High Particle Volume Fractions Research Articles

Investigating the Impact of Flow Profile on Heat Transfer in Nanofluid Flow: A Numerical Study

Assessing the profitability of an energy system requires careful consideration of various factors, including fluid characteristics, geometry shape, and operating conditions. This study investigates the influence of sinusoidal rib shapes, with different space ratios (e/b) ranging from 0 to 1, on heat transfer in nanofluid flow. The channel's upper surface is subjected to a uniform heat flux, employing Al2O3 nanofluid as the working fluid and varying Reynolds numbers from 5000 to 20000. Additionally, the effect of aluminum nanoparticle volume fraction, ranging from 0 to 6%, is analyzed. Simulation results indicate that the performance of the corrugated surface in the channel is significantly influenced by rib shapes and their geometrical parameters. The highest Performance Evaluation Criteria (PEC) index is achieved for ribs with a space ratio (e/b) of 0 at Reynolds number of 5000 and a volume fraction of 6% nanoparticles. Furthermore, the average Nusselt number shows an increasing trend with higher particle volume fraction and Reynolds numbers.

Experimental study of a dense stream of particles impacting on an inclined surface

We present measurements of the impact process of gravity-driven flows of particles against inclined surfaces for a range of particle loadings in the four-way coupled regime. These were investigated with particle image velocimetry (PIV) for a range of particle curtain thicknesses, volume fractions, velocities and mass flow rates using a series of hoppers of various aperture sizes, following impact on a steel plate at a range of angles from 15° to 75°. In addition, micro-shadowgraphy was used to provide new details of the region of impact, providing the first direct evidence of the formation of layer of particles that slide along the surface for sufficiently high particle volume fractions. Statistical measurements of particle impact and rebound velocities, together with rebound angle and coefficient of restitution, were obtained from the PIV analysis, which revealed that an increase in particle loading leads to increase in the probability that a layer of particles at the surface is formed at the point of impact. This layer acts to absorb energy for all impact angles, thereby influencing the rebound processes. For impact angles close to the normal direction the layer also increases the range of rebound angles, while it causes particles to slide along the surface as a chute-like flow for large inclination angles.

A Novel Shrink–Expand–Shrink Method for Modeling Composites with Ultrahigh Volume Fractions of Pre-Graded and Gradient-Distributed Particles

An advanced shrink–expand–shrink method is proposed in this paper for efficiently modeling concrete-like particle-reinforced composites with ultrahigh volume fraction of aggregates. The gradation of the aggregates can be pre-given and the aggregate spatial distribution can be nonuniform. By this method, the shrunk aggregates are first generated in the model space, and then expanded to jostle each other, afterwards they are shrunk again to normal size to obtain the final mesostructure models. Any high volume fractions of particles even more than 90% can be easily achieved by adjusting the shrinkage level during this process. Besides, a gradient distribution algorithm is established to conform to the aggregate segregation during the actual pouring process, and the corresponding periodic boundaries can also be generated to quickly create large specimens. Finally, the compression processes of polymer concrete with different aggregate packing densities are calculated via a finite element method. The shrink–expand–shrink method and the corresponding numerical models have memorable importance in the performance analysis and material design of particle-reinforced composites.

Dry sliding wear characteristics of in-situ hybrid (Al3Ni + Al2O3)–Al composites

Dry sliding wear characteristics of in-situ hybrid (Al3Ni + Al2O3)/Al composites synthesised by solid-state combustion of Al– xNiO ( x = 8 wt%, 10 wt%, 12 wt% and 15 wt%) system were investigated. The results indicate that the in-situ hybrid Al3Ni + Al2O3 particles can markedly improve wear performance of composites. The increase of wear resistance with an increase in NiO content at room temperature can be attributed to high hardness offered by high volume fraction of hard Al3Ni and Al2O3 particles. While, at 200 °C, the increase in wear resistance of composites with higher NiO contents can be attributed to the high thermal stability of composites. With the increase of NiO content from 8% to 15%, the wear mode of composites also is discussed at room temperature and 200 °C.

Damage behavior of high particle volume fraction composites with initial damage by finite element meso-modeling

Analyzing the deformation and damage processes of high particle volume fraction composites (HPVFCs) based on the meso-structure modeling plays an important role in their service lifetime and functional reliability. The reported mesoscopic structural models are usually ideal models without initial damage and apply to composites with the volume fraction of particles less than 80%. This work provides a new mesoscopic structure model called aggregate settlement-merging model to successfully simulate the mechanical behavior and damage mechanism of the polymer-bonded explosive (PBX) under quasi-static loads, and the macroscopic response is accurately predicted by the cohesive finite element method. This simulation commendably predicts the failure modes including interface debonding, binder filaments, and grain failure, which is consistent with the observation in micromechanical experiments. The multi-scale relationship between macroscopic mechanical properties and microscopic initial interface damage is quantitatively established in a form of an exponential function. This study shows a relationship between meso-structures and performance and provides strategy for designing next generation high-performance HPVFCs.

Effect analysis on the flow uniformity and pressure drop characteristics of the rotary diesel particulate filter for heavy-duty truck

In this work, a three-dimensional simulation model and mathematical models of the rotary diesel particulate filters (R-DPF) for heavy-duty truck was established to investigate its velocity and total pressure distribution characteristics. The effect mechanism of the particle load, particle volume fraction, inlet velocity, porosity and micropore diameter on the flow uniformity and pressure drop characteristics of the R-DPF under different conditions were obtained through the simulation. The results indicate that the change in the airflow direction and the osmotic resistance from the filter material are the main reasons for the flow uniformity and pressure drop characteristics of the R-DPF for heavy-duty truck. The R-DPF with higher particle load or inlet velocity has better flow uniformity. Whereas the R-DPF with higher particle volume fraction has worse flow uniformity. The increase in the porosity and micropore diameter make the flow uniformity of the R-DPF show a trend of decreasing first and then increasing. Besides, the increase of the particle load, particle volume fraction and inlet velocity will enlarge the total pressure drop of the R-DPF, while the increase of the porosity and micropore diameter will lower the total pressure drop of the R-DPF.

Multiphase simulation of sustainable nanoenhanced ionic liquid coolants for improved thermal performance in Ti–6Al–4V alloy drilling

Extensive research has been conducted by the manufacturing industry to enhance the efficiency of drilling processes by focusing on the utilization of nanoenhanced cutting fluids that possess excellent heat conductivity. Due to their eco-friendliness and adaptability of physical and chemical properties, ionic fluids offer enormous potential for application as cutting fluids. This study investigates the computational fluid dynamics analysis of the heat transfer performance of various ionanofluid pairs dispersed with nanoparticles as cutting fluids in the drilling process using Ansys Fluent software. For this purpose, 1-Hexyl-3-methyl-imidazolium-tetrafluoroborate is considered the ionic fluid, and its thermal behavior is examined by dispersing it with nanoparticles of copper, silver, and multiwalled carbon nanotubes (MWCNT) at different particle volume fractions and Reynolds numbers. The workpiece is composed of an alloy of titanium Ti–6Al–4V, while the drill bit is made of tungsten carbide-cobalt. It is observed that the ionic nanocoolant mist emanates from the spray tip and moves towards the drill bit-workpiece interface. Initially, the coolant's velocity is greatest close to the orifice, and as time passes, it approaches the drilling space. The data indicates that the spraying velocity of the coolant augments over time and that it disperses heat at the tool-chip interface. The results help us validate the flow and interaction of ionanocoolant with the drilling zone. With a rise in the volume fraction of added nanoparticles and Reynolds number, the results indicated a significant decrease in the drilling temperature. With a higher particle volume fraction, the MWCNT-ionic coolant combination decreases the drilling temperature of pure ionic liquid by 25.64 %. The copper, silver, and MWCNT ionanofluids enhance the average heat transfer coefficient of pure ionic coolant by 35.14 %, 47.42 %, and 62.75 %, respectively. In addition, MWCNT nanocoolants demonstrated improved thermal performance and heat removal rate in comparison to copper and silver ionanocoolants.

Structure and tribological properties of “carbon steel – VC containing coating” compositions formed by non-vacuum electron-beam surfacing of vanadium-containing powder mixtures

Vanadium carbides are known as efficient reinforcing components for surface layers of steel parts. They provide good tribological characteristics of material under sliding friction conditions and against dry abrasive particles. Since wear conditions usually are quite severe, a thick coating is needed to provide the reliable protection of the part. The majority of currently used surfacing technologies ensure the coatings of insufficient thickness. For this reason, the non-vacuum electron beam cladding which is able to provide thick coatings is a reasonable alternative to other surfacing methods. This paper presents an analysis of the microstructure, micro- and nanohardness, elasticity, and tribological characteristics of coatings formed by non-vacuum electron-beam surfacing of vanadium and graphite-containing powder mixtures. In the course of the study, coatings with the average thicknesses of 1.8, 2.5, 2.6, 3.4 mm were formed on the surface of blanks made of medium‑carbon steel. Hardness of cladding layers was significantly improved compared to the low-carbon steel (by the factor of 5–7). The wear resistance of coatings under conditions of sliding friction was 5.8 times higher than the wear resistance of the base metal after quenching and tempering; the wear resistance of coatings under conditions of friction against fixed and loosely fixed abrasive particles was ∼2 and ∼4.8 times higher, respectively. Surface hardening of steel alloyed with vanadium and carbon derives from the presence of a martensitic matrix, ledeburite precipitates and a high volume fraction (14.5–29.5 vol%) of vanadium carbide particles of a dendritic form, which act as rigid barriers.

Factors influencing the low-temperature fluidity of kerosene-based magnetic fluid: Surfactant, volume fraction

To explore the effect factors on the fluidity of magnetic fluid at low temperatures, the magnetic fluid was prepared by dispersing stearic acid and oleic acid-modified particles into kerosene, the effect of volume fraction was also considered. The microstructure and morphology of the particles were obtained by XRD and TEM, and the surfactant content on the surface of the particles was determined by TG. The hysteresis loops of magnetic particles and magnetic fluids were obtained by VSM. As the temperature decreased, it was observed that the magnetic fluid with a high volume fraction of stearic acid-modified particles would lose fluidity quickly under a non-magnetic field. But it could respond to a certain magnetic field, and the fluidity could be restored quickly by increasing the temperature. The rheometer measured the viscosity of the magnetic fluid-dependent temperature, shear rate, and magnetic field. When a high melting point of stearic acid coated particles, the stearic acid with adequate chain length could affect the fluidity of magnetic fluid with a volume fraction greater than 5% at roughly 20 °C. The free surfactant formed micelle, the base carrier was restricted in the particles adhering to the micelles, eventually causing the magnetic fluid to lose fluidity.

Thermal Stress Formation in a Functionally Graded Al2O3-Adhesive Single Lap Joint Subjected to a Uniform Temperature Field

This study investigates the strain and stress states in an aluminum single lap joint bonded with a functionally graded Al2O3 micro particle reinforced adhesive layer subjected to a uniform temperature field. Navier equations of elasticity theory were designated by considering the spatial derivatives of Lamé constants and the coefficient of thermal expansion for local material composition. The set of partial differential equations and mechanical boundary conditions for a two-dimensional model was reduced to a set of linear equations by means of the central finite difference approximation at each grid point of a discretized joint. The through-thickness Al2O3-adhesive composition was tailored by the functional grading concept, and the mechanical and thermal properties of local adhesive composition were predicted by Mori–Tanaka’s homogenization approach. The adherend–adhesive interfaces exhibited sharp discontinuous thermal stresses, whereas the discontinuous nature of thermal strains along bi-material interfaces can be moderated by the gradient power index, which controls the through-thickness variation of particle amount in the local adhesive composition. The free edges of the adhesive layer were also critical due to the occurrence of high normal and shear strains and stresses. The gradient power index can influence the distribution and levels of strain and stress components only for a sufficiently high volume fraction of particles. The grading direction of particles in the adhesive layer was not influential because the temperature field is uniform; namely, it can only upturn the low and high strain and stress regions so that the neat adhesive–adherend interface and the particle-rich adhesive–adherend interface can be relocated.

Oxidation mechanism of high-volume fraction SiCp/Al composite under laser irradiation and subsequent machining

Conventional mechanical machining of a composite material comprising an aluminum matrix reinforced with a high volume fraction of SiC particles (hereinafter referred to as an SiCp/Al composite) faces problems such as rapid tool wear, high specific cutting force, and poor surface integrity. Instead, a promising method for solving these problems is laser-induced oxidation-assisted milling (LOAM): under laser irradiation, the local workpiece material reacts with oxygen, thus forming loose and porous oxides that are easily removed. In the present work, the oxidation mechanism of SiCp/Al irradiated by a nanosecond pulsed laser is studied to better understand the laser-induced oxidation behavior and control the characteristics of the oxides, with laser irradiation experiments performed on a 65% SiCp/Al composite with various laser parameters and auxiliary gases (oxygen, nitrogen, and argon). With increasing laser pulse energy density, both the ablated groove depth and the width of the heat-affected zone increase. When oxygen is used as the auxiliary gas, an oxide layer composed of SiO2 and Al2O3 forms, and CO2 is produced and escapes from the material, thereby forming pores in the oxides. However, when nitrogen or argon is used as the auxiliary gas, a recast layer is produced that is relatively difficult to remove. Under laser irradiation, the sputtered material reacts with oxygen to form oxides on both sides of the ablated groove, and as the laser scanning path advances, the produced oxides accumulate to form an oxide layer. LOAM and conventional milling are compared using the same milling parameters, and LOAM is found to be better for reduced milling force and tool wear and improved machined surface quality.

Experimental and Numerical Investigation of Flow Structure and Heat Transfer Behavior of Multiple Jet Impingement Using MgO-Water Nanofluids.

Nanofluids have attracted significant attention from researchers due to their ability to significantly enhance heat transfer, especially in jet impingement flows, which can improve their cooling performance. However, there is a lack of research on the use of nanofluids in multiple jet impingements, both in terms of experimental and numerical studies. Therefore, further investigation is necessary to fully understand the potential benefits and limitations of using nanofluids in this type of cooling system. Thus, an experimental and numerical investigation was performed to study the flow structure and heat transfer behavior of multiple jet impingement using MgO-water nanofluids with a 3 × 3 inline jet array at a nozzle-to-plate distance of 3 mm. The jet spacing was set to 3, 4.5, and 6 mm; the Reynolds number varies from 1000 to 10,000; and the particle volume fraction ranges from 0% to 0.15%. A 3D numerical analysis using ANSYS Fluent with SST k-ω turbulent model was presented. The single-phase model is adopted to predict the thermal physical nanofluid. The flow field and temperature distribution were investigated. Experimental results show that a nanofluid can provide a heat transfer enhancement at a small jet-to-jet spacing using a high particle volume fraction under a low Reynolds number; otherwise, an adverse effect on heat transfer may occur. The numerical results show that the single-phase model can predict the heat transfer trend of multiple jet impingement using nanofluids correctly but with significant deviation from experimental results because it cannot capture the effect of nanoparticles.

A numerical study of the spectral radiative properties of packed bed with mixed bauxite and silica spheres

Bauxite and silica particles have gained increasing attention for applications in the field of concentrated solar power. In this work, a Monte Carlo ray-tracing simulation is performed to predict the radiative properties (absorptance, reflectance, and transmittance) of packed beds with mixed bauxite and silica spherical particles at wavelengths of 0.5, 2.6, 9.0, and 9.35 µm. These wavelengths are representative for the visible, near-infrared, and the mid-infrared regions that are important for solar and thermal radiation, respectively. A repeating unit-column approach is used to mathematically represent the particle bed. The effects of particle mixing ratio, volume fraction, and wavelength on the predicted radiative properties are examined. The obtained radiative properties are inputted to an inverse method to retrieve the effective absorption and scattering coefficients as well as the scattering albedo, which may be later used in a continuous-scale radiative heat transfer analysis. Furthermore, the independent scattering model is used to obtain the absorption coefficient and scattering albedo based on the absorption and scattering cross sections predicted by a Monte Carlo algorithm for a single particle. It is shown that the independent scattering model underpredicts the scattering coefficient for opaque particles but overpredicts the scattering coefficient for semitransparent particles for sufficiently high particle volume fractions. The radiative properties calculated from the independent scattering model are compared to the full Monte Carlo simulation of the particle bed to examine the influence of particle mixing on dependent scattering.

Modulus Estimation of Composites with High Porosity, High Particle Volume Fraction, and Particle Eigenstrain: Application to the LIB Active Layer with a Bridged-Particle Mesostructure

Due to the complex mesostructure and components of composite active layers in lithium-ion battery (LIB) electrodes, coupled with the concentration-dependent material properties and eigenstrains, efficiently estimating the effective modulus of the active layers remains a great challenge. In this work, the classic Mori–Tanaka method is found to be unable to estimate the modulus of the active layer. By realizing the importance of the mesostructure feature, a rod-rod model is proposed. The resulting modulus is expressed analytically. It is shown that the rod-rod model can accurately estimate the modulus evolution of the active layer if the material properties of the components and the evolution of volume fractions are known in advance. Moreover, a simplified rod-rod model is also developed to reduce the complexity of the proposed method. By knowing the volume fractions at two arbitrary states of charge and subsequently determining two constants, the simplified model can estimate the modulus efficiently. Considering both its accuracy and its simplicity, the simplified rod-rod model is the most suitable for the estimation. Thus, the methods developed in this work provide a new perspective for analyzing the material properties of composite active layers in LIB electrodes.

Deformation behavior of cell walls in an additively manufactured hybrid metallic foam

A hybrid Al-Al3Ni metallic foam was synthesized in-situ via directed energy deposition (DED) of Ni-coated Al 6061 powder, without the need for a foaming agent. Three-dimensional characterization via X-ray computed tomography shows that the foam contains approximately 61.5 % porosity and a high volume fraction of the Al3Ni phase (∼60 vol. %) within the cell walls. This microstructure is notably distinct from the eutectic structure that is typically observed in conventionally processed Al-Ni alloys. To investigate the mechanical properties and deformation mechanisms of the Al-Al3Ni cell walls, in-situ micro-pillar compression was performed, and the results reveal a notable yield strength of 560 MPa and a compressive strain that exceeds 30 %. These properties are attributable to the presence of a high volume fraction of Al3Ni particles, in combination with the characteristics of the Al/Al3Ni interfaces. To provide insight into the deformation mechanisms in the cell walls we used in-situ mechanical testing, transmission electron microscopy and precession electron diffraction characterization, together with molecular dynamics simulations. Our results reveal two distinct mechanisms: uniform dislocation-based deformation of the Al alloy phase and localized deformation within the slip bands in the Al3Ni phase. The results further highlight the importance of the Al/Al3Ni interfaces in mechanical strengthening and the transfer of plastic deformation from the Al phase to the Al3Ni phase. At high mechanical loads, cracks form due to the large stress concentration at the slip bands and slip band intersections in the Al3Ni, giving rise to intragranular fracture of Al3Ni and finally interfacial debonding and cracking.

3D-printable, lightweight, and electrically conductive metal inks based on evaporable emulsion templates jammed with natural rheology modifiers

Conductive metal inks with 3D-printable rheological properties have gained considerable attention, owing to their potential for manufacturing 3D electronics. Typically, such inks are formulated with high volume fractions of metal particles to achieve both rheological and electrical percolation. However, this leads to a high product cost and weight, making this approach potentially undesirable for practical application. In this study, naturally occurring ingredients, i.e., bee pollen microparticles (BPMPs) and citric acids (CAs), are used to produce a jammed hexane-in-aqueous suspension-type emulsion with controllable viscoelasticity as a template for conductive metal particles. Correspondingly, it is possible to develop 3D-printable, lightweight, and conductive inks. The BPMPs and CAs, as rheology modifiers, facilitate the 3D printability of the ink. After drying, the ink forms 3D networks without macroscopic discontinuities. Hexanes co-dispersed with BPMPs and CAs in the aqueous continuous phase improve the ink rheological processability and create internal macropores within the 3D-printed structure upon evaporation under ambient conditions, decreasing the product density. A conductive copper ink based on the emulsion template shows excellent 3D printability and electrical percolation at low metal loadings (<10 vol%); moreover, the printed ink with the optimized formulation has a remarkably low density (<2 g ∙ cm−3).

Experimental Analysis on the Dependence of the Capacitance of Magnetorheological Fluids on Frequency

The capacitance characteristics of magnetorheological fluids (MRFs) were studied experimentally based on simulation analysis. The nonlinear relationship between the capacitance of MRFs and electric field frequency was measured by the self-made circuit device. The effects of magnetic induction intensity and particle volume fraction on the capacitance characteristics of MRFs were investigated. The results show that the nonlinear dependence of the capacitance of an MRF on frequency induced by the effect of tunnel current decreases with the increase in frequency. The capacitance of an MRF is directly related to the particle structures. The capacitance of the MRF with a chain particle structure is greater than that of the MRF with a random particle distribution. The smaller the clearance between the adjacent particles, the greater the capacitance of the MRF. The network particle structure and particle contact will reduce the capacitance of an MRF. The capacitance of an MRF increases with the increment of external magnetic field. The capacitance of an MRF with higher particle volume fraction is smaller than that of an MRF with lower particle volume fraction.

Turbulent channel flow of a binary mixture of neutrally buoyant ellipsoidal particles

A turbulent channel flow of a binary mixture of finite-size neutrally buoyant ellipsoidal particles is studied by using a parallel direct-forcing fictitious domain method at a friction Reynolds number of 180 and the particle aspect ratios of 1/3 (oblate particle) and 4 (prolate particle), respectively. The total particle volume fraction is fixed at 14.16% and 21.24% and the relative fraction of prolate and oblate particles are varied. The mean velocity profile normalized by the bulk velocity shows a strong difference between the single phase flow and particulate flow, while showing a small difference between mono-disperse and bi-disperse particulate flow cases. The lower fluid and particle Reynolds stress contributions in a high oblate particle volume fraction ratio cases lead to the drag reduction. Under moderate particle volume concentration, the oblate and prolate particle results show self-similarity of velocity profiles of solid phase, including particle velocity, concentration, orientation, and rotation. Both kinds of ellipsoidal particles tend to align their long axes with the streamwise direction in the whole channel.

Validation of a two-layer depth-averaged model by comparison with an experimental dilute stratified pyroclastic density current

Numerical results of a two-layer depth-averaged model of pyroclastic density currents (PDCs) were compared with an experimental PDC generated at the international eruption simulator facility (the Pyroclastic flow Eruption Large-scale Experiment (PELE)) to establish a minimal dynamical model of PDCs with stratification of particle concentrations. In the present two-layer model, the stratification in PDCs is modeled as a voluminous suspended-load layer with low particle volume fractions (lesssim {10}^{-3}) and a thin basal bed-load layer with higher particle volume fractions (sim {10}^{-2}) on the basis of the source condition in the experiment. Numerical results for the suspended load quantitatively reproduce the time evolutions of the front position and flow thickness in the experimental PDC. The numerical results of the bed-load and deposit thicknesses depend on an assumed value of settling speed at the bottom of the bed load ({W}_{mathrm{sH}}). We show that the thicknesses of bed load and deposit in the simulations agree well with the experimental data, when {W}_{mathrm{sH}} is set to about 1.25times {10}^{-2} m/s. This value of the settling speed is two orders of magnitude smaller than that predicted by a hindered-settling model. The small value of {W}_{mathrm{sH}} is considered to result from decreasing in the effective deposition speed due to the erosion process accompanied by saltating/rolling of particles at the bottom of the bed load.

The Investigation of Proppant Particle-Fluid Flow in the Vertical Fracture with a Contracted Aperture

Summary Proppant placement plays a crucial role in maintaining the conductivity of fractures after a hydraulic fracturing treatment. The process involves the transport of particles by fluid flow in complex fractures. Many studies have focused on proppant transport and distribution in the fracture with a constant aperture, but relatively few studies have investigated the proppant-fluid flow in a vertical fracture with a contracted aperture. In this work, we examine experimentally proppant transport in a fracture with a contracted aperture. The objective is to evaluate the distribution of particle beds in the contracted fracture at different flow conditions. In this paper, particle-fluid flow in the contracted fracture is studied experimentally by a laboratory size slot. A planar slot with a constant width is used to benchmark the experimental results, and a published correlation validates the bed equilibrium heights in the planar slot. Six types of particles are chosen to simulate the effects of particle density and size. The proppant distribution is evaluated by the bed height when the bed reaches the equilibrium states. The effects of fluid velocity, fluid viscosity, particle density, particle size, and particle volume fraction on particle distribution are investigated. The results confirm that the proppant particle-fluid flow in the contracted slot is more complicated than that in the planar slot. The phenomena of particle vortices and resuspension were observed at the contraction of the cross-section. The shape on the top of the bed is like a descending stair in which the height gradually decreases in the length direction. The bed height in the contracted slot is lower and more irregular than that in the planar slot at the same flow conditions. Smaller sands injected at a high flow rate and fluid viscosity can form a lower bed. The trend would be reversed by using denser particles and high particle volume fraction. A reliable model expressed by four dimensionless numbers is developed by the linear regression method for predicting the bed equilibrium height. The model and experimental results provide directions to quantitatively evaluate the particle transport and distribution in a fracture with a contracted aperture.