2 mm and 3 mm. Their geometry is seldom a perfectly defined sphere, contrary to the
usual assumption; instead, flat cylinders and ellipsoids are the most common shapes and
floc dimensions and shape are of capital importance to the assessment of De. Particularly,
if the aggregates are assumed to be spherical or cylindrical, the determination of their
diameter is crucial for the final result (Hamdi, 1995). One of the best ways to do this non-
destructively is by image analysis. Such a technique applied to yeast flocs has been
developed by Vicente et al. (1996), where both the floc size distribution of different
populations and the number of flocs present in the treated samples have been determined
by fitting a Gauss curve to the experimental data. From there, the values of the average
floc size and their respective standard deviation can be calculated.
Most studies use the second approach mentioned above to calculate De. Some authors
use non-reactive solutes: Libicki et al. (1988) calculated the effective diffusivity of
nitrous oxide, a non-reactive solute, within cell aggregates of Escherichia coli. The
effective diffusivity was found to decrease with increasing cell volume fraction. Other
authors use inactivated cells, as is the case of Ananta et al. (1995), who measured the
oxygen transfer characteristics of aggregates of Solanum aviculare with 3 mm to 12.5
mm in diameter. Effective diffusivity of oxygen in deactivated aggregates was found to
increase with particle diameter varying between 2% and 40% of the molecular diffusivity
of oxygen in water at the same temperature. The authors considered, therefore, that
severe oxygen limitations occurred in the aggregate; nevertheless, one should pay
attention when interpreting the results, since the measurements were made with
inactivated cells, which may have a different behaviour from that of active cells.
Vicente et al. (1998) studied mass transfer characteristics (effective diffusivity, De,
and external mass transfer coefficient, Kc) of glucose and oxygen in flocs (0.90 mm to
2.42 mm in diameter) of S. cerevisiae using inactivated cells but a different technique. A
modified diffusion cell (Figure 13.3) (Vicente et al., 1997) was used in order to avoid
floc destruction. De and Kc were calculated using two methods: a classical one, based on
analytical solutions of Fick’s law of diffusion and a numerical one, based on general
mass balances of a component in flocs and bulk phase. Diffusion coefficients were found
to be, for glucose, 17% of the diffusivity in water and, for oxygen, between 0.2% and 1%
of the diffusivity in water, which is in agreement with the data from Ananta et al. (1995)
if the size is considered. Kc values increased with the agitation rate, as expected, and have
values which range from 7.5×10−^9 m·s−^1 to 15×10−^9 m·s−^1. These values indicate that not
only the mass transfer inside flocs, but also the one outside them, may be a limiting step
in this process.
It is possible to calculate a floc critical diameter, defined as the diameter at which
solid phase diffusion limitations become more important than liquid phase diffusion
limitations (Hamdi, 1995). A floc diameter greater than its critical value can have
consequences such as the presence of useless biomass inside the reactor, undesirable
metabolite production by inactive biomass, or changes in chemical and biochemical
characteristics of medium and microorganisms. The author, therefore, suggests a
continuous disintegration of the flocs into a smaller size. In the same line, Webster (1981)
developed criteria allowing the assessment of the importance of substrate diffusional
limitations within cell flocs, depending on the rate law used to describe substrate
consumption.
Multiphase bioreactor design 400