14 Numerical Methods for Coupled Population Balance Systems …
479
phenomena are dominant. They have to be identified and integrated in the PBS as
shown in (1.4), while other terms may be neglected. The resulting coupled PBS needs
to be parameterized. For that, benchmark problems are required. Here, a growth
dominated crystallizer is selected.
As mentioned in the previous section, the crystal mass can be used as internal
property coordinate of the PBS. The goal of the presented benchmark problem is
to intensify a process to grow faceted crystals shape-selectively. Hence, a measure
of crystal size is applied as internal coordinate. To determine the crystal size distribution (CSD), 3d-crystal shapes are estimated from 2d-projections of the observed
crystals following the methods by [9–11]. The shape is described by the perpendicular distances of the crystal faces to the crystal center. It is sufficient to consider one
perpendicular distance x (m) for each face type to describe the full symmetry of an
ideal crystal. Potassium aluminum sulfate dodecahydrate, also called potash alum,
crystallizes predominantly as octahedron in aqueous solution. Hence, its shape can
be characterized by one face type. The resulting crystal distribution is uni-variate.
The benchmark problem is of high dimension. There are four dimensions in
time and space and one internal coordinate. Further, the solid and liquid phases
are coupled.
2.2 Helically Coiled Flow Tube Crystallizer
2.2.1 Setup and Process
Growth-dominated experiments are realized in a helically coiled flow tube (HCT)
crystallizer. The crystallization is temperature controlled. For the experiments, solution is pumped from a reservoir to the HCT, as depicted in Fig. 1. The solution passes
a degasifier before crystal seeds are added, where the seeds are of a defined size fraction. The suspension is cooled in the HCT to grow. At the outlet of the HCT, the
crystal population is imaged by a flow-through microscope. Finally, the crystals are
dissolved in a reservoir.
Seeds are sieved in different size fractions. The seed fractions are applied for residence time experiments without growth and for growth experiments. In the experiments, several process parameters can be varied systematically: helix orientation,
average fluid flow rate, crystal seed fraction, feed concentration, and temperature
[66, 67]. Selected results are shown for an HCT crystallizer with a coil diameter of
0.11 m and an inner tube diameter of 0.006 m at laminar flow rates.
2.2.2 Residence Time Distribution
In residence time experiments for the dispersed phase, a sieved crystal size fraction
was added within 10 s at the inlet. The solution was saturated and isothermal conditions were applied to avoid crystal growth. The residence time was estimated from
479
phenomena are dominant. They have to be identified and integrated in the PBS as
shown in (1.4), while other terms may be neglected. The resulting coupled PBS needs
to be parameterized. For that, benchmark problems are required. Here, a growth
dominated crystallizer is selected.
As mentioned in the previous section, the crystal mass can be used as internal
property coordinate of the PBS. The goal of the presented benchmark problem is
to intensify a process to grow faceted crystals shape-selectively. Hence, a measure
of crystal size is applied as internal coordinate. To determine the crystal size distribution (CSD), 3d-crystal shapes are estimated from 2d-projections of the observed
crystals following the methods by [9–11]. The shape is described by the perpendicular distances of the crystal faces to the crystal center. It is sufficient to consider one
perpendicular distance x (m) for each face type to describe the full symmetry of an
ideal crystal. Potassium aluminum sulfate dodecahydrate, also called potash alum,
crystallizes predominantly as octahedron in aqueous solution. Hence, its shape can
be characterized by one face type. The resulting crystal distribution is uni-variate.
The benchmark problem is of high dimension. There are four dimensions in
time and space and one internal coordinate. Further, the solid and liquid phases
are coupled.
2.2 Helically Coiled Flow Tube Crystallizer
2.2.1 Setup and Process
Growth-dominated experiments are realized in a helically coiled flow tube (HCT)
crystallizer. The crystallization is temperature controlled. For the experiments, solution is pumped from a reservoir to the HCT, as depicted in Fig. 1. The solution passes
a degasifier before crystal seeds are added, where the seeds are of a defined size fraction. The suspension is cooled in the HCT to grow. At the outlet of the HCT, the
crystal population is imaged by a flow-through microscope. Finally, the crystals are
dissolved in a reservoir.
Seeds are sieved in different size fractions. The seed fractions are applied for residence time experiments without growth and for growth experiments. In the experiments, several process parameters can be varied systematically: helix orientation,
average fluid flow rate, crystal seed fraction, feed concentration, and temperature
[66, 67]. Selected results are shown for an HCT crystallizer with a coil diameter of
0.11 m and an inner tube diameter of 0.006 m at laminar flow rates.
2.2.2 Residence Time Distribution
In residence time experiments for the dispersed phase, a sieved crystal size fraction
was added within 10 s at the inlet. The solution was saturated and isothermal conditions were applied to avoid crystal growth. The residence time was estimated from