Unifying the Theory of Thickening, Filtration, and Centrifugation

Abstract

Solid/liquid separation operations involve relative movement of solids and liquids in both slurries and porous compactible beds. Thickening, filtration, and centrifugation are governed by the simultaneous flow of liquid through and compaction of porous paniculate beds. In the relative motion, the liquid may have a higher velocity than the solids as in filtration or the reverse as in thickening and sedimentation. Movement of liquid is accompanied by a simultaneous collapse of the structure of the cakes and sediments under mechanical, centrifugal, gravitational, and fluid induced frictional stresses. A unified approach to solid-liquid separation with cake formation requires the solution of two simultaneous equations. The first rate equation (Darcy-Shirato) involves the gradient of the liquid pressure and the velocity of the liquid relative to the solid. The second stress balance equation contains the gradients of both the liquid pressure and the induced stress on the solid matrix. Neither of these equations can be solved independently. Elimination of the liquid pressure gradient between the equations leads to the particulate structure equation whose solution establishes the cake structure and leads to formula for calculating solid and liquid rates. If gravitational or centrifugal sedimentation precedes cake formation, a first order hyperbolic partial differential equation governs the suspension concentration and the flux of solids at the cake surface.