Mass Transfer
457
surface or a pellet surface;
cA
is the concentration in the bulk liquid.
ks
is the mass transfer
coefficient, which is a function of the physical properties of the considered system and of the liquid
flow rate around the pellet. For pellets in suspension the value of
ks
can be estimated using
correlations of the type given by Eq. 10.28.
A certain concentration difference between the bulk phase and the pellet surface is obviously
necessary to obtain a mass transfer of substrate to the pellet, and the reaction will thus take place at
a slightly lower concentration than the bulk phase concentration. The question is now how much
lower than the bulk concentration the surface concentration will be, and how that in turn will affect
the conversion rate. Consider a system where cells are present only at the surface of a pellet. To
emphasize that A is a substrate, its concentration is denoted
sA
in the bulk liquid phase and
sAs
at the
pellet surface. The mass balance for the transport and reaction of the substrate at the pellet surface
is given by
where
-qA
is the volumetric substrate consumption rate. At steady state and assuming Monod
kinetics we get
Note that the biomass concentration should be given with respect to the same volume as
a,
e.g.
g (L medium)'1.
Equation 10.36 can be rendered dimensionless by the following substitutions:
(10.35)
k M
S A ~ S aJ
= YxsH
(10.36)
s
Y u
x
_
-tj “ max
(10.37)
giving
1
- S A =
Da
SA+a
(10.38)
Da is called the
Damkohler number.
The solution of the algebraic Eq. (10.38) is
(10.39)
where