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Aggregation Models

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

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

The aggregation crystallization model describes the evolution of the particle number density \(n\) driven by particle aggregation. Here, we consider binary particle aggregation based on particle size \(x\), which is called internal coordinate.

The size-based aggregation equation roots in the classical volume based Smoluchowski coagulation equation, which we transform to a size-based equation using the identity

()\[\begin{aligned} n_v \mathrm{d} (k_v x^3) &= n \mathrm{d} x, \end{aligned}\]

which identifies the change of particle count within a volume interval \([k_vx^3, k_vx^3+d(k_vx^3)]\) with the change of particle count within a size interval \([x,x+dx]\).

Size-based binary aggregation is governed by the integro-differential equation

()\[\begin{split}\begin{aligned} \frac{\partial n(x)}{\partial t} &= \frac{x^2}{2} \int_{x_c}^x \frac{\beta \left((x^3-\lambda^3)^\frac{1}{3},\lambda \right)}{(x^3-\lambda^3)^\frac{2}{3}} n\left( (x^3-\lambda^3)^\frac{1}{3}\right) n(\lambda) \mathrm{d} \lambda \\ &\phantom{=} - n(x) \int_0^{x_\mathrm{end}} \beta(x,\lambda) n(\lambda) \mathrm{d} \lambda . \end{aligned}\end{split}\]

Here, \([x_{c}, x_{\mathrm{end}}]\) is the considered particle size interval, \(\beta(x,\lambda)\) is the aggregation kernel specific to the underlying aggregation mechanism. The aggregation kernels considered here are based on a aggregation rate constant \(\beta_0 > 1\):

Constant kernel

\[\beta(x, \lambda) = \beta_0\]

Brownian kernel

\[\beta(x, \lambda) = \beta_0 \frac{(x + \lambda)^2}{x \lambda}\]

Smoluchowski kernel

\[\beta(x, \lambda) = \beta_0 (x + \lambda)^3\]

Golovin kernel

\[\beta(x, \lambda) = \beta_0 (x^3 + \lambda^3)\]

Differential force kernel

\[\beta(x, \lambda) = \beta_0 (x + \lambda)^2 (x^2 - \lambda^2)\]

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