Hello friends of Steemit! In this second installment of the series of posts on simulation of diffusion in porous media, I will explain some of the computational techniques that I have used in the design and/or generation of the models of porous media used in the simulations.
In the works I have done, and of which I will show some results in the following posts, the porous media used have been designed theoretically and computationally. To date, the models generated represent porous media two-dimensional (2D), not consolidated and biphasic, that is, composed of two phases, a solid dispersed phase (phase σ) represented by Square or circular flat particles, although other forms have also been used, and for a non-solid phase or continuous fluid (phase β) represented by the pores that remain between the particles. Some works has also been done with consolidated porous media.
Up to now, only two-dimensional porous media have been designed to simplify the study and optimize computational calculation time based on the available calculation capacity. Although, in nature, the existing porous media are three-dimensional, these first works represent a first approach to the study of diffusion in porous media.
In addition, the designed porous media are periodic, that is, they are formed by the spatial repetition of a portion called unit cell, like the minimal units that form a crystal when it is spatially repeated. This additional feature facilitates the simulation of the phenomenon, since we work with the unit cell instead of the complete medium, and the results are valid for all the porous medium.
Figure 1: Mosaic of non-consolidated two-dimensional porous media models.
According to the distribution of the dispersed or discontinuous phase in the unitary cells of the models, two types of porous media have been generated: some so-called ordered and others called disordered.
Ordered Porous Media
The unit cells of such means are square, in which the discontinuous phase is represented by polygons or flat figures of any type such as, for example, squares, rectangles, circles, ellipses, among others, centered in the cells.
In the following figure, a square unit cell centered by a square particle and the corresponding model of periodic porous medium formed by the spatial repetition of the cell is shown.
Figure 2: (a) Square unit cell centered by a square. (b) Model of periodic medium resulting from the spatial repetition of cell 2(a).
The figure 3 shows a square unit cell centered by a circular particle and the periodic porous media model formed by the spatial repetition of the cell.
Figura 3: (a) Square unit cell centered by a circle. (b) Model of periodic medium resulting from the spatial repetition of cell 3(a).
Disordered Porous Media
The unit cells of the models of these porous media used during the investigation were obtained from a computational algorithm, whose formation parameters can be controlled and, therefore, the structure or geometry of the same.
One of the computational algorithms used to generate the unit cells of these media is the Random Sequential Absorption Method, called the RSA Method. In this method, the position of a first particle is randomly selected in a plane (the unit cell). Next, a second particle is generated and the distance between the first and second particles is compared to verify that the particles do not overlap. If the second particle does not overlap with the first, its position is fixed; if on the contrary it is superimposed, it is eliminated. This process is continued until the desired porosity is found, or until there is no other particle in the unit cell.
The desired porosity is obtained by controlling the number of particles present in the system. For the case of circular particles, the porosity is given by the expression:
where:
is the porosity of the medium.
is the number of particles.
is the radius of the particles.
is the length of the side of the unit cell of the porous medium.
In figure 4, the unitary cells of two disordered porous media models are shown, one of porosity 90% and another of porosity 60%.
Figure 4: Unitary cells of disordered media with porosity (a) 90% and (b) 60%.
In the next installment of the series, I will talk about the structural characterization of the porous media models generated and used in my research work in the area.
Thanks for your kind reading. I hope this post was liked. I wait for you in the next installment.
If you wish to obtain additional information about the phenomenon of diffusion in porous media, I invite you to read the previous posts of the series:
Diffusion in Porous Media – A world of applications – Part 1
Diffusion in Porous Media – A world of applications – Part 2
Diffusion in Porous Media – A world of applications – Part 3
Diffusion in Porous Media – A world of applications – Part 4
Diffusion in Porous Media – Simulation – Part 1: An Introduction
Note: All figures are self-made.
Sources:
Borges da Silva, E. A.; Souza, D. P. y Ulson da Souza, A. A., 2007. Prediction of effective diffusivity tensors for bulk diffusion with chemical reactions in porous media. Brazilian Journal of Chemical Engineering, 24, pp. 47-60.
Dullien, F. A. L., 2000. Porous Media: Fluid transport and pore structure, 2da edic., Academic Press Inc., Londres, Reino Unido, pp. 6-110, 288-297, 501-562.
Torquato, S., 2002. Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Editorial Springer, USA.
Whitaker, S., 1967. Diffusion and dispersion in porous media, AIChE (Am. Inst. Chem. Eng.), J. 13, pp.420-426.
Zhang, Z. P.; Yu, A. B. y Dodds, J. A., 1997. Analysis of the Pore Characteristics of Mixture of Disks. Journal of Colloid and Interface Science, 195, pp. 8–18.