What is the Thiele modulus and effectiveness factor in catalytic reactions?

Thiele modulus (phi) is dimensionless group comparing reaction rate to diffusion rate: phi = L*sqrt(k/De), where L is characteristic length, k is rate constant, and De is effective diffusivity. High phi means diffusion-limited; low phi means kinetically-limited. Effectiveness factor (eta) is actual rate divided by rate without diffusion limitation. For first-order reaction in sphere: eta = (3/phi)*(coth(phi) - 1/phi). When phi < 0.4, eta approaches 1 (kinetic control); when phi > 4, eta approaches 3/phi (diffusion control).