*4.2. Breakage*

To verify the proposed new method for solving the breakage PBE, a simplified process of milling a two-component blend was simulated. Figure 8 shows its flowsheet structure. Two feeders supply material with di fferent particle sizes and API concentrations into the system. Their mixture, shown in Figure 9, enters the mill unit to be mixed with the holdup material and crushed afterwards.

**Figure 8.** Flowsheet structure of the breakage process.

The initial blend entering mill is a mixture of smaller particles with a higher API concentration and larger particles with a lower API concentration. Both distributed parameters are given as Gaussian normal distributions with parameters from Table 3. The mill is initially filled with 50 kg of the same blend (Figure 9).

**Figure 9.** Initial distribution for the breakage process.


**Table 3.** Model parameters used for breakage process.

The particle size is described between 0 to 4 × 10−<sup>3</sup> mm<sup>3</sup> with 100 equidistant classes, whereas the mass content of an active ingredient is distributed over 500 equidistant classes ranging from 0 to 10%. AllmodelparametersusedtosimulatebreakageprocessarelistedinTable3.

The steady-state is reached in 25 min of the process time. Figures 10 and 11 show the distribution of the product after this time point.

**Figure 10.** Simulation results of the breakage process.

**Figure 11.** Breakage diagram.

Due to the constant supply of material during the whole process time, initial distributions *A* and *B* (Figure 11) are still clearly visible in the product. At the same time, it is evident from Figure 10 that the reduction in particle size caused by the breakage process does not lead to changes in the content of the active component: the distribution of the API concentration in the outlet stream remains the same as in the initial distribution. Both *A* and *B* do not mix, but only change particle size gradually grinding down from *A* to *C* and from *B* to *D*.

Thus, with the help of transformation matrices, it was possible to avoid mixing the secondary parameters, despite the fact that the model of the mill unit was developed considering only the particle size distribution.

## *4.3. Coupled Agglomeration and Breakage*

Having both models of agglomerator and mill, it is possible to simulate part of a complex pharmaceutical process with a closed circuit and an external classification of the material, as it is shown in Figure 12. New material enters the process through two feeders, supplying particles with different sizes and concentrations of API, which mixture is shown in Figure 13. This blend is mixed with the holdup inside the agglomerator, where the growth of particles occurs. The two-compartment screen unit separates material leaving the agglomerator into three fractions. Oversized particles are crushed using the mill unit, and afterwards combined with the undersized particles and sent back to the agglomerator by mixing with external material from feeders. The middle-sized agglomerates are considered as product. Initially, both agglomerator and mill are filled with the same material as it comes from feeder 1.

**Figure 12.** Flowsheet structure of a coupled agglomeration and breakage process.

All model parameters used to simulate the process from Figure 12 can be found in Table 4. Initial distributions of material are given by Gaussian function.


**Table 4.** Model parameters used for coupled agglomeration and breakage process.

**Figure 13.** Initial distribution for the coupled agglomeration and breakage process.

The flowsheet was simulated until a steady state was reached, which occurred after 40 min of the process time. The distribution of the product is shown in Figure 14. There are two peaks for particles with a higher API concentration. One is mainly due to the presence of the source blend from feeder 2 (its coarse part obtained after the sieving), mixed with new particles formed in agglomerator and mill. The second peak is the result of agglomeration of particles with the same high API concentration with each other. For a low API content, only one peak is observed, since all smaller fractions (from agglomeration of the primary material from feeder 1 and milling results) were cut off by the screen unit. At the same time, a mixture of two initial blends, formed as a result of the agglomeration of materials with different API content, is present in a significant amount, but does not dominate the material with initial concentrations.

**Figure 14.** Distribution of the material in the product at steady-state.

In general, using the description of granular materials with multidimensional distributions and units developed on the basis of transformation matrices, it is possible to perform simulations of complex processes involving materials distributed over a large number of dimensions. Thus, in this example, it is possible to assess the homogeneity of the material that appears at the output of the simulated process, since this parameter is crucial in pharmaceutical production.
