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. 2024 Feb 22;9(9):10459–10467. doi: 10.1021/acsomega.3c08519

Could the Rebound Characteristics of Oblique Impact for SiO2 Particles Represent the Ash Particles?

Xue Li †, Jun Xie ‡,*, Ming Dong §, Sheng Chen ∥, Wenjie Dong †
PMCID: PMC10918674  PMID: 38463255

Abstract

graphic file with name ao3c08519_0012.jpg

Collisions between particles or with a surface have been widely applied, in which the restitution coefficients are the important parameter to describe the particle rebound behavior. SiO2 particles are often used instead of ash particles in theoretical analyses; however, whether this is justifiable has not been confirmed. This paper compares the rebound characteristics of oblique impact for SiO2 particles and ash particles by experimental and theoretical analyses. Based on the rigid-body theory, the tangential restitution coefficients, rebound angle-particle center, and reflection angle-contact path predicted by SiO2 particles are basically in agreement with the experimental results for ash particles, especially at large impact angles. However, there is a slight error at 2.2 m/s as the velocity approaches the critical capture velocity.

1. Introduction

In the chemical and steel industry, coal is the main energy for production. Ash, as a combustion product of coal, constitutes a primary influencing factor on the heat transfer efficiency of the heat exchanger and causes environmental pollution. The ash deposition on the heat transfer surfaces is one of the primary problems relevant to pulverized coal boiler operation and design and fuel selection.1−3 It has been reported that excess ash deposition in convective coolers not only leads to unplanned shutdowns but also causes a reduction in heat transfer or even causes the corrosion of the boiler heat exchange tube.4 Therefore, it is necessary to control the ash deposition. This led to advances in developing models to predict ash deposition behavior.

The impact between a sphere and a substrate is one of the basis mechanisms in the formation of ash deposition, especially the restitution coefficient of the impact is a relevant boundary condition that defines the particle fate within computational fluid dynamics models for predicting the ash deposition process.5−7 There are many factors that influenced the impact results, such as the particle velocity, impact angle, particle radius, particle shape, property parameters of the particle and surface, reaction atmosphere, coal type, flow dynamics, and so on.8−11 Many researchers have intensively investigated the particle impact through experiments and proposed many models to predict the rebound behaviors of the particle.

One of the earliest experiments that show the direct measurement of particle velocity for normal impact under vacuum conditions was conducted by Dahneke.12 The restitution coefficient for normal impact was presented, and the critical capture velocity that occurs when the particle motion stops upon impact with a surface was discovered; however, its accurate value was not achieved in those work. In order to obtain the accurate value, Rogers and Reed13 measured the critical capture velocities of the impact for glass, copper, and stainless-steel particles with a high-speed camera. The effect of the particle material on the critical capture velocity was achieved. For other factors, Wall et al.14 researched the effect of particle diameters, particle and surface material, and impact velocity on critical capture velocity by experiments. Furthermore, the process of particle energy change and the normal restitution coefficient under different collision conditions were investigated.

For oblique impacts, tangential force causes sliding or localized slip, which modifies the rebound characteristic of particles. Researchers found that particles can still bounce when the normal incident velocity was less than the critical capture velocity.15,16 The effect of impact angle on normal restitution coefficient (en), tangential restitution coefficient (et), and total restitution coefficient (e) had been researched by experimental and theoretical methods.10 It was found that en fluctuates little with the impact angle, while et varies greatly. A classical rigid-body theory based on the impulse theorem and the momentum theorem was developed to predict the particle rebound characteristics.17 According to the classical rigid-body theory, Wu et al.18,19 analyzed the particle motions after the elastoplastic particles obliquely impact the elastoplastic surface by the finite element method and propose a dimensionless analysis method to predict et, with en. Based on the dimensionless analysis method, Tomar and Bose20 compared the restitution coefficient of glass and steel particles with diameters between 1 and 6 mm. It was found that et is a function of en and impact angle; sticking occurs at larger impact angles, while sliding dominates at lower impact angles. During elastic collision, the critical impact angle that distinguishes between sticking and sliding is determined by Maw et al. and a method for deriving the coefficient of sliding friction was obtained.21,22

For real ash particles, the rebound parameters are different to predict due to multiple factors affecting the collision process, for example, carbon conversion, temperature, impact velocity, surface properties, and structure.23,24 A 2D model was developed to predict the rebound characteristic of ash particles in the boiler.25,26 Research found that the shape and size of ash particles are practical reasons for deposition. Yang et al.27 considered the influence of impact angle, incident velocity, particle properties, and furnace operation conditions on particle deposition. The maximum deposition efficiency approaches that of small particles with an increased incidence velocity. For near-wall studies, Troiano et al.28,29 carried out experiments on oblique impact of coal, char, and ash particles with a flat surface. The effect of carbon conversion, impact velocity, and surface properties on the restitution coefficient had been investigated. They also pointed out that the particle material and the surface properties have little effect on en during oblique collision. The variation of the restitution coefficient of ash particles with the impact angle and incident velocity has been obtained experimentally.30,31 However, the above studies quantitatively describe only the rebound parameters of ash particles. Compared to ash particles, the rebound behavior of SiO2 particles with a regular single property are easier to obtain. Therefore, it is more meaningful to predict the rebound characteristics of ash particles by SiO2 particles.

Above all, the experimental and theoretical investigations of SiO2 particles and ash particles that obliquely impact the stainless-steel surface are presented. The variations of restitution coefficient, rebound angle, and dynamic friction coefficient are determined by experiments. The emphasis is on using the rebound parameter from regular SiO2 particles to predict ash particles’ behavior after oblique impact with a surface. Based on the rigid-body theory, the rebound angle velocities and tangential restitution coefficients of SiO2 particles were obtained, and the rebound parameters of ash particles were further predicted.

2. Experimental System

The experimental facility for the oblique impact between the particle and stainless-steel plate is shown in Figure 1. The experimental facility mainly includes the inlet system, a collision unit, and a high-speed camera system. The nitrogen as the transport gas drives the particles into the collision unit. The airflow is adjusted by the mass flow meter and consequently changes the particle incident velocity. The particle is injected using a particle generator. The surface roughness of the stainless-steel plate is kept below 0.2 μm. The surface inclination angle varies from 15 to 75° at an interval of 15°. The high-speed camera system consists of a light source and a Phantom v12.1 digital high-speed camera with a 10× lens and a computer. The captured particles are in the focus plane of the camera for high-resolution images. The shooting parameters of the digital high-speed camera are shown in Table 1. The time interval of particle images in Figure S1 was two frames. The incident velocity of the SiO2 particle was 4.23 m/s with an impact angle of 15°.

Figure 1.

Figure 1

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Experimental system of particle oblique impact.

Table 1. Technology Parameters of Phantom v12.1.

item parameter
lens VS-M0910 M × 10 microscopic
frame rate 66,037 frames/s
exposure time 6.22 μs
resolution 256 × 256
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To ensure the accuracy of the experiment, no less than 30 experimental groups are conducted at the same condition. The incident and rebound velocities of the particles are obtained by averaging the velocities between the captured images of the particle before and after the collision. During high-speed camera shooting, the random error of the sampling is ±0.5%, and the position uncertainty of the particle in the image is ±l pixel. Therefore, the measured mean displacement error is ±0.5 pixel. In this experiment, the random error is 0.38 at a 4.3 m/s incident velocity, and the relative error is ±8.8%. The deviations of the impact angle and particle diameter are ±1.5° and ±2.75 μm, respectively.

The deposition of ash particles on the heat exchanger surface is the main factor affecting the heat exchange efficiency in the boiler heat exchanger. The majority of the ash deposition on the heat exchange surface is from the inertial impaction of particles with diameters larger than 10 μm.32 For different types of coal, the ash content varies. The predominant component in the ash is always SiO2.33,34 Therefore, the SiO2 and ash particles with diameters of 25 μm are selected. The morphology of the SiO2 particle is obtained by SEM, as shown in Figure 2. The diameter distribution of the SiO2 particle in Figure S2 is measured by a laser particle size analyzer (Mastersizer 2000, Malvern Instruments Ltd., Malvern, UK). The ash particles used in this study were from an actual power plant, as imaged by SEM in Figure 3. The bulk of the diameters is in the range of 5–30 μm. A particle size of 25 μm was selected for both SiO2 and ash particles in the experiment.

Figure 2.

Figure 2

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Morphology of SiO2 particles imaged by SEM.

Figure 3.

Figure 3

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Ash particle imaged by SEM.

This article assumes that the crystalline phase of ash only consists of quartz and mullite. Through XRD analysis, the density and Young’s modulus of ash particles are calculated as follows, and the results are listed in Table 2.

2. 1
2. 2

where ρ is the density, E is the Young’s modulus, V is the volume fraction, and subscripts g, q, and m denote glass, quartz, and mullite, respectively.

Table 2. Physical Parameters of Ash Particles Are Derived by XRD.

item total crystalline phase quartz mullite ash SiO2
ratio % 44.90 6.80 38.10    
density (kg/m3) 2500 2650 2800 2370 2320
Young’s modulus (GPa) 73 94 230 127 75
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3. Theoretical Considerations

The schematic of the particle oblique impact with the surface is shown in Figure 4. Note that V, v, and w are the velocities of the sphere center, the translational velocities at the contact path, and the angular velocities, respectively. The subscripts i and r indicate the incidence and rebound phases, respectively.

Figure 4.

Figure 4

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Schematic of the particle obliquely impacting the surface.

As the particle obliquely impacts with stainless-steel plate, en, et, and e are given by

3. 3
3. 4
3. 5

where θi is the impact angle and the subscripts n and t represent normal and tangential directions, respectively.

The impulse ratio is used to describe the correlation between the tangential and normal interactions during an impact. According to Newton’s second law and the conservation of angular momentum about point C, the impulse ratio f, normal impulse Pn, the tangential impulse Pt, and the angular momentum Pw are as follows

3. 6
3. 7
3. 8
3. 9

where m, I, and ω are the particle quality, moment of inertia about the spherical center, and particle rotation angle, respectively. For a solid sphere, there is I = 2mR2/5. The rebound angular velocity, tangential restitution coefficient, and the rebound translational velocities at the contact path can be written as

3. 10
3. 11
3. 12

The dimensionless method is introduced to analyze the oblique impact process. The dimensionless rebound angular velocity Φr, incident angular velocity Φi, the dimensionless rebound tangential surface velocity at the contact path Λr, and the dimensionless impact angle Θ are defined as follows35

3. 13
3. 14
3. 15
3. 16

where R is the particle radius and μf is the dynamic friction coefficient. Hence, eqs 10–12 can be rewritten as

3. 17
3. 18
3. 19

From eqs 17–19, parameters Φr, et, and Πr are functions of Φi and Θ. When f/μf is determined, the parameters Φr, et, and Λr can be obtained.

4. Results and Discussion

4.1. Effect of Impact Angle on the Restitution Coefficient

The restitution coefficient is used to determine the energy loss during a collision, which is defined as the ratio of the rebound velocity to the incident velocity. Figure 5 shows the variation of en, et, and e with the impact angle under the incident velocities of 4.26, 3.24, and 2.47 m/s. The deviation of the incident velocity is ±0.05 m/s. The error bars are not indicated in the graph for clarity. In this experiment, the maximum error of restitution coefficient is less than 10%. As the impact angle increases, the incident kinetic energy is converted into friction and rolling losses. The en fluctuates between 0.35 and 0.45 and tends to be constant when the impact angle increases to 45°.

Figure 5.

Figure 5

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Restitution coefficient versus impact angle at different incident velocities.

This variation of en is inconsistent with the single normal impact process. During oblique impact, the similar regulation of en has been obtained by.10,36 The reason is that only a vertical contact displacement exists between the particles and the contact surface during normal collisions. The maximum normal contact displacement δn is reached. For oblique collisions, however, sliding and rolling occurs between the two contacting surfaces.21,22 The contact surface changes constantly. Then, the maximum normal contact displacement δon in an oblique collision does not reach δn in a normal collision. The normal energy dissipation is a positive correlation with the maximum normal contact displacement.37 Therefore, en in an oblique collision is larger than that in a normal collision. The energy dissipation can be divided into viscoelastic and plastic deformation losses between the contact surfaces.38en is not linearly related to the normal incident velocity but remains constant during the viscoelastic loss stage. Consequently, the variation of en is small in the experiment.

The trend of et decreases first and then increases with the increasing impact angle. At 45°, et reaches a minimum of 0.412 at an incident velocity of 2.47 m/s in the experiment. When the impact angle is larger than 45°, et increases with increasing impact angle. This is because the tangential kinetic energy increases and the normal load decreases. Therefore, the frictional losses decrease and et rises.

The variation of e decreases and then increases with the increasing impact angle. At a low impact angle, e is close to en. As impact angle increases, e tends to et, which corresponds to eq 5.

4.2. Effect of Impact Angle on Rebound Angle

The rebound angle-particle center θr is shown in Figure 6, which is defined as

4.2. 20

Figure 6.

Figure 6

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Rebound angle-particle center vs impact angle at different incident velocities.

θr increases and is greater than the impact angle as the impact angle increases. For different incident velocities, the difference between rebound angles is small.

The reflection angle-contact path is hard to obtain directly through experiments, which is calculated from experimental data. tan θcr (see Appendix) is calculated as

4.2. 21

θcr first decreases and then increases with increasing impact angle, as shown in Figure 7. When the impact angle is less than 60°, θcr is almost less than zero. At 75°, θcr is positive and increases rapidly. The slipping state reduces the friction loss during a collision, which makes et and e increase.

Figure 7.

Figure 7

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Reflection angle-contact path vs impact angle at different incident velocities.

4.3. Dynamic Friction Coefficient

During an oblique collision, slipping will cause friction. There is a critical incident angle αi (αi = 90°–θi) below which a particle in the collision process is the gross slipping state, otherwise it is in the rolling state.21,22 The dynamic friction coefficient μf in tangential force Ft = μfFn is defined as

4.3. 22

Figure 8 illustrates the et versus Inline graphic, and the slope of the curve represents the dynamic friction coefficient. In the experiment, the dynamic friction coefficient is 0.417. When the impact angle is greater than 45°, the experimental values are in good agreement with the diagonal line, which indicates that the whole collision process is in a gross slipping state.

Figure 8.

Figure 8

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Tangential restitution coefficient versus (1 + en)/tan Θi at different incident velocities.

4.4. Predictions of Rebound Characteristics of Oblique Impact

4.4.1. SiO2 Particles

The variation of f/μf with dimensionless impact angle Θ from Figure S3 is obtained. The dynamic friction coefficient μf is a constant that is calculated in Section 4.3. The impulse ratio f increases with the dimensionless impact angle Θ until reaching a certain value. The fitting curve introduces a hyperbolic tangent function, which is expressed as

4.4.1. 23

When f/μf is 1, the particles are grossly sliding from the beginning of contact with the plate to leaving the plate. As the dimensionless impact angle increases, the deviation among the three sets becomes significant. With the increasing impact angle, the shooting process becomes more challenging due to the increased proportion of the flat surface in the captured image. The location of impact on the flat surface exhibits a greater degree of randomness. Therefore, the deviation between the three sets of experimental data from the fitting curve increases.

The critical dimensionless impact angle Θc of complete sliding is defined as follows35,39

4.4.1. 24
4.4.1. 25

where G is the shear modulus and υ is the Poisson’s ratio. Subscripts 1 and 2 are for the particle and stainless-steel plate, respectively. The critical dimensionless impact angle Θc is 5.855. The ratio of f/μf reaches a critical value with the increasing dimensionless impact angle Θ. The critical dimensionless impact angle calculated by eq 23 is 5.749°. In contrast to Θc obtained by eq 24, the error is 1.81%.

When ignoring the incident angular velocity, the parameters for the rebound characteristics of SiO2 particles by eqs 17–19 are as follows

4.4.1. 26
4.4.1. 27
4.4.1. 28

The dimensionless rebound angle velocities (Φr) versus the impact angle from G4 are illustrated. During an oblique collision, the tangential force is opposite to the movement of the particles. Therefore, the minus sign indicates the direction. The absolute value of dimensionless rebound angle velocities increases with the increase in incident velocity. When the impact angle is greater than the critical value, Φr is constant. The rebound angle velocity will not increase indefinitely. Particles with large velocities have greater inertia and shorter contact times;10 thus, Φr decreases.

The tangential restitution coefficient versus impact angle from G5 is shown. When the impact angle is less than 15°, et increases. With increasing impact angle, et decreases and then increases, reaching the minimum value of 0.431. et shows a delay with increasing incident velocity. The larger the incident velocity is, the larger the impact angle corresponding to the minimum et. The range of impact angles is 15–75° in the experiment. The variation of et in Section 4.1 conforms to this pattern.

4.4.2. Prediction of Ash Particles

Figure 9 shows the tangential restitution coefficient versus impact angle. et of SiO2 can be roughly predicted by the et of ash particles. At an incident velocity of 2.2 m/s, the difference in et is more obvious. This is because the restitution coefficients fluctuate more as the velocity approaches the critical capture velocity. As the incident velocity increases, et fits well, especially at large impact angles. The e is closer to et at large impact angles, as shown in Figure 5. Therefore, the total energy loss is in the tangential direction. By comparison of the physical properties of SiO2 and ash particles, the main difference is Young’s modulus. Young’s modulus reflects the ability to resist elastic deformation; thus, the larger the value of Young’s modulus, the more the energy loss. Therefore, et of ash particles is lower than et of SiO2 at speeds of 3.4 and 4.22 m/s. In order to obtain better fitting results, justification is provided by correcting the physical parameters of en in the dimensionless impact angle, as shown in eq 16.

Figure 9.

Figure 9

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Comparation of the tangential restitution coefficient of SiO2 and ash particles vs impact angle.

Figures 10 and 11 illustrate rebound angle-particle center and reflection angle-contact path versus impact angle, respectively. At 2.2 m/s, the differences between θr and θcr for SiO2 and ash particles are large, but the overall trends are more similar and can better predict the movement of the particles. At small impact angles, θr fit well. Since positive and negative impact angles indicate direction, θr and θcr of ash particles are much larger. This is because of the low sphericity of the ash particles and therefore the large rebound angular dispersion.

Figure 10.

Figure 10

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Comparation of the rebound angle-particle center of SiO2 and ash particles versus the impact angle.

Figure 11.

Figure 11

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Comparation of the reflection angle-contact path of SiO2 and ash particles vs impact angle.

5. Conclusions

An experimental study was performed using SiO2 and ash particles that obliquely impact the stainless-steel surface under different incident velocities. The rebound behavior of SiO2 and ash particles was compared by the rigid-body theory. This led to the following conclusions:

  • (1)

    In the impact angle range from 15 to 75°, the normal restitution coefficient fluctuates between 0.35 and 0.45 under different incident velocities. As the tangential contact changes from rolling to sliding, the tangential restitution coefficient decreases first and then increases with a minimum impact angle of 45° in the experiment.

  • (2)

    The rebound angle-particle center is always larger than the impact angle. At impact angle of 75°, the reflection angle-contact path grows rapidly and greater than 0, which is due to the gross slipping between two contacting surfaces. The dynamic friction coefficient between SiO2 particles and the stainless-steel surface is 0.417.

  • (3)

    By comparison of experimental with theoretical predictions, SiO2 particles can basically replace ash particles, especially at large impact angles. Except for the slight error at a 2.2 m/s incident velocity, the tangential restitution coefficient, rebound angle-particle center, and reflection angle-contact path are well fitted to the experiment.

Acknowledgments

The authors acknowledge the Foundation of State Key Laboratory of Coal Combustion (no. FSKLCCA2209), Foundation of Wuhan Institute of Technology (no. K2021079), Key Laboratory of Advanced Reactor Engineering and Safety, Ministry of Education (no. ARES202104), and National Natural Science Foundation of China (no: 12302335).

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.3c08519.

  • Formula derivation of tan θcr from the Appendix (PDF)

  • Diameter distribution of the SiO2 particles, the images of particle collision, f/μf versus Θ at different incident velocities, dimensionless rebound angle velocities versus the impact angle, and tangential restitution coefficient versus impact angle (PDF)

The authors declare no competing financial interest.

Supplementary Material

ao3c08519_si_001.pdf (51.6KB, pdf)
ao3c08519_si_002.pdf (331.4KB, pdf)

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