Rapid Flow Regime Research Articles

Remarks on Constitutive Modeling of Granular Materials

In this paper, we provide a brief overview of certain fundamental concepts which can be used to derive constitutive relations for the stress tensor of granular materials. These include concepts such as dilatancy, cohesion, yield criterion, shear banding, etc. The focus will be on the constitutive relations which are used in the so-called ‘frictional flow’ or ‘slow flow’ regime as opposed to the rapid flow regime; in the slow flow regime the material is about to yield or has just yielded and the flow has been initiated. This type of flow occurs in the storage of grains, etc., in silos and bins or hoppers after the valves/gates are opened. The techniques of continuum mechanics are used to discuss constitutive relations where the effects of non-linearities such as yield stress, dilatancy, density gradients, etc., are important.

The role of previously glaciated landscapes in spatiotemporal variability of streamflow in snow-dominated watersheds: British Columbia, Canada

Study regionTwenty-six glaciated basins on the Interior Plateau (56° N, 125° W) in British Columbia, Canada. Study focusLandscape and streamflow characteristics in glaciated basins show high spatiotemporal variability, while the linkages among runoff characteristics, landscapes, and glacial history remain unclear. We compiled a 10-year dataset of daily climate and streamflow observations, and 54 landscape attributes for 26 glaciated basins to explore the relationship between landscape, streamflow and glacial history. A Hierarchical Multiple Factor Analysis was performed to analyze landscapes. The hydrological signatures were selected to describe streamflow characteristics. A correlation analysis and machine learning models were used to explore the relationships between the glacial legacy and streamflow variation. New hydrological insights for the region(1)The landscape is well-structured. The principal components of PC1 and PC2 explain 44% of the total landscape variance, PC1 representing basins with steep, high-relief terrain and alpine glaciation and PC2 representing large, elongated basins with moraine and glacial lakes; (2) The landscape heterogeneity has a key role in accurate streamflow predictions. Most hydrological signatures are controlled by landscape components PC1 and PC2 and can be well predicted by Random Forest using landscape features as inputs. (3) Alpine glaciation results in rapid runoff generation and variable flow regime in mountains, while valley glaciation leads to smooth hydrographs with large baseflows and mild and infrequent extreme flow events in plateaus.

Mechanism of loop seal clearance occurring at U-shaped pipe with air-water two-phase flow

Main purposes of the LOCAL (LOop seal Clearance separate effect test of ATLAS) facility are to investigate the effect of cross-over leg (COL) diameter at the loop seal clearance (LSC), to evaluate the two-phase flow regime through high speed visualization, and to suggest the mechanistic models for the LSC prediction at Small Break-Loss of Coolant Accident (SB-LOCA) in Light Water Reactors (LWRs). In this paper, we introduce a key mechanism of the onset of the LSC occurring at U-shaped COL with air-water two-phase flow condition. The onset of LSC is coupled with the flow regime transition at vertical upward and horizontal section of COL. Slug flow with single Taylor bubble results in decrease of pressure difference at vertical upward section and causes rapid flow regime transition at horizontal section from the stratified to the slug. The critical velocity of gas leading to flow regime transition at horizontal section can explain the onset of LSC and high speed visualization can provide the clear understanding of LSC phenomena at the local point of view.

A new kinetic theory model of granular flows that incorporates particle stiffness

Granular materials of practical interest in general have finite stiffness; therefore, the particle collision is a process that takes finite time to complete. Soft-sphere Discrete Element Method (DEM) simulations suggest that there are three regimes for granular shear flows: inertial regime (or rapid flow regime), elastic regime (or quasistatic regime), and the transition regime (or elastic-inertial regime). If we use tf to represent the mean free flight time for a particle between two consecutive collisions and tc to represent the binary collision duration, these regimes are implicitly related to the ratio tc/tf. Granular flows can be successfully predicted by the classical Kinetic Theory (KT) when they are in the inertial regime of low particle-particle collision frequencies and short time contacts (tc/tf ≈ 0). However, we find that KT becomes less accurate in the transition regime where the collision duration tc is no longer small compared with the collision interval tf (tc/tf > 0.05). To address this issue, we develop a soft-sphere KT (SSKT) model that takes particle stiffness k as an input parameter since tc/tf is mainly determined by k. This is achieved by proposing a modified expression for the collision frequency and introducing an elastic granular temperature Te. Compared with the classical KT that only considers the kinetic granular temperature Tk, a redefined total granular temperature (Tg = Tk + Te/3) that takes both kinetic fluctuation energy and elastic potential energy into consideration is used in the SSKT model. The model is developed for identical frictionless particles with the linear-spring-dashpot collision scheme; however, it can be extended to frictional systems as well after the modification of constitutive equations. We show that the proposed SSKT extends the applicability of the KT framework to the transition regime without losing significant accuracy. The rheological crossover has been explained physically, and the regime boundaries that separate the inertial regime and the elastic regime are quantitatively determined, showing good agreement with the previous regime map that was based on the DEM simulations. Our SSKT predictions also show that for unsteady flows such as homogeneous cooling, the particle stiffness could have a large impact on the granular flow behavior due to the energy transfer between Te and Tk.

Characterising powder flowability at high shear rates by the ball indentation method

Unreliable powder flow is a major problem during processing of powders. The shear cell is the most widely used method for powders subjected to moderate or high stresses, and under quasi-static conditions, with established methods for designing large bins and hoppers based on the measurement. However, this method is not suitable for measuring the flowability of dynamic systems, such as powder mixing.Here, the ball indentation method is investigated as a technique for evaluating powders in the intermediate and dynamic regime of flow. The method, which simply consists of dropping a ball onto a cylindrical bed of powder previously consolidated, directly measures hardness, which is related to the unconfined yield stress of the powder by the constrain factor (Hassanpour and Ghadiri, 2007). The impact of the ball on the bed is recorded with a high-speed camera to determine velocity and penetration depth. The shear rate is varied by using a range of indenter materials and sizes, and a range of drop heights. The hardness against the strain rate is considered for several materials.It was found that the indenter size does not influence the hardness results, which are consistent with the flowability evaluation achieved with the rheometer. Furthermore the hardness, which is independent of the strain rate in quasi-static conditions, becomes shear rate dependent in intermediate regime of flow. Further work is needed to evaluate hardness in the rapid granular flow regime.

Effect of particle stress tensor in simulations of dense gas–particle flows in fluidized beds

A two-fluid model based on the kinetic theory of granular flow for the rapid-flow regime and the Coulomb friction law for the quasi-static regime is applied to predict the hydrodynamics of dense gas–particle flow in a three-dimensional fluidized bed. Two different models for the particle stress tensor that use different constitutive equations in the elastic-inertial regime are examined to assess their ability to predict bed dynamics. To understand how particle stress models affect structural features of the flow, a quantitative analysis is performed on some important aspects of the mechanics of bubbling beds that have received relatively little attention in the literature. Accordingly, different flow regimes are identified in the context of fluidized beds through the dimensionless inertial number, and the main characteristics of each regime are discussed. In addition, how the particle stress tensor manifests itself in the bubble characteristics, natural frequency of the bed, and particle Reynolds stress are investigated, all of which help to better understand the complex dynamics of the fluidized bed. The numerical results are validated against published experimental data and demonstrate the significant role of the stress tensor in the elastic-inertial regime.

Computations of Fluid Dynamics of a 50 MWe Circulating Fluidized Bed Combustor

Gas−particle two-phase turbulent flows are numerically studied in a 50 MWe circulating fluidized bed combustor. The dense flow of particles is modeled by the frictional stress models adopted from solid mechanics theory, and the dilute flow in the so-called rapid granular flow regime is modeled from the kinetic theory of granular flow. At low concentrations the viscosity due to the effect of the presence of particles is modeled by means of a semiempirical viscosity for fluidized catalytic cracking particles. The distribution of the velocity and concentration of particles in a circulating fluidized bed combustor is predicted. Simulations indicate that the dense regime with a high concentration of particles is in the bottom part and the dilute regime with a low concentration in the upper part of the furnace. The effect of secondary air on the flow behavior is analyzed, and the penetration length of the secondary air jet is computed in a 50 MWe circulating fluidized bed combustor.

A model for gas–solid flow in a horizontal duct with a smooth merge of rapid–intermediate–dense flows

Granular flow in the rapid flow regime is dominated by particle–particle collisions and the constitutive relations for the solid stress are obtained from the classic kinetic theory of granular flow. In the dense flow regime, on the other hand, particles interact via enduring contacts and the solid stress can be deduced from soil mechanics theories. In this paper, constitutive equations, recently proposed by Tardos et al. [2003. Slow and intermediate flow of frictional bulk powder in the Couette geometry. Powder Technology 131, 23–39.] has been incorporated in the simulation of gas–solid flow in a horizontal duct. These equations smoothly merge the rapid granular flow solution with the so-called “intermediate” regime (where both kinetic/collisional and frictional contributions might play a role) and reduce to Coulomb yield condition for slow frictional flow (shear rate → 0). The results of this new modelling approach have shown good qualitative agreement with the reported experimental observation on wide range of gas–solid flow conditions. In this study, we also present the definition of boundaries between rapid–intermediate–dense flow regimes based on the dimensionless shear rate ( λ ), and a modified Reynolds number ( Re). We have shown that the intermediate flow regime can be classified at approximately 0.1 < λ < 1.0 and 100 < Re < 3000 .

Granular material flows – An overview

The paper attempts to give a critical overview of the field of granular flow with attention both to the history and the underlying physics that govern the field. It starts with a discussion of the basic transport mechanisms in a granular flow. It continues with a discussion of contact mechanics – the way that individual particles see each other mechanically. It then discusses the historical limiting regimes of granular flow, the Quasistatic and the Rapid-Flow regimes. Finally, it concludes with a review of the Elastic picture of granular flow, which both unifies the Quasistatic and Rapid regimes and fills in the intervening space. It shows that the rheological behavior of granular systems changes with system scale constraints, and, in particular, that the materials behave differently under controlled-stress and controlled-concentration conditions. The Elastic model defines an entire flowmap of granular flow and thus allows one to place boundaries on where the Quasistatic and Rapid-Flow models (sometimes called kinetic theory models) are something of a red herring and cannot be applied to common granular flows.

Brazil nut effect and excluded volume attraction in vibrofluidized granular mixtures

A two dimensional bi-disperse vibrofluidized granular mixture is studied in the rapid flow regime, where particle interactions occur due to instantaneous collisions. Both experiments and simulations are carried out, and these show the existence of two phenomena which have been observed only in very dense granular flows or in equilibrium systems. The Brazil nut phenomenon, which involves the rise of larger particles in a granular mixture upon vibration, has been observed in dense systems due to the percolation of small particles though the interstitial spaces between the large particles, or due to convection rolls. In the present case, where neither effect is present, it is observed that the fluidization of the smaller particles by vibration results in an exponentially decaying density profile, at heights large compared to the particle diameter, and thereby a pressure field that decreases with height. The larger particles, suspended in this decaying pressure field, experience a larger pressure at the bottom and a smaller pressure on top, and they rise to a height where the net force caused by the decreasing pressure is balanced by the weight of the particle. An attractive force between the large particles, similar to the entropic attraction effect in mixtures of colloids and polymers, is also observed in this nonequilibrium system, because when the distance between the large particles is less than the small particle diameter, the pressure between the large particles is smaller than that on the outside. Analytical results are derived for each of these effects, and these are in agreement with the experimental and simulation results.

Elastic Granular Flows

Large scale landslide simulations provided the surprising result that the effective friction coefficient (the ratio of shear to normal forces at the base of the slide) increased with the shear rate. This might possibly explain the effect of slide volume on the runout of large landslides, but it also indicates that landslides operated in an entirely new and unexplored flow regime.Previously, granular flows had been divided into (1) the slow, quasistatic regime, in which the effective friction coefficient is taken to be a material property and thus constant, and (2) the fast, rapid-flow regime, where the particles interact collisionally, but which scales in such a way that the effective friction coefficient is independent of the shear rate. Both indicate that the effective friction coefficient is independent of the shear rate. Consequently the landslides operated in a separate intermediate regime.This talk will discuss detailed computer simulation studies into this intermediate regime and into the transitions between regimes. In this way, it is possible to draw the entire flowmap connecting the quasistatic and rapid-flow regimes. The key was to include the elastic properties of the solid material in the set of rheological parameters; in effect this put solid properties into the rheology of granular solids, properties that were unnecessary in previous theories as a result of the plasticity and kinetic theory formalisms on which quasistatic and rapid-flow theories are respectively based. Granular flows are then divided into two broad categories, the Elastic Regimes, in which the particles are locked in force chains and interact elastically over long duration contact with their neighbors and the Inertial regimes, where the particles have broken free of the force chains. The Elastic regimes can be further subdivided into the Elastic-Quasistatic regime (the old quasistatic regime) and the Elastic-Inertial regime. The Elastic-Inertial regime is the “new” regime observed in the landslide simulations, in which the inertially induced stresses are significant compared to the elastically induced stresses. The Inertial regime can also be sub-divided into an Inertial-Non-Collisional where the stresses scale inertially, but the particles interact through long duration contacts, and the Inertial-Collisional or the old rapid-flow regime.Finally, it will be shown that Stress-Controlled flows are rheologically different from Controlled-Volume flows. Physically, there is a range of dense concentrations (solid volume fractions between 0.5 and 0.6) in which it is possible to form force chains and thus to demonstrate elastically. But there are conditions under which force chains do not form at the same average concentrations. (In other words it is possible for the material to exhibit two different states at the same concentration.) By forcing the material to support an applied loads across force chains, Stress-Controlled flows generally behave elastically through this range of concentrations under the same conditions where Controlled-Volume flows behave inertially.

Granular Flows and Gas Fluidization

This presentation examines the roles that granular flow theory might play in the modeling the solid-phase stresses in fluidized beds. It begins with a discussion of the historical limiting cases of quasistatic and rapid-flow regimes. It then reviews the elastic granular theories that unify those two regimes. Finally, it shows that measurements of solid-phase pressures cannot be explained by commonly used rapid-flow theories, and most likely lie in the elastic flow regimes.

Dense, rapid flows of inelastic grains under gravity.

The standard hydrodynamic description does not apply to the rapid flow regime of inelastic grains in the dense limit. Emphasizing the role of inelastic loss and collapse, we propose a new approach relying on a nonlocal dissipation scheme. Our model succeeds in accounting qualitatively and quantitatively for the linear profile of velocity found in experiments on dense gravity-driven flows.

Slow and intermediate flow of a frictional bulk powder in the Couette geometry

Most current research in the field of dry, non-aerated powder flows is directed toward rapid granular flows of large particles. Slow, frictional, dense flows of powders in the so-called quasi-static regime were also studied extensively using Soil Mechanics principles. The present paper describes the rheological behavior of powders in the “intermediate” regime lying between the slow and rapid flow regimes. Flows in this regime have direct industrial relevance. Such flows occur when powders move relative to solid walls in hoppers, bins and around inserts or are mixed in high and low shear mixers using moving paddles. A simple geometry that of a Couette device is used as a benchmark of more complicated flows. The constitutive equations derived by Schaeffer [J. Differ. Equ. 66 (1987) 19] for slow, incompressible powder flows were used in a new approach proposed by Savage [J. Fluid Mech. 377 (1998) 1] to describe flows in the intermediate regime. The theory is based on the assumption that both stress and strain-rate fluctuations are present in the powder. Using Savage's approach, we derive an expression for the average stress that reduces to the quasi-static flow limit when fluctuations go to zero while, in the limit of large fluctuations, a “liquid-like”, “viscous” character is manifested by the bulk powder. An analytical solution of the averaged equations for the specific geometry of the Couette device is presented. We calculate both the velocity profile in the powder and the shear stress in the sheared layer and compare these results to experimental data. We show that normal stresses in the sheared layer depend linearly on depth (somewhat like in a fluid) and that the shear stress in the powder is shear rate dependent. We also find that the velocity of the powder in the vicinity of a rough, moving boundary, decays exponentially so that the flow is restricted to a small area adjacent to the wall. The width of this area is of the order of 10–13 particle diameters. In the limit of very small particles, this is tantamount to a shear band-type behavior near the wall.

Scaling relationships for rotating drums

Rotating drums are extensively used in the chemical and process industries as kilns, mixers, dryers and reactors. Despite challenges from the development of newer and more specialised technologies such as fluidised beds, rotating drums continue to find applications. This is mainly due to their ability to handle varying feed stocks, in particular granular materials with broad particle size distribution and significant difference in physical properties. However, the scale-up methodology for such devices is still largely empirical and no general and systematic method has been established. This work is therefore devoted to the development of a set of scaling relationships which can be used to properly design rotating drums. The scaling relationships are obtained by non-dimensionalising the differential equations governing the behaviour of solids motion in rotating drums. The derived dimensionless groups include Froude number, pseudo-Reynolds number, pseudo-Euler number, drum geometric ratios, drum inclination, drum fill percentage, size distribution of particles, and physical properties of the particles and drum wall such as restitution coefficient and elasticity modulus. At relatively low pseudo-Reynolds numbers, the effect of velocity fluctuation and hence the granular temperature can be neglected, the granular flows are thus in the quasi-static regime. At relatively low pseudo-Euler numbers, the effect of the frictional contribution to the total stress tensor is negligible and the flow is in the rapid granular flow regime. The upper limit of the quasi-static regime is evaluated and the adequacy of the application of granular flow kinetic theory to rotating drums is discussed. It is shown that granular flows in large drums operated at low to medium rotational speeds are often in the quasi-static regime, whereas those in small drums operated at medium rotational speeds may be in the transition flow regime. Preliminary experiments have been carried out with the aim of reducing the total number of controlling dimensionless groups. The dimensionless groups investigated include the drum length-to-diameter ratio, rotational Froude number, fill percentage, particle-to-drum diameter ratio, particle restitution coefficient, and the relative drum wall roughness. It is shown that the Froude number, particle-to-drum diameter ratio, drum fill percentage, and particle restitution coefficient can be combined to give a single dimensionless parameter if drums are operated in a rolling mode and the Froude number is greater than 0.003.

Computer simulation of wet granulation

Agglomeration of fine particles in wet granulation is achieved by introducing a binder fluid onto a shearing mass of powder. Owing to the viscosity and the surface tension of the fluid, powder particles are bound together to form larger aggregates. Despite its widespread use in the chemical, pharmaceutical and food industries, little effort has gone into comprehensive modeling of the overall process from first principles. Modeling is important however, if one needs to estimate a-priori agglomerated granule characteristics such as size, shape and density, from knowledge of operating conditions and powder and binder physical and chemical properties. In this work, we present a model of wet granulation that is essentially a computer simulation of shear flows of solid particles, some of which are wet (covered by binder and therefore sticky) while the rest are dry. While simulations of shear flows of dry solid particles have earlier been reported in the literature, present work takes this simulation a step further and introduces a liquid surface layer to some particles in the domain. The additional force experienced by two relatively moving particles interacting via their binder-covered layers is modeled by using results from lubrication theory and Stokesian dynamics. The numerical simulations reflect two distinct regimes of agglomeration: that of granule growth and that of granule breakup. The granule growth regime takes place in all granulators including low and high shear machines while granule break-up is mainly characteristic of medium and high-shear devices in which agitation as generated by some mechanical means. During granule growth-simulations, the movement of sticky (binder covered) particles is studied in a constant shear, rapid granular flow regime. From these simulations, final granule size, shape and size distributions were obtained using a pattern-recognition technique. A second kind of simulation, also using rapid granular flow modeling, follows the deformation and break-up of an agglomerate made from particles held together by a liquid, viscous binder. Results from these simulations yield critical values of a dimensionless parameter that contains inertial and viscous dissipation effects (the so-called Stokes number). Below a critical value of the Stokes number, agglomerates are stable and only rotate in response to shear while above the critical value they break into several pieces. Around the critical value, they attain a steady elongation. These simulations allow one to obtain correlations between critical sizes, i.e., granules that deform somewhat but do not break, and different parameters of the problem.

Granular flows

We review both experimental and theoretical works concerning granular flows. We successively address the regime of slow deformations, which is mainly governed bystericinteractions and frictionforces, thenthe rapid flow regime, whichdeals with inelastic collisions, and lastly the regime of intermittent avalanches.

Granular Flows in the Elastic Limit

ABSTRACTThis paper describes recent computer simulation studies into the rheological behavior of granular materials in the regime that lies between the quasistatic and rapid-flow regime. This investigation was prompted by studies of landslides, hopper flows and the “phase change” (i.e. the change between solidlike and fluid-like behavior) all of which indicated that the shear-to-normal stress ratio (the effective friction coefficient for the material) increased with shear rate. The results presented herein do demonstrate that the stress ratio varies with a dimensionless parameter created by scaling the shear rate with the stiffness of the interparticle contacts. In dense regimes, the stresses themselves scale with the stiffness indicating that they are generated by the elastic response of particle networks. Such speculation is supported by studies that show that the normal stresses are strongly dependent on the interparticle friction coefficient which affects the ability of internal elastic particle structures to support load and by the time variation of stress, which shows a spiky behavior as the structures form and break. However, analyses also indicate that these observations cannot explain the hopper, landslide and phase-change as these systems operate in regimes different from those in which the effect was observed. Finally, the effects of non-linear contacts are investigated and an appropriate scaling that takes the non-linearity into account is proposed.

Analyses of slow high-concentration flows of granular materials

A theory intended for slow, dense flows of cohesionless granular materials is developed for the case of planar deformations. By considering granular flows on very fine scales, one can conveniently split the individual particle velocities into fluctuating and mean transport components, and employ the notion of granular temperature that plays a central role in rapid granular flows. On somewhat larger scales, one can think of analogous fluctuations in strain rates. Both kinds of fluctuations are utilized in the present paper. Following the standard continuum approach, the conservation equations for mass, momentum and particle translational fluctuation energy are presented. The latter two equations involve constitutive coefficients, whose determination is one of the main concerns of the present paper. We begin with an associated flow rule for the case of a compressible, frictional, plastic continuum. The functional dependence of the flow rule is chosen so that the limiting behaviours of the resulting constitutive relations are consistent with the results of the kinetic theories developed for rapid flow regimes. Following Hibler (1977) and assuming that there are fluctuations in the strain rates that have, for example, a Gaussian distribution function, it is possible to obtain a relationship between the mean stress and the mean strain rate. It turns out, perhaps surprisingly, that this relationship has a viscous-like character. For low shear rates, the constitutive behaviour is similar to that of a liquid in the sense that the effective viscosity decreases with increasing granular temperature, whereas for rapid granular flows, the viscosity increases with increasing granular temperature as in a gas. The rate of energy dissipation can be determined in a manner similar to that used to derive the viscosity coefficients. After assuming that the magnitude of the strain-rate fluctuations can be related to the granular temperature, we obtain a closed system of equations that can be used to solve boundary value problems. The theory is used to consider the case of a simple shear flow. The resulting expressions for the stress components are similar to models previously proposed on a more ad hoc basis in which quasi-static stress contributions were directly patched to rate-dependent stresses. The problem of slow granular flow in rough-walled vertical chutes is then considered and the velocity, concentration and granular temperature profiles are determined. Thin boundary layers next to the vertical sidewalls arise with the concentration boundary layer being thicker than the velocity boundary layer. This kind of behaviour is observed in both laboratory experiments and in granular dynamics simulations of vertical chute flows.

Dynamics of granular stratification

Spontaneous stratification in granular mixtures has been recently reported by H. A. Makse et al. [Nature 386, 379 (1997)]. Here we study experimentally the dynamical processes leading to spontaneous stratification. Using a high-speed video camera, we study a rapid flow regime where the rolling grains size segregate during the avalanche. We characterize the dynamical process of stratification by measuring all relevant quantities: the velocity of the rolling grains, the velocity of the kink, the wavelength of the layers, the rate of collision between rolling and static grains, and all the angles of repose characterizing the mixture. The wavelength of the layers behaves linearly with the thickness of the layer of rolling grains (i.e., with the flow rate), in agreement with theoretical predictions. The velocity profile of the grains in the rolling phase is a linear function of the position of the grains along the moving layer. We also find that the speed of the upward-moving kink has the same value as the mean speed of the downward-moving grains. We measure the shape and size of the kink, as well as the profiles of the rolling and static phases of grains, and find agreement with recent theoretical predictions.