**5. Conclusions**

The use of transformation matrices is a prerequisite for working with multidimensional distributed parameters of solids. This approach allows for proper handling of all parameters of granular materials, even those which are not directly handled by the model. Consequently, the number of distributed parameters in the model and in its environment may not match. At the same time, the development of each individual model becomes more complicated due to the fact that in contrast to the traditional approach, when the output distribution is explicitly calculated and set to the outlet, it is required to derive the laws of material transformation between inputs and outputs for each time step.

In this article, the applicability of transformation matrices to the description of processes of agglomeration and crushing, which are usually formulated by the population balance equations, was shown. The proposed new methods allow for extracting information about the transformation laws from PBEs and calculating the transformation matrices on that basis. The approach uses the finite

volume method for spatial discretization and the second-order Runge–Kutta method for obtaining the complete discretized form of the population balance equations.

To perform the test studies, the derived methods have been implemented in the dynamic flowsheet simulation environment Dyssol. Some exemplary production processes and their parts were simulated using the derived models to prove the applicability and advantages of presented approaches for modelling of granular materials described by multidimensional distributed parameters.

As case studies have shown, the use of transformation matrices enables the correct calculation of the dependent secondary parameters, avoiding their mixing during the simulation. The proposed method can be extended with additional laws for dependent parameters, for example, for mixing of materials. This will significantly expand its application scope and will allow simulating more sophisticated processes. In general, the proposed approach allows usage of a more complex description of materials with the help of several distributed parameters, without the need to adjust each specific model, and transferring the task of their correct calculation to the modelling system. In this way, the new method of reformulating population balance equations greatly increases the possibilities for creating generally applicable systems for the simulation of solid phase processes.

**Author Contributions:** Conceptualization: M.D. and J.K.; methodology: M.D., N.D., and V.S.; software: V.S.; validation: V.S. and N.D.; formal analysis: V.S.; investigation: V.S. and N.D; data curation: V.S.; writing—original draft preparation: V.S. and N.D.; writing—review and editing: V.S., N.D., M.D., S.H., and J.K.; visualization: V.S.; supervision: S.H., J.K., and M.D.; project administration: M.D.; funding acquisition: S.H., M.D., and J.K.

**Funding:** Authors gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within the priority program SPP 1679 "Dynamic simulation of interconnected solids processes DYNSIM-FP"; of the German Academic Exchange Service (DAAD) via Research Grants—Bi-nationally Supervised Doctoral Degrees, 2017/18 (57299293); of the Alexander von Humboldt Foundation (Research Group Linkage Programme); and of the DFG—Projektnummer 392323616 and the Hamburg University of Technology (TUHH) via the funding programme "Open Access Publishing".

**Conflicts of Interest:** The authors declare no conflict of interest.
