(RANS) approaches. DNS yields to solve each individual turbulent structure,
whereas LES only directly solves the large energy-containing scales, while the
effects of the more universal small scales are modeled. RANS comprises timeaveraged flow equations which allow to simulate small- and large-scale eddies
with a minimum but still challenging computational efforts (for more details, see
[2]). Consequently, RANS simulations are often favored. They only require 1/100 to
1/10 computational efforts compared to LES [3] or DNS. Although RANS models
require several modeling assumptions and approximations, their predictive power is
sufficient for providing insight in reactor-scale substrate concentration gradients [4].
Many research projects have shown that cells react in a multiresponse, multilayer
fashion comprising the on- and offset of transcriptional regulation programs, as well
as proteomic and metabolic changes [5–11]. The latter are subject of state-of-the-art
approaches mirroring the instantaneous metabolic response on extracellular heterogeneities [12–16]. However, the consideration of transcriptional and translational
effects introduces different timescales of cellular response. Consequently, initiation
and execution may be spatially disconnected which differs fundamentally from the
instantaneous metabolic responses studied so far [17, 18]. Coupling cellular kinetics
with fluid dynamics consolidates the knowledge about microbial cell kinetics with
fluid dynamics and mass transfer in industrial scale bioreactors yielding to optimize
the design of both [16].
To investigate the consequences of environmental heterogeneities, proper modeling frameworks should link local variations with cellular and subcellular kinetics.
Mainly two different methods exist in computational fluid dynamic (CFD) to display
the adaptation of the cellular behavior to the environment: (1) population balance
models (PBM) [19–21] and (2) the Euler-Lagrangian method (EL) [12, 13, 15, 22–
25]. Both methods rely on the grid-based Euler approach to simulate the continuum
background, meaning the fluid surroundings of the particles. The grid-based method
relies on spatial volume discretization. Adjacent volumes are connected via transport
equations and are considered as homogeneous. Consequently, the following rule of
thumb holds true: the smaller the volumes or the finer the grid, the more realistic is
the output. An extensive study of mesh dependency on biological output has been
conducted by Kuschel and Takors [26].
In the PBM approach, microorganisms are considered as part of the continuum
with no erratic changes allowed [19, 27, 28]. Particles are grouped in classes, and a
predefined distribution range of particles is implemented via distribution density
functions. These equations are useful to determine relevant macroscopic properties
such as the interfacial area or the biomass-specific growth rate [12, 14, 29]. Hence,
PBMs represent a powerful modeling framework for the description of fundamental
properties that are characterized by distributions in a coarse timescale. However,
cellular adaptations may happen on different timescales than macroscopic fluctuations and may show much more interactions than implemented in common PBMs.
To overcome those limitations, the Lagrangian method may be applied. Individual properties are assigned to each moving particle (e.g., biological cell). Noteworthy, these traits cause interactions with the environment receiving equal
environmental feedback. In essence, Euler-Lagrange approach (EL) tracks a given
232
C. S. S. Hajian et al.
whereas LES only directly solves the large energy-containing scales, while the
effects of the more universal small scales are modeled. RANS comprises timeaveraged flow equations which allow to simulate small- and large-scale eddies
with a minimum but still challenging computational efforts (for more details, see
[2]). Consequently, RANS simulations are often favored. They only require 1/100 to
1/10 computational efforts compared to LES [3] or DNS. Although RANS models
require several modeling assumptions and approximations, their predictive power is
sufficient for providing insight in reactor-scale substrate concentration gradients [4].
Many research projects have shown that cells react in a multiresponse, multilayer
fashion comprising the on- and offset of transcriptional regulation programs, as well
as proteomic and metabolic changes [5–11]. The latter are subject of state-of-the-art
approaches mirroring the instantaneous metabolic response on extracellular heterogeneities [12–16]. However, the consideration of transcriptional and translational
effects introduces different timescales of cellular response. Consequently, initiation
and execution may be spatially disconnected which differs fundamentally from the
instantaneous metabolic responses studied so far [17, 18]. Coupling cellular kinetics
with fluid dynamics consolidates the knowledge about microbial cell kinetics with
fluid dynamics and mass transfer in industrial scale bioreactors yielding to optimize
the design of both [16].
To investigate the consequences of environmental heterogeneities, proper modeling frameworks should link local variations with cellular and subcellular kinetics.
Mainly two different methods exist in computational fluid dynamic (CFD) to display
the adaptation of the cellular behavior to the environment: (1) population balance
models (PBM) [19–21] and (2) the Euler-Lagrangian method (EL) [12, 13, 15, 22–
25]. Both methods rely on the grid-based Euler approach to simulate the continuum
background, meaning the fluid surroundings of the particles. The grid-based method
relies on spatial volume discretization. Adjacent volumes are connected via transport
equations and are considered as homogeneous. Consequently, the following rule of
thumb holds true: the smaller the volumes or the finer the grid, the more realistic is
the output. An extensive study of mesh dependency on biological output has been
conducted by Kuschel and Takors [26].
In the PBM approach, microorganisms are considered as part of the continuum
with no erratic changes allowed [19, 27, 28]. Particles are grouped in classes, and a
predefined distribution range of particles is implemented via distribution density
functions. These equations are useful to determine relevant macroscopic properties
such as the interfacial area or the biomass-specific growth rate [12, 14, 29]. Hence,
PBMs represent a powerful modeling framework for the description of fundamental
properties that are characterized by distributions in a coarse timescale. However,
cellular adaptations may happen on different timescales than macroscopic fluctuations and may show much more interactions than implemented in common PBMs.
To overcome those limitations, the Lagrangian method may be applied. Individual properties are assigned to each moving particle (e.g., biological cell). Noteworthy, these traits cause interactions with the environment receiving equal
environmental feedback. In essence, Euler-Lagrange approach (EL) tracks a given
232
C. S. S. Hajian et al.