The generic solution of the above system is provided by a diffusion-reaction model in which metal removal from the bulk liquid may be mass transport limited during high biomass output and reaction rate limited during inhib­ited growth. The mass balance across the bulk liquid compartment (Fig. 15.11a) is given by the following equation:

15.8

15.9

where rfl = dissolved species removal rate (ML^T1); ju = Dwidujdz, dissolved species flux rate (ML^T-1); u = {P, U, C}, is the vector for dissolved species concentration, where P = phenol concentration (ML-3), U = metabolites concentration (ML-3), and C = metal pollutant concentration (ML-3); rXB = cell growth rate in the bulk liquid (ML-2^1); x = {Xp, XE, X/}, is the vector of biomass concentrations, where Xp = viable cell concentration (ML-3) of organic compound degraders, XE = viable cell concentration (ML-3) of toxic metal reducers, and Xj = inert cell concentration (ML-3); bx = {bxp, bxE}, is the vector of cell death coefficients (T^1) for the respective cell types; and X = {V, XE, X/}, is a vector of cell detachment rate coefficients (7"1). The reaction rate and cell growth terms can be formulated based on the enzy­matic removal of substrates as shown in the derivation of Equation 15.5 and the Monod kinetics, respectively.

Biofilm mass balance

The removal of the dissolved species and cell growth is represented by a set of diffusion-reaction partial differential equations (PDEs). The PDEs represent a mass balance across an infinitesimal biofilm section (Sz) parallel to the substratum surface (Fig. 15.11b):

15.10

15.6 where: jx = Dwx3(x)/3z, mass flux rate of biomass (ML-2^1), ruf = the vector of removal rates of dissolved species in the biofilm (ML-3^1), rxf = the vector of biomass production rates in the biofilm zone (ML-3^1), and є = is a biofilm porosity constant (Vfvods/Vftotal). The movement of cells across the biofilm is induced by physical displacement due to growth whereas dis­solved species are transported by diffusion. Thus, the values of the j terms for cells are expected to be lower (by orders of magnitude) than the j terms for dissolved species in the biofilm.