Power Law Flow Behaviour Index Research Articles

Power-law fluid annular flows between concentric rotating spheres subject to hydrodynamic slip

ABSTRACT We report analytical solutions to the problem of non-Newtonian power-law fluid flows in the annular space between a pair of concentric spherical surfaces rotating at distinct angular velocities with the inner and outer wall boundaries subject to general asymmetric hydrodynamic slip conditions. Analytical solutions are possible because of assuming constant valued apparent hydrodynamic slip lengths in the linearized kinematic slip conditions, and our solutions can be validated against the limiting results of Newtonian fluids, no-slip conditions or a single rotating sphere reported in previous literature. Comprehensive systematic parametric studies show that (additional to the power-law fluid flow behavior index) the degree of hydrodynamic slip at the inner surface is the dominant factor that determines the limiting values of the viscous torque exerted on the inner sphere as the outer-to-inner radius ratio assumes significantly large values. Nonetheless, the flow behavior index and outer slip length prove to be the crucial key parameters responsible for a variety of torque responses, which can be categorized by a compact analytical expression, as the outer-to-inner radius ratio is increased in the small to moderate regime. We propose a criteria which identifies the proper slip length and outer-to-inner radius ratio combinations for a given power-law flow behavior index such that the hydrodynamic slip wall effects of the outer surface can be minimized or eliminated. A simple method is also presented to characterize and quantify the apparent hydrodynamic slip effects by use of the concentric rotating spheres viscometer.

Improvement of Local (Umuna) Clay for the Production of Drilling Mud

The continuous importation of foreign bentonite in the presence of proved and potential clay deposits spread across states in the country is not consistent with the government desire and commitment for the development of local content. This research explores the possibility of improving the Umuna local clay samples for the production of drilling mud. The clay samples were experimentally analysed in accordance with API standard of measurement. The obtained results show a marked trend as most of the considered rheological parameters such as yp, PV, gel strength, yp/PV, viscosity @600rpm and fluid loss were far away from satisfying the minimum required API standard specification for drilling mud. While parameters such as mud density, n-factor, K-factor, specific gravity and filter cake all satisfy the minimum required API standard specification for drilling mud production. The rheological properties of the local clays were significantly improved after beneficiation with 2g PAC-R. These obtained result after beneficiation now equated to that of the imported bentonite and also satisfy the required minimum API standard specification. The local clay fell under the inactive soil type based on its average activity value of 0.52. The presence of oxide compositions and cation exchange capacity (CEC) range of 70 – 150 Meq/100g suggests and confirms that the analyzed clay is of the montmorillonite family, with traces of illite which is good for drilling mud production. The obtained average thinner cake value (0.35) for the local clay is more desirable in drilling operations for reducing the risk of pipe sticking. This property with the combination of high value (29.39av) of consistency factor (k) and low value (0.28av) of the power law flow behavior index (n) portrays the clay sample been very much adequate for effective hole cleaning due to its high viscosity resulting from high consistency factor (k) and enough shear thinning property derived from the low power law flow behavior index (n).

Finite double layer and non-Newtonian power-law effects on electrokinetic diffusioosmotic flows in parallel plate microchannels

Diffusioosmotic flows of electrolytic non-Newtonian power-law liquids in parallel plate microchannels are theoretically investigated by considering the finite thickness of the interfacial electrical double layers. Semi-analytical or numerical solutions to the shear stress, flow velocity, effective viscosity, and volume flow rate are obtained as functions of the spatial coordinate, half channel width to Debye length ratio, interfacial zeta potential, diffusivity difference parameter, and the power-law flow behavior index. The border curves of zero flow rate obtained herein delineating the parametric regimes in which the non-Newtonian diffusioosmotic flow rate is directed towards downstream or upstream are distributed differently on the zeta potential versus diffusivity difference parameter map as compared to those obtained using thin double layer theory. As compared to Newtonian liquids, the rheology of dilatant and pseudoplastic liquids respectively enlarges and reduces the parametric regimes in which flow rate towards downstream occurs, which is a result exactly opposite to that obtained based on the thin double layer theory reported in recent literature.

Similarity solution of axisymmetric non-Newtonian wall jets with swirl

The similarity solution of axisymmetric wall jets with swirl on bodies of revolution for non-Newtonian power-law fluids is presented. The physical and geometrical meaning of all parameters appearing in the course of similarity procedure is treated in detail. Especially, the functional dependence of length, velocity, and pressure similarity scales on the shape and swirl parameters and the power-law flow behaviour index is determined. The already published results related to the similarity solution obtained are discussed.

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.

New model for single spherical particle settling velocity in power law (visco-inelastic) fluids

Particle settling in a non-Newtonian power law fluid is of interest to many industrial applications, including chemical, food, pharmaceutical, and petroleum industry. Conventionally, the Newtonian model for the drag coefficient prediction is extended to non-Newtonian fluids. The approach of merely replacing a viscosity term in Newtonian correlation with a power law apparent viscosity is reported to be inadequate. In this investigation, the inadequacy of the Newtonian model to correlate the data of single solid spherical particle moving in power law liquids is demonstrated. An approach presented earlier by Shah has been adopted to re-analyze the previously published data of particle settling in various non-Newtonian fluids from five different investigations. The particle settling velocity data have been correlated with two dimensionless quantities – drag coefficient C d and particle Reynolds number Re – as C d 2 - n Re 2 versus Re, rather than the conventional correlation of C d versus Re. A new model to predict the settling velocity of a spherical particle moving in inelastic power law liquids is presented, which reduces to the expected Newtonian fluid limit. It is shown that the Shah’s method predicts the particle settling velocity data much closer to the experimental data than the Newtonian standard drag curve that has been widely used by many researchers. The new model is valid for a wide range of power law flow behavior index n (0.281–1.0) and particle Reynolds number Re (0.001–1000). The paper is concluded by presenting an illustrative example to calculate the settling velocity of a spherical particle in non-Newtonian liquid.

Drag on non-spherical particles in power law non-Newtonian media

The free settling velocity of cylinders and disks falling in quiescent Newtonian and power law liquids has been measured over wide ranges of experimental conditions of the particle Reynolds number (10 − 5 –∼300), power law flow behaviour index (0.31–1) and the length-to-diameter ratio, ∼0.4–∼14. The corresponding range of sphericity is 0.62 to 0.86. An existing drag expression which has been tested extensively for spherical particles falling in Newtonian and in power law fluids has been slightly modified here for non-spherical particles. In particular, the use of this drag expression necessitates a knowledge of an equal volume sphere diameter (to evaluate the Reynolds number and drag coefficient) and the ratio of the surface area to the projected area of a non-spherical particle. With these modifications, the approach outlined here reproduces the present and the literature data for a wide range of non-spherical particles including cones, prisms, needles, cylinders settling in both Newtonian and power law fluids with reasonable levels of accuracy.

Sedimentation of a circular disk in power law fluids

The continuity and momentum equations have been solved numerically for the two-dimensional steady flow of power law fluids over a thin circular disk oriented normal to the direction of flow. Extensive results on the individual and total drag coefficients are obtained as functions of the power law flow behavior index ( 0.4 ⩽ n ⩽ 1.0 ), Reynolds number ( 1 ⩽ Re ⩽ 100 ) and the blockage ratio, disk-to-cylinder diameter ratio ( 0.02 ⩽ e ⩽ 0.5 ), which can be used to estimate the settling velocity of a circular disk. The numerical predictions of drag are consistent with the scant experimental results available in the literature.

Convective heat transfer for power law fluids in packed and fluidised beds of spheres

The forced convection heat transfer characteristics for an incompressible and steady flow of power law liquids in fixed and extended beds of spherical particles has been studied numerically. The sphere–sphere hydrodynamic interactions have been accounted for by using a simple cell model. Within the framework of such a cell model, the momentum and energy equations have been solved using a finite difference method to obtain the velocity and temperature fields. Extensive numerical estimates of the local and average Nusselt numbers as functions of the physical, rheological and kinematic variables have been presented and discussed for the two commonly employed thermal boundary conditions. In broad terms, the Nusselt number for power law fluids (both shear-thinning and shear-thickening conditions) normalized with respect to the corresponding value for a Newtonian fluid shows weak additional dependence on the power law flow behaviour index. The shear-thinning behaviour is seen to promote heat transfer and as expected the shear-thickening behaviour impedes heat transfer in fixed and fluidised beds. All in all, the present results encompass wide ranges of conditions as follows: Reynolds number: 1–500; Peclet number: 1–500; bed voidage: 0.4–0.8 and the flow behaviour index: 0.5–1.8 thereby covering extremely shear-thinning and shear-thickening types of fluid behaviours. The paper is concluded by presenting detailed comparisons with the limited analytical and/or experimental results available for liquid–solid mass transfer in such systems.

A note on pressure drop for the cross-flow of power-law liquids and air/power law liquid mixtures past a bundle of circular rods

The steady flow of non-Newtonian polymer solutions on their own and together with air transverse to a bundle of circular rods, has been studied experimentally. In particular, the frictional pressure drop has been measured as a function of non-Newtonian power-law constants, flow rates, voidage of bundles and the input fraction of air over wide ranges of conditions as: power-law flow behaviour index, 0.5 to 1; voidage values of 0.78 and 0.87–0.88; air velocity (superficial), 0.09 to 0.28 m/s and liquid velocity (superficial) 0.018 to 0.5 m/s. While the single phase flow results for power-law liquids are in excellent agreement with an existing equation, the two-phase pressure drop results are also in accordance with the form of the well-known Lockhart–Martinelli correlation. Based on these results, it is suggested that a value of 5–10 for the Reynolds number based on rod diameter marks the limit of the laminar flow in such systems.

EFFECT OF PROCESSING, PRESERVATION AND STORAGE ON RHEOLOGY OF GUAVA PULP

ABSTRACTRheological behavior of white and pink guava pulps was investigated using Rheotest 2 coaxial cylinder viscometer over shear rates of 0.6 to 145.8 s−1. The shear rate‐shear stress data followed the power law, Casson, Herschel‐Bulkley, Michaels and Bolger models. The power law flow behavior index values were in the range of 0.14 to 0.19. Presence of measurable yield stress indicated Bingham model of the pulps. Storage of pulps for 30 weeks had negligible influence on flow behavior index values. The consistency index and yield stress values decreased with extending storage from 0 to 30 weeks. The Michaels and Bolger model distinctly described the non‐Newtonian nature of cold‐ and hot‐break pulps as well as different behavior of the pink guava pulps. The structure shear resistance values for hot‐break pulps at zero shear rate were considerably higher than those of the cold‐break, indicating their high pseudoplasticity.

Power law fluid flow over a bundle of cylinders at intermediate Reynolds numbers

Extensive theoretical estimates of pressure, friction and total drag coefficients for the cross flow of power law fluids normal to an array of long circular cylinders are reported in this paper. The equations of continuity and of momentum have been solved numerically for the unknown velocities and pressure which, in turn, have been processed further to infer the values of drag coefficients. The inter-cylinder interactions have been approximated by the two commonly employed “concentric cylinders” cell models. The results reported herein embrace the following ranges of physical and kinematic conditions: power law flow behaviour index: 0.5, 0.6, 0.8 and 1; Reynolds number, 1–500 and the voidage of 0.4 and 0.5, albeit limited results are also presented for voidage values of 0.6 and 0.9. The accuracy of the numerical solution procedure has been established by carrying out extensive comparisons with the previously available analytical and/or numerical results for the flow of Newtonian and power law fluids. Similarly, the applicability of this simple approach of the cell model has been demonstrated by presenting detailed comparisons between the present predictions and the limited experimental results available in the literature for the flow of shear-thinning polymer solutions over banks of cylinders.

Computational fluid dynamics model for predicting flow of viscous fluids in a large fermentor with hydrofoil flow impellers and internal cooling coils

Considerable debate has occurred over the use of hydrofoil impellers in large-scale fermentors to improve mixing and mass transfer in highly viscous non-Newtonian systems. Using a computational fluid dynamics software package (Fluent, version 4.30) extensive calculations were performed to study the effect of impeller speed (70-130 rpm), broth rheology (value of power law flow behavior index from 0.2 to 0.6), and distance between the cooling coil bank and the fermentor wall (6-18 in.) on flow near the perimeter of a large (75-m3) fermentor equipped with A315 impellers. A quadratic model utilizing the data was developed in an attempt to correlate the effect of A315 impeller speed, power law flow behavior index, and distance between the cooling coil bank and the fermentor wall on the average axial velocity in the coil bank-wall region. The results suggest that there is a potential for slow or stagnant flow in the coil bank-wall region which could result in poor oxygen and heat transfer for highly viscous fermentations. The results also indicate that there is the potential for slow or stagnant flow in the region between the top impeller and the gas headspace when flow through the coil bank-wall region is slow. Finally, a simple guideline was developed to allow fermentor design engineers to predict the degree of flow behind a bank of helical cooling coils in a large fermentor with hydrofoil flow impellers.

Thermal and normal squeezing effects in lubrication of rollers by a power-law fluid

A theoretical study of hydrodynamic lubrication in pure rolling of identical rigid cylinders has been carried out using the usual two-parameter power-law fluid model. Normal squeezing motion together with cavitation has been investigated. In this study the effects arising from the variation of viscosity with hydrodynamic pressure and temperature have been incorporated in the analysis. The resulting governing equations have been transformed to a set of ordinary differential equations which have been solved numerically using the Adams-Moulton method. A non-uniform grid was employed to achieve more accurate predictions of pressure and temperature, especially in the high pressure region. A detailed examination of the theoretical results obtained herein seems to suggest that the temperature dependence of the lubricant viscosity causes a reduction in both the load-carrying capacity and the surface traction of the system, whereas the normal squeezing motion (in a direction perpendicular to that of the fluid flow) leads to a substantial increase in pressure and also displaces the pressure peak towards the centre-line of contact. This observation appears to be true for all the values of the power-law flow behaviour index studied herein.

X-RAY FLOW VISUALISATION OF FLOW PATTERNS DURING THE MIXING OF YIELD STRESS, NEWTONIAN AND DILATANT FLUIDS

Abstract This paper describes the use of an X-ray flow visualisation technique to visualise flow patterns in opaque fluids. Flow patterns were observed in a pseudoplastic fluid possessing a yield stress, in a dilatant clay suspension and in a Newtonian fluid for comparison. Impeller Reynolds numbers were in the range 4.5 ≤ Re≤ 74. The flow patterns observed in the fluid possessing a yield stress indicate that the predominant motion in the mixed cavern surrounding the impeller is circular in nature. Comparison of the flow patterns observed in the dilatant suspension, power law flow behaviour index n ≅ 3, with those in the Newtonian fluid al the same Reynolds number, shows that bulk motion throughout the vessel is greater but extensional flows seem reduced. The observed flow patterns support the prediction that when mixing a power law fluid with n > 2 an increase in impeller speed will lead to a reduction in impeller Reynolds number.