## Net Accumulation of Biofilm

As shown in Figure 1, development of a bacterial biofilm comprises the following fundamental processes: (1) substratum preconditioning by adsorption of fluid-phase organic molecules; (2) bacterial cell deposition to the conditioned substratum (cell transport to the surface, cell adhesion to the substratum, cell desorption); (3) bacterial metabolism (cell substrate conversion; cell growth and replication; extracellular exopolymer production; cell starvation, death, lysis); (4) biofilm removal (cell and biofilm de

tachment, biofilm sloughing). Naturally, the relative influence of each process to the overall rate of accumulation is dependent upon the specific system, the prevailing environmental conditions, and biological changes throughout the lifetime of the biofilm.

Simple unstructured models of biofilm formation can be written. Biofilm formation is assumed to occur on surfaces exposed to a well-mixed bulk fluid phase, thus tacitly neglecting any concentration gradients in the bulk liquid. Any mass transfer limitations in the bulk liquid will only complicate the already difficult task of determining the kinetics of a heterogeneous reaction. Suspended biomass in the bulk liquid arises due to either cell growth and replication or to detachment of biofilm material. Planktonic cells leave the liquid phase by either the effluent liquid leaving the reactor volume or by cell deposition onto the reactor surfaces. A single, sterile growth-limiting substrate, S, enters the reaction volume, where it is consumed by either the suspended or biofilm-bound cells. Any reasonable measure of both biofilm and suspended cell concentrations is acceptable (cell number, biomass dry weight, biomass organic carbon). However, to complete material balances over the entire system, one most know the stoi-chiometric relationship between changes in the suspended and attached cell concentrations due to growth and the limiting substrate utilized.

Based on these assumptions, material balances for both the suspended cell biomass and single growth-limiting substrate can be written as equations 10 and 11:

Suspended biomass balance: VdX/dt =

FX (Rg V)suspended + RdetA RdepA (10)

where X is the suspended cell biomass (ML-3); S is the limiting substrate concentration (ML-3); Vis the reaction volume (L3); (Rg)suspended is the rate of suspended cell growth (MbiomassL—3t—1); Rdet is the rate of biofilm detachment (MbiomassL—2t— 1); Rdep is the net rate of bacterial cell deposition at a substratum (MbiomassL —2t—1); (Rg)biofilm is the rate of biofilm cell growth (MbiomassL—2t— 1); Y is the yield coefficient (Mbiomass/Msubstrate); and A is the reaction area (L2). Rate expressions for the process rates (Rg, Rdet, Rdep) may depend directly on such changing parameters as biofilm amount, limiting substrate, and suspended biomass concentration.

Biofilm net accumulation can be described by equation

Biofilm net accumulation:

Because rate expressions for both planktonic and biofilm growth are nonlinear saturation kinetic functions of the instantaneous substrate concentration and first-order functions of Xand B, respectively, equations 10-12 are coupled, nonlinear ordinary differential equations that require either simultaneous numerical integration (for the transient situation) or simultaneous algebraic solution under steady-state conditions (i.e., dS/dt, dX/dt, and dB/dt = 0). Numerous biofilm accumulation studies have been carried out that can be mathematically described by this set of equations.

Should the reactor be operated at a hydraulic residence time shorter than the generation time of the cells, then one can assume the term (RgsV) is negligible; however, the corresponding term in equation 11, representing suspended cell substrate metabolism (RgsV/YX/S), cannot be disregarded unless in the specific system this term is determined to be small compared to the other terms, or substrate is not supplied.

In a simple bacterial adhesion study, experiments can be carried out without exogenous substrate, thus the Rg terms in equations 10-12 could be ignored. Thus, equation 12 becomes dBdt - Rdep - Rdet

and is valid provided the adherent bacterial replication is zero.

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