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

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

MULTI-SCALE MODELING OF HIGH-SHEAR GRANULATION

High shear wet granulation is a particle design process used to increase the size of a primary powder material through the addition of liquid binder. This thesis focuses on the multi-scale nature of high-shear granulation in order to understand behavior changes due to coupled consolidation and coalescence as well as operational changes that occur during the scale up of horizontal ploughshare mixer granulators. The methodology relied on a micro-scale model for coalescence, a meso-scale model to describe flow within the granulator and a macro-scale population balance to describe the whole system

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

Modelling of biological systems using multidimensional population balances

Biological systems are intrinsically heterogeneous and, consequently, their mathematical descriptions should account for this heterogeneity as it often influences the dynamic behaviour of the individual cells. For example, in the cell cycle dependent production ofproteins, it is necessary to account for the distribution of the individual cells with respect to their position in the cell cycle as this has a strong influence on protein production. A second notable example is the formation of cancerous cells. In this case, the failure of regulatory mechanisms results in the transition of somatic cells to their cancerous state. Therefore, in developing the corresponding mathematical model, it is necessary to consider both the different states of the cells as well as their regulation. In this regard, the population balance equation is the ideal mathematical framework to capture cell population heterogeneity as it elegantly takes into account the distribution of cell populations with respect to their intracellular state together with the phenomena of cell birth, division, differentiation and recombination. Recent developments in solution algorithms together with the exponential increase in computational abilities now permit the efficient solution of one-dimensional population balance models which attribute the heterogeneity of cell populations to differences in the age or mass of individual cells. The inherent complexity of biological systems implies that the differentiation of cells based on a single characteristic alone may not be sufficient to capture the underlying biological phenomena. Therefore, current research is focussing on the development of multi-dimensional population balances that consider the differentiation of cells based on multiple characteristics, most notably, the state of cells with respect to key intracellular metabolites. However, conventional numerical techniques are inefficient for the solution of the formulated population balance models and this warrants the development of novel, tailor-made algorithms. This thesis presents one such solution algorithm and demonstrates its application to the study of several biological systems. The algorithm developed herein employs a finite-volume technique to convert the partial-differential equation comprising the population balance model into a set of ordinary differential equations. A two-tier technique based on the solution technique for inhomogeneous differential equations is then developed to solve the system of ordinary differential equations. This approach has two main advantages: (a) the decomposition technique considerably reduces the stiffness of the system of equations enabling more efficient solution, and (b) semianalytical solutions for the integrals employed in the modelling of cell division and differentiation can be obtained further reducing computation times. Further improvements in solution efficiency are obtained by the formulation of a two-level discretisation algorithm. In this approach, processes such as cell growth which are more sensitive to the discretisation are solved using a fine grid whereas less sensitive processes such as cell' division - which are usually more computationally expensive - are solved using a coarse grid at a higher level. Thus, further improvements are obtained in the efficiency of the technique. The solution algorithm is applied to various multi-dimensional population balance models of biological systems. The technique is first demonstrated on models of oscillatory dynamics in yeast glycolysis, cell-cycle related oscillations in eukaryotes, and circadian oscillations in crayfish. A model of cell division and proliferation control in eukaryotes is an example of a second class of problems where extracellular phenomena influence the behaviour of cells. As a third case for demonstration, a hybrid model of biopolymer accumulation in bacteria is formulated. In this case, cybernetic modelling principles are used to account for intracellular competitions while the population balance framework takes into consideration the heterogeneity of the cell population. Another important aspect in the formulation ofmulti-dimensional population balances is the development of the intracellular models themselves. While research in the biological sciences is permitting the formulation of detailed dynamic models of various bioprocesses, the accurate estimation of the kinetic parameters in these models can be difficult due to the unavailability of sufficient experimental data. This can result in considerable parametric uncertainty as is demonstrated on a simple cybernetic' model of biopolymer accumulation in bacteria. However, it is shown that, via the use of systems engineering tools, experiments can be designed that permit the accurate estimation of all model parameters even when measurements pertaining to all modelled quantities are unavailable.Imperial Users onl

Multi-scale population balance modelling and controllability of granulation processes

Imperial Users onl