scholarly journals Approximations for the Concentration and Effectiveness Factor in Porous Catalysts of Arbitrary Shape: Taylor Series and Akbari-Ganji’s Methods

2021 ◽  
Vol 8 (4) ◽  
pp. 527-537
Author(s):  
Ramu UshaRani ◽  
Lakshmanan Rajendran ◽  
Marwan Abukhaled
Keyword(s):  
Taylor Series ◽  
Arbitrary Shape ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Effectiveness Factor ◽  
Michaelis Constant ◽  
Thiele Modulus ◽  
Porous Catalysts ◽  
Analytical Approaches ◽  
Analytical Expressions

A mathematical model of reaction-diffusion problem with Michaelis-Menten kinetics in catalyst particles of arbitrary shape is investigated. Analytical expressions of the concentration of substrates are derived as functions of the Thiele modulus, the modified Sherwood number, and the Michaelis constant. A Taylor series approach and the Akbari-Ganji's method are utilized to determine the substrate concentration and the effectiveness factor. The effects of the shape factor on the concentration profiles and the effectiveness factor are discussed. In addition to their simple implementations, the proposed analytical approaches are reliable and highly accurate, as it will be shown when compared with numerical simulations.

Mass Transfer with Chemical Reactions in Porous Catalysts: A Discussion on the Criteria for the Internal and External Diffusion Limitations

2013 ◽  
Vol 334-335 ◽  
pp. 279-283
Author(s):  
S. Gültekin
Keyword(s):  
Mass Transfer ◽  
Chemical Reactions ◽  
Effective Diffusivity ◽  
Effectiveness Factor ◽  
Internal Mass ◽  
Thiele Modulus ◽  
Porous Catalysts ◽  
Diffusion Limitations ◽  
External Diffusion ◽  
Internal Mass Transfer

In this study, criteria for internal mass transfer given in the literature were investigated by considering tortuosity (τ) in the porous catalysts. Uncertainties in τ, which may have values between 2 to 7, have a big impact on the effective diffusivity (Deff) which also affects the Thiele Modulus (Φ). Since effectiveness factor (η) is function of Φ, then the criteria given for limitations are questionable. The value of Deff, and in turn the value of Φ calculated for the τ=2 to 7. At low Φ the effects are very small, but when the Φ increases the effect becomes more pronounced. As a result, when using internal mass transfer limitations, one has to be very careful not to get trapped by the disguised kinetics, results of which may end up with disaster.

scholarly journals Reduced basis approximations of the solutions to spectral fractional diffusion problems

2020 ◽  
Vol 28 (3) ◽  
pp. 147-160
Author(s):  
Andrea Bonito ◽  
Diane Guignard ◽  
Ashley R. Zhang
Keyword(s):  
Computational Cost ◽  
Fractional Power ◽  
Reaction Diffusion ◽  
Fractional Diffusion ◽  
Diffusion Problem ◽  
Quadrature Point ◽  
Reduced Basis ◽  
Fast Evaluation ◽  
Bottle Neck ◽  
Diffusion Problems

AbstractWe consider the numerical approximation of the spectral fractional diffusion problem based on the so called Balakrishnan representation. The latter consists of an improper integral approximated via quadratures. At each quadrature point, a reaction–diffusion problem must be approximated and is the method bottle neck. In this work, we propose to reduce the computational cost using a reduced basis strategy allowing for a fast evaluation of the reaction–diffusion problems. The reduced basis does not depend on the fractional power s for 0 < smin ⩽ s ⩽ smax < 1. It is built offline once for all and used online irrespectively of the fractional power. We analyze the reduced basis strategy and show its exponential convergence. The analytical results are illustrated with insightful numerical experiments.

scholarly journals Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms

2016 ◽  
Vol 16 (4) ◽  
pp. 609-631 ◽  
Author(s):  
Immanuel Anjam ◽  
Dirk Pauly
Keyword(s):  
Error Control ◽  
A Posteriori Error Estimates ◽  
Reaction Diffusion ◽  
Posteriori Error ◽  
Diffusion Problem ◽  
Mixed Formulation ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Linear Partial Differential Equations ◽  
Primal And Dual

AbstractThe results of this contribution are derived in the framework of functional type a posteriori error estimates. The error is measured in a combined norm which takes into account both the primal and dual variables denoted by x and y, respectively. Our first main result is an error equality for all equations of the class ${\mathrm{A}^{*}\mathrm{A}x+x=f}$ or in mixed formulation ${\mathrm{A}^{*}y+x=f}$, ${\mathrm{A}x=y}$, where the exact solution $(x,y)$ is in $D(\mathrm{A})\times D(\mathrm{A}^{*})$. Here ${\mathrm{A}}$ is a linear, densely defined and closed (usually a differential) operator and ${\mathrm{A}^{*}}$ its adjoint. In this paper we deal with very conforming mixed approximations, i.e., we assume that the approximation ${(\tilde{x},\tilde{y})}$ belongs to ${D(\mathrm{A})\times D(\mathrm{A}^{*})}$. In order to obtain the exact global error value of this approximation one only needs the problem data and the mixed approximation itself, i.e., we have the equality$\lvert x-\tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-% \tilde{y}\rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}=\mathcal{M}(% \tilde{x},\tilde{y}),$where ${\mathcal{M}(\tilde{x},\tilde{y}):=\lvert f-\tilde{x}-\mathrm{A}^{*}\tilde{y}% \rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. Our second main result is an error estimate for all equations of the class ${\mathrm{A}^{*}\mathrm{A}x+ix=f}$ or in mixed formulation ${\mathrm{A}^{*}y+ix=f}$, ${\mathrm{A}x=y}$, where i is the imaginary unit. For this problem we have the two-sided estimate$\frac{\sqrt{2}}{\sqrt{2}+1}\mathcal{M}_{i}(\tilde{x},\tilde{y})\leq\lvert x-% \tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-\tilde{y}% \rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}\leq\frac{\sqrt{2}}{% \sqrt{2}-1}\mathcal{M}_{i}(\tilde{x},\tilde{y}),$where ${\mathcal{M}_{i}(\tilde{x},\tilde{y}):=\lvert f-i\tilde{x}-\mathrm{A}^{*}% \tilde{y}\rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. We will point out a motivation for the study of the latter problems by time discretizations or time-harmonic ansatz of linear partial differential equations and we will present an extensive list of applications including the reaction-diffusion problem and the eddy current problem.

Evaluating the effectiveness factor from a 1D approximation fitted at high Thiele modulus: Spanning commercial pellet shapes with linear kinetics

2009 ◽  
Vol 64 (11) ◽  
pp. 2762-2766 ◽  
Author(s):  
Néstor J. Mariani ◽  
Clarisa Mocciaro ◽  
Sergio D. Keegan ◽  
Osvaldo M. Martínez ◽  
Guillermo F. Barreto
Keyword(s):  
Effectiveness Factor ◽  
Thiele Modulus ◽  
Linear Kinetics
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