Application of the direct quadrature method of moments to polydisperse gas-solid fluidized beds

Most of today's computational fluid dynamics (CFD) calculations for gas-solid flows are carried out assuming that the solid phase is monodispersed, whereas it is well known that in many applications, it is characterized by a particle size distribution (PSD). In order to properly model the evolution of a polydisperse solid phase, the population balance equation (PBE) must be coupled to the continuity and momentum balance equations. In this work, the recently formulated direct quadrature method of moments (DQMOM) is implemented in a multi-fluid CFD code to simulate particle aggregation and breakage in a fluidized-bed (FB) reactor. DQMOM is implemented in the code by representing each node of the quadrature approximation as a distinct solid phase. Since in the multi-fluid model, each solid phase has its own momentum balance, the nodes of the DQMOM approximation are convected with their own velocities. This represents an important improvement with respect to the quadrature method of moments (QMOM) where the moments are tracked using an average solid velocity. Two different aggregation and breakage kernels are tested and the performance of the DQMOM approximation with different numbers of nodes are compared. These 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.     

Development of a multi-fluid model for poly-disperse dense gas-solid fluidised beds, Part II: Segregation in binary particle mixtures

Particle mixing and segregation rates in a bi-disperse freely bubbling fluidised bed have been studied with a new multi-fluid model (MFM) based on the kinetic theory of granular flow for multi-component systems (see Part I). The MFM simulation results have been compared with digital image analysis experiments obtained by Goldschmidt et al. [2003. Digital image analysis of bed expansion in dense gas-fluidised beds. Powder Technology 138, 135-159] for bi-disperse mixtures of glass beads. In strong contrast to MFMs previously described in the literature, that strongly overestimate the segregation rates, the new MFM seems to underestimate the segregation rates at longer times. This underprediction of the segregation rate is probably related to the neglect of frictional stresses associated with long-term multiple-particle contacts resulting in an overestimation of the mobility of the emulsion phase, which is corroborated by discrete particle simulations without friction between the particles and the particles and the wall. The level of the granular temperature of the segregating system, as computed with the new MFM, compares reasonably well with the granular temperatures found in the DPM simulation.     

Numerical Simulation of Particle Segregation in Bubbling Gas‐Fluidized Beds

A multi‐fluid Eulerian model incorporating the kinetic theory of granular flow is used for the simulation of bubbling fluidized beds containing a binary mixture of Geldart B particles at low gas velocities. The cases of density, size and combined density/size segregation are investigated using computational fluid dynamic simulations. Various expressions for the drag force are evaluated for predicting different segregations. The simulation results show that summation of the particle‐particle drag force, i.e., the “hindrance effect” term, and the Stokes drag of particles, which is modified based on the Wen‐Yu drag model can be used for accurate simulation of a binary mixture of particles differing in size, density, or both. Bed expansion and dimensionless axial segregation profiles of CFD results are compared with the experimental data and good agreement is found.     

基于EMMS介尺度模型的双分散鼓泡流化床的模拟

双分散气固鼓泡流化床中颗粒通常具有不同粒径或密度,导致产生颗粒偏析等现象,影响传递和反应行为。颗粒分离和混合与气泡运动密不可分,其中相间曳力起关键作用。最近Ahmad等提出了一种基于气泡结构的双分散介尺度曳力模型,能成功预测双分散鼓泡流化床的床层膨胀系数。本研究耦合该曳力模型与连续介质方法,模拟了两种不同的双分散鼓泡流化床,通过分析不同流化状态下的气泡运动、颗粒浓度比的轴向分布等参数,进一步检验模型的适用性。研究表明,当双分散颗粒处于完全流化状态时,耦合双分散介尺度曳力模型可合理预测不同颗粒的分离现象;而其处于过渡流化状态时,新曳力模型和传统模型均无法获得合理结果,此时调节固固曳力可改进模拟结果。     

Solution of population balance equations using the direct quadrature method of moments

The implementation of a population balance equation (PBE) in computational fluid dynamics (CFD) represents a crucial element in the simulation of multiphase flows. Some of the available methods, such as classes methods (CM) and Monte Carlo (MC) methods, are computationally expensive and simulation of real cases of practical interest requires intractable CPU times. On the other hand, other methods such as the method of moments (MOM) are computationally affordable but have proven to be inaccurate for a number of cases. In recent work a new closure, the quadrature method of moments (QMOM), has been introduced, applied and validated. In our earlier work, QMOM was shown to be an efficient and accurate method for tracking the moments of the particle size distribution (PSD) in a CFD simulation. However, QMOM presents two main disadvantages: (i) if applied to multi-variate distributions it loses simplicity and efficiency, and (ii) by tracking only the moments of the PSD, it does not represent realistically polydisperse systems with strong coupling between the internal coordinates and phase velocities. In order to address these issues, in this work the direct quadrature method of moments (DQMOM) is formulated, validated, and tested. DQMOM is based on the idea of tracking directly the variables appearing in the quadrature approximation, rather than tracking the moments of the PSD. Nevertheless, for monovariate cases we show that QMOM and DQMOM yield identical results. In addition, we show how it is possible to extend the DQMOM to multivariate cases and some of relevant theoretical and numerical issues are discussed. These issues are discussed in the present work for homogeneous and one-dimensional flows. References to recent CFD applications of DQMOM to multiphase flows are provided as further proof of the utility of the method.     

Hydrodynamic modeling of particle rotation for segregation in bubbling gas-fluidized beds

A multi-fluid Eulerian model has been improved by incorporating particle rotation using kinetic theory for rapid granular flow of slightly frictional spheres. A simplified model was implemented without changing the current kinetic theory framework by introducing an effective coefficient of restitution to account for additional energy dissipation due to frictional collisions. Simulations without and with particle rotation were performed to study the bubble dynamics and bed expansion in a monodispersed bubbling gas-fluidized bed and the segregation phenomena in a bidispersed bubbling gas-fluidized bed. Results were compared between simulations without and with particle rotation and with corresponding experimental results. It was found that the multi-fluid model with particle rotation better captures the bubble dynamics and time-averaged bed behavior. The model predictions of segregation percentages agreed with experimental data in the fluidization regime where kinetic theory is valid to describe segregation and mixing.     

Particle mixing/segregation in gas-solid fluidized bed of ternary mixtures

The particle concentration profiles and minimum fluidizing velocity of ternary mixtures were investigated experimentally. All the experiments were carried out in an 88 mm I.D. transparent acrylic column containing fluidized beds of ternary particles of different sizes, densities and shape. Mixing/segregation patterns were visualized under different operating conditions. The experimental results are compared with empirical relationships for mixing index and take-off velocity. A new definition of the mixing index, including particle minimum fluidizing velocity and density predicted the mixing/segregation behaviour reliably. The proposed correlation of take-off velocity agrees with the experimental results very well.     

Species segregation of binary mixtures and a continuous size distribution of Group B particles in riser flow

Experiments involving a gas–solid, pilot-scale circulating fluidized bed (CFB) have been carried out, with a focus on species segregation measurements in a riser. Three mixtures were considered: (i) a binary mixture with particles of different sizes (d_ave) but same material density (ρ_s), (ii) a binary mixture with particles of different material densities (ρ_s) but same size (d_ave), and (iii) a continuous particle size distribution (PSD). Local measurements of the composition (i.e., species segregation) of each mixture were obtained over a range of operating conditions. Similar to previous works, the results show that the more massive species (i.e., greater d_ave or ρ_s) preferentially segregates toward the wall in all cases. Several new trends were also observed. First, for the binary mixtures, composition of the more massive species increases with riser height at the wall under some operating conditions. The operating conditions that cause this phenomenon are mutually exclusive for the size-difference and density-difference systems. Second, for the continuous PSD, radial segregation is observed even when there is a net positive flux in the annular region, contrary to previous findings which indicated segregation only for conditions leading to a net downward flux in the annular region. Finally, two qualitative differences between the binary and continuous mixtures were noted: (i) a monotonic decrease in species segregation is observed for the binary mixtures with an increase in the solid loading (m), while a non-monotonic trend is observed for the continuous PSD, and (ii) while the shape of the radial segregation profile is flattest at the riser bottom for the binary mixtures, the flattest radial profile is at the riser top for the continuous PSD.     

A particle-to-particle heat transfer model for dense gas-solid fluidized bed of binary mixture

A particle-to-particle collisional heat transfer model in the frame of Eulerian-Eulerian approach was proposed in this paper. By incorporating it into the multi-fluid model to close the enthalpy equations, the heat transfer between different particle classes in a gas bubbling fluidized bed of binary mixture was investigated, based on the CFD simulations of particle mixing in literature (Cooper and Coronella, 2005). The results showed that the particle-to-particle heat exchange coefficient between different particle classes increases with increasing the size of large particle class and the superficial gas velocity. The ratios of the particle-to-particle heat transfer to the gas-to-particle heat transfer range from 8.04% to 15.0% for various calculating conditions. In order to better understand the heat transfer behavior in a dense gas-solid fluidized of binary mixture, it is important to take the particle-to-particle heat transfer into account.     

Impact of material property and operating conditions on mass flux profiles of monodisperse and polydisperse Group B particles in a CFB riser

Experiments directed at understanding local mass flux behavior of Geldart Group B materials in the riser of a gas-solids circulating fluidized bed (CFB) have been carried out. Three monodisperse materials (with differences in particle size and/or material density), two binary mixtures (one with only a particle size difference between the species and the other with only a material density difference), and one continuous particle size distribution (PSD) have been investigated at four operating conditions. Results show that the riser axial position has the greatest influence on mass flux behavior, especially near the top of the riser, where profile shapes consistently have an inverted U-shape or V-shape. The material type (i.e., monodisperse materials of different particle sizes and/or particle densities or different types of polydispersity) and operating conditions effects are secondary but more apparent at the riser bottom. An interesting observation involving binary mixtures is that while the mass flux profiles of the density-difference binary mixture mimics that of one of its (monodisperse) constituent components, the size-difference binary mimics neither of its two monodisperse components.     

DEM study on the discharge characteristics of lognormal particle size distributions from a conical hopper

This study employs the discrete element method (DEM) to investigate the impact of the widths of lognormal particle size distributions (PSDs) with the same mean particle diameter on hopper discharge behaviors, namely, discharge rate, particle velocities, and size‐segregation. Results reveal that (i) the hopper discharge rate decreases as PSD width increases; (ii) the mean discharge rates are constant with time, but the fluctuations increase as the PSD width increases; (iii) the overall size‐segregation increases with PSD width; (iv) the overall mean particle diameters of the narrower PSDs do not exceed the initial mean of 5 mm, whereas that of wider ones do; (v) the relationship between PSD width and particle velocities is non‐monotonic with no consistent trends; and (vi) no direct correlation exists between particle velocity and size‐segregation. The results here provide valuable insights on the behavior of the prevalent polydisperse mixtures in hoppers. © 2017 American Institute of Chemical Engineers AIChE J, 64: 1174–1190, 2018     

Modeling,Simulation, and Experimental Evaluation of Particle Size Distribution in Stabilized Fluidized Bed Adsorption System

Fluidized beds containing solid particles of a wide size distribution is of significant practical importance. In such systems, the overall behavior depends on the opposing effect of mixing and classification. In the present work, the mixing and segregation behavior of liquid fluidized beds containing particles of different sizes is described mathematically. Particle size distribution (PSD) is studied in a glass column of 5 cm internal diameter and 250 cm length. Ion exchange resins were used as a solid phase with a particle size range of 50 to 650 µm. The bed was fluidized at constant and low water flow velocity and the particle size measurement was carried out at different locations over the column length by analytical scanning electron microscope. Particle size fractionation data obtained by Malvern Mastersizer-2000, version 5.4, was utilized in the solution of developed model equations to obtain PSD. It is apparent that the mixing model along with the classification model represents better results than any other model given by various researchers in the literature. The proposed model is in good agreement with the PSD data given by Malvern particle size analyzer.     

Polydisperse granular flows in a bladed mixer: Experiments and simulations of cohesionless spheres

The flow and segregation of polydisperse, spherical particle mixtures in a bladed mixer was investigated using experimental and computational techniques. Discrete element simulations were able to reproduce the qualitative segregation profiles and surface velocities observed experimentally. For a binary system with a 2:1 size ratio, segregation by size occurs due to a sieving mechanism. Segregation in the binary system is fast, with a fully segregated system observed after just 5 revolutions. However, the numerical simulations showed that the extent of segregation in the bladed mixer can be reduced by introducing intermediate particle sizes in between the smallest and the largest particles. Addition of intermediate particle sizes increases convective and diffusive particle motion promoting a mixing mechanism that reduces segregation via the sieving mechanism. Void fraction within the bladed mixer increases as the degree of polydispersity is increased allowing the particles to move more freely throughout the particle bed. Higher void fractions also increase the ability of large particles to penetrate deeper into the particle bed. Normal and shear stresses are also affected by particle size distributions, with lower average values obtained for the system with the largest number of particle species. Differences in the amount of stress generated by each particle species were observed. However, the difference in stresses is reduced as the number of particle species in the system is increased.     

Effect of particle shape on liquid-fluidized beds of binary (and ternary) solids mixtures: segregation vs. mixing

Experiments were carried out in water-fluidized binary (and ternary) mixtures of teflon spheres, discs and rods. All particles had the same volume, while the discs and rods had nearly the same sphericity. It is shown that segregation can occur by shape, with similar segregated and mixed zones as when binary mixtures of different size or density are fluidized. The model of Pruden and Epstein (1964; Stratification by size in particulate fluidisation and in hindered settling. Chemical Engineering Science 19, 696), in which the degree of segregation depends on the bulk density difference(s) of the corresponding monocomponent beds at the same liquid velocity, is vindicated qualitatively for each system, but sphericity is not sufficient as a single shape factor to yield a single quantitative correlation of the transitions between segregation patterns for the different systems. Segregation by shape of non-isometric particles appears to require higher reduced density differences than sizing of spheres, probably because of the greater bed instabilities generated by the non-isometric particles. Overall bed voidage is predicted well by the serial model of Epstein et al. (1981; Liquid fluidisation of binary particle mixture- I.Overall bed expansion. Chemical Engineering Science 36, 1803).     

All the Brazil nuts are not on top: Vibration induced granular size segregation of binary,ternary and multi-sized mixtures

Vertically vibrated systems of granular materials have been used to gain insight on granular segregation for decades. However, the majority of studies have focused on the rise of a single large “intruder” particle in an otherwise monodisperse bed, or binary mixtures. As most industrially relevant granular materials are characterized by some degree of polydispersity, a study of granular mixtures with additional particle sizes is warranted to determine the role of polydispersity in granular mixing and segregation. In this work, the segregation of binary, ternary and polydisperse mixtures of nuts and spheres in a vertically vibrated cylinder is studied experimentally and computationally. We find that the presence of the other species – besides the smallest and largest size – is responsible for dramatic reductions in the final degree of segregation compared to a binary mixture. In addition, we quantify orders of magnitude reduction in the segregation rate of a mixture of polydisperse spheres compared to a binary mixture. This reduction in segregation coincides with an increase in diffusive and convective mixing and correlates with a lower average system density. Voidage distributions demonstrate the bed packing structure plays an important role in enabling multi-sized systems to remain in a more mixed state as compared to binary systems. Our observations show that this prototypical segregation experiment provides as much insight into why materials do not segregate as why they do.     

Particle segregation in a spouted bed of binary mixtures of particles

Minimum spouting velocity and segregation behaviour of binary mixtures of particles differing in size have been studied. The experiments were carried out in a bed of 20 cm diameter at superficial gas velocities up to 1.3 U_ms by use of silica sand of four different particle sizes from 0.655 to 2.23 mm. An empirical equation was proposed for the minimum spouting velocity of binary mixtures. The effects of the particle size difference and the superficial gas velocity on segregation were investigated. Results showed that considerable radial segregation as well as axial segregation occurred even for high gas velocity under the condition of large particle size difference.     

Dynamics of segregation and fluidization of binary mixtures in a cylindrical fluidized bed

Dynamics of segregation and fluidization of unary particles and binary mixtures in a cylindrical fluidized bed is investigated using temporally– and spatially–resolved measurements of solids volume fraction (α_s) performed using Electrical Capacitance Tomography (ECT). Through the comparison with high-speed imaging, we have shown that ECT can be used to measure the segregation behavior in cylindrical fluidized beds quantitatively. ECT measurements have been used further to quantify the effects of mixture composition, particle–diameter ratio, and superficial gas velocity on the bed segregation behavior. Dynamics of fluidization behavior is characterized using the time–evolution of local α_s fluctuations, corresponding frequency distribution, and bubble size distribution. Further, a relation between the measured variance of α_s fluctuations at different radial locations and corresponding flow structures under different fluidization conditions is established. The present work helps to understand dynamics of segregation and fluidization of binary mixtures and to provide a database for validation of Eulerian multifluid CFD models.     

Bubbling fluidization of biomass and sand binary mixtures: Minimum fluidization velocity and particle segregation

Understanding the minimum fluidization velocity of biomass and sand mixtures is fundamental to ensuring the optimal performance of fluidized beds in a thermo-conversional process, such as fast pyrolysis. The present work aimed to determine the minimum fluidization velocity of binary mixtures using the characteristic diagram of pressure drop in the bed and to develop an experimental correlation for the minimum fluidization velocity of biomass and sand mixtures. Three types of biomass (sweet sorghum bagasse, waste tobacco and soybean hulls) and four sands with different sizes were investigated. The results showed that the fluid dynamic behavior of binary mixtures is directly related to the biomass size and shape. For sweet sorghum bagasse (more irregular particles), higher biomass percentages led to lower minimum fluidization velocities, which differed from the behaviors observed for waste tobacco and soybean hulls. The diameter ratio inert/biomass effectively influenced the segregation, with a higher ratio causing more pronounced bed segregation. A good fluidization regime (with little segregation) for biomass and sand mixtures was obtained using the smallest sand (d₅₀ = 0.35). Considering the studied operating conditions, the proposed correlation can be used satisfactorily to predict the minimum fluidization velocities for mixtures of biomass and sand in fluidized beds.     

Hydrodynamic modeling of the entrainment of Geldart A group particles in gas-solid fluidized bed: The effect of column diameter

A multi-fluid Eulerian computational fluid dynamics (CFD) model is used to simulate the entrainment of fluid catalytic cracking (FCC) particles in gas-solid fluidized beds. Entrainment of Geldart A group particles was studied because of their wide range of industrial use. The model was based on the kinetic theory of granular flow. The CFD model was used to investigate the effect of column diameter on the entrainment flux of particles in a binary mixture. Two different sizes of particles were used because many engineering applications deal with binary mixture of particles in fluidized beds. Various column diameters, including 38 mm, 76 mm, 114 mm, 152 mm, and 190 mm, were investigated. The entrainment flux of particles was increased with decreasing column diameter. The effect of column diameter was not significant for column diameters larger than 114 mm. Furthermore, increasing the superficial gas velocity increased the entrainment flux of particles. Model predictions were also compared with experimental findings.     

Segregation and intermixing in polydisperse liquid–solid fluidized beds: A multifluid model validation study

Multifluid model (MFM) simulations have been carried out on liquid–solid fluidized beds (LSFB) consisting of binary and higher-order polydisperse particle mixtures. The role of particle–particle interactions was found to be as crucial as the drag force under laminar and homogenous LSFB flow regimes. The commonly used particle–particle closure models are designed for turbulent and heterogeneous gas–solid flow regimes and thus exhibit limited to no success when implemented for LSFB operating under laminar and homogenous conditions. A need is perceived to carry out direct numerical simulations of liquid–solid flows and extract data from them to develop rational closure terms to account for the physics of LSFB. Finally, a recommendation flow regime map signifying the performance of the MFM has been proposed. This map will act as a potential guideline to identify whether or not the bed expansion characteristics of a given polydisperse LSFB can be correctly simulated using MFM closures tested.