• biofilm model;
  • effectiveness factor;
  • continuum heterogeneous biofilm;
  • multiple-substrate limitation;
  • Monod kinetics


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/m3. For Case II, ammonium was the limiting substrate for a biofilm surface concentration, CNs, ≤13.84 g of N/m3. At these concentrations ammonium was limiting inside the biofilm, and oxygen was fully penetrating. Conversely, for CNs > 13.84 g of N/m3, 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. Biotechnol. Bioeng. 2014;111: 2252–2264. © 2014 Wiley Periodicals, Inc.