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Particle Resolved DNS and URANS Simulations of Turbulent Heat Transfer in Packed Structures
... Specifically, interface-resolved Lagrangian methods are generally termed particulate direct numerical simulation. 4 The no-slip boundary condition at the particle surface allows particles to have an impact on the surrounding fluid, while the fluid itself applies an integrated hydrodynamic force to the particles. Characterizing the internal mechanics of fluid-solid interaction is made easier by this coupling mechanism. ...
A moving multiblock (MMB) grid refinement method is developed for lattice Boltzmann modeling of fluid–solid flows. This method addresses the need for high resolution near freely moving bodies, particularly in pore-scale simulations of porous particles. The MMB method is an adaptation of the traditional static multiblock (SMB) scheme, where adjacent subdomains overlap by one coarse mesh unit to facilitate efficient information exchange. However, the computationally intensive temporal interpolation used in the SMB method is replaced by spatial interpolation in the MMB. Additionally, each grid block begins to move collectively following a single time step evolution of the coarsest grid block, which is inspired by the moving domain method. Consequently, only the buffer layer of fine grids that migrates toward the coarse grid side needs to be rebuilt, which lowers the computational costs associated with spatial interpolation while maintaining method accuracy. The second-order accuracy of the method is verified through simulation of Poiseuille flow. The method is subsequently applied to simulate particle motion in Poiseuille and Couette flows, the sedimentation of an ellipse under gravity in a vertical channel, and harmonic oscillation of a cylinder in a stationary fluid. The flow field exhibits smoothness across boundaries, and the obtained results correlate well with established findings in the literature, demonstrating the method's feasibility and accuracy for fluid-particle flows. We examine pore-scale simulations of a permeable particle translating inside channel flow as a particular application. Results indicate that porous particles migrate toward an equilibrium position between the channel wall and centerline.
Packed bed reactors are widely used to perform solid‐catalyzed gas‐phase reactions and local turbulence is known to influence heat and mass transfer characteristics. We have investigated turbulence characteristics in a packed bed of 113 spherical particles by performing particle‐resolved Reynolds‐averaged Navier–Stokes (RANS) simulations, Large Eddy Simulation (LES), and Direct Numerical Simulation (DNS). The predictions of the RANS and LES simulations are validated with the lattice Boltzmann method (LBM)–based DNS at particle Reynolds number (Rep) of 600. The RANS and LES simulations can predict the velocity, strain rate, and vorticity with a reasonable accuracy. Due to the dominance of enhanced wall‐function treatment, the turbulence characteristics predicted by the ε‐based models are found to be in a good agreement with the DNS. The ω‐based models under‐predicted the turbulence quantities by several orders of magnitude due to their inadequacy in handling strongly wall‐dominated flows at low Rep. Using the DNS performed at different Rep, we also show that the onset of turbulence occurs between 200≤Rep≤250.
Particle‐resolved CFD simulations provide detailed insights into fixed‐bed reactors. One of the remaining challenges is the generation of high‐quality computational cells, which is especially challenging near particle‐particle contacts. This contribution presents a generalized and automated contact modification independent of the particle shape by introducing a defined gap of 1 % of the particle diameter. This approach is successfully demonstrated for non‐spherical cylinders, hollow and multi‐hole cylinders, and a novel particle shape in terms of radial void fraction, pressure drop and an illustrative heat transfer study.
Slender packed beds are widely used in the chemical and process industry for heterogeneous catalytic reactions in tube-bundle reactors. Under safety and reaction engineering aspects, good radial heat transfer is of outstanding importance. However, because of local wall effects, the radial heat transport in the vicinity of the reactor wall is hindered. Particle-resolved computational fluid dynamics (CFD) is used to investigate the impact of internal heat fins on the near wall radial heat transport in slender packed beds filled with spherical particles. The simulation results are validated against experimental measurements in terms of particle count and pressure drop. The simulation results show that internal heat fins increase the conductive portion of the radial heat transport close to the reactor wall, leading to an overall increased thermal performance of the system. In a wide flow range (100<Rep<1000), an increase of up to 35% in wall heat transfer coefficient and almost 90% in effective radial thermal conductivity is observed, respectively.
Catalytic fixed‐bed reactors with a low tube‐to‐particle diameter ratio are widely used in industrial applications. The heterogeneous packing morphology in this reactor type causes local flow phenomena that significantly affect the reactor performance. Particle‐resolved computational fluid dynamics has become a predictive numerical method to analyze the flow, temperature, and species field, as well as local reaction rates spatially and may, therefore, be used as a design tool to develop new improved catalyst shapes. Most validation studies which have been presented in the past were limited to simple particle shapes. More complex catalyst shapes are supposed to increase the reactor performance. A workflow for the simulation of fixed‐bed reactors filled with various industrially relevant complex particle shapes is presented and validated against experimental data in terms of bed voidage and pressure drop. Industrially relevant loading strategies are numerically replicated and their impact on particle orientation and bed voidage is investigated.
Turbulent flows in porous media occur in a wide variety of applications, from catalysis in packed beds to heat exchange in nuclear reactor vessels. In this review, we summarize the current state of the literature on methods to model such flows. We focus on a range of Reynolds numbers, covering the inertial regime through the asymptotic turbulent regime. The review emphasizes both numerical modeling and the development of averaged (spatially filtered) balances over representative volumes of media. For modeling the pore scale, we examine the recent literature on Reynolds-averaged Navier–Stokes (RANS) models, large-eddy simulation (LES) models, and direct numerical simulations (DNS). We focus on the role of DNS and discuss how spatially averaged models might be closed using data computed from DNS simulations. A Darcy–Forchheimer-type law is derived, and a prior computation of the permeability and Forchheimer coefficient are presented and compared with existing data.
Expected final online publication date for the Annual Review of Fluid Mechanics, Volume 52 is January 5, 2020. Please see full paper at: https://www.annualreviews.org/doi/abs/10.1146/annurev-fluid-010719-060317
Direct numerical simulations (DNS) are performed in a triply periodic unit cell of a face-centred cubic (FCC) lattice covering the unsteady inertial, to fully turbulent, flow regimes. The DNS data are used to quantify the flow topology, integral scales, turbulent kinetic energy (TKE) transport and anisotropy distribution in the tortuous geometry. Several unique flow features are observed within this low porosity configuration, where the mean flow undergoes strong acceleration and deceleration regions with presence of three-dimensional helical motions, weak wake-like structures behind spheres, stagnation and jet-impingement-like flows together with merging and spreading jets in the main pore space. The jet-impingement and weak wake-like flow structures give rise to regions with negative total TKE production. Unlike flows in complex shaped ducts, the turbulence intensity levels in the cross-stream directions are found to be larger than those in the streamwise direction. Furthermore, due to the compact nature and confined geometry of the FCC packing, the turbulent integral length scales are estimated to be less than 10 % of the bead diameter even for the lowest Reynolds number studied, indicating the absence of macroscale turbulence structures for this configuration. This finding suggests that even for the highly anisotropic flow within the pore, the upscaled flow statistics are captured well by the representative volumes defined by the unit cell.
Convective heat transfer in turbulent flow in porous media is, in general, less understood compared to laminar and creeping flows. It is broadly integrated into industrial applications even though a lack of understanding in microscale. In the present study, we report a direct numerical simulation (DNS) based microscopic analysis of heat and fluid flow in synthetic porous media with an aim to link the macroscopic statistics and qualitative observations with pore-scale signatures of the same. DNS is conducted with two computational codes with a cautious validation on the microscale. The conventional finite-volume solver shows an excellent agreement with high-order spectral/hp element solvers. The synthetic porous media are comprised of square cylinders in a staggered array. Macroscopic observation indicates an increased pressure loss and increased Nusselt number (N u) as a result of high Reynolds numbers. However, a further examination of the ratio of Stanton number to skin friction coefficient St/C f reveals that increasing Reynolds numbers lead to more pressure loss than improved heat transfer. Microscopic analysis shows that N u is highest on the impinging surface, whereas lowest on the wake surface. In addition, it tends to be homogeneously distributed on surface area in the high porosity case. In the low porosity case, N u on the sharp corners, is significantly higher than on other locations. Probability density function (PDF) of streamwise velocity fluctuation u shows Gaussian like distribution within two standard deviation. The high porosity group owns a spike around zero abscissa on PDF of both u and temperature fluctuation T , which is an indication of less chaotic flow motions.
The pressure drop of a packed bed thermal energy storage system with irregular shaped solid pellets and tank-to-particle diameter ratio of 10.4 is investigated. The bed height to diameter ratio is 2. The particle sphericity is calculated and used to compare pressure drop correlations to the measured values in the particle Reynolds number range of 353 ≤ Rep ≤ 5206.
A transport equation for the turbulent viscosity is assembled, using empiricism and arguments of dimensional analysis, Galilean invariance, and selective dependence on the molecular viscosity. It has similarities with the models of Nee and Kovaszany, Secundov et al., and Baldwin and Barth. The equation includes a non-viscous destruction term that depends on the distance of the wall.
This document describes the current formulation of the SST turbulence models, as well as a number of model enhancements. The model enhancements cover a modified near wall treatment of the equations, which allows for a more flexible grid generation process and a zonal DES formulation, which reduces the problem of grid induced separation for industrial flow simulations. Results for a complete aircraft configuration with and without engine nacelles will be shown.
Computational fluid dynamics as a simulation tool allows obtaining a more detailed view of the fluid flow and heat transfer mechanisms in fixed-bed reactors, through the resolution of 3D Reynolds averaged transport equations, together with a turbulence model when needed. In this way, this tool permits obtaining of mean and fluctuating flow and temperature values in any point of the bed. An important problem when modeling a turbulent flow fixed-bed reactor is to decide which turbulence model is the most accurate for this situation. To gain insight into this subject, this study presents a comparison between the performance in flow and heat transfer estimation of five different RANS turbulence models in a fixed bed composed of 44 homogeneous stacked spheres in a maximum space-occupying arrangement in a cylindrical container by solving the 3D Navier–Stokes and energy equations by means of a commercial finite volume code, Fluent 6.0®. Air is chosen as flowing fluid. Numerical pressure drop, velocity and thermal fields within the bed are obtained. In order to judge the capabilities of these turbulence models, heat transfer parameters (Nuw, kr/kf) are estimated from numerical data and together with the pressure drop are compared to commonly used correlations for parameter estimations in fixed-bed reactors.
A new k-epsilon eddy viscosity model, which consists of a new model dissipation rate equation and a new realizable eddy viscosity formulation, is proposed. The new model dissipation rate equation is based on the dynamic equation of the mean-square vorticity fluctuation at large turbulent Reynolds number. The new eddy viscosity formulation is based on the realizability constraints: the positivity of normal Reynolds stresses and Schwarz' inequality for turbulent shear stresses. We find that the present model with a set of unified model coefficients can perform well for a variety of flows. The flows that are examined include: (1) rotating homogeneous shear flows; (2) boundary-free shear flows including a mixing layer, planar and round jets; (3) a channel flow, and flat plate boundary layers with and without a pressure gradient; and (4) backward facing step separated flows. The model predictions are compared with available experimental data. The results from the standard k-epsilon eddy viscosity model are also included for comparison. It is shown that the present model is a significant improvement over the standard k-epsilon eddy viscosity model.
Particle-resolved computational fluid dynamics simulations of flow and pressure drop in fixed beds of spheres give poor accuracy at higher Reynolds number (Re) under turbulent conditions. The influence of meshing is investigated for two cases, a face centered cubic packing with no wall effects, and a narrow bed of tube-to-particle diameter ratio three with strong wall effects. Local approaches to meshing at contact points were compared, using both surface flattening (caps) and bridging between spheres. Accurate results for pressure drop at moderate to high Re > 1000 required computationally expensive fine uniform meshes, however at highest Re pressure drop appeared to be low. The use of boundary adaption improved results with less expensive meshes. The aspect ratio of surface prisms and the sizes of the bridges used at contact points had strong effects. Velocity profiles were much less sensitive to the mesh refinement and could be obtained at reasonable cost.
Microscopic analysis of turbulence topology in a regular porous media is presented with a series of direct numerical simulation (DNS). The regular porous media are comprised of square cylinders in a staggered array. Triply periodic boundary conditions enable efficient investigations in a representative elementary volume (REV). Three flow patterns channel with sudden contraction, impinging surface and wake are observed and studied quantitatively in contrast to the qualitative experimental studies reported in literature. Among these, shear layers in the channel show the highest turbulence intensity due to a favorable pressure gradient and shed due to an adverse pressure gradient downstream. Turbulent energy budget indicates a strong production rate after the flow contraction and a strong dissipation on both shear and impinging walls. Energy spectra and pre-multiplied spectra detect large scale energetic structures in the shear layer and a breakup of scales in the impinging layer. However, these large scale structures break into less energetic small structures at high Reynolds number conditions. This suggests an absence of coherent structures in densely-packed porous media at high Reynolds number. Anisotropy analysis with a barycentric map shows that the turbulence in porous media is highly isotropic in the macro-scale, which is not the case in the micro-scale. In the end, proper orthogonal decomposition (POD) is employed to distinguish the energy-conserving structures. The results support the pore scale prevalence hypothesis (PSPH). However, energetic coherent structures are observed in the case with sparsely-packed porous media.
In 2006, Dixon et al. published the comprehensive review article entitled “Packed tubular reactor modeling and catalyst design using computational fluid dynamics.” More than one decade later, many researchers have contributed to novel insights, as well as a deeper understanding of the topic. Likewise, complexity has grown and new issues have arisen, for example, by coupling microkinetics with computational fluid dynamics (CFD). In this review article, the latest advances are summarized in the field of modeling fixed-bed reactors with particle-resolved CFD, i.e. a geometric resolution of every pellet in the bed. The current challenges of the detailed modeling are described, i.e. packing generation, meshing, and solving with an emphasis on coupling microkinetics with CFD. Applications of this detailed approach are discussed, i.e. fluid dynamics and pressure drop, dispersion, heat and mass transfer, as well as heterogeneous catalytic systems. Finally, conclusions and future prospects are presented.
A promising workflow for computational generation and meshing of resolved-particle randomly packed beds of arbitrary-shaped particles is described. The workflow is based on an automated package that is developed for the Bullet Physics Library for packed bed generation, and on the shrink-wrap method for handling the meshing challenges of the particle contact areas. Packed bed properties such as radial voidage and particle angle distributions are validated against existing experimental data and theoretical methods in the literature. Furthermore, the accuracy of the generated mesh using the shrink-wrap technique is evaluated by calculating the bulk and radial voidage distribution, the pressure drop, and by velocity profile and heat transfer analysis using computational fluid dynamics (CFD) simulations. It is shown that the proposed workflow can speed up the process of the geometry and mesh generation while it maintains the structural features of the bed. Finally, applications of such geometries to fixed bed reaction engineering problems are presented to illustrate simulations that can be achieved using the new workflow.
Particle resolved direct numerical simulation (PR-DNS) has been used extensively to obtain closures for heat transfer from static particle arrays. However, most of the currently available closure models are valid for packings of spherical particles only. We present closure models for momentum and heat transfer in densely packed cylindrical particle assemblies of different aspect ratios (2, 4 and 6). Our packings are generated using the Discrete Element Method (DEM). Subsequently, the void space is meshed with a high quality computational grid, and steady-state DNS simulations are completed to provide insight into the local heat transfer and pressure drop characteristics. The variation observed in the values for the local heat transfer rates from our PR-DNS study implies the necessity of specifying confidence intervals when reporting a correlation for the corresponding Nusselt number. Our newly developed correlations are applicable to densely packed beds of cylindrical particles in the porosity range (0.405 < ε < 0.539), and allow the estimation of the variability of the Nusselt number.
Various models are available for simulating turbulent flows in porous media.Models based on the eddyviscosity assumption are often adopted to close the Reynolds stress term. In order to validate the assumptions behind such turbulencemodels, we studied the dynamics of macroscopic momentum and turbulence kinetic energy in porous media flows by utilizing Direct Numerical Simulation (DNS). The generic porous matrix is composed of regularly arranged spheres. The resulting periodic porous medium is bounded by two walls. The DNS analyses with a Lattice Boltzmann method were performed for various values of the applied pressure gradient, pore size to channel width ratio, and porosity. The DNS results were averaged over time and volume to obtain macroscopic results. The results show that the macroscopic shear Reynolds stress in all Representative Elementary Volumes (REVs), independent of their location, is negligibly small, although the mean velocity gradient takes nonzero values near the wall. The turbulence kinetic energy production rate is generally balanced by the dissipation rate in each REV. The DNS results support a zero-equation turbulencemodel that accounts for the fact that turbulent structures are restricted in size by the pore scale. The DNS results also suggest that the Brinkman term, which expresses the diffusion of momentum, has an important effect near the wall where the gradient of the shear stress is large. Therefore, the Brinkman term should be taken into account in the macroscopic momentum equation as a component of the total drag. A preliminary macroscopic model for calculating turbulentporous media flows has been proposed and compared with our DNS results.
This paper aims to develop a generic framework for detecting contact between cylindrical particles in discrete element modelling based on a full exploitation of the axi-symmetrical property of cylinders. The main contributions include: (1) A four-parameter based local representative system is derived to describe the spatial relationship between two cylinders so that the 3D cylinder–cylinder intersection problem can be reduced to a series of 2D circle–ellipse intersections, which considerably simplifies the contact detection procedure. (2) A two-stage contact detection scheme is proposed in which no-overlap contact pairs are identified in the first overlap check stage, and then the actual overlap region is determined in the second resolution stage and represented by two schemes: the layered representation which is generic, and the edge representation which is numerically more efficient but less accurate. (3) The most significant contribution is the development of two theorems that establish a fundamental relationship between the contact point and contact normal of two contacting cylinders, offering a simple approach to determining the normal direction based on the contact point and vice versa. These theorems are valid not only for cylinders, but also for any axi-symmetrical shapes and their combinations. Some numerical issues are discussed. Numerical examples are presented to illustrate the capability and applicability of the proposed methodologies.
When a turbulent flow in a porous medium is determined numerically, the crucial question is whether turbulence models should account only for turbulent structures
restricted in size to the pore scale or whether the size of turbulent structures could
exceed the pore scale. The latter would mean the existence of macroscopic turbulence
in porous media, when turbulent eddies exceed the pore size. In order to determine
the real size of turbulent structures in a porous medium, we simulated the turbulent
flow by direct numerical simulation (DNS) calculations, thus avoiding turbulence
modelling of any kind. With this study, which for the first time uses DNS calculations,
we provide benchmark data for turbulent flow in porous media. Since perfect DNS
calculations require the resolution of scales down to the Kolmogorov scale, often only
approximate DNS solutions can be obtained, especially for high Reynolds numbers.
This is accounted for by using and comparing two different DNS approaches, a finite
volume method (FVM) with grid refinement towards the wall and a lattice Boltzmann
method (LBM) with equal grid distribution. The solid matrix was simulated by a
large number of rectangular bars arranged periodically. The number of bars in the
solution domain with periodic boundary conditions was reduced systematically until
a minimum size was found that does not suppress any large-scale turbulent structures.
Two-point correlations, integral length scales and energy spectra were determined in
order to answer the question of whether or not macroscopic turbulence can be found
in porous media.
High temperature reactors (HTR) are being considered for deployment around the world because of their excellent safety features. The fuel is embedded in a graphite moderator and can sustain very high temperatures. However, the appearance of hot spots in the pebble bed cores of HTR's may affect the integrity of the pebbles. A good prediction of the flow and heat transport in such a pebble bed core is a challenge for available turbulence models and such models need to be validated. In the present article, quasi direct numerical simulations (q-DNS) of a pebble bed configuration are reported, which may serve as a reference for the validation of different turbulence modeling approaches. Such approaches can be used in order to perform calculations for a randomly arranged pebble bed. Simulations are performed at a Reynolds number of 3088, based on pebble diameter, with a porosity level of 0.42. Detailed flow analyses have shown complex physics flow behavior and make this case challenging for turbulence model validation. Hence, a wide range of qualitative and quantitative data for velocity and temperature field have been extracted for this benchmark. In the present article (part I), results related to the flow field (mean, RMS and covariance of velocity) are documented and discussed in detail. Moreover, the discussion regarding the temperature field will be published in a separate article. (c) 2012 Elsevier B.V. All rights reserved.
Turbulent heat transfer is an extremely complex phenomenon, which has challenged turbulence modellers over various decades. The limitations of the commonly used eddy diffusivity approach have become more evident specially for innovative nuclear reactor applications with low-Prandtl fluids like liquid metals. One of the objectives of the THINS (Thermal Hydraulics of Innovative Nuclear Systems) project sponsored by the European Commission is to push forward the validation and adoption of more accurate closures for single-phase turbulence for innovative reactors in engineering codes. As a part of this THINS project, CD-adapco has implemented in its commercial code STAR-CCM+ an algebraic turbulent heat flux model. In the present work, this implemented model has been widely tested and further calibrated for the application to natural, mixed and forced convection flow regimes at low-Prandtl number. As an outcome, a modelling correlation is proposed in combination with a newly proposed set of model coefficients. This proposed correlation shows dependency of Reynolds and Prandtl numbers in a logarithmic manner to accommodate the wall-normal temperature gradient for the heat flux term. The use of this correlation brings significant improvements in the prediction of heat transfer in liquid metals in all flow regimes.
Design of industrial scale packed bed reactors can be aided by using computational fluid dynamics (CFD)-discrete element method (DEM) simulation for understanding the transport phenomena. However, conducting CFD-DEM simulation for the whole large-scale packed bed reactor is computationally prohibitive, which limits the usage of this tool. Hence, a methodology has been developed to identify the segments of the bed, which can serve as a good representative for the CFD simulation of large industrial scale packed bed reactors. Two segments, a cylindrical cut-segment and a wall-segment, have been used to represent the core central region of packed bed and the near wall region of packed bed, respectively. The methodology determines the size (diameter and height) of the cylindrical bed cut-segment that can be a good representative for the large-scale packed bed reactor. The segments are then used to obtain the information on transport phenomena (friction factor coefficients (pressure drop) and the heat transfer/mass-transfer coefficients) for packed beds made of spherical particles and non-spherical particles. CFD simulations have been conducted on the DEM generated packing segments for a wide particle Reynolds number range (from laminar to turbulent regime) to develop the correlations. The proposed methodology (on cut-segment and wall-segment) has been validated for spherical particle packed bed by comparing results to well-established correlations (such as Ergun equation for pressure drop, Wakao correlation and multiparticle Ranz–Marshal correlation for particle heat transfer coefficient, and Dixon–Lubua and Colledge–Paterson correlation for wall heat transfer coefficient). The methodology is then applied to understand the transport phenomena and to develop correlations for long cylindrical pellet (aspect ratio 7) and fluted-ring. The pressure drop correlation obtained for the long cylindrical pellet (aspect ratio of 7) packed bed has been compared to the correlation for cylindrical pellet with aspect ratio of 5.77 (obtained by Nemec; Nemec, D.; Levec, J. Chem. Eng. Sci. 2005, 60, 6947). The comparison shows that the effect of increasing aspect ratio on pressure drop is captured as per the expected trend. The methodology has been applied in a chemical looping combustion reactor based on the pressure drop and heat transfer results. Thus, the proposed CFD-DEM methodology offers a computationally efficient way of understanding the transport phenomena in an industrial scale reactor through simulating a methodically selected segment inside the reactor. This enables design and performance assessment of such reactors.
This paper reports large eddy simulation results for a randomly stacked bed of spherical pebbles, using a second-order accurate, cell-centered finite volume method on an unstructured polyhedral mesh. The selected flow configuration represents the core of a high temperature reactor, in which nuclear fuel is embedded in the pebbles. The geometrical arrangement consists of approximately 30 pebbles, which are randomly stacked and in contact with each other. Extensive analyses of flow and thermal fields are performed to derive valuable insights on the flow characteristics. The predicted flow-field is fairly complex and exhibits highly unsteady and three-dimensional turbulent behavior with strong rotational and cross flow regions. The flow, while moving over the pebbles shows attenuation and enhancement in the turbulence levels, and eventually yields to flow separation and its subsequent reattachment. Consequently, a wide range of temperatures over the pebbles is observed; this includes the appearance of hot-spots especially around contact areas. The occurrence of high- and low-velocity streak regions, and the corresponding temperature fields are examined and quantified. The reported analyses can be used for validation of low-order turbulence modeling applications to complex pebble flow configurations.
High Temperature Reactors (HTRs) are being considered all over the world for deployment because of their excellent safety features. A particular inherent safety advantage of HTR designs is related to the very high temperature that the fuel can sustain basically preventing the fuel from melting even in the event of loss of cooling. Generally, the core can be designed using a graphite pebble bed. Test reactors have shown safe and efficient operation, however questions have been raised about possible occurrence of local hot spots in the pebble bed which may affect the pebble integrity. A good prediction of the flow and heat transfer phenomena in the pebble bed core of an HTR is a challenge for available turbulence models, which still require to be validated for pebble bed applications. In the present study large eddy simulation (LES) of a well-defined single face cubic centered pebble bed is performed. The obtained results are extensively compared with a quasi-direct numerical simulation (q-DNS) database in order to analyze the prediction capabilities and feasibility of used LES approach for such a complex flow regime. The simulations are performed at a Reynolds number of 3088, based on pebble diameter, with a porosity level of 0.42. The obtained results are found to be in good agreement with the q-DNS results.
The published heat transfer data obtained from steady and nonsteady measurements are corrected for the axial fluid thermal dispersion coefficient values proposed by Wakao[1].The corrected data in the range of Reynolds number from 15 to 8500 are correlated by the analogous form of the mass correlation proposed by Wakao and Funazkri[2]:Nu= 2 + 1.1 Pr Re0.6
A new meshing method for fixed beds consisting of monodisperse spherical particles is presented. The particles are flattened near the particle–particle and particle–wall contact points, respectively, to avoid bad cell qualities. Compared to known methods from literature the modifications are so small that a subsequent correction of these modifications is not necessary. CFD simulations are performed for tube to particle diameter ratios of 3≤D/d≤10 in the laminar, transitional and turbulent flow regime and are compared with results from literature concerning porosity and pressure drop. The flow pattern within the bed is also investigated. The random fixed bed is generated with a DEM-code and the fluid domain is meshed with the commercial CFD code STAR-CCM+. The focus of this work is to verify, that physically correct results as well as meshes with a low number of cells and therewith short calculation time can be obtained with this new meshing method.
Fixed bed reactors are among the most important equipment in chemical industries as these are used in chemical processes.
An accurate insight into the fluid flow in these reactors is necessary for their modeling. The pressure drop and heat transfer
coefficient have been studied for the fixed bed reactor with tube to particle diameter ratio (N) of 4.6 and comprising 130
spherical particles using computational fluid dynamics (CFD). The simulations were carried out in a wide range of Reynolds
number: 3.85≤Re≤611.79. The RNG k-ɛ turbulence model was used in the turbulent regime. The CFD results were compared with empirical correlations in the literature.
The predicted pressure drop values in laminar flow were overestimated compared with the Ergun’s [27] correlation and underestimated
in the turbulent regime due to the wall friction and the flow channeling in the bed, respectively. It was observed that the
CFD results of the pressure drop are in good agreement with the correlations of Zhavoronkov et al. [28] and Reichelt [29]
because the wall effects have been taken into account in these correlations. Values of the predicted dimensionless heat transfer
coefficient showed better agreement with the Dixon and Labua’s [32] correlation. This is explained by the fact that this correlation
is a function of the particle size and shape in the bed.
In this work, the influence of the confining walls on pressure drop in packed beds is studied numerically for moderate tube/particle diameter ratios. Two different configuration types are investigated, a regular type and an irregular one. The regular configurations follow the atomic body-centered cubic and face-centered cubic structure in ideal crystals, whereas the modified ballistic deposition method is employed to generate the irregular configurations. To validate the simulation results, four experimental pressure drop correlations are used, namely by Ergun (1952) [1], by Carman (1937) [2], by Zhavoronkov et al. (1949) [3] and by Reichelt (1972) [4]. Simulation results for the regular configurations are in a good agreement with the Carman correlation. For the irregular configurations, which are closer to real packed beds, agreement with the correlations of Zhavoronkov et al. (1949) [3] and Reichelt (1972) [4] is better. This is explained by the fact that the latter two correlations take the wall effect into account. CFD simulations provide useful data on the flow field inside packed beds which can be used for the improvement and further optimization of the design and operation of packed bed reactors.
The distinct element method is a numerical model capable of describing the mechanical behavior of assemblies of discs and spheres. The method is based on the use of an explicit numerical scheme in which the interaction of the particles monitored contact by contact and the motion of the particles modelled particle by particle. The main features of the distinct element method are described. The method is validated by comparing force vector plots obtained from the computer program BALL with the corresponding plots obtained from a photoelastic analysis.
A discussion of modeling passive scalar transport in turbulent flows is given. Several methods employed to close the scalar-flux term u′′ that arises during Reynolds averaging are provided. Alternatives and improvements to the gradient diffusion hypotheses are addressed, most notably, the need for an alternative to the global constant turbulent Schmidt and Prandtl numbers. The reader is given a brief history covering methods used to predict turbulent Schmidt and Prandtl numbers, along with recommendations for future research, based partially on studies by Professor Stuart Churchill. More detailed formulations of turbulent Schmidt or Prandtl numbers will enable better approximations of the influence of turbulence in models of passive scalar flows using the gradient diffusion hypothesis.
As in the first and second editions, the book revolves around the fact that turbulence modeling is one of three key elements in CFD. Very precise mathematical theories have evolved for the other two, viz., grid generation and algorithm development. By its nature, i.e., creating a mathematical model that approximates the physical behavior of turbulent flows, far less precision has been achieved in turbulence modeling. This text addresses the problem of selecting/devising such models. The fundamental premise is, in the spirit of G. I. Taylor, an ideal model should introduce the minimum amount of complexity while capturing the essence of the relevant physics. The text begins with the simplest models and charts a course leading to some of the most complex models that have been applied to a nontrivial flow. Along the way, a systematic methodology is presented for developing and analyzing turbulence models. The methodology makes use of tensor calculus, similarity solutions, singular perturbation methods, and numerical procedures. The text stresses the need to achieve a balance amongst the physics of turbulence, mathematical tools required to solve turbulence-model equations, and common numerical problems attending their use (i.e., what good is a model if it makes your program crash?). Several user friendly programs and detailed user's guides are provided on the Compact Disk that accompanies the text. Many of the applications are used throughout the text to permit comparison of complicated models with simpler models. A completely objective point of view is taken in assessing the merits of models and their range of applicability. The text includes an extensive Bibliography, a detailed Index and well thought out homework problems of varying degrees of difficulty.
In this study, we investigate the transport and transfer properties inside packed beds of spherical particles by means of CFD simulations. Heat and mass transfer properties have been computed in packing configurations of increasing complexity at low to moderate Reynolds numbers (1<Re<80). Only liquid-phase flows are studied (300<Sc<1000). The problem of contact points between particles, which is inherent to finite-volume numerical methods, is solved by applying a contraction of 2% on all the particles of the bed. We show that this treatment has very little influence on the results when analyzed with dimensionless numbers as Nu=f(Re, Pr) or Sh=f(Re, Sc). Finally, a very dense packing of spheres was built using a Discrete Element Method and used to represent the real granular media. Transfer predictions by the model are very realistic. Longitudinal and transverse dispersion coefficients are determined inside geometries containing hundreds of particles. Predictions of dispersion coefficients are consistent with literature, but a correction is applied to improve results, because the bed contraction leads to the underestimation of the transverse dispersion coefficient. The model is found to be very promising to study the “near wall channelling” phenomena inside small packed columns induced by the heterogeneity of the porosity profile close to the wall.
Randomly packed fixed-bed reactors are widely used in the chemical process industries. Their design is usually based on pseudo-homogeneous model equations with averaged semi-empirical parameters. However, this design concept fails for low tube-to-particle diameter ratios (=aspect ratios) where local phenomena dominate. The complete three-dimensional (3D) structure of the packing has therefore to be considered in order to resolve the local inhomogeneities.New numerical methods and the increase of computational power allow us to simulate in detail single phase reacting flows in such reactors, exclusively based on material properties and the 3D description of the geometry, thus without the use of semi-empirical data. The successive simulation steps (packing generation, fluid flow and species calculation) and their validation with experimental data are described in this paper. In order to synthetically generate realistic random packings of spherical particles, we apply a Monte-Carlo method. The subsequent numerical simulation of the 3D flow field and coupled mass transport of reacting species is done by means of lattice Boltzmann methods. The simulation results reveal that not only the local behaviour but also integral quantities like the pressure drop depend remarkably on the local structural properties of the packings, a feature which is neglected when using correlations with averaged values.
A new k-[epsilon] eddy viscosity model, which consists of a new model dissipation rate equation and a new realizable eddy viscosity formulation, is proposed in this paper. The new model dissipation rate equation is based on the dynamic equation of the mean-square vorticity fluctuation at large turbulent Reynolds number. The new eddy viscosity formulation is based on the realizability constraints; the positivity of normal Reynolds stresses and the Schwarz' inequality for turbulent shear stresses. We find that the present model with a set of unified model coefficients can perform well for a variety of flows. The flows that are examined include: (i) rotating homogeneous shear flows; (ii) boundary-free shear flows including a mixing layer, planar and round jets; (iii) a channel flow, and flat plate boundary layers with and without a pressure gradient; and (iv) backward facing step separated flows. The model predictions are compared with available experimental data. The results from the standard k-[epsilon]
A new approach to modeling the effects of a solid wall in one-point second-moment (Reynolds-stress) turbulence closures is presented. The model is based on the relaxation of an inhomogeneous (near-wall) formulation of the pressure–strain tensor towards the chosen conventional homogeneous (far-from-a-wall) form using the blending function α, for which an elliptic equation is solved. The approach preserves the main features of Durbin’s Reynolds-stress model, but instead of six elliptic equations (for each stress component), it involves only one, scalar elliptic equation. The model, called “the elliptic blending model,” offers significant simplification, while still complying with the basic physical rationale for the elliptic relaxation concept. In addition to model validation against direct numerical simulation in a plane channel for Reτ = 590, the model was applied in the computation of the channel flow at a “real-life” Reynolds number of 106, showing a good prediction of the logarithmic profile of the mean velocity.
Modeling of packed bed reactors: Hydrogen production by the steam reforming of methane and glycerol
- A G Dixon
- B D Macdonald
- A Olm