Calibration of parameters for DEM simulations of solar particle receivers by bulk experiments and surrogate functions

Journal Pre-proof Calibration of parameters for DEM simulations of solar particle receivers by bulk experiments and surrogate functions

J. Grobbel, S. Brendelberger, M. Henninger, C. Sattler, R. PitzPaal PII:

S0032-5910(19)30987-8

DOI:

https://doi.org/10.1016/j.powtec.2019.11.028

Reference:

PTEC 14910

To appear in:

Powder Technology

Received date:

22 April 2019

Revised date:

24 October 2019

Accepted date:

11 November 2019

Please cite this article as: J. Grobbel, S. Brendelberger, M. Henninger, et al., Calibration of parameters for DEM simulations of solar particle receivers by bulk experiments and surrogate functions, Powder Technology(2019), https://doi.org/10.1016/ j.powtec.2019.11.028

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© 2019 Published by Elsevier.

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Calibration of parameters for DEM simulations of solar particle receivers by bulk experiments and surrogate functions J. Grobbela,, S. Brendelbergerb, M. Henningera, C. Sattlerb, R. Pitz-Paalb a b

Institute of Solar Research, German Aerospace Center (DLR), Jülich, Germany Institute of Solar Research, German Aerospace Center (DLR), Köln, Germany

Abstract

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Bauxite particles known as proppants from the fracking industry are an emerging heat transfer

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medium for solar thermal power plants. The discrete element method (DEM) has just recently been applied to these systems and the information on parameters for contact force models is scarce;

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calibrated parameters cannot be found. In this work a novel, three-stage calibration approach based

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on multiple bulk experiments and surrogate functions is presented. The experiments comprise the angle of repose, the residence time on a horizontal conveyor and the impact on an inclined plate.

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Particle-particle and particle-wall parameters for the Hertz contact force model and the modified linear spring dash-pot rolling friction model are determined for five different proppants and two

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wall materials. The assumptions of the calibration procedure and the calibration results themselves are checked and discussed on the basis of a sensitivity analysis. It is shown that almost all

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parameters can be constricted to unique values by the approach and that most parameters only gradually differ between particle types. Additionally, the influence of coarse graining on the

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parameters is examined. The behavior of the restitution coefficients is explained in detail by kinetic gas theory. It is found that only the particle-wall restitution coefficient is invariant to coarse graining, but not the particle-particle restitution coefficient, which must decrease with coarse-graining factor. Keywords: calibration, contact parameters, discrete element method, solar energy, particle receiver, coarse graining



Corresponding author. Tel.: +49 2203 601 4416

Email address: [email protected] (J. Grobbel)

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1.

Introduction Central receiver solar thermal power plants usually employ a molten salt mixture as heat

transfer medium, which is heated by concentrated solar radiation in the so called receiver [1]. The hot salt is stored in large tanks and released from them when desired to drive a steam cycle via a heat exchanger. In this way the plant can provide dispatchable power even during off-sun hours, which leads to a high capacity factor [2] and the ability to support grid stability with a renewable resource [3]. The usage of molten salt has some drawbacks: to prevent freezing and degradation it must be held in a temperature range between 290 °C and 580 °C [4], it is corrosive and it is kept in

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tubes, so that it can only be heated indirectly by the solar radiation. In contrast, ceramic particles as

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an emerging alternative heat transfer medium can withstand very high temperatures above 1000 °C and can be irradiated directly. Thus higher thermal receiver efficiencies can be reached [5] and the

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storage size can be smaller due to the bigger temperature range [6]. Additionally ceramic particles

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are non-corrosive, cheap and have a high heat capacity [7].

Consequently, various prototype particle receivers are tested by different groups [8, 9, 10,

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11, 13]. In most of the systems, bauxite particles originally designed for the fracking industry and also known as proppants, are used [12, 9, 11, 10]. They have good flow characteristics due to their

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high sphericity, are cheaply available in large amounts and have good optical properties [7]. The Discrete Element Method (DEM) was only recently applied to these solar particle

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receivers and only a few publications on such topic can be found. In most of these studies, there was no special attention on the contact parameters for the DEM force models. For example,

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particle friction was completely neglected in a study on a moving bed receiver [14]. In a series of publications on fluidized bed solar receivers [15, 16, 17, 18] parameters from a pioneering paper on fluidized beds by Tsuji et al. [19] were used even though the contact model and the particle type differed. In a preliminary DEM study of a falling film receiver some parameter variations were made, but the authors mention that calibration has been postponed to future work [20]. In context of an investigation on a receiver prototype employing a wire-mesh screen a rigorous sensitivity analysis investigating the influence of the contact parameters on the mass flow through the screen was performed [21, 22]. However, only one parameter was changed at a time and no parameter set was calibrated. One difficulty in the determination of contact parameters by fitting the contact parameters to bulk experiments is that the outcome of the DEM simulations is dependent on multiple

Journal Pre-proof parameters. In a naïve approach one would need to cover the whole parameter space with DEM simulations and choose the parameter set which fits best to the experiments. But if many parameters are involved, the parameter space is very large and this approach is not feasible. More intelligent approaches are based on neural network [23] or genetic algorithms [24]. A detailed review mentioning also direct measurement methods is given in [25]. Inspired by the work of Rackl et al. [26], where an optimization with surrogate models built by Latin Hypercube Sampling (LHS) was used, a modified version of this approach is developed in this work. It is then applied to five bauxite proppants differing in size and obtained from two manufacturers. Contact model

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parameters for these particles are determined. As coarse-graining is considered to be necessary to

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simulate real-size solar particle receivers, its impact on the contact model parameters is discussed

Materials and Methods

2.1.

Investigated Particles

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2.

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as well.

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The five different bauxite proppants investigated in this work are shown in figure B.1. They were provided by Saint Gobain (”SG”) and by CarboCeramics (”Carbo HSP”) and have a

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density of around 3560 kg m−3, differing by less than 2% between the samples. Two types (SG10H and Carbo HSP13) were already used in other studies and the naming was adopted from there.

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To determine the size and shape of the particles, they were poured on a cardboard with an attached ruler. Digital images were taken from the top and analyzed with the free image processing

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software ImageJ [27]. The images were converted to grayscale ones, the particles were separated from the background by setting an appropriate threshold for the grayscale value, touching particles were separated by the watershed algorithm and then the projected areas and perimeters of the detected particles were exported and further processed. The diameter of particle i is calculated from the projected (index P ) surface (index s ) area:

d Ps,i =

4 AP,i



(1)

The circularity is defined as the ratio between the perimeter of a circle with radius dPs,i and the detected perimeter U

Journal Pre-proof  Ps,i =

 d Ps,i

.

U

(2)

From these two quantities, characteristic particle statistics were deduced, which are diameters at the end of the 10%, 50% and 90% quantiles of the particle size distribution, named d10,0 , d50,0 and d90,0 [ad110] d10,0 10 quantile of particle size distribution and the Sauter diameter [28]

d 32

d = d

3 Ps,i

i

2 Ps,i

(3)

.

i

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These diameters and the arithmetic mean of the circularity 10 are reported in table B.1.

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For the Carbo HSP particles the manufacturer also provided a sieving analysis, which

Wall Materials

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2.2.

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agreed very well with the results gathered with the optical method.

Particle-wall contact parameters depend on the wall material, which is not the same for all

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solar thermal particle receivers. Here we investigate two very common ones: smooth stainless steel

2.3.

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and alumina insulation board (AL-30AA from Circar Ceramics).

Selected Models for the Discrete Element Method

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The selected models in the Discrete Element Method determine which contact parameters

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are required. Therefore the models are described in detail in the following. In the DEM, the force

mi si =

Ncon

Ncon

j =1

j =1

Fji,n 

F

j i ,t

 mi g

(4)

and torque

I i i =

Ncon

r

ij ,c

 Fj i ,t  Ti ,roll

(5)

j =1

balances describe the motion of each particle i with its mass mi and moment of inertia Ii . Within the scope of this study, the relevant forces are the gravitational force mi g and contact forces Fj i exerted from particle or wall j on particle i . They are divided into a normal (index n ) and tangential (index t ) component. Electrostatic, cohesive and fluid induced forces are

assumed to be negligible in context of this work and therefore omitted in equation (4) and (5) for

Journal Pre-proof clarity. As in real-size solar receivers the particle number will be in the order of several millions, the particles are modeled by mono-sized spheres to reduce computation time. In this case normal contact forces go through the center of gravity of the particle and only the tangential part induces a torque in equation (5), acting at the point of contact described by the vector rij ,c , originating from the center of the particle. To compensate for the lack of rolling resistance of the spherical particles, an artificial torque Ti ,roll is added which should account for the shape of the real particle [29]. To obtain particle velocities, positions, angular velocities and orientations, the particle lateral and

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angular acceleration vectors si and i are integrated. The time step for the explicit integration scheme was kept below a fifth of the Rayleigh time step, which is calculated from particle radius

R



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tRayleigh =

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R , density  , shear modulus G and Poisson’s ratio  according to [30]

0.8766  0.1631

G

.

(6)

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All simulations were performed in the open-source software LIGGGHTS [31], version 3.8.0. In

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the following, the selected contact force and rolling friction models are described based on their

2.3.1. Contact model

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implementation in this code, which is available online [32].

To compute the contact forces, the particles are allowed to overlap as depicted in figure

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B.2. The resulting force is modeled by a spring for the elastic and a damper for the dissipative part

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of the contact. Thus the force Fj i ,n in normal direction en is

Fj i ,n = kn n  cn vn,rel  en

,

(7)

where  n denotes the normal overlap and vn,rel the relative normal velocity of the two particles i and j in contact. A variety of models for the spring stiffness kn and the damping parameter cn exists. For the mentioned studies on solar particle receivers linear spring dashpot (LSD) models [14, 15] or Hertz models [20, 21] have been applied. The Hertz model predicts a more realistic contact area than the LSD models. This is considered important for later heat transfer simulations, so that the Hertz model was selected here. It is a nonlinear model, as the spring stiffness kn and damping parameter cn are themselves functions of the normal overlap:

Journal Pre-proof 4 kn ( n ) = Yeff Reff  n 3

cn ( n ) = 2

5 ln(e) 6 ln 2(e)   2

(8)

meff 2Yeff Reff  n

(9)

They also depend on the effective Young’s modulus Yeff , the effective particle radius Reff , effective particle mass meff and restitution coefficient e . The equations for the effective properties are found in Appendix A. The ”tangential history model” was selected for the tangential force component. In this

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model the shear vector  t is defined as the sum of the incremental displacements since the initial

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contact at time tc,0 given by the relative tangential velocity at the contact point v tr and the time

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increment dt :

 t =  v tr dt . t

(10)

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tc ,0

From that, the tangential contact force is calculated according to

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if kt  t   Fn

if kt  t >  Fn (sliding)

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Fj i ,t

kt t  ct v tr  =  t     Fn t 

(11)

During sliding, the tangential displacement vector is rescaled to have a magnitude of max

=

 Fn kt

.

(12)

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t

The tangential spring stiffness k t and the tangential damping parameter ct are again functions of the normal overlap  n :

kt = 8Geff Reff  n

ct = 2

5 ln(e) 6 ln 2(e)   2

(13)

meff kt

(14)

It should be noted here that our description of the tangential history model in equation (11) differs from the usually cited equation given by Kloss et al. [31], because we think it describes the actual implementation in LIGGGHTS 3.8.0 more precisely in situations with small shear and high relative tangential velocity.

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2.3.2. Rolling friction model In this study the modified elastic-plastic spring dashpot (EPSD2) model [29, 33] was chosen as a rolling friction model after some initial tests with angle of repose simulations. They showed creeping bulk behavior for the constant directional torque (CDT) model, as reported in the literature [29, 33]. The name of the EPSD2 model in LIGGGHTS is misleading, as in the modified version there is no dashpot part as in the original model but only a spring term. The rolling friction is calculated by the numerical integration of

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Troll = kR rel

kR = kt ( Reff )2

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with the torsional spring stiffness

(15)

(16)

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and the relative rotation of the contact partners rel . As the artificial torque is equivalent to a

force in the contact model in equation (12):

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tangential force for a spherical particle, it is reasonable to limit it in a similar way as the tangential

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| Troll | R Reff Fn

.

(17)

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In the EPSD2 model the rolling friction coefficient R therefore only affects the maximum possible rolling friction torque; below this limit, it is only influenced by the tangential spring

Calibration Approach

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2.4.

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stiffness k t .

The parameters required for the contact models and the rolling friction model are summarized in table B.2.

As particle radius we use half the Sauter diameter d 32 and as density the one provided by the manufacturers. Poisson’s ratio  is set to 0.3 , a typical value for bauxite [34]. The modulus Y is set to 5 MPa, the minimum value allowed in the DEM software. This is 4 to 5 magnitudes

below the actual modulus of the particles, which is expected to be similar to the one of alumina (several 100GPa). This softening of the particles is a common way to reduce the simulation time [35], as it increases the Rayleigh time step in equation (6). The quality of the softening simplification will be checked later in a sensitivity analysis.

Journal Pre-proof To determine the remaining six restitution and friction coefficients, there are basically two approaches: measure them directly on a particle scale (Direct Measuring Approach) or calibrate them to the outcome of bulk experiments (Bulk Calibration Approach) [25, 36]. As the bauxite proppants are rather small and no perfect spheres, direct measurement is considered to be difficult [36]. In addition, parameters obtained by the bulk calibration approach can be seen as ”adjustment parameters” [25] and can compensate for model inaccuracies, for example for the assumption of spherical particles. Direct measurement of the rolling friction coefficient is not possible at all because it is a ”purely empirical parameter” [25]. This motivated us to develop a calibration

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approach based on bulk experiments. The following ones were chosen:

Angle of repose (AOR)  glued on particles laminated on cardboard

(b)

Angle of repose (AOR)  S or Al2O3 on a flat surface (steel or Al2O3)

(c)

Particle transport on a horizontal conveyor (HC) plate laminated with particles: time

t100glued to transport 100 g

Particle transport on a horizontal conveyor (HC) plate (steel or Al2O3): time t100S or

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(d)

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(a)

(e)

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t100Al2O3 to transport 100 g

Impact of particles on an inclined plate (steel or Al2O3): masses mi in boxes

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positioned at different distances The parameters are obtained from these experiments by the procedure depicted in figure

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B.3. It consists of three stages, in each of these stages two of the six missing contact parameters are determined. Each stage starts with a latin hypercube sampling (LHS) of the respective DEM models of the experiments. In the LHS, the possible range of each parameter is split into the same number of subintervals and samples are randomly created in a way that each subinterval contains only one sample point. In two dimensions like here, it can be visualized by a chess board with rooks, which cannot take each other. From the results at the sample points surrogate models of the DEM models, more precisely polynomial Kriging functions of the contact parameters p , are created. From these surrogate models fi (p) , the relative residuals Resi to the experiment for the i -th target value yExp,i are constructed:

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Resi =

fi (p)  yExp,i

(18)

.

yExp,i

Their quadratic sum is then minimized to give the contact parameters at each stage. In the first stage, these contact parameters are the particle-particle sliding and rolling friction coefficients pp and  R,pp . The angle of repose and horizontal conveyor experiments with the glued particles (a) and (c) are assumed to be mainly dependent on these parameters, so that their results are taken as target values and surrogate functions of DEM models of these experiments are created. Since particles are glued to walls in these experiments, the particle-wall

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parameters play no role. The only remaining parameter needed for the simulations of the

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experiments is the particle-particle restitution coefficient, which is kept constant at this stage as it is expected to have a minor influence.

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In the following second stage the angle of repose and the horizontal conveyor experiment on a flat surface are used to determine the particle-wall rolling and sliding friction coefficients

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pw and R,pw . The particle-particle friction coefficients are taken from the outcome of the

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previous stage and the restitution coefficients are assumed to have no influence, so that their values are fixed.

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In the final third stage, the plate impact experiment delivers the particle-particle and

previous two stages.

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particle-wall restitution coefficients epp and epw . The friction coefficients are taken from the

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The presented calibration approach is similar to the one proposed by Rackl et al. [26], who also worked with Kriging functions in a first calibration step. In fact, we utilized their software DEcalioc to generate the Kriging functions for our work. However, they calibrated all parameters simultaneously, which we also tried initially. We found that the calibration in stages has the following advantages over the simultaneous calibration: •

Significantly less simulations are needed to build accurate two-dimensional surrogate models than 6D models.

•

The 2D surrogate models can be visualized easily which is not possible with 6D models.

•

The global search for initial values for the minimization of the residuals is computationally very expensive in six dimensions.

Journal Pre-proof The only disadvantage of the calibration in stages is the assumption of certain parameters to be constant in each stage. For example, the particle-particle restitution coefficient might have an influence on the outcome of the angle of repose and horizontal conveyor simulations, but is kept constant in the first two stages. Besides this, for both the simultaneous and the stage approach the use of surrogate models instead of the DEM models for the minimization introduces an error. Therefore, at the end of the calibration process, the results are checked by running the DEM

2.5.

Calibration Experiments and DEM Setups

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models with the gathered parameters and the deviations from the target values are determined.

the previous section are described in detail.

2.5.1 Angle of repose experiments (a) and (b)

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In this section the experimental setups and the corresponding DEM-models mentioned in

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The setup of the angle of repose experiments (a) and (b) is depicted in figure 3. The static

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angle of repose was measured by pouring the particles through a funnel (4) on a plate of the respective contact material (2), taking a picture with a digital camera (6) and analyzing this picture

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via image processing with the opencv package in Python. To conduct experiment (a), the plate of contact material was replaced by a cardboard covered by a layer of particles, fixed by double-sided

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duct tape and shown in figure B.5. The whole setup including the funnel was reproduced in LIGGGHTS. In the simulations of experiment (a), the sliding and rolling friction coefficients of

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the contact material were set to the extremely high value 1000 to mimic the glued particles on the cardboard. A typical measurement overlayed with lines from the image processing is depicted in figure B.6 alongside with the image processing of the DEM result.

2.5.2.

Conveyor Plate Experiments (c) and (d) Experiments (c) and (d) are both conducted with the apparatus shown in figure B.7. It

consists of a horizontally oscillating plate (1) mounted on a slider (5), which is driven by a stepper motor (9) along a rail (6). A particle mass of 300 g is poured continuously through the funnel (3) onto the plate (1), meanwhile the particles are transported alongside the plate by the oscillation until they fall onto the scale (2). The time between opening the funnel and the moment when 100 g of the particles arrived on the scale is measured and denoted as t100 . The contact surfaces

Journal Pre-proof attached to the plate were exchanged to characterize the parameters for the different wall materials. In case of experiment (c) the cardboard is laminated with particles and fixed on the plate as shown in figure B.8. The acceleration and position profile of the plate is shown in figure B.9. During the slowly accelerated forward motion, the particles should stick to it and move forward. Then the plate is stopped and moved back with high acceleration, so that the particles ideally slip on the plate. Their forward motion is decelerated; depending on the accelerations and the particle friction they could even be transported backwards at some point in this stage. However, over an entire cycle the net

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motion is positive and they are transported forward. To check if the plate follows the desired

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motion, high speed camera videos were analyzed for the case of experiment (c), where the plate is laminated with particles as shown in figure B.8. The extracted plate and glued particle positions

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are also shown in figure B.9, alongside the desired position profile which was given to the controller. The plate follows the desired profile with some minor deviations, mainly the period is a

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little longer than defined. The glued particles follow the same motion as the plate, so that they are

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considered to be glued sufficiently to the plate. In LIGGGHTS, the motion profile is defined as a sum of cosine functions. That is why there is a slight deviation between the desired profile and the profile of the plate in the DEM. To realize the lamination in LIGGGHTS, the particle-wall sliding

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friction and rolling friction coefficients were set to the value of 1000, so that the particles basically make the same motion as the plate in the DEM (dotted lines). After the plate motion had been

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activated, the plug closing the funnel was removed and the weight on the scale was recorded over

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time. Since the funnel mass flow influences the outcome of the experiment, it was measured beforehand and used for the simulations. The measured mass flows are documented in Appendix B. In the experiments it was seen that a few particles never leave the oscillating plate even though they are not glued to it. These particles usually reside in between glued ones. A removal of these particles before each new experiment without removing some of the glued particles is difficult and could potentially change the plate’s position. Instead an initial state with these particles was created by doing a preliminary experiment, where particles were poured on the plate and it was kept running until no particles left anymore. The same was done in the simulation.

2.5.3. Plate Impact Experiment (e) Figure B.10 shows the apparatus of experiment (e) on the left and the corresponding DEM

Journal Pre-proof model with the dimensions on the right. The particles to be tested are filled into the funnel (1), fall through the pipe (2) and hit a 45° inclined plate of the wall material (3). The particles are deflected by the plate and fall either in one of the four plastic boxes (4) or beside them. The masses in the respective boxes are weighted.

3.

Results and Discussion

3.1.

Sensitivity Analysis

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To check the assumptions for the three-stage calibration approach presented in section 2.4, sensitivity studies with the DEM models of each experiment were conducted with a particle size of

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1.595 mm (Carbo HSP13 particles). The parameters were set to the baseline values in table B.3

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and then each one of the parameters was varied while the other parameters were kept constant.

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3.1.1. Angle of repose

The sensitivity of the angle of repose on the contact parameters is shown in figure B.11 for

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the case where particles are glued to the ground as in experiment setup (a) (left) and for a flat ground as in setup (b) (right). With glued particles on the ground the number of variables reduces

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from six to three, because wall-related parameters are irrelevant. All curves are not very smooth due to a natural variation in the angle of repose, which occurs as single particles can cause

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avalanches slightly changing the resulting angle [37].

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In both the glued and not glued case, the restitution coefficients epp and if applicable epw have a small influence, as it was assumed for the first two stages of the calibration process. The angle of repose  tends to increase with rolling and static friction coefficients, which is in agreement with the literature [29, 38]. The slope of the curves of these coefficients is steep in figure B.11, which indicates a high sensitivity and a good suitability of the angle of repose experiments for calibration stages one and two. On the flat surface it can be noted that  is not influenced by pw and R,pw above a certain value. This is because the base of the pile becomes stable and the angle is mainly influenced by pp and  R,pp like in the case with glued particles on the ground. This explains the qualitatively similar sensitivity on the particle-particle coefficients

pp and R,pp in the glued and not glued case. The rolling friction was found to be essential to build a pile, which is also reported by

Journal Pre-proof Grima et al., who state that  R,pp needs to be larger than 0.2 [39]. We also found that on flat ground a minimum particle-wall friction coefficient pw is required to provide the base of the pile as described above. In the majority of calibration studies reviewed by Coetzee [25], this is neglected and only the particle-particle contact parameters are calibrated while the particle-wall values are set to some value. Our sensitivity study shows that this could be problematic if the surface is not rough enough.

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3.1.2. Horizontal conveyor Figure B.12 shows the sensitivity of the transport time t100 of the horizontal conveyor for

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experiment setup (c) with the particles glued to the oscillating surface (left) and for setup (d) with a smooth surface (right).

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For both the glued and not glued case the restitution coefficients epp and epw show

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almost no influence, as it was assumed for calibration stages one and two. If particles are glued to

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the plate, increasing the particle-particle sliding friction pp or rolling friction  R,pp leads to an increasing particle residence time on the plate, whereas these coefficients only barely have an influence if particles are transported on a flat surface. In this case the strong sensitivity on the

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particle-wall friction coefficient pw is obvious. The other friction coefficients show very little

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influence, mostly the particle-wall rolling friction R,pw in the range between 0.1 and 0.3. These results are reasonable, as particles on oscillating plates often behave similar to a

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solid body [40]. If the plate surface is smooth, the bulk of particles slides along this surface, so that mostly the particle-wall coefficient pw influences the transport speed. In case of a rough surface or with particles glued to the ground, a layer of particles on the wall is established and the remaining bulk of particles glides on this first layer, so that the particle-particle rolling and sliding friction coefficients pp and  R,pp influence the result. The sensitivities show that the conveyor experiments are well suited to determine the friction coefficients together with the AOR experiments in calibration stages one and two.

3.1.3. Plate impact The sensitivity study of the plate impact experiment shows how the masses in the boxes

Journal Pre-proof placed at different distances to the impact plate change when the contact parameters vary. The sensitivity on the restitution coefficients is clearly visible, as depicted in figure B.13. For the mass in the first box, both the particle-particle restitution coefficient epp and the particle-wall restitution coefficient epw have a strong influence, while for the masses in the other boxes epp has a negligible influence in contrast to epw . This behavior makes sense, as above the first box particle-particle contacts are significantly more frequent than above the other bins. The pure change of direction caused by a particle-particle hit makes it very unlikely for a particle to reach

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boxes 2-4, regardless of epp , whereas for box one the epp clearly affects whether the particle will

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fall into the box or not. One can also see a steep peak in the particle-wall restitution curves for the masses in box 2-4. With increasing epw the particles first do not reach the box until they hit the

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nearer box wall and some particles fall into the box. This corresponds to the left side of the peak.

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When the particles hit the center of the box, the top of the peak is reached and on the decreasing slope the particles tend to hit rather the more distant wall of the box and more and more particles

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fly too far.

Besides the particle-wall restitution coefficient, for boxes 2-4 a strong dependence on the

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particle-wall sliding friction coefficient pw stands out, which is not the case for the first bin. A possible reason for this behavior is the change of energy, which is dissipated both at the walls of

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the tube during the fall of the particles and at the plate during the impact. However, this strong dependence is only seen for very low friction values, which lay in a rather unrealistic low range, as

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one can expect from the horizontal conveyor sensitivity study. Therefore we have four target values depending on the two restitution coefficients in different ways so that this experiment is well suited to isolate them in step three of the calibration procedure.

3.1.4.

Influence of Young’s modulus and Poisson’s ratio Figures B.11, B.12 and B.13 also show the sensitivity of the numerical experiments on the

modulus Y and on Poisson’s ratio  . For the angle of repose setup, the sensitivity is not exceeding the natural variation caused by the avalanches and also for the other setups the sensitivity to these parameters is small. Therefore the softening of the particles is a valid approach to reduce the simulation time and it is justified to exclude Y and  from the calibration.

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3.2.

Experiments In the following the experimental results are presented and discussed.

3.2.1. Angle of repose The angles of repose  measured in setup (a) and (b) are listed in table B.4. Experiment (a) was repeated five times and experiment (b) 15 times to obtain the standard error of the mean, which is below 0.66 for all reported angles. Due to the higher wall friction, the angles of repose on the alumina board are in general

of

higher than on the smooth steel plate. An exception are the Carbo HSP30/60 (the smallest)

ro

particles where the angle on both surfaces is about the same. Because of the small particle size, the base of the pile seems to be stable on the smooth steel surface already, so that the angle is

-p

controlled by particle-particle friction only and a similar angle like on the alumina board is

re

obtained. This interpretation is backed by the case when particles are glued to the wall. Here the angles of repose are similar to the ones on the alumina board except for the Carbo HSP13 (the

lP

largest) particles, where the angle is about 2 higher. The further stabilizing effect on the base particle layer by the glue only has an effect on the largest particles, for the other particles the base

ur

3.2.2. Horizontal conveyor

na

particle layer was already at maximum stability on the alumina board.

In experiments (c) and (d) the mass coming off the horizontal conveyor is measured over

Jo

time as depicted in figure B.14. It shows one experiment with Carbo HSP13 particles and particles glued to walls. After the first particles reach the end of the plate, the mass on the scale almost linearly increases. The slope of the line shows that the mass flow is only slightly lower than the one coming out of the funnel. The results of all experiments are summarized in table B.5 alongside with the standard errors of the mean which were gathered by repeating all experiments five times. The comparison between the contact surfaces clearly shows that they have an influence on the transport speed. The residence time of the particles increased for all particle types with increasing roughness of the plate. It should be noted that the Al2O3 surface is an insulation board, which is perceptibly rougher than the steel plate. Since the mass flow through the funnel is different for all particle types, it is difficult to generally draw conclusions from comparisons

Journal Pre-proof between particle types. However, the tendency that t100 decreases with particle size could be caused by an increased mass flow through the funnel.

3.2.3. Plate impact The results of the plate impact experiment (e) are listed in table B.5. With the porous alumina insulation board only the HSP13 particles were tested to have a comparison to the steel values. For all particle types the amount of particles in the boxes decreases with the distance of the boxes to the impact plate. The low standard errors of the mean show the high reproducibility of the

of

experiments. The ratio of the masses varies between the particle types, i.e. the ratio of the masses

ro

in the first two bins is m2 / m1 = 0.28 for the big Carbo HSP13 particles, while it is m2 / m1 = 0.76

-p

for the small Carbo HSP30/60 particles.

re

Results of plate impact experiments

Particles

Wall

Initial mass [g]

Carbo HSP13

steel

SG10H

steel

Carbo

steel

m3 [g]

m4 [g]

68.73

30.21  0.15

8.5  0.09

2.74  0.04

0.55  0.03

33.79

12.79  0.09

6.3  0.05

1.58  0.05

0.25  0.01

34.83

10.90  0.18

6.34  0.04

2.57  0.14

0.61  0.05

na

lP

m2 [g]

Carbo

steel

30.87

12.47  0.09

4.94  0.02

1.42  0.03

0.26  0.01

HSP20/40

Jo

ur

HSP16/30

m1 [g]

steel

17.64

5.84  0.08

4.46  0.03

1.17  0.07

0.14  0.01

Al2O3

68.73

38.12  0.24

7.89  0.08

1.26  0.03

0.30  0.03

Carbo HSP30/60 Carbo HSP13

3.3.

Parameter Calibration The target values gained from the experiments were used to perform the calibration based

on the three-stage approach described in section 2.4. The results of each stage are presented and discussed in the following.

Journal Pre-proof 3.3.1 Calibration stage one: pp and  R,pp In stage one, a LHS of the angle of repose and horizontal conveyor DEM-model with glued particles

was

conducted

with

100

sampling

points

in

the

parameter

range

[pp , R,pp ] [0.1,0.8] [0.05,0.6] to obtain the Kriging functions shown in figure B.15 for the Carbo HSP13 particle size.

The contour plot of the angle of repose simulation is similar to the one shown by Wensrich

of

[29]. As expected, the angle of repose increases with both rolling and sliding friction coefficients, even extreme angles can be obtained with high values of these coefficients. The horizontal

ro

conveyor mainly shows a dependence on the sliding friction coefficient as expected from the sensitivity study. However, for higher sliding friction coefficients the rolling friction coefficients

-p

have more and more importance.

re

The contour line of the experimental target value is highlighted with a dashed red line. The intersection of these lines from the angle of repose and the horizontal conveyor experiment gives

lP

the rolling and sliding friction coefficients between the particles, indicated by a red dot. The curves only intersect at one point and measurement errors would not alter this intersection point very

na

much. It can be concluded that this first calibration stage with the glued particle surfaces is well

ur

suited to determine pp and  R,pp .

Jo

3.3.2. Calibration stage two: pw and R,pw The experiments and models for the second stage are the same, except that no particles are glued to the contact surfaces. This brings the sliding and rolling friction coefficients pw and

R,pw between particles and walls into play. Based on the LHS[eLHS]LHSLatin Hypercube Sampling of the DEM models, the Kriging functions of these two variables were created and are shown in figure B.16 for the Carbo HSP13 particle size. The particle-particle parameters are the fixed values from the previous stage and the restitution coefficients were set to the fixed value of 0.6, as they have minor influence. The contour plot of the angle of repose shows some different characteristics than the previous plot for a glued particle surface and the particle-particle coefficients. In general, if both the particle-wall sliding and rolling friction coefficient are high, the base of the pile is stable and

Journal Pre-proof the angle of repose is given by the particle-particle friction coefficients pp and  R,pp . Here, for example, the maximum angle is just above 38°, which is the target value of calibration stage one for the particle-particle coefficients. If one decreases either pw or R,pw below a certain value, the base of the pile breaks down and the wall parameters have an impact on the angle of repose. This happens most noticeably for sliding friction coefficients pw < 0.3 , where the rolling friction has minor influence. Above 0.3, the impact of the sliding friction is not so dominant anymore and the rolling friction stronger influences the angle of repose.

of

The horizontal conveyor simulation is barely influenced by the particle-wall rolling

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friction R,pw if the sliding friction pw is below about 0.35. If the sliding friction is above this value, increasing the rolling friction leads to a higher particle residence time on the plate and the

-p

sliding friction has only minor influence.

The DEM models are the same for different contact surfaces, so that both the target values

re

for steel and alumina are drawn as dashed lines into the same plot. Both intersection points are in a

lP

region where the gradient of both the angle of repose and the residence time on the plate is larger in the direction of the sliding friction coefficient than in the direction of the rolling friction

na

coefficient. Therefore the reliability of the gathered sliding friction coefficient can be considered higher than the one of the rolling friction coefficient. Nevertheless, the rolling friction value is almost the same for both contact materials, which can make sense, as both contact surfaces are flat

Jo

ur

and the particle shape is also the same.

3.3.3. Calibration stage three: epp and epw In the third calibration stage, a LHS of the plate impact simulation was performed with restitution coefficients [epp , epw ] [0.2,0.9] [0.2,0.9] . For the Carbo HSP13 particle size this leads to the Kriging functions of the box masses shown in figure B.17. As expected from the sensitivity analysis, the Kriging surface of the first box looks essentially different from the Kriging surfaces of the other boxes. In the entire parameter range, the mass in the first box is reduced if either the particle-particle or the particle-wall restitution coefficient is increased. This is not the case for the other boxes; here the particle-wall restitution coefficient epw is the deciding parameter. Also, there is a steep peak in the Kriging surfaces for theses boxes. The experimental

Journal Pre-proof target value, in this case for steel and indicated by a dashed red line, is always close to this peak and could be on either side of it. To decide on which side, two of the boxes 2-4 need to be included in the analysis. Here boxes 2 and 3 were taken and it is clear, that this leads to a particle-wall restitution coefficient around 0.43. As the mass in box 1 is sensitive to epp , it was possible to determine both restitution coefficients with this experiment.

0.30788

0.45895

0.43295

0.29667

0.12274

0.42278

0.76948

0.44398

0.8945

0.45067

0.82056

0.44292

0.4 †

0.30147

0.30374

0.5*

0.29240

0.087756

na

0.3628

0.39274

ur

0.71654

of 0.37928*

0.16226

0.38452

0.7058

Al2O3

0.29985

0.407383

Al2O3

HSP30/60 Steel

0.7756

0.37547

0.61156

Al2O3

0.27059

0.085561

Al2O3

HSP20/40 Steel

0.43895

0.30725

0.52994

HSP16/30 Steel

0.74152

0.69445

ro

Steel

0.27395

R,pp

Al2O3 SG10H

epw

pw

re

Steel

epp

pp

lP

HSP13

R,pw

-p

Calibration Results

0.12984

0.45* 0.49223 †

0.41975

Jo

* : no intersection of contour lines

† : simulation of AOR was in all points too low

3.3.4. Comparison between particle types The calibration results of all particles are summarized in table B.7. The rolling friction values of all particles are by about a factor of 3 lower between the particles than between a particle and a wall. This makes sense, as one can imagine that a particle rolls easier on the convex surface of another particle than on flat ground. As mentioned, the particle wall rolling friction R,pw is hard to determine in calibration stage two; above a value of about 0.3 the simulations are not very sensitive to this parameter anymore, which is considered the reason of the variation in R,pw between particle types. In some cases, the contour lines of the angle of repose and the horizontal

Journal Pre-proof conveyor simulation did not intersect in calibration stage two. The values of pw and R,pw are obtained by taking the closest distance between the contour lines in this case. Due to the high sensitivity of the horizontal conveyor experiment to pw (almost vertical line in figure B.16) its value could still be determined accurately. There was also no intersection point or better not even an experimental AOR contour line when the target angle of repose from the experiment could not be reached by the surrogate model. This happened when the measured angle of repose on the alumina board was slightly higher than on the surface with the glued particles. In both of these experiments the base of the pile is stable and the pile angle is only controlled by particle-particle

of

friction. As pp and  R,pp come from the first calibration stage, the maximum angle of repose in

ro

the second calibration stage ideally is the target angle from stage one.

-p

The sliding friction coefficient to a wall pw is very similar among different particle types. With a value between 0.36 and 0.41 it is for all particles higher on the alumina insulation

re

board than on steel with a value between 0.29 and 0.31. This was expected from the sensitive

lP

roughness of the materials. In literature, the particle-wall friction coefficient is sometimes measured directly: particles are glued together to form a plate, which is then placed on another

na

plate of the wall material. This plate is slowly tilted and the inclination angle at which the particles start to slide is noted, which in turn gives the friction coefficient [44]. To check whether our calibration results are plausible, we conducted such an experiment for the largest particles and

ur

steel as contact partner and obtained a particle-wall friction coefficient of 0.36, which is above the

Jo

values we got from our calibration procedure. While the static friction coefficient is determined in the direct measurement, our calibration procedure pw rather gives a dynamic friction value, as it mainly originates from the horizontal conveyor experiment with its gliding particles. As static friction is usually higher than dynamic friction, the values obtained by both approaches seem to be reasonable and the deviation between them makes sense. The sliding friction between the particles pp is essentially higher than between particles and both wall materials pw . This could be assumed from the horizontal conveyor experiments without glue, because no fixed first particle layer developed. It is hard to tell what causes the large difference between pp and pw . The particles have a ceramic surface as the alumina board, probably their shape has an impact on the sliding friction coefficient, too. Interesting to note is the

Journal Pre-proof relatively large difference in pp between the SG10H particles from the company Saint Gobain to the other particles produced by CarboCeramics. This is attributed to the different surfaces of the particles, which can be seen in the microscopic images in figure B.1. The particle-wall restitution coefficients epw on steel are with values between 0.43 and 0.45 very similar between the different particle types. As epw did not change significantly between particle types for steel as contact parameter, for the alumina insulation board it was decided to calibrate it only for the Carbo HSP13 particles. On the insulation board epw has a value

of

of 0.3, which is clearly lower than on steel with 0.44. The results are in accordance with

ro

preliminary high-speed camera tests with single particles, which also showed a lower restitution coefficient on the insulation board.

-p

The particle-particle restitution epp shows more variation between particle types than epw

re

. There is a significant difference between the SG10H particles from Saint Gobain and the other particles from Carbo Ceramics: The Carbo Ceramics particles have a particle-particle restitution

lP

coefficient between 0.74 and 0.89, while it is 0.46 for the SG10H particles. This probably relates to the different surface as discussed above for the particle-particle friction coefficient. Ideally, both

na

the calibration with steel and with the insulation board as contact partner should result in the same particle-particle restitution coefficient epp . Here the value of 0.776 for the Carbo HSP13 particles

ur

on the insulation board is close to 0.742 on steel, which confirms the results.

Jo

3.3.5. Final check with DEM models Since the calibration was done in stages and based on surrogate models, the results were finally checked with the DEM models. The relative deviations to the experimental values are listed in table B.8. For the angle of repose and the horizontal conveyor experiments they are below 4%. It must be noted that the angle of repose has some natural variance, as avalanches can be formed and single particles can cause the angle to differ noticeably. The mass in box 1 in the plate impact experiment also shows little deviation, whereas the masses in boxes 2 and 3 significantly differ, in one case by almost 60. This is caused by the high gradient with respect to the wall restitution coefficient epw in the DEM model for boxes 2 and 3, which can be seen in figure B.17. A small change in epw causes a large increase or decrease of the mass in the boxes. However, this high

Journal Pre-proof sensitivity to epw also means that a deviation in the mass within the box does not affect epw much, so that it does not question the calibration results and overall the calibration can be considered successful.

t100

m1

m2

m3

1.43

-1.30

0.07

-0.20

0.71

40.48

-52.25

-2.87

0.09

-0.44

-19.21

2.10

-3.48

1.50

-3.39

59.15

14.04

0.97

12.41

0.54

-5.21

-30.05

-5.56

-10.22

-19.29

-10.09

Al2O3 SG10H

Steel

3.10

-1.19

Al2O3 HSP16/30

Steel

-3.04 1.21

-1.29

Al2O3 HSP20/40

Steel

-1.43

-4.01

Steel

0.10

Al2O3

Coarse-Graining

1.96

-3.52

-0.17

-0.03

2.04

-8.44

0.09

-6.18

1.11

-1.71

-0.09

ur

3.4.

-2.76

na

HSP30/60

-0.08

-3.31

lP

Al2O3

ro



-p

Steel

t100glued

re

HSP13

 glued

of

Final check of calibration results, deviations to experimental values

Jo

A particle receiver with several meters of aperture size may contain several millions of particles. This causes long run time of the DEM simulations with real size particles, which is often not feasible. To circumvent this problem, a typical approach is to use larger particles in the simulation than in reality, so called coarse graining. The contact parameters must be adapted for these larger particles. This was done here by increasing the particle diameter in the DEM setups leading to the Kriging functions in the calibration. Some phenomena are difficult to describe by coarsed grained particles, especially the flow within narrow gaps, for example through the funnel of the angle of repose experiment. In the simulations the funnel was therefore replaced by an insert region with the same mass flow and exit velocity as the funnel. All simulations were performed with exactly the same masses and mass flows. If the coarse-graining factor

Journal Pre-proof * dSim d p CG = = d 32 d p

(19)

is high, it causes some inaccuracies in the post processing, for example in the angle of repose determination as the pile surface is not so smooth anymore. Therefore we investigated only moderate CG factors: particle diameters d p* of 2 mm, 2.5 mm and 3 mm were used for the calibration of the Carbo HSP13 particles instead of the real size diameter d p of 1.6 mm. The resulting parameters visualized in figure B.18 show that the particle-particle sliding friction

of

* coefficient pp drops with increasing CG factor. This is intuitive as bigger particles harder glide

above each other in the laminated horizontal conveyor experiment, similar to a surface with higher

ro

* coarseness. To match the target residence time of the real sized particles, pp has to decrease with

-p

the CG factor accordingly. To still meet the angle of repose then, the particle-particle rolling

re

* friction coefficient  R,pp has to increase, which is overall the case. In another coarse-graining

lP

* study where the dynamic angle of repose was measured, pp could remain unchanged for

moderate coarse-graining ratios [45]. However, as the study used a linear contact model, non-spherical particles and no rolling friction model, it is questionable if the results should be

na

compared to the current study.

ur

* The particle-wall sliding friction coefficient  pw basically does not change if coarse

graining is applied. This was expected, since this coefficient also barely changed between particle

Jo

sizes; the particles behave in an arrangement similar to a rigid body on the horizontal conveyor and * the size of the particles therefore has little influence on  pw . Like in the case of real size particles, * the particle-wall rolling friction coefficient R,pw is hard to calibrate exactly because the

* simulations are barely sensitive to R,pw for values above 0.3, so that the trend in the points for * R,pw has no special physical meaning. * The particle-particle restitution coefficient epp drops with the CG factor due to less

frequent collisions in the coarse-grained case, which should be explained by the kinetic gas theory in the following. The collision frequency of a single particle is [41]

z =  dp2 v

N V

,

(20)

Journal Pre-proof where v is the average particle velocity and N the number of particles in a volume V . The average particle velocity is assumed to be the same in the coarse-grained simulation to preserve kinetic energy. Since d p scales with CG and N with 1/ CG3 , z scales with 1 / CG . The collision frequency of all N particles is then proportional to 1/ CG 4 . The energy dissipation over time due to particle-particle collisions is proportional to the collision frequency of all 2 particles and the energy dissipation per collision, which scales with the 1  epp and the mass of the

particle. It follows 2 1  epp

,

of

energy dissipation due to p-p collisons time

CG

(21)

ro

because the mass of each particle scales with CG3 . If one assumes that energy dissipation over

-p

* time should remain the same for coarse-grained simulations, epp has to follow the relationship

,

(22)

re

* 2 epp = 1  (1  epp )  CG2

which is indicated by a dotted line in figure B.18. The same expression was derived by Lu et al.

lP

* with similar reasoning [44, 45]. The formula depicts a decline of epp , but less pronounced as it is

na

seen in the curve obtained by the calibration. By the collision frequency with a wall it can be * shown in a similar way, that the particle-wall restitution coefficient epw should be independent of

ur

* CG . This is also seen in the results, epw does not alter noticeably when coarse-graining is

applied. It should be noted here that in many other studies the restitution coefficient is considered

Jo

invariant [46, 47], following the derivation from Bierwisch [48]. In this derivation it is stated that the number of collisions per volume and time scales with 1/ CG3 , which is questionable. The decreasing particle-particle restitution coefficient with coarse graining factor in our work supports the expectation mentioned in [49] and the derivation by Lu et al.[42, 43]. For a coarse-graining factor of 1.9, the solution of the optimization problem for the plate impact simulation jumps from the valley between peaks of masses m2 and m3 , which look * * similar to the ones shown in figure B.17, to a lower value of epw , because further lowering epp is

* * not possible. It is very questionable if epp and epw would describe other experiments well in this

case, so that their values are shown as unconnected, empty markers in figure B.18.

Journal Pre-proof

4.

Conclusion In this study, a three stage calibration approach based on bulk flow experiments (angle of

repose, horizontal conveyor and plate impact) and surrogate models (Kriging polynomials) was developed. It was applied successfully to determine DEM parameters for a Hertz contact model and a rolling friction model for two wall materials and five different sorts of bauxite particles, which are considered for application in solar thermal particle receivers. Only moderate variations among particle types could be seen in the resulting parameters, in particular the sliding friction

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pw and particle-wall restitution epw barely differed. With ceramic insulation board as contact partner, the particle-wall friction was higher and the particle-wall restitution coefficient lower than

ro

on steel, as it was expected. The results were further discussed based on a sensitivity analysis of the different numerical experiments and contour plots of their surrogate models. We found that

-p

rolling friction is necessary to describe the motion behavior of the investigated particles and that

re

the rolling friction to a wall R,pw is confined to a value above  0.3 , but a precise value can hardly be determined with our calibration approach. In general, the obtained parameters are

lP

considered to be reliable due to the consistency of the calibration results with the sensitivity study and findings from literature.

na

Additionally a study was conducted to provide parameters for coarse-grained simulations. The particle-wall sliding friction pw and the particle-wall restitution coefficient epw were found

ur

to be invariant to the coarse-graining, while particle-particle friction pp and particle-particle

Jo

restitution epp decreased with increasing coarse-graining factor. As the restitution coefficient epp is often assumed to be invariant to coarse graining in the literature, our results and their successful explanation by kinetic particle theory induce further investigation of this topic. The determined parameters for both the real size and coarse-grained particles can pave the way for solar particle receiver simulations via DEM, as no calibrated parameters for bauxite particles have been published before. If other than Bauxite particles should be used, the developed calibration procedure can be applied to them as well. In the future the implications of coarse-graining should be investigated in more detail; not only on the particle motion, but also on the heat transfer models for solar particle receiver simulations.

Journal Pre-proof

Appendix A. Further Constants in Contact Model (A.1)

2(2 1 )(1 1 ) 2(2  2 )(1  2 ) 1 =  Geff Y1 Y2

(A.2)

1 1 1 =  Reff R1 R2

(A.3)

1 1 1 =  meff m1 m2

(A.4)

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Appendix B Funnel Mass Flows

of

1 1  12 1  22 =  Yeff Y1 Y2

-p

The funnel mass flows influence the outcome of the horizontal conveyor experiments (c) and (d), so that they were measured before the actual experiment. The results are listed in table

re

B.9. The funnels all had the same nominal size (outlet diameter 7.5 mm) but showed different mass flows, while the mass flow of each funnel was highly reproducible. This is explained by

lP

production tolerances or previous usage of the funnels, where they might have been deformed. The

C. K. Ho, Advances in central receivers for concentrating solar applications, Solar Energy

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(2014) 4197–4214 (2014).

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List of Figures B.1 Microscope images of investigated bauxite particle types. The white alumina particle between the SG10H particles is a tracer used for particle tracking

44

B.2 Two particles in contact, overlapping by  n in normal direction. The normal contact force is represented by a network of a spring and a dashpot

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B.3 Calibration procedure

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B.4 Experimental setup of angle of repose experiment

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B.5 Cardboard laminated with SG10H particles

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B.6 Analysis of the experimental (top) and simulated (bottom) angle of repose of CarboHSP

B.7 Setup of horizontal conveyor experiment

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particles on steel

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B.9 Motion profile of horizontal conveyor

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B.8 horizontal conveyor laminated with SG10H particles

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B.11 Sensitivity of angle of repose simulation to contact parameters

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B.10 Setup of plate impact experiment (left) and simulation (right)

B.12 Sensitivity of horizontal conveyor simulation to contact parameters. Left: with particles

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glued to the plate, right: particles moving on a flat surface

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B.13 Sensitivity study for plate impact DEM simulation, experimental value of CarboHSP13 on

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steel

B.14 Mass on scale, HSP13 particles on laminated plate

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B.15 Contour lines of Kriging functions, generated from 100 sample points for Carbo HSP13 particle diameter. Intersection point of dashed experimental target value contour lines marked with a dot

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B.16 Contour lines of Kriging functions of angle of repose and horizontal conveyor simulations, generated from 100 sample points. Intersection point of dashed experimental target value contour lines marked with a dot

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B.17 Contour lines of Kriging functions for the masses in the boxes of the plate impact simulation, generated from 100 sample points 60 B.18 Calibration results with coarse-grained particle size

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Figure B.1: Microscope images of investigated bauxite particle types. The white alumina particle

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Figure B.2: Two particles in contact, overlapping by  n normal direction. The normal contact

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Figure B.3: Calibration procedure

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Figure B.4: Experimental setup of angle of repose experiment

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Figure B.5: Cardboard laminated with SG10H particles

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Figure B.6: Analysis of the experimental (top) and simulated (bottom) angle of repose of

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CarboHSP particles on steel

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Figure B.7: Setup of horizontal conveyor experiment

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Figure B.8: horizontal conveyor laminated with SG10H particles

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Figure B.9: Motion profile of horizontal conveyor

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Figure B.10: Setup of plate impact experiment (left) and simulation (right)

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Figure B.11: Sensitivity of angle of repose simulation to contact parameters

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Figure B.12: Sensitivity of horizontal conveyor simulation to contact parameters. Left: with

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CarboHSP13 on steel

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Figure B.14: Mass on scale, HSP13 particles on laminated plate

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Figure B.16: Contour lines of Kriging functions of angle of repose and horizontal conveyor simulations, generated from 100 sample points. Intersection point of dashed experimental target

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Figure B.18: Calibration results with coarse-grained particle size

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List of Tables 60

B.2 Parameters required for DEM contact model

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B.3 Baseline parameters for sensitivity study of calibration experiments

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B.4 Results of angle of repose experiments

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B.5 Results of horizontal conveyor experiments

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B.6 Results of plate impact experiments

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B.7 Calibration Results

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B.8 Final ckeck of calibration results

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B.1 Particle size statistics

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Journal Pre-proof Table B.1: Particle size statistics, diameters in μm

d32

Name

d10,0

d50,0

10

d90,0

Carbo HSP13*

1595 1344 1547 1774 0.941

SG10H‡

1201 1019 1176 1322

Carbo HSP16/30

1101

917

1052 1247 0.948

Carbo HSP20/40

826

703

800

919

0.954

Carbo HSP30/60

603

496

578

677

0.957

0.94

* : used in the CentRec prototype receiver [8]

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Journal Pre-proof Table B.2: Parameters required for DEM contact force and rolling friction models Value

R

Particle radius

0.5 d32



Particle density

3560 kg m3

Y

Young’s modulus

5 MPa



Poisson’s ratio

0.3

epp

Particle-particle restitution coeff.

calibrated

epw

Particle-wall restitution coeff.

calibrated

pp

Particle-particle sliding friction coeff.

pw

Particle-wall sliding friction coeff.

R,pp

Particle-particle rolling friction coeff.

calibrated

R,pp

Particle-wall rolling friction coeff.

calibrated

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Name

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Journal Pre-proof Table B.3: Baseline parameters for sensitivity study of calibration experiments

E



epp

5 MPa

0.3

0.6 0.6

pp

pw

R,pp

R,pp

0.35 0.35

0.2

0.2

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Journal Pre-proof Table B.4: Mean value of angle of repose measurements and standard error of the sample mean. Angle on steel  S , on alumina Al2O3 and on glued particles  glued

Al2O3

S

Name

 glued

34.10°±0.16° 36.20°±0.26° 38.28°±0.61°

SG10H

41.71°±0.29° 46.20°±0.31° 45.65°±0.65°

Carbo HSP16/30

38.63°±0.23° 40.36°±0.36° 40.98°±0.57°

Carbo HSP20/40

35.14°±0.52° 38.09°±0.22° 37.80°±0.43°

Carbo HSP30/60

40.12°±0.12° 40.19°±0.46° 40.89°±0.54°

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Carbo HSP13

Journal Pre-proof Table B.5: Results of horizontal conveyor experiments. Time to reach 100 g on scale for a plate surface made of steel ( t100S ), made of alumina ( t100Al2O3 ) and made of fixed particles ( t100glued )

t100glued

t100S

t100Al2O3

Carbo HSP13

10.13±0.06 s 11.59±0.10 s 17.90±0.02 s

SG10H

9.99±0.06 s

12.71±0.11 s 15.99±0.12 s

Carbo HSP16/30

9.73±0.07 s

11.87±0.04 s 15.55±0.13 s

Carbo HSP20/40

9.31±0.07 s

11.18±0.02 s 14.22±0.06 s

Carbo HSP30/60

9.18±0.01 s

11.40±0.07 s 13.39±0.06 s

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Journal Pre-proof Table B.6: Results of plate impact experiments Wall

Initial mass [g]

m1 [g]

m2 [g]

m3 [g]

m4 [g]

Carbo HSP13

steel

68.73

30.21 ±0.15

8.5 ±0.09

2.74 ±0.04

0.55 ±0.03

SG10H

steel

33.79

12.79 ±0.09

6.3 ±0.05

1.58 ±0.05

0.25 ±0.01

Carbo HSP16/30

steel

34.83

10.90 ±0.18

6.34 ±0.04

2.57 ±0.14

0.61 ±0.05

Carbo HSP20/40

steel

30.87

12.47 ±0.09

4.94 ±0.02

1.42 ±0.03

0.26 ±0.01

Carbo HSP30/60

steel

17.64

5.84 ±0.08

4.46 ±0.03

1.17 ±0.07

0.14 ±0.01

Carbo HSP13

Al2O3

68.73

38.12 ±0.24

7.89 ±0.08

1.26 ±0.03

0.30 ±0.03

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R,pw

R,pw

epp

epw

0.69

0.31

0.09

0.27

0.74

0.44

0.27

0.78

0.30

0.38*

0.46

0.43

0.77

0.44

0.89

0.45

0.82

0.44

Al2O3 SG10H

Steel

0.38 0.53

Al2O3 HSP16/30

Steel

Steel

0.61

Steel

0.30

0.40† 0.12

0.38 0.71

Al2O3 HSP30/60

0.16

0.41

Al2O3 HSP20/40

0.31

0.29 0.36

0.72

0.30 0.09

0.13

0.39

* : no intersection of contour lines

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† : simulation of AOR was in all points too low

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0.45* 0.49†

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Al2O3

0.30

0.42

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pw

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HSP13

pp

0.50* 0.42

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Table B.8: Final check of calibration results, deviations to experimental values t100

m1

m2

m3

1.4%

–1.3%

0.1%

–0.2%

0.7%

40.5%

–52.3%

–2.9%

0.1%

–0.4%

–19.2%

2.1%

–3.5%

1.5%

–3.4%

59.2%

14.0%

–3.0%

–0.1%

–3.3%

2.0%

1.0%

12.4%

0.5%

–3.5%

–0.2%

–5.2%

–30.1%

–5.6%

–10.2%

–19.3%

–10.1%

Al2O3 SG10H

Steel

3.1%

–1.2%

Al2O3 HSP16/30

Steel

1.2%

–1.3%

Al2O3 HSP20/40

–1.4%

Steel

–4.0%

Al2O3 HSP30/60

Steel

0.1%

–2.8%

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2.0%

–8.4%

0.1%

–6.2%

1.1%

–1.7%

–0.1%

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Al2O3

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

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t100glued

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HSP13

 glued

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Experiments (c)

Carbo HSP13

24.16 g/s

18.97 g/s

SG10H

23.78 g/s

20.15 g/s

Carbo HSP16/30

23.89 g/s

21.27 g/s

Carbo HSP20/40

25.83 g/s

22.99 g/s

Carbo HSP30/60

29.0 g/s

26.12 g/s

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Highlights  Three-stage calibration procedure based on bulk experiments and surrogate functions  Sensitivity study of angle of repose, horizontal conveyor and plate impact tests  Calibration of DEM parameters for 5 different bauxite proppants for solar receivers  Coarse-graining study providing parameters for scale-up  Particle-Particle restitution not invariant as often assumed in literature

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Johannes Grobbel: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Project administration, Software, Supervision, Visualization, Writing – original draft, Writing – review&editing Stefan Brendelberger: Supervision, Writing – review & editing Matthias Henninger: Investigation, Writing – review & editing Christian Sattler: Funding acquisition, Supervision, Project administration, Resources Robert Pitz-Paal: Funding acquisition, Supervision, Project administration, Resources

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