PDF method for population balance in turbulent reactive flow

Turbulent reactive flows with particle formation, such as soot formation and precipitation, are characterized by complex interactions between turbulence, scalar transport, particle formation and particle transport and inter-particle events such as coagulation. The effect of formation, growth and coagulation on the particle size distribution (PSD) must be modelled by the population balance equation (PBE). While the PBE has been studied extensively in homogeneous systems and, recently, in simple flows, its coupling with turbulent reactive flows poses a wealth of new questions. Processes such as nucleation, growth and coagulation are described by kinetic laws that link them to the local concentrations of the reactive scalars, which are random in a turbulent flow. This accounts for additional mechanisms that induce randomness and fluctuations to the particle concentration and PSD. Furthermore, conventional RANS closure of the coagulation term PDE (which describes the evolution of the PSD) leads to unknown correlations. In this work a new pdf approach is developed, based on the transport of the joint pdf of reactive scalars and particle number densities at different sizes, which overcomes the additional closure problems. It is also shown how the pdf method can be solved numerically via Monte-Carlo methods, and this is demonstrated via two applications in a partially stirred reactor: precipitation via nucleation-growth and coagulation. In each case the pdf method is compared with models that neglect correlations at various levels, and it is demonstrated that the interaction of turbulence with particle formation mechanisms accounts for significant deviations in the PSD.     

Improved orthokinetic coagulation model for fractal colloids: Aggregation and breakup

An improved discretized population balance equation (PBE) is proposed in this study. This improved discretized PBE has new probability distribution functions for aggregates produced in non-uniform discrete coagulation modeling. The authors extended an improved particle coagulation model previously developed to an adjustable geometric size interval (q), where q is a volume ratio of class k+1 particles to class k particles (υ_k+1/υ_k=q). This model was compared with exact numerical solutions of continuous (uniform discretized) PBEs and applied to simulate the particle aggregation and breakup with fractal dimensions lower than 3. Further, comparisons were made using the fractal aggregate collision mechanisms of orthokinetic coagulation with the inclusion of flow induced breakup.In the course of the investigation the new algorithm was found to be a substantial improvement in terms of numerical accuracy, stability, and computational efficiency over the continuous model. This new algorithm makes it possible to solve fractal particle aggregation and breakup problems with high accuracy, perfect mass conservation, and exceptional computational efficiency, thus the new model can be used to develop predictive simulation techniques for the coupled coagulation using computational fluid dynamics (CFD) and chemical reaction modeling.     

Mathematical Modeling of Double-Stage Electrostatic Precipitators Based on a Modified Eulerian Approach

A new mathematical model has been developed to predict the performance of a double-stage electrostatic precipitator. This model is based on the Eulerian approach for particle dispersion taking into account the particle size distribution. In this model, by calculating the frequency size distribution of particles and particle concentration distribution simultaneously through a modified particle dispersion equation, the CPU time for solving the governing equations has been reduced significantly. In order to evaluate velocity distribution of the fluid, the k     

A discrete-sectional model for particle growth in aerosol reactor: Application to titania particles

A one-dimensional discrete-sectional model has been developed to simulate particle growth in aerosol reactors. Two sets of differential equations for volume and surface area, respectively, were solved simultaneously to determine the size distributions of agglomerates and primary particles. The surface area equations were derived in such a way that the coagulation integrals calculated for the volume equations could be used for the surface area equations as well, which is new in this model. The model was applied to a production of TiO₂ particles by oxidation of titanium tetrachloride. Model predictions were compared with experimental data and those of a two-dimensional sectional model. Good agreement was shown in calculated particle size distributions between the present model and the two-dimensional model, which is more rigorous but demands a large amount of computer time and memory. Compared to experimental data, the primary particle size calculated by the model was more sensitive to the variation of reactor temperature.     

Computer simulation of the polydispersive sol coagulation process

This work reports on a computer‐simulated investigation of the coagulation rate for a system comprising spherical sol and coagulant particles. The discussed experiment positively verified the functioning of the simulated coagulation system, where the aggregation process proceeds in line with the particle‐cluster model as a rapid and perikinetic coagulation process that satisfies the Smoluchowski equation. The rate of the simulated coagulation process satisfies kinetic equations for both first‐order and second‐order reactions. Selected concepts and models of the coagulation theory have been also verified. © 2012 Canadian Society for Chemical Engineering     

A new splitting wavelet method for solving the general aerosol dynamics equation

In this paper, a new and robust splitting wavelet method has been developed to solve the general aerosol dynamics equation. The considered models are the nonlinear integro-partial differential equations on time, size and space, which describe different processes of atmospheric aerosols including condensation, nucleation, coagulation, deposition, sources as well as turbulent mixing. The proposed method reduces the complex general aerosol dynamic equation to two one-dimensional splitting equations in each time interval, and further the wavelet method and the upstream finite difference method are proposed for solving the particle size directional and the spatial directional splitting equations. By the method, the aerosol size spectrum is represented by a combination of Daubechies’ wavelets and substituted into the size-directional splitting equation at each time step. The class of Daubechies’ wavelets in the wavelet-Galerkin scheme as trial and weight functions has the advantages of both compact support and orthonormality which can efficiently simulate the sharp shape distribution of aerosols along the particle size direction. Numerical experiments are given to show the efficient performance of the method.     

Modelling study of emulsion latex coagulation processes in coagulators

This paper presents a numerical study of emulsion latex coagulation processes in continuous coagulators based on the full computational fluid dynamics approach. The RANS approach together with the k‐ε turbulence model was used to describe the detailed flow field in the coagulators. The coagulant mixing process was modelled by the convection‐diffusion equation and the emulsion latex coagulation process was formulated by the population balance equation of the particle size with a coagulation kernel including a perikinetic and orthokinetic combined mechanism. The flow and coagulation models were independently validated by means of comparing simulated results to the relevant experimental data from the literature. A series of simulations were carried out to study the effects of coagulator bottom shape, salt solution feeding location, residence time and agitation speed, as well as the influence of four typical scale‐up criteria on the latex particle coagulation process. The presented results would be helpful for the relevant process design, development, and scale‐up of continuous latex coagulators.     

Two‐dimensional model of methane thermal decomposition reactors with radiative heat transfer and carbon particle growth

A two‐dimensional model of methane thermal decomposition reactors is developed which accounts for coupled radiative heat and polydisperse carbon particle nucleation, growth, and transport. The model uses the Navier–Stokes equations for the fluid dynamics, the radiative transfer equation for methane and particle species radiation absorption, the advection–diffusion equation for gas and particle species transport, and a sectional method for particle species nucleation, heterogenous growth, and coagulation. The model is applied to a tubular laminar flow reactor. The simulation results indicate the development of a reaction boundary layer inside the reactor, which results in significant variation of the local particle size distribution across the reactor. © 2011 American Institute of Chemical Engineers AIChE J, 58: 2545–2556, 2012     

A radiometric method for the characterization of particulate processes in colloidal suspensions Part 1. Theoretical approach

A theoretical approach to a new radiometric method for the characterization of particulate processes in stable colloidal suspensions is given. Following Rajagopal's studies of Brownian coagulation, the change of particle volume during the Brownian coagulation of sols is correlated with the particle size before the coagulation process. It is proved that the ratio (F_b/F_a)_o of mean particle size before the coagulation process is equal to the ratio (^*F_b/F_a)_tE of imaginary particle size. The ratio of imaginary particle sizes is calculated from heterogeneous exchange fractions attained in sols during the exchange fractions attained in sols during the exchange process (proceeding simultaneously with the coagulation process) using the graphic form of Wagner's solution of differential equation of diffusion. It is shown that this method is suitable for determining the change of the relative mean particle size during the ageing of systems, and thus for the characterization of particulate processes in colloidal suspensions.     

SOLUTION OF POPULATION BALANCE EQUATIONS IN EMULSION POLYMERIZATION USING METHOD OF MOMENTS

In this work, the method of moments is used for solution of population balance equations appearing in modeling of emulsion polymerization (EP). The zero-one model without coagulation effect and the pseudo-bulk model including coagulation effect are investigated as two common approaches for modeling EP processes. The fixed quadrature method is used to close the set of moment equations, and the maximum entropy approach is applied to reconstruct the particle size distribution from a finite number of its moments. Comparing the results with those obtained by the high-precision finite volume technique indicates that, despite the low computational load of the moment method, it has an acceptable accuracy. These features support use of the moment technique for other applications such as on-line control or optimization in particulate processes.     

Direct numerical simulation of nanoparticle coagulation in a temporal mixing layer via a moment method

Direct numerical simulations of coagulating aerosols in two-dimensional, incompressible, iso-thermal mixing layers are performed. The evolution of the particle field is obtained by utilizing a moment method to approximate the aerosol general dynamic equation. We use a moment method which assumes a lognormal function for the particle size distribution and requires the knowledge of only the first three moments. This approach is advantageous in that the number of equations which are solved is greatly reduced. A Damköhler number is defined to represent the ratio of convection to coagulation time scales. Simulations are performed for three flows: Damköhler numbers of 0.2, 1, and 2. The spatio-temporal evolution of the first three moments along with the mean diameter and standard deviation are discussed.     

A new method for the reconstruction of the particle radius distribution function from the sedimentation curve

A new method for the reconstruction of the particle radius distribution function from the sedimentation curve is proposed. This method permits us to obtain a continuous smooth distribution function. Two approaches are compared. The first approach is based on the calculation of the second derivative from the sedimentation curve. The second one is based on the solution of the original integral equation which describes a sedimentation process. Both of these approaches can be reduced to the problem of the solution of the Fredholm integral equation of the first kind. From the theory of integral equations, it is known that this problem is ill-posed. The usual methods lead to unstable solutions and we are forced to use special regularizing algorithms. In this paper, the Tikhonov regularization method is used to stabilize the solution of the integral equation. It is shown that the accuracy of both methods is higher than the accuracy of the graphical method, but the approach based on the solution of the original integral equation gives a more stable solution than that based on the derivative. The accuracy of the new method permits us to reconstruct the fine structure of the particle radius distribution function. Such an analysis cannot be carried out with the rough bar diagram obtained from the graphical method. The new method is absolutely indispensable in technology for controlling the degree of powder fineness.     

EVALUATION OF NUMERICAL METHODS FOR SIMULATING AN EVOLVING PARTICLE SIZE DISTRIBUTION IN GROWTH PROCESSES

The particle growth term renders hyperbolic the dynamic population balance equation. Problems associated with the numerical solution of hyperbolic partial differential equations with stationary grid methods are well known. Moreover in the common case of combined molecular particle growth and coagulation, the convolution integral of the coagulation terms makes the moving grid methods computationally intractable. To cope with practical problems, this work is focused on numerical solution methods, of the population balance equation, characterized by relatively small computational cost and fair accuracy - comparable to that of relevant experimental data. For this purpose, previous work on appropriate discretization of the coagulation terms is extended for the growth terms. Several numerical methods are systematically evaluated and further extended. Recommendations are made concerning the best method, by taking into account the nature of the problem, the prevailing physical conditions, and the main quantity of interest; i.e. certain specific moments, or the entire size distribution     

A Mathematical Model for Ultrafine Iron Powder Growth in a Thermal Plasma

ABSTRACT

A two-dimensional model is developed for the growth of ultrafine metal powders in a thermal plasma reactor. The model accounts for particle formation by nucleation, and growth by condensation and Brownian coagulation. Transport of particles occurs by convection, thermophoresis, and Brownian diffusion. The conservation equations for the moments of the particle size distribution are solved, coupled to the equation for the conservation of metal vapor. Elliptic conservation equations result from the consideration of both axial and radial diffusion of the particles. This allows for simulations in complex, recirculating flows, which are likely to occur for numerous reactor configurations and parameters. A progressive grid refining technique is used to accelerate convergence. The model is applied to the case of a typical thermal plasma reactor for the production of ultrafine iron powders. The fields of the macroscopic properties of the aerosol population and the contribution of the different mechanisms are analyzed in various conditions, some of which involve important recirculations. The effect of operating parameters on the properties of the powder generated is studied. The results are compared for some of the conditions to those obtained numerically and experimentally by Girshick et al. (1993).     

Generalized linear driving force approximation for adsorption of multicomponent mixtures

This work deals with the development of efficient numerical tools for the solution of diffusion dominated parabolic partial differential equations. This study finds its application in the modeling of the intraparticle mass balance necessary for describing dynamic adsorption processes.The orthogonal collocation method is proposed as the basis for developing generalized linear driving force approximations for adsorption and desorption of multicomponent mixtures in a single particle, independently of the mass transport model adopted. Based on this approach, it was possible to derive some approximations previously obtained from the analytical series of the homogeneous diffusion equation.Orthogonal collocation is also compared to other numerical methods found in the literature, using both the homogeneous diffusion and the dusty-gas mass transport models. The results show that orthogonal collocation is the more consistent approach.     

The Effects of Differential Diffusion on Nanoparticle Coagulation in Temporal Mixing Layers

The effects of size-independent diffusive transport on nanoparticle growth is studied by performing direct numerical simulation of nanoparticle coagulation in temporal mixing layers. The flow field is obtained by solving the incompressible Navier-Stokes equations, while the evolution of the particle field is obtained by using a nodal approach to approximate the aerosol general dynamic equation. Simulations are performed where particles diffuse according to their size and also where all particles have the same diffusivity. For the latter, the model assumes that all particles of different sizes have the same diffusivity as the smallest particles. The advantage of the second approach is the length scales that need to be resolved are larger, facilitating more affordable computations. Simulations are performed at two volume fractions to assess the effects of the models under different growth rates. The results indicate the use of size-independent diffusion coefficients predicts particle sizes and geometric standard deviations that are larger than those obtained with size-dependent diffusion coefficients.     

Modeling Critical Air Exchange Rates (CAERs) for aerosol number concentrations from nano-particle sources using an “effective coagulation coefficient” approach

An important issue in the context of air pollution by indoor combustion sources pertains to the joint effect of source strength, coagulation, and ventilation rate on the ultrafine particle exposure metrics. It was recently predicted by detailed numerical analysis of the Smoluchowski coagulation equation with continuous source and sink terms that the ultrafine particle number, mass, and surface area concentrations do not monotonically decrease with increasing air exchange rate, but display peak concentrations at certain Critical Air Exchange Rates (CAERs). As these results are of considerable significance for exposure assessment as well as for implementing particle control technologies, it is necessary to assess the CAER for different aerosol characteristics. Given the fact that the numerical method of solving coagulation equation with realistic Fuchs kernel is computationally intensive, simpler semi-analytical approaches are desired for providing reasonable estimates of CAER and clearer insight into the counter-intuitive, peaking behavior. In this article, we present such an approach by replacing the Fuchs kernel by a spectrum-averaged effective coagulation coefficient, within the framework of the steady-state model. The effective coagulation coefficient is size independent but depends implicitly on the aerosol concentration thus capturing the combined effect of coagulation and removal processes. The number concentrations obtained from this method have been compared and validated against the numerical solutions. The model predicts more pronounced effects on the peaking behavior as well as larger CAER values for fractal particles as compared to compact particles. The results are further discussed.

Copyright © 2017 American Association for Aerosol Research     

Vibrated beds of powders: A new mathematical formulation

A new mathematical formulation was made to deal with the compressible gas model that represents the vibrated particle bed. A novel boundary condition, which incorporates the equation of motion of the bed and the equation of continuity in the air gap between the bed and the vessel base, was introduced. This made it possible to reduce three differential equations, hitherto treated as governing equations that characterize the vibrated particle bed, to a single partial differential equation with pertinent initial and boundary conditions. Experiments were also conducted to assess the validity of the model. Satisfactory agreements between the predicted and measured values have been observed within limited parameter ranges.     

A Bimodal Integral Solution of the Dynamic Equation for an Aerosol Undergoing Simultaneous Particle Inception and Coagulation

A model is developed from the general aerosol dynamic equation, using a bimodal integral formulation that includes particle formation and growth by coagulation in the free molecular regime. The particle inception mode accounts for the introduction of newly formed particles which, through coagulative collisions with one another, constitute the source of the particles in the growth mode. A numerical solution for the system of the first three moments of the particle volume distribution function is discussed, under the assumption of a logarithmic-normal behavior of the two modes of the size distribution function. The bimodal integral solution is subject to a detailed comparison with the MAEROS sectional model for the case of an aerosol that undergoes free molecular coagulation occurring simultaneously with particle formation by a Gaussian source pulse, under flamelike conditions.     

Eulerian Model for Prediction of Particle Transport and Deposition in Turbulent Duct Flows with Thermophoresis

The Reynolds-averaged equations for turbulent particle population/transport in an Eulerian framework must be closed by specifying models for several terms: a turbophoretic force; a turbulent thermophoretic force; and a turbulent particle-diffusion term. In this article, new models are proposed for the turbophoretic term, as a particle-size dependent extrapolation of the corresponding turbulent fluid-velocity correlation, and for the turbulent thermophoretic term as an eddy-viscosity-scaled multiple of the corresponding mean thermophoretic term, appropriate for small low-inertia particles with τ⁺_p < 10. When the turbophoresis model is incorporated in a system of equations that describes particle motion within the surrounding fluid, it predicts particle deposition velocities that are in good agreement with experimental data over a range of particle sizes. When this equation system is included in a computational model to predict particle transport in turbulent pipe flows, the efficiency of particle deposition in pipes with upstream heating and downstream cooling is found to be in fair agreement with experimental measurements at two different Reynolds numbers, and over a range of particle sizes and temperature differences.

Copyright 2015 American Association for Aerosol Research