C

kAparticle( a w(particle)-aw(air)) aw(air)

equivalent to a single pseudo-homogeneous phase with average properties of the solids and air (heat capacity, density etc).

Fig. 13.3. The consequences, for model complexity, of a desire to model all the macroscale phenomena. (a) If the bed is mixed but is not well mixed, it is a complex matter to model the flow patterns. In the example on the right there is a region of circulating particles and a region of relatively-static particles. (b) Inclusion of the bioreactor wall as a separate subsystem complicates the model, as shown on the right. (c) If the substrate and air phases can be assumed to be in equilibrium, this simplifies the description of the system, as shown on the left surr

• To neglect the bioreactor wall. SSF bioreactors commonly contain several phases that could be treated as separate subsystems (Chap. 2). Equations must be written that describe the changes within each of these subsystems and the exchanges between subsystems. In order to reduce the overall number of equations, it is often decided not to treat the bioreactor wall as a separate subsystem (Fig. 13.3(b)). It may be simply ignored or it may be lumped together with the substrate bed. This removes the ability of the model to describe the changes in the temperature of the bioreactor wall, which might in fact have important influences on the process. Even when the bioreactor wall is recognized as a separate subsystem, typically it is assumed that the whole bioreactor body is at the same temperature, in order to not have to use partial differential equations to characterize the temperature gradients within it.

• To treat the substrate bed as a single pseudo-homogeneous phase. The substrate bed is often treated as though, at any particular point in the bed, the air and solid at that point were in equilibrium. The advantage is that in this case it is not necessary to describe the solids and inter-particle air as separate phases. Rather, the bed can be treated as though it were a single phase with the average properties of the air and solid (Fig. 13.3(c)). If this is done, the air and solid phases are assumed to have the same temperature and it is possible simply to write an equilibrium relationship to relate the air humidity with the temperature and the water activity of the solids. Of course, the suitability of this simplification depends on whether in actual practice this equilibrium is approached. The alternative is to treat the moist substrate particles and the inter-particle air as separate phases. This implies that equations must be written to describe heat and mass transfer between these two phases. Further, it implies that the solids-to-air heat and mass transfer coefficients must be determined.

• To limit the number of key state variables. It is possible to simplify the model by minimizing the number of macroscale state variables it describes. For example, the simplest models concentrate only on the substrate temperature, assuming that water levels are automatically controlled within the bioreactor. Some models include both energy and water balances. In some cases an O2 balance might also be done. Of course models that contain balances for all three quantities (energy, O2, and water) will be most flexible in describing what controls the rate of growth under a wide range of operating conditions.

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