Design of Fermentation Processes
379
which is excessively expensive when an almost quantitative conversion of substrate is desired.
The solution that gives minimum total residence time is, however, still the chemostat-plug flow reactor
combination discussed in Example 9.9.
r = 1 + 0.2068 = 1.207
(
5
)
The choice between two reactors in series (T =1.21 h) or one reactor (T = 1.26 hr) with a recirculation pump
is difficult. The first design is somewhat more complicated, while the pumping cost may be significant in
the second design
(R
= 2.5).__________________________________________________________________
In the present treatment of the tubular reactor, only the simplest form of the mass balance has been
used. Much more complicated forms are found in the literature. Thus, if a gaseous substrate is fed
in cross-flow (e.g., from a sparger mounted in the reactor axis), a transport term
kla(ci
- c,), where
is the interfacial saturation concentration of the substrate, appears on the right-hand side of Eq.
(9.54). Another term, the so-called axial dispersion term
D.
dz2
(9.84)
is also added on the right-hand side of Eq. (9.67). A theoretical foundation for this term can be
deduced for unaerated fermentations. With the generally low reaction rates of bioreactions, the fluid
has to move very slowly through the reactor and the velocity profile is probably parabolic
[v. =v
0
( l
- / - 2
/
R2
)]. Here radial concentration gradients exist, and through a perturbation solution
of the underlying parabolic partial differential equation that describes the steady-state profile in the
z and
r
directions, the influence of the radial transport is converted to a perturbation term like Eq.
(9.84) in a one-dimensional differential equation [see, e.g., Villadsen and Michelsen (1978)]. Here
DefT
is related to the molecular diffusivity of
c,
in the liquid phase. In the design of industrial plug
flow or loop reactors the overall mixing effect of gas bubbles, baffles, static mixers, and the like can
be described in terms of the axial dispersion coefficient Defr. Correlations for the effect of scale-up
on reactor performance are often based on the concept of axial dispersion as discussed in Chapter
11. One may, on the other hand, have some reservations concerning the widespread use of axial
diffusion terms in scientific studies of tubular (bio-)reactor performance. The simple model of Eq.
(9.67) brings forward the main aspect of the subject without obfuscation by the numerical and
conceptual difficulties tied in with the conversion of Eq. (9.67) from a first-order initial-value
problem to a second-order boundary-value problem. The dispersion model contains more
parameters and may give a better fit to a given set of laboratory data, but the theoretical basis of the
model is weak, and lack of detail in the transient description of the microbial kinetics easily
invalidates any attempt to give a physical interpretation of the model parameters.
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