CL: Diffusion with (generic) Non-zero Bulk Flow

In addition to diffusion through a stagnant gas, there are a variety of other scenarios from which non-equimolar counter diffusion can arise (i.e., where we might obtain a bulk flow). A good example is heterogeneous catalysis problems (i.e. reactions on a surface).

Consider the reaction:

$2A \rightarrow 2B+C$

If this reaction occurs between gas molecules on a solid surface:

we clearly have a net molar flux away from the surface as two moles/molecules of A is making a total of three moles of product (although our net mass flux would still be zero).

In solving the problem for the total flux of species A, we again start with the definition of the total flux:

$\displaystyle{N_A = -c_{total}D\nabla y_A+c_AV}$

In this case our Vis given as

$\displaystyle{V = \frac{c_Av_A+c_Bv_B+c_Cv_C}{c_{total}} = \frac{N_A+N_B+N_C}{c_{total}}}$

where we can use stoichiometry to relate NA to the other N's to get:

$\displaystyle{\frac{1}{2}N_A = -\frac{1}{2}N_B = -N_C}$

NOTE:

Be careful not to trick yourself into saying 2NA = -2NB = -NC. Since there are already 2 moles of A forming 1 mole of C, this would actually be backward and make our answer really wrong.

Plugging this into NA we get

$\displaystyle{N_A = -c_{total}D\nabla y_A + y_A\left( N_A-N_A-\frac{1}{2}N_A\right)}$

$\displaystyle{N_A = -c_{total}D\nabla y_A - \frac{y_A}{2}N_A}$

solving for NA gives

$\displaystyle{N_A = -\frac{c_{total}D\nabla y_A}{1+\frac{y_A}{2}} }$

which in one dimension becomes

$\displaystyle{N_A = -\frac{c_{total}D\frac{dy_A}{dz}}{1+\frac{y_A}{2}} }$

OUTCOME:

Calculate the magnitude of diffusive mass flow/flux for systems with non-zero bulk flow