Modeling the mechanical behavior of sodium borohydride (NaBH4

Materials and Design 108 (2016) 240–249

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Modeling the mechanical behavior of sodium borohydride (NaBH4) powder Yakir Nagar a, Alex Schechter b, Boaz Ben-Moshe c, Nir Shvalb d,⁎ a

Department of Electrical Engineering, Ariel University, Israel Department of Chemical Sciences, Ariel University, Israel c Department of Computer Science, Ariel University, Israel d Department of Industrial Engineering, Ariel University, Israel b

a r t i c l e

i n f o

Article history: Received 2 January 2016 Received in revised form 19 June 2016 Accepted 20 June 2016 Available online 27 June 2016 Keywords: Powder simulated behavior Discrete element simulation NaBH4 mechanical parameters Genetic algorithm as a search method

a b s t r a c t This paper addresses the numeric optimization for NaBH4 powder flow which is commonly used for hydrogen gas production. During the motion process of the powder, a high number of collisions occur between particles constituting the powder. This paper focuses on modeling and finding the parameters that govern these collisions. 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 normal-shear force graphs were extracted. Analytical estimations were formulated for each of the experiments. These graphs were then compared with graphs generated by 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. Lastly, an angle of repose test validated our model. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Sodium borohydride, NaBH4, is an air stable salt considered as a promising hydrogen storage material for energy conversion in fuel cells. Hydrogen can be generated from NaBH4 by heating-pressure cycles or via hydrolysis reaction with water over a catalytic surface. These methods require the transport of the solid salt into the reaction zone. Yet the flow of the powder is a non-trivial task due to forces and interactions between the particle grains: 1.1. Motivation and related works Fuel cells are electrochemical devices which convert chemical energy directly to usable electrical energy in a most efficient and highly clean manner. When operated with hydrogen fuel and oxygen from air, a fuel cell produces clean water and some heat in addition to electrical power. In recent years the development of polymer electrolyte fuel cells (PEMFC) for low temperature all-electrical vehicles has made impressive progress towards commercialization with respect to cost, durability and power density (size).

⁎ Corresponding author. E-mail address: [email protected] (N. Shvalb).

http://dx.doi.org/10.1016/j.matdes.2016.06.077 0264-1275/© 2016 Elsevier Ltd. All rights reserved.

However, hydrogen generation remains a critical barrier for large-scale commercialization of the PEMFC technology, especially for portable applications. The need to store hydrogen safely and efficiently has provided a vast domain for considerable research [1]. In general, hydrogen can be stored using several methods by varying their volumetric and gravimetric hydrogen densities. These methods include high pressure gas cylinders, liquid hydrogen in cryogenic tanks [2], adsorbed hydrogen on materials with a large specific surface area (e.g., intermetallic compounds) [3], metal hydrides [4,5], carbon nanotubes [6,7], metal-organic framework [8], reforming of hydrocarbons, hydrolysis of reactive metals (e.g., Al , Li , Na) and metal hydrides (e.g., LiH , MgH 2 , LiAlH4) with water [9]. The last method has the potential to provide the highest gravimetric hydrogen storage density. Metal hydride reacting with protons from the water molecules releases highly pure hydrogen gas. For example, MgH2, NaAlH4, LiBH4 and NaBH4 have a hydrogen gravimetric density of 7.6% , 7.4% , 18.4% and 10.8% respectively [10]. Among these materials, sodium borohydride, NaBH4, is an air stable, commercially available salt, considered as a promising hydrogen storage material for energy conversion in fuel cells. Review papers have been dedicated to various aspects of hydrogen production from NaBH4 [11,12,13,14] describing the properties, challenges and current development status of this technology. In general, hydrogen can be generated from NaBH4 by heating-

Y. Nagar et al. / Materials and Design 108 (2016) 240–249

pressure cycles or via hydrolysis reaction with water over a catalytic surface. The total reaction is described below (Eqs. (1), (2)): NaBH4 þ2H2 O→4H2 þNaBO2 ΔH o ¼ −216:7

kJ NaBH4 mol

ð1Þ ð2Þ

To date, the large majority of reactors are based on flow of sodium borohydride solution stabilized by NaOH solution (5–10 wt%) over selected catalysts in a fixed bed reactor. The rate of hydrogen production is influenced by the catalyst performance, NaBH4 concentration, stabilizer concentration, reaction temperature, complex kinetics and excess water requirement [14]. Despite the fact that the aqueous solution of NaBH4 is the most convenient storage form, which greatly simplifies the reactors, it has severe restrictions. The main one is NaBH4 solubility. It is 55 g (NaBH4) per 100 g (H2O) [15] at 25 °C, that translates into a gravimetric hydrogen storage capacity of 7.5 wt%. However, in the presence of NaBO2 product, with lower solubility of 28 g (NaBO2) per 100 g (H2O), also affects the solubility of NaBH4. Therefore the maximal concentration of NaBH4 aqueous solution applicable in these flow reactors is limited to 25 wt% of NaBH4 to prevent clogging of the reactor. Under this concentration, the gravimetric hydrogen storage capacity is only around 3.5 wt% [16]. We have considered a new method that applies transport of the solid NaBH4 salt into a water solution reaction zone. Thus, fresh solution is constantly produced in the reactor. Nevertheless, the flow of the powder is a non-trivial task, due to forces and interactions between the particle grains. Often the solid feed is disturbed by gumming of the solid fine powder exposed to high humidity released in the exothermic reaction. Understanding the mechanical properties of this salt is crucial for designing efficient hydrogen generators based from NaBH4. 1.2. Discrete element method 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 powder's external texture, and its density are just a few factors affecting flow. In addition, there are parameters that vary over the course of the flow, for example, the powder's density, which creates internal instability. In order to resolve the uniformity problem, empirical, analytic, and numeric models were developed to demonstrate the character of the flow of powder under geometric constraints. Discrete element method is a numeric model based on Cundall's and Strack's theory [17] which models the interaction between 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 numerical discrete element method derive from the relative simplicity of the particle motion solution even subjected to changes in the systems boundary conditions. This can be achieved by changing geometric measurements during the motion of particle flow regimes due to external forces and particle velocity. The ability to build experimental arrays within a reasonable time frame was also a factor. The main limitation of the discrete element method is the demand for computer resources which translates into longer processing-time as more particles are fed into the system. As described in Section 3, each calculation set in our optimization scheme requires 3 shear simulations and an additional 3 tension tests performed 100 times overall. Since pretests indicated that NaBH4 is characterized by a very high stiffness values (compared with in Y. Shigeto et al. [18] where stiffness is 103 times lower), we could estimate that each simulation may last up to 5 h. Many studies concentrate on particle size based on simulation results for particles that are larger than the originals [19,20]. These reports have shown a low deviation between simulation results (but see the coarse grain model, [21] for fluidized beds). Still, in order to make sure

241

our grain size assumptions are valid we have conducted a set of partial shear experiments using grain sizes 5, 30, 50, 70, 75, 90 and 110 times the original grain size; it was observed that grain of size 75 times the original size behavior was sufficiently close to that of 5 times the original grain size. Nevertheless, “to be on the safe side” we used particles of size 50 times larger than the original particle size. Therefore, we can extend the radii of the particles in order to shorten the processing time. We can also see that particle shape has an effect on simulation results. 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. The studies conducted [20,22] showed that decreasing the particle circularity can reduce the particle flow by up to 30%. In the discrete element method, two spheres can be attached (a clump), thereby creating complex shapes that differ from symmetrical circles. When the particles geometries in the simulation approach what is found in reality, we see an improvement in the simulation results. Good correlation between the mechanical properties of the particle in motion in the simulation should take into account the interactions occurring between particles and between the particles and the surface. The mathematical model for describing the interaction uses a KelvinVoight element (simultaneous spring and damping): The spring connecting two particles radially demonstrates their Young modulus, i.e., their elasticity; the spring connecting two particles tangentially demonstrates the friction between them [23]; and Cundall and Strack propose calculating the damping constant values as per a secondorder spring and damp system multiplied by the damping coefficient determined by the trials. In addition, a spring stretched radially creates an attraction force between the particles, demonstrating the cohesion between them. A number of studies have shown 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 with accuracy to half a millinewton [24]. From a shear force graph, we can derive important mechanical properties such as elasticity parameters and a restitution coefficient.1 In order to calculate the mechanical parameters of NaBH4 particles that are approximately 50 μm in size (see Section 2), we shall make use of measurements of micro-properties of the particles by means of their macro properties such as can be seen in [26], i.e., the penetration 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.

2. Methods 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 Newton's second law, where, on the one hand, the forces between particles and, on the other, between particles and the interior surfaces are derived from deformations, i.e., in every iteration, the overlap depth 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 the following: a mechanism whose function it is to express the radial forces 1 Another method for measuring the 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 μm. 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 [25]).

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Fig. 3. A pair of particles depicted as overlapping squares are described as follows: (a) radial overlap and (b) tangential overlap. Fig. 1. Photograph of NaBH4 grains under an optical microscope.

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. Note that in most cases particle shapes are non-spherical (as in the case of sodium borohydride) in such cases one should utilize clump shaped particles. The radial rigidity between the bodies is demonstrated by pressure spring kn related to the particles elasticity module E (Young modulus). The rigidity in the tangential direction is demonstrated by ks. These parameters govern the shear stiffness and the normal stiffness respectively. Some studies (see for example [26]) assume proportionality between these constants (Poisson ratio ν); on the other hand, we know that the Poisson ratio for powders ranges from 0.2 to 0.3 (iron is 0.27; sand is [0.2, 0.4]); As will be described later on, we assumed these as equal for running time reasons. But, in light of the above upon reaching the desired parameters' values some simple simulation tests were conducted which indicated that changing shear as well as the normal stiffness ratio (from 1 to 0.2) maintained the results. A pair of particles is exposed to force proportional to their relative accelerations and opposing the direction of their motion. Viscous damping occurs in both the tangential and normal directions as μg. The Coulomb damping μs is given by F c ¼ −μ s N

ð3Þ

in the tangential direction (with N the normal force). kp is the cohesion parameter which results with a constant cohesion force Akp as long as the distance between the two particles is less than 1.1 times the radius of the particles (i.e. the harmonic mean of the radii). Here A is the contact area, which is a function of the grains overlap. Such cohesion between two grains and between a grain and an interior surface is caused by a number of forces. First, among the most significant forces is electrostatic interaction between localized surface partial charges on two

adjacent particles, often referred as a van-der-Waals forces [27]. Second, cohesion is caused by 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; rather, it depends on the number of collisions. A third cause of cohesion is that of liquid bridges formed between two adjacent particles when exposed to high levels of humidity [28]. Two such particles will experience a capillary attractive force, formed by a liquid solution bridge saturated with NaBH4 dissolved salt (in our case). Obviously, the larger the number of wetting-evaporation cycles, the larger the number of permanent bonds between particles in the substance. In this case the dried agglomerate can be considered as a larger particle. In general, the importance of van-der-Waals forces is obvious in particles smaller than a few micrometers, this is also supported by the fact that in high Relative Humidity regimes (N 40%), the capillary force is dominant. Capillary phenomena between particles are relevant when the size is smaller than 500 μm, and forces resulting from absorption of vapor occur in particles smaller than 80 μm. Based on this assumption, we have decided to approximate our model by taking into account the capillary forces and neglect van-der-Waals. Using an optical microscope, the size of NaBH4 grains was measured, and an average value of 50 μm (see Fig. 1) was found. Therefore, in NaBH4 powder in a closed system with high humidity levels (100% R.H) it can be assumed that the van der Waals forces are negligible due to the size of the grains and low electrostatic forces are 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 the literature as interlocking phenomena (see [29]). Here, we assume such an evaporation process does not occur so we shall not include permanent bond phenomena in our model. There are two approaches for modeling the interaction between particles: describing collision between two particles as momentary

Fig. 2. Interaction model between two particles.

Fig. 4. A clumped particle.


243

Fig. 5. Overall scheme for finding physical parameter.

(referred to as a “hard” contact model) or as a continuing process occurring in several time steps (referred to as a “soft” contact model). For the latter, forces in such a model are induced as a function of mutual penetration. In choosing the model we considered: 1. Most of the DEM formulations follow the “soft” contact model of Cundall and Strack. This simplistic model, which we later on describe in detail, has been thoroughly studied and compared with experiments. 2. When modeling spherical particles one may introduce a non-linear Hertzian contact model - this is not the case here. Accordingly we applied a linear contact model. 3. It is commonly agreed that for fine particles (or particles surrounded in fluids) damping should be incorporated. 4. The tangential spring stiffness may be different from the normal; still, since it is hard to obtain the tangential coefficient we shall not pursue it here and set k =ks =kn. 5. Capillary phenomena between particles are expected to be relevant at all sizes. Forces resulting from absorption of vapor at high humidity levels (100% R.H) present during the reaction may lead to the capillary effect. This is likely to occur in particles of intermediate size of 80 μm and below Nevertheless, it should be noted that the models pose inevitable inaccuracies since the stiffness coefficient may be non-constant and geometry dependent. The model we shall use is depicted in Fig. 2 - a pair of particles that come into contact with each other at a certain overlap, so that forces act upon them to separate them radially and tangentially. For simplicity's sake, let us imagine the collision of two particles that occurs with some geometrical overlap, so that forces act upon them to separate them radially and tangentially (Fig. 3 depicts two particles as squares only for the purpose of clearly differentiating between the tangential and the radial directions). The radial and tangential forces are: F n ¼ −kn hn

ð4Þ

Fs ¼

ks hs Fnμs

ks hs ≤μ F n ks hs Nμ F n

ð5Þ

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 μs. Note that this model is a hysteretic one [30], 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. Energy loss will be expressed as μg in all directions (tangential and normal). 2. Based on Coulumb friction in the direction opposite to movement will be denoted by μs. The governing parameters of the selected model are therefore: 1. 2. 3. 4.

The stiffness coefficient k (see Eqs. (4), (5)) The friction angle μs (see Eq. (5)). _ The viscous damping coefficient μg (exerting −μ g y). The cohesion constant kp (the cohesion force is assumed to be proportional to kp [31] and is applied when interaction occurs. Interaction between particles takes place when the distance is less than 1.1 times the radius of the particle).

Other parameters affecting the results of the simulation are, as aforementioned, the size of the particles, their shapes, and their spatial distribution. As mentioned, in order to shorten the calculation time, the size of the particles was increased 50 times NaBH4 particles actual size (50 μm). The particle shape was determined as a pair of clumped particles, where the distance between the spheres' centers D ranged from 0.5R to R in a uniform value distribution (Fig. 4). To enable diffusion of particles similar to what is found in reality, we produce the particle cloud at a distance from the lower portion of the box such that when

Fig. 6. The shear strength experiment.

244


Fig. 7. A typical Shear experiment: (1) The lower right sub figure: represents a resulting linear function. The lower line represents real experiment results, upper line corresponds to the simulations; (2) Other three sub figures: depicts wedge experiments with normal forces 2 N; 4 N; 8 N. the simulation results are presented together with their spline approximations respectively. Note the plateau slope as movement initiates, this is attributed to a stage in which small chain forces break and particles rearrange. Since the shear experiment may have several apparent break points (like in the lower left sub figure), the break points were manually estimated. The spline lines were calculated for convenience alone. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

we activate the gravitational forces on the particles, they will arrange themselves freely on the lower portion of the box. Y. Tsuji et al. [32] demonstrated the importance of the chosen time step in DEM. Here we shall use an optimized calculated time step implemented in YADE DEM (the average time step was 10−5 s). 2.1. Obtaining the governing parameters Fig. 5 depicts the outline of our study. To find the mechanical parameters of NaBH4 powder, a series of experiments were conducted (see Section 3). Different experimental setups enabled us to distinguish between the various parameters in order to obtain their preliminary values. As mentioned, 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 particle size will change the effect on the experimental results. Therefore, to minimize the effect of particle size thereon, we shall conduct small-scale wedge penetration

and shear experiments. The further we reduce the experimental model, the smaller the number of powder particles is involved. Yet, when reducing the model, we need a higher resolution 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 for the quality of results. Using the discrete-element software YADE-DEM for each real experiment conducted, the simulation was built to represent the real experiment. In addition, results and graphs are produced from the simulation 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 estimation of the parameters. The resulting (reduced) parameter-space was found to be as follows: the stiffness coefficient [60 ⋅ 106, 900 ⋅ 106]N/m; the friction coefficient [0.01,1.5]; the damping coefficient [0.05,0.5]; and the normal cohesion [5,1500]Pa. Next, we conducted an optimization using a genetic algorithm (GA) to obtain better fit.

Fig. 8. Wedge experiment. lower curves indicate real test and polynomial approximation of the simulation results are indicated by the upper curves. The displacement ranges are set from zero to the wedge structure lengths. Here, the stiffness coefficient is 635 ⋅ 106 N=m; the friction angle 0:28 rad; the damping coefficient 0:356; and the normal cohesion 534 Pa. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)


245

2.2. Generic genetic algorithm In this subsection we present a generic framework of a genetic algorithm (see Algorithm 1) for optimizing a multi-variable cost function which is mainly used for global optimization problems. We have used such a framework in order to estimate the governing parameters of the powder by minimizing the differences between the simulated data and experimental results. The main idea behind the algorithm is to mimic the natural selection process. A genome corresponds to a set of parameters and the fitness corresponds to a cost function to be optimized. Algorithm 1. A general framework for genetic algorithm.

Fig. 10. Normal and shear displacement due to horizontal wedge displacement.

.

a finer optimization step later on. This resulted in a further reduced parameter space: the stiffness coefficient ranged over [60 ⋅ 106, 650 ⋅ 106] N/m; the friction coefficient ranged over [0.01,1]; the damping coefficient [0.1,0.5] and the normal cohesion [50,1000] Pa. Finaly, this was followed by a manual search accompanied with the analytic estimation introduced in Section 5. 3. Experiments

The basic algorithm uses two operators: (1) A crossover operator that mates a pair of genes to form a new one, by mixing their genes in much the same way it is done in nature. (2) A mutation operator which is a simple perturbation operator which is applied over a small subset of the gene population. 2.3. Genetic algorithm implementation During the GA optimization stage all parameters were given the same weight. We used Python's Pygen library in order to implement our simulation. The algorithm starts by constructing a set of 90 simulation tasks (created for the discrete element program). Using a parallelism method each CPU computes single task at each time. At the end of the tasks a fitness function is evaluated by estimating the RMS (root mean square) error between the simulation and the real experiment graph for each genome. The five genomes with the higher fitness rate create the next generation. A genome was defined as a set of 3 shear test simulations and 3 additional penetration tests. A generation was constructed from 15 genomes. This amounted to 180 simulations in the course of two generation runs. Since each of the simulation tasks takes approximately 8 h using an Intel Xeon E5-2680 v2 processor. We considered only the three penetration tests at first and conducted

3.1. The shear strength experiment Provided information on friction and cohesion of the particles [29, 33]. In this experiment, a NaBH4 powder was inserted into two cylinders measuring 2 cm in diameter and 1 cm high that were placed one atop the other (Fig. 6). The upper cylinder contained a plate measuring 2 cm in diameter that applied normal, steady pressure, compressing the powder. We used a TestometricM250 − 3CT Materials Testing Machine with accuracies ±0.1 N and ±0.1 mm 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 multiple times, and between experiments, we increased the normal pressure and measured the shearing obtained in the failure. Recall that when force is applied upon a powder spontaneous force chains are created (i.e. graphs of grain contacts, see [34]). By failure we refer to a collapse of these chains. Such a phenomenon will result in a sudden change in the force slope (as the function of the tangential shift). Thus the expected shear stress τ and normal stress σ behavior is: τ ¼ σ tanð f Þ þ C

ð6Þ

where C is the powder cohesion obtained at the cutoff point with the axis of shear 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 Fig. 7). 3.2. The wedge penetration experiment The wedge penetration experiment was conducted using the same Materials Testing Machine to lunge three wedges, the areas of whose

Fig. 9. (a) Particle arrangement in the box. (b) Forces applied on the third particle.

Fig. 11. A schematic view of the shear test.

246


4.2. The wedge penetration simulation This experiment was conducted using 6000 particles instead of 50,000 to decrease time consumption. For each of the wedges, a penetration simulation was conducted up to a depth of 20 mm and 17 mm, commensurate with the real experiment. The force-displacement graphs of the simulation results are presented in Fig. 8. Note that due to the difference in the size of the particles in the simulation and the real experiments, the number of 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. 5. Analytical formulation Fig. 12. (a) Particle geometry. (b) Particle forces.

bases measure 10 × 15 mm, with differing head angles: 30° and 90° and 0° (horizontal plate). The wedges penetrated into a cube with a crosssectional area measuring 50 × 30 mm and 100 mm deep, containing NaBH4 powder. The (filtered) force-displacement graphs are depicted as blue lines in Fig. 8 (i.e. a polynomial approximation). Note that the experiments are valid to the point where the edge structures end (the 30°, 90° were 20 mm long, while the flat wedges are slightly shorter 17 mm).

4. DEM experiments We used open-source “YADE-DEM” (Discrete Element Method software). For the three real experiments conducted, each had its own correspondingly designed simulation. All experiments were conducted using randomly dispersed clump particles. Particles were constructed such that their center distances varied from 0.5R to R with uniform distribution values.

4.1. The shear strength simulation This was carried out using 10,000 particles which were randomly pre-packed in a cylinder. Three different tests were conducted with 2 , 4 and 8 N normal force applied on top of the upper cylinder. In each simulation the break shear point was extracted. Locating them on a shear-normal force coordinate system yielded an approximated linear function, as can be seen from Fig. 7.

To enable an educated manual search (as described in Section 1) it is essential to have a rough estimate of the main physical factors affecting each of the aforementioned experiment results. Note that the damping effect may be of importance in dynamic cases, however, for quasi-static simulation such as ours the effect of damping will not be significant (see [35]). 5.1. Wedge penetration equations We shall first describe our attempt to estimate the behavior of the force reaction in the 0° wedge experiment. We consider a simplified pack as depicted in Fig. 9(a). Focusing our attention only for the vicinity of the depicted configuration we can write: F ¼ F n1 sinðθ1 Þ þ F s1 cosðθ1 Þ þ F n2 sinðθ2 Þ þ F s2 cosðθ2 Þ

ð7Þ

For simplicity's sake, we assume symmetry (i.e. θ2 = π− θ1), so: F ¼ kn hn1 sinðθ1 Þ þ ðμ s hn1 kn þ ks hs1 Þ cosðθ1 Þþ þ kn hn2 sinðθ1 Þ þ ðμ s hn2 kn þ ks hs2 Þ cosðθ1 Þ

ð8Þ

Here kn , ks are the normal and shear elasticities respectively, hn , hs are the normal and shear displacements and μs is the friction coefficient. The shear and normal displacements due to the vertical wedge displacement are demonstrated in Fig. 10 which implies hn = h cos (θ)and hs = hsin(θ). Substituting into the original equation yields: F ¼ cosðθ1 Þ sinðθ1 Þð2kn h þ 2ks hÞ þ 2μ s hkn cos2 ðθ1 Þ

Fig. 13. (a) Micro particle displacement hn, hs and output displacement Hn,Hs. (b) Zoom out view.

ð9Þ


247

Fig. 14. Wedge penetration experiments: second GA attempt. Upper curves indicate the simulation results, lower curves are a polynomial t of degree 9 of the real experiments. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The angle between particles in a random pile is distributed uniformly (see [36]). So for a three dimensional flat wedge penetration experiment the expectancy reaction force on the wedge due to infinitesimal displacement is given by integrating over all angles 0 b θ b pi/2. Eð F Þ ¼ hðkn þ ks þ πμ s kn =2Þ

ð10Þ

Note that this is merely a magnitude estimation.

and F n2 ¼ kp sinðθÞ

So the shear and normal forces equations resulting from normal and shear infinitesimal displacements Fs(Hn, Hs, θ), Fn(Hn, Hs, θ) can be obtained. To eliminate particle angle dependence we make use of the uniform distribution property. The integration split the two cases in the interaction angle 7π/20, this yields:

5.2. Shear test equations

Z

Consider a simplified particle pack which under shear rearranges as shown in Figs. 11 and 12. As a result an upper row particle shown in Fig. 12 is subjected to the set of forces. We denote Fsin − Fsout the normal forces at the shear direction and at the opposite direction respectively. Fn2 is the cohesion force, Fn is the normal force applied and Fn1 , Fs1 are the normal and shear forces respectively. Thus: F n ¼ F n1 sinðθ1 Þ−F s1 cosðθ1 Þ þ F n2 cosðθ1 =2Þ

ð11Þ

F nin −F nout ¼ F n1 cosðθ1 Þ−F s1 sinðθ1 Þ þ F n2 sinðθ1 =2Þ

ð12Þ

Here Fs1 = μshnkn + kshs and Fn1 =knhn. Recall that we assume the cohesion force Fn2 is proportional to the constant kp and is applied when interaction occurs. Interaction between particles takes place when the distance is less than 1.1R (in what follows a simplified model having a unit contact area between grain). The distance h between particles 1 and 3, Fig. 12(a) is given by h = 4R sin (θ1/2)(when the distance is 1.1R the corresponding angle is 7π/20). Note that the particle's normal and shear micro displacements are a function of the normal and shear output macro displacements (Fig. 13). hs = Therefore, hn = (Hn tan (θ) + Hs) cos (θ)and (Hn tan (θ) + Hs) sin (θ) − Hn/ cos (θ). Substituting these into Eq. (11) and solving for local forces yields: F n1 ¼ kn ðH n tanðθÞ þ Hs Þ cosðθÞ; F s1 ¼ μ s kn cosðθÞðH n tanðθÞ þ Hs Þ þ ks ððH n tanðθÞ þ H s Þ sinðθÞ−H n = cosðθÞÞ;

ð13Þ

ð14Þ

Eð F s Þ ¼

7π=20 π=3

Z F s dθþ

2π=3 7π=20

F s −kp sinðθ=2Þdθ ¼ 0:026kp

ð15Þ

þ 1:04Hs kn þ 0:95μ s H n kn Z Eð F n Þ ¼

7π=20 π=3

Z F n dθþ

2π=3 7π=20

F n þ kp cosðθ=2Þdθ

ð16Þ

−0:732kp þ 0:05Hn kn þ 0:02μ s H n kn −0:011μ s H s kn To pursue the shear stress intersection line (denoted by C in Eq. (6)) we set the macro displacement Hn to zero in the shear force equation. Assuming kn = ks we get: C ¼ 0:26kp þ 1:04H s kn þ

0:95μ s kn 175:7kp þ 2:86μ s Hs kn 12:57kn −5:26μ s kn

ð17Þ

The slope of the linear line (in Eq. (6)) can now be estimated as the partial derivative: ∂Eð F n Þ −10:45μ 2s þ 20:8μ s þ 0:52 ¼ μ s ð0:55μ s −22Þ ∂Eð F s Þ

ð18Þ

So the slope is positively dependent friction angle (this was later observed in most experiment ranges, but not over all range). 6. Results To initiate our calculations an initial parameter range was needed. For that end we ran the wedge simulation with an estimated parameter set. Here the shear test did not performed well.

Fig. 15. Wedge penetration experiments: manual refining according to Equation 10 reduces friction angle and raises stiffness coefficient. Upper curves indicate the polynomial t of the simulation results while the lower curves represents real experiment results. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 16. Angle of repose tests. (a) Experiment results and (b) simulation result. Notice that in the experiment a vertical slope in the top region of the powder bulk was created. This was not captured in the simulation since our model does not include wall-particle cohesion.

We implemented the above to the first GA optimization stage in which the fitness function was only for the wedge penetration experiment (due to reasons of time constraint). This stage resulted in the following: the stiffness coefficient 635 ⋅ 106 N/m; the friction angle 0.28 rad; the damping coefficient 0.356; and the normal cohesion 534 Pa. This set of parameters yielded the following: 1. A very good fit for the wedge simulation see Fig. 8. 2. A linear shear graph of y =0.472x + 0.49, while the real experiment linear graph y = 0.125x + 0.16. So, the error in both the internal friction angle and powder cohesion is about 400%. A second GA optimization procedure was then performed with a reduced parameter range considering both shear and wedge penetration tests. The resultant parameters are as follows: the stiffness coefficient 620 ⋅ 106 N/m; the friction angle 0.315 rad; the damping coefficient 0.327; and the normal cohesion 688 Pa. This set of parameters yielded the following: 1. A good correlation was obtained between the wedge penetration experiment and that of the simulation. The wedge experiment results are depicted in Fig. 14. 2. However, the shear simulation resulted with a 250% difference from the experimental

Fig. 17. Comparison between penetration tests. Gray line corresponds to a 6000 particles test, dark line corresponds to a 50,000 particles test; both grain sizes macroscopic forces are of the same scale.

Next, we implemented our analytic analysis to reduce the shear experiment error. Note that Eq. (10) implies that simply reducing the friction angle parameter will result in changing the wedge penetration as well. Accordingly, in order to maintain the wedge penetration results while lowering the linear shear slope, we: (1). Reduced the friction angle by 8%. (2). Raised the stiffness coefficient by 6%.

Hence, the tested parameters were as follows: the stiffness coefficient 658 ⋅ 106 N/m; the friction angle 0.2901 rad; the damping coefficient 0.327; and the normal cohesion 688 Pa resulting with: 1. A good correlation was obtained between the wedge penetration experiment and that of the simulation. The results are depicted in Fig. 15. 2. The linear shear graph was found to be y = 0.2164x + 0.27 (a 70% slope error). 7. Validation In order to verify the parameter results, we implemented an angle of repose scheme. We conducted an experiment in which an open cylinder filled with NaBH4 powder was positioned on a horizontal table. The cylinder slowly moved upward forming a pile as the powder poured out. The pile angle (depicted in Fig. 16) depended on the mechanical parameters of the powder, mostly by the friction coefficient [35]. The test was performed using a 50 mm high cylinder having a 20 mm radius and the upward speed was set to 20 mm/s. The DEM simulation corresponding the test was performed on 3000 clumped particles with a 35 mm radius each. The cylinder speed and geometry were identical in the simulation and the real experiment. Fig. 16 depicts the repose angles from both the experiments and simulation tests. A good correlation was obtained as the angles range from 35° to 45° in the real experiment and from 37.5° to 40° 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 environmental variables in the former. Recall that due to the time required for running the set of wedge penetration tests in the optimization step, these tests were performed using a reduced number of particles (6000 instead of 50,000). Therefore, we needed to verify that the parameters we obtained above suited the wedge penetration tests as well. In order to do so, we used the parameter set obtained (stiffness coefficient 620 ⋅ 106 N/m, friction angle 0.315 rad, damping coefficient 0.327 and the normal cohesion 688 Pa),


in Section 6 on a 50,000 clumped particle tests, simulating 90° wedge. The results are presented in Fig. 17. 8. Summary We have conducted a set of experiments to pursue the values of the physical governing parameters of NaBH4 powder. Accordingly, we have changed the governing parameters of a suitable DEM simulation. The experiments included were: (1) A wedge penetration: in order to filter out the friction coefficient and the elasticity we conducted three experiments with three different wedges with varying angles; (2) A set of shear tests corresponding to a varying normal force. The critical shear force was recorded. We have implemented a genetic algorithm to optimize the parameter fit. Since this scheme is time consuming we have formulated a statistical estimation for each of the experiments. These were, then, applied in order to manually fine-tune the governing parameters. In the course of our manual search it was found that the statistical equations above allowed for more realistic powder behavior to be simulated which is reflected in the graph (e.g. Fig. 15). Due to time consumption considerations we applied a reduced number of 6000 particles to the wedge tests. Thus, to verify our results we have conducted two additional tests: an angle of repose experiment and wedge penetration tests with 50,000 particles. Our project results and validation experiments show that NaBH4 can be fairly simulated using the mechanical model of Fig. 2 with the stiffness coefficient k = 658 ⋅ 106 N/m, the friction angle μs = 0.2901 rad, the viscous damping coefficient μg = 0.327 and the cohesion constant kp = 688 Pa. The authors believe that the overall research scheme may be used to extract the mechanical parameters of the other powders. When using clumped particles, the rotation should be neglected and so we did for the sodium borohydride particles (which are non-symmetric by nature). Although the shape of the resulting clumped particles is close to that of the NaBH4, the authors believe that this should be further investigated (see [20]). In their future work the authors would like to implement the coarse grain DEM model. Future work will implement larger spherical particles (which will introduce rotation) with rotational friction. This is believed to simulate the non-spherical smaller particles. Lastly, the authors would like to apply their findings for the design of a unique hydrogen generator. References [1] Y.S.H. Najjar, Hydrogen safety: the road toward green technology, Int. J. Hydrog. Energy 38 (25) (2013) 10716–10728. [2] N. Bimbo, W. Xu, J.E. Sharpe, V.P. Ting, T.J. Mays, High-pressure adsorptive storage of hydrogen in mil-101 (cr) and ax-21 for mobile applications: cryocharging and cryokinetics, Mater. Des. 89 (2016) 1086–1094. [3] G. Sandrock, G. Thomas, The iea/doe/snl on-line hydride databases, Appl. Phys. A Mater. Sci. Process. 72 (2) (2001) 153–155. [4] T. Tamura, Y. Tominaga, K. Matsumoto, T. Fuda, T. Kuriiwa, A. Kamegawa, H. Takamura, M. Okada, Protium absorption properties of Ti–V–Cr–Mn alloys with a bcc structure, J. Alloys Compd. 330 (2002) 522–525. [5] K.J. Gross, G.J. Thomas, C.M. Jensen, Catalyzed alanates for hydrogen storage, J. Alloys Compd. 330 (2002) 683–690. [6] L.H. Kumar, C.V. Rao, B. Viswanathan, Catalytic effects of nitrogen-doped graphene and carbon nanotube additives on hydrogen storage properties of sodium alanate, J. Mater. Chem. A 1 (10) (2013) 3355–3361. [7] D.-W. Lim, S.A. Chyun, M.P. Suh, Hydrogen storage in a potassium-ion-bound metal– organic framework incorporating crown ether struts as specific cation binding sites, Angew. Chem. Int. Ed. 53 (30) (2014) 7819–7822.

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