Reaction rate calculations for cosubstrates diffusing into catalyst layer from opposite sides

Abstract

The effects of diffusion on a reaction taking place in a permeable catalyst are examined theoretically for the case where the reaction has two substrates supplied from opposite sides of a catalytic slab. The solutions of the reaction-diffusion equation for combinations of zeroth-and first-order kinetics are given in terms of an effectiveness factor and a parameter describing the position in the layer where the reaction occurs. In these terms, the results vary only weakly with reaction order. The use of the exact solutions for a reaction that is zeroth order in both sub strates is proposed as a general rule for estimating the reaction rate and the reaction position.

Biofilm development in a membrane-aerated biofilm reactor: effect of intra-membrane oxygen pressure on performance

“…MABR performance has previously been modelled by Casey et al (1999a) considering only the growth reaction by adapting a model proposed by Karel and Robertson (1987) for cosubstrates diffusing from opposite sides of a catalyst. However, in the model described here, the Karel and Robertson (1987) model (referred to here as the`growth only model') is only one step in the solution of a combined model of growth and endogenous decay. The algorithm for the numerical solution is described in the appendix, and the parameters used are given in Table 2.…”

“…The solution to the mathematical model is based on the model of Karel and Robertson (1987) using the same assumptions as those proposed by Casey et al (1999a) for the MABR system used in these experiments, i.e., zero order kinetics for both limiting substrates, a steady state with respect to reaction and diffusion and a structurally homogeneous bio®lm. However, in the revised model presented here, the boundary of the biocatalyst used was not the entire annular bio®lm, but the annular section of bio®lm excluding the endogenous layer.…”

Model‐based comparative performance analysis of membrane aerated biofilm reactor configurations

“…In the case of dual-limited biofilms it is useful to introduce an additional dimensionless parameter c, originally defined by Karel and Robertson (1987) and represents a ratio describing the relative availability of colimiting substrates.…”

“…Unlike zero order kinetics where the rate is limited by the biomass or the amount of substrate present, with Monod kinetics as the concentration falls close to the Monod constant, the rate becomes dependant upon substrate concentration. For co-limiting substrates (Karel and Robertson, 1987) defined four substrate limitation regimes; (A) dual limitation where the concentration of both substrates reach zero within the biofilm, (B) oxygen limitation, where the carbon source fully penetrates, (C) carbon substrate limitation, whereby oxygen fully penetrates, (D) growth rate limitation, that is, both substrates fully penetrate the biofilm and reaction rate is limited only by intrinsic kinetics. In the present study, to distinguish between the four limitation regimes, A-D, the mathematical model was used to define transition boundaries for varying values of c and f, whereby limitation was defined according to the concentration of a substrate within the biofilm relative to the value of its Monod constant.…”

Membrane‐Aerated Biofilm Reactor Behaviors for the Treatment of High‐Strength Ammonium Industrial Wastewater

“…In some cases, cell detachment due to shear stresses has been associated with the hydrodynamic conditions in the bioreactor. The homogeneous model [21] considers the steady-state behavior of the countercurrent diffusion of substrates coupled to a zero or first-order kinetics in a permeable catalyst material. The extractive membrane bioreactor (EMB) model was developed for the treatment of wastewater containing dichloro-ethane (DCE) [22][23].…”