Processes 2018, 6,82
takes advantage of Pontryagin’s maximum principle and maximizes the control Hamiltonian over
the set of all admissible controls [62]. The application of this approach becomes difficult for larger
systems with state constraints [63], so direct approaches based on parameterization of variables
have been preferred [64,65]. The method of collocation has been proposed for parameterization of
variables [66,67]. The stiffness of the system of ordinary differential equations (ODE) is evaluated for
different initial values. The stiffness ratio is calculated using the eigenvalues of the Jacobian matrix
at different points in time [68]. It is observed that the order of magnitude of the stiffness ratio varies
between 6 and 28 throughout the integration. Therefore, an approach based on the discretization of
time using fixed step sizes does not provide a good approximation of the solution unless the step size
is very small, i.e., less than 10 −9 h. Instead, integration is carried out using appropriate solvers for stiff,
nonlinear ODEs and the problem is formulated for the application of the interior point method [69].
Equations (8)–(11) represent the solution to the mathematical optimization problem. T i and C i are the
time and composition of the ith feeding. In addition to initial nutrient concentrations, schedule, and
composition of feeding, it also finds optimal criteria for setting aeration and agitation rates. Aeration
is stopped or started based on the DO level. For the adjustment of impeller rotation speed, a measure
of homogeneity is defined based on the relative standard deviation (RSD) of distribution of cells over
compartments.
maximize
C 0 ,T i ,C i ,RSD cri ,DO cri
t f
0
biomassdt
(8)
Subject to:
Process Model: Equation (6)
Control Bounds:
0 < ... < T i−1 < T i < T i+1 < ... < t f
(9)
C imin ≤ C i ≤ C imax
(10)
0% ≤ RSD cri ,DO cri ≤ 100%.
(11)
3. Case Study
A 3 L bioreactor with Rushton impeller and sparger is considered. The operation lasts for
two weeks. The dimensions of the reactor are shown in Figure 3. It is operated at liquid fill levels of
130, 155, 180, and 205 mm. Feeding is simulated by change of liquid fill level. So, overall, three steps of
feeding are allowed. For the discretization of space into computational cells, the space is divided into
several meshing zones. This helps Fluent to generate hexahedral cells where possible. It also reduces
the total number of computational cells and improves the convergence of the solution. Meshing for
the highest operating volume results in 363,175 computational cells, 947,988 faces, and 226,971 nodes.
Agitation rates of 150, 225, and 300 RPM and two options for aeration are considered: sparging air at
0.01 vessel volume per minute (vvm), and no aeration. Table 1 shows the physical properties calculated
from CFD simulations. The reactor is divided into compartments as shown in Figure 4. The data
exported from CFD simulations are used to calculate flow matrices for all states of operation. The
unstructured model developed by Xing et al. [60] is adapted to predict the behavior of the bio-phase.
The metabolic model captures the effects of concentrations of glucose, glutamine, lactate, and ammonia
on cellular rates of growth and death in a CHO culture. The rate of utilization of glutamine for essential
metabolic functions, i.e., maintenance, is calculated from Equation (12). The values of the model
parameters are shown in Table 2.
m Gln =
a 1 [Gln]
a 2 +[Gln]
(12)
122
takes advantage of Pontryagin’s maximum principle and maximizes the control Hamiltonian over
the set of all admissible controls [62]. The application of this approach becomes difficult for larger
systems with state constraints [63], so direct approaches based on parameterization of variables
have been preferred [64,65]. The method of collocation has been proposed for parameterization of
variables [66,67]. The stiffness of the system of ordinary differential equations (ODE) is evaluated for
different initial values. The stiffness ratio is calculated using the eigenvalues of the Jacobian matrix
at different points in time [68]. It is observed that the order of magnitude of the stiffness ratio varies
between 6 and 28 throughout the integration. Therefore, an approach based on the discretization of
time using fixed step sizes does not provide a good approximation of the solution unless the step size
is very small, i.e., less than 10 −9 h. Instead, integration is carried out using appropriate solvers for stiff,
nonlinear ODEs and the problem is formulated for the application of the interior point method [69].
Equations (8)–(11) represent the solution to the mathematical optimization problem. T i and C i are the
time and composition of the ith feeding. In addition to initial nutrient concentrations, schedule, and
composition of feeding, it also finds optimal criteria for setting aeration and agitation rates. Aeration
is stopped or started based on the DO level. For the adjustment of impeller rotation speed, a measure
of homogeneity is defined based on the relative standard deviation (RSD) of distribution of cells over
compartments.
maximize
C 0 ,T i ,C i ,RSD cri ,DO cri
t f
0
biomassdt
(8)
Subject to:
Process Model: Equation (6)
Control Bounds:
0 < ... < T i−1 < T i < T i+1 < ... < t f
(9)
C imin ≤ C i ≤ C imax
(10)
0% ≤ RSD cri ,DO cri ≤ 100%.
(11)
3. Case Study
A 3 L bioreactor with Rushton impeller and sparger is considered. The operation lasts for
two weeks. The dimensions of the reactor are shown in Figure 3. It is operated at liquid fill levels of
130, 155, 180, and 205 mm. Feeding is simulated by change of liquid fill level. So, overall, three steps of
feeding are allowed. For the discretization of space into computational cells, the space is divided into
several meshing zones. This helps Fluent to generate hexahedral cells where possible. It also reduces
the total number of computational cells and improves the convergence of the solution. Meshing for
the highest operating volume results in 363,175 computational cells, 947,988 faces, and 226,971 nodes.
Agitation rates of 150, 225, and 300 RPM and two options for aeration are considered: sparging air at
0.01 vessel volume per minute (vvm), and no aeration. Table 1 shows the physical properties calculated
from CFD simulations. The reactor is divided into compartments as shown in Figure 4. The data
exported from CFD simulations are used to calculate flow matrices for all states of operation. The
unstructured model developed by Xing et al. [60] is adapted to predict the behavior of the bio-phase.
The metabolic model captures the effects of concentrations of glucose, glutamine, lactate, and ammonia
on cellular rates of growth and death in a CHO culture. The rate of utilization of glutamine for essential
metabolic functions, i.e., maintenance, is calculated from Equation (12). The values of the model
parameters are shown in Table 2.
m Gln =
a 1 [Gln]
a 2 +[Gln]
(12)
122