Dynamic permeability of an assemblage of soft spherical particles

Under oscillatory Stokes flow, dynamic permeability of assemblage of soft spherical particles is derived. For the bed of soft particles, the fluid‐particle system is represented as an assemblage of uniform permeable spheres fixed in space. Each sphere, with a surrounding envelope of fluid, is uncoupled from the system and considered separately. This model is popularly known as cell model. Oscillatory Stokes equations are employed inside the fluid envelope, and oscillatory Brinkman equations are used inside the porous region. Four known boundary conditions namely: Happel's, Kuwabara's, Kvashnin's, and Cunningham's are considered on the outer boundary and results are compared. The behavior of dynamic permeability is analyzed with various parameters such as Darcy number (Da), frequency parameter (?), porosity (φ), and viscosity ratio (δ). Copyright © 2013 John Wiley & Sons, Ltd.     

Stokes flow of an assemblage of porous particles: stress jump condition

The present article investigates the overall bed permeability of an assemblage of porous particles. For the bed of porous particles, the fluid-particle system is represented as an assemblage of uniform porous spheres fixed in space. Each sphere, with a surrounding envelope of fluid, is uncoupled from the system and considered separately. This model is popularly known as cell model. Stokes equations are employed inside the fluid envelope and Brinkman equations are used inside the porous region. The stress jump boundary condition is used at the porous-liquid interface together with the continuity of normal stress and continuity of velocity components. On the surface of the fluid envelope, three different possible boundary conditions are tested. The obtained expression for the drag force is used to estimate the overall bed permeability of the assemblage of porous particles and the behavior of overall bed permeability is analyzed with various parameters like modified Darcy number (Da*), stress jump coefficient (??), volume fraction (??), and effective viscosity.     

Bed of polydisperse viscous spherical drops under thermocapillary effects

Viscous flow past an ensemble of polydisperse spherical drops is investigated under thermocapillary effects. We assume that the collection of spherical drops behaves as a porous media and estimates the hydrodynamic interactions analytically via the so- called cell model that is defined around a specific representative particle. In this method, the hydrodynamic interactions are assumed to be accounted by suitable boundary conditions on a fictitious fluid envelope surrounding the representative particle. The force calculated on this representative particle will then be extended to a bed of spherical drops visualized as a Darcy porous bed. Thus, the “effective bed permeability” of such a porous bed will be computed as a function of various parameters and then will be compared with Carman–Kozeny relation. We use cell model approach to a packed bed of spherical drops of uniform size (monodisperse spherical drops) and then extend the work for a packed bed of polydisperse spherical drops, for a specific parameters. Our results show a good agreement with the Carman–Kozeny relation for the case of monodisperse spherical drops. The prediction of overall bed permeability using our present model agrees well with the Carman–Kozeny relation when the packing size distribution is narrow, whereas a small deviation can be noted when the size distribution becomes broader.     

Effective medium model for flow through beds of porous cylindrical fibres

We have used effective medium model for beds of circular cylindrical porous fibres in order to estimate the overall bed permeability (OBP). It is assumed that a representative circular porous cylindrical fibre is inside a fluid envelope beyond which effective medium is used. Both inside the cylindrical fibre and in the effective medium, Brinkman equation is used, however of different permeabilities and in the fluid envelope Stokes equation is used. The OBP corresponding to the porous fibres is estimated when the flow direction is perpendicular to the axis of the cylindrical fibres as well as parallel to the fibres. This in turn is used to estimate the OBP corresponding to a collection of porous cylindrical fibres that are randomly oriented. We have compared the results with some existing literature.     

Stokes flow past an assemblage of slip eccentric spherical particle‐in‐cell models

The quasisteady axisymmetrical flow of an incompressible viscous fluid past an assemblage of slip eccentric spherical particle‐in‐cell models with Happel and Kuwabara boundary conditions is investigated. A linear slip, Basset type, boundary condition on the surface of the spherical particle is used. Under the Stokesian approximation, a general solution is constructed from the superposition of the basic solutions in the two spherical coordinate systems based on the particle and fictitious spherical envelope. The boundary conditions on the particle's surface and fictitious spherical envelope are satisfied by a collocation technique. Numerical results for the normalized drag force acting on the particle are obtained with good convergence for various values of the volume fraction, the relative distance between the centers of the particle and fictitious envelope and the slip coefficient of the particle. In the limits of the motions of the spherical particle in the concentric position with cell surface and near the cell surface with a small curvature, the numerical values of the normalized drag force are in good agreement with the available values in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

Simulation of Wooden Particle Drying

With regard to future calculations of the combustion inside a packed bed of wooden particles, the drying process of a single wet particle is simulated in the present study. Essentially, conservation equations in spherical coordinates are solved by a finite volume approach for the interior of the single wooden particle. It is shown that a thin transition zone between the wet and the dry area exists during evaporation. The fundamental reasons are pointed out. Furthermore, the high velocity values in the transition zone are explained physically. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Motion of a distant solid particle in a shear flow along a porous slab

The motion of a solid and no-slipping particle immersed in a shear flow along a sufficiently porous slab is investigated. The fluid flow outside and inside of the slab is governed by the Stokes and Darcy equations, respectively, and the so-called Beavers and Joseph slip boundary conditions are enforced on the slab surface. The problem is solved for a distant particle with length scale a in terms of the small parameter a/d where d designates the large particle–slab separation. This is achieved by asymptotically inverting a relevant boundary-integral equation on the particle surface, which has been recently proposed for any particle location (distant or close particle) in Khabthani et al. (J Fluid Mech 713:271–306, 2012). It is found that at order O(a/d) the slab behaves for any particle shape as a solid plane no-slip wall while the slab properties (thickness, permeability, associated slip length) solely enter at O((a/d)²). Moreover, for a spherical particle, the numerical results published in Khabthani et al. (J Fluid Mech 713:271–306, 2012) perfectly agree with the present asymptotic analysis.     

Stokes flow past a porous sphere using Brinkman's model

A general non-axisymmetric Stokes flow past a porous sphere in a viscous, incompressible fluid is considered. The flow inside the sphere is governed by Brinkman's equations. A representation for velocity and pressure for the Brinkman's equations is suggested and a method of finding the flow quantities is given. Faxén's laws for drag and torque for the flow past a porous sphere are also given.     

Expansions at small Reynolds numbers for the flow past a porous circular cylinder

The purpose of this article is to use the method of matched asymptotic expansions (MMAE) in order to study the two-dimensional steady low Reynolds number flow of a viscous incompressible fluid past a porous circular cylinder. We assume that the flow inside the porous body is described by the continuity and Brinkman equations, and the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. Formal expansions for the corresponding stream functions are used. We show that the force exerted by the exterior flow on the porous cylinder admits an asymptotic expansion with respect to low Reynolds numbers, whose terms depend on the characteristics of the porous cylinder. In addition, by considering Darcy's law for the flow inside the porous circular cylinder, an asymptotic formula for the force on the cylinder is obtained. Also, a porous circular cylinder with a rigid core inside is considered with Brinkman equation inside the porous region. Stress jump condition is used at the porous–liquid interface together with the continuity of velocity components and continuity of normal stress. Some particular cases, which refer to the low Reynolds number flow past a solid circular cylinder, have also been investigated.     

Stokes flow inside a porous spherical shell: Stress jump boundary condition

An arbitrary Stokes flow of a viscous, incompressible fluid inside a sphere with internal singularities, enclosed by a porous spherical shell, using Brinkmans equation for the flow in the porous region is discussed. At the interface of the clear fluid and porous region stress jump boundary condition for tangential stresses is used. The drag and torque are found by deriving the corresponding Faxens laws. It is found that drag and torque not only change with the varying permeability, but also change for different values of stress jump coefficient. Critical permeability is found for which drag and torque change their behavior. As a limiting case the corresponding Faxens laws for the rigid spherical shell with internal singularities has been obtained.Received: December 17, 2002; revised: February 3, 2004     

Modelling of displacement washing of pulp fibers using orthogonal collocation on finite elements

Flow of fluid through packed bed of porous particles is modelled with the help of Peclet number (Pe) and Biot number (Bi). Packed bed is divided into three zones, flowing liquor, intrapore solute present in pores of particles and solute adsorbed on particle surface. Langmuir isotherm is used to describe the relationship between intrapore solute concentration and concentration of solute adsorbed on particle surface, whereas the bulk fluid concentration and the intrapore solute concentration are related by linear adsorption isotherm. Model predicted values are also compared with the experimental values. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

缓慢调制波在多孔海床上的演化

本文采用多重尺度法分析了具有缓慢调制的波列在多孔海床上的演化问题.海床上部波浪采用了势流理论,海床下部的渗流采用了Darcy定律.两者在海床面上进行衔接,从而导出了上部波浪的波幅一阶和二阶的调制方程,并求出了相应的解,下部渗压场的解亦同时给出.     

Numerical simulation of dense fluid-particle flows using computational fluid dynamics (CFD) and discrete element method (DEM)

Discrete Element Method (DEM) has been successfully coupled with Computational Fluid Dynamics (CFD) in the framework of OpenFOAM an open source CFD simulation code. In the current study, at first the model is validated with the simple test case of spherical particle comparing the results with the analytical solution. Then the simulation of a gaseous fluidized bed is considered. The coupled mass and momentum balance equations are used to calculate the flow behavior, particle fluidization and bubble formation. The dimensions of the simulation domain are similar to Link et al. (2005) but with different stiffness of particles. The higher velocity of gaseous fluid relative to particles entering through a jet causes the particles to fluidize. The particles behavior, fluidization, bubble formation and the velocity vectors of particles show a good agreement with the literature. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solutions to a class of transport problems with radially dominant convection

For fluid systems dealing with drops and bubbles, there are many situations in which the flow is dominated by a radial field. An analysis is carried out for a general class of problems, in which the primary flow is a purely radial type in a spherical geometry and the secondary flow is a perturbation on it. In particular, the flow solutions are obtained for a particle in extensional flow, rotating particle, and a particle in a linear shear flow. In addition, the steady state heat/mass flow equations with radial convection are solved in a fairly general form for spherical boundaries. The solutions lead to a new class of polynomials for the radial functions of the separated solutions. Some of the fundamental properties of these polynomials have also been derived.     

The propagation of elastic waves in a viscous-fluid-saturated porous medium

Elastic shock waves in a viscous-fluid-saturated porous medium are investigated. The porosity is only taken into account with respect to pores communicating with one another, and isolated pores are considered as elements of the elastic part of the porous skeleton. It is shown, using the theory of discontinuity, that in such a medium there are two types of vortex-free waves and one equivoluminal wave. Differential equations and their solution for determining the change in the wave-front intensity are obtained. The effect of the fluid viscosity and porosity on the propagation of spherical waves is demonstrated using an example.     

Influence of radiation on MHD free convection from a vertical flat plate embedded in porous media with thermophoretic deposition of particles

Magneto-hydrodynamics and thermal radiation effects on heat and mass transfer in steady laminar boundary layer flow of a Newtonian, viscous fluid over a vertical flat plate embedded in a fluid saturated porous media in the presence of the thermophoresis particle deposition effect is studied in this paper. The governing equations are transformed by special transformations. Brownian motion of particles and thermophoretic transport are considered in the flow equations. The magnetic field is considered to be applied. Rosseland approximation is used to describe the radiative heat flux in the energy equation. The resulting similarity equations are solved numerically by the fourth-order Runge–Kutta method with shooting technique. Many results are obtained and representative set is displayed graphically to illustrate the influence of the various parameters on the wall thermophoretic deposition velocity, concentration, temperature and velocity profiles.     

Flow of viscous stratified fluid of variable viscosity past a porous bed

To study the effects of stratification and slip velocity on the flow of fluid of variable viscosity over a permeable bed, we divide the flow into two zones called zone 1 and zone 2. Zone 1 pertains to the flow called the free flow governed by the Navier-Stokes equations in the region between the impermeable upper plate and the porous bed.. Zone 2 pertains to the flow in the bed governed by the modified Darcy Law. Using the slip velocity boundary condition, velocity distributions in zones 1 and 2 are obtained and are matched at the interface. The boundary layer thickness just beneath the permeable interface and the friction factor are also obtained.     

Stochastic modeling of non-equilibrium multiphase flow in porous media

Non-equilibrium phenomena arising from the pore scale dynamics can have considerable effect on the large scale dynamics of multiphase flow in porous media. In such cases, the relative permeabilities and capillary pressure curves are not just simple functions of phase saturations, rather are dynamic functions of space and time. The present work proposes a stochastic approach in which particle mobility can be modelled based on the actual conductance of fluid volume inside a pore space. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Chemical waves and localized chaos in a model of heterogeneous catalytic reaction in a porous particles

A mathematical model of an oscillatory chemical reaction in a porous catalyst particle is considered. The model describes an oscillatory medium uniformly distributed throughout the volume of a spherical particle. The dynamical interaction of the reaction with the diffusive flow of the gaseous reagent inside the pores generates nonstationary dissipative structures in the oscillatory medium on the surface of the catalyst. Depending on the pressure in the gaseous phase, the model produces specific chemical waves and localized spatio-temporal chaos. The study was partially supported by the Russian Foundation for Basic Research (grant No. 96-03-34427a). Translated from Chislennye Metody i Vychislitel'nyi Eksperiment, Moscow State University, pp. 31–43, 1998.     

Green’s function,temperature in a convectively cooled sphere with arbitrarily located spherical heat sources

Steady state heat conduction in a convectively cooled sphere having arbitrarily located spherical heat sources inside is treated with the method of Green’s function accompanied by a coordinate transform. Green’s function of the heat diffusion operator for a finite sphere with Robin boundary condition is obtained by spherical harmonics expansion. Verification of the analytical solution is exemplified in some generic cases related to the pebbles of South-African PBMR as of year 2000 with 268 MW thermal power. Analytical results for different sectors of the sphere (pebble) are compared with the results of computational fluid dynamics code FLUENT^™. This work is motivated through a modest effort to assess the stochastic effects of distribution and volumetric effects of fuel kernels within the pebbles of future-promising pebble bed reactors.