ABSTRACT
For type II models, the specific rate σ of substrate utilization is directly proportional to the specific rate of product formation,
σ = a (12.1)
where a is a positive stoichiometric constant while µ and are allowed to have general dependence on S and P . It can be noted that when both the maintenance terms ms andmp of the previous type I model (Equations (11.411.5)) are negligible then type I model becomes a special case of type II model, since in this case the rates σ and are related by
σ = Y xs
Yxp (12.2)
Type II models are, therefore, more general since independent expressions can be assumed for µ and . The model Equations (11.1-11.3) can be suitably rendered dimensionless by introducing the following variables,
S¯ = S
Sref , X¯ =
aX
Sref , P¯ =
P
Pref , D¯ =
D
µref (12.3)
t¯ = tµref , µ = µref µ¯, = ref ¯, λ1 = ref µref
λ2 = refSref aµrefPref
(12.4)
where Pref , Sref , ref , and µref are reference terms for P , S, , and µ, respectively. The model in dimensionless form is given by
dX¯
dt¯ = µ¯X¯ − D¯X¯ (12.5)
dS¯
dt¯ = D¯(S¯f − S¯)− λ1¯X¯ (12.6)
dP¯
dt¯ = λ2¯X¯ − D¯P¯ (12.7)
We assume that the rates µ¯ and ¯ are given by the following forms:
µ¯(S¯, P¯ ) = µ¯1(S¯)µ¯2(P¯ ) (12.8)
¯(S¯, P¯ ) = ¯1(S¯)¯2(P¯ ) (12.9)
As was mentioned in the previous chapter, this class of models was used in the literature to model a number of bioprocesses such as ethanol fermentation of cellulose-hydrolystate [126, 272, 357].
