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.^[1] If the criterion is satisfied, pore diffusion limitations are negligible. The criterion is
${\displaystyle N_{W-P}={\dfrac {{\mathfrak {R}}R_{p}^{2}}{C_{s}D_{eff}}}\leq 3\beta }$
Where ${\displaystyle {\mathfrak {R}}}$ is the reaction rate per volume of catalyst, ${\displaystyle R_{p}}$ is the catalyst particle radius, ${\displaystyle C_{s}}$ is the reactant concentration at the particle surface, and ${\displaystyle D_{eff}}$ 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,${\displaystyle \eta }$, and reaction order, n, the quantity ${\displaystyle \beta }$ is defined by the equation:
${\displaystyle \eta ={\dfrac {3}{R_{p}^{3}}}\int _{0}^{R_{p}}[1-\beta (1-r/R_{p})^{n}]r^{2}\ dr}$
for small values of beta this can be approximated using the binomial theorem:
${\displaystyle \eta =1-{\dfrac {n\beta }{4}}}$
Assuming ${\displaystyle \eta =0.95}$ with a reaction order ${\displaystyle n=2}$ gives value of ${\displaystyle \beta }$ equal to 0.1. Therefore, for many conditions, if ${\displaystyle N_{W-P}\leq 0.3}$ then pore diffusion limitations can be excluded.^[2]

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