CL: Diffusion through Cylindrical or Spherical Shells

If we examine the same heterogeneous catalysis problem as before, for reaction,

$2A \rightarrow 2B+C$

but now use a spherical catalyst:

our problem changes slightly.

We still have a net molar flux away from the surface, and must still begin with the definition of the total flux (if solving for species A):

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

Again, 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:

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

Plugging this into NA we get

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

solving for NA gives

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

If our one dimension now is spherical coordinates we get

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

NOTE:

While the flux was constant in a planar system (since the area didn't change with position), only the flow is constant in cylindrical or spherical coordinates! This changes how we integrate this equation.

Writing this in terms of the molar flow, we get

$\displaystyle{\dot M_A = -4\pi r^2c_{total}D\frac{\frac{dy_A}{dr}}{1+\frac{y_A}{2}}}$

Taking ctotal to be constant and rearranging, we can integrate between the surface of the catalyst (where yA = 0, if the reaction is very fast) and "far away" from the surface (where we expect to regain the bulk concentration):

$\displaystyle{\dot M_A\int_R^\infty \frac{dr}{r^2} = -4\pi r^2c_{total}D\int_0^{y_{A_{bulk}}}\frac{dy_A}{1+\frac{y_A}{2}}}$

whose solution is

$\displaystyle{\dot M_A = -8\pi Rc_{total}D\ln{\left (1+\frac{y_{A_{bulk}}}{2}\right )}}$

(Recall that the negative sign means that the flow of A is toward the sphere surface.)

NOTE:

The approach for cylindrical geometries is the same, but a solution for a known concentration at infinity is not possible.

OUTCOME:

Calculate the magnitude of diffusive mass flow/flux through a cylindrical and spherical shells