460
Chapter 10
S/i = 2 + 0 .1 3 G r^ S c 3
(6)
Suppose the mass transfer coefficient for glucose is needed, and that the solution is very dilute. We have
DK
lllC03e = 0.69 10'9 m
2
s ' (at 25 °C). Assuming a viscosity and density of 10
3
kg m
'1
s
1
and P = 1000 kg m
'3
(close to water) we get
lCT
3
Sc
= -------—
----------= 1449
0.69 10
"9
-1000
(7)
Inserted into Eq.
6
, we get
Sh ~
60 and
,
60* 0.69 10
“9
k ,
= — -
--
-
-
-
-
-
-
--
-
-
-
-
-
-=2*10
J
2
- i
0 ”3
(
8
)
For a medium containing filamentous fungi, the viscosity is normally much higher than 10° kg m
1
s ', and
for these systems the Grashof number may be smaller than 18. The above described correlations hold only
for suspended pellets. For, e.g. packed beds, other correlations need to be used (see, e.g., Moo-Young and
Blanch, 1981)._____________________________________________________________________________
10.2.2. Intraparticle Diffusion
When microbial cells (or enzymes) are present within a pellet, the substrates have to be transported
not only to the pellet, but also into the pellets, and metabolic products formed by the cells have to
be transported out of the pellet. The convective transport inside a pellet is normally small, and the
transport can therefore be well described by diffusion only. For diffusive transport to occur, a
concentration gradient is needed, i.e. the concentration of a substrate inside the pellet must be
smaller than at the surface for any transport to take place. The cells placed close to the center of a
pellet will therefore not be exposed to the same substrate concentration as its more fortunate
relatives located close to the pellet surface. If the substrate concentration difference is too large,
starvation for carbon source, or anaerobic conditions, may occur in the center1. Again, it is
important to analyze and compare the rate of reaction and the rate of the transport processes. The
transient mass balance for substrate A can be written
^ =
D A eg V sA
+
qA
(10.41)
where
DA
eff
is the so-called
effective dijfusivity,
^ is the Laplacian operator, and
qA
is the
volumetric reaction rate. (Note that the volume is here the pellet volume).
1
The fate of the cell at the pellet cotter can be compared to that of an unfortunate dinner guest placed at the very end of a long table.
In the worst-case scenario, the plates may well be emptied before they reach the end of the table, particularly if all the dinner guests
are very hungiy.
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