title: The continuum heterogeneous biofilm model with multiple limiting substrate Monod kinetics.
abstract: We describe a novel procedure to estimate the net growth rate of biofilms on multiple substrates. The approach is based on diffusion-reaction mass balances for chemical species in a continuum biofilm model with reaction kinetics corresponding to a Double-Monod expression. This analytical model considers a heterogeneous biofilm with variable distributions of biofilm density, activity, and effective diffusivity as a function of depth. We present the procedure to estimate the effectiveness factor analytically and compare the outcome with values obtained by the application of a rigorous numerical computational method using several theoretical examples and a test case. A comparison of the profiles of the effectiveness factor as a function of the Thiele modulus, φ, revealed that the activity of a homogeneous biofilm could be as much as 42% higher than that of a heterogeneous biofilm, under the given conditions. The maximum relative error between numerical and estimated effectiveness factor was 2.03% at φ near 0.7 (corresponding to a normalized Thiele modulus φ* = 1). For φ < 0.3 or φ > 1.4, the relative error was less than 0.5%. A biofilm containing aerobic ammonium oxidizers was chosen as a test case to illustrate the model's capability. We assumed a continuum heterogeneous biofilm model where the effective diffusivities of oxygen and ammonium change with biofilm position. Calculations were performed for two scenarios; Case I had low dissolved oxygen (DO) concentrations and Case II had high DO concentrations, with a concentration at the biofilm-fluid interface of 10 g O2 /m(3) . For Case II, ammonium was the limiting substrate for a biofilm surface concentration, CNs , ≤13.84 g of N/m(3) . At these concentrations ammonium was limiting inside the biofilm, and oxygen was fully penetrating. Conversely, for CNs  > 13.84 g of N/m(3) , oxygen became the limiting substrate inside the biofilm and ammonium was fully penetrating. Finally, a generalized procedure to estimate the effectiveness factor for a system with multiple (n > 2) limiting substrates is given.
authors.affiliation: INIQUI (CONICET)-Facultad de Ingeniería, Universidad Nacional de Salta, Av. Bolivia 5150, Salta, 4400, Argentina.
authors.forename: Elio Emilio
authors.initials: EE
authors.lastname: Gonzo
authors.forename: Stefan
authors.initials: S
authors.lastname: Wuertz
authors.forename: Veronica B
authors.initials: VB
authors.lastname: Rajal
mesh_terms: D064751:Ammonium Compounds / Q000378:metabolism; D001420:Bacteria, Aerobic / Q000187:drug effects* / Q000378:metabolism / Q000502:physiology*; D018441:Biofilms / Q000187:drug effects* / Q000254:growth & development*; D010084:Oxidation-Reduction; D010100:Oxygen / Q000378:metabolism
chemical_list: D064751:Ammonium Compounds; D010100:Oxygen
keywords: Monod kinetics; biofilm model; continuum heterogeneous biofilm; effectiveness factor; multiple-substrate limitation