Continuum modelling of segregating tridisperse granular chute flow

Abstract

Segregation and mixing of size multidisperse granular materials remain challenging problems in many industrial applications. In this paper, we apply a continuum-based model that captures the effects of segregation, diffusion and advection for size tridisperse granular flow in quasi-two-dimensional chute flow. The model uses the kinematics of the flow and other physical parameters such as the diffusion coefficient and the percolation length scale, quantities that can be determined directly from experiment, simulation or theory and that are not arbitrarily adjustable. The predictions from the model are consistent with experimentally validated discrete element method (DEM) simulations over a wide range of flow conditions and particle sizes. The degree of segregation depends on the Péclet number, Pe, defined as the ratio of the segregation rate to the diffusion rate, the relative segregation strength κ_ij between particle species i and j, and a characteristic length L, which is determined by the strength of segregation between smallest and largest particles. A parametric study of particle size, κ_ij, Pe and L demonstrates how particle segregation patterns depend on the interplay of advection, segregation and diffusion. Finally, the segregation pattern is also affected by the velocity profile and the degree of basal slip at the chute surface. The model is applicable to different flow geometries, and should be easily adapted to segregation driven by other particle properties such as density and shape.

1. Introduction

Segregation of multidisperse granular materials is common in natural and industrial processes, such as landslides and processing of ores and polymers [1–3]. In gravity-driven flows of initially well-mixed particles of different sizes, smaller particles fall through the interstices between larger particles causing size segregation. This mechanism is commonly referred to as particle percolation. In this paper, we use a multidisperse segregation model to better understand the segregation of multidisperse mixtures in developing chute flow, and validate the model using discrete element method (DEM) simulations over a wide range of flow conditions. While other researchers [4] have considered multidisperse developing segregation from a theoretical standpoint, here we use a segregation length scale unique to developing segregation, introduce a new non-dimensionalization scheme to accommodate multidisperse mixtures and quantitatively compare the model results to DEM simulations. We further examine the effects of particle size, operating conditions (using dimensionless parameters), and the velocity profile on segregation in chute flow.

Most theoretical approaches to modelling segregation in granular flows have focused on bidisperse mixtures in order to understand the underlying mechanisms and develop predictive frameworks [5–18]. Although not as commonly studied, segregation in multidisperse (multi-component) systems made up of several discrete particle sizes and in polydisperse systems containing a range of particle sizes characterized by a continuous probability distribution, such as a log-normal distribution have been explored both in experiments [19–22] and in simulations using the DEM [21–24] for different flow geometries. In spite of this previous research, a generally applicable model for predicting multidisperse or polydisperse segregation is only now developing.

The simple geometry of chute flow makes it well suited for the study of multidisperse and polydisperse flows. Over the past three decades, there has been substantial study of segregation in chute flow; see table 1 for a sample of past research on the topic. For example, Gray & Ancey [4] developed a continuum model to describe the segregation of flowing multidisperse mixtures based on the interaction of advection, segregation and diffusion in the flowing layer. Marks et al. [23] developed a continuum model for polydisperse segregation that relies on a fitting parameter determined from DEM simulations. Recently, a stochastic lattice model incorporating the effects of segregation, mixing and crushing was used to predict steady-state grain size distributions in uniformly sheared granular flows [25]. Although this approach can connect micro- and macroscale advection-driven processes for polydisperse systems, it provides only qualitative agreement with experimentally measured grain size distributions. These and other models often include adjustable parameters, some of which are challenging to directly relate to physical quantities, making it difficult to apply the results to specific particle sizes, shapes and densities or to generalize the model to a continuous distribution of particle sizes.

Table 1.

A sample of past dense granular segregation research in chutes.

references	methods	dispersity
Savage & Lun [6]	theory and experiment	bidisperse
Dolgunin et al. [7]	theory and experiment	bidisperse
Dolgunin & Ukolov [5]	theory and experiment	bidisperse
Gray & Thornton [8]	theory	bidisperse
Gray et al. [9]	theory	bidisperse
Gray & Chugunov [10]	theory	bidisperse
Thornton et al. [11]	theory	bidisperse
Wiederseiner et al. [13]	theory and experiment	bidisperse
Thornton et al. [15]	DEM	bidisperse
Tunuguntla et al. [17]	theory and DEM	bidisperse
Larcher & Jenkins [16]	theory and DEM	bidisperse
Larcher & Jenkins [18]	theory and DEM	bidisperse
Gray & Ancey [4]	theory	multidisperse
Marks et al. [23]	theory and DEM	polydisperse
Bhattacharya & McCarthy [22]	DEM and experiment	multi- and polydisperse
Marks & Einav [25]	theory	polydisperse
Schlick et al. [26]	theory and DEM	multi- and polydisperse

Recently, a continuum-based segregation model for bidisperse mixtures that was successfully applied to gravity-driven flows in quasi-two-dimensional bounded heaps [27] and circular tumblers [28] was extended to fully developed multi- and polydisperse segregation, where it demonstrated quantitative agreement with DEM simulation results [26]. The modelling approach uses segregation parameters and material-dependent scalings obtained from DEM simulations that are directly connected to the physics of segregation and are applicable to a wide range of flow rates, particle size ratios, particle size distributions, flow geometries and even density-driven segregation [29]. Here, we use this continuum modelling framework to examine the segregation of tridisperse (rather than bidisperse) granular mixtures for developing segregation (rather than fully developed segregation) in chute flow resulting from the interaction between advection, segregation and diffusion. We further introduce a new non-dimensionlization scheme for multidisperse systems incorporating a streamwise segregation length scale that characterizes developing segregation.

As granular material flows down a chute, segregation drives the formation of sublayers enriched in a particular particle size while, simultaneously, collisional diffusion acts to remix particles. The velocity field of the flow is determined by the chute’s surface roughness and inclination angle as well as the volumetric flow rate. In steady state, the total streamwise flux of each species remains constant along the entire length of the chute. Sufficiently far downstream, fully developed segregation occurs in which there is a balance between segregation and diffusion, and, consequently, the concentration profiles of the material through the depth of the flowing layer remain constant.

Here, we compare the theoretical predictions of our continuum modelling framework for tridisperse chute flow to results from equivalent DEM simulations and explore the model predictions for a range of parameters to gain insight into multidisperse segregation in developing flows. Even though the model and parameters can be easily extended to more than three particle species, the tridisperse system provides an important stepping stone between bidisperse and polydisperse systems and, at the same time, allows easy data visualization.

2. Segregation model

We consider size segregation of tridisperse granular materials flowing down a chute, shown schematically in figure 1a. The streamwise and normal directions are x and z, respectively, and the origin is at the most upstream position of the bottom of the flowing layer. A δ thick mixture of three particle sizes flows at volumetric flow rate, Q, down a chute inclined at angle θ with respect to the horizontal with a fully developed velocity profile u(z), that is assumed, to first order, to be independent of the local particle concentration. The gap thickness between the two side walls of the chute is T, and the two-dimensional flow rate is q=Q/T. While there can be streamwise variation in the flowing layer thickness under certain conditions [13,30], here we assume that δ is constant for the purpose of simplicity, similar to previous studies [4,8,13]. In §4a, we show that a constant value of δ reproduces spatial concentration fields consistent with DEM simulations.

Figure 1.

(a) Sketch of quasi-two-dimensional chute flow showing the flow geometry, evolution of the segregation and a schematic streamwise velocity profile. (b) The characteristic streamwise segregation length scale, L, is defined as streamwise distance travelled by a small particle as it moves from the free surface to the bottom of a flowing layer composed of large particles at a characteristic streamwise surface velocity u_surf. (Online version in colour.)

The model is based on the scalar advection diffusion transport equation for the concentration of species i:

$$\frac{\mathrm{\partial }{c}_i}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot ({\mathit{u}}_i{c}_i)=\mathrm{\nabla }\cdot (D\mathrm{\nabla }{c}_i),$$ 2.1

where u_i is the velocity of species i, c_i is the volume concentration of species i and D is the scalar collisional diffusion coefficient. For quasi-two-dimensional chute flow, we assume no net motion of species in the spanwise (y) direction (i.e. zero spanwise velocity v_i=0). Consequently, ${\mathit{u}}_i=u_i\hat{x}+w_i\hat{z}$ with streamwise and normal velocity components u_i and w_i, respectively, for species i. The streamwise velocity component of species i is written most generally as u_i=u+u_p,i, where u_p,i is the streamwise component of the gravity-driven percolation velocity of species i relative to the mean streamwise flow velocity u. However, for this case and most other free surface flows, u_p,i≪u so that u_i can be accurately approximated as equal to u. The normal velocity component of species i is written as w_i=w+w_p,i, where w_p,i is the percolation velocity of species i relative to the mean normal flow velocity w due to segregation. With these assumptions, the transport equation (2.1) can be written as [27,28,31]

$$\frac{\mathrm{\partial }{c}_i}{\mathrm{\partial }t}+\frac{\mathrm{\partial }(u{c}_i)}{\mathrm{\partial }x}+\frac{\mathrm{\partial }(w{c}_i)}{\mathrm{\partial }z}+\frac{\mathrm{\partial }(w_{p,i}{c}_i)}{\mathrm{\partial }z}=D\frac{{\mathrm{\partial }}^2{c}_i}{\mathrm{\partial }{x}^2}+D\frac{{\mathrm{\partial }}^2{c}_i}{\mathrm{\partial }{z}^2}.$$ 2.2

Although diffusion can be anisotropic [32], for simplicity we assume here that D is isotropic and homogeneous. This simplified assumption has yielded results that reproduce data from experiment and simulation in a variety of situations including plug flow [10], chute flow [13,15,23,26] and bounded heap flow [26,27,29].

This approach to modelling segregation in flowing granular materials was first suggested by Bridgwater et al. [33] in a slightly different form. Over the last decades, many researchers have used similar approaches with a variety of expressions for the segregation flux, which is the last term of the left-hand side of equation (2.2) [4,5,7,8,17,18,23]. Here, we follow the form used by Fan et al. [27] and Schlick et al. [26,28,31]. The key to the subsequent formulation of the model is the semi-empirical relation for the component of the segregation velocity normal to the free surface, w_p,i, which is derived from size bidisperse mixtures composed of particles with diameters α_i and α_j. The segregation velocity of species i depends on the local shear rate, $\dot{\gamma }$, and the concentration of the other species, c_j, as:

$$w_{p,i}=S({\alpha }_i,{\alpha }_j)\dot{\gamma }{c}_j,$$ 2.3

where S is an empirically determined segregation length scale [27]. Equation (2.3) can be obtained by linearizing a more complicated expression for size segregation of bidisperse particles derived by Savage & Lun [6]. Gray & Ancey [4] used an approach similar to equation (2.3) except that their formulation did not explicitly include shear rate, $\dot{\gamma }$, while the Savage & Lun [6] model requires to account for local effects due to relative flow that allow percolation to occur. We further note that equation (2.3) is applicable only to free surface flowing layers where the relatively small effects of lithostatic pressure on segregation are included in the coefficient S(α_i,α_j), unlike other situations where there is a large overburden [34]. For spherical glass particles, the segregation length scale can be expressed as a function of the particle size ratio:

$$S({\alpha }_i,{\alpha }_j)=Bmin({\alpha }_i,{\alpha }_j)\ln \left(\frac{{\alpha }_i}{{\alpha }_j}\right),$$ 2.4

where $\ln$ is the natural logarithm and B is a constant dependent on the intrinsic properties of the particles [31]. Based on DEM simulations of spherical glass particles in bidisperse bounded heap flow, B=0.26 for particle size ratios in the range 1/3≤α_i/α_j≤3 and 1 mm≤α_i≤3 mm [31]; we use this value of B for the rest of the paper.

Equations (2.3) and (2.4) capture the downward percolation of small particles and the upwards movement of large particles in a flowing mixture. Equation (2.3) is based on the local shear rate and local particle concentrations, and, thus, is valid everywhere in dense, bidisperse and gravity-driven granular surface flows provided $\dot{\gamma }$ and c_j are known. Using equations (2.3) and (2.4) in the framework described above, quantitative agreement between segregation in the model, DEM simulations and experiments has been demonstrated for size bidisperse flow in a bounded heap [27] and in a circular tumbler [28], and for density bidisperse flow in a bounded heap, where particle sizes α_i and α_j are replaced by particle densities ρ_i and ρ_j [29]. We note that there is a slight asymmetry in the segregation velocity for shear flows in which a small particle surrounded by large particles moves downwards faster than a large particle surrounded by small particles moves upwards [6,34–36]. However, equation (2.3) captures the leading-order behaviour and is sufficient to accurately predict the overall behaviour of segregating flows [26–29]. It is possible to extend equations (2.3) and (2.4) to a wider size ratio as long as S(α_i,α_j) can be measured from DEM simulations. Furthermore, for size ratios greater than approximately 6, ‘free sifting’ occurs, in which small particles percolate between large particles without being significantly influenced by the large particle flow [6,37–39]. The model described here will not properly account for the physics in this situation.

The expression for the segregation velocity in bidisperse mixtures (equation (2.3)) was recently generalized to describe multidisperse (more than two different particle sizes) and polydisperse (continuous size distribution) segregation [26]. This generalization assumes that the segregation velocity of particle i depends on a linear combination of the segregation length scales S(α_i,α_j) weighted by the corresponding concentration of each surrounding species j, and neglects higher-order interactions. For n distinct particle sizes in a mixture, each with diameter α_j and local concentration c_j, the percolation velocity is generalized to

$$w_{p,i}=\sum _{j=1}^{n}S({\alpha }_i,{\alpha }_j)\dot{\gamma }{c}_j.$$ 2.5

This equation differs from the approach used by Gray & Ancey [4] for a tridisperse mixture (n=3) by virtue of the crucial explicit inclusion of the shear rate in equation (2.5) that allows the expression to describe the local flow effects. As will be shown later, it is necessary to include the shear rate to account for the impact of different velocity profiles that can occur in chute flow on the segregation. Equation (2.5) has a further advantage over the approach of Gray & Ancey [4] in that S(α_i,α_j) can be defined purely in terms of the particle sizes (equation (2.4)), whereas the Gray & Ancey [4] approach provides no means to determine their equivalent parameters. Equation (2.5) also differs from the approach proposed by Marks et al. [23], which includes a free parameter C (eqn 3.2 of [23]), which was postulated to be related to particle shape, surface roughness and the shear rate profile.

Substituting equation (2.5) into equation (2.2) yields the full scalar transport equations for particle mixtures with n distinct particle sizes:

$$\frac{\mathrm{\partial }{c}_i}{\mathrm{\partial }t}+\frac{\mathrm{\partial }(u{c}_i)}{\mathrm{\partial }x}+\frac{\mathrm{\partial }(w{c}_i)}{\mathrm{\partial }z}+\sum _{j=1}^{n}S({\alpha }_i,{\alpha }_j)\frac{\mathrm{\partial }}{\mathrm{\partial }z}(\dot{\gamma }{c}_j{c}_i)=D\frac{{\mathrm{\partial }}^2{c}_i}{\mathrm{\partial }{x}^2}+D\frac{{\mathrm{\partial }}^2{c}_i}{\mathrm{\partial }{z}^2}.$$ 2.6

Later in this paper we demonstrate that the streamwise velocity profile, u, can be accurately assumed to be fully developed and independent of the local particle concentration or position along the length of the chute. Through conservation of mass, the overall mean particle velocity normal to the free surface, w, is zero. Consequently, the transport equation in the chute can be expressed as

$$\frac{\mathrm{\partial }{c}_i}{\mathrm{\partial }t}+u\frac{\mathrm{\partial }{c}_i}{\mathrm{\partial }x}+\sum _{j=1}^{n}S({\alpha }_i,{\alpha }_j)\frac{\mathrm{\partial }}{\mathrm{\partial }z}(\dot{\gamma }{c}_j{c}_i)=D\frac{{\mathrm{\partial }}^2{c}_i}{\mathrm{\partial }{x}^2}+D\frac{{\mathrm{\partial }}^2{c}_i}{\mathrm{\partial }{z}^2}.$$ 2.7

This system of n coupled partial differential equations can be simplified to (n−1) partial differential equations because the sum of the species concentrations $\sum _{i=1}^n{c}_i=1$. Equation (2.7) describes how the local volume concentration of species i is determined by advection due to the mean flow, segregation due to percolation and diffusion due to random particle collisions. It includes the dependence of the segregation velocity on both the spatially varying local shear rate and the local particle concentrations, as well as the dependence of concentration on advection via the streamwise velocity u and diffusion via the diffusion coefficient D.

Equation (2.7) is non-dimensionalized using characteristic streamwise and normal length scales L and δ, respectively, and a characteristic streamwise surface velocity scale 2q/δ as follows:

$$\tilde{x}=\frac{x}{L},\tilde{z}=\frac{z}{\delta },\tilde{t}=\frac{t}{\delta L/2q},\tilde{u}=\frac{u}{2q/\delta }\text{and}\tilde{\dot{\gamma }}=\frac{\dot{\gamma }}{2q/{\delta }^2}.$$ 2.8

The dimensionless governing equation for the concentration of species i is then:

$$\frac{\mathrm{\partial }{c}_i}{\mathrm{\partial }\tilde{t}}+\tilde{u}\frac{\mathrm{\partial }{c}_i}{\mathrm{\partial }\tilde{x}}+\left(\frac{L}{{\delta }^2}\right)\hspace{0.17em}[\sum _{j=1}^{n}S({\alpha }_i,{\alpha }_j)\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{z}}(\tilde{\dot{\gamma }}{c}_j{c}_i)]=\left(\frac{\delta }{2qL}\right)\hspace{0.17em}D\frac{{\mathrm{\partial }}^2{c}_i}{\mathrm{\partial }{\tilde{x}}^2}+\left(\frac{L}{2q\delta }\right)\hspace{0.17em}D\frac{{\mathrm{\partial }}^2{c}_i}{\mathrm{\partial }{\tilde{z}}^2}.$$ 2.9

The streamwise segregation length scale, L, characterizing the developing segregation is based on the streamwise distance travelled by the smallest species (figure 1b). The time for a small particle to move from the top to the bottom of a flowing layer is of order

$$t_{\mathrm{s}}\propto \frac{\delta }{w_{p,i}}.$$ 2.10

From equation (2.3), a small particle in a flowing layer composed entirely of large particles has the characteristic segregation velocity

$$w_{p,i}\propto ⟨\dot{\gamma }⟩S_{\mathrm{l},\mathrm{s}},$$ 2.11

where $⟨\dot{\gamma }⟩$ is the depth-averaged shear rate, and the subscripts l and s refer to the largest and smallest particles, respectively. We define the characteristic length scale, L, as the idealized streamwise displacement of the small particle as it moves from the free surface to the bottom of a flowing layer composed only of large particles:

$$L\propto u_{\text{surf}}{t}_{\mathrm{s}}\propto ⟨\dot{\gamma }⟩\delta \frac{\delta }{⟨\dot{\gamma }⟩S_{\mathrm{l},\mathrm{s}}}\propto \frac{{\delta }^2}{S_{\mathrm{l},\mathrm{s}}},$$ 2.12

where we assume that, regardless of its depth, the particle continues in the streamwise direction at the surface velocity, u_surf, which is the maximum velocity in the flowing layer. As L is based on segregation for the smallest and largest species, it represents the shortest segregation length scale in a multidisperse or polydisperse system and is independent of particle species concentration. A similar expression could be achieved using similar logic for a large particle to move from the bottom to the top of a flowing layer ignoring asymmetric segregation effects. An alternative definition characterizes the average velocity in the flowing layers as u_surf/2, assuming a velocity profile linear in depth. However, the difference is only a factor of 2, which we prefer to avoid for simplicity later on. The influence of L on the model predicted segregation is discussed in §5.

Substituting expression (2.12) into equation (2.9) yields

$$\frac{\mathrm{\partial }{c}_i}{\mathrm{\partial }\tilde{t}}+\tilde{u}\frac{\mathrm{\partial }{c}_i}{\mathrm{\partial }\tilde{x}}+\sum _{j=1}^n{\kappa }_{i,j}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{z}}(\tilde{\dot{\gamma }}{c}_j{c}_i)={\left(\frac{S_{\mathrm{l},\mathrm{s}}}{\delta }\right)}^2\frac{1}{Pe}\frac{{\mathrm{\partial }}^2{c}_i}{\mathrm{\partial }{\tilde{x}}^2}+\frac{1}{Pe}\frac{{\mathrm{\partial }}^2{c}_i}{\mathrm{\partial }{\tilde{z}}^2},$$ 2.13

where κ_i,j=S_i,j/S_l,s and Pe=2qS_l,s/(δD). As S(α_i,α_j)=−S(α_j,α_i), it follows that κ(α_i,α_j)=−κ(α_j,α_i), κ_l,s=1 and κ_s,l=−1 by definition. Pe is the ratio of the diffusion time scale, δ²/D, to the segregation time scale L/(2q/δ)=δL/2q=δ³/2qS_l,s, and κ(α_i,α_j) is the ratio of the segregation length for species i and species j to the segregation length for the smallest and largest species pair. κ_i,j and Pe depend only on particle and flow properties, which are either determined directly by the control parameters of the problem (α_i and q) or can be directly measured from experiments or simulations (δ, S and D).

S(α_i,α_j) is $O(\overline{\alpha }/10)$ [28], where $\overline{\alpha }$ is the average particle diameter, and δ is generally $O(10\overline{\alpha })$ [13,30]. Consequently, the ratio, (S_l,s/δ)², in the streamwise diffusion term is O(10⁻⁴). Thus, the first term on the right-hand side of equation (2.13) can be safely ignored. The non-dimensional multidisperse scalar transport equations can then be written as

$$\frac{\mathrm{\partial }{c}_i}{\mathrm{\partial }\tilde{t}}+\tilde{u}\frac{\mathrm{\partial }{c}_i}{\mathrm{\partial }\tilde{x}}+\sum _{j=1}^n{\kappa }_{i,j}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{z}}(\tilde{\dot{\gamma }}{c}_j{c}_i)=\frac{1}{Pe}\frac{{\mathrm{\partial }}^2{c}_i}{\mathrm{\partial }{\tilde{z}}^2}.$$ 2.14

The boundary conditions for this equation are based on balancing the segregation flux with the diffusive flux at the top and bottom of the flowing layer ($\tilde{z}=1$ and 0) [9]. Thus, at these two boundaries, the boundary conditions can be expressed as

$$\sum _{j=1}^n{\kappa }_{i,j}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{z}}(\tilde{\dot{\gamma }}{c}_j{c}_i)=\frac{1}{Pe}\frac{{\mathrm{\partial }}^2{c}_i}{\mathrm{\partial }{\tilde{z}}^2}.$$ 2.15

Particles enter at the upstream end of the domain and exit at the downstream end of the domain via advection due to the streamwise velocity. Because κ_i,j=−κ_j,i by definition, we use κ_i,j, where κ_i,j≥0, and replace κ_j,i with −κ_i,j in the rest of this paper.

Lastly, because $\sum _{j=1}^n{c}_j=1$, equations (2.14) for a tridisperse mixture (n=3) can be expressed as

$$\begin{array}{rl}& \tilde{u}\frac{\mathrm{\partial }{c}_{\mathrm{s}}}{\mathrm{\partial }\tilde{x}}-{\kappa }_{\mathrm{l},\mathrm{s}}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{z}}[\tilde{\dot{\gamma }}(1-c_{\mathrm{m}}-c_{\mathrm{s}})c_{\mathrm{s}}]-{\kappa }_{m,s}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{z}}(\tilde{\dot{\gamma }}{c}_{\mathrm{m}}{c}_{\mathrm{s}})=\frac{1}{Pe}\frac{{\mathrm{\partial }}^2{c}_{\mathrm{s}}}{\mathrm{\partial }{\tilde{z}}^2}\\ \text{and}& \tilde{u}\frac{\mathrm{\partial }{c}_{\mathrm{m}}}{\mathrm{\partial }\tilde{x}}+{\kappa }_{m,s}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{z}}(\tilde{\dot{\gamma }}{c}_{\mathrm{s}}{c}_{\mathrm{m}})-{\kappa }_{\mathrm{l},\mathrm{m}}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{z}}[\tilde{\dot{\gamma }}(1-c_{\mathrm{s}}-c_{\mathrm{m}})c_{\mathrm{m}}]=\frac{1}{Pe}\frac{{\mathrm{\partial }}^2{c}_{\mathrm{m}}}{\mathrm{\partial }{\tilde{z}}^2},\end{array}\}$$ 2.16

for steady flow, where s, m and l represent small, medium and large size particle species, respectively.

They can be solved using standard initial boundary-value routines for systems of parabolic equations. Here, the pdepe routine in Matlab is used. The inlet concentration profiles can be arbitrarily specified (e.g. perfectly mixed or perfectly segregated) or can be based on concentration profiles measured from experiments or DEM simulations. The streamwise velocity profile $\tilde{u}$ is assumed to be fully developed and independent of the local particle concentration or position along the length of the chute, though these assumptions are not required to apply the formalism. The boundary conditions (equations (2.15)) at the free surface and the bottom of the flowing layer can be specified in terms of the fluxes of small and medium particles as

$$\begin{array}{rl}& -{\kappa }_{\mathrm{l},\mathrm{s}}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{z}}[\tilde{\dot{\gamma }}(1-c_{\mathrm{m}}-c_{\mathrm{s}})c_{\mathrm{s}}]-{\kappa }_{\mathrm{m},\mathrm{s}}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{z}}(\tilde{\dot{\gamma }}{c}_{\mathrm{m}}{c}_{\mathrm{s}})=\frac{1}{Pe}\hspace{0.17em}\frac{{\mathrm{\partial }}^2{c}_{\mathrm{s}}}{\mathrm{\partial }{\tilde{z}}^2}\\ \text{and}& {\kappa }_{\mathrm{m},\mathrm{s}}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{z}}(\tilde{\dot{\gamma }}{c}_{\mathrm{s}}{c}_{\mathrm{m}})-{\kappa }_{\mathrm{l},\mathrm{m}}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{z}}[\tilde{\dot{\gamma }}(1-c_{\mathrm{s}}-c_{\mathrm{m}})c_{\mathrm{m}}]=\frac{1}{Pe}\hspace{0.17em}\frac{{\mathrm{\partial }}^2{c}_{\mathrm{m}}}{\mathrm{\partial }{\tilde{z}}^2}.\end{array}\}$$ 2.17

Numerically, the inlet values for concentration are integrated forward from $\tilde{x}=0$ to the furthest downstream extent of the flow to determine the concentrations of small and medium particle species c_s and c_m, respectively. The large particle concentration is c_l=1−c_s−c_m.

3. Discrete element method simulations and flow kinematics

To validate the predictions of the model and to acquire necessary kinematic information for the model, DEM simulations of a tridisperse mixture of spherical glass particles in a developing chute flow are performed based on the flow geometry shown in figure 2. Particles are dispensed from a vertical channel at volumetric flow rate Q. At a given chute angle, θ, and below a critical flow rate, q_c, the flowing layer thickness, δ, is essentially constant [30]. For the DEM simulation example presented below, an equal volume by component particle mixture with mean particle diameters, α_i=2, 3.2 and 5 mm (corresponding to size ratios of R=1:1.6:2.5 with respect to the smallest particle) and density ρ=2500 kg m⁻³ flows down a 2.5 cm wide chute inclined at an angle of θ=26.5° with a simulated flowing length l=0.7 m. To reduce particle ordering, the diameter of each species is distributed uniformly between 0.95 α_i and 1.05 α_i for all DEM simulations. The parameter κ_ij is based on the mean diameters. The bottom wall boundary condition is set such that particles that initially contact the flat bottom wall are immobilized to increase surface roughness and friction. Thereafter, all particles flow over these immobilized particles. The smooth, flat vertical side wall uses the same parameters as the particle–particle interaction. The DEM simulation parameters used here have been validated by numerous experiments using spherical glass particles; further details of the DEM methodology can be found in related publications [26,27,29,40–43] and appendix A.

Figure 2.

(a) Sketch of quasi-two-dimensional segregating chute flow with transverse gap thickness T and chute angle θ. Granular material is fed into the chute at a volumetric feed rate Q. (b) Sketch of the computational domain in a rotated coordinate system.

To implement the segregation model for tridisperse mixtures, the flowing layer depth, velocity profile and diffusion coefficient are needed. Here, these characteristics of the flow are measured from the DEM simulations. However, DEM simulations are not necessary to determine the parameters used in the model. For instance, the diffusion coefficient can come from correlations based on shear rate and particle size [31,44]; the velocity profile can be based on theory, such as the Bagnold profile [45,46], and so on. In all cases, the model results are calculated using parameters derived directly from the flow. Parameter values used in the model are either measured from DEM simulations, use values from previous studies based on bidisperse mixtures of particles or are based on standard velocity profiles typical of chute flow. In other words, if S, D and the velocity profile are known, whether from simulations, theory or experiment, the model can be used to predict the segregation.

From previous studies [47–49], the streamwise velocity in the flowing layer can be approximated as

$$u(z)=Uf(z),$$ 3.1

where f(z) characterizes the depth dependence with f(δ)=1 so that u(δ)=U, and U is determined by q and f(z) as

$$U=\frac{q}{{\int }_0^{\delta }f(z)\hspace{0.17em}\mathrm{d}z}.$$ 3.2

The segregation model used here allows any functional form f(z) to characterize the velocity profile, so that later in this paper we can consider how the segregation pattern depends on the form of the velocity profile. To obtain the velocity field to compare the model results to DEM simulations, a bin-averaging method was used to characterize streamwise velocity profiles at different streamwise locations as shown in figure 3a. A ‘relaxed Bagnold’ expression f(z)=1−(1−z/δ)^β [49] provides a reasonable approximation to the velocity profile. This expression becomes the classic Bagnold velocity profile when β=1.5. Combining equations (2.8), (3.1) and (3.2), the dimensionless expression for the mean velocity field is

$$\tilde{u}(\tilde{z})=\frac{u(z/\delta )}{2q/\delta }=\frac{\beta +1}{2\beta }[1-{(1-\tilde{z})}^{\beta }].$$ 3.3

Fitting data from the DEM simulation to this equation gives β=1.37, which is similar to other numerical simulations [47,48]. Hence, this value is used for the velocity profile in the continuum model for comparison to DEM simulations, in §4a. The streamwise velocity profiles and free surface velocities are nearly constant along the entire length of the flow domain, as shown in figure 3a,b, supporting the assumption that the velocity field is fully developed. The scaled normal velocity in figure 3c is approximately 100 times smaller than the characteristic streamwise free surface velocity. Within the flow domain, advection dominates in the streamwise direction, so we assume that diffusion is important only in the normal direction. To obtain the diffusion coefficient, the mean squared displacement normal to the free surface for the entire flow domain, 〈ΔZ(Δt)²〉, was measured over nine time intervals, each with a duration of Δt=0.3 s. The details concerning how the diffusion coefficient is obtained from 〈ΔZ(Δt)²〉 are included in appendix B. Assuming diffusion to be homogeneous and isotropic within the entire flow domain as in earlier work [26–29], the diffusion coefficient in the normal direction is approximated as D=〈ΔZ(Δt)²〉/2Δt. The resulting diffusion coefficient is D=5.1 mm² s⁻¹, consistent with the value D=6.22 mm² s⁻¹, calculated using a semi-empirical relation for the bidisperse quasi-two-dimensional bounded heap [31]. The difference between these two values of D is probably due to the different velocity fields for each flow, correspondingly different mean shear rates , and the fact that the value for D in this study is for tridisperse flow, while the previous study considered bidisperse flow. Even though there is 20% difference between D obtained from the two different methods, segregation is relatively insensitive to D, as will be shown later for different values of Pe.

Figure 3.

Kinematics of a particle mixture (α_i=2,3.2 and 5 mm) at q= 3400 mm² s⁻¹ from DEM simulations. (a) Averaged streamwise velocity profile versus depth at different streamwise locations. Black curve is a least squares fit of the data to a ‘relaxed’ Bagnold velocity profile; see equation (3.3). (b) Surface velocity u_s is nearly independent of streamwise position. (c) Scaled average normal direction velocity w/(2q/δ) is approximately zero through the depth of the flowing layer at different streamwise locations. (d) Normal direction mean squared displacement, 〈ΔZ(Δt)²〉, over the entire flow domain versus time for nine different time intervals (colours) (see text). Error bars in (a–c) indicate one standard deviation. (Online version in colour.)

4. Model predictions

(a). Comparison with discrete element method simulations

To validate the continuum transport model for tridisperse segregation, we compare steady-state concentration fields predicted by equation (2.16) (using the velocity profile obtained from fitting the DEM data to equation (3.3) and D=5.1×10⁻⁶ m² s⁻¹) with DEM simulation results at the same operating conditions (feed rate, flowing layer thickness and inlet condition). In DEM simulations, the particles segregate slightly before leaving the vertical feed channel, so $c_{\mathrm{s}}(0,\tilde{z})\ne c_{\mathrm{m}}(0,\tilde{z})\ne c_{\mathrm{l}}(0,\tilde{z})\ne 1/3$. To account for this, the inlet concentration profile at the upstream end of the computational domain for the model was matched to the corresponding profile in the DEM simulation. Figure 4 compares results from the model and DEM simulation for an equal volume concentration mixture of 2, 3.2 and 5 mm diameter particles (for which we use the shorthand for the ratios R=1:1.6:2.5) at q=3400 mm² s⁻¹ and θ=26.5°, which results in δ=22 mm in the DEM simulation. In the model, the values of the four dimensionless parameters κ_l,s, κ_l,m, κ_m,s and Pe were calculated based on the imposed feed rate (q), δ and D from DEM simulations, and the semi-empirical expression for the segregation length scale (S) for bidisperse spherical glass particles obtained from DEM simulations of bounded heap flow [31].

Figure 4.

Comparison of concentration from DEM simulation and continuum model for tridisperse chute flow. (a) Particle concentration from DEM simulation. (b) Particle concentration from model. (c) Particle concentration profiles from simulation (markers) and model (curves) at different streamwise positions, x/L, from 0 to 0.7, where yellow, red and blue(light grey, dark grey and black in greyscale) represent the concentration of small, medium and large particles, respectively. S_l,s=0.48 mm, S_l,m=0.37 mm, S_m,s= 0.24 mm, D=5.1 mm² s⁻¹, δ=22 mm, q= 3400 mm² s⁻¹, θ=26.5°, κ_l,s=1.0, κ_l,m=0.78, κ_m,s=0.52, Pe= 29.

The volume concentration contours of the three different species for both the DEM simulation and the model are shown in figure 4a,b. Each image represents the flow domain extending from the free surface at the top of the image to the bottom of the chute and from the domain inlet on the left to 0.7L on the right, where the segregation is nearly fully developed. The DEM simulation results exhibit a slight decrease in the flowing layer thickness with the downstream position, reflected as a step change in the upper surface of figure 4a, which is not included in the model. Blue, red and yellow represent pure concentrations of large, medium and small particles, respectively. Solid curves in figure 4b indicate the c_i=0.6 contour for each species. Where species mix, the corresponding colour is also mixed based on the concentrations of each species. Thus, an equal mixture of large (blue) and medium (red) particle species is represented by purple; an equal mixture of medium (red) and small (yellow) particle species is represented by orange; and an equal mixture of all three species is represented by brown. Figure 4a demonstrates tridisperse segregation similar to the qualitative prediction by Gray & Ancey [4] using their multicomponent particle size segregation model (figs 15 and 16 [4]). It is evident that segregation occurs in the direction normal to the free surface of the flowing layer. Large particles segregate towards the free surface and form a layer of segregated large particles at the top of the flowing layer (blue region). Small particles percolate downwards to the bottom of the flowing layer and form a layer of segregated small particles just above the bottom surface of the chute (yellow region). Medium particles segregate upwards from small particle-rich regions and percolate downwards from large particle-rich regions to form a middle layer of segregated medium particles (red region). Because the flux of any species i, ${\int }_0^{\delta }u(x,z)c_i\hspace{0.17em}\mathrm{d}z$, is conserved at any streamwise position x, the upper layer of large particles is thinner than the lower layer of small particles due to the higher velocity near the surface. Figure 4b demonstrates how the model accurately predicts the tridisperse segregation pattern, similar to the general pattern from the approach used by Gray & Ancey [4], but with physical parameters that are easily connected to actual flow conditions. The reason behind the close match between DEM results and the model is twofold. First, as the tridisperse system starts to segregate, two bidisperse systems form locally: a large and medium bidisperse flow in the upper portion of the flow, and a medium and small bidisperse flow in the bottom portion of the flow. Second, it has been shown that segregation is not strongly affected by the shear rate dependence of the diffusion coefficient [27]. Thus, a constant diffusion coefficient is sufficient to generate quantitative agreement between model and DEM simulations.

Compared with previous work [4], the physical parameters (S_ij and D) needed for the model are reduced. In Gray & Ancey [4], the segregation parameters equivalent to those used in our approach (S_νμ in their work) were not specified. In fact, the number of segregation parameters needed, N_n, grows quadratically for a mixture of n components ($N_n=\frac{1}{2}n(n-1)$). While it is possible to apply the model proposed by Gray & Thornton [8] and Gray & Ancey [4] to a bidisperse case (N_n=1) by manually fitting model parameters to experimental results as done by Wiederseiner et al. [13], this only achieves qualitative comparison with the bidisperse case. This is further complicated by the increased number of parameters for three or more components, which makes the model proposed by Gray & Ancey [4] even more challenging to apply.

The segregation predicted using the model matches DEM simulation results not only at the end of the simulated region, where the three components of the mixture are nearly fully segregated though there is ongoing remixing between species because of collisional diffusion, but also at different streamwise positions where segregation is still developing. This agreement is remarkable considering the simplifying assumptions incorporated in the model. First, the model uses a single velocity profile that is assumed constant at all streamwise positions, independent of the concentration. Second, the model uses a single averaged diffusion coefficient even though the diffusion coefficient depends on local particle size and shear rate [31,44]. Finally, the model uses a segregation coefficient derived from an entirely different flow geometry (bounded heap flow) for bidisperse (not tridisperse) segregation. Nevertheless, concentration profiles from the model still match well with the DEM simulation at different streamwise positions, as shown in figure 4c, demonstrating not only the accuracy of the model but also its robust character. Note that the DEM data are plotted in figure 4c such that the free surface data are always at $\tilde{z}=1$ to account for the small decrease in surface height in the flow direction. As the dimensionless parameters Pe, κ_ij, and characteristic segregation length scale L depend on physical control parameters, we further validate the predictions of the model by comparing with DEM simulations over a wider range of flow operating conditions in the electronic supplementary material.

(b). Influence of α_i, Pe and κ_i,j on segregation

Having validated the continuum model against DEM simulation, we now use the model to systematically investigate the effects of α_i and Pe on segregation for a well-mixed inlet. We first illustrate the effect of α_m on segregation for fixed α_s and α_l at different Pe, but a similar analysis can be performed for any of the species. We use a no-slip linear velocity profile because of its simplicity and ability to accurately approximate the velocity field for flow depths less than 20 particle diameters thick [46,47]. Figure 5 depicts an array of concentration contour maps, as in figure 4b, each corresponding to the flowing region for $0\le \tilde{x}\le 1$ and $0\le \tilde{z}\le 1$ over a wide range of Pe and α_m. Here, we set α_s=1 mm and α_l=3 mm to investigate the effects of α_m and Pe. The values of κ_i,j are calculated based on the corresponding set of particle diameters. At high Pe, segregation dominates, resulting in segregated layers, while at low Pe, diffusion dominates, resulting in mixed particles (brown in the concentration contour maps). In the high Pe regime, the borders between different species are relatively sharp for 1.4 mm≤α_m≤2.6 mm, indicating a distinct transition from one pure species to another. However, the smaller difference in particle size for small α_m at high Pe in the upper left of figure 5 results in reduced segregation of the small and medium particles (orange region). For Pe<100, the small size difference between medium and small particles (α_s=1 mm and α_m=1.4 mm) along with strong diffusion results in no fully segregated layers of small and medium particles at one characteristic length scale downstream. As Pe increases, even a small size difference produces segregated layers of small and medium particles. The model predictions for the tridisperse case collapse to the bidisperse case when α_m is close to either its lower bound, α_s (left column) or its upper bound, α_l (right column). When α_l≫α_m≈α_s (left column), large particles quickly separate from medium and small particles, which remain in a mostly mixed state (orange region) unless the diffusion is minimal so that segregation dominates (high Pe). When α_s≪α_m≈α_l (right column), small particles quickly separate from medium and large particles, which remain mostly mixed (purple region) unless segregation dominates (high Pe). An interesting observation is that, in the high Pe region, the segregation between small and medium particles is stronger when α_m approaches α_s (upper left corner) than the segregation between medium and large particles when the size of the medium particles approaches the size of the large particles (upper right corner). There are two reasons for this phenomenon. First, the segregation length scale, S, in the empirical expression in equation (2.4) depends logarithmically on the size ratio, and the size ratio is larger when the medium particle size approaches the small particle size than when the medium particle size approaches the large particle size. Second, the flow is slower in the lower portion of the flowing layer giving small and medium particles more time to segregate. Note that Pe measured from the DEM results presented in previous sections are generally less than 100. However, this does not imply that Pe is restricted to be less than 100 for all operating conditions. Considerable insight can be gained from the high Pe region (Pe≈10³) in figure 5 where the impact of the size ratio R is clearer.

Figure 5.

Tridisperse particle concentration contours in the flowing layer from the continuum model for different α_m and Pe for a well-mixed inlet condition with a no-slip linear velocity profile ($u=u_{\text{surf}}\tilde{z}$), α_s=1 mm, and α_l=3 mm. Each small box shows the entire flowing layer domain $(0\le \tilde{x}\le 1$, $0\le \tilde{z}\le 1)$. Colours are as in figure 4.

In addition to the effects of α_i and Pe on segregation, the model can also reveal the effects of κ_i,j on segregation where κ_i,j is the relative segregation strength between particle species i and j. Because κ_l,s=1 by definition, the remaining parameters are κ_l,m and κ_m,s. While similar analysis can be done for systems with more than three particle species, tridisperse flow provides an intermediate step between bidisperse and polydisperse while allowing easy data visualization for aiding physical insight. Figure 6 shows an array of concentration contour maps for a well-mixed inlet condition over a range of κ_l,m and κ_m,s, again assuming a no-slip linear velocity profile. The magnitude of κ_l,m controls the segregation between large and medium species. As κ_l,m increases at fixed κ_m,s (rows), the segregation between the large and medium species increases. On the other hand, the magnitude of κ_m,s controls the segregation between medium and small species. As κ_m,s increases at fixed κ_l,m (columns), the segregation between medium and small species increases. Restrictions imposed by particle size (α_s<α_m<α_l) and the empirical expression for S (equation (2.4)) limit the physically accessible range of κ_l,m and κ_m,s for 1–3 mm glass spheres to the unshaded region in figure 6. This is because the size of the medium particle bounds S_m,s relative to S_m,l. To explain, when κ_m,s is near 1, α_m must be close to α_l so that S_m,s≈S_l,s. But when this is the case, κ_l,m is necessarily small. Likewise, when κ_l,m is near 1, α_m must be close to α_s, so κ_m,s is necessarily small. Consequently, only the two extreme cases located at the upper left and lower right corners in figure 6 are possible for κ_m,s=1 and κ_l,m=1, respectively. When κ_l,m=0 and κ_m,s=1 (upper left), the model collapses back to the bidisperse case because the large and medium species are the same size. The physical context corresponding to this scenario is medium and large particles so similar in size that they are inseparable, while small particles segregate from medium and large particles. A similar situation occurs when κ_l,m=1 and κ_m,s=0 (lower right). In this case, the medium and small species are the same size and do not segregate.

Figure 6.

Tridisperse particle concentration contours in the flowing layer for different κ_l,m and κ_m,s for the well-mixed inlet condition with a no-slip linear velocity profile and Pe=100. Each small box represents the entire flowing layer domain $(0\le \tilde{x}\le 1$, $0\le \tilde{z}\le 1)$. The unshaded boxes represents the physically accessible region for spherical glass particles with $\frac{1}{3}\le {\alpha }_i/{\alpha }_j\le 3$ and 1 mm≤α_i≤3 mm. Colours are as in figure 4.

As κ_i,j is defined as the ratio of S_i,j to S_l,s, which may depend not only on the size ratio, but also on other particle properties such as density ratio, shape difference, surface roughness, and so on, the empirical expression for the segregation length scale (equation (2.4)) may be different. Consequently, the value and functional dependence of S_i,j could be different and, along with it, the κ_i,j space. For example, based on equation (2.4), there is a small region where S_l,m is slightly larger than S_l,s because of the dependence of S on the actual particle size. This results in a small region where κ_l,m is slightly greater than 1 (not shown in figure 6). Relationships for S other than the logarithmic expression in equation (2.4) result in a different physically accessible space than that shown in figure 6.

5. Interplay of advection, segregation and diffusion

The dimensionless governing equation (2.13) indicates that the dimensionless parameters κ_i,j and Pe control the concentration field through the interplay of segregation and diffusion in segregating chute flow. However, advection also plays a role via the streamwise segregation length, L. This is readily apparent when one considers the time scales for segregation, t_s=δ/w_p, diffusion, t_d=δ²/D and advection, t_a=l/u_surf, where l is the streamwise length of the domain. Segregation dominates advection within the domain when t_s<t_a, which after some simple manipulation can be shown to be equivalent to L<l. By considering the analytical solution of equation (2.14) in the absence of advection and segregation, the dimensionless form of the diffusion time scale, ${\tilde{t}}_{\mathrm{d}}$, can be estimated as ${\tilde{t}}_{\mathrm{d}}=Pe/{\pi }^2\approx Pe/10$. The competition between different time scales is evident in figure 5. If the flow domain is at least L long ($\tilde{x}\ge 1$), then particles remained mixed when diffusion dominates (small Pe) and segregate when diffusion is weak (large Pe). If l≪L ($\tilde{x}\ll 1$ in figure 5), then the strong advection preserves the inlet condition to a significant extent.

To better understand the competition between different time scales, it is helpful to consider an ‘inverted’ segregated inlet condition, where small particles are initially at the top, medium particles at the centre and large particles at the bottom of the flowing layer, such that

$$\begin{array}{rl}{c}_{\mathrm{s}}(0,\tilde{z})& =\{\begin{array}{ll}1& 0.817\le \tilde{z}\le 1,\\ 0& \tilde{z}<0.817,\end{array}\\ c_{\mathrm{m}}(0,\tilde{z})& =\{\begin{array}{ll}1& 0.577\le \tilde{z}\le 0.817,\\ 0& \tilde{z}<0.577\end{array}\\ \text{and}{c}_{\mathrm{l}}(0,\tilde{z})& =\{\begin{array}{ll}1& \tilde{z}\le 0.577,\\ 0& 0.577<\tilde{z}\le 1.\end{array}\end{array}\}$$ 5.1

With this inlet concentration boundary condition and a no-slip linear velocity profile, the fluxes of small, medium and large particles entering the chute at $\tilde{x}=0$ are approximately equal. Figure 7 shows an array of Pe−α_m contour maps (similar to figure 5) for an ‘inverted’ inlet condition with a no-slip linear velocity profile. Consistent with physical insight gained from figure 5 for a mixed inlet condition, a strongly segregated state occurs at high Pe, and a well-mixed state occurs at low Pe. As Pe is increased, the situation is more complicated than for the mixed inlet. At high Pe and low α_m (upper left), segregation between medium and small particles is smaller than between medium and large particles. Consequently, medium particles stay below the small particles but large particles quickly rise through the layers of medium and small particles to the top of the flowing layer. As α_m approaches α_l (upper right), segregation between medium and small particles is stronger than between medium and large particles. In this case, medium particles stay above the large particles but the small particles quickly percolate through the medium and large particle layers to the bottom of the flowing layer. Thus, varying α_m changes the trajectory of the medium particle species to downwards at small α_m and to upwards at large α_m. At moderate values of α_m, medium particles that are initially close to large particles, first sink together with the small particles to the bottom of the flowing layer and then move upwards away from the small particles. By contrast, medium particles that are initially close to small particles, first rise together with large particles and then later percolate downwards to form a layer rich in medium particles below the large-particle layer and above the small-particle layer. Of course, if advection is significantly faster than segregation, corresponding to small $\tilde{x}$ (or l≪L), then the inlet condition is preserved.

Figure 7.

Tridisperse particle concentration contours for the ‘inverted’ inlet condition for different α_m and Pe with a no-slip linear velocity profile, α_s= 1 mm, and α_l=3 mm. Each small box shows the entire computational domain $(0\le \tilde{x}\le 1$ and $0\le \tilde{z}\le 1)$. Colours are as in figure 4.

6. Influence of the velocity profile on segregation

Previous studies have demonstrated that segregation patterns in granular flow depend on the depthwise variation of the streamwise velocity [27]. Using an information entropy approach, Savage & Lun [6] predicted the dependence of the segregation velocity on the local shear rate $\dot{\gamma }$, leading, in part, to the percolation model in equation (2.3) where the local percolation velocity depends on the velocity profile through the local shear rate. For a no-slip linear velocity profile, which has been assumed to this point, the local shear rate is constant and the segregation velocity only depends on the local concentration. For a nonlinear velocity profile (e.g. Bagnold or exponential), the local shear rate is a function of depth. As a result, the segregation velocity varies with depth even if the local particle concentrations are the same. This can significantly change particle distributions. In this section, we consider several different velocity profiles to demonstrate the dependence of the concentration field on the velocity profile.

To explore the effects of the velocity profile on segregation, we consider no-slip linear, Bagnold (β=1.5 in equation (3.3)) and exponential velocity profiles with the same flow rate q for a mixed inlet condition with equal volumes of small, medium and large particles. Combining equations (2.8), (3.1) and (3.2) with f(z)=e^kz/δ, a dimensionless expression for the exponential velocity profile is given by

$$\tilde{u}(\tilde{z})=\frac{k}{2({\mathrm{e}}^k-1)}\hspace{0.17em}{\mathrm{e}}^{k\tilde{z}},$$ 6.1

where k determines the slip velocity at the wall. When k=3, the slip at the wall is 5% of the surface velocity.

The dimensionless velocity profiles $\tilde{u}(\tilde{z})$ and the corresponding dimensionless local shear rates $\dot{\gamma }$ are shown in figure 8a–c for the three velocity profiles. The corresponding segregation patterns for identical Pe and κ_i,j are shown in figure 8d–f. Near the free surface of the flow, the exponential velocity profile produces the strongest and fastest segregation of medium and large particles compared to the other profiles, while the Bagnold profile results in the weakest and slowest segregation. This is because the exponential profile has the largest shear rate in the upper part of the flow, while the Bagnold profile has the smallest shear rate. Near the bottom of the chute, the Bagnold and no-slip linear velocity profiles have faster initial segregation than the exponential velocity profile because of their relatively large shear rates near the bottom of the flowing layer. Moving upwards in the flowing layer, the shear rate for the exponential velocity profile becomes large compared to the no-slip linear or Bagnold profiles, leading to stronger overall segregation and a thicker layer of segregated small particles further downstream.

Figure 8.

Comparision of tridisperse particle concentration fields for no-slip linear (a,d), Bagnold (b,e) and exponential (c,f) velocity profiles (solid curves). For (a–c), solid curves represent the dimensionless streamwise velocity (bottom axis) and dashed curves represent the dimensionless shear rate magnitude (top axis). For (d–f), solid curves indicate the c_i=0.75 contour for each species. Pe=100 and κ_l,s=1.0,κ_m,s=0.63, κ_l,m=0.74, which correspond to α_s=1 mm,α_m= 2 mm and α_l=3 mm, respectively. Colours are as in figure 4.

The roughness of the bottom wall of the chute also affects the velocity profile. With a rough bottom wall, the basal velocity ($\tilde{u}(0)$) is approximately zero [30,48,50]. However, with a smooth bottom wall, slip is possible at the bottom wall [47,51,52], which can be expressed as a percentage of the free surface velocity. Recent work suggests that slip or non-slip boundary conditions can be predicted by their proposed base roughness parameter which is a function of basal particle size ratio and basal particle construction [53,54]. Figure 9 demonstrates how slip affects particle segregation in the flowing layer for an exponential velocity profile with constant streamwise flux. As noted above, k=3 provides a slip of 5%; when k=2.3, the slip is 10%; and when k=1.61, the slip is 20%. The velocity profiles for these three values of k are plotted in figure 9a–c along with the corresponding local shear rate profiles which vary substantially with the degree of slip, particularly the magnitude of the local shear rate near the top of the flowing layer. The corresponding solutions of the continuum model are shown in figure 9d–f. Because the 5% slip profile has the largest average $\dot{\gamma }$ throughout the flowing layer, the overall degree of segregation is the strongest as shown by the rapid flattening of the concentration contours. The 20% slip case has the smallest average $\dot{\gamma }$ throughout the flowing layer, and, consequently, the overall degree of segregation is so weak that the concentration profile is still developing at the end of one characteristic length. The segregation of the small particles is particularly slow in this case because of the low shear for $0\le \tilde{z}\le 0.5$. From a practical standpoint, segregation in a chute can be reduced by a large slip velocity.

Figure 9.

Effect of bottom boundary slip condition on tridisperse particle concentration in chute flow for exponential velocity profiles with 5%, 10% and 20% slip. For (a–c), solid curves represent the dimensionless streamwise velocity (bottom axis) and dashed curves represent the dimensionless shear rate magnitude (top axis). For (d–f), solid curves indicate the c_i=0.75 contour for each species. Pe= 100,κ_l,s=1.0,κ_m,s=0.63,κ_l,m=0.74, which correspond to α_s=1 mm,α_m= 2 mm and α_l=3 mm, respectively. Colours are as in figure 4.

7. Conclusion

Tridisperse flow provides an important connection between bidisperse and polydisperse flow because it tests the pairwise interaction hypothesis of the percolation model and provides more straightforward physical insight than is possible with polydisperse systems. This paper uses a continuum transport model to explore segregation of tridisperse granular material in a gravity-driven flow. The approach accurately models tridisperse chute flow as indicated by the close agreement between its predictions and results from DEM simulation over a wide range of flow conditions including different incline angles, particle size distributions, flow rates and flowing layer thicknesses. The model is based on an understanding of the kinematics of granular flow and has no arbitrarily adjustable fitting parameters, though the dependence of the segregation length scale on the size ratio must be known for bidisperse mixtures. Particle segregation depends on Pe=2qS_l,s/(δD), which is the ratio of the segregation rate to the diffusion rate, κ_i,j=S_i,j/S_l,s, which is the relative segregation strength between particle species i and j, and L=δ²/S_l,s, which is a characteristic length scale based on the segregation between the smallest and largest particle.

The advantage of this model is that it allows exploration of the effects of relative particle size, diffusion and the velocity profile on the segregation characteristics. Unlike cases where particles flow in a thin layer down a surface of other particles at a natural dynamic angle of repose, chute flow is affected by the roughness of the surface on which the particles flow and the angle of the surface. These effects can be included in the model via the velocity profile, provided it can be determined from experiments, simulations or theory. An important aspect of chute flow is that the flowing layer thickness or velocity profile may change with streamwise position. While this complicates the analysis, it is possible to model these effects provided the streamwise variation in the velocity profile and flowing layer thickness are known, again from experiments, simulations or theory. Also, the velocity profile appears to couple only weakly to the degree of segregation for the cases we have studied, which is also the case for the other free surface granular flows we have studied [27,29].

While we have demonstrated the effectiveness of the tridisperse modelling framework for spherical particles segregating by size, it is likely that it can be readily adapted to other types of particle dispersity (e.g. particle density or shape), provided that relations for the dependence of the segregation velocity and the diffusion on the kinematics and concentration can be determined either through experiment or DEM simulation. The framework also captures how different velocity profiles and the degree of slip affect the segregation pattern in a chute flow. However, challenges remain. For example, we have not considered the coupling between the segregation and rheology or velocity feedback from the segregation, which is likely to be important when size ratios are large or the degree of segregation is high. These mechanisms can potentially be incorporated into the formalism by deriving the velocity field from a momentum equation with a rheology model (for example, μ(I) rheology [50,55–58]) dependent on the local particle concentration. Also, recent work indicates that, for a small friction coefficient (≈0.1), the segregation mechanism can be different [59], which is a possibility not considered in this work. These extensions will further complement the current model and advance it towards becoming part of a general framework for modelling segregating particle flows.

Appendix A. discrete element method simulations

In DEM simulations, the translational and rotational momenta of each particle are tracked by integrating Newton’s Second Law. For simplicity, a linear-dashpot model for particle interaction forces is used, which, nonetheless, allows accurate simulation of ensembles of spherical glass particles for dense granular flow [26,27,29,40–43]. (For a more dilute system such as granular gas, a more advanced force model is needed to be used to account for the velocity-dependent restitution coefficient [60].) For two contacting particles i and j, the normal force is ${\mathit{F}}_{ij}^{\mathrm{n}}=-k_{\mathrm{n}}{\mathit{\delta }}_{\mathrm{n},ij}-{\eta }_{\mathrm{n},ij}{\mathit{v}}_{\mathrm{n},ij}$, and the tangential force is ${\mathit{F}}_{ij}^{\mathrm{t}}=-k_{\mathrm{t}}{\mathit{\delta }}_{\mathrm{t},ij}-{\eta }_{\mathrm{t},ij}{\mathit{v}}_{\mathrm{t},ij}$. Here, k_n and k_t are the normal and tangential spring stiffnesses, respectively, η_n,ij and η_t,ij are the normal and tangential damping coefficients, respectively which can be calculated using the effective mass of the two contacting particles, or a particle contacting an infinite mass wall, and the restitution coefficient [61]; δ_n,ij is the normal displacement between two particles, and δ_t,ij is the tangential displacement, which is measured by ${\mathit{\delta }}_{\mathrm{t},ij}={\int }_{t_0}^t{\mathit{v}}_{\mathrm{t},ij}\hspace{0.17em}\mathrm{d}t$, where t₀ is the initial contact time between two particles; and v_n,ij and v_t,ij are the relative velocities of particles in the normal and tangential direction, respectively. When the relation |F_t,ij|>μ|F_n,ij| is satisfied, the Coulomb friction model for sliding is used to calculate the tangential contact force as F_t,ij=−μ|F_n,ij|δ_t,ij/|δ_t,ij|, where μ=0.3 is the friction coefficient. The tangential stiffness is $k_{\mathrm{s}}=(\frac{2}{7})k_{\mathrm{n}}$, where k_n=1400 N m⁻¹, which ensures that the normal and tangential oscillation frequencies are equal in the zero damping limit [62]. The damping coefficients are ${\eta }_{\mathrm{n},ij}=2|\ln e|\sqrt{k_{\mathrm{n}}{m}_{\text{eff}}/({\pi }^2+{\ln }^{2}e)}$ and ${\eta }_{\mathrm{t},ij}=(\frac{2}{7}){\eta }_{\mathrm{n},ij}$, where m_eff=m_im_j/(m_i+m_j) is the effective mass of the two contacting particles or a particle contacting an infinite mass wall, and e=0.9 is the restitution coefficient [61,62]. The symplectic Euler integration algorithm [63–65] is used to update particle positions and velocities. The simulation time step is chosen to be it is smaller than the critical time step, which is one-tenth of the minimal natural oscillation period of the spring-mass system [41,66–68].

Appendix B. Diffusion coefficient

The spatio-temporal average of the diffusion coefficient, D, of the flow is calculated only in the normal direction. To do so, the time evolution of the non-affine trajectory component in the normal direction of every particle, $\mathrm{\Delta }Z(\mathrm{\Delta }t)=z(t_0+\mathrm{\Delta }t)-z(t_0)-{\int }_{t_0}^{t_0+\mathrm{\Delta }t}w(t)\hspace{0.17em}\mathrm{d}t$, is used to calculate the mean squared displacement in the normal direction, 〈ΔZ(Δt)²〉 [69,70]. Here, w(t) is the local mean normal velocity at time t, and 〈*〉 denotes the ensemble average. The diffusion coefficient is then calculated as the slope of the equation 〈ΔZ(Δt)²〉=2DΔt for the range Δt where the relationship is linear (here, 0 to 0.3 s) [32].

Data accessibility

All data are based on model and DEM simulation results.

Authors' contributions

Z.D. carried out the DEM simulations and theoretical calculations, and drafted the manuscript. P.B.U., J.M.O. and R.M.L. designed the study, guided the research and edited the manuscript.

Competing interests

The authors have no competing interests.

Funding

Z.D. was partially supported by NSF grant no. CBET-1511450.