Processes 2018, 6,82
Flow matrices contain information on inlet and outlet fluxes, gas volume fraction, dissipation rate
of mechanical energy, and gas superficial velocity of compartments and characterize the flow inside
the reactor under specific operating conditions. The change in operating conditions is simulated by
replacing the flow matrices of the current state of operation with those associated with the new state.
This is based on the assumption that the time for the flow to reach a new steady state as a result of
a change in the operational conditions is negligible compared to the total processing time. It should be
noted that compartmental modeling provides an approximation of the solution and as such predicts
more homogenous distribution for species since each compartment is homogeneous, and does not
account for diffusive mass transfer. On the other hand, it makes it possible to take into account
hydrodynamics in dynamic analysis of reactor performance and couple the model with optimization
solvers. In this work, integration of hydrodynamics with metabolism refers to capturing the effects
of dissolved oxygen (DO) concentration, bubbles, and turbulent eddies on the metabolic activities
and viability of cells. Biological processes are captured through unstructured modeling. This utilizes
a reduced number of reactions to macroscopically capture cellular kinetics.
2.1. Development of CFD Simulations
Computational fluid dynamics (CFD) simulations are developed in ANSYS ® Fluent ® 15.0.7 for the
prediction of spatial variations of environmental parameters. Conservation laws of mass, momentum,
and energy are usually used to describe a single phase flow, gas or liquid. If the thermodynamic,
transport, and chemical properties of a component need to be specified, the field equations may be
accompanied by the constitutive equations of state, stress, chemical reactions, etc. The presence of
interfacial surface in a multi-phase flow complicates the mathematical formulation of the problem.
To derive the field and constitutive equations of a multi-phase flow, such as inside a bioreactor, local
characteristics have to be considered. This is not straightforward due to unknown motions of multiple
deformable interfaces, variable fluctuations due to turbulence and moving interfaces, and discontinuity
of properties at the interface. Obtaining local mean values of flow properties has been shown to be
an efficient way to eliminate instantaneous fluctuations. Three averaging methodologies have been
developed: Eulerian, Lagrangian, and Boltzmann statistical averaging. In the Eulerian approach,
time and space coordinates are independent and other variables are expressed with respect to them.
In the Lagrangian averaging methodology, particle coordinates replace spatial coordinates. If the
purpose of modeling is studying the group behavior of particles, the Eulerian approach is preferred.
However, if the behavior of individual particles is of interest, the Lagrangian description has a clear
advantage [45]. Tracking individual bubbles increases the computation. Additionally, it would only
improve model predictive power if the extent of the interactions between individual bubbles and the
liquid phase could be quantified. These interactions involve growth, breakage, and agglomeration of
bubbles and energy dissipation due to bubble rupture. Therefore, in this study gas and liquid phases
are treated as continua and Eulerian averaging is used. The Eulerian multiphase model creates sets of
momentum and continuity equations for each phase and couples them through exchanging pressure
and interphase coefficients [46]. Turbulence of flow is calculated using the k-ε viscosity model, which
has been widely used for stirred tanks [47]. It is a robust model that gives reasonably accurate results
for a wide range of turbulent flows [48]. A k-ε model consists of two transport equations, one each
for the turbulent kinetic energy (k) and the energy dissipation rate (ε). The motion of the impeller
is captured using a multiple reference frame (MRF). To implement the MRF model, the geometry is
broken up into stationary and moving zones. The MRF model approximates the flow in the moving
zone around the impeller by freezing the motion of the moving part in a specific position and observing
the instantaneous flow field. To use flow variables of one zone for calculation of fluxes at the boundary
of the adjacent zone, a local reference frame transformation is performed at the interface between cell
zones. In the absence of large-scale transient effects due to weak impeller–wall interactions, the MRF
approach provides a reasonable approximation of the flow [48].
118
Flow matrices contain information on inlet and outlet fluxes, gas volume fraction, dissipation rate
of mechanical energy, and gas superficial velocity of compartments and characterize the flow inside
the reactor under specific operating conditions. The change in operating conditions is simulated by
replacing the flow matrices of the current state of operation with those associated with the new state.
This is based on the assumption that the time for the flow to reach a new steady state as a result of
a change in the operational conditions is negligible compared to the total processing time. It should be
noted that compartmental modeling provides an approximation of the solution and as such predicts
more homogenous distribution for species since each compartment is homogeneous, and does not
account for diffusive mass transfer. On the other hand, it makes it possible to take into account
hydrodynamics in dynamic analysis of reactor performance and couple the model with optimization
solvers. In this work, integration of hydrodynamics with metabolism refers to capturing the effects
of dissolved oxygen (DO) concentration, bubbles, and turbulent eddies on the metabolic activities
and viability of cells. Biological processes are captured through unstructured modeling. This utilizes
a reduced number of reactions to macroscopically capture cellular kinetics.
2.1. Development of CFD Simulations
Computational fluid dynamics (CFD) simulations are developed in ANSYS ® Fluent ® 15.0.7 for the
prediction of spatial variations of environmental parameters. Conservation laws of mass, momentum,
and energy are usually used to describe a single phase flow, gas or liquid. If the thermodynamic,
transport, and chemical properties of a component need to be specified, the field equations may be
accompanied by the constitutive equations of state, stress, chemical reactions, etc. The presence of
interfacial surface in a multi-phase flow complicates the mathematical formulation of the problem.
To derive the field and constitutive equations of a multi-phase flow, such as inside a bioreactor, local
characteristics have to be considered. This is not straightforward due to unknown motions of multiple
deformable interfaces, variable fluctuations due to turbulence and moving interfaces, and discontinuity
of properties at the interface. Obtaining local mean values of flow properties has been shown to be
an efficient way to eliminate instantaneous fluctuations. Three averaging methodologies have been
developed: Eulerian, Lagrangian, and Boltzmann statistical averaging. In the Eulerian approach,
time and space coordinates are independent and other variables are expressed with respect to them.
In the Lagrangian averaging methodology, particle coordinates replace spatial coordinates. If the
purpose of modeling is studying the group behavior of particles, the Eulerian approach is preferred.
However, if the behavior of individual particles is of interest, the Lagrangian description has a clear
advantage [45]. Tracking individual bubbles increases the computation. Additionally, it would only
improve model predictive power if the extent of the interactions between individual bubbles and the
liquid phase could be quantified. These interactions involve growth, breakage, and agglomeration of
bubbles and energy dissipation due to bubble rupture. Therefore, in this study gas and liquid phases
are treated as continua and Eulerian averaging is used. The Eulerian multiphase model creates sets of
momentum and continuity equations for each phase and couples them through exchanging pressure
and interphase coefficients [46]. Turbulence of flow is calculated using the k-ε viscosity model, which
has been widely used for stirred tanks [47]. It is a robust model that gives reasonably accurate results
for a wide range of turbulent flows [48]. A k-ε model consists of two transport equations, one each
for the turbulent kinetic energy (k) and the energy dissipation rate (ε). The motion of the impeller
is captured using a multiple reference frame (MRF). To implement the MRF model, the geometry is
broken up into stationary and moving zones. The MRF model approximates the flow in the moving
zone around the impeller by freezing the motion of the moving part in a specific position and observing
the instantaneous flow field. To use flow variables of one zone for calculation of fluxes at the boundary
of the adjacent zone, a local reference frame transformation is performed at the interface between cell
zones. In the absence of large-scale transient effects due to weak impeller–wall interactions, the MRF
approach provides a reasonable approximation of the flow [48].
118