New quadrature-based moment method for the mixing of inert polydisperse fluidized powders in commercial CFD codes

To describe the behavior of polydisperse multiphase systems in an Eulerian framework, we solved the population balance equation (PBE), letting it account only for particle size dependencies. To integrate the PBE within a commercial computational fluid dynamics code, we formulated and implemented a novel version of the quadrature method of moments (QMOM). This no longer assumes that the particles move with the same velocity, allowing the latter to be size-dependent. To verify and test the model, we simulated the mixing of inert polydisperse fluidized suspensions initially segregated, validating the results experimentally. Because the accuracy of QMOM increases with the number of moments tracked, we ran three classes of simulations, preserving the first four, six, and eight integer moments of the particle density function. We found that in some cases the numerics corrupts the higher-order moments and a corrective algorithm, designed to restore the validity of the moment set, has to be implemented

Computational Fluid Dynamics of Catalytic Reactors

Today, the challenge in chemical and material synthesis is not only the development of new catalysts and supports to synthesize a desired product, but also the understanding of the interaction of the catalyst with the surrounding flow field. Computational Fluid Dynamics or CFD is the analysis of fluid flow, heat and mass transfer and chemical reactions by means of computer-based numerical simulations. CFD has matured into a powerful tool with a wide range of applications in industry and academia. From a reaction engineering perspective, main advantages are reduction of time and costs for reactor design and optimization, and the ability to study systems where experiments can hardly be performed, e.g., hazardous conditions or beyond normal operation limits. However, the simulation results will always remain a reflection of the uncertainty in the underlying models and physicochemical parameters so that in general a careful experimental validation is required. This chapter introduces the application of CFD simulations in heterogeneous catalysis. Catalytic reactors can be classified by the geometrical design of the catalyst material (e.g. monoliths, particles, pellets, washcoats). Approaches for modeling and numerical simulation of the various catalyst types are presented. Focus is put on the principal concepts for coupling the physical and chemical processes on different levels of details, and on illustrative applications. Models for surface reaction kinetics and turbulence are described and an overview on available numerical methods and computational tools is provided

Solution of bivariate aggregation population balance equation: a comparative study

The present work shows a comparative study of two different numerical methods for solving bivariate aggregation population balance equations. In particular, we summarize the cell average technique (Kumar et al. in Comput Chem Eng 32(8):1810-1830, 2008) and the finite volume scheme (Singh et al. in J Comput Appl Math 308:83-97, 2016) for solving pure aggregation population balance equation. The qualitative and quantitative numerical results of various order moments and number density functions are compared with the exact results for analytically tractable kernels. The results reveal that the finite volume scheme approximates the results more accurately and efficiently as compared to the cell average technique. With respect to mixed moments, in particular, the total variance of excess solute (Matsoukas et al. in AIChE J 52(9):3088-3099, 2006), the finite volume scheme is superior to the cell average technique. Additionally, it is also shown that bicomponent moments are more sensitive to the selection of the grid and require finer discretization to reduce errors

Numerical Methods for Coupled Population Balance Systems Applied to the Dynamical Simulation of Crystallization Processes

Uni- and bi-variate crystallization processes are considered that are modeled with population balance systems (PBSs). Experimental results for uni-variate processes in a helically coiled flowtube crystallizer are presented.Asurvey on numerical methods for the simulation of uni-variate PBSs is provided with the emphasis on a coupled stochastic-deterministic method. In this method, the equations of the PBS from computational fluid dynamics are solved deterministically and the population balance equation is solved with a stochastic algorithm. With this method, simulations of a crystallization process in a fluidized bed crystallizer are performed that identify appropriate values for two parameters of the model such that considerably improved results are obtained than reported so far in the literature. For bi-variate processes, the identification of agglomeration kernels from experimental data is briefly discussed. Even for multi-variate processes, an efficient algorithm for evaluating the agglomeration term is presented that is based on the fast Fourier transform (FFT). The complexity of this algorithm is discussed as well as the number of moments that can be conserved