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

A comparative study of numerical approximations for solving the Smoluchowski coagulation equation

In this work, numerical approximations for solving the one dimensional Smoluchowski coagulation equation on non-uniform meshes has been analyzed. Among the various available numerical methods, finite volume and sectional methods have explicit advantage such as mass conservation and an accurate prediction of different order moments. Here, a recently developed efficient finite volume scheme (Singh et al., 2015) and the cell average technique (Kumar et al., 2006) are compared. The numerical comparison is established for both analytically tractable as well as physically relevant kernels. It is concluded that the finite volume scheme predicts both number density as well as different order moments with higher accuracy than the cell average technique. Moreover, the finite volume scheme is computationally less expensive than the cell average technique

Extension of moment projection method to the fragmentation process

© 2017 Elsevier Inc. The method of moments is a simple but efficient method of solving the population balance equation which describes particle dynamics. Recently, the moment projection method (MPM) was proposed and validated for particle inception, coagulation, growth and, more importantly, shrinkage; here the method is extended to include the fragmentation process. The performance of MPM is tested for 13 different test cases for different fragmentation kernels, fragment distribution functions and initial conditions. Comparisons are made with the quadrature method of moments (QMOM), hybrid method of moments (HMOM) and a high-precision stochastic solution calculated using the established direct simulation algorithm (DSA) and advantages of MPM are drawn

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

Behavior of the dispersed phase in a biphasic liquid-liquid Comportement de la phase dispersé dans un contacteur diphasique liquide-liquide

In this work two major hydrodynamic parameters: the holdup of the dispersed phase and the Sauter diameter are considered. In the first part, this is done for describing the hydrodynamics of interacting liquid–liquid dispersions with using different drop breakup, coalescence and growth models in a droplet population balance model. Based on the variational iteration method, different process cases have been performed and, it is possible to find the exact solution or a closed approximate solution of a problem. For the simultaneous growth and the coalescence terms a comparison between the present method and projection method which include discontinuous Galerkin and collocation techniques are made respectively. The results are encouraging and the new method has proven to be suitable to predict holdup and Sauter diameter profiles. In the second part, we extended the dual quadrature method of generalized moments (DuQMoGeM) to solve the population balance model for the hydrodynamics of liquidliquid extraction columns using a multi-compartment model. The DuQMoGeM results were compared to analytical solutions for batch and continuous well-mixed vessels and extraction columns, showing that it is accurate for predicting the evolution of the low order moments and the drop number distribution along with the column height. We also modeled a K¨ uhni column for which the simulation accurately predicted the steadystate experimental holdup, encouraging the DuQMoGeM usage to solve the population balance equation for heterogeneous systems and different columns

Bivariate extension of the moment projection method for the particle population balance dynamics

This work presents a bivariate extension of the moment projection method (BVMPM) for solving the two-dimensional population balance equations involving particle inception, growth, shrinkage, coagulation and fragmentation. A two–dimensional Blumstein and Wheeler algorithm is proposed to generate a set of weighted particles that approximate the number density function. With this algorithm, the number of the smallest particles can be directly tracked, closing the shrinkage and fragmentation moment source terms. The performance of BVMPM has been tested against the hybrid method of moments (HMOM) and the stochastic method. Results suggest that BVMPM can achieve higher accuracy than HMOM in treating shrinkage and fragmentation processes where the number of the smallest particles plays an important role

Computational fluid dynamics simulation of fluidized bed polymerization reactors

In this research, a CFD algorithm for simulation of fluidized bed polymerization reactors is described. In order to properly model the evolution of a polydisperse solid phase, population balance equation (PBE) must be solved along with other transport equations. A novel approach---DQMOM is applied to polydisperse fluidized bed to simulate particle aggregation and breakage in the reactors. Two different aggregation and breakage kernels are tested and the performance of the DQMOM approximation with different numbers of nodes are compared. Results show that the approach is very effective in modeling solid segregation and elutriation and in tracking the evolution of the PSD, even though it requires only a small number of scalars. After successfully developed DQMOM-multi-fluid CFD model, the multi-fluid model is validated with available experiments and discrete particle simulation (DPS). The results show good agreements with experiment data for binary system and DPS reults, and the simulations can describe segregation and mixing behavior in the fluidized bed;After the model development and validation, 2D and 3D simulations are conducted for a pilot-scale polymerization fluidized bed at operating conditions. Significant differences are observed between 2D and 3D simulations. The results shows that, for an industrial-scale fluidized bed, only 3D simulations are able to match the statics (bed height and pressure drop) and the dynamics (pressure power spectra) properties of the bed. The residence time for a polyethylene pilot reactor is on the order of hours, and the time scale for the fluid dynamics in the bed is on the seconds. It is impossible to run a three-dimensional simulation for hours using current CFD codes. Due to the time scale problem, a chemical reaction engineering model based on the age of particles is combined with multi-fluid model to initialize the fluidized bed to a steady state. Direct quadrature method of moments (DQMOM) is used to simulate the particle size distribution in the bed. The hot spots in the fluidized bed are also investigated using CFD simulations

Modeling flocculation and deflocculation processes of cohesive sediments

The transport and fate of cohesive sediments are responsible for many engineering, environmental, economic and policy issues that relate to, for example, siltation and dredging in navigation channels, water quality, water turbidity, pollutant transports, and biological ecosystem responses. Our current understanding, however, is insufficient to conduct accurate quantitative predictions of these processes. This is because the cohesive particles in natural waters will flocculate, which determines the settling, and thus the deposition behaviors. The simulation of flocculation processes is a primary challenge since the time variation of Floc Size Distribution (FSD) is controlled by a partial differential equation that also contains the integration of FSD itself. Previous models either address less characteristic sizes, which produce biased FSDs, or are incapable of modeling a relative large study domain in order to better express the FSDs with more size groups. In this study, a cohesive sediment flocculation model developed based on the framework of Population Balance Model (PBM) is solved by the Quadrature Method of Moments (QMOM). This PBM�QMOM flocculation model has reasonably compromised by both the model robustness and model efficiency. The former lies in the capability of describing the time evolution of the FSDs with a maximum of eight size classes, and the latter is reflected in its efficiency to solve PBM with transport terms and the potential to be coupled in a flow-mud estuary model. The model predictions are compared to both the analytical (or trusted class method) results for general PBMs (i.e., beyond the scope of specific research field), and the published experimental results of kaolinite suspension and colloidal montmorillonite. After that, an experimental activity has been carried out to develop a Sony NEX-5R camera system (with extension tubes and close-up) to automatically acquire floc images under various controlled environments, and to use MATLAB software to process the FSDs. This process is validated by the results of two set of sample particles. The validated camera system is first applied in a five liter mixing chamber to investigate the effects of salinity and selected organic matters on kaolinite flocculation. Then, the camera system is improved and assembled in a waterproof house for underwater use to provide data for a conceptual one-dimensional application in a relatively large turbulence tank. The flow field of the tank is measured by an acoustic Doppler velocimetry. The flocculation processes in the mixing chamber or cylindrical tank are modeled by PBM�QMOM and validated by camera statistical FSDs. While chemical and biological effects are not explicitly included in PBM�QMOM (implicitly included in fitting parameters) at this time to address the basic mechanisms of flocculation, these effects can be further extended when the process itself is better understood through other laboratory experiments or field measurements

Population balance model-based optimal control of batch crystallisation processes for systematic crystal size distribution design

During recent years crystallisation has found applications in many chemical industries, such as pharmaceutical, petrochemical, micro-electronics and food industries. Crystallisation is a basic step for purification or separation for a large variety of organic, inorganic and pharmaceutical compounds. Most of the product qualities are directly related to the shape of the crystal size distribution (CSD). The main difficulty in batch crystallisation processes is to accomplish a uniform and reproducible CSD. On-line control during the process allows for improved crystalline product quality, shorter process times and reduction or elimination of compromised batches. The actual prediction and estimation of the shape of the distribution at the end of the batch can provide useful information for monitoring or designing the operating curve for the supersaturation controller. Model-based approaches provide consistency of the CSD, can be used for better control and also for product design by reverse engineering the process to achieve the desired CSD and shape. This research presents a novel methodology for solving the population balance equation (PBE) for the estimation of the shape of the crystal size distribution for batch crystallisation processes. The approach combines the quadrature method of moments (QMOM) and the method of characteristics (MOCH), and provides a computationally efficient technique for the reconstruction of the whole crystal size distribution. The technique was used to estimate the kinetic parameters for the size-dependent growth and secondary nucleation, for potash alum-water system using industrial pilot plant data provided by BASF, Chemical Company. The combined technique was also used to estimate the size-dependent dissolution parameters for potash alum-water system, using laboratory scale data. The QMOM-MOCH solution approach is evaluated in a model-based dynamic optimization study, with the aim to obtain the optimal temperature profiles, which drive the system in both the supersaturated and under-saturated region, to achieve desired target CSD. Using growth, dissolution and nucleation parameters the technique was used to optimise the temperature trajectories to obtain bimodal and mono-modal distributions. The technique can serve as a soft sensor for predicting the CSD, or as a computationally efficient algorithm for off-line design or on-line adaptation of operating policies based on knowledge of the full CSD data. Additionally, the PBE model was solved using the method of characteristics under the assumption of constant supersaturation. At constant supersaturation growth is the dominating phenomenon, yielding a simplified analytical expression for the prediction of the CSD. The research presents the new methodology for the systematic design of the setpoint operating curves for supersaturation controlled crystallisation processes, which produces a desired target crystal size distribution (CSD) at the end of the batch. A design parameter, was introduced as a function of the supersaturation and time, and is evaluated for supersaturation controlled processes. Based on the design parameter and the simplified analytical model, the supersaturation setpoint and batch time are determined using an optimisation approach to obtain a target distribution with a desired shape. Two additional methods are also proposed that use the seed in conjunction with the supersaturation setpoint design, and analytical CSD estimator for shaping the product CSD. The first approach designs a seed recipe as a mixture of crystals resulting for example from standard sieve analysis. In this approach the seed was introduced at the beginning of the batch. The second approach introduces the dynamic seeding concept, which allows an easily implementable methodology to achieve complex target CSDs using seed with mono-modal distribution as a process actuator. These methodologies were validated for potassium dichromate-water system. Size-dependent growth kinetic parameters for the potassium dichromate-water system were identified using as experimental setup developed at Loughborough University. The experiments presented in the thesis also illustrates the simultaneous application of in situ Process Analytical Technology (PAT) tools, such as focused beam reflectance measurement (FBRM) for nucleation detection, attenuated total reflection (ATR) UV/Vis spectroscopy for concentration monitoring, as well as the in-line use of a Mastersizer for real-time CSD measurement in the case of the potassium dichromate in water system. The approaches provide a comprehensive framework for model-based dynamic optimisation of crystallisation processes, which combines efficient numerical solution approaches of the PBE with the formulation of novel optimisation problems. The techniques presented include controlled dissolution, simultaneous optimisation of operating policies and seed recipes and dynamic seeding. Simulation and experimental evaluations of the proposed approaches demonstrate the potential of the techniques to provide significant improvement in the current state-of-the-art in crystallisation control

Numerical aspects of population balance equations coupled to computational fluid dynamics

It can be the motion of clouds, the movement of a smoke plume, or the dynamics of fluids in processes which are interesting to food, petroleum, chemical, pharmaceutical and many other industries; they are all governed by the same physical laws: fluid dynamics and population balances. Numerical solution of Population Balance Equations (PBE) coupled to Computational Fluid Dynamics (CFD) is a promising approach to simulate liquid/gas-liquid dispersed flows, for which the governing physical phenomena are breakup and coalescence of bubbles/droplets, additional to transport phenomena of fluids. In the literature, there are many breakup and coalescence models to close the PBE. Unfortunately, there is no unified framework for these closures; and, it is one of our objectives: to determine appropriate coalescence and breakage kernels for liquid/gas-liquid dispersions. Another objective is to investigate numerical techniques for one-way coupled CFD and PBE, and to develop a computational tool. The developed tool is based on the incompressible flow solver FeatFlow which is extended with Chien's Low-Reynolds number k-epsilon turbulence model and PBE. The presented implementation ensures strictly conservative treatment of sink and source terms which is enforced even for geometric discretization of the internal coordinate. The validation of our implementation which covers a wide range of computational and experimental problems enables us to proceed into three-dimensional applications as, turbulent flows in a pipe and through static mixers. Regarding the studies on static mixers, not only we have obtained numerical results; we have conducted comprehensive experimental studies in the Sulzer Chemtech Ltd. laboratories (Winterthur, Switzerland). The inclusive experimental results has offered a good ground for verifying the adopted mathematical models and numerical techniques. The obtained satisfactory results in the studies for one-way coupled CFD and PBE has motivated us to study two-way coupled CFD-PBE models. The so far developed numerical recipe of which main ingredients are the method of classes, positivity-preserving linearization and the high-order FEM-AFC with FeatFlow including the standard k-epsilon solver has been extended to cover bubble induced turbulence and mixture-model with algebraic slip relation. A smart algorithm is developed, offering a compromise between the computational cost and the accuracy. Numerical simulation of air-in-water dispersed phase systems in a flat bubble column which is, numerically, a very challenging case-study and is experimentally studied by Becker et al. has been performed with the developed computational tool. The dynamic movement of the bubble swarm which is observed in the experiments have been successfully simulated. Keywords: computational fluid dynamics (CFD), population balances, coalescence, breakage, numerical solution, method of classes, parallel parent daughter classes, simulation, static mixers, multiphase flows