You're reading the documentation for a development version. For the latest released version, please have a look at v5.0.4.

Aggregation Models

For detailed information on the crystallization models implemented in CADET, including aggregation, please refer to [30].

The aggregation model can be combined with Primary Particle Formation and/or Fragmentation Models. Further, it can be applied in any of the unit operations, specifically in a STR or DPFR.

The aggregation is governed by the Smoluchowski coagulation equation and describes the evolution of the particle number density \(n\) changes due to agglomeration/aggregation. Here, we consider binary particle aggregation based on particle size \(x\), which is called internal coordinate.

Fig. 9 Particle agglomeration/aggregation considered in the Smoluchowski coagulation equation.

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

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

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 Smoluchowski coagulation equation

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

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 of your system.

Five kernels are already implemented in CADET. They include:

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

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

where \(\beta_0 > 0\) is the aggregation rate constant. Please reach out to us for customized aggregation kernels.

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