Abstract

Most research on sedimentation-consolidation processes of flocculated suspensions, summarized in [3], has been concerned with one-dimensional batch settling models and their extensions to continuous thickening, while industrial thickeners require an at least two-dimensional treatment. However, most 1-D sedimentation models can not be extended to multidimensions in an obvious simple way. The authors have recently proposed a general phenomenological theory of sedimentation-consolidation processes, based on the theory of mixtures, which yields a complete set of model equations in multidimensions [2]. This note is a brief outline of that theory. We assume that the solid particles are small with respect to the sedimentation vessel and have the same density rho(s); that the constituents of the suspension are incompressible; that the suspension is completely flocculated before the sedimentation begins; and that there is no mass transfer between the solid and the fluid. Then the mixture can be described by the local solids volume fraction phi, the solid and fluid phase velocities v(s) and v(f) and Cauchy stress tensors T(s) and T(f), the gravity force b = -gk where k is the upwards-pointing unit vector, and the solid-fluid interaction force per unit volume m. Using the volume-average velocity q = phi v(s) + (1- phi)v(f), the local mass balances for the solid and for the mixture are partial derivative(t)phi + del . (phi v(s)) = 0 and del . q = 0, respectively. The respective solid and liquid component linear momentum balances are rho(s)phi D(t)(s)v(s) = del . T(s) + rho(s)phi b + m and rho(f)(1 - phi)D(t)(f)v(f) = del . T(f) + rho(f)(1 - phi)b - m.