for a 3.5 mm bubble (van der Pol et al. 1992), the contribution by the rise-dependent
killing-volume would become significant only over a distance of one meter or more. In
the presence of protective additives less attachment occurs and higher columns may be
required for the rise-dependent killing volume to become significant, although also the
rise-independent part will become smaller. Since most, if not all, studies on cell death are
done in columns shorter than one meter, an effect of reactor height remains unnoticed.
Meier et al. (1999) experimentally showed that for serum-free medium using columns of
0.34 m and 1.42 meter the killing volume is dependent on reactor height.
Bavarian et al. (1991), citing work of Kelly and Spottiswood on ore flotation,
suggested that the rate of cell attachment is equal to the product of three factors being (i)
the rate of collision, (ii) the probability of adhesion, (iii) the probability that an attached
cell will not be detached. Sutherland (taken from Bavarian et al. 1991) derived the
following equation for this:
(19)where p is the probability per unit time of stable cell attachment to a bubble, v (m·s−^1 ) is
the relative particle-bubble velocity, C (particles·m−^3 ) is the particle concentration, and ps
is the probability of a particle remaining attached to a bubble.
Apart from the possibility of detachment of attached cells, this equation is almost
identical to the solution of Meier et al. (1999) for the potential- flow situation. Meier et
al. (1999), however, stated that potential flow is in general not applicable, since
surfactants are present that make the interface more rigid. Bavarian et al. (1991) use for
part of their study bubbles with diameters smaller than 1 mm in the absence of
surfactants, which in fact is in between the potential- and creeping-flow regime.
Furthermore, as discussed by Bavarian et al. (1991), for large bubbles in the presence of
surfactants a third hydrodynamic regime may occur referred to as large-wake
hydrodynamics (Andrews et al., 1988). In this situation cells may be captured in a wake
behind the bubble without being attached. In terms of the model of Meier this would
mean that the actual projected radius is larger than supposed or, in other words, cells that
would normally get passed the bubble are now also captured in the wake. Consequently,
effects of reactor height on the killing volume may become significant at lower heights.
All models assume that attachment of cells leads to cell death. As will be discussed
further on, this is likely to occur when the bubble breaks up at the surface. Cell death at
the surface would obscure any death of attached cells occurring during the rise of the
bubble. Assuming a medium viscosity of 10−^3 N.s.m−^2 , a rising velocity of 0.25 m·s−^1 and
a cell diameter of 15 μm a shear stress of 15 N·m−^2 can be calculated (Tramper et al.,
1986). This stress is of the order of magnitude to cause damage for exposure times of
minutes. Using a more detailed analysis Bavarian et al. (1991) estimated the maximum
shear on a cell is about 0.5 N·m−^2 during the rise period of 17–270 s. For these exposure
times this value is on the low side for shear stresses causing cell damage.
In conclusion, during rise of a bubble cells adsorb to its surface. In order to get
adsorbed, cells must be in contact with the bubble for a certain required contact time,
which depends on the type and concentration of protective additives used. Since cells are
likely to be killed at the surface, the killing volume may become height dependent. In the
absence of protective additives this dependence becomes only significant for reactors
Multiphase bioreactor design 472