Spectral Finite Element Methods for solving Fractional Differential Equations with applications in Anomalous Transport
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Quantifying species transport rates is a main concern in chemical and petrochemical industries. In particular, the design and operation of many large-scale industrial chemical processes is as much dependent on diffusion as it is on reaction rates. However, the existing diffusion models sometimes fail to predict experimentally observed behaviors and their accuracy is usually insufficient for process optimization purposes. Fractional diffusion models offer multiple possibilities for generalizing Fick’s law in a consistent manner in order to account for history dependence and nonlocal effects. These models have not been extensively applied to the study of real systems, mainly due to their computational cost and mathematical complexity. A least squares spectral formulation was developed for solving fractional differential equations. The proposed method was proven particularly well-suited for dealing with the numerical difficulties inherent to fractional differential operators. The practical implementation was explained in detail in order to enhance reproducibility, and directions were specified for extending it to multiple dimensions and arbitrarily shaped domains. A numerical framework based on the least-squares spectral element method was developed for studying and comparing anomalous diffusion models in pellets. This simulation tool is capable of solving arbitrary integro-differential equations and can be effortlessly adapted to various problems in any number of dimensions. Simulations of the flow around a cylindrical particle were achieved by extending the functionality of the developed framework. A test case was analyzed by coupling the boundary condition yielded by the fluid model with two families of anomalous diffusion models: hyperbolic diffusion and fractional diffusion. Qualitative guidelines for determining the suitability of diffusion models can be formulated by complementing experimental data with the results obtained from this approach.
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Carella, Alfredo R.; Dorao, Carlos A.. Migration of Species into a Particle Under Different Flow Conditions. Computational Fluid Dynamics 2010 - Proceedings of the Sixth International Conference on Computational Fluid Dynamics: 869-871, 2011. 10.1007/978-3-642-17884-9_112.
Carella, Alfredo Raúl; Dorao, Carlos Alberto. Least-Squares Spectral Method for the solution of a fractional advection-dispersion equation. Journal of Computational Physics. (ISSN 0021-9991), 2012. 10.1016/j.jcp.2012.04.050.
Carella, Alfredo Raul; Dorao, Carlos Alberto. Modeling of fractional diffusion on a catalytic particle under different flow conditions. DIFFUSION IN MATERIALS - DIMAT 2011: 121-126, 2012. 10.4028/www.scientific.net/DDF.323-325.121.
Carella, A.R.; Dorao, C.A.. N-dimensional Least Squares Spectral Method formulation for the general Fractional Diffusion Equation. .