Details
For the first-order reaction A ? B with rate law rA = kCA, the differential equation that describes diffusion and reaction in the catalyst pellet is:
\frac{d^2C_A}{dr^2} + \frac{2}{r}\frac{dC_A}{dr} - \frac{kC_A}{D_e} = 0
where r is the radius of the catalyst pellet (cm), C_A is the concentration in the catalyst (mmol/cm³), k is the first-order rate constant (1/s), $D_e = \frac{D_{AB}\phi\sigma}{\tau}$ is the effective diffusivity (cm²/s), ? is the pellet porosity, ? is the constriction factor, ? is tortuosity; and ?, ?, and ? are set based on typical values for these variables and are unitless.
The boundary conditions for the differential equation in spherical coordinates
are:
C_A = finite at r = 0
C_A = C_{A,s} at r = R
where R is the pellet radius (cm) and C_{A,s} is the concentration (mmol/cm³) at r = R.
