...

23.- Modeling Batch Sedimentation Processes

by user

on
Category: Documents
1

views

Report

Comments

Transcript

23.- Modeling Batch Sedimentation Processes
Case 23
Modeling Batch Sedimentation Processes
A homogeneous solid-liquid mixture consist of dense solid particles suspended in a fluid.
The solid particle concentration is independent fo height. If such a mixture contained inside
a tall container is the allowed to settle, solid particles descend driven by their own weight
and they become concentrated in the lower end of the container. This process is called
sedimentation and has multiple applications in process and extractive metallurgy.
The objective of sedimentation process analysis is to determine the rate at which solid
particles accumulate in the lower end of the container. The sedimentation process has been
empirically investigated and the results of such studies are summarized in the so-called
Kynch curve. Iso-solid particle concentration curves are drawn inside the container. The
displacement of such curves with time determines the sedimentation rate.
In this study the particle method has been used to mathematically investigate the sedimentation process. In the particle method considers the system as constituted by a number
of mutually interacting computational particles Each particle interacts with all other particles in the assembly and the net force acting on the particle must be computed by summing
the interactions with all particles, i.e. for a system containing n particles, the force on the
i−th particle is
Fi =
N
X
Fij
j6=i=1
To represent a sedimenting two phase mixture constituted by materials with two different
densities, computational particles with two different masses are used. Moreover, one can
introduce different interaction force parameters among the particles.
In practice, an assembly of particles placed initially at a selected set of individual locations
will in general not be in equilibrium. Particles will either be too close or too far from each
other and will tend to repel/attract. Therefore, proper initialization of a particle method
calculation requires a period of relaxation to allow interparticle separations to reach their
equilibrium values.
The numerical computation of the time evolution of the system of particles is based on
the (Newton) equations of motion for the particles, i.e.
mi
dri
= Fi
dt2
1
for i = 1, 2, ..., N .
Specifically, the calculation starts by computing the net force acting on each particle is
the assembly resulting from the interaction with all other particles. To reduce the expense of
computations an empirically determined cutoff radius is introduced and only the interaction
forces produced by those particles located relatively close to the given particle are accounted
for in the sum. For time advancement, the so called modified Verlet algorithm has proven
most stable and accurate. In this algorithm one considers a given particle of mass m at three
consecutive time levels, t − ∆t, t and t + ∆t, where ∆t is a small, suitably chosen time step
and then computes position at t + ∆t by
r(t + ∆t) = 2r(t) − r(t − ∆t) + ∆t2 F(t)/m
The velocity at t is computed by
v(t) =
r(t + ∆t) − r(t − ∆t)
2∆t
and finally, the velocity at t + ∆t is computed by
v(t + ∆t) = v(t) + ∆t[F(t + ∆t) + F(t)]/2
The above calculation is performed for all particles in the assembly to complete the
advancement of the computation by one time step from t to t + ∆t and the process is simply
repeated as many times as desired.
Selected computed results for the sedimenting two phase system are shown in the attached
figures.
2
Fly UP