Several characteristics common to many SSF processes have implications for the balance/transport part of SSF bioreactor models:
• The vast majority of SSF processes are batch processes. Therefore almost all the models that have been developed to describe SSF bioreactors are dynamic models. This means that equations are written as differential equations and therefore need to be solved by numerical integration. Pseudo-steadystate models, in which the differential term is equated to zero and which can therefore be solved algebraically, are only rarely used.
• The presence of moisture and thermal gradients within the bed in many bioreactors means that in most cases it is necessary to write the model equations with differential terms that include both space and time, that is, as partial differential equations. These partial differential equations make the model more difficult to solve (see Chap. 13.5), but they are unavoidable if it is intended that the model describe bed heterogeneity.
• Even when bioreactors are mixed, the mixing will typically not be perfect. It is possible to model the mixing of a bed of solid particles using "discrete particle models", which describe the movement of individual particles as a result of the forces acting upon them and track the position of a population of thousands of particles (see Fig. 8.10(a)). However, it can take days for the model to solve even with a number of particles much smaller than a bioreactor really contains. Even if a general circulation pattern can be assumed (Fig. 13.3(a)), it is not a simple matter to characterize the flow pattern and to express the effectiveness of mixing as a function of operating conditions. Therefore, in those bioreactors in which the bed is mixed, it is usual to assume that the bed is well mixed.
As a result of these complicating factors, especially the appearance of partial differential equations in models of heterogeneous beds, the following decisions are often made in order to develop models that can be solved by a desktop computer within seconds to minutes, that is, fast-solving models:
Well-mixed: Therefore uniform across the bed
Well-mixed: Therefore uniform across the bed
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