2.2  Total Diffusion Flux

David B. McWhorter

2.2 Total Diffusion Flux

The flux calculated by Equation 2 is referred to as a viscous flux because the resistance to the fluid motion results from viscous shear. Diffusive motion is not resisted by viscous shear. Diffusion of a component in a gas mixture is defined as the increment of motion that is in addition to that imparted by the viscous motion. Thus, the overall or total flux of an individual constituent is given by Equation 3.

$\displaystyle N_{i}=N_{i}^{D}+N_{i}^{v}$ , $\displaystyle i=A,B$

(3)

Equation 3 defines diffusion as the increment of motion of constituents that is in addition to the motion imparted by advection in viscous flow. The first term on the right of Equation 3 is referred to as the total diffusion flux of component i and the second is the contribution to the flux of component i that results from advection via the viscous contribution to the motion of the phase. We will learn in later developments that the total diffusion term includes advection of species i by the diffusion-generated contribution to bulk flow.

The sum of N_i for i=A, B gives the flux of the gas as a whole as expressed in Equation 4.

$\displaystyle N=N^{D}+N^{v}$

(4)

The flux of the gas phase as a whole is due to both diffusion and viscous flow. Again, those who are used to thinking of binary diffusion as a process in which the diffusion fluxes of the constituents are of equal magnitude and opposite in sign will find Equation 4 unfamiliar.

Readers familiar with the traditional treatment of transport of dissolved constituents in groundwater may wonder why Equation 3 doesn’t contain a term representing mechanical dispersion. Non-uniform advection at the pore scale creates pore-scale concentration gradients that are manifest at the macroscopic scale by spreading or dispersal of species that is in addition to that attributable to diffusion alone. This extra increment of spreading is known as mechanical dispersion and is readily observable in solute transport in groundwater. However, multi-dimensional diffusion at the pore scale acts to smooth the variable concentration created by non-uniform advection and, thus, reduces mechanical dispersion. In the case of gases, where diffusion coefficients are typically 1000 times greater than for solutes in liquids, mechanical dispersion is not likely to be an important spreading mechanism relative to macroscopic diffusion. For this reason, a term representing mechanical dispersive flux is not included in Equation 3.

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