A problem of widespread industrial and theoretical importance is the separation of fine solids from liquids. The suspended solids are consolidated under the influence of a body force applied to the particles, for example, gravitational force in gravity thickening or an applied pressure in a pressure filter. Such mechanical dewatering processes use much less energy than evaporative drying.
Gravitational thickening, both batch and continuous, has received quite alot of attention in the literaturel-10 . However little mathematical analysis has been done on the other common type of dewatering process, pressure filtering. This technique relies on the removal of liquid by expression, that is, compression of suspended material with drainage. A simple example is a cylinder containing the material, compressed by a piston at one face and only the liquid is allowed to pass through a porous membrane at the other face. Specifying the fluid expression rate or applied pressure are two modes of operation of such filters. Some simple linear models 11-12 for constant expression rate and constant applied pressure, and a nonlinear problem for stable suspensions have been studied. Models incorporating the yielding solid rheological properties of flocculated suspensions 2,14,15 will be discussed here. A full understanding of the dependence and sensitivities of filter presses to the various physical parameters in the process will be of value in designing more efficient presses, and ultimately to optimizing the performance of pressure filters. Further, some of the material properties of these suspensions, for example the yield stress, may be able to be determined from a simple test experiment using filter presses, as has been described for batch centrifuge settling2.
Buscall and White2 discussed the rhealogical properties of concentrated suspensions and they and Howells et al7 applied it to settling under gravity of a flocculated suspension in a closed bottom container. Such suspensions have also been studied by us in a gravity thickeners. The particles of the suspension interact directly with one another to give rise to a local particle pressure ps , which is the effective stress tensor. When electrolyte or polymer flocculants have ‘been added to the suspension connected aggregate structures of many particles are produced held together be van der Waals or polymer - bridging forces. Once the average particle volume fraction is high enough that a network of connected particles is formed, the suspension takes on the properties of a solid (albeit flimsy) structure. In particular, compressive stresses on the suspension can be transmitted via the network throughout the system and the structure then possesses the ability to support itself. In a flocculated system above this volume fraction, the particle pressure ps should be more properly thought of as a network pressure. When such a network has formed’ throughout the system, we are free to increase ps by applying some sort of external compression to the network for example, push on it with a piston, or increase the gravitational forces in a centrifuge. As this process is applied, the network structure will resist further compression and ps will increase until the compressive forces become so strong that the structure will begin to deform irreversibly. The rheological property to describe this is the compressive yield stress P,(Q), which is defined as the value of the network pressure at which the flocculated suspension at volume fraction o will no longer resist compression elastically, and will start to yield and so irreversibly consolidate.
This compressive yield stress PY(o) is an implicit function of the strength of the interparticle bridging forces and possibly the previous shear history of the system, which will determine the primary floe size and internal structure. In Buscall and Whitez, it is , shown how the equilibrium bed height measurehents in a centrifuge can be used to measure PY(o). Power law curves of the type
with various values of n or m have been fitted to experimental systems14-15. Here og is called the gel point, and is the value below which P,(Q) cannot be experimentally distinguished from zero. It may be considered the volume fraction at which all the primary floes become interconnected This concept of a network pressure is used later in the kinetic description of consolidation.
A one-dimensional model for cylindrical falter presses will be established, using the rheological properties discussed above. The analysis can then be divided into the two distinct modes of operation - specified fluid expression rate or specified applied piston pressure. In the first mode approximate analytic results can be obtained for typical P,(o) profiles. When the applied piston pressure is specified, the fluid flux and piston position must -be solved.for as part of the moving boundary problem. Some analytic results for small times along. with a numerical method for solving the full transient solution are given. These two modes of operation show fundamental differences and these are discussed in the conclusion. Depending on the qualities of the filter cake and the speed of operation, one mode of operation may be more suitable than another. The case in which the suspension is initially fully networked (o0 > og ), and the other case when it is initially unnetworked (o0 >og) are both considered. The properties exhibited by the two cases are quite different and will be discussed separately.