Terminal velocity of porous spheres

Terminal velocity of porous spheres was experimentally measured for a Reynolds number range of 0.2 to 120 for a normalized sphere radius, β = R/R of 15.6 to 33, where R and k are the sphere radius and permeability, respectively. The drag coefficient for 15 < β < 33 was found to be C_D = 24Ω/Re [1 + 0.1315 Re^{(0.82 - 0.05w)}] for 0.1 < Re ≤ 7 and C_D = 24Ω/Re [1 + 0.0853 Re^{(1.093 - 0.105w)}] for 7 < Re < 120 with w = log₁₀Re where Re is the sphere Reynolds number and Ω=2β² [1 - (tanh β/β)] / 2β² + 3[1 - tanh β/β)] At high Reynolds numbers, it was found that the porous sphere terminal velocity was less affected by the container walls than for the case of an impermeable sphere. However, at very low Reynolds numbers, the wall effects were found to be similar for both the permeable and the impermeable spheres.     

Mass transfer in gas-fluidized beds: measurement of actual driving forces

Concentration profiles of naphthalene vapors were measured in the flow of air through fluidized beds of naphthalene spheres. Seven different sizes of spheres were used in this study and ranged from 0.02480 to 0.2000 cm. To avoid saturation conditions in the air leaving the bed, the naphthalene spheres were diluted in a matrix of inert beads (styrene divinylbenzene copolymer) of the same size and density as the naphthalene spheres. The established concentration profiles were graphically integrated to produce from them actual mass transfer coefficients. These transfer coefficients were then used to calculate j_d, mass transfer factors, for this fluidized bed system and also the product j_dRe_f which has been found to depend on Re_mf, the corresponding Reynolds number at minimum fluidization conditions. Actual driving forces, (Δp)_a were normalized with the corresponding log-mean values, (Δp)_m, to produce the driving force factor, F = (Δp)_a/(Δp)_m. This factor has been correlated with the parameters of the system and can be predicted with a certainty of 10.5%.     

Coupled CFD–DEM of particle-laden flows in a turning flow with a moving wall

The nature of the particle–solid interactions and particle–fluid interactions in rectangular duct bend geometry with/without a moving wall is studied, taking into account particle collision, colloidal, and hydrodynamic forces, and four way coupling between the fluid flow and particles. The focus is on systems where particles and fluid phase have similar length scales, fluid Reynolds number (Re_f) ∼ 1, and particle's Stokes number (St) ∼ 1. Particles move toward the walls of the channel near the bend, and have long residence times in these regions. Buoyancy force has negligible effect on particle motion, where adhesion and drag forces lead to particle motion and agglomeration patterns. The effect of a free surface on agglomeration sites in the turning flow is elucidated.     

Effect of Drag–Reducing Polymers on the Rate of Liquid-Solid Mass Transfer in Fixed Beds of Spheres under Forced Convection Conditions

The effect of drag–reducing polymers on the rate of liquid – solid mass transfer in a packed bed reactor under forced convection conditions was studied by measuring the rate of diffusion–controlled dissolution of copper spheres in acidified chromate solutions. The variables investigated were superficial liquid velocity, sphere diameter, bed height, and polymer concentration. The mass transfer coefficient was found to increase with increasing superficial liquid velocity. Increasing both sphere diameter and bed height were found to decrease the mass transfer coefficient. Polymer addition was found to decrease the rate of mass transfer by an amount ranging from 29.2 to 56.9% depending on superficial liquid velocity and polymer concentration. Mass transfer data were correlated in absence and in the presence of drag–reducing polymer, using the following equations, respectively: J_d = 3.71Re^–0.54 and, J_d = 2.5 Re^–0.61where J_d is mass transfer J-factor and Re is the Reynolds number.     

Effect of blockage on drag and heat transfer from a single sphere and an in-line array of three spheres

The effect of blockage ratio on the steady flow and heat transfer characteristics of incompressible fluid over a sphere and an in-line array of three spheres placed at the axis of a tube has been investigated numerically. The Navier-Stokes and thermal energy equations have been solved numerically using FLUENT for the following ranges of parameters: for a single sphere, 2 ≤ β ≤ 10; 1 ≤ Re ≤ 100; for the three-sphere system, for two values of sphere-to-sphere distance, namely s = 2 and 4. All computations were carried out for two values of the Prandtl number, i.e., 0.74 and 7, corresponding to the flow of air and water respectively. Extensive results on streamline patterns, wake characteristics (angle of separation and recirculation length), drag coefficient and Nusselt number are presented to elucidate the interplay between the blockage and the Reynolds number and their influence on drag and Nusselt number.     

Drag coefficients of irregularly shaped particles

The steady-state free-fall conditions of isolated groups of ordered packed spheres moving through Newtonian fluids have been studied experimentally. Measurements of the drag coefficients are reported in this paper for six different geometrical shapes, including isometric, axisymmetric, orthotropic, plane and elongated conglomerates of spheres. From these measurements, a new and accurate empirical correlation for the drag coefficient, C_D, of variously shaped particles has been developed. This correlation has been formulated in terms of the Reynolds number based on the particle nominal diameter, Re, the ratio of the surface-equivalent-sphere to the nominal diameters, d_A/d_n, and the particle circularity, c. The predictions have been tested against both the experimental data for C_D collected in this study and the ones reported in previous works for cubes, rectangular parallelepipeds, tetrahedrons, cylinders and other shapes. A good agreement has been observed for the variously shaped agglomerates of spheres as well as for the regularly shape particles, over the ranges 0.15<Re<1500, 0.80<d_A/d_n<1.50 and 0.4<c<1.0.     

Lift force in bubbly flow systems

From the significance of three-dimensional simulation of dispersed flow systems in many engineering fields, extensive study was conducted for lift force in a single particle system as well as a multiparticle system. In this study, the lift force in a single particle system was modeled by considering the effect of bubble deformation on the lift force. The model was finalized based on existing data obtained in the range of particle Reynolds number from 3.68 to 78.8, viscous number from 0.0435 to 0.203 and Eötvös number from 1.40 to 5.83. The viscous number is defined by where μ_f, ρ_f, σ, g and Δρ are, respectively, fluid viscosity, fluid density, surface tension, gravitational acceleration and density difference between phases. The applicability of the model to higher particle Reynolds number system such as an air-water system was qualitatively examined. The lift force model developed in a single particle system was extended to a multiparticle system. The applicability of the extended lift force model was qualitatively examined. The similarity between drag and lift forces were also discussed.     

Hydrodynamics of inhomogeneous agglomerates for a fractal dimension of 2.5 at intermediate Reynolds numbers: Effect of agglomerate sizes and Reynolds numbers

In this study, the effect of agglomerate sizes for a fractal dimension (D_f) of 2.5 on the hydrodynamics at intermediate Reynolds numbers (Re) of 1–120 was assessed. The results show that a core behaves like a solid sphere that exists in the central region inside the agglomerate. In addition, increasing the agglomerate diameter represents adding an extra permeable layer outside the agglomerate. For a larger Re or a smaller agglomerate diameter, the fluid can enter and penetrate through the agglomerate more easily, and the hydrodynamic characteristics of agglomerates deviate more from those of solid spheres. The effect of diameters on the velocity and pressure profiles becomes less significant with the increase in the diameter. Based on the simulated results, the drag ratio has an approximately linear relationship with Re, and its intercept has an exponential relationship with the dimensionless agglomerate diameter. Compared with homogeneous porous spheres, the drag ratio of the agglomerate is different. The effect of diameters on the drag ratio decreases as the diameter increases. It should be noted that the effect of radially varying permeability on inhomogeneous agglomerates should not be ignored and that the effect weakens as Re increases.     

Velocity histories of vaporizing fuel drops moving through stagnant gas

Newton's second law of motion has been applied here to a vaporizing drop being depleted according to the ‘d²-relation’ in order to determine the velocity histories of drops injected vertically into stagnant gas. In the evaluation of the drag forces, three possible values of drag coefficient, namely Stokes value based on instantaneous velocity and diameter, Stokes value based on average velocity and initial diameter, and a constant value of 1.0 independent of Reynolds number (for which there is experimental justification at Reynolds numbers not too small), have been employed. The second and third cases result in non-linear differential equations for the velocity histories, and have been solved using the Runge-Kutta-Nyström method. The effects on the histories of the initial velocity with which the drops are injected into the ambient gas, either vertically down or up, have been investigated for all three drag values. The results have been plotted in terms of normalized coordinates which (unlike others in the literature) are shown to have physical significance. The concept of a terminal velocity does not apply for a particle whose size is decreasing with time. The velocity histories for the two Stokes drag coefficients are quite similar, despite the considerably different manner in which the Reynolds number is defined in the two cases; this vindicates the common practice of calculating and correlating experimental values of (Re) and C_D on the basis of initial diameter and average velocity for the sake of convenience and practicality. For large downward injection velocities, the gravitational and buoyancy forces can be neglected, especially for the two cases of Stokes C_D; but the errors for all three cases of upward injection are too significant to be neglected.     

A numerical study of the accelerating motion of a dense rigid sphere in non-newtonian power law fluids

The equations of motion of an accelerating sphere falling through non-Newtonian fluids with power law index n in the range 0.2 ≤ n ≤ 1.8 were integrated numerically using the assumption that the drag on the sphere was a function of both power law index and terminal Reynolds number, Re_t For 10^?2 ≤ Re_t ≤ 10³ both dimensionless time and distance travelled by the sphere under transient conditions showed a much stronger dependence on the flow behaviour index, n, for shear-thinning than for shear-thickening fluids. The form of this dependence is investigated here. Furthermore, results in four typical shear-thinning fluids suggested a strong correlation between the distance and time travelled by the sphere under transient conditions and the value of the fluid consistency index. The analysis reported herein is, however, restricted to dense spheres falling in less dense fluids, when additional effects arising from the Basset forces can be neelected.     

The voidage function and effective drag force for fluidized beds

Here, an experimental investigation on the effective drag force in a conventional fluidized bed is presented. Two beds of different particle size distribution belonging to group B and group B/D powders were fluidized in air in a diameter column. The drag force on a particle has been calculated based on the measurement of particle velocity and concentration during pulse gas tests, using twin-plane electrical capacitance tomography. The validity of the voidage function “correction function”, (1−ε_s)ⁿ, for the reliable estimation of the effective drag force has been investigated. The parameter n shows substantial dependence on the relative particle Reynolds number , and the spatial variation of the effective static and hydrodynamic forces. It is also illustrated that, a simple correlation for the effective drag coefficient as function of the particle Reynolds number (Re_p), expressed implicitly in terms of the interstitial gas velocity, can serve in estimating the effective drag force in a real fluidization process. Analysis shows that, the calculated drag force is comparable to the particle weight, which enables a better understanding of the particle dynamics, and the degree of spatial segregation in a multi-sized particle bed mixture. The analogy presented in this paper could be extended to obtain a generalized correlation for the effective drag coefficient in a fluidized bed in terms of Re_p and the particle physical properties.     

Gas desorption from liquids: Mass transfer and drag coefficients for single bubbles in free rise through newtonian liquids

Small CO₂ bubbles have been generated in quiescent water and 8.0 and 21.0 wt. % aqueous sucrose solutions supersaturated with CO₂. Measurements have been made of the free rise velocity and growth rate of the bubbles in the range 60–1170 μm in diameter by a photographic method. The experimental results for the CO₂-water system agree well with theoretical and empirical equations for rigid spheres in creeping flow when d_B is less than about 150 μm and begin to approach the equations for partially contaminated or clean fluid spheres as d_B or Re increases. Similar results are obtained for CO₂ bubbles in aqueous sucrose solutions. The critical Reynolds number at which the bubble surface begins to move decreases with increases in liquid viscosity and this can be interpreted in terms of the stagnant cap model. As Re further increases, experimental results agree well with the predictions for spherical bubbles free from surfactant in potential flow before finally beginning to deviate from them.     

The Voidage Function for the Drag Force Ratio in a Liquid‐Fluidized Bed

Di Felice (1994) has shown that the ratio of the drag coefficient, C_D, on a sphere in a liquid‐fluidized bed of uniform spheres to the drag coefficient, C_DS, on the same sphere in isolation and subjected to the same superficial liquid velocity, u, is given by a function ?^?β, where β was expressed as an empirical function of the particle Reynolds number, Re = duρ/µ. Here it is shown that C_D/C_DS is well approximated by ?^?mm, where the Richardson‐Zaki index n is a function of the terminal free‐settling Reynolds number, Re_t = du_tρ/µ, and m is 2 plus the slope of the standard log C_DS vs. log Re plot at plot at Re = Re_t. The present model, using the best experimentally confirmed equation for n and a new simple equation for and a new simple equation for m, is compared with that of Di Felice in their respective abilities to predict liquid‐fluidized bed expansion.     

Behavior of Compact Nonspherical Particles in the TSI Aerodynamic Particle Sizer Model APS33B: Ultra-Stokesian Drag Forces

The aerodynamic behavior of aggregates consisting of uniform polystyrene latex (PSL) spheres and unaggregated cuboidal Natrojarosite particles in a TSI aerodynamic particle sizer (Model APS33B) has been studied. In initial tests, monodisperse PSL micro-spheres ranging from 0.3 to 7 μm in geometric diameter were generated from aqueous suspensions using a Lovelace nebulizer. APS33B responses for these uniform-sized particles showed multiple peaks. The major (primary) peak, which resulted from the smallest particle, corresponded to the unaggregated single spheres (singlets); the second, third, and fourth peaks were identified as doublets, triangular triplets, and tetrahedral quadruplets, respectively. Both doublets and triplets moved with their long axes in perpendicular (maximum drag) orientation to the flow direction in the APS33B. In contrast, the tetrahedral particles were isometric and had the same dynamic shape factor (drag resistance) for all three primary orientations. The particle Reynolds numbers (Re _p) for these particles were calculated and ranged from 0.2 to 30 in the sensing volume of the APS33B detector (i.e., ultra-Stokesian conditions). Ultra-Stokesian drag forces for all three types of aggregates were, therefore, estimated and expressed as a function of an empirical factor (1 + aRe ^b _p) to the Stokesian drag force. The ultra-Stokesian drag of a Natrojarosite particle was measured in the range 20 Re _p < 50 and could be described with a similar expression. This approach facilitates the study of the dynamic behavior of nonspherical particles and yields new information about the characteristics of drag forces in the ultra-Stokesian regime     

Coke formation on the surface of α-Al2O3 in the catalytic pyrolysis of naphtha

The catalytic pyrolysis of naphtha has been carried out in a quartz reactor loaded with 5 mm α-A1₂0₃ spheres. The yields of ethylene and propylene exhibit about 10% and 5% higher values compared to the thermal pyrolysis in the absence of α-A1₂0₂ spheres at the same reaction conditions. The coke formation on α-A1₂0₃ spheres increased continuously along with the axial length of the reactor as well as with reaction time. Results suggest that the concentration of the coke precursors in the gas phase may increase along with the axial length of the reactor. Coke filled up completely the internal pore of the sphere near the exit of the reactor after reaction for 4 hr. The coke film on the external surface of the sphere grew continuously thicker. The coke formation was influenced strongly by the physical properties of the α-Al₂O₃ spheres. Coke deposition was the least on the α-A1₂0₃ sphere with the lowest surface area and pore volume among the tested α-A1₂0₃ spheres.     

Finite element simulation of deep bed filtration

In this paper, the numerical model for separation efficiency and transport in periodic porous media is studied. Finite element method was used to simulate the development of a predictive model of behavior of porous media during injection of particles. This paper describes the effects of injected particle size, Reynolds number and particle drag coefficient. The numerical results show that the separation efficiency increased with injected particle size increase. The separation efficiency is found to increase with increasing Reynolds number. For the effect of drag force, C_D, in porous media, numerical results show that for C_D<10 and C_D>100, the separation efficiency is not affected by drag coefficient in the range of drag coefficient from 10 to 100, and the separation efficiency significantly depends on the Reynolds number.     

Inertial migration and multiple equilibrium positions of a neutrally buoyant spherical particle in Poiseuille flow

The radial migration of a single neutrally buoyant particle in Poiseuille flow is numerically investigated by direct numerical simulations. The simulation results show that the Segré and Silberberg equilibrium position moves towards the wall as the Reynolds number increases and as the particle size decreases. At high Reynolds numbers, inner equilibrium positions are found at positions closer to the centerline and move towards the centerline as the Reynolds number increases. At higher Reynolds numbers, the Segré and Silberberg equilibrium position disappears and only the inner equilibrium position exists. We prove that the inner annuluses in the measurements of Matas, Morris & Guazzelli (J. Fluid Mech. 515, 171–195, 2004) are not transient radial positions, but are real equilibrium positions. The results on the inner equilibrium positions and unstable equilibrium positions are new and convince us of the existence of multiple equilibrium radial positions for neutrally buoyant particles.     

Drag Force and Slip Correction of Aggregate Aerosols

The drag force on aggregate particles of uniform spheres was measured in a Millikan apparatus as a function of Knudsen number. Our experiment was designed to study the effect of particle orientation on the slip correction factor of nonspherical particles. The velocities of charged particles in a gravitational field with and without an applied electrical field were measured. An electrical field strength of 2000 V/cm was used to align doublet and triplet particles. Results showed that an aggregate particle moved in random orientation while in the gravitational field. The same particle moved with its polar axis parallel to the electric field (doublets) or with its plane of centers parallel to the electrical field (triangular triplets). Using a nonlinear regression method, both the dynamic shape factor and slip correction factor could be determined separately from the data. The dynamic shape factors at different orientations were in good agreement with those obtained previously in a sedimentation tank. The slip correction factor of singlet particles agreed with results previously obtained by Allen and Raabe for latex particles. Slip correction factors of doublets and triangular triplets can also be expressed in the Knudsen-Weber form: 1 + 2λ/d _a [1.142 + 0.558 exp(?0.999 d _a/2λ)]. The adjusted sphere diameter d _a was 1.21 d ₁ (primary diameter) for doublets moving parallel to the flow and 1.31 d ₁ for doublets randomly oriented. These results show that the slip correction factor of a nonspherical particle depends on the orientation and confirm the theory proposed by Dahneke.     

Sedimentation in a dilute emulsion

In the present work the method of Batchelor[1] is adopted to calculate the statistical mean speed of sedimentation in a dilute dispersion of identical spherical droplets. The mean settling speed is shown to be

where U₀ is the settling velocity of a single fluid sphere falling under gravity through an unbounded quiescent fluid, σ is the ratio of the viscosities μ′/μ where the upper prime indicates the disperse phase, c is the volume fraction of the spheres, and K (σ) is a numerical factor whose values are displayed in Table 1.An exact evaluation of K (σ) requires knowledge of the solutions for the drag force components acting on two fluid spheres in the directions of their line of centre and perpendicular to it. Such a solution for the latter case is not yet available, hence, an approximate solution based on the ‘method of reflections’ is used. This causes some error in the numerical results for K (σ), the magnitude of which is examined and discussed.