Design of Fermentation Processes
361
A mass balance for cells at the hydrocyclone gives
v(l +
R)x = vxe +
R v x r
(9.42)
or
1
+ R = f + Rß
or
f = \- R ( ß - l)
The fraction of cells recirculate! to the reactor is
_
R
v x r
_ R/3
v(l + ii)x
1 +
R
and the cell concentration in the reactor is
(9.43)
(9.44)
X
=
\- R { ß -\)
(9-45)
Next consider the ultrafilter version in figure 9.4 B:
A mass balance for biomass is
Vft(s) x = VjX or ft
= v; /
V
=
(v —
v2) / V ~ D (1 —
Q)
(9.46)
where
D
is the filter separation factor defined as
v2
/v and
D = v/V
as usual.
It is seen that unit B can work at a dilution rate (1- D
) '1
higher than the ordinary continuous
stirred reactor for which
D(Q=
0) =
ft.
Since p is chosen to be the same with or without recycling
of biomass, the biomass productivity is increased by a factor (1 - J2)1. The biomass appears in a
usually small stream v t = (
1
-
33
) v, and with a much higher concentration
x
=
x(D=0)
(1-33
) 1
than
without recycling. The product is in a cell-free stream v
2
Also here the productivity has increased by a factor (1 - 32)'1
if the product from vt can be
retrieved. If the stream v, is regarded as waste the productivity of
P
is increased by a factor
'Q
(1
-3Q
)'1
We can shortly describe the functioning of the cell recirculation set up as follows:
(i and consequently the effluent concentrations of substrate and product, which are both functions
ofp alone are chosen by the designer of the reactor at values which are optimal by some criterion
-it may be that a low effluent s-value is needed, or the yield o f product may be small if s is too
high. To obtain a satisfactory volumetric production rate recirculation by either scheme A or
scheme B is chosen. The two design parameters
if
in A, and D in B) are chosen such that the
value of
D
(chosen to satisfy a given production in the reactor) matches the desired value of
ft.
In
both schemes the higher productivity is due to the higher cell density in the reactor. This result is
at least in theory independent of whether there is a loss of substrate due to maintenance.