Semianalytical expression of effectiveness factor for Michaelis‐Menten kinetics

Abstract

To design a heterogeneous chemical reactor it is common to employ the concept of an effectiveness factor to determine whether the reactions in a catalyst particle are limited by pore diffusion. This concept can also be applied to design a biochemiFormulation cal catalyst particle may be written as,The dimensionless mass balance of the susbtrate in a spheri--0 cal reactor where the enzymes are immobilized on the internal surface of a porous support (Bailey and Ollis, 1977).Most of the biochemical reaction systems are frequently characterized by the Michaelis-Menten type of kinetics. Due to their nonlinear nature, numerical methods in general are required to calculate the effectiveness factor of an immobilized enzyme catalyst particle, such as the orthogonal collocation method used by Ramachandran (1975), and a special numerical scheme for high the corresponding boundary conditions areThiele moduli proposed by Chang (1982). Obviously if an approximate solution giving the desired accuracy is available, the computation effort can be drastically reduced. Some approximate solutions are available in the literature, such as the empirical solutions suggested by Atkinson (1971) and Bailey and Ollis (1977), and the weighting-factor methods proposed by Moo-Young and Kobayashi (1972), Gondo et al. (1974), and Kobayashi et al. (1 976). The weighting-factor-methods employ a weighting function to weight the effectiveness factors for reactions in zeroth-and first-order regions, while the modification of Thiele modulus is required, for example, using the generalized Thiele modulus concept (Bischoff, 1965). Among those, only the model proposed by Kobayashi et ai. (1976) could generate a large range of the effectiveness factors over a large range of the existing parameters with errors less than 2.0 percent. But the weighting function of this model is strictly empirical.The purpose of this article is to propose a semianalytical expression of an effectiveness factor for the Michaelis-Menten type kinetics. This expression satisfies the asymptotic behaviors of the reaction system and is applicable over the whole range of existing parameters.Cormpondena concerning this paper should be addressed to Chung-Sung Tan Equation 1 contains two asymptotic behaviors which are:1 . Zeroth-order reaction when p is large 2. First-order reaction when p is small. The effectiveness factors at these extreme conditions are well documented in the literature (Hill, 1977 and Smith, 1971) The real solution to Eq. 1 therefore should exist between these two asymptotic solutions, which may be expressed by where qo and q , are the effectiveness factors for zeroth-and firstorder reactions, respectively; $ is the modified Thiele modulus; and W is the weighting factor which is the function of @ and 0.In order to seek the suitable expressions of $ and W, the analytical solutions a t small and large Thiele moduli are first derived. This can be done by using the regular perturbation method for small 9 and the singular perturbation method for large (Finlayson, 1980)

A Review on the Mathematical Modeling of Biofilm Processes: Advances in Fundamentals of Biofilm Modeling

“…Recent trends in single-substrate biofilm modeling include the development of simplified algebraic expressions for biofilm kinetics [8±10], the development of pseudoanalytical solutions for substrate flux into steady-state biofilm [11±14] and the development of effectiveness factor expressions for biofilm processes with Michaelis-Menten kinetics [15], Monod kinetics with and without product inhibition [16,17]. The salient features of these models will be discussed in part II of the paper.…”