Validation study on spatially averaged two‐fluid model for gas‐solid flows: II. Application to risers and fluidized beds

In our prior study (Schneiderbauer, AIChE J. 2017;63(8):3544–3562), a spatially averaged two‐fluid model (SA‐TFM) was presented, where closure models for the unresolved terms were derived. These closures require constitutive relations for the turbulent kinetic energies of the gas and solids phase as well as for the subfilter variance of the solids volume fraction. We had ascertained that the filtered model do yield nearly the same time‐averaged macroscale flow behavior in bubbling fluidized beds as the underlying kinetic‐theory‐based two‐fluid model, thus verifying the SA‐TFM model approach. In the present study, a set of 3D computational simulations for validation of the SA‐TFM against the experimental data on riser flow and bubbling fluidized beds is performed. Finally, the SA‐TFM predictions are in fairly good agreement with experimental data in the case of Geldart A and B particles even though using very coarse grids. © 2018 The Authors AIChE Journal published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers AIChE J, 64: 1606–1617, 2018     

亚格子过滤双流体模型的网格依赖性

基于双流体方程和颗粒动力学理论的计算模型被广泛应用于流化床的气固两相流数值计算,高精度网格是其准确计算流动的必要条件。一些经典的微尺度阻力模型,其网格尺度决定其模拟结果的精度。亚格子过滤双流体模型是一种有效的适用于粗糙网格的计算模型,其包含的气固相间作用力和颗粒相应力本构方程是在高精度网格条件下,以微尺度双流体方程和颗粒动力学理论计算得到的气固流场为基础,对计算结果进行小尺度过滤后得出。使用亚格子过滤双流体模型替换基于颗粒动力学理论的双流体模型,针对同一物理问题,在不同网格尺度下进行了数值计算,结果表明此计算模型相比经典阻力模型具有较好的网格无关特性,并且和实验结果较为一致。同时也对颗粒动力学理论与之相结合进行了尝试,即仅使用亚格子过滤阻力模型,颗粒相应力仍然使用颗粒动力学模型,其计算结果的网格无关性及与实验值的吻合程度优于经典模型。     

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

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

A spatially‐averaged two‐fluid model for dense large‐scale gas‐solid flows

We present a spatially‐averaged two‐fluid model (SA‐TFM), which is derived from ensemble averaging the kinetic‐theory based TFM equations. The residual correlation for the gas‐solid drag, which appears due to averaging, is derived by employing a series expansion to the microscopic drag coefficient, while the Reynolds‐stress‐like contributions are closed similar to the Boussinesq‐approximation. The subsequent averaging of the linearized drag force reveals that averaged interphase momentum exchange is a function of the turbulent kinetic energies of both, the gas and solid phase, and the variance of the solids volume fraction. Closure models for these quantities are derived from first principles. The results show that these new constitutive relations show fairly good agreement with the fine grid data obtained for a wide range of particle properties. Finally, the SA‐TFM model is applied to the coarse grid simulation of a bubbling fluidized bed revealing excellent agreement with the reference fine grid solution. © 2017 American Institute of Chemical Engineers AIChE J, 63: 3544–3562, 2017     

Bubble dynamics in a 3‐D gas–solid fluidized bed using ultrafast electron beam X‐ray tomography and two‐fluid model

Bubble characteristics in a three‐dimension gas‐fluidized bed (FB) have been measured using noninvasive ultrafast electron beam X‐ray tomography. The measurements are compared with predictions by a two‐fluid model (TFM) based on kinetic theory of granular flow. The effect of bed material (glass, alumina, and low linear density polyethylene (LLDPE), d_p ～1 mm), inlet gas velocity, and initial particle bed height on the bubble behavior is investigated in a cylindrical column of 0.1‐m diameter. The bubble rise velocity is determined by cross correlation of images from dual horizontal planes. The bubble characteristics depend highly upon the particle collisional properties. The bubble sizes obtained from experiments and simulations show good agreement. The LLDPE particles show high gas hold‐up and higher bubble rise velocity than predicted on basis of literature correlations. The bed expansion is relatively high for LLDPE particles. The X‐ray tomography and TFM results provide in‐depth understanding of bubble behavior in FBs containing different granular material types. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1632–1644, 2014     

On the effects of the flow macro-scale over sub-grid modelling in dense gas–solid fluidized flows

Multiscale modelling of gas–particle fluidized flows is frequently approached by means of sub-grid modelling, which provides constitutive closures for filtered formulations applied to large scale simulations. A widely practiced procedure for the derivation of sub-grid models consists of filtering over predictions from highly resolved simulations under two-fluid modelling. The present work is intended as a contribution in this field by providing new supporting evidence for the enhancement of sub-grid closure models. Most of the efforts in the area have been directed to providing sub-grid models dependent on meso-scale filtered effects alone, and under low gas Reynolds number suspension conditions. In this work, macro-scale conditions are added to the analysis thereby accounting for flow topology, particularly for dense gas–solid fluidized flows. Two macro-scale variables are considered in the simulations, namely the domain average solid volume fraction and the domain average gas Reynolds number. So, in addition to the usual meso-scale filtered markers, relevant filtered parameters are also related to those macro-scale conditions. The filtered parameters of interest here are the effective interphase drag coefficient and filtered and residual stresses in both of the phases. Various domain average solid volume fractions and domain average gas Reynolds numbers were enforced, thereby providing for a variety of macro-scale dense conditions. It was found that both these macro-scale parameters considerably affect the meso-scale and the resulting filtered parameters of dense gas–solid flows, even though this occurs in a milder way when compared to results for dilute flow conditions available in the literature.     

Verification of filtered two‐fluid models for gas‐particle flows in risers

The effect of solid boundaries on the closure relationships for filtered two‐fluid models for riser flows was probed by filtering the results obtained through highly resolved kinetic theory‐based two‐fluid model simulations. The closures for the filtered drag coefficient and particle phase stress depended not only on particle volume fraction and the filter length but also on the distance from the wall. The wall corrections to the filtered closures are nearly independent of the filter length and particle volume fraction. Simulations of filtered model equations yielded grid length independent solutions when the grid length is ～half the filter length or smaller. Coarse statistical results obtained by solving the filtered models with different filter lengths were the same and corresponded to those from highly resolved simulations of the kinetic theory model, which was used to construct the filtered models, thus verifying the fidelity of the filtered modeling approach. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

Comparative analysis of subgrid drag modifications for dense gas‐particle flows in bubbling fluidized beds

Many subgrid drag modifications have been put forth to account for the effect of small unresolved scales on the resolved mesoscales in dense gas‐particle flows. These subgrid drag modifications significantly differ in terms of their dependencies on the void fraction and the particle slip velocity. We, therefore, compare the hydrodynamics of a three‐dimensional bubbling fluidized bed computed on a coarse grid using the drag correlations of the groups of (i) EMMS, (ii) Kuipers, (iii) Sundaresan, (iv) Simonin, and the homogenous drag law of (v) Wen and Yu with fine grid simulations for two different superficial gas velocities. Furthermore, we present an (vi) alternative approach, which distinguishes between resolved and unresolved particle clusters revealing a grid and slip velocity dependent heterogeneity index. Numerical results are analyzed with respect to the time‐averaged solids volume fraction and its standard deviation, gas and solid flow patterns, bubble size, number density, and rise velocities. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4077–4099, 2013     

Euler–euler anisotropic gaussian mesoscale simulation of homogeneous cluster‐induced gas–particle turbulence

An Euler–Euler anisotropic Gaussian approach (EE‐AG) for simulating gas–particle flows, in which particle velocities are assumed to follow a multivariate anisotropic Gaussian distribution, is used to perform mesoscale simulations of homogeneous cluster‐induced turbulence (CIT). A three‐dimensional Gauss–Hermite quadrature formulation is used to calculate the kinetic flux for 10 velocity moments in a finite‐volume framework. The particle‐phase volume‐fraction and momentum equations are coupled with the Eulerian solver for the gas phase. This approach is implemented in an open‐source CFD package, OpenFOAM, and detailed simulation results are compared with previous Euler–Lagrange simulations in a domain size study of CIT. The results demonstrate that the proposed EE‐AG methodology is able to produce comparable results to EL simulations, and this moment‐based methodology can be used to perform accurate mesoscale simulations of dilute gas–particle flows. © 2017 American Institute of Chemical Engineers AIChE J, 63: 2630–2643, 2017     

Quantifying the uncertainty introduced by discretization and time‐averaging in two‐fluid model predictions

The two‐fluid model (TFM) has become a tool for the design and troubleshooting of industrial fluidized bed reactors. To use TFM for scale up with confidence, the uncertainty in its predictions must be quantified. Here, we study two sources of uncertainty: discretization and time‐averaging. First, we show that successive grid refinement may not yield grid‐independent transient quantities, including cross‐section–averaged quantities. Successive grid refinement would yield grid‐independent time‐averaged quantities on sufficiently fine grids. Then a Richardson extrapolation can be used to estimate the discretization error, and the grid convergence index gives an estimate of the uncertainty. Richardson extrapolation may not work for industrial‐scale simulations that use coarse grids. We present an alternative method for coarse grids and assess its ability to estimate the discretization error. Second, we assess two methods (autocorrelation and binning) and find that the autocorrelation method is more reliable for estimating the uncertainty introduced by time‐averaging TFM data. © 2017 American Institute of Chemical Engineers AIChE J, 63: 5343–5360, 2017     

Validation study on spatially averaged two‐fluid model for gas–solid flows: I. A priori analysis of wall bounded flows

In our prior study (Schneiderbauer, AIChE J, 2017;63(8):3544–3562), we presented a spatially averaged two‐fluid model, where closure models for the unresolved terms were derived. These closures require constitutive relations for the turbulent kinetic energies (TKEs) of the gas and solids phase as well as for the sub‐filter variance of the solids volume fraction (VVF). In this study, we have performed highly resolved TFM simulations of a set of three‐dimensional wall dominated periodic channels. An a priori analysis shows that these closures are able to correctly predict the wall profiles of the sub‐grid drag modification, the TKEs, the turbulent viscosities and the VVF without requiring special wall corrections. Solely the mixing lengths, which is required by the closures, has to be adapted in the vicinity of wall similar to single‐phase turbulence; in particular, the minimum of the filter size and the distance to the wall should be used. © 2018 The Authors AIChE Journal published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers AIChE J, 64: 1591–1605, 2018     

Conventional and data-driven modeling of filtered drag,heat transfer,and reaction rate in gas–particle flows

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.     

A system‐size independent validation of CFD‐DEM for noncohesive particles

For the first time, CFD‐DEM simulations of small‐scale fluidized beds are quantitatively validated against large‐scale experiments. Such validation is possible via the identification of a measurement independent of system size, namely defluidization. CFD‐DEM inputs (particle properties and operating conditions) are measured directly. Sphericity is found to be critical, even for highly spherical particles. This size‐independent method of validation is valuable since it allows for validation of CFD‐DEM models without restrictions on system sizes or particle sizes. © 2015 American Institute of Chemical Engineers AIChE J, 61: 4051–4058, 2015     

Assessment of mesoscale solid stress in coarse‐grid TFM simulation of Geldart A particles in all fluidization regimes

This study focused on assessing the effect of mesoscale solid stress in the coarse grid two‐fluid model (TFM) simulation of gas–solid fluidized beds of Geldart Group A particles over a broad range of flow regimes, including bubbling, turbulent, fast, and pneumatic transport fluidization regimes. Particularly, the impact of mesoscale solid pressure, mesoscale solid viscosity, and mesoscale solid stress anisotropy were investigated by comparing six different coarse‐grid TFM settings. Compared with the available experimental data, it is found that both the kinetic theory‐based TFM with only drag correction and the filtered TFM can predict the flow behavior in all fluidization regimes. Mesoscale solid pressure and viscosity have the opposite impact on flow hydrodynamics; they compete and offset each other, which confirms the assumption employed in many previous studies that the mesoscale solid stress could be neglected in coarse‐grid TFM simulation. Published 2018. This article is a U.S. Government work and is in the public domain in the USA. AIChE J, 64: 3565–3581, 2018     

Solids transport models comparison and fine‐tuning for horizontal,low concentration flow in single‐phase carrier fluid

We determined and fine‐tuned the solids transport models appropriate for predicting the single‐phase carrier fluid velocity to transport solid particles in conduits for horizontal, low concentration flow. A database with 538 experimental data points was compiled. A literature review was performed to determine the data ranges, forces, and mechanisms used to develop 44 models, and their velocity predictions were compared against the database using statistics. Using the dimensionless forms of the models and the data, the model parameters were adjusted to improve their accuracy and identify the dominant forces. At low concentrations: for liquid/solid flow from a bed of solids and gas/solid flow from the bottom of pipelines, the particle weight, and inertial and viscous forces dominate; for gas/solid flow from a bed of solids, the particle weight, and inertial, viscous, and adhesive forces play a role; and gaps exist in the data for large‐diameter pipes and high‐density gases. © 2013 American Institute of Chemical Engineers AIChE J, 60: 76–122, 2014     

An Improved LES on Dense Particle‐Liquid Turbulent Flows Using Integrated Boltzmann Equations

An improved large eddy simulation (LES) using a dynamic second‐order subgrid stress (SGS) model has been developed for simulating dense particle‐liquid two‐phase turbulent flows. The governing equations of each phase are obtained from a microscopic point of view, using the kinetic theory of molecular gas. They are derived by multiplying the Boltzmann equation of each phase by property parameters and integrating over the velocity space. An inter‐particle collision term is included in the governing equation of the particle phase. Assuming a Maxwellian distribution of the velocity for particle‐phase, an inter‐particle collision term is derived.     

Comparison of Two‐Fluid and Discrete Particle Modeling of Dense Gas‐Particle Flows in Gas‐Fluidized Beds

Both two‐fluid models embedding the kinetic theory of granular flow for particulate phase stress (TFM) and discrete particle models (DPM) are widely used for the numerical simulation of gas fluidization. In this study, a detailed comparison between results obtained from both TFM and DPM is reported, including axial and radial solid concentration profiles, solids circulation patterns, pressure drop and its standard deviation and granular temperature. It was shown that good agreement can be obtained even in cases of low restitution coefficient, which suggests the possible applicability of kinetic theory of granular flow beyond its nominal range of validity and clearly indicates that the continuum treatment of the solids phase in TFM provides a good approximation of its discrete nature.     

Verification of Eulerian–Eulerian and Eulerian–Lagrangian simulations for turbulent fluid–particle flows

We present a verification study of three simulation techniques for fluid–particle flows, including an Euler–Lagrange approach (EL) inspired by Jackson's seminal work on fluidized particles, a quadrature–based moment method based on the anisotropic Gaussian closure (AG), and the traditional two‐fluid model. We perform simulations of two problems: particles in frozen homogeneous isotropic turbulence (HIT) and cluster‐induced turbulence (CIT). For verification, we evaluate various techniques for extracting statistics from EL and study the convergence properties of the three methods under grid refinement. The convergence is found to depend on the simulation method and on the problem, with CIT simulations posing fewer difficulties than HIT. Specifically, EL converges under refinement for both HIT and CIT, but statistics exhibit dependence on the postprocessing parameters. For CIT, AG produces similar results to EL. For HIT, converging both TFM and AG poses challenges. Overall, extracting converged, parameter‐independent Eulerian statistics remains a challenge for all methods. © 2017 American Institute of Chemical Engineers AIChE J, 63: 5396–5412, 2017     

Pore scale inertial flow simulations in 3‐D smooth and rough sphere packs using lattice Boltzmann method

Pore‐scale inertial flows in periodic body centered cubic (BCC) arrays of smooth and rough sphere packs were simulated using lattice Boltzmann method. Computed velocity fields were visualized and averaged to calculate macroscopic flow parameters characteristic of porous media such as permeability, tortuosity, and β factor as well as the transition Reynolds number values and compared well with established correlations. Furthermore, hemispherical depositions on the smooth spheres in the regular BCC array were used to calculate roughness induced changes in macroscopic flow parameters. As the next step toward simulating inertia dominated flow in natural porous media, simulations were validated for low Reynolds number flow in a three‐dimensional (3‐D) CT image of irregular pack of uniform diameter spheres. This work aims to define 3‐D canonical studies for roughness induced inertial flows in porous media and to assess the capability of LBM for simultaneous prediction of absolute permeability and β factor. © 2013 American Institute of Chemical Engineers AIChE J, 59: 4858–4870, 2013