The fundamental modes of mixing of granular materials in slowly rotating cylinders

A mathematical model is derived for the mixing of a uniform granular materi al in a slowly rotating cylinder. The model assumes a plane failure surface at a constant dynamic angle of repose, and uniform mixing within the lower half of the failure surface. This leads to an integral equation which desc ribes concentrations of components at the lower half of the failure surface . The equations for the fundamental eigenvalues and eigenfunctions of this integral equation are derived, and shown to provide an adequate approximati on to available experimental data. The rate of mixing is determined typical ly by the real part of the lowest eigenvalue, and the oscillatory nature of mixing is determined typically by the imaginary part of the first eigenval ue. A closed-form expression is obtained for the corresponding eigenfunctio ns, which are typically somewhat radial in nature, and have as many maxima and minima as the order of the eigenfunction. Significant improvement (of a bout 60%) in mixing efficiency is possible if preconditioning allows remova l of most of the first eigenfunction, but whether this can be achieved prac tically remains an open question.