Graphical processing unit (GPU) acceleration for numerical solution of population balance models using high resolution finite volume algorithm

© 2016 Elsevier LtdPopulation balance modeling is a widely used approach to describe crystallization processes. It can be extended to multivariate cases where more internal coordinates i.e., particle properties such as multiple characteristic sizes, composition, purity, etc. can be used. The current study presents highly efficient fully discretized parallel implementation of the high resolution finite volume technique implemented on graphical processing units (GPUs) for the solution of single- and multi-dimensional population balance models (PBMs). The proposed GPU-PBM is implemented using CUDA C++ code for GPU calculations and provides a generic Matlab interface for easy application for scientific computing. The case studies demonstrate that the code running on the GPU is between 2–40 times faster than the compiled C++ code and 50–250 times faster than the standard MatLab implementation. This significant improvement in computational time enables the application of model-based control approaches in real time even in case of multidimensional population balance models

Quadrature-based models for multiphase and turbulent reacting flows

The simulation of physical systems requires accurate and robust methods with relatively low cost and it is still the challenge in many applications of engineering processes, specifically in multiphase flow systems. Soot formation, distribution of the aerosols in the atmosphere, reactive precipitation, and combustion modeling are some examples of these processes. Computer simulations of theses systems require a model that can be adapted to that reality. In this study, a quadrature based method of moments (QBMM) is used to address the problems related to the reactive multiphase flow systems. First, the log-normal kernel density function is implemented into the extended quadrature method of moments (Ln-EQMOM). Ln-EQMOM is verified reconstructing the NDF and calculating the moments of a distribution obtained by the linear combination of two log-normal distributions. Later, this numerical procedure is used for problems of aggregation and breakup of fine particles to solve the population balance equation (PBE). The results are compared to the rigorous solutions reported for the cases under consideration \citep{vanni2000}. Finally, the method is verified using two analytically known problems (\textit{e.g.} coalescence and condensation). In comparison to EQMOM with $\Gamma$ kernel density function \citep{yuan2012}, Ln-EQMOM is faster in terms of computations and it preserves the moments more accurately. Then EQMOM with $\beta$ kernel density function is implemented to approximate the solution of the transport equation for the composition probability density function (PDF) of a passive scalar using the Fokker-Planck model to treat the molecular mixing term. The results then compared in a similar condition to those obtained with direct numerical simulation (DNS). The $L_2$ norm of the PDF is reported for two test cases that have been considered. Later the new approach is introduced to address the problems includes the mixing and reaction. Conditional quadrature method of moments (CQMOM) and using the joint composition PDF for the mixture fraction and progress variables, it is possible to address the problems with two consecutive competitive reactions, one reaction and fast reaction, all including the mixing of reactants. direct quadrature method of moments (DQMOM) also expressed for the joint composition PDF. Results obtained with CQMOM and DQMOM are compared with each other. Finally, the CQMOM approach for mixing problems was tested considering two consecutive competitive reactions to verify the implementation and validate the proposed approach. Coupled mixing-PBE approach was then used to investigate polymer aggregation in a multi-inlet vortex reactor (MIVR), typically used to perform flash nanoprecipitation for the production of nanoparticles used in pharmaceutical applications

Strongly coupled fluid-particle flows in vertical channels. II. Turbulence modeling

In Part I, simulations of strongly coupled fluid-particle flow in a vertical channel were performed with the purpose of understanding, in general, the fundamental physics of wall-bounded multiphase turbulence and, in particular, the roles of the spatially correlated and uncorrelated components of the particle velocity.The exact Reynolds-averaged (RA) equations for high-mass-loading suspensions were presented, and the unclosed terms that are retained in the context of fully developed channel flow were evaluated in an Eulerian–Lagrangian (EL) framework. Here, data from the EL simulations are used to validate a multiphase Reynolds-stress model (RSM) that predicts the wall-normal distribution of the two-phase, one-point turbulence statistics up to second order. It is shown that the anisotropy of the Reynolds stresses both near the wall and far away is a crucial component for predicting the distribution of the RA particle-phase volume fraction. Moreover, the decomposition of the phase-average (PA) particle-phase fluctuating energy into the spatially correlated and uncorrelated components is necessary to account for the boundary conditions at the wall. When these factors are properly accounted for in the RSM, the agreement with the EL turbulence statistics is satisfactory at first order (e.g., PA velocities) but less so at second order (e.g., PA turbulent kinetic energy). Finally, an algebraic stress model for the PA particle-phase pressure tensor and the Reynolds stresses is derived from the RSM using the weak-equilibrium assumption

Discussion on DQMOM to solve a bivariate population balance equation applied to a grinding process

A bivariate population balance equation applied to a grinding process is implemented in a model (PBM). The particles are simultaneously characterized by their size and their mechanical strength, expressed here by the minimum energy needed to break them. PBM is solved by the Direct Quadrature Method of Moments (DQMOM). The mixed moments of the distribution are expressed by the quadrature form of the population density defined for one order (N) and incorporating the weights and the abscissas defined for the two properties. The effect of the quadrature order (N = 2,3,4) and the selected set of the 3N moments needed to solve the system on the accuracy of the results is discussed. For a given order of the quadrature, the selected set of the initial mixed moments slightly affects first the weights and abscissas derived from the initial particle distribution. The set of moments also affects the precision of the moments calculated versus time but only those having high orders in relation with the respective range of the solid properties considered. Problems of convergence and significant differences in the predicted mixed moments are also observed when the order of the quadrature is equal to 2. However, the changes of a bivariate distribution versus time applied to a grinding process are well predicted using the DQMOM approach, choosing a number of nodes equal to 3, associated with a smart selection of the moment set, incorporating all the moments of interest

Optimal Moment Sets for Multivariate Direct Quadrature Method of Moments

The direct quadrature method of moments (DQMOM) can be employed to close population balance equations (PBEs) governing a wide class of multivariate number density functions (NDFs). Such equations occur over a vast range of scientific applications, including aerosol science, kinetic theory, multiphase flows, turbulence modeling, and control theory, to name just a few. As the name implies, DQMOM uses quadrature weights and abscissas to approximate the moments of the NDF, and the number of quadrature nodes determines the accuracy of the closure. For nondegenerate univariate cases (i.e., a sufficiently smooth NDF), the N weights and N abscissas are uniquely determined by the first 2N non-negative integer moments of the NDF. Moreover, an efficient product-difference algorithm exists to compute the weights and abscissas from the moments. In contrast, for a d-dimensional NDF, a total of (1 + d)N multivariate moments are required to determine the weights and abscissas, and poor choices for the moment set can lead to nonunique abscissas and even negative weights. In this work, it is demonstrated that optimal moment sets exist for multivariate DQMOM when N ) nd quadrature nodes are employed to represent a d-dimensional NDF with n ) 1-3 and d ) 1-3. Moreover, this choice is independent of the source terms in the PBE governing the time evolution of the NDF. A multivariate Fokker-Planck equation is used to illustrate the numerical properties of the method for d ) 3 with n ) 2 and 3

Comminution process modeling based on the monovariate and bivariate direct quadrature method of moments

Population balance equations (PBE) applied to comminution processes are commonly based on selection and breakage functions, which allow the description of changes of particle‐size distributions vs. time. However, other properties, such as the particle strength, also influence the grinding kinetics. A bivariate PBE was developed and resolved by the direct quadrature method of moments. This equation includes both the particle size and strength, the latter of which is defined as the minimum energy required for breakage. The monovariate case was first validated by comparing the predicted moments with those calculated from the size distributions given by an analytical solution of the PBE derived for specific selection and breakage functions. The bivariate model was then compared with a discretized model to evaluate its validity. Finally, the benefit of the bivariate model was proven by analyzing the sensitivity of some parameters and comparing the results of the monovariate and bivariate cases

Numerical Investigation of Particles Breakage and Growth in Gas-Solid Processes: Spiral Jet Milling and Polyolefin Polymerization in Fluidized Bed Reactors

L'abstract è presente nell'allegato / the abstract is in the attachmen