Continuum heterogeneous biofilm model—A simple and accurate method for effectiveness factor determination

Abstract

We present a novel analytical approach to describe biofilm processes considering continuum variation of both biofilm density and substrate effective diffusivity. A simple perturbation and matching technique was used to quantify biofilm activity using the steady-state diffusion-reaction equation with continuum variable substrate effective diffusivity and biofilm density, along the coordinate normal to the biofilm surface. The procedure allows prediction of an effectiveness factor, η, defined as the ratio between the observed rate of substrate utilization (reaction rate with diffusion resistance) and the rate of substrate utilization without diffusion limitation. Main assumptions are that (i) the biofilm is a continuum, (ii) substrate is transferred by diffusion only and is consumed only by microorganisms at a rate according to Monod kinetics, (iii) biofilm density and substrate effective diffusivity change in the x direction, (iv) the substrate concentration above the biofilm surface is known, and (v) the substratum is impermeable. With this approach one can evaluate, in a fast and efficient way, the effect of different parameters that characterize a heterogeneous biofilm and the kinetics of the rate of substrate consumption on the behavior of the biological system. Based on a comparison of η profiles the activity of a homogeneous biofilm could be as much as 47.8% higher than that of a heterogeneous biofilm, under the given conditions. A comparison of η values estimated for first order kinetics and η values obtained by numerical techniques showed a maximum deviation of 1.75% in a narrow range of modified Thiele modulus values. When external mass transfer resistance, is also considered, a global effectiveness factor, η(0) , can be calculated. The main advantage of the approach lies in the analytical expression for the calculation of the intrinsic effectiveness factor η and its implementation in a computer program. For the test cases studied convergence was achieved quickly after four or five iterations. Therefore, the simulation and scale-up of heterogeneous biofilm reactors can be easily carried out.

The continuum heterogeneous biofilm model with multiple limiting substrate Monod kinetics

“…The continuous heterogeneous biofilm considers that both substrate diffusivity and biofilm density change with depth in the direction normal to the biofilm surface (Gonzo et al, ). For dual‐substrate Monod kinetics, the concentration profiles of both substrates, A and B (Fig.…”

“…Yet the growing number of multidimensional mathematical models used to describe mass transfer and reactions in biofilms are often inaccessible to experimentally oriented researchers (Wanner et al, ; Wuertz et al, ) because they (i) describe hypothetical rather than experimental biofilms, (ii) the physical and hydrodynamic conditions in experimental biofilms are insufficiently characterized, or (iii) the numerical simulation techniques are not available to the experimentalists. It is, therefore, of interest to consider analytical approaches to advanced biofilm modeling (Gonzo et al, ) that may be more amenable to many researchers, especially those in the life sciences.…”

“…In the continuum heterogeneous model, the biofilm is treated as a continuous phase where the phenomena mediated by microorganisms, densely packed in micro‐colonies separated by interstitial voids, are lumped in an effective mass transfer (effective diffusivity) and a net biofilm growth rate. This model considers that both the effective diffusivity of the substrate and the biofilm density vary along the coordinate normal to the biofilm surface (Beyenal and Lewandowski, ; Gonzo et al, ).…”

“…Therefore, by calculating the effectiveness factor, it is possible to estimate the net rate of substrate consumption and the net biofilm growth rate. In a previous work (Gonzo et al, ), we developed a procedure to estimate analytically the effectiveness factor, considering different biofilm heterogeneities as well as different values for the Monod half‐maximum‐rate concentration constant for a single substrate. This approach allows easy computation of η in a computer program and, consequently, the rate of substrate utilization can be determined under many different conditions resulting from biofilm heterogeneity.…”

Growth of Myxococcus xanthus in Continuous-Flow-Cell Bioreactors as a Method for Studying Development

“…However, growing M. xanthus as a biofilm adhering to a solid surface under continuous-medium-flow conditions allows for stricter control of constant environmental parameters. Biofilms have been studied extensively in numerous bacterial systems, including Pseudomonas aeruginosa, Bacillus subtilis, and Staphylococcus aureus, among others (7)(8)(9)(10)(11)(12). While biofilms of M. xanthus are less well studied, recent work has begun to expand our knowledge of biofilm formation and how it relates to nutrient sensing and development (13)(14)(15).…”