Scaling of bulk material properties and equipment

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Delft University of Technology

FACULTY MECHANICAL, MARITIME AND MATERIALS ENGINEERING

Department Marine and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl

This report consists of ## pages and # appendices. It may only be reproduced literally and as a whole. For commercial purposes only with written authorization of Delft University of Technology. Requests for consult are only taken into consideration under the condition that the applicant denies all legal rights on liabilities concerning the contents of the advice.

Specialization: Transport Engineering and Logistics Report number: 2012.TEL.7739

Title: Scaling of bulk material properties and equipment

Author: R.J.W. van Gils

Title (in Dutch) Schalen van bulk materiaal eigenschappen en werktuigen

Assignment: Literature Confidential: no

Initiator (university): Ir. S.W. Lommen and Dr.ir. D.L. Schott Supervisor: Ir. S.W. Lommen and Dr.ir. D.L. Schott Date: December 3, 2012

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Delft University of Technology

FACULTY OF MECHANICAL, MARITIME AND MATERIALS ENGINEERING

Department of Marine and Transport Technology Mekelweg 2 2628 CD Delft the Netherlands Phone +31 (0)15-2782889 Fax +31 (0)15-2781397 www.mtt.tudelft.nl

Student: R.J.W. van Gils Assignment type: Literature Mentor: Ir. S.W. Lommen Report number: 2012.TEL.7739

Specialization: TEL Confidential: No

Creditpoints (EC): 10

Subject: Scaling of bulk material properties and equipment

With performing tests on scale with dry bulk material, the question is always how to design the experiments on scale to have the outcome that is required. There are many variables playing a role within the material itself, such as particle properties (e.g. size, volume, density), bulk properties (e.g. angle of repose, shearstrength, particle size distribution), interparticle forces (e.g. liquid bridges, electrostatic forces), etc.

Depending on the objective of an experiment, one might scale the equipment size (e.g. 1:10). That means that the volume of the equipment will be scaled 1:103. This leads to questions such as: Do we have to scale the particle size or volume with the same ratio?

What will happen with the bulk material behavior in that case? Or what happens to the interparticle forces, and the forces on the equipment. How are they being scaled?

Some work was done in our group using dimension analysis to assess scaling of equipment and structures (e.g. V.A. Ritman (report number 92.3.TT.3029)).

Your assignment is to investigate and make an overview of the literature about scaling of bulk material properties and equipment. That comprises amongst others the following:

Describe the variables and parameters that play a role in scaling of experimental setups for bulk material handling (from particle properties to drive power).

How do these variables and parameters relate to each other.

What scaling possibilities are there, and what are the consequences for the other parameters and variables.

What has been published in this field of interest, how do researchers deal with scaling? Elaborate the options with 2 examples, e.g. grab.

Classify and compare the literature found

It is expected that you conclude with a recommendation for further research based on the results of this study.

The report should comply with the guidelines of the section. Details can be found on blackboard. The mentors,

Ir. S.W. Lommen Dr.ir. D.L. Schott

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Summary

Scaling is a widely used concept in engineering. It is mainly used to predict the outcome of full scale behaviour by performing experiments on a small scale model and convert the results to full scale. In the research into bulk solids properties and the interaction with equipment, scaling is not as well implemented as in other engineering fields. However, scaling is an effective and low cost concept for the design of bulk solids handling equipment and for the research into the behaviour of bulk solids materials. Therefore the main objective of this research is making an overview of the literature regarding scaling of bulk properties and equipment. The most important findings in the literature considering this subject together with relevant background information are discussed in this report.

Scaling of bulk material and equipment can be divided in two groups: scaling regarding physical experiments and scaling regarding computer simulations with the discrete element method (DEM). DEM is a numerical method based on a simple mathematical model, described by the laws of motion. Within these groups two kinds of models can be distinguished, small scale models and full scale models.

In experiments small scale models are mainly used, since experiments at full scale are expensive. The majority of the studies concerning small scale experiments resulted in empirical scaling laws. These are deduced from a series of experiments and describing the measurements in the investigated region. Extrapolation to full scale is necessary, however, in studies this is often not included. Scaling laws which are deduced from an analytical description of the process imply that also material properties should be scaled, causing problems in selecting the proper material. Computer simulations with the discrete element method allow the use of full scale models. However, due to the proportionality between the number of particles and the computational time, the number of particles is the limiting factor. Therefore scaling in DEM simulation mainly concerns the up-scaling of the particles. Scaling laws for the up-scaling process can be deduced from the equations of motion used in DEM. The link between physical experiments and DEM simulation is made by the tests, like the shear test and the angle of repose test, which are used in the calibration and validation process of a DEM simulation.

Recommendations for further research concerning DEM simulations are to verify the used scaling laws, investigate the limited particle size with respect to equipment dimensions and to standardize the calibration and validation process. Concerning physical experiments, recom-mendation are to investigate the validity of the empirical scaling laws outside the investigated regions.

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Summary (in Dutch)

Schalen is een veel gebruikt begrip in de techniek. Het wordt voornamelijk gebruikt om het gedrag op ware grootte te voorspellen met behulp van schaal experimenten waarvan de resultaten worden vertaald naar ware grootte. Er is nog geen algemene toepassing van schalen in onderzoek naar het gedrag van vaste bulk materialen in interactie met werktuigen. Echter onderzoeken met schaalmodellen zijn geschikt en kosten besparend voor onderzoek naar het gedrag van bulk materialen en het ontwerp van werktuigen. Zodoende is het hoofddoel van deze literatuur studie het maken van een overzicht van de beschikbare literatuur welke betrekking heeft op het schalen van bulk materiaal eigenschappen en werktuigen. Dit rapport bevat de belangrijkste bevindingen samen met relevante achtergrond informatie.

Het schalen van bulk materiaal en werktuigen kan hoofdzakelijk worden verdeeld in twee groepen, namelijk schalen in fysieke experimenten en schalen in computersimulaties met de discrete elementen methode (DEM). DEM is een numerieke methode die is gebaseerd op een eenvoudig wiskundig model, gebasseerd op de bewegingsvergelijkingen. Binnen deze groepen zijn twee soorten modellen te onderscheiden, (kleine) schaal modellen en ware grootte modellen.

In experimenten wordt vooral met schaal modellen gewerkt, aangezien experimenten op ware grootte kostbaar zijn. De meerderheid van de onderzoeken met schaal modellen hebben geresulteerd in empirische schalingswetten. Deze zijn afgeleid van een serie van experimenten en beschrijven alleen het onderzochte gebied. Extrapolatie naar ware grootte is vereist, maar dit is vaak niet een onderdeel van het onderzoek. Schalingswetten die zijn afgeleid van een analytische beschrijving van het proces impliceren dat ook materiaal eigenschappen moeten worden geschaald, dit resulteerd in problemen bij het selecteren van het juiste materiaal voor de experimenten.

Computersimulaties met de discrete elementenmethode maken het gebruik van ware grootte modellen. Vanwege de proportionaliteit tussen het aantal deeltjes in een simulatie en de rekentijd, is het aantal deeltjes een beperkende factor. Daarom betreft schalen in DEM simulaties voornamelijk het opschalen van de deeltjes. Schalingswetten voor het opschaal proces kunnen worden afgeleid uit de in de DEM gebruikte bewegingsvergelijkingen. Testen, zoals afschuiftest en de rusthoek test, zijn de schakel tussen fysieke experimenten en DEM simulatie, ze worden gebruikt voor de kalibratie en validatie van een DEM simulatie.

Aanbevelingen voor verder onderzoek met betrekking tot DEM simulaties zijn het controleren van de gebruikte schaalwetten, het onderzoeken naar de maximale deeltjesgrootte met betrekking tot de afmetingen van de werktuigen en een standaard voor het kalibratie en validatie proces. Met betrekking tot fysieke experimenten is de aanbeveling om de geldigheid van de empirische schaalwetten buiten de onderzochte gebieden te onderzoeken.

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List of symbols

a Acceleration m · s−2

c Damping coefficient [kg · s−1]

Cr Coefficient of restitution [−]

dp Particle diameter [m]

ds Diameter of a sphere with the same surface as the particle [m]

dv Diameter of a sphere with the same volume as the particle [m]

D, Do Silo diameter, Outlet orifice diameter [m]

E Modulus of elasticity N · m−2

f Inter-granular friction [−]

Fn Froude number [−]

F Contact force vector [N ]

Fcoh Cohesive force vector [N ]

g, g Gravitational acceleration (vector) m · s−2

G Shear modulus N · m−2

h Granular bed thickness [m]

I Inertia moment kg · m2

k Stiffness coefficient N · m−1

k The Beverloo constant [−]

L Length [m]

m, M Mass [kg]

n Rotational speed rev · min−1

n Normal unit vector [−]

r, ˙r, ¨r Position, velocity and acceleration vector [m] ,m · s−1_{, m · s}−2

R Radius [m]

t, T Time [s]

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t Tangential unit vector [−]

T Torque vector [N · m]

v Velocity m · s−1

w Work of adhesion J · m−2 = kg · s−2

W Flow rate kg · s−1

α Geometric scaling factor [−]

β Compressibility factor [−]

β Damping [−]

γ Hopper angle, Inclination angle [◦] δ Effective internal friction angle [◦]

δ Overlap [m]

δa Arch shape coefficient [−]

Porosity [−]

θ, θh, θo Angle of repose, heap, outflow [◦]

θ, ˙θ, ¨θ Angular position, velocity and acceleration vector [rad] ,rad · s−1_{, rad · s}−2 µd Coefficient of dynamic friction [−]

µr Coefficient of rolling friction [−]

µs Coefficient of static friction [−]

ν Poisson’s ratio [−]

π Dimensionless factor [−]

ρb, ρp Bulk density, particle density kg · m−3

σ Stress tensor N · m−2

σy unconfined yield stress N · m−2

σ1 Major principle stress N · m−2

φ Internal friction angle [◦]

φw Wall friction angle [◦]

ψ Sphericity [−]

ω Rotational speed rad · s−1

Some symbols are provided with subscripts or superscripts, these are explained here. The subscriptsn andtrefer to the normal and tangential components,i andj refer to particles i and

j. Reference values are indicated by a0 as subscript and the superscript∗ implies an equivalent

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Introduction

Scaling is a widely used concept in engineering. It is mainly used to predict the outcome of full scale behaviour by performing experiments on a small scale model and convert the results to full scale. The main advantage of scaling is the reduction cost during the design stage. Building a prototype on full scale is significantly more expensive than one in model scale, e.g. cars, air planes or ships. A fault in the design can be easily adjusted and tested again. However, scaling has some difficulties, because some model parameters are not, or limited scalable. In the design of a ship for example, gravity and the viscosity of water are hard to scale. Due to this kind of problems, results of model experiments cannot directly convert to full scale quantities, usually. Scaling laws are required and the results sometimes have to be adapted with correction factors for full scale.

In the field bulk solids, scaling is not as well implemented as in other engineering fields. One of the reasons for this is the fact that many scaling laws are based on an analytical description of the process. But for many of the processes in the field of bulk solids there is no analytical model available. The scaling laws which are already known are mostly empirical ones. With arise of the discrete element method (DEM) by Cundall and Strack (1979), a discrete model for the behaviour of bulk solids, the opportunities for conducting scaling laws have grown. Nevertheless there are still no widely adopted scaling laws available. Due to the DEM, it is possible to simulate the statics and dynamics of bulk solids on computers. Simulating full scale applications requires a lot of computational power, which is not (yet) available or cannot be done in reasonable time. Therefore, engineers try to reduce the calculation time by up-scaling the particle size in the simulation. The effects of this approach and its limitations needs further research.

The main objective of this research is making an overview of the literature regarding scaling of bulk properties and equipment. The most important findings in the literature considering this subject are discussed. Relevant background information is given in chapter 2, including important properties in the interaction between bulk solids and equipment, an introduction to the discrete element method and an explanation on dimensional analysis. Scaling in the field of bulk solids is mainly divided in two categories, physical experiments and discrete element method (DEM) simulations. In Chapter 3 the different ways of scaling within these two categories, and the problems which arise with it are discussed. Chapter 4 discusses a dimensional analysis for DEM, together with a general investigation procedure for a simulation. Finally in chapter 5 some conclusions will be conducted and a recommendation for further research is given.

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Chapter 2

Background

Before scaling can be discussed in detail, some background knowledge is provided in this chapter. In section 2.1 all relevant properties in the interaction between bulk solids materials and equipment are presented. Also some experimental methods for the determination of these properties are discussed in section 2.2. The discrete element method (DEM) is used for computer simulations of bulk solids, an introduction is given in section 2.3. In section 2.4 a comprehensive explanation is given on dimensional analysis, which are used for the derivation of scaling laws. This chapter concludes with a discussion in section 2.5.

2.1 Bulk material and equipment properties

The properties which play a role in scaling will be discussed in this section. Starting in section 2.1.1 with considering the bulk solids as whole, also called bulk of macro scale. Bulk or macro behaviour is caused by particle or micro properties, this can be divided in properties of the particle itself in section 2.1.2 and inter-particle properties, section 2.1.3. A piece of equipment used in the handling of bulk solids can be described by parameters,which are listed in section 2.1.4. Properties which describe the interaction between bulk solids and equipment are discussed in section 2.1.5.

2.1.1 Bulk properties

Density

The bulk density ρb kg · m−3, is defined by McGlinchey (2005) as the ratio between the mass

of the particles in a sample and the volume of the sample, i.e. ρb =

masssolids

volumetotal

(2.1) The size of the voids (the spaces) between the particles, depends on the compaction. The ratio between these voids and the total volume of the bulk solids is described by the porosity [−]

= volumespaces/voids volumetotal

(2.2) Increasing the compaction results in a decreasing porosity and a increasing bulk density. Cohesion

The stickiness of bulk materials with itself is called cohesion, it is caused by inter-particle forces as described in section 2.1.2. A cohesive bulk solid does gain strength on compaction. Non-cohesive bulk solids are also called free-flowing.

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Angle of repose

The angle of inclination of the free surface of a heap of bulk solids, has a certain maximum. Attempting to form a heap with a steeper inclination angle will result in a collapse of the heap. This maximum angle is called the angle of repose θ [◦]. Depending on the measurement method, slightly different angles can be found one material. In section 2.2 an apparatus is discussed which measures two angles of repose, the angle of repose heap θh, and angle of repose outflow θo.

Shear strength

A bulk solid has a resistance to shear, which is indicated by the shear strength. It is caused by friction and the interlocking of particles.

Effective internal friction angle

The effective angle of internal friction δ follows from measurements with the shear cell tester, see section 2.2. The presence and size of the non-flowing areas in silos is heavily influenced by this angle. It should be noted that δ is not a real physical angle in the bulk solid, it is defined as the tangent of the ratio between the shear stress and the normal stress (Rhodes, 2008)

Internal damping

Energy dissipation in bulk solids is caused by the damping properties of the particles. Damping arises if the coefficient of restitution is smaller than one.

2.1.2 Particle properties

Size and particle size distribution

The definition of a granular medium as given by Forterre and Pouliquen (2009) is “A collection of macroscopic particles, their size being typically greater than 100 µm”. This limitation in size corresponds to a limitation in the type of interaction between particles. A granular medium means non Brownian particles, which interact solely by friction and collision. If the particles are smaller, Baxter et al. (2000) calls it a powder. For smaller particles, other interactions like Van der Waals forces, like humidity, start to play a role. Rhodes (2008) gives various methods for determining the particle size of non-spherical particles. Often the size distribution of a bulk solids can be described by a statistical distribution, like a normal distribution. The average aspect ratio, which is the ratio between the largest and smallest diameter, indicates the width of the particle size distribution.

Shape

There are several ways to define the shape of a particle, one way is defining the sphericity ψ [−] as by Shamlou (1988) as follows:

ψ = surface area of a sphere having the same volume as the particle surface area of the particle

Which is equal to

ψ = dv ds

2

where dv [m] is the diameter of a sphere having the same volume as the particle and ds [m] is

the diameter of a sphere having the same surface as the particle. Another way is the shape-factor, which is te ratio between the (mean) particle diameter and the diameter of a sphere having the same volume.

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Density

In agreement with equation 2.1 the particle density ρp kg · m−3 is given by

ρb=

masssolids

volumeparticles

(2.3) The relation between the bulk density and the particle density is then written by

ρb = ρp(1 − ) (2.4)

Modulus of elasticity

The modulus of elasticity E N · m−2_{described the tendency of a particle to deform temporary}

due to an applied force. Poisson’s ratio ν [−] describes the relation between the transverse strain to the axial strain. The shear modulus G N · m−2_{is the tendency of a particle to shear when}

contrary force are applied. The relation between E, ν and G can be written as follows: G = E

2 (1 + ν) (2.5)

2.1.3 Inter-particle properties

Coefficient of restitution

The coefficient of restitution Cr [−] describes the damping during a collision. A Cr equal to one

indicates that there is no energy dissipation at all, on the other hand, a Cr equal to zero implies

that all the energy is dissipated. Coefficient of static friction

The relation between the maximum tangential force and the normal force between two particles is described by the coefficient of static friction µs [−].

Coefficient of dynamic friction

The dynamic friction coefficient µd [−] is the friction coefficient of two particles during sliding.

Coefficient of rolling friction

A torque acting on a particle will result in a roll motion, this will induce a resistance torque in opposite direction. The relation between these two torques are related by the coefficient of rolling friction µr [−].

Cohesive forces

Liquid bridges, electrostatic forces and van der Waals forces causes cohesive bulk behaviour. For many bulk solids the cohesive forces are negligible, and those materials are called non-cohesive.

2.1.4 Equipment properties

Type, shape and size

In the handling of bulk solids many types of equipment are used, like silos, belt conveyors, grabs and mills. The shape and size description depends on the equipment type, this is illustrated with an example of a silo. In Holdich (2002) the shape of a silo is described by “A hopper is the conical, or converging part of a powder storage vessel; the bin is the parallel sided section,

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usually cylindrical or rectangular, and the word silo is used to cover the entire vessel.”1 _Resultant

descriptive dimensions are, height, width or diameter, the angle of the hopper and the length and width or diameter of the silo outlet orifice.

Surface roughness

The surface roughness is a measure of the texture of the surface. The friction coefficient between the particles and the surface is also dependent on the surface roughness.

Drive power

Equipment like conveyor belts can be characterise by their drive power. The power which is needed to drive the equipment is indicating for the size of the equipment and amount of bulk solids which is handled.

2.1.5 Bulk-equipment interaction properties

Size ratio

The size ratio between the particles and equipment is often defined as the ratio between a characteristic length of the equipment and the mean or maximum particle diameter (in the case of spheres). A characteristic length, is a dimension of the equipment which is important in the relation between the particle and equipment piece. The orifice diameter is the characteristic length in case of a silo, because the relationship to the maximum particle diameter is important for arching of the bulk material above the orifice.

Coefficient of static friction

The relation between the maximum tangential force and the normal force between a particle and a piece of equipment is described by the coefficient of static friction µs[−].

Coefficient of dynamic friction

The dynamic friction coefficient µd [−] is the friction coefficient of a particle sliding on a surface.

Adhesion

The stickiness of bulk material to other materials is called adhesion. Adhesion between bulk solids and equipment is mainly caused by interlocking.

Wear rate

Different types of wear can be distinguished, in the handling of bulk solids impact wear and abrasive wear frequently occur. Material properties of the bulk solids and the equipment, and the (relative) velocity are important parameters.

Induced velocity and acceleration

The induced velocity and acceleration is the change in velocity or acceleration caused by the interaction between the particle and the equipment.

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2.2 Determining properties

Experimental tests are small tests for the determination of particle and bulk properties, like the angle of repose or the shear strength. The results of these tests can be used for the design of bulk handling equipment. A variety of methods for the characterisation of bulk solids materials are described in the literature such as by McGlinchey (2005) and Rhodes (2008). Important parameters in characterisation are, shape, shape factors, size, size distribution and porosity, which are all illustrated in section 2.1.2.

The shear strength of a bulk solids can be tested with a shear tester. The Jenike shear cell is a frequently used shear tester and is described by i.a. Rhodes (2008) and Schulze (2008) and is depicted in figure 2.1. For a bulk sample with a certain density under a normal load the shear

Figure 2.1: Principle of the Jenike shear cell (dimensions in [mm]) (Schulze (2008)) force can be measured, which results in the shear stress. Performing tests for a bulk solids with varying density and normal load results in a data set which can be used for the design of silos. A comparable test is used for the determination of the wall friction angle φw. In that case the base

ring in figure 2.1 is replaced by a sample of the wall material.

As discussed in section 2.1.1 the exact angle of repose depends on the measurement technique. Bierwisch et al. (2009) describes the angle of repose box, depicted in figure 2.2, which can be

Figure 2.2: Angles of repose box (Bierwisch et al. (2009))

used for the measurement of the angle of repose outflow θo and the angle of repose heap θh.

The upper box is filled with a sample of the bulk solids, the bottom of the box is pulled away (indicated by the arrow in figure 2.2) after which both angles are formed and can be measured.

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2.3 Introduction to the discrete element method

The discrete element method is a numerical method for calculating the interaction and motion of a system containing discrete particles. The DEM, founded by Cundall and Strack (1979), is extensively discussed in the literature by i.a. Zhou et al. (2002), Radja¨ı and Dubois (2011) and O’Sullivan (2011). An overview of is given in figure 2.3. The main part is the calculation cycle, starting with detection of contacts (as well as particle-particle contacts as particle-equipment contacts), then calculate the interaction forces, update the particle positions and start over. The behaviour of a DEM model crucially depends on the applied force laws, the particle shape

Figure 2.3: Overview of the discrete element method (Raji and Favier (2004))

and the size distribution (Bierwisch et al. (2009)). Often a simple contact model between two particles is used as can be seen in figure 2.4. The fundamental equation of the translation of a

Figure 2.4: Simple contact model for two spherical particles with springs, dashpots, a friction slider and a cohesion bond.

spherical particle i with radius Ri[m] is expressed as follows:

mir¨i = ki

X

j=1

(Fn,ij+ Ft,ij) + mig (2.6)

where mi [kg] and ¨ri [m] are the mass and acceleration of particle i at time t [s]. The involved

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and the inter-particle contact forces between particle i and j in normal and tangential direction, Fn,ij [N ] and Ft,ij [N ]. Because these contact forces act on the contact point between particles i

and j, a torque Ti [N · m] is generated. The equation for the rotational motion is then described

by: Iiθ¨i = ki X j=1 Tij (2.7)

where ¨θi rad · s−2 is the angular acceleration and Ii kg · m2 is the inertia moment, for a

sphere given by Ii= 2₅miRi2. According to the contact model presented in figure 2.4, the contact

forces and torque can be written as:

Fn,ij = knδn· nij + cn· ˙rn,ij+ Fcohesion (2.8)

Ft,ij = ktδt· tij+ ct· ˙rt,ij (2.9) Tij = (Ft,ij· Ri) × nij − µr|Fn,ij| Ri ˙ θi ˙ θi (2.10)

with in these equations:

|δt,ij| if r˙t,ij= 0 and δt,ij6= 0

0 Otherwise

(2.18)

where k N · m−1_{, c [kg · s}−_{1], δ [m], µ}

r [−] and ˙r m · s−1, are stiffness coefficient, damping

coefficient, overlap, rolling friction coefficient and velocity, respectively. The unit normal and tangential vector at the contact point are indicated by n and t. In preceding equations the subscripts n and t indicate the normal and tangential components, subscripts i and j refer to particle i and j. The cohesive force Fcoh needs only taken into account when the bulk solid

material is cohesive, and can be written as (Bierwisch et al. (2009)) Fcoh=

√

8πwE∗_(R∗_δ

n)3/4· nij (2.19)

in which w J · m−2 is the work of adhesion per unit of contact area. The tangential force is limited by Coulomb’s laws, according to:

Ft,ij ≤ µs|Fn,ij| (2.20)

where µs [−] is the static friction coefficient. The stiffness and damping coefficients are given by:

kn= 4₃E∗ p R∗_δ n kt= −2 q 5 6β √ m∗ q 2E∗p R∗_δ n (2.21) cn= −8G∗ p R∗_δ n ct= −2 q 5 6β √ m∗ q −8G∗p R∗_δ n (2.22)

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Where R∗ [m], m∗ [kg], E∗ N · m−2_{and G}∗ _{N · m}−2_{are the equivalent values of the radius,}

mass, elasticity and shear modulus, respectively. The damping is described by β [−]. These values can be calculated as follows:

R∗ = RiRj Ri+ Rj (2.23) m∗ = mimj mi+ mj with mi = 4₃πR3iρp (2.24) 1 E∗ = 1 − νi2 Ei + 1 − νj 2 Ej (2.25) 1 G∗ = (2 − νi) Gi +(1 − νj) Gj (2.26) β = _p ln Cr ln2Cr+ π2 (2.27)

where Cr [−] is the restitution coefficient and ν [−] Poisson’s ratio. The variables E, ν, ρp, µs,

µr and Cr are referred to as the DEM parameters (in case of a non-cohesive bulk solid). Values

of the DEM parameters may differ when considering particle-equipment contact. Calculating ¨ri

and ¨θi with equations 2.6 and 2.7, and assuming that velocities and accelerations are constant

over each time step in the simulation, the new particle position can be determined with: ˙ri t +1₂∆t = ˙ri t − 1₂∆t + ¨ri(t) ∆t (2.28)

ri t +1₂∆t = ri(t) + ˙ri t + 1₂∆t ∆t (2.29)

˙

θi t +1₂∆t = ˙θi t − 1₂∆t + ¨θi(t) ∆t (2.30)

θi t +1₂∆t = θi(t) + ˙θi t +1₂∆t ∆t (2.31)

Other integration schemes than the one used above, are also possible. To ensure a stable numerical process the time step ∆t [s] should be smaller than the critical time step tc [s]:

tc=

πRmin

0.163ν + 0.8766 r ρp

G (2.32)

In which Rmin is the minimum radius. This equation is deduced from the Rayleigh wave speed

of force transmission on the surface of elastic bodies (Grima and Wypych (2011)).

The DEM parameters, must be calibrated in order to get meaningful bulk behaviour in a simulation. The most common way of calibration, is to model experimental tests, as discussed in section 2.2, with a DEM package, these tests are called simulation tests. The experimental tests are performed in a laboratory with the real bulk material while measuring and observing it’s behaviour. The same tests are modelled with DEM and the DEM parameters are calibrated until the desired behaviour is reached. To verify the obtained, calibrated, DEM parameters, another test needs to be performed.

Lommen et al. (2010) investigated the calibration of a penetration test. The experimental tests results of a specific shaped tool are compared to DEM simulation results. Using varying values for the DEM parameters static friction (inter-particle and equipment-particle) and rolling friction, they are able to define four sets with different friction coefficients which approximate the experimental results the best. An experimental test and a DEM simulation with another shaped tool is used to select the best set of parameters. Finally a angle of repose DEM simulation test has been done to compare the four parameters sets. The results showed large differences in the angle of repose, which indicates the importance of the calibration process.

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2.4 Dimensional analysis

Scaling laws can be deduced from dimensionless numbers, the Buckingham’s-Pi theorem is used to derive dimensionless numbers from the relevant system variables. This theorem is briefly discussed in this section, an extensive explanation can be found in Pankhurst (1964), Gerstel (2003) and Ritman (1993).

Assume that a system can be described by the variables Q1, Q2, ... ,Qn. Each variable Qj

(j = 1..n) can be written as a product of power terms of the basic variables A1, A2, ... ,Am

(there must hold m < n). The definition of a basic variable is that its unit can not be expressed by units of other basic variables. Length L [m], mass M [kg] and time T [s] are examples of basic variables, so Qj = m Y i=1 Aaij i j = 1..n (2.33)

According to Buckingham’s-Pi theorem for a system with n variables and m basic variables, the minimum number of dimensionless factors is equal to r = n − m. The dimensionless factors πq

(q = 1..r) can be written as a product of power terms of the variables Qj, so

πq= n Y j=1 Qkjq j q = 1..r (2.34)

Together with equation 2.33 this results in: πq= n Y j=1 m Y i=1 Aaij i !kjq (2.35) Or, πq= [Aa111· A a21 2 · ... · Aamm1]k1q· [Aa112· A a22 2 · ... · Aamm2]k2q· ... · [Aa11n · A a2n 2 · ... · Aammn]knq (2.36)

with q = 1..r. The factors πq must be dimensionless, so there must hold:

πq= A01· A02· ... · A0m (2.37)

This results for each πq in a set of m equations, reading:

a11· k1q+ a12· k2q+ ... + a1n· knq = 0 a21· k1q+ a22· k2q+ ... + a2n· knq = 0 .. . am1· k1q+ am2· k2q+ ... + amn· knq = 0 (2.38)

In total there are m × r equations. In matrix form it becomes:      a11 a12 . . . a1n a21 a22 . . . a2n .. . ... . .. ... am1 am2 . . . amn      ·      k11 k12 . . . k1r k21 k22 . . . k2r .. . ... . .. ... kn1 kn2 . . . knr      = 0 (2.39)

This can be written as:    a11 . . . a1,n−r .. . . .. ... am1 . . . am,n−r   ·    k11 . . . k1r .. . . .. ... kn−r,1 . . . kn−r,r   +    a1,n−r+1 . . . a1n .. . . .. ... am,n−r+1 . . . amn   ·    kn−r+1,1 . . . kn−r+1,r .. . . .. ... kn1 . . . knr   = 0 (2.40)

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Or,

A0· K0+ A1· K1 = 0 (2.41) Choosing K0 to be the unit matrix, ensures that the number of occurrences of each parameters Qj in πq is as low as possible. The solution for K1 can be found by solving:

K1 = A1−1

· −A0

(2.42) The solutions for the r dimensionless factors πq can be read from:

K =K

0

K1

(2.43) Finally, scaling laws can be deduced from the obtained dimensionless numbers. It should be noted that interchanging the columns of matrix A will change the dimensionless factors π, but not the scaling laws.

2.5 Discussion

In order to get usable results form scaling experiments, all the properties that influences the process needs to be considered. Of each property the effect of scaling needs to be known. Properties which are dimensionless, like friction coefficients, are not influenced by scaling, these are equal in both systems (the original and the scaled system). Other properties are affected by scaling, how these are affected, depend on the scaling laws. In the case of physical scaling experiments some properties are more easily to scale than others. Making a small scale model of a equipment piece is not so hard, this could also be thought of the particle size, but if the particles are already of the order of micrometers, breaking becomes an issue. In the case of DEM simulation, almost any property is scalable, but the particle shape can only be approximated, because the simulation must be made in a reasonable time frame. Bulk behaviour is caused by various properties, scaling a property can have an influence on the bulk behaviour, which is not always desirable. Therefore, the effects of scaling properties, during experiments and simulations, should be monitored.

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Chapter 3

Scaling

Scaling plays an important role during research and design in all fields of engineering, likewise its used in the research into the behaviour of bulk materials and in the design of bulk handling equipment. This chapter starts with an explanation of the concept of scaling, scaling laws and scaling factor in section 3.1. Research into the behaviour of bulk materials with the help of scaling, can be performed in two ways: By means of physical experiments (from here called experiments) as discussed in section 3.2, and by means of a discrete element method simulation on a computer (from here called DEM simulation) as discussed in section 3.3. Each of these two sections are divided in 3 subsections, treating small scale mode, large scale models and testers. Section 3.4 will discuss some remarkable findings about the literature presented in this chapter.

3.1 Definition

In order to get usable information from a system at scale model, the scale model must be similar to the original system. This similarity means that the dynamic properties of both systems are equal. Results of experiments with scale models must be converted to full scale quantities by means of scaling laws. Scaling laws can be conducted from a theoretical description of the system, a useful tool is dimensional analysis as is investigated by Ritman (1993). The theoretical model is often a simplification of reality, which results in scaling errors. If a theoretical description of the system is not available, empirical scaling laws can be conducted from a series of slightly different experiments. When empirical scaling laws are, for example, extrapolated to a much larger scale, the applicability is in question.

The (geometric) scaling factor α is such defined that a measure of length at model scale equals α times the same measure of length at full scale, or

x0 = αx (3.1)

In which x is a measure of length at full scale and x0 is (the same) measure length at model scale. All quantities at model scale will be denoted with an accent (’). Usually α is smaller than one, and so the scale model is smaller than the original system. Sometimes up-scaling an existing system is desired, in that case α will be bigger than one.

3.2 Experiments

Research to the scalability of bulk solids with the aid of experiments is discussed in here. In section 3.2.1 the state of the art small scale model experiments are discussed. A small scale model is mostly a simplification of the full scale equipment, this causes scaling errors. To acquire knowledge about these errors, measurements at full scale equipment have been performed, as is discussed in section 3.2.2.

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3.2.1 Small scale models

A model is an simplification of the reality, so is a small scale model. When extracting knowledge from small scale experiments one have to take into account the assumptions that are made. If the assumptions are known, the effect of these assumptions are well understood and the acquired results are explainable, it is possible to say something about the full scale system.

The research of Beverloo et al. (1961) is the first one that investigated the outflow rate of silos extensively. Flat-bottomed silos were used with varying the diameter, filling height, orifice outlet size and shape. Seven different seeds (numbers 1 to 7 in figure 3.1) and 3 sand fractions where tested. Bulk and particle properties of which the writers thought it may influence the flow are: Bulk density of the packing, true specific density, void fraction, angle of repose, number of particles per gram, the particle size distribution, filling height of the silo and a shape-factor (see section 2.1.2). After the outflow has become stationary, the outflow rates were determined in gram per minute. All experiments were repeated 3 times in order to guarantee repeatability. The experiments indicated that the filling height and diameter of the silo have no influence on the flow-rate. Experiments with the sand fractions revealed that the influence of the particle size on the outflow rate is negligible when the ratio between the orifice diameter Do and the particle

diameter dp is larger than 20. In figure 3.1 the outflow rates W are plotted as a function of the

circular orifices area, all materials give a straight line. Using dimensional analysis the authors conducted a general formula for the flow-rate through circular orifices. This empirical formula, which is known as Beverloo’s equation, reads:

W = 0.58ρb

√

g(Do− kdp)2.5 (3.2)

in which W is the flow rate [kg/s]1, g the gravitational acceleration [m/s2] and k [−] is a constant which is called the Beverloo constant. For all investigated seeds, the Beverloo constant was equal to 1.4, only the sand fractions gave another value of 2.9, but the authors were unable to give an explanation for this. With six other experiments a comparison is made between the calculated rates and the experimental results. The mean deviation is 5% with a maximum of 12.5%, so the presented formula is said to be good for a “calculated estimation”. Also the formula is in good agreement with comparable results in earlier papers.

Figure 3.1: Influence of orifice area on flow-rate of various materials (Beverloo et al. (1961)) Johanson (1972) discusses scale-up parameters for silos, to obtain knowledge for meaningful small scale experiments. The flow of bulk material is mainly depended on properties of the bulk material and on the compacting pressure (which is not a property of the material). The effect of pressure on the most important bulk and particle properties were discussed. These properties are:

1

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Effective angle of internal friction δ, angle of internal friction φ, wall friction angle φw, unconfined

yield stress σy, compressibility factor β and bulk density ρb. For scale up, dimensionless numbers

in the scale model must be the same as in full scale. The only property of the aforementioned which is not dimensionless is σy, so the author defines a new dimensionless number σ_σ1_y in which

σ1 is the major principal stress. Each of these dimensionless numbers change with the pressure,

the change of these numbers over the range of pressures must be equal in the model as in real silo. The pressure range in a cylindrical silo of which the filling height is larger than the diameter is given by: Doρbg 2 < σ1 ≤ Dρbg 4 tan φwk (3.3) The factor k is equal to 1−sinδ_1+sinδ. Depended on the purpose of the experiments the condition of equality for all dimensionless numbers can sometimes be relaxed. With 3 numerical examples the author shows that in the most cases when using the same bulk material in small scale as in full scale, it is not possible to keep the earlier defined ratio σ1

σy the same in small and full scale.

Hereby small scale experiments can produce completely different behaviour than would exhibit in the full scale silo. The author also states that to assure the same dynamical effects during outflow in the model silo as in the full scale one the acceleration of the bulk solids must be the same. The acceleration (in [m/s2]) in the conical part of a cylindrical silo is given by

a = v2 4 Do

tan γ (3.4)

where v is the average velocity [m/s] and γ is the hopper angle [◦]. The equation for the outflow rate W kg · s−1 reads (in case of a circular orifice)

W = ρb1₄πDo2v (3.5)

Assuming that the small scale and full scale silo meet the relation given by equation 3.1, combining equations 3.4 and 3.5 results in

W W0 = ρb ρ0_b Do D0 o 2.5 = ρb ρ0_b 1 α 2.5 (3.6) which assures equal accelerations in small and large scale models. If the same bulk material in the scaled system as in the original system is used (ρ0_b = ρb), the scaling law for the outflow rate

will become W0 = α2.5W .

A new developed model for silo outflow for silos with a conical hopper is presented by Oldal et al. (2012). The outflow is considered to be a process of the formation and collapse of arches in the hopper. From this model an equation similar to Beverloo’s equation (equation 3.2) is conducted, however, without empirical constants. The equation reads

W = π 6ρb

q

2gδa(Do− dp)5 (3.7)

in which δa is the shape coefficient of the arch above the outflow orifice and depends on the

material in the silo. The factor 6 arises from averaging the velocity field of the particles. The equation holds for orifices diameters of

7dp ≤ Do≤ 0.6D

Since this model uses the phenomena of arching, it can only be used in case of funnel flow (in mass flow there are no arches). An experiment with wheat was performed to determine the influence of the hopper angle γ on the outflow rate W . For each hopper angle the orifice diameter was the same. The measured and calculated outflow rates are presented in figure 3.2, also the

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Figure 3.2: Measured and predicted discharge rates (Oldal et al. (2012))

prediction according to equation 3.2 (Beverloo et al. (1961)) and equation 3.5 (Johanson (1972)) are displayed.

The thick red line in figure 3.2 represents the transition from mass flow to funnel flow, as observed during the experiments. The three predictions are all only valid for funnel flow. The agreement between the prediction by the authors and the one by Beverloo et al. (1961) can be clearly seen and agrees with the measurements. However, Johanson (1972) is only applicable for a small range of hopper angles.

A prediction of the discharge mass flow rate of coarse granular material from mass flow hoppers (bigger than 0.5 [mm]) in terms of frictional properties of the material and the effect of wall friction is made by Williams (1977). From the momentum equations, which are not completely solved, a lower and an upper limit is defined for the flow rate. Because these limits differ only circa 20% it is said that these limits can be used for design purposes. Experiments demonstrate a good agreement for coarse particles. On the other hand, for finer materials, the predictions that are made are about 30% too high. A possible explanation here for is that the air drag during outflow, which is not incorporated in the model, becomes significant and results in a reduction of the outflow rate.

Based on the similitude of Froude, which is commonly used in hydraulics, Weber (1968) derives a set of scaling rules. He assumes that elastic deformations are negligible, the bulk material obeys strictly Coulomb’s law and is non-cohesive, and the porosity remains equal. Scaling geometry and particles with α, Weber’s scaling laws are defined as follows:

ρ0_b = ρb E0 = αE ν0 = ν σ0= ασ

t0 =√αt v0 =√αv a0 = a f0 = f (3.8) In which E is the elasticity modulus, ν Poisson’s ratio, σ the stress tensor, t the time, v the velocity, a the acceleration and f is the inter-granular friction which obeys Coulomb’s law (this relation holds for the static and dynamic friction coefficient).

With the introduction of the discrete element method by Cundall and Strack (1979) a new theoretical description of a granular system is available. P¨oschel et al. (2001) uses the viscoelastic grain model to investigate the interacting between contacting particles. They found out that “naive scaling”, scaling the construction and the particle size with α, might cause wrong results. In order to guarantee identical dynamical properties between the model and the original system the material properties has to be scaled too. In table 3.1 these scaling laws are given, the particle density ρp is not scaled. With the assumption that particles are spherical, this model is not very

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original system scaled system all lengths x αx time t √αt elastic constant _ρ E p(1−ν2) α E ρp(1−ν2)

dissipative constant A √αA

Table 3.1: Scaling laws given by P¨oschel et al. (2001)

shaped particles. Even if the particles are spheres, a material has to be found which satisfies the scaled material properties. In section 3.3.1 the results of DEM simulation tests with these proposed scaling laws will be discussed.

Due to the dependency of the strength of fine and cohesive bulk solids on the consolidation stress, small scale experiments with cohesive bulk solids in silos are not possible in a natural gravity field. Considering the bulk solids as a continuum, Molerus and Sch¨oneborn (1976) (which is more extensive than the English version Molerus and Sch¨oneborn (1977)) deduces a scaling expression for small scale experiments with the use of a bunker centrifuge for silos with a conical hopper. In a bunker centrifuge the gravitational acceleration g is replaced by a centrifugal acceleration a (both in [m/s2]). The centrifugal acceleration must be 1/α times the gravity acceleration for a model with scaling factor α. The scaling law for the silo geometry becomes

x0 = g

ax ⇒ α = g

a (3.9)

The particle properties and sizes remains the same, as long as the mean particle diameter is significantly smaller than the orifice diameter. The authors state that the ratio between the orifice Do and the particle diameter dp must be larger than 100 (this is deducted from the Jenike

shear cell tester). An advantage of experiments with a bunker centrifuge is that not the orifice diameter is varied, but the centrifugal acceleration. The construction of a bunker centrifuge

Figure 3.3: Bunker centrifuge, measurements in mm (Molerus and Sch¨oneborn (1976)) is depicted in figure 3.3. To reduce the radial dependence of the centrifugal forces, the ratio between the measurements in figure 3.3 Rm and δR can be increased. At the beginning of each

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run, the silo (in figure 3.3 bunker named), is filled in vertical position. Next, the silo (with a closed orifice) is set in the bunker centrifuge. The material is consolidated at a rotational speed nc in [rev/min], afterwards the centrifuge is stopped and the orifice is opened. Subsequently the

rotational speed is increased until the material starts to flow out of the silo, this speed is listed as nf. Two states of stress can be distinguished in a silo, during filling a active state of stress is

generated which is rearranged to a passive state of stress during emptying. In order to guarantee corresponding states of stresses during filling and emptying between model and full scale silo, the multiples of the centrifugal acceleration during consolidation ac in [m/s2], must be equal to

the centrifugal acceleration during outflow af. Because the rotational speed is proportional to

the centrifugal acceleration this holds that the rotational speeds must be equal to full fill the condition of corresponding stress states, so

ac= af ⇒ nc= nf (ac/f = rω2= r 2πnc/f 60 ) (3.10) Where ω is the rotational speed in [rad/s] and r is the distance to the rotational axis in [m]. From the theory with regard to funnel flow, equation 3.10 is found to be the limiting condition to avoid rat holing in large scale funnel flow silos. By performing several experiments, the correct centrifugal acceleration (or rotational speed) can be found by which the minimum required orifice diameter can be calculated for full scale. However this does not apply for mass flow. Due to the existence of the two states of stress, the required orifice diameter in the filling state is larger than in the discharge state, as is discussed in Grostck and Schwedes (2004, 2005). With a series of two experiments the correct orifice diameter can be established.

In the handling of bulk solids, flow down a inclined plane, like chute flow, is common. The flow of a thin layer of material down a smooth inclined plane is investigated by Augenstein and Hogg (1974). From a theoretical consideration with a simple friction model the authors suppose that the effective coefficient of resistance to flow µ between the plane and a particle is independent of the velocity and of the inclination of the plane γ. The velocity vof the particles is increasing with a constant acceleration, which is dependent on γ and µ. They presume that when v2 is plotted as a function of the distance along the inclined plane z, this would result in a straight line. From the slope of the line, the value of µ can be obtained. Results from experiments with sand confirm the theory, as can be seen in figure 3.4. The maximum variations in the

Figure 3.4: Variation in particle velocity for composite sand along a stainless steel surface (Augenstein and Hogg (1974))

measurement where 2%. The lines in figure 3.4 indicate that for some inclination angles the particles accelerate, however, when the inclination angle is too small, the particles will decelerate. Indicating that there is an angle for which the acceleration will be zero, which is useful when

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controllability of the process is required. Further, the influence of the particle size on the friction coefficient has been investigated. It was found that for the stainless steel surface with the sand µ was decreasing with the particle size. Similar experiments were performed in which the sand flowed down a plane consisting of a stationary layer of the same material, which resulted in a constant friction coefficient. In a succeeding article Augenstein and Hogg (1978) expanded this work with taking into consideration the thickness of the bed of particles. For smooth surfaces the friction coefficient turns out to be also independent of the bed thickness, so the acceleration is constant over the bed thickness. On the other hand, for rough surfaces, the acceleration varies with the bed depth. Two rough beds where considered, rough bed 1 consist of a stationary layer of exactly the same sand and, rough bed 2 consists of a stationary layer of finer sand. In figure 3.5 the results of one of the experiments, of sand flowing down a rough bed 1, is showed. The velocity squared is plotted against the relative bed depth h∗, which is made dimensionless

Figure 3.5: Variation of velocity with distance along flow lines of constant h∗ (Augenstein and Hogg (1978))

with the locally maximum bed depth. The lines indicates constant relative bed depths, which are called flow lines. Two flow regions can be distinguished, the flow at a certain distance h∗ away from the plane is accelerating, and the acceleration increases with h∗. Close to the plane, depending on the roughness of the surface, two phenomena are observed. The plane with rough bed 1 resulted in a constant velocity region, which indicates a fully developed flow. On the other hand, the plane with rough bed 2 resulted in a region with a constant acceleration. The findings are summarized in a empirical scaling law

a(h∗) = ao 0 ≤ h∗≤ h∗b a(h∗) = am " 1 − ( 1 − h ∗ 1 − h∗_b 1.5# h∗_b ≤ h∗≤ 1 (3.11)

In which a0 and am are empirical constants. a0 is equal to zero for rough bed 1, which indicates a

fully developed flow, and constant for rough bed 2, which indicates slip at the surface of the plane. The boundary between the flow regions is indicated by h∗_b, which was constant for experiments with the same roughness. The empirical scaling law (equation 3.11) is only valid for the sand used in these experiments.

The aim of the research by Pouliquen (1999) was to experimentally investigate influence of the inclination angle γ, the roughness of the plane and, the bed thickness h. The particles used

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sizes in Bulk Surface [mm] solids roughness system 1 0.5 ± 0.04 0.5 ± 0.04 system 2 1.3 ± 0.13 1.3 ± 0.13 system 3 1.15 ± 0.1 1.3 ± 0.13 system 4 0.5 ± 0.04 1.3 ± 0.13 Table 3.2: Glass beads used for bulk mate-rial and for the rough surface for different systems studied Pouliquen (1999)

Figure 3.6: The Froude number as a func-tion of h/hstop(γ) for the four systems of

beads and for different inclination angles (Pouliquen (1999))

for the bulk solids flow, as well as the ones used for the roughness of the plane, are spherical glass beds. In table 3.2 the 4 different experiments are shown, each tested for 7 inclination angles (22◦− 28◦). The dimensionless velocity, also called Froude number Fn, is defined as

Fn=

v pgdp

(3.12) In which v is the mean velocity of the granular layer. Plotting the Froude number as a function of _hh

stop for all obtained data resulted in a single straight line, as can be seen in figure 3.6. The

layer thickness is indicated by h, hstop(γ) is the critical thickness when the flow stops, which is

dependent on γ. An explanation for the deviation of one measurement of system 2 was not given. The results can be expressed in a single scaling law

u pgdp

= β h hstop(γ)

(3.13) with β = 0.136 and is independent of the inclination angle, the size of the beads and the plane roughness. From this relation a empirical relation for the dynamic friction coefficient between plane and bulk solids is conducted.

3.2.2 Full scale models

Experiments at full scale are seldom performed, mainly because of the high costs. Nevertheless the results can be very useful in gaining knowledge and validating small scale experiments, experimental tests and design methods.

As a consequence of scaling errors, recommendations relating to the design based on results obtained with small scale experiments, could turn out to be wrong in full scale. Which is demonstrated by W´ojcik et al. (2012). Four different inserts, varying in size and location, were tested at full scale. The authors compared their work with various small scale experiment with inserts in silos at small scale, performed by others. It was found that small scale results of stresses cannot transferred directly to full scale as a result of size effects. These are mainly caused by the pressure level, but also due to difference in the size ratio between the particles and the silo, and the difference in particle size distributions.

Because measurement techniques for flow patterns in silos used for small scale models are not viable at full scale, Chen et al. (2005) describes a technique for flow measurements in full scale silos. A full scale experimental silo filled with iron ore pellets was seeded with radio frequency tags. During outflow the tags were detected at the orifice, from which the residence time could be deducted. With a number of experiments the author proved the reliability and effectiveness of the measurement technique without interfere with the silo’s structure or operation.

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3.3 DEM simulation

Simulation with the discrete element method provides the possibility of modelling equipment at full scale with reduced costs in comparison to full scale experiments. Also detailed information about the process can be obtained, because measurements can be performed at any position. In each time step in a DEM simulation, the force on each individual particle are determined. Depending on the particle shape, size and the amount of bulk solids in a simulated system, the total number of particles can become in the order of millions or even more. Even with the most advanced computers of nowadays, the calculation time would become unacceptably long. To reduce calculation time, particle size and shape are often adjusted, this is the subject of section 3.3.1. In the literature is nothing found about DEM simulations with small scale models. Because this has some potential, it is discussed in section 3.3.2.

3.3.1 Full scale models

In a DEM simulation it is possible to model a piece of bulk solids handling equipment as full scale. It would be also possible to approximate a particle shape quite closely by building-up a particle with multiple spheres. The main problem with these models would be the computational time of the simulation, which can become several years in a large system. Reducing the computational time is a very important subject concerning full scale DEM simulations.

The best option to reduce the computational time in a DEM simulation is to reduce the number of particles by enlarging them. Achmus and Abdel-Rahman (2002) investigated the influence of using one set of calibrated DEM parameters for different scaling factors. The used scaling factors α where 20, 25 and 30, with respect to particle size distribution of Karlsruhe medium sand. A triaxial test was used to calibrate the DEM parameters. Despite of the differences found between the results of the different scaling factors, the authors extracted a single set of DEM parameters. The results of a odometer test showed a remarkable scale effect, from which was concluded that for each scaling factor, a calibration procedure should be carried out.

An investigating to the influence of DEM parameters on the angle of repose was made by Zhou et al. (2002). The container which was used to measure the angle of repose outflow θo and heap

θh is comparable with the one in figure 2.2. The DEM parameters particle diameter, rolling and

sliding friction (both particle-particle and particle-wall), density, Poisson ratio, Young’s modulus, damping coefficient and the container thickness were varied. The size of the container was dependent on the particle size, which keeps the number of particles constant (2000). The DEM results where compared with results from experiments with glass beads with equal diameters as in the computer model. The main conclusion of this research was that the angle of repose increases when either the rolling or sliding friction coefficient is increased. A increase in particle size results in a decrease of the angle of repose. From the results an empirical formula was conducted.

The effect of the particle size during outflow from a wedge-shaped hopper is investigated by Balevi˘cius and Ka˘cianauskas (2008). The only variable in this research was the mean particle diameter dp, the ratios of the minimum and maximum diameter to their mean diameter was

kept constant, just as all the DEM parameters. The mean particle diameters and its number are given in figure 3.7. The particles in this figure are coloured by using a scalair f ,

fi =

X

i6=j

| F_ij | (3.14)

which describes the state of the granules after filling by representing the sum of the absolute inter-particle contact forces Fij acting on a particle. The outflow simulations showed that the

model with the least number of particles had the lowest discharge rate and on the hopper wall the normal stress was lower and the shear stress was higher. The authors concluded that the

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(a) dp= 0.0298[m] (b) dp= 0.0378[m] (c) dp= 0.0648[m]

Figure 3.7: State of granules. Models contain respectively 20400, 10000 and 1980 particles (Balevi˘cius and Ka˘cianauskas (2008))

increase of the particle diameter induces a material homogeneity causing an artificial friction which results in an increased dissipation.

Sakai and Koshizuka (2009) proposed a model, called coarse grain model (CGM), in which coarse particles represent a group of original particles. This model is applied to a three-dimensional pneumatic conveying system. With the assumption that the kinetic energy of a CGM particle should agree with that of a group of original particles, the contact force is modelled. By balancing the original particles with the CGM particle, the drag force and other forces are modelled. The volume of a CGM particle, which is α times larger than a original particle, is α3 times as big. The velocity of the CGM particle is assumed to be equal to the average velocity of the group of particles (averages are indicated with an overline), as can be seen in figure 3.8. Agreement in

Figure 3.8: Coarse grain model. (a) Translation; (b) rotation; (Sakai and Koshizuka (2009)) kinetic energy results in the following relation between the two models (original and scaled):

1 2m 0_r_˙02₊1 2I 0_θ_˙02_{= α}3 1 2m ˙r 2 +1₂I ˙θ2 (3.15)

In which m, ˙r, I and ˙θ are representing the mass, velocity, moment of inertia and angular velocity receptively. The contact force between two particles is estimated to be α3 times larger in the

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case of the CGM particle. Considering the rotational motion, in accordance with equation 2.10 (without rolling friction), the following relation is deduced:

¨ θi 0 = T 0 ij I_i0 = (R0_i· Ft,ij0) × n0_ij I_i0 = αRi· α3Ft,ij × nij α5_I i = α 4_T i α5_I i = 1 αθ¨i (3.16) where T is the torque vector, ¨22 the angular acceleration, R the particle radius, Ftthe tangential

contact force component and n the unit normal vector. The normal and tangential component of the contact force can be written, in accordance with equation 2.8 and 2.9 and assuming that all original particles behave the same during collision, as follows:

F_n0 = α3Fn= α3 knδn· nij + cn· vij = α3 knδn0 + cnvn,ij0

(3.17) F_t,ij0 = α3Ft= α3 ktδt· nij + ct· v0t,ij = α3 ktδt0· nij+ ct· vt,ij0

(3.18) in which δ, k and c are the overlap, stiffness and damping coefficient, respectively. This model is extended for drag and external forces. Especially the drag forces is comprehensive and will not be explained further in detail, because this is out of the scope of this research, nevertheless, the used method can be used in future research. The authors verified the coarse grain model with a series of simulations of a pneumatic conveying system. The used scaling factors where α = 1, 2, 3 and showed that with less particles the system was still accurately modelled. This model is extended in Sakai et al. (2010) for a fluidized bed, and applied in a two-dimensional case. The results were in good agreement with the simulation of the original particles. For the scaling factors α = 2 and 3 the speed-up was 3.0 and 4.3 respectively.

Kloss et al. (2012) presented three dimensionless numbers on the basis of the same idea, as shown in figure 3.8. The equation of motion is equal to equation 2.6, only cohesion and rotation are neglected. The derivation of dimensionless numbers is not mentioned in the article, but its reproduced here according to the theory explained in section 2.4. The relevant variables are kn,

cn, Ri, ρp and v0 (n = 5), in which the latter is a reference velocity. The units of these variables

are described by the units of the m = 3 basic variables, which are mass M ([kg]), length L ([m]) and time T ([s]), in table 3.3. The entities of the table are equal to the entities in matrix A

kn cn Ri ρp v0

M [kg] 1 1 0 1 0 L[m] 0 0 1 -3 1 T [s] -2 -1 0 0 -1

Table 3.3: The units of the relevant variables used in the dimensional analysis made by Kloss et al. (2012), expressed in the units of the basic variables

in equation 2.39. There are n − m = 5 − 3 = 2 dimensionless factors πq to find. According to

equation 2.41, this problem can be written as follows:   1 1 0 0 −2 −1  · k11 k12 k21 k22 +   0 1 0 1 −3 1 0 0 −1  ·   k31 k32 k41 k42 k51 k52  = 0 (3.19)

The solution is found using equation 2.41, which results in:

K =K 0 K1 =       k11 k12 k21 k22 k31 k32 k41 k42 k51 k52       =       1 0 0 1 −1 −2 −1 −1 −2 −1       (3.20)

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The dimensionless factors can be read from matrix K: π1 = knk11· ckn21· Rki31· ρ k41 p · vk051 ⇔ π1 = kn Ri· ρp· v20 π2 = kkn12· ckn22· R k32 i · ρkp42· v k52 0 ⇔ π2 = cn R2_i · ρ_p· v₀ (3.21)

From these dimensionless numbers the scaling laws can be deduced. The scaling laws for the radius (lengths), the density and the (reference) velocity are predefined:

R0_i = αRi ρ0p = ρp v00= v0 (3.22)

The other scaling laws are of the form

k0_n= αknkn c

0

n= αcncn (3.23)

In which αkn and αcn are the scaling factors that needs to be defined with the use of the

dimensionless numbers (equation 3.21): k_n0 R0_i· ρ0 p· v00 2 = kn Ri· ρp· v2₀ ⇒ αknkn αRi· ρp· v2₀ = kn Ri· ρp· v2₀ ⇒ α_k_n = α c0_n R02_i · ρ0 p· v00 = cn R2 i · ρp· v0 ⇒ αcncn αR2 i · ρp· v0 = cn R2 i · ρp· v0 ⇒ αcn = α (3.24) R0 = αR, ρ0_p = ρp, µ0s= µs, µ0r= µr, kn0 = αkn (3.25)

The order of magnitude of the speed up is α3, because the number of particles is α3 times smaller. For a model of a chute, the abrasion was simulated. The scaling rules were extended for the used wear model. Three simulations were made, one with the original particles, one with enlarged particles (without considering scaling rules), and one using the scaling rules, the results can be seen in figure 3.9. The colour of the chute indicates the wear, the particle colour indicate the

Figure 3.9: Simulation results of chute abrasion (Kloss et al. (2012))

particle velocity. The original model and the coarse grained model are in close agreement. A part of the research done by Bierwisch et al. (2009) concerns coarse graining. The proposed idea for scaling is that the scaled particles and the original ones, incorporates the same energy density and evolution of energy density. This is done by adjusting the contact force, equations 2.8 and 2.9, in an appropriate way. If the particle density and the porosity remains equal in both systems, then the density of potential energy will be independent of the particle diameter, so

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If the scaling is not affecting velocities, the kinetic energy density is preserved. Kinetic energy is lost during collision, the dissipated energy is equal to:

∆Ekin= 1₂m∗ ˙ rb_n,ij 2 − ˙ra_n,ij2 (3.27) in which the superscripts b and a imply the values of before and after the collision. The effective mass scales with α3, the number of collisions per unit of time and volume, with (1/α)3. When the coefficient of restitution remains the same, the energy dissipation per unit of time and volume is unaffected. The equations of motion, in normal and tangential direction, during collision are given by m∗r¨n,ij = 4₃E∗ √ R∗_δ3/2 n · nij− γn p R∗_δ n· ˙rn,ij+ √ 8πwE∗_(R∗_δ n)3/4· nij (3.28) m∗r¨t,ij = −min " µs 4 3E ∗√ R∗δ3/2_n · nij − γn p R∗δn· ˙rn,ij , κt r δn R∗|δt| # · δt |δt| (3.29) in which γn √

R∗δn is equal to cn in equation 2.8 and κtpδn/R∗ is equal to kt in equation 2.9.

The damping in tangential direction and the rotation is neglected. Using the equations of motion

Figure 3.10: Dependence of angle of repose on the grain diameter (Bierwisch et al. (2009)) (equations 3.28 and 3.29) as the base for a dimensional analysis, the authors derived the scaling

laws. This derivation is reproduced below, according to the theory in section 2.4. Table 3.4 shows the units of the relevant parameters (horizontally) (n = 6) expressed in the units of the basic variables (vertically) (m = 3). The number of dimensionless factors is r = n − m = 3. Similar to

w γn κt R∗ E∗ ρp

M [kg] 1 1 1 0 1 1 L[m] 0 -1 0 1 -1 -3 T [s] -2 -1 -2 0 -2 0

Table 3.4: The units of the relevant variables used in the dimensional analysis made by Bierwisch et al. (2009), expressed in the units of the basic variables

equation 2.41 this problem can be written as:   1 1 1 0 −1 0 −2 −1 −2  · K0+   0 1 1 1 −1 −3 0 −2 0  · K1 = 0 (3.30)

Scaling of bulk material properties and equipment

Summary

Summary (in Dutch)

List of symbols

Contents

Chapter 1

Introduction

Chapter 2

Background

2.1

Bulk material and equipment properties

2.2

Determining properties

2.3

Introduction to the discrete element method

2.4

Dimensional analysis

2.5

Discussion

Chapter 3

Scaling

3.1

Definition

3.2

Experiments

3.3

DEM simulation