2.1 The Core Idea of Lifeline Analysis
After different methods have been addressed in the previous section, the focus is
now on implementing the EL framework in CFD. The basis for each simulation is
the geometry and the spatial discretization of its volume. To obtain high-quality
discretization, a so-called mesh, the geometry needs to be simplified by removing
edges, surface filets, and nonessential components. Accordingly, agitators and
baffles may be replaced by zero-thickness walls [34]. Commercial codes often
make use of orthogonal quality, aspect ratio, and skewness of individual grid cells
to qualify the mesh.
To simulate the continuum environment of a bioreactor, the Eulerian approach is
used together with a case-specific model. Most commonly, the RANS standard k-ε
model is chosen for stirred tank reactors [12, 14, 15, 35, 36]. As known, standard
RANS models work well in flow scenarios with lowly frequent changing mean flows
compared to high turbulence frequencies. However, the turbulent dissipation rate (ε)
is underestimated near the impeller tip but shows decent agreement in the bulk
[37]. Consequently, new methods of turbulence modeling combining advantages of
different models have been proposed, recently [38, 39]. In essence, they apply zonal
methods such as the RANS/LES approach. RANS and a subgrid-scale model (LES)
are used in different domains separated by sharp or dynamic interfaces. In general,
such hybrid models are useful to increase the accuracy where necessary but keeping
computational demands limited.
Convergence of the solution can be assumed if all residuals are below 10
À5 and
oscillation in the mean velocity magnitude and bubble diameter (Parameter of
interest depend on the individual simulation) is below 1% [12, 14]. When steady
state is reached zero-thickness walls can be converted from wall to interface/interior
boundary conditions to prevent particle trapping when the frozen flow field assumption is made. Although the solution is converged, the results might not be accurate.
To improve the accuracy of the simulation, the results need to be mesh-independent.
That means the results (e.g., ε, power number or velocity magnitude for stirred tank
reactors) do not vary significantly (below 5%) when the mesh density is increased.
Now, the mesh-independent mixing time may be used to validate the resulting flow
patterns, the simulated impeller power numbers, k, and ε profiles in the discharge
stream of the impeller. Noteworthy, simulated mixing times of non-aerated scenarios
are expected to be shorter than true mixing times of aerated processes.
As a result, a steady-state converged flow field is calculated that is not updated
during the particle tracking phase. Basically, the procedure mirrors the simplifying
assumption of massless cells neither interacting with each other nor with the
continuum surrounding. Accordingly, the history of cellular experiences recorded
in particle lifelines only monitors the interaction with fixed hydrodynamics and
concentration profiles. The latter may be established using black-box cellular uptake
kinetics [12, 14, 15]. Using nonstructured Monod-type kinetics, the biomass-specific
substrate uptake rate instantaneously adapts to the local concentrations. Hence,
reaction kinetics may even be embedded in the liquid phase to save computation
234
C. S. S. Hajian et al.
After different methods have been addressed in the previous section, the focus is
now on implementing the EL framework in CFD. The basis for each simulation is
the geometry and the spatial discretization of its volume. To obtain high-quality
discretization, a so-called mesh, the geometry needs to be simplified by removing
edges, surface filets, and nonessential components. Accordingly, agitators and
baffles may be replaced by zero-thickness walls [34]. Commercial codes often
make use of orthogonal quality, aspect ratio, and skewness of individual grid cells
to qualify the mesh.
To simulate the continuum environment of a bioreactor, the Eulerian approach is
used together with a case-specific model. Most commonly, the RANS standard k-ε
model is chosen for stirred tank reactors [12, 14, 15, 35, 36]. As known, standard
RANS models work well in flow scenarios with lowly frequent changing mean flows
compared to high turbulence frequencies. However, the turbulent dissipation rate (ε)
is underestimated near the impeller tip but shows decent agreement in the bulk
[37]. Consequently, new methods of turbulence modeling combining advantages of
different models have been proposed, recently [38, 39]. In essence, they apply zonal
methods such as the RANS/LES approach. RANS and a subgrid-scale model (LES)
are used in different domains separated by sharp or dynamic interfaces. In general,
such hybrid models are useful to increase the accuracy where necessary but keeping
computational demands limited.
Convergence of the solution can be assumed if all residuals are below 10
À5 and
oscillation in the mean velocity magnitude and bubble diameter (Parameter of
interest depend on the individual simulation) is below 1% [12, 14]. When steady
state is reached zero-thickness walls can be converted from wall to interface/interior
boundary conditions to prevent particle trapping when the frozen flow field assumption is made. Although the solution is converged, the results might not be accurate.
To improve the accuracy of the simulation, the results need to be mesh-independent.
That means the results (e.g., ε, power number or velocity magnitude for stirred tank
reactors) do not vary significantly (below 5%) when the mesh density is increased.
Now, the mesh-independent mixing time may be used to validate the resulting flow
patterns, the simulated impeller power numbers, k, and ε profiles in the discharge
stream of the impeller. Noteworthy, simulated mixing times of non-aerated scenarios
are expected to be shorter than true mixing times of aerated processes.
As a result, a steady-state converged flow field is calculated that is not updated
during the particle tracking phase. Basically, the procedure mirrors the simplifying
assumption of massless cells neither interacting with each other nor with the
continuum surrounding. Accordingly, the history of cellular experiences recorded
in particle lifelines only monitors the interaction with fixed hydrodynamics and
concentration profiles. The latter may be established using black-box cellular uptake
kinetics [12, 14, 15]. Using nonstructured Monod-type kinetics, the biomass-specific
substrate uptake rate instantaneously adapts to the local concentrations. Hence,
reaction kinetics may even be embedded in the liquid phase to save computation
234
C. S. S. Hajian et al.