Mass Transfer
469
With slab geometry and first-order kinetics, the overall effectiveness factor can be calculated by
d>
o
2
-------------
1
------_
tanh(<t>)
Bi
rji
1
O
2
Veff
Bi
(10.58)
where
Bi
is the so-called
Biot number fo r mass transport
which expresses the ratio between the
characteristic film transport rate and the characteristic intraparticle diffusion rate. For slab geometry
Bi =
k.L
D
(10.59)
A,tff
Eq. (10.58) may approximately be applied to other geometries as well (by defining A = V/A as
described for the generalized Thiele modulus). For a first order reaction we have
O
2
A
k,
Bi
ks
(10.60)
Clearly, this term is related to the external mass transfer, and it will only be of importance if the rate
of reaction is large compared to the mass transfer rate. In the absence of intraparticle mass transfer
resistance in a porous particle there is no reason to consider an effect of external resistance.
The mathematical treatment of both external and internal mass transfer is described extensively in
chemical engineering textbooks due to their importance in traditional catalysis (see e.g. Levenspiel,
1999 or Fogler, 1999). For the purpose of immobilized cells, however, a detailed quantitative
analysis of effectiveness factors serves little purpose. The kinetic expressions are often derived for
suspended cell cultures and may therefore not be accurate, or even adequate, for immobilized cells.
It is, furthermore, very likely that major changes in the metabolism sets in if oxygen depletion
occurs inside the pellet (see also discussion in Section 11.4). The main reason for still taking the
trouble of analyzing mass transfer processes into pellets is to make sure that the pellet size is small
enough to avoid severe mass transfer limitations. For that purpose approximate kinetic expressions
will do the trick.
PROBLEMS
Problem 10.1 Determination of
kp
in a pilot plant bioreactor
The aim of a study of penicillin fermentation by Christensen (1992) and by Pedersen (1992) was to set up
models for the microbial and reactor dynamics. A major purpose of this study was to see how mass transfer
affects the overall performance of the process. A series of experiments was therefore designed for this
purpose. The bioreactor can be assumed to be ideal, i.e., there are no concentration gradients in the medium.
It is assumed that process air has the composition 20.95% Cb, 0.030 % CCb, and 79.02% N2. The total
volume of the bioreactor was V = 15 L. The equilibrium concentration of 0
2
in solutions of sulphite or
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