Finite volume approach for fragmentation equation and its mathematical analysis

peer-reviewedThis work is focused on developing a finite volume scheme for approximating a fragmentation equation. The mathematical analysis is discussed in detail by examining thoroughly the consistency and convergence of the numerical scheme. The idea of the proposed scheme is based on conserving the total mass and preserving the total number of particles in the system. The proposed scheme is free from the trait that the particles are concentrated at the representative of the cells. The verification of the scheme is done against the analytical solutions for several combinations of standard fragmentation kernel and selection functions. The numerical testing shows that the proposed scheme is highly accurate in predicting the number distribution function and various moments. The scheme has the tendency to capture the higher order moments even though no measure has been taken for their accuracy. It is also shown that the scheme is second-order convergent on both uniform and nonuniform grids. Experimental order of convergence is used to validate the theoretical observations of convergence

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

Numerical approximations of population balance equations in particulate systems

Magdeburg, Univ., Fak. für Mathematik, Diss., 2006von Jitendra Kuma

Population balance modeling of crystallization for monitoring and optimal control

The Population Balance Equation (PBE) describe the change in the particle size distribution occasioned by a variety of mechanisms. This study evaluate the PBE with nucleation, growth and dissolution. The nucleation with growth process is known to be problematic due to the sharp profiles that it causes. The dissolution process requires the knowledge of the number of particles at the minimal stable size, such that unstable particles are removed from the particulate phase. These conditions require the use of specialized numerical method for mathematical modeling. In this thesis, an emphasis is given in the Moving Sectional Method, which combines the method of classes with the method of characteristics to mitigate numerical diffusion errors. The work focused on the PBE application for the enantioselective crystallization of racemic compounding forming systems. Initially, the conservation of the moments of the distribution was analyzed for the MSM with growth and nucleation mechanisms and methods were proposed for the addition of new elements of the particle size mesh. The estimation of kinetic parameters was approached for the dissolution of NaCl in solutions of monoethyleneglycol (MEG) from data of color patterns (RGB), which can be obtained by experimental apparatus of low cost. A method for determining the operating conditions for the batch crystallizer based on the ternary diagram is described. Subsequently, it is shown that the information obtained in the ternary diagram, such as the maximum yield obtained by the process due to thermodynamics, can be used to formulate constraints for a control method based on non-linear optimization to obtain the desired characteristics of the product.A Equação do Balanço da População (PBE) descreve a mudança na distribuição do tamanho de partícula. Este estudo avalia a PBE com nucleação, crescimento e dissolução. Processos com nucleação e crescimento são conhecidos por serem problemáticos devido aos perfis descontínuos que causam. A dissolução requer o conhecimento do número de partículas no tamanho estável mínimo, de modo que as partículas instáveis sejam removidas da fase particulada. Essas condições requerem o uso de métodos numéricos especializados para sua modelagem matemática. Nesta tese, é dada ênfase ao Moving Section Method (MSM) que combina o método das classes com o método das características para mitigar erros de difusão numérica. O trabalho concentrou-se na aplicação do PBE para a cristalização enantiosseletiva de sistemas formadores de compostos racêmicos. Inicialmente, a conservação dos momentos da distribuição foi analisada para o MSM com mecanismos de crescimento e nucleação e propôs-se métodos para a adição de novos elementos da malha do tamanho de partículas. A estimação de parâmetros cinéticos foi abordada para a dissolução de NaCl em soluções de monoetilenoglicol (MEG) a partir de dados de padrões de cores (RGB), o que pode ser obtido por aparato experimental de baixo custo. Um método para determinar as condições de operação para o cristalizador em batelada com base no diagrama ternário é descrito. Posteriormente, mostram-se que as informações obtidas no diagrama ternário, como o rendimento máximo obtido pelo processo devido à termodinâmica, podem ser usadas para formular restrições para um método de controle baseado em otimização não linear para obter as características desejadas do produto

Simulations of Nanoparticle Synthesis in Laminar Stagnation Flames

A new implementation of a multidimensional solver for studying nanoparticle synthesis in laminar flames is presented. The governing equations are convective-diffusive-reactive partial differential equations that are discretised using the finite volume method. Detailed chemical source terms and transport coefficients are used to close the equations. The implementation of these governing equations is discussed and the numerical algorithm used to solve them is presented. The new solver is verified against analytic solutions and numerical solutions from 1D models for counterflow diffusion flames. The new solver was used to calculate the flame location, shape and temperature of laminar premixed ethylene jet-wall stagnation flames when the equivalence ratio, exit gas velocity and burner-plate separation distance are varied. The simulation results were compared to new experimental 2D measurements of CH* chemiluminescence and temperature. The 2D simulations showed excellent agreement, and correctly predicted the flame shape, location and temperature as the experimental conditions were varied. The new solver was used to study growth of inorganic nanoparticles in premixed, jet-wall stagnation flames. Titanium dioxide, also known as titania and TiO2, is a white powder than has many uses as a pigment, including in paper and cosmetics, and was selected as the system to apply the new solver. TiO2 nanoparticles formed from titanium tetraisopropoxide (TTIP) were simulated using a two step methodology, which enabled insight into the variations of particle properties as a function of the deposition radius. Two different TTIP loadings (280 and 560~ppm) were studied in two flames, a lean flame (equivalence ratio 0.35) and a stoichiometric flame (equivalence ratio 1.0). First, the growth of particles was described with a spherical particle model fully coupled to the conservation equations of chemically reacting flow. Second, particle trajectories were extracted from the 2D simulations and post-processed using a detailed particle model solved with a stochastic numerical method. The simulation produced gas phase predictions of flame location that are in good agreement with available literature. The particle morphologies and size distributions were examined and found to be dependent on the deposition radius. Particles began to have different size distributions at a deposition radius of approximately one and a half times the nozzle radius (1.0 cm), which should be kept in mind when synthesising and modelling nanoparticles for novel applications. This coincided with the growth of total residence time along particle trajectories. It is suggested that experiments critically examine the radially uniformity of deposited particles do not affect the performance for their intended application.Gates Cambridge Foundation, Gates Cambridge Scholarship (OPP1144