Page 295 - Process Modelling and Simulation With Finite Element Methods

282 Process Modelling and Simulation with Finite Element Method5

In solid particle dynamics, the population balance equations also are
expressible in terms of some convolution integrals. Nicmanis and Hounslow
[27] used FEM to describe an integro-differential equation with convolution-type
integral terms, where the major processes of this type are aggregation and
breakage. The collision rules for bubbles and particles depend substantially on
the physicochemical properties of the liquid medium. Traces of flocculent and
coagulent effect the probability of bubble-particle agglomeration and floc
formation. However, in terms of particle dynamics, the collision rules can be
formulated to match observed kinetic rates. Thus, a semi-empirical approach to
population balance equations, fitting the coefficients of the aggregation,
breakage, and growth models, is a successful technique in characterizing particle
processes. Randolph and Larson [28] cite the change in number density n(v) of
particles with volume n in the product stream of a continuous mixed-suspension,
mixed-product removal crystallizer in which these three processes are occurring
from an inlet stream with feed population ni,(v):

(' ) - nin (' + d (v ) ~1 (v )) = b (v ) - d (v ) (7 27)
(G
z dv
where T is the residence time in the crystallizer, G(v) is a volume-dependent
growth function and the number density of nuclei is incorporated into the
equation as a boundary condition n(O)=no. b(v) and d(v) are suggestively
denoted as the birth and death terms for the volume fraction of size v. In
general, there are contributions to both terms from both aggregation and
breakage. Case 1 considered by [27] is a purely aggregation model, so we will
cite only the forms derived by Hulburt and Katz [29] for aggregation.

%
b(v) = p (v - w, w)n(v - w)n( w)dw
-
U (7.28)
d(v) = n(v)p (v,w)n(w)dw
0
Succinctly, birth by aggregation is due to the probability of combining particles
with volumes which sum to v (and sticking). Death is by the probability of a
particle of volume v participating in a collision (and sticking).