Fractal characteristics of crushed particles of coal gangue under compaction

Powder Technology 305 (2017) 12–18

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Fractal characteristics of crushed particles of coal gangue under compaction Jixiong Zhang, Meng Li ⁎, Zhan Liu, Nan Zhou Key Laboratory of Deep Coal Resource Mining, School of Mines, Ministry of Education of China, China University of Mining and Technology, Xuzhou 221116, China

a r t i c l e

i n f o

Article history: Received 28 April 2016 Received in revised form 9 August 2016 Accepted 22 September 2016 Available online 23 September 2016 Keywords: Coal gangue Crushed particles Fractal characteristics Particle breakage Compaction test

a b s t r a c t The surface morphology of compacted and crushed gangue presents self-similarity and other fractal characteristics. This research constructed a fractal model for the particle size of compacted and crushed gangue based on fractal theory and particle size distribution information. To investigate the fractal characteristics of compacted gangue, compaction experiments were carried out under varied stresses and with different particle sizes. Results showed that the particle size distributions of crushed gangue specimens with two distinct lithology exhibited fractal characteristics. Fractal dimension of each crushed specimen ranged from 0.352 to 2.654, with increase of stress, the particle size of each specimen tended to be distributed in a more dispersed fashion. Meanwhile, the fractal dimension increased with increased content of small particles, and tended to be a definite value. While the fractal dimension decreased with increased rock strength for the same initial particle size gradation and stress. © 2016 Elsevier B.V. All rights reserved.

1. Introduction In coal mining, gangues which are piled up on the surface after discharge pose a threat to the environment around mining areas [1–4]. Coal mining by solid backfill mining technology, as a green mining method was proposed to solve problems such as pressed coal and gangue deposition presented by traditional mining technologies [5–7]. Using this method, solid wastes such as gangues can be used to backfill mined-out areas directly to replace the coal removed, protect surface construction from subsidence, and reduce environmental harm [8,9]. However, gangues backfilled in mined-out areas are likely to crush due to the overburden pressure. Then the crushed particles result in changes in their gradation which affect the strength and deformation of such gangue backfills [10,11]. Thus, research into the relationship between particle breakage and pressure for compacted gangues will provide useful guidance for mining practitioners using solid backfill mining technology. Scholars have carried out a series of studies on gangue particle breakage. Zhang et al. [12] performed compaction experiments on loose gangues, and obtained a relationship between strain, expansion coefficient, and compactness, as well as some relevant characteristics relating to compaction time. Shi and Cheng [13] established the fractal model of rockfill material based on four groups of triaxial test with high confining pressure. Jiang et al. [14] discussed the relationship between the compactness and ⁎ Corresponding author. E-mail address: [email protected] (M. Li).

http://dx.doi.org/10.1016/j.powtec.2016.09.049 0032-5910/© 2016 Elsevier B.V. All rights reserved.

breakage of coal gangues using compaction experiments by dividing the compaction into two stages: crushed and consolidation. Su et al. [15] obtained compaction characteristics for crushed gangues with three distinct lithologies and showed that the expansion coefficients increased with increasing particle size. Moreover, the rest of the measured expansion coefficients were little influenced by lithology or particle size. Liu et al. [11,16] performed experimental research to investigate the basic mechanical properties of gangues for road-use in Northern Xuzhou city, China and refined current thinking on the issue of particle breakage. Zhou et al. [17] studied the compressive deformation and energy dissipation of gangue in the loading process under conditions of different particle sizes, loading rates and first-time stress loads. Most existing studies investigated the breakage phenomenon and factors influencing gangue particle breakage. Few studies have been made on the variation of fractal dimension of crushed particles with different compaction stresses and gradations. This research established a model for the crushed particle size of compacted gangue based on fractal theory and particle size distribution information. In addition, the authors explored the fractal characteristics of particle breakage for compacted gangue by conducting compaction experiments under varied stresses and with different particle sizes. 2. Fractal model for crushed particles of coal gangue The compaction and breakage of particle packs may be considered as an energy dissipation process, characterised by self-similarity. Therefore, the authors applied a fractal model to describe the particle size distribution of gangue after compaction. Based on the association of

J. Zhang et al. / Powder Technology 305 (2017) 12–18

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Fig. 1. Schematic illustration of experiment system.

particle numbers and characteristic scale, the basic fractal definition is: N∝r−D

ð1Þ

where N refers to the number of particles with characteristic scale (particle radius) bigger than r, D is the fractal dimension, rm. denotes the minimum radius, Nm is the number of particles with radius bigger than rm., also called the total number. According to Eq. (1): N ¼ Nm




r rm

−D

ð2Þ

Similarly, after differentiation of Eq. (1): dN∝r −D−1 dr

ð6Þ

The relationship between particle numbers and mass is: dN∝r −3 dM

ð7Þ

According to Eq. (5) and Eq. (6): α ¼ 3−D

Based on the particle size and quantitative frequency distribution of corresponding particles, the fractal dimension D can be obtained. However, screen testing usually calculates particle size gradation according to mass proportion. It is inconvenient here to derive statistics for particle numbers corresponding to each particle size. Therefore, the association of particle size and mass has to be found. Turcotte [18] assumed that mass is subject to a Weibull distribution:
 α  M ðr Þ r ¼ 1− exp − MT σ

ð8Þ

The fractal dimension D is then calculated. The derivation above, suggested by Turcotte, includes the assumption of r/σ ≪ 1, which possibly limits its feasibility. To exclude this assumption, Eq. (1) is rewritten based on the fractal relationship of particle numbers to particle size. Therefore, the number of particles with size larger than d become: −D

ð3Þ

where M(r) is the particle mass with radius smaller than r, MT is the total mass, and σ is associated with the average size. Suppose that σ b b 1, after series expansion, Eq. (3) may be simplified to:  r α

M ðr Þ ¼ MT σ

ð4Þ

Differentiating Eq. (4): dM∝r α−1 dr

ð5Þ

ð9Þ

NðxNdÞ ¼ Cd

where C is a coefficient of proportionality, the mass of particles with size less than d is: Z Md ðxbdÞ ¼

d dm

sρx3 dNðxÞ

ð10Þ

where s is the shape factor of the particles; ρ is their density; and dm is the minimum particle size. Since there is: dNðxÞ ¼ CDx−D−1 dx

ð11Þ

Table 1 Mechanical properties of coal gangue. Lithology

Density ρ (kg/m3)

Elastic modulus E/GPa

Poisson μ

Compressive strength σc/MPa

Tensile strength σt/MPa

Cohesion c/MPa

Internal friction angle φ/°

Sandstone Sandy mudstone

2940 2800

75.68 17.98

0.19 0.32

135.85 67.61

18.89 6.73

12.86 10.61

37.55 35.49

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Fig. 2. Grading curves of crushed gangues.

By substituting Eq. (11) into Eq. (10): Md ðxbdÞ ¼

 CDsρ  3−D 3−D −dm d 3−D

ð12Þ

dM is denoted as maximum particle size, the total size of the specimen is: MT ¼ Md ðxbdM Þ ¼

 CDsρ  3−D 3−D dM −dm 3−D

ð13Þ

The accumulated mass proportion on the grading curve of particle size is: Md ðxbdÞ ¼ MT

3−D d −dm 3−D 3−D dM −dm

Fig. 3. Calculation of the sample height before compaction.

flange. The tube was 125 mm inner-diameter, 12 mm in thickness, and 305 mm in height. The loading was performed by the loading plate and dowel bar supplied with the steel cylinder in the test system. The schematic illustration of experiment system is shown in Fig. 1. 3.2. Preparation of specimens

3−D

ð14Þ

Eq. (15) indicates that the slope of the lg(Md/MT) − lg(d/dM) curve is 3-D in double logarithmic coordinates. Then D is obtained by fitting the lg(Md/MT) − lg(d/dM) line using the available data.

Gangues obtained from the No. 12 Pingdingshan coal mine, China, were used as the experimental material. The gangues presented two distinct lithologies: sandstone and sandy mudstone. The mechanical parameters are listed in Table 1. During mining with solid backfill, the particle size of gangues in mined-out areas was no larger than 50 mm. Therefore, both gangues were mashed to crushed gangue specimens with particle sizes not larger than 50 mm. Then the specimens were screened, grade-by-grade, using square-holed stone screens with nominal aperture sizes of 2.5 mm, 10 mm, 16 mm, 20 mm, 31.5 mm, and 40 mm. The mass after screening was recorded and the mass proportion of the specimens on each screen surface of the total mass was calculated. Grading curves are shown in Fig. 2.

3. Compaction experiments of coal gangue

3.3. Experimental schemes

3.1. Experimental apparatus

To study the influences of axial stress, lithology, and gradation on the compaction characteristics of crushed gangues, compaction experiments were conducted on each crushed specimen for each gradation and under varied axial stress conditions. The samples were all kept with their natural water content, and no other water was added during the test process. The total of 36 experiments are summarised in Table 2. The crushed gangues were manually mixed to form crushed specimens with varied gradations according to the ratios in Table 2. Uniformly mixed specimens were prepared with equal ratios of constituents, and manual crushed specimens were prepared according to the particle size range 2.5–50 mm. To reduce the friction between the specimens and the inner wall of the steel cylinder, grease was smeared on the inner wall. Then the mixed specimens were poured into the compaction cylinder. After the specimens were levelled and covered, the height of the specimens in the cylinder was measured. Before compacting the samples, their initial height was calculated. When the samples were placed into the compaction device, according to the height h1 of the steel cylinder, the height h2 of the dowel bar,

It is assumed that the minimum particle size is dm = 0, Eq. (14) is converted to: Md ðxbdÞ ¼ MT




d dM

3−D ð15Þ

The loading equipment used in the experiments was a YAS-5000 electro-hydraulic servo-motor test system (Changchun Kexin Instruments Company, Changchun, China). The system delivered a maximum axial force of 5000 kN over a stroke up to 250 mm. Compaction was performed using a steel cylinder designed by the authors. The cylinder was made from Q235 seamless steel tube and connected to a base by a Table 2 Compaction testing schemes of crushed coal gangue. Lithology

Particle size range/mm

Axial stress/MPa

Sandstone and sandy mudstone

2.5–50 manual crushed gradation 2.5–16 uniform mixing 20–31.5 uniform mixing 31.5–50 uniform mixing

2 5 10 15 20 – – 10 15 20 2 5 10 15 20 2 5 10 15 20

Note: Uniform mixing means the same content of each particle size.

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Fig. 4. Stress-strain curves: (a) Sandstone; (b) Sandy mudstone.

the thickness h3 of the loading plate, and the height h4 of the dowel bar exceeding the steel cylinder, the height of samples before being compacted was thus h0 = h1 + h4 − h2 − h3. However, as shown in Fig. 3, h1, h2 and h3 were given, thus h0 can be calculated accurately. Specific experimental steps were as follows: 1. Samples were weighed and put into the compaction device. 2. Samples were weighed for each group and then placed into the compaction device in several layers. Afterwards, the sample surface was struck-off until smooth. 3. We calculated the height h0 of samples before compaction. 4. After the samples were put into the steel cylinder, the loading plate was placed on the surface of samples so as to make contact with the samples, and then the height of the sample before compaction was calculated. 5. We placed the compaction device into the testing machine to apply load to the samples. 6. After testing, the specimens were taken out and re-screened, gradeby-grade, using the square-holed stone screens. The mass retained on each screen was recorded and the particle size gradation after compaction was thus acquired. 4. Experimental results and discussions 4.1. Stress-strain relationship during compaction When the crushed sandstone and sandy mudstone were compacted under a stress of 20 MPa, the stress-strain curves were as shown in Fig. 4. Fig. 4 shows that:

(1) With increased stress, the change in strain was rapid, stable, and smooth, as it increased. (2) The particle size gradation had little influence on the compaction strain of crushed sandstone but did influence the compaction deformation of the sandy mudstone. For the two specimen types, the compaction deformation of the uniformly mixed specimen (manually crushed and degraded) and the specimen with all particle sizes b16 mm was smaller than that of its 20 mm down counterpart. Meanwhile, the smaller the rock strength, the larger the influence thus exerted. (3) For the specimens with two distinct lithologies, both compaction-induced strains under the influence of manual crushed gradation were less than those for each single gradation specimen and larger than that arising in the 16 mm down sample. This phenomenon indicated that smaller particle sizes underwent the least compaction deformation and the addition of small particles improved resistance to the applied stress during compaction.

4.2. Breakage of coal gangues during compaction Particle breakage frequently occurred in each compaction experiment. Even sandstone particles with their uniaxial compressive strength of 135.85 MPa broke under a confined compression test at 2 MPa, as shown in Fig. 5. According to the particle mass remaining on each screen after compaction, the particle size grading curves were drawn in double logarithm coordinates with aperture size as the horizontal coordinate and passing rate (the ratio of specimen mass passing through a certain

Fig. 5. Post-compaction particle breakage in sandstone: (a) Pre-compaction; (b) Post-compaction (stress of 2 MPa).

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J. Zhang et al. / Powder Technology 305 (2017) 12–18

Fig. 6. Particle size grading curves of sandstone before and after compaction: (a) Manual crushed gradation of 2.5 to 50 mm; (b) Uniformly mixed specimens of 2 to 16 mm; (c) Uniformly mixed specimens of 20 to 31.5 mm; (d) Uniformly mixed specimens of 31.5 to 50 mm.

grade of screen to the total specimen mass) as the vertical coordinate. The grading curves are shown in Fig. 6 and Fig. 7. Fig. 6 and Fig. 7 show that the particle size grading curves of all specimens after compaction lay further up the y-axis compared to those before compaction. This indicated that the content of smaller particles increased and particle breakage occurred. In addition, the larger the stress, the more extensive the particle breakage, and the lower the uniaxial compressive strength of the particles, the greater the extent of the breakages was. 4.3. Fractal dimension of particle breakage In this work, the compaction-induced breakage of the specimens was analyzed using a fractal model. By fitting the rearranged particle size gradation data using Eq. (15), the fractal dimension D of the gangue specimens under different compaction stresses was calculated. The relationship between fractal dimension D and stress is shown in Fig. 8 and Fig. 9. Fig. 8 and Fig. 9 show that: (1) The particle size gradation of each crushed specimen showed fractal characteristics. The particle size gradation of the two gangue specimens exhibited fractal characteristics after manual crushed, and the fractal dimension of each crushed specimen ranged from 0.352 to 2.654. (2) Regarding sandstone and sandy mudstone, the fractal dimension D of specimens with particle sizes from 20 to 31.5 mm and 31.5 to 50 mm rose quickly at stresses lower than 10 MPa and tended to be smooth at stresses N10 MPa. (3) With similar particle size gradation and stress, D increased as the

compressive strength of the gangue specimens decreased. (4) As the stress increased, the fractal dimension D of all the specimens increased as it tended to a value which was approximately 2.5. This implied that, with increased stress, the particle size gradation of all specimens tended to be consistent with only moderate amounts of particle breakage. This was because, when the particle breakage reached a certain extent, the particle size gradation of the specimens approximated to its ideal distribution and the specimens were subsequently further compacted. In this situation, the particles made sufficient contact with each other. Meanwhile, the change of particle shape after breakage inhibited further particle breakage. In addition, the close contact between particles hindered their rearrangement and the overall deformation of the specimens increased slowly thereafter.

5. Conclusions The authors conducted basic mechanical property-measuring experiments on intact and crushed gangue after compaction respectively. The relationship between compaction strain and stress for the specimens with their differing lithologies and particle size gradations was obtained. Moreover, the gradation fractal characteristics of specimens with different lithologies and particle size gradations at varied stress levels were analyzed. The conclusions were: (1) Compaction strain-stress curves rose rapidly over the range 0 to 2 MPa, and even faster from 2 to 5 MPa; while they increased slowly from 5 to 10 MPa, and tended to be smooth at stresses N 10 MPa. Particle size gradation exerted little influence on the

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Fig. 7. Particle size grading curves of sandy mudstone before and after compaction: (a) Manual crushed gradation of 2.5 to 50 mm; (b) Uniformly mixed specimens of 2 to 16 mm; (c) Uniformly mixed specimens of 20 to 31.5 mm; (d) Uniformly mixed specimens of 31.5 to 50 mm.

compaction strain when crushed sandstones; the compaction deformations when crushed sandy mudstone and coal samples were mostly affected by particle size gradation. For the two specimens with different lithologies, the compaction deformation of the uniformly mixed specimen comprising a manually crushed specimen and that with a particle size of b16 mm was smaller than that of the 20 mm down sample.

Fig. 8. Variation of fractal dimension of sandstone particle size gradation with stress.

(2) After being compressed, the particle size gradation of each crushed specimen displayed fractal characteristics. The particle size gradation of the specimens with two different lithologies showed fractal characteristics merely after manual crushed. The fractal dimension of all specimens ranged from 0.352 to 2.654. (3) With increased stress, the particle sizes of each specimen tended to be less uniform (i.e. more widely graded). Meanwhile, the

Fig. 9. Variation of fractal dimension of sandy mudstone particle size gradation with stress.

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fractal dimension D increased with increasing small particle content, and it tended to a definite value. Additionally, it decreased with increasing rock strength for the same initial particle size gradation and stress.

Acknowledgements This research was supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (51421003) and the Foundation for Distinguished Professor of Jiangsu Province (2015(29)). References [1] N. Adibee, M. Osanloo, M. Rahmanpour, Adverse effects of coal mine waste dumps on the environment and their management, Environ. Earth Sci. 70 (2013) 1581–1592. [2] G.W. Fan, D.S. Zhang, X.F. Wang, Reduction and utilization of coal mine waste rock in China: a case study in Tiefa coalfield, Resour. Conserv. Recy. 83 (2014) 24–33. [3] K.H. Zheng, C.L. Du, J.P. Li, B.J. Qiu, D.L. Yang, Underground pneumatic separation of coal and gangue with large size (≥50 mm) in green mining based on the machine vision system, Powder Technol. 278 (2015) 223–233. [4] J.X. Zhang, N. Zhou, Y.L. Huang, Q. Zhang, Impact law of the bulk ratio of backfilling body to overlying strata movement in fully mechanized backfilling mining, J. Min. Sci. 47 (2011) 73–84. [5] M. Junker, H. Witthaus, Progress in the research and application of coal mining with stowing, Int. J. Min. Sci. Tech. 23 (2013) 7–12.

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