All current mathematical models for biofilms assume that the electron donor, the electron acceptor, and all nutrients are transported to the biomass within the biofilm by diffusional processes alone. In addition, consideration must also be given to the transport of those constituents from the bulk fluid to the biofilm. In this section, we will examine these processes and the techniques used to model them. This examination will be limited to transport of a single electron donor, i.e., the substrate, to one type of biomass. It will be assumed that the electron acceptor and all other nutrients are provided in sufficiently high concentration in the bulk liquid so as not to be limiting within the biofilm.
Consider a flat plate covered with a base biofilm. If this plate is placed into a substrate solution, the concentration of the substrate at the surface of the biofilm will be less than the concentration in the bulk of the fluid because of the substrate consumption by the microorganisms within the biofilm. Furthermore, because of that consumption, the substrate concentration will continue to drop with depth in the biofilm. In order for the consumption to continue, substrate must be transported from the bulk fluid to the liquid-biofilm interface by molecular and turbulent diffusion. It must also be transported within the biofilm. As discussed above, although both diffusion and advection are involved in internal transport, the phenomenon is modeled as if it were due to diffusion alone. Nevertheless, the net effect of these events is to cause a substrate concentration profile that looks something like the one in Figure 15.4. In this situation, the observed substrate consumption rate depends on the rate of mass transport external to and within the biofilm as well as on the true, intrinsic substrate consumption rate of the biomass, i.e., the true reaction rate without any mass transfer limitations. Consequently, if one were to observe the substrate consumption rate of a biofilm as a function of the substrate concentration in the bulk liquid, it would differ from the intrinsic relationship between substrate consumption rate and substrate concentration that could be measured when the microorganisms were dispersed throughout the liquid phase (thereby eliminating mass transfer effects). Thus, the effects of mass transfer obscure the true reaction rate relationship in a biofilm and any attempt to model the situation without incorporating the effects of mass transfer would be futile.
External mass transfer is typically modeled by idealizing the substrate concentration profile in the bulk liquid as shown in Figure 15.7. The variation in substrate concentration is restricted to a hypothetical stagnant liquid film of thickness L> through which substrate must be transported to reach the biofilm. As a consequence, the substrate concentration throughout the remaining fluid, i.e., the bulk liquid phase, is constant. All resistance to mass transfer from the bulk fluid to the biofilm is assumed to occur in the stagnant liquid film.
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