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.     

Wall effects on the velocities of a single sphere settling in a stagnant and counter-current fluid and rising in a co-current fluid

Experimental results were obtained on the steady settling of spheres in quiescent media in a range of cylindrical tubes to ascertain the wall effects over a relatively wide range of Reynolds number values. For practical considerations, the retardation effect is important when the ratio of the particle diameter to the tube diameter (λ) is higher than about 0.05. A new empirical correlation is presented which covers a Reynolds number range Re = 53-15,100 and a particle to tube diameter ratio λ < 0.88. The absolute mean deviation between the experimental data and the presented correlation was 1.9%. The well-known correlations of Newton, Munroe and Di Felice agree with the presented data reasonably well. For steady settling of spheres in a counter-current water flow, the slip velocity remains practically the same as in quiescent media. However, for rising spheres in a co-current water flow, the slip velocity decreases with increasing co-current water velocity, i.e., the wall factor decreases with increasing co-current water velocity. Consequently, the drag coefficient for rising particles in co-current water flow increases with increasing water velocity.     

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.     

The influence of fluid elasticity on wall effects for creeping sphere motion in cylindrical tubes

Experimental results are presented of wall effect for the slow motion of spheres in elastic, constant-viscosity liquids. The results are correlated in terms of diameter ratio for d/D < 0.3, and Weissenberg number We < 5. Weissenberg number is defined as We = 2θV_m/d, with θ the Maxwellian relaxation time (θ = N₁/2τγ). The wall effect is found to be adequately described by Newtonian expressions for small Weissenberg number, We < 0.01. For larger values of the Weissenberg number, We > 0.2, virtually no wall effect is discernible; the small effect observed is correlated by the wall factor expression The wall effect observed is ascribed to the influence of fluid elasticity alone, since all the fluids used were elastic to a greater or lesser extent, but showed no shear thinning.      

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.     

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.     

Wall effects on terminal velocity of small drops in newtonian and non-newtonian fluids

This work investigated the extent of the wall effects on the free falling velocity of fluid spheres in quiescent Newtonian and pseudoplastic non-Newtonian media. The terminal velocity has been measured as a function of the physical properties of the both dispersed and continuous phases, and of falling tube diameter. It is shown that inthecreeDing flow region (Re « 1) and for the conditions when the viscosity of the dispersed phase is much smaller khan khat of khe continuous phase, the extent of wall effect is determined only by the ratio of the sizes of the settling fluid sphere and of the vessel. The same analytical relation correlates well the data for both Newtonian and non Newtonian continuous media in the range d/D < 0.45.     

ANIONIC POLYMERIZATION OF LACTAMS IN MICRODISPERSION. I. KINETIC ASPECTS OF THE PROCESS OF CAPROLACTAM POLYMERIZATION

Kinetics of the anionic polymerization of caprolactam (CL) in microdispersion was investigated at temperatures between 90 and 130°C and at stirring rates between 500 and 2000 rpm, respectively. For the investigated range of temperature data analysis showed pseudo-first order kinetics with respect to the monomer concentration. The rate constants are strongly influenced by temperature, increasing with the increase in temperature. For the associated activation energy, a value of 22.51 kJ/mol was obtained. The reaction order is also influenced by the hydrodynamic regime. It seems that below a threshold value of Reynolds Number, the rate of the reaction is independent of the concentration of CL, following pseudo-zero-order kinetics, which is replaced by first-order kinetics above this threshold value of N _Re. However, in the second region of N _Re, the values of rate constant are almost independent of N _Re.     

The ratio of terminal velocity to minimum fluidising velocity for spherical particles

The analogy between the states of a particle falling at its terminal velocity in a fluid and that of a particle in a bed, at incipient fluidization by the same fluid, suggests the possibility of a correlating minimum fluidizing and terminal velocities and of predicting the minimum fluidizing velocity. A semi-theoretical curve has been obtained, relating (Ret/Re_mf) to Fn = gp_F (rH_S – rH_F) d³/μ² and it has been compared with new experimental data collected for this purpose in the range 10³<Fn<10⁸. Analytical expressions for (Re_t/Re_mf) are proposed.     

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     

Prediction of bubble terminal velocity in surfactant aqueous solutions

Bubble terminal velocity has a significant effect on gas holdup, residence time, and efficiency of the interface transfer. Surfactant is often required to generate small and stable bubbles in gas-liquid two-phase devices. In this paper, bubble terminal velocities were obtained for different surfactant aqueous solutions using high-speed CCD (charged couple device) system and digital image analysis technology. Experimental results showed that available correlations were not able to accurately predict terminal velocity of bubbles rising in surfactant aqueous solutions. Thus, a new correlation is proposed based on experimental data and it provides an accurate approximation of bubble terminal velocity. The average relative error for the proposed correlation is determined to be 7.2% in MIBC aqueous solutions, 4.5% in OP-10 aqueous solutions, and 4.6% in 2-octanol aqueous solutions. The proposed correlation agrees well with experiment data from literate within ranges of the Morton number, Mo, the bubble Reynolds number, Re, and the Eötvös number, Eo: 3.29 × 10⁻¹¹<Mo<4.29 , 0.08<Re<1062 , 0.04<Eo<91.16 .     

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.     

Settling of cylinders in power law liquids

New measurements on drag coefficient and wall effects on the free settling motion of cylinders in two shearthinning polymer solutions are reported. The ranges of conditions covered in this study are: 1 < Re_∞∞ < 40; 0.25 < (LID) < 2; 0.079 < dID < 0.4, and n = 0.65 and 0.74. Both wall correction factor and drag coefficient results are in line with the Newtonian behavior provided a modified Reynolds number is used to represent the results.     

Coalescence of two bubbles rising in line at low reynolds numbers

Bubble coalescence phenomena have been examined at Reynolds numbers of 0·5–80 for five different classes of bubbles. The approach velocity of the following bubble (u₂) is experimentally obtained and its behaviour with respect to the leading bubble velocity (u₁) is examined. The coalescence of bubbles with Re < 7 follows the analysis of weightless solid spheres. Bubbles having toroidal wakes (Re > 7), coalesce with two additional velocities imparted due to the wake structure. Equations are developed to predict u₁/u₂ during coalescence.     

Particle-assisted dynamic wetting in a suspension liquid jet impinged onto a moving solid at different flow rates

We provide evidence that the introduction of solid spheres assists the dynamic wetting in suspension jet flows. The onset velocities of air entrainment in the suspension were measured and compared with those in the filler-free fluids to directly access the particle-assisted dynamic wetting. The measured onset velocity increased with increasing the particle contents at Re<20 where the dynamic contact line is located beneath the impinging liquid. The variation in the onset velocity agrees with a simple lubrication theory based on the film-splitting hypothesis. However, the onset velocities of air entrainment become independent of the particle contents at Re>20, higher liquid viscosities or increased jet lengths, showing different air entrainment behaviors from other low-Reynolds number coating flows.     

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.     

CFD evaluation of internal manifold effects on mass transport distribution in a laboratory filter-press flow cell

The internal manifold geometry strongly influences the flow distribution inside an electrochemical reactor. The mass transport coefficient is a function of the flow pattern and is a key parameter in successful electrochemical reactor design and scale-up. In this work, a commercial computational flow dynamics (CFD) package was used to describe the flow pattern in the FM01-LC reactor at controlled volumetric flow rates (corresponding to mean linear flow velocities past the electrode surface between 0.024 and 0.11 m s^?1). Numerical Re numbers were obtained for each local flow velocity at different positions in the reactor channel. From a known mass transport correlation (based on dimensionless groups, i.e. Sh, Re, Sc), numerical k _m values were obtained (in the range 200 < Re < 1,000) at different positions in the reactor channel. Computed k _m numbers are compared against experimental values. This computational approach could be useful in reactor design or selection since it facilitates a fast, preliminary reactor flow and mass transport characterisation without experimental electrochemical measurements.     

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.     

Effect of mechanical and air-particle cleansing protocols of provisional cement on immediate dentin sealing layer and subsequent adhesion of resin composite cement

Immediate dentin sealing (IDS) could avoid contamination of dentin from impression material and provisional cement but prior to final cementation of indirect restorations, removal of the provisional cement may damage the IDS. The objectives of this study were to investigate the effect of mechanical and air-particle cleansing protocols of provisional cement on IDS layer and subsequent adhesion of resin composite cement. The cuspal dentin surfaces of human third molars (N = 21, n_quadrant = 84) were exposed by a low-speed diamond saw under water cooling and conditioned with an adhesive system based on the three-step etch and rinse technique (OptiBond FL). Provisional cement (Freegenol) was applied on each specimen. They were then randomly divided into six subgroups where the provisional cement was removed either by (1) air-borne particle abrasion with 50-μm Al₂O₃ particles at 2 bar (AL2), (2) air-borne particle abrasion with 50-μm Al₂O₃ particles at 3.5 bar (AL3.5), (3) air-borne particle abrasion with 30-μm SiO₂ particles at 2 bar (SL2), (4) air-borne particle abrasion with 30-μm SiO₂ particles at 3.5 bar (SL3.5), (5) prophylaxy paste (Cleanic) (PP) or (6) pumice-water slurry (PW) at 1500 rpm for 15 s. The dentin surface on each tooth was assigned to four quadrants and each quadrant received the cleansing methods in a clockwise sequence. The non-contaminated and non-cleansed teeth acted as the control (C). Two separate teeth, contaminated and cleansed according to six cleansing protocols, were allocated for scanning electron microscopy (SEM) analysis (×2000). The dentin surfaces in each quadrant received resin composite luting cement (Variolink II, Ivoclar Vivadent) incrementally in a polyethylene mould (diameter: 1 mm²; height: 4 mm) and photopolymerized. The specimens were stored in distilled water for 24 h at 37 °C until the testing procedures and then shear force was applied to the adhesive interface until failure occurred in a universal testing machine (0.5 mm/min). Microshear bond (μSBS) was calculated by dividing the maximum load (N) by the bonding surface area of the resin cement. Failure types were analysed using optical microscope and SEM. Data (MPa) were analysed using one-way ANOVA (α = 0.05). Two-parameter Weibull distribution values including the Weibull modulus, scale (m) and shape (₀), values were calculated. Mean μSBS results (MPa) showed a significant difference between the experimental groups (p = 0.011) and were in a descending order as follows: C (8 ± 2.3)^a < AL2 (6.7 ± 2.4)^b < PP (6.9 ± 2)^b < PW (6.5 ± 2.1)^b < AL3.5 (5.8 ± 1.1)^b < SL2 (5.3 ± 1)^b < SL3.5 (5.2 ± 1)^b. Failure types were predominantly mixed failure type between the dentin and the adhesive resin which is a combination of adhesive and cohesive failures in the adhesive resin. Cohesive failure in the dentin was not observed in any of the groups. Weibull distribution presented lower shape (₀) for C (3.9), AL2 (3.2), PP (3.5) and PW (3.6). SEM analysis showed rough surfaces especially in the air-abraded groups whereas mechanical cleansing methods presented smoother surfaces and partially covered by particle remnants all of which occluded the dentin tubuli.     

RTD studies in trickle bed reactors packed with porous particles

The residence time distribution (RTD) for liquid phase in a trickle bed reactor (TBR) has been experimentally studied for air-water system. Experiments were performed in a 15.2 cm diameter column using commerical alumina extrudates with D/d_p ratio equal to 75 to eliminate the radial flow differences. The range of liquid and gas flow rates covered was 3.76 < Re_L < 9.3 and 0 < Re_G < 2.92. The axial dispersion model was used to compute axial dispersion coefficient. The effect of liquid and gas flow rates on total liquid holdup and axial dispersion was investigated. The total liquid holdup has been correlated to liquid and gas flow rates.