Three viewpoints used by our group provide an efficient and useful method of analysis for many problems of powder processing: a geometrical approach, a kinematic approach, and a micro-mechanical approach.
Geometrical
An important feature of many powder
processing applications is that the geometry of the system dictates the
flow of the material. Because of this, it is sometimes possible to
derive a substantial amount of insight from purely geometrical
considerations. For example, in a slowly rotated tumbler mixer --
where mixing proceeds via avalanches down the free surface -- the
geometrical and dynamical components of the mixing can be decoupled.
For a non-segregating mixture, neglecting the detailed dynamics and
examining only the motion of the avalanches yields much information and
agrees with experiment both qualitatively and quantitatively. By
decomposing the problem in this way, it is possible to isolate the
geometrical aspects and to add complexities in a controlled fashion.
This allows even seemingly difficult problems to be probed in a simple
and methodical way.
Kinematic
Another method for studying
particulate systems is the kinematic approach. Take a continuously
flowing two-dimensional tumbler mixer as an example. In this mixer,
the rotation produces closed streamlines which trace out a region of
solid body rotation and one of down-surface flow. Applying mass and
momentum balances to the region of down-surface flow yields a
quantitatively accurate model of the entire flow field of the mixer.
Mixing and segregation within this device are then incorporated into
the model by the addition of mass flux terms; the exact form and
parameter values for these fluxes can be obtained from a
micro-mechanical approach. In the case of a non-segregating mixer of
circular cross-section, the mixing is well described by radial
circulation-time differences as well as diffusion within the region of
down-surface flow. In more complicated geometries -- such as an
ellipse, mixing can also occur via chaotic trajectories induced by an
effective time-variation of the mixer geometry.
Micro-Mechanical
The interactions of individual
grains and surfaces has long been an area of research, both
experimentally and theoretically, and much has been learned in this
area. Because of this and the ever increasing power and efficiency of
modern computers, researchers have recently turned to micro-mechanical
methods of investigation for the study of granular materials. This
approach determines the macroscopic behavior of the flow through
simulation of many simultaneous microscopic collision events. This
type of model has been used with success for a variety applications,
such as: pile formation, shaking simulation, or tumbler mixers.
