Drag Force Model Research Articles

Computational Fluid Dynamics–Discrete Element Method Numerical Investigation of Binary Particle Mixing in Gas–Solid Fluidized Bed with Different Drag Models

The fluidized bed is a critical reactor in the energy and chemical industries, where the mixing and agglomeration behaviors of binary particles significantly influence both the efficiency of reaction processes and the uniformity of final products. However, the selection of appropriate drag force models remains a subject of debate due to the variability in particle properties and operating conditions. In this study, we investigated the fluidization behavior of binary mixtures composed of two different sizes of Geldart-D particles within a fluidized bed, evaluating nine distinct drag force models, including Wen and Yu; Schiller and Naumann; Ergun; Gidaspow, Bezburuah, and Ding; Huilin and Gidaspow; De Felice; Syamlal and O’Brien; and Hill, Koch, and Ladd. We focused on four key parameters: particle mixing degree, migration characteristics, temperature variation, and mean pressure drop. Simulation results revealed that the choice of drag model markedly affected mixing behavior, migration dynamics, and temperature distribution. Notably, the Ergun; Gidaspow, Bezburuah, and Ding; and Hill, Koch, and Ladd models exhibited superior particle mixing uniformity. While the drag model had a relatively minor impact on particle temperature changes, its selection became critical in simulations requiring high-temperature precision. Regarding pressure drop, the Huilin and Gidaspow and Gidaspow, Bezburuah, and Ding models demonstrated smaller and more stable pressure drop fluctuations. These findings offer valuable theoretical insights into gas–solid two-phase flow under binary particle mixing and provide practical guidance for the design and operation of fluidized bed reactors.

Development of a XGBoost-based drag force model for freely evolving particle suspensions

An XGBoost-based drag model is developed using data from Particle Resolved Simulations (PRS) of freely evolving spherical particle suspensions, encompassing Reynolds numbers from 10 to 300, solid volume fraction between 0.1 and 0.4, and particle-to-fluid density ratio of 2, 10 and 100. Drag force data from 150 continuous time instances in PRS are divided into two sets: the first set that includes data from the initial 120 instances is used for training the model and interpolation testing, while the second set comprises drag forces from the final 30 instances is used exclusively for extrapolation testing. Both interpolation and extrapolation tests demonstrate significantly improved accuracy compared to traditional drag correlations. Notably, the model achieves its highest prediction accuracy for particles with density ratios of 100, which is attributed to the increased influence of unsteady drag forces at lower density ratios that cannot be fully captured by instantaneous particle distributions alone.

Numerical Investigation into Gas-Solid Fluidized Bed Reactors for Activating the Iron-based Fischer-Tropsch Synthesis Catalyst

Abstract The present work conducted experimental and computational fluid dynamics simulation studies on the hydrodynamic characteristics of gas-solid fluidized bed reactors for activating the iron-based Fischer-Tropsch synthesis catalyst. In order to provide reliable guidance for the design and scale-up of the reactor, the present study applied experimental data from a pilot fluidized bed device to verify the accuracy of the drag force model used in the two-fluid model and further used the validated drag force model to simulate an industrial-scale fluidized bed activation reactor. It was found that the bubbling-EMMS drag model can accurately simulate the pilot-scale fluidized bed in the flow regimes ranging from bubbling to turbulent fluidization. Simulation studies on the industrial-scale fluidized bed activation reactor showed that the reactor was in a turbulent fluidization regime under the designed operating gas velocity, with high gas-solid dispersion efficiency, reasonable utilization of reactor space, and low gas-solid separation load.

Modeling the impact of terrain surface deformation on drag force using discrete element method and empirical formulation

Understanding motion-induced terrain deformation is essential for improving predictive models in geotechnical engineering, infrastructure development, and robotics. The existing analysis assumes a uniform and unchanged environment and lacks a description of the terrain deformation induced by motion, which is particularly significant when the grains are loosely packed. To address this gap, we performed numerical investigations to mathematically model the changes in the surrounding environment and their corresponding effects on resistance. Specifically, we examined the response of granular materials and the energy variation of the particles during motion. Increased energy in the direction of motion was categorized as influences above or below the medium surface. Aligned with the influencing factors, two variables were considered to quantify the impact on the drag force. Building upon the energy analysis and the framework of Resistive Force Theory, we proposed a drag force model for buried motion in loose granular terrain. Compared with the experimental data, the average predictive error decreased from 17.8% to less than 2.7%. Our study provides a feasible approach for incorporating the effects of terrain deformation into a drag force model, thereby enhancing the accuracy and contributing to the development of granular locomotion.

Computation of SWCNT/MWCNT-doped thermo-magnetic nano-blood boundary layer flow with non-Darcy, chemical reaction, viscous heating and Joule dissipation effects

CNTs have been shown to exhibit exceptional electrical and thermal conductivities, chemical and mechanical stability and physiochemical reliability in wide-ranging applications covering biomedicine, energy and aerospace. Motivated by emerging applications in nano-drug delivery in medicine and radiative ablation therapy, a steady-state mathematical model is established for magnetohydrodynamic boundary-layer transport of chemically reactive carbon nanotubes (CNTs)-doped electrically conducting blood along a porous stretching wall of a blood vessel. Multiple and single walls CNTs are considered using the model developed by Xue (2005). Heat generation, radiative flux, wall mass (concentration) slip, viscous heating and Joule magnetic dissipation are incorporated. Chemical reaction effects are addressed utilizing homogenous first-order model. The Darcy-Forchheimer drag force model is utilized for bulk matrix and inertial drag effects in the porous medium. Radiative heat-transport is studied considering Rosseland's diffusion model. The governing partial differential conservation equations for mass, momentum, energy and species are formulated in a two-dimensional Cartesian coordinate system with allied wall and free-stream boundary conditions. Via scaling transformations, a nonlinear boundary value problem is derived. Numerical computations are achieved through bvp4c scheme. The impact of selected parameters on dimensionless quantities (velocity, temperature, concentration, skin friction, local Nusselt and Sherwood numbers) are computed and elaborated through tables and graphs. A good correlation of the numerical solutions with earlier simpler models from the literature is included. The simulations show that with augmentation in Hartmann magnetic number and Forchheimer inertial drag number, significant damping in the nano-doped blood flow is produced for both SWCNTs and MWCNTs. Temperature is elevated with increment in dissipation effect i. e. Eckert number. Higher values of mass (solutal) slip parameter induce a depletion in the concentration. Nusselt number i. e. heat transfer rate to the blood vessel wall is improved with greater values of solid volume fraction for both CNTs. The present model is relevant to radiative ablation therapy combined with nano-medical treatments in blood vessels wherein constant mechanical loading via blood pressure and flow can elevate internal stresses (circumferential wall stress and endothelial shear stress) and induce morphological alterations in blood vessel wall (stretching) and endothelium in conjunction with biochemical reactions.

Dynamic evaporation characteristics of liquefied natural gas droplets

To delve into the intricate evaporation and dispersion mechanisms of dense droplets formed in the vicinity of liquefied natural gas (LNG) accidental releases, it is imperative to first examine the evaporation dynamics of individual moving LNG droplets. This paper presents a visual experimental setup designed to scrutinize the temporal evolution of diameter and displacement of single free-falling LNG droplets. Additionally, eight typical drag force models used for droplet motion state calculations were assessed. The optimal drag force models were selected to accurately predict the displacement of LNG droplets in the wide range of 100 < Re < 10 000. Moreover, eight typical gas phase models applied to predict heat and mass transfer were evaluated, revealing that none accurately capture the dynamic evaporation of free-falling LNG droplets. Subsequently, a new gas phase model suitable for predicting LNG droplet evaporation behavior is proposed. Furthermore, the periodic oscillation behavior of LNG droplet shape during the falling process is uncovered. The oscillation amplitude and dominant frequency of droplets are quantitatively investigated using the aspect ratio of droplets. Finally, an in-house program is developed to comprehensively analyze the evaporation characteristics of LNG droplets under different initial droplet diameters, velocities, and ambient temperatures. Based on gray relational analysis, the relative importance of three impacting factors on the evaporation coefficient is ranked.

Numerical Simulation Study of Gas-Liquid Two-Phase Flow in a Pressurized Leaching Stirred Tank

The gas-liquid flow and oxygen content in a pressurized leaching stirred tank significantly influence the chemical reaction rates, while the specific dynamics of gas-liquid flow in the sulfuric acid system remain largely unexplored. In this study, a mathematical model of gas-liquid flow within a stirred tank is developed using the Euler-Euler approach, with the turbulence and drag force models being validated against experimental data. Utilizing this validated and reliable model, this study investigates the impacts of the sulfuric acid concentration, baffles, air inlet velocity, and bubble diameter on the flow field and gas holdup in a two-phase system consisting of a sulfuric acid solution and oxygen. The findings indicate that introducing a specific concentration of sulfuric acid decreases the solution velocity and increases the gas holdup within the tank. However, once the sulfuric acid concentration reaches a certain threshold, further increases have a diminished effect on the gas-liquid phases. The installation of baffles enhances the turbulent kinetic energy and increases the gas holdup while only resulting in a minimal 1.2% increase in power consumption. Additionally, the inlet velocity and bubble diameter have a relatively minor impact on the tank’s flow field. However, increasing the inlet velocity significantly boosts the gas holdup, whereas an increase in the bubble diameter marginally reduces it. Furthermore, introducing a sulfuric acid solution into the tank can enhance the gas holdup when the gas inlet velocity is low. Conversely, when the gas inlet velocity is high, the addition of sulfuric acid results in a decrease in the gas holdup. The conclusions from this study contribute to enhancing the mixing effectiveness and oxygen content within the tank, providing a substantial theoretical basis for optimizing the design and operating conditions of pressurized leaching stirred tanks.

Neural-network-based drag force model for polydisperse assemblies of irregular-shaped particles

In this study, a drag force model of irregular-shaped particles in gas-solid flows is investigated using neural network approaches. Pre-developed variational autoencoder, multi-layer perceptron model, and transpose convolutional neural-network model are utilized to obtain intermediate parameters of a pair-wise interaction point-particle (PIEP) model. These parameters are utilized to predict individual drag force coefficients of polydisperse, particulate flows. To apply the PIEP-based model to medium-concentration flows, the average drag force coefficients are obtained through an in-house particle-resolved direct numerical simulation, and a regressed model is developed to predict solid volume fraction factors. With the regression model, the PIEP-based model using the neural-network approaches shows good prediction in particulate systems from low to intermediate concentrations. This study provides computationally efficient models of practical, realistic particulate systems for large scale fluid dynamics simulation, in that it allows predicting the drag forces of polydisperse particles with a small amount of data.

A Drag Force Model of Vertical Penetration into a Granular Medium Based on DEM Simulations and Experiments

The force exerted on a cylindrical intruder as it penetrates a granular medium was analyzed utilizing both experiments and the discrete element method (DEM). In this work, a series of penetration experiments were performed, considering cylindrical intruders with different nose shapes. We found that the drag force of the intruder with a hemispherical nose is close to that of those with conical noses with apex angles of 53° and 90°. The drag force of the blunt-nosed intruder is bigger; the drag force of the conical-nosed intruder with an apex angle of 37° is the smallest. We studied the interplay between the drag force on an intruder with a hemispherical nose and key variables—the penetration velocity (V), penetrator’s diameter (di), and friction coefficient (μ). From this analysis, two piecewise functions were derived: one for the average drag force versus the penetration velocity, and the other for the scaled drag force versus the friction coefficient. Furthermore, the average drag force per contact point, Fa/P, can be succinctly represented by two linear relationships: Fa/P = 0.232μ + 0.015(N) for μ<0.9, and Fa/P = 0.225(N) for μ≥0.9.

Computational analysis of magnetized Casson liquid stretching flow adjacent to a porous medium with Joule heating, stratification, multiple slip and chemical reaction aspects

This article aims to investigate the characteristics of thermo-solutal magnetohydrodynamic (MHD) non-Newtonian smart coating boundary layer flow of a stretching substrate adjacent to a porous medium, considering the influence of chemical reactions and thermal radiation subject to a transverse static magnetic field. A non-Darcy drag force model is deployed to capture both Darcy bulk drag and inertial Forchheimer (quadratic) drag effects. A diffusion flux model is deployed for radiative heat transfer. The Casson viscoplastic model has been utilized to simulate rheological characteristics. Due to polymeric slip effects, three slip phenomena are included at the wall (hydrodynamic, thermal and concentration) in the formulation. Furthermore, viscous dissipation and Ohmic heating (Joule dissipation) are also included. Robust scaling similarity variables are deployed to transform the governing partial differential equations into ordinary differential equations. Subsequently, the emerging dimensionless coupled nonlinear boundary value problem is solved utilizing the Bvp4c method in MATLAB version 2022. This numerical approach allows for a logical parametric examination of all key control parameters on the transport phenomena, enabling a comprehensive understanding of the system behavior. Validation with previous studies is included. Detailed graphical and tabular computations are included for velocity, temperature, concentration, skin friction, Nusselt number and Sherwood number, for the influence of Darcian parameter, Forchheimer inertial parameter, mixed convection, velocity (momentum) slip, magnetic number, Casson parameter, nonlinear thermal convection parameter, nonlinear concentration convection parameter, radiation parameter, thermal stratification parameter, Prandtl number, heat source/sink parameter, Eckert number, thermal slip parameter, Schmidt number, chemical reaction, solutal stratification parameter and solutal slip parameter. Detailed interpretation of the physics associated with these multiple effects is included. The simulations provide further insight into the transport characteristics of electromagnetic viscoplastic coating material manufacturing.

Theoretical derivation and a priori validation of a new scalar variance-based sub-grid drag force model for simulation of gas–solid fluidized beds

A novel sub-grid drag force model is proposed for coarse-grid Euler–Euler simulation of gas–solid fluidized beds. Starting from a transport equation for the drift velocity, an equilibrium condition is used as a basis to derive a new algebraic drift velocity model. The sub-grid correlations that show up are closed by a large-eddy PDF approach inspired from LES of turbulent reacting flows. The new analytical model only depends on the resolved slip velocity and on a few sub-grid moments of the solid volume fraction. Then, a conditional averaging procedure shows that the new model can be properly captured by a simple functional expression that only requires a closure for the sub-grid variance of the solid volume fraction. A priori validation studies show that the drift velocity is predicted with high accuracy (R2>0.90) for a large range of filter widths and for both Geldart A and Geldart B particles.

Construction and Analysis of the Mesoscale Drag Force Model Based on Machine Learning Methods

Construction and Analysis of the Mesoscale Drag Force Model Based on Machine Learning Methods

A Filtering Approach to the Estimation of Model Parameters in the Vegetative Canopy Model

The drag-force model for canopy flows has difficulty in preparing appropriate drag coefficients CD to the canopy layer, even if its geometrical structure is quite simple. To overcome this problem, we employed a unique approach in our past studies where a microscopic fully-resolved simulation of the target flow on a fine grid (called preliminary analysis) is performed to extract a proper CD for the target canopy layer in advance of the macroscopic prediction on a coarse grid together with the drag-force model (main analysis). This approach can clarify what is essential or less essential information in the model parameters for a better prediction of canopy flows. In the present study, under the framework of this approach, we try to incorporate the dependence of the canopy density X on the spatial resolution of the main analysis into the drag force model. The influence becomes especially significant in sparse canopies. Here, we introduce two modifications to our past approach. First, we choose CDX instead of CD as the model parameter to be extracted from the preliminary analysis results. Second, in the model parameter extraction process, we apply spatial filtering to the data obtained from the preliminary analysis to remove high-wavenumber components that cannot be represented on the coarse grid. Application of the method has demonstrated an improved accuracy in the prediction of an open-channel flow with patched vegetation zones.

High speed impact on granular media: breakdown of conventional inertial drag models.

In this study, we extensively explore the impact process on granular media, particularly focusing on situations where the ratio of impact speed to acoustic speed is on the order of 0.01-1. This range significantly exceeds that considered in existing literature (0.0001-0.001). Our investigation involves a comprehensive comparison between our simulation data, obtained under high-speed conditions, and the established macroscopic drag models. In the high-speed regime, conventional drag force models prove inadequate, and the drag force cannot be separated into a depth-dependent static pressure and a depth-independent inertial drag, as suggested in previous literature. A detailed examination of the impact process in the high-speed limit is also presented, involving the spatio-temporal evolution of the force chain network, displacement field, and velocity field at the particle length scale. Unlike prior works demonstrating the exponential decay of pulses, we provide direct evidence of acoustic pulses propagating over long distances, reflecting from boundaries, and interfering with the original pulses. These acoustic pulses, in turn, induce large scale reorganization of the force chain network, and the granular medium continuously traverses different jammed states to support the impact load. Reorientation of the force chains leads to plastic dissipation and the eventual dissipation of the impact energy. Furthermore, we study the scaling of the early stage peak forces with the impact velocity and find that spatial dimensionality strongly influences the scaling.

Application of a subgrid-scale urban inundation model for a storm surge simulation: Case study of typhoon Haiyan

This study evaluated the individual drag force model (iDFM), a subgrid-scale model for simulating tsunami inundation in coastal urban areas. The iDFM was then applied to characterize Typhoon Haiyan-induced storm surge inundations in 2013. Using a post-typhoon survey dataset containing inundation depths at different survey points with inundation limits, the performance of iDFM was validated, and the results confirmed that iDFM accurately estimates the storm surge inundation depths but overestimates the limit of inundation. In addition, the results revealed that the iDFM flow velocity was a dominant factor, and the inclusion of buildings in the simulated affected the arrival of the leading edge of inundation. Furthermore, the reduction in flow velocity was sensitive to the subgrid-scale parameters, and the spatial distribution pattern of buildings located in the inundated area affected the relative difference in the maximum flow velocity with the regional roughness model. The inundation characteristics of the Typhoon Haiyan storm surge models were compared with a historical tsunami event. The inundation characteristics were similar and model differences were observed mainly in the fluid velocity and momentum flux rather than the inundation depth. However, the magnitudes of relative differences observed in Tacloban were larger for fluid velocity and momentum flux, respectively, than in the tsunami event. The effect of building drag on advection was also substantial in Tacloban due to the differences in the inflow mass and momentum fluxes.

A corrected filtered drag model considering the effect of the wall boundary in gas-particle fluidized bed

The databases used to construct traditional mesoscale drag models is usually obtained from fine-grid simulation results in the full-periodic domains, ignoring the function of the wall boundary. In actual fluidized bed reactors, the presence of wall boundary can significantly affect the heterogeneous structures in near-wall areas, thereby affecting the drag force. Therefore, this work aims to further enhance the performance of the prediction results of existing filtered drag model by introducing the influence of the wall boundary. Based on the mesoscale drag force model published in previous literature, a correction coefficient considering the influence of the distance from the grid to the wall boundary is proposed. The applicability of the new corrected model is further enhanced by introducing the influence of solid volume fractions and slip velocities. Simple posteriori validations are systematically conducted in various common fluidized beds including the bubbling bed, the turbulent bed, and the fast bed to assess the predictive capability of the newly proposed correction drag model.

Coarse-graining dense discrete phase model for modeling particle dynamics in a 3D tapered fluidized bed coater: Analysis of different drag models

Fluidized bed coating (FBC) is widely employed in the food industry to microencapsulate food powder ingredients, primarily using a top-spray design in a tapered fluidized bed (TFB). A comprehensive understanding of particle dynamics is crucial for successfully designing FBC processes. This study focuses on developing and analyzing the coarse-graining dense discrete phase model (CG-DDPM) to investigate particle fluid dynamics in tapered fluidized bed coater (TFBC). Following the successful development of the CG-DDPM, we conducted a sensitivity analysis of drag force models due to the unique operating conditions involved. Five drag models were investigated, including four classic homogeneous drag models (Gidaspow, Wen-Yu, Syamlal-O'Brien, and Huilin-Gidaspow) and one advanced energy-minimization-multiscale bubbling (EMMS-B) heterogeneous model. While all drag models exhibited reasonable qualitative behavior, the EMMS-B model demonstrated superior quantitative performance, particularly at a gas velocity (Ug) of 1.75 m/s. However, at Ug = 1 m/s, none of the drag models perfectly resolved the axial and lateral solids concentration profiles. This study's findings contribute to identifying the best-performing drag model and highlight the need for more appropriate drag models to represent particle fluid dynamics in TFBC accurately. Moreover, employing the EMMS-B model in the top-spray design of TFBC can enhance the microencapsulation of food powder ingredients due to its superior particle dynamics predictions compared to homogeneous drag models.

Numerical investigation of mixing process of high-pressure jet-cutting clay by water–air coaxial nozzle considering soil rheological properties

Simulation and accurate modeling of the mixing process of the high-pressure jet-cutting clay by the water–air coaxial nozzle is significantly important for the performance optimization of the triple fluid jet grouting. In this paper, a numerical model considering the soil rheological properties is proposed to investigate the mixing process of the high-pressure jet-cutting clay. The cohesive force model of clay is obtained based on the solution of the power law index and consistency factor by coupling the Herschel-Bulkley and soil logarithmic models. The interaction model among the gas phase, the liquid phase, and the clay medium is further established through use of the drag force model. A laboratory device of high-pressure jet-cutting transparent clay is developed to prove the feasibility of the proposed model for the mixing process of the high-pressure jet-cutting clay. Finally, using the validated numerical model, the mixing process of the high-pressure jet-cutting clay by the water–air coaxial nozzle with varying radial spacings between the air nozzle and water nozzle is numerically investigated, and the axial stability of the jet, the width of the cross-sectional profile, and the variation of the central axis velocity field of the mixing process are analyzed. Results demonstrated that the variation trend of the jet in both simulations and experiments is consistent, and the maximum error in jet depth is better than 3.3%, validating the accuracy of the numerical model of the mixing process of the high-pressure jet-cutting clay. The optimal radial spacing size for a water–air coaxial nozzle in high-pressure jetting of clay medium is 1.4 mm, which provides the best axial stability, the narrower jet cross-section, and the slowest decay of jet velocity along the central axis.

Iterative screening methodology for optimal modeling of bubble coalescence/breakup and interphase force in CMFD simulation of flow boiling

The Euler–Euler two fluid approach has been widely adopted in computational multi-fluid dynamic (CMFD) simulation of nucleate boiling flow. Accurate prediction of bubble size and interphase forces are desired to properly model interphase momentum transfer in the CMFD simulation. Bubble dynamics, including nucleation, coalescence/breakup and condensation, should be modeled to calculate bubble size variation. In the meantime, modeling of interphase forces is crucial to predict distribution of void fraction. Numerous closure models have been developed for bubble dynamics and interphase forces. Diverse model combinations were reported depending on the specific application. As a step forward to converge the diversity, efforts are dedicated to establish a methodology to iteratively screen the closure models for nucleate flow boiling at elevated pressures based on the DEBORA experiment. In the proposed methodology, the closure models for each physics are first evaluated and sorted based on the predicted value in fixed-point analysis and its effect on CMFD results observed in separate-effect analysis. The second part of the methodology is iterative screening. To achieve accurate bubble size prediction, the optimal coalescence model is identified for each breakup model. With the optimal coalescence counterpart, the optimal breakup model is then identified. For the interphase force models, the optimal lift force model is first determined based on the CMFD simulation with the same drag force model and without wall lubrication force model. The optimal wall lubrication force and drag force model are selected in sequence. Utilized one single case of DEBORA experiments, the optimal set of the coalescence/breakup models, i.e., Prince et al.’s and Luo et al.’s models, and the interphase models, i.e., Tomiyama et al.’s models for drag and lift force, Lubchenko et al.’s model for wall lubrication force and Burns et al.’s model for turbulence dispersion force has been selected. Validation of the model combination based on more DEBORA experiment datasets and Bartolomei et al.’s experiment shows promising prediction accuracy.

Unsteady flow of a ternary nanofluid over a slow-rotating disk subject to uniform suction and backpropagated neural network

The purpose of the proposed research work is to explore the heat source or sink impact on the unsteady three-dimensional flow of ternary-hybrid nanofluid through a rotating disk. The magnetohydrodynamic flow of ternary-hybrid nanofluid under the impact of radiative heat transfer and uniform suction is also discussed in this study. The partial differential equations of the flow problem are reduced into ordinary differential equations by employing apt similarity transformation and solved numerically using the Runge–Kutta Fehlberg fourth–fifth order method. The various nondimensional parameters’ effects on velocity and thermal profiles are illustrated using graphs. In addition, a Levenberg Marquardt backpropagated neural network is employed for determining the Nusselt number and skin friction model. The outcomes of the developed Levenberg Marquardt backpropagated neural network models are indicated through the performance metrics. Result reveals that a rise in the suction parameter decreases the velocity profiles. The thermal profile increases with higher values of thermal radiation and heat source/sink parameters. In addition, the presented Levenberg Marquardt backpropagated neural network models’ scheme is found to be a perfect tool for estimating heat transfer and surface drag force models.