A mathematical model for sedimentation-consolidation processes

Most research on sedimentation-consolidation processes of flocculated suspe nsions, summarized in [3], has been concerned with one-dimensional batch se ttling models and their extensions to continuous thickening, while industri al thickeners require an at least two-dimensional treatment. However, most 1-D sedimentation models can not be extended to multidimensions in an obvio us simple way. The authors have recently proposed a general phenomenologica l theory of sedimentation-consolidation processes, based on the theory of m ixtures, 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 sedimentat ion vessel and have the same density rho(s); that the constituents of the s uspension are incompressible; that the suspension is completely flocculated before the sedimentation begins; and that there is no mass transfer betwee n the solid and the fluid. Then the mixture can be described by the local s olids 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- ph i)v(f), the local mass balances for the solid and for the mixture are parti al derivative(t)phi + del . (phi v(s)) = 0 and del . q = 0, respectively. T he 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.