The removal of substrates and other materials from solution in a fixed-film system involves complex physical and chemical processes which include: film transport, pore surface transport, adsorption reaction, and growth of biofilm and suspended biomass (Thacker et al., 1981; Weber and Chakravorti, 1974). Expressions derived from biofilm kinetics are best solved by simpli­fication by using empirical approaches. The mechanism of substrate removal by fixed-biofilm is shown in the simplified model of biofilm-on-inert-media in Fig. 15.11. The complexity of resolving mathematical equations expressed by the biofilm model lies in recognizing the critical features, namely:

(i) The moving boundary problem

The biofilm thickness (Lf) can change while substrate diffuses into the biofilm during transient-state when the rate of substrate utilization is

not constant. The moving boundary problem is analogous to the problem of heat transfer during freezing and melting of ice in ther­modynamics (the “Stefan” problem, Danckwerts, 1950), and the diffu­sion of oxygen in absorbing tissue (Crack and Gupta, 1972); and (ii) Diffusion with nonlinear reaction

At any particular time, the rate of substrate diffusion across the liquid/ biofilm interface as described by Fick’s law, is equal to the rate of substrate utilization in the biofilm governed by the nonlinear Monod kinetics.

Figure 15.11 represents removal of dissolved species y in a multispecies biofilm contain microbial cells x, where x and y are vectors of cell types and dissolved species. In the above example, an organic compound P is used as primary carbon source to support a culture containing metal reducers XE and the organics degrader Xp. In this particular system, metabolites U are produced inside the biofilm to feed the metal reducers. The metal reducers are assumed to be unable to grow on the primary supplied carbon source P.