Modeling of the Spray Zone for Particle Wetting in a Fluidized Bed

In a spray agglomeration process the particle wetting influences the agglomerate growth and particle dynamics in the granulator. The mass of binder liquid that is deposited on single particles affects the amount of energy dissipation during particle contacts. For the agglomeration of colliding particles the whole impact energy has to be dissipated due to viscous and capillary adhesion forces in the liquid film and plastic deformation of the material. Therefore, a detailed knowledge of the particle wetting is necessary to model the agglomeration process. This contribution uses a coupled DEM‐CFD approach to describe the spray zone of a two‐fluid nozzle in a fluidized bed agglomerator. Droplets modeled as discrete elements showed the formation of a spray zone with a conical shape. Simulations of the spray zone and the wetting of single particles are in good agreement with experimental results.     

超疏水表面液滴合并诱导弹跳现象分析

在紫铜基表面上制备了具有微纳结构的十八烷基硫醇自组装超疏水表面,采用高速摄像和显微技术研究了水平表面上液滴合并运动特性。实验结果表明,超疏水表面上液滴合并过程中释放出的表面能可以克服表面黏附作用诱发液滴弹跳现象,液滴越小,弹跳高度越大。考虑液滴表面黏附功、液滴运动引起的黏性耗散能等,根据能量守恒原理进行了理论分析。理论分析结果表明,液滴合并诱导弹跳现象存在最小液滴半径约为39.9 μm;随着液滴半径的增大,液滴弹跳高度增大,液滴半径为83.7 μm时达到弹跳最大高度点,随着液滴半径继续增大,液滴弹跳高度减小,直至最大半径约3.9 mm时,无液滴弹跳现象发生而是在合并位置发生脉动。     

疏水表面液滴合并弹跳过程的数值模拟

液滴合并弹跳对强化热泵空调系统中的凝结传热及防结霜、除霜等方面均有良好的应用前景。在综合考虑固-液、气-固和气-液表面自由能,重力势能,液滴内部黏性耗散功及表面黏附功的基础上建立了液滴合并及弹跳的分阶段能量模型,并进行了超疏水表面不同半径液滴合并弹跳时的模型模拟与实验验证,得到较好的吻合。基于该模型研究了液滴数量、半径均匀性及不同表面状态对液滴合并弹跳过程的影响规律。结果表明,液滴数量增加时,合并阶段临界接触角由120°减小至105°,半径尺寸均匀性增加时,弹跳阶段临界接触角从140°减小至130°。当表面接触角大于140°时,固液接触系数影响微乎其微。可见,液滴数量的增多及液滴尺寸均匀性的提升有利于合并弹跳过程的发生,固液接触系数对合并弹跳过程的影响程度随表面接触角的增大而减小。     

Segregation of cohesive powders in a vibrated granular bed

The effects of small amounts of added liquid on the segregation behavior of a granular system under vertical vibration by DEM simulation are investigated in this study. The cohesive forces of grains are incorporated into DEM simulations via a simplified dynamic liquid bridge force model. The simulation results show that capillary forces in addition to viscous forces have an important effect on the segregation phenomenon. The segregation rate of larger intruder rises to the top of the bed is found to depend on the liquid content. The segregation rate is sharply increased when a small amount of liquid is added to granular system. A transition to the reduction of segregation rate occurs at a critical liquid content. It has shown that this transition can be interpreted as the increase of attractive force between grains due to viscous force. The viscous forces make the particles stick more tightly to each other and retard the movement of particles, thus reducing the segregation rate. The segregation rate is also related to the convection motion of the granular system. The presence of convection enhances the segregation rate of wet granular materials.     

Effects of capillary force and surface deformation on particle removal in turbulent flows

A new rolling detachment model for particle removal in the presence of capillary forces based on the maximum adhesion resistance was developed. The new model uses an effective thermodynamic work of adhesion model that includes the effects of capillary forces generated by the formation of liquid meniscus at the interface. The JKR and DMT models for elastic particle and surface deformations and the Maugis and Pollock model for the plastic deformation were extended to include the effect of capillary forces. Under turbulent flow conditions, the criteria for incipient rolling detachments were evaluated. The turbulence burst model was used to evaluate the air velocity near the substrate. The critical shear velocities for resuspension of particles of different sizes were evaluated and the results were compared with those without capillary force. The model predictions were compared with the available experimental data and good agreement was found.     

An experimental study and model assessment of polymer sintering

Experiments were conducted using pairs of particles inside a hot stage microscopy setup with the ultimate objective to study the coalescence, which is a crucial stage in the rotational molding process. It was found that the geometry of the particles had no significant effect on the sintering rate. The sintering rate increases as the viscosity of the resin decreases. However, this effect became less important as the particle size decreased. The experimental results of this study have been compared with the available mathematical models based on balance of viscous and surface tension forces. The model developed by Frenkel and a corrected version by Eshelby predict a faster coalescence than observed experimentally. However, Hopper's model is in relatively good agreement with the present experimental data. Yet there is evidence that mechanisms other than Newtonian viscous flow may play a role in polymer sintering.     

The capillary break-up of a binary liquid column inside a tube

The break-up of a stationary viscous liquid cylinder surrounded by an immiscible viscous liquid contained in a circular tube by capillary forces at the fluid interface is considered. It was found that the maximum growth-rate parameter f* depends upon η'/η, the ratio of the viscosity of the inner liquid to that of the outer liquid and a/S, the ratio of the radius of the inner liquid cylinder to that of the tube. Both f* and corresponding wavelength parameter (2πa/λ)* have been calculated theoretically; measured values of both parameters are in good agreement with the theory.     

Large dynamic contact angles

The relation between the dynamic contact angle and the capillary number is explored by extending de Gennes’ model where viscous dissipation is equated to the surface work. A cutoff is used to eliminate the contact line singularity. The aim has been to increase the applicability of existing results to include higher contact angles. The flow near the contact line is assumed to be in the creeping flow regime. The results uphold Voinov–Hoffman–Tanner rule at small contact angles but only in the liquid–air system and not in the liquid–liquid system. The more prevalent liquid–air system shows a fold at a dynamic contact angle of 180°. This indicates that at 180° the system jumps out of the dynamic contact line arrangement to an entrained system that has no contact line. Liquid–liquid systems have folds at much lower contact angles. Comparisons with experiments show very good agreement up to contact angles of 120° but the comparisons show only qualitative agreement at large contact angles. Possible explanations have been provided for the latter.     

A thermodynamically consistent phase-field model for viscous sintering

A thermodynamically consistent phase-field model for viscous sintering is proposed. It is based on an energetic variational formulation that allows the governing equations to be analytically derived from a defined energy law. The conservation of mass is satisfied through the incompressibility assumption and the assumption that mass density is uniform initially within the particle compact while the balance of linear momentum is formulated from an energy dissipation law. The morphological changes of particles are described by the temporal and spatial evolution of a phase-field variable governed by a modified Cahn-Hilliard equation, and the motion of viscous mass flow is controlled by the Stokes equation incorporating the surface tension effect. The application of the phase-field model is illustrated by examining the effect of particle shape, initial contact angle and rearrangement effects on viscous sintering.     

同向双螺杆挤出过程中聚合物颗粒熔融问题探讨Ⅰ—相变分数法与实验验证

在双螺杆挤出过程中，聚合物颗粒挤出熔融过程中较为常见的一种固、液两相共存的形式可以用“海－岛”模型来描述。提出了“海－岛”模型的一种简化模型－中间模型，通过引入“相变分数”对粘性耗散热进行分配，得出了固相分数及熔体平均温度沿螺槽方向的变化规律。通过实验对相关结果进行了验证，对比表明该方法基本能反映聚合物颗粒熔融的真实过程。     

Mass transfer rates from bubbles in stirred tanks operating with viscous fluids

In this paper, the role of liquid viscosity on the mass transfer rates in stirred tank reactors has been theoretically studied. Liquid viscosity affects liquid diffusivity and bubble size distribution by defining bubble stability in the flow. A population balance, taking into account the effect of liquid viscosity on the coalescence and break-up closures, has been combined with Higbie–Kolmogorov's theory to predict the effect of liquid viscosity on the mass transfer rates. Experimental results from the literature for stirred tanks operating with one single Rushton turbine have been used as comparison. Different moderately viscous aqueous solutions (glucose, glycerol and millet-jelly) have been considered. Bubble break-up depends on the critical deformation of the bubbles in the continuum phase. A correlation between the Weber critical number and the liquid viscosity has been found. Once the bubble distribution is accurately determined, the volumetric mass transfer rate in viscous solutions can be predicted theoretically.     

Granular pressure and particle velocity fluctuations prediction in liquid fluidized beds

A numerical modelling approach for the dynamic simulation of solid-liquid fluidized beds is evaluated. This approach is based on an Eulerian two-fluid formulation of the transport equations for mass, momentum and fluctuating kinetic energy. The solid-phase fluctuating motion model is derived in the frame of granular medium kinetic theory accounting for the viscous drag influence of the interstitial liquid phase. Solid-liquid fluidized bed two-dimensional simulations were performed for flow configurations taken from the experimental work of Zenit et al. [1997. Collisional particle pressure measurements in solid-liquid flows. Journal of Fluid Mechanics 353, 261-283], for three types of solid particles of contrasted inertia in water at high particle Reynolds number (nylon, glass and steel beads). Experimental and numerical granular pressures exhibit a satisfactory agreement with both low and high inertia particles, the best level of prediction being observed with the most inertial particles. Sensitivity of the predictions to the phenomenological laws used in the model is also presented and it appears that, due to non-linear correlations, the average granular pressure in the bed is a less sensitive variable than the fluctuating kinetic energy (or granular temperature). The transport mechanisms of the mean granular temperature in the bed are therefore analyzed as a function of the solid fraction and the particle inertia. At low and moderate Stokes number (nylon and glass beads) and in all range of solid-phase fraction, the dominant production mechanism of fluctuating kinetic energy is due to the mean velocity gradient, whereas the main dissipation term is that induced by the viscous drag. At higher Stokes number (steel beads) and concentration, the production of the granular temperature is controlled by the compressibility effects via the granular pressure. In this case, the dissipation is mainly provided by inter-particle collisions.     

Population balance modelling of granulation with a physically based coalescence kernel

It was previously published by the authors that granules can either coalesce through Type I (when granules coalesce by viscous dissipation in the surface liquid layer before their surfaces touch) or Type II (when granules are slowed to a halt during rebound, after their surfaces have made contact) (AIChE J. 46 (3) (2000) 529). Based on this coalescence mechanism, a new coalescence kernel for population balance modelling of granule growth is presented. The kernel is constant such that only collisions satisfying the conditions for one of the two coalescence types are successful. One constant rate is assigned to each type of coalescence and zero is for the case of rebound. As the conditions for Types I and II coalescence are dependent on granule and binder properties, the coalescence kernel is thus physically based. Simulation results of a variety of binder and granule materials show good agreement with experimental data.     

A two-dimensional study of the rupture of funicular liquid bridges

An analysis of the rupture behaviour of liquid bridges in simple granular systems is reported. Wet granules present a complex structure in which primary particles are held together at a microscopic scale by means of capillary forces mediated by a liquid binder. These capillary interactions control the mechanical properties of the particle assemblies as well as the kinetics and pathways of granule growth. The evaluation of the interaction between contacting particles by means of the interstitial liquid binder in its various possible configurations has recently been reported for two-dimensional systems of contacting particles (J. Colloid Interface Sci. 220 (1999) 42). In the current work particle separation forces and bridge rupture have been examined for the symmetric separation pathways in all possible wetted states of triplets of particles. The separation force and the energy of the system due to the liquid capillary action have been calculated during the process of separation, which is assumed to occur slowly through a sequence of equilibrium configurations. A wide range of behaviour is found for the rupture processes of the different liquid configurations.     

Viscoelasticity and Kinetics of Wetting on Rubber

The kinetics of spreading of a liquid drop is usually controlled by conversion of capillary potential energy into viscous dissipation within the liquid when the solid is rigid. However, if the solid is soft, a “wetting ridge” near the solid/liquid/vapour triple line can also be a dissipative sink as the wetting front moves. As a consequence, the kinetics of wetting of rubber may be controlled essentially by viscoelastic losses in the polymer rather than by viscous losses in the liquid drops. Therefore, a direct analogy between the kinetics of wetting and adhesion, respectively, for a liquid and a solid on an elastomeric substrate has been recently proposed. In this paper, the superposition of viscoelastic braking and moderate rubber swelling in the drop spreading phenomenon is considered.     

Viscoelasticity and Kinetics of Wetting on Rubber

The kinetics of spreading of a liquid drop is usually controlled by conversion of capillary potential energy into viscous dissipation within the liquid when the solid is rigid. However, if the solid is soft, a “wetting ridge” near the solid/liquid/vapour triple line can also be a dissipative sink as the wetting front moves. As a consequence, the kinetics of wetting of rubber may be controlled essentially by viscoelastic losses in the polymer rather than by viscous losses in the liquid drops. Therefore, a direct analogy between the kinetics of wetting and adhesion, respectively, for a liquid and a solid on an elastomeric substrate has been recently proposed. In this paper, the superposition of viscoelastic braking and moderate rubber swelling in the drop spreading phenomenon is considered.     

CFD modeling of slurry bubble column reactors for Fisher-Tropsch synthesis

Industrial bubble column reactors for Fischer-Tropsch (FT) synthesis include complex hydrodynamic, chemical and thermal interaction of three material phases: a population of gas bubbles of different sizes, a liquid phase and solid catalyst particles suspended in the liquid. In this paper, a CFD model of FT reactors has been developed, including variable gas bubble size, effects of the catalyst present in the liquid phase and chemical reactions, with the objective of predicting quantitative reactor performance information useful for design purposes. The model is based on a Eulerian multifluid formulation and includes two phases: liquid-catalyst slurry and syngas bubbles. The bubble size distribution is predicted using a Population Balance (PB) model. Experimentally observed strong influence of the catalyst particles concentration on the bubble size distribution is taken into account by including a catalyst particle induced modification of the turbulent dissipation rate in the liquid. A simple scaling modification to the dissipation rate is proposed to model this influence in the PB model. Additional mass conservation equations are introduced for chemical species associated with the gas and liquid phases. Heterogeneous and homogeneous reaction rates representing simplified FT synthesis are taken from the literature and incorporated in the model.Hydrodynamic effects have been validated against experimental results for laboratory scale bubble columns, including the influence of catalyst particles. Good agreement was observed on bubble size distribution and gas holdup for bubble columns operating in the bubble and churn turbulence regimes. Finally, the complete model including chemical species transport was applied to an industrial scale bubble column. Resulting hydrocarbon production rates were compared to predictions made by previously published one-dimensional semi-empirical models. As confirmed by the comparisons with available data, the modeling methodology proposed in this work represents the physics of FT reactors consistently, since the influence of chemical reactions, catalyst particles, bubble coalescence and breakup on the key bubble-fluid drag force and interfacial area effects are accounted for. However, heat transfer effects have not yet been considered. Inclusion of heat transfer should be the final step in the creation of a comprehensive FT CFD simulation methodology. A significant conclusion from the modeling results is that a highly localized FT reaction rate appears next to the gas injection region when the syngas flow rate is low. As the FT reaction is exothermal, it may lead to a highly concentrated heat release in the liquid. From the design perspective, the introduction of appropriate heat removal devices may be required.     

Modeling the evolution and rupture of stretching pendular liquid bridges

A simplified mathematical model and numerical simulations of the governing Navier–Stokes equations are used to predict the shape evolution, rupture distance, and liquid distribution of stretching pendular liquid bridges between two equal-sized spherical solid particles. In the simplified model, the bridge shape is approximated with a parabola, and it is assumed that the surface tension effects dominate the viscous, inertial, and gravitational effects. For the numerical simulations, a commercial Computational Fluid Dynamics (CFD) software package – FLUENT – is used. The rupture distance predictions obtained with both models are compared with experimental data and a reasonable agreement is found. The results of the numerical investigations show that for simulations with negligible viscous, inertial, and gravitational effects, the rupture distance approaches an asymptotic value, which is close to the value predicted by the simplified model. The bridge profiles predicted using the simplified model and the numerical simulation are compared. It is found that a second-order polynomial appropriately represents the stable bridge shape for particles with identical contact angles; however, for liquid bridges between particles with different contact angles, the numerical simulations of the governing Navier–Stokes equations should be used.     

AGGREGATION AND BREAKAGE OF SOLID PARTICLES IN SUSPENSIONS AGITATED IN A VIBRATING MIXER: A NON-FRACTAL APPROACH

Aggregation and breakage of solid particles or aggregates suspended in liquid can be found in numerous industrial processes. It is crucially important to predict the evolution of aggregates at any moment during the process. This aim can be achieved by modeling based on solid phase population balances. A lumped discrete population balance model has been selected for the verification of the experimental data. The experiments have been carried out in a laboratory-scale vibrating mixer equipped with a disc-type reciprocating impeller. A reasonably good agreement between the computational and experimental results of PSD data has been obtained. Dependencies between aggregation and breakage rate coefficients and the average energy dissipation rate in the mixer have been shown.