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It is now only necessary to specify one kinetic equation, which can be for qs of p. In practice the Monod kinetic equation for i (p = pmCs/(Cs + Ks) is often chosen. This choice leads, however, to a very nasty inconsistency. According to Monod, l = 0 if the concentration of substrate Cs becomes 0. Substituting this result in the Herbert-Pirt equation shows, however, that (forp = 0) qs = ms. The inconsistency is then that in the absence of substrate (Cs = 0, p = 0) there is still consumption of substrate for maintenance (v = — qs = ms); this is very strange indeed. These problems can be avoided in a most simple way by introducing the kinetics of the substrate consumption rate in the most simplistic way (a Michaelis-Menten type of relation, see Figure 8):

The kinetic relations 14a and 14b contain four model parameters: Fi^X, Ks, ms, and qDax. Figure 8 shows howi and qs depend on Cs.

Clearly, there is a maximal substrate uptake rate qDax, but also for i a maximal value (umax) is seen at high Cs. The value of imax follows from equation 14b by taking the and the result is limit Cs > Ks

In addition, it is clear from Figure 8 that there is a minimal substrate concentration (Cs,min) at which i = 0. For Cs < Cs,min, the growth rate l becomes negative. An expression for Cs,min can be found from equation 14b by putting i = 0, which results in

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