4.2 Effect of Solid Particles Embedded in the Gas
Diffusing species in a gas-particle mixture experience resistance that is the sum of that due to molecule-molecule collisions and molecule-solid collisions. Equation 14 adds the rate of momentum loss due to molecule-particle collisions (second term on the right) to the rate of momentum loss due to molecule-molecule collisions (first term on the right).
(14) |
where:
= | total diffusive mole flux of component A (mol / L^{2}) | |
= | effective Knudsen diffusion coefficient for component A (L^{2}/T) |
Equation 14 is of central importance to the developments in this book and warrants further discussion.
Note from Equation 10 that the flux, J_{A}, appearing on the right side of Equation 14, is a function of the mole fluxes of both constituents A and B; hence, it can be concluded that the rate of momentum change for species A due to intermolecular collisions depends upon the flux of both constituents. Now suppose that we regard the solid particles to be giant, stationary molecules, as did the authors of the Dusty Gas Model. Then the second term on the right is conceptually the same as the first term, but with the flux of one constituent, the giant molecules (solid particles), equal to zero (Cunningham and Williams, 1980).
We use the effective diffusion coefficient, D, in the first term on the right of Equation 14 because particles are obstructing the diffusion space in this case. The parameter appearing in the second term is known as the effective Knudsen diffusion coefficient in honor of Martin Knudsen, a pioneer in the study of molecule-solid collisions. As is the case for the effective molecular diffusion coefficient, the effective Knudsen diffusion coefficient is the macroscopic manifestation of molecular-scale momentum exchange, but in this case due to molecule-solid collisions instead of intermolecular collisions. Note that the Knudsen coefficient is species specific, i.e., the effective Knudsen diffusion coefficient for species A is different than for species B. This is because the molecule-particle collisions of a particular species are independent of the presence of other diffusing species. The determination of effective molecular and Knudsen diffusion coefficients is discussed in the section on parameter estimation.
It might seem that conservation of momentum should dictate zero loss of momentum when the molecules elastically collide with massive immobile particles. Indeed, such would be the case for smooth particles on which the angle of reflection is equal to the angle of incidence for all collisions (specular reflection). On the other hand, Cunningham and Williams (1980) argue that reflection of molecules impinging on a macroscopic element of rough surface (rough at the molecular scale) are chaotically reflected (diffuse reflection) when viewed at the scale of the surface element as a whole. According to these authors “… in contrast to the situation with smooth walls, the gas will lose momentum as it flows along a rough wall”.