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, *V*is 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 N_{A} to the
other N's to get:

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

Plugging this into *N*_{A} 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 *N*_{A} 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

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

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 c_{total} to be constant and rearranging, we can
integrate between the surface of the catalyst (where y_{A}
= 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.)

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

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