428
Chapter 10
(ca ~ ca ) =
( C A,inla
C A
)
A,outlet
C A
)
^ ^ A . i n l e t ~ C A
) —
^ ( C A.outlet ~ C
'a )
(
10
.
8
)
Example 10.2. Requirements for
ha
in a laboratory bioreactor
We now return to the experimental data analyzed in Example 3.5. We want to calculate the minimum value
of
h a
needed to maintain the dissolved oxygen concentration in the medium above a desired value. Since
the dissolved oxygen concentration in aqueous solutions is very low, the oxygen consumption
approximately equals the mass transfer of oxygen at steady state, i.e.,
~<lo=<lo = k ia(c0 - co)
(1)
or
kta =
(co
co
)
(
2
)
If the dissolved oxygen concentration is to be kept at 60% of the saturation value obtained by sparging with
air containing 20.95% oxygen, we find
(co ~ co)
= c o(1 - 0-6)
(
3
)
and if the bioreactor is operated at 1 atm, we find from Table 10.1
*
0.2095
,
co =
790(T = O'265' 10
moles L'
(4)
Inserting this value in Eq. (3) and using the calculated value of
-q0
in Example 10.1, we find by using Eq.
(
2
)
kp
= 748 h ‘ = 0.208 s'1
(5)
This
k/a
value can normally be obtained in a well-stirred laboratory bioreactor. For larger stirred tank
bioreactors, it may be difficult to obtain a
ha
value above 500 h'1
and oxygen limitation may become a
problem._________________________________________________________________________________
10.1.1,
Models for ft/
The idea of expressing the mass transfer across surfaces by an overall mass transfer coefficient
multiplied by a concentration difference is clearly a simplification of a complicated physical reality.
It is for many reasons desirable to find a relation between the mass transfer coefficient and known
physical variables such as e.g. the diffusivity of the solute. The first model of this kind was due to
Nemst (1904), who assumed that mass transfer occurred by diffusion through a stagnant film.