Study of the discharge behavior of Rosin-Rammler particle-size distributions from hopper by discrete element method: A systematic analysis of mass flow rate, segregation and velocity profiles

Journal Pre-proof Study of the discharge behavior of Rosin-Rammler particle-size distributions from hopper by discrete element method: A systematic analysis of mass flow rate, segregation and velocity profiles

Raj Kumar, Arun K. Jana, Srikanth R. Gopireddy, Chetan M. Patel PII:

S0032-5910(19)30771-5

DOI:

https://doi.org/10.1016/j.powtec.2019.09.044

Reference:

PTEC 14718

To appear in:

Powder Technology

Received date:

4 March 2019

Revised date:

13 September 2019

Accepted date:

14 September 2019

Please cite this article as: R. Kumar, A.K. Jana, S.R. Gopireddy, et al., Study of the discharge behavior of Rosin-Rammler particle-size distributions from hopper by discrete element method: A systematic analysis of mass flow rate, segregation and velocity profiles, Powder Technology(2019), https://doi.org/10.1016/j.powtec.2019.09.044

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© 2019 Published by Elsevier.

Journal Pre-proof

Study of the discharge behavior of Rosin-Rammler particle-size distributions from hopper by discrete element method: A systematic analysis of mass flow rate, segregation and velocity profiles.

Depart ment of Chemical Engineering, Sardar Vallabhbhai Nat ional Institute of Technology Surat -395007,

Gu jarat, India,

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Raj Kumara , Arun K. Janaa , Srikanth R. Gopireddyb, Chetan M. Patela*

Daiichi-Sankyo Europe GmbH, Pharmaceutical Develop ment, Luitpoldstrasse 1, 85276

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Pfaffenhofen, Germany

*Corresponding author. E-mail address: [email protected], [email protected]

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Tel.: +91-261-2201647, Fax: +91-261-2227334

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Journal Pre-proof ABSTRACT In the present work, the discrete element method (DEM) is used to investigate the behaviour of Rosin-Rammler (R-R) particle size distribution (PSD) during hopper discharge. The study attempts to identify mass flow rate (MFR) and segregation as a function of spread parameter (PSD width) and location parameter, which includes detailed analysis of bulk density, segregation and velocity profile of particles. The applicability of the widely used Beverloo

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correlation for prediction of MFR for studied R-R PSDs is also investigated. The results show

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that, (1) MFR increases as PSD width (at fixed location parameter) increases, whereas MFR

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decreases with increase in location parameter at fixed PSD width; (2) the studied R-R PSDs exhibit a monotonic trend of flow bulk density and MFR with respect to PSD width and

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location parameter; (3) the effect of PSD width on overall segregation is found to be more

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than the location parameter; (4) the velocity profiles for considered R-R PSDs vary along the height of hopper, but remain uniform in shape and follow a pure parabolic increase from the

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hopper wall; (5) DEM flow rate prediction shows a good agreement with the Beverloo

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correlation by applying the concept of flow bulk density.

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Keywords: Discrete element method; Rosin- Rammler particle size distribution; PSD width; Mass flow rate; Segregation; Velocity profiles

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Journal Pre-proof 1. Introduction

Granular or powder materials storing and handling is considered to be the most important step in every chemical process industry. The biggest challenge faced by chemical researchers is to produce sufficient quantity and quality of product at low cost witho ut wasting these particulate or granular materials [1]. The behavior of granular materials during discharge from hoppers or bins plays a crucial role in many industrial processes suc h as chemical,

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agricultural, food and pharmaceutical industries [2]. For example, uninterrupted powder flow from hopper is vital for successful tableting process as it is the rate limiting factor for high

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tablet production, where with the modern tableting machines it is possible to produce up to

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1.6 M (Mega) tablets per hour [3]. Granular flows can be extremely complex and in general are not well understood. The granular material cannot be considered as solid, liquid or gas to

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fully characterize their flow state. In fact they are characterized as the fourth state of matter

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due to its puzzling nature and behavior [4,5]. The key understanding towards the behavior of

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granular material in a hopper not only aids in obtaining the correct flow, but also to ensure that the hopper is properly designed to deliver quality product during processing time [6].

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The prediction of mass flow rate (MFR) from hoppers or bins is both a classical problem regarding unpredictable behaviour of granular material and a problem of engineering concern. Numerous correlations were proposed over many decades and are used to predict discharge rates of granular materials from different shape hoppers [7–12]. Some of them were based on the dimensional analysis and others using experimental results. From applicability point of view, Beverloo equation [7] is a dominating choice in predicting MFR from hoppers having circular outlets. It takes the form: ˙

√ (

)

; where ˙ is the mass flow

rate, C and k are the empirical discharge and shape coefficients respectively which need to be determined experimentally, ρ is bulk density, D is the hopper outlet diameter, and d is mean particle diameter. There have been several attempts to improve the Beverloo correlation 4

Journal Pre-proof which include the effects such as fill height [13–16], friction between granular material and hopper wall [17,18], hopper outlet diameter [10,19], particle properties [16,20–23] on MFR. Nedderman et al. [18] claimed that material dilation during discharge is a strong function of voidage characteristics of the flowing material. Following this line of reasoning, they defined a term 'flow bulk density' and found a much restricted range of discharge coefficient, C between 0.575 and 0.59. Humbly and co-workers [16] also studied the discharge behaviour from flat bottom hopper and proposed modification in Beverloo equation by incorporating the

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concept of flow bulk density. But the use of such concept to MFR prediction somewhat requires a precise set of experiments having high reproducibility. The discharge dynamics

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from hopper depends upon several factors which include material properties, processing

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conditions and hopper geometry. The flow dynamics gets completely changed due to change

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in any one of these parameters, which may results in non-uniformity in product stream termed as segregation [24,25]. The degree of segregation is greatly affected by factors such as

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particle size distribution width (span), mass fraction of individual components, mean particle

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diameter, filling method as well as the hopper angle [26–28].

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Over the last few decades, Discrete Element Method (DEM) has alleviated some of the pitfalls associated with processing granular mixtures [29,30] and processing conditions in terms of segregation [31,32], mass discharge rate [33,34], flow mode [30,35,36] and velocity profiles [37–40]. The utility of DEM in particle technology sector has increased due to its ability to track the each particle [41] and studying variety of processes at experimental scale [42]. Anand et al. [33] utilized DEM to study the discharge behaviour of granular material having different size distribution from rectangular/wedge-shaped hopper. They suggested that filling of voids by fines particles results in increase in flow bulk density and hence MFR. Ketterhagen et al. [31] also assessed binary-size mixtures and found that the greatest extent of segregation is at the fine mass fraction of 5%, above which the segregation intensity 5

Journal Pre-proof decreases as the mass fraction of fines increases. In the recent study Zhao et al. [43,44] employed DEM to explore the effect PSD width of lognormal PSDs on the discharge rate and segregation from 3D hopper. They reported that continuous lognormal PSDs with the same mean having different widths exhibit distinctly different behaviors. Generally, the shape of PSD curve is best outlined by a continuous function [45] and different PSD functions (i.e. normal, log-normal and Rosin- Rammler) are used to characterize polydispersed systems like powders [46]. In this category, the applicability of Rosin- Rammler (R-R) PSD in different

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sectors (i.e. particle technology, spray technology, fragmentation processes etc.) is increased in recent years. The R-R PSD was originally developed to describe the distribution of coal

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particles [47], and it has been shown that the R-R PSD equation is one of the well suited

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distribution to represent powders made by grinding, milling, and crushing operations [48,49].

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The R-R PSD is generally expressed by two parameters, location parameter (x′, the particle size corresponding to 0.632 cumulative distribution undersize i.e. mass fraction undersize)

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and PSD width (n, spreadness of PSD), which can be easily accessed by linear regression

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[45]. Djamarani and Clark characterized R-R PSD based on fines and coarse fractions and

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expressed the PSD in terms of two parameters (i.e. fines/ coarse and coarse + fines fractions) [48]. The R-R PSD has been accepted as a robust distribution in broad range of applications, but the behaviour of R-R PSDs during hopper discharge process is still in its infancy. The research literature on the behavior of particulate solids in hoppers is vast and varied [10,22,23,50–53], but an understanding and predictive ability of continuous R-R PSD in terms of R-R parameters (i.e. x′ and n) is still an open and unexplored topic. In order to bridge the gap regarding the impact of the width of R-R PSDs and location parameter on hopper discharge characteristics, this study uses DEM to investigate the common continuous Rosin- Rammler PSDs. The objective of the current study is to study the effects of Rosin-Rammler PSD width and location parameter on the hopper discharge 6

Journal Pre-proof characteristics, namely, mass flow rate, size-segregation and velocity profiles. These investigations will be useful in understanding hopper discharge characteristics that are difficult to study in a controlled manner experimentally. 2. Methodology This section presents all the details about mathematical approach, studied particle size

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distributions (PSDs) and geometrical configuration.

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2.1. Discrete element method (DEM)

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The LIGGGHTS (LAMMPS Improved for General Granular and Granular Heat Transfer

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Simulations) code is used to perform the all simulations [54]. The simulations are carried out using the three-dimensional DEM, which computes the trajectories of each and every particle

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using Newton’s second law of motion. This is a well-established approach to study the

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granular material flow dynamics and the details about this method and theory can be found elsewhere [55]. Hertz and Mindlin & Deresiewicz theories are used to compute the normal

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and tangential forces, respectively [44,56]. 2.2. Particle-size distribution

In the present work, Rosin- Rammler particle size distributions (R-R PSDs) is considered having cumulative-mass form [57] as follows: (

[ ( ) ])

where, R is cumulative percentage undersize mass fraction distribution function, x is particle size (x ≥ 0), x′ is the characteristic particle size (location parameter) of the distribution, i.e., the particle size corresponding to 0.632 cumulative distribution undersize, and n is spread parameter (PSD width) of the distribution (n > 0). 7

Journal Pre-proof Figs. 1 (a) and (b) show cumulative mass distributions (Fig. 1a) and the corresponding probability density distributions functions (Fig. 1b), for five values of PSD width (n), viz., 3, 3.5, 4, 4.5 and 5 at fixed location parameter (x′) of 5 mm (see Fig. 1a at a cumulative mass fraction of 0.632). It can be seen by Fig. 1 (b) that the width (n) of R-R PSD decreases with increasing value of n. Furthermore, the number of components comprising the mixture or size range also increases with decrease in the value of n (low value of n indicates wider PSD). Similarly Figs. 1 (c) and (d) show cumulative mass distributions (Fig. 1c) and the

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corresponding percentage mass distributions (Fig. 1d), for five values of location parameter (x′), viz., 3, 3.5, 4, 4.5 and 5 (see Fig. 1c at a cumulative mass fraction of 0.632, represented

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by black dashed lines) at fixed PSD width (n=5). It is clear from the figure that with decrease

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PSD is shifted to lower value (Fig. 1d).

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in location parameter value at constant PSD width (n), the average particle size of considered

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2.3. Geometrical setup

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Fig.1

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The discharge characteristic of lognormal PSDs in a conical hopper has been extensively described by Zhao et al. [43,44]. In extension to [43,44], the focus of present study is to understand the behaviour of continuous R-R PSDs during hopper discharge process as a function of PSD width and location parameter. Therefore different simulations were performed for Rosin-Rammler PSDs with varying PSD width and location parameter (Section 2.2). The geometrical configuration consists of a conical hopper identical to [41,43,44], and is depicted in Fig. 2. Fig.2

8

Journal Pre-proof The hopper geometry was created with the open source finite element mesh tool known as Gmsh [58] and later imported as physical boundaries into LIGGGHTS. 2.3. Model Validation The present DEM model is validated against the experimental and numerical study of Li et al. [41] and Zhao et al. [43] respectively. Li et al. [41] in their experimental study investigated discharge characteristics of monodisperse glass beads having particle diameter of 10 mm in

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semi-cylindrical hopper assembly. Hence, hopper geometry is cut in the half to visualize the flow mode. First comparison is carried out in terms of hopper bed heights between above

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mentioned studies [41,43] and our simulation work at different time instances, and is depicts

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in Fig. 3.

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Fig. 3

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Specifically, each column represents a specific study, while each row represents the discharge time namely, t = 0, 6, 12 , and 18 sec. At each time instance, the hopper bed heights between

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simulations and past studies is excellent and indicates that present DEM model is able to

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quantitatively reproduce the experiments of Li et al. [41] during hopper discharge process. Second comparison is carried out in terms of mass flow rate (MFR) of monodisperse glass beads particles (5 mm) used in simulation work of Zhao et al. [43] and present simulation work, which is shown in Fig. 4. Fig.4 It can be seen clearly that discharge mass with time remains relatively same between numerical study of Zhao et al. [43] and our present work, which further confirms the proper selection of DEM parameters. The material properties along with the other DEM parameters adopted in the present work are given in Table 1. 9

Journal Pre-proof Table 1 4. Results and Discussion 4.1. Mass flow rate (MFR) This section explores the impact of PSD width (n) and location parameter (x′) on MFR for studied R-R PSDs. Fig. 5 (a) illustrates the continuous discharged mass from the hopper with

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Fig. 5.

f

time as a function of PSD width (n) having constant location parameter (x′).

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It is found that the MFR increases with the increase in PSD width (low value of n indicates

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wider PSD). This increase in MFR is due to increase in the mass fraction of the finer particles smaller than 5 mm (location parameter, x′) with increase in PSD width. These results

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resemble with the different studies on binary mixtures [21,33], where increasing mass

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fraction of finer species results in increased MFR. In the latest study, Zhao et al. [44] investigated mass discharge characteristics of lognormal particle size distributions as a

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function PSD width which is defined as the ratio of standard deviation and mean of

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distribution. They found that MFR decreases as the PSD width increases. Such discrepancy could be due to definition of PSD width, which leads to different inter-particles interactions among the various size particles. On the other hand, at fixed PSD width having different location parameter, the MFR is decreased in the following order; x′ =3 > x′ =3.5 > x′ = 4 > x′ =4.5 > x′ =5 (see Fig. 5 (b)). It is evident that with increase in location parameter value, the mean diameter of considered PSD is shifted to higher value (Fig. 1(d)) and hence reduced MFR is observed. A similar behavior was reported, wherein increase in mean diameter resulted in decrease in MFR [14]. But the effect of increasing PSD width at constant locatio n parameter (x′) on MFR is less as compared to increase in location parameter at constant width, which can be clearly seen by comparing the slopes of Fig. 5 (a) and Fig. 5 (b). 10

Journal Pre-proof 4.1.1. Static bulk density or Flow bulk density As has already been shown in the previous section, increase in PSD width (at constant location parameter) and decrease in location parameter (at constant PSD width) results in increased MFR. In order to study the dependency of initial packing density (whic h is termed as static bulk density in the present work) on MFR, static bulk density is estimated at different heights before discharge. The average of 12 measurements is used to examine the

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relationship between MFR and static bulk density for studied PSDs. Fig. 6 (a) and 6 (b) show

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the variation of static bulk density and MFR with PSD width (n) and location parameter (x′)

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respectively. Each graph comprises two y-axis namely MFR (marked in black) and static bulk

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density (marked in red), plotted against n and x′ respectively.

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Fig. 6.

It is evident from the plot that the increment in PSD width (n) results in increase MFR and

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shows a positive relationship with static bulk density (Fig. 6 (a)). On the other hand, a

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negative relation between static bulk density and MFR is observed for constant width PSD having different location parameter (see Fig. 6 (b)). As per Beverloo equation, MFR is

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positively related to the static bulk density, but in present work contrary nature of static bulk density with MFR is observed for constant width PSDs. In general, static bulk density is strongly affected by particle size and decreases with decreasing particle size [59]. As mentioned earlier, reduction in location parameter at constant width results in reduced mean diameter of mixture. It can be one of the reasons that for PSD having different location parameter, increased MFR is obsevered even though low value of static bulk density is noticed. From the comparison of results (Fig. 6), two interesting observations are clear, (1) the effect of PSD width on static bulk density is found to be more as compared to location parameter, whereas the trend reverse for MFR, (2) it is not necessary that the mixture or PSD

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Journal Pre-proof having high static bulk density will always have more MFR and it purely depends upon the PSD and its arrangement. During granular discharge, several particles mechanisms such as percolation or sifting due to which particles segregate, where fine particles tend to move in the direction of gravitational acceleration relative to the larger particles. Humbly et al. [16] studied the discharge characteristics of binary granular mixtures and defined flow bulk density (ρd ) which takes the

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form, ρ d = ρp (1-εd), where, ρp and εd are the particle density and voids respectively. It is clear

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from the equation that flow bulk density is a strong function of void fraction generated during

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hopper discharge which in fact depends upon PSD. Anand et al. [33] reported that flow bulk

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density is a dominant parameter to increase MFR in binary mixtures by filling voids between coarser particles with finer one. To verify such a concept, a quantitative approach to calculate

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flow bulk density is carried out for investigated R-R PSDs. For that, particles movement at center region at 52 mm above outlet (i.e., cone height) is tracked during whole discharge

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time. The flow bulk density is defined as the ratio of total mass of the particles inside the

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selected region to volume of that region. It is worthwhile to note that such a flow bulk density

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calculation is time dependent because the total mass inside that region changed during each simulation time step. Fig. 7 (a) and Fig. 7 (b) show the variation of flow bulk density with time for considered PSDs as a function of PSD width and location parameter respectively. Fig. 7. For constant location parameter (x′) having different PSD width (n), flow bulk density increases with increase in PSD width (Fig. 7 (a)). For all considered PSD, the flow bulk density decreases drastically during initial discharge and fluctuates around a constant value. The flow bulk density is expectedly higher for the wider PSD and shows a positive relationship with MFR. Basically, with increase in PSD width, the tendency of particles to fill the available voids increases which results in dense packed structure during discharge. On the 12

Journal Pre-proof other hand the flow bulk density values for constant PSD width having different location parameter (x′) remains quite close to each other (see Fig. 7 (b)). It is due to less scope of particles packing as the PSD width remains co nstant. These results are consistent with the previous findings on binary and ternary mixtures, where increase flow bulk density is claimed for high discharge rate by filling the available voids between larger particles with smaller ones [33,50]. Fig. 8 (a) and Fig. 8 (b) also plot the average flow bulk density and MFR as a

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Fig. 8.

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function of PSD width (n) and location parameter (x′) respectively.

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The three main results are: (1) the flow bulk density shows a positive relation with MFR for all considered R-R PSDs; (2) the flow bulk density is expectedly higher for the wider PSD at

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constant location parameter as compared with the constant width PSDs having different

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location parameter (x′) and (3) the difference in the flow bulk density remains approximately

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small with respect to location parameter at constant PSD width.

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In the latest study, Zhao et al. [44], showed a lack of relationship between bulk density and MFR for lognormal PSDs as a function of PSD width. But in the present study of R-R PSDs,

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MFR is positively related to the flow bulk density as per Beverloo equation [7]. 4.1.2 Beverloo correlation

For several decades, Beverloo correlation has been the dominant choice for prediction of MFR through an orifice. Mankoc et al. [9] checked the accuracy of Beverloo correlation for different outlet sizes. They modified the Beverloo correlation for all outlet size by introducing an exponential term which was tied with local density variations near the outlet. As seen in previous section, static and flow bulk density of considered PSDs show distinct behaviour towards MFR. Hence choice of static bulk density (ρ) in Beverloo correlation, while calculating the discharge rate raises interestingly questions. Basically, density term in 13

Journal Pre-proof Beverloo correlation was incorporated to provide dimensional consistency. But, as the hopper discharge begins, density within a hopper varies with respect to both time and position. In this section a compressive analysis of static and flow bulk density towards prediction of MFR is performed. Fig. 9 (a) and 9 (b) shows the variation of static density and flow bulk density for considered PSDs.

Fig. 9.

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It can be seen that the flow densities are in general lower than the initial packing densities (static densities) for all considered PSD, and having maximum value for wider PSD (n=3).

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This may be tied to the extent of dilation among the different ranges of particle sizes (Fig. 1),

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which leads to the behavior illustrated in Figure 9 (a). Similar findings were also reported by

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Arteaga and Tuzun [21] in their study in which they studied effect of fines percentage on binary system. To check the applicability of flow bulk density in MFR prediction, MFR

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obtained by DEM simulations are compared with Beverloo correlation by incorporating the

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flow density concept which takes the form:

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˙

In the above equation, the term particle density and

(

(

)√ (

)

;

) represents flow bulk density, where

is the

is void fraction during discharge. The discharge coefficient (C) is the

function of packing fraction of particles and value of C lies between 0.575 and 0.59 by incorporating the concept of flow bulk density as suggested by Nedderman and co-workers [18]. In the present study, value of C is taken as 0.58 [60]. Fig. 10 shows the prediction of MFR with DEM and Beverloo correlation (with static and flow bulk density) for considered PSDs.

Fig. 10. 14

Journal Pre-proof The results indicate that MFR increases as the PSD width broadens (Fig. 10 (a)) and this profile is very well captured by Beverloo and DEM results. From these results, it may be concluded that, DEM has reproduced the MFR for different PSDs quite well. The good agreement between the results obtained by Beverloo correlation (with flow bulk density) and DEM is observed for almost all the considered PSDs, whereas the Beverloo correlation (with static bulk density) overestimates the MFR in these cases. Such a good agreement for prediction of MFR with flow bulk density suggests use of flow bulk density instead of static

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bulk density.

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4.2. Segregation analysis

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This section further explores ineluctable size segregation during discharge for studied R-R

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PSDs. Size-based particle segregation during hopper discharge is frequently reported in the literature (see, for example, [21,27,31,35,61]. To the authors’ knowledge, the existing

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literature on segregation is limited to bidiseperse and tri-disperse particles system, where

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smaller size particle (fines) is used to characterize segregation phenomena. Since present

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PSDs contain various sizes of particles and concept of single size particle (which is termed as fines in past) segregation is not suitable to quantify the segregation in the present scenario. Xu et al. [62] in their work quantitatively characterize and investigate the transient local segregation behavior of the binary particles by making use of square grids. In the latest study, Zhao et al. [43] investigated segregation of continuous lognormal particle size distributions by dividing the PSD in three ranges along with the mean diameter of mixture. In particular to hopper discharge, reference is missing when it comes to understand size segregation of R-R PSD having different poly-disperse nature. In the present work, location parameter (x′, particle sizes less than 63.2% cumulative distribution undersize) is taken as reference point and the particle sizes below location parameter is termed as fines. Hence, extent of

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Journal Pre-proof segregation is quantitatively described by the mean diameter of mixture, fines and behaviour of different size fractions during hopper discharge. 4.2.1. Variation in mean diameter of mixture Fig. 11 (a) illustrates the mass-based normalized mean diameter (i.e., ratio of the mean particle diameter of considered PSD inside hopper during particular time instant to the initial mean particle diameter. A value of normalized mean diameter inside hopper equal to 1 show

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completely mixed state, whereas low and high value shows coarse depleted and coarse rich

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material inside hopper respectively) with time as a function of PSD width (n).

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Fig. 11.

The mean diameter inside hopper roughly remains constant during initially discharge period

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of discharge up to 3 sec for all considered PSDs. As the flow proceeds, the interaction

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between different sizes particles is obvious, where small size particle species continuously percolate through the spaces between the large particles. For narrower PSD (n=5), normalized

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mean diameter inside hopper remain equal to 1 during entire discharge process, showing

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completely mixed state. On the other hand, with increase in PSD width (low value of n indicates wider PSD), segregation phenomena becomes more prevalent, whic h can be clearly seen by increase in mean diameter inside hopper. Since fines are continuously discharged from hopper and it corresponds to a state (at approximately 14 sec), where coarse particles occupy most of the centerline channel. It results in accumulation of fines at the side walls of hopper and as the flow further proceeds; these accumulated fines are discharged. Hence, by changing the PSD width from wider (n=3) to narrower (n=5), the percolation or shifting effects diminish leaving well mixed mixture inside hopper. Earlier studies permit to conclude that increasing fines content in mixture decreased the intensity of particle segregation [21,25,28,31] by filling the available voids during discharge. But for considering case, two 16

Journal Pre-proof interesting observations are worthwhile to note, (1) the mass fraction of the finer species smaller than location parameter (5 mm) increases with increase in PSD width (n), and hence disagree with the results of past studies on binary mixture [32,63], whereby segregation decreases as the mass fraction of the finer species increases, (2) the number of size components increases with increase in PSD width, hence the results also disagree with another study on multi-component mixtures (binary, ternary and q uaternary) in which extent of segregation decreases as the number of size components increases [63,64]. Fig. 11 (b) also

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shows the change in the normalized mean diameter inside hopper as a function of location parameter (x′) having constant PSD width (n=5), with each trendline ending at different time

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instants due to the varying MFR as depicted in section 4.1 (Fig. 5 (b)). For constant width

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PSD (n=5) having high value of location parameter (x′=5), the normalized mean diameter

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remains constant to value of 1 for maximum time. On the other hand, with decrease in location parameter, the decrease in mean diameter inside hopper happen much early, with a

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greater decrease for the PSD having low value of location parameter (x′=3). This suggests

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that low location parameter PSD is likely to induce segregation much earlier during discharge

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and can be controlled by increasing the mean diameter of mixture. As the mass fraction of smaller particles (fines) less than 5 mm increases with decrease in location parameter at constant PSD width, the results here also contradicts with different studies on bi-disperse mixtures [21,32,63], where segregation decreases with increase in mass fraction of fines. Such discrepancy is due to different inter-particles interactions and percolation phenomena for studied R-R PSDs as compared to past studies on mono-diseperse, bidiseperse and tridisperse particles system. Since these observations completely depend upon the choice of fines, we further try to investigate segregation by tracking the behaviour of different size faction for considered PSDs which are discussed in the next section. 4.2.1. Behaviour of different size fractions 17

Journal Pre-proof In this section, behaviour of three different size fractions (i.e. size fraction below location parameter, at location parameter and above location parameter) of considered PSDs is separately analyzed during discharge process. It provides an understanding of the PSD width and location parameter's effect on overall segregation. Fig. 12 (a) shows the change in the mass fraction of three considered size fractions (normalized with initial mass fraction of fractions) as the hopper discharges for R-R PSDs having constant location parameter (x′=5)

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Fig. 12

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with different PSD width (n=3, 4 and 5).

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The mass fraction of fines species (size fraction below location parameter) inside hopper for

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narrower PSD (n=5) is relatively uniform in composition until the end of discharge where a

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slight percolation of fines is observed. However, as the PSD width increases, fines percolation becomes more frequent, which further confirms the results obtained in Fig. 11 (a).

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On the other hand, the mass fraction of fines species (size fraction below location parameter)

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inside hopper is relatively uniform in composition for high value of location parameter PSD

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(x′=5) and tend to fluctuates with decrease in value of location parameter. Fig. 13.

4.3. Velocity analysis

As a qualitative comparison, the velocity of the particles for considered PSD is also observed during hopper discharge. Fig. 14 depicts the velocity profile from DEM simulations for R-R distribution at constant location parameter (x′ =5) having PSD width (n) of 3, 4 and 5 at four time instances (namely, t = 0, 5, 10, and 15 sec). The geometries in Fig. 14 are cut in the half in order to view the velocities in the center of the hopper. The formation of a core with higher

18

Journal Pre-proof velocities above the outlet near the center can be observed and this core region tends to decrease with decrease PSD width (see at 10 sec). Fig. 14. Thus it can be concluded that the wider PSD imparts fluidity characteristics and majority of the particles were in motion during discharge, whereas the materials with narrower PSD move at a very low speed with a gradient across the geometry cross section. This corresponds

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to the discharge state in the upper region of hopper which shows no movement of particles on

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the surface (indicated by blue region in Fig. 14 (c) at discharge time of 10 sec). Fig. 15 also

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shows snapshots of the instantaneous velocity profiles at four time instances (namely, t = 0, 5, 10, and 15 sec) for R-R PSDs at constant PSD width (n) having location parameter of 3, 4

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and 5 respectively. During discharge process, the particle velocities throughout the particle

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bed increases and the different particle sizes tend to flow at different rates. The velocity distributions are markedly different for PSDs having constant width, which is due to

al

increases in mean diameter with increase in location parameter. At t =5 sec (second column

rn

of Fig. 15), the center region velocities near the hopper outlet is comparatively highest for

Jo u

PSD having low value of location parameter, which keep on decreasing with increase in location parameter.

Fig. 15.

As the discharge further increases, a center V-channel is formed where fines species continuously percolating through the vicinity of coarser one. This V-channel tends to decrease with increase in location parameter, which can be clearly seen by comparing the velocity profile at 10 sec (see third column of Fig. 15). The early discharge of PSD having low value of location parameter (Fig. 15 (a), at 15 sec) further confirms the results obtained in Fig. 5 (b).

19

Journal Pre-proof 4.3.1. Local particles velocity In this section, a comprehensive analysis of local velocity of particles is presented to elucidate the effect of PSD width (n) and location parameter (x′) on velocity profiles. To examine this, particles velocity are tracked at different dimensionless radial positions (r/R = 0 to 1) of hopper. Here, R is the radius of cylindrical section of hopper and r is the radial position assessed, such that r/R = 0 and 1 represent, respectively, at the center axis and wall.

f

The spatial averages are taken 20 mm vertically and radially at different heights (named as

pr

Fig. 16.

oo

section 1, section 2, section 3, section 4 and section 5) as shown in Fig. 16.

e-

Fig. 17 (a) displays the averaged velocity as a function of PSD width (n) having constant

Pr

location parameter (x′=5) at the center (r/R = 0) in section 1. The values are time-averaged for

al

sample time of 0.5 sec.

rn

Fig. 17.

Jo u

Clearly, the relationship between PSD width and particle velocities is generally nonmonotonic with no consistent trends. These results are surprising one, since MFR is found to maximum for wider PSD. This indicates a negative relationship between MFR and center velocity as a function of PSD width. Similar findings were also reported by Zheo et al. [43,44] in their study in which they studied discharge characteristics of lognormal PSD as a function of PSD width. On the other hand, for constant width (n=5) PSDs having different location parameter (x′ =3, 3.5, 4, 4.5, 5), a direct correlation between velocity and MFR is observed at the center (Fig. 17 (b)). The center hopper velocity decreases sharply with increase in location parameter (x′) and have minimum center velocity for monodisperse particle system. This implies that for fixed PSD width, reducing location parameter and hence

20

Journal Pre-proof mean diameter of PSD, results in increased centerline velocity and MFR as they experience low resistance to flow compared to higher location parameter PSD. Furthermore a detailed microscopic analysis on the average fields of the velocity for considered R-R PSDs at different heights is performed. Fig. 18 shows the average velocity field as a function of PSD width at different radial positions (r/R = 0, 0.5, 1, -0.5, -1) of the hopper for considered sections (Fig. 16) at different time instances.

oo

f

Fig. 18.

pr

Specifically, each column represents a specific time instant (namely 5, 10 and 15 sec), while each row represents a specific section (represented by different colors, grey for section 1,

e-

yellow for section 2 , red for section 3, blue for section 4 and green for section 5). By

Pr

visualizing the velocity fields, material can be divided into three different regions during whole discharge, (1) plug- flow region (section 5 (marked by green) and section 4 (marked by

al

blue)); (2) transition region (section 3, marked by red) and (3) converging flow region

rn

(section 2 (marked by yellow) and section 1 (marked by grey)). As for plug- flow region

Jo u

(section 5 and 4), the material in cylindrical section of a hopper moves in plug- flow fashion, having flatter velocity profile during whole discharge. On the other hand, velocity profile somewhat differ in lower part (section 3) during whole discharge. In this transition region, the top material surface continues to decline towards outlet during whole discharge process (see Fig. 18 (c, h, m)). On moving further down the hopper bed (section 2 and 1, converging flow region), the centerline velocity becomes greater and a depression is formed in the top material surface (see Fig. 14 (d, i, n, e, j and o)). It can be seen that the magnitude of the velocity decreases with radial positions and is relatively insignificant in the region close to the wall. Such a bell shaped velocity profile is also observed by different authors [65,66]. Regarding the impact of PSD width on velocity profile, the spatial distribution of the velocity

21

Journal Pre-proof magnitudes seems to be unaffected by the PSD width during whole discharge and is almost overlapping on one another in each of the section. Agreeing with Fig. 17 (a), Fig. 18 shows clearly that velocity is not a dominant parameter for increased MFR with PSD width. On the other hand, the velocity profile in each section is directly affected by the location parameter (see Fig. 15). Fig. 19.

oo

f

For PSD having low value of location parameter, the velocity profile is extended downwards with time in each of the section and tends to decrease or flatter when the value of location

pr

parameter is increased. These flatter profiles have lower centerline velocities and agree well

e-

with the trend for MFR (Fig. 5 (b)). Furthermore, the variation of particle velocities caused

Pr

by changing location parameter (x′) at constant width are greater as compared to change in PSD width (n) at constant location parameter (x′). Collectively, Fig. 18 and 19 show that the

al

radial velocity profiles of the particles are different for different PSDs (having different PSD

rn

widths and location parameter), and velocity of particles to enhance MFR purely depends

studied PSDs.

Jo u

upon the type of PSD and hence not found to be a dominant parameter affecting MFR for all

4.3.2. Velocity profiles

The prediction of velocity field inside hopper has become a bone of contention for the research fraternity dealing with powder materials. Albaraki and Antony [37] in their studies investigated spatial and temporal distributions of the velocity fields inside different internal angles hoppers using digital particle image velocimetry (DP IV) and high speed videography. Although they studied dynamic flow trajectories and velocity profiles in detail, but no direct expression between hopper parameters and velocity field was obtained. Maiti et al. [40] studied the discharge behaviour of granular materials (sand grains and glass beads) through 22

Journal Pre-proof the eccentric opening of rectangular silos and extended the kinematic model for granular discharge. Magalhães et al. [38] have studied the effect of orifice size and hopper opening angle on the velocity fields of monodisperse spherical grains and concluded the power law dependence of the distance from the orifice. In the latest study, Wang et al. [39] have presented the expression for velocity profile in a quasi-two-dimensional wedge-shaped hopper. But the applicability of such expression on 3D geometry and polydisperse particle

f

system is still a largely unexplored topic.

oo

Hence, in the present work, an attempt is made to understand the velocity profile of

pr

considered R-R PSDs flowing out from three-dimensional conical hopper. Fig. 20 shows a

e-

typical velocity profile for studied PSDs superimposed on section 1 (Fig.16) of hopper.

Pr

Fig. 20.

Specifically, y denotes the radial distance from the wall to the center and R is the radius of

al

cylindrical section of hopper. It can be seen that the velocity u increases from the hopper wall

rn

to a certain value at the center of hopper. By scaling the radial velocity (u) by center velocity

Jo u

(umax ), the velocity profile can be expressed in parabolic form:

=( )

It is interesting to note that the fit of the rescaled velocity profiles to a power law hold good for y ≥ D (outlet diameter). Hence the velocity profiles of considered R-R PSDs can be expressed by two parameters of center velocity (umax ) and radius of cylindrical section (R) of hopper. 4.4. Comparison between lognormal and R-R PSDs

23

Journal Pre-proof In this section, a direct comparison of DEM results with R-R and lognormal PSDs is presented. The spreadness of lognormal PSD is governed by standard deviation lnσ, which has a same role as the width (n) in the R-R PSDs. Generally, parameter n and lnσ are inversely proportional to each other [67]. In order to match R-R PSDs with lognormal PSDs, the d0.5 of studied R-R PSDs are calculated and used to generate the respective lognormal PSDs. To achieve the same initial mass, the numbers of glass beads used are 292500, 371491, 414979, 522473 respectively, for the widths (lnσ) of 0.20, 0.25, 0.28 and 0.33. Fig. 21 depicts

PSDs.

e-

pr

Fig. 21.

oo

f

the comparative mass based frequency distributions between R-R and generated lognormal

It can be seen that overall lognormal PSD shapes are similar; however, there is shift to left as

Pr

compared to R-R PSDs having similar spread (see Fig. 21 (a-d)). Moreover, the content of fines percentage is increased in lognormal PSDs at same spreadness of R-R PSDs.

al

In order to observe such difference on MFR and segregation, DEM simulations are performed

rn

with generated lognormal PSDs (i.e. lnσ = 0.33, 0.28, 0.25 and 0.20). Table 2 shows a

Jo u

comparative MFR prediction by DEM simulations for both the distributions. Table 2

From these observations two points are worthwhile to note, (1) MFR increases with increase in PSD width (i.e. n and lnσ) for both types of considered PSDs, which is tied to increase in mass fraction of the finer particles, (2) for same spreadness, lognormal PSDs shows improvement in MFR as compared to R-R PSDs lognormal. Such a significant impact of type of PSD on MFR as seen in Table 2 warrants a close look at segregation behaviour. For that, mass-based normalized mean diameter (index of segregation) is tracked for each lognormal PSD and compare with R-R distribution. Fig. 22 compares the segregation results between lognormal and R-R PSDs. 24

Journal Pre-proof Fig. 22. As the PSD width is increasing, the segregation intensity is increased for both PSDs, but R-R PSDs suffers more segregation as compared to lognormal PSDs. But the concept of increasing fine content to reduce the segregation does not hold true for both PSDs. Conclusions Despite the ubiquity of Rosin- Rammler particle-size distributions (R-R PSDs) in particle

f

technology sectors, there has been little research focused on mass discharge characteristics of

oo

particles following continuous R-R PSDs from hopper, a gap we address in this study. The

pr

discharge behaviour of R-R PSDs is modeled numerically using the discrete element method (DEM). The impact of the PSDs widths and location parameters of R-R PSDs on mass flow

e-

rate (MFR), bulk density, segregation and local particles velocity are analysed during hopper

Pr

discharge. Firstly, the location parameter of R-R PSDs is found to be significantly impacting the MFR where it is shown that MFR increases with decrease in PSD location parameter at

al

constant PSD width. Next, for PSD having constant location parameter, it is reported that

rn

wider PSD have larger MFR as compared to narrower PSD. The effect of change in PSD

Jo u

width (at constant location parameter) on segregation is more than the change in location parameter (at constant PSD width) for considered PSDs. The relationship between PSD width and particle velocities is generally non- monotonic with no consistent trends. The velocity profile for studied R-R PSDs can be expressed by two parameters expression, which exhibits parabolic nature. From the analysis of results, it is found that high flow bulk density is favorable to increase MFR. Furthermore, for the studied R-R PSDs, use of flow bulk density term in place of the bulk density in Beverloo correlation increase the accuracy of MFR predictions to within +7% of measured DEM discharge rates. A comparative analysis between R-R and lognormal PSDs shows that for the same spreadness, R-R PSDs suffers more segregation as compared to lognormal PSDs. The past observations on bidiseperse and 25

Journal Pre-proof tri-disperse particles system, contradicts for studied distributions where extent of segregation increased with increase in mass fraction of the finer species. The applicability and experimental investigation of present PSDs particularly with smaller real powder fractions is an open area of research. These different factors will be carefully examined in the future.

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30

Journal Pre-proof Figures Captions Fig. 1. (a) R-R PSDs (cu mu lative mass fraction distributions) for five values of PSD width (n) i.e. 3, 3.5, 4, 4.5 and 5; the location parameter x′ of the five distributions is 5 mm, (b) Mass based R-R PSDs fo r five values of n (3, 3.5, 4, 4.5 and 5), (c) R-R PSDs (cu mulat ive mass fraction d istributions) for five values of location parameter (x′) i.e. 3, 3.5, 4, 4.5 and 5; the PSD width (n) of the five distributions is 5 mm, (d) Mass based R-R PSDs for five values of x′ i.e. 3, 3.5, 4, 4.5 and 5. Fig. 2. Schematic of the computational 3D conical hopper. Fig. 3. Co mparison of the instantaneous bed height of the experimental [41], numeric study of Zha o et al. [43] and present simulation at time instances of (a) 0, (b) 6, (c) 12, and (d) 18 sec.

f

Fig. 4. Co mparison of continuous discharge mass versus time profiles for monosize part icle systems (5 mm) between simulation work of Zhao et al. [43] and present study.

pr

oo

Fig. 5. Discharged mass versus time profiles for (a) R-R PSDs having constant location parameter (x′=5) with five values of PSD width (n) i.e 3, 3.5, 4, 4.5, and 5, (b) R-R PSDs having constant PSD width (n=5) with five values of location parameter(x′) i.e. 3, 3.5, 4, 4.5, and 5.

e-

Fig. 6. Relationship between static bulk density and MFR for investigated R-R PSDs as a function of, (a) PSD width (n), (b) Location parameter (x′). Fig. 7. Flo w bulk density variations for investigated R-R PSDs during discharge time as a function of, (a) PSD

Pr

width (n), (b) location parameter (x′).

Fig. 8. Relat ionship between flow bulk density and MFR for investigated R-R PSDs as a function of, (a) PSD width (n), (b) Location parameter (x′).

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Fig. 9. Variation in static and flo w bulk density for (a) PSD with constant location parameter having different PSD width (n), (b) PSD with constant PSD width having different location parameter (x′).

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Fig. 10. MFR predicted by the DEM simu lation and Beverloo correlation (with static and flow bulk density) for considered R-R PSD, (a) as a function of PSD width (n), (b) as a function of location parameter ( x′). Fig. 11. Normalized mass based mean particle diameter inside hopper with respect to time for, (a) R-R distribution at constant location parameter (x′=5) having different PSD width, (b) R-R d istribution at constant PSD width (n=5) having different location parameter. Fig. 12. Normalized mass-based particles fraction inside hopper during discharge process for constant location (x′=5) R-R distribution having different PSD width, (a) n=3, (b) n=4 and n=5. Fig. 13. Normalized mass-based particles fraction inside hopper during discharge process for constant width (n=5) R-R distribution having different location parameter, (a) x′=3, (b) x′=4 and (c) x′=5. Fig. 14. Velocity profile at four time instances (namely, t = 0, 5, 10, and 15 s) for R-R d istribution at constant location parameter having PSD width of, (a) x′ =3, (b) x′=4, (c) x′=5. Fig. 15. Velocity profile at four time instances (namely, t = 0, 5, 10, and 15 s) for R-R distribution at constant location parameter having PSD width of, (a) x′ =3, (b) x′=4, (c) x′=5. Fig. 16. Sketch of the different sections for velocity analysis. Fig. 17. Behaviour of studied PSD during hopper discharge on (a) average particle velocity at center as a function of PSD width, and (b) average part icle velocity at center as a function of location parameter. (Spatial averages are taken every 20 mm vertically and radially for sample time of 0.5 sec).

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Journal Pre-proof Fig. 18. Analysis of radially velocity profiles as a function of PSD width for studied R-R PSDs in d ifferent section of hopper during hopper discharge process. The five rows represent different sections of hopper that are defined in Fig. 12, section 5 (a, f and k), section 4 (b, g, and l), section 3 (c, h and m), section 2 (d, i and n) and section 1 (e, j and o), while the three colu mns represent specific t ime instant namely 5 sec (a -e), 10 sec (f-j) and 15 sec(k-o). Fig. 19. Analysis of radially velocity profiles as a function of location paramter for studied R-R PSDs in different section of hopper during hopper discharge process. The five rows represent different sections of hopper that are defined in Fig. 12, section 5 (a, f and k), section 4 (b, g, and l), sectio n 3 (c, h and m), section 2 (d, i and n) and section 1 (e, j and o), wh ile the three colu mns represent specific time instant namely 5 sec (a -e), 10 sec (fj) and 15 sec(k-o). Fig. 20. Rescaled velocity profiles, (a) as a function of PSD width, (b) as a fun ction of location parameter for studied R-R PSDs. The solid lines are the fits of the rescaled velocity profiles to a power law as u/u max=(y/R) 2 for y ≥ D (outlet diameter).

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Fig. 21. Mass based frequency distribution for lognormal and R-R PSDs having parameters, (a) n=3 and lnσ = 0.33, (2) n=3.5 and lnσ = 0.29, (3) n=4 and lnσ =0.25 and (4) n=5 and lnσ =0.2.

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Fig. 22. Comparative analysis of segregation of lognormal and R-R PSDs having parameters, (a) n=3 and σ = 0.33, (2) n=3.5 and σ = 0.29, (3) n=4 and σ =0.25 and (4) n=5 and σ =0.2.

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Fig. 1. (a) R-R PSDs (cu mu lative mass fraction distributions) for five values of PSD width (n) i.e. 3, 3.5, 4, 4.5 and 5; the location parameter x′ of the five distributions is 5 mm, (b ) Mass based R-R PSDs for five values of n (3, 3.5, 4, 4.5 and 5), (c) R-R PSDs (cu mulat ive mass fraction d istributions) for five values of location

i.e. 3, 3.5, 4, 4.5 and 5; the PSD width (n) of the five distributions is 5 mm, (d) Mass based R-R PSDs for five values of x′ i.e. 3, 3.5, 4, 4.5 and 5. parameter (x′)

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Fig. 2. Schematic of the computational 3D conical hopper.

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Fig. 3. Co mparison of the instantaneous bed height of the experimental [41], numeric study of Zhao et al. [43] and present simulation at time instances of (a) 0, (b) 6, (c) 12, and (d) 18 sec.

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Fig. 4. Co mparison of continuous discharge mass versus time profiles for monosize part icle systems (5 mm) between simulation work of Zhao et al. [43] and present study.

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Fig. 5. Discharged mass versus time pro files for (a) R-R PSDs having constant location parameter (x′=5) with five values of PSD width (n) i.e. 3, 3.5, 4, 4.5, and 5, (b) R-R PSDs having constant PSD width (n=5) with five values of location parameter(x′) i.e. 3, 3.5, 4, 4.5, and 5.

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Fig. 6. Relationship between static bulk density and MFR for investigated R-R PSDs as a function of, (a) PSD width (n), (b) Location parameter (x′).

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Fig. 7. Flo w bulk density variations for investigated R-R PSDs during discharge time (Spatial averages are taken for sample time of 0.5 sec) as a function of, (a) PSD width (n), (b) location parameter (x′).

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Fig. 8. Relat ionship between flow bulk density and MFR for investigated R-R PSDs as a function of, (a) PSD width (n), (b) Location parameter (x′).

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Fig. 9. Variaion of static and flow bulk density for investigated R-R PSDs as a function of, (a) PSD width (n), (b) Location parameter (x′).

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Fig. 10. MFR predicted by the DEM simu lation and Beverloo correlation (with static and flow bulk density) for considered R-R PSD, (a) as a function of PSD width (n), (b) as a function of location parameter ( x′).

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Fig. 11. Normalized mass based mean particle diameter inside hopper with respect to time for, (a) R-R distribution at constant location parameter (x′=5) having different PSD width, (b) R-R distribution at constant PSD width (n=5) having different location parameter.

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Fig. 12. Normalized mass-based particles fraction inside hopper during discharge process for constant location (x′=5) R-R distribution having different PSD width, (a) n=3, (b) n=4 and n=5.

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Fig. 13. Normalized mass-based particles fraction inside hopper during discharge process for constant width (n=5) R-R distribution having different location parameter, (a) x′=3, (b) x′=4 and (c) x′=5.

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Fig. 14. Velocity profile at four time instances (namely, t = 0, 5, 10, and 15 sec) for R-R distribution at constant location parameter having PSD width of, (a) n=3, (b) n=4, (c) n=5.

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Fig. 15. Velocity profile at four time instances (namely, t = 0, 5, 10, and 15 s) for R-R distribution at constant location parameter having PSD width of, (a) x′ =3, (b) x′=4, (c) x′=5.

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Fig. 16. Sketch of the different sections for velocity analysis.

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Fig. 17. Behaviour of studied PSD during hopper discharge on (a) average particle velocity at center as a function of PSD width, and (b) average part icle velocity at center as a function of location parameter. (Spatial averages are taken every 20 mm vertically and radially for sample time of 0.5 sec).

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Fig. 18. Analysis of radially velocity profiles as a function of PSD width for studied R-R PSDs in d ifferent section of hopper during hopper discharge process. The five rows represent different sections of hopper that are defined in Fig. 12, section 5 (a, f and k), section 4 (b, g, and l), section 3 (c, h and m), section 2 (d, i and n) and section 1 (e, j and o), while the three colu mns represent specific t ime instant namely 5 sec (a -e), 10 sec (f-j) and 15 sec(k-o).

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Fig. 19. Analysis of radially velocity profiles as a function of location parameter ( x′) for studied R-R PSDs in different section of hopper during hopper discharge process. The five rows represent different sections of hopper that are defined in Fig. 11, section 5 (a, f and k), section 4 (b, g, and l), section 3 (c, h and m), section 2 (d, i and n) and section 1 (e, j and o), wh ile the three colu mns represent specific time instant namely 5 sec (a -e), 10 sec (fj) and 15 sec(k-o).

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Fig. 20. Rescaled velocity profiles, (a) as a function of PSD width, (b ) as a function of location parameter for studied R-R PSDs. The solid lines are the fits of the rescaled velocity profiles to a power law as u/u max=(y/R) 2 for y ≥ D (outlet diameter).

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Fig. 21. Mass based frequency distribution for lognormal and R-R PSDs having parameters, (a) n=3 and lnσ = 0.33, (2) n=3.5 and lnσ = 0.29, (3) n=4 and lnσ = 0.25 and (4) n=5 and lnσ =0.2.

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Fig. 22. Comparative analysis of segregation of lognormal and R-R PSDs having parameters, (a) n=3 and ln σ = 0.33, (2) n=3.5 and lnσ = 0.29, (3) n=4 and lnσ =0.25 and (4) n=5 and lnσ =0.2.

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Journal Pre-proof Table 1. Parameters considered in simulations [43]. Parameter

Value

Spread parameter or PSD width (n)

3, 3.5, 4, 4.5, 5

Location parameter (x′)

3, 3.5, 4, 4.5, 5

Density, ρp (kg/m3 )

2460

Young’s modulus, Poisson’s ratio,

(

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0.1

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0.25 0.6

Coefficient of static friction of particle - wall (-)

0.6

Coefficient of rolling friction of particle–particle (-)

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Coefficient of rolling friction of particle–wall (-)

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Coefficient of static friction of particle - particle (-)

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Coefficient of restitution of particle–particle (-) 0.99

0.05 0.99 0.99

Time step, Δt (s)

2 x 105

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Table 2. MFR comparison of lognormal and R-R PSDs. Particle size distribution

PSD parameter

MFR (Kg/s) 70

Journal Pre-proof (PSD) R-R Lognormal R-R Lognormal R-R Lognormal R-R Lognormal

1.64 1.70 1.60 1.65 1.57 1.63 1.54 1.59

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n=3 lnσ= 0.33 n=3.5 lnσ= 0.29 n=4 lnσ= 0.25 n=5 lnσ= 0.20

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Journal Pre-proof Highlights 

Discharge behavior of Rosin-Rammler particle-size distributions was studied by DEM

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Flow rate increases as PSD width increases and decrease with location parameter

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The PSD width affects segregation more than location parameter for considered PSDs

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The velocity profile of Rosin-Rammler PSDs represents typical parabolic nature.

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DEM mass flow rate prediction shows a good agreement with the Beverloo

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correlation

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