How Unconfined Yield Strength is Effected by Particle Shape?

One of the properties that influences a multitude of processes in the handling of bulk powder materials is unconfined yield strength. An understanding of the particle scale properties that have an effect on strength assists engineers to design more efficient products before production, thereby limiting the possibility of costly mistakes and enhancing productivity.

This article explores the relationship between particle shape and bulk unconfined yield strength. As an experimental work, this research suggests that the key parameters that influence bulk strength are the number of particle contacts per adjacent particle and the direction of these contacts. Further, this article recommends a way of incorporating shape effects in predictive models that relate strength to particle scale parameters.

Introduction

It is critical to understand bulk flow properties in order to get a better understanding of process behavior. Generally speaking, there are three distinct phenomena that tend to cause trouble when it comes to industrial processes.

In the first case, cohesive materials create flow stoppages which are stagnant regions in process equipment. Moreover, there is the possibility of material segregation, which creates a mixture of varying quality during the flow of material from process equipment. Lastly, there is also the tendency for material to flow at erratic or uncontrolled rates from process equipment.

However, there is one common flow property that simultaneously targets all three of the above process problems. In other words, this flow property is the unconfined yield strength, defined as the major principle stress causing an unconfined bulk material to fail in shear. A material’s tendency of arching over outlets and forming ratholes in process equipment directly depends on the unconfined yield strength [1].

In other words, unconfined yield strength dictates the stress holding material together on a free surface. It functions as the major principle stress acting in parallel to the free surface, which supports the external forces that tend to tear the surface apart [2]. This free surface spans the outlet in the case of an arch.

However, in a rathole, this free surface is the pipe shaped channel’s surface formed during discharge. The rathole remains stable provided that the strength is large enough to support the stress around the perimeter of the rathole. This causes material to cling to the surface of the container, resulting in a significant stagnant region forming around the central flow channel.

Moreover, when it comes to process equipment, there is a tendency to form piles during filling discharge. This can also occur during operation, in some cases, such as in a rotary shell blender [3]. Normally, a pile is a free surface, and the thickness of the avalanche layer depends on the unconfined yield strength of the bulk material [4].

As a result, processes forming piles are partially controlled by the bulk material’s degree of cohesion. What’s more, often, during pile formation, segregation occurs [5]. In other words, whatever controls the pile formation also tends to affect segregation.

A given material’s ability to stick together often mitigates its tendencies for segregation [6]. Often, erratic flow problems in process equipment are a result of excess air stored in the material [7], the collapse of the rathole [8], or the sudden movement of stagnant material.

Moreover, the ability of gas counter-flow to fluidize material is dependent on the cohesiveness of the material. In essence, Geldhart class C materials are cohesive and difficult to fluidize, since they form channels and not bubbles [9] [10] [11].

In addition to this, unconfined yield strength can also be conceptualized as an assembly or particles’ resistance to shear. There have been several models proposed, which relate the particle scale properties to the bulk unconfined yield strength. As such, these are summarized in Table 1 below.

Table 1. Relationship Between Unconfined Yield Strength and Particle scale Properties

Model Mechanism Source
Van der Waals forces Mollerus [12]
Capillary forces Rabinovich [13]
Elastic fracture Rumpf [14]
Plastic-elastic fracture Specht [15]

However, it is important to note that the relationships depicted in Table 1 are for perfectly spherical particles that have a uniform particle size. In reality, none of these relationships contain the effect of particle shape or size distribution.Therefore, this article explores the effect of particle shape on the bulk unconfined yield strength.

Experimental Methods

When conducting a study of how shape affects yield strength, one of the main challenges is the ability to obtain a consistent sample of distinct shapes that poses strength as a bulk. This study uses plastic pellets of different shapes (round, heart, and stars), that are coated with soft Tacky Wax from Yaley Enterprises, to render them cohesive (see Figure 1).

Typical Pellets Used in this Analysis

Figure 1. Typical Pellets Used in this Analysis

Using pre-measured samples of three different shaped pellets, a prescribed amount of Tacky Wax was applied. These pellets and wax were heated to 50 °C and subsequently mixed for approximately 30 minutes, thus creating three distinct mixtures with 2.02% + 0.06% by weight Tacky Wax. Then, these mixtures were cooled to a temperature of 21 °C, prior to measuring the bulk strength.

Each pellet shape’s uniformity of coating was then calculated by measuring the deviation in weight of 100 coated pellets for each of the three shapes and subsequently comparing this value to the deviation in the weight of 100 non-coated pellets. This analysis resulted in the wax coatings on the pellets being nearly uniform.

Take, for instance, the heart-shaped pellets. These coated pellets varied in weight by 3.37% and the non-coated pellets varied in weight by 3.19%, on average. An implication of this difference in pellet variation is that the coating of Tacky Wax creates an additional variation of about 0.18% in pellet to pellet weights, in the case of heart-shaped pellets.

Such a variation represents below 10% of the total concentration of the 2.02% Tacky Wax on the sample. What’s more, the two other pellet shapes displayed similar results, suggesting that the coated materials had a reasonably uniform coating of Tacky Wax on each particle. It also implied that the 30-minute mixing time was sufficient to create a representative sample.

It must be noted that there are a variety of test techniques to measure bulk unconfined yield strength [16], [17], [18], [19]. For instance, direct shear methods such as the Schulze method or Jenike method, need materials with small particle sizes for the generation of good data.

Moreover, the Johanson uniaxial tester is equipped to give an approximation of the unconfined yield strength by using just one sample, with the ability to generate reasonable results when using larger particles.

As per extant literature, the standard 5 cm diameter test cell works well with 0.5 cm diameter particles. In addition, the Johanson uniaxial test method was utilized for a 10 cm test cell, thereby suggesting the possibility of reasonable results with particles as large as 1 cm in diameter.

Further, Johanson’s uniaxial method also allows for excellent control over the stress level applied to the material. To confirm this, ten repeat measurements of unconfined yield strength at a series of consolidation pressures were taken, averaged, and then plotted as a function of compaction pressure (see Figure 2). For clarity, error bars for data in this plot were omitted but have been used in figures later in this research.

Unconfined yield strength of various shaped particles as a function of consolidation pressure

Figure 2. Unconfined yield strength of various shaped particles as a function of consolidation pressure

As the consolidation pressure increases, the round particles seem to gain in strength quickly, but level off at higher consolidation pressures. At low consolidation pressures, the hearts appear to have very similar strength values to the round particles, but exhibit larger strength values at higher consolidation pressures.

In contrast, the stars seem to have the same strength as the round particles when faced with low consolidation pressures, but increase in strength more rapidly than the round particles as the consolidation pressure increases. Further, they are shown to level off to about the same value as the round particles at high consolidation pressures.

This article’s main aim is to examine the differences between these strength values and determine whether certain particle shape characteristics can explain this behavior.

It is important to note one of the most obvious potential differences between these particles, that is, the probability of multiple contacts between the same two particles for non-spherical particles. Figure 3 displays the presence of more than one contact point cementing the particles together for two adjacent particles, especially in the case of non-spherical particles.

Typical particle contacts between adjacent particles

Figure 3. Typical particle contacts between adjacent particles

It can be seen from these particles that there is only one contact point between particles one and two. Nevertheless, there exists two contact points between particles two and three. One of the interpretations of unconfined yield strength posits that strength equals the initial resistance of a bulk material to shear, which is caused by the integrated effect of all the individual forces acting between adjacent particles in the shear zone.

There are two categories of forces that exist between particles. A few of those forces are adhesive forces while some are frictional forces, as detailed in Figure 4. Furthermore, Figure 4 shows how the particle above these two adjacent particles moves to the left during shear, which induces a friction force (Ffrict) on the moving particle.

Typical forces acting on a particle in shear

Figure 4. Typical forces acting on a particle in shear

Moreover, there is also a normal force (Fnorm) which acts upon the moving particle at the frictional contact point. In addition, there are two external forces (Fx and Fy) acting in the particle’s x and y direction, which are a direct result of other particles in the vicinity. Finally, adhesion forces (or Fad) act to bind adjacent particles together, therefore providing a pulling resistance as the top particle moves towards the left.

Assuming strength is a product of the number and type of adhesion points between the adjacent particles. If it is possible to count the number of contacts between the adjacent particles and estimate the relative magnitude of contact forces, then it is easy to compare the strength of particles with single contacts between particles, as well as the strength of particles that have multiple contacts between adjacent particles.

However, the fact that these three systems’ coating is the same, implies that the contact forces on all particles are roughly the same (this excludes any particle curvature issues). Then, it would be expected that the unconfined yield strength would scale with the number of contacts in a given unit volume.

In other words, if an estimate of the number of total contacts in a given unit volume could be arrived at, then a correction factor accounting for the number of contacts could be utilized to relate the spherical particle unconfined yield strength to the non-spherical particle unconfined yield strength.

Here, the challenge lies in subjecting material to shear (inter-particle motion) during the strength measurement. What’s more, the strain imposed during shear testing using the uniaxial strength tester was around 16%. Such a situation is further compounded by the fact that this strain occurs at a prescribed stress condition. In an ideal scenario, it would be prudent to measure the number of contacts as a function of both the stress and strain placed on the material.

Thus, a special test cell was constructed, allowing material to be strained at a prescribed contact pressure. The cell mentioned above comprised of a series of hinged plates placed in between two sheets of glass, thus forming a box that was shaped like a rhombus (refer to Figure 5).

Pure Shear Box (front view)

Figure 5. Pure Shear Box (front view)

Additionally, material was placed in the box in the middle of these two glass sheets, while a piston was placed on top of this material. While the bottom of the box oscillated back and forth, to stimulate a situation of strain, a load was simultaneously applied to the box. As such, the piston managed to provide the pivot point, thereby allowing the box to transform from a square into a rhombus and back again.

Equation 1 represents the total shear ( ) in the box, which was computed from the maximum extension angle (w) of the sidewalls and the number of cycles (Ncyc).

In this test cell, the pellets were positioned, and a constant load was then placed on the piston while a strain was induced. The visualization of the pellets in the tester was permitted through the front of the test cell, which was kept clear.

Once the material was subjected to a given strain at a prescribed consolidation load, images were clicked of the particle assembly in the tester, while the number of contacts between the adjacent particles was recorded through manual visual inspection of these pictures.

For approximately 100 particles of each particle shape, contact information was thus measured. It was noted that strength is a factor that depends on the forces causing adhesion between particles.

When two or more contacts exist between two adjacent non-spherical particles, the forces cannot act along the axis joining the two particle centers.

These are termed vector forces. These two contact forces could be replaced using a single force that would act through the center of the particle, or perhaps even an external moment or screw term. The forces are not required to pass through the center of the particle, resulting in a net moment acting about some axis in space, thus causing this extra moment term.

For now, this moment or screw term was neglected, but the contact data was easily adjusted to account for the fact that only the component of the normal adhesion forces acts in a direction parallel to a line, connecting two adjacent particles together. The net effect of this was the reduction in the pull-off force between two adhering particles (detailed in Figure 6).

Effective contact force correction for multiple contacts

Figure 6. Effective contact force correction for multiple contacts

In addition, the images collected of the particle assembly were also optically analyzed, through the use of a software called ImageJ, to determine the angle of these contacts relative to the particle-to-particle centroids.

Assuming that each normal contact results in identical forces, it follows that Equation 2 defines the average correction factor for force in the direction of particle-to-particle contact (along the centroid axis). Thus, the number of contacts would be multiplied by this correction factor in order to obtain the effective number of contacts for computing strength correction terms.

In this case, the average number of contacts per adjacent particle was measured to be a function of stress at a strain value of approximately 16% (about 0.23 cycles at an extension angle of 10 degrees). Each contact’s angles (1 and 2), relative to the centroid axis, were measured, and the correction factor (Cf) was computed. Once all of these contact values were computed, they were all averaged using the imaged particles.

Figures 6 and 7 display the net result of the above method; the relationship between the stress level and the effective number of contacts between particles. Such a number obtained must be greater than 1.0, if it is to suggest the existence of more than one particle contact per adjacent particle on average.

Number of contacts as a function of stress at 16% strain for heart shaped particles

Figure 7. Number of contacts as a function of stress at 16% strain for heart shaped particles

However, it is important to note that the number of contacts per adjacent particle for round shaped particles is always equal to 1.0. Having said that, different shapes can have contact numbers that exceed 1.0.

For instance, it is evident from Figure 6 that the stress level does not change the number of contacts per adjacent particle in the case of heart-shaped particles, thus yielding approximately 15 to 20% more contacts than would be expected from a spherical system.

Considering only the number of contacts, it is evident that the average number of contacts per adjacent particle occupies a range of between 1.2 and 1.15. In the case of heart-shaped particles, the exact value depends on the stress level applied. If the analysis includes the direction of these contacts, it follows that the number of effective contacts per adjacent particle decreases to between 1.15 and 1.11. Again, this depends on the stress level.

Using star-shaped particles, the exact same analysis was carried out. Considering only the number of star particle contacts, it resulted in the average number of contacts per adjacent particle being a range of between 1.42 and 1.14.

Once again, this depended on the stress level applied to the star-shaped particles. Considering the direction of these contacts for the analysis, the resultant number of effective contacts per adjacent particle varies from 1.30 and 1.10, depending on the stress level.

It is important to note that this material shows a large effect of stress on the number of contact points per adjacent particles. This effect occurs because, in the case of the star particle system, there are two preferred structures. The first structure causes the stars to line up with the flat star surfaces parallel to each other, thus resulting in a single contact point per adjacent particle.

On the other hand, the other stable configuration occurs when the tips of the stars interlock, forming multiple contacts. As a result, the imposed shear creates a rotation in the particles, thereby forcing a predominance of flat-to-flat contacts with every increase of shear and stress level.

Finally, once the flat star surfaces come into contact with each other, it is difficult to produce further rotation. Thus, as seen in Figure 8, the effective number of contacts decreases at high-stress levels.

Number of contacts as a function of stress at 16% strain for star shaped particles

Figure 8. Number of contacts as a function of stress at 16% strain for star shaped particles

In order to estimate the strength of a non-spherical particle system considering the data from round particles, the above correction factors can be used as direct multiplicative factors. Once the strength measured for the round particles is multiplied by the effective number of contacts in the heart-shaped particle system, it produces an approximation of the data measured from the uniaxial shear cell, as evident in Figure 9.

Correction factor applied to round particle strength to correct for the number of contacts per adjacent particle for heart shaped particles

Figure 9. Correction factor applied to round particle strength to correct for the number of contacts per adjacent particle for heart shaped particles

While the computed heart strength curve fits the data well at high consolidation pressures, it still shows some deviation at lower consolidation pressures. However, when the direction of heart particle contacts is included in the computation, the resulting data over the entire range of stress levels compares well with experimental measurements, as shown in Figure 10.

Correction factor applied to round particle strength to correct for the number of contacts and the direction of the contact per adjacent particle for heart shaped particles

Figure 10. Correction factor applied to round particle strength to correct for the number of contacts and the direction of the contact per adjacent particle for heart shaped particles

Therefore, the implication is that the unconfined yield strength acts as a function of two factors, both as the number of particle contacts per adjacent particle, as well as the direction of those particle contacts.

Finally, the identical analysis was performed using star-shaped particles. Figure 11 shows how the multiplication of the strength measured for round particles by the effective number of contacts in the star-shaped particle system produces an approximation of the data measured from the uniaxial shear cell. However, one limitation is that the computed star strength curve does not fit the data well and predicts high values.

Correction factor applied to round particle strength to correct for the number of contacts per adjacent particle for star shaped particles

Figure 11. Correction factor applied to round particle strength to correct for the number of contacts per adjacent particle for star shaped particles

The inclusion of the direction of star-shaped particle contacts aids in lowering the prediction, resulting in a prediction that befits the experimental results for most of the lower solid stress level region. In addition, there seems to be a degree of deviation at the larger stress levels. This simply suggests that the above simplistic analysis requires a level of additional modification to explain the observed data (see Figure 12).

Correction factor applied to round particle strength to correct for the number of contacts and the direction of the contact per adjacent particle for star shaped particles

Figure 12. Correction factor applied to round particle strength to correct for the number of contacts and the direction of the contact per adjacent particle for star shaped particles

Moreover, it can be observed that the fit is good enough to defend the suggestions of two key parameters that influence the strength of a bulk powder system. The first being the number of contact per adjacent particles, while the latter is the direction of these contacts relative to the centroid axis between the adjacent particles.

Essentially, further work needs to be done if this model is generalized to any system having variable particle shapes. As a result, this can be taken up in further research as the subject of another paper.

Summary

This article produces several important implications. Firstly, it demonstrates that the number of contacts per adjacent particle as well as the direction of these contacts are key factors that influence the bulk strength of material.

Furthermore, this work also suggests that simple models that include the effects mentioned therein may be useful to predict bulk unconfined yield strength for particle scale properties. Future models that describe the yield strength of bulk materials should consider incorporating these above-mentioned effects.

However, a major limitation of this approach is that it requires the measured strength data in the case of an ideal system. However, if this ideal system undergoes further characterization, it follows that the technique outlined above can be implemented in order to predict non-ideal systems.

Conversely, this approach also has the following strength. In developing models that predict strength in ideal systems that use only particle scale properties, the approach outlined in this research would be beneficial because it would extend the ideal system to non-ideal conditions.

Thus, in other words, this approach could help bridge the gap between materials in the real world and idealized systems. As a result, the logical next step encompasses the extension of this analysis to apply it to smaller particle systems and general shape systems.

References and Further Reading

  1. Jenike, A.W., Storage and flow of solids, Bulletin No. 123, University of Utah Engineering Experiment Station, November 1964.
  2. Jenike, A.W., Gravity flow of bulk solids, Bulletin No. 108, University of Utah Engineering Experiment Station, 1961.
  3. Moakher, M., T. Shinbrot and F.J. Muzzio, Experimentally validated computations of flow, mixing and segregation of non-cohesive grains in 3D tumbling blenders, Powder Technology, 109 (1-3), p. 58-71, 2000.
  4. Chaudhuri, Bodhisattwa, Amit Mehrotra, Fernando J. Muzzio, M. Silvina Tomassone, Cohesive effects in powder mixing in a tumbling blender, Powder Technology, 165, p.105–114, 2005.
  5. Johanson, Kerry, Chris Eckert, Dev Ghose, Millorad Djomlija, Mario Hubert, Quantitative measurement of particle segregation mechanisms, Powder Technology, 159 1 – 12, 2005.
  6. Johanson, Jerry R., Ph.D., Solids segregation: causes and solutions, Powder and Bulk Engineering, August, 1988.
  7. Rathbone, T. and R. M. Nedderman, The deaeration of fine powders, Powder Technology, 51, 115, 1987.
  8. Bell, Timothy A., Challenges in the scale-up of particulate processes – an industrial perspective, Powder Technology, Volume 150, Issue 2, p 60-71, February 2005.
  9. Wes, G. W. J., S. Stemerding and D. J. van Zuilichem, Control of flow of cohesive powders by means of simultaneous aeration, and vibration, Powder Technology Volume 61, Issue 1, p. 39-49, April 1990.
  10. Brunia, Giovanna, Paola Lettieria, David Newtonb, and Diego Barletta, An investigation of the effect of the interparticle forces on the fluidization behaviour of fine powders linked with rheological studies, Chemical Engineering Science, Volume 62, Issues 1-2, p. 387-396, January 2007.
  11. Wang, Zhaolin, Mooson Kwauk, and Hongzhong Li, Fluidization of fine particles, Chemical Engineering Science, Volume 53, Issue 3, p. 377-395, February 1998.
  12. O. Molerus, The role of science in particle technology, Powder Technology, Volume 122, Issues 2-3, 22, p. 156-167, January 2002.
  13. Rabinovich, Y., M. S. Esayanur, K. Johanson, B. Moudgil, Oil mediated particulate adhesion and mechanical properties of powder, Proceedings World Congress on Particle Technology: paper 370, 2002.
  14. Rumpf, H., Particle Technology. New York: Chapman and Hall, 1990.
  15. Specht, D., Caking of granular materials: an experimental and theoretical study, Thesis, University of Florida, 2006
  16. de Silva, S.R., Characterization of particulate materials – a challenge for the bulk solid fraternity, Powder Handling & Processing, December 2000.
  17. Van der Kraan, M., Techniques for the measurement of flow properties of cohesive powders, Thesis, University of Delft, 2000.
  18. Johanson, J.R., The Johanson indicizer system vs. the Jenike shear tester, Bulk Solids Handling, 12(2), p. 237-240, 1992.
  19. Peschl, I.A.S.Z., Quality control of powders for industrial application, Proceedings of 14 th annual Powder & Bulk Solids Conference, Chicago, p. 517-536, 1989.

This information has been sourced, reviewed and adapted from materials provided by Particulate Systems.

For more information on this source, please visit Particulate Systems.

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