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ABSTRACT
The classical Fick’s law with a constant coefficient is not suitable for gas diffusion in fractal porous media. This study analyzes non-steady-state gas diffusion through a fractal porous media with fractal pore models used to create natural porous media. The governing equations for the molecular motion are then used to model the molecular diffusion in the fractal pore model. Extensive numerical simulations are then used to develop a general diffusion equation. The predictions of the diffusion equation agree well with numerical simulations and experimental data. The diffusion equation reproduces the classical Fick’s law when the fractal pore structures have a weak effect on the diffusion. The results show that non-steady-state diffusion depends on the history of the diffusion process and that the diffusion rate increases with increasing porosity, decreasing specific surface area, and decreasing fractal dimension.
Funding
The work was financially supported by the National Natural Science Foundation of China (No. 51176096).
Nomenclature
| = | arbitrary constant | |
| = | gas concentration (mol/m3) | |
| = | initial gas concentration in the simulations (mol/m3) | |
| = | gas concentration in the first space in the experimental setup (mol/m3) | |
| = | gas concentration in the second space in the experimental setup (mol/m3) | |
| = | gas concentration in the first porous media sample (mol/m3) | |
| = | gas concentration at steady state (mol/m3) | |
| = | diffusion coefficient (m2/s) | |
| = | fractal dimension based on mercury intrusion measurements | |
| = | spatial dimension | |
| = | fractal dimension | |
| = | mean pore diameter (m) | |
| = | diameter of the porous media samples (cm) | |
| = | anomalous diffusion exponent | |
| = | parameter in Eq. (2) | |
| = | height of the porous media samples (cm) | |
| = | loop variable | |
| = | element number in the x direction | |
| = | number of particle boundary element in the +x direction | |
| = | number of particle boundary element in the –x direction | |
| = | gas flux through each cross section (mol/m2/s) | |
| = | diffusion flux into the first space in the experimental setup (mol/m2/s) | |
| = | diffusion flux into the first porous media sample from the first space (mol/m2/s) | |
| = | element number in the y direction | |
| = | element number in the z direction | |
| = | side length of an element in the fractal pore model (m) | |
| = | total number of molecules in one element | |
| = | initial total number of molecules in one element | |
| = | total number of molecules diffusing out of one element per unit time | |
| = | final pressure in the experiment (Pa) | |
| = | initial pressures in the six spaces in the experimental setup (Pa) | |
| = | universal gas constant | |
| = | mean-square displacement of a random walker (m2) | |
| = | pore radius (m) | |
| = | mean pore radius (m) | |
| = | accumulated pore surface area (m2) | |
| = | pore surface area per unit solid volume (m2/m3) | |
| = | specific surface area of the porous media samples (m2/g) | |
| = | experimental temperature (K) | |
| = | time (s) | |
| = | time step (s) | |
| = | volumes of the six spaces in the experimental setup (cm3) | |
| = | mean molecular velocity (m/s) | |
| = | distance (m) | |
| = | infinitesimal distance (m) | |
| = | coordinate variable | |
| = | coordinate variable | |
| = | coordinate variable | |
| = | parameter related to the pore structure | |
| = | ||
| = | mean free path of the gas molecules (m) | |
| = | circumference-to-diameter ratio | |
| = | porosity | |
| = | parameter very close to 0 | |
| = | density of the porous media samples (g/cm3) | |
| = |