The rheology of suspensions of porous zeolite particles in polymer solutions

We present data and predictive models for the shear rheology of suspended zeolite particles in polymer solutions. It was found experimentally that suspensions of zeolite particles in polymer solutions have relative viscosities that dramatically exceed the Krieger–Dougherty predictions for hard sphere suspensions. Our investigations show that the major origin of this discrepancy is due to the selective absorption of solvent molecules from the suspending polymer solution into zeolite pores. The effect raises both the polymer concentration in the suspending medium and the particle volume fraction in the suspension. Consequently, both the viscosity of the polymer solution and the particle contribution to the suspension viscosity are increased. We propose a predictive model for the viscosity of porous zeolite suspensions by incorporating a solvent absorption parameter, α, into the Krieger–Dougherty model. We experimentally determined the solvent absorption parameter by comparing viscosity data for suspensions of porous and nonporous MFI zeolite particles. Our results are in good agreement with the theoretical pore volume of MFI particles.     

Some further observations on the creeping motion of spheres through Ellis model fluids

Using the formulation of Hopke and Slattery, upper and lower bounds on the drag coefficient of a sphere moving slowly in Ellis model fluids have been calculated, over wide range of conditions, and compared with the suitable experimental data available in the literature. C _D drag coefficient - d diameter of sphere - El Ellis number - Re Reynolds number based on zero-shear viscosity - V terminal falling velocity of a sphere - X drag correction factor - Ellis model parameter - ₀ zero-shear viscosity - _1/2 Ellis model parameter     

The squeezing flow of a model suspension fluid

The paper is concerned with the squeezing flow of a model suspension fluid. The numerical solution obtained by a time-dependent Boundary Element Method is compared to an asymptotic solution at large radius. It is found that the kinematics are Newtonian in character, and the fibres quickly align themselves radially. Consequently, the squeezing force is only weakly dependent on the initial orientations of the fibres and the device can be used for measuring the effective viscosity of the suspension. The effective viscosity found from the squeezing flow agrees surprisingly well with experimental data and numerical data derived from the falling sphere geometry at low volume fractions ( < 0.1).     

An experimental study of multimodal glass suspension rheology to test and validate a polydisperse suspension viscosity model

Experimental measurements of non-colloidal multimodal suspension viscosities are performed over a wide range of mixing ratios and used to test the robustness and predictive capability of a recent viscosity model (Mwasame et al. in Phys Fluids 28:061701, 2016b), subsequently referred to as the MWB model. Three unimodally distributed particle suspensions with narrow size distributions are blended to make the bimodal and trimodal suspensions used in the rheological experiments. We demonstrate how predictions for mixture viscosities can be made using the MWB model only requiring the volume-weighted average particle sizes and viscosity correlations of the individual unimodal suspensions comprising the blend. The resultant model predictions are found to be in good agreement with measured bimodal and trimodal viscosity data to within expected experimental uncertainty. The datasets provided here can be used to validate future modeling efforts, and the MWB model can be used to optimize the viscosity of multimodal suspension mixtures for specific performance criteria.     

Rheology of low viscosity,high concentration brown coal suspensions

The paper describes how the theology of low concentration brown coal suspensions can be exploited to produce high concentration, low viscosity suspensions which are attractive as a potential coal-water fuel. Brown coal suspensions with solid concentrations approaching those of bituminous black coal have been prepared. The high inherent water content ( 60 wt %) and macroporosity of the brown coal have been reduced by thermal and chemical means. The hydrophobicity of the coal surface has been increased sufficiently to reduce the tendency for swelling and water uptake. This, together with densification, has allowed the solids content to be progressively improved from 30 wt % solids with raw coal to 65 wt% solids with modified coals while maintaining the viscosity of the suspension at a low level. The high solid concentration was achieved without additives.     

The physical interaction of spherical particles in suspensions

Summary An experimental examination of the interaction between a falling sphere and an aqueous suspension of nonsettling spheres through which it is passing is described. It is shown that if theReynolds number is small the falling speed of a sphere is unaffected by the size of the suspended spheres, provided that the volume concentration of the suspension is unchanged. This is shown to be true even when the falling sphere is of the same size as the suspended spheres. At highReynolds numbers, however, the falling speed of a sphere diminishes as its size approaches that of the suspended particles. The significance of these results to the theory of the viscosity of suspensions is discussed.     

Yield stress and maximum packing fraction of concentrated suspensions

The yield stress and features of the structure of concentrated suspensions based on silica flour, with particles of average diameter around 4 m, were investigated in terms of a phenomenological model. The yield stress of a concentrated suspension of known solid volume concentration is estimated by employing a shear-dependent maximum packing fraction _m which is obtained by model fitting equilibrium viscosity data, and by incorporating a first-order kinetic equation. The model proposed was examined by using several mineral suspensions in which silica flour was mixed with metal oxide particles so that microstructural features of the suspensions could be adjusted. A cocoa fat suspension was also used as a test sample having radically different chemistry. The agreement between the model prediction and independently obtained experimental evidence is acceptable. Furthermore, a qualitative explanation is obtained by a scaling analysis in an effort to relate the model parameters with the suspension structure that stems from interactions among the suspension constituents.     

A boundary element simulation of the unsteady motion of a sphere in a cylindrical tube containing a viscoelastic fluid

A boundary element method is used to simulate the unsteady motion of a sphere falling under gravity along the centreline of a cylindrical tube containing a viscoelastic fluid. The fluid is modelled by the upper-convected Maxwell constitutive equation. Results show that the viscoelasticity of the liquid leads to a damped oscillation in sphere velocity about its terminal value. The maximum sphere velocity, which occurs in the first overshoot, is approximately proportional to the square root of the Weissenberg number when the ratio of the sphere radius to the tube radius is sufficiently small. Particular attention is also paid to the wall effects. It is shown that a closer wall reduces the oscillatory amplitude of the sphere velocity but increases its frequency. The results suggest that the falling-ball technique, which is now widely used for viscosity measurement, might also be used for the determination of a relaxation time for a viscoelastic fluid.     

Investigation of dense slurry suspensions with coaxial mixers: Influences of design variables through tomography and mathematical modelling

The coaxial mixers enhance the suspension of concentrated slurries in an agitated reactor. In this research work, the complex slurry suspension and dissemination behavior in a coaxial slurry mixing system (comprised of a close clearance anchor rotating with a low speed and an inner axial impeller rotating with a high speed) was analyzed employing ERT (electrical resistance tomography, a non-intrusive flow visualization technique), and computational fluid dynamics (CFD). The numerical models were validated by comparing the axial solid concentration profiles generated using the ERT data and the CFD simulation results. The influences of various important parameters such as the diameter of the inner axial impeller, the inner impeller type, and the inner impeller spacing on the hydrodynamic characteristics of the slurry suspensions in a coaxial mixing vessel were thoroughly analyzed. The radial and axial velocity profiles of solid particles were generated using the validated mathematical models. The assessment of energy loss due to the solid–solid collisions, the particle–fluid frictions, and the particle–vessel wall collisions was conducted. The evaluation of optimum inner impeller clearance and inner impeller diameter is essential to attain a high degree of solids suspension and dissemination in a coaxial slurry mixing system.     

Estimation of zero-shear viscosity of polymer solutions from falling sphere data

Summary Extrapolation methods for the determination of zero-shear viscosity from falling sphere tests are compared with each other and in particular with dicrect viscometric measurements of this parameter. It is found that all methods of extrapolation overestimate the true zero-shear viscosity and that the discrepancy depends on the degree of shear thinning encountered by the falling spheres. Falling sphere tests only yield the true zero-shear viscosity when the spheres fall in the lower Newtonian region of fluid behaviour. In most instances a suitable combination of sphere properties to achieve this can only be found in the case of very viscous fluids which can in any case also be characterized by direct viscometric measurements in this region.If sphere fall data must be extrapolated, methods based on shear rate rather than shear stress appear preferable since they generally yield lower values of zero-shear viscosity, which are therefore nearer to the true value.

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Zusammenfassung Verschiedene Extrapolationsmethoden zur Bestimmung der Null-Viskosität mit Hilfe von Kugelfallversuchen werden miteinander und insbesondere mit der direkten viskosimetrischen Messung dieses Parameters verglichen. Es wird gefunden, daß alle Extrapolationsmethoden den wahren Wert der Null-Viskosität überschätzen und daß der Unterschied vom Grad der Scherentzähung abhängt, der beim Kugelfall vorliegt. Kugelfallversuche liefern nur dann die wahre Nullviskosität, wenn diese im unteren newtonschen Bereich durchgeführt werden. In den meisten Fällen kann eine geeignete Kombination von Kugeleigenschaften zur Realisierung dieser Bedingungen aber nur bei sehr viskosen Flüssigkeiten gefunden werden, die dann genausogut durch direkte viskosimetrische Messungen in diesem Bereich gekennzeichnet werden können.Wenn Kugelfalldaten extrapoliert werden müssen, scheinen Methoden der Auftragung gegen die Schergeschwindigkeit besser geeignet zu sein als solche gegen die Schubspannung. Im ersten Fall werden nämlich durchweg niedrigere Werte der Null-Viskosität erhalten, die somit näher bei den wahren Werten liegen.

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Nomenclature c a constant [—] - d sphere diameter [m] - D container diameter [m] - v sphere fall velocity in an infinite medium [ms^–1] - shear rate [s^–1] - shear rate for = 0.95 ₀ [s^–1] - apparent viscosity [kg m^–1 s^–1] - ₀ zero shear viscosity [kg m^–1 s^–1] - f fluid density [kg m^–3] - _p particle density [kg m^–3] With 8 figures and 2 tables     

Shear and extensional rheology of carbon nanofiber suspensions

The morphology and rheology of carbon nanofiber (CNF) suspensions were studied. The CNFs, produced by decomposing organic vapors at elevated temperature in the presence of metal catalysts, have characteristic diameter and length of 100 nm and 20–100 m, respectively. The CNFs, as delivered, have a strong tendency to clump into mm-sized agglomerates. The efficacy of CNF/glycerol-water suspensions was studied vs. two processing parameters: mechanical sonication and chemical treatment. Experimental measurements revealed that sonication alone reduces the size of CNF clumps from millimeter to micrometer scale, but cannot achieve uniform dispersion. The chemically untreated sonicated suspensions contain clumps of nanofibers with a characteristic size of 20×50 m, together with smaller aggregations of partially dispersed nanofibers. In response to this unsatisfactory dispersion, the effect of acid treatment before dispersion was investigated. This acid treatment, which makes the surface of the CNFs more hydrophilic, greatly improves dispersion in the aqueous solution: treatment followed by sonication results in a uniform dispersion of individual nanofibers. At the same time, however, we observed that surface treatment and subsequent sonication greatly shorten the nanofibers.The rheology of CNF/glycerol-water suspensions is highly non-Newtonian both in shear and extensional flows, with strong dependence on the dispersion, particle length, and concentration of the CNFs. As the solvent is Newtonian, all of the elastic and strain-rate dependent behavior in the CNF/aqueous suspensions derives from the addition of nanofibers. The steady shear viscosity of the untreated-sonicated (poorly dispersed, with longer fibers) suspensions is highly shear thinning with a viscosity that increases three orders of magnitude as concentration varies from 0.5 wt% to 5 wt%. Beyond 5 wt% the suspensions are too viscous to be effectively mixed by sonication. When the CNFs are chemically treated and then sonicated (resulting in much better dispersion but shorter fibers), the viscosity exhibits little shear thinning, and only varies by a factor of two from pure solvent to 5 wt%. In small amplitude oscillatory shear measurements, we found strong indications of elastic behavior in both the treated and untreated suspensions, with elastic modulus G greater than loss modulus G. In particular, for both systems G exhibits a low-frequency plateau when nanofiber concentration is 3 wt% or above, a characteristic of elastic solidlike response. Again, there is a strong dependence on CNF dispersion and fiber length: At low frequencies, the elastic modulus of the 5 wt% untreated suspension (with agglomerates and longer fibers) is four orders of magnitude larger than that of the 5 wt% treated suspension (with uniformly dispersed, shorter fibers). In addition, G of untreated suspensions is a much stronger function of concentration than that of treated suspensions, indicative of network formation.The rheology and morphology of nanofiber suspensions were related by identifying morphology of the suspensions with the assumptions of the kinetic theory-based elastic and rigid dumbbell constitutive models; the approach is to specify the parameters in the kinetic theory models in terms of microscale morphological features measured in the SEM. Of those investigated, the elastic dumbbell model with anisotropic hydrodynamic drag is the most successful, effectively modeling the small amplitude oscillatory shear and steady shear behavior of the treated sonicated suspensions. As for the treated unsonicated and untreated sonicated suspensions, which contain mesoscale agglomerates not present in the underlying assumptions of the dumbbell models, it is discovered that the elastic dumbbell with parameters assigned from morphological measurements predicts the correct trends in the steady shear experiments, but fails to accurately predict the small amplitude oscillatory shear experiments.     

A bootstrap mechanism for non-colloidal suspension viscosity

The role of friction in non-colloidal suspensions is examined with a model which splits the viscosity into a frictionless component (τ*) plus a frictional component which depends on the ratio of the particle pressure (P) to the shear stress (τ). The model needs the input by computation of τ* and P and a suitable choice of particle friction coefficient (μ). It can be extended to elongational flows and cases where sphere roughness is important; volume fractions up to 0.5 are considered. It is shown that friction acts in a feedback or “bootstrap” manner to increase the suspension viscosity. The analysis is also useful for deducing the friction coefficient in suspensions from experimental data. It was applied to several sets of experimental data and reasonable correlations of the viscosities were demonstrated. An example of the correlation for spheres in a silicone oil is shown for volume fractions 0.1–0.5.

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Graphical abstract

     

Elongational flows of some non-colloidal suspensions

The rheology of a system must be explored not only in viscometric flows, but also in other flow classes, and so, we present some results for the axisymmetric elongational flow of non-colloidal suspensions of spheres. We compare our results with data from shear flows using the same matrices and spheres. We have experimented with non-colloidal suspensions of 40-μm diameter polystyrene spheres with volume fractions (?) varying from 0.3 to 0.5. Two matrix fluids were used—one was a near-Newtonian polydimethyl siloxane of 12 Pa-s viscosity and the other was a variant of the M1 Boger fluid sample of Sridhar which we call M1*. We did not find that the Trouton ratio for either of these fluids was 3; generally, the ratio was larger. We investigated the role of sphere roughness using spheres roughened to 5.3 % of the radius in a 50 % suspension in silicone oil and found an increase of elongational viscosity of about 65 % which is comparable with the 60 % increase in shear viscosity with roughness noted previously. For the silicone oil matrix, we found no rate effect, with very little strain-hardening. By contrast, the M1-type matrix suspensions showed strain-hardening and an increase of elongational viscosity with elongation rate.     

Sedimentation of particles in shear flows of viscoelastic fluids with fibers

We study the effect of fiber additives on rheology and sedimentation of particle suspensions in a base viscoelastic suspending fluid in the case when the suspension is subjected to shear flow. We found experimentally that fiber additives (3–6 mm in length and 8–12 μm in diameter at a mass fraction of 0–0.4%) increase the suspension viscosity and retard the particle sedimentation significantly. At the same mass concentration, long and thin fibers reduce the sedimentation velocity and increase the viscosity to a much greater extent than short and thick fibers. We revealed that both rheology and sedimentation are controlled by a single conformational parameter (overlap parameter) defined as the number of fibers per unit volume multiplied by fiber length cubed.     

Simultaneous measurement of size and velocity of microbubbles moving in an opaque tube using an X-ray particle tracking velocimetry technique

An X-ray particle tracking velocimetry (PTV) technique was developed to simultaneously measure the sizes and velocities of microbubbles in a fluid without optical aberration. This technique is based on a combination of in-line X-ray holography and PTV. The X-ray PTV technique uses a configuration similar to that of conventional optical imaging techniques, and is easy to implement. In the present work, microbubbles generated from a fine wire by electrical heating were used as tracer particles. The X-ray PTV technique simultaneously recorded size and velocity data for microbubbles (_b=10–60 m) moving upward in an opaque tube (inner diameter =2.7 mm). Due to the different refractive indices of water and air, phase contrast X-ray images clearly show the exact size and shape of overlapped microbubbles. In all of the working fluids tested (deionised water and 0.01 M and 0.10 M NaCl solutions), the measured terminal velocity of the microbubbles rising through the solution was proportional to the square of the bubble diameter. The proposed technique can be used to extract useful information on the behaviour of various bio/microscale fluid flows that are not amenable to analysis using conventional methods.     

Three-dimensional cellular structure of detonations in suspensions of aluminium particles

Recently, we have used scarce available data on the detonation cell size in suspensions of aluminium particles in air and oxygen to adjust the kinetic parameters of our two-phase model of detonations in these mixtures. The calculated detonation cell width was derived by means of two-dimensional (2D) unsteady simulations using an assumption of cylindrical symmetry of the flow in the tube. However, in reality, the detonation cells are three-dimensional (3D). In this work, we have applied the same detonation model which is based on the continuous mechanics of two-phase flows, for 3D numerical simulations of cellular detonation structures in aluminium particle suspensions in oxygen. Reasonable agreement on the detonation cell width was obtained with the aforementioned 2D results. The range of tube diameters where detonations in $\text{ Al/O}_2$ mixture at a given particle size and concentration would propagate in the spinning mode has been estimated (these results make a complement to our previous analysis of spinning detonations in Al/air mixtures). Coupling these results with the dependencies of detonation cell size on the mean particle diameter is of great interest for the understanding of fundamental mechanisms of detonation propagation in solid particle suspensions in gas and can help to better guide the experimental studies of detonations in aluminium suspensions. It is shown that the part of detonation wave energy used for transverse kinetic energy of both gas and particles is quite small, which explains why the propagation velocity of spinning and multi-headed detonations reasonably agrees with the ideal CJ values.     

Experimental investigations on two-phase flow past a sphere using digital Particle-Image-Velocimetry

Digital Particle-Image-Velocimetry was applied to investigate particle trajectories in a gas flow past a sphere. The particle displacement was determined by autocorrelation analysis of image sections. To enhance the signal/noise ratio a synthetic image with idealized particle pictures was generated from the real image. The autocorrelation function (ACF) was calculated using the Fast Hartley Transformation (FHT). The desired secondary maximum of this function was detected by an algorithm with subpixel resolution. A data validation step testing the plausibility of the velocity vectors completes the image analysis. Particle trajectories are traced with help of the particles' velocity vectors. The particle deposition on a sphere can be deduced from the course of these trajectories.List of symbols Cu Cunningham correction - e double distance between the limiting particle trajectory and the stagnation point axis - f focal length - H Hartley-Transform - M enlargement factor - S interference band spacing - x coordinate - y coordinate - x _p particle diameter - x _T droplet or sphere diameter - V image as grey value function - _.rel face velocity of droplet or sphere - particle image displacement - x particle image displacement in x-direction - y particle image displacement in y-direction - collection efficiency - wavelength of the laser light - _L fluid viscosity - _L fluid density - _p particle density     

Numerical study of concentrated fluid–particle suspension flow in a wavy channel

The particle migration effects and fluid–particle interactions occurring in the flow of highly concentrated fluid–particle suspension in a spatially modulated channel have been investigated numerically using a finite volume method. The mathematical model is based on the momentum and continuity equations for the suspension flow and a constitutive equation accounting for the effects of shear‐induced particle migration in concentrated suspensions. The model couples a Newtonian stress/shear rate relationship with a shear‐induced migration model of the suspended particles in which the local effective viscosity is dependent on the local volume fraction of solids. The numerical procedure employs finite volume method and the formulation is based on diffuse‐flux model. Semi‐implicit method for pressure linked equations has been used to solve the resulting governing equations along with appropriate boundary conditions. The numerical results are validated with the analytical expressions for concentrated suspension flow in a plane channel. The results demonstrate strong particle migration towards the centre of the channel and an increasing blunting of velocity profiles with increase in initial particle concentration. In the case of a stenosed channel, the particle concentration is lowest at the site of maximum constriction, whereas a strong accumulation of particles is observed in the recirculation zone downstream of the stenosis. The numerical procedure applied to investigate the effects of concentrated suspension flow in a wavy passage shows that the solid particles migrate from regions of high shear rate to low shear rate with low velocities and this phenomenon is strongly influenced by Reynolds numbers and initial particle concentration. Copyright © 2008 John Wiley & Sons, Ltd.     

The effect of sphere–wall interactions on particle motion in a viscoelastic suspension of FENE dumbbells

A new method for simulating the motion of particles in viscoelastic Boger fluids is extended to problems with bounded geometries. Viscoelasticity is incorporated into the Stokesian dynamics method by modeling a viscoelastic fluid as a suspension of finite-extension nonlinear-elastic (FENE) dumbbells. Wall–particle and wall–bead interactions are included by using the image system method of Blake; particle–particle and particle–bead interactions are also modified by the presence of the wall. The method of incorporating sphere–wall interactions is verified by doing calculations for several problems involving particle–wall interactions in Newtonian fluids. The method is then used to study particle–wall interactions in viscoelastic dumbbell suspensions by examining several problems of interest: the sedimentation of a spherical particle near vertical and tilted walls; the sedimentation of a nonspherical particle between two flat plates; and the migration of a neutrally buoyant sphere in plane Poiseuille flow. We find that a single sphere falling near a wall moves toward the wall and exhibits anomalous rotation. When the wall is tilted by an amount less than a few degrees, the sphere still moves toward the wall, but tilting the wall greater than an angle of approximately 1.5° results in the sphere falling away from the wall. A nonspherical particle settling in a channel exhibits an oscillatory motion, but ultimately becomes centered in the channel with its long axis parallel to gravity. Finally, it is shown that a neutrally buoyant sphere in plane Poiseuille flow migrates to the channel center in wide channels, but migrates to the walls when the sphere is sufficiently large relative to the channel width.     

On the viscosity of suspensions of solid spheres

A cell theory is used to derive the dependence of the zero-shear-rate viscosity on volume concentration for a suspension of uniform, solid, neutrally buoyant spheres. This result reduces to Einstein's solution at infinite dilution and to Frankel and Acrivos's expression in the limit as the concentration approaches its maximum value. Good agreement is found between the solution and the available data for the entire concentration range, provided that the maximum concentration is determined from the viscosity data themselves.Nomenclature a radius of sphere - d the distance separating the sphere surfaces measured parallel to the line connecting the sphere centers - E energy dissipation rate in one-half the liquid volume separating the spheres - E _cell total energy dissipation rate in the cell - E _homogeneous energy dissipation rate in the cell of a hypothetical one phase fluid - E _interaction energy dissipation rate in the cell due to sphere interactions - E _sphere energy dissipation rate in the cell due to the sphere at the cell center - F force on one sphere - h minimum separation distance between two spheres - J (1/2)d = one-half the distance separating the sphere surfaces measured parallel to the line connecting the sphere centers - p pressure - W velocity of one sphere in squeezing flow between two spheres relative to the midpoint of the line connecting the sphere centers - _i unit vectors in thei-th direction - elongation rate - viscosity of the suspending fluid - _r */ = relative viscosity - * viscosity of the suspension - the total stress tensor - the part of the total stress tensor that vanishes at equilibrium - volume fraction of spheres