Discrete particle simulation in horizontally rotating drum:

Parametric uncertainty quantification of physical parameters

from granular material

Pedro Miguel Andrade Fernandespedro.m.andrade.fernandes@tecnico.ulisboa.pt

Instituto Superior Tecnico, Universidade de Lisboa, Portugal

June 2017

Abstract

This work deals with the dynamics of the granular flows in rotating cylindrical drums using thediscrete element method (DEM) to create a three-dimensional computational model. The methodhas the capacity to track each particle that is in the drum and determine the dynamic state ofeach one. This method use a robust way to calculate the interactions and the movements of theparticles, however, it has a high computational cost, especially when the systems are large (highnumber of particles). The study performed in this work was focused on the uncertainties from theinput parameters of the computational model. The analysed input parameters are related to thephysical characteristics of the particles that influence the existent interactions on the granular flows,such as friction coefficients, restitution coefficients, Young’s modulus, Poisson’s ratios and others inputparameters. The propagation of uncertainty from the model input parameters to the output solutionsare quantified using a non-intrusive spectral projection, based on the polynomial theory of the chaosexpansion that allows the determination of stochastic solutions from a set of deterministic simulationof the computational model. With a defined range of uncertainty that are introduced in the inputparameters, this method determines the propagation of the uncertainty throughout the model andthe stochastic solutions of the model which allows a uncertainty quantification with the capacity tounderstand how the input parameters has impact on the granular flows.

Keywords: Discrete element method; Rotational drum; Granular flows; Uncertaintyquantification; Stochastic analyses.

1. Introduction

Granular media exhibit a variety of dynamic be-haviours on numerous industrial processes and alsoon natural phenomena. Granular materials are thesecond-most used material in industry processes af-ter water, according to [1]. The common industriesthat manipulate powder and granular media are,for example, the pharmaceutical industries whereit is used the process of mixing and coating toproduce pills and others pharmaceutical products[2, 3, 4]. The food industry has some processes likehandling and mass transport [5]. Some treatmentsand incineration of municipal solid waste have todeal with granular media too [6]. There are manyother processes that need to handling with granu-lar material like chemistry, metallurgic and manyothers processes [7, 8, 9, 10]. Naturals phenom-ena like avalanches and landslides are generatedby the granular media dynamics and properties,these phenomena are often the cause of many catas-

trophes and a better approach of these problemscan help to give more safety to the society. It isvery easy to enumerate processes and phenomenawhich the granular materials behaviour is predomi-nant. Throughout this work the study is focused onthe behaviour of particles within a rotating drum,where many of the industries mentioned above usethis type of configurations for their processes. Inthis way, a deeper study about the granular mate-rials dynamics and about the parameters that haveimpact on their behaviour, can provide a develop-ment in the mentioned subjects.

Granular flows have complex behaviours which isa issue for the study of granular dynamics. Thegranular behaviour depends on the way that it ishandled and Pouliquen et al. [11] explain howthese behaviours can be divided on different dy-namics. When strongly agitated, the granular me-dia behaves like a gas. On the opposite situation,when the granular flow is very slow and the defor-

1

mation occurs at low velocity, it is named of quasi-static regime. Between these two regimes the gran-ular material have the behaviour of a liquid flow.Some studies were developed in order to describeand predict granular dynamics on liquid regime(dense flow) with the implementation of Navier-Stokes equations combined with rheology methods[12, 13] and also free-mesh models were developed.The results of these models were satisfactory, how-ever it has good results just for the dense regime.

The Discrete Element Method - DEM was ini-tially developed by Cundall [14] where the methodwas applied to rocks and other granular materialslike sand. This method is simple but very effectiveand robust, where the trajectory of each particle in-side the system is calculated based on all the forcesacting on it and integrating with the Newton’s sec-ond law of motion and the kinematic equation forthe position. The forces considered are usually thegravitation, contact forces due to collisions, solid-solid interactions such as electrostatics, Van derWaals or cohesive forces and fluid-solid interactionsin multiphase flows. The limitations present in themethods presented above do not exist in DEM, be-cause it is possible to study either systems wherethe granular flow is dense, such as in dispersed sys-tems where the collisions are more frequent [15].However, when the system have a large amount ofgranular material, the computation of the model be-comes heavy. The model requires a set of physicalparameters to determine the interactions behaviourlike the materials proprieties (e.g. Young’s modulusand Poisson’s ratio) and interaction proprieties (e.g.friction and restitution). Usually these parametersand others assumptions are introduced in computa-tional models as ideal assumptions. If studies takein account these assumptions, the results will havea higher impact and a better perception of the sys-tems behaviour.

The uncertainty quantification has been appliedin several studies that involve computational mod-els, with the aim of understanding how the uncer-tainties that are introduced through the input pa-rameters of the model propagate through the pro-cesses. In computational simulations the uncer-tainty can have origin from several factors such asmodel structure, modelling assumptions, constitu-tive laws, model parameters, inputs, domain geome-try and boundary/initial conditions. These sourcesintroduce uncertainty into the computational modelor into input parameters that, consequently, propa-gate the uncertainty to the output solutions. Thus,it is necessary to quantify and understand the ef-fect of this propagation from stochastic solutions.One of the methods that is frequently used to ob-tain the uncertainty quantification is the MonteCarlo method, which perform a statical analysis

on deterministic solutions from the computationalmodel with randomly selected conditions. MonteCarlo technique is a robust and generic method,besides it has a high computational cost even witha low number of parameters [16, 17]. An alternativemethod approach to Monte Carlo method is basedon the Polynomial Chaos (PC) expansion [18] topredict the uncertainty propagation along the timeand the physical space. This methodology presentstwo models of spectral projection, Intrusive Spec-tral Projection (ISP) and Non-Intrusive SpectralProjection (NISP) that consists in analytical andnumerical projections of the stochastic equationsinto an orthogonal basis of the polynomials. TheISP method have the advantage of being efficienton solving problems with stochastic variables, butit is necessary to reformulate the governing equa-tions of the deterministic model. The NISP ap-proach have the advantage of using the original de-terministic model, where throughout a set of simu-lations at specifics collocation points, it is obtainedthe stochastic solutions [19].

The main objective of this work is evaluate theinfluence of the uncertainty into the physical param-eters defined as the inputs parameters on granularflow behaviour, in a horizontal rotating drum. Ini-tially the objective is to develop a computationalmodel of the system and preform its validation, inorder to have a deterministic model to obtain thenecessary solutions to determine the stochastic so-lutions and consequently the statical results such asprobability density function (PDF) and confidenceintervals.

This article is organized in the following manner:in the next section it is exposed the models usedin the development of the work and the conditionsof the computational model, such as physical inputsand geometric proprieties. The principal results ob-tained are exposed in the Section 3 and the mainconclusions are presented in the Section 4.

2. ModelsThe study to be carried out in this dissertation

has as main subjects the dynamics of the granularmedium in rotating drums, modelling of the systemand the quantification of the uncertainty. This sec-tion describes the background of each subject suchas governing equations and computational methodsapplied in this work.

2.1. Discrete Element Method - DEMThe process of particles dynamics calculation

with DEM models consists in determine the initialsettling state (positions, velocities and contacts ofthe granular particles). Then, it is applied move-ment at the boundaries that originate new forcesbetween the particles and the walls of the system,where the model uses the governing equations to

2

determine the new state. This cycle is performedat each time-step until the defined simulation timeis achieved [20]. For the deterministic model a com-mercial software (Star-CCM+) was used which thephysics and numerical procedure will be presented.The linear momentum of particles is defined by thesecond Newtons Law that allows it to calculate theposition and velocity of each particle through theapplied forces:

mid~vidt

=∑

~Fi (1)

with the mass of the particle mi and the velocityof the particle ~vi. ~Fi is a force that each particleis subjected to, which can be a gravitational force,contact one or others. The angular momentum isdefined by the following equation:

Iid~ωidt

=∑

~Ti (2)

where Ii is the moment of inertia and ~ωi is the angu-lar velocity. ~Ti is the torque that is applied in theparticle caused by contact forces during collisionsand it is expressed as follows:

~Ti = ~rc × ~Fc − µr|~rc||~Fc|~ωi|~ωi|

(3)

where the vector ~rc represents the vector from theparticle center of gravity to the contact point andthe coefficient of rolling resistance is represented byµr. This coefficient is related to the resistance thata body offers to start the rotation around the pointof contact with a surface.

𝑑𝑣𝑖

𝑑𝑡

𝐶𝑀𝑖

𝑑𝜔𝑖

𝑑𝑡

𝐹𝑐

𝑇𝑖

Figure 1: Basics of the body motion.

In the Fig.1 is shown schematically the balanceof momentum of a body. It is shown, in a represen-tative way, the vectors of force, torque and acceler-ations associated with the movements and externalforces that a body typically suffer in granular flows.

The granular material considered in this studyis composed of spherical particles. Thus, the col-lisions are computationally calculated through thesoft sphere approach. In this collision model thewhole process of collision or contact is solved bynumerical integration of the equations of motion

where the deformation during the collisions doesnot exist. The deformation is replaced with theoverlap δ of two particles as is shown schematicallyin Fig.2(a). This method use a system of spring-

(a)

𝐹𝑛

𝐹𝑡

(b)

Figure 2: Scheme of a collision between two parti-cles.

dashpot in order to represent the usual effect thatexists on the collisions, where the repulsive forcegenerated by the elastic deformation is expressedby equating the overlap distance with compressionof a spring. During the real collisions occurs anenergy loss due to the deformation process and itdepends on the deformation speed. Thus, this ef-fect of dissipative energy is expressed by a dash-pot[21].

The formulation is based on the Hertz-Mindlintheory of contact that it is a non-linear extensionof the spring-dashpot model [22, 23]. The force ofcontact is divided in two components, normal andtangential as is presented in Fig.2(b):

Fcontact = Fn + Ft (4)

The normal force between to particles ~FnABis for-

mulated as follows:

~FnAB= (−knδnAB

− ηn~vnAB~nAB)~nAB (5)

where the relative velocity vector between the twoparticles in collision is represented by ~vnAB

(~vnAB=

~vnA− ~vnB

), δnABis the displacement of particle

caused by the normal force, kn is the normal com-ponent of the spring stiffness and ηn is the normalcomponent of the damping coefficient. ~nAB is theunit vector in the direction of the line from the cen-ter of the two particles. The tangential force ~FtAB

is formulated as follows:

~FtAB= −kt~δtAB − ηt~vsAB

(6)

where kt, ηt and ~δtAB corresponds to the stiffnessof the spring, damping coefficient and displacement,respectively, on the tangential direction. ~vsAB

is theslip velocity of the contact point. The stiffness forthe model are calculated from physical properties,such as Young’s modulus E and Poisson’s ratio ν.Thus, the stifness kn is determined as follows:

kn =4

3Eeq

√Req (7)

3

where Eeq and Req are the equivalent Young’s mod-ulus and the equivalent radius, respectively, andthey are calculated through the following expres-sions:

Eeq =1

1−ν2A

EA+

1−ν2B

EB

(8)

Req =1

1RA

+ 1RB

(9)

The tangential stiffness kt is obtained as follows:

kt = 8Geq√Reqδ

1/2n (10)

where Geq is the equivalent shear modulus that iscalculated by:

Geq =1

2(2−νA)(1+νA)EA

+ 2(2−νB)(1+νB)EB

(11)

The damping coefficients calculations are related tothe coefficient of restitution e which is regarded asa physical parameter of the material [24]. Thus, thedamping coefficients are obtained by the followingexpression:

η =−ln(e)√π2 + ln(e)2

√5Meqk (12)

where Meq is the equivalent particle mass and it isdefined as:

Meq =1

1MA

+ 1MB

(13)

where MA and MB are the particles mass of eachparticle in collision. The Eq.12 is used by the modelto reach both tangential and normal components.These expressions presented are the governing equa-tions for the collisions particle-particle. When thecollisions are between particles and the walls, theexpressions to use are the same, besides the equiva-lent mass and radius are assumed as Req = Rp andMeq = Mp.

2.2. Simulation ConditionsThe three-dimensional deterministic model con-

sist in a cylindrical rotating drum horizontallyorientated within granular material composed byspherical particles. The geometric and physicalproperties are based on the work of Yang et al. [25].

Z

𝐃𝐂

L

Figure 3: Scheme of the drum.

The scheme of the drum is presented at Fig.3and the geometric parameters used are given inTab.1. The drum ends caps have a preponderant

Properties

Dc [mm] 100

Dp [mm] 3

L [mm] 16

Filling level [%] 35

Table 1: Geometric model properties.

effect on granular flows in horizontal drums [26]. Toavoid this effect it would be necessary a drum withhigher dimension and consequently a higher num-ber of granular material which it would increase thecomputational cost. In order to overcome this is-sue, periodic boundaries at the ends caps to achievethe same results of a higher drum are imposed, be-cause of the free movement along the axial direc-tion [27]. The time-step defined for this model is0.001 seconds which it was verified that it is anenough time-step to perform the simulation. Be-sides that the model uses substeps to iterate thecollisions calculus. The physical properties of the

Properties

Friction coefficient (µ) [-] 0.5

Restitution Coefficient (e) [-] 0.9

Rolling Resistance Coefficient (µr) [-] 0.002

Young’ Modulus (E) [Pa] 1 x 107

Poisson’s Ratio (ν) [-] 0.29

Density (ρ) [kg/m3] 2500

Table 2: Physical properties of the granular media[25].

granular material used in this work is presented atthe Tab.2. These properties are inputs of the de-terministic model. The drum is modelled with alu-minium properties which the Young’s modulus Ecis 68 GPa, the Poisson’s ratio νc is 0.33 and thedensity ρc is 2702 kg/m3.

2.3. Non-Intrusive Spectral Projection - NISPThe Polynomial Chaos (PC) expansion is a non-

sampling based method that uses a spectral pro-jection of the random variables to determine theevolution of uncertainty in a dynamical system. Inthe NISP method the output stochastic process usesa set of deterministic solutions from the computa-tional model. The following mathematical formula-tion is based on the works of Reagnan et al. [28]and Najm et al. [19]. First, it is necessary to de-scribe each uncertainty parameter explored in thisstudy with a probability density function (PDF),

4

namely characterize the uncertainty parameter witha mean, a standard deviation and a PDF type.

Let Xi be a model uncertain parameter and f acorresponding solution variable. For a defined PDFof the variable Xi, X(ξi) can be expressed with aPC expansion as follows:

Xi(ξi) =

p∑j=0

cXij Ij(ξi) i = 1, ..., N (14)

where cXij are the known expansion coefficients and

Ij are the orthogonal polynomials of order, for j =0, ..., p. This method can be generalized for N inde-pendent random variables (X1, , XN ), for each vari-able there will be an associated stochastic dimen-sion ξi, i = 1, ..., N . This forms a multi-dimensionalstochastic space where the stochastic solution f(~ξ)can be represented by the following PC expansion:

f(~ξ) =

P∑j=0

cfjΦj(~ξ) (15)

where cfj are the unknown PC expansion mode co-

efficients of f(~ξ) and Φi are the multi-dimensionalorthogonal polynomials. The total number of termsP + 1 in the PC expansion is given by:

P + 1 =(N + p)!

(N !p!)(16)

where p is equal to the order of the maximum poly-nomial order. The inner product between two or-thogonal polynomials is expressed as follows:

〈ΦiΦj〉 =

∫ΦiΦjW (~ξ)d~ξ = 〈Φ2

j 〉δij (17)

where W (~ξ) is the weighting function and δij isthe Kronecker delta function. The expansion co-efficients cfj can be formulated through a Galerkinprojection in Eq.15 doing the inner product at bothsides of the expression by Φj , the expression ob-tained is:

cfk =〈f(~ξ)Φk〉〈Φ2

k〉=

∫f(~ξ)ΦkW (~ξ)d~ξ

〈Φ2k〉

, k = 0, ..., P

(18)In the NISP approach the deterministic modelsolution fd is evaluated for different values of(X1(ξ1), ..., XN (ξ1)). The coefficients cfj can be cal-culated from the set of deterministic solutions us-ing the Eq.18 and the PC expansion f(~ξ) can bereconstructed with Eq.15. The integral in the nu-merator of Eq.18 is numerically solved by a Gaussquadrature. Thus, each random variable ξi mustbe sampled on Si collocation points (e.g. Gauss),therefore the required multi-dimensional samples is

defined as follows:

S =

N∏i=1

Si (19)

Finally, the coefficients cfj are obtained by:

cfk ≈Σ

S1,...,SNj1,...,jN

fd(Xj1,...,XjN

)Φk(ξj1 ,...,ξjN )∏N

i=1 qji〈Φ2

k〉(20)

where k = 0, ...P , ξji and qji , j = 1, .., Si arethe Gauss quadrature points and correspondingweights, sampled on the random variable ξi.

3. ResultsIn this section is exposed the validation results,

where the deterministic model is compared to liter-ature sources. Besides that, the uncertainty quan-tification results are also presented in this part ofthe work.

3.1. ValidationThe validation of the computational model is re-

quired, in order to get confidence in the results. Thevalidation process consists in achieve consistent re-sults with the work of Yang et al. [25]. The outputsanalysed of the model are focused on the dynamicsof the granular flow for different speed rotations.The particle velocity υ is normalized by the transi-tional velocity of the drum wall as follows:

υ∗ =2× υω ×Dc

(21)

where ω is the rotational velocity and Dc is thedrum diameter. The Fig.4 shows the results ob-tained by the present computational model and theresults of the literature. The Fig.4(a) presents theevolution of the mean normalised velocity with timefor a rotation speed of 5 rpm and it is observedthat the behaviour of the actual model have a sat-isfactory approach. On the model developed on the

R. Y. YangComputational Model

υ*

1

2

3

4

5

Time (s)0 2 4 6 8 10 12 14 16 18

(a) Evolution of the meannormalised velocity with timefor a rotation speed of 5 rpm.

R.Y. YangLog-NormalComputational Model

P(υ*)

1

2

3

4

υ*0 0.25 0.5 0.75 1 1.25 1.5

(b) Distribution of normal-ized particle velocities for arotation speed of 80 rpm.

Figure 4: Comparison between the own model re-sults and the literature results.

present work, the distributions is obtained with a

5

Parameters Mean Value Variation +\− [%] Standard Deviation

µpp\µpw [-] 0.5 20 0.05

e [-] 0.9 10 0.045

µr [-] 2.00 x 10−3 20 0.2 x 10−3

E [Pa] 1.00 x 107 20 0.1 x 107

ν [-] 0.29 20 0.029

Table 3: Input physical parameters of the granular material and its uncertainty proprieties.

mean in time of the distributions where the simu-lation time is 100 seconds and at each 0.1 secondit is obtained a instantaneous distributions of nor-malized particle velocities. The Fig.4(b) shows thedistribution of the normalized particles velocity fora rotation speed of 80 rpm. In this figure are pre-sented the results obtained in the literature (R. Y.Yang and Log-Normal) and the results of the devel-oped model. The approximation obtained is satis-factory, although the little difference noticed can beexplained by the fact that the model developed hasmore points in the curve and the simulation time is100 seconds. The actual model is in this way vali-dated which can be used for the remaining study.

3.2. Uncertainty Quantification

This section is focused to the study of uncertaintyand its influence on granular flow dynamics. Forthis study it is necessary to define the uncertaintyof the output parameters with mean values, stan-dard deviation and distribution type. The param-eters and its uncertainty proprieties are exposed atthe Tab.3, where the PDF defined for all the pa-rameters was a semi-circular Beta PDF distribu-tion. For this type of distribution the numerical ap-proach to determinate the coefficients cfj should bea Gauss-Jacobi quadrature points. The stochasticsolutions presented in this section are approximatewith five points of integration and with a fourth-order polynomial. The output results consideredin this section are the distribution of normalizedparticle velocities, the behaviour of the mean veloc-ity along the time and the angle θ that is definedby the slope that the granular bed exhibits. Withthe stochastic solution it is possible create staticaldata such as confidence intervals (CI) of 95% andPDF which characterizes the influence of the un-certainty on the granular flow dynamics o rotatingdrums. For a higher consistent results, the actualstudy considers two different regime flows (two ro-tation velocities), rolling (40 rpm) and cascading(150 rpm).

The particle velocity distribution is a method ofanalysis that allows to analyse the dynamic be-haviour of the granular flow inside the horizontal ro-tating drum. From the introduction of uncertaintyin the parameters, the distribution is analyzed by

creating a relation between the input parametersand the velocity of the particles in the drum throughconfidence intervals. The Fig.5 presents the influ-ence of each input parameters on the distributionthrough the CI and the stochastic mean for therolling regime. It is necessary to take into account

P(υ*)

1

2

3

4

υ*0 0.25 0.5 0.75 1 1.25 1.5

Stochastic MeanCI

(a) µpp

P(υ*)

1

2

3

4

υ*0 0.25 0.5 0.75 1 1.25 1.5

Stochastic MeanCI

(b) µpw

P(υ*)

1

2

3

4

υ*0 0.25 0.5 0.75 1 1.25 1.5

Stochastic MeanCI

(c) e

P(υ*)

1

2

3

4

υ*0 0.25 0.5 0.75 1 1.25 1.5

Stochastic MeanCI

(d) µr

P(υ*)

1

2

3

4

υ*0 0.25 0.5 0.75 1 1.25 1.5

Stochastic MeanCI

(e) E

P(υ*)

1

2

3

4

υ*0 0.25 0.5 0.75 1 1.25 1.5

Stochastic MeanCI

(f) ν

Figure 5: Stochastic Mean and CI of the distribu-tion of normalized particle velocities for each pa-rameter with a rotation velocity of 40 rpm.

the meaning of the CI for these distributions, sinceit is a normalized distribution, i.e. the integral ofthe distribution curve must be equal to 1. The only

6

curve respecting the normalization is the stochasticmean curve, whereas the limits defined by the CIdo not respect this normalization. Thus, the limitsof the shaded region (CI limits) indicates where thedistribution line may be, but always respecting thenormalization. In other words, when it is occurs asituation that the distribution line, at some velocityregion, is close to the lower CI limit there will be an-other region where the curve will be close to the up-per limit, in order to maintain the integration equalto 1. These results shows that some parametershave a higher influence than others. In the Fig.5(a)is shown the CI calculated from the uncertaintyof the friction coefficient between particle-particleµpp which shows a distribution with a higher ICcompared to the other distributions. The µpw alsoproves to be an influential parameter, see Fig.5(b),compared to the other results, Fig.5(c)-(f).

P(υ*)

1

2

3

4

υ*0 0.25 0.5 0.75 1 1.25 1.5

Stochastic MeanCI

Figure 6: Stochastic Mean and CI of the distribu-tion of normalized particle velocities for the frictioncoefficient particle-particle parameter µpp with a ro-tation velocity of 150 rpm.

The Fig.6 shows the CI and stochastic mean ofthe distributions for the friction coefficient particle-particle parameter µpp at the cascading regime. It isobserved that this CI is lower than the one obtainedat rolling regime. Which means that the increaseof the rotation velocity of the drum, decreases theuncertainty influence on the distribution.

The evolution of the mean normalised velocityoutput is analysed with the same statical process asthe shown above for the distributions, with stochas-tic means and CI. The Fig.7 presents the stochasticmean and CI of the mean particle velocity alongthe time that were obtained on the study of theboth frictions coefficients for the rolling regime. Itis clearly verified that the impact on this outputis higher for the friction between particle-particle(Fig.8(a)), where the CI have a more pronounced

υ*

0.2

0.4

0.6

0.8

1

t [s]0 20 40 60 80 100

Stochastic MeanCI

(a) µpp

υ*

0.2

0.4

0.6

0.8

1

t [s]0 20 40 60 80 100

Stochastic MeanCI

(b) µpw

Figure 7: Stochastic Mean and CI of the evolutionof the mean normalized velocity with time for a ro-tation speed of 40 rpm.

area compared to the friction between particle-wall(Fig.8(b)). Continuing the analysis of the Fig.8(a),it is also seen that, in general, the distance betweenthe stochastic mean and the upper CI limit is higherthan the distance to the lower CI limit. For thecascading regime the results are shown in the Fig8.The CI of both results are smaller than the rollingregime which means that for the uncertainty de-fined by the influence of the frictions in the meannormalized velocity of the particles is reduced withthe increase of the rotation velocity. This aspecthas already been evidenced in the distributions ofnormalized particle velocities.

υ*

0.1

0.2

0.3

0.4

0.5

t [s]0 20 40 60 80 100

Stochastic MeanCI

(a) µpp

υ*

0.1

0.2

0.3

0.4

0.5

t [s]0 20 40 60 80 100

Média EstocásticaIC

(b) µpw

Figure 8: Stochastic Mean and CI of the evolutionof the mean normalised velocity with time for a ro-tation speed of 150 rpm.

7

The configuration of the granular flow inside thedrum is also an important data to be analysed. Away to study this information is obtaining the an-gle of the bed θ. The aim of this result is to un-derstand the effect of the uncertainty on the angle.As in the previous results the particle-particle fric-tion coefficient µpp exhibited a more relevant impactin relation to the other parameters, it is showedin the Fig.9 the PDF distribution of the angle θfor the the uncertainty input parameter µpp in thetwo considered regimes. Firstly, it is observed that

P(θ)

0.05

0.1

0.15

0.2

0.25

θ [º]21 22 23 24 25 26 27 28 29 30 31

(a) 40 RPM

P(θ)

0.1

0.2

0.3

0.4

θ [º]32 33 34 35 36 37 38 39 40 41 42

(b) 150 RPM

Figure 9: PDF distribution of the angle θ for theuncertainty parameter µpp.

in the rolling regime (Fig.9(a)) the granular bedpresents the mean angle of 25.5◦ and the cascad-ing regime (Fig.9(b)) have a mean angle of 36.5◦,approximately 10 degrees of difference between thetwo regimes. In addition to the differences in themeans of the angle, the shape of the curves arealso different for each regime. In the distributionobtained in the rolling regime the range of angles

covered by the curve is approximately between 21◦

and 30◦, while in the cascade regime, the distribu-tion curve has a range between 34◦ and 39◦. Onceagain it is verified that the increase of the rotationvelocity has impact on the uncertainty influence.

In order to be more aware of the granular flowdynamics and the impact of the uncertainty, a lo-cal analysis is performed. This analysis is focusedon the behaviour of the particles along the drumradius. The Fig.10 shows a scheme of the stud-ied area. This study consists of determining ve-

Figure 10: Scheme of the drum region in study.

locity profiles along the drum radius through av-erages in space and time, applying the uncertaintyin the input parameters and verifying the influencethrough the stochastic solutions. The Fig.11 showsthe stochastic mean and CI for the velocity profile(normalized velocity) over the drum radius obtainedin the study of the friction coefficient µpp, for theboth regimes in study. The resolution of the curvesare apparently low, due to the fact that the sphereshave a diameter of 0.003 m and increasing the res-olution would introduce an excess of noise. Fromthe results presented it is possible to verify that theparticles have a higher velocity on the surface ofthe agglomerate and near the walls of the drum,whereas in the inner region of the agglomerate theparticles velocity are minimal. It is also verifiedthat the rolling regime has a higher CI, Fig.11(a)relative to the IC obtained for the cascade regime,Fig.11(b). These results intensify the analysis ofprevious results which it was observed a decreaseon the impact of the uncertainty when the drumvelocity increase. In both cases the size of the ICvaries along the radius, where the higher values aresituated in the region of the surface and near thewalls, i.e. where the particles velocity are high.

The main results of this work were exposed inthis section. In the next section is given a briefsummary of the obtained results and the importantinformations to be retained from this work.

4. Conclusions

The developed work consisted of a study on thebehaviour of granular flow inside a cylindrical ro-tary drum, with the introduction of uncertainty

8

Nor

mal

ized

Vel

ocity

, υ*

0

0.2

0.4

0.6

0.8

1

Radius [m]0 0.01 0.02 0.03 0.04 0.05

Stochastic MeanCI

(a) 40 RPM

Nor

mal

ized

Vel

ocity

, υ*

0

0.2

0.4

0.6

0.8

1

Radius [m]0 0.01 0.02 0.03 0.04 0.05

Stochastic MeanCI

(b) 150 RPM

Figure 11: Stochastic mean and CI for the velocityprofile over the drum radius obtained in the studyof the friction coefficient µpp.

parameters. The uncertainty quantification wasperformed with the NISP which introduces uncer-tainty to the input parameters from a computa-tional model also developed in this work. Withthe DEM it was possible to create a system to ob-tain deterministic solutions, to be used in the NISPmodel and determine the stochastic solutions. Thestochastic solutions obtained allowed to perform asatisfactory uncertainty quantification with signifi-cant results. In this work it was possible to under-stand how the granular flow behave inside a drumfor different regimes. The influence of the physi-cal parameters were evidenced through the intro-duction of the uncertainty on the inputs. It wasconcluded that not all physical parameters understudy have the same impact on the granular flowdynamics. For the defined distribution of the uncer-tainty parameters, it was observed that the param-eter that have a significant effect was the frictioncoefficient, principally between the particle-particleinteractions. The drum velocity showed to be im-portant in the uncertainty effect which the increaseof the drum rotation provides a decrease of the un-certainty influence. For high rotation velocities themovement transmitted to the granular material isdominant and the uncertainty is less perceptible, forthe degree of uncertainty used in this work.

As future work there are multiple subjects thatcan be explored. The study developed in this workcan be applied in more specifics processes that usesgranular materials, such as processes existing on

industrial process like pharmaceutical coating orheating processes. A better vision of the uncer-tainty impact on specifics operations can helpingto improve and develop solutions to the process.Another idea to the future work is introduce moreuncertain variables to the system or create sys-tems with higher complexity (e.g. heat transfer andchemical reactions).

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