## Modeling Particle Size Changes

Particle size reduction may occur as a result of the growth process. For example, if the microorganism degrades a polymer that is responsible for the particle structure, then the particle size will decrease. It may be of interest to describe the reduction in particle size within the bioreactor model. For example, the reduction in particle size might be used to estimate the decrease in overall bed volume.

To date the modeling of particle size reduction has not been done in association with bioreactor models, although there is no reason why it could not be done.

17.3.1 An Empirical Equation for Particle Size Reduction

Nandakumar et al. (1994) derived an equation for the length of the residual substrate particle flakes. The equation was derived on the basis of the assumption that the consumption of the substrate particle at the substrate/biomass interface was limited by diffusion of O2 through the biomass film, which was assumed to grow in such a manner as to keep the overall particle size constant. Their assumptions are unlikely to be true in practice and therefore the equation should be considered as empirical. Other empirical equations might also adjust well to experimental data. The equation was (Nandakumar et al. 1994):

where L is the initial particle length, lc is the residual particle length at time t, and T is the time for complete particle degradation. Based on their model, they showed how T could be expressed in terms of fundamental constants, however, in practice they determined T by regression of Eq. (17.10) against experimental data for residual particle size versus time. This regression can be done by treating t as the dependent variable and the fractional particle length (X=lJL) as the independent variable, in which case the equation is:

To measure residual particle size experimentally, it is necessary first to remove the biomass layer. Nandakumar et al. (1994) simply sieved their wet fermented substrate. The process organism was a bacterium, and they claimed that it was easily removed during the wet sieving. Such studies would be more difficult with a fungus, which would bind more tightly to the residual substrate particle.

17.3.2 How to Model Particle Size Changes in Bioreactor Models?

As noted above, it may be interesting to model particle size changes in bioreactor models in order to predict changes in the overall bed volume. This requires an understanding of various factors:

• degradation of the residual substrate particle;

• expansion of the biofilm, and how this is affected by reduction in the size of the residual substrate particle;

• interactions between particles caused by biomass and gravity and how this affects bed structure.

Particle degradation has not yet been taken into account in bioreactor models in SSF. More understanding of the phenomenon and how it affects the bed structure is needed before it can be incorporated into a model in a meaningful way. A useful model would not only predict changes in bed volume but also in the bed porosity. Bed porosity is important for bioreactors that are forcefully aerated, since it is one of the factors that determine the pressure drop through the bed.

Such models may or may not attempt to model the development of the biomass structure above the particle surface. The work of Rajagopalan et al. (1997) gives an idea of how this might be done, at least for a residual substrate particle of constant size. They modeled the expansion of a biofilm of constant density, for a single particle. Applying such a model to the situation in a bioreactor would be more complex since the spatial distribution of particles would prevent the biomass from expanding freely. Such a model could predict the filling in of void spaces, and therefore would be useful for predicting changes in bed porosity.