*4.1. Agglomeration*

A simple pharmaceutical process of agglomerating blends with two different concentrations of the active pharmaceutical ingredient (API) was simulated. The process structure is illustrated in Figure 4. The solid material is described using two interdependent distributed parameters:


**Figure 4.** Flowsheet structure of the agglomeration process.

The initial blend entering the agglomerator is a mixture of materials from two feeders (Figure 5):


**Figure 5.** Input distribution for the agglomeration process.<sup>2</sup> To improve readability, in all the distributions shown here and further, values of mass fractions less than 10−<sup>5</sup> were cut <sup>o</sup>ff.

Both particle sizes and API concentrations are given in terms of the Gaussian normal distribution with mean values μ and standard deviations σ shown in Table 1. Feeders continuously supply the material at a rate of 0.25 kg/s each. The final distribution after their mixing is shown in Figure 5. The agglomerator is initially filled with 200 kg of the same blend. All model parameters used to simulate the agglomeration process are listed in Table 1.



Figures 6 and 7 show the distribution of the product stream in steady-state, which was reached after 25 min of process time. Clearly visible individual peaks, which were formed as a result of the agglomeration of particles with different sizes and different API concentrations. Since the initial blend is supplied continuously at a constant rate during the entire agglomeration process, the initial distributions *A* and *B* (Figure 7) are still distinguishable in the final mixture. Agglomeration of material *B* with itself gives *F*, which has a doubled mean size and the same API concentration as *B*, as expected. In turn, agglomeration of *B* with *F* (having the same concentrations) gives a peak at *M*, which does not alter API content and has a particle size equal to (*B* + *F)*. The same is true for the material from feeder 1, agglomerating with itself according to the schema: (*A* + *A*) = *C*, (*A* + *C*) = *E*, and (*A* + *E*) + (*C* + *C*) = *H*.

**Figure 7.** Agglomeration diagram.

More complex behaviour is observed if particles with different API concentrations agglomerate. In this case, the particle sizes are added up and the concentration is averaged according to Equation (59). For example, agglomeration of particles from *A* and *B* gives *D*. Further, G is formed as a mixture of (*A* + *D*) and (*B* + *C*). The result of the agglomeration of *B* and *C* is shifted below the concentration value of 0.05 due to the mutual influence of two factors. First, the introduction of the size-dependent growth rate β∗ (Equation (57)) leads to faster growth of smaller particles. Second, averaging by the second distribution takes into account the actual amount of material being agglomerated in selected classes (see Equations (59) and (60)).

The complete scheme of material transitions during the agglomeration, shown in Figure 7, is represented in Table 2.


**Table 2.** Material transitions during agglomeration.

Thus, information on the change in particle size during the agglomeration process, obtained from the generated transformation matrix, made it possible to take into account and correctly process the dependent distributed parameters of the agglomerated material.
