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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. [ABSTRACT FROM AUTHOR]

A mathematical model of the magnetohydrodynamic free convective flow of a viscous incompressible uid, which is based on a system of coupled steady-state nonlinear deferential equations, is discussed. A new approach of the homotopy perturbation method is employed to derive analytical expressions of the uid velocity, uid temperature, and species concentration. The efficiency and accuracy of the derived results are tested against highly accurate and widely used numerical methods. The obtained analytical expressions are employed to study the effects of the magnetic field, chemical reaction, and other relevant flow parameters on uid velocity, uid temperature, and species concentration. Sensitivity analysis of these parameters is also presented. [ABSTRACT FROM AUTHOR]

• Analytical expressions for the electric potential for planar, spherical and cylindrical geometries are derived used a simple series method. • The proposed mathematical approach is simple and reliable and can be easily implemented by researchers from various science disciplines. • A study of the effect of various parameters such as grain size, grain shape and doping level is conducted. • Analytical formula for the density of oxygen ion is derived. • The maximum surface potential is determined. In this article, the mathematical model presented by Barami and Ghafarinia (Sensors and Actuators B: Chemical 293 (2019) 31–40) for metal oxide grains is discussed. The nonlinear governing model, which is a Poisson-Boltzmann-type equation, is solved by a simple and efficient method. Analytical expressions for the electrical potential and oxygen ion density within the planar, cylindrical and spherical metal oxide grains are expressed in terms of the absolute value of the electric potential and grain thickness. In addition, an investigation of the effects of parameters such as grain diameter, grain thickness, and uniform distribution of single donors on the performance. The validity of the derived analytical expression is established by direct comparison with computer-generated numerical simulations in addition to previously available limiting cases' results (low and high oxygen adsorption on the grain). [ABSTRACT FROM AUTHOR]