An aerogel catalyst support is a sol-gel network of nanometre struts that packs an enormous internal surface (often 400–1000 m²/g) into a small pellet, giving active-metal nanoparticles a huge area to disperse on. But reactant gas has to diffuse down the mesopores to reach the catalytic sites buried deep inside — and in pores this narrow, molecules collide with the pore walls far more often than with each other, so transport is Knudsen diffusion, not bulk (Fick) diffusion:
D_K = (d_p / 3) · √(8RT / πM)
Comparing the pore-diffusion rate to the reaction rate for a spherical pellet of radius R gives the Thiele modulus φ. Solving the steady-state diffusion–reaction equation in spherical coordinates for a first-order reaction yields the concentration profile and the effectiveness factor η (the fraction of the pellet's intrinsic activity actually realised, once internal diffusion limits access):
φ = R √(k / D_eff)
C(r)/C_s = (R/r) · sinh(φ·r/R) / sinh(φ)
η = (3/φ²) · [φ·coth(φ) − 1]
When φ is small (fine mesopores, thin pellet, slow kinetics) diffusion is fast relative to reaction, C(r) stays close to the surface value everywhere, and η → 1 — every catalytic site is used. When φ is large (wide pores gone macroporous and slow, or a big pellet with fast intrinsic kinetics) the reactant is consumed before it reaches the core, C(r) collapses toward zero at the centre, and η → 3/φ — most of that expensive high-surface-area interior sits unused. This is exactly why real aerogel catalyst supports are engineered as thin monoliths, beads or coatings rather than large blocks: past a certain size the extra surface area stops paying for itself.
- Pellet radius R — larger pellets raise φ and push the profile toward a diffusion-starved core.
- Pore diameter dp — wider pores raise the Knudsen diffusivity DK, speeding transport, but a fixed skeletal volume then packs less surface into the same mass (BET area ≈ 6/(ρs·dp) for a cylindrical-pore model).
- Rate constant k — models catalyst loading/activity or temperature (via the Arrhenius law); higher k raises φ for the same geometry.
- Node colour on the pellet = local C(r)/Cs from the analytical profile above — dark blue is depleted, yellow-white is near the bulk surface concentration.