*Article* **Application of Transformation Matrices to the Solution of Population Balance Equations**

#### **Vasyl Skorych 1,\*, Nilima Das 2, Maksym Dosta 1, Jitendra Kumar 2 and Stefan Heinrich 1**


Received: 25 July 2019; Accepted: 11 August 2019; Published: 14 August 2019

**Abstract:** The development of algorithms and methods for modelling flowsheets in the field of granular materials has a number of challenges. The difficulties are mainly related to the inhomogeneity of solid materials, requiring a description of granular materials using distributed parameters. To overcome some of these problems, an approach with transformation matrices can be used. This allows one to quantitatively describe the material transitions between different classes in a multidimensional distributed set of parameters, making it possible to properly handle dependent distributions. This contribution proposes a new method for formulating transformation matrices using population balance equations (PBE) for agglomeration and milling processes. The finite volume method for spatial discretization and the second-order Runge–Kutta method were used to obtain the complete discretized form of the PBE and to calculate the transformation matrices. The proposed method was implemented in the flowsheet modelling framework Dyssol to demonstrate and prove its applicability. Hence, it was revealed that this new approach allows the modelling of complex processes involving materials described by several interconnected distributed parameters, correctly taking into consideration their interdependency.

**Keywords:** population balance equation; dynamic flowsheet simulation; transformation matrix; process modelling; agglomeration; milling; solids; multidimensional distributed parameters
