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An analysis has been completed for a comprehensive set of vertical, dilute-phase pneumatic conveying pressure drop data from an investigation by Flatow. The data were collected in the established flow region for eight different materials conveyed in 0.05-, 0.10-, 0.20-m internal diameter, 20-m tall steel risers. Particle velocities derived from the pressure drop data were used to develop an equation of motion that includes terms for pipe diameter, terminal velocity, coefficient of restitution, and particle shape. The best data fit was achieved using the actual gas density and the actual gas velocity adjusted for voidage. Adjusting the terminal velocity for voidage, an approach recommended by many investigators, did not improve the fit for reasons identified by the present research. Using the equation of motion, particle velocities were predicted and used to calculate total pressure drops that are within ±15% of the measured values. The calculated values also produce the characteristic trough-shaped total pressure drop curves allowing the minimum pressure drop gas velocity to be determined without recourse to a separate correlation. A comparison with other studies using shorter risers indicates that data from these studies likely include acceleration effects. A separate study will investigate this observation further.

In the present paper, the three dimensional (3D) settlement process of two side-by-side free-falling spheres in a water tank was numerically investigated using coupled Smoothed Particle Hydrodynamics (SPH) - Discrete Element Method (DEM) method. The maximum particle Reynolds number, calculated based on the terminal velocity and sphere diameter, is R e m = 2 . 4 × 1 0 4 . The accuracy of the model was firstly validated by simulating the single sphere water-entry, settlement of two circular cylinders arranged in series and two identical spheres settlement side by side problems, respectively. The simulation results exhibited a satisfactory agreement with the available numerical and experimental data in the literatures. Then, the settlement process of two spheres with different sizes and transverse distances were simulated. In particular, the motion patterns of two interacting spheres with different initial gaps and diameter ratios, including motion trajectory, vertical falling distance and flow field, were systematically investigated. The results demonstrate that when initial gap S 0 ∗ S 0 ∗ > 0.75, the interaction between the two spheres are negligible. In addition, the interaction between particles would slow down the settling velocity and affect the terminal settling distance.

Empirical correlations and experimental works relating the drag coefficient (CD) of spheres to the Reynolds number (Re) and Re to the Archimedes number (Ar) were reviewed with a focus on the correlations applicable to the entire Re range. In addition, experiments with various spherical particles and fluids for the entire range of Re were conducted using high-speed video and added to the literature data. One of the correlations from the literature for Re-Ar and a modified correlation for CD–Re were found to best fit the experimental results. The analysis also established new correlations for Re–Ha (a non-dimensional number for velocity not including particle size) and CD–Ar. The latter is easier to use than the common CD–Re curve.

The management of fluids in space is complicated by the absence of relevant buoyancy forces. This raises significant technical issues for two-phase flow applications. Different approaches have been...

To enable the mechanistic description of the mixing of larger particles in gas-fluidized beds in models (e.g. fuel particles in combustors), knowledge about the rheology of the bed emulsion is required. Here, it is crucial to determine the drag on large fuel-alike particles. This work presents the experimental work on the fate of 13 different solid spheres falling or rising through a bed of air and glass beads at minimum fluidization. The trajectories of the tracer are highly resolved (sampling rate of 200 Hz) by means of magnetic particle tracking, this previously unmet accuracy allows disclosing the complex rheological behavior of gas-solids fluidized bed emulsions in terms of drag on immersed objects. The trajectories reveal that none of the tracers reach terminal velocity during their fall and rise through the bed. The shear stress is obtained through the drag force by solving the equation of motion for the tracer. The data reveal particularities of the bed rheology and clear differences of its effect on rising and falling particles. When studying the shear stress over the characteristic shear rate of each tracer, it can be seen that the stress of the bed on the tracers is dominated by a yield stress, with a somewhat smaller contribution of the shear stress. For rising tracers this last contribution is almost negligible. The falling tracers show strong interaction with the bed emulsion, resulting in a fluctuating shear stress, which increases with tracer size and density. The stagnation of some tracers at low shear rates reveals a viscoplastic behavior of the bed emulsion, exhibiting a typical yield stress that showing a clear dependence on the tracer diameter and buoyant density. The concept of yield gravity is used in order to introduce a normalized shear stress which provides additional verification of the experimental observations in relation to the influence of tracer size and relative density on the shear stress.

This numerical study investigates the dynamics around a single Taylor bubble rising with small-dispersed bubbles in a vertical pipeline. The Volume-of-fluid (VOF) method in Ansys Fluent 18.1 is used to track the Taylor bubble, and the Discrete Particle Method (DPM) is used to track the small-dispersed bubbles, for the following parameter range; $$812 \leqslant \text{Re} \leqslant 5684$$ , $${\kern 1pt} Eo = 94$$ and $$\log (Mo) = - 10.6$$ . The coupling strategies for both the continuous gas–liquid phase as well as the discrete bubble phase are all accounted for in the numerical calculations. The dispersed bubbles significantly affect the behaviour of the rising Taylor bubble in numerous ways. The terminal velocity increases with an increase in the number of dispersed bubbles. The rise velocity is, however directly correlated with the deformation of the nose of the bubble. The flow dynamics around the nose of the Taylor bubble are stronger for low velocities due to lower inertia. The computations also demonstrated that there are both acceleration and deceleration effects on the Taylor bubble. Our predictions are in quantitative agreement with published experimental results by Cerqueira and Paladino (Int. J. Multiph. Flow, vol. 133, p. 103,450, Dec. 2020).

In particle-laden flows, a turbulent field can be produced in the carrier phase by the movement of the particle/spray cloud. In this study, the intensity and the integral length scale of the particle-induced turbulence are studied using a simple mechanistic model with comparison to experimental data and numerical simulations for large-scale numerical applications. The experimental results of DynAsp are investigated with numerical simulation results. Out of the spray nozzle, two regions can be distinguished for the spray dynamics: an inertial zone and an equilibrium zone. It is found that the initial injection velocity of the cloud has little effect on the terminal slip-velocity of the particles in the equilibrium zone far from the injection region. The turbulent kinetic energy is closely related to the particle slip-velocity and shows a maximal value when particles reach their terminal velocity inside the equilibrium zone. The integral length scale depends mainly on three parameters: particle slip-velocity, particle size and volume fraction. Combined with the terminal slip-velocity correlation, the reduced-order mechanistic model can give a reasonable estimation of the turbulent kinetic energy as well as the integral length scale of the particle-laden flow in large-scale configurations.

In this study, the interaction between a spherical solid particle descending in a fine-bubble plume and the fine-bubbles around the particle is investigated experimentally. The fine-bubbles, which are formed by water electrolysis and have an average diameter of 0.037 mm, rise via buoyancy to form bubble plumes. The average bubble velocity in the conduit is 0.05 mm/s. A spherical solid particle with a diameter of 11.1 mm and a density of 1130 kg/m3 is dropped into the fully-developed bubble plume. The particle falls in a meandering motion in the fine-bubble plume. In this process, the terminal velocity of the particles falling in the fine-bubble plume is almost identical to that of a particle falling in the quiescent water. Additionally, after the fine-bubbles have been separated from the particle surface, they flow into the wake of the particle, forming a stagnant region. The particle wake diameter increases threefold, as compared with its value in the downstream direction of the fine-bubble plume.

This work demonstrates that an additional resistance term should be included in the Navier–Stokes equation when fluids and objects are in relative motion. This is based on an observation that the effect of the microscopic molecular random velocity component parallel to the macroscopic flow direction is neglected. The two components of the random velocity perpendicular to the local mean flow direction are accounted for by the viscous resistance, e.g., by Stokes’ law for spherical objects. The relationship between the mean- and the random velocity in the longitudinal direction induces differences in molecular collision velocities and collision frequency rates on the up- and downstream surface areas of the object. This asymmetry therefore induces flow resistance and energy dissipation. The flow resistance resulting from the longitudinal momentum transfer mode is referred to as thermal resistance and is quantified by calculating the net difference in pressure up- and downstream the surface areas of a sphere using a particle velocity distribution that obeys Boltzmann’s transport equation. It depends on the relative velocity between the fluid and the object, the number density and the molecular fluctuation statistics of the fluid, and the area of the object and the square root of the absolute temperature. Results show that thermal resistance is dominant compared to viscous resistance considering water and air in slow relative motion to spherical objects larger than nanometer-size at ambient temperature and pressure conditions. Including the thermal resistance term in the conventional expression for the terminal velocity of spherical objects falling through liquids, the Stokes–Einstein relationship and Darcy’s law, corroborates its presence, as modified versions of these equations fit observed data much more closely than the conventional expressions. The thermal resistance term can alternatively resolve d’Alembert’s paradox as a finite flow resistance is predicted at both low and high relative fluid–object velocities in the limit of vanishing fluid viscosity.

An experimental study for the drag of solid spheres in drag reducing Fluids is presented. Polyacrylamide solutions of different concentrations are considered. The drag coefficient is measured by tie terminal velocity of the spherical particle falling in stagnant fluid using high speed camera, Compared to Newtonian Fluid results, the drag of solid particle in polymer solutions exhibit reduced values. However, a drag increase Is exhibited at low Reynolds number. Analysis of the data show the presence of a critical Reynolds number that distinguish between the drag increase and drag reduction regions. It has been postulated that the measured drag is a result of two opposing effects of polymer additives. One is flow resistance increase due to increased extensional viscosity. The other, is the reduction in the form drag due to the influence of these additives on vortex shedding and boundary layer separation. It has been found that the experimental data of drag coefficient is best correlated with the polymer -particle parameter ᴓ or Weissenberg number Ws as well as Reynolds number.

Fine kaolinite particles are mineral particles that are found in mine wastewater. The particles' shape is one of the parameters that causes a significant change in the sedimentation dynamics in water environments. In this work, experimental methods and Computational Fluid Dynamics - Discrete Element Method (CFD-DEM) methods served to investigate the dynamic characteristics of fine kaolinite particle sedimentation. The Results of statistical analyses show that the length-width ratio of fine kaolinite particles is 1–3 and the average simplified spherical coefficient is 0.625 in the 50–500 μm particle size range. The simplified spherical coefficient formula proved to be effective according to experimentation and simulations. Moreover, the effects of particle size, liquid viscosity, and liquid velocity on kaolinite particle sedimentation dynamic characteristics were numerically studied in detail by using the modified spherical coefficient. The simulation revealed that an increase in liquid viscosity resulted in a declining particle terminal velocity. However, the sensitivity of the particle terminal velocity affected by liquid viscosity fell when the particle size also declined. When an increase in particle size occurred, the sensitivity of the particle terminal velocity to the influence of upwelling water decreased.

The terminal velocity of cloud drops and raindrops used in numerical model calculations can significantly affect weather predictions. Current formulations rely on laboratory experiments made in the 1940s and 1960s. Because these experiments were performed only at typical environmental conditions of 20°C and 1013 hPa, parameterizations have been introduced to deduce the terminal velocity aloft without rigorous evaluation. In this study, an incompressible two-phase flow direct numerical simulation model is used to calculate the free-falling motion of axisymmetric drops with diameters between 0.025 and 0.5 mm to study the terminal fall velocity. Simulated terminal fall velocities of free-falling drops at 20°C and 1013 hPa agree within 3.2% with the previous empirical parameterization (Beard formula), and 4.5% with existing laboratory data in the diameter range between 0.3 and 0.5 mm. The velocities converge to the analytic Hadamard–Rybczynski solution within 2% for small Reynolds numbers, demonstrating the robustness of our simulations. Simulations under various atmospheric conditions show that existing empirical parameterizations that account for the air density dependence of the terminal velocity have errors up to 11.8% under the conditions examined in this study. We propose a new empirical formula that describes the air density dependence of the terminal velocity. It is also shown that the falling speed of a small drop is not sensitive to shape oscillation, and the terminal velocity decreases by only less than 1.3% when the axis ratio increases by 12% with reduced surface tension. Internal circulation within falling drops is also presented and compared with previous studies.

Single bubbles rising in a confined geometry – a parallelepidic bubble column with a 4 mm thickness – in the presence of a non-Newtonian liquid phase are studied using a shadowgraphy technique. The context is the cultivation of microalgae at high cell concentrations. In fact, it necessitates the reduction of the bioreactor thickness to allow light penetration through the whole culture. In the same time, the medium becomes shear-thinning for biomass concentrations higher than 30 g L−1. A 2 g L−1 carboxymethyl cellulose (CMC) and a 1 g L−1 Xanthan Gum (XG) aqueous solutions are studied to mimic the behavior of 42 g L−1 Chlorella vulgaris cultures. Besides, a comparison with water, representing the Newtonian liquid phase at low culture concentrations, is proposed. The bubble equivalent diameter is varied between 1.34 and 3.36 mm using different capillary sizes to generate the bubbles. Results show the complex and coupled effects of confinement and shear-thinning behavior of the liquid phase on bubbles terminal velocity, shape and trajectory. In fact, compared to the case of an infinite liquid medium, confinement increases wall friction on the bubbles, while reduces the liquid phase apparent viscosity in its neighborhood due to increased shear rate. This is why, depending on the bubble size, the bubble terminal velocity in shear-thinning fluids can be higher or lower than in unconfined conditions. For bubbles rising in shear-thinning fluids, contrary to water, it is observed that confinement induces a transition from spherical shape to ellipsoidal shape that occurs sooner in terms of bubbles diameter or Eo number. A correlation for the bubble drag coefficient in confined conditions in presence of shear-thinning fluids is also proposed for confinement ratios db/e between 0.34 and 0.84.

The results of experimental studies of the terminal velocity of peat dust in an air flow with different size distributions of dust particles are presented, and the particle sizes and shapes were examined by microscopic methods. The aerodynamic drag of peat particles of various sizes in an air flow was studied, and a criterion relationship between the drag coefficient CD and the Reynolds number (Re) was determined. The experimental results on the terminal velocity of peat dust were compared with data calculated using well-known methods, and recommendations for their application in the design of pneumatic fuel supply systems were given.

Ember transport and the subsequent development of spot fires is a significant mode of wildfire spread, particularly in extreme conditions. An important simplifying assumption made in early research into ember transport is the terminal-velocity assumption, in which embers are assumed to always fly at their terminal velocity relative to the wind field. With increases in computational power, it is now possible to directly simulate the atmospheric conditions resulting from wildfires and such simulations can resolve the larger of the turbulent processes involved. Because of the time-scales at which these processes occur, the terminal-velocity assumption may not be justified when modelling ember transport using these simulations. In this study we use a large eddy simulation of a turbulent plume to examine the validity of the terminal-velocity assumption when modelling the long-range transport of non-combusting embers. The results indicate that the use of the terminal-velocity assumption significantly overestimates the density of ember landings at long range, particularly for embers with higher terminal fall speeds.

One of the most important design parameters for the inkjet printing device is the inkjet drop velocity, which changes along the falling path from the ejection out of a nozzle to the impact on a substrate. Therefore, in this study, the droplet velocity of the piezoelectric drop-on-demand inkjet was measured. The nozzle diameter was 70 µm and polyethylene glycol aqueous solution was used as ink. The inkjet droplets were generated within a range from 500 to 10000 Hz of frequencies at a reference piezoelectric input voltage. The successive inkjet droplets were captured by the high-speed camera with 100000 fps, then the inkjet droplet velocities were analyzed visually at each falling location. The velocity change of a drop up to its terminal velocity was calculated theoretically to be compared with the experimental results. This study found that the velocity of the continuously ejected droplets depended on the interval of the droplets, as well as the density and diameter of the droplet which are essential factors of the general velocity change of a falling drop. The inkjet droplets with the long interval can be considered as an independent droplet so that it can follow the terminal velocity curve of the general drop. However, the inkjet droplets with short interval have the higher velocity than the general terminal velocity curve. It is considered that each droplet gets into the wake field of the former droplet so that its drag force can decrease and the velocity can increase.

Mineral dry density separation behavior (mineral beneficiation) involves segregation of high density particles and low density particles in the feed, i.e., in nutshell, mineral dry density beneficiation is density segregation in physics terms. In dry beneficiation, this density segregation is achieved using air as fluidized medium. Density segregation on dry separators can be achieved in two ways: (1) using minimum fluidization velocity differences and (2) using terminal velocity difference between different density particles. Dry separators based on minimum fluidization velocities follow two steps for dry beneficiation: (1) vertical density stratification and (2) horizontal segregation. Particles due to buoyancy, drag and gravitational forces get vertically density stratified and later frictional, gravity and external (vibrational, etc.) forces horizontally segregate different density particles. Dry separators based on terminal velocities follow single step for dry beneficiation of mineral, and air velocity in this type of dry density separators is maintained between terminal velocity of high density and low density particles. Velocity of air is higher for low density particles, which leads to higher drag and buoyancy forces to get pushed out of system with air and get collected at further distance than high density particles leading to density segregation. Mathematical models have been developed using force balance on particles and simulated in MATLAB for both type of separators to explain the physics of separation. Solutions to the simulated mathematical model equations are the particle trajectories of different density particles in the two dry separators. Iron ore is used as feed for studying separation futures in mathematical modeling on dry separators.

A novel theoretical model which takes into account the particle property, equipment parameters, particle terminal velocity and Reynolds number was established to describe the separation of particles under critical conditions. Meanwhile, this established model was used to predict the superficial velocity of particle separation in the different spacing of inclined channels. A system was developed to test and verify the model. The correlation coefficient (R2) and Sum of Squares for Error (SSE) were adopted to evaluate the experimental results. When the ratio of fluid Reynolds number to particle Reynolds number was less than 43, the novel model was proved to be highly accurate in the prediction of the superficial velocity. The results suggested that the novel model provided much better conclusions than does a tradition empirical model. This study provides a theoretical and experimental foundation for the design of industrial inclined channel structure and operating parameters.

Two-phase liquid-gas injection constitutes an important industrial process that is used in most separators. At the early step of injection, a cylindrical bubble is formed. As time elapses, the bubble shape becomes more complex and very difficult to analyze. In this study, a simple analytical model is developed to explain bubble shape changes. The analytical model was developed based on water flow inertia that continually pushes the bubble while the drag force resists it so that the frontal area of the bubble increases. The bubble size and frontal area were estimated using the assumption of the equilibrium between inertia force and drag force neglecting viscous force. From the estimation, the role of the vortex ring from the difference between theoretical and experimental results can be identified. The analytical model was verified through experimental data collected on the shape deformation induced by bubble motion at the beginning of injection. The experimental data used as verification were measured from the bubble nose image with ten times repetition having the uncertainty of ±6%. The experimental method is conducted by injecting a bubble along the horizontal direction into a water pool. The inertial force of the water flow in front of the bubble nose generates the bubble. The bubble suddenly changes its shape, moves in the form of a bubble jet, and undergoes gradual shape changes. The frontal area of the bubble increases and reaches a maximum at the terminal velocity point. The bubble shape deformation is affected by the inertial force of the water flow that pushes the bubble forward. Accordingly, the bubble changes its shape from cylindrical to spherical, and then to an ellipsoidal disk. When the bubble attains terminal velocity, the inertial force becomes equal to the drag force. The edge of the ellipsoidal disk bubble exhibits increased surface tension. The difference between experimental data and the analytical model is due to the complex fluid and dynamic flow surrounding the bubble. The mathematical framework proposed in this work is envisaged to be an important tool for the prediction of the bubble frontal area

Numerical simulations are performed to investigate the effect of electric field and electrification process on dynamic, microphysical, and precipitation in thunderclouds. A three-dimension (3D) convective cloud model with electrification and discharge scheme is used to explore the differences between kinematic, microphysical process, precipitation, and lightning activity. The results show that the electrification process significantly enhances the updraft before the first discharge in thundercloud, but the electric field force has a significant effect on the updraft after the first discharge. The influence of electric field force on the formation of precipitation is greater than that of electrification process, and the increase of ice crystal terminal velocity is helpful to the formation of liquid precipitation, and then the increase of graupel terminal velocity is beneficial to the development of solid precipitation. The results also suggest that the new scheme of graupel terminal velocity enhances the high-concentration region of graupel before the first discharge, and the region lasts as long as 10 min, which effectively promotes the production of raindrop in the whole troposphere at specific time. Additionally, the electric field has the most significant effect on the velocity of smaller particles, and an increasing of ice crystal fall speed above 0 °C is also helpful to the formation of lightning. Moreover, it is found that the electric field force enhances the collision and coalescence between cloud droplets/raindrops and graupel during the mature period of thundercloud.

Knowledge on regime transition from conventional to circulating fluidization is important to scale-up and operation of liquid-solid fluidized systems in industrial applications. Previous experimental studies reported that the onset velocity measured by bed empty time test is pertinent to the physical properties of particle and liquid, and independent of solid inventory, bed geometry, and configuration. The effect of liquid properties is, however, not clear yet since tap water is usually used as the working fluid in laboratory test for convenience. To address this problem, a Computational Fluid Dynamics-Discrete Element Method (CFD-DEM) model is applied to numerically measure the onset velocity of particles in different liquid media. The model is first validated in terms of the experimental data of bed expansion in literature. Then several kinds of particles and three kinds of liquid media of different densities and viscosities are further simulated. We find that the Reynolds number based on onset velocity and Archimedes number follows a power-law relationship. With the decrease of Archimedes number, the discrepancy of Reynolds number based on onset velocity and that on particle terminal velocity becomes highly significant. The ratio of the onset velocity to particle terminal velocity for various particles in one liquid medium is not a constant, and instead dependent on both the particle and liquid properties.

The paper presents an analysis of the effect of collisions between coarse and fine particles in the vertical flow of a stream of fine particles suspended in gas on the exchange of momentum in these conditions. It was found that the phenomenon of double collisions of fine particles with the surface of coarse particles can be observed during the flow of a stream of fine particles suspended in gas around moving and stationary coarse particles. This phenomenon changes the nature of the relationship between the force of mutual interactions and mass flow rate of fine particles. The proposed corrective relationship significantly improved the consistency of the value of coarse particle terminal velocity depending on mass flow rate of fine particles suspended in the gas flowing around coarse particles. The range of importance of the relationships concerning the terminal velocity of coarse particles was determined.

The settling of well-characterised fractal aggregates has been investigated numerically. The Immersed Boundary Method coupled with the Lattice Boltzmann Method (IBM-LBM) was developed to fully solve the liquid flow passing through the porous aggregate. Aggregates with prescribed structures covering the large spectrum of fractal dimension were fabricated and the benchmark experimental data have been collected to extensively validate the numerical model. Subsequently, the settling dynamics of fractal aggregates in liquid in terms of orientation evolution, time dependent drag force and terminal orientation was investigated and correlated to their morphological characteristics. Different equations have been developed for predicting the terminal settling velocity of fractal aggregates. A correlation that only requires inputs of primary particle size and fractal dimension is proposed for predicting the terminal velocity of fractal aggregates. Deviations less than 7% were obtained when applying the model for aggregates with the primary particle size ranging between 1.5 mm and 5 mm. The model's applicability to small aggregates comprised of primary particles less than 1 mm has also been demonstrated and discussed.

Industrial processes do not often deal with a pure fluid. Impurities can affect physical properties of gas-liquid systems, resulting in change of gas bubbles velocity. Purpose of the study ...

Porous particles are widely used in liquid–solid and gas-liquid-solid reactors, and accurate prediction of drag coefficients ( C D ) for porous particles is essential for designing and optimizing these reactors. The drag coefficients for porous particles, however, have been mainly obtained via theoretical analysis and numerical simulations. Only three experimental studies were carried out to study the drag force on porous spherical particles due to the difficulties in the preparation of model porous spheres with a homogeneous porosity and homogeneous surface properties. In this paper, a method to prepare model porous spheres which have a homogeneous porosity and homogeneous surface properties with a good sphericity for a wider range of relative permeability β has been presented. The details of this method were documented, and it showed that the porous spheres constructed could withstand rigid collisions at the maximum terminal velocity of 1 m/s and keep the structure intact with a negligible quantity of binders. Then gravity-driven settling experiments of a single porous sphere by use of particle tracking velocimetry (PTV) method were carried out to evaluate the feasibility of the use of the prepared porous spheres. Experimental results for 12 porous spheres, with particle Reynolds number ( R e ) ranging from 1 to 108 and relative permeability β from 21.6 to 315.6, agreed well with the previous results in the literature. The porous particles prepared by our proposed method can be potentially used for measuring drag coefficients of the porous permeable spheres in a wider range of particle Reynolds number and permeability and studying the complex interaction between the porous particles and the surrounding fluid.

Data in the literature on the influence of water temperature on the terminal velocity of a single rising bubble are highly contradictory. Different variations in bubble velocity with temperature are reported even for potentially pure systems. This paper presents a systematic study on the influence of temperature between 5 °C and 45 °C on the motion of a single bubble of practically constant size (equivalent radius 0.74 ± 0.01 mm) rising in a clean water and n-pentanol solution of different concentrations. The bubble velocity was measured by a camera, an ultrasonic sensor reproduced in numerical simulations. Results obtained by image analysis (camera) were compared to the data measured by an ultrasonic sensor to reveal the similar scientific potential of the latter. It is shown that temperature has a significant effect on the velocity of the rising bubble. In pure liquid, this effect is caused only by modifying the physicochemical properties of the water phase, not by changing the hydrodynamic boundary conditions at the bubble surface. In the case of the solutions with surface-active substances, the temperature-change kinetics of the dynamic adsorption layer formation facilitate the immobilization of the liquid/gas interface.

The motion of solids in a viscous fluid with nonlinear complex behavior has important applications in different industries and engineering sciences. This work experimentally investigated the free-fall behavior of rotating and non-rotating spherical particles in a viscous fluid. The effects of physical parameters, including particle size, particle density, fluid viscosity, and rotational velocity, on the terminal velocity of the falling spherical particles were examined. A high-speed camera was employed to capture the phenomena and study the free-fall behavior of the spherical solid particles in a viscous fluid. The studied spherical particles had five different densities of 2800, 7800, 7900, 8660, and 8960 kg/m3 and three varying dimensions of 4, 5, and 6 mm. To explore the effect of fluid viscosity on the terminal velocity of particles, two types of fluids with the viscosities of 0.012 and 0.014 Pa.s were used. The initial rotational velocities were 0, 600, 900, and 1200 rpm for the falling particles. The obtained results revealed that increases in the particle diameter, density, and rotational velocity, along with a decrease in fluid viscosity, accelerated the terminal velocity and shortened the time of reaching this velocity. Furthermore, an increase in the Reynolds number gave rise to a decrease in the lift coefficient. On the other hand, the accelerated dimensionless rotational velocity increased the lift coefficient.

A two-fluid model is used to study the effect of point particles on the Rayleigh-Benard stability threshold in a laterally unbounded cell. Equal particles are steadily and uniformly introduced at the top plate at their terminal velocity with a fixed temperature. Both velocity and temperature are allowed to vary while, falling, the particles interact with the fluid. This interaction is modulated by the ratio of the particle density and heat capacity to those of the fluid. Particles are found to have a stabilizing effect, which increases with their concentration and density up to several orders of magnitude above the single-phase stability threshold. This result is primarily a consequence of the particle mechanical, rather than thermal, coupling with the fluid. The particle initial temperature has a strong effect on the undisturbed temperature distribution in the cell, which is a significant factor for the stability of the system. The addition of particles greatly increases the dimension of the parameter space necessary to characterize the flow.

Distinct from solid particle, bubble is soft matter, which tends to adjust its shape according to motion, and according to the work of Tomiyama, for bubble with diameter larger than 1.3mm, the rising path would change from rectilinear to zig-zag or helical, which make the determination of the terminal velocity more difficult. Even for the bubble with diameter larger than 0.075 mm reported by Pawliszak, terminal velocity scatter started. The existence of surfactants or contaminants and different initial deformations of bubbles introduced by varying capillary or nozzle diameters were generally considered to be the two main reason account for the terminal velocity scatter. In this paper, the rising characteristics of small bubbles with 1 < Re < 100 were investigated. Especially the motion characteristic right after the departure. Comparisons were made between three bubbles with almost the same diameter (0.58 ± 0.008mm), the same aspect ratio (0.98 ± 0.01), it was found that the only differences lied on oscillation frequencies of aspect ratio. And the higher oscillation frequency of aspect ratio was, the higher steady-state velocity the bubble would achieve. Therefore, it was thought that the deformation is one of the ways for bubble to accelerate itself, i.e., bubbles accelerated itself through the oscillations of aspect ratio. The oscillation of aspect ratio is of high possibility to be one reason for velocity scatter.

This paper extends the study by Tee et al. (2020) to investigate the effect of large coherent structures on motion of spheres with specific gravities of 1.006 (P1) and 1.152 (P3) at Reτ = 670 and 1300 (d+ = 56 and 116). The sphere and fluid motions are tracked simultaneously via 3D particle tracking and stereoscopic particle image velocimetry over the streamwise-spanwise plane, respectively. With sufficient mean shear, sphere P1 lifts off of the wall upon release before descending back towards the wall at both Reτ. It typically accelerates strongly over a streamwise distance of less than one boundary layer thickness before approaching an approximate terminal velocity. By contrast, the denser sphere P3 does not lift off upon release but mainly slides along the wall. At lower Reτ where wall friction is stronger, this sphere translates with unsteady velocity, significantly lagging the local fluid. The streamwise velocities of both spheres correlate strongly with the fast- and slow-moving zones that approach and move over them. In most runs, both spheres lag the local coherent structures and travel with either fast- or slow-moving zones throughout the observed trajectories. Vortex shedding, which is most prevalent for sphere P3 at Reτ = 670, is also important. The sphere spanwise motion is prompted by wall friction, spanwise fluid motion, and/or meandering of the coherent structures, and spheres do not appear to migrate preferentially into slow-moving zones.

In present work, we report an experimental setup that has been developed to characterize highly under-expanded helium and nitrogen jets confined in a vacuum chamber. The evolution of Zone of Silence (ZOS) is studied and found that experimentally measured ZOS for helium and nitrogen jets are in agreement with empirical relation and mathematical model. The velocity of the jet inside ZOS is measured using a developed time of fight (TOF) probe without using a skimmer. We are able to measures the velocity of the gas jet itself compared to a skimmer based method which measures the velocity of gas that is subjected to further expansion after the skimmer. Moreover, this method is free from skimmer interference and calibration involved with it. We observe that the measured velocity for helium is smaller than the terminal velocity. This can be attributed to losses which can occur due to nozzle geometry. Simulations are carried out using available DS2V code. Experimentally observed velocities of jets inside ZOS are compared with simulated values and are found to be in agreement within experimental uncertainties.

A new theoretical formula can be used to calculate superficial velocity for particle separation with a critical condition, and it is based on the size, density, terminal velocity, Reynold number of the particle and equipment parameters. In a continuous system, a series of fractionation experiments were conducted to quantify the separation performance for 0–2 mm coal particles under various factors, including channel spacing, solids throughput and split fluidization rate. The separation performance considers the ash content and yield of products varying with the particle size. A great fluidization environment can be created by the 6 mm channel spacing. For the solids throughput of 10.20 t/(m2h) provides a well effective separation in the narrow channels. In addition, the split fluidization of 0.0058 m/s can produce a higher shear induce force in the inclined channels to prevent the low-density particles from being lost in the underflow. The theoretical superficial velocity calculated by the new formula is 0.042 m/s, which can report the particles with the upper and lower size reach at 9-fold to the overflow, it is almost the same as the actual separation fluidization velocity of 0.04 m/s. Meanwhile, the data obtained under each separation condition is basically consistent with data of the sink-float mothed, which shows a good separation performance for the Reflux Classifier.

The dynamics of rising bubbles in a liquid sodium pool plays a significant role in the estimation of the radiological source term during anticipated transients and normal operation of a fast reactor. Sodium is chemically reactive with air and water. Therefore, carrying out experiments using sodium on a reactor scale level requires extra precautions. We report an in-house experimental technique to depict the bubble dynamics along with numerical support to establish similarity behavior between water and liquid sodium, to reduce the number of experiments in sodium systems for in-depth knowledge of bubble dynamics in the case of the fast breeder reactor. In the present work, 3-D bubble dynamics is studied in the OpenFOAM platform and the validated numerical model is used to study bubble dynamics in sodium to establish similitude between the water and sodium systems. The bubble aspect ratio of 0.9 cm diameter bubbles in the sodium system is found to be in close agreement with that of 0.5 cm bubbles in the water system. Experimental and numerical findings suggest similarity of the aspect ratio between water and scaled-up sodium systems. The similarity criteria established between water and sodium are found to be very useful in conducting water-based experiments to study bubble dynamics in sodium systems.

The freezing of a stream of free-falling monodispersed droplets was simulated through the development of numerical models in this work. Prediction of the freezing time and temperature transition of a single droplet is beneficial for optimisation of novel continuous spray freeze drying (cSFD) processes. Estimations of the vertical free-falling distance of the droplets in a slip stream and predictions of the chances of droplet coalescence greatly enhance process understanding and can be leveraged to direct equipment design and process development. A design space of droplet diameters in the range from 100 μm to 400 μm and ambient temperatures from −120 °C to −40 °C was explored. The rate of supercooling within the design space was predicted to range from 48 to 830 °C s−1 depending on the ambient temperature and droplet size. A comparison of the vertical free-falling distances of solitary droplets and streams of droplets at different temperatures showed that the terminal velocity of a vertically falling stream of droplets is always in excess of the terminal velocity of a solitary droplet of the same size. A difference of 1.35 m was predicted for the free-falling distance of a 400 μm droplet compared to a stream of droplets at −42 °C. A comparison between flow rates for consecutively generated 100 μm droplets showed that droplet coalescence was predicted at 0.05 Lh−1, whilst at 0.02 Lh−1 a separation distance of 23 μm was maintained thus preventing coalescence.

In the present research, the rising behavior of air bubble in a viscous liquid is investigated experimentally. Aqueous solutions of glycerol and CMC were used as the Newtonian and shear-thinning non-Newtonian viscous liquids, respectively. The bubble is formed via injection of air by a syringe pump and rises in the quiescent viscous liquid. The process was captured using a high-speed camera (1000 fps) and was post processed to obtain the bubble characteristics such as the center of mass and aspect ratio. The experimental results were verified using the existing literatures and the non-dimensional numbers were reduced to two (Velocity number and Flow number) by lumping the parameters. In addition, an empirical correlation was presented for the Velocity number at the moment of detachment based on the Flow number. In our experiments, three regimes were observed during the bubble rise after its detachment. In regime 1, the bubble shape oscillates between disk and hemisphere. In regime 2, the bubble gradually deforms into an ellipsoidal shape and in regime 3, the bubble shape remains nearly unchanged. Furthermore, the results indicated that the velocity behavior is influenced by the shape of the bubble: in regime 1, the velocity fluctuates. In regime 2, the velocity decreases temporarily and then increases toward the terminal velocity and finally in regime 3, the velocity increases or decreases monotonically toward the terminal velocity. Also, a criterion was found for the Velocity number in order to simultaneously predict the regime of bubble rise in a viscous liquid for Newtonian and non-Newtonian fluids.

The surface tension of seawater becomes increasingly important on the construction of ocean environment and the analysis of ocean characteristics. A non-invasive measurement method of seawater surface tension coefficient across a salinity range of 10 g/kg ≤ S ≤ 30 g/kg and a temperature range of 25.83 °C ≤ t ≤ 37.79 °C was carried out through the visual measurement system in this paper. The implicit representation of the surface tension coefficient and bubble terminal rising velocity was established on the basis of the bubble force balance theory. The motion characteristics of the bubble were calculated by high-speed photography. Additionally, the intrinsic and extrinsic parameters of the camera, as well as the coefficients of lens distortion were estimated. A new correlation of the surface tension coefficient was developed as a function of the salinity and temperature. The quantitative correlations analysis shows that the experimental results are in good agreement with the predicted values by the typical correlations, and the maximum deviation is less than 0.28%, which indicates that the method is valid and has high precision.

Flow at the molecular level induces shear-induced unfolding of single proteins and can drive their assembly, the mechanisms of which are not completely understood. To be able to analyze the role of flow on molecules, we present uniform-flow molecular dynamics simulations at atomic level. The pull module of the GRoningen MAchine for Chemical Simulations package was extended to be able to force-group atoms within a defined layer of the simulation box. Application of this external enforcement to explicit water molecules, together with the coupling to a thermostat, led to a uniform terminal velocity of the solvent water molecules. We monitored the density of the whole system to establish the conditions under which the simulated flow is well-behaved. A maximal velocity of 1.3 m/s can be generated if a pull slice of 8 nm is used, and high velocities would require larger pull slices to still maintain a stable density. As expected, the target velocity increases linearly with the total external force applied. Finally, we suggest an appropriate setup to stretch a protein by uniform flow, in which protein extensions depend on the flow conditions. Our implementation provides an efficient computational tool to investigate the effect of the flow at the molecular level.

The Maxey–Riley equation has been extensively used by the fluid dynamics community to study the dynamics of small inertial particles in fluid flow. However, most often, the Basset history force in this equation is neglected. Analytical solutions have almost never been attempted because of the difficulty in handling an integro-differential equation of this type. Including the Basset force in numerical solutions of particulate flows involves storage requirements which rapidly increase in time. Thus the significance of the Basset history force in the dynamics has not been understood. In this paper, we show that the Maxey–Riley equation in its entirety can be exactly mapped as a forced, time-dependent Robin boundary condition of the one-dimensional diffusion equation, and solved using the unified transform method. We obtain the exact solution for a general homogeneous time-dependent flow field, and apply it to a range of physically relevant situations. In a particle coming to a halt in a quiescent environment, the Basset history force speeds up the decay as a stretched exponential at short time while slowing it down to a power-law relaxation, ${\sim}t^{-3/2}$, at long time. A particle settling under gravity is shown to relax even more slowly to its terminal velocity (${\sim}t^{-1/2}$), whereas this relaxation would be expected to take place exponentially fast if the history term were to be neglected. An important mechanism for the growth of raindrops is by the gravitational settling of larger drops through an environment of smaller droplets, and repeatedly colliding and coalescing with them. Using our solution we estimate that the rate of growth rate of a raindrop can be grossly overestimated when history effects are not accounted for. We solve exactly for particle motion in a plane Couette flow and show that the location (and final velocity) to which a particle relaxes is different from that due to Stokes drag alone. For a general flow, our approach makes possible a numerical scheme for arbitrary but smooth flows without increasing memory demands and with spectral accuracy. We use our numerical scheme to solve an example spatially varying flow of inertial particles in the vicinity of a point vortex. We show that the critical radius for caustics formation shrinks slightly due to history effects. Our scheme opens up a method for future studies to include the Basset history term in their calculations to spectral accuracy, without astronomical storage costs. Moreover, our results indicate that the Basset history can affect dynamics significantly.

This paper presents the description of the experimental apparatus, instrumentation, and processing techniques developed for the study of quasi-real slug flow using Particle Image Velocimetry, High-Speed Camera and Laser Diode Photocell techniques. Most experimental studies of the flow fields around Taylor bubbles, aiming a deeper insight into slug flow pattern, analyze the situation of isolated Taylor bubbles rising in stagnant or co-current liquid flow. Furthermore, most of these studies, consider highly viscous fluids (or, in general, low values of the inverse viscosity number, N f = ρ 2 gD 3 / μ 2 ), to get “well behaved” Taylor bubbles, where flow fields are more easily measured. However, in most industrial situations slug flows are characterized by the presence of small dispersed bubbles in the liquid stream flowing together with large Taylor bubbles, and high values of the inverse viscosity number. On the other hand, the chaotic nature of a real slug flow would not allow an adequate characterization of the flow around single Taylor bubbles, mainly, due to the lack of the repeatability of the flow field measurement, making difficult the calculation of the averaged fields. Thus, in this work, a specific experimental apparatus and instrumentation were developed, which allows the study of the flow around Taylor bubbles with and without dispersed small bubbles in the liquid stream under controlled conditions. This allows the PIV measurements, around Taylor bubbles in flows with high N f numbers, be done under repeatable conditions and thus, the determination of the averaged flow fields. The laser diode photocell technique is used to synchronize the PIV system with the passage of Taylor bubbles at the test section as well as to measure the terminal velocity and length of Taylor bubbles. A dynamic masking procedure was developed to mask out the Taylor bubbles noses and tails in the PIV images, since, for high values of the inverse viscosity number ( N f ≈ 13 , 275 ) the interface fluctuations are strong and, therefore, no fixed masking can be used.

Bubble detachment, rise, and bouncing upon impact with a free surface is studied experimentally in variable gravity conditions. Previous investigations focused on the effects of fluid properties such as viscosity or surface tension on the rise and bouncing dynamics. Gravity force is a crucial factor in the detachment, rise and bouncing processes. However, the effect of different gravity levels has never been studied experimentally. In this paper we analyze the role of gravity in the detachment, rise velocity and bouncing motion of millimetric bubbles colliding with a free surface. Single air bubbles in ethanol are detached from a nozzle by the buoyancy force. After reaching a terminal velocity, the rising bubble interacts with the free surface in a bouncing process prior to coalescence. The equivalent bubble diameter at detachment decreases as the gravity level increases, in agreement with the theoretical prediction. An expression for the terminal velocity as a function of gravity is proposed. The terminal velocity is found to increase with the gravity level, although bubbles are smaller at higher values of gravity. The bouncing process has been modelled by a damped oscillator, in which the free surface acts as an elastic membrane. An expression for the frequency of bouncing as a function of gravity has been obtained, showing a good agreement with the experimental results. The motion of the bubble during the bouncing process can be approximated by an underdamped oscillator even if viscosity is negligible. Therefore, viscosity is not the main responsible for damping, which is probably due to energy transfer from the bubble to the fluid in the form of vortex and surface waves generation.

In a vast number of industrial applications, particle-laden flows consist of irregularly-shaped particles with various sizes and shapes and thus, the drag coefficient is strongly affected and different for each particle. The key issue is to incorporate these characteristics within numerical calculations. Therefore, a sample-set of irregularly-shaped combustion residues or rather clinker flakes, with sizes between 50 μm and 250 μm was experimentally and numerically investigated. Particle geometry, terminal velocity measurements and drag analysis were carried out within the present work. Areflected-light microscope and an optical particle analyzer were used for geometry analysis. Terminal velocities were measured in a water filled glass column with a commercially available digital camera with a recording rate of 1000 frames per second and a frame-by-frame tracking method. At first a simple particle classification method based on this geometrical analysis and terminal velocity measurements in a static fluid was introduced. As a result, several particle groups were defined based on the obtained data. As a second step, this data were used as input parameters for two simple drag models from literature. The required particle shape parameters were determined by velocity recalculations and data fitting. Furthermore, the applicability of these drag models in combination with the experimentally determined particle shape descriptors was evaluated within a simple test rig, where freely falling particles are exposed to a lateral compressed airflow. The particles would be deflected and settle at different positions at the bottom of the test rig. Numerical calculations of the test rig were performed using a commercial Computational Fluid Dynamics (CFD) code and a numerically efficient Euler-Lagrangian approach in order to predict the accurate motion of particles. The particle mass dispersion at the bottom of the test rig was compared to numerical results. It was shown that the used drag models in combination with the experimentally determined particle shape descriptors improved the prediction of particle drag coefficients and trajectories significantly without an increase of computational cost and time.

Laboratory measurements of drop fall speeds by Gunn–Kinzer under still air conditions with pressure corrections of Beard are accepted as the “gold standard”. We present measured fall speeds of 2 and 3 mm raindrops falling in turbulent flow with 2D-video disdrometer (2DVD) and simultaneous measurements of wind velocity fluctuations using a 3D-sonic anemometer. The findings based on six rain events are, (i) the mean fall speed decreases (from the Gunn–Kinzer terminal velocity) with increasing turbulent intensity, and (ii) the standard deviation increases with increase in the rms of the air velocity fluctuations. These findings are compared with other observations reported in the literature.

The evolution of a single raindrop falling below a cloud is governed by fluid dynamics and thermodynamics fundamentally transferable to planetary atmospheres beyond modern Earth's. Here, we show how three properties that characterize falling raindrops -- raindrop shape, terminal velocity, and evaporation rate -- can be calculated as a function of raindrop size in any planetary atmosphere. We demonstrate that these simple, interrelated characteristics tightly bound the possible size range of raindrops in a given atmosphere, independently of poorly understood growth mechanisms. Starting from the equations governing raindrop falling and evaporation, we demonstrate that raindrop ability to vertically transport latent heat and condensible mass can be well captured by a new dimensionless number. Our results have implications for precipitation efficiency, convective storm dynamics, and rainfall rates, which are properties of interest for understanding planetary radiative balance and (in the case of terrestrial planets) rainfall-driven surface erosion., Comment: submitted to JGR: Planets; 40 pages, 8 figures, 3 tables, 4 appendices; supporting information with 9 pages, 8 figures, 1 table; associated code at https://github.com/kaitlyn-loftus/rainprops

A new type of liquid–solid circulating fluidized bed, the conventional circulating fluidized bed (CCFB), which operates at a superficial liquid velocity lower than the particle terminal velocity, was numerically studied by the Eulerian-Eucerin two-fluid model. The flow structures in the upward CCFB where the liquid and solids flow upward by using heavy particles with a density higher than the liquid, and in the inverse CCFB where liquid and solids flow downward with light particles are compared in this work. Time-averaged axial and radial profiles of the liquid velocity, particle velocity, and solid holdup from the numerical results are presented. Instantaneous profiles representing the local flow structures are also provided.

In this work, we present an alternative methodology to solve the particle-fluid interaction in the resolved CFDEM ® coupling framework. This numerical approach consists of coupling a Discrete Element Method (DEM) with a Computational Fluid Dynamics (CFD) scheme, solving the motion of immersed particles in a fluid phase. As a novelty, our approach explicitly accounts for the body force acting on the fluid phase when computing the local momentum balance equations. Accordingly, we implement a fluid-particle interaction computing the buoyant and drag forces as a function of local shear strain and pressure gradient. As a benchmark, we study the Stokesian limit of a single particle. The validation is performed comparing our outcomes with the ones provided by a previous resolved methodology and the analytical prediction. In general, we find that the new implementation reproduces with very good accuracy the Stokesian dynamics. Complementarily, we study the settling terminal velocity of a sphere under confined conditions.

Droplet sliding naturally happens with practical significance in developing artificial self-cleaning surfaces or impermeable barriers. On water-repellent soil surfaces, such processes evolve at very small scales, typically at the particle level. To address this, this paper presents a two-dimensional Lattice Boltzmann (LB) study on the droplet sliding dynamics on a layer of regularly arranged particles with varying size and contact angle (CA) aimed at mimicking conditions comparable to those of real soils. The numerical droplet is initialized above the inclined granular surface with different lifting distances and deposited by gravity. The droplet hits the surface with different impacting velocities and subsequently slides down the slope. Four droplet-sliding behaviors were observed: a droplet sticks to the granular surface, a droplet moves by pinning and depinning of its interface (“stick-slip”), a droplet undergoes periodic elongation and shortening during sliding, and a droplet lifts off the granular surface and may be ruptured. For a droplet that displays the “stick-slip” behavior, the sliding velocity reaches a converged terminal velocity, which increases with a higher CA, a more inclined slope, and a smaller particle size. However, nonunique terminal velocities were identified to be affected by the impacting velocities, but their correlation is not continuous and may not be positive. Finally, we propose to quantify the rotational or translational movement by effective kinematic ratio (EKR), which is defined as the translational kinematic energy divided by the total kinematic energy. The unique relation between the EKR and the terminal velocity is suggested to be one practical indicator to intrinsically characterize the water repellency at the particle level.

The settling dynamics of falling spheres inside a Laponite suspension is studied. Laponite is a colloidal synthetic clay that shows physical aging in aqueous suspension due to the spontaneous evolution of inter-particle electrostatic interactions. In our experiments, millimeter-sized steel balls are dropped in aqueous Laponite suspensions of different ages (i.e., time elapsed since sample preparation). The motion of the falling balls are captured using a high-speed camera and the velocities of their centroids are estimated from the images. Interestingly, we observe that balls of larger diameters fail to achieve terminal velocity over the entire duration of the experiment. We propose a mathematical model that accounts for rapid structural changes (expected to be induced by the falling ball) in Laponite suspensions whose aging time scales are much slower than the time of fall of the ball. For a range of ball sizes and Laponite suspension ages, our model correctly predicts the time-dependence of the ball velocity. Furthermore, fits to our model allow us to estimate the rates of destructuring of the thixotropic suspensions due to the passage of the falling ball., 9 pages, 6 figures