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Fragmentation Models

For detailed information on the crystallization models implemented in CADET, including aggregation, please refer to [26].

The fragmentation/breakage model considered here can be combined with Population Balance Models and / or Aggregation Models. Further, it can be applied in any of the unit operations, specifically in a tank or DPFR.

The fragmentation/breakage crystallization model describes the evolution of the particle number density \(n\) driven by particle fragmentation. Here, we consider multiple fragmentation, i.e. the general breakage of a particle into a particle size distribution, based on particle size \(x\), which is called internal coordinate.

Size-based fragmentation is governed by the integro-differential equation

()\[\begin{aligned} \frac{\partial n(x)}{\partial t} &= - S(x) n(x) + \int_x^{x_{\mathrm{max}}} S(\lambda) b(x | \lambda) n(\lambda) d\lambda. \end{aligned}\]

Here, \(x_{\mathrm{end}}\) is the maximal considered particle size, \(b(x | \lambda)\) is the probability density function for the generation of a particle of size :math`:x from breakage of a particle of size \(\lambda\), and \(S\) is the selection function which determines the rate of fragmentation.

()\[\begin{aligned} S(x) := S_0 x^{3 \alpha}, \end{aligned}\]

where \(\alpha > 0\) reckons the breakage rate as a function of particle volume.

The propability breakage function is defined as

()\[\begin{aligned} b(x | \lambda) := 3 x^2 \frac{\gamma}{\lambda^3} \left( \frac{x^3}{\lambda^3} \right)^{\gamma - 2}, \end{aligned}\]

where \(\gamma > 1\) determines the average number of daughter particles into which a mother particle breaks. Further, \(b\) satisfies

()\[\begin{split}\begin{aligned} \int_{x_\mathrm{c}}^\lambda x^3 b(x | \lambda) dx &= \lambda^3, \\ N(\lambda) = \int_{x_\mathrm{c}}^\lambda b(x | \lambda) dx &\geq 1, \end{aligned}\end{split}\]

where \(N(\lambda)\) is the total number of daughter particles that a mother particle of size \(\lambda\) generates on average.

For information on model parameters and how to specify the model interface, see Crystallization / Precipitation models.