Diffusion and Reaction in a Catalyst Pellet

The overall rate of reaction in an isothermal, porous, spherical catalyst pellet is calculated for a first-order, gas-phase reaction that is limited by diffusion in the catalyst pores. This Demonstration plots the reactant concentration inside the catalyst pellet versus the pellet radius. Use the sliders to set the pellet radius, diffusivity and reaction rate constant. The Thiele modulus is a dimensionless number that represents the ratio of reaction rate to diffusion rate. The effectiveness factor is the overall rate of reaction divided by the rate of reaction if the entire catalyst were at CA,s = 0.04 mmol/cm3 (the external surface concentration).

Download the CDF file to view the simulation using the free Wolfram CDF player.

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:
where r is the radius of the catalyst pellet (cm), CA is the concentration in the catalyst (mmol/cm3), k is the first-order rate constant (1/s),   is the effective diffusivity (cm2/s), φ is the pellet porosity, σ is the constriction factor, τ is tortuosity; and φ, σ, and τ are set based on typical values for these variables and are unitless.

The boundary conditions for the differential equation in spherical coordinates are: 
where R is the pellet radius (cm) and CA,s is the concentration (mmol/cm3) at r = R.

The differential equation in dimensionless form is:
where Ψ = CA/CA,s and λ =r/R . The Thiele modulus Φ is the dimensionless ratio of the surface reaction rate to the diffusion rate.

The boundary conditions in dimensionless form are:
The solution to the differential equation in dimensionless form is:
where c1 and c2 are constants that are obtained using the boundary conditions. The solution becomes:  
and substituting the dimensionless variables into the solution yields:

The overall rate of reaction MA (mmol/s) is:

substituting this into the solution yields: 
The effectiveness factor η is the reaction rate in the pellet divided by the reaction rate at the surface:
Thus, η is: