number of particles while fluctuating in the bioreactor, thereby recording interactions
with the environment. The dispersed phase can exchange momentum, mass, and
energy with the fluid phase. The approach renders considerably simpler when
particle-particle interactions may be neglected. This requires a fully diluted (volume
fraction <10%) dispersed second phase. Considering the case of biological cells as
moving “particles,” further simplifications are often assumed: As the observation
window of fluctuating cells is smaller than time constants of physical changes inside
the grids, cellular impacts on physical states inside the grid are often neglected.
Trajectories of an individual particle are predicted by integrating particle forces
which is coded in a Lagrangian reference frame. In essence, particle inertia is
balanced with forces acting on the particle. Typically, the Stokes number for a
microorganism (diameter: 5 Â 10
À6 m) is <0.01 giving rise to the fair assumption
that particles move with the flow possessing a negligible mass [12, 14]. Hence,
massless particles can be treated as ideal flow followers and immediately adapt to the
local flow velocity, and no force balance has to be solved, reducing the computation
time considerably [30]. However, if bulk velocity is the only motional source of
particles, they tend to get stuck in poorly or steadily mixed zones. To be precise, in
order to prevent trapping close to reactor surfaces or inside eddies, particleturbulence is modeled via the discrete random walk (DRW). Mimicking characteristic circulation times in small eddies, random velocity is activated to enable particle
escape. The approach leads to temporary constant functions mirroring fluctuating
velocity impacts [30, 31].
A major challenge of the EL method is the significant computational burden. As
discussed in [32, 33], the computation time depends on the required time resolution
and the number of tracked particles. First, the broad range of timescales for metabolic reaction and cellular adaptation [6, 17, 18] requires simulation of flow with fine
temporal resolution. For instance, time steps of milliseconds are needed to track
particles properly especially in highly agitated flows. Second, a large number of
particles may be required to fulfill the so-called ergodicity constraint, a prerequisite
for a sound biological readout. Too low numbers of tracked particles lead to artificial
spatial variations finally causing nonrealistic interpretations.
PBM and EL frameworks are usually applied for monophasic conditions
represented by the Euler phase. The dispersed phase typically comprises either
bubbles or particles, but not both in combination. However, tracking individual
cells in bubbled (aerated) bioreactors should be an attractive goal for future application to analyze aerobic cultivations [11]. This may require a combination of both
methods outlined above, leading to an Euler-Euler-Lagrange formulation (EEL). In
detail, the fluid surrounding the bubbles and particles is calculated via the Euler
approach, while the interfacial area of the bubbles (for mass transfer) is derived with
the population balance method. Additionally, the history of cells may be recorded
via Lagrangian particle tracking.
Euler-Lagrangian Simulations: A Proper Tool for Predicting Cellular Performance. . .
233
with the environment. The dispersed phase can exchange momentum, mass, and
energy with the fluid phase. The approach renders considerably simpler when
particle-particle interactions may be neglected. This requires a fully diluted (volume
fraction <10%) dispersed second phase. Considering the case of biological cells as
moving “particles,” further simplifications are often assumed: As the observation
window of fluctuating cells is smaller than time constants of physical changes inside
the grids, cellular impacts on physical states inside the grid are often neglected.
Trajectories of an individual particle are predicted by integrating particle forces
which is coded in a Lagrangian reference frame. In essence, particle inertia is
balanced with forces acting on the particle. Typically, the Stokes number for a
microorganism (diameter: 5 Â 10
À6 m) is <0.01 giving rise to the fair assumption
that particles move with the flow possessing a negligible mass [12, 14]. Hence,
massless particles can be treated as ideal flow followers and immediately adapt to the
local flow velocity, and no force balance has to be solved, reducing the computation
time considerably [30]. However, if bulk velocity is the only motional source of
particles, they tend to get stuck in poorly or steadily mixed zones. To be precise, in
order to prevent trapping close to reactor surfaces or inside eddies, particleturbulence is modeled via the discrete random walk (DRW). Mimicking characteristic circulation times in small eddies, random velocity is activated to enable particle
escape. The approach leads to temporary constant functions mirroring fluctuating
velocity impacts [30, 31].
A major challenge of the EL method is the significant computational burden. As
discussed in [32, 33], the computation time depends on the required time resolution
and the number of tracked particles. First, the broad range of timescales for metabolic reaction and cellular adaptation [6, 17, 18] requires simulation of flow with fine
temporal resolution. For instance, time steps of milliseconds are needed to track
particles properly especially in highly agitated flows. Second, a large number of
particles may be required to fulfill the so-called ergodicity constraint, a prerequisite
for a sound biological readout. Too low numbers of tracked particles lead to artificial
spatial variations finally causing nonrealistic interpretations.
PBM and EL frameworks are usually applied for monophasic conditions
represented by the Euler phase. The dispersed phase typically comprises either
bubbles or particles, but not both in combination. However, tracking individual
cells in bubbled (aerated) bioreactors should be an attractive goal for future application to analyze aerobic cultivations [11]. This may require a combination of both
methods outlined above, leading to an Euler-Euler-Lagrange formulation (EEL). In
detail, the fluid surrounding the bubbles and particles is calculated via the Euler
approach, while the interfacial area of the bubbles (for mass transfer) is derived with
the population balance method. Additionally, the history of cells may be recorded
via Lagrangian particle tracking.
Euler-Lagrangian Simulations: A Proper Tool for Predicting Cellular Performance. . .
233