Drag Coefficient Of Particles Research Articles

Modeling drag coefficients of spheroidal particles in rarefied flow conditions

Transport of particles in flows is often modeled in a combined Eulerian–Lagrangian framework. The flow is evaluated on an Eulerian grid, while particles are modeled as Lagrangian points whose positions and velocities are evolved in time, resulting in particle trajectories embedded in the time-dependent flow field. The method essentially resolves the flow field in complex geometries in detail but uses a closure model for the particle dynamics aimed at including the essential particle–fluid interactions at the cost of detailed small-scale physics. Rarefaction effects are usually included through the phenomenological Cunningham correction on the drag force experienced by the particles. In this Lagrangian point-particle approach, any explicit reference to the finite size and the shape of the particles, and their local orientation in the flow field, is typically ignored. In this work we aim to address this gap by deriving, from fully-resolved Direct Simulation Monte Carlo (DSMC) studies, heuristic or closure models for the drag force acting on prolate and oblate spheroidal particles with different aspect ratios, and a fixed orientation, in uniform ambient rarefied flows covering the transition regime between the continuum and free-molecular limits. These closure models predict the drag in the transition regime for all considered parameter settings (validated with DSMC data). The continuum limit is enforced a priori and we retrieve the free-molecular limit with reasonable accuracy (based on comparisons with literature data). We also include in the models the capability to predict effects related to basic gas-surface interactions via the tangential momentum accommodation coefficient. We furthermore assess the validity of the proposed closure model for particle dynamics in proximity to solid walls. This investigation extends our previous work, which focused on small aspect ratio spheroids with exclusively diffusive gas-surface interactions [see Livi et al. (2022)]. The derived models are obtained for isothermal, subsonic flows relevant for particle contamination control in semiconductor manufacturing.

Experimental and numerical estimation of wall effect and drag coefficient on hollow frustum settling in annular and non-annular channels

The wall effect and drag coefficient of hollow frustum particles (Tapered hollow cylinders) settling in cylindrical non-annular and annular channels were experimentally investigated. The terminal velocity, V, of the hollow frustum particles is measured by considering flow channels of different diameters filled with various working fluids. The present work deals with the study of drag coefficient for a range of 0.12 ≤ deq/D ≤ 0.44, 0.22 ≤ deq/L ≤ 0.43, and 0.22 ≤ di/do ≤ 0.77. The hollow frustum particle's terminal velocity, V, decreases with increased blockage ratios and di/do ratios. The experiment revealed that the frustum attains a steady-state orientation. The hollow frustum particle is affected by the wall's retardation, which is described by the wall factor. The hollow frustum particle's wall factor (f) depends on the Reynolds number (Re) and blockage ratio. The wall factor increases with increasing Reynolds number up to a certain point called critical Reynolds number, Rec; after that, the wall factor remains constant with increasing Reynolds number. The hollow frustum particles settling in the cylindrical non-annular channel were more subjected to wall effects than those in the annular channel. Simple correlations between the wall factor and the blockage ratio are developed. The current research also focuses on the relationship between Reynolds number (Re) and drag coefficient (CD). The wall effect is considered while estimating the hollow frustum particle's drag coefficient (CD). The numerical estimate of the experimental drag coefficient is made using Ansys- fluent 18.1, and the experimental and numerical results are consistent with each other.

Hydrodynamic force and torque models for cylindrical particles in a wide range of aspect ratios

During the pneumatic conveyance of biomass in a coal-fired power station boiler, biomass particles have cylindrical shapes with different aspect ratios. They move through the fluid at any angle and rotate strongly. However, highly accurate and general models of the drag, lift, and torque coefficients (CD, CL, and CT) for biomass particles in a wide range of aspect ratios, especially the CT model and the high aspect ratios, are currently lacking. This paper presents detailed direct numerical simulations of the flow around cylindrical cylinders with varying aspect ratios (6 ≤ AR ≤ 22), Reynolds numbers (100 ≤ Re ≤ 2000), and angles of incidence (0° ≤ θ ≤ 90°). The simulation was conducted using the OpenFOAM solver with the body-fitted mesh method. The flow characteristics and force coefficients of cylindrical particles with different AR were systematically analyzed. New functional correlations between CD, CL, and CT and AR, Re, and θ values were established. The mean squared errors for CD, CL, and CT were 8.8 × 10–2, 2.4 × 10–2, and 4.7 × 10–2, with average relative errors of 5.8%, 3.5%, and 8.17%, respectively. A comparison of the results with other experimental and simulation data in previous literatures showed that the new CD and CL models have considerable higher predictive ability. The generality of the new CD model expanding to low ARs of 1.5 and 3 is verified finally. The new force and torque models are expected to improve the accuracy of Eulerian–Lagrangian simulations of various cylindrical particle-laden flows in the utility of biomass energy.

A New Model for Predicting Drag Coefficient and Settling Velocity of Coarse Mineral Particles in Newtonian Fluid

Efficient transport in vertical pipeline hydraulic lifting systems is vital for coarse-grained ore, necessitating a deep comprehension of the settling traits of coarse mineral particles. In this study, we conducted a series of settling experiments on individual coarse particles in Newtonian fluids with varying viscosities, employing a self-designed and manufactured settling apparatus. A total of 133 sets of experimental data on the free settling of coarse particles in Newtonian fluids were obtained by recording the particle settling process with a high-speed camera and applying image processing techniques. A mechanical model was employed to perform statistical analysis on the experimental data and establish a predictive model for the drag coefficient and an explicit predictive model for the settling terminal velocity of coarse-grained ore in Newtonian fluids. The average relative errors between the predicted values and experimental values of the drag coefficient and settling terminal velocity models are 4.26% and 7.34%, respectively. This confirms the reliability of the provided predicted model, providing a theoretical foundation for determining the hydraulic lifting speed of coarse mineral particles in vertical pipelines for deep mining.

DCC-DNN: A deep neural network model to predict the drag coefficients of spherical and non-spherical particles aided by empirical correlations

The particle drag force coefficient is a critical variable in modeling multiphase energy systems. Even though various empirical drag models with limited conditions have been published before, developing a general drag model is still an important topic. Existing analytical and modeling techniques of particle drag are incapable of supporting the application’s complexity due to their limitations to very specific conditions. This paper proposes the Drag Coefficient Correlation-aided Deep Neural Network (DCC-DNN) architecture to predict the particle drag force coefficient from various single-particle experimental data. Beyond sphericity and Reynolds number, the proposed approach includes an expanded set of features supported by the literature. Simultaneously, model regularization and meta-learning help train a generalized and more reliable drag model, despite the limited data available and the variance exhibited in individual single-particle studies. The presented model applies to spherical and non-spherical particles, providing much-needed generality and reliability for industrial applications.

Drag, lift and torque correlations for axi-symmetric rod-like non-spherical particles in locally linear shear flows

This paper derives new correlations to predict the drag, lift and torque coefficients of axi-symmetric non-spherical rod-like particles for several fluid flow regimes and velocity profiles. The fluid velocity profiles considered are locally uniform flow and locally linear shear flow. The novel correlations for the drag, lift and torque coefficients depend on the particle Reynolds number Rep, the orientation of the particle with respect to the main fluid direction θ, the aspect ratio of the rod-like particle α, and the dimensionless local shear rate G̃. The effect of the linear shear flow on the hydrodynamic forces is modeled as an additional component for the resultant of forces acting on a particle in a locally uniform flow, hence the independent expressions for the drag, lift and torque coefficients of axi-symmetric particles in a locally uniform flow are also provided in this work. The data provided to fit the coefficient in the new correlation are generated using available analytical expressions in the viscous regime, and performing direct numerical simulations (DNS) of the flow past the axi-symmetric particles at finite particle Reynolds number. The DNS are performed using the direct-forcing immersed boundary method. The coefficients in the proposed drag, lift and torque correlations are determined with a high degree of accuracy, where the mean error in the prediction lies below 2% for the locally uniform flow correlations, and bwlow 1.67%, 5.35%, 6.78% for the correlation accounting for the change in the drag, lift, and torque coefficients in case of a linear shear flow, respectively. The proposed correlations for the drag, lift and torque coefficients can be used in large-scale simulations performed in the Eulerian-Lagrangian framework with locally uniform and non-uniform flows.

Direct numerical simulation of the drag, lift, and torque coefficients of high aspect ratio biomass cylindrical particles

Biomass straw fuel has the advantage of low-carbon sustainability, and therefore, it has been widely used in recent years in coupled blending combustion with coal-fired utility boilers for power generation. At present, the drag force FD, the lift force FL, and the torque T evaluation model are very limited. In this study, within a wide range of Reynolds numbers (10 ≤ Re ≤ 2000) and incident angles (0° ≤ θ ≤ 90°), the computational fluid dynamics open source code OpenFOAM-body-fitted mesh method is used to carry out the direct numerical simulation of the flow characteristics of large cylindrical biomass particles with a high aspect ratio of L/D = 9:1. The results show that (1) the projected area of the cylinder begins to decrease after reaching the maximum at θ = 15°, while the change in the incident angle causes the formation of a smaller recirculation zone on the leeward side of the structure, and the effect of the pressure difference on the drag coefficient (CD) is reduced. (2) The lift coefficient (CL) displays a parabolic symmetric distribution when θ = 45°, and then the distribution becomes asymmetrical when Re > 100. The torque coefficient (CT) exhibits a similar trend. (3) Based on the simulation data and the literature data, new models for CD, CL, and CT for cylinders with L/D = 9:1, 10 ≤ Re ≤ 2000 and 0° ≤ θ ≤ 90° are obtained, and the mean square errors are 2.4 × 10−2, 1.4 × 10−2, and 6.4 × 10−2, respectively. This new model can improve the accuracy and adaptability of the universal model of gas–solid dynamics for biomass particles.

A Novel Prediction Model of the Drag Coefficient of Irregular Particles in Power-Law Fluids

The settlement drag coefficient of irregular particles in power-law fluids is a crucial parameter in the field of petroleum engineering. However, the irregular shape of the particle and the non-Newtonian rheological properties of the fluid make it challenging to predict the settlement drag coefficient. In this study, the spherical and irregular particle sedimentation processes in power-law fluids have been analyzed using a visual device and a high-speed camera system. A mechanical model dependent on the force balance of settlement particles was adopted to conduct a detailed statistical analysis of 114 spherical particle experimental results, and a prediction model of the drag coefficient of spherical particles in the power-law fluid was established with a mean relative error of 3.85%. On this basis, considering the influence of geometric shape on the law of particle sedimentation, a new irregular particle sedimentation resistance coefficient model in power-law fluid is established via the incorporation of the parameter circularity of 2D shape description c into the spherical particle sedimentation resistance coefficient predictive model. The parameters in the new irregular particle sedimentation resistance coefficient predictive model can be obtained via nonlinear data fitting of the 211 groups of irregular particles using experimental results in the power-law fluid. The model has high prediction accuracy for the drag coefficient of irregular particles in power-law fluid, with a mean relative error of 4.47, and expands the scope of engineering applications, which is of great significance for fracturing scheme design and wellbore cleaning.

Sphericity based correlations for flow resistance coefficients of non-spherical particles of irregular shape beyond the Stokes regime

In this brief paper, a novel set of correlations for the drag, lift and torque coefficients for non-spherical particles of irregular shape is proposed. Such correlations are developed for a sphericity range between 0.7 and 0.95 and for intermediate Reynolds numbers in the range 1,200, common in industrial and environmental processes. The proposed expressions are derived by fitting the results obtained by means of Particle-Resolved Direct Numerical Simulations (PR-DNS) for a uniform flow around different sets of irregular particles and considering a large number of random orientations. The resulting flow resistance coefficients can be approximated by normal distributions whose first and second order statistical moments (i.e., mean and standard deviation) are described in this work by fitting functions depending of particle Reynolds number and sphericity. The derived correlations are simple, can be easily implemented and allow the tracking of irregular particles in a Lagrangian stochastic framework.

Wall effect on cluster particle's settling terminal velocity and drag coefficient in Newtonian and non-Newtonian fluid medium

The experimental investigation of the wall effect on the cluster particle settling in the Newtonian and non-Newtonian fluid medium is carried out in three different diameter flow channels. The cluster usually forms during the sedimentation of particles in a fluid medium, so it becomes necessary to study the behavior of the cluster particles. Different cluster particles are considered depending on the number of spheres (N) attached to the cluster and the cluster's shape. The present experiment covers the following range of conditions: 0.05 ≤ deq/D ≤ 0.24, 2 ≤ N ≤ 7, 0.64 ≤ n ≤ 1, and 0.14 ≤ K ≤ 1.81. The results reveal that the terminal velocity varies with the blockage ratio (deq/D), N, and cluster shape. For a particular deq/D ratio and the same N, the terminal velocity of polyhedron particles is high compared to the planar and chain shape particles. The blockage ratio and the Reynolds number affect the wall factor of the cluster particles. However, from the experiment, it is observed that the wall effect also depends on the orientation of the particles. The impact of the wall on the cluster particle is high in Newtonian fluid when compared to cluster particles in non-Newtonian fluid. The present work additionally investigates the influence of drag on cluster particles in the presence and absence of the wall effect. The numerical relationships are developed to predict experimental results in Newtonian and non-Newtonian fluid mediums.

Simulation and verification of particle flow with an elastic collision by the immersed edge-based smoothed finite element method

The immersed edge-based smoothed finite element method (IES-FEM) is proposed for the study of elastic collision particulate flow. Particle collision becomes more realistic by using the penalty function and the hyperelastic constitutive model. The effects of grid resolution and Reynolds numbers on particle terminal velocity and drag coefficient are discussed to verify the calculation accuracy and stability. Single-particle collisions with the bottom and side walls are analyzed and experimentally verified. Results show that the calculation error of IES-FEM is less than 0.6% when the fluid grid size is 0.5 times the particle mesh size and the time step is 10–4 s. Particle drag coefficient and flow characteristics agree well with the published models and experiment results. To demonstrate the capabilities of IES-FEM in complex elastic particle systems, the collision and rebound of multiple particles are determined, including the drafting–kissing–tumbling of two circular particles; the chase, collision, and deformation of rectangular particles; and the repeated formation and separation of particle clusters. This work extends the application of IES-FEM in particle-resolved direct numerical simulation methods, which will provide an optional tool for future elastic blood cell flow and collision.

Density-dependent drag coefficient for gas-adsorbed particles in free-molecule flows

Gas adsorption by the spherical particles in gas–particle flows has been recently studied by Yu et al. [“Direct simulation Monte Carlo of the gas adsorption of particles in gas–particle flows,” Phys. Fluids 34, 083302 (2022)]. However, the gas-adsorption distribution on the particle surface has heretofore remained unknown. This paper addresses this knowledge gap by introducing a numerical method to calculate the gas-adsorption distribution for ellipsoidal particles in gas–particle flows. We split the particle surface into internal flat plates and calculate the gas adsorption for each internal flat plate in the gas flow. Based on this numerical method, the gas adsorption distribution for the ellipsoidal particles is reconstructed by using the direct simulation Monte Carlo method. The results show that the average adsorption by prolate particles with particle eccentricity reverses as the particle temperature increases. Moreover, we show that the density-dependent drag coefficient for particle motion in the free-molecule flow may evince gas adsorption at the particle surface. Those points could inspire the studies of dust physics in rarefied gas spaces.

Impacts of Particle Shape, Skeletal Porosity and Density on the Settling Velocity of Gravel‐Size Coral Debris

AbstractKnowledge of the hydrodynamic behavior of carbonate bioclasts is essential for solving coastal morphodynamic problems in marine carbonate depositional environments. Marine carbonate grains are produced within the sedimentary system itself and are deformed by subsequent hydrodynamics, resulting in particle densities, sizes and shapes that are distinct from those of terrigenous sediments. In this study, we investigated the shapes and settling characteristics of gravel‐size natural coral debris collected from the coasts of Okinawa Prefecture, Japan. Based on the shape parameters and settling velocities obtained using image analysis techniques and settling tube experiments, respectively, a correlation model for predicting the settling velocity of coral gravel particles was developed by extending the previous formula of Dietrich (1982, https://doi.org/10.1029/wr018i006p01615) over particle Reynolds number values ranging from 384.23 to 4439.76. The model determines the settling velocity as a function of particle nominal diameter, form, roundness, and surface texture parameters that were selected after examining several other shape descriptors. Compared with other models, the proposed model showed better settling velocity predictive skills for coral gravel particles classified into four shape types. The proposed model also determines how the drag coefficient of coral gravel particles should be modified from that for spherical particles, by accounting for the abovementioned coral gravel particle shape parameters. The obtained data set of settling velocity and modification of the drag coefficient can be used for quantitative interpretation of suspension, transport, and sedimentation processes in marine carbonate depositional environments.

Experimental estimation of the settling velocity and drag coefficient of the hollow cylindrical particles settling in non‐Newtonian fluids in an annular channel

AbstractThe drag coefficient data of particles settling in an annular channel is very much essential for designing different solid–fluid handling equipment, such as the fluidized bed. Experimental settling velocity, wall factor, and drag coefficient data of the hollow‐cylinder particle are presented. Carboxymethyl cellulose solution has been used as the working fluid with a flow index of 0.64 ≤ n ≤ 0.91 and a consistency index of 0.31 ≤ K ≤ 1.81. The experimental results covered a wide diameter ratio range (0.14 ≤ deq/L ≤ 0.46), hollow cylinder inner to outer diameter ratio (0.2 ≤ di/do ≤0.8), and Reynolds number (0.05 ≤ Re ≤ 51 and 0.09 ≤ Re∞ ≤ 55). deq, di, and do are the equivalent inner and outer diameters of the particle, L is the annular gap, and Re and Re∞ are the Reynolds numbers in the presence and absence of the wall effect, respectively. The wall factor decreased, and the drag coefficient increased with deq/L and di/do ratios. The above parameters declined with the Reynolds number. The hollow cylinder experienced a lesser wall effect than the spherical particles settling in a non‐annular channel. In some cases, the wall factor of the hollow cylinder is found to be equal to the spherical particles settling in an annular channel. The developed correlations have successfully predicted the drag coefficients of the hollow cylinder.

Review—Drag Coefficients of Non-Spherical and Irregularly Shaped Particles

AbstractThe knowledge of simple and relatively accurate closure equations for the drag coefficients of nonspherical particles is very important for Eulerian multiphase numerical codes that require such inputs for the solution of the momentum equation. However, the existing correlations for the drag coefficients are laden with high uncertainty, which propagates in the numerical results. This review paper examines critically the state of our knowledge of the drag coefficients of nonspherical particles starting with our understanding of the “size” and “shape” of particles. Lengthscales and other parameters that have been used to characterize sizes of nonspherical particles are presented. Shape factors and shape descriptors that have been used in past correlations are also presented together with an appraisal of their applicability. Several correlations that have been developed for the determination of the drag coefficients of nonspherical particles with regular and irregular shapes are also presented together with available information on their accuracy and applicability. Based on the review of the available correlations general recommendations are offered on the use of the correlations as closure equations and inputs in computatioal fluid dynamics (CFD) codes.

Special Issue on the 2022 Fluids Engineering Division Summer Meeting

The ASME Journal of Fluids Engineering (JFE) and the ASME Fluids Engineering Division (FED) present a Special Issue of papers that were presented at the 2022 Fluids Engineering Division Summer Meeting (FEDSM). The FED conference organizers were Professor Philipp Epple (chair) and Professor Kamran Siddiqui (co-chair), whose efforts helped the community assemble in the first in-person FED meeting since the pandemic. The conference was successful due to the hard work and dedication of the conference chairs, technical committee chairs, symposia organizers and other FED volunteers. The 2022 FEDSM was held August 3–5, 2022 at the Intercontinental Toronto Center, Toronto, ON, Canada and included three plenary speakers plus technical presentations and 145 attendees.The Special Issue is pleased to include the review paper by Professor Efstathios (Stathis) Michaelides (Texas Christian University) and Professor Zhigang Feng (University of Texas at San Antonio) titled “Drag Coefficients of Non-spherical and Irregularly-Shaped Particles.” The other paper topics include cardiovascular flow in the aorta, boundary layer response to plasma actuators, thermally stratified turbulent boundary layer flows, flapping foils, and unsteady propulsion.The Special Issue also presents the winners of the Flow Visualization competition, organized by Prof. Philipp Epple. The competition is an initiative that begin in 2017 and was expanded to include both still images and videos. The 2022 FEDSM winners in the video category were:In closing, my sincerest gratitude is extended to JFE Associate Editors, Professor Costanza Aricò, Professor Arindam Banerjee, Professor Luigi P. M. Colombo, Professor Philipp Epple, Professor Pierre Sullivan, and Professor Qianhong Wu, who shared in the responsibilities. I would also like to thank the JFE Editorial Assistant, Ms. Colette Montague, who helped ensure timely processing, and the support of the ASME staff, especially Ms. Beth Darchi, Ms. Erica Hodge, Ms. Tamiko Fung, and Ms. Jenna Seyer. Finally, thank you to our colleagues who served as reviewers and of course the authors.

High fidelity adaptive Cartesian octree grid computations of the flow past a Platonic polyhedron up to a Reynolds number of 200

We use a high fidelity non-boundary fitted Distributed Lagrange Multiplier/Fictitious Domain method implemented on an adaptive Cartesian octree grid to investigate the unbounded flow of a Newtonian fluid past a stationary Platonic polyhedron. We consider five Platonic polyhedrons and a sphere as an asymptotic polyhedron featuring an infinite number of faces. We explore the impact of angularity on the drag and lift coefficients for a range of Reynolds numbers Re∈[1,200] over which the flow is steady or only mildly unsteady. In order to reduce the parameter space, we consider three specific angular positions of the polyhedron: face facing the flow, edge facing the flow and vertex facing the flow. The effects of these angular positions, notably on drag and lift coefficients, are discussed. For low Reynolds numbers, the particle crosswise cross-sectional surface area has a prominent influence on the drag coefficient while for higher Reynolds numbers the impact of the particle angular position becomes more significant. We also observe that the change in the symmetry of the wake region and the corresponding regime transitions are strongly related to the lift coefficient evolution with Re. Finally, we evaluate the accuracy of various non-spherical particle drag coefficient correlations available in the literature and provide new insights into the understanding of unbounded laminar flows past a non-spherical angular particle.

Experimental Assessment of Drag Coefficient for Quasi-Radially-Symmetric Microplastic Particles Sinking in Water Stream

Microplastics (MPs) are found to be omnipresent in marine environments and their production and disposal are constantly increasing. In order to control and mitigate their negative effects on the environment, it is important to understand the processes involved in their transport and sinking. Most of the conducted experimental studies on MP in water have focused on exploring the settling behavior of MP particles under static flow conditions. This paper presents the results of an experimental study of MP particles in a horizontal stream of water. A series of laboratory experiments were carried out to measure the drag coefficient of quasi-radially-symmetric MP particles with different shapes (spheres, cubes, and crosses) and sizes (1.5 to 3.0 mm). In the study, the relationship between the drag coefficient and flow conditions represented by the local Reynolds number is investigated. The experimental procedure was validated, and a total of 400 measurements were performed. Values of the measured drag coefficient ranged from 0.8 to 2.5 for the values of Re from 40 to 300. The results show that the shape of MP particles in the water stream considerably influences the particle drag coefficient, with a rather complex relationship between the drag coefficient and flow conditions.

A Review of Drag Coefficient Models in Gas‐Liquid Two‐Phase Flow

AbstractAlthough great efforts have been made in modeling the drag force in gas‐liquid two‐phase flow, the research about that in both theory and method is at the stage of development due to its diversity and complexity, and previous summary work about drag is usually based on a certain aspect. In this review, a unified framework that guides the drag coefficient model in gas‐liquid two‐phase flow is provided. From simple to complex, this paper introduces the drag coefficient of solid particles, single bubbles, bubble swarms, and bubbles in turbulence moving in liquid in turn. The variables affecting the drag coefficient in each of the above situations and the relationship between the drag coefficient and these variables are mentioned.

Direct Numerical Simulation of Fluid Flow and Heat Transfer of a Reactive Particle Layer with Stefan Flow in Supercritical Water

Supercritical water gasification is an efficient and clean way of energy conversion. The research on different scales, such as the system, reactor, and particle, has different temporal and spatial significance. A study on particle–particle and particle–fluid–particle interaction on the particle scale has a fundamental guiding value for revealing gasification performance on the reactor scale. Reactive particles such as coal are pyrolyzed and gasified in a high-temperature and high-pressure reactor to form Stefan flow, which affects the mass, momentum, and energy transfer between particles and supercritical water. In this paper, a particle-resolved direct numerical simulation study of a reactive particle layer in supercritical water is carried out to investigate the effect of different particle layer solid holdups and Stefan flow intensities and distributions on the flow and heat transfer process between the particle layer and supercritical water. This work analyzes the pressure and friction drag coefficients to which the particles are subjected and specifies the flow, velocity, and temperature distribution inside and around the particle layer. The results show that the drag coefficient and Nusselt number of particles in the particle layer decrease gradually along the flow direction, and the presence of particle Stefan flow further reduces the drag force and Nusselt number of particles. With the increasing solid holdup of the particle layer, the particle–fluid–particle interaction becomes more intense, and the effect of Stefan flow cannot be negligible.