As pointed out in Chap. 12, a mathematical model of an SSF bioreactor requires two sub-models, a sub-model that describes the growth kinetics of the microorganism and a sub-model that describes the energy and mass balances and transport phenomena. Each of these sub-models is written at an appropriate level of detail, depending on what simplifications and assumptions have been made. Chapter 13 argued for the use of simple empirical equations within the kinetic sub-model, in order not to make it too difficult to solve the bioreactor model. Chapters 14 to 17 address various questions related to the establishment of kinetic sub-models of this type (Fig. 14.1).
The aim is to write a kinetic equation in which the change in the amount of biomass, or a variable associated with it, is described by a differential equation, with the parameters of this differential equation taking into account the effect on growth of the key state variables that will be included in the bioreactor model, such as the temperature and water activity of the substrate bed. This is achieved as shown in Fig. 14.2. Note that the experiments done for the purpose of selecting the kinetic equation should be done after some efforts have been made to find a medium on which the organism grows well and to identify the optimal environmental conditions. This book does not address the optimization of the medium and environmental conditions (see the further reading section at the end of this chapter). A kinetic profile is constructed by measuring the biomass, or some indirect indicator of the biomass, in samples removed over the time course of the fermentation (Fig. 14.2(a)). Various kinetic equations are fitted to the data by regression and the one that fits best to the data is selected. Later, experiments are done in which different environmental conditions are imposed, such that, after analysis of the growth profile in each condition, plots can be made that relate the parameters of the kinetic equation to the environmental variable (Fig. 14.2(b)). Each kinetic parameter will then be expressed as an empirical function of the environmental parameter.
The current chapter covers some of the issues that must be addressed before beginning the process of kinetic modeling and then goes on to explain how the basic kinetic equation is selected.
• adjust the curve by regression, giving the parameters of the equation
Kinetic sub-model to be incorporated into the bioreactor model
• write the growth equation in differential form
• incorporate the effects of environmental conditions on growth ^
• relate other phenomena to growth (See Chap. 17)
Fig. 14.2. The traditional approach to establishing the growth-kinetic equation. (a) A growth curve is established under optimal conditions and an empirical kinetic equation is selected that describes the curve well. (b) The parameters of the equation are expressed in terms of key environmental conditions (e.g., temperature) by repeating the growth curve experiment in different conditions, determining the growth parameters for each curve, and expressing the parameters as empirical functions of the environmental variable
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