Application of Transformation Matrices to the Solution of Population Balance Equations

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.

Special Issue on “Chemical Process Design, Simulation and Optimization”

“…It is applied to the classical Tennessee Eastman process. Skorych et al [6] study the solution of population balance equations. This type of equation occurs in the description of the distributions of sizes of granular materials, such as polymers and crystals.…”

Integrated process simulation of porcelain stoneware manufacturing using flowsheet simulation

“…Particles can be distributed along various property coordinates, such as size, density, moisture content, etc. Thus, to perform a correct numerical calculation of material streams described with multidimensional parameters the special approaches like transformation matrices should be applied [18]. For dynamic and steady-state flowsheet calculations of solids processes an open-source simulation framework Dyssol has been developed [17,19].…”

Flowsheet simulation of solids processes: Current status and future trends

“…are neglected or estimated only partially. From the computational point of view there already exist methods to treat such information such as multidimensional population balance models or transformation matrices [14]. The main challenge is in the derivation of mechanistic or first-principles models, where knowledge of microscale processes like a formation of nuclei, their further agglomeration, densification [15], etc.…”

“…Such implicit calculation makes it possible to extend applicability of model significantly. In this case the models developed for strictly limited dimensions of parameters space can be used for a larger number of dimensions [14].…”

“…• development of numerical methods for proper treatment of multidimensional distributed parameters. This can include solvers of multidimensional PBM, like finite volume method [65], calculation approaches like transformation matrices [14] or alternative methods like Monte Carlo approach [66];…”