## The Kinetic Submodel Is Based on a Differential Growth Equation

Chapters 14 and 15 have shown how experiments should be done in order to select an appropriate empirical equation to describe the growth kinetics. This involves working with experimental growth curves and fitting the integrated form of the appropriate kinetic equation, which could be, for example, one of the equations from Table 14.1. However, the integrated form is not appropriate for direct incorporation into the bioreactor model. This can be understood by considering a simple model such as that shown in Fig. 12.4. In the logistic equation used to describe the growth kinetics in this model, the parameter n is expressed as a function of temperature. If the integrated form of the equation were to be used (Eq. (14.6) in Table 14.1), then n would have to be maintained constant, which is not consistent with the fact that the temperature and therefore |i vary during the fermentation. On the other hand, this does not present any problem for the numerical integration of the differential equation (Eq. (16.3) in Table 16.1), since n can take on a new value for each step in the integration process.

The current chapter concentrates on how the differential form of the kinetic equation is incorporated into the bioreactor model. The kinetic sub-model expresses the various parameters in the growth equation as functions of the local conditions: This is the link that allows the bioreactor model to describe how growth is restricted by poor macroscale transport, since such transport limitation will lead to unfavorable local conditions for growth. The manner in which this is done is covered in the current chapter. The question of how to describe the manner in which growth in turn affects the local conditions is considered in Chap. 17.

Note that this chapter and the next consider growth in terms of the dry biomass itself. However, if the kinetic equation is determined in terms of a biomass component, the same considerations can be applied. Of course, the units must be changed appropriately. For example, biomass has the units of g-dry-biomass g-substrate-1. If glucosamine were used, it would be necessary to write a term for it with units of mg-glucosamine g-substrate-1, and this will affect the significance and units of other parameters, such as yield coefficients.