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... In spatially periodic domains (Agrawal et al., 2001;Igci et al., 2008;Jiang et al., 2020a;Liu et al., 2020b;Ozel et al., 2016;Radl and Sundaresan, 2014;Rauchenzauner and Schneiderbauer, 2022;Rubinstein et al., 2017;Tang et al., 2015;Zhou and Fan, 2015), the scale separation is implied such that the mesoscale behavior is independent of the macroscale flow/boundary con-straints, as indicated in Mouallem et al. (Mouallem et al., 2018) In other words, the mesoscale structure formed in the periodic domain is assumed to be typical of any fluidized bed with specific macroscale conditions. However, granular flows and gas-fluidized beds are basically far from equilibrium states and characterized by the lack of scale separation (Goldhirsch, 2003;Wang et al., 2018), although a near equilibrium state with scale separation is possible in certain simple granular systems (Ojha et al., 2004;Weinhart et al., 2013). ...
... where the averaged drag force (also referred to as the filtered drag force (Ozel et al. 2016 ...
... In both CFD-DEM and TFM simulations, the quantitative results are normally sensitive to the grid size. In CFD-DEM simulations, grids as small as 1.75 d p may be needed to get grid-independent statistics in very dilute suspensions (Capecelatro et al., 2014) and 3 d p is fine enough in cases at higher solids volume fraction (Ozel et al., 2016). In TFM simulations, the grid size achieving gridindependent results is commonly considered to be between 3 and 10 times the particle diameter (Agrawal et al., 2001;Jiang et al., 2020a;McCulloch and Pitts, 1943;Ouyang et al., 2021;Ozel et al., 2013;Parmentier et al., 2012;Sarkar et al., 2016). ...
Article
To understand the effects of macroscale constraints on fluidization, we investigate the differences between the periodic domain and realistic bed by using fine-grid simulations. The differences in these two systems are highlighted by identifying three force-balance conditions with respect to the gasphase, solid-phase and the mixture, respectively. Specifically, these three conditions are not satisfied at the microscale, but satisfied at the macroscale in both systems; Over the mesoscale, the gas-phase force balance is established in the periodic domain and in the fluidized bed at bubbling states, but not at turbulent states, whereas the solid-phase and the mixture force balance conditions are established only in the periodic domain, not in the realistic bed. The influence of the bounding wall can be implicitly included by introducing the gas-phase pressure gradient, turbulent kinetic energies (TKEs) of gas and solids phases, and drift velocity as the markers for the drag force.
... It is widely reported that when multiphase models based on Kinetic Theory of Granular Flow (KTGF) (and the DDPM is one of them) are used with coarse grids, then using common homogeneous drag models leads to failure in the prediction of the multiphase flow hydrodynamics [42,43]. This failure is due to the unrealistic prediction of drag force caused by the inability of coarse grids to resolve mesoscale structures (such as particle strands in cyclones) [24,[42][43][44][45][46][47][48][49][50]. To overcome the deficiencies of homogeneous drag models for industrial-scale simulations heterogeneous drag models (also called sub-grid drag models) should be used to account for the effect of small mesoscale and heterogeneous structures on the drag correlations [24,[42][43][44][45][46][47][48][49][50]. ...
... This failure is due to the unrealistic prediction of drag force caused by the inability of coarse grids to resolve mesoscale structures (such as particle strands in cyclones) [24,[42][43][44][45][46][47][48][49][50]. To overcome the deficiencies of homogeneous drag models for industrial-scale simulations heterogeneous drag models (also called sub-grid drag models) should be used to account for the effect of small mesoscale and heterogeneous structures on the drag correlations [24,[42][43][44][45][46][47][48][49][50]. ...
... (2) to (11): 2)-(10)) [42]. Note that using sub-grid drag models constituted based on Eulerian-Eulerian simulations (such as the revised Sarkar et al. drag model) in an Eulerian-Lagrangian model (such as DDPM in present work) is an approximation as the particle information needs be mapped to the Eulerian grid before using the correlations [48,88]. An approximation of this kind is reasonable and the drag models deduced for Eulerian-Eulerian simulations can be used in Eulerian-Lagrangian simulations (such as DDPM and MP-PIC) and the other way around [48,88] (Particularly, the revised Sarkar et al. drag model has been used successfully in Eulerian-Lagrangian models in the literature [50,84]). ...
Article
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A hybrid multiphase model (Dense Discrete Phase Model, DDPM) coupled with an agglomeration model and a sub-grid drag model was developed for simulation of industrial-scale cyclones with high solid loading. The model is validated by experimental results of pressure drop and separation efficiency from a pilot-scale cyclone with a diameter of 1.6 m. Key trends such as improvement in separation efficiency and reduction in pressure drop of cyclone due to an increase in particle load are well captured by the model. It is concluded that including the agglomeration model is crucial in particular in cases involving very fine particles (d < 15 μm) for accurate predictions of pressure drop and separation efficiency, while using the sub-grid drag modification improves the prediction of separation efficiency.
... Therefore, either the need for grid size scaling or sub-grid corrections was recognized and pointed out [78][79][80], as a common requirement also for Eulerian-Eulerian and MP-PIC simulations on coarse grids (Figure 8a). For example, Radl and Sundaresan [78] examined the vertical upflow in periodic domains at different Reynolds' number and solids concentration, later focusing specifically on fluid and particle coarsening for parcel-based simulations [81]. Figure 8a shows graphically a typical effect of the use of coarse-grained parcels, which requires fluid grid coarsening. ...
... (a) Graphical example of the fluid-particle coarsening sequence leading to large cell sizes compared to the actual particles; (b) effect of the solid volume fraction ( = 1 − ) and number of particles per grain ( = in the present notation) on the normalized filtered drag coefficient . Reprinted from [81], with permission from Elsevier. ...
... Several periodic systems were investigated, mostly with the objective to improve understanding, characterize the influence of the coarse graining degree and grid size. Examples of such analyses are [68,78,80,81]. ...
Article
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In the last decade, a few of the early attempts to bring CFD-DEM of fluidized beds beyond the limits of small, lab-scale units to larger scale systems have become popular. The simulation capabilities of the Discrete Element Method in multiphase flow and fluidized beds have largely benefitted by the improvements offered by coarse graining approaches. In fact, the number of real particles that can be simulated increases to the point that pilot-scale and some industrially relevant systems become approachable. Methodologically, coarse graining procedures have been introduced by various groups, resting on different physical backgrounds. The present review collects the most relevant contributions, critically proposing them within a unique, consistent framework for the derivations and nomenclature. Scaling for the contact forces, with the linear and Hertz-based approaches, for the hydrodynamic and cohesive forces is illustrated and discussed. The orders of magnitude computational savings are quantified as a function of the coarse graining degree. An overview of the recent applications in bubbling, spouted beds and circulating fluidized bed reactors is presented. Finally, new scaling, recent extensions and promising future directions are discussed in perspective. In addition to providing a compact compendium of the essential aspects, the review aims at stimulating further efforts in this promising field.
... [41][42][43][44][45][46][47][48][49] It has to be noted that most of those models have been developed in the context of the Euler-Euler approach but it is agreed that these are also apply to Euler-Lagrange simulations. [40,50] Recently, we have presented a novel approach for deriving constitutive relations for the unresolved terms appearing in coarse grid simulations. Thereby, we advanced a spatially-averaged two-fluid model (SA-TFM), which is based on the concepts of turbulence modelling. ...
... [43,65] Euler-Lagrange methods commonly require at least a grid spacing, which is approximately 3-4 times the particle diameter. [38,50] It is, therefore, straightforward to apply coarse grids, which allow a considerable reduction of the computational effort. ...
... The solids phase is discretized by employing 3,636,983 parcels, while we use an average grid spacing of 0.1 to solve the gas-phase equations. This grid spacing is approximately 100 times larger than the resolution requirement for DDPM, [50] which implies a performance gain of about 4 orders of magnitude in the case of SA-DDPM. Figure 3 shows snapshots of the solids volume fraction. The figure clearly unveils that the bubbling/slugging regime, which is observed in the experiments, [68] is predicted by the SA-DDPM approach. ...
Article
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Detailed simulations of industrial‐scale fluidized beds such as the FINEX process are still unfeasible due to the wide range of spatial scales. Due to the computational limitations it is common to apply coarse grids, which do not resolve all relevant structures. In our previous study (Schneiderbauer, AIChE J., 63:3544‐3562, 2017), we have presented sub‐grid models, which enable the coarse grid simulation of dense large‐scale gas‐solid flows. In this paper, we apply these corrections to a parcel‐based the dense discrete phase model (DDPM), allowing to study the hydrodynamics of the FINEX® process. Furthermore, the parcel approach is augmented by an unreacted shrinking core model (USCM) to account for the direct reduction of the iron ore particles by the reducing agents of H2 and CO. This DDPM model is tested first for a cold pilot‐scale fluidized bed and second, the USCM approach is validated for the direct reduction in a lab‐scale fluidized. Finally, the model is applied to the FINEX process. The results show fairly good agreement with measurements of the average bed voidage and with experimentally determined particle size distributions. The results further indicate that fines are immediately reduced while the reduction of the largest ore grains takes considerably longer. This article is protected by copyright. All rights reserved.
... This is because, compared with TFM, CFD-DEM directly solves the motion and heat transfer of individual particles and hence produces more accurate results. Studies even show that CFD-DEM using a grid size around 3d p is able to obtain accurate results consistent with particle resolved-direct numerical simulation (PR-DNS), 35,36 where d p is the particle diameter. ...
... In this stage, the relevant contents related to the microscopic correlation such as two-point structure and the behavior at material front will be ana- A grid size of 2.5 d p is adopted since studies have shown that CFD-DEM using a grid size around 3d p is able to give grid-independent results. 35,36 With respect to the computational domain size, one can refer to Capecelatro et al., 38 which shows that more than 32 times the characteristic cluster length is needed in the gravity-aligned direction for obtaining domain size-independent statistics. However, in this work, as can be seen from Figure 1 For the discretization and solution method of the governing equations of the gas and solid phases, readers can refer to Yu et al. 39 The adopted domain-averaged solid volume fraction ⟨ϕ s ⟩ is 0.05, the particles are randomly distributed in the computational domain initially. ...
Article
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Computational fluid dynamics–discrete element method is used to simulate gas–solid flows with heat transfer. The strategy of repetitively resetting the gas phase temperature is adopted to maintain the temperature difference between the gas and solid phases. The difference between the heat transfer characteristics produced by this strategy and the traditional heat source method is analyzed. It is found that heat transfer in dense regions is artificially enhanced by the heat source method. Therefore, the results obtained by the temperature reset method are filtered to develop a three‐marker model for the filtered interphase heat transfer coefficient (IHTC). In the developed model, the filtered gas–solid temperature difference is nondimensionalized by the average gas–solid temperature difference in surrounding grids, which is shown to have an impact on the filtered IHTC. The developed model is demonstrated to be effective and accurate in both a prior and a posterior tests.
... This necessitates the need for the coarsegraining of the standard CFD-DEM method to make it capable of simulating industrial systems. Over last 10-13 years, there has been a continued interest on the development of coarsegrained CFD-DEM methods [15][16][17][18][19][20][21][22][23][24]. In the coarse-grained CFD-DEM method, a specified number of original particles are represented by a relatively larger particle or parcel. ...
... This approach was later extended by Tauscendschon et al. [20] to model the fluidization of the cohesive particles in a fluidized bed dryer. Research work was also done on the development of appropriate interphase drag model for the coarse grained CFD-DEM method [21,22]. Coarse-grained CFD-DEM model was used to investigate the coal gasification [23] and biomass gasification [24] in a fluidized bed reactor. ...
Article
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Fluidized bed dryer often used in the pharmaceutical industry for drying of wet granules. Coupled computational fluid dynamics (CFD) – discrete element method (DEM) is frequently used to model the drying process because of its ability to obtain the relevant information at the particle level. However, it becomes almost impossible to model the industrial scale fluidized bed dryer using the coupled CFD-DEM method because of the presence of large number of particles (1068)\left(\sim {10}^{6-8}\right). To reduce the number of particles to be tracked in the simulation, coarse grained coupled CFD-DEM method was developed by researchers where a certain number of particles of the original system was represented by a relatively bigger particle in the coarse-grained system. The appropriate scaling of the particle–particle and particle–fluid interaction forces is necessary to make sure that the dynamics of the coarse-grained particles/parcels accurately represent the dynamics of the original particles. The coarse-graining of the drying process of pharmaceutical granules during fluidization needs systematic coarse-graining of the momentum, heat, and solvent vapor transfer process. A coarse grained coupled CFD-DEM method was used to model the momentum and heat transfer during the fluidization of pharmaceutical granules. It was shown that the heat transfer during the fluidization of large number of particles could be predicted by simulating a smaller number of bigger particles with appropriate scaling of particle–particle heat and momentum transfer, and particle–fluid heat and momentum transfer at significantly smaller computational time. This model can be further extended by including a coarse-grained moisture transport model in future.
... TFM simulations have difficulties resolving particle size distributions (PSD) and capturing the influence of particle shapes while the Euler-Lagrangian methods can directly resolve them 49 . Current research indicates that the sub-grid drag models are also important to the coarse-grained Euler-Lagrangian simulations 29,[50][51][52] . Similar to the filtering of fine-grid TFM, the filtered drag model can also be constructed from the filtering of fine-grid CFD-DEM simulations [53][54][55] . ...
... The sub-grid gas velocity was also reconstructed from the gridscale pressure balance equation according to weighted local porosities 56 or simplified momentum equations 54 . However, due to the limited research of filtered CFD-DEM drag models, many of the coarse-grained Euler-Lagrangian simulations are still using the drag models derived from fine grid TFM 51,57,58 . Recent research 59 showed that the filtered TFM drag model overcorrects the influence of sub-grid structures if used for simulations with only fluid coarse-graining. ...
Article
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The accuracy of coarse-grained Euler-Lagrangian simulations of fluidized beds heavily depends on the mesoscale drag models to account for the influences of the unresolved sub-grid structures. Traditional filtered drag models are regressed with mesoscale markers such as voidage and slip velocities. In this research, a filtered drag was regressed with both mesoscale and macro-scale markers using fine grid Computational Fluid Dynamics-Discrete Element Method (CFD-DEM) simulations. The traditional non-linear regression method was compared with machine learning regression using an Artificial Neural Network (ANN) implemented in PyTorch and coupled with MFiX. The new drag showed higher accuracy than the Wen-Yu drag and another filtered drag derived from the two-fluid model. The nonlinear regression shows slightly better results than ANN regression in cases with similar R-squared values. The utilization of the gas inlet velocity as an additional macro-scale marker reduced the errors by up to 55.3% in the tested cases.
... However, DEM is not suitable for industrial systems, typically consisting of more than a billion particles because of high computational time. Several researchers worked on the coarse-graining of the DEM (and CFD-DEM) models to simulate the industrial systems relevant to different applications in a reasonable time [164][165][166][167][168][169][170] . In the coarse-grained CFD-DEM, a specified number of original particles are represented by a large particle or parcel [164][165][166][167][168][169][170] . ...
... Several researchers worked on the coarse-graining of the DEM (and CFD-DEM) models to simulate the industrial systems relevant to different applications in a reasonable time [164][165][166][167][168][169][170] . In the coarse-grained CFD-DEM, a specified number of original particles are represented by a large particle or parcel [164][165][166][167][168][169][170] . The parcel-parcel contact forces and the parcel-fluid interaction forces are modified to ensure that the coarse-grained system represents the original system. ...
Article
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The fluidized bed is an essential and standard equipment in the field of process development. It has a wide application in various areas and has been extensively studied. This review paper aims to discuss computational modeling of a fluidized bed with a focus on pharmaceutical applications. Eulerian, Lagrangian, and combined Eulerian-Lagrangian models have been studied for fluid bed applications with the rise of modeling capabilities. Such models assist in optimizing the process parameters and expedite the process development cycle. This paper discusses the background of modeling and then summarizes research papers relevant to pharmaceutical unit operations.
... By spatially averaging the highly resolved fine-grid simulation results, the constitutive relations used for coarse-grid simulations can be extracted. So far, numerous contributions [14][15][16][17][18][19][20][21][22][23][24][25] proposed a novel meso-scale drag model for coarse-grid E-L simulations. They indicated that the optimization of the viscous term is of importance for improving the predictability of the scaled drag force. ...
... Generally, the macroscopic phenomena are not sensitive to variations in Young's modulus commonly ranging from 10 6 to 10 9 Pa for cold-flow in the literature. 18,19,47,48 A sensitivity analysis of Young's modulus is included in the SI to evaluate the effect in a thermal gas-particle system. In order to ensure the accuracy of the generated model and to take into account the computational expenses, the Young's modulus is assigned to 10 8 Pa. ...
Article
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Accurately predicting the complex inhomogeneous heat transfer behavior in gas–solid fluidized beds is of fundamental importance. In this work, we constitute an enhanced filtered interphase heat transfer coefficient (IHTC) closure by systematically filtering the dataset from highly resolved three‐dimensional (3D) computational fluid dynamics–discrete element model simulations. Particularly, effects of several potential filtered variable markers on filtered IHTC predictions are examined by statistical analysis. We reveal the formulated filtered IHTC correction closure manifests a systematic dependence on filtered interphase temperature difference as an additional marker. The proposed closure shows good agreement with the filtered fine‐grid simulation data in an a priori analysis. Moreover, the difference of filtered IHTC corrections deduced from 3D Euler–Euler and Euler–Lagrange simulations is quantified. Finally, the comparative analysis between our proposed filtered IHTC formulation and those in literature is implemented. This work holds a potential to facilitate the development of thermal gas–solid flow modeling.
... Moreover, many articles have largely studied the influence of different drag models on direct macroscale flow parameters such as the gas/particle velocity and volume fraction (Xu et al., 2018;Bian et al., 2019Bian et al., , 2020 while relatively few studies have been focused on such influence on filtered mesoscale quantities such as filtered slip velocity, filtered correction factor and filtered mesoscale drag force (Zhou and Wang, 2014;Zhu et al., 2018). Besides, relatively few filtered SGM investigations have employed "more ideal" DNS-based HDCs as a sub-input in constructing filtered mesoscale drag closures (Radl and Sundaresan, 2014;Ozel et al., 2016). Radl et al. (2012) found that the microscopic HDCs of Wen and Yu (1966) and Beetstra et al. (2007) are of little impact on filtered drag predictions for dilute flows (εs≤0.15). ...
... (2) 16.32 and 10.44% for <εs>=0.25. Noting that Δgrid=~3ds was used in mesoscale modeling of inhomogeneous gas-particle flows in the literature, for example, Radl and Sundaresan (2014), Ozel et al. (2016) and Yu et al. (2020a). Therefore, the selection of Δgrid=3.3ds ...
Article
Filtered mesoscale model can be formulated from highly-resolved continuum or discrete simulations. The embedded microscopic homogeneous drag closure (HDC) is of key importance in determining the reliability and accuracy of such simulations. This work investigates the effects of sub-input HDCs on filtered mesoscale predictions using highly-resolved simulations. Quantitative comparisons directly reveal that there are significant differences between the commonly-practiced Wen-Yu drag closure and the direct numerical simulation (DNS) based HDCs, especially for moderate and dense gas-particle flows. Moreover, the HDCs from DNS of static particles agree better with the benchmark data from DNS of dynamic gas-particle flows at very low Reynolds numbers for εs>0.05∼0.10 while Wen-Yu drag is more applicable for the remaining range. Regarding that DNS is commonly implemented over a specific range of operating conditions, an enhanced HDC via refitting more elaborate high-fidelity DNS data (εs=[0.01, 0.65], Res=[1, 1000]) from literature is proposed and analyzed.
... In 2016, Ozel et al. [45] studied the effect of drag forces in E-E (Euler-Euler) and MP-PIC models. The process involved conducting fine-mesh simulations in the E-L (Euler-Lagrange) model to extract corrections for mesh dependency and applying them to coarsemesh simulations. ...
Article
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In recent years, the fast growth of computational power has allowed the application of computational fluid dynamics (CFD) in a wide range of areas of interest, such as gas–solid unit operations. In this context, the multiphase particle-in-cell (MP-PIC) method appears as an option to represent fluid–particle and particle–particle interactions, avoiding the complexity of tracking each particle and the high computational cost derived from this. The MP-PIC method can represent the particles as a group with the same characteristics, allowing the simulation of gas–solid systems at different scales. To achieve this, the particle–particle interactions are simplified using the solid stress tensor to represent them; this does not require explicit expressions. This approach has a low computational cost, allowing the simulation of industrial cases using just workstations. This paper provides a review of the literature on the solid stress tensor and its commercial and non-commercial applications, including its historical and mathematical development in the description of particle–particle interactions. In addition, to consolidate the knowledge and advancing understanding in this crucial aspect of multiphase flow simulations, this review identifies the current challenges and opportunities for future research in multiphase systems based on the solid stress tensor. In addition, this review identifies the current challenges and opportunities for future research in multiphase systems based on the solid stress tensor.
... The grid was a 5 mm regular hexahedral mesh, approximately eight times the maximum diameters of the real particle. This size is adequate to ensure results are independent of the mesh size. 1, 31,32 The particles enter the fluidized bed from the inlet with a certain mass fraction (as shown in Table 2), and then they flow into the fluidized bed under gravity. After fluidization, the particles follow the blue dotted line in Figure 1a in sequence through chamber 1, the elutriation pipe, chamber 2, and then flow out through the outlet. ...
... It, therefore, requires prohibitively high computational demands to solve the equations of motion for each particle. Thus, considerable effort is made in developing parcel-based (coarse-grained) [57][58][59][60] or MP-PIC (multiphase particle-in-cell) [61][62][63][64][65][66][67] based DEM methods to bring particle-based approaches further to spatially larger scales. ...
Article
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In this review paper, we provide a brief overview of the recent advances in the continuum modeling of gas–particle flows. First, we focus on the kinetic theory-based two-fluid models, which have become a valuable tool to investigate small-scale moderately dense turbulent gas–particle flows. Second, the continuum description is quite restrictive with respect to the maximum grid spacing, and large-scale simulations usually employ coarse mesh resolutions to keep the analyses practicable. Such coarse-graining inevitably neglects the small unresolved scales, which requires additional modeling. Here, filtered two-fluid models have been applied successfully to a variety gas–solid flow problems. Finally, we give a condensed outline about future research challenges for the continuum modeling of gas–particle flows.
... Algebraic models for the drift velocity have been proposed in several studies. 7,8,19 Rauchenzauner and Schneiderbauer 20 expressed the drift velocity in terms of the subgrid turbulent kinetic energy of the gas and the scalar variance of the particle volume fraction, which were determined by solving corresponding dynamic transport equations. In a recent study, Hardy et al. ...
Article
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Gas-particle flows are commonly simulated through a twofluid model at the industrial scale. However, these simulations need a very fine grid to have accurate flow predictions, which is prohibitively demanding in terms of computational resources. To circumvent this problem, the filtered two-fluid model has been developed, where the large-scale flow field is numerically resolved and small-scale fluctuations are accounted for through subgrid-scale modeling. In this study, we have performed fine-grid two-fluid simulations of dilute gas-particle flows in periodic domains and applied explicit filtering to generate data sets. Then, these data sets have been used to develop artificial neural network (ANN) models for closures such as the filtered drag force and solid phase stress for the filtered two-fluid model. The set of input variables for the subgrid drag force ANN model that has been found previously to work well for dense flow regimes is found to work as well for the dilute regime. In addition, we present a Galilean invariant tensor basis neural network (TBNN) model for the filtered solid phase stress, which can nicely capture the anisotropic nature of the solid phase stress arising from subgrid-scale velocity fluctuations. Finally, the predictions provided by this new TBNN model are compared to those obtained from a simple eddy-viscosity ANN model.
... Several attempts aimed at speeding up calculations have been reported in the literature such as coarse-graining in which the flow is calculated on a coarser grid while simultaneously switching to different, or filtered, models, as done e.g. in switching from a Direct Numerical Simulation of a turbulent flow to a Large Eddy Simulation, or in coarsening approaches of gas-solids flow simulations such as reported by Ozel et al. [42] or Lei et al. [38] The price is a loss of precisely the details of the flow we are interested in. Further options are switching to GPU computing or simplifying the models for the forces [43]. ...
... Algebraic models for the drift velocity have been proposed in several studies. 7,8,20 Rauchenzauner and Schneiderbauer 21 expressed the drift velocity in terms of the subgrid turbulent kinetic energy of the gas and the scalar variance of the particle volume fraction, which were determined by solving corresponding dynamic transport equations. In a recent study, Hardy et al. 22 found that the drift velocity could be expressed in terms of the scalar variance of the particle volume fraction, with the same model applying to all filter sizes. ...
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Gas-particle flows are commonly simulated through two-fluid model at industrial-scale. However, these simulations need very fine grid to have accurate flow predictions, which is prohibitively demanding in terms of computational resources. To circumvent this problem, the filtered two-fluid model has been developed, where large-scale flow field is numerically resolved and small-scale fluctuations are accounted for through 1 subgrid-scale modeling. In this study, we have performed fine-grid two-fluid simulations of dilute gas-particle flows in periodic domains and applied explicit filtering to generate datasets. Then, these datasets have been used to develop artificial neural network (ANN) models for closures such as the filtered drag force and solid phase stress for the filtered two-fluid model. The set of input variables for the subgrid drag force ANN model that has been found previously to work well for dense flow regimes is found to work as well for the dilute regime. In addition, we present a Galilean invariant tensor basis neural network (TBNN) model for the filtered solid phase stress which can capture nicely the anisotropic nature of the solid phase stress arising from subgrid-scale velocity fluctuations. Finally, the predictions provided by this new TBNN model are compared with those obtained from a simple eddy-viscosity ANN model.
... Although a smaller system relative to actual experimental system was simulated using the present coupled CFD-DEM model to avoid the difficulty due to the large number of particles, long duration of the complete drying process associated with the full experimental system, and smaller value of Δt DEM required to simulate the full experimental system because of smaller diameter of the actual granules, coarse graining of the present CFD-DEM model may in future allow us to simulate the full experimental system in a reasonable time duration. Coarse-grained CFD-DEM can considerably reduce the simulation time, and many researchers are working on this topic (45)(46)(47)(48). However, no work has been reported on the computational modeling of complete drying of wet granules in a fluidized bed dryer using coarse-grained CFD-DEM method till now to the best of our knowledge. ...
Article
Drying of wet granules in a fluidized bed dryer is an important part of the pharmaceutical tablet manufacturing process. Complicated gas-solid flow patterns appear in the fluidized bed dryer, and interphase momentum, heat, and mass transfer happen during the drying process. A coupled computational fluid dynamics (CFD)-discrete element method (DEM)-based approach was used to model the drying process of pharmaceutical wet granules in a fluidized bed dryer. The evaporation of water from the surfaces of the particles and the cohesion force between the particles due to the formation of liquid bridges between the particles were also considered in this model. The model was validated by comparing the model predictions with the experimental data available from the literatures. The validated model was used to investigate the drying kinetics of the wet granules in the fluidized bed dryer. The results from numerical simulations showed that the dynamics and rate of increase of temperature of wet particles were considerably different from those of dry particles. Finally, the model was used to investigate the effects of inlet air velocity and inlet air temperature on the drying process. The model predicted increase in drying rate with the increase of inlet air velocity and inlet air temperature. This model can help not only to understand the multiphase multicomponent flow in fluidized bed dryer but also to optimize the drying process in the fluidized bed dryer.
... Analogous to the case of the heterogeneity index, ∇pg has also been recommended as an essential marker for correlating the drift velocity 37,38,51 ; it is hereinafter expressed as dg = dg ( s , slip , ∇ g )| Δ f =const . Ozel et al. 40 applied structural models and determined the drift velocity as functions of non-local coarse-grid quantities 26 . Recently, Rauchenzauner and Schneiderbauer 52 proposed a dynamic anisotropic spatially averaged TFM, where udg was closed by anisotropic components of the gas-phase turbulent kinetic energy and the scalar variance of the solid volume fraction. ...
Article
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Multiscale simulations of fluidized beds should account for the effect of sub‐grid structures on drag. How to extract the features of these structures in the form of proper finite quantities, namely markers, has posed great challenges in mesoscale drag modeling. The choice of markers has seldom been investigated in terms of their rationality and adequacy. This article introduces a two‐step scheme that is applied in the classic experimental approach to reflect on the choice of markers. The steady‐state definitional relations of the drag correction factor are deduced from force balance equations, with emphasis on the difference between definitional relations and constitutive relations. A comparison between common drag models obtained from fine‐grid simulations and corresponding definitional relations shows that the challenge in developing a general mesoscale drag model cannot be circumvented by correlating the heterogeneity index or drift velocity with the solid volume fraction, slip velocity, and gas‐phase pressure gradient.
... Fluid transport through the airways was solved by the volume-filtered mass and momentum conservation equations (Anderson and Jackson, 1967;Capecelatro and Desjardins, 2013) implemented in OpenFOAM v2.2 (Weller et al., 1998). Particle and fluid phases were coupled through a version of the CFDEMcoupling platform (Kloss et al., 2012) modified to benefit from faster two-way coupling by Ozel et al. (2016). ...
Article
For many of the one billion sufferers of respiratory diseases worldwide, managing their disease with inhalers improves their ability to breathe. Poor disease management and rising pollution can trigger exacerbations that require urgent relief. Higher drug deposition in the throat instead of the lungs limits the impact on patient symptoms. To optimise delivery to the lung, patient-specific computational studies of aerosol inhalation can be used. However in many studies, inhalation modelling does not represent situations when the breathing is impaired, such as in recovery from an exacerbation, where the patient’s inhalation is much faster and shorter. Here we compare differences in deposition of inhaler particles (10 and 4 micron) in the airways of three patients. We aimed to evaluate deposition differences between healthy and impaired breathing with image-based healthy and diseased patient models. We found that the ratio of drug in the lower to upper lobes was 35% larger with a healthy inhalation. For smaller particles the upper airway deposition was similar in all patients, but local deposition hotspots differed in size, location and intensity. Our results identify that image-based airways must be used in respiratory modelling. Various inhalation profiles should be tested for optimal prediction of inhaler deposition.
... The time-to-solution can be further reduced by coupling GPU accelerations with coarse-grained simulations (Lu et al., 2016). As illustrated in Fig. 22, in coarse-grained CFD-DEM simulation, the fluid coarse-graining is the utilization of coarse CFD grid while the particle coarse-graining is the lumping of physical particles into numerical parcels (Ozel et al., 2016;Yu et al., 2020). To account for the unresolved sub-grid structures due to coarse-graining, the EMMS drag (Yang et al., 2003;Lu et al., 2017b) or filtered drag (Igci et al., 2008;Gao et al., 2018;Yu et al., 2020) was used. ...
Article
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In this research, a Graphical Processing Unit (GPU) accelerated Discrete Element Method (DEM) code was developed and coupled with the Computational Fluid Dynamic (CFD) software MFiX to simulate granular and multiphase flows. The Fortran-based CFD solver was coupled with CUDA/C++ based DEM solver through inter-process pipes. The speedup to CPU is about 130 to 243 folds in the simulation of particle packings. In fluidized bed simulations, the DEM computation time is reduced from 91% to 17% with a speedup of 78 folds. The simulation of Geldart A particle fluidization revealed a similar level of importance of both fluid and particle coarse-graining. The filtered drag derived from the two-fluid model is suitable for Euler-Lagrangian simulations with both fluid and particle coarse-graining. It overcorrects the influence of sub-grid structures if used for simulations with only fluid coarse-graining.
... Among all the proposed methods, two methods of Energy Minimization Multi Scale (EMMS) and filtering have attracted attentions of majority of the researchers. Gao et al. [132] classified the heterogeneous drag models into four types based on the derivation methods: -Heterogeneous drag models derived from fine-grid kinetic theory of granular flow in TFM simulations (such as [133][134][135]) -Heterogeneous drag models derived from fine-grid CFD-DEM simulations (such as [136,137]) -Heterogeneous drag models derived from particle-resolved direct numerical simulations (such as [138]) -Heterogeneous drag models derived from meso-scale structure-based methods such as EMMS (such as [139][140][141]) ...
Article
Drag force models are one of the most important factors that can affect TFM and CFD-DEM simulation results of two-phase systems. This article investigates the accuracies, implementation issues and limitations of the majority of the drag models for spherical, non-spherical and systems with size distribution and evaluates their performance in various simulations. Around 1888 data points were collected from 19 different sources to evaluate the drag force closures on mono-dispersed spherical particles. The Reynolds number and fluid volume fraction ranges were between 0.01 and 10,000 and between 0.33 and 1, respectively. In addition, 776 data points were collected from seven different sources to evaluate the drag force closures on poly-dispersed spherical particles. The Reynolds numbers were between 0.01 and 500, fluid volume fractions between 0.33 and 0.9, and diameter ratios up to 10. A comprehensive discussion on the accuracy and application of these models is given in the article.
... The developed correction considering the influence of mesoscale structures could be used to improve the predictions of coarse-grid simulations. [9][10][11][12][13][14] To acquire an understanding of the underlying physics from the filtered dataset, both the conventional modeling (i.e., data fitting via an algebraic function form) and artificial neural network (ANN)-based data-driven modeling (DDM) can be adopted. The key to the success of the conventional method is to select the best function form and the optimal subgrid quantity markers. ...
Article
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This study presents conventional and artificial neural network‐based data‐driven modeling (DDM) methods to model simultaneously the filtered mesoscale drag, heat transfer and reaction rate in gas–particle flows. The dataset used for developing the DDM is filtered from highly resolved simulations closed by our recently formulated microscopic drag and heat transfer coefficients (HTCs). Results reveal that the filtered drag correction is nearly independent of filter size when including the filtered gas phase pressure gradient. We further find that the filtered HTC correction critically depends on the added filtered temperature difference marker while the filtered reaction rate correction shows weak dependence on the additional markers. Moreover, compared with conventional correlations, DDM predictions agree better with filtered resolved data. Comparative analysis is also conducted between existing HTC corrections and our work. Finally, the applicability of conventional and data‐driven models coupled with coarse‐grid computational fluid dynamics simulations for pilot‐scale (reactive) gas–particle flows is validated comprehensively.
... Filtering analysis of fine-grid simulation data has access to both F d and F d , and so F d,sgs can be found at each location in the bed for the chosen filter size. It has been established previously that F d,sgs can be expressed as (Parmentier et al., 2012;Ozel et al., 2013Ozel et al., , 2016Ozel et al., , 2017Rubinstein et al., 2017;Cloete et al., 2018a;Neau et al., 2020;Jiang et al., 2020): ...
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Simulations of large-scale gas-particle flows using coarse meshes and the filtered two-fluid model approach depend critically on the constitutive model that accounts for the effects of sub-grid scale inhomogeneous structures. In an earlier study (Jiang et al., 2019), we had demonstrated that an artificial neural network (ANN) model for drag correction developed from a small-scale systems did well in both a priori and a posteriori tests. In the present study, we first demonstrate through a cascading analysis that the extrapolation of the ANN model to large grid sizes works satisfactorily, and then performed fine-grid simulations for 20 additional combinations of gas and particle properties straddling the Geldart A-B transition. We identified the Reynolds number as a suitable additional marker to combine the results from all the different cases, and developed a general ANN model for drag correction that can be used for a range of gas and particle characteristics.
... In coarse grid simulation, the heterogeneous drag model should be used to account for the effect of the mesoscale structure. The heterogeneous drag model can be derived from fine grid two-fluid model simulations [31][32][33][34][35][36][37][38][39][40], fine grid CFD-DEM model simulations [41,42], particle-resolved direct numerical simulations (PR-DNS) [43], and mesoscale-structure-based methods. [44] In our previous work [36], a new filtered drag model was developed based on three-dimensional fine grid TFM simulations. ...
Article
Computational fluid dynamics (CFD) is a powerful tool for prediction and analysis of complex multiphase flow hydrodynamics and residence time distribution (RTD) in chemical reactors. This study presented the validation and application of a filtered drag model for solid RTD prediction in a pilot-scale fluid catalytic cracking (FCC) circulating fluidized bed (CFB) riser with Geldart A particles. First, the filtered drag model implemented in the open-source MFiX-TFM solver was validated for flow hydrodynamics simulation in a FCC CFB riser. After that, the model was further employed to validate its ability for solid RTD prediction in the same riser by comparing the simulation results with the experimental data. The simulation with the filtered drag model well reproduced the tracer response experiment which is more accurate than that with the Gidaspow drag model. Simulations with both pulse and step tracer injection methods were compared, which reveals the limitation of solid RTD measurement using a pulse tracer injection method in the experiment.
... For each phase, a set of conservation equations are solved and closed with the stability condition. Both of these types of sub-grid scale model have been tested for use in gas-solid fluidized bed applications with Eulerian-Eulerian [7,8,57,58], CFD-DEM [59][60][61][62], and MP-PIC [30,34,35,63] In this work, our aim is to explore the validity of the DDPM with EMMS/bubbling drag for the two-dimensional (2D) hydrodynamic modeling of bubbling fluidized beds consisting of Geldart A, A/B, and B particles. The sensitivity of various modeling parameters (grid size, drag force, particle number per parcel, turbulence, and particleparticle restitution coefficients) and their effects on bubbling bed hydrodynamics are comprehensively investigated and validated using experimental measurements. ...
Article
The fundamental problem encountered in the bubbling fluidized bed reactors is the presence of multiscale structures which cannot be resolved by the conventional drag models. In this study, a novel hybrid Eulerian-Lagrangian dense discrete phase model (DDPM) based on energy minimization and multiscale (EMMS) drag is proposed for the first time to analyze the hydrodynamics of bubbling fluidized bed reactors with Geldart A, A/B, and B particles. By comprehensive and comparative investigations of a number of key modeling parameters (grid size, drag force, particle number per parcel, turbulence, and particle-particle restitution coefficients), our proposed DDPM-EMMS model stands out of the currently-available counterparts in terms of improved grid-independency, multiscale structures resolvability with coarser grids, better parcel-independency, and better performance with laminar treatment against turbulence.
... The use of CFD-CGDEM inherently includes a coarsening of the mesh. This can result in unresolved flow features, which leads to an inaccurate drag prediction [73]. In analogy to MP-PIC and filtered-KTGF, the use of sub-grid drag models may improve the accuracy of CFD-CGDEM results in those cases. ...
Article
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Coarse‐Grained DEM is a relatively new, promising and, efficient method for the numerical simulation of particulate systems. The accuracy of the method relies on appropriate scaling rules for contact and fluid‐particle interaction forces. A variety of scaling rules is reported in the literature, including the additional use of drag correction models. To shed some light on the topic, different contact scaling rules are used and compared with DEM results regarding mean and RMS values of pressure drop and average particle height in a fluidized bed. The results indicated that in terms of averaged values as mean particle position and voidage profile the CFD‐CGDEM approach leads to accurate results for low scaling factors. With increasing scaling factors, deviations get higher. Regarding the particle dynamics, the approach leads to an underestimation of RMS values of particle position indicating a loss of particle dynamics in the system due to coarse‐graining. Furthermore, the results show that temporal discretization and contact scaling rule is not as important as expected. The impact of cell cluster size on drag force calculation is studied by comparing numerical results with experiments. The use of Energy Minimization Multi‐Scale drag correction is investigated, and a reduced mesh dependency and good accuracy are observed.
... Without taking into account the meso-scale structures, traditional drag models, such as Wen and Yu model [17] and Gidaspow model [2], overestimate the drag force [3]. By comparison, meso-scale drag models can be divided into two main groups according to their derivation method: EMMS drag model [18,19] based on meso-scale structure analysis and filtered drag model based on fine-grid two-fluid/ CFD-DEM simulations [20][21][22][23][24]. In this work we focus on EMMS-based method for MP-PIC simulations [11,12,25]. ...
Article
To account for the sub-grid heterogeneity of bubbling gas-solid flows in the multi-phase particle-in-cell (MP-PIC) method, a bubble-based EMMS/DP drag model is proposed in this work by taking into account the particle/parcel position information. The local, inter-parcel porosity is calculated by using the method of smoothed particle hydrodynamics (SPH), while the intra-parcel porosity is closed by using the energy-minimization multi-scale (EMMS) model. The local velocities of gas phase and different solid parcels are determined by using the constraint of pressure drop balance between phases as in the EMMS/bubbling model. The drag force acting on each parcel is then determined with the sub-grid information of velocities and porosities. Two cases of bubbling fluidized bed are simulated to validate this drag model. And the results show good agreement with experimental data.
... This is of interest in the context of industrialscale fluidized beds, where applications of both fluid and particle coarsening are inevitable. An alternative to these kinds of models are the filtered drag models (31,33). These offer a correction to regular drag laws and are obtained by performing simulations of particles falling in a fluid on successively coarser grids for different flow conditions and solids-phase fractions. ...
Article
Fluid–solid systems play a major role in a wide variety of industries, from pharmaceutical and consumer goods to chemical plants and energy generation. Along with this variety of fields comes a diversity in apparatuses and applications, most prominently fluidized and spouted beds, granulators and mixers, pneumatic conveying, drying, agglomeration, coating, and combustion. The most promising approach for modeling the flow in these systems is the CFD-DEM method, coupling computational fluid dynamics (CFD) for the fluid phase and the discrete element method (DEM) for the particles. This article reviews the progress in modeling particle–fluid flows with the CFD-DEM method. A brief overview of the basic method as well as methodical extensions of it are given. Recent applications of this simulation approach to separation and classification units, fluidized beds for both particle formation and energy conversion, comminution units, filtration, and bioreactors are reviewed. Future trends are identified and discussed regarding their viability. Expected final online publication date for the Annual Review of Chemical and Biomolecular Engineering, Volume 11 is June 8, 2020. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
... The specific calculation method is shown in Table 1. ② Surface force related to particle surface area [35], including drag and lift, etc.. Equivalent to the centroid of the particle, the surface resultant force A F and surface resultant torque A M are: ...
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When the unresolved CFD-DEM (Computational Fluid Dynamics and Discrete Element Method) method is used to solve two-phase flow (composed of fracturing fluid and quartz sand) problems in a pipe, although the collision and accumulation of particles can be described in mesoscopic scale, the method has a serious shortcoming: the contradiction between computational efficiency and computational accuracy. In the current study, an improved CFD-DEM method based on time roundabout increment way is proposed. By improving the solution strategy of unresolved CFD-DEM in terms of time incremental way, the iterative convergence criteria and time advancement algorithm of fluid-particle coupling have been established. The improved CFD-DEM method can automatically adjust the CFD time steps according to the convergence criteria. At each time step, the particle iteration updates the force between the fluid and the particle. Also the fluid can be solved in a roundabout way based on the convergence criteria. A computational example is given by applying the improved CFD-DEM method to analyze a two-phase flow (fracturing fluid and quartz sand) in a contraction-expansion pipe. The result shows that the new method can simulate particle collisions efficiently, and calculate the pressure loss of the two-phase flow accurately. It is found that when the sand ratio is increased from 0 to 56%, the accumulation of particles at the outer edge of the sudden contraction/expansion section is more pronounced, while the force chain is much easier to be formed. When the diameter ratio increases from 0.3 to 0.7, the particle accumulation is weakened and the particle collision force chain is evenly distributed. Our research work provides a highly effective computational method for the solution of solid-liquid two-phase flow, and can be applied for coupled dynamic analysis of particles and fluids.
... While the heterogeneity based sub-grid models, such as EMMS [23,24], make some specific assumptions about the form of the unresolved clusters and streamers, the filtered approach aims to find appropriate constitutive relation for the unresolved terms appearing due to filtering of the balance equations. Closure models are commonly deduced from highlyresolved simulations (either Euler-Euler or Euler-Lagrange [25,26]), which are filtered using filters of different sizes. Different markers, such as solids volume fraction and slip velocity, are then used to classify the sub-filter scale state and averaged to obtain statistics of the filtered quantities. ...
Article
Two different methods for closure modeling of the unresolved terms appearing in the filtered two-fluid model (fTFM) are discussed and compared. I) The spatially-averaged two-fluid model (SA-TFM) is based on generalizing the concepts of large eddy simulation to gas-particle flows. II) In the ADM-TFM approach the unresolved terms are modeled by an approximate deconvolution model, where an approximation of the unfiltered solution is obtained by repeated filtering. Finally, these models are applied to a lab-scale and a pilot-scale fluidized bed. Both approaches yield fairly good agreement with a highly resolved reference simulation as well as with experimental data. Additionally, both methods deliver reasonably grid independent solutions up to a grid resolution of 2 cm in the case of Geldart type A particles.
... We asses these (effectively two) different coarse-graining rules using fluidization of particles in a periodic domain as a test problem. As fluid grid resolution is known to have a large effect on predictions [36], we use the same fluid grid cell in the original and coarse-grained systems so that we can focus exclusively on particle coarsening. The CFD-DPM approach requires smoothing of exchange fields (e.g., particle volume fraction, drag force) that are projected on the fluid grid, since parcel size could become comparable to or even larger than a fluid cell depending on the degree of particle coarsening employed (when one chooses not to coarsen the fluid grid). ...
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Although Euler-Lagrange simulations can be performed with millions of particles, particle coarsening where particles are replaced by parcels is necessary for simulation of large particulate and fluid-particle flows. The present study examines coarsening strategies for cohesive particles, where cohesion arises through either van der Waals interaction or liquid bridges between particles. In the latter case, the dynamics of liquid transfer between particles is also taken into account. Strategies based on matching dimensionless overlap, stress and effective coefficient of restitution are shown to lead to same coarse graining rules, while that based on matching the Bond number yields a different set of rules. Test simulations involving fluidization of cohesive particles reveal that the stress-based coarse graining rules provide better approximation of the average slip velocity between the gas and the particles.
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In dilute turbulent particle-laden flows, such as atmospheric dispersion of pollutants or virus particles, the dynamics of tracer-like to low inertial particles are significantly altered by the fluctuating motion of the carrier fluid phase. Neglecting the effects of fluid velocity fluctuations on particle dynamics causes poor prediction of particle transport and dispersion. To account for the effects of fluid phase fluctuating velocity on the particle transport, stochastic differential equations coupled with large eddy simulation are proposed to model the fluid velocity seen by the particle. The drift and diffusion terms in the stochastic differential equation are modelled using neural networks ('neural stochastic differential equations'). The neural networks are trained with direct numerical simulations (DNS) of decaying homogeneous isotropic turbulence at low and moderate Reynolds numbers. The predictability of the proposed models are assessed through a priori analyses and a posteriori simulations of decaying homogeneous isotropic turbulence at low-to-high Reynolds numbers. Total particle fluctuating kinetic energy are under-predicted by 40% with no model, compared to the DNS data. In contrast, the proposed model predictions match total particle fluctuating kinetic energy to within 5% of the DNS data for low to high-inertia particles. For inertial particles, the model matches the variance of uncorrelated particle velocity to within 10% of DNS results, compared to 60-70% under-prediction with no model. It is concluded that the proposed mathematical and computational framework is applicable for flow configurations involving tracer and inertial particles, such as transport and dispersion of pollutants or virus particles.
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Computational fluid dynamics coupled with discrete element method (CFD‐DEM) has been widely used to understand the complicated fundamentals inside gas–solid fluidized beds. To realize large‐scale simulations, CFD‐DEM integrated with coarse‐grain model (CG CFD‐DEM) provides a feasible solution, and has led to a recent upsurge of interest. However, when dealing with large‐scale simulations involving irregular‐shaped particles such as biomass particles featuring elongated shapes, current CG models cannot function as normal because they are all developed for spherical particles. To address this issue, a CG CFD‐DEM for nonspherical particles is proposed in this study, and the morphology of particles is characterized by the super‐ellipsoid model. The effectiveness and accuracy of CG CFD‐DEM for nonspherical particles are comprehensively evaluated by comparing the hydrodynamic behaviors with the results predicted by traditional CFD‐DEM in a gas–solid fluidized bed. It is demonstrated that the proposed model can accurately model gas–solid flow containing nonspherical particles, merely the particle dynamics are somewhat lost due to the scaleup of particle size. Finally, the calculation efficiency of CG CFD‐DEM is assessed, and the results show that CG CFD‐DEM can largely reduce computational costs mainly by improving the calculation efficiency of DEM. In general, the proposed CG CFD‐DEM for nonspherical particles strikes a good balance between efficiency and accuracy, and has shown its prospect as a high‐efficiency alternative to traditional CFD‐DEM for engineering applications involving nonspherical particles.
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In dilute turbulent particle-laden flows, such as atmospheric dispersion of pollutants or virus particles, the dynamics of tracer-like to low inertial particles are significantly altered by the fluctuating motion of the carrier fluid phase. Neglecting the effects of fluid velocity fluctuations on particle dynamics causes poor prediction of particle transport and dispersion. To account for the effects of fluid phase fluctuating velocity on the particle transport, stochastic differential equations coupled with large-eddy simulation are proposed to model the fluid velocity seen by the particle. The drift and diffusion terms in the stochastic differential equation are modelled using neural networks ('neural stochastic differential equations'). The neural networks are trained with direct numerical simulations (DNS) of decaying homogeneous isotropic turbulence at low and moderate Reynolds numbers. The predictability of the proposed models are assessed against DNS results through a priori analyses and a posteriori simulations of decaying homogeneous isotropic turbulence at low-to-high Reynolds numbers. Total particle fluctuating kinetic energy is under-predicted by 40% with no model, compared to the DNS data. In contrast, the proposed model predictions match total particle fluctuating kinetic energy to within 5% of the DNS data for low to high-inertia particles. For inertial particles, the model matches the variance of uncorrelated particle velocity to within 10% of DNS results, compared to 60-70% under-prediction with no model. It is concluded that the proposed model is applicable for flow configurations involving tracer and inertial particles, such as transport and dispersion of pollutants or virus particles.
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Accurate modeling and simulation of irregular shape sand particle fluidization are challenging as particle shape-resolved simulation is complex and time-consuming. In this study, the coarse-grained discrete element model (CG-DEM)-computational fluid dynamics (CFD) was adopted to improve the computational feasibility of traditional CFD–DEM and simplify the mathematical modeling by lumping a certain amount of physical particles into a numerical parcel. 3D CGDEM–CFD simulated the fluidization behavior of irregular shape sand particles in a fluidized bed, and several key parameters or model settings were assessed to identify critical model factors. Moreover, the impact of the model parameters on hydrodynamics was quantified by comparing the simulation results with the experimental data. The results demonstrate that applying a rolling friction model can increase the simulation accuracy of the fluidization behavior of non-spherical sand particles.
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Different reaction technology determines different catalytic materials and solid density affects the hydrodynamics behavior in a fluidized bed. An efficient hydrodynamics study is, therefore, highly desirable, even for evaluating the assumptions through pilot-scale investigation. To develop new fluidized bed technology that requires low-density particles (e.g., the vapor-phase Beckman rearrangement to produce ε-caprolactam), the authors thoroughly studied the hydrodynamics of low-density particles (true density 750 kg/m³, packing density 450 kg/m³) in a turbulent fluidized bed. Pilot-scale cold experiments were conducted in an ∅800 mm column with optical fiber probes. The TFM was adopted, where the Wen&Yu model, EMMS model, and six filtered models were coupled, respectively. The solution of the EMMS model is often challenging because of the iterative solution of nonlinear equations. In this work, the calculation process of the EMMS model was optimized, saving one global search for every possible state solution, and a set of nonlinear equations can be linearized. Comparisons between experiment and simulation results revealed that the EMMS models gave the best prediction. This work can provide a research model and guidance for the design of industrial fluidized bed reactors using low-density particles.
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Euler-Lagrange method is powerful for studying dense gas-solid flow, where the Eulerian grid is typically 3–5 times of particle diameter to ensure the accuracy of information mapping between gas and solid phases. This condition limits its applications in simulating reactors with complex geometries that require the use of Eulerian grids that are comparative to or smaller than particle diameter. In this study, eight methods for mapping discrete particle information to continuous fluid field were compared first via the simulation of a packed bed, their pros and cons were analyzed and discussed in detail; A kernel function method (the normalized kernel function method II, NKFMII) was then selected to simulate the hydrodynamics and heat transfer of gas-solid bubbling fluidized beds, the results not only validated NKFMII for interphase information mapping but also indicated that a grid-size-independent solution can be achieved; Finally, in order to balance the computational efficiency and accuracy a hybrid particle centroid method (PCM) and NKFMII method was proposed to achieve the interphase information mapping and then to simulate the hydrodynamics of a fluidized bed with immersed tubes, the ability of local grid refinement and the effectiveness of the coupled PCM and NKFMII method were demonstrated. This study proved that Euler-Lagrange simulation with local grid refinement is able to simulate gas-solid reactors with complex geometries.
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Most natural resources are processed as particle-fluid multiphase systems in chemical, mineral and material industries, therefore, discrete particle methods (DPM) are a reasonable choice of simulation method for engineering the relevant processes and equipment. However, direct application of this method is challenged by the complex multiscale behavior of such systems, which leads to enormous computational cost or otherwise qualitatively inaccurate description of the mesoscale structures. The coarse-grained DPM based on the energy-minimization multi-scale (EMMS) model, or EMMS-DPM, was proposed to reduce the computational cost by several orders while maintaining an accurate description of the mesoscale structures, which paves the way for its engineering applications. Further empowered by the highly-efficient multi-scale DEM software DEMms and the corresponding customized heterogeneous supercomputing facilities with GPUs, it may even approach realtime simulation of industrial reactors. This short review will introduce the principles of DPM, in particular, EMMS-DPM, and the recent developments in modeling, numerical implementation and applications of large-scale DPM which aims to reach industrial scale on one hand and resolves mesoscale structures critical to reaction-transport coupling on the other hand. This review finally prospects on the future developments of DPM in this direction.
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The Spatially-Averaged Two-Fluid Model (SA-TFM) is a functional multiphase turbulence model aimed at predicting the influence of unresolved meso-scale structures on the macro-scale flow properties in coarse-grid simulations of gas-particle flows. In this study, we highlight the general applicability of the SA-TFM model due to the dynamic estimation of the model coefficients through test-filters, the anisotropic treatment of the Reynolds-stresses and the drag force correction, and additional wall-friction boundary conditions for the particle-phase velocity , Reynolds-stress and turbulent kinetic energy. The dynamic approach derives information on the unresolved meso-scale flow properties from the resolved macro-scale variables without the need for any intensive prior considerations of the specific flow structure. Thereby, the drift velocity, a measure for the drag-reduction due to the presence of meso-scale structures is estimated from the turbulent kinetic energies of the phases and the variance of the solids volume fraction. We validate the dynamic anisotropic SA-TFM against highly resolved fine-grid Two-Fluid Model simulation data and experimental measurements of Geldart type A and B particles in bubbling to turbulent flow regimes. In the course of this extensive study, we find that the predictions for the macro-scale flow properties, such as slip-velocity, bed expansion, volume fraction distribution, and mass-flux, are in good agreement with the experimental and fine-grid simulation data in a vast number of cases, ranging from unbound fluidization to pilot-scale fluidized beds, thus, implying a wide applicability of the dynamic model independent of the underlying grid-size.
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The coarse-grained discrete element method (CGDEM) is promising for its ability to reduce computational cost. However, compared with discrete element method (DEM), CGDEM often overpredicts the granular temperature in simulations of systems with frictional particles. This is partially due to the fact that traditional coarsening strategies only account for the correction to energy dissipation due to inelastic collisions. This work proposes two types of new coarsening strategies that also make the correction to energy dissipation caused by the frictional force between particles. CGDEM simulations of homogeneous cooling systems (HCSs) and two bubbling fluidized beds are executed to evaluate the performance of the proposed strategies. It is found that, relative to CGDEM with traditional coarsening strategies, CGDEM with the proposed strategies gives a more accurate prediction of the instantaneous granular temperature in HCSs. They also better reproduce the time-Averaged fields obtained by DEM simulations for the considered bubbling fluidized beds. This demonstrates the necessity of considering both inelastic and frictional origins of energy dissipation in the coarsening strategy.
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Simulations of strongly coupled (i.e., high-mass-loading) fluid-particle flows in vertical channels are performed with the purpose of understanding the fundamental physics of wall-bounded multiphase turbulence. The exact Reynolds-averaged (RA) equations for high-mass-loading suspensions are presented, and the unclosed terms that are retained in the context of fully developed channel flow are evaluated in an Eulerian–Lagrangian (EL) framework for the first time. A key distinction between the RA formulation presented in the current work and previous derivations of multiphase turbulence models is the partitioning of the particle velocityfluctuations into spatially correlated and uncorrelated components, used to define the components of the particle-phase turbulent kinetic energy (TKE) and granular temperature, respectively. The adaptive spatial filtering technique developed in our previous work for homogeneous flows [J. Capecelatro, O. Desjardins, and R. O. Fox, “Numerical study of collisional particle dynamics in cluster-induced turbulence,” J. Fluid Mech. 747, R2 (2014)] is shown to accurately partition the particle velocityfluctuations at all distances from the wall. Strong segregation in the components of granular energy is observed, with the largest values of particle-phase TKE associated with clusters falling near the channel wall, while maximum granular temperature is observed at the center of the channel. The anisotropy of the Reynolds stresses both near the wall and far away is found to be a crucial component for understanding the distribution of the particle-phase volume fraction. In Part II of this paper, results from the EL simulations are used to validate a multiphase Reynolds-stress turbulence model that correctly predicts the wall-normal distribution of the two-phase turbulence statistics.
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Euler-Lagrange simulations of fluidized beds of Geldart Group A particles containing different levels of fines are performed in periodic domains with various domain-averaged solid volume fractions. Bubble-like voids readily form when no fines are added. Introducing fines does not reduce bubble sizes if van der Waals force between particles is not accounted for. In contrast, the addition of van der Waals force produces significant changes. With no fines, bubbles are found to be suppressed at sufficiently high solid volume fractions, corresponding to the expanded bed regime for Group A particles. With the addition of fines, bubbles can be suppressed at lower solid volume fractions. With more fines added, bubbles can be suppressed at even lower solid volume fractions. When bubbles are suppressed, the system is found to be in a stable solid-like regime. In this regime, forces on each particle are balanced, and the particle velocity fluctuations are dampened. This article is protected by copyright. All rights reserved.
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Simulations of wet fluidized beds of particles in small periodic domains have been carried out using a CFD-DEM approach. A liquid bridge forms upon particle-particle collision, which then ruptures when the particle separation exceeds a critical distance. The simulations take into account only the surface tension force of attraction due to the liquid bridge. Increasing the strength of cohesion leads to larger agglomerates, and correspondingly, higher gas velocities are required to fully support the particles. The slip velocity results from the simulations have been correlated in terms of a Bond number characterizing the strength of cohesion, volume of liquid in the bridge, and particle volume fraction. The CFD-DEM results are systematically coarse-grained to expose the dependence of the filtered drag coefficient on Eulerian filter size, surface tension forces, liquid loading, and solids loading in wet gas-solid fluidized beds.
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In a fluidized bed, the drag force acts to oppose the downward force of gravity on a particle, and thus provides the main mechanism for fluidization. Drag models that are employed in large-scale simulations of fluidized beds are typically based on either fixed-particle beds or the sedimentation of particles in liquids. In low-Reynolds-number ( Re ) systems, these two types of fluidized beds represent the limits of high Stokes number ( St ) and low St , respectively. In this work, the fluid–particle drag behaviour of these two regimes is bridged by investigating the effect of St on the drag force in low- Re systems. This study is conducted using fully resolved lattice Boltzmann simulations of a system composed of fluid and monodisperse spherical particles. In these simulations, the particles are free to translate and rotate based on the effects of the surrounding fluid. Through this work, three distinct regimes in the characteristics of the fluid–particle drag force are observed: low, intermediate and high St . It is found that, in the low- Re regime, a decrease in St results in a reduction in the fluid–particle drag. Based on the simulation results, a new drag relation is proposed, which is, unlike previous models, dependent on St .
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Useful as a reference for engineers in industry and as an advanced level text for graduate engineering students, Multiphase Flow and Fluidization takes the reader beyond the theoretical to demonstrate how multiphase flow equations can be used to provide applied, practical, predictive solutions to industrial fluidization problems. Written to help advance progress in the emerging science of multiphase flow, this book begins with the development of the conservation laws and moves on through kinetic theory, clarifying many physical concepts (such as particulate viscosity and solids pressure) and introducing the new dependent variable-the volume fraction of the dispersed phase. Exercises at the end of each chapterare provided for further study and lead into applications not covered in the text itself.
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Granular flow of three experiments are predicted by a computational particle fluid dynamic CPFD numerical scheme in three-dimension using the true particle size distribution. The experiments are simple which show the distinct characteristics of particle flow which differs from fluid flow. The experiments are flow of particles in sedimentation, a U-tube and from a hopper. The CPFD method models the fluid as a fluid and models the particles as discrete particles (material description). The CPFD method is a form of discrete element method, where each particle has three dimensional forces from fluid drag, gravity, static-dynamic friction, particle collision and possibly other forces. However, unlike DEM models which calculate particle-to-particle force by a spring-damper model and direct particle contact, the CPFD method models collision force on each particle as a spatial gradient. The CPFD numerical method predicts all three experiments.
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Predictions of simulations based on filtered Two-Fluid Models (TFMs) with constitutive relations for filtered fluid–particle drag coefficient and filtered stresses proposed by Igci and Sundaresan [Ind. Eng. Chem. Res. 50 (2011) 13190–13201] and Milioli et al. [AIChE J. 59 (2013) 3265–3275] were compared against experimental data from a bubbling fluidized bed challenge problem put forward by the National Energy Technology Laboratory and Particulate Solids Research Inc. It is found that the most important correction to filtered models is a modification to the drag, and filtered stresses play a secondary role at best. As expected, coarse grid simulations using the kinetic-theory based TFM over-predicted the gas–particle drag force, yielding unphysical bed expansion. The filtered fluid–particle drag model proposed by Igci and Sundaresan that classifies the inhomogeneity in sub-filter scale flow structures using filter size and filtered particle volume fraction as markers also predicted unphysical bed expansion. Refined filtered drag models proposed by Milioli et al. based on filtered fluid–particle slip velocity as an additional marker led to good agreement with experimental data on bed expansion and the time-averaged gas pressure gradient. It was also observed that inadequate grid resolution in the region between gas distributor and the adjacent cylindrical wall of the test unit could lead to spurious asymmetric gas–particle flow predictions. With the inclusion of adequate inflation layer elements in that region, flow predictions became nearly symmetric with little to no effect on bed expansion predictions. However, it dramatically and qualitatively altered the details of gas–particle structures in the bed.
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Significance The quantitative ability of a kinetic‐theory‐based, two‐fluid model is demonstrated in a clustering (unstable) gas‐solid system via highly resolved simulations. Unlike previous works, this assessment is validated against ideal computational fluid dynamics‐discrete element method data to minimize sources of discrepancy. Overall, good agreement in mean‐slip velocities is observed with relative errors less than 20% over a mean solids concentration range of 0.02–0.25. Local concentration gradient distributions are also studied, showing a distinct shift toward higher gradients at higher mean solids concentrations which is proposed as the bottleneck in obtaining grid‐independence rather than the cluster length scale. © 2015 American Institute of Chemical Engineers AIChE J, 62: 11–17, 2016
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In this work, we present the development of an open-source software for modeling granular material by means of the Discrete Element Method. LIGGGHTS (www.liggghts.com) stands for 'LAMMPS Improved for General Granular and Granular Heat Transfer Simulations' and is based on LAMMPS ('Large-Scale Atomic/Molecular Massively Parallel Simulator'), a successful open source Molecular Dynamics code written and distributed by Sandia National Laboratories for massively parallel computing on distributed memory machines. We first give a brief overview of implemented models and features. These comprise CAD geometry import, features for particle insertion and packing, contact models, wallstress analysis and wear prediction, moving mesh capability, a 6 DOF feature and non-spherical particle handling. Finally, we would like to focus on the simulation of coupled granular-fluid systems with the CFD-DEM approach.
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A discrete particle model for flows of Group A particles in Geldart's classification is studied. In general, Group A particles are fine and light, so that the adhesion force has a strong effect on their fluidization behavior. Inter-particle adhesion force of Group A particle was measured first. Secondly, the DEM–CFD coupling simulation with the measured adhesion force was performed, and the simulated results were compared with the experimental data about a small-scale fluidized bed for verification of the simulation. It was found from the results that there were considerable differences between their flow patterns. In order to reveal the cause of the differences, the effect of the adhesion force in the contact force model was studied on the motion of a single particle colliding with a wall.
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CFD calculations for multiphase gas-particle flows would benefit from a multiphase turbulence theory which would allow the direct calculation of an average solution. A slightly truncated form of the two fluid equations, which are the basis of the MFIX code, have been formally averaged, introducing Favre-like averages for the phasic velocities. This process results in a set of equations which can be solved directly for the average variables describing the flow. This equation set has significantly fewer closure terms than would have occurred using standard Reynolds averaging. The full set of required closure relations is enumerated. Some simple forms for these closures are suggested, although no attempt is made to establish their validity.
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In this study a prismatic spouted bed was characterized experimentally and modelled by means of 3D CFD-DEM simulations. The main focus was on the investigation of the influence of the gas flow rate on the bed dynamics and spouting stability. Pressure drop time series obtained at different gas velocities were used for the identification of flow regimes by means of the frequency domain and of chaotic properties such as the correlation dimension and Kolmogorov entropy. The gas and particle dynamics were investigated through simulations of different operational regimes: the spouting onset, as well as stable and instable regimes. A 3-D bed behaviour, typical for slot-rectangular beds, was found. A good agreement between simulations and experiments in the particle flow patterns, bed expansion and dynamics of characteristic gas pressure fluctuations was achieved. The particle dynamics as a function of the gas velocity was investigated for the entire bed. For one of the stable regimes, the bed regions showing different particle dynamics (spout, fountain and annulus) were characterized in detail. A regime map showing the stable operational window in dependence on an inlet-to-bed size ratio and gas velocity is also provided.
Conference Paper
Coarse-grid simulations of gas-solids flows are common practice with continuum-based models, which is necessary to obtain numerical results quickly enough to be useful. Such coarse-grid simulations are able to qualitatively resolve large-scale behaviors, such as bubbles and large clusters of solids that have been observed experimentally. It is well known that the continuum approach fails to capture the small-scale heterogeneous flow structures unless very high grid resolutions are employed. Such grid refinement studies have also led to the development of sub-grid models that can be used with coarse-grid continuum simulations and result in faster simulations of gas-solids flows [1, 2]. Discrete particle models that track the trajectory and collisions of every particle are more accurate than continuum models but are not economical for many practical gas-solids flow systems due to the large amount of particles involved. A more affordable discrete technique, called multiphase particle-in-cell (or MP-PIC) is based on lumping many particles together in a parcel in order to track fewer particles and replaces the collisional interactions with a continuum pressure gradient [3]. The MP-PIC technique still requires coarse computational grids so that fewer parcels can be used and the computational cost remains affordable. In this study we examine if the MP-PIC approach, when applied over coarse grids, requires similar sub-grid corrections as continuum models. To address this issue, we have conducted continuum and MP-PIC simulations of gas-particle flows in a 2D periodic domain at several grid resolutions. The simulations assumed that the particles were all of the same size. The constitutive models for the continuum approach was chosen to match those employed in the MP-PIC approach; specifically, both of these modeling approaches used the same solids pressure model and the Wen & Yu [4] drag law. Both continuum and MP-PIC simulations using coarse grids manifested nearly homogeneous flows where the domain-average slip velocity was essentially the same as that for homogeneous flow. When the grid was refined, the numerical results for both approaches manifested heterogeneous flow structures that increased the domain-average gas-particles slip velocity to 2.5-3 times that for homogeneous flow; importantly, both approaches manifested very similar flow structures and yielded essentially the same domain-average slip velocity. These results confirm the inherent equivalence between the two approaches. From a practical point of view, our study indicates that the sub-grid models that are being developed for the continuum approach can (and should) be imported to the MP-PIC approach. References: [1] Andrews, A. T.; Loezos, P. N.; Sundaresan, S. Coarse-grid simulation of gas-particle flows in vertical risers. Ind. Eng. Chem. Res. 2005, 44, 6022. [2] Igci, Y.; Andrews, A. T.; Sundaresan, S.; Pannala, S.; O'Brien, T. Filtered two-fluid models for fluidized gas-particle suspensions. AIChE J. 2008, 54, 1431. [3] Snider, D.M. An incompressible three-dimensional multiphase particle-in-cell model for dense particle flows. J. Comput. Phys. 2001, 170, 523549. [4] Wen, C. Y.; Yu, Y. H. Mechanics of Fluidization. Chem. Eng. Prog. Symp. Ser. 1966, 62, 100-111.
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The thirteen chapters of this book are introduced by a preface and followed by five appendices. The main chapter headings are: motion and forces at a point of contact; line loading of an elastic half-space; point loading of an elastic half-space; normal contact of elastic solids - Hertz theory; non-Hertzian normal contact of elastic bodies; normal contact of inelastic solids; tangential loading and sliding contact; rolling contact of elastic bodies; rolling contact of inelastic bodies; calendering and lubrication; dynamic effects and impact; thermoelastic contact; and rough surfaces. (C.J.A.)
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We present an Eulerian-Lagrangian numerical simulation (LNS) scheme for particulate flows. The overall algorithm in the present approach is a variation of the scheme presented earlier. In this numerical scheme we solve the fluid phase continuity and momentum equations on an Eulerian grid. The particle motion is governed by Newton’s law thus following the Lagrangian approach. Momentum exchange from the particle to fluid is modeled in the fluid phase momentum equation. Forces acting on the particles include drag from the fluid, body force and the interparticle force that prevents the particle volume fraction from exceeding the close-packing limit. There is freedom to use different models for these forces and to introduce other forces. In this paper we have used two types of interparticle forces. The effect of viscous stresses are included in the fluid phase equations. The volume fraction of the particles appear in the fluid phase continuity and momentum equations. The fluid and particle momentum equations are coupled in the solution procedure unlike an earlier approach. A finite volume method is used to solve these equations on an Eulerian grid. Particle positions are updated explicitly. This numerical scheme can handle a range of particle loadings and particle types. We solve the fluid phase continuity and momentum equations using a Chorin-type fractional-step method. The numerical scheme is verified by comparing results with test cases and experiments.
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Starting from a kinetic theory (KT) model for monodisperse granular flow, the exact Reynolds-averaged (RA) equations are derived for the particle phase in a collisional fluid particle flow. The corresponding equations for a constant-density fluid phase are derived from a model that includes drag and buoyancy coupling with the particle phase. The fully coupled macroscale/hydrodynamic model, rigorously derived from a kinetic equation for the particles, is written in terms of the particle-phase volume fraction, the particle-phase velocity and the granular temperature (or total granular energy). As derived from the hydrodynamic model, the RA turbulence model solves for the RA particle-phase volume fraction, the phase-averaged (PA) particle-phase velocity, the PA granular temperature and the PA turbulent kinetic energy of the particle phase. Thus, unlike in most previous derivations of macroscale turbulence models for moderately dense granular flows, a clear distinction is made between the PA granular temperature Theta(p), which appears in the KT constitutive relations, and the particle-phase turbulent kinetic energy k(p), which appears in the turbulent transport coefficients. The exact RA equations contain unclosed terms due to nonlinearities in the hydrodynamic model and we briefly discuss the available closures for these terms. Finally, we demonstrate by comparing model predictions with direct numerical simulation results that even for non-collisional fluid particle flows it is necessary to provide separate models for Theta(p) and k(p) in order to correctly account for the effect of the particle Stokes number and mass loading
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We present a computational study of cluster-induced turbulence (CIT), where the production of fluid-phase kinetic energy results entirely from momentum coupling with finite-size inertial particles. A separation of length scales must be established when evaluating the particle dynamics in order to distinguish between the continuous mesoscopic velocity field and the uncorrelated particle motion. To accomplish this, an adaptive spatial filter is employed on the Lagrangian data with an averaging volume that varies with the local particle-phase volume fraction. This filtering approach ensures sufficient particle sample sizes in order to obtain meaningful statistics while remaining small enough to avoid capturing variations in the mesoscopic particle field. Two-point spatial correlations are computed to assess the validity of the filter in extracting meaningful statistics. The method is used to investigate, for the first time, the properties of a statistically stationary gravity-driven particle-laden flow, where particle–particle and fluid–particle interactions control the multiphase dynamics. Results from fully developed CIT show a strong correlation between the local volume fraction and the granular temperature, with maximum values located at the upstream boundary of clusters (i.e. where maximum compressibility of the particle velocity field exists), while negligible particle agitation is observed within clusters.
Article
New constitutive relations for filtered two‐fluid models (TFM) of gas‐particle flows are obtained by systematically filtering results generated through highly resolved simulations of a kinetic theory‐based TFM. It was found in our earlier studies that the residual correlations appearing in the filtered TFM equations depended principally on the filter size and filtered particle volume fraction. Closer inspection of a large amount of computational data gathered in this study reveals an additional, systematic dependence of the correction to the drag coefficient on the filtered slip velocity, which serves as a marker for the extent of subfilter‐scale inhomogeneity. Furthermore, the residual correlations for the momentum fluxes in the gas and particle phases arising from the subfilter‐scale fluctuations are found to be modeled nicely using constitutive relations of the form used in large‐eddy simulations of single‐phase turbulent flows. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3265–3275, 2013
Data
This paper aims at investigating the capability of numerical models to ac- curately capture the physical characteristics of particle clustering in vertical risers. Within the energy sector, particle clustering in vertical risers of circu- lating fluidized bed reactors are known to play a key role in the multiphase dynamics as well as secondary processes such as catalytic conversion and heat transfer. Recent experiments suggest that particle clustering is most significant in the fully developed flow region of the riser, hence this study focuses on this region. To explore such flows, a high-fidelity large-eddy sim- ulation framework is combined with a Lagrangian particle tracking solver to simulate statistically stationary gravity-driven risers in vertical pipes for a large range of Archimedes numbers. The walls of the reactor are modeled using a conservative immersed boundary scheme integrated with the La- grangian particle tracking framework. A structure tracking algorithm akin to particle image velocimetry is used to accumulate statistics on individual clusters. Cluster descent velocities display excellent agreement with experi- mental measurements for the range of flow conditions considered. Predicted volume fraction fluctuations and mean solid concentration within the clusters also match experimental correlations. The probability distribution function of solid concentration and radial distribution function provide insight on the degree of clustering and the characteristic cluster length scale. The degree of particle clustering is found to be independent of the Archimedes number, and models for the volume fraction distribution are discussed. Statistics on the solid concentration and phase velocities for two- and three-dimensional configurations are compared, and the ramifications of simulating risers in two dimensions are discussed.
Article
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 sub-grid 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.