Semi-analytical expressions for the concentrations and effectiveness factor for the three general catalyst shapes

Exploring Nonlinear Reaction-Diffusion in Enzyme Immobilized Systems: Integer and Fractional Order Modeling

This paper presented a kinetic model of the Langmuir-Hinshelwood-Hougen-Watson (LHHW) type for porous catalysts with simple one-dimensional geometry, including spheres, infinite cylinders, and flat pellets. The model was applied to systems involving immobilized enzymes, where enzymes are attached to porous support materials to enhance stability and reusability. The LHHW model provided a tool for understanding and modeling reaction kinetics in heterogeneous porous catalysts and immobilized enzymes. A nonlinear reaction-diffusion equation was generated using finite-range Fickian diffusion and nonlinear reaction kinetics, crucial for accurately modeling the behavior of immobilized enzymes. This research addressed a gap in the existing literature by introducing fractional derivatives to investigate enzyme reaction kinetics, capturing the complex dynamics of substrate interaction and reaction rates within the porous matrix. An approximation method based on Lucas wavelets was employed to find solutions for substrate concentration and effectiveness factors across various parameter values. The analytical solutions derived from the Lucas wavelet method (LWM) were evaluated against the fourth-order Runge-Kutta method, showing great agreement between the LWM solutions and numerical counterparts. These results optimized diffusion and reaction kinetics, paving the way for advancements in biocatalysis and efficient enzyme reactor design.

Theoretical Analysis of Mass Transfer Behavior in Fixed-Bed Electrochemical Reactors: Akbari-Ganji’s Method

The theoretical model for a packed porous catalytic particle of the slab, cylindrical, and spherical geometries shape in fixed-bed electrochemical reactors is discussed. These particles have internal mass concentration and temperature gradients in endothermic or exothermic reactions. The model is based on a nonlinear reaction–diffusion equation containing a nonlinear term with an exponential relationship between intrinsic reaction rate and temperature. The porous catalyst particle’s concentration is obtained by solving the nonlinear equation using Akbari-Ganji’s method. A simple and closed-form analytical expression of the effectiveness factor for slab, cylindrical, and spherical geometries was also reported for all values of Thiele modulus, activation energy, and heat reaction. The accordance with results of a reliable numerical method shows the good accuracy that their approximate solution yields.

Modelling of Biotrickling Filters for Treatment of NOx Analytical Expressions for the NOx Concentration in Both Gas and Biofilm Phases

A mathematical model of an ideal biotrickling filter (BF) system that inoculates a recently identified strain of Chelatococcus daeguensis TAD1 and brings about efficient nitrogen oxide treatment is discussed. The proposed model is based on nonlinear mass transport equations at the gas–biofilm interface. Using Akbari–Ganji’s technique, approximate analytical expressions for the nitric oxide concentration in the gaseous and biofilm phases were developed for all feasible system parameters. In addition, to investigate the dynamic behaviour of the system, a numerical analysis of the problem is provided using MATLAB tools. To demonstrate this new approach, graphical data are provided and quantitatively discussed. This theoretical result has good agreement with the numerical simulation (MATLAB) results for the experimental values of parameters.