Prediction of segregation in funnel and mass flow discharge

In this paper we present a model to predict the onset and evolution of segregation during the discharge of binary mixtures of granular materials. The model accounts for the multi-phase and multi-component nature of the granular mixtures, to simulate the main flow regimes occurring in the discharge of silos (funnel and mass flow) and how they affect segregation. The new comprehensive model for segregation follows a continuum Eulerian approach and results from the coupling between an ad-hoc rheology for granular flow and a percolation model for multi-component mixtures. Predictions are compared with independent literature experimental data, for short and tall silos and prove to be quite accurate, after a tuning of the percolation flux sub-model. The larger segregation in short flow paths with smaller amount of fines reported by the experiments is quantitatively predicted. The model also predicts the three phases observed in experiments during the discharge of tall silos.


Introduction
Several industries process granular materials routinely, including chemical, food and pharmaceutical to mention a few.Often, granular solids are mixtures of particles, differing in size, shape, density and other features.Because of these differences, the homogeneity of the granular mixtures can be compromised by segregation during handling operations.Segregation is the tendency of particles with different physical properties to separate from the mixtures; it can be determined by several mechanisms: sterical, inertial, frictional and related to local velocity gradients.Segregation may induce a degree of heterogeneity in the powders incompatible with the industrial process requirements or with the final products quality standards.The development of tools for prediction of segregation helps to control and minimize its occurrence and impact.
Segregation onset and drawbacks are well known in the industrial practice since a long time; notwithstanding several studies, segregation remains an open issue.
One of the oldest studies in the literature (Brown, 1939) considers a free surface flow on inclined chutes.Brown suggests that the main mechanism of segregation is the inter-particle strongly connected with the free surface displacement, a convective equation for air is included in the comprehensive model, to track the air/pseudo-continuous granular material interface.Two types of flow regimes in silos are studied: funnel and mass flow.These determine different segregation patterns and intensity, as demonstrated by Sleppy and Puri (1996) in their experiments on sugar mixtures.The experiments by Ketterhagen et al. (2007) and Arteaga and Tüzün (1990) nicely prove that the funnel flow regime promotes segregation, at the discharge point, whereas mass flow regime keeps segregation bounded.These experiments were used to critically evaluate our model predictions and to assess its reliability on different scales and geometries.

Model outline 2.1 A comprehensive model of segregating granular flow with a free surface
The simulation of bi-dispersed granular flows, allowing for a free, moving surface was achieved by an Eulerian, continuum approach to the two phase flow.Accordingly, the static computation domain included two regions, accounting for the air and the solids, the latter treated as a pseudofluid with ad-hoc rheology.The extension and movement of the two regions were tracked by means of their interface.
The local pressure, p, and velocity field, u, were determined by the conservation equations, applied to the total mass and the momentum, for incompressible fluids: The local density, ρ, and viscosity, η, were very different in the air and solids domains.The values used in the unique momentum balance were determined as (3) where the volumetric fraction of solids,  was used and the subscripts a and g indicate respectively the air and the granular material.It determined the extent of the solid domain, treated as a nonporous pseudo-fluid.For numerical stability  was treated as a continuous variable which changed from 0 (air) to 1 (solid) in a very limited layer of finite thickness at the interface between the two phases.It was determined by the convective equation for a passive scalar: This approach to two phase flow is referred to as Level Set method (Olsson and Kreiss, 2005).It allows tracking the interface motion.Its implementation may require the addition of a small, artificial diffusion flux, to cope with the sharp discontinuity at the interface, Г.The densities of air and of granular material, as well as the viscosity of air, were assumed constant.The local viscosity of the granular pseudo-fluid was the key to determine its movement.It was defined through a sub-model, detailed in Artoni et al. (2009), based on a non-local rheology (Hennan et al., 2013).Shortly, the local viscosity was determined by the local granular temperature, , defined as the mean square fluctuating velocity (Jenkins and Savage, 1983), as: where  0 is the viscosity coefficient parameter,   is a granular temperature scale, d g is the average particle diameter and  is the granular temperature.The granular temperature scale   depends on the diameter of the particle and it is defined as  k  g d g where k  is a dimensionless parameter.
 can be considered as a measure of the local mobility of the particles and calculated through the translational kinetic energy balance: where П is the deviatoric stress tensor, q T is the fluctuating energy flux and z T is the dissipation rate.
The simulations of segregation required specifying the local fraction of different species.In this study we focused on bi-dispersed mixtures of solids, where the size was the only feature that differentiated the species.The local mass fraction of species i in the mixture,  i , was given by: that assumed a convective-segregating transport mechanism, where the segregating flux, J s , described the separation of fines from coarse material caused by percolation; it acted in the direction of gravity (Christakis et al., 2005), assumed to be y.We then neglected the segregation flux in x direction.The model was an original combination of literature models (Fan et al., 2014;Hajra et al., 2012).The particle diameter ratio, used by Hajra et al., 2012 , and the shear rate used by Fan et al. (2014), were considered as the two most important factors in the percolation mechanism.The segregation velocity of small particles was assumed as: where ρ ( ∂u ∂t +u⋅ ∇u ) =−∇p +∇⋅ [ η( ∇u+( ∇u) T ) ] +ρ g was the magnitude of the shear rate, d f and d c the small and the large particle diameter, respectively.K was a model parameter discussed later.Hajra et al. (2012) develop their model assuming that the segregation velocity is proportional to the difference between the diameter of small (the segregating particle) and the average diameter: The average diameter <d> is the mass averaged particle size defined as: Expanding <d>, the term contained in (8) is obtained.Hajra et al. (2012) state that the segregation velocity of each specie is proportional to the concentration of the other specie as previously found also by Savage and Lun (1988).This dependence is in agreement with the physical evidence that percolation of a fine particle increases when larger amount of coarse particles surround it because larger inter-particle void are available.Fan et al. (2014) introduce in the segregation velocity, the dependence on the shear rate, since percolation occurs when the material is dilated due to flow.With the increasing of the shear rate, larger inter-particle voids are created and fine particles can percolate more easily.This is however a necessary condition for percolation but not a sufficient one.It is also required a sufficient difference between the size of coarse and fine particles according to Hajra et al. (2012) model.The combination of the two conditions was obtained in the present percolation model through the product of the two independent sources of segregation proposed by Fan et al. (2014) and Hajra et al. (2012) respectively.It has to be considered just as a first attempt to tackle a complex phenomenon using an hydrodynamic continuum rheology in a two phase unsteady system (solid and air) where the solid phase is a binary mixture of particles (coarse and fine).Further refinements may indeed include inertial effects occurring in the free surface flows and segregation flux in the x direction.The segregation flux was obtained as the product between the segregation velocity and the mass fraction of small particles, as: and J s , c =−J s , f .According to eq. ( 9), percolation does not occur in the absence of bulk deformation, differently from the model of Christakis et al. (2002).The solution of eq. ( 7) may require some artificial diffusion for convergence, particularly when sharp composition differences arise locally.It has no physical meaning, nor quantitative relevance; and it is a purely numerical artefact.All the model eqs.( 1), ( 2), ( 4), ( 6), and ( 7) have the general structure of n-dimensional, advection and diffusion conservation equations, whose generalized numerical solution is implemented in several commercial and open source codes, using state-of-the-art numerical methods, and versatile pre-and post-processors to manage complex geometries.We based our implementation of the model equations on COMSOL Multiphysics (ver.4.3), an established, wide spread FEM code that is flexible enough to implement the granular flow model.The individual species transport equation was implemented through a general coefficient form for PDEs.Note that eqs.( 6) and ( 7) were solved only in the solids region, according to the indication of .

Initial and Boundary Conditions
The initial configuration assumed zero velocity everywhere, a specified partitioning of the domain between air and granular material (better specified in each example, see Figures 1 and 2) and a given distribution of fines consisting of a uniform initial concentration at a specified mass fraction.
Boundary conditions for eqs.( 1) and (2) were the following: atmospheric pressure on the open top of the silo, slip at the walls where there was air and partial slip where the granular material was flowing.The partial slip condition at the wall set the ratio of the tangential velocity to its gradient in the normal direction, by means of a constant slip length (Artoni et al., 2009).It allowed for a limited tangential velocity at the walls, as common in granular flow in contact to many comparatively smooth solid surfaces.At the silos discharge point, the flow rate was fixed, as common in industrial practice where rotary, screw or belt feeders withdraw material at a constant mass flow rate.This was simply expressed as a constant, plug flow velocity at the outlet.Among several options, depending on the extractor, in our computations we set the outlet velocity according to the Beverloo's equation (Neddermann, 1992), valid for gravity discharge of granular materials through an orifice.
Two different flow rates were imposed at the outlet: 2.2•10 -5 m 3 /s for the funnel flow silo and 1.5•10 -4 m 3 /s for the mass flow silo.
Boundary conditions for the phase variable  were no flux at the walls and for numerical reason the value is not equal to 0, but it is a very small number, i.e 10 -29 ) at the inlet boundary.For the translational kinetic energy, insulation was imposed at the walls whereas at the open top boundary a constant granular temperature was set, with a value equal to the average temperature of Ketterhagen and co-worker (2007) use two transparent, acrylic silos, quite short, with very different hoppers to achieve distinct discharge regimes.They are reported in Figure 1, with the original dimensions.One operates in funnel flow (Figure 1a) while mass flow regime prevails in the taller (Figure 1b), with a longer, slender hopper.The silos are filled with binary mixtures of glass beads; the level of filling is determined to load the same amount of material in both silos.The experiments use different loading policies.In this work, however, only the experiments starting from a homogeneous filling are considered.The silos are discharged with a discontinuous start-stop method, following the ASTM Standard for evaluating the segregation propensity of the materials.18 samples of 55 ml are withdrawn after each stop.The material collected in each sample is sieved to determine the mass fraction of each species.The experimental data from Ketterhagen et al. (2007), selected for model validation, include 2 types of regimes (mass and funnel flow), with mixtures of 2 different size ratios (1.9 and 4.3) and 3 different mass fractions (5, 20 and 50% of fines).Hence, a total of 12 experiments are simulated.Simulations have been carried out in 2D.

Experimental studies on tall silos
In the industrial practice silos have a longer cylindrical section, compared to those used by Ketterhagen et al. (2007) which follow the ASTM standard to measure segregation.To consider a more realistic configuration and study the interactions between the upper cylindrical and the lower conical section during segregation, the experiments by Arteaga and Tüzün (1990) are considered.They use two silos as well, made of Perspex, shown in Figure 2 with dimensions.The first one operates in funnel-flow regime (Figure 2a) and the second one in mass-flow regime (Figure 2b).These silos are loaded with a mixture of nearly spherical granules, made of ABS or acrylic particles, turnip and radish seeds.The mixtures initially loaded in silos are homogeneously premixed.Here we considered the experiments with mixtures of radish seeds and acrylic particles (see Table 1).Specifically, a mixture having 20% of fines with a coarse/fine diameter ratio equal to 4 was simulated.

Short silos
The evolution of uneven distribution of fines from initially uniform mixtures was simulated and shown in Figures 3 and 4, in case of funnel and mass flow, respectively.Conditions reflected the experiments of Ketterhagen et al. (2007).The time marks suggested that the segregation evolved quite rapidly.In funnel flow regime, Figure 3, the small particles accumulated below the shearing regions, shown in Figure 5. Shear concentrated below the free surface, particularly approaching the outlet; the material was mainly stagnant in the other regions of the hopper, close to the walls and the corners.The finest material migrated downwards where the shear was higher, then it remained trapped near the hopper wall.Since the nature of the funnel flow discharge is first in-last out, small particles rapidly accumulated in the hopper and they remained there until the end of discharge.This created an inhomogeneous outlet flow composition during time, where the coarse particles preferentially exited first.
On the contrary, mass flow, Figure 4, followed a first in-first out discharge pattern.The material was mainly sheared at the walls, as shown in Figure 6, contrary to the funnel flow where shear developed within the bulk (Figure 5).Fines accumulation occurred along the whole length of the inclined walls, where the shear was larger.Segregation appeared to be less localized than in funnel flow regime, with a large core region that became almost devoid of fines.This was consistent with some shear to propagate throughout the whole hopper section (Figure 6), because of the converging flow, giving a chance for percolation of fines everywhere in the hopper.However, since all the material was flowing, including the most peripheral one, the average composition across the outlet section remained close to the initial value, even during the initial transient, when fines accumulated at the walls.
The composition maps shown in Figure 3 and 4 are intuitive and useful to understand the nature of the discharge mechanism and qualitatively visualize the effect of flow field on segregation.However, a more quantitative comparison is needed.It is formulated in the next section.
Mimicking the experimental procedure, we calculated the average composition in samples of fixed volume, collected at the outlet of the silo over discrete time intervals.Following the representation of the experimental results in the original literature, we reported the progress of discharge as normalized fines mass fraction ( = fines mass fraction in each sample / initial fines mass fraction) vs. fractional mass discharged, up to a given time, for each experimental case analysed.

Tuning of the segregation flux
In this work the segregation process was modelled through the species material balance, where the transport mechanisms were convection and percolation.
Convection was ruled by the granular flow model, already validated in our previous works.The precise rheology of the granular materials in the experiments considered, could not be validated because neither the velocity, nor the stress fields inside the silos or at the walls are reported in the referenced literature.Still, we could successfully reproduce the discharge regime experimentally observed, either funnel or mass flow, at changing hopper geometry.This was a solid confirmation of the capability of our continuous granular flow model.Finally, the discharge velocity was always imposed in the experiments, further reducing the importance of a precise definition of the rheological parameters, as far as the flow regime was captured.It should be underlined that a good prediction of the local granular flow determined the accuracy of the segregation model, which was affected by the local shear rate.It impacted on the segregation flux through eq. ( 9), where the relative importance of was ruled by ∂ϕ ∂t +u⋅∇ϕ=0 .The segregation flux was proportional to the shear rate through a parameter K.However, a constant K led to predictions that were not sufficiently accurate, when compared to the experimental data, particularly in the funnel flow.It was observed that different values of K were required to fit the data satisfactorily when the initial amount of fines varied f   = 0.05, 0.2, 0.5).Optimal, constant K linearly correlated with f  , as shown in Figure 7.The correlation recalls the findings of Hajra et al. (2012), that suggest a linear dependence of K() with an intrinsic-and a concentrationdependent component, K T and K S , respectively, as in eq. ( 10).The linear correlation found could be recast as K = 2.4-7.9(1-f 0 ), which was consistent with eq. ( 10) when K T /d c = 2.4 and K S /d c =-7.9.Adopting these values of parameters K T and K S in eq. ( 10) to determine K as a function of the local composition, even if estimated from the initial, average composition, led to a good agreement between the predictions and the experiments, during all the discharge, as shown in Figure 8 for a single case.Others will follow.The values of root mean square error, normalized with respect to the average value of the experimental fraction of fines (NRMSE), reported in Table 2, gave general information about the performance of the model.The implementation of the concentration-dependent K in eq. ( 8) gave a general improvement of results in most of the cases examined Enhancements were more significant with initial mixtures where fines were more abundant ( f 0 = 0.2 and 0.5).The NRMSE values, calculated considering concentration-dependent K, resulted less than 10%, except for some critical cases (d c /d f =4.3 and  f 0 = 0.05) that will be discussed later.

The effect of the particle size ratio and initial fines concentration
In the first mixture analysed, the fines mass fraction was 0.05. Figure 9 shows the comparison between experiments and simulations for two different particle size ratios, d c /d f =1.9 and 4.3.As a general comment we observe that the extent of composition inhomogeneity at the outlet increased with the difference in the two particle sizes.For the largest particle size ratio, large voids were expected between the particles so that fines could percolate more easily, giving a greater tendency for the mixture to size segregate.The variations were larger in funnel flow compared to mass flow.
With the largest size ratio d c /d f =4.3, an excess of fines at the beginning of the discharge is reported in the literature experiments, both for mass and funnel flow (Figure 9b).It appears that even if the Authors try to start the discharge with a homogeneous mixture, it is impossible in practice to obtain such initial condition.The size ratio is so large that the small glass beads apparently percolate during silos loading, yielding an uneven initial mixture, before the material is set in motion.The difference in the initial composition between those assumed (i.e.implemented in the model) and those actually obtained experimentally weakened the validation, in the case of the largest size ratio.This was particularly evident for the mass flow regime.Clearly this represents a bias of the experimental data.After the initial phase, the qualitative trend was well reproduced by simulations and for the size ratio of 1.9 also a quantitative agreement with the experiments was obtained.With a larger initial amount of fines,  f 0 = 0.2, the intensity of segregation decreased, as apparent from Figure 10.The qualitative agreement between numerical results and experimental data remained quite good, including some fluctuations of composition during the discharge, particularly in the case of funnel flow.As expected, for the largest size ratio the difficulty to experimentally obtain an initially homogeneous composition is evident, jeopardizing the agreement with simulations for the mass flow regime.
Figure 10.Comparison with experimental data of mass (⊲) and funnel flow (⎕) in short silos  f 0 = 0.2; solid lines are numerical predictions.
However, for mass flow discharge and 4.3 diameter ratio the predictions were better than the more diluted case of  f 0 = 0.05.With a further increase of the initial amount of fines,  f 0 = 0.5, the segregation dropped dramatically, as evident from Figure 11, in both flow regimes.This was well captured by the simulations.Contrary to what occurred in the previous cases, also the mixture with the larger particle size ratio remained homogeneous during the filling stage, and the numerical results were consistent with the initial phase of the discharge.The results of Figures 9 to 11 confirmed the qualitative observations suggested by the concentration maps, Figures 3 and 4. It could be observed that at the end of any funnel flow discharge, an increase of fines was observed at the outlet, more evident in the experimental data; this was not the case in the mass flow.
It was an expected consequence of the first in-last out nature of funnel flow; the residual portion of fines accumulated at the wall left the hopper towards the end of the discharge.

Tall silos
Tall silos experiments by Arteaga and Tüzün (1990) were used for comparison.We kept the same parameters (see Table 1) used for simulating the granular flow in short silo experiments, for consistency.Also the same K( f ) developed above for short silos was used for this configuration as well.We found that a constant linear dependence on ηg=η0ρgdg 2 exp ( θ * θ ) was adequate for short silos, while a stronger dependence, Segregation was less intense than in short silos, as clear from Figure 12.However, there were quite significant deviations from the initial composition, particularly at the end of the discharge, that were well predicted by the model.Notwithstanding the crude choice of the parameters the agreement was good; the NRMSE values were always less than 10%, between 8.2 % and 7 %.
The experimental data suggested that three different stages could be identified during the discharge: an initial transient, a pseudo-steady state and a final transient.According to Arteaga and Tüzün (1990) the initial transient, with an enrichment of fines in the outlet flow is caused by the formation of a flame-shaped central core region, due to the granular bed dilation in the initial stages of the flow.In the dilated zone, percolation is favoured and accompanied by high value of shear rate in proximity of the outlet.The second stage, observed at 18% of the total discharge, is a pseudosteady state in which no segregation is reported at the outlet.This is due to the absence of significant radial gradients of velocity in all the cylindrical section, except at the shear region adjacent to the walls.The third stage, finally, starts at approx.78% of the discharge and is characterized by a larger segregation, greater in the funnel flow regime.Experiments and simulations showed that the segregation patterns on the short silos were similar to those observed in tall ones, in both discharge regimes, excepted for the absence of the pseudosteady stage which was missing in short silos.All these stages were well captured by the simulations, thanks to a proper description of the segregation mechanism.

Conclusions
In this work segregation in granular flow of bi-dispersed mixtures was simulated, using a continuum approach.Granular flow was described with a rheological model for free-flowing materials and adapted to multi-phase and multi-component conditions.The simulation of bidispersed granular flow, allowing for a free, moving surface was achieved by an Eulerian, continuum approach to the two phase flow.Segregation was modelled with a mass transport equation applied to a bi-dispersed mixture of particles of different sizes.The model included the effects of two different mechanisms: advection due to flow and segregation due to percolation.Two different types of flow regime occurring in silos were investigated: the funnel and the mass flow regimes.
Simulations were compared with independent experimental data found in the literature, for short and tall silos.The simulations were carried out taking into account different size ratio (from 1.9 to 4.3) of the binary mixture and different initial fines concentration (from 5 to 50%).The segregation model, which primarily took into account the difference in size of the particles, was sensitive to the local mixture composition and the local shear rate, provided by the velocity field.The model was predictive for the spatial distribution of the mass fraction of the fines (or coarse) inside the silo at different stages of the discharge.Furthermore, it allowed to evaluate the fines weight fraction across the outlet with respect to the fractional mass discharged.
The segregation model proposed was simple and its strength was the coupling with a rheological model able to predict the flow rate and a model able to follow the interface between air and bidispersed solid.Notwithstanding the simplicity of the proposed segregation model and the crude estimation of the parameters, the qualitative and quantitative agreement with experimental data was good.In tall silos the model correctly predicted the three different stages of discharge: initial transient, pseudo-steady state and final transient.
Further work is however required to extend the model to more complex cases.These include additional percolation mechanisms originated by other physical properties such as differences in particle density and introducing inertial effects which typically occur at the free surface.Further refinements include considering poly-disperse systems, where different size classes are present.So far, we implemented a one-way coupling between the granular flow and the segregation model, where the flow structure affected the segregation, but the local composition did not modify the rheology.Further development is therefore required on the two-way coupling, to also take into account the effect of local size difference on the flow patterns.

Figure 1 .
Figure 1.Silo geometries used in Ketterhagen et al. (2007) experiments, operating in (a) funnel flow regime and (b) mass flow regime.Γ indicates the initial interface between the two phases.The measures are given in cm.

Figure 2 .
Silo geometries used inArteaga and Tüzün (1990) experiments, operating in (a) funnel flow regime and (b) mass flow regime.Γ indicates the initial interface between the two phases.The measures are given in cm.

Figure 3 .
Figure 3. Evolution of segregation in funnel flow regime for  f °= 0.05; d c /d f = 4.3.The color indicates the fines mass fraction.

Figure 4 .
Figure 4. Evolution of segregation in mass flow regime for  f °= 0.05; d c /d f = 4.3.The color indicates the fines mass fraction.

Figure 7 .
Figure 7. Optimal K values as a function of the initial fraction of fines.

Figure 8 .
Figure 8.Comparison with experimental data of mass (⊲) and funnel flow (⎕) regimes in short silos after K( f ) optimization with  f 0 = 0.5; solid lines are numerical predictions.

Table 2 .
NRMSE for all tests in silos of Figure1for constant K and variable K.