Radial slip velocity and solid velocity are formulated by using the energy-minimization multi-scale (EMMS) model

leading to analytical solutions for radial distribution of voidage

fluid velocity

cluster diameter and particle mass flux

and indicating the existence of radial heterogeneity and annulus/core structure. Model calculation is well consistent with experimental data. The model overcomes the difficulty in solving the original EMMS model

and makes it possible to predict patterns in gas-solid two-phase systems without using adjustable parameters.