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Since the bed is not well mixed, conditions such as bed temperature are different at different positions in the bed, affecting /Umax, such that:

• the same set of microscale equations must be applied to each different position within the substrate bed.

• the solution of the microscale equations will be different for each position within the bed

Fig. 13.1. The consequences, for model complexity, of a desire to model growth as a function of the concentration of a soluble nutrient. (a) A well-mixed submerged-liquid fermentation; (b) Solid-state fermentation in a static substrate bed

13.2.2 Is It Worthwhile to Describe the Spatial Distribution of the Biomass at the Microscale?

Modeling the spatial distribution of the biomass could potentially bring benefits. For example, a model that described how the microorganism grew into the inter-particle spaces could be used to predict how the pressure drop through the bed would change during the fermentation. However, any attempt to describe the spatial distribution of the biomass at the microscale will greatly increase the complexity of the model. Even for the relatively simple situation involving the growth of biofilms it would be necessary to describe the three dimensional arrangement of the particles. In fact, many SSF processes involve fungi, and it is not a simple matter to describe their growth in three dimensions (Fig. 13.2). One way to do this would be to describe the extension and branching of individual hyphae, however, it would be necessary to know the statistical distributions of branch frequencies and branch angles and a very large number of individual hyphae would need to be described. Furthermore, in an agitated bed, the effect that agitation would have on the spatial distribution of the biomass is not sufficiently understood to be able to describe it mathematically.

As a result of these complexities, and our current poor quantitative understanding of them, typically the biomass concentration is simply treated in bioreactor models as a global average over the particle, being expressed with units of "g dry biomass per g of dry solids".

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