Weisz–Prater criterion

The Weisz–Prater criterion is a method used to estimate the influence of pore diffusion on reaction rates in heterogeneous catalytic reactions. If the criterion is satisfied, pore diffusion limitations are negligible. The criterion is
[math]\displaystyle{ N_{W-P} = \dfrac{\mathfrak{R} R^2_p}{C_s D_{eff}} \le 3\beta }[/math]
Where [math]\displaystyle{ \mathfrak{R} }[/math] is the reaction rate per volume of catalyst, [math]\displaystyle{ R_p }[/math] is the catalyst particle radius, [math]\displaystyle{ C_s }[/math] is the reactant concentration at the particle surface, and [math]\displaystyle{ D_{eff} }[/math] is the effective diffusivity. Diffusion is usually in the Knudsen regime when average pore radius is less than 100 nm.
For a given effectiveness factor,[math]\displaystyle{ \eta }[/math], and reaction order, n, the quantity [math]\displaystyle{ \beta }[/math] is defined by the equation:
[math]\displaystyle{ \eta = \dfrac{3}{R^3_p} \int_{0}^{R_p} [1-\beta (1-r/R_p)^n] r^2\ dr }[/math]
for small values of beta this can be approximated using the binomial theorem:
[math]\displaystyle{ \eta = 1-\dfrac{n \beta}{4} }[/math]
Assuming [math]\displaystyle{ \eta = 0.95 }[/math] with a reaction order [math]\displaystyle{ n = 2 }[/math] gives value of [math]\displaystyle{ \beta }[/math] equal to 0.1. Therefore, for many conditions, if [math]\displaystyle{ N_{W-P} \le 0.3 }[/math] then pore diffusion limitations can be excluded.