Kinetic theories for the coagulation and sedimentation of particles

The coagulation and sedimentation of particles is central to many environmental and industrial processes. Kinetic descriptions of this process can be divided into two approaches. Population balance equation (PBE) theory accounts for mass transfer between particle size classes by coagulation and loss of particle mass by sedimentation. Its practical application is limited by the fact that the rate constants, or kernels, for coagulation between all combinations of cluster sizes are unknown, and only a few solutions are available for simplified forms of the coagulation kernel. A second approach involves the use of simple rate expressions for the loss of particle mass with time, where the order of the reaction is determined by the mathematical properties of the coagulation kernel and the sedimentation term. Two different theories (similarity theory, ST, and the quasi-steady-state hypothesis, QSSH) give conflicting estimates for the order of the reaction. In this paper, we derive a PBE solution for the choice of a constant kernel and a particle removal rate that increases linearly with cluster volume. The kinetics of mass removal predicted by this solution are then compared directly to generalized forms of ST and QSSH which we also derive. We find that the PBE solution does not rigorously conform to either ST or QSSH, although the predictions of ST may be close enough for practical applications. This paper presents the first rigorous comparison of PBE, ST, and QSSH descriptions of particle coagulation and sedimentation. (C) 2001 Academic Press.