Investigation of the Dominant Effects of Non-Spherical Particles on Particle–Wall Collisions

Abstract

A deep understanding of the particle–wall collision (PWC) behaviors of non-spherical particles is important for managing gas–solid flows in industrial applications. It is important to identify the dominant parameters and to develop the common PWC prediction models for typical non-spherical particles. In this paper, different types of non-spherical particles were used to conduct the fundamental experiments. The effects of key parameters such as particle size, non-sphericity, wall roughness, and impact angle were analyzed. The results show that the trends of the collision coefficients with the impact angle for all non-spherical particles are similar. The dominant factors of particle–wall collisions are particle sphericity and wall roughness. A model with four parameters was fitted from the experimental data. The model can predict the collisions of non-spherical particles on rough steel walls with sizes ranging from 50 to 550 microns.

1. Introduction

The particle–wall collision phenomena have an important impact on the behavior of gas–solid two-phase flows, which are frequently found in energy, mechanical, and other industrial applications. For instance, solid fuel particles are generated in pulverizers to provide suitable particles for the coal powder furnace. More than 70% of the fine particles are smaller than about 74 μm, as coarse particles lead to slagging and incomplete combustion [1,2]. In typical types of particle classifiers such as cyclones, which are commonly used in chemical engineering, particle–wall collisions occur frequently and affect the separation efficiency and consequently the fineness of the particle sizes and instrument efficiency [3,4]. Therefore, it is important to understand the particle–wall collision behaviors of these types of particles, for better design of pulverizers, classifiers, and cyclones. Computational fluid dynamics (CFD) is an effective tool for the analysis of gas–solid two-phase flows. In order to produce accurate CFD simulations of such flows with wall interactions, particle–wall collision models are needed to predict the particle motions correctly. A model derived from impulsive equations and Coulomb’s friction law is commonly used [5]. It describes the motion with the restitution coefficient e, which is related to the energy changes by plastic deformation, and the friction coefficient f, which represents the relative amount of friction force. These coefficients are not material properties and can be affected by many parameters such as the particle diameter and shape, the impact angle, the impact velocity, and the wall roughness. To apply the model in simulations, reasonable values of the coefficients must be provided.

Many experimental studies have been carried out on particle–wall collisions to analyze the impact behavior and to obtain the collision coefficients. For normal collisions with the impact angle α₁ = 90°, the friction coefficient f has no effect, while the restitution coefficient e is often expressed as a function of the impact velocity v₁ [6,7]. With v₁ increasing from the critical velocity, e increases from 0 to the maximum value as the adhesive effects decline relatively. Then, e starts to decrease with a very large v₁ as plastic deformation occurs [8]. For oblique collisions, both e and f generally have negative correlations with the impact angle α₁, which is the angle between the impact velocity direction and the wall [9,10,11]. If α₁ is sufficiently small, the particles may be able to gain a normal velocity component, causing e to increase over 1 [12]. Model equations of the collision coefficients based on α₁ have been proposed [13,14]. The relationships between e, f, and other conditions such as the particle diameter can also be found in previous studies [6,15,16,17,18,19,20,21,22].

For spherical and relatively large particles, constant collision coefficient values can offer a sufficient approximation to produce accurate predictions [5,23,24]. On the other hand, small non-spherical particles often exhibit irregular rebound behaviors [9]. The non-spherical shapes of the particles are known to alter the probabilities of different impact conditions on the local wall structure, leading to a smaller restitution coefficient e, and a larger friction coefficient f when the impact angle is small [25]. Additionally, many existing experimental studies have shown scattered results of e and f [26,27,28,29,30]. For non-spherical particles, the restitution coefficient’s distribution is concentrated at small values. While the presence of a rough wall generally spreads the distribution of e, non-spherical particles’ e is less affected by the wall roughness [25]. Therefore, these types of particles require more complex expressions of the collision coefficients, involving parameters that represent the shape factor, for accurate predictions of the rebound motions [31,32].

The particle velocities are often measured by optical methods, such as high-speed photography [18,33,34]. Particles that are easy to capture optically, such as glass beads, are commonly used in the experiments. On the other hand, typical non-spherical fuel particles such as coal and biomass have received less focus in existing studies. An experimental study by K.-P. Schade et al. investigated the impact behavior for coal particles and steel walls, focusing on the erosion effects [35]. Constant coefficient values and polynomial equations of the modified impact angles were provided, and the relations to the impact angle and velocity were shown. Experimental data of coke, polyethylene, and polystyrene particles reported by L. M. Gibson et al. [32]. revealed relations between the restitution coefficients and the rebound angles. Regression models involving the particle sphericity were proposed to describe the restitution coefficient components based on analysis of variance (ANOVA). More recently, M. Troiano et al. conducted experiments using coal, char, and ash particles with fixed impact angles, where the effects of impact velocities and carbon conversion degrees in cold and hot conditions were analyzed [36,37]. Rebound analyses have been successfully applied in the modeling of entrained-flow slagging gasifiers [38,39]. J. Xie et al. presented experimental results of using fly ash particles with a stainless-steel wall, and the effects of the impact velocity and angle on the rebound motion were reported [40]. R. Darko et al. presented the effects of an anisotropic virtual rough wall on the rebound motion [41]. W. Jacob et al. reported the importance of the particle shape for the rebound behavior [42]. However, the existing data for typical non-spherical particles are still limited. The collision behavior differences between those particles and regularly shaped particles have not been thoroughly analyzed.

Our previous work [25] analyzed and improved the virtual wall model for particle–wall collision, enabling it to better fit experimental results of spherical and non-spherical glass beads and glass powders colliding with steel walls. However, the real particles that have been widely used in real industrial applications vary in material and shape. In order to clarify the dominant effects on particle–wall collisions and to provide a model widely applicable to various types of non-spherical particles, the behaviors of typical non-spherical particles such as glass powder, coal, and biomass particles’ collisions with a polished/unpolished steel wall were studied experimentally through optical measurements. The effects of the key parameters on the collision coefficients, such as the particle material and non-sphericity, wall roughness, impaction velocity, particle size, and impact angle, were analyzed. In particular, the effect of sphericity was investigated through a thorough experimental comparison among different particle shapes, including spherical glass beads, Shepherd’s purse seed with a similar ellipsoid shape and narrow sphericity, and coal/glass powder/biomass particles with wide range of sphericities. The common behaviors of these non-spherical particles were clarified through data analysis since the particle parameters such as particle size, particle shape, and impact velocity vary significantly and are hard to measure for each particle in industrial applications. An impact model based on the statistical parameters of a group of particles is thus meaningful for real applications; therefore, for this experiment, we chose particles with a large span of statistical characteristics, which is the same as in the industrial applications, and the average parameter value, such as that for particle size or particle sphericity, was used as the parameter input. Data analysis was based on a large amount of experimental statistical data, and it was carried out to obtain statistical characteristics with industrial application value and to analyze their dominant effects. An empirical model was proposed based on the experimental data, which can be used for CFD simulations of two-phase flows involving typical non-spherical particles with the size range of 50 μm to 550 μm.

2. Experimental Facility

Particle–wall collision experiments were conducted using different types of particles, expanding on the previous works with only glass beads [25]. The setup of the experimental facility is shown in Figure 1. The gas cylinder releases pressurized nitrogen into the pipe, which passes through the sonic nozzle flow meter to the particle feeder. The particles are pushed into the flow continuously by the piston driven by a motor, and the speed of the motor can be adjusted to control the rate. During the experiments, a very small constant motor speed was used to ensure a dilute flow. The particle feeding rate was not measured quantitively as the particle–wall interactions were not dependent on it in the dilute flow. After accelerating in a thin pipe with a diameter of 2 mm, the gas–solid flow is conveyed into the collision section where a rotatable block is installed. One of the rotatable block’s vertical faces is used as the target wall for particle collision. The impact angle was partially controlled by turning the block’s axle horizontally in the bearing. The particle–wall collisions occurred in the horizontal plane, illuminated by a laser sheet generator, and the high-speed camera captured particle trajectories from the top of the section. The photos were then stored by the computer to obtain the particle motion data.

The resolution width of the high-speed camera (IDT NR3-S2, San Jose, FL, USA) was 256 pixels, and the frame rate was 4125 frames per second. For particles with diameter d = 110 μm and velocity v = 6 m/s, the product of the resolution width and the frame rate was more than 19v/d. As a result, with the pixel’s physical size smaller than d, the image’s physical width could be large enough to contain the rebound trajectories. The calibration factor was 62 um/pixel. The uncertainties that could lead to errors represented by pixels were related to the frame rate and the particle velocity. For example, given the frame rate of 4125 Hz and the particle velocity of 6 m/s, the velocity error caused by the pixels in this study was no more than 1.42%. The resolution height was adjusted for each case so that the entire particle trajectories were covered. The camera exposure was 117 μs, about half of the frame interval. The laser generator (GL 532-100HP) produced a continuous laser sheet of 20 mm×1 mm. Its wavelength was 532 nm and the power was 11 W. If the particle trajectory was out of the probe volume, it became invisible in the images and would be discarded during post-processing. Therefore, all the identified collisions occurred in the two-dimension plane, and the influence of three-dimensional rebound effects was relatively small. Given the frame rate of 4125 Hz and the particle velocity of 6 m/s, the velocity error caused by projecting three-dimensional motion onto a two-dimensional plane does not exceed 2.66%.

The photos captured of the particles (see Figure 2 for examples) were post-processed by a self-developed algorithm through a MATLAB program, as illustrated in Figure 3. The algorithm of the program was the same as that used in the previous work [25]. The wall boundary (Figure 3a) was located by applying a Hough transform. The particles’ locations (Figure 3b) were found by subtracting the next image in the time sequence, applying a threshold, and then finding the connected components. There were usually multiple particles in one image, and we sought to correctly identify the same particle in successive images. For this purpose, it was assumed that the particle motion without collisions was almost uniformly linear, because the frame rate was high enough and the force effects would be ignored during the short period. The combinations of the particle locations in consecutive images were all tested; if a certain combination conformed with the assumption (forming a straight line with almost equal spacing), the locations would be matched and treated as the same particle. The resulting straight-line trajectories are shown in Figure 3c,d. Finally, particle–wall collisions were identified by matching the trajectories to and from the wall. An example of the post-processing result is shown in Figure 3e. The impact angle and impact velocity for each particle collision could then be identified through the post processing. The particles’ velocity magnitudes and directions were derived from the post-processing results. For each collision instance, the data of the second to the fifth particles after collision and prior to collision were included. This was to avoid the strong reflective lights near the wall, and to keep the measured time small so that the assumption of uniformly linear motion held. With the drag force governed by Stokes’ law, $F_d=3\pi \mu d\cdot v$, the particle velocity relative to the flow v_r was proportional to −dv_r/dt, and v_r was subject to exponential decay with a very large time constant. For example, for glass particles with d = 160 μm, the time constant τ was 192 ms. Within the measurement time (3× frame time), the relative error of v_r was only $1-\mathrm{e}\mathrm{x}\mathrm{p}(-t⁄\tau )=0.38\%$ [25]. The total error of the three types mentioned above was $1.42\%+2.66\%+0.38\%=4.46\%$.

All of the experimental conditions are shown in

Table 1. Experimental conditions.

No.	Particle	d (μm)	v1 (m/s)	Sphericity	Target Wall	Identified Collisions
1~2	Coal A	110	3/9	0.69	Steel	9867
3	Coal A	110	6	0.69	Polished Steel
4~6	Coal A	60/110/220	6	0.69	Steel
7	Coal B	150	6	0.64	Steel	2154
8~9	Biomass	310/550	6	0.23	Steel	3556
10~12	Glass beads	85/130/165	6	~0.95	Steel	26,676
13~15	Glass beads	85/130/165	6	~0.95	Polished Steel
16~17	Glass powder	165/205	6	~0.75	Steel	8605
18~19	Glass powder	165/205	6	~0.75	Polished Steel
20	Shepherd’s purse seed	500	6	0.78	Steel	1441

. The target walls were unpolished and polished steel. The roughness levels Ra of the walls were 2.3 μm and 0.4 μm, respectively. Considering the widespread application of non-spherical particles, representative particles with a wide range of sphericity were selected, including two types of coal particles, one type of biomass particle, and glass powders. The sphericity is the ratio of the diameter of the equivalent circle of the two-dimensional projection area to the diameter of the smallest circle covering the projection. Since there was a significant difference in sphericity between non-spherical and spherical particles, Shepherd’s purse seed with a more uniform sphericity was selected to fill the gap. The coal particle types had different Hardgrove Grindability Index (HGI) values; the majority of the cases involved Coal A with HGI = 50, while the softer Coal B with HGI = 91 was used in one case. In addition, the greater-hardness glass particles and lesser-hardness biomass particles were involved to investigate the effect of hardness on particle collision. The particles were obtained by grinding the coal or biomass and then sieving them to different sizes. The diameters d listed in Table 1 were the median values of the measured size distributions in laser diffraction analysis. The carrier gas and the wall were at room temperature; as the effects of the temperature have been covered by previous studies, this work primarily focused on the particle non-sphericity instead.

At the outlet of the acceleration pipe, the particles’ actual velocities and angles varied from the gas velocity and the injection angle, and the exact values could not be controlled individually. Therefore, the results were regrouped by the measured velocity and angle to analyze the collisions statistically [25]. In Table 1, the median impact velocities v₁ of the conditions are listed. For each case, the experiments were repeated with different injection angles, and the measured impact angles were regrouped into 20°, 30°, 40°, 50°, 60°, and 70°. These conditions were designed to analyze the wall collision characteristics of the non-spherical particles, as well as the effects of wall roughness, impact velocity, particle size, and particle hardness. Figure 4 shows different particles. The detailed conditions are listed in Table 1; the data set included 52,299 valid collisions for all experimental cases.

3. Results and Discussion

3.1. Effects of Key Parameters on Particle Collision

The results for the particle velocities were analyzed by applying the model with the restitution coefficient e and the friction coefficient f. These coefficients were calculated as [5,25]

$$e=v_{2\mathrm{n}}/v_{1\mathrm{n}}$$

(1)

$$f=\frac{v_{1\mathrm{t}}-v_{2\mathrm{t}}}{\left(1+e\right)v_{1\mathrm{n}}}$$

(2)

where v_1n and v_1t are the normal and tangential impact velocities, and v_2n and v_2t are the normal and tangential rebound velocities. Similar to the non-spherical glass particles, the rebound motions of the non-spherical particles with different materials and shapes were scattered with large variance. The data were filtered by an outlier detection algorithm based on the interquartile range IQR. The values outside [Q_L − 1.5 IQR, Q_U + 1.5 IQR] were excluded, where Q_L and Q_U are the lower and upper quartiles of the results data [43]. In the experiment, the impact angles were regrouped and it was difficult to accurately control the impact velocity, so the filtered data were analyzed statistically, to produce results revealing the statistical collision behaviors, with use in real industrial applications.

Figure 5 shows the mean values of e and f for the coal particles colliding with the steel wall, with different impact angles α₁. The impact velocities of the cases are in the 6 m/s group unless otherwise noted. The dependencies of e and f on α₁ of the coal particles follow the same patterns as for the glass particles; e decreases with decreasing slopes when α₁ becomes larger, and f decreases with increasing α₁ approximately linearly. For the coal A particles with HGI = 50, both e and f are almost the same as those of the non-spherical glass particles with the steel wall. Unlike the glass particles, the coal A particles with different diameters have similar values of collision coefficients. Although glass and coal have distinct material properties, their rebound motions on rough walls can be very similar due to their irregular particle shapes. For the softer coal B particles with HGI = 91, e is larger than coal A for medium to large impact angles α₁ ≥ 40°, while f has the same value as coal A. There are complex relationships between the HGI and the particles’ mechanical properties such as the compressive ductility and the ultimate stress [44,45], and it is unknown which property contributes most to the different behaviors during collision. However, such differences are smaller than the differences caused by the particle sphericity.

The effects of the impact velocity on the collision coefficients are shown in Figure 6. Coal A particles and the steel wall were used in these cases. With the impact velocity increasing, small decreases in e and increases in f were observed. The results were consistent with the existing theories about velocity’s effects at slow speeds [6]. And the effects of collision velocity were limited, too.

Figure 7 shows the results of the coal A particles with the steel wall and the polished steel wall. For the polished wall with lower roughness, the dependence of e on the impact angle α₁ was smaller, and e became smaller than that for the rough wall with a small α₁. This was caused by the particles’ direction changes for glancing impacts; for coal A particles (d = 110 μm) within the α₁ = 20° group, the mean rebound angle with the rough steel wall was 5° larger than that with the polished steel wall. These effects are often attributed to the rough wall’s local inclinations [9,10]. The coal particles’ e can be as low as 0.45 with α₁ → 20° when colliding with the polished wall, which is lower than that of the non-spherical glass particles. On the other hand, the f of coal particles has little difference from that of the non-spherical glass particles, and both cases have lower f values than when colliding with the rough wall, when α₁ is small.

For applications of fuel particles such as thermal power generation, the effects of the temperature are important for the particle–wall collision behaviors. To analyze such effects, the collision coefficients of the coal particles were compared with the results in Troiano et al., 2017 [36] (Figure 7). At room temperature, the cited case with the brass wall resulted in the same collision coefficients as the steel wall in this paper, due to the similar wall material types. The refractory wall had much greater roughness so its friction coefficient f was larger. As the temperature rose to 1380 °C, the refractory wall’s f became even larger as the particles’ tangential rebound velocities decreased [36].

There are few existing studies about thermal effects on particle–wall collision [36,37,46]. Dong et al. 2014 [46] showed that when there are temperature gradients between the inlet and the wall, the effects of the thermophoretic force are important for the wall collisions of small (d = 20 μm) SiO₂ particles. In this paper, the coal particle’s size was much larger, reducing the influences of external forces. Troiano et al. 2017 [36] showed that collision behavior changes under hot conditions with char/slag transition, which can be explained by changes in mechanical properties. For example, ash particles’ rebound velocities decrease at high temperatures, as the particle stickiness increases significantly [47]. But the results in Figure 7 show that temperature has limited effects on particle collision if the mechanical properties of the particle do not have an apparent change. Especially for non-spherical particles, even with material changes (e.g., from coal to char), the particle wall behaviors for a similar wall roughness show great similarities. The results in Figure 7 also show that wall roughness has an apparent effect on the friction coefficient.

We take the biomass particle–steel wall collision data with different particle sizes as an example, as shown in Figure 8. Similar to the case with coal A particles, the collision coefficients of these biomass particles are close to those of the non-spherical glass particles with the rough steel wall. The results prove that changing the particle diameter causes only small variations in e for large impact angles, and it has nearly no effect on f for non-spherical particles with low sphericity.

3.2. Similarities of the Non-Spherical Particle Collisions

For a thorough comparison among the different particle types, Figure 9 and Figure 10 show the collision coefficients of all the cases. In Figure 9a, the restitution coefficients of the spherical glass particles have significantly different values and trends from those of the non-spherical particles. With large impact angles α₁, the e of glass beads can be up to 0.6~0.8, while all the non-spherical particles have much lower e results, ranging from 0.2 to 0.4. Specifically, the restitution coefficients for non-spherical particles of coal, glass powder, and biomass impacting with unpolished steel are shown as the shaded region in Figure 9a; it is interesting that the region is concentrated, although the materials are very different, as well as the particle size and particle sphericity. Meanwhile, these regions for the restitution coefficients of coal and glass powder impacting with polished steel are more scattered than that for steel cases. This difference shows that sphericity and wall roughness have more important effects on particle wall collision behaviors compared to the other collision parameters. It can also be observed that the restitution coefficient of Shepherd’s purse seed is larger than that of all other non-spherical particles, but is lower than those of most of the spherical particles. Based on these comparisons, it is clear that sphericity plays a dominant role in setting particle impaction behavior when it is less than 0.7, and that such an effect makes the particle behavior similar. Such a trend is shown clearly in Figure 9b with α₁ = 60°, where the spherical glass particles’ e is positively correlated with the particle diameter d. On the other hand, the coal and biomass particles’ e values are close to a horizontal line at around 0.3 regardless of d. The results of the non-spherical particles with the unpolished steel wall are highlighted with shaded areas in both figures. The areas are also very small and concentrated.

The friction coefficients f as functions of α₁ and d are plotted in Figure 10a,b. As f → 0 for normal collisions (α₁ = 90°), the differences among the conditions become clear with small impact angles. With α₁ = 20°, the glass beads’ f is generally lower than those of the coal and biomass particles. Similar observations can be made based on Figure 10b concerning the correlations of f with d. The shaded areas containing the fuel particles’ results with the steel wall have limited ranges. Therefore, it can be concluded that the particle non-sphericity and the wall roughness are the major influencing factors of collisions. In particular, the collisions between non-spherical particles and the rough steel wall behave similarly, and it is possible to develop a simple particle-collision model for a wide range of non-spherical particles with impaction angles ranging from 0 degrees to 90 degrees, sizes ranging from 50 μm to 550 μm, material HGI hardness values ranging from 50 to over 100, and collision velocities ranging from 3 m/s to 10 m/s or higher.

In order to directly show the effect of sphericity on both e and f, the restitution coefficients and the friction coefficients of different particle types as functions of sphericity under the fixed impact angle of 40° and impact velocity of 6 m/s are shown in Figure 11. It is clear that e decreases first when sphericity decreases from 1.0 to 0.75, and then e reaches a steady narrow range from 0.3 to 0.45 as sphericity decreases to very small values such as 0.23. On the contrary, f increases with the decrease in sphericity.

3.3. Modeling

The results show that all non-spherical particles exhibit similar stochastic behaviors during particle–wall collisions, and their rebound velocities and collision coefficients have wide distributions. Figure 12 shows the scatter plot of the velocity ratio e_v = v₂/v₁ for the coal A particles (d = 110 μm) with the steel wall. The value e_v is chosen instead of e = v_2n/v_1n for model validation here, because e_v is still applicable with α₁ → 0 and v_1n → 0 unlike e. With both a small α₁ (grazing impacts) and large α₁ (near-normal impacts), the distribution of e_v has large variations. To build a model for such collisions, it is necessary to introduce parameters representing the variations for both cases. Based on Wang’s work [31], the model with four parameters is used here (modeled restitution coefficient e_m, standard deviation of the modeled restitution coefficient Δe_m, modeled friction coefficient f_m, standard deviation of the random wall angle Δγ). The subscript “m” of e_m and f_m is used to differentiate them from the apparent coefficients e and f based on the impaction angle without a consideration of wall roughness. The modeled restitution coefficient, e_m is assumed to follow a normal distribution N(e_m, Δe_m²), f_m is set to a constant f_m, and the collision plane is assumed to have an inclination angle γ~N(0, Δγ²) against the wall [31]. Given the four parameters e_m, Δe_m, f_m, and Δγ, the stochastic collision results are predicted as follows:

• Generating random e and γ from the corresponding distributions;
• Calculating the rebound velocity (v_2n, v_2t), based on the virtual wall with an inclination of γ against the apparent wall. The equations are [31]

  $$v_{2\mathrm{n}}=-e{v}_{1\mathrm{n}}$$

  (3)

  $$v_{2\mathrm{t}}=\left\{\begin{array}{c}\left(5v_{1\mathrm{t}}+d{\omega }_1\right)/7, \text{non-sliding}\\ v_{1\mathrm{t}}-f\left(1+e\right){\epsilon }_0{v}_{1\mathrm{n}}, \mathrm{sliding}\end{array}\right.$$

  (4)

  where ω₁ is the impact angular velocity, and ε₀ is the sign function of the velocity direction of the contact surface. It should be noted that the non-sliding equation in (4) results in a different f from the provided f_m, and the presence of γ strongly affects the rebound behavior for glancing impacts. Therefore, although the model parameters e_m and f_m are independent of α₁, the predicted e and f will depend on α₁, which is consistent with the experimental data.

The parameters e_m, Δe_m, f_m, and Δγ are fitted from the experimental data, using the median and lower and upper quartiles of e_v and the rebound angle α₂, for each group of impact angle α₁. For the coal A particles with the steel wall, the results are shown in

Table 2. Model parameters for fuel particles with steel wall based on Coal A data.

Particle Type	em	Δem	fm	Δγ	Relative Error
Coal A	0.28	0.13	0.44	9.66	6.0%

. The model’s goodness of fit was evaluated by using the relative velocity distribution error R: [31]

$$\delta ={\left(\frac{e_{v,modal}}{e_{v,data}}-1\right)}^2+{\left({\alpha }_{2,modal}-{\alpha }_{2,data}\right)}^2$$

(5)

$$R=\sqrt{\sum _{{\alpha }_1=20°,30°,\dots ,90°}{\delta }_{{\alpha }_1,median}+\frac{{\delta }_{{\alpha }_1,25\mathrm{\%}}+{\delta }_{{\alpha }_1,75\mathrm{\%}}}{2}}$$

(6)

This is composed of e_v’s relative error and α₂′s absolute error, as they both contributed to the rebound velocity’s relative change. The distribution’s median, lower quartile (25%), and upper quartile (75%) are all considered. The relative error R is 6.0% for the case in Table 2, making it possible to apply the model in the simulations.

Considering the similarity of the collision behaviors of all the non-spherical particles with the unpolished wall, it is expected that a single set of parameters in Table 2 can fit multiple conditions’ results. Figure 13 compares the predicted e_v by the model with these parameters, and the experimental data of three different cases. When colliding with the rough steel wall, both the 110 μm coal A particles and the 310 μm biomass particles show good matches between the model and the data. Only slight deviations are observed for the 150 μm coal B particles. Nevertheless, e_v values of the coal A particles with the polished steel are larger and have narrower distributions for grazing impacts, deviating from the predictions of the same model parameters.

The results in Figure 13 indicate that this model can be applied to typical non-spherical particles impacting rough steel walls, which are frequently found in industrial uses. Even without experimental data for the specific particle–wall combination, the set of parameters in Table 2 can provide reasonable accuracy, as the collision behaviors mostly depend on the particle non-sphericity and the wall roughness. By sampling the model, it is possible to predict the non-spherical particles’ rebound motions and improve CFD simulations of related gas–solid flows.

4. Conclusions

In this study, particle–wall collision experiments were carried out with typical non-spherical particles, including glass powders, coal, and biomass particles, colliding with polished and rough walls. The results of the collision coefficients were determined, and the effects of various parameters were analyzed. It has been concluded that

• The dominating factors of particle–wall collisions are the particle non-sphericity and the wall roughness. Most of the cases with non-spherical particles and a rough steel wall show similar collision behaviors.
• The effects of the particle diameter on the collision coefficients of non-spherical particles can be ignored, unlike for spherical particles.
• It is possible to predict the particles’ rebound motions for a wide range of typical non-spherical particles with a four-parameter model. Using the fitted values in Table 2, collisions of typical non-spherical particles with sizes ranging from 50 μm to 550 μm on steel walls can be predicted with reasonable accuracy.