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

Analytical solution for a three-dimensional non-homogeneous bivariate population balance equation---a special case

There has been a dramatic increase in the number of research publications using the population balance equation (PBE). The PBE allows the prediction of the spatial distribution of the dispersed phase size for an accurate estimation of the flow fields in multiphase flows. A few recent studies have proposed new efficient numerical methods to solve non-homogeneous multivariate PBE and implemented the same in computational fluid dynamics (CFD) codes. However, these codes are generally benchmarked against other numerical methods and applied without verification. To address this gap, an analytical solution for a three-dimensional non-homogeneous bivariate PBE is presented here for the first time. The method of manufactured solutions (MMS) has been used to construct a solution of the non-homogeneous PBE containing breakage and coalescence terms, and an additional source term appearing as a result of this method. The analytical solution presented in this work can be used for the rigorous verification of computer codes written to solve the non-homogeneous bivariate PBE. Quantification of the errors due to different numerical schemes will also become possible with the availability of this analytical solution for the PBE

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

A framework for polydisperse pulp phase modelling in flotation

Froth flotation is one of the most widely-used mineral processing operations. The pulp zone in flotation tanks is polydisperse in general and serves as a medium for the interaction between the solid particles and the gas bubbles in a liquid continuum, leading to particle–bubble attachment/detachment and bubble coalescence/breakage phenomena. To better predict the hydrodynamics and inform the design of e cient flotation equipment, it is therefore important to accurately model and simulate the evolution of the size distribution of the dispersed phases. This has created an urgent need for a framework that can model the pulp phase in an e cient manner, which is not currently available in the literature. The available software products are not e cient enough to allow for a tractable modelling of industrial-scale flotation cells and in some cases they cannot model the polydispersity of the dispersed phase at all. This work presents an e cient numerical framework for the macroscale simulation of the polydisperse pulp phase in froth flotation in an open-source finite element computational fluid dynamics (CFD) code that provides an e cient solution method using mesh adaptivity and code parallelisation. A (hybrid finite element–control volume) finite element framework for modelling the pulp phase has been presented for the first time in this work. An Eulerian–Eulerian turbulent flow model was implemented in this work including a transport equation for attached and free solid particles. Special care was taken to model the settling velocity of the free solids and the modification of the liquid viscosity due to the presence of these particles. Bubble polydispersity was modelled using the population balance equation (PBE), which was solved using the direct quadrature method of moments (DQMOM). Appropriate functions for bubble coalescence and breakage were chosen in the PBE. Mesh adaptivity was applied to the current problem to produce fully-unstructured anisotropic meshes, which improved the solution e ciency, while all simulations were executed on a multicore architecture. The model was validated for 2D simulations of a bubble column against experimental results available in the literature. After successful validation, the model was applied to the simulation of the pulp phase in a flotation column for monodisperse and polydisperse solids. Polydispersity of the solids was modelled for the first time in this work using three separate solid size classes. A clear dependence of the flotation rate on the particle size was noticed and the monodisperse solids simulations were shown to over-predict the flotation rate. Other than flotation, this open-source framework can be used for the simulation of a variety of polydisperse multiphase flow problems in the process industry

Population balance modelling of polydispersed particles in reactive flows

Polydispersed particles in reactive flows is a wide subject area encompassing a range of dispersed flows with particles, droplets or bubbles that are created, transported and possibly interact within a reactive flow environment - typical examples include soot formation, aerosols, precipitation and spray combustion. One way to treat such problems is to employ as a starting point the Newtonian equations of motion written in a Lagrangian framework for each individual particle and either solve them directly or derive probabilistic equations for the particle positions (in the case of turbulent flow). Another way is inherently statistical and begins by postulating a distribution of particles over the distributed properties, as well as space and time, the transport equation for this distribution being the core of this approach. This transport equation, usually referred to as population balance equation (PBE) or general dynamic equation (GDE), was initially developed and investigated mainly in the context of spatially homogeneous systems. In the recent years, a growth of research activity has seen this approach being applied to a variety of flow problems such as sooting flames and turbulent precipitation, but significant issues regarding its appropriate coupling with CFD pertain, especially in the case of turbulent flow. The objective of this review is to examine this body of research from a unified perspective, the potential and limits of the PBE approach to flow problems, its links with Lagrangian and multi-fluid approaches and the numerical methods employed for its solution. Particular emphasis is given to turbulent flows, where the extension of the PBE approach is met with challenging issues. Finally, applications including reactive precipitation, soot formation, nanoparticle synthesis, sprays, bubbles and coal burning are being reviewed from the PBE perspective. It is shown that population balance methods have been applied to these fields in varying degrees of detail, and future prospects are discussed

Numerical modelling of polydispersed flows using an adaptive-mesh finite element method with application to froth flotation

An efficient numerical framework for the macroscale simulation of three-phase polydispersed flows is presented in this thesis. The primary focus of this research is on modelling the polydispersity in multiphase flows ensuring the tractability of the solution framework. Fluidity, an open-source adaptive-mesh finite element code, has been used for solving the coupled equations efficiently. Froth flotation is one of the most widely used mineral processing operations. The multiphase, turbulent and polydispersed nature of flow in the pulp phase in froth flotation makes it all the more challenging to model this process. Considering that two of the three phases in froth flotation are polydispersed, modelling this polydispersity is particularly important for an accurate prediction of the overall process. The direct quadrature method of moments (DQMOM) is implemented in the Fluidity code to solve the population balance equation (PBE) for modelling the polydispersity of the gas bubbles. The PBE is coupled to the Eulerian--Eulerian flow equations for the liquid and gas phases. Polydispersed solids are modelled using separate transport equations for the free and attached mineral particles for each size class. The PBE has been solved using DQMOM in a finite element framework for the first time in this work. The behaviour of various finite element and control volume discretisation schemes in the solution of the PBE is analysed. Rigorous verification and benchmarking is presented along with model validation on turbulent gravity-driven flow in a bubble column. This research also establishes the importance of modelling the polydispersity of solids in flotation columns, which is undertaken for the first time, for an accurate prediction of the flotation rate. The application of fully-unstructured anisotropic mesh adaptivity to the polydispersed framework is also analysed for the first time. Significant improvement in the solution efficiency is reported through its use.Open Acces

Doctor of Philosophy

dissertationFroth flotation is a highly complex, multiphase, and multiscale process that is usually performed in large tanks called mechanical flotation cells. The aim of this research is to investigate the single and multiphase flow hydrodynamics in lab scale flotation cells by decoupling the hydrodynamics from physicochemical effects. Both experimental and numerical approaches are used to study the behavior of flows in lab and pilot scale flotation cells. Nonintrusive experimental techniques such as particle image velocity (PIV) and electrical resistance tomography (ERT) techniques are used to measure flow velocities, solids holdup, mixing efficiency, and to interpret flow pattern. Eulerian-Eulerian computational fluid dynamics (CFD) models are developed and tested for solid-liquid (slurry) and gas-liquid flows in stirred tanks and flotation cells. Using single phase CFD simulations, the effect of flotation specific impeller blade shape and impeller size on mean flow and pumping behavior is tested in lab scale flotation cells for the first time. In the absence of a stator, the mean flow is found to transition from radial to axial type flow when the off-bottom clearance is below the critical value. This prediction is experimentally verified using time averaged PIV data. Based on the analysis of pumping and power number data, the rectangular shaped blade design is found to be the most efficient. The impeller blade shape is found to critically affect the flow in the vicinity of the impeller and a design with the largest surface area is needed to create an intense turbulence zone, needed for mixing and dispersion of incoming air. Eulerian-Eulerian CFD model is used to study the solid phase suspension and mixing characteristics for monosized silica particles. Experimental comparison with the results from the literature for stirred tanks and in-house ERT measurements suggest that the model performs reasonably well. Population balance equation model (PBM) is coupled with CFD to study gas dispersion, mixing, and local bubble size distribution in the stirred tank and flotation cell using quadrature method of moments (QMOM) approach in ANSYS Fluent solver. The default QMOM model in Fluent is found to be inaccurate due to independent solution of moment transport equations and therefore is supplied with a moment correction algorithm from the literature to successfully identify and correct the invalid moment sequence during the CFD simulation. The new model is found to be superior to the current models in its ability to satisfactorily predict the overall gas holdup and local bubble size distribution for stirred tanks under moderate aeration and agitation rates. This model is extended to study the development of flow regimes based on the gas dispersion pattern in a generic flotation cell. Though highly useful, the coupled CFD-PBM approach is computationally intensive and requires considerable effort to achieve an accurate solution. This motivated us to develop a PBM based on the high-order moment conserving method of classes (HMMC) approach for a pilot scale XCELL flotation cell for frother concentration over critical coalescence concentration, thus, only considering breakage of bubbles. Nonlinear optimization solvers in Matlab are used to calculate the point estimates of adjustable parameters in breakage models. The 95% bootstrap calculated using empirical bootstrap indicates very high confidence in estimated parameters. The HMMC model provides an accurate description of steady state bubble size distribution and the mean number diameters only using overall gas holdup and specific energy as inputs

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