(537b) Applications of Fractional Calculus to Anomalous Mass and Heat Transfer Phenomena In Complex Media

Many problems of mass and heat transfer in complex media exhibit what is called anomalous transport phenomena. Anomalous phenomena are all those process that don’t obey the classical models for transport (e.g., Fick and Fourier’s laws for mass and heat, respectively), describing this kind of processes with anomalous diffusion is a problem that has led investigators to formulate corrections of the classical laws. In this challenge of making successful corrections authors have commented on the necessity of including memory as an important factor to be considered when working in complex media^[1].  Two of such approaches include the introduction of Levy- like processes and the Continuous-Time Random Walks (CTRW) theory, both of which arrive to an Integro-differential equation called space-time fractional diffusion equation (STFDE) (which is applicable for both mass and heat transfer).^[2]

In this work analytic and numerical methods are shown for solving specific boundary value problems on the time fractional diffusion equation (TFDE) compared with experimental data for mass and heat transport in complex media – solute diffusion in a gel matrix and heat conduction in granular media, respectively. The analytical treatment uses a combined transform method were both Laplace and Fourier Transforms are applied. Evaluation of the analytical solution is made by numerical Laplace inversion and the deformation of contours method developed by Luchko.^[3] The experimental parameters for the TFDE are found by procedures similar to those implemented by Kosztołowicz et al. ^[4] using a penetration distance function.

For the numerical solution of the TFDE a finite differences method has been developed that evaluates fractional derivates of the Caputo kind, computational implementations for both mass and heat are developed and are applied to different boundary conditions.

Finally this work makes a comparison of classical and fractional models for transport phenomena in complex media, observing a significant advantage of using the fractional approach, especially when predicting outside of the time interval used for determination of the parameters (Diffusivity for classical models and fractional derivative and Diffusivity for the TFDE).

References:

1. W. L. Vargas, J. J. McCarthy, Stress effects on the conductivity of particulate beds. Chemical Engineering Science E 57, (2002).
2. D. Hernandez, C. Varea, and R. A. Barrio. Dynamics of reaction-diffusion systems in a subdiffusive regime. Physical Review E 79, 026109 (2009).
3. Y. Luchko. Algorithms for evaluation of the Wright function for the real arguments’ values. Fractional Calculus & Applied Analysis V 11, N 1 (2008).
4. T. Kosztołowicz, K. Dworecki, and St. Mrówczyñski, Measuring subdiffusion parameters. Physical Review E 71, 041105 (2005).