Drag on Freely Falling Cones in Newtonian and in Power Law Fluids

New extensive data on the terminal falling velocities of conical shaped bodies in scores of Newtonian and power law fluids are reported. Altogether, 11 Newtonian and 11 non‐Newtonian test liquids together with 33 cones made from four different materials and 14 spheres of three different materials have been used to gather 486 individual data points covering wide ranges of conditions as follows: Reynolds number 0.0019 to 507; power law flow behaviour index 0.4 to 1, the value of sphericity 0.59 to 0.79; and the cone‐to‐fall tube diameter ratios up to 0.264 to assess the extent of wall effects. A simple expression is developed to estimate the terminal falling velocity of a cone from a knowledge of its dimensions, and the terminal velocity of an equivalent sphere. A generalized drag equation applicable to both Newtonian and power law liquids is also presented.     

Power‐Law Fluid Flow Over a Sphere: Average Shear Rate and Drag Coefficient

Based on the consideration of the rate of mechanical energy dissipation, an expression for the average shear rate for a sphere falling in a power‐law fluid in the creeping flow regime has been deduced. The average shear rate in a power‐law fluid (n<1) appears to be higher than that in an equivalent Newtonian fluid. This in turn has been combined with the numerical predictions of drag coefficient (up to Reynolds number of 100) of a sphere to develop a generalized drag correlation for power‐law liquids encompassing both n > 1 and n < 1 which appears to apply up to much higher values of the Reynolds number. The available experimental data have been used to demonstrate the reliability and accuracy of the new correlation for shearthinning liquids. Also, in the limit of n = 1, this expression reproduces the standard drag curve with a very high accuracy.     

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.     

An explicit equation for the terminal velocity of solid spheres falling in pseudoplastic liquids

An explicit equation is proposed which predicts directly the terminal velocity of solid spheres falling through stagnant pseudoplastic liquids from the knowledge of the physical properties of the spheres and of the surrounding liquid. The equation is a generalization of the equation proposed for Newtonian liquids. By properly defining the dimensionless diameter, d_*, a function of the Archimedes number, Ar, and the dimensionless velocity, U_*, a function of the generalized Reynolds number, Re, to account for the non-Newtonian characteristics of the liquid, the final equation relating these two variables has similar form to the Newtonian equation. The predictions are very good when they are compared to 55 pairs of Re—C_D for non-Newtonian data and 37 pairs for Newtonian data published previously. The root mean square error on the dimensionless velocity is 0.081 and much better than the only other equation previously proposed.     

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.     

Flow through porous media of a shear-thinning liquid with yield stress

Darcy's law for the laminar flow of Newtonian fluids through porous media has been modified to a more general form which will describe the flow through porous media of fluids whose flow behavior can be characterized by the Herschel-Bulkley model. The model covers the flow of homogeneous fluids with a yield value and a power law flow behavior. Experiments in packed beds of sand were carried out with solutions of paraffin wax in two oils and with a crude oil from the Peace River area of Canada. The model fitted the data well. A sensitivity analysis of the fitting parameters showed that the model fit was very sensitive to errors in the flow behavior index, n , of the Herschel-Bulkley model. A comparison of the “n” values calculated from viscometer measurements and from flow measurements agreed well. A more general Reynolds number for flow through porous media, which includes a fluid yield value, was developed. The data were fitted to a Kozeny-Carman type equation using this Reynolds number. The constant in the Kozeny-Carman equation was determined for the two packed beds studied using Newtonian oils. The data could all be represented, within the experimental error, by the relationship f^* = 150/Re^*. Since the mean volume to surface diameter of the packing was determined by the measurement of its permeability to a Newtonian oil, assuming C' = 150, the new definition of the Reynolds number allows the direct use of the Kozeny-Carman equation with Herschel-Bulkley type fluids.     

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.     

Breakup of jets in power law non-newtonianndashnewtonian liquid systems

Non-Newtonian effects on the breakup of jets into drops were experimentally studied for power law fluidNdashNewtonian systems under the condition of zero jet velocity relative to the continuous phase. While laminar breakup lengths of Newtonian jets in non-Newtonian liquids agree with the prediction from the stability analysis for Newtonian systems, non-Newtonian jets in Newtonian liquids are less stable than Newtonian jets. Experimental diameters of drops formed from jets in NewtonianNdashnon-Newtonian and in non-NewtonianNdashNewtonian systems are in good agreement with the prediction based on stability analysis for Newtonian systems.     

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.     

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.     

Newtonian power curve and drop size distributions for vibromixers

Power input data are presented for a twin flat disk up-and-down moving (vibromixer) impeller operating in a small vessel with a range of Newtonian liquids. Vibromixer power number and Reynolds number are defined and are used to establish the Newtonian power curve for this type of mixer. Drop size distributions are presented for xylene-in-water dispersions under turbulent flow conditions in the vibromixer and are shown to vary with the maximum velocity of the disk (2πAf). The Sauter mean drop diameter of the distribution is related to the vibromixer Weber number, (We =ρ(2πAf)²D/σ), by an equation of the type d₃₂/D = C (We)^?3/5 with the coefficient C = 0.37.     

Some qualitative and quantitative features of die swell

Upper and lower bounds for the die swell ratio are derived for all simple fluids in the presence of surface tension and under the assumptions that the flow remains viscometric up to a finite distance from the exit and that bounds on the shear stress distribution in this length exist. Both circular and plane jets are considered and the bounds so derived depend on the normal stress on the wall in the viscometric region. For Newtonian and power law fluids, this stress can be determined, a priori, from existing theories. However, for viscoelastic liquids, it is an open question at present.     

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.     

Three‐Dimensional Simulation of Bubble Formation Through a Microchannel T‐Junction

Three‐dimensional simulations of bubble formation in Newtonian and non‐Newtonian fluids through a microchannel T‐junction are conducted by the volume‐of‐fluid method. For Newtonian fluids, the critical capillary number Ca for the transition of the bubble breakup mechanism is dependent on the velocity ratio between the two phases and the microchannel dimension. For the power law fluid, the bubble diameter decreases and the generation frequency increases with higher viscosity parameter K and power law index n. For a Bingham fluid, the viscous force plays a more important role in microbubble formation. Due to the yield stress τ_y, a high‐viscous region is developed in the central area of the channel and bubbles deform to a flat ellipsoid shape in this region. The bubble diameter and generation frequency are almost independent of K.     

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.     

A rotating sphere viscometer

The use of a rotating sphere viscometer for the measurement of parameters in the flow curves of inelastic as well as viscoelastic liquids is examined. An experimental investigation of the primary flow around a sphere rotating in Newtonian and viscoelastic liquids is carried out by using a new “three-dimensional particle technique.” Currently available theoretical analyses of rotation of a sphere in viscoelastic liquids are shown to be inadequate to describe the experimental primary velocity distribution data. Theoretical results for the primary distribution derived on the basis of a creeping flow of a power law liquid are found to describe the experimental data well. This distribution is then used to derive torque–angular velocity relationships, which are then confirmed experimentally for both inelastic and viscoelastic liquids. The results of this work justify the use of a rotating sphere viscometer as a useful tool for the measurement of parameters of flow curves of inelastic and viscoelastic liquids.     

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.     

Gas-liquid mass transfer in unbaffled dual-impeller mixers

The gas-liquid mass transfer performance of different unbaffled dual-impeller mixers was investigated experimentally using low- and high-viscosity Newtonian and non-Newtonian liquids. The tested configurations were composed of two Rushton turbines located at two different height levels in the vessel in centered or off-centered positions. Various mixers were compared based on their respective mass transfer performance measured by means of two dissolved oxygen probes located at different levels in the vessel. k_la correlations and an axial homogeneity criterion were established for the different configurations and liquids used. It was concluded that the dual shaft mixer consisting of two off-centered shafts appears to be a very interesting and flexible configuration to deal with Newtonian liquids in a large range of viscosities. However, concerning the gas-liquid dispersion in non-Newtonian liquids, it is shown that off-centered shaft configurations have to be avoided.     

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.     

A local power‐law approximation to a smooth viscosity curve with application to flow in conduits and coating dies

Coating dies distribute liquid into a uniform layer for coating onto a moving substrate. A die comprises one or two cavities spanning the coating width and adjoining narrow slots of much higher resistance to flow. In modeling coating dies, the flows in the slots and cavities are often approximated as one‐dimensional to achieve a fully geometrically parameterized model of low computational load suitable for optimizing design for multiple liquids and flow rates. The power‐law model is mathematically efficient for one‐dimensional flows of shear‐thinning liquids but does not include limiting viscosities at low and high shear rates that are frequently present. In previous work, the truncated power‐law model, which is terminated at the limiting Newtonian viscosities, was used to alleviate this shortcoming without sacrificing the mathematical advantages. In this work, the Carreau–Yasuda model replaces the truncated power‐law model as an improvement. For flows in slots and cavities, the Carreau–Yasuda model can be approximated accurately by a local power‐law model with little increase in computational load over the truncated power‐law model. In the transition regions of the Carreau–Yasuda model between Newtonian and power‐law behavior, the local power‐law model gives more accurate results than the truncated power‐law model. POLYM. ENG. SCI., 54:2301–2309, 2014. © 2013 Society of Plastics Engineers