Settling length for turbulent flow of air in an annulus

The settling length, or distance downstream from the entrance required for the development of the velocity profile, has been determined for four square entrance sectioned annuli, with diameter ratios of 0.2, 0.3, 0.5 and 0.7. Using air, a Reynolds number range of 5,000 to 50,000 was covered, and correlations have been obtained relating the settling length, equivalent diameter, Reynolds number and diameter ratio. The expression L/De = 0.795 R_e³⁷⁴(D₁/D₂)^?0.60 correlated the data for Reynolds numbers lower than 22,000 and for Reynolds numbers greater than this the relation was L/D_e = 15.96 R_e^.077(D₁/D₂)^?624.     

Quantification of laminar mixing in the kenics static mixer: An experimental study

The laminar flow field in a Kenics KM static mixer has been studied using laser induced fluorescence and digital image analysis. Mixing was quantified by measurement of the number average striation thickness, variance of striation widths and interfacial area, for elements of length to diameter (L/D) ratios of 0.8, 1.0, 1.5 with 90° twist per element. From flow visualisations, transitions were observed in the flow where vortices developed above the first and second elements at Reynolds numbers of 43 and 90 for L/D = 0.8 and Reynolds numbers of 55 and 105 for L/D = 1.0. It was found that these vortices did not appreciably enhance mixing based on striation thickness and variance of striation widths measurements after 4 to 5 elements. The influence of viscosity ratio showed a viscosity ratio (dyed stream/bulk stream) of I had faster interfacial area growth and created more uniform mixtures compared to a viscosity ratio of 0.2 for flow rate ratio of 0.2.     

Wall effect for the fall of spheres in cylindrical tubes at high reynolds number

New experimental results on the wall effect for sphere motion in cylindrical tubes are presented and discussed for the conditions d/D ≤ 0.9 and Re_m ≤ 20000. Extensive comparisons with previous studies have been carried out to evaluate their predictability and to demonstrate the utility of the present results. The wall factor, defined as the ratio of settling velocity in an unbounded medium to that measured in a cylindrical tube, is found to depend on sphere-to-tube diameter ratio and on sphere Reynolds number. However, for small values of the Reynolds number (Re ≤ 0.5), as well for large values (Re ≥ 1000), the Reynolds number dependence of the wall factor disappears; in these regions, only the dependence on diameter ratio remains.     

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.     

Hindered settling in non-newtonian power law liquids

New experimental results on the hindered settling of model glass bead suspensions in non-Newtonian suspending media are reported. The data presented encompass the following ranges of variables: 7.38 × 10^?4 ≤ Re_1∞ ≤ 2; 0.0083 ≤ d/D ≤ 0.0703; 0.13 ≤ C ≤ 0.43 and 1 ≥ n ≥ 0.8. In these ranges of conditions, the dependence of the hindered settling velocity on concentration is adequately represented by the corresponding Newtonian expressions available in the literature. The influence of the power law flow behaviour index is completely embodied in the modified definition of the Reynolds number used for power law liquids.     

Prediction of single particle settling velocities through liquid fluidized beds

Single particle settling velocities through water fluidized beds of mono-sized glass spheres (d_p = 0.645, 1.20, 1.94, 2.98 and 5 mm in diameter) were studied experimentally using a column, 40 mm in diameter. The settling spherical particles (D_p = 10 and 19.5 mm) had different densities (1237 to 8320 kg/m³), while the settling particles (D_p = 5 and 2.98 mm) were glass spheres. The pseudo-fluid model, which considers a liquid fluidized bed as a homogenous pseudo-fluid, predicts single particle settling velocities quite well if the ratio D_p/d_p is larger than about 10. With decreasing ratio D_p/d_p, the overall friction between the settling particle and the fluidized media increases. A method for predicting single particle settling velocities through a liquid fluidized bed is proposed and discussed. Following the approach of Van der Wielen et al. [L.A.M. Van der Wielen, M.H.H Van Dam, K.C.A.M. Van Luyben, On the relative motion of a particle in a swarm of different particles, Chem. Eng. Sci. 51 (2006) 995-1008], the overall friction is decomposed into a particle-fluid and a particle-particle component. The effective buoyancy force is calculated using the transition function proposed by Ruzicka [M.C. Ruzicka, On buoyancy in dispersion, Chem. Eng. Sci. 61 (2006) 2437-2446]. A simple model for predicting the collision force is proposed, as well as a correlation for the collision coefficient. The mean absolute deviation between the experimental and calculated slip velocities was 5.08%.     

On the Settling Velocity in a Nonstationary Atmosphere

A general drag coefficient has been used in the equation of motion for solid spherical particles. The time constants, stopping times, and settling velocities in a still atmosphere are computed for a wide range of Reynolds numbers. The settling times are compared with the times calculated when a particle is falling in a fluctuating atmosphere. It is found that such particles will get significantly longer settling times owing to an enhancement in the drag coefficient caused by an increase of the relative velocity between the particle and the fluid. Surprisingly, this enhancement is present for a horizontal wind field due to a coupling between particle motion in different directions, but it is also present for a vertical field. The effect is most pronounced in the intermediate Reynolds number region, slightly above the Stokes range, where the increase in settling time can be more than 10% for certain fluctuation frequencies and amplitudes. This indicates that such particles must be carefully treated when they are falling in a nonstationary medium     

Mist flow visualization for round jets in Aerosol Jet® printing

With the microdroplets of water serving as light scattering particles, the mist flow patterns of round micro-jets can be visualized using the Aerosol Jet^® direct-write system. The visualization images show that the laminar mist jet (with sheath-to-mist ratio Y?=?1:1) appears to extend to more than 20 times the diameter of nozzle orifice D for jet Reynolds number Re?<?600, especially with D?=?0.3?mm and less. For smaller jets (e.g., with D?=?0.15?mm), laminar collimated mist flow might be retained to 40×D for Re?<?600 and for Re ～1500 within 20×D from the nozzle. The laminar part of mist flow associated with larger jets (e.g., with D?=?1.0?mm for Re?<?600) tends to exhibit noticeable gradual widening due to viscous diffusion. For free jets, their breakdown length—the distance from nozzle where transition from laminar to turbulent mist flow takes place as signaled by a rapid widening of mist stream—is shown to decrease with increasing Re. The presence of impingement wall tends to prevent turbulence development, even when the wall is placed further downstream of the free-jet breakdown length for a given Re. The critical Re for an impinging jet to develop turbulence increases as the standoff S is reduced. The mist flow of impinging jet of D?=?1.0?mm seems to remain laminar even for Re?>?4000 at S?=?12?mm.

Copyright © 2018 American Association for Aerosol Research     

Sedimentation of a sphere along the axis of a long square duet filled with non-newtonian liquids

Based on extensive experimental results, it is shown that the retardation effect caused by the confining walls on the free settling velocity of a sphere is smaller with square walls than that with cylindrical boundaries. This is true for both Newtonian and power law fluids, provided the particle Reynolds number is small (< about 5). The values of the wall factor for Newtonian liquids are in excellent agreement with theory (up to R / L ≤ 0.1) while those for power law fluids have been correlated empirically via a linear relationship. The results reported here encompass the following ranges of conditions: 1 ≥ n ≥ 0.7; Re < 15 and 0.024 < R/L < 0.238.     

Terminal settling velocity and drag coefficient of biofilm‐coated particles at high Reynolds numbers

The drag force (F_d) on bio‐coated particles taken from two laboratory‐scale liquid–solid circulating fluidized bed bioreactors (LSCFBBR) was studied. The terminal velocities (u_t) and Reynolds numbers (Re_t) of particles observed were higher than reported in the literature. Literature equations for determining u_t were found inadequate to predict drag coefficient (C_d) in Re_t > 130. A new equation for determining F_d as an explicit function of terminal settling velocity was generated based on Archimedes numbers (Ar) of the biofilm‐coated particle. The proposed equation adequately predicted the terminal settling velocity of other literature data at lower Re_t of less than 130, with an accuracy >85%. © 2010 American Institute of Chemical Engineers AIChE J, 2010     

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.     

Caffeine Molecular Imprinted Microgel Spheres by Precipitation Polymerization

Imprinted uniform microgel spheres were prepared by precipitation polymerization. Acetonitrile was used as the dilute solvent with MAA as the monomer, EDMA as the crosslinker and caffeine as the print molecule. Comparison of caffeine adsorption on molecular imprinted and blank microgel spheres was made. Langmuir model was used to fit the adsorption data. It was found that the caffeine imprinted microgel spheres show specific binding sites to the target molecules. A binding study of caffeine on imprinted microgel spheres was made by Scatchard analysis; the dissociation constants (K_D) and the maximum binding capacity were K_D= 1.84×10⁻⁴mol/L,Q _max = 16.98 μmol/g for high affinity binding site and K_D=1.33×l0⁻³ mol/L, Q_max=46.84 μmol/g for lower affinity binding site, respectively This microgel spheres can be useful affinity adsorbents in further applications.     

Liquid phase residence time distribution for gas–liquid flow in Kenics static mixer

The residence time distribution (RTD) of the liquid phase for co-current gas–liquid upflow in a Kenics static mixer (KSM) with air/water and air/non-Newtonian fluid systems was investigated. The effect of liquid and gas superficial velocities on liquid holdup and Peclet number was studied. Experiments were conducted in three KSMs of diameter 2.54 cm with 16 elements and 5.08 cm diameter with 8 and 16 elements, respectively, of constant L_e/D_e = 1.5 for different liquid and gas velocities. A correlation was developed for Peclet number, in terms of generalized liquid Reynolds number, gas Froude number and liquid Galileo number, where as for liquid holdup, a correlation was developed as a function of gas Reynolds number. The axial dispersion model was found to be in good agreement with the experimental data.     

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.     

Electromigration of sodium ions and electro-osmotic flow in water-saturated, compacted Na-montmorillonite

Sodium ions spiked with ²²Na as a tracer were migrated by electromigration and electro-osmosis in the water-saturated compacted Na-montmorillonite at dry densities 1.0×10³ kg m⁻³, under an electric potential gradient. Dissolved helium was also migrated by electro-osmosis in the montmorillonite. After migration, concentration profiles of the sodium ions and helium were obtained by γ-spectrometry and mass-spectrometric methods, respectively. From the profiles of both chemical species, not only migration due to electrokinetic phenomena but also mechanical dispersion was observed in the montmorillonite. The dispersion coefficients, D_i, and apparent migration rates, U_i^a, of ²²Na and helium were found in the compacted Na-montmorillonite at 1.0×10³ kg m⁻³. The migration of helium in the montmorillonite under an electric potential gradient reflects that of water because helium migrates as an electrically neutral species. The parameters D_He^m, U_He^a, and α_He correspond to those of water. The mechanical dispersion coefficients, D_Na^m, of ²²Na⁺ ions are much smaller than those of water obtained by helium. The dispersivity parameters, α_Na, for ²²Na⁺ obtained from these D_Na and U_Na^a values are 10⁻⁵ m and those for water (α_He) are 10⁻³ m. This indicates that ²²Na⁺ ions migrate in different spaces than water in the compacted montmorillonite under a potential gradient. This finding suggests that the migration of Na⁺ ions occurs in the interlayer and/or on the outer surfaces of the montmorillonite; whereas dissolved helium migrates in the pore water.     

Hydrodynamic transport of cylindrical capsules in a vertical pipeline

Data are presented on the flow phenomena associated with the hydraulic transport of right cylindrical capsules, having d/D and L/d ratios, and relative density in the range of 0.49 to 0.82, 4 to 14, and 1.15 to 2.7 respectively, in a 7.6 cm 1.D. vertical pipeline with flow Reynolds numbers up to 3.6 × 10⁵. On the basis of the experimental results the effect of the individual pertinent variables which affect the velocity ratio Rv, pressure gradient ratio Rp and the unit energy requirement E for the capsule are discussed. Semi-empirical correlations between the capsule velocity and the capsule gradient and the pertinent variables of the system are also presented.     

Measurements of pollen grain dispersal in still air and stationary, near homogeneous, isotropic turbulence

The dispersal of ragweed, pine and corn pollen as well as polystyrene spheres in still air and stationary, near homogeneous, isotropic turbulence (HIT) was investigated using high-speed, digital inline holographic cinematography enabling Lagrangian tracking of the particles. Mean still air settling velocities were similar as reported literature values. Small discrepancies were most likely related to species/size differences and water content of the grains. Near-HIT was generated by loudspeakers mounted on the corners of a 40 cm³ chamber and the turbulent flow field at the center of the chamber was validated using stereoscopic Particle Image Velocimetry (PIV). Results showed near homogeneity and near isotropy with mean velocities 5–10 times smaller than the corresponding rms values of velocity fluctuations. The turbulent kinetic energy dissipation rate was determined from the PIV data sets and used to calculate the Kolmogorov scales and Taylor microscales. Experiments were carried out for two different loudspeaker amplifications corresponding to Taylor microscale Reynolds numbers, R_λ=144 and 162, respectively. The mean settling velocity in turbulent conditions was in all cases higher than the corresponding still air value, the difference becoming smaller as particle Stokes numbers increased. For the present conditions, the still air particle settling velocity was lower than the rms values of air fluctuating velocities. As a result, dispersion was dominated by inertia and for a given R_λ, particle fluctuating velocity autocorrelations fell more rapidly as the particle Stokes number decreased; corresponding particle diffusion coefficients also decreased. Transverse particle diffusion coefficients were lower than those in the direction of gravity in agreement with the continuity effect. Under the present range of experimental parameters, results showed that inertial particles (0.6<St<11) in highly turbulent conditions disperse more effectively than the air.     

Electrochemical wall mass transfer in liquid particulate systems

Ionic mass transfer coefficients between the wall of a 2.081 inch tube and liquid fluidized beds of lead glass, soda glass and lucite spheres have been measured using the diffusion-controlled reduction of ferricyanide ion at a nickel cathode for porosities 0.90 to 0.45 and Schmidt numbers 580 to 2100. The developed fluidization mass transfer coefficient for 41 < D_T/d_p < 105 were correlated by i_D E = 0.274 Re_H^?0.38 for 10 < Re_H <1600 and by J_D E = 0.455 Re_H^?0.44 for 16.7 < D_T/d_p < 27 and 50 < Re_H < 3500. Re_H is the hydraulic Reynolds number = d_H up/μ_E and d_H is D_T E/[1 + (3/2) ((1–E)) (D_T/d_p)). The distinct effect of D_T/d_p ratio is attributed to wall effects and the non-particulate behaviour of the fluidized bed for D_T/d_p < 27. Measurements in the open pipe and packed bed agreed very well with literature values. The packed bed gives highest mass transfer coefficients at given Re_H.     

Mass transfer characteristics of low H/D bubble columns

The mass transfer characteristics of 0.2, 0.6 and 1.0 m diameter bubble columns having a low height to diameter ratio (0.6 < H/D < 4) and operated at low superficial gas velocities (0.01 < V_G < 0.08 m/s) were investigated. Different types of spargers were used to study their effect on the column performance. The values of effective interfacial area, a , and volumetric mass transfer coefficient, k_L a , were measured by using chemical methods. The values of a and k_L a were found to vary from 40 to 420 m²/m³ of clear liquid volume and from 0.01 to 0.16 s^?1, respectively, in the range of V_G, and V_L covered in this investigation. The value of the liquid-side mass transfer coefficient, k_L, was found to vary from 3 × 10^?4 to 7 × 10⁴ m/s. The effect of the physical properties of the system on the values of a was also investigated. The height to diameter ratio and the column diameter did not have significant effect on the values of gas holdup, a and k_L a . It was found that the sparger design is not of critical importance, provided multipoint/multiorifice gas spargers are used. The comparative performance of bubble columns having low H/D with horizontal sparged contactors and tall bubble columns has been considered.     

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

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