R.P. Chhabra

Bubbles, Drops, and Particles in Non-Newtonian Fluids

Preface Preface to the First Edition Acknowledgements About the Author INTRODUCTION, SCOPE, AND ORGANIZATION NON-NEWTONIAN FLUID BEHAVIOR Definition of a Newtonian Fluid Non-Newtonian Fluids: Definition, Examples of Different Types, Mathematical Models Dimensional Considerations in the Mechanics of Viscoelastic Fluids Experimental Techniques: Rheometry RIGID PARTICLES IN TIME-INDEPENDENT FLUIDS WITHOUT A YIELD STRESS Governing Equations for a Sphere Spherical Particles in Newtonian Fluids Spheres in Shear-thinning Fluids Spheres in Shear-thickening Fluids Light Spheres Rising in Pseudoplastic Media Pressure Drop due to a Settling Sphere Non-Spherical Particles RIGID PARTICLES IN VISCOPLASTIC FLUIDS Static Equilibrium of Particles Flow Field: Shape and size of flow zones Drag Force Role of Values of Yield stress used in correlations Time Effects RIGID PARTICLES IN VISCOELASTIC FLUIDS Flow over a sphere Flow over a cylinder Other Studies Involving Interactions Between Non-Newtonian Characteristics, Particle Shape, Flow Field, etc. FLUID PARTICLES IN NON-NEWTONIAN MEDIA Formation of Fluid Particles Shapes of Bubbles and Drops in Free Rise or Fall Terminal Velocity-Volume Behavior in Free Motion Drag Behavior of Single Particles Bubble and Drops Ensembles in Free Motion Coalescence of Bubbles and Drops Breakage of Drops Motion and Deformation of Bubbles and Drops in Confined Flows NON-NEWTONIAN FLUID FLOW IN POROUS MEDIA AND PACKED BEDS Porous Medium Definition, Examples and Characterization Flow of Newtonian Fluids Flow of Non-Newtonian Fluids Miscellaneous Effects Two Phase Gas/Liquid Flow FLUIDIZATION AND HINDERED SETTLING Two-Phase Fluidization Sedimentation or Hindered Settling Three Phase Fluidized Beds MOMENTUM, HEAT AND MASS TRANSFER IN BOUNDARY LAYER FLOWS Boundary Layer Flows Viscoelastic Effects in Boundary Layers Mass Transfer from Bubbles Mass Transfer from Drops Mass Transfer from Ensembles of Bubbles and Drops Heat and Mass Transfer in Fixed Beds Heat and Mass Transfer in Fluidized Beds Heat and Mass Transfer in Three Phase Fluidized Beds Heat Transfer from Tube Bundles WALL EFFECTS Definition For Rigid Spheres For Non-Spherical Particles For Drops and Bubbles FALLING OBJECT RHEOMETRY Falling Ball Method Rolling Ball Method Rotating Sphere Viscometer Falling Cylinder Viscometer *All Chapters contain Introduction, Summary and Nomenclature sections References Subject index Author index

Drag on non-spherical particles: an evaluation of available methods

In this work, a selection of widely used correlations have been critically evaluated for estimating the drag coefficient of non-spherical particles in incompressible viscous fluids. Experimental results have been culled from 19 independent studies embracing wide ranging particle shapes including cylinders, needles, cones, prisms, discs, rectangular, parallelepiped and cubes. The resulting data base consisting of 1900 data points encompasses wide ranges of physical and kinematics conditions as: sphericity, 0.09 to 1 and the Reynolds number ranging from 10−4 to 5×105. In particular, the performance of five methods has been critically examined. The best method appears to be that of Ganser which uses the equal volume sphere diameter and the sphericity of particle. The resulting overall mean error is about 16%, though maximum error can be as large as ∼100%. In general, the lower the sphericity, the poorer is the prediction.

Flow of non-Newtonian fluids in fixed and fluidised beds

Abstract An attempt has been made to reconcile and to critically analyze the voluminous literature available on the flow of rheologically complex fluids through unconsolidated fixed beds and fluidised beds. In particular, consideration is given to the prediction of macro-scale phenomena of flow regimes, pressure drop in fixed and fluidised beds, minimum fluidisation velocity, dispersion and liquid–solid mass transfer. Available scant results seem to suggest that flow patterns qualitatively similar to that observed for Newtonian fluids, can be expected for the flow of purely viscous non-Newtonian fluids. A Reynolds number based on the effective pore size and pore velocity is seen to be a convenient parameter for the delineation of these flow regimes. Out of the four approaches available, the generalisation of the capillary model, due to Comiti and Renaud ( Chem. Engng. Sci. 44 (1989) 1539–1545), appears to be the best for the estimation of the pressure drop through fixed beds. This method requires the flow rate – pressure drop data for the flow of a Newtonian fluid, such as air or water, through the same bed to evaluate the two key parameters, namely, the tortuosity and the dynamic surface area. While this approach can accommodate non-spherical particle shape and the wall effects and encompasses all possible flow regimes, it is limited to the situations where the polymer–wall interactions are negligible. Similarly, based on a combination of the capillary and drag models, satisfactory expressions have been identified for the prediction of the minimum fluidising velocity and velocity-voidage behaviour of uniformly expanded fluidised beds for power-law liquids and beds of spherical particles. Little is known about the effect of particle shape and column walls on these parameters. Even less work has been reported on dispersion and liquid–solid mass transfer in packed and fluidised beds, and no theoretical or experimental results seem to be available on heat transfer in these systems. Therefore, the expressions for the prediction of Peclet and Sherwood numbers presented herein must be regarded as somewhat tentative at this stage. Finally, little definitive and quantitative information is available on the role of viscoelasticity and of the effects arising from polymer/wall interactions, polymer retention, etc.

Transport processes in bubbles, drops and particles

1. Constitutive Moedling of Bubbly Liquids K.R. Rajagopal and L. Tao 2. Analytical Expressions E.E. Michaelides 3. Electrokinetic- and Thermocapillary-Flow-Driven Aggregation of Particles and Bubbles on Surfaces P.J. Slides, J.L. Anderson, H. Kasumi, S.A.Guelcher, and Y.E. Solomentsev 4. Recent Developments in the Bubble Velocity Jump Discontinuity D. Rodrigue and D. DeKee 5. Cavitation and Bubble Dynamics P.R.Williams 6. Foam Darinage, Coasrening and Evaporation S.A.Magrabi and B.Z. Dlugogorski 7. Interphase Mass and Heat Transfer in Gas-Fluidized Beds J.R.Grace 8. Progress in an Industrial Application of Fluidized Beds: Advances in the Sand Coremaking Process S.I.Bakhtiyarov, R.A.Overfelt and D.A. Siginer 9. Sedimentation and Fluidization of Solid Particles in LiquidsR. Di Felice 10. Determination of the Constitutive Relationship for Filer Cakes in Cake Filtration Using the Analogy between Filtration and Diffusion B.V. Ramarao, C. Tien and C.N. Styadev 11.Tracer Dispersion in Fluid Motion through a Porous Medium J.P. du Plessis 12. Flows of Concentrated Granular Mixtures P. Coussot 13. Wall Effects on Spheres Falling Axially in Cylindrical Tubes R.P. Chhabra 14. Steady and Transient Motion of Spherical Liquids G.H. McKinlye 15. Particle Deposition in Membrane Systems V. Chen and D.E. Wiley 16. Rheological Properties of Concentrated Suspensions P.J. Carreau and F. Cotton

CREEPING SPHERE MOTION IN HERSCHEL-BULKLEY FLUIDS - FLOW-FIELD AND DRAG

The results of a comprehensive experimental study on the creeping sphere motion in Herschel-Bulkley model fluids are reported and discussed in this paper. A series of aqueous solutions of different grades of Carbopol resin have been used to encompass wide ranges of rheological model parameters. In particular, the size and shape of the sheared cavity surrounding a moving sphere and the drag behaviour have been investigated in this study. A laser-speckle tracer method was used to obtain information on the structure of the flow field about a sphere which, in turn, shows the size and shape of the underformed regions in the flow domain. The measured terminal velocity data together with the physical properties and dimensions have been used to deduce the values of drag coefficient as a function of the pertinent dimensionless groups. The paper is concluded by presenting extensive comparisons, of both sheared cavity characteristics and drag behaviour, between the present results and the literature correlations and/or data.

The influence of fluid elasticity on the drag coefficient for creeping flow around a sphere

Drag coefficients are measured for the creeping motion of a sphere in nonshear-thinning elastic fluids. The data obtained cover a Weissenberg number range of 1.66 × 10−4 to 2.02. For 0 ⩽ We ⩽ 0.1 no significant deviation from the Stokes drag is observed as a result of fluid elasticity. For We > 0.1 a continuous reduction in drag below the Stokes value is observed until an asymptotic reduction of 26% is reached for We >/ 0.7. Existing theoretical analyses are inadequate for predicting the reduction in drag observed and its asymptotic value. The effect of fluid elasticity on the drag coefficient in the absence of any significant shear-thinning effects is clearly demonstrated.

Effects of Reynolds and Prandtl Numbers on Heat Transfer Across a Square Cylinder in the Steady Flow Regime

ABSTRACT The effects of Reynolds and Prandtl numbers on the heat transfer characteristics of an isolated square cylinder have been investigated for the range of conditions 1 ≤ Re ≤ 45 and 0.7 ≤ Pr ≤ 4,000 (the maximum value of Peclet number being 4,000) in crossflow. Heat transfer correlations are obtained for the constant cylinder temperature and constant heat flux boundary conditions on a solid square cylinder in the steady flow regime. In addition, the variation of local Nusselt number on each face of the obstacle and representative isotherm plots are presented to elucidate the role of Prandtl number on heat transfer in the steady flow regime.

Rheological behaviour of aqueous suspensions of laponite: new insights into the ageing phenomena

In this paper, the ageing behaviour of suspensions of laponite with varying salt concentration is investigated using rheological tools. It is observed that ageing is accompanied by an increase in the complex viscosity. The creep experiments performed at various ages show damped oscillations in the strain. The characteristic time scale of the damped oscillations, the retardation time, shows a prominent decrease with increasing age of the system. However, this dependence weakens with an increase in the salt concentration, which is known to change the microstructure of the system from glass like to gel like. We postulate that a decrease in the retardation time can be represented as a decrease in the viscosity (friction) of the dissipative environment surrounding the arrested entities that oppose elastic deformation of the system. We believe that ageing in colloidal glass leads to a greater ordering that enhances relative spacing between the constituents, thereby reducing the frictional resistance. However, since a gel state is inherently different in structure (fractal network) to that of a glass state (disordered), ageing in the gel does not induce ordering. Consequently, we observe an inverse dependence of retardation time on age, which becomes weaker with an increase in the salt concentration. We analyse these results from the perspective of ageing dynamics of both glass and gel states of laponite suspensions.

Forced convection heat transfer in tube banks in cross flow

Abstract The forced convection heat transfer characteristics for an incompressible, steady and Newtonian fluid flow over a bundle of circular cylinders has been investigated numerically. The inter-cylinder hydrodynamic interactions have been approximated by employing a simple cell model. The momentum and energy equations have been solved by using a finite difference based numerical solution procedure for a range of physical and kinematic conditions. Furthermore, the role of the type of thermal boundary condition, namely, a constant temperature or a constant heat flux, imposed on the surface of the cylinder has also been elucidated. Extensive results on the temperature fields, and on the variation of the Nusselt number on the surface of a typical cylinder in the assemblage have been obtained for two values of the Prandtl number (corresponding to air and water). The Reynolds number of flow was varied in the range 1–500 and the voidage of the assemblage ranged from 0.4 to 0.99 thereby covering the entire range of interest as encountered in tubular heat exchangers and in fibrous beds. The paper is concluded by presenting extensive comparisons with the limited analytical/numerical and/or experimental results available in the literature for the case of a single cylinder as well as that for tube bundles.

Wall effect for spheres falling at small reynolds number in a viscoplastic medium

Abstract The wall factor, f = u / u ∞ , for spheres falling slowly under gravity through viscoplastic media has been measured. A critical value of sphere-to-tube diameter ratio, ( d / D ) crit , is found to exist, below which no wall effect is descernible, f = 1. The critical diameter ratio is correlated in terms of a dimensionless yield-gravity group Y = τ y / gd (ϱ s − ϱ); it appears to be related to the dimensions of the sheared fluid envelope surrounding the sphere. For diameter ratios larger than the critical value, the behaviour of the wall factor is very similar to that found for shear-thinning fluids without yield stress.