13 Morphological Modelling and Simulation …
457
experimental conditions. That is, provided the size of the particles remains similar to
the particles in the training set. It was proposed to optimize the developed procedure
so that it works for differently sized particles, too.
4 Modelling of Crystallization with Consideration
of Morphology
The previous sections explain how crystal morphology can be mathematically
described or classified into categories. This section focuses on the effects of complex
morphology on growth of single crystals and agglomerates.
For simulations of crystallization processes it is common to assume populations
of crystals that can be described with only one size parameter (e.g. diameter for
spherical particles) [36]. However, Fig. 11 shows that this a bad assumption for
complex crystalline structures and complex single crystals (Fig. 3 left-hand side).
The problem with this assumption is that one can often not correctly describe both the
volume and the surface area of crystals at the same time with only one size parameter.
Authors then often chose volume-equivalent diameters to model e.g. growth of the
crystals because so at least the mass is conserved. However, growth depends on
the surface area that is available for growth. Therefore, significant inaccuracies are
introduced by simplifying the morphology of crystals.
4.1 Modelling of the Growth of Faceted Crystals
Reinhold and Briesen [8] addressed the growth of potash alum crystals under consideration of their morphology. The behavior of populations—such as crystals in
suspensions—is frequently modeled with population balance models [37]. A simplified version that considers only growth of a constant number of particles can be
written as [8]:
∂n
∂t
+ ∇(gn) = 0
(29)
where n is the number density distribution of a population of, in this case, crystals and
g is the growth rate of crystal surfaces. Equation (29) describes the problem studied
by Reinhold and Briesen [8]: 26-faced potash alum crystals (as shown in Fig. 1)
were considered to grow in an ideally mixed batch crystallizer. A constrained H C -
representation was employed to reduce the dimension of the face distances to seven
entries in h C . This means that the growth rate g in Eq. (29) was actually a vector
with seven entries that each described the growth rate of one crystal face group.
Therefore, the number density distribution also features seven internal coordinates.
457
experimental conditions. That is, provided the size of the particles remains similar to
the particles in the training set. It was proposed to optimize the developed procedure
so that it works for differently sized particles, too.
4 Modelling of Crystallization with Consideration
of Morphology
The previous sections explain how crystal morphology can be mathematically
described or classified into categories. This section focuses on the effects of complex
morphology on growth of single crystals and agglomerates.
For simulations of crystallization processes it is common to assume populations
of crystals that can be described with only one size parameter (e.g. diameter for
spherical particles) [36]. However, Fig. 11 shows that this a bad assumption for
complex crystalline structures and complex single crystals (Fig. 3 left-hand side).
The problem with this assumption is that one can often not correctly describe both the
volume and the surface area of crystals at the same time with only one size parameter.
Authors then often chose volume-equivalent diameters to model e.g. growth of the
crystals because so at least the mass is conserved. However, growth depends on
the surface area that is available for growth. Therefore, significant inaccuracies are
introduced by simplifying the morphology of crystals.
4.1 Modelling of the Growth of Faceted Crystals
Reinhold and Briesen [8] addressed the growth of potash alum crystals under consideration of their morphology. The behavior of populations—such as crystals in
suspensions—is frequently modeled with population balance models [37]. A simplified version that considers only growth of a constant number of particles can be
written as [8]:
∂n
∂t
+ ∇(gn) = 0
(29)
where n is the number density distribution of a population of, in this case, crystals and
g is the growth rate of crystal surfaces. Equation (29) describes the problem studied
by Reinhold and Briesen [8]: 26-faced potash alum crystals (as shown in Fig. 1)
were considered to grow in an ideally mixed batch crystallizer. A constrained H C -
representation was employed to reduce the dimension of the face distances to seven
entries in h C . This means that the growth rate g in Eq. (29) was actually a vector
with seven entries that each described the growth rate of one crystal face group.
Therefore, the number density distribution also features seven internal coordinates.