Monodisperse Suspensions Research Articles

Exact Solution for Deep Bed Filtration with Finite Blocking Time

An initial-boundary value problem for a quasilinear system describing deep bed filtration of a monodisperse suspension in a medium with pores of various sizes is investigated analytically. The filt...

Filtration model with multiple particle capture

Grout filtration in porous soil is used in construction industry to create underground waterproof walls. When the suspension flows through the pores, various forces act on the suspended particles, blocking them on the frame of the porous medium. A one-dimensional model of deep bed filtration for a monodisperse suspension in a porous medium with several particle capture mechanisms is considered. The mathematical model includes the equation of mass balance of suspended and retained particles and the kinetic equation of deposit growth with a piecewise-smooth linear-constant filtration function and a nonlinear concentration function. The solution of the nonlinear model is obtained by the finite difference method using an explicit difference scheme with second-order approximation. To construct the asymptotics of a complex model, the solutions of simplified linear and semilinear models and their combination are used. In the zone of a smooth filtration function, the best approximation of the solution of a complex model is determined by a certain linear combination of simple solutions. In another area, solution of a simplified problem with a piecewise-smooth filtration function and a linear concentration function is closest to the solution of a nonlinear model. Calculations show that in the zone of a smooth filtration function, a combination of simple solutions defines a solution approximation with second-order of smallness. In the area where it is necessary to take into account the non-smoothness of the filtration function, the approximation of a solution has a first order of smallness.

Radial patterns and velocity field of non-Brownian suspensions in a fully filled horizontal rotating cylinder

This work reports radial patterns and velocity profiles of fluid-particle suspensions rotating in a fully filled horizontal cylinder using the technique of Particle Image Velocimetry. Dilute and mono-dispersed suspensions of non-Brownian settling as well as floating particles were considered. An analysis of the effect of particle shape on the flow patterns is provided. This was carried out by comparing the suspensions of spherical and cylindrical particles. Impact of particle shape on the particle-fluid interaction in determining the flow structure is more pronounced at lower speeds. Analysis of the velocity profiles showed a resemblance of the buoyant and settling suspensions phases. Despite the similar features observed, a difference was noted when the centrifugal forces dominate. Our experimental results are in good qualitative agreement with the previous experiments and simulations. Through a detailed investigation of velocity field we have established the strong relationship between the phase transition and rotational speed.

Modelling Of Transport And Capture Of Particles In A Porous Medium

The study of the transport and capture of particles moving in a fluid flow in a porous medium is an important problem of underground hydromechanics, which occurs when strengthening loose soil and creating watertight partitions for building tunnels and underground structures. A one-dimensional mathematical model of long-term deep filtration of a monodisperse suspension in a homogeneous porous medium with a dimensional particle retention mechanism is considered. It is assumed that the particles freely pass through large pores and get stuck at the inlet of small pores whose diameter is smaller than the particle size. The model takes into account the change in the permeability of the porous medium and the permissible flow through the pores with increasing concentration of retained particles. A new spatial variable obtained by a special coordinate transformation in model equations is small at any time at each point of the porous medium. A global asymptotic solution of the model equations is constructed by the method of series expansion in a small parameter. The asymptotics found is everywhere close to a numerical solution. Global asymptotic solution can be used to solve the inverse filtering problem and when planning laboratory experiments.

Particle Capture in Porous Medium

Filtration problems in porous media are important for studying the movement of groundwater in porous formations and the spreading of liquid concrete injected into porous soil. Deep bed filtration of a monodisperse suspension in a homogeneous porous medium with two simultaneously acting particle capture mechanisms is considered. A mathematical model of suspension flow through porous medium with pore blocking by size-exclusion and arched bridging is developed. Exact solutions are obtained on the concentration front and at the porous medium inlet. For the linear filtration function, exact and asymptotic solutions are constructed.

Advantages of the cross-correlation method for estimating nanoparticle sizes in suspensions

This article presents a cross-correlation method for estimation the size of nanoparticles in monodisperse colloidal solutions and suspensions. The original scheme of the experimental setup for carrying out cross-correlation experiments is presented. The results of measurements for a liquid sample with dispersed particles are presented for the cross-correlation method and the classical dynamic light scattering method, in which the autocorrelation function is calculated instead of the cross-correlation function of the scattering signals. The results obtained demonstrate the fact that for highly diluted samples, autocorrelation and cross-correlation produce the same result. Since the cross-correlation method is also applicable to turbid samples, this gives it a significant advantage over the dynamic light scattering method.

Modeling of Cellular Detonation in Gas Suspensions of Submicron and Nano-Sized Aluminum Particles

Cellular detonation in monodisperse suspensions of submicron and nano-sized aluminum particles is numerically simulated. Approaches of mechanics of heterogeneous media are applied. The transition from the continuum to free-molecular regime of the flow around the particles is taken into account in the processes of interphase interaction. Particle combustion is described within the framework of the semi-empirical model developed previously. Results calculated for two-dimensional flows in a plane channel for suspensions of aluminum particles with the particle size ranging from 1 µm to 100 nm are presented. The regular structure of cellular detonation is found to transform to an irregular structure as the particle size decreases. An increase in the peak pressure and enlargement of the detonation cell are also noted, which is attributed to enhancement of the activation energy of reduced kinetics caused by the transition to the kinetic regime of combustion of aluminum particles.

Hydrodynamic interactions and the diffusivity of spheroidal particles.

It is intuitive that the diffusivity of an isolated particle differs from those in a monodisperse suspension, in which hydrodynamic interactions between the particles are operative. Batchelor [J. Fluid Mech. 74, 1-29 (1976) and J. Fluid Mech. 131, 155-175 (1983)] calculated how hydrodynamic interactions influenced the diffusivity of a dilute suspension of spherical particles, and Russel et al. [Colloidal Dispersions (Cambridge University Press, 1991)] and Brady [J. Fluid Mech. 272, 109-134 (1994)] treated nondilute (higher particle volume fraction) suspensions. Although most particles lack perfect sphericity, little is known about the effects of hydrodynamic interactions on the diffusivity of spheroidal particles, which are the simplest shapes that can be used to model anisotropic particles. Here, we calculate the effects of hydrodynamic interactions on the translational and rotational diffusivities of spheroidal particles of arbitrary aspect ratio in dilute monodisperse suspensions. We find that the translational and rotational diffusivities of prolate spheroids are more sensitive to eccentricity than for oblate spheroids. The origin of the hydrodynamic anisotropy is that found in the stresslet field for the induced-dipole interaction. However, in the dilute limit, the effects of anisotropy are at the level of a few percent. These effects have influence on a vast range of settings, from partially frozen colloidal suspensions to the dynamics of cytoplasm.

Boundary-induced inhomogeneity of particle layers in the solidification of suspensions.

When a suspension freezes, a compacted particle layer builds up at the solidification front with noticeable implications on the freezing process. In a directional solidification experiment of monodisperse suspensions in thin samples, we evidence a link between the thickness of this layer and the sample depth. We attribute it to an inhomogeneity of particle density that is attested by the evidence of crystallization at the plates and of random close packing far from them. A mechanical model based on the resulting modifications of permeability enables us to relate the layer thickness to this inhomogeneity and to select the distribution of particle density that yields the best fit to our data. This distribution involves an influence length of sample plates of about 11 particle diameters. Altogether, these results clarify the implications of boundaries on suspension freezing. They may be useful to model polydisperse suspensions with large particles playing the role of smooth boundaries with respect to small ones.

The Numerical Prediction of Flow of Branched Polymer Melts Through Planar Contraction Channel by Finite Element Method

In this work we present an investigation of a complex viscoelastic flow through an abrupt expansion with expansion ratio D/d = 4, where the Oldroyd-B model was consider as a constitutive equation. The governing equations are solved using the Finite Element Method. The polymer is modeled as a monodisperse suspension of “Oldroyd-B” molecules, which provides a direct link the molecular topology and the flow properties of the melt. The branching produces an enhancement in the size of upstream vortex in creeping flows. Also was found that including inertia forces will give the intensity of lip vortex increases, while the size of corner vortex will initially decreased with growth of Weissenberg number but with more increase (We>4), the size of corner vortex will begin to increase in size. The effect arises from the differing responses in shear and extensional flows of polymer melts.

Effects of Polydispersity on Structuring and Rheology in Flowing Suspensions

The size and distribution of particles suspended within a fluid influence the rheology of the suspension, as well as strength and other mechanical properties if the fluid eventually solidifies. An important motivating example of current interest is foamed cements used for carbon storage and oil and gas wellbore completion. In these applications, it is desired that the suspended particles maintain dispersion during flow and do not coalesce or cluster. This paper compares the role of mono- against polydispersity in the particle clustering process. The propensity of hard spherical particles in a suspension to transition from a random configuration to an ordered configuration, or to form localized structures of particles, due to flow is investigated by comparing simulations of monodisperse and polydisperse suspensions using Stokesian dynamics. The calculations examine the role of the polydispersity on particles rearrangements and structuring of particles due to flow and the effects of the particle size distribution on the suspension viscosity. A key finding of this work is that a small level of polydispersity in the particle sizes helps to reduce localized structuring of the particles in the suspension. A suspension of monodisperse hard spheres forms structures at a particle volume fraction of approximately 47% under shear, but a 47% volume fraction of polydisperse particles in suspension does not form these structures.

Boundary and concentration effects on the creeping motions of core-shell particles

In this study, the creeping motions of core-shell particles at low-Reynolds numbers were analyzed and discussed in three cases: (i) a freely translating particle; (ii) a particle in a concentric hole; (iii) particles in a monodisperse suspension. The core and shell were assumed concentric and consist of immiscible liquids. The results revealed the particle motions were significantly influenced by the radius (or volume fraction) and viscosity of the shell and core, especially the shell viscosity. In the limiting situations, our general results for two-phase particles could be simplified to the studies of single-phase particles (solid, liquid or gas bubble) in literature. For the case of a freely suspending particle with the same volume fraction of shell and core, the modified coefficient was reduced by more than 30% as the dimensionless shell viscosity reduced from 100 to 0.01 at ηc∗=1; however, the coefficient was only slightly reduced by less than 1.5% as the dimensionless core viscosity reduced from 100 to 0.01 at ηs∗=1. For the translation of a particle in a concentric hole, the boundary effect of the cavity wall performed to significantly hinder the particle motion. When the volume fraction of the particle in the hole was 0.01 and 0.1, in the case of a21 = 1.25, ηc∗=1 and ηs∗=0.01, the boundary effect retarded the particle velocity to 2/3 and 1/3 of the free particle velocity, respectively. In a monodisperse system, the mean translational velocity of particles was hindered by the hydrodynamic interaction between the particles, and the hindrance effect increased as the particle concentration increased. Particle concentration and shell viscosity were the main properties affecting the fluid dynamics of the suspension.

Dynamics of bidensity particle suspensions in a horizontal rotating cylinder.

We report Stokesian dynamics simulations of bidensity suspensions rotating in a horizontal cylinder. We studied the phase space and radial and axial patterns in settling as well as floating systems. Each system was composed of particle mixtures of two different densities. As many as eight unique phases are identified for each system along the radial plane. The bidensity system shows similarity to the monodisperse case only when the radial distribution of the particles is completely uniform. Characteristic behavior of the bidensity systems is identical at low rotation rates and contrasting when centrifugal force dominates. Expressing the phase boundaries in terms of dimensionless parameters U_{s}/(ΩR) and g/(Ω^{2}R) gives a linear fit unifying the data in the gravity-dominated regime. At high rotation rates, the behavior is opposing for either system though linear in nature. In the axial direction, number density profiles of both systems affirm the phenomenon of band formation. Location of the axial bands remains the same for heavy and light particles in both systems. We have also reestablished that an inhomogeneous particle configuration in the radial plane induces growing instabilities in the axial plane which amplify to form particle bands similar to monodisperse suspensions.

Global asymptotics of filtration in porous media

Filtration problems are actual for the design of underground structures and foundations, strengthening of loose soil and construction of watertight walls in the porous rock. A liquid grout pumped under pressure penetrates deep into the porous rock. Solid particles of the suspension retained in the pores, strengthen the loose soil and create watertight partitions. The aim of the study is to construct an explicit analytical solution of the filtration problem. A one-dimensional model of deep bed filtration of a monodisperse suspension in a homogeneous porous medium with size-exclusion mechanism of particles retention is considered. Solid particles are freely transferred by the carrier fluid through large pores and get stuck in the throats of small pores. The mathematical model of deep bed filtration includes the mass balance equation for suspended and retained particles and the kinetic equation for the deposit growth. The model describes the movement of concentrations front of suspended and retained particles in an empty porous medium. Behind the concentrations front, solid particles are transported by a carrier fluid, accompanied by the formation of a deposit. The complex model has no explicit exact solution. To construct the asymptotic solution in explicit form, methods of nonlinear asymptotic analysis are used. The new coordinate transformation allows to obtain a parameter that is small at all points of the porous sample at any time. In this paper, a global asymptotic solution of the filtration problem is constructed using a new small parameter. Numerical calculations are performed for a nonlinear filtration coefficient found experimentally. Calculations confirm the closeness of the asymptotics to the solution in the entire filtration domain. For a nonlinear filtration coefficient, the asymptotics is closer to the numerical solution than the exact solution of the problem with a linear coefficient. The analytical solution obtained in the paper can be used to analyze solutions of problems of underground fluid mechanics and fine-tune laboratory experiments.

Numerical solution of filtration in porous rock

The filtration problem is one of the most relevant in the design of retaining hydraulic structures, water supply channels, drainage systems, in the drainage of the soil foundation, etc. Construction of transport tunnels and underground structures requires careful study of the soil properties and special work to prevent dangerous geological processes. The model of particle transport in the porous rock, which is based on the mechanical-geometric interaction of particles with a porous medium, is considered in the paper. The suspension particles pass freely through large pores and get stuck in small pores. The deposit concentration increases, the porosity and the permissible flow of particles through large pores changes. The model of one-dimensional filtration of a monodisperse suspension in a porous medium with variable porosity and fractional flow through accessible pores is determined by the quasi-linear equation of mass balance of suspended and retained particles and the kinetic equation of deposit growth. This complex system of differential equations has no explicit analytical solution. An equivalent differential equation is used in the paper. The solution of this equation by the characteristics method yields a system of integral equations. Integration of the resulting equations leads to a cumbersome system of transcendental equations, which has no explicit solution. The system is solved numerically at the nodes of a rectangular grid. All calculations are performed for non-linear filtration coefficients obtained experimentally. It is shown that the solution of the transcendental system of equations and the numerical solution of the original hyperbolic system of partial differential equations by the finite difference method are very close. The obtained solution can be used to analyze the results of laboratory research and to optimize the grout composition pumped into the porous soil.

Problem of Deep Bed Filtration in a Porous Medium with the Initial Deposit

The macroscopic model of long-term deep-bed filtration flow of a monodisperse suspension through a porous medium with size-exclusion particle-capture mechanism and without retained-particle mobilization is considered. It is assumed that the pore accessibility and the fractional particle flux depend on the deposit concentration and at the initial time the porous medium contains a nonuniformly distributed deposit. The aim of the study is to find the analytical solution in the neighborhood of a mobile curvilinear boundary, namely, of the suspended-particle concentration front. The property of having fixed sign is proved for the solution. The exact solution of the filtration problem on the curvilinear front is found in explicit form. The sufficient condition of existence of the solution on the concentration front is obtained. An asymptotic solution is constructed in the neighborhood of the front. The time interval of applicability of asymptotics is determined from the numerical solution.

Filtration in porous rock with initial deposit

The problems of underground fluid mechanics play an important role in the design and preparation for the construction of tunnels and underground structures. To strengthen the insecure soil a grout solution is pumped under pressure in the porous rock. The liquid solution filters in the pores of the rock and strengthens the soil after hardening. A macroscopic model of deep bed filtration of a monodisperse suspension in a porous medium with a size-exclusion mechanism for the suspended particles capture in the absence of mobilization of retained particles is considered. The solids are transported by the carrier fluid through large pores and get stuck at the inlet of small pores. It is assumed that the accessibility factor of pores and the fractional flow of particles depend on the concentration of the retained particles, and at the initial moment the porous medium contains an unevenly distributed deposit. The latter assumption leads to inhomogeneity of the porous medium. A quasilinear hyperbolic system of two first-order equations serves as a mathematical model of the problem. The aim of the work is to obtain the asymptotic solution near the moving curvilinear boundary - the concentration front of suspended particles of the suspension. To obtain a solution to the problem, methods of nonlinear asymptotic analysis are used. The asymptotic solution is based on a small-time parameter, measured from the moment of the concentration front passage at each point of the porous medium. The terms of the asymptotics are determined explicitly from a recurrent system of ordinary differential and algebraic equations. The numerical calculation is performed by the finite difference method using an explicit TVD scheme. Calculations for three types of microscopic suspended particles show that the asymptotics is close to the solution of the problem. The time interval of applicability of the asymptotic solution is determined on the basis of numerical calculation. The constructed asymptotics, which explicitly determines the dependence on the parameters of the system, allows to plan experiments and reduce the amount of laboratory research.

Steady shear viscosity of semi-dilute bubbly suspensions

The steady shear viscosity of bubbly suspensions is known to depend upon the suspending fluid viscosity, the bubble volume fraction and a dimensionless number (the capillary number) accounting for the deformability of the bubbles under a viscous stress. However, experimental data on bubbly suspensions in the literature are scarce and present two main shortcomings: (i) the studied systems are polydisperse, which leads to arbitrariness in the definition of a capillary number, and (ii) there is a lack of data for capillary numbers of order unity, where bubbles are slightly deformable. This leads to the absence of clear conclusion on the relevance of the existing theoretical models. In order to make significant progress, an original device is designed to produce monodisperse suspensions of bubbles in a highly viscous Newtonian fluid. The steady shear viscosity of the bubbly suspensions is measured in the dilute and semi-dilute concentration regimes (volume fraction of bubbles between 2% and 18%) using a Couette geometry with a 20-bubbles-wide gap, for capillary numbers ranging between 0.01 and 10. The new obtained data are shown to be much more accurate than data previously reported in the literature. Experimental data are in very good agreement with the model of Frankel and Acrivos (1970), originally developed for dilute suspensions, in the whole range of studied volume fractions.

Model of the solar-driven reduction of cobalt oxide in a particle suspension reactor

We develop a model to investigate the impact of volume fraction and extent of mixing on the thermal efficiency with which a cavity-based particle suspension solar reactor reduces Co3O4 into CoO and O2. Thermal efficiency is defined as the fraction of solar energy used to drive the endothermic reduction. In the model, particles move continuously through the reactor in either mixed or plug flow—mixing conditions that give rise to isothermal and nonisothermal suspensions, respectively—and reduce at a rate governed by shrinking core kinetics, the parameters of which are determined using thermogravimetric data. Radiation is simulated using the Monte Carlo Ray Tracing technique. The model is applied to a reactor heated by 4 kW of point-focused solar radiation with a mean concentration ratio of 1400 suns containing monodisperse suspensions of 40 μm diameter particles with volume fractions between 1×10−5 and 1×10−2. Thermal efficiency is insensitive to mixing for the two conditions considered. The maximum thermal efficiency obtained for mixed flow with an isothermal suspension is 34.1% at 102 g min−1 and a volume fraction of 2×10−4. At the same volume fraction, the maximum thermal efficiency for nonisothermal plug flow is 33.2% at 94 g min−1. Thermal efficiency is more sensitive to the volume fraction, but only below a threshold value of 2×10−4. Thus, from the perspective of coupling heat transfer to the chemical reaction, design and operational efforts of particle suspension reactors for the reduction of cobalt oxide should focus on generating suspensions of at least this threshold value rather than on mixing the particles within the suspension.

Solution of the multiple light scattering problem in turbid suspensions via cross-correlation processing

The original scheme of the cross-correlation technique for assessing the size of colloid particles in suspensions of high turbidity without any dilution is proposed. In the designed experimental scheme the illumination with two in-phase coherent laser beams are used. The scattered radiation signalsare registered by the square-law photodetectors and processed without any special correlators on a computer. It allowed us to simplify the experimental setup. Parameters of the setup were established by calibrating experiments. Cross-correlation functions were obtained for monodisperse suspensions of latex microspheres, and their sizes were assessed.