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.     

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.     

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.     

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.     

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.     

Free convection in power-law fluids from a heated sphere

In this work, the governing field equations describing heat transfer from a heated sphere immersed in quiescent power-law fluids have been solved numerically. In particular, consideration has been given to elucidate the role of Grashof number (Gr), Prandtl number (Pr) and power-law index (n), on the value of the Nusselt number (Nu) for a sphere in the natural convection regime. Further insights are provided by presenting streamline and constant temperature contours. The results presented herein encompass the following ranges of conditions: 10≤Gr≤10⁷; 0.72≤Pr≤100 and 0.4≤n≤1.8 thereby covering both shear-thinning and shear-thickening types of fluid behaviours. Broadly, all else being equal, shear-thinning behaviour can enhance the rate of heat transfer by up to three-fold where as shear-thickening can impede it up to ～30?40% with reference to that in Newtonian fluids. The paper is concluded by presenting detailed comparisons with the scant experimental data and the other approximate treatments of this problem available in the literature.     

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.     

Falling ball method for determining zero shear and shear-dependent viscosity of polymeric systems: Solutions,melts, and composites

The falling ball method (FBM) is one of the well-established techniques for measuring the viscosity of Newtonian liquids at the room as well as at elevated temperatures and pressures. Owing to its simplicity and low cost, the possibility of extending its range of application to non-Newtonian systems including virgin and filled polymer melts, composites, polymer-solutions, and so forth, is explored here, In this work, theoretical results for the flow of power-law fluids past a sphere have been used to extract the values of the zero-shear viscosity and shear-dependent viscosity in the low-shear rate limit. The theoretical scheme outlined here has been validated by presenting comparisons with experimental results for scores of polymer solutions for which both falling sphere and rheological data are available in the literature. Indeed, the good correspondence obtained between these two independent data is encouraging and it is thus possible to use the FBM for shear-thinning systems when the resulting Reynolds numbers are such that the flow is viscosity-dominated, and the inertial effects are negligible. This implies that the Reynolds number should be ≤ ~1 for shear-thinning fluids and ≤ ~10⁻⁵ for shear-thickening fluids.     

The effect of mesh refinement on the computation of entrance effects for shear-thinning fluids

The effect of mesh refinement on the finite element computation of velocity and pressure is assessed for the creeping flow of a power law fluid and a Carreau fluid through an axisymmetric sudden contraction. The emphasis is placed on the determination of the entrance pressure correction n_ent. The augmented Lagrangian formalism is used and the discretization of the variational problem is based on the Crouzeix-Raviart triangular element. For power-law fluids, it is found that fine meshes yield values of n_ent which are significantly lower than those obtained in previous numerical studies, especially for high shear-thinning fluids. A similar trend is observed in the Carreau model.     

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.     

Effect of blockage on heat transfer from a cylinder to power law liquids

The influence of planar confining walls on the steady forced convection heat transfer from a cylinder to power-law fluids has been investigated numerically by solving the field equations using FLUENT (version 6.2). Extensive results highlighting the effects of the Reynolds number (1?Re?40), power-law index (0.2?n?1.8), Prandtl number (1?Pr?100) and the blockage ratio (β=4 and 1.6) on the average Nusselt number have been presented. For a fixed value of the blockage ratio, the heat transfer is enhanced with the increasing degree of shear-thinning behaviour of the fluid, while an opposite trend was observed in shear-thickening fluids. Due to the modifications of the flow and temperature fields close to the cylinder, the closely placed walls (i.e., decreasing value of the blockage ratio) further enhance the rate of heat transfer as the fluid behaviour changes from Newtonian to shear-thickening fluids (n>1), the opposite influence is seen with the decreasing value of the flow behaviour index (n) in shear-thinning (n<1) fluids. Finally, the functional dependence of the present numerical results on the relevant dimensionless parameters has been presented in the form of closure relationships for their easy use in a new application.     

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.     

A new investigation of the metzner-otto concept for anchor mixing impellers

Anchor impellers are commonly used for the homogenization of non-Newtonian fluids, often in association with a set of coaxial turbines. The optimal design of such mixers relies on the knowledge of power drawn by the individual impellers. In non-Newtonian mixing, this can be readily obtained using the Metzner-Otto (1957) concept. In this article, the Metzner-Otto concept and the determination of the constant K_s for anchor impellers have been revisited using numerical and experimental techniques for the case of shear-thinning and shear-thickening fluids. Contrary to literature findings, it is shown that the constant K_s does not vary strongly with the power law index and that, for mixer design purposes, the use of a constant value of K_s for each of the rheological behaviors considered is adequate.     

An analytical model for the prediction of power consumption for shear-thinning fluids with helical ribbon and helical screw ribbon impellers

An approximate analytical model has been developed to predict power consumption for the mixing of shear-thinning fluids with helical ribbon and helical screw ribbon impellers in the laminar flow regime. Extensive data on power input measurements embracing a wide range of flow behaviour index, with strong (n<0.4) and weak (0.4<n<1) shear-thinning fluid characteristics, available in the literature have been used to demonstrate the applicability of the present model for a wide range of helical ribbon mixer configurations. The model is able to explain the differences in the data reported in the existing literature and to successfully predict the complex dependence of power consumption on the fluid properties and the system geometry. Finally, the proposed correlation only requires a knowledge of the flow behaviour index of the fluid and of the geometrical parameters of the mixing systems (wall clearance, number of ribbons, pitch and width of the ribbons) and one characteristic parameter K_p of the mixing system which can be obtained from a single measurement of power for Newtonian liquids in the laminar regime.     

Three-dimensional finite element analysis of polymeric fluid flow in an extrusion die. Part I: Entrance effect

The Galerkin finite element method has been applied to study the three-dimensional flow field of power-law fluids inside an extrusion die. Two inlet designs, i.e., center-fed and end-fed, have been considered. The effects of inertial force as represented by the Reynolds number Re, inlet geometry, and the power-law index n on lateral flow uniformity and vortex formation in the entrance region have been examined. A flow visualization technique has been carried out to experimentally verify the theoretical prediction of the three-dimensional flow field inside a die. It has been found that increasing Re or decreasing n will deteriorate flow uniformity. Depending on the direction of the inlet jet stream, the inertial force may create a flow peak in the central region of a center-fed die, or the maximum flow rate will appear close to the end of the die for an end-fed die. For highly shear-thinning fluids, lower flow rates are always observed close to the end of the dies. It is concluded that creating a plug flow in the inlet tube of the extrusion die is advantageous for both center-fed and end-fed designs.     

Unified entry length for newtonian and power–law fluids in laminar pipe flow

A numerical method based on finite differencing is used for investigating the steady–state entrance region laminar flow of incompressible Newtonian and power–law fluids in a circular pipe. The Solution method is validated by comparing the results for Newtonian fluids with those reported in the literature. For power–law fluids, the entry length results are compared with other approximate solutions in the literature. On the basis of the calculated results, a generalized entry length ξ99 = 0.056 is shown to be valid for the laminar flow at Re > 200 of both Newtonian and power–law fluids with 0.75 < n < 1.5.     

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.     

Continuous UV cure monitoring and dielectric properties of photocross-linked epoxy–acrylate resins

The changes in viscosity of epoxy–acrylate resins during the photocross-linking process have been monitored continuously. The curing show change of shape and a distinct peak especially at high intensities. The intensity dependence of gel time follows a superlinear power law of the type t_pαI^?1.6 rather than the ½ power law. The dielectric relaxation studies on these photocross-linked films having different compositions of epoxy and n-butyl methacrylate reveal a particular composition range in which various dielectric parameters pass through critical values. These various findings have been explained on the basis of phase separation taking place during the UV curing process but at a submicron level leading to a very small domain size. © 1994 John Wiley & Sons, Inc.     

Calendering of non-newtonian fluids

The calendering of non-Newtonian fluids by two rotating cylinders to produce thin films of fluids finds wide application in polymer sheet-making and food-drying industries. Theoretical work has previouly been devoted to the symmetrical case where the cylinders are of equal diameters rotating at the same speed. The present work proposes a new one-film theory of calendering of power law fluids for unequal radii and surface velocities of the calendering cylinders. The relationship between the dimensionless thickness of the calendered fluid, Δ_e^* and that of the incoming fluid, Δ_i^* is shown to be a function of the ratio of the surface velocities of the cylinders and the power law index. The result further shows that Δ_e^* tends to asymptote after the second decade of Δ_i^*