according to this theory the killing volume would be proportional to the square of the
bubble diameter, as is the case for the model of Wu and Goosen (1995a).
Meier et al. (1999) incorporated the adsorption of cells to bubbles in another way.
They divided the hypothetical killing volume into two contributions: a rise-dependent (Vd,
rise (m
(^3) )) and a rise-independent (V
dā² (m
(^3) )) term:
(15)
Cells are assumed to attach irreversibly to the bubble if they have been in contact with the
bubble longer than a required contact time, which is in accordance with the experimental
data of Michaels et al. (1995a). As the bubble rises from the sparger to the top of the
reactor, it sweeps a certain volume in which the cells come in contact with the bubble for
the desired time. This volume is given by
(^) (16)
where Rp (m) is an effective radius projected by the rising bubble as shown in Figure
15.3. The projected radius is next calculated from the stream functions for creeping
(equation 17) flow with a non-slip boundary condition and for potential flow (equation
18):
(17)
(18)
Lethal effects of bubbles in animal-cell culture 469