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9 ................................................................................................................ . 1 9 ........................................................................................................... 1.1 10 ..................................................................................................... 1.2 Discrete Element Method :) ............................. 14) 1.3

NaBH4 ......................................................................... 17 . 2 17 .................................................................................................. 2.1 22 ............................................................................................... 2.2

direct shear test): ................................................................... 22) 2.2.1 27 ...................................................................................... 2.2.2

32 ......................................................................................... 2.3 44 .............................................................................. 2.4

44 ............................................................................... 2.4.1 44 ............................................................................................... 2.4.2 47 .................................................................................................... 2.4.3 54 ................................................................................................... 2.4.4

56 ..................................................................................................... . 3 56 ............................................................................................................. 3.1 57 .............................................................................................. 3.2

58 ............................................................................................ 3.2.1 60 ............................................................................................................. 3.3

61 ............................................................................................................ 63 ............................................................................................. ' 68 ................................................................................... ' Arduino ............................................................................................... 73 ' Matlab ...................................................................................................... 76' 78 ......................................................................................................... '

5

PEM ........................................................................ 10 : 1 12 ........................... [.1] : 2

12 ......................................................... [.2] / :3 13 ................................................ [.3] : 4 14 ......................................................... [.4] : 5 20 .......................................................... . NaBH4 : 6 R2 ................................. 20 - R1 : 7 21 .......... . un - us . : 8 22 .................................................................................... : 9

23 ...... . s n :10 : 11

24 ................................................................................................... . : 12

............................................................................................................................. 25 25 ................................................. : 13 26 .................... . : 14 . : 15

27 ............................................................................... . 27 ............................................ .0 - 90 30 : 16 28 ................................................................. : 17 test 3 test 4)) . 29 90 -( test 1 test 2) 0 - : 18 29 ........................................................................... ' : 19

30 .......................................................................... ' : 20 30 .......................................................................... ' : 21 31 ...................................... . : 22 31 ................................ test 4,5 - test 1,2,3: 23 32 .................... 4 - . - :: 24 Yade-DEM ................. 33 : 25 : 26

............................................................................................................................. 33 fz2 fz :27 fz3. ....................................................................................................................... 34

34 ................................... : 28 44 ,( ) ( . ) : 29 44 ................................................................ . : 30 44 ................................................................................. : 31 44 ..................... . ( , ) ( : )32 Hn ............................................. 44 - Hs :33 44 ......................................................................... : 34 44 ...................................................................................... : 35 44 ................................................. - : 36 . . 090 . . 00 . : 37 47 .................................................................................................................. .030

. . 090 . . 00 . : 38 47 .................................................................................................................. .030

. . 090 . . 00 . : 39 48 .................................................................................................................. .030

48 ..................................................... :40 49 .......................................... 41: 49 ................................. . : 42 50 ............. 090 :43 51 ........ - - :44 :45

52 ........................................................ . 53 ........ : 46

6

54 ........................ : 47 56 ............................................................................. : 44 56 .................................................................................... : 45 57 ........................................................................ : 46 58 ................................................................. . : 47 3 1 : 48

59 ................................................................... . 8-, 6, 4 : 49

59 ................................................................................................................ . 60 ............. : 50 60 ................................................................... : 51 Yade Dem ....................................................... 65 : 59 67 ..................................................................... . mesh : 60 68 ............................................................................................. : 52

69 ......................... . b PWM : 53 69 ............................................ : 54

70 ................................................................................. : 55 71 .................................................................... : 56 Matlab GUI (Graphical User Interfaces) .................................................... 72: 57 72 .................................... NaBH4 : 58 76 ............................................................... : 61 76 .................................................................... :62 77 .............................. : 63 77 ....................................... :64 77 ................................................ : 65

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61

Larminie, James, Andrew Dicks, and Maurice S. McDonald. Fuel cell systems

explained. Vol. 2. New York: Wiley, 2003.

[1]

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[2]

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[3]

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[4]

Barbir, Frano. PEM fuel cells. Springer London, 2006. [5]

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[6]

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[7]

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[8]

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[9]

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Cleary, Paul W., and Mark L. Sawley. "DEM modelling of industrial granular flows: 3D case

studies and the effect of particle shape on hopper discharge."Applied Mathematical Modelling 26.2

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[14]

Price, Mathew, Vasile Murariu, and Garry Morrison. "Sphere clump generation and trajectory

comparison for real particles." Proceedings of Discrete Element Modelling 2007 (2007).

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Garcia-Rojo, R., S. McNamara, and H. J. Herrmann. "Influence of contact modelling on the

macroscopic plastic response of granular soils under cyclic loading." Mathematical Models of

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Wojewoda, Jerzy, et al. "Hysteretic effects of dry friction: modelling and experimental

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Margenau, H. "Van der Waals forces." Reviews of Modern Physics 11.1 (1939): 1. [21]

Chareyre, B., L. Scholts, and F. Darve. "Micro-statics and micro-kinematics of capillary

phenomena in dense granular materials." Powders and Grains 2009 (Golden, USA) (2009).

[22]

Baxter, J., et al. "A DEM simulation and experimental strategy for solving fine powder flow

problems." Chemical Engineering Research and Design 78.7 (2000): 1019-1025.

[23]

Mitarai, Namiko, and Franco Nori. "Wet granular materials." Advances in Physics 55.1-2 (2006):

1-45.

[24]

Johnson, C. E., et al. "Shear measurement for agricultural soils--a review."Trans. ASAE 30.4

(1987): 935-938.

[25]

Mohamed, A. M. O. "Determination of in situ parameters of sandy soils for off-road vehicle

mobility." Journal of Terramechanics 40.2 (2003): 117-133.

[26]

Tulluri, Sai S. Analysis of Random Packing of Uniform Spheres Using the Monte Carlo

Simulation Method. Diss. New Jersey Institute of Technology, Department of Mechanical

Engineering, 2003.

[27]

Bonacucina, Giulia, et al. "Mechanical characterization of pharmaceutical solids: A comparison

between rheological tests performed under static and dynamic porosity conditions." European

journal of pharmaceutics and biopharmaceutics 67.1 (2007): 277-283.

[28]

Mani, Roman, Dirk Kadau, and Hans J. Herrmann. "Liquid migration in sheared unsaturated

granular media." Granular Matter 15.4 (2013): 447-454.

[29]

Gladkyy, Anton, and Rdiger Schwarze. "Comparison of different capillary bridge models for

application in the discrete element method." Granular Matter16.6 (2014): 911-920.

[30]

S.C. Amendola, S.L. Sharp-Goldman, M.S. Janjua, N.C. Spencer, M.T. Kelly, P.J. Petillo, M.

Binder, Int. J. Hydrogen Energy 25 (2000) 969975.

[31]

S.C. Amendola, S.L. Sharp-Goldman, M.S. Janjua, M.T. Kelly, P.J. Petillo, M. Binder, J. Power

Sources 85 (2000) 186189.

[32]

H. Dong, H. Yang, X. Ai, C. Cha, Int. J. Hydrogen Energy 28 (2003) 10951100. [33]

Y. Kojima, K. Suzuki, K. Fukumoto, M. Sasaki, T. Yamamoto, Y. Kawai, H. Hayashi, Int. J.

Hydrogen Energy 27 (2002) 10291034.

[34]

Y. Kojima, Y. Kawai, H. Nakanishi, S. Matsumoto, J. Power Sources 135 (2004) 3641. [35]

milauer, Vclav, and Bruno Chareyre. "Yade dem formulation." Yade Documentation (2010). [36]

63

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NaBH4: 64

73

Arduino '

/*

1. function to first comunicate with matlab

the function establish the first contact with matlab

start void loop

2. readnig parameters from matlab (execute only one)

reading the intial running parameters

-calibrate the desire parameters due to there own sensor

3. sending data if the matlab request by sending something [slave]

if matlab send any signal to the buffer the arduino will send the data.

4. reading values from analog pins

-read the value from the analog pins

5. main motor control, if the pressure is high it will stop the motor!

turn and stop the powder pouring

6. valve control (due to inside pressure)

control a delicate movement of the servo valve to higher and lower the hydrogen flow.

7. dead end valve (open end close)

control the dead end valve opening time and close time

normally we need 10 to 1 seconds

*/

#include

Servo myservo1; // create servo object to control a servo Servo myservo2; // create servo object to control a servo int red = 7; // the red light indicate parameters transfer int green = 8; // the green light blinking indicate that main loop is working int pwm_a = 3; //PWM control for motor outputs 1 and 2 is on digital pin 3 int pwm_b = 5; //dir control for motor outputs 1 and 2 is on digital pin 12 char conversionBuffer[5];// for reading the buffer string int w=0; // for the reading parameters loop

void setup() { Serial.begin(9600); myservo1.attach(6); // attaches the servo on pin 5 to the servo object myservo2.attach(12); // attaches the servo on pin 6 to the servo object pinMode(red, OUTPUT); pinMode(green, OUTPUT); pinMode(pwm_a, OUTPUT); pinMode(pwm_b, OUTPUT); analogWrite(pwm_a, 0); //set both motors to run at (100/255 = 39)% duty cycle (slow) analogWrite(pwm_b, 0); //set both motors to run at (100/255 = 39)% duty cycle (slow)

establishContact(); // send a byte to establish contact until receiver responds }

////////////////////// setup for the program //////////////////////

float flow_th=20; // thresh-hold for the hydrogen flow

///////////////////// declaring is needed////////////////////////

float pos=60; float current=0;// will be connect to A1 float flow =0;/// flow float inside_pressure=0;// will be connect to A3 float inside_pressure_th=0; int valve_time=0; int pressure=0; int Starttime = millis(); int connection_time=0; int dead_end_open_time=1; int motor_spin=0; float motor_speed=0; float maximum_motor_current=0; unsigned long int revers_time=0; unsigned long int time=0; //////////////////////// override by matlab/////////////////////////

float flow_desire=0; float inside_pressure_max=0; float inside_pressure_min=0;

74

float dead_end_close_time=0; //[seconds]

void loop() { ////////////// reading parameters from matlab (execute only one) ////////////

while (w==0) { digitalWrite(red, HIGH); flow_desire = Serial.parseFloat(); inside_pressure_max = Serial.parseFloat(); inside_pressure_min = Serial.parseFloat(); maximum_motor_current = Serial.parseFloat(); dead_end_close_time = 9; digitalWrite(red, LOW); flow_th=0.05*flow_desire; if( flow_desire != 0 && inside_pressure_max != 0 && inside_pressure_min != 0 &&

dead_end_close_time != 0 ){ w=1; Serial.println(5); digitalWrite(red, HIGH); delay(2000); digitalWrite(red, LOW); flow_desire=flow_desire*1; //need to be calibrate to a flow sensor inside_pressure_max=inside_pressure_max*1; //need to be calibrate with sensor inside_pressure_min=inside_pressure_min*1; //need to be calibrate with sensor dead_end_open_time=10-dead_end_close_time;

} } //////// sending data if the matlab request by sending something [slave] ////////

if (Serial.available() > 0) { Serial.read(); Serial.println(1010); Serial.println(inside_pressure); Serial.println(current); //Serial.println(pos);

Serial.println((pos-10)/11);// position of the control valve Serial.println(flow); //Serial.println(100); connection_time=millis(); delay(10); digitalWrite(green, HIGH); delay(10); digitalWrite(green, LOW); }

///////////// reading values from analog pins //////////////////

current=analogRead(A1)*(0.18/50);//amper, volteg on the shunt calibration. flow=analogRead(A2); // need to be divided in R [I=V/R] flow=(flow-37.96)/0.8356; inside_pressure=analogRead(A3); //// pressure 180=0.4 bar inside_pressure=(inside_pressure*0.264-10.81)/100; inside_pressure_th=0.1*inside_pressure_min;

/////// main motor control, if the pressure is high it will stop the motor! //////

/////////////// MOTOR CONDITION ///////////////

// if the pressure in the tank is high we stop the motor spin

if (inside_pressure >= inside_pressure_max+inside_pressure_th) { motor_spin=0; motor_speed=0;} // if the pressure in the tank is low we turn the motor on to build pressure

if (inside_pressure = maximum_motor_current && motor_spin==1) { motor_spin=-1; revers_time = millis();} // if the number of second of revers didn't pass

time=millis(); if(time -revers_time

75

if(motor_spin==1){ if(motor_speed=-100) motor_speed=motor_speed-0.1;}

if(motor_speed=0){ analogWrite(pwm_a, abs(motor_speed)); analogWrite(pwm_b, 0);}

///////// valve control (due to inside pressure) //////////

//if the pressure is below 0.4 bar close the valve

if (inside_pressure < inside_pressure_min-inside_pressure_th) {

pos=10; myservo1.write(pos); } // 180=0.4 bar

//if the pressure is higher start open the valve

if (inside_pressure >= inside_pressure_min+inside_pressure_th) { if ( flow = (flow_desire+flow_th) ) { pos=pos-0.01; } if (pos=120) { pos=120; } myservo1.write(round(pos)); // tell the servo to go to the

position in variable 'pos'

} //////////// dead end valve (open end close) //////////////

int dead_end_open_time=10-dead_end_close_time;

valve_time=millis() ; if (valve_time < Starttime+(dead_end_close_time*1000)) {

myservo2.write(15); } // close if (valve_time >= Starttime+(dead_end_close_time*1000)) {

myservo2.write(120);} // open

if (valve_time >=

Starttime+(dead_end_close_time*1000)+(dead_end_open_time*1000)) {

Starttime = millis(); }

} // end main loop

///// function to first comunicate with matlab void establishContact() { //Serial.flush(); while (Serial.available()

76

Matlab '

. Arduino - matlab

.

.

()establishContact

' 4(. 'void_setup )

.

///// function to first comunicate with matlab void establishContact() { //Serial.flush(); while (Serial.available() 0)

test=fscanf(s1,'%s');

if(test=='5')

display(['good contact']);

flushinput(s1);

w=0;

end

end

end : 66

. string .

' 4 ' w . string

77

while (w==0) { flow_desire = Serial.parseFloat(); inside_pressure_max = Serial.parseFloat(); inside_pressure_min = Serial.parseFloat(); maximum_motor_current = Serial.parseFloat(); if( flow_desire != 0 && inside_pressure_max != 0 && inside_pressure_min != 0 &&

dead_end_close_time != 0 ){ w=1; Serial.println(5);

: 67

.' 5 '

. n/ string string ()Serial.parseFloat

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.

'. 4 '' 4 . '

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pause(0.2)

fprintf(s1,'%s\n','4');

pause(0.5)

if(s1.BytesAvailableFcnCount >0)

test_byte=str2double(fscanf(s1,'%s'));

end

if(test_byte==1010)

test_byte=0;

av1=str2double(fscanf(s1,'%s'));

av2=str2double(fscanf(s1,'%s'));

pwm=str2double(fscanf(s1,'%s'));

av0=str2double(fscanf(s1,'%s'));

flushinput(s1);

end : 68

if (Serial.available() > 0) { Serial.read(); Serial.println(1010); Serial.println(inside_pressure); Serial.println(current); Serial.println(pos); Serial.println(flow); }

: 69

78

'

Calculating the mechanical parameters governing sodium borohydride

) powder4NaBH(

iv

and N. Shvalb iii

Moshe-,B. Benii

hechterc, A. SiY. Nagar

Abstract

The paper addresses the numeric optimization for NaBH4 powder flow which is

commonly used for hydrogen gas production. During the process of the powder motion,

high number of collisions occurs between particles constituting the powder. These

collisions are characterized by physical constants such as drag coefficients and restitution

coefficients. This paper issues these parameters. We use a discrete element method

to model the powder and assume that the powder is composed of tiny spheres interacting

according to a specific spring damping model. In a series of appropriate physical wedge

penetration experiments, force-displacement graphs were measured. In Addition a set of

shear tests were conducted from which a normal-shear forces graphs were extracted. We

formulate analytical estimations for each of the experiments. We then, compared these

graphs with graphs generated by their corresponding simulation tests. Using Genetic

Algorithm optimization we obtained a set of governing parameters that best fits the

powder behavior. In order to refine our results we have used our analytical formulations

to manually search the parameter space for a better fit.

Keywords: Discrete Element Method, powder, adhesion, Sodium borohydrid, mechanical

parameters, wedge insertion test, direct shear test, angle of repose.

1. Introduction

Calculating the character of the motion of powder under various geometric constraints

is a complex task. The lack of uniformity in the size of the particles, the powders external

texture, and its density are just a few factors affecting its flow. In addition, there are

parameters that vary over the course of the flow, for example, the powders density,

which creates internal instability, i.e., the character of the powders motion changes over

the course of its flow.

In order to resolve uniformly problem, empirical, analytic, and numeric models have

been developed to demonstrate the character of the flow of powder under geometric

constraints. Discrete element method is a numeric model based on 1Cundals and Strasss

Preprint submitted to Comput. Methods Appl. Mech. Engrg.

April 4, 2015

79

[1] theory, which objective is to demonstrate particle motion taking into account the

interaction between particles; between the particles and the interior surfaces, as well as

the effect of various gravitational and acceleration forces that apply to the particles.

The advantages of the numeric discrete element method derive from the relative

simplicity of the particle motion solution even under changes in the systems boundary

conditions. This is carried out by changing the geometric measurements during the

motion of the particles; by changing the flow regimes due to external forces at work; by

changing the velocities of the particles; as well as the by ability to build experimental

arrays within a reasonable time frame.

The main limitation of the discrete element method is the demand for computer

resources which translate into longer calculation as the more particles are fed into the

system, this difficulty can be overcome by defining a property that lends this method its

advantage over analytical empirical models.

In order for the model to demonstrate the particle motion with high accuracy, there is

a need to introduce each particles size, which demands long calculation time, even for

few thousand particles. Therefore such calculation cannot be carried out in a reasonable

time by todays processors, Many studies concentrate on particle size on simulation

results for particles larger than the originals [2,3]. These reports have shown a low

deviation between simulation results. Therefore, we can extend the radii of the particles

in order to shorten the calculation time.

We can also see [3] that the particles shapes have an effect on simulation results as

well: When the particles are modeled as symmetrical circles, there is a decrease in their

shear force, in turn affecting the character of the particle flow. In studies conducted [4,2],

we can see that the particles circularity is decreased, affecting the particle flow up to 30%

lower. In the discrete element method, two spheres can be attached (clump), thereby

creating complex shapes that differ from symmetrical circles. When the particles

geometries in the simulation approach that in reality, we see an improvement in the

simulation results.

Good correlation between the mechanical properties of the particle in motion in the

simulation take into account the interactions occurring between particles and between

the particles and the surface. The mathematical model for describing the interaction uses

a Kelvin-Voight element (simultaneous spring and damping): The spring connecting two

particles radially demonstrates their yang module, i.e., their elasticity; the spring

connecting two particles tangentially demonstrates the friction between them [5];

Cundall and Strack propose calculating the damping constant values as per a second-order

spring and damp system multiplied by the damping coefficient determined by the trials.

80

In addition, a spring stretched radially creates attraction forces between the particles,

demonstrating the cohesion between them.

A number of studies have demonstrated how the mechanical constants of various

materials are calculated, such as producing a strain graph between two particles using an

instrument that draws together and applies pressure on two particles that is accurate to

a nanometer; and a dynamometer that calculates the force between them accurate to

half a millinewton [6]. From a shear force graph, we can derive important mechanical

properties such as an elasticity model and a restitution coefficient.

Remark. Another method for measuring restitution coefficient is to use accurate

measurement tools that examine the particle motion before and after collision with the

surface. Such methods necessitate the use of costly measurement tools and are

applicable when the size of the particles is larger than 500 micrometers. On the other

hand, there are analytical formulae connecting spring constants and normal and

tangential damping to the tangential and the normal restitution coefficients of the

particles (see [9]).

In order to calculate the mechanical parameters of NaBH4 particles whose size is

approximately 50 micrometers (see below), we shall make use of measurement of micro-

properties of the particles by means of a macro property such as can be seen in [7], i.e.,

the insertion of wedges at various head angles and shear experiments in order to study

the effect of the cohesion and damping friction parameters on the various graphs

obtained.

1.1 The Mechanical model

The discrete element method is a numeric method that demonstrates the behavior of a

substance composed of a large number of particles. Each iteration calculates the forces

acting on every particle as a result of contact with adjacent particles and external forces.

A balance force calculation is written based on Newtons second law, where the forces

between particles and between the particles and the interior surfaces are derived from

the deformations, i.e., in every iteration, the overlap depth nu (see Fig. 3) between a pair

of particles and a particle and the interior surface. The larger the overlap depth, the

stronger the forces between the bodies, and commensurately, the stronger the forces

developing between them.

Every interaction between particles and between a particle and an interior surface

includes: a mechanism whose function it is to express the radial forces between two

bodies at the point of contact between them; an element that expresses the tangential

forces between them; and an element whose function it is to express the energy loss to

the system. In detail: The radial rigidity between the bodies is demonstrated by pressure

spring nk related to the particles elasticity module E (Young modulus). The rigidity in the

tangential direction is demonstrated by sk . These parameters govern the shear stiffness

81

and the normal stiffness respectively. Some studies (see for example [7]) assume

proportionality between these constants (Poisson ratio ); on the other hand, we know

that the Poisson ratio for powders ranges from 0.3 to 0.2 (iron is 0.27; sand is 0.2-04);

therefore, here, we shall determine each separately.

Over the course of the particles motion, there is energy loss (dissipation) deriving from

particle-to-particle collision and other processes in the system, such as friction in the

temperature range. In order to model these processes, we include a damping model, of

which there are two common ones: The viscous model, wherein a pair of particles are

exposed to force proportional to their relative velocities and opposing the direction of

their motion sF cy ; The Coulomb (non-viscous) model, which models classic frictional

force [8] wherein a pair of particles are exposed to force proportional to their relative

accelerations land opposing the direction of their motion . Viscous damping occurs in

both the tangential and normal directions set as g . The Coulomb damping s is in the

tangential direction. Spring constant pk demonstrates the cohesion force between two

bodies. Such cohesion between two grains and between a grain and a interior surface are

caused by a number of forces: (1) Among the most significant is electrostatic interaction

between localized surface partial charges on two adjacent particles, this is often referred

to as a van der Waals forces [16]. The power of the van der Waals forces depend in

descending order: The matters molar mass, i.e., the larger the electron cloud, the greater

the forces between atoms in polarity, i.e., attraction between constant dipole in two

molecules is stronger than that of instantaneous dipole forms on other molecules; A

larger molecular surfaces result in stronger interaction. (2) Full charge electrostatic forces

formed by friction of particles during their collision. This may be attractive or repulsive,

and does not require proportional contact between the particles but depends on the

number of collisions. (3) Liquid bridges between particles. Such a bridge is formed

between to adjacent particles exposed to high level of humidity [11]. Two such particles

will experience a capillary attractive force, formed by a liquid solution bridge saturated

with NaBH4 dissolved salt (in our case). The longer the distance between particles, the

greater the contact angle between the liquid and the particle, and the thickness of the

bridge decreases. The bridge is broken when the segregation force between the particles

is stronger than the capillary phenomenon between them. A permanent bond between

soluble particles can form during evaporation of the liquid bridge. The larger the wetting-

and-evaporation cycles, the larger the number of permanent bonds between particles in

the substance.

In general, the importance of van der Waals forces is obvious in particles smaller than

1-2 micrometers; Capillary phenomenon between particles is relevant when the size is

smaller than 500 micrometers and forces resulting from absorption of vapor occurs in

particles smaller than 80 micrometers.

82

Fig. 1: Photograph of NaBH4 grains under an optical microscope.

Using an optical microscope, the size of NaBH4 grains was measured, and an average

value of 50 micrometers (see Fig. 1) was found. Therefore, in NaBH4 powder in a closed

system with high humidity levels (100% R.H), it can be assume that the van der Waals

forces are negligible due to the size of the grains, and low electrostatic forces due to

humidity levels.

Remark. A permanent bond between soluble particles can form during evaporation of

the liquid bridge. Such a bond is referred to in literature as interlocking phenomenon (see

[12]). Here, we assume such an evaporation process does not occur so we shall not

include permanent bond phenomena in our model.

Fig. 2: Interaction model between two particles.

The model we shall use is depicted in Fig. 2. Imagine pair of particles that come into

contact with each other at a certain overlap, so that radial forces act upon them to

separate them radially and tangentially, drawing them closer thereby. For simplicity let

us imagine a pair of identical square particles that seek to bond on a common edge as

depicted in Fig 3.

83

(a) (b)

Fig. 3: A pair of particles that depicted as overlapping squares describe: (a) Radial

overlap, and (b) tangential overlap.

The radial and tangential forces are

(1) n n nF k u

(2) s s s s n

s

n s s s n

k u k u FF

F k u F

The tangential friction model can be thought of as mass connected to a spring. The mass

is subjected to Coulomb friction and is dragged )pushed or pulled) while connected to a

spring at its other end. As the exterior force remains inside a friction cone the mass is

static. The angle of the cone is the Coulumb friction coefficient s . Note that this model is

a hysteretic one [8], wherein mass is given to cyclic loading.

To express the energy loss to the system we shall use two damping mechanisms:

1. Based on the viscous model. These will be expressed as g in all direction

(tangential and normal).

2. The second based on Coulumb friction in the opposite movement direction and

set as s

The governing parameters of the selected model are therefore: (add units)

(1) The stiffness coefficient k .

(2) The friction angle s .

(3) The viscous damping coefficient g .

(4) The cohesion coefficient pk .

Other parameters affecting the results of the simulation are, as aforementioned, the

size of the particles, their shapes, and their diffusion in space. As aforementioned, in order

to lengthen the calculation time, the size of the particles was increased 50 times NaBH4

particles actual size (50 microns). The particles shape was determined as pair of clumped

particles, where the distance between the circles centers D ranged from 0.5 R to R in a

uniform value distribution (Fig. 4). To enable diffusion of particles similar to that in reality,

we produce the particle cloud at a distance from the lower portion of the box such that

84

when we activate the gravitational forces on the particles, they will arrange themselves

freely on the lower portion of the box.

Fig. 4: Two particles bonded using the clumping function.

1.2 Obtaining the controlling parameters

Figure 5 depicts the outline of our study. To find the mechanical parameters of NaBH4

powder, a series of experiments were conducted (see section 2), each of which is

influenced in a differing way thereby. The lack of dependency obtained between the

experiments enabled us to distinguish between the various parameters in order to obtain

their preliminary values. We shall conduct the experiments measuring force and travel.

As aforementioned, the size of the particles constitutes an effect on the results of the

simulation. In the shear experiment, for example, while the main parameters affecting

the shear strength are the friction and the cohesion between the particles, the particles

size will change the effect on the experiment results. Therefore, to minimize the effect of

the particles size thereon, we shall conduct small-scale insertion and shear experiments.

The further we reduce the experimental model, the smaller the number of powder

particles involved. Yet, when reducing the model, we need a higher resolution of the

measuring instrument, and in addition, the experiment will be more sensitive to

disruptions and inaccuracies. Therefore, while there is a need for minimization of the

experimental models, we need to maintain accuracy the quality of results.

Fig. 5: Overall scheme for finding physical parameter.

Using the discrete-element software YADE-DEM, for each real experiment conducted, a

simulation was built that is identical to the real experiment. In addition, results and graphs

are produced from the simulation as well, and compared to the real experiment. In the first

phase of finding the parameters, we have conducted a set of simulated experiments and a

thorough literature review to find a rough magnitude estimation of the parameters. The

resulting (reduced) parameter-space was found to be: the stiffness coefficient

85

6 660 10 900 10 ; the friction angle 0.01-1.5 rad; the damping coefficient 0.05 0.5 ; and the normal cohesion 5 1500 . Next, we conducted an optimization using a genetic algorithm (GA) to obtain better fit.

Since each of the simulation tests takes approximately eight hours in an Intel Xeon E5-

2680 v2 processor. We considered only the three penetration tests at first and conducted

a finer optimazation step. This resulted in a further reduced parameter space: the stiffness

coefficient ranged over 6 6600 10 650 10 ; the friction angle ranged over 0.01-1 rad; the

damping coefficient 0.1 0.5 and the normal cohesion 50 1000 .

Finaly, this was followed by a manual search accompanied with the analytic estimation

introduced in section 4

2. Supporting experiments

The shear strength experiment: results give us information on friction and cohesion of

the particles [13, 14]. In this experiment, we inserted the NaBH4 powder into two

cylinders measuring 2 cm in diameter and 1 cm high that were placed one atop the other

(Fig. 5). The upper cylinder contained a plate measuring 2 cm in diameter that applied

normal, steady pressure, compressing the powder. We used Testometric M250-3 CT

Materials Testing Machine with accuracies 0.1 Newton and 0.1 millimeter to conduct

our experiment. We applied horizontal pressure that produced shearing on the plane

between the cylinders, which was measured until motion was produced between the

cylinders. This experiment was run a number of times, where between experiments, we

increased the normal pressure and measured the shearing obtained in the failure.

Fig. 5: The shear strength experiment.

Failure here refers to a collapse of spontaneous trusses created in the along the powder

grains. Such a phenomenon will result in a sudden change in the force slope (as the

function of the tangential shift). So the expected shear stress and normal stress

behavior is:

(3) =tan(f)+C Where C is the powder cohesion obtained at the cutoff point with the axis of shear

86

strength; and f is the internal friction angle of the powder obtained from the slope of the

resulting line (depicted as a blue line in Figure 8).

The wedge insertion experiment: was conducted using the same Materials Testing

Machine to insert three wedges, the areas of whose bases measure 10 x 15 mm, with

differing head angles: 30o and 90o and 00 (horizontal plate) as shown in Fig. 7 . The wedges

penetrated into a cube with a cross-sectional area measuring 50 x 30 mm and 100 mm

deep, containing NaBH4 powder.

Fig. 7: Wedge insertion experiment with head angles measuring

30o and 90o and a horizontal plate.

The force-displacement graphs are depicted as blue lines in Fig. 9. Note that due to the

difference in the size of the particles in the simulation and the real experiments, the

particles involved in the contact points between the plate and the substance differ. These

differences resulted in a higher level of oscillation in the force-displacement graph for the

simulation results.

3. DEM experiments

We used open-source Yade - Discrete Element Method software. For the three real

experiments conducted, a simulation was built simulating for each. All experiments were

conducted using clump particles dispersed randomly. The particles were constructed such

that their centers distances varied from 0.5 R to R with uniform distribution values.

The shear strength simulation was carried out using 10,000 particles; these were

randomly pre-packed in a cylinder. Three different tests were conducted with 2, 4 and 8

Newton normal force applied on the top of the upper cylinder. In each simulation the

break shear point was extracted. Locating them on a shear-normal force coordinates

system yielded an approximated linear function see Figure 8.

87

Fig. 8: shear experiment. The lower right corner presents a resulting linear function: a

blue line represents the real experiment's results red line represents simulation results

with optimized set of parameters.

The wedge insertion: was conducted using 20,000 particles. For each of the wedges, an

insertion simulation was conducted up to a depth of 20 mm, commensurate with the real

experiment. The force-displacement graphs of the simulation results are presented in Fig.

9.

Fig. 9: Wedge experiment. Blue line indicates real test and polynomial simulation

result in red

4. Analytical formulation

To enable an educated manual search (as described in Section 1) it is essential to have a

rough estimate of what are the main physical factor affecting each of the aforementioned

experiments results. Note that the damping effect may be of importance in dynamic cases,

however, for quasi-static simulation (see [19]) such as ours the effect of damping will not be

significant (see [20]).

88

4.1 Wedge insertion experiment

We shall first describe our attempt to estimate the behavior of the force reaction in the

00 wedge insertion experiment. We consider a simplified pack as depicted in Figure 11a.

(a) (b)

Fig. 10: (a) Particle arrange in the box. (b) Forces apply on particle number 3

Focusing our attention only for the vicinity of the depicted configuration we can write:

n1 1 S1 1 n2 2 s2 2F=F sin( )+F cos( )+F sin( )-F cos( )

For simplicity we assume symmetry (i.e. 2 1 =180- ), so:

n n1 1 s n1 n s s1 1

n n2 1 s n2 n s s2 1

F=k h sin( )+ h k +k h cos( )+

+k h sin( )+ + h k +k h cos( )

Here n sk ,k are the normal and shear elasticity respectively, nh ,hs are the normal/shear

displacements and s is the friction coefficient. The shear and normal displacements in

the vertical wedge movement displacement is demonstrated in Fig. 11 which implies

n sh =hcos(), h =hsin() .

Fig. 11: Normal and shear displacement due to horizontal wedge displacement.

Substituting into the original equation yields:

89

(2) 2

1 1 n s s n 1F=cos( )sin( ) 2k h+2k h +2 hk cos ( )

The angle between particles in random pile is distributed uniformly (see [17]).

So for a three dimensions flat wedge insertion experiment the expectancy reaction force

on the wedge due to infinitesimally displacement is:

(3) n s nE(F)

h(2k +2k + k )=

2

s

Note that this is merely magnitude estimation.

4.2 shear test experiment

Consider a simplified particle pack as shown in Fig. 15a which under shear rearranges as

shown in Fig. 12b and Fig 13a.

Fig. 12: Shear test demonstration

As a result an upper row particle shown in Fig 13b is subjected to the set of forces. We

denote s(in) s(out)F -F the normal forces at the shear direction and at the opposite direction

respectively. n2F is the cohesion force, nF is the normal force applied and n1 s1F , F are the

normal and shear forces respectively.

(b) (a)

Fig. 13: (a) Particle distance calculation. (b) Particle forces apply.

Thus:

(4) 1

n n1 1 s1 1 n2

F =F sin( ) F cos( )+F sin( - )

2 2

90

(5) 1

s(in) s(out) n1 1 S1 1 n2

F -F =F cos( )+F sin( )+F cos( - )

2 2

Here s1 s n n s sF = h k +k h , n1 n nF =k h . We assume the cohesion force n2F is proportional to

the constant pk [18] and can be apply when interaction occur. Interaction between

particles tack place when the distance is less than 1.1R .The distance h between particle

1 and 3 (see Fig. 13a) is given by:

1

1

2Rsin( )h=

sin( - )

2 2

For the distance of 1.1R the corresponding angle is 720

.

Note that the particle normal and shear micro displacement are function of the normal

and shear output macro displacements (Fig 14):

(a) (b)

.n sH ,Hfollowing output displacement n sh ,h: (a) micro particle displacement 14. Fig

(b) zoom for displacement in (a)

Therefore:

nn s

H h =( +H )sin( -)

2tan( -)

2

n ns s

H Hh =( +H )cos( -)-

2tan( -) sin( -)

2 2

Substituting into Eq 4. and solving for local forces yields:

91

n1 n

Hn F =k ( +Hs)sin( -)

2tan( -)

2

s1 s n s

Hn Hn HnF = k sin( -)( +Hs)+k ( +Hs)cos( -)-

2 2tan( -) tan( -) sin( -)

2 2 2

n2

F =k cos( - )

2 2p

So the shear and normal forces equations resulting from normal and shear

infinitesimally displacement are:

(6)

s s n s n s n

s n s n p

F (H ,H ,)=k cos()(H cos()+H sin())-sin()(k (H cos()-

-H sin())-k (H cos()+H sin())+k sin( )

2

s

(7)

n n s s n s n s n

n s n p

F (H ,H ,)=cos()(k (H cos()-H sin()-k (H cos()+H sin()))+

+k sin()(H cos()+H sin())-k cos( )

2

To eliminate particle angle dependence we make use of the uniform distribution

property. The integration split the two cases in the interaction angle 720

.

7 2

20 3

s s s p

7

3 20

EF = F d + F k sin( )d 0.026 1.04 0.95

2p s n n nk H k H k

7 2

20 3

n n n p

7

3 20

EF = F d + F k cos( )d 0.05 0.732 0.02 0.011

2n n p n n s nH k k H k H k

To pursue the shear stress intersection line (denoted by C in Eq. 1) we set the macro

displacement nH in the shear force equation. After assuming kn=ks we get:

(8) n n

s n

n

0.95k (175.7 2.86H k )C=0.26k 1.04H k +

12.57 5.26k

p s

p

n

k

k

The slope of the linear line (in Eq. 1) can now be estimated as the partial derivative:

n n n n

S S

E F E(Fs(EF ))-E(Fs(EF +EF ))=

E F E(F )

92

2

n n

EF EF 6.986(32.86 +66.19 157.9)=

EF EF (5.267 12.5)

s s s n

s n

H H

H H

(9)

26.986(32.86 +66.19 157.9) =

(5.267 12.5)

5. Results

For the first GA optimization stage the fitness function calculated only for the wedge insertion

stiffness coefficientthe experiment (for time consumption reasons). This stage resulted with:

the normal ad; the damping coefficient 0.356; andr 0.28friction angle the; 6635 10

to the compare( 0.472 0.49y x graphshear linear ing the followingyield534 cohesion

results are nt he wedge insertion experimeT). 0.125 0.16y x experiment linear graph real

depicted in Fig 15

Fig. 15: wedge insertion experiments: first GA stages results. Red line indicate the

simulation results.

Next, a GA optimization procedure was performed again with a reduced parameter range

considering both shear tests and wedge insertion tests. The resultant parameters are: the stiffness

coefficient6620 10 ; the friction angle 0.315 rad; the damping coefficient 0.327; and the normal

cohesion 688. The resulting linear shear graph is: 0.335 0.2y x .

Fig. 16: wedge insertion experiments: second GA stages results. Red lines

indicate the simulation results.

A good correlation was obtained between the wedge insertion experiment and that of the

simulation. However the two linear shear equation slopes were 250% different from one

another. Recall that this represents the powder's overall friction. Naively this could be fixed

by simply reducing the friction angle parameter. However, Eq. 3 exemplifies that changing

93

the friction angle will result in changing the wedge insertion as well. Accordingly, in order to

maintain the wedge insertion results while lowering the linear shear slope we reduced the

So, the parameters tested were: d raised the stiffness coefficient in 6%.friction angle in 8% an

rad; the damping coefficient 0.2901the friction angle ; 6658 10stiffness coefficientthe

is: 0.327; and the normal cohesion 688 resulting with the linear shear graph

0.2164 0.27y x

Fig 17: wedge insertion experiments: manual refining according to Eq.3 reduce friction

angle and raise stiffness coefficient. Red line indicate the simulation results.

6. Validation

In order to verify the parameters result we implemented an angle of repose scheme. We

have conducted an experiment in which an open cylinder filled with NaBH4 powder was

positioned on a horizontal table. The cylinder slowly moves upward forming a pile as the

powder pours. The pile angle (depicted in Figure 18) depends on the mechanical parameters

of the powder, mostly by the friction coefficient [21]. The test was performed using a 50mm

high cylinder having a 20mm radius and the upward speed was set to 20 mm/sec.

The DEM simulation corresponding test was performed on 3000 clumped particles with a

35mm radius each. The cylinder speed and geometry was were taken identical in the

simulation and the real experiment. Fig. 18 depicts the repose angles from both the

experiments and simulation tests. A good correlation was obtained as the angle ranges

between 350

to 450

in the real experiment and between 37.50

to 400

in the simulation. Note

that range of angles obtained in the simulation was smaller than that of the real experiment.

This could be attributed to the homogeneity of the particle parameters and the lack of

environment variables in the former.

Recall that due to the time required for running the set of wedge insertion tests in the

optimization step, these tests were performed using a reduced number of particles (6000

instead of 50,000). So, it is required to verify that the parameter we obtained above suit the

wedge penetration tests as well. In order to do so, we used the parameter set obtained

(stiffness coefficient 6620 10 , friction angle 0.315 rad, damping coefficient 0.327 and the

normal coheion 688), in Section 5 on a 50,000 clumped particle tests, simulating 900

wedge.

The results presented in Figure 19.

94

Fig 18: Angle of repose tests. Left: Experiment results. Right: simulation results

Fig 19: Instron comparison Left: 6000 particle Right: 50000 particle

7. Conclusions

We have conducted a set of experiments to pursue a set of physical governing parameters in

NaBH4 powder behavior. We used GA optimization to significantly reduce the parameter

space to search within. In order to conduct a manual search we have formulated analytic

equations that describe the experiments. In the course of our manual search it was found

that these described the general dependencies for example see Fig. 17 As discussed above

all measured parameters were found to be very close in the simulation to those measured in

the real set of experiments with the measured friction and experimented one having the same

order of magnitude. The particles shape is known [2] to affect the resulting friction.

Specifically, Asaf et al. found that the non-dimensional distance d/R between two clumped

particles is significantly effects the friction, as mentioned above we considered a normally

distributed d/R [0.5,1] clumped particles. The authors believe that for NaBH4 powder

were cohesion plays a central role the particle's shape will play a role though it is expected to

be of less importance, nevertheless this should be investigated.

95

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