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.     

Experimental investigations of non‐Newtonian/Newtonian liquid‐liquid flows in microchannels

The plug flow of a non‐Newtonian and a Newtonian liquid was experimentally investigated in a quartz microchannel (200‐µm internal diameter). Two aqueous glycerol solutions containing xanthan gum at 1000 and 2000 ppm were the non‐Newtonian fluids and 0.0046 Pa s silicone oil was the Newtonian phase forming the dispersed plugs. Two‐color particle image velocimetry was used to obtain the hydrodynamic characteristics and the velocity profiles in both phases under different fluid flow rates. The experimental results revealed that the increase in xanthan gum concentration produced longer, bullet‐shaped plugs, and increased the thickness of the film surrounding them. From the shear rate and viscosity profiles, it was found that the polymer solution was in the shear‐thinning region while the viscosity was higher in the middle of the channel compared to the region close to the wall. Circulation times in the aqueous phase increased with the concentration of xanthan gum. © 2017 The Authors AIChE Journal published by Wiley Periodicals, Inc. on behalf of American Institute of Chemical Engineers AIChE J, 63: 3599–3609, 2017     

Bubble Volume and Aspect Ratio Generated in Non‐Newtonian Fluids

Bubble formation from an orifice submerged in quiescent polyacrylamide aqueous solution was investigated numerically with a sharp‐interface coupled level‐set/volume‐of‐fluid method based on the rheological characteristics of the fluid. In both non‐Newtonian fluids and Newtonian fluids, the numerical approach was able to capture accurately the deformation of the bubble surface, validated by comparison with experimental results. The effects of orifice diameter, solution mass concentration, and gas flow rate on bubble volume and aspect ratio were evaluated. Both the instantaneous and detached volume decrease with the orifice diameter but increase with mass concentration and gas flow rate. The aspect ratio at the departing point tends to rise with the orifice diameter and mass concentration and falls with the gas flow rate.     

Computational Fluid Dynamics Simulation of Generation and Coalescence of Bubbles in Non‐Newtonian Fluids

The successive generation and coalescence behaviors of bubbles from two parallel nozzles in non‐Newtonian fluids were numerically simulated by using the volume of fluid (VOF) method. Three flow patterns for bubbles and the related flow regime transition lines were obtained. Two critical nozzle intervals exist: one for the bubble coalescence before pinch‐off, and another for alternating bubble formation then in‐line coalescence under different conditions. Two correlations were proposed to predict the dimensionless critical nozzle intervals for the transition of bubble‐flow patterns. The influences of nozzle diameter, gas flow rate, nozzle interval, and rheological properties of fluid on bubble‐flow patterns were investigated systematically.     

Breakup dynamics of slender bubbles in non‐newtonian fluids in microfluidic flow‐focusing devices

This study aims to investigate the breakup of slender bubbles in non‐Newtonian fluids in microfluidic flow‐focusing devices using a high‐speed camera and a microparticle image velocimetry (micro‐PIV) system. Experiments were conducted in 400‐ and 600‐μm square microchannels. The variation of the minimum width of gaseous thread with the remaining time before pinch‐off could be scaled as a power‐law relationship with an exponent less than 1/3, obtained for the pinch‐off of bubbles in Newtonian fluids. The velocity field and spatial viscosity distribution in the liquid phase around the gaseous thread were determined by micro‐PIV to understand the bubble breakup mechanism. A scaling law was proposed to describe the size of bubbles generated in these non‐Newtonian fluids at microscale. The results revealed that the rheological properties of the continuous phase affect significantly the bubble breakup in such microdevices. © 2012 American Institute of Chemical Engineers AIChE J,, 2012     

Coaxial Mixer Hydrodynamics with Newtonian and non‐Newtonian Fluids

The performance of several combinations of a wall scraping impeller and dispersing impellers in a coaxial mixer operated in counter‐ and co‐rotating mode were assessed with Newtonian and non‐Newtonian fluids. Using the power consumption and the mixing time as the efficiency criteria, impellers in co‐rotating mode were found to be a better choice for Newtonian and non‐Newtonian fluids. The hybrid impeller‐anchor combination was found to be the most efficient for mixing in counter‐rotating or co‐rotating mode regardless of the fluid rheology. For both rotating modes, it was shown that the anchor speed does not have any effect on the power draw of the dispersing turbines. However, the impeller speed was shown to affect the anchor power consumption. The determination of the minimum agitation conditions to achieve the just suspended state of solid particles (N_js) was also determined. It was found that N_js had lower values with the impellers having the best axial pumping capabilities.     

Power Requirement when Mixing a Shear‐Thickening Fluid with a Helical Ribbon Impeller Type

The optimal design of close clearance impellers requires the knowledge of the power demand of the mixing equipment. In non‐Newtonian mixing, this can be readily obtained using the Metzner and Otto concept [1]. In this work, this concept and the determination of the K_s value for an atypical helical agitator (PARAVISC system from Ekato firm) have been revised in the case of shear‐thinning fluids and a shear‐thickening fluid. For poor shear‐thinning fluids, it has been shown that for our mixing system the K_s value does not vary strongly with the flow behavior index, and may be regarded as a constant for the mixing purpose design. By contrast, for the shear‐thickening fluid, power consumption measurements indicate that the relationship between the K_s values and the flow behavior index is much more complex due to a partial solidification of the product around the impeller.     

Energy Dissipation Rates of Newtonian and Non‐Newtonian Fluids in a Stirred Vessel

Energy dissipation rates of water and glycerol as Newtonian fluids and carboxyl methyl carbonate solution as non‐Newtonian fluid in a stirred vessel are investigated by 2D particle image velocimetry and compared. Mean velocity profiles reflect the Reynolds (Re) number similarity of two flow fields with different rheological properties, but the root mean square velocity profiles differ in rheology at the same Re‐number. Energy dissipation rates are estimated by direct calculation of fluctuating velocity gradients. The varying energy dissipation rates of Newtonian and non‐Newtonian fluids result from the difference in fluid rheology and apparent viscosity distribution which decides largely the flow pattern, circulation intensity, and rate of turbulence generation.     

Power Requirement for Mixing Shear‐Thinning Fluids with a Planetary Mixer

The optimal control of processes dealing with non‐Newtonian liquids requires the knowledge and control of the power demand of the mixing equipment. In this context, an extension of the Metzner and Otto concept to planetary mixers is proposed to adapt this concept to planetary mixers. The theoretical part of this work defines modified expressions of Reynolds and power numbers. These definitions introduce a characteristic velocity u_ch that is used to define the parameter K_s. A planetary mixer is employed to experimentally ascertain this guideline. Power consumption measurements carried out by mixing shear‐thinning fluids permit to determine the K_s factor. This factor varies only slightly with the flow behavior index and may be regarded as a defined constant for this geometry. Finally, experiments with an additional shear‐thickening fluid confirm the validity of this approach.     

Natural convection in a vertical porous cavity filled with a non‐newtonian binary fluid

Numerical and analytical study of natural convection in a vertical porous cavity filled with a non‐Newtonian binary fluid is presented. The density variation is taken into account by the Boussinesq approximation. A power‐law model is used to characterize the non‐Newtonian fluid behavior. Neumann boundary conditions for temperature are applied to the vertical walls of the enclosure, while the two horizontal ones are assumed impermeable and insulated. Both double‐diffusive convection (a = 0) and Soret‐induced convection (a = 1) are considered. Scale analysis is presented for the two extreme cases of heat‐driven and solute‐driven natural convection. For convection in a thin vertical layer (A ? 1), a semianalytical solution for the stream function, temperature, and solute fields, Nusselt and Sherwood numbers are obtained using a parallel flow approximation in the core region of the cavity and an integral form of the energy and constituent equations. Numerical results of the full governing equations show the effects of the governing parameters, namely the thermal Rayleigh number, R_T, the Lewis number, Le, the buoyancy ratio, φ, the power‐law index, n, and the integer number a. A good agreement between the analytical predictions and the numerical simulations is obtained. © 2012 American Institute of Chemical Engineers AIChE J, 58: 1704–1716, 2012     

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.     

Influence of impeller type on hydrodynamics and gas‐liquid mass‐transfer in stirred airlift bioreactor

The influence of impeller type in a mechanically stirred airlift bioreactor was analyzed in relation to the non‐Newtonian viscous fluids. The agitation was carried out through a marine impeller (axial impeller) and a paddle impeller (radial impeller) located along with the gas sparger in the region comprised by the riser. The bioreactor was sparged with air under different velocities (0.036–0.060 m s^?1). Carboxymethylcellulose 1.94% and xanthan 1.80% were used as a fluid model. The gas holdup and volumetric mass‐transfer coefficient increased in up to five and three times, respectively, when compared to a conventional airlift bioreactor; however, better results were obtained when the straight paddle impeller type was used. The results suggest that the studied bioreactor can be used successfully in viscous fluid, and it can be more efficient than conventional airlift bioreactors. The results obtained suggest the use of radial impellers. © 2015 American Institute of Chemical Engineers AIChE J, 61: 3159–3171, 2015     

Viscoelastic Liquids in Stirred Vessels – Part I: Power Consumption in Unaerated Vessels

Different shear‐thinning and elastic fluids (STE fluids) have been stirred under unaerated conditions, in vessels equipped with Rushton disc turbines. Their power consumption has been evaluated over a wide range of stirring rates and their Metzner‐Otto constant (k_s) has been measured. A correlation has then been proposed to predict k_s values for a Rushton turbine operating in non‐Newtonian solutions. Power curves of STE fluids have been drawn and compared with reference curves (Newtonian, shear‐thinning inelastic and elastic with constant shear viscosity fluids). The STE fluids have thus been divided into two categories. The STE fluids of the first category (STE I fluids), which are concentrated viscous solutions of polymers (guar, CMC) reducing the power consumption at the beginning of the transitional region and connecting with the Newtonian reference at higher Reynolds numbers. In contrast, STE solutions of the second category (STE II fluids), which are solutions of drag reducing polymers (PAA), are less viscous and more elastic. They reduce the power consumption at the end of the transitional region and do not connect with the Newtonian reference, at least until Re = 6000. A general correlation has finally been proposed to model the power curve of STE fluids stirred by a Rushton turbine from the laminar to the turbulent regions, as a function of their elasticity.     

Predicting Power for a Scaled‐up Non‐Newtonian Biomass Slurry

High‐solids biomass slurries exhibit non‐Newtonian behavior with a yield stress and require high power input for mixing. The goals were to determine the effect of scale and geometry on power number P₀, and estimate the power for mixing a pretreated biomass slurry in a 3.8 million L hydrolysis reactor of conventional design. A lab‐scale computational fluid dynamics model was validated against experimental data and then scaled up. A pitched‐blade turbine and A310 hydrofoil were tested for various geometric arrangements. Flow was transitional; laminar and turbulence models resulted in equivalent P₀ which increased with scale. The ratio of impeller diameter to tank diameter affected P₀ for both impellers, but impeller clearance to tank diameter affected P₀ only for the A310. At least 2 MW is required to operate at this scale.     

Characterization of Bubble Shapes in Non‐Newtonian Fluids by Parametric Equations

Based on experiments with single air bubbles rising in stagnant non‐Newtonian fluids, an innovative model containing the aspect ratio (E) and two parameters (α, β) was proposed and proved to be capable of characterizing the bubble shape from spherical/ellipsoidal to prolate/oblate‐tear with good accuracy. Several impacts on bubble deformation were investigated, involving the rheological properties of the fluids and different forces exerted on the bubble, which were quantified by multiple dimensionless numbers (e.g., Reynolds, Eötvös, and Deborah number). Within a wide range, the empirical correlations were obtained for parameter β, and between α and β. Together with the shape model, a complete system was set up for bubble shape characterization and prediction that will provide new ideas for future studies on bubble hydrodynamics.     

Hydrodynamic Performance of a Ring-Style High-Shear Impeller in Newtonian and Shear-Thinning Fluids

Laminar mixing of Newtonian and shear-thinning fluids induced by a Hockmeyer®-type impeller was investigated. Two unbaffled tanks at three impellers off-bottom clearances (c) were studied. Six geometric combinations, i.e., two d/T and three c/T, were examined where d and T are the impeller and tank diameters, respectively. Determination of the Metzner-Otto constant (K_s) was undertaken. The effects of d/T, c/T, and fluid rheology on K_s, power demand, pumping, shear and viscous dissipation were analyzed. The evaluated geometric ratios and rheology do not significantly affect K_s and power demand, only the rheology had an impact on the remaining hydrodynamic parameters. Pumping was favored with the Newtonian fluid, and shear and viscous dissipation increased with the shear-thinning fluid.     

Pressure drop in a split‐and‐recombine caterpillar micromixer in case of newtonian and non‐newtonian fluids

Microreactors are very promising tools for the design of future chemical processes. For example, emulsions of very narrow size distribution are obtained at much lower energy consumption than the one spent with usual processes. Micromixers play thereby an eminent role. The goal of this study is to better understand the hydrodynamic properties of a split‐and‐recombine Caterpillar micromixer (CPMM) specially with regard to handling viscoelastic fluids, a topic hardly addressed so far in the context of micromixers in general, although industrial fluids like detergent, cosmetic, or food emulsions are non‐Newtonian. Friction factor was measured in a CPMM for both Newtonian and non‐Newtonian fluids. For Newtonian fluids, the friction factor in the laminar regime is f/2 = 24/Re. The laminar regime exists up to Reynolds numbers of 15. For shear‐thinning fluids like Carbopol 940 or viscoelastic fluids like Poly Acryl Amide (PAAm) aqueous solutions, the friction factor scales identically within statistical errors up to a generalized Reynolds number of 10 and 0.01, respectively. Above that limit, there is an excess pressure drop for the viscoelastic PAAm solution. This excess pressure drop multiplies the friction factor by more than a decade over a decade of Reynolds numbers. The origin of this excess pressure drop is the high elongational flow present in the Caterpillar static mixer applied to a highly viscoelastic fluid. This result can be extended to almost all static mixers, because their flows are generally highly elongational. © 2013 American Institute of Chemical Engineers AIChE J, 59: 2679–2685, 2013     

The effect of surfactants on deformation of falling non‐Newtonian drops in a Newtonian liquid

Deformation of settling non‐Newtonian ellipsoidal drops in a Newtonian liquid was experimentally observed. Corn oil was used as the Newtonian phase and solutions of polyacrylamide in aqueous glycerine as the non‐Newtonian phase. The shear‐thinning behaviour of the drops fluid was controlled by the amount of polymer dissolved, while the effect of interfacial tension was examined using different concentrations of sodium dodecyl sulphate (SDS). In the range of 1 < E < 2.9, 0.2 < Eo, < 23, and 0 < Ma < 17.2, drop eccentricity increased linearly with a modified Eötvös number taking into account the effect of surfactants. For the range of experimental conditions tested, drop deformation was mainly controlled by viscous and interfacial tension forces, while shear‐thinning and inertia effects were negligible.     

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.     

Pressure Drop Models for Gas/Non‐Newtonian Power‐Law Fluids Flow in Horizontal Pipes

Models commonly used in literature are evaluated versus 696 data points to predict the pressure drop of gas/non‐Newtonian power‐law fluids flow in horizontal pipes. Suitable models are recommended. A new correlation is developed by ignoring the pressure drop across the gas slug and adopting the liquid slug holdup of gas/non‐Newtonian fluid flow into the homogeneous model. The theoretical curves can capture the test data trends and the overall agreement of predicted values with experimental data is sufficient to be practically applied in industry.