When Pwu is large, Xm{a is much larger than 1 + X0, 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, (P^ ~ Pq ) / s0Ysp < 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 Ya and vary with M due to maintenance demands, the mass balances in Eqs. (9.3) - (9.5) for the batch reactor are modified to x(t = 0) = x0 (9.12)

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 ms and mp (e.g., ms = 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 maintenance has little influence when the specific growth rate is high since the ratio between ms and ¡^ is the relevant parameter in the dimensionless substrate balance in Eq. (4) of Example 9.2. Consequently, it is in the continuous stirred tank reactor or in the fed batch reactor rather than in the batch reactor that one should be worried about maintenance losses (that is, if biomass and not the metabolite is the desired product).

Example 9.2 Effect of maintenance on distribution of substrate between biomass and product Assume that M = Mm»* throughout the fermentation. Whether this approximation is reasonable or not depends on the value of a = Ks /s0. If a is less than 0.01, virtually all biomass is formed with a constant specific growth rate. With the typical values for kinetic parameters and operating conditions used in Example 9.1, the maximum possible increase of biomass concentration is i^o = 2 g L" . When 98% of the available substrate has been consumed, S is still 20 times as large as a (i = 200mg L~l and Ks =10mg L1), and the approximation M=Mm** is reasonable. Whether M is constant or starts to decrease for smaller values of s has no visible influence on the total biomass yield, which can at most increase by a further 2%-and probably considerably less, since much of the remaining substrate goes to maintenance.

With substitution of for Ysx in the definition of the dimensionless variables given by Eq. (9.9) we have:

for which the solution is

X = X0 exp(Ö) = X0 expiez) S = l-jro(l + ô)[exp(0)-l]

where

Similarly, for a metabolic product,

Table 9.2. Changes in the distribution of a given amount of consumed substrate between biomass and producta for various values of mr._

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