348
Chapter 9
0 =
(Po+Xm - X
o)
l + -r„ + a
J x _
-*■„(!+ -Ï.)
U »
a
ln(X0 + l~X )
x ^ - l- x 0-a
J (X^ -X)
j
- I - * o )
l ( ^ - ^ o ) J .
When
pnn
is large,
Xm<a
is much larger than 1 +
X
0, the largest value obtained by complete conversion of
substrate to biomass. In this case the inhibition term is negligible for all
t,
and Eq. (15) degenerates to Eq.
(9.10). If, however,
(Pnax
~
Pq
)
I SqYsp
< 1, the fermentation stops when
X = X ^
and the corresponding
biomass concentration
x
can be found from Eq. (9.7a) with
p = p ^ .
The two examples illustrate that an analytical solution of the mass balances is possible for a number of M
expressions of the form of Eq. (1). The analytical solution has several appealing features; the influence of
key parameter combinations is emphasized, and limiting solutions are often extractable. The quest for an
analytical solution should, however, not be pursued too ardently, because standard computer programs
easily find the solution to the mass balances just as accurately as insertion in the final expressions.
Furthermore—as will become evident in most of the examples of this chapter—it is rarely possible to obtain
an analytical solution at all.__________________________________________________________________
When the yield coefficients
and
Yv
vary with
M
due to maintenance demands, the mass balances
in Eqs. (9.3) - (9.5) for the batch reactor are modified to
dx
i r
^
*
o
X
II
©
II
(9.12)
^ = - < C V + '" . ) * ;
at
©
N
S'
II
(9.13)
^ = ( C “> + " 0 * ;
dt
v
M
p}
p(t
= 0
) = p 0
(9.14)
where the maintenance kinetics of Eqs. (7.24) and (7.25) have been inserted for
rs
and
rp.
Quite
often there is a simple relation between
m,
and
mp
(e.g.,
rns
=
mp
for homofermentative, anaerobic
lactic acid fermentation, since a given amount of sugar used for cell maintenance is retrieved
quantitatively as the product, lactic acid).
It is not possible as in Eqs. (9.6) - (9.8) to rewrite Eqs. (9.12) - (9.14) as one differential equation in
x and two explicit expressions for
s
and
p.
Certain important results concerning the total yield of
biomass that can be obtained from a given amount of substrate can, however, as illustrated in
Example 9.2, be calculated approximately. One qualitative result of general validity is that
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