Settling velocity of cubes in Newtonian and power law liquids

New extensive data on the free settling velocity of thirty cubes of various densities and sizes falling in scores of Newtonian and Power law liquids are reported herein to supplement the existing data, for there is very little prior data on cubes in power law liquids. The new data embrace the range of conditions as follows: sphericity of 0.805; power law index, 0.61 to 1 and consistency index, 0.0078-15.31 Pa sⁿ; Reynolds number, 0.0013 to 860. The new results are shown to be consistent with an existing drag correlation which has been tested extensively using the literature data for spherical and non-spherical particles falling in Newtonian and power law liquids with acceptable levels of accuracy.     

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.     

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.     

A simple correlation for gas bubbles rising in power‐law fluids

Recently, the author (Rodrigue, 2001 a, b) proposed a generalized correlation for the steady rise of gas bubbles in uncontaminated viscous Newtonian fluids of infinite extent. It is the purpose of this note to show that this model can be modified for inelastic non‐Newtonian power‐law fluids. Using data taken from six different studies, it is shown that the modified model can predict quite nicely the bubble velocity for these non‐Newtonian inelastic fluids.     

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.     

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     

Wall Effects for the Free Rise of Solid Spheres in Moderately Viscous Liquids

For the first time ever, the influence of wall effects on the free rise of a buoyant solid sphere in a square column in a non‐Newtonian (pseudoplastic) fluid was determined. It was found that in most cases, as the column width decreased, the terminal rise velocity of the sphere would decrease as well. It was also discovered that wall effects could change the rising sphere trajectory. Spheres that displayed a spiraling trajectory in larger columns would display a more linear trajectory in smaller columns. Occasionally, due to this change in trajectory, the solid sphere terminal velocity would increase as the column width decreased. This phenomenon (the positive wall effect) has not been reported for falling spheres in Newtonian or non‐Newtonian fluids.     

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.     

Local hydrodynamic parameters of bubble column reactors operating with non‐Newtonian liquids: Experiments and models development

The effects of liquid phase rheology on the local hydrodynamics of bubble column reactors operating with non‐Newtonian liquids are investigated. Local bubble properties, including bubble frequency, bubble chord length, and bubble rise velocity, are measured by placing two in‐house made optical fiber probes at various locations within a bubble column reactor operating with different non‐Newtonian liquids. It was found that the presence of elasticity can noticeably increase the bubble frequency but decreases the bubble chord length and its rise velocity. The radial profiles of bubble frequency, bubble chord length, and bubble rise velocity are shown to be relatively flat at low superficial gas velocity while they become parabolic at high superficial gas velocity. Moreover, the bubble size and gas holdup are correlated with respect to dimensionless groups by considering the ratio between dynamic moduli of viscoelastic liquids. The novel proposed correlations are capable of predicting the experimental data of bubble size and gas holdup within a mean absolute percentage error of 9.3% and 10%, respectively. © 2015 American Institute of Chemical Engineers AIChE J, 62: 1382–1396, 2016     

Integral methods for flow in a conical centrifuge

A set of through-thickness averaged equations for momentum and strain rate are derived for the problem of axi-symmetric free-surface flow within a spinning cone. The expressions are independent of the choice of constitutive law and can therefore be used for modeling the flow of a variety of materials within an industrial conical centrifuge. By assuming a through-thickness velocity profile the distribution of flow thickness and average velocity over the internal surface of the cone can be obtained. The approach has been validated for thin Newtonian viscous flow by comparison with full three-dimensional solutions of the Navier–Stokes equations obtained with a commercial CFD package. The averaged equations provide an accurate prediction of the flow thickness, velocity and the length of the zones of influence of inlet and outlet boundary conditions.     

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.     

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.     

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.     

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.     

THEORETICAL PREDICTION OF GAS HOLDUP IN BUBBLE COLUMNS WITH NEWTONIAN AND NON-NEWTONIAN LIQUIDS

A simple model based on an energy balance which takes into account the friction losses at the gas-liquid interface and the slip velocity of single bubble is used to simulate the gas holdup in bubble columns containing Newtonian and non-Newtonian liquids which circulate in both laminar and turbulent flows. Experimental data available from the literature for bubble columns up to 7 m height and 1 m diameter with water and glycerol as Newtonian liquids and different solutions of CMC in a wide range of concentrations as non-Newtonian liquids are simulated with good agreement despite the simplifications made to describe the gas liquid flow regimes. Most of the differences between experimental and calculated gas holdup are justified on the basis of the simplifying assumptions.     

An experimental study of motion of cylinders in newtonian fluids: Wall effects and drag coefficient

The terminal velocity of several cylinders (of glass, perspex and stainless steel) falling with their axis parallel to the direction of motion has been measured in a series of Newtonian fluids embracing a 40-fold variation in liquid viscosity. The measurements have been carried out in fall tubes of four different diameters to elucidate the importance of wall effects. The experimental results encompass the following ranges of conditions: cylinder to fall tube diameter ratio: 0.08 to 0.4; length to diameter ratio: 0.05 to 2 and Reynolds number varied from 0.2 to 180. The wall effects are discussed in a manner analogous to those for spherical particles. Terminal velocity data are analysed using two approaches, namely, drag coefficient-Reynolds number relationship and a dimensionless velocity ratio denoting the departure from the behaviour of an equivalent sphere. Predictive equations have been developed using both schemes.     

Elastic liquid jet impaction on a high‐speed moving surface

In the railroad industry a friction‐modifying non‐Newtonian liquid, showing elastic behavior, may be applied to the rail in the form of a liquid jet. The interaction of this elastic liquid jet and the moving surface—specifically whether it splashes or adheres without splash—is important in this industrial application. Twelve different elastic liquids with widely varying relaxation times were tested to isolate the effect of elasticity from other fluid properties. Using high‐speed imaging, the interaction between the impinging jet and the moving surface could be captured and analyzed. Although similar to Newtonian jets, for which the Reynolds number plays a major role, the Deborah number was also salient to the splash of elastic liquids. At the elevated Weber numbers of the testing, the Weber number had a much smaller impact on splash than did the Reynolds or Deborah numbers. The ratio of the surface velocity to the jet velocity has only a small effect on the splash. © 2012 American Institute of Chemical Engineers AIChE J, 2012     

Generalized slit flow of an ellis fluid

Isothermal, steady‐state and fully developed flows of Ellis fluids in planar and annular slits are discussed. The flow equations derived for Ellis fluids describe also the flows of Newtonian and power law fluids as specific cases. The most important flows resulting from thp general theory, i.e., the pressure flows in flat and annular slits for stationary channel walls and at transverse and longitudinal movements of a wall, are analyzed in detail. Numerical verification of the results leads to the conclusion that the planar and annular flows of Ellis and power law fluids are qualitatively similar The quantitative differences resulting from the slit curvature and the type of constitutive equation are relatively large only for flows of strongly non‐New‐tonian liquids.     

Flow of generalized newtonian liquids through fixed beds of nonspherical particles

The flow of generalized Newtonian liquids through a random fixed bed of particles has been investigated and a universal method of calculation of the creeping flow was suggested. The usefulness of this method has been verified experimentally for the flow of power law, Ellis and Sutterby liquids through fixed beds of different nonspherical particles.     

Terminal velocity of non-spherical particles falling through a Carreau model liquid

A procedure has been proposed for the estimation of the terminal falling velocity of non-spherical particles moving in a Carreau model fluid in the transition flow region. The procedure is based on a modification of the relationship formerly developed for the fall of spherical particles including the particle dynamic shape factor. The suitability of the proposed procedure has been confirmed by good agreement between experimental and calculated terminal falling velocity data. In the experiments, the terminal falling velocity of short cylinders and rectangular prisms in polymer solutions of different measure of shear thinning and elasticity has been measured.