14 Numerical Methods for Coupled Population Balance Systems …
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diameter at the final time, than the results for σ = 0.1. There is a particularly good
agreement with respect to the second issue for κ = 8000, where the average diameter
is in the interval [232, 260] μm.
3 Bi- and Multi-variate Processes
The evolution of the crystal population is defined in (1.4) for a d int -dimensional
internal property coordinate. The internal coordinates are estimates for the crystal
size and shape. In Sect. 2.1, the description of a uni-variate substance was introduced.
An example of a bi-variate substance is potassium dihydrogen phosphate. Also for
the bi-variate system, the 3d-crystal shape of single crystals can be determined with
high accuracy [15]. For agglomerated particles, further descriptors can be selected
to describe the size and shape of a particle that is composed of several primary
particles. Primary and agglomerated potash alum crystals are depicted in Fig. 7. The
descriptors for agglomerates may again be based on a shape estimation, e.g., the
projections may be fitted to geometrical polytopes [61]. There is a large number of
further shape descriptors, such as the Feret diameter [23], the length of the boundary
curve of a projected particle, the projection area, the area of the convex hull of the
projection, the diameter, perimeter, and volume of a circle of the same projected
area, the widths of the major and minor axes of an ellipse, the convexity [23], the
eccentricity [71], the sphericity, and the fractal dimension [65]. Here, the volume of
a sphere of equivalent diameter is chosen since the agglomerates in the considered
benchmark process are compact. The volume is used to calculate the mass of a crystal.
The mass is assumed to be an additive property.
3.1 Agglomeration Kernel Identification from Experiments
In Sect. 2.5, an agglomeration dominated crystallizer was presented. Agglomeration
depends on the local distribution of crystals in the fluidized bed crystallizer (FBC),
which is determined by the fluid dynamics in the FBC [6]. The particle movement was
therefore simulated as described in Sect. 2.5. In the agglomeration term (1.5) of the
PBE, the agglomeration kernel κ agg determines the rate of agglomeration. Required
agglomeration kernels can be identified from measurement data and numerical simulations by solving inverse problems. An example of such measurement data is shown
schematically in Fig. 14.
The agglomeration kernel is usually an unknown functional relation of the volumes of agglomerating particles. Thus, the identification of the kernel is an illconditioned problem. To solve the problem, two approaches can be applied. A set of
unknown parameters can be identified using measurement data. For this approach,
the structure of the kernel has to be known, which can be estimated from modeling
the agglomeration process [51] or from known approaches [17]. A second approach
503
diameter at the final time, than the results for σ = 0.1. There is a particularly good
agreement with respect to the second issue for κ = 8000, where the average diameter
is in the interval [232, 260] μm.
3 Bi- and Multi-variate Processes
The evolution of the crystal population is defined in (1.4) for a d int -dimensional
internal property coordinate. The internal coordinates are estimates for the crystal
size and shape. In Sect. 2.1, the description of a uni-variate substance was introduced.
An example of a bi-variate substance is potassium dihydrogen phosphate. Also for
the bi-variate system, the 3d-crystal shape of single crystals can be determined with
high accuracy [15]. For agglomerated particles, further descriptors can be selected
to describe the size and shape of a particle that is composed of several primary
particles. Primary and agglomerated potash alum crystals are depicted in Fig. 7. The
descriptors for agglomerates may again be based on a shape estimation, e.g., the
projections may be fitted to geometrical polytopes [61]. There is a large number of
further shape descriptors, such as the Feret diameter [23], the length of the boundary
curve of a projected particle, the projection area, the area of the convex hull of the
projection, the diameter, perimeter, and volume of a circle of the same projected
area, the widths of the major and minor axes of an ellipse, the convexity [23], the
eccentricity [71], the sphericity, and the fractal dimension [65]. Here, the volume of
a sphere of equivalent diameter is chosen since the agglomerates in the considered
benchmark process are compact. The volume is used to calculate the mass of a crystal.
The mass is assumed to be an additive property.
3.1 Agglomeration Kernel Identification from Experiments
In Sect. 2.5, an agglomeration dominated crystallizer was presented. Agglomeration
depends on the local distribution of crystals in the fluidized bed crystallizer (FBC),
which is determined by the fluid dynamics in the FBC [6]. The particle movement was
therefore simulated as described in Sect. 2.5. In the agglomeration term (1.5) of the
PBE, the agglomeration kernel κ agg determines the rate of agglomeration. Required
agglomeration kernels can be identified from measurement data and numerical simulations by solving inverse problems. An example of such measurement data is shown
schematically in Fig. 14.
The agglomeration kernel is usually an unknown functional relation of the volumes of agglomerating particles. Thus, the identification of the kernel is an illconditioned problem. To solve the problem, two approaches can be applied. A set of
unknown parameters can be identified using measurement data. For this approach,
the structure of the kernel has to be known, which can be estimated from modeling
the agglomeration process [51] or from known approaches [17]. A second approach