5.2 Molecular Regime – Non-uniform Pressure

The total flux of a component is the sum of the total diffusive flux and the advective flux resulting from viscous flow (Equation 3). The advective flux resulting from viscous flow is the product of the mole fraction and viscous flux, as explained previously. Then our task is reduced to determining the total diffusive flux affected by non-uniform pressure. The spadework pertinent to this task has already been accomplished. Neglect the second term on the right side of Equation 22 because we are considering the molecular regime and substitute Equation 20 for N_{B}^{D} in the remaining term. Upon rearrangement we have Equation 24.

\displaystyle N_{A}^{D}=-\frac{DC\ dx_{A}/dl+\left ( D+D_{B}^{K} \right )x_{A}dC/dl}{1-\left ( 1-M_{AB}^{0.5} \right )x_{A}} (24)

Equation 24 uses dCA = CdxA xAdC. When the ideal gas law is used to replace the total molar concentration gradient with the gradient of gas pressure, we see that the second term in this result calculates the effect of pressure gradient on diffusion. This effect is sometimes referred to as pressure diffusion. Recall the discussion in Section 3 in which we identified the pressure gradient in the bulk gas as a driving force for diffusion of individual species, as well as for viscous flow.

The flux equation for species A (interchange subscripts for species B) affected by both diffusion and advection via viscous flow is obtained by simply adding the viscous advective flux to Equation 24. We then have Equation 25.

\displaystyle N_{A}=-\frac{DC\ dx_{A}/dl+\left ( D+D_{B}^{K} \right )x_{A}dC/dl}{1-\left ( 1-M_{AB}^{0.5} \right )x_{A}} \displaystyle -x_{A}\frac{k_{g}p}{\mu }\frac{dC}{dl} (25)

This equation is readily reduced to the simpler expression of Equation 26 for the condition D_{B}^{K}\mu /k_{g}p< < 1.

\displaystyle N_{A}=-\frac{DC\ dx_{A}/dl}{1-\left ( 1-M_{AB}^{0.5} \right )x_{A}} \displaystyle -x_{A}\frac{k_{g}p}{\mu }\frac{dC}{dl} (26)

This simplification is tantamount to assuming that dCA ≈  CdxA (i.e., pressure diffusion is negligible) and that the total diffusion flux is satisfactorily approximated by Equation 23. Note that the product DC is independent of pressure, owing to the fact that the effective diffusion coefficient is inversely proportional to pressure. Click on these exercise links to view example problems Exercise 3 and Exercise 4.


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