For designing bubble columns and airlift reactors the hypothetical-killing-volume theory
of Tramper can be used. The killing volume can than be split into a rise-dependent and
rise-independent part as in equations (14) and (15). The rise-dependent part will usually
be negligible. The relation between the rise independent killing volume and different
parameters is still not exactly known. From a mechanistic point of view it would be
logical to assume that cells in the bubble film and in a small liquid layer around the
bubble cavity are killed upon bubble rupture:
(30)with k (–) being a proportionality constant containing the fraction of cells killed and the
concentration factor, A (m^2 ) the surface area, and h (m) the film thickness, where cell
death occurs. The surface area is proportional to the square of the bubble diameter
meaning larger bubbles are more detrimental. On the other hand the forces associated
with cavity collapse, which enter the above equations in the fraction killed and the
thickness of the layer at the cavity wall, increase exponentially with decreasing bubble
size. Thus, below a critical bubble diameter smaller bubbles can be more damaging.
However, experimental proof for this is limited to one observation by Handa et al.
(1985). In addition to a decrease in bursting forces, the proportion of the bubble that is
above the surface at rupture will increase with increasing bubble diameter. Based on this
a proportionality of the rise-independent killing volume to the square of the bubble
diameter seems more logical than to the third power. As long as exact information on the
mechanism is lacking we just assume that cells in a layer of thickness h around a bubble
are killed. This results in
(31)From this equation it can be seen that for larger reactors the rise-dependent killing
volume becomes relatively more important. The volumetric oxygen demand of the cells
determines the minimal amount of sparging that is required:
(32)Where qo (mol.cell−^1 .s−^1 ) is the specific oxygen consumption rate, kol (m.s−^1 ) is the mass-
transfer coefficient, A (m−^1 ) is the specific surface area, C*ol (mol.m−^3 ) is the saturation
concentration of oxygen in medium, and Col (mol.m−^3 ) is the oxygen concentration in
medium. For bubble columns A is given by
(33)
Combining equations (32) and (33), the gas flow rate required for a given cell
concentration is given by
(34)
Multiphase bioreactor design 484