Given the difficulties in modeling biomass growth in SSF systems even with simple empirical equations, it is not surprising that much attention has been paid to overcoming these difficulties, and that little attention has been paid to modeling the kinetics of product formation.
In fact, it may be very difficult to use other than simple empirical equations for product formation within a bioreactor model, especially for products like enzymes or secondary metabolites. The production of these products can depend on variables such as the rate of nutrient uptake. As argued in Sect. 13.2, nutrient uptake is often controlled by the rate at which the nutrient diffuses to the surface, and this can only be predicted by a model that describes intra-particle diffusion processes. Such models are too complex to include in fast-solving bioreactor models.
The empirical equation of Leudeking and Piret might be used (Ooijkaas et al. 2000):
where rP is the overall rate of product formation (kg h-1). YPX is the yield of product from the growth reaction (kg-product kg-dry-biomass-1) and mP is the coefficient for product formation related to maintenance metabolism (kg-product kg-dry-biomass-1 h-1).
In order to determine the yield and maintenance coefficients of Eq. (17.12) it may be necessary to fit an integrated version of this equation to the product profile, in a manner similar to that shown in Fig. 17.2. As in that case, it is necessary to substitute the integral and differential versions of the kinetic equation into Eq. (17.12). For example, with logistic growth kinetics, it is possible to derive the following integrated equation (Ooijkaas et al. 2000):
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