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.     

Drag on chains and agglomerates of spheres in viscous newtonian and power law fluids

New experimental data on the free settling velocity of straight chains (up to twenty spheres) and planar clusters of touching spheres in Newionian and power law media are reported. The results embrace the following ranges of conditions: 0.65 ≤ n ≤ 1; Re < - 2.5 and 1.22 < m < 48.87 Pa·sⁿ. The straight chain drag measurements are in line with theoretical predictions for Newtonian fluids. The present results in power law fluids seem to suggest that it is possible to express the drag on a straight chain of spheres in terms of that on a single sphere of equal volume. Limited results with planar clusters are satisfactorily correlated using a volume equivalent sphere diameter.     

Pressure drop in two phase cocurrent upward flow in packed beds: Air/non-newtonian liquid systems

New data on the two phase pressure drop for the concurrent upflow of air-liquid (Newtonian and non-Newtonian) mixtures through packed beds of spherical and non-spherical particles are presented. The results for single phase flows and for the air-Newtonian liquid mixtures have been used both to gauge the overall accuracy of the present experimental methods and to evaluate the validity of the predictive expressions available in the literature. The two phase pressure drop has been measured as a function of the liquid and gas flow rates, column diameter and the power law model constants. Depending upon a suitable combination of the gas and liquid fluxes and the power law index, the two phase pressure drop may be less than its value for the flow of liquid alone. A simple expression is proposed which correlates the present set of experiments (nearly 500 data points) with satisfactory levels of accuracy over the following ranges of conditions: 0.54 ≤ n ≤ 1; 0.001 ≤ Re_L^* ≤ 50; 3.7 ≤ Re_G ≤ 177 and 0.9 ≤χ (Lockhart-Martinelli parameter) ≤ 104.     

Laminar dispersion of polymer solutions in helical coils

In the present study the step response experiments were carried out with power law fluids in two helical coils to examine the suitability of axial dispersed plug flow model in describing the laminar dispersion of non-Newtonian fluids in helical coils. The ranges of variables covered are 10 ≤ λ ≤ 100,0.01 ≤ N_Regen ≤ 2.5,0.001 ≤ N_De ≤ 0.77 and 0.035 ≤ τ ≤ 1.33. It is found that coiling results in reduced dispersion to that in a straight tube.     

An experimental study of non-newtonian fluid flow through fixed and fluidized beds of non-spherical particles

Extensive measurements of pressure drop in fixed beds, minimum fluidization velocity and expansion characteristics for beds of non-spherical particles are reported in the following ranges of conditions: 10^-3 ≤ Re ≤ 20; 0.66 ≤ n ≤ 1 and 0.41 ≤ ? ≤ 0.75. Based on an analysis of these results, it is illustrated that the existing frameworks originally developed for Newtonian fluid flow through beds of spherical particles are also satisfactory for power law fluid flow through beds of non-spherical particles, provided a volume equivalent diameter modified by a sphericity factor and a modified Reynolds number are used instead of their usual definitions.     

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.     

Reactivity of disulfide peptide l-cystine with iodido (diethylenetriamine)platinum(II) and synthesis of products in the presence of an iodide ion

Substitution reactions of complex [PtI(dien)]⁺ (where, dien=diethylenetriamine ) with sulfur-containing peptide l-cystine has been studied in 1.0×10^?1 M aqueous perchlorate or acetate medium between 298≤T (K)≤323 and 2.30≤pH≤4.45 using a UV–visible spectrophotometer. Products obtained have characterized from their physico-chemical and spectroscopic methods at various pH and temperatures. From this characterization, products have indicated that [PtI(dien)]⁺ has formed a complex with l-cystine and acts as a bidentate ligand, through Pt–S bond at 2.30≤pH≤3.30 and through Pt–N and Pt–S bond of cystine in 3.95≤pH≤4.45. At 2.30≤pH≤3.30, ring opening and closing of dien have occurred at 308 and 323 K, respectively, and the same has happened at pH≥3.95. All reactions have followed the rate law – d[mixture]/dt=(k ₁+k ₂ [cystine]) [Pt(II)], where k ₂ denotes the second-order rate constant. Activation parameters E _a , Δ H ^# and Δ S ^# have been determined. Product formation and reversible and forward reaction rate constants have also been evaluated.     

Sedimentation of supsensions: implications of theories of hindered settling

Expressions are derived for the parameters A in Steinour's theory and n Richardson and Zaki's expression dealing with the hindered settling of powdered materials.It is shown that n is simply related to ?₁, the initial liquid volume fraction of a uniformly mixed suspension for which [Q(1 — ?)] was maximum value, Q being the linear settling rate of the suspension/supernatant interface and ? the initial liquid volume fraction of the suspension. For systems obeying the Richardson-Zaki expression, the settling rate at ?₁ approaches the limit V_s exp^?1, where V_s is the estimated Stokes' law limiting velocity for the system, when ?₁ approaches unity (i.e. infinite dilution).Highly hindered systems have large values for n, and ?₁ values approaching unity; for such systems Q_?1 ? V_s exp^?1. It is suggested that such behaviour implies the existence of relatively long-range forces within suspensions, hindering settling, and that particle-liquid (including particle-liquid-particle) forces are of importance in addition to particle-particle interactions.Evidence is presented that hindrance to settling is directly proportional to the polarity of the solid/liquid system and that the density of charge on the superficial particle surface is the dominant factor in determining hindrance. Maximum reduction of surface polarity without causing flocculation is suggested as the most efficient condition for separation by settling under gravity and pumping off supernatant liquor.n and ?₁ can be considered as useful indices of hindrance to sedimentation. ?₁ has particular significance as the initial liquid volume fraction at which solids flux has maximum value (for any suspension which obeys the Richardson-Zaki equation).     

Steady Flow of Power Law Fluids across a Circular Cylinder

The momentum equations describing the steady cross‐flow of power law fluids past an unconfined circular cylinder have been solved numerically using a semi‐implicit finite volume method. The numerical results highlighting the roles of Reynolds number and power law index on the global and detailed flow characteristics have been presented over wide ranges of conditions as 5 ≤ Re ≤ 40 and 0.6 ≤ n ≤ 2. The shear‐thinning behaviour (n < 1) of the fluid decreases the size of recirculation zone and also delays the separation; on the other hand, the shear‐thickening fluids (n > 1) show the opposite behaviour. Furthermore, while the wake size shows non‐monotonous variation with the power law index, but it does not seem to influence the values of drag coefficient. The stagnation pressure coefficient and drag coefficient also show a complex dependence on the power law index and Reynolds number. In addition, the pressure coefficient, vorticity and viscosity distributions on the surface of the cylinder have also been presented to gain further physical insights into the detailed flow kinematics.     

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.     

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.     

Lattice dynamics of yttria: A combined investigation from spectrum measurements and first-principle calculations

C-type Y₂O₃ ceramics (relative density ~94%) were prepared at 1500 °C for 2 hours with 1% wt. ZnO as sintering aid. The cell parameters of Y₂O₃ from Rietveld refinements are a = 10.6113(1) Å, V = 1194.8(1) ų. The vibrational modes / lattice dynamics of Y₂O₃ were investigated using vibrational spectra (Raman and infrared reflection spectra) and first-principle (DFT) calculations. Eight of the 22 predicted first-order Raman modes and 12 of 16 predicted IR modes are observed and reliably assigned. For the observed vibrational modes, an excellent linearity (f_exp = 1.023f_theo, R² = 0.9999) between frequency from calculations (f_theo) and that from measurements (f_exp) is observed. Accordingly, the corrected frequency (f_cor) of vibrational modes, phonon band structure, and density of phonon states (DOPS) of Y₂O₃ are presented, in which, the frequency of phonons of Y₂O₃ is ≤625.2 cm⁻¹ (wavelength ≥16.0 μm) with a gap of 30.6 cm⁻¹ from 486.0 to 516.6 cm⁻¹ (wavelength 20.6 - 19.4 μm) at room temperature. The modes with f_theo ≥292.5 cm⁻¹ (f_cor ≥299.2 cm⁻¹) are dominated by the vibrations of O²⁻ (light atom vibrations) and the vibrational modes with f_theo ≤239.0 cm⁻¹ (f_cor ≤244.5 cm⁻¹) are dominated by the vibrations of both Y³⁺ and O²⁻ (co-vibrations). The three modes T_u⁽⁷⁾ at 301.6 cm⁻¹, T_u⁽¹⁰⁾ at 333.7 cm⁻¹, and T_u⁽¹²⁾ at 369.7 cm⁻¹ of Y-O stretch vibrations dominate the phonon dielectric constant and dielectric loss of Y₂O₃ with more than 85% contributions.     

FLOW OF NON-NEWTONIAN POWER LAW LIQUIDS THROUGH PACKED AND FLUIDIZED BEDS

The creeping flow of power law liquids through assemblages of spherical particles has been studied theoretically. The inter-particle interactions are modelled via the zero vorticity cell model. The governing equations have been solved numerically to obtain the theoretical estimates of the drag force experienced by an assemblage placed in a streaming power law fluid. The results reported herein encompass wide ranges of fluid behaviour (values of power law index) and bed voidage thereby covering packed and fluidized bed conditions. Detailed comparisons with experimental data suggest that this theory can be used to predict pressure drop for power law fluid flow in packed beds as well as velocity-bed expansion characteristics for a fluidized bed. By analogy with the Newtonian case, intuitively, one would expect these results to be applicable to hindered settling in power law fluids, and indeed this is borne out by the limited amount of data covering the range 1≥n≥0.8 available in the literature.     

Effect of particle shape on hindered settling in creeping flow

Solid particles of uniform size and shape were used to study the effect of particle shape on hindered settling in creeping flow (Re_o ? 0.2), where fluid flow patterns are independent of Reynolds number and the effect of shape is most prominent. The particles of different shape studied were spherical glass beads, cubical sodium chloride crystals and ABS plastic pellets, brick-like sugar crystals, and angular (imperfect octahedral) mineral silicate crystals. The liquids used were aqueous polyethylene glycol solutions and various blends of hydrocarbon oils. Two particle sizes on the average were investigated for each particle shape, and five settling column diameters were employed, so that the overall range of column-to-particle diameter ratio covered was 22 – 226.A Richardson—Zaki type equation of the form u = u_i?ⁿ was found to correlate the constant settling rate data for each particle size and shape over the voidage range ? = 0.65 – 0.9. However, the wall effect on hindered settling rate was found in most cases to be considerably smaller than that predicted by Richardson and Zaki. The term u_i, obtained by linearly extrapolating the settling velocity u (below ? = 0.9) to ? = 1 on a log—log plot of u versus ?, was found to be measurably lower than the corresponding free settling velocity. The index n varied from an average value of 4.8 for the smooth spheres to 5.4 for the cubes to 5.8 for both the brick-like and the angular particles. These values graphically display a definite trend with settled bed voidage, ?_b, which is shape-dependent and easily measured, and may therefore be a convenient parameter for taking account of shape variation generally.The method proposed by Beranek and Klumpar for correlating fluidization data on different shaped particles, which depends on ?_b, was found to be moderately successful in correlating the present settling data for different shapes.     

Estimation of hindered settling velocity of suspensions

Four effective-medium models (EM-I, II, III, IV) are utilized and compared for determining hindered settling velocity of equi-sized particles in a viscous fluid. Among the models, EM-IV model is found to accurately predict the effective viscosity and the hindered settling velocity of monodisperse suspensions. In EM-IV model which was developed for determining the diffusivity of proteins in a biological membrane by Dodd et al. [T.L. Dodd, D. A. Hammer, A.S. Sangani, D.L. Koch, J. Fluid Mech. 293 (1995) 147], the effective-medium region begins at the distance R = a[(1 ? S(0))/?]^1/3 from the origin where the center of the test particle is located, where a is the radius of the particle, ? is the volume fraction of the particles in the suspension, and S(0) is the zero wavenumber limit of the structure factor. The estimations by EM-IV model agree very well with the exact calculations and the experimental observations. The hindered settling velocity U of the particles is given, in Richardson–Zaki form, by U/U₀ = (1 ? ?)^5.5, where U₀ is the settling velocity for an isolated particle.     

Elutriation characteristics of solids from liquid–solid fluidized bed systems Part III: A study of the possible causes of non-linearity in the elutriation of fine particles from fluidized beds

The “critical concentration” of solids, defined by Trawinski as the break-point from linearity in the plot of log (C/C_o) versus θ in any elutriation process, represents the attainment of a sudden increase in bed viscosity which finally leads to a retentive capacity of the bed with respect to the fines, characterized by the “equilibrium bed concentration”, C_E. Experimental data on C_c have been correlated empirically with various system parameters as The ranges of the parameters are 1.1 ≤ G/G_me ≤ 1.4; 1.28 ≤ d₁/d₂ ≤ 2.85; 1.0 ≤ ρ₁/ρ₂ ≤ 2.6; 0.50 ≤ w/W ≤ 0.90; and 2.7 ≤ H_a/H_s ≤ 17.7.     

Flow of non‐Newtonian polymeric solutions through fibrous media

The equations of motion (continuity and momentum) describing the steady flow of incompressible power law liquids in a model porous medium consisting of an assemblage of long cylinders have been solved numerically using the finite difference method. The field equations as well as the pertinent boundary conditions have been re‐cast in terms of the stream function and vorticity. The inter‐cylinder interactions have been simulated using a simple “concentric cylinders” cell model. Extensive information on the detailed structure of the flow field in terms of the surface vorticity distribution, streamlines, and viscosity distribution on the surface of the solid cylinder as well as on the values of the pressure and friction drag coefficients under wide ranges of physical (0.4 ≤ ϵ ≤ 0.95; 1 ≥ n ≥ 0.4) and kinematic (0.01 ≤ Re ≤ 10) conditions have been obtained. The numerical results presented herein have been validated using the experimental results for the flow of Newtonian and power law fluids available in the literature; the match between the present predictions and the experiments was found to be satisfactory. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 76: 1171–1185, 2000     

Scale‐Free Power‐Law Distribution of Emulsion‐Polymerized Branched Polymers: Power Exponent of the Molecular Weight Distribution

Summary: The molecular weight distribution (MWD), formed in emulsion polymerization that involves the polymer transfer reaction during Interval II, may approach the power‐law distribution as polymerization proceeds. The power exponent, α, of the weight fraction distribution W(M) = M^?α conforms to the relationship, α = 1/P_b, where P_b is the probability that the chain end is connected to a backbone chain. The MWD of emulsion‐polymerized polyethylene reported in literature agrees reasonably well with the relationship, W(M) = M^?α with α = 1/P_b. This simple relationship could be used to estimate the P_b value from the MWD data, possibly leading to determining the polymer transfer constant under well‐designed experimental conditions. Because α > 1, the number‐average MW always approaches a finite value, but the weight‐ and higher order‐averages of MWD may continue to increase as the particle grows without limit depending on the magnitude of P_b. The power‐law distributions are self‐similar, possessing the nature of fractals and lacking a characteristic scale. The i‐th moment of the MWD for the present reaction system continues to increase without limit during Interval II for P_b ≥ 1/i.

Molecular weight distribution of the emulsion‐polymerized polyethylene.     

Ferroelectric to Relaxor Transition in BaTiO3–Bi(Zn2/3Nb1/3)O3 Ceramics

The (1?x)BaTiO₃–xBi(Zn_2/3Nb_1/3)O₃ (x = 0.01–0.30) ceramics were synthesized by solid‐state reactions. The solubility limit was determined to be x = 0.20. A systematic structural transition from a tetragonal phase (x ≤ 0.034), to a mixture of tetragonal and rhombohedral phases (0.038 ≤ x ≤ 0.20), and finally to a pseudocubic phase (x ≥ 0.22) at room temperature was identified. Dielectric measurement revealed a ferroelectric (x ≤ 0.04) to relaxor (x ≥ 0.06) transition with permittivity peak broadening and flattening, which was further verified by Raman spectroscopy and differential scanning calorimetry (DSC). Activation energies obtained from the Vogel–Fulcher model displayed an increasing trend from ~0.03 eV for x ~ 0.05, to unusually high values (>0.20 eV) for the compositions with x ≥ 0.15. With the increase in Bi(Zn_2/3Nb_1/3)O₃ content, the polarization hysteresis demonstrated a tendency from high nonlinearity to sublinearity coupled with the reduction in remnant polarization and coervice field. The deconvolution of the irreversible/reversible polarization contribution was enabled by first‐order reversal curve distributions, which indicates that the decreasing polarization nonlinearity with the increase in Bi(Zn_2/3Nb_1/3)O₃ concentration could be related with the change from the ferroelectric domain and domain wall contributions to the weakly coupled relaxor behaviors.     

Hindered settling of coagulating suspensions

The concentration of the potential-determining ions (H⁺ and OH^?) markedly affects the settling (under hindered conditions) of quartz suspensions. High collision frequency between particles under hindered conditions makes such a system extremely sensitive to the collision efficiency given by the energy of interaction between particles.Mean equivalent Stokes diameters calculated from hindered settling experiments with three size fractions of quartz correspond well to the diameters determined by the use of a sedimentation balance only when the hindered settling experiments are performed at pH values above 5. For the wide size fractions containing coarse particles, the initial solid concentration required for settling experiments must be higher than that for fractions containing finer particles.In the pH range 2 – 4 the experiments detect coagulation, indicating clearly that the pH of the iso-electric points is in this range.