A two-scale numerical model of mass transport and adsorption in a cylindrical radial-flow filter

A two-scale (macro–micro) mathematical model and a verified numerical method are developed for mass transport and adsorption in an annular cylindrical filter under inward radial flow. At the macro scale, an axisymmetric advection–dispersion equation with the interstitial velocity |uᵣ| ∝ 1/r is coupled, through external film transfer and local Langmuir equilibrium, to intraparticle surface diffusion inside spherical granules resolved by the homogeneous surface diffusion model. The film coefficient and the radial dispersion are made velocity-dependent through standard Sherwood and dispersivity correlations. The complete dimensionless problem is derived and the governing groups (radial Peclet number, capacity ratio, Biot number, granule diffusion time, and a radius-dependent Damköhler number) are evaluated for a representative granular-activated-carbon case; for these parameters the process is intraparticle-diffusion limited, with Daₚ ∼ 10²–10³. The model is reduced to the classical linear-driving-force law in the fast-diffusion, negligible-film limit. The numerical method combines a conservative finite-volume discretisation on an equal-volume radial grid, Strang operator splitting, and a spectral (modal) reduction of the granule problem. Grid, time-step, and modal convergence, machine-precision global mass conservation, solution positivity, and agreement with the linear-driving-force limit are demonstrated. A controlled comparison against a uniform-velocity bed of equal residence time shows that, at constant intraparticle diffusivity, the radial velocity field mainly redistributes the internal loading while leaving the outlet breakthrough close to the uniform-velocity case.