drag force | [] | |
Lift force | [] | |
Turbulent dispersion force | [] | |
Virtual mass force | [] | |
Wall lubrication force | [] | |
g | Gravitational acceleration | [m s−2] |
Null hypothesis | [-] | |
k | Turbulent kinetic energy | [m2 s−2] |
Liquid side mass transfer coefficient | [m2 s−1] | |
Volumetric oxygen mass transfer coefficient | [h−1] | |
m | Molecular weight | [] |
N | Total number of particles | [-] |
N | Total number of cells | [-] |
n | Number of particles per unit volume | [m−3] |
Unit normal pointing away from the wall | [-] | |
OTR | Oxygen transfer rate | [mol L−1 h−1] |
OUR | Oxygen uptake rate | [mol L−1 h−1] |
P | Power | [] |
p | Pressure | [] |
p | Exponent of error reduction | [-] |
p | p-value | [-] |
Q | Aggregation frequency | [s−1] |
Cell specific oxygen uptake rate | [mol cells−1 h−1] | |
r | Refinement factor | [-] |
T | Temperature | [] |
t | Time | [] |
Velocity | [m s−1] | |
V | Volume | [m3] |
Representative volume for the ith size range | [m3] | |
Data to be transformed | [-] | |
Greek symbols | ||
Volume fraction | [-] | |
Significance level | [-] | |
Dimensionless daughter size distribution | [-] | |
Breakup frequency | [m3 s−1] | |
Turbulent energy dissipation rate | [m2 s−3] | |
Estimated discretisation error | [-] | |
Factor defined by eq. 10 | [-] | |
Cell specific constant | [-] | |
Power parameter | [-] | |
Dynamic viscosity | [] | |
Kinematic viscosity | [m2 s−1] | |
Density | [kg m−3] | |
Surface tension | [N m−1] | |
Reynolds stress tensor | [] | |
Sub- and Superscripts | ||
a | Air | |
atm | Atmospheric | |
b | Bubble | |
br | Breakup | |
coal | Coalescence | |
g | Gas | |
i | Index | |
l | Liquid | |
t | Turbulent | |
w | Water |
Appendix A
Field | Fixed Walls (Vessel, Sparger) | Moving Walls (Stirrer, Shaft) | Inlet | Outlet |
---|---|---|---|---|
alpha.air | zeroGradient | zeroGradient | fixedValue (uniform 1) | inletOutlet |
alphat. | compressible:: alphatWallFunction | compressible:: alphatWallFunction | calculated | calculated |
epsilon | epsilonWall-Function | epsilonWall-Function | fixedValue (uniform 0.00015) | zeroGradient |
k | kqRWallFunction | kqRWallFunction | fixedValue (uniform 3.75 · 10−5) | zeroGradient |
nut. | nutkWallFunction | nutkWallFunction | fixedValue (uniform 1 · 10−8) | inletOutlet |
p | calculated | calculated | calculated | calculated |
p_rgh | fixedFluxPressure (uniform 1 · 105) | fixedFluxPressure (uniform 1 · 105) | fixedFlux-Pressure | prghPressure |
Theta | zeroGradient | zeroGradient | fixedValue (uniform 1.0 · 10−7) | inletOutlet |
T. | zeroGradient | zeroGradient | fixedValue (uniform 298.15) | inletOutlet |
U.air | fixedValue | movingWall-Velocity | fixedValue uniform (x 0 0) | pressureInlet-OutletVelocity |
U.water | fixedValue | movingWall-Velocity | fixedValue uniform (0 0 0) | pressureInlet-OutletVelocity |
Short Biography of Authors
| Stefan Seidel holds a Bachelor’s degree in biotechnology and a Master’s degree in applied computational life sciences from Zurich University of Applied Sciences. His focus is on the classical process engineering characterization of bioreactors as well as on multiphase CFD simulations and their validation using shadowgraphy and particle image velocimetry. Since 2017, he has been a research assistant in the Competence Center for Biochemical Engineering and Cell Cultivation Techniques at ZHAW and since 2020, a Ph.D. student at the Technical University of Berlin. |
| Dieter Eibl has held an engineering degree in food technology since 1981 and a Ph.D. in biotechnology from the Technical University in Köthen since 1986. Since 1991, Prof. Eibl has been working at the University of Applied Sciences in Wädenswil as a lecturer. He is the head of the Competence Center for Biochemical Engineering and Cell Cultivation Techniques and working group leader for biochemical engineering. Prof. Eibl brings more than 25 years of professional expertise in upstream process development, scaling-up, project, and team management. |
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Water | |
---|---|
Property | Value |
Density | m−3 |
Kinematic viscosity | 0.8927 · 10−6 m2 s−1 |
Molecular weight | 18 |
Prandtl number | |
Rheology model | Newtonian fluid |
Specific heat capacity | 4182 J kg−1 K−1 |
Temperature | |
Air | |
Property | Value |
Density | 1.1839 kg m−3 |
Kinematic viscosity | 1.579 · 10−9 m2 s−1 |
Molecular weight | |
Prandtl number | |
Rheology model | Newtonian fluid |
Specific heat capacity | 1007 J kg−1 K−1 |
Temperature | |
General | |
Property | Value |
Atmospheric pressure | |
Gravitational acceleration g | m s−2 |
Surface tension | N m−1 |
Interfacial Force Models | |
---|---|
Interfacial Force Coefficient | Model |
Drag | Schiller and Naumann [76] Schiller and Naumann with swarm correction [76] Ergun [77] Ishii and Zuber [78] Tomiyama correlated [79] Gidaspow, Ergun, Wen, and Yu [80] |
Lift | Tomiyama [81] Legendre and Magnaudet [82] Constant lift coefficient Neglecting the lift force |
Virtual mass | Lamb [83] Constant virtual mass coefficient Neglecting the virtual mass force |
Turbulent dispersion | Gosman [64] Neglecting the turbulent dispersion force |
Breakup and coalescence models | |
Phenomenon | Model |
Breakup | Exponential kernel Laakkonen, Alopaeus, and Aittamaa with DSD [84] Laakkonen, Alopaeus, and Aittamaa without DSD [84] Lehr, Millies, and Mewes [32] Luo and Svendsen [85] |
Coalescence | Coulaloglou and Tavlarides [86] Luo [87] Prince and Blanch [88] Lehr, Millies, and Mewes [32] |
Mesh | Cells | Average Non-Orthogonality | Mass Transfer Coefficient [m s ] | Discretization Error [%] |
---|---|---|---|---|
Mesh 1 | 256,253 | 5.024 | 2.33 · 10−4 | 4.4 |
Mesh 2 | 807,957 | 4.119 | 2.42 · 10−4 | 4.0 |
Mesh 3 | 1,155,407 | 3.808 | 2.39 · 10−4 | 2.9 |
Mesh 4 | 1,842,135 | 3.478 | 2.43 · 10−4 | 3.3 |
Model | a [m ] | [m s ] | [h ] | [%] |
---|---|---|---|---|
Johnson and Hunag [113] | 1.83 · 10−5 | |||
Prasher and Wills [109] | 1.38 · 10−5 | |||
Perez and Sandall [110] | 5.41 · 10−3 | |||
Akita and Yoshida [112] | 1.83 · 10−4 | |||
Kawase and Moo-Young [115] | 2.43 · 10−4 | |||
Garcia-Ochoa and Gómez [111] | 9.11 · 10−4 | |||
Brüning [114] | 2.94 · 10−4 |
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Seidel, S.; Eibl, D. Influence of Interfacial Force Models and Population Balance Models on the kLa Value in Stirred Bioreactors. Processes 2021, 9, 1185. https://doi.org/10.3390/pr9071185
Seidel S, Eibl D. Influence of Interfacial Force Models and Population Balance Models on the kLa Value in Stirred Bioreactors. Processes. 2021; 9(7):1185. https://doi.org/10.3390/pr9071185
Chicago/Turabian StyleSeidel, Stefan, and Dieter Eibl. 2021. "Influence of Interfacial Force Models and Population Balance Models on the kLa Value in Stirred Bioreactors" Processes 9, no. 7: 1185. https://doi.org/10.3390/pr9071185