Click OK to input the entered parameters, and then click Run in the DiffuDict section to start the Knudsen Diffusion command. The results are immediately shown in the opening Result Viewer after the process is finished, the screenshots below show the results obtained for the MPL model (Example 3).

The Report Tab

In the Results - Report subtab, the following results are reported:

• The dimensionless Relative diffusivity as defined through equation (145) .
• The Effective diffusivity in porous media. These values are obtained by multiplying the diffusivity of the particles with the porosity.
• The Particle diffusivity computed by equation (143) using either the user defined mean thermal velocity or the default velocity of 1 voxel/s.
• The resulting Knudsen tortuosity including both Tortuosity factors and Tortuosity for each space direction as defined through equation (146) .

Then, some calculation information is given:

• The computed Characteristic Length .
• The corresponding reference diffusivity computed through (144) .
• The overall runtime and memory usage.

Afterwards, some statistical information about the performed random walk method is reported:

• The average number of collisions with a pore wall a random walker experienced during the simulated time.
• The average distance that a random walker traveled.
• If Auto Distance, Check Convergence was chosen, the initial and final average travel distance used to check the convergence for .
• The number of random walks evaluated to achieve the requested tolerance.
• As a check for plausibility of the results: the percentage of random walkers that have not collided with any pore wall during the whole simulation time. If this value is not equal to 0 %, consider increasing the Average Travel Distance!
• The criterion that caused the simulation to finish.

Instead of the computed characteristic length, a user defined characteristic length can be used to determine the relative diffusivity by entering its value in the left panel and clicking the Apply… button.

Changing the characteristic length will also change the computed Relative diffusivity and the Knudsen tortuosity.

The random walk algorithm is done using a velocity of 1 voxel per second. However, the results can be rescaled to any given mean thermal velocity by entering a User Defined Thermal Velocity.

Changing the thermal velocity will also change the computed Particle diffusivity and the Effective diffusivity. To get the diffusivity in the structure for a specific diffusivity as input (e.g. for oxygen at 20°C), define the corresponding thermal velocity (for oxygen at 20°C: 440.4 m/s). The Particle diffusivity describes how quickly an oxygen molecule will travel through the pores of the structure by diffusive reflection at the pore walls. The Effective diffusivity describes how quickly a concentration of species will move through the homogenized porous media by diffusion.

For a cylindrical pore, the Particle diffusivity is expected to match , where is the diameter of the pore, and the mean thermal velocity. To achieve this result for a cylindrical pore, the pore diameter has to be entered manually as User Defined Characteristic Length.

In this example, for a pore diameter of 200 nm and a mean thermal velocity of 464 m/s, a diffusivity of approx. 3.07e-5 m²/s is computed, which is very close to the expected value of 3.09e-5 m²/s. For cylindrical pores, the Knudsen tortuosity factor is 1 in the direction of the pore, when using the pore diameter as characteristic length.

The Plots Tab

In the Results – Plots subtab, the determined Rel. Diffusivity Max.(which is the largest relative diffusivity value achieved in X-, Y-, or Z-direction) is shown over the average travel distance, starting from the initial average travel distance to the final average travel distance. These values are computed during the computation to check for convergence, and the last point corresponds to the final result. Be aware that this plot does not change when the relative diffusivity changes in the report tab because of a user defined characteristic length, i.e. it always shows the relative diffusivity values corresponding to the computed characteristic length.