Where for spherical pellets of radius r_s , for a first-order reaction is given as

(13)

k₁ is rate constant, ρ_p is pellet density; D_e is effective diffusivity.

Hence, to calculate Thiele modulus, both effective diffusivity D_e and rate constant k₁ need to be known. Fig 3 shows the typical plot of η vs .

Fig. 3. Typical plot effectiveness factor η as function of Thiele modulus φ_s

As the figure shows, for small values of . Equation (13) shows that small values of are obtained when the:

1. 1. pellets are small

2. diffusivity is large

3. reaction is intrinsically slow ( catalysts of low activity).

For , can be approximated as . For such large , intra-particle diffusion has a large effect on the rate. At this condition, diffusion into the pellet is relatively slow and reaction occurs before the reactant has diffused far into the pellet. When , rate for the whole pellet is the same as the rate at surface concentration C_s and the entire catalyst surface is fully active.