time. In essence, this simplification mirrors the assumption of homogeneously
dispersed bacteria [12, 14]. Other more complex approaches using a structured
kinetic model coupled to the gradient in the Euler phase to directly determine the
intracellular state of the microbial cell have been applied but are limited due to their
computational demand [30, 36]. Monod-type kinetics are mainly valid for long-term
balanced growth which might hold true for a lab-scale chemostat to characterize a
strain but falls short when offering practical solution for industrial processes like a
substrate limited fed-batch [40–42]. In the case of overflow metabolism for baker’s
yeast, several structured models were employed in various scenarios some of which
include glycolytic oscillations in a population of yeast cells [36], continuous fermentation of Saccharomyces cerevisiae [41], fed-batch fermentation with emphasis
on ethanol concentration of oxygen uptake rate, and questioning pyruvate dehydrogenase’s role as a bottleneck enzyme [43], a cybernetic model for batch fermentation
[44]. Availability of such information encourages researchers to employ more
sophisticated kinetic models in their CFD simulations [32, 45].
To qualify whether or not statistically sufficient particles (cells) are tracked, the
ergodic theorem is typically applied. The criterion follows the fundamental idea that
a stochastic process proceeding in time converges to the same average as the average
of the entire system’s space. In other words, averages of state variables recorded in
lifelines should be the same irrespective whether one single particle is tracked for
endless time or a proper number is monitored for shorter intervals. In practice,
typically 10
5 particles are added (well distributed in space) and mixed, at least for
the duration of the mixing time to ensure a homogeneous distribution. Individual
experiences of each particle are recorded for further analysis.
Depending on the goal and timescale of the process, the readout frequency must
be appropriate to provide reasonable resolution (usually in the range of 10–30 ms).
As outlined above, particle slip with the convective fluid may be neglected which
basically reflects the very low Archimedes numbers of microbial cells [12, 14]. By
analogy, momentum transfer between the particles and the fluid phase is not
considered either, again reflecting the small masses of the cells (quasi-single
phase, [46]). Accordingly, so-called one-way coupled EL approaches are today’s
standard. On contrast, two-way coupled simulations considering the particlecontinuum interactions require very high computational demands that are hardly
applicable yet [47].
Even with many assumptions made, the system of equations resulting from
discretization is tremendous due to the facts that (a) it is a three-dimensional problem
and (b) a sufficiently large number of particles and time steps are required to achieve
a realistic and statistically sound description of the population. To keep the computational cost-feasible assumptions on operation type of the process, kinetics are taken
into account. To reduce the computational burden for simulation, multiphase simulation is done in Eulerian frame [12, 14, 15, 36], or if it is known that oxygen is found
in excess quantities all over reactor and its volume fraction does not exceed 10%,
simpler approaches can be employed [13, 22–24]. Currently, the common approach
is to apply one-way coupling meaning that particles are affect by the flow field and
not the other way around [15, 25, 36]. Haringa et al. [13, 22–24] considered two-way
Euler-Lagrangian Simulations: A Proper Tool for Predicting Cellular Performance. . .
235
dispersed bacteria [12, 14]. Other more complex approaches using a structured
kinetic model coupled to the gradient in the Euler phase to directly determine the
intracellular state of the microbial cell have been applied but are limited due to their
computational demand [30, 36]. Monod-type kinetics are mainly valid for long-term
balanced growth which might hold true for a lab-scale chemostat to characterize a
strain but falls short when offering practical solution for industrial processes like a
substrate limited fed-batch [40–42]. In the case of overflow metabolism for baker’s
yeast, several structured models were employed in various scenarios some of which
include glycolytic oscillations in a population of yeast cells [36], continuous fermentation of Saccharomyces cerevisiae [41], fed-batch fermentation with emphasis
on ethanol concentration of oxygen uptake rate, and questioning pyruvate dehydrogenase’s role as a bottleneck enzyme [43], a cybernetic model for batch fermentation
[44]. Availability of such information encourages researchers to employ more
sophisticated kinetic models in their CFD simulations [32, 45].
To qualify whether or not statistically sufficient particles (cells) are tracked, the
ergodic theorem is typically applied. The criterion follows the fundamental idea that
a stochastic process proceeding in time converges to the same average as the average
of the entire system’s space. In other words, averages of state variables recorded in
lifelines should be the same irrespective whether one single particle is tracked for
endless time or a proper number is monitored for shorter intervals. In practice,
typically 10
5 particles are added (well distributed in space) and mixed, at least for
the duration of the mixing time to ensure a homogeneous distribution. Individual
experiences of each particle are recorded for further analysis.
Depending on the goal and timescale of the process, the readout frequency must
be appropriate to provide reasonable resolution (usually in the range of 10–30 ms).
As outlined above, particle slip with the convective fluid may be neglected which
basically reflects the very low Archimedes numbers of microbial cells [12, 14]. By
analogy, momentum transfer between the particles and the fluid phase is not
considered either, again reflecting the small masses of the cells (quasi-single
phase, [46]). Accordingly, so-called one-way coupled EL approaches are today’s
standard. On contrast, two-way coupled simulations considering the particlecontinuum interactions require very high computational demands that are hardly
applicable yet [47].
Even with many assumptions made, the system of equations resulting from
discretization is tremendous due to the facts that (a) it is a three-dimensional problem
and (b) a sufficiently large number of particles and time steps are required to achieve
a realistic and statistically sound description of the population. To keep the computational cost-feasible assumptions on operation type of the process, kinetics are taken
into account. To reduce the computational burden for simulation, multiphase simulation is done in Eulerian frame [12, 14, 15, 36], or if it is known that oxygen is found
in excess quantities all over reactor and its volume fraction does not exceed 10%,
simpler approaches can be employed [13, 22–24]. Currently, the common approach
is to apply one-way coupling meaning that particles are affect by the flow field and
not the other way around [15, 25, 36]. Haringa et al. [13, 22–24] considered two-way
Euler-Lagrangian Simulations: A Proper Tool for Predicting Cellular Performance. . .
235