3.1 Polydisperse segregation in steady-state chute flow

To demonstrate the application of the model (3.1) to polydisperse segregation, we consider chute flow in the limit of an infinitely long chute, in which the segregation of a polydisperse mixture is assumed to have reached steady state. Although the method can be applied to other dense gravity-driven flows, chute flow has the advantage of being a relatively simple flow in steady state, and it has applications to both industrial processes and geophysical debris flows (Hutter et al. Reference Hutter, Svendsen and Rickenmann1996). A DEM simulation for polydisperse segregation in chute flow is shown in figure 2. In the simulation, particle sizes are modelled using a truncated log-normal distribution by volume with mean particle diameter ${\it\alpha}=2$ mm and minimum and maximum particle diameters of 1 mm and 4 mm, respectively. In an idealized chute flow, particles flow continuously down an infinitely long incline with a constant particle depth ${\it\delta}$ and $z=0$ located at the free surface. In the DEM simulation, periodic boundary conditions were implemented in the streamwise ( $x$ ) direction to simulate a fully developed, steady-state condition and to minimize the computational domain. Frictional sidewalls constrained the simulation domain in the spanwise ( $y$ ) direction, with a gap thickness  $T=1.76$ cm and a particle–wall Coulomb sliding friction coefficient ${\it\mu}_{pw}=0.1$ . The bottom boundary consisted of a stationary, regular packing of small particles with diameters uniformly distributed between 1.0 and 1.5 mm, simulating a rough wall composed of glued spheres. These boundary conditions produce a steady-state velocity profile that is approximately linear through the depth of the flowing layer.

Previous DEM simulations (Silbert et al. Reference Silbert, Ertas, Grest, Halsey, Levine and Plimpton2001; Rognon et al. Reference Rognon, Roux, Naaim and Chevoir2007) and theory (Rognon et al. Reference Rognon, Roux, Naaim and Chevoir2007) have approximated the velocity profile for chute flows using a Bagnold-type rheology, giving rise to a profile of the form: $\boldsymbol{u}\sim [{\it\delta}^{3/2}-(-z)^{3/2}]\hat{\boldsymbol{x}}$ . More complex profiles have been suggested by others to account for discrepancies in experimentally observed profile shapes across a wide variety of flowing regimes (Ancey Reference Ancey2001). Adding to this complexity are the coupled effects of size segregation on the velocity profile in non-monodisperse mixtures, which at present has received little investigation except for recent work by Marks et al. (Reference Marks, Rognon and Einav2012). However, in most studies of monodisperse flows down rough inclines, the velocity is well approximated by a linear profile for sufficiently shallow flow depths ( ${<}$ 20 particle diameters thick) (Silbert et al. Reference Silbert, Ertas, Grest, Halsey, Levine and Plimpton2001; Midi Reference Midi2004). Consistent with the DEM results, we use a linear velocity profile in the model: $\boldsymbol{u}=\dot{{\it\gamma}}\left(z+{\it\delta}\right)\hat{\boldsymbol{x}}$ , where $\dot{{\it\gamma}}\equiv \partial u/\partial z$ is the (constant) shear rate and ${\it\delta}$ is the particle depth. We note that steady-state concentration distributions for a linear velocity profile (presented later in this section) were very similar to those for a Bagnold velocity profile.

For steady-state chute flow, $c(\boldsymbol{x},t,{\it\alpha})=c(z,{\it\alpha})$ , since the particle concentrations are independent of the streamwise position $x$ . Therefore, equation (3.1) becomes

for $-{\it\delta}\leqslant z\leqslant 0$ . The boundary conditions for (3.1) are the segregation flux equals the diffusion flux at the boundaries ( $z=-{\it\delta},0$ ). In addition, the overall particle size distribution, $f({\it\alpha})$ , is specified such that

where $\int _{{\it\alpha}_{min}}^{{\it\alpha}_{max}}f({\it\alpha})\,\text{d}{\it\alpha}=1$ . Note that the particle concentrations are weighted by the velocity, so that the total flux of particles through $-{\it\delta}\leqslant z\leqslant 0$ is specified. Integrating (3.3) with respect to $z$ and applying the no flux boundary condition yields

for $-{\it\delta}\leqslant z\leqslant 0$ . Note that the governing (3.5) specifies that, in steady state, the segregation flux equals the diffusion flux in the entire domain ( $-{\it\delta}\leqslant z\leqslant 0$ ), which is the same as the boundary condition at $z=-{\it\delta},0$ . This makes sense physically since, in steady state, particles do not on average move in the $z$ direction.

To non-dimensionalize (3.5), let $\tilde{z}=z/{\it\delta}$ , $\tilde{{\it\alpha}}={\it\alpha}/d_{0}$ , $\tilde{c}=d_{0}c$ , $\tilde{S}=S/d_{0}$ and $\tilde{f}=d_{0}f$ , where $d_{0}$ is a characteristic length associated with the particle size. Substituting these dimensionless quantities in (3.5) yields

for $-1\leqslant \tilde{z}\leqslant 0$ , where ${\it\Gamma}=d_{0}\dot{{\it\gamma}}{\it\delta}/D$ . Using the same dimensionless quantities as above and the linear velocity profile, equation (3.4) becomes

with $\int _{\tilde{{\it\alpha}}_{min}}^{\tilde{{\it\alpha}}_{max}}\tilde{f}(\tilde{{\it\alpha}})\,\text{d}\tilde{{\it\alpha}}=1$ .

Equation (3.6) is similar to the steady-state advection–segregation–diffusion equation presented in Schlick et al. (Reference Schlick, Fan, Umbanhowar, Ottino and Lueptow2015b ) in that the product ${\it\Gamma}\tilde{S}=\dot{{\it\gamma}}{\it\delta}S/D$ represents the relative importance of segregation compared to diffusion. For bidisperse segregation in a bounded heap or rotating tumbler (Fan et al. Reference Fan, Schlick, Umbanhowar, Ottino and Lueptow2014; Schlick et al. Reference Schlick, Fan, Umbanhowar, Ottino and Lueptow2015b ), this ratio can be expressed as the product of a dimensionless segregation parameter, ${\it\Lambda}$ , representing the ratio of segregation to advection, and the Péclet number, $Pe$ , representing the ratio of advection to diffusion, such that ${\it\Lambda}Pe=\dot{{\it\gamma}}{\it\delta}S/D$ . Of course, there is no streamwise advective time scale in fully developed chute flow. Thus, the product ${\it\Lambda}Pe$ is analogous to ${\it\Gamma}\tilde{S}$ in that it represents the relative importance of segregation compared to diffusion. However, the model for polydisperse segregation differs from previous results for bidisperse systems (Fan et al. Reference Fan, Schlick, Umbanhowar, Ottino and Lueptow2014; Schlick et al. Reference Schlick, Fan, Isner, Umbanhowar, Ottino and Lueptow2015a ,Reference Schlick, Fan, Umbanhowar, Ottino and Lueptow b ) in that the non-dimensional percolation length scale $\tilde{S}$ is not a constant. Hence, (3.6) represents a natural non-dimensionalization, where $\tilde{S}$ remains inside the integral, as it must. Equation (3.6) can be solved for $\tilde{c}(\tilde{z},\tilde{{\it\alpha}})$ using a finite difference method with the midpoint rule to approximate the integral, one-sided differencing in the $\tilde{z}$ direction, and the method of successive iterations (Ames Reference Ames1977) to account for the nonlinearity. For the remainder of this work, tildes are dropped and dimensional quantities are denoted with a subscript ‘ $d$ ’.

While (3.6) can be applied to any particle size distribution, here we primarily consider log-normal distributions, which are prevalent in many different scientific areas (Limpert et al. Reference Limpert, Stahel and Abbt2001). In granular systems, particle sizes have been shown to be log-normally distributed in iron–copper alloys fabricated using a coevaporation technique (Peng & Dai Reference Peng and Dai1994), aggregates of snow and ice from snow avalanches (Bartelt & McArdell Reference Bartelt and McArdell2009), urea from an industrial fluidized bed granulator (Bertin et al. Reference Bertin, Cotabarren, Bucula and Pina2011) and aerobic granules cultivated in a sequencing batch reactor (Su & Yu Reference Su and Yu2005).

The variable ${\it\alpha}_{d}$ is log-normally distributed with median $d_{0}$ and geometric standard deviation ${\it\sigma}_{g}$ when $\ln ({\it\alpha}_{d}/d_{0})$ is normally distributed with mean 0 and standard deviation ${\it\sigma}=\ln {\it\sigma}_{g}$ . Therefore,

Note that $d_{0}$ is notationally the same as $d_{50}$ , a measure of the median particle size as commonly reported in industrial applications.

3.2 Validation of polydisperse model by DEM simulation in chute flow

To validate the theoretical model, we compare to DEM simulations of polydisperse chute flow. Figure 2 shows results from a DEM simulation for the initial condition and at a later segregated state. Particle sizes follow an overall log-normal distribution by volume according to (3.8) with ${\it\sigma}=0.3$ and ${\it\alpha}_{d}$ non-dimensionalized by $d_{0}=2$ mm. Since in DEM simulations it is computationally expensive to have a large particle size ratio, particle sizes were restricted to $0.5<{\it\alpha}<2$ resulting in the particle size distribution (dashed curve) shown in figure 3 (or, since $d_{0}=2$ mm, particle diameters are between 1 and 4 mm). Note that this size range slightly extrapolates the applicability of (2.3), but this has negligible impact on the results because pairs of particles with a large size ratio ( $3<{\it\alpha}_{i}/{\it\alpha}_{j}<4$ ) interact infrequently due to their small numbers (95 % of the particles in the log-normal distribution are in the size range $0.6\leqslant {\it\alpha}\leqslant 1.8$ , well within the range of applicability of (2.3)). To verify that the impact of applying (2.3) outside the range where it is accurate is minor, the model was solved using both (2.3) and a more accurate approximation to the percolation length scale data in Schlick et al. (Reference Schlick, Fan, Isner, Umbanhowar, Ottino and Lueptow2015a ) for size ratios $1/4<{\it\alpha}_{i}/{\it\alpha}_{j}<4$ . Since only a negligible difference in concentration distributions was observed, we use the simpler (2.3) for convenience, even for size ratios slightly exceeding ${\it\alpha}_{i}/{\it\alpha}_{j}>4$ , which occurs for some of the examples presented in the next section. In this simulation, the particle depth was ${\it\delta}=6.2$ cm and the surface velocity was $u(x,0)=1.9~\text{m}~\text{s}^{-1}$ . Since the velocity field in the DEM simulation is approximately linear, the shear rate can be estimated as $\dot{{\it\gamma}}\approx u(x,0)/{\it\delta}=31~\text{s}^{-1}$ . The diffusion coefficient was obtained from the DEM simulation, which varied slightly with depth. For simplicity, we used the average value, $D=2.2\times 10^{-5}~\text{m}^{2}~\text{s}^{-1}$ , throughout the domain. (Alternatively, the diffusion coefficient could be estimated as a function of the local shear rate and mean particle size but this has been shown to not significantly improve the model prediction for bidisperse segregation in heap flows (Fan et al. Reference Fan, Schlick, Umbanhowar, Ottino and Lueptow2014).) These parameters result in ${\it\Gamma}=\dot{{\it\gamma}}d_{0}{\it\delta}/D=175$ .

Figure 3.

Comparison of steady-state particle size distributions for the model (curves) and the DEM simulation for the conditions in figure 2(b) (data points) in terms of spatially averaged concentrations in the bottom half $\left<c({\it\alpha})\right>_{b}$ (grey, blue online, ○) and top half $\left<c({\it\alpha})\right>_{t}$ (grey, red online, ▫) of the flow (see text). For the model, ${\it\Gamma}=175$ . The particles have an overall truncated log-normal distribution, $f({\it\alpha})$ , with ${\it\sigma}=0.3$ (3.8), as indicated by the dashed curve (model) and triangles (DEM, black, ▵). (Inset) Mean particle diameter as a function of depth for the model (black, dashed curve) and DEM simulation (light grey, orange, ▫). All DEM data are averaged between 30 and 60 s of simulated time, during which the system is in the fully developed segregated state shown in figure 2(b).

To compare results from the DEM simulation to the model, a subtle precaution is needed to ensure that the calculated $c(z,{\it\alpha})$ (from the model) and the measured $c(z,{\it\alpha})$ distributions are equivalent. Since the DEM simulation uses periodic boundary conditions in the streamwise direction instead of prescribed flux boundary conditions typical of an actual chute flow, the set of particles is fixed throughout the simulation. Consequently, the total distribution of particles by volume satisfies

Hence, for the case of chute flow with periodic boundary conditions in the streamwise direction, we numerically solve (3.6) subject to (3.9) instead of (3.7). For the purposes of comparison between the model and simulation, particle concentrations are spatially averaged in the top and bottom halves of the chute as

where $\left<c({\it\alpha})\right>_{t}$ is the average concentration of particles in the top half of the domain and $\left<c({\it\alpha})\right>_{b}$ is the average concentration in the bottom half of the domain.

The model and the DEM simulation are compared in figure 3 for $\left<c({\it\alpha})\right>_{t}$ and $\left<c({\it\alpha})\right>_{b}$ . The model agrees remarkably well with the DEM simulation, given the simplicity of the model and the simplifying assumptions made for the chute flow. Smaller particles tend toward the bottom portion of the flowing layer while larger particles tend toward the top, consistent with the image in figure 2(b). The inset of figure 3 shows the depth profile of the mean particle size as measured from the simulation and predicted by the model. The local mean particle size $\langle {\it\alpha}\rangle$ agrees quite well through the entire depth of the flowing layer, with only slight deviations near the top and bottom boundaries of the flowing layer, perhaps due to differences between the assumed linear velocity profile and the actual slightly nonlinear velocity profile, the estimation of $S$ or the assumption of a constant value for $D$ . Nevertheless, the match between the model and DEM simulations is quite good for most of the flowing layer thickness, demonstrating the effectiveness of the approach.

3.3 Effects of ${\it\Gamma}$ and $f({\it\alpha})$ in chute flow

With the validation of the theoretical model by DEM simulation, we now explore the relative effects of segregation and diffusion on the steady-state particle distribution by varying the dimensionless parameter, ${\it\Gamma}$ . Figure 4 shows $c(z,{\it\alpha})$ as a function of ${\it\alpha}$ at different depths in the domain for several values of ${\it\Gamma}$ . Particle sizes are distributed log normally according to (3.8) with ${\it\sigma}=0.4$ . At ${\it\Gamma}=3$ , the distributions at the top, middle and bottom of the domain are nearly identical to the overall particle size distribution $f({\it\alpha})$ (i.e. $c(z,{\it\alpha})\approx f({\it\alpha})$ ), indicating that little segregation occurs. As ${\it\Gamma}$ becomes larger (less collisional diffusion), stronger segregation occurs. Near the bottom of the domain ( $z=-0.9$ ) there are mostly smaller particles (smaller ${\it\alpha}$ ). Moving higher in the domain, there are fewer smaller particles, so the peak of the distribution shifts to the right. Moreover, toward the bottom of the flowing layer, the distribution of particle sizes becomes narrower, as evident in figure 4(d).

Figure 4.

Steady-state probability density functions $c(z,{\it\alpha})$ for various ${\it\Gamma}$ at depths $z=-0.9$ (dark grey, blue online), $z=-0.5$ (grey, red online), and $z=-0.1$ (light grey, green online), showing increased segregation as ${\it\Gamma}$ increases: (a)  ${\it\Gamma}=3$ , (b)  ${\it\Gamma}=10$ , (c)  ${\it\Gamma}=30$ , (d)  ${\it\Gamma}=100$ . Particle sizes are distributed log normally according to (3.8) with the overall size distribution $f({\it\alpha})$ shown as a dashed curve.

Figure 5.

Steady-state probability density functions $c(z,{\it\alpha})$ with an overall log-normal size distribution $f({\it\alpha})$ (dashed curve), for various ${\it\sigma}$ (see (3.8)) and ${\it\Gamma}=100$ : (a)  ${\it\sigma}=0.2$ , (b)  ${\it\sigma}=0.8$ . Here $z=-0.9$ (dark grey, blue online), $z=-0.5$ (grey, red online), and $z=-0.1$ (light grey, green online).

The effect of the variance ${\it\sigma}^{2}$ of the overall (log-normal) particle size distribution for ${\it\Gamma}=100$ is shown in figure 5. For small ${\it\sigma}$ , as in figure 5(a), particles are similar in size, so little segregation occurs. As ${\it\sigma}$ increases, segregation increases, and $c(z,{\it\alpha})$ varies substantially for different values of $z$ . For example, in figure 5(b), $c(-0.9,{\it\alpha})$ has much less spread than $c(-0.1,{\it\alpha})$ due to the long tail of the log-normal distribution. Thus, for large ${\it\sigma}$ , there is a wider range of large particle sizes ( $1<{\it\alpha}<3$ ) near the free surface.

While log-normal particle size distributions are prevalent in many natural and industrial settings, other particle size distributions provide further insight into polydisperse segregation and demonstrate the broad applicability of our approach. In figure 6, six different distributions (summarized in table 1) are compared for ${\it\Gamma}=100$ at different depths in the domain. All six distributions have similar qualitative behaviour: small particles percolate toward the bottom of the flowing layer while large particles segregate toward the top.

Figure 6.

Effect of different overall particle size distributions $f({\it\alpha})$ on steady-state probability density functions $c(z,{\it\alpha})$ for ${\it\Gamma}=100$ at $z=-0.9$ (dark grey, blue online), $z=-0.5$ (grey, red online), and $z=-0.1$ (light grey, green online): (a) log normal, (b) normal, (c) uniform, (d) exponential, (e) triangle, (f) gamma. The dashed curves represent $f({\it\alpha})$ . The six different overall particle size distributions are summarized in table 1. Note the different scales for each plot.

Table 1.

Particle size distributions considered in figure 6.

The log-normal and the normal particle size distributions in figure 6 generate segregated distributions $c(z,{\it\alpha})$ that are approximately log-normally and normally distributed, respectively, at each depth $z$ . For the uniform distribution and the exponential distribution, the segregated distributions are quite different at each depth. Near the bottom of the flow, $c(z,{\it\alpha})$ decays rapidly as ${\it\alpha}$ increases, and $c(z,{\it\alpha})$ has a maximum at ${\it\alpha}={\it\alpha}_{min}$ for both distributions. Conversely, near the middle or top of the flow, $c(z,{\it\alpha})$ is similar to a normal distribution. The triangular distribution results in segregated distributions similar to the normal distribution as well, except that $c(z,{\it\alpha})$ is skewed on different sides of the distribution at different depths, and $c(z,{\it\alpha})=0$ at ${\it\alpha}=1$ and ${\it\alpha}=5$ , since $f(1)=f(5)=0$ . The gamma distribution has a similar shape to the log-normal distribution, and thus $c(z,{\it\alpha})$ is similar for both distributions. Some of the overall size distributions in figure 6 exceed the range of applicability of the relation for the percolation length scale (2.3) but, as noted previously, the bulk of the particles are within the range of applicability, particularly at steady state.

Other particle size distributions can also be considered. Those depicted in figure 6 are only meant to give a flavour of the different particle segregation patterns possible and to demonstrate the potential that the modelling approach holds for describing other polydisperse mixtures.