Critical comparison of hydrodynamic models for gas-solid fluidized beds—Part I : bubbling gas-solid fluidized beds operated with a jet

In the Eulerian approach to model gas-solid fluidized beds closures are required for the internal momentum transfer in the particulate phase. Firstly, two closure models, one semi-empirical model assuming a constant viscosity of the solid phase (CVM) and a second model based on the kinetic theory of granular flow (KTGF), have been compared in this part in their performance to describe bubble formation at a single orifice and the time-averaged porosity profiles in the bed using experimental data obtained for a pseudo two-dimensional fluidized bed operated with a jet in the center. Numerical simulations have shown that bubble growth at a nozzle with a jet is mainly determined by the drag experienced by the gas percolating through the compaction region around the bubble interface, which is not much influenced by particle-particle interactions, so that the KTGF and CVM give very similar predictions. However, this KTGF model does not account for the long term and multi particle-particle contacts (frictional stresses) and under-predicts the solid phase viscosity at the wall as well as around the bubble and therefore over-predicts the bed expansion. Therefore, in the later part of the paper, the bubble growth at a single orifice and the time-averaged porosity distribution in the bed predicted by the KTGF model with and without frictional stresses are compared with experimental data. The model predictions by the KTGF are improved significantly by the incorporation of frictional stresses, which are however strongly influenced by the empirical parameters in this model. In Part II the comparison of the CVM and KTGF with experimental results is extended to freely bubbling fluidized beds.     

Practical validation of the two-fluid model applied to dense gas-solid flows in fluidized beds

This paper discusses the simulation of bubbling gas-solid flows by using the Eulerian two-fluid approach. Predictions of particle motion, bed expansion, bubble size and bubble velocity in bubbling beds containing Geldart B particles are compared with experimental results and correlations found in the literature. In addition, gas mixing in a bed of Geldart A particles is investigated.An in-house code has been developed based on the finite-volume method and the time-splitting approach using a staggered grid arrangement. The velocities in both phases are obtained by solving the 2D Reynolds-averaged Navier/Stokes equations using a partial elimination algorithm (PEA) and a coupled solver. The k-ε turbulence model is used to describe the turbulent quantities in the continuous phase.In general, the model predictions are in good agreement with experimental data found in the literature. Most important observations are: the level of the restitution coefficient was found to be crucial in order to obtain successful results from 2D axisymmetric simulations of a system containing Geldart B particles. Bubble size and bubble rise velocities are not as sensitive to the restitution coefficient. The turbulence model is of outmost importance concerning gas mixing in a fluidized bed of Geldart A particles.From these numerical analyzes an optimized granular flow two-fluid model can be designed for the purpose of simulating reactive systems in fluidized bed reactors.     

Bubble characteristic in three phase fluidized beds

Bubble size and rising velocity in freely bubbling liquid-gas and three phase fluidized beds have been measured by means of movie photography. Three solids (1–6 mm), a variety of liquids, and air were employed as the three phases.Both bubble size and rising velocity were found to increase with gas velocity but were relatively insensitive to the liquid velocity, viscosity, and surface tension. At the high gas rates employed, the bubble characteristics were independent of particle size. Correlations are presented for calculating bubble size and rising velocity.     

Filtered and heterogeneity‐based subgrid modifications for gas–solid drag and solid stresses in bubbling fluidized beds

Two different approaches to constitutive relations for filtered two‐fluid models (TFM) of gas–solid flows are deduced. The first model (Model A) is derived using systematically filtered results obtained from a highly resolved simulation of a bubbling fluidized bed. The second model (Model B) stems from the assumption of the formation of subgrid heterogeneities inside the suspension phase of fluidized beds. These approaches for the unresolved terms appearing in the filtered TFM are, then, substantiated by the corresponding filtered data. Furthermore, the presented models are verified in the case of the bubbling fluidized bed used to generate the fine grid data. The numerical results obtained on coarse grids demonstrate that the computed bed hydrodynamics is in fairly good agreement with the highly resolved simulation. The results further show that the contribution from the unresolved frictional stresses is required to correctly predict the bubble rise velocity using coarse grids. © 2013 American Institute of Chemical Engineers AIChE J, 60: 839–854, 2014     

Assessment of gas-particle flow models for pseudo-2D fluidized bed applications

The aim of this work is to provide more insight into the general modeling criteria for simulating pseudo-2D bubbling fluidized beds. For this purpose, two experimental-based problems are studied. First, a fluidized bed with a high-speed central jet problem is analyzed. A qualitative study of the first bubble indicates that the bubble shape prediction is highly sensitive to the frictional model adopted. The most accurate results in terms of bubble shape and detachment time are given by a frictional model that relates the strain-rate fluctuations with the granular temperature. Second, a uniformly fluidized bed problem in bubbling regime is considered. For this case, the drag models and boundary conditions for the particulate phase are investigated. Time-averaged solid phase velocity profiles are compared with the results of the literature where it is found that no-slip conditions (or partial slip with a high specularity coefficient) are more appropriate than slip conditions at the walls for these regimes. Regarding the drag force, although none of the models presented could match the experimental velocity predictions for low gas velocities at the lower region of the bed, the Di Felice model produces the most accurate results for the whole range of regimes considered.     

Flow structure of the solids in gas-solid fluidized beds

Existence of clusters in dense fluidized beds was investigated by analyzing the time-position data of a tracer obtained in several radioactive particle tracking experiments. It was found that in the case of sand particles, more gas passes through the bed as bubbles with increasing the superficial gas velocity and in the case of FCC powder, flow of the gas through the bed as bubbles does not increase in the turbulent fluidization regime. Cluster diameters were estimated from their velocities and found that descending clusters are generally larger than ascending ones and the size of both increases with increasing the superficial gas velocity. Bubble velocities evaluated in this work are in good agreement with the correlations in the bubbling regime of the fluidization available in the literature.     

The clustered dense phase model for group A fluidization: I. Dense phase hydrodynamics

A new hydrodynamic model is proposed to represent the gas flow in the dense phase of fluidized Group A powders. The model views that the particles form clusters under the influence of inter-particle forces, giving rise to the formation of a heterogeneous void structure consisting of clusters of particles and interstitial cavities. The model contains two parameters, one representing the intrinsic void structure of the clusters and the other representing their size.The model predictions have been tested against reported experimental dense phase hydrodynamic data of Group A powders, both in bubble-free beds and in freely bubbling beds. The results show 2-3 particle clusters in bubble-free beds, but considerably larger clusters, containing 100 particles or so, in the dense phase of freely bubbling beds. The model also provides predictions for bubble through-flow, bubble splitting from below, dense-phase solids circulation, and interstitial gas bypassing in freely bubbling beds.     

Non-intrusive monitoring of bubbles in a gas-solid fluidized bed using vibration signature analysis

A reliable method was developed to study bubble behavior by analysis of vibration signals in fluidized beds. The advantage of this method is that the vibration probe is in indirect contact with the process. Accelerometers were used to record vibration signals generated by particle flow through the fluidized bed at various superficial gas velocities and particle sizes. Measurement of vibration signals, sampled at 25 kHz for 30 s, enabled investigation of changes in flow structure related to flow regime transitions. To study bubble behavior under different conditions, different particle sizes were used in the experiments. The measurements were extensively analyzed using wavelet and fast Fourier transforms. Results indicate that the vibration frequency generated by bubbles is between 1000 and 3000 Hz. The vibration analysis was effectively used to detect minimum fluidization and transition from bubbling to slugging in gas-solid fluidized beds.     

Heat transfer within dynamically structured bubbling fluidized beds subject to vibration: A two-fluid modeling study

Bubbling fluidized beds are often used to achieve a uniform particle temperature distribution in industrial processes involving gas and particles. However, the chaotic bubble dynamics pose significant challenges in scale-up. Recent work (Guo et al., 2021, PNAS 118, e2108647118) has shown that using vibration can structure the bubbling pattern to a highly predictable manner with the characteristic bubble properties independent of system width, opening opportunities to address key issues associated with conventional bubbling fluidized beds. Herein, using two-fluid modeling simulations, we studied heat transfer characteristics within the dynamically structured bubbling fluidized bed and compared to unstructured bubbling fluidized beds and packed beds. Simulations show that the structured bubbling fluidized bed can achieve the most uniform particle temperature distribution because it can achieve the best particle mixing while maintaining a global heat transfer coefficient similar to that of a freely bubbling fluidized bed.     

Flow of gas and particles in a bubbling fluidized bed with a filtered two-fluid model

Numerical simulations of gas-particles flow in a bubble fluidized bed with two large eddy simulations of gas and solid phases are presented. For gas phase and solid phase, the sub-grid scale model for the viscosity is based on the Smagorinsky form. The sub-grid model for the particle pressure proposed by Igci et al. (2008) is modified by replacing the minimum fluidization velocity. The collisional interaction of particles is considered by the kinetic theory of granular flow. Flow behavior of gas and particles is performed by means of these two sub-grid scale models. The subgrid closure for the particle phase viscosity and pressure led to a qualitative change in the simulation results. Predictions are compared with experimental data measured by Yuu et al. (2000) and Taghipour et al. (2005) in the bubbling fluidized beds. The distributions of concentration and velocity of particles are predicted in the bubbling fluidized bed. The predicted filtered particle phase pressure increases and the filtered particle phase viscosity decreases with the increase of particle concentration. The qualitative importance of the model constant c_s of particles is demonstrated.     

Effect of bed size on hydrodynamics in 3‐D gas–solid fluidized beds

It is well known that hydrodynamics observed in large scale gas–solid fluidized beds are different from those observed in smaller scale beds. In this article, an efficient two‐fluid model based on kinetic theory of granular flow is applied, with the goal to highlight and investigate hydrodynamics differences between three‐dimensional fluidized beds of diameter 0.10, 0.15, 0.30, 0.60, and 1.0 m, focusing on the bubble and solids flow characteristics in the bubbling regime. Results for the 0.30 m diameter bed are compared with experimental results from the literature. The bubble size evolution closely follows a correlation proposed by Werther for small beds, and a correlation proposed by Darton for sufficiently large beds. The bubble size increases as the bed diameter is increased from 0.10 to 0.30 m, and remains approximately constant for bed diameters from 0.30 to 1.0 m. Concurrently, an increase in bubble rise velocity is observed, with a much high bubble rise velocity in the largest bed of diameter 1.0 m due to gulf stream circulations. The dynamics in shallow and deep beds is predicted to be different, with marked differences in bubble size and solids circulation patterns. © 2015 American Institute of Chemical Engineers AIChE J, 61: 1492–1506, 2015     

Experimental validation of CFD models for fluidized beds: Influence of particle stress models, gas phase compressibility and air inflow models

This work compares numerical simulations of fluid dynamics in fluidized beds using different closure models and air feed system models. The numerical results are compared to experiments by means of power spectral density distributions of fluctuating pressure signals and bubble statistics obtained from capacitance probe measurements. Two different particle rheology models are tested in combination with two different values of the maximum particle volume fraction. The first particle model predicts the particle pressure by an exponential power law and assumes a constant particle viscosity (CPV), and the second model predicts the stresses using the kinetic theory of granular flow (KTGF). Furthermore, two model approaches for the air inflow are evaluated. The first inflow model includes the coupling between the air-feed system and the fluidized bed in the simulation, and the second model assumes a constant mass flow of gas into the fluidized bed. Finally, the influence of the compressibility of the gas phase on the numerical predictions is investigated. The numerical simulations are made using the CFX-4.4 commercial flow solver.The simulations show that the KTGF model gives a more evenly distributed bubble flow profile over the bed cross-section, while the CPV model gives a more parabolic bubble flow profile, with a higher bubble flow in the central part of the bed. This work shows that the KTGF model results are in significantly better agreement with the experiments. It is furthermore shown that the modelling of the air-feed system is crucial to for predicting the overall bed dynamic behaviour.     

Investigation into the hydrodynamics of gas–solid fluidized beds using particle image velocimetry coupled with digital image analysis

The hydrodynamics of a freely bubbling, pseudo 2‐D fluidized bed has been investigated experimentally for different bed aspect ratios at different superficial gas velocities by using Particle Image Velocimetry (PIV) combined with Digital Image Analysis (DIA). Coupling of both non‐invasive measuring techniques allows us to obtain information on both the bubble behaviour and emulsion phase circulation patterns simultaneously. In particular, the combination of DIA with PIV allows to correct for the influence of particle raining through the roof of the bubbles on the time‐averaged emulsion phase velocity profiles.     

Hydrodynamics of gas-solid flow around immersed tubes in bubbling fluidized beds

A numerical study was conducted based on the gas-solid two-fluid model using the body-fitted coordinate system to analyze the behavior of particles and bubbles flow in bubbling fluidized beds without and with immersed tubes. The kinetic theory of granular flow was implemented in the model. The images of simulated instantaneous particle concentration and velocity gave the process of the formation, coalescence and eruption of bubbles. The effects of the tube pitch and superficial gas velocity on the fluidization in a bubbling fluidized bed were investigated. Calculated bubble frequencies without and with immersed tubes were in agreement with previous experimental and simulation findings. The wavelet multi-resolution analysis was used to analyze the simulated data of instantaneous particle concentration. From the random-like particle concentration fluctuations, the fluctuating components due to particle flow and bubble motion can be extracted based on the wavelet multi-resolution analysis over a time-frequency plane.     

A critical comparison of frictional stress models applied to the simulation of bubbling fluidized beds

The behaviour of a gas-solid flow in a bubbling fluidized bed operated near the minimum fluidization condition is strongly influenced by the frictional stresses between the particles, these being highly concentrated and their motion dominated by enduring contact among them and with the walls.The effect of the introduction of frictional stresses in a Eulerian-Eulerian two fluid model based on the kinetic theory of the granular flow is evaluated. The models of Johnson and Jackson [1987. Frictional-collisional constitutive relations for granular materials, with application to plane shearing. Journal of Fluid Mechanics 176, 67-93], Syamlal et al. [1993. Mfix documentation: volume I, theory guide. Technical Report DOE/METC-9411004, NTIS/DE9400087, National Technical Information Service, Springfield, VA], and Srivastava and Sundaresan [2003. Analysis of a frictional-kinetic model for gas-particle flow. Powder Technology 129, 72-85] are compared with the kinetic theory of the granular flow and with experimental data both in a bubbling fluidized bed with a central jet and in a bubbling fluidized bed with a porous distributor. The predicted evolution of the bubble diameter along the height of the fluidized beds is examined, the shapes of the bubbles predicted by the models are compared and the evolution in time of the bubbles is shown. In the case of the bed with a central jet, the bubble detachment time is also calculated. The results show that the introduction of a frictional stress model improves the prediction of the bubbles diameter in a bubbling fluidized bed with a central jet and positively affects the bubbles diameter distribution in a uniformly fed bubbling fluidized bed. The high sensitivity of the model to the value of the particulate phase fraction at which frictional stresses start to be accounted for is pointed out through a sensitivity analysis performed on the Srivastava and Sundaresan [2003. Analysis of a frictional-kinetic model for gas-particle flow. Powder Technology 129, 72-85] model.     

Insights in distributed secondary gas injection in a bubbling fluidized bed via discrete particle simulations

Experiments have shown that distributed secondary gas injection via a fractal injector in fluidized beds can significantly reduce the bubble size, and may also decrease the bubble fraction. In order to gain insight into the distribution of the gas between the phases and the mechanisms behind these effects simulations of small bubbling fluidized beds with one or two secondary gas injection points were carried out using a discrete particle model. Although the systems are very small, so that wall effects cannot be excluded, the model predicts that the bubble size and bubble fraction both decrease with secondary gas injection, while the gas flow through the dense phase increases. The secondary gas tends to stay in the dense phase, which limits the amount of gas available to form bubbles and is the main contributor to the decrease in the bubble size and fraction. The gas-solid contact improves as a result.     

A new model for ejected particle velocity from erupting bubbles in 2-D fluidized beds

A new model is proposed for obtaining the velocity profile of the particle ejected from the bubble dome in a freely bubbling 2-D fluidized bed. Its basis is the supposition that the initial velocity of the ejected particles, with a direction perpendicular to the dome contour, depends on bubble velocity and bubble growth velocity. This model differs from those previously appearing in the literature in that it is valid not only for vertical-ascent circular bubbles.Experiments were carried out in a freely bubbling 2-D fluidized bed using a high-speed video camera to measure the velocity profile. Upon comparing these results with the proposed model, it was established that, excepting some isolated cases, the model properly predicts the magnitude and direction of the maximum particle ejection velocity and the velocity profile.Using the work of Shen et al. (2004. Digital image analysis of hydrodynamics two-dimensional bubbling fluidized beds. Chemical Engineering Science 59, 2607-2617), we obtain two general equations for the bubble velocity and the bubble growth velocity in a 2-D fluidized bed. These expressions, together with the proposed model, can be used to calculate the initial velocity of the ejected particles.     

不等粒径流化床的软球模拟

基于颗粒轨道模型,提出了粒径分别服从均匀分布与正态分布的软球方法,其中流体运动用Navier-Stokes方程描述,颗粒运动服从牛顿第二定理.模拟了不等粒径流化床中的气泡和节涌现象,并分别研究了表观气速、颗粒刚度系数、粒径分布不同时,固相颗粒的速度分布规律.其研究结果表明：分布板结构和表观气速对气泡行为有一定影响,随表观气速的增加,气泡形成、上升、破裂的速度加快;并且刚度系数越大,颗粒轴向速度随时间衰减越快;宽粒径分布的颗粒轴向速度大于窄粒径分布的颗粒轴向速度.     

Novel phenomenological discrete bubble model of freely bubbling dense gas–solid fluidized beds: Application to two‐dimensional beds

A phenomenological discrete bubble model has been developed for freely bubbling dense gas–solid fluidized beds and validated for a pseudo‐two‐dimensional fluidized bed. In this model, bubbles are treated as distinct elements and their trajectories are tracked by integrating Newton's equation of motion. The effect of bubble–bubble interactions was taken into account via a modification of the bubble velocity. The emulsion phase velocity was obtained as a superposition of the motion induced by individual bubbles, taking into account bubble–bubble interaction. This novel model predicts the bubble size evolution and the pattern of emulsion phase circulation satisfactorily. Moreover, the effects of the superficial gas velocity, bubble–bubble interactions, initial bubble diameter, and the bed aspect ratio have been carefully investigated. The simulation results indicate that bubble–bubble interactions have profound influence on both the bubble and emulsion phase characteristics. Furthermore, this novel model may become a valuable tool in the design and optimization of fluidized‐bed reactors. © 2012 American Institute of Chemical Engineers AIChE J, 2012