Processes 2018, 6,82
2.2. Development of the Integrated Model
The state of operation is defined by operating volume, impeller rotation speed, and gas sparging
flowrate. For every admissible state a separate CFD simulation is developed to obtain the steady-state
flow under associated operating conditions. After the solution is converged, the flow information of
computational cells is extracted. Compartments are formed through agglomeration of computational
cells. Inlet and outlet fluxes, gas volume fraction, dissipation rate of mechanical energy, and gas
superficial velocity of compartments are calculated using data obtained from CFD simulations.
The surface between two neighbor compartments is composed of faces of computational cells.
The mass flowrates reported for these faces are summed to calculate the flowrates between neighbor
compartments. The net flow between two neighbors is not necessarily zero, but the net flow for
each compartment should be zero in order to maintain mass conservation. Fluent does not report
flowrates for faces at the boundaries. Boundary faces are located at the surface of meshing zones.
Although very small, these missing values cause mass imbalance and introduce errors. These errors are
smoothed out by making the smallest possible changes to the calculated flowrates. It is assumed that
the calculated value for mass flowrate from compartment j to i, F ij , has error ε ij . The sum of squared
errors is minimized by solving the problem explained by Equations (1) to (3). The second constraint
makes sure that zero elements will remain zero, i.e., flowrates between non-neighbor compartments
remain zero. The effect of agitation on distribution of cells between compartments is only understood
if the sedimentation of cells is captured. The rate of sedimentation is estimated as r 2 /4 (mm/h) and r
is cell radius in µm[49]. Cell radius is estimated assuming spherical shape, density equal to that of
water, and average mass of 1.1165 × 10 −6 mg [20].
minimize
ε ij
∑
i
∑
j
ε ij
2
(1)
subject to:
∑
i
F ij − ε ij = ∑
i
F ji − ε ji
(2)
− F ij ≤ ε ij ≤ F ij
(3)
It has been reported that cells can be damaged when the power input is greater than
22,500 W·m −3 [50,51]. Power input is calculated by multiplying the density of the liquid phase
(kg/m 3 ) by the turbulent energy dissipation rate (m 2 /s 3 ). The turbulent energy dissipation rate is
the rate of absorption of kinetic energy that breaks up large eddies. This is then converted to heat
by viscous forces [52]. A rate of cell damage of 3.4% min −1 has been reported for cells in high shear
regions [51]. The volume fraction of high shear region in compartments is calculated using the data
obtained from CFD simulations. Since cells are assumed to be homogenously distributed inside
compartments, the volume fraction of the high shear region is equal to the fraction of cells exposed
to shear beyond their tolerable threshold. Therefore, the rate of loss of viable cells under operating
condition op and in compartment c is calculated using Equation (4):
k d,shearop,c = 0.034. volume f raction o f high shear region op,c .
(4)
Interaction with bubbles has also been reported as one of the sources of cell loss. Cells attach
to bubbles, rise with them to the surface, become trapped in the foam layer, and perish. Also,
the maximum energy dissipated due to bubble rupture is two or three orders of magnitude higher than
the tolerable threshold for cells [51]. No value has been reported in the literature for the rate of cell loss
due to interaction with bubbles. One motive for integrated modeling is to capture uncertainty where it
occurs. The rate of cell loss due to interaction with bubbles is estimated by assuming an interaction
vicinity around bubbles. It is assumed that a fraction of the cells in a compartment that are in the
vicinity of bubbles are lost over the average lifespan of a bubble in the compartment. The volume
119
2.2. Development of the Integrated Model
The state of operation is defined by operating volume, impeller rotation speed, and gas sparging
flowrate. For every admissible state a separate CFD simulation is developed to obtain the steady-state
flow under associated operating conditions. After the solution is converged, the flow information of
computational cells is extracted. Compartments are formed through agglomeration of computational
cells. Inlet and outlet fluxes, gas volume fraction, dissipation rate of mechanical energy, and gas
superficial velocity of compartments are calculated using data obtained from CFD simulations.
The surface between two neighbor compartments is composed of faces of computational cells.
The mass flowrates reported for these faces are summed to calculate the flowrates between neighbor
compartments. The net flow between two neighbors is not necessarily zero, but the net flow for
each compartment should be zero in order to maintain mass conservation. Fluent does not report
flowrates for faces at the boundaries. Boundary faces are located at the surface of meshing zones.
Although very small, these missing values cause mass imbalance and introduce errors. These errors are
smoothed out by making the smallest possible changes to the calculated flowrates. It is assumed that
the calculated value for mass flowrate from compartment j to i, F ij , has error ε ij . The sum of squared
errors is minimized by solving the problem explained by Equations (1) to (3). The second constraint
makes sure that zero elements will remain zero, i.e., flowrates between non-neighbor compartments
remain zero. The effect of agitation on distribution of cells between compartments is only understood
if the sedimentation of cells is captured. The rate of sedimentation is estimated as r 2 /4 (mm/h) and r
is cell radius in µm[49]. Cell radius is estimated assuming spherical shape, density equal to that of
water, and average mass of 1.1165 × 10 −6 mg [20].
minimize
ε ij
∑
i
∑
j
ε ij
2
(1)
subject to:
∑
i
F ij − ε ij = ∑
i
F ji − ε ji
(2)
− F ij ≤ ε ij ≤ F ij
(3)
It has been reported that cells can be damaged when the power input is greater than
22,500 W·m −3 [50,51]. Power input is calculated by multiplying the density of the liquid phase
(kg/m 3 ) by the turbulent energy dissipation rate (m 2 /s 3 ). The turbulent energy dissipation rate is
the rate of absorption of kinetic energy that breaks up large eddies. This is then converted to heat
by viscous forces [52]. A rate of cell damage of 3.4% min −1 has been reported for cells in high shear
regions [51]. The volume fraction of high shear region in compartments is calculated using the data
obtained from CFD simulations. Since cells are assumed to be homogenously distributed inside
compartments, the volume fraction of the high shear region is equal to the fraction of cells exposed
to shear beyond their tolerable threshold. Therefore, the rate of loss of viable cells under operating
condition op and in compartment c is calculated using Equation (4):
k d,shearop,c = 0.034. volume f raction o f high shear region op,c .
(4)
Interaction with bubbles has also been reported as one of the sources of cell loss. Cells attach
to bubbles, rise with them to the surface, become trapped in the foam layer, and perish. Also,
the maximum energy dissipated due to bubble rupture is two or three orders of magnitude higher than
the tolerable threshold for cells [51]. No value has been reported in the literature for the rate of cell loss
due to interaction with bubbles. One motive for integrated modeling is to capture uncertainty where it
occurs. The rate of cell loss due to interaction with bubbles is estimated by assuming an interaction
vicinity around bubbles. It is assumed that a fraction of the cells in a compartment that are in the
vicinity of bubbles are lost over the average lifespan of a bubble in the compartment. The volume
119