time-averaged values of the eddy velocity. For eddies of scale much smaller than the primary eddies but
larger than the terminal eddies (of scale
the time-averaged velocity of the eddies is given by
where u(L) is the velocity for a length scale L.
The shear stress acting on a bubble is largely determined by the velocity of eddies of about the same size as
the bubble diameter. For
we therefore use
Inserting Eq. (2) in Eq, (3), we obtain Eq. (10.22) with
Description of energy input to a process by the theory of isotropic turbulence is valuable for understanding
bubble behavior in a bioreactor. The theory may also be used to predict the influence of energy input on the
fragmentation of hyphal elements and mycelial pellets. Local isotropic turbulence is, however, an
idealization not always realized in practice. Furthermore, in the derivation of Eq. (2), the energy dissipated
per unit liquid volume is used regardless of the means by which that energy is delivered (mechanical
agitation or injection of compressed gas). This is an idealization, but an acceptable approximation for many
of bubbles can be considered as a three-step process (Moo-Young and Blanch, 1981).
Bubbles occasionally come into contact with each other within the liquid phase. This
contact is characterized by a flattening of the contact surfaces, leaving a thin liquid film
separating liquid continuously decreases
approximately 10-6 cm (Tse
Finally, the film ruptures, which completes the coalescence process.
The entire process occurs in the milliseconds range, and the last step is practically instantaneous.
What determines if coalescence will occur, is therefore if the time constant of the second step is
smaller than the contact time of the bubbles. The rate of the second step is controlled by the
properties of the liquid film. In a multicomponent liquid phase, interaction between molecules of
different species leads to an enhanced concentration in the film layer of one or more of the soluted
species. This results in an increased repulsion between two bubbles and therefore in a reduced
coalescence. Especially with surface-active compounds, the enrichment in the film layer is
considerable, even for very small concentrations in the bulk liquid. The influence of inorganic ions
on the coalescence is illustrated in Fig. 10.5 where the mean Sauter diameter is shown as a function
of the ionic strength of the electrolyte solution. Alcohols and other organic compounds have a
similar influence on the coalescence. Here small alcohols are less efficient as coalescence reducers
than larger alcohols, e.g., methanol has less influence on the mean bubble diameter than octanol at
= p lu(L = d