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Morphological Characterization of Superfine Pulverized Coal Particles. 1. Fractal Characteristics and Economic Fineness Jiaxun Liu,*,† Xiumin Jiang,‡ Xiangyong Huang,‡ and Shaohua Wu† †

School of Energy Science and Engineering, Harbin Institute of Technology, West Straight Street, Harbin 150001, China and ‡ School of Mechanical Engineering, Shanghai Jiao Tong University, Minhang District, Shanghai 200240, China Received August 31, 2009. Revised Manuscript Received October 20, 2009

Superfine pulverized coal combustion is a new pulverized coal combustion technology which has better combustion stability, higher combustion efficiency, lower NOx and SO2 emissions, and higher comprehensive efficiency than when using conventional particle sizes. In this paper, we novelly applied both the classical single power law fractal dimension and a piecewise approach to analyze the particle-size distribution (PSD) of superfine pulverized coal particles. In addition, we introduced the fractal theory into scanning electron microscopy image analysis by adopting the slit island method. The grey relational analysis was used to study the degree of relative importance of the influencing factors about fractal dimensions. Furthermore, the piecewise characteristics of the PSD of coal particles were studied in detail. All curves on the log-log scale can be approximated by two intersecting lines, and each curve can be quantified by four variables. The grey relational analysis was also used for further study on the relationships between the four parameters. Finally, a new method for identifying the economic granule size of pulverized coal particles, that is, economic fineness based on the power consumption of coal mills, E2, was proposed by a utilizing neural network method. Final results indicate that the economic fineness of Shenhua pulverized coal particles based on E2 is about 18.94, while that of Neimenggu pulverized coal particles is about 36.13. This provides some reference for a coal-fired power plant to confirm the economic granule size, which has certain guidance meaning for economical operation and low power consumption.

was found that the technique of superfine pulverized coal particle combustion has many advantages, such as better stability, higher combustion efficiency, lower NOx and SO2 emission, and higher comprehensive efficiency than when using conventional particle sizes. The physical structure of pulverized coal particles is an important factor for the coal combustion process because of its significant influence on the heat/mass transfer rate and reaction surfaces. The fractal concept was originally introduced by Mandelbrot (1967) and has since been generalized for describing the geometrical properties of irregular settings or fragments.13 Fractal theory is one of the methods of nonlinear mathematics and has become increasingly popular in social and natural science as a means for characterizing intricate phenomena. Fractal theory makes it easy to quantitatively describe complex self-similar geometry, which is difficult with Euclidian geometry.14 Sun et al.14 introduced fractal theory into scanning electron microscopy (SEM) image analysis and utilized the particle surface fractal dimension (Dps) and internal surface fractal dimension (Ds) to quantitatively describe the fractal character of coal and char particles. The results show that Dps increases at first and then decreases with the increase of the carbon burnoff ratio, while Ds shows a different development trend which can reflect the character of the inner pore. Nakagawa et al.15 studied the change of fractal properties of different

1. Introduction 1

Coal is the major energy resource in China. Coal accounts for more than 70% of the total energy consumption, and emissions from coal combustion are the major anthropogenic contributors to air pollution in China.2 The proposal of superfine pulverized coal particle combustion provided a new way to understand the effect of particle size on combustion.3,4 Studies5-9 show that pulverized coal particle size and its distribution influence the ignition temperature, char burnout, flame stability, NOx and SOx emission, and comprehensive operation costs. Through our previous research,10-12 it *To whom correspondence should be addressed. Tel.: þ86 21 3420 6052. Fax: þ86 21 3420 5521. E-mail: [email protected]. (1) Li, Y. W.; Zhao, C. S.; Wu, X.; Lu, D. F.; Han, S. Korean J. Chem. Eng. 2007, 24, 319–327. (2) Chan, C. K.; Yao, X. H. Atmos. Environ. 2008, 42, 1–42. (3) Nakamura, M.; Takashi, K.; Kuwahara, M.; Watanabe, H.; Kitamura, R.; Tanaka, T. Int. Conf. Power Eng. 1997 1997, 2, 453–458. (4) Jiang X. M.; Li J. B.; Qiu J. R. Proc. CSEE 2000, 20, 71-74 (in Chinese). (5) Wigley, F.; Williamson, J.; Gibb, W. H. Fuel 1997, 76, 1283–1288. (6) Nugroho, Y. S.; McIntosh, A. C.; Gibbs, B. M. Fuel 2000, 79, 1951–1961. (7) Man, C. K.; Jacobs, J.; Gibbins, J. R. Fuel Process. Technol. 1998, 56, 215–227. (8) Murillo, R.; Navarro, M. V.; Lopez, J. M.; Garcia, T.; Callen, M. S.; Aylon, E.; Mastral, A. M. J. Anal. Appl. Pyrolysis 2004, 71, 945–957. (9) Kucuk, A.; Kadoglu, Y.; Gulaboglu, M. S. Combust. Flame 2003, 133, 255–261. (10) Jiang, X. M.; Zheng, C. G.; Yan, C.; Liu, D. C.; Qiu, J. R.; Li, J. B. Fuel 2002, 81, 793–797. (11) Jiang, X. M.; Zheng, C. G.; Qiu, J. R.; Li, J. B.; Liu, D. C. Energy Fuels 2001, 15, 1100–1102. (12) Zhang, C. Q.; Jiang, X. M.; Wei, L. H.; Wang, H. Energy Conv. Manag. 2007, 48, 797–802. r 2009 American Chemical Society

(13) Lu, P.; Jefferson, I. F.; Rosenbaum, M. S.; Smalley, I. J. Eng. Geol. 2003, 69, 287–293. (14) Hu, S.; Li, M.; Xiang, J.; Sun, L. S.; Li, P. S.; Su, S.; Sun, X. X. Fuel 2004, 83, 1307–1313. (15) Nakagawa, T.; Komaki, I.; Sakawa, M.; Nishikawa, K. Fuel 2000, 79, 1341–1346.

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coals with the help of SAXS and proved that the inner pore network of coal has fractal characteristics. Furthermore, it was found that Ds greatly decreased with an increase of temperature, which was presumed to be caused by the mobile components liberated from the cohesion structure of coal. Friesen and Mikula16 described the measurements of the pore volume of a number of coal and char samples using mercury intrusion porosimetry and concluded that the fractal dimensions of the porous surfaces range from 2.6 for some coals to 3 for the chars. Xu et al.17 obtained the fractal dimensions of the coal micropore surface using gas adsorption data, which are in the admissible range 2-3. Mahamud and Novo18 successfully employed the method proposed by Friesen and Mikula, the procedure of Neimark, and the methodology of Zhang and Li to analyze mercury porosimetry data from a fractal perspective. Final results showed that the oxidation of the pore surface tends to lower the fractal dimension while the access of mercury to previously nonaccessible regions tends to increase this dimension. Fractal methodology is also applied to other techniques of textural characterization such as transmission electron microscopy, SEM, or optical microscopy image analysis. Physical adsorption of gases and SAXS are also used for fractal determinations.19 From the brief review above, we notice that the fractal properties of coals have long been a research subject. However, most the studies focused on the surface characteristics, especially on the micropore structure of coal. It is known that pulverized coal particle size has a great impact on the performance of coal combustion. However, there has not been much research on the fractal characteristics of pulverized coal, not to mention superfine pulverized coal particles. In recent years, the formalism of fractal geometry has attracted great attention as a powerful tool for describing various complex natural phenomena, especially in the mechanics and physics of rocks and soils.20 Particle-size distribution (PSD) is of paramount importance in understanding soil physical properties, as it can be used to estimate soil hydraulic properties such as the water-retention curve and saturated or unsaturated conductivities. There has been much research on this subject. Prosperini and Perugini21 analyzed the particlesize distributions by applying fractal geometry methods, and results indicated that two fractal scaling domains are identified, characterizing different ranges of particle sizes. In detail, regarding D1 (for smaller particle sizes), values range from 2.507 to 2.958, whereas D2 is between 2.913 and 2.999. The fitted values of fractal dimensions obey in all cases the relation D1 < D2. Gimenez et al.22 reviewed several different fractal models for the soil-water retention curve and the hydraulic conductivity-water content function. Experimental evidence suggested that the morphology of soil structure is fractal within some scale limits, but there is no clear understanding of the relationship among different kinds of fractal dimensions of soil structure. Xiao et al.23 investigated the particlesize distribution of interlayer shear zone material and its

Figure 1. PSD of SH bituminous coal specimens.

implications in geological processes. The results showed that all curves on the log-log scale can be approximated by two intersecting lines. It was found that the value of the breaking point implies the degree of subsequent secondary geological processes, and the larger the value of D1 is, the higher the degree of interlayer shear experienced. Lu et al.24 found the particle-size distribution of fine particles subjected to comminuting in the laboratory statistically exhibited fractal behavior in which the dimension lied within 2.2-2.6 and suggested a multifractal character. Millan et al.25 adopted a segmented fractal model (piecewise) to evaluate for detecting more than one scaling domain within particle-size distribution which is commonly used in soil science studies. They draw the conclusion that fractal domains for PSD were within the ranges 0-0.2 mm and 0.2-2 mm, for which D1 correlated with soil clay content while the second domain behaved as fractal for clay loam, loam, and sandy soils. More and more studies have been conducted on piecewise characteristics of PSD, ever since Grout et al.26 proposed multifractal techniques as a promising alternative to a single fractal dimension in 1998. In this paper, we novelly applied both the classical single power law fractal dimension and a piecewise approach to analyze the PSD of superfine pulverized coal particles. In addition, we introduced fractal theory into SEM image analysis by adopting the slit island method (SIM). The grey relational analysis (GRA) was used to study the degree of relative importance of the influential factors about fractal dimensions. The grey relational analysis was also used for further study on the relationships between the four parameters. Finally, a new method for identifying the economic granule size of pulverized coal, that is, economic fineness, was proposed through analyzing Yc (the breaking points of piecewise fractal model) and the mean particle size of the coal samples with the help of a neural network method. 2. Experimental Section Shenhua (SH) and Neimenggu (NMG) bituminous coals of China were chosen for the experiments. The coal samples were pulverized into eight different mean particle sizes using a jet mill. And then, the particle-size distributions of the particles were analyzed using a Malvern MAM5004 Laser Mastersizer made in

(16) Friesen, W. I.; Mikula, R. J. J. Colloid Interface Sci. 1987, 120, 263–271. (17) Xu, L. J.; Zhang, D. J.; Xian, X. F. J. Colloid Interface Sci. 1997, 190, 357–359. (18) Mahamud, M. M.; Novo, M. F. Fuel 2008, 87, 222–231. (19) Mahamud, M. M. Appl. Surf. Sci. 2007, 253, 6019–6031. (20) Borodich, F. M. J. Mech. Phys. Solids 1997, 45, 239–259. (21) Prosperini, N.; Perugini, D. Geoderma 2008, 145, 185–195. (22) Gimenez, D.; Perfect, E.; Rawls, W. J.; Pachepsky, Y. Eng. Geol. 1997, 48, 161–183. (23) Xiao, Y. X.; Lee, C. F.; Wang, S. J. Eng. Geol. 2002, 66, 221–232.

(24) Lu, P.; Jefferson, I. F.; Rosenbaum, M. S.; Smalley, I. J. Eng. Geol. 2003, 69, 287–293. (25) Millan, H.; Gonzalez-Posada, M.; Aguilar, M.; Dominguez, J.; Cespedes, L. Geoderma 2003, 117, 117–128. (26) Grout, H.; Tarquis, A. M.; Wiesner, M. R. Environ. Sci. Technol. 1998, 32, 1176–1182.

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Figure 3. Single power law fractal dimensions of SH PSD.

Figure 2. PSD of NMG bituminous coal specimens.

structure. Three different methods have been used in this paper to calculate fractal dimensions on the basis of single power law fractal theory, a piecewise approach and the slit island method. 3.1. Single Power Law Fractal Dimension Describing PSD. Mandelbrot (1982)28 suggested that fractal fragmentation could be quantified by measuring the fractal dimension through the equation

Table 1. Mean Diameter of Tested Coal Samples coal samples SH NMG

mean diameter (μm) 14.705 12.561

17.439 14.999

21.3 25.862

44.264 52.778

Table 2. Ultimate and Proximate Analyses of the SH and NMG Coals proximate analysis (wt %) (ad)

ultimate analysis (wt %) (ad)

SH

moisture (mass %) volatile (mass %) ash (mass %) Fixed carbon (mass %)

11.5 24.22 10.7 53.58

NMG

moisture (mass %) volatile (mass %) ash (mass %) Fixed carbon (mass %)

14.72 35.69 10.64 38.95

C H O N S C H O N S

Nðr > RÞ ¼ CR -D 63.13 3.62 9.94 0.70 0.41 54.82 4.39 14.58 0.63 0.22

ð1Þ

where D is the fragmentation fractal dimension, N(r > R) is the total number of particles with a linear dimension r that is greater than a given comparative size, R, and C is a proportionality constant. The pore solid fractal (PSF) model presents symmetry between distributions of pores and solids.29 Bird et al.30 made use of the cumulative solid mass distribution of the PSF model and applied it to PSD: Ms ðdedi Þ ¼ cdi 3 -Ds

the U. K., and the results are shown in Figures 1 and 2. The equivalent mean particle sizes of SH samples were 14.705, 17.439, 21.3, and 44.264 μm, while those of NMG samples were 12.561, 14.999, 25.862, and 52.778 μm, which are listed in Table 1. The properties of the coals are presented in Table 2. The ultimate analysis data were obtained on a LECO CHN 600 (America) and a sulfur analyzer, and then the oxygen content was obtained by difference. The proximate analysis was done on a LECO MAC 500 (U. S. A.). From the proximate analysis, it can be inferred that both SH and NMG coal samples are medium volatile bituminous coals with common ash. In addition, from the ultimate analysis, it is indicated that the low levels of sulfur and nitrogen make them particularly desirable for power generation and the synthesis of clean conversion products. The surfaces of coal specimens were mapped using a scanning electron microscope (HITACHI S-2150, Japan) to show the microstructure characteristics. The digital images were collected when the system was working stably under conditions of 15 keV accelerating voltage and 1000 magnification.

ð2Þ

where di is the upper limit diameter of the particles, Ms (d e di) represents the cumulative mass of elements, Ds is the fractal dimension, and c is a composite scaling constant. Logarithmic transformation of eq 2 results in a linear relationship between Ms (d e di) and di for a scale-invariant (i.e., fractal) PSD: log½Ms ðdedi Þ ¼ k log di þ log c

ð3Þ

If there exists a linear relationship between Ms (d e di) and di in double logarithmic coordinates, then it suggests that the PSD of particles is fractal. Assuming that the slope of the line is k, the fractal dimension can be expressed as ð4Þ Ds ¼ 3 - k The single power law fractal dimension of the PSD can be obtained according to eq 3, and the results are displayed in Figures 3 and 4. The Ds for each coal sample was calculated by determining the slope of the best-fit line through the data points using linear regression. It shows that the correlation coefficients of the lines are all greater than 0.9, and it the linearity of fit can be considered good.31 Therefore, it was

3. Results and Discussion Superfine pulverized coal has a complex surface structure. The routine parameters of the pulverized coal physical structure such as the microporosity, specific surface area, porosity ratio, and pore volume cannot easily describe the surface characteristics objectively and comprehensively.27 Fractal dimensions can give useful information about the pulverized coal physical

(28) Mandelbrot B. B. The Fractal Geometry of Nature; Freeman: New York, 1982. (29) Perrier, E.; Bird, N.; Rieu, M. Geoderma 1999, 88, 137–164. (30) Bird, N. R. A.; Perrier, E.; Rieu, M. Eur. J. Soil Sci. 2000, 51, 55– 63. (31) Jiang X. M.; Yang H. P.; Li Y.; Liu H. J. China Coal Soc. 2003, 28, 414-418 (in Chinese).

(27) Liang D.; Wang Y. H.; Xiao S. H. J. HeiLongjiang Inst. Sci. Technol. 2007, 1 8-10 (in Chinese).

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Figure 4. Single power law fractal dimensions of NMG PSD. Table 3. Fractal Dimensions Calculated from Power Law Theory SH_14.705 SH_17.439 SH_21.3 SH_44.264 NMG_12.561 NMG_14.999 NMG_25.862 NMG_52.778

fractal dimensions (Ds)

correlation coefficients (R2)

2.25946 2.24008 2.16622 2.12553 2.23868 2.14945 2.09861 2.0364

0.909 0.921 0.948 0.96 0.911 0.939 0.963 0.979

Figure 6. Surface structure of SH bituminous coal particles.

Figure 7. Surface structure of NMG bituminous coal particles. Figure 5. Influence of the coal particle size on the fractal dimensions.

dimensions will be. However, no such trend can be observed between the different coals. For example, the fractal dimension of SH_14.705 is larger than that of NMG_12.561. This is because the fractal dimension is also related with coal qualities such as fixed carbon and volatility, as explained in detail in the grey relational analysis section. On the other hand, the value of Ds can also reflect the irregularity of a granular system.24 The higher the value of Ds is, the more regular the particle-size distribution appears. Otherwise, the more irregular it tends to be. This can be seen clearly from Figures 1 and 2. 3.2. Single Fractal Dimension of Coals Applying the Slit Island Method. Mandelbrot first applied the slit island method in 1984 when the fracture toughness of 300-grade maraging steel was studied and the fractal dimension was correlated with the impact energy. Mandelbrot derived a perimeter-area relationship when applied to fractured surfaces, which is called the ‘‘slit island method’’.34

concluded that the single power law fractal theory was applicable. In each figure, linear equations inferred from eq 3 are given for every coal sample, and from the slopes of these equations, the corresponding fractal dimensions are calculated by eq 4, which are summarized in Table 3 with the R2 values. From Table 3, it is revealed that all of the fractal dimensions range from 2.04 to 2.26, which is consistent with the former studies32,33 in which, usually, from comminution experiments, the values belong to the interval 2 < Ds < 3 and only infrequently fall outside of said interval. Furthermore, the fractal dimension decreases with the increase of the pulverized coal particle size, see Figure 5. It indicates that the more fine particles the sample contains, the higher the degree of the particle fragmentation and the larger the fractal (32) Carpinteri, A.; Pugno, N. Geomech. 2002, 26, 499–513. (33) Carpinteri, A.; Pugno, N. Powder Technol. 2003, 131, 93–98.

(34) Bigerelle, M.; Iost, A. Eng. Fract. Mech. 2004, 71, 1081–1105.

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Figure 8. Fractal dimension of SH bituminous coals using the SIM method.

The relation between perimeter and area is given by eq 5:

The image processing system of particle measurement was built applying the MATLAB software. Image segmentation is one of the most important steps during particle image recognition and analysis processing. There are hundreds of image segmentation algorithms, and two of them, namely thresholding and edge detection, are usually used. In this paper, the edge detection method was adopted. There are mainly three steps in the edge detection algorithm, which are smoothing the input image, regulating the gradient of the smooth image, and extracting the image edges. First, the Photoshop software was introduced to realize image preprocessing such as image denoising, intensity, and binarization processing. The coal surface images from SEM were loaded into the Photoshop software. After each image was calibrated, a grey threshold was selected to clearly distinguish the coal particles from the background by changing the grey scale graph range from 0 to 255, that is, binarization processing.14,35 Then, the treated images were calculated by the segmentation algorithms, which were implemented by utilizing MATLAB language. In the end, after this particle image recognition process, the software of IMAGE-PRO was used to analyze the information such as the perimeter and the area of the particles in the binary images converted from Figures 6 and 7. The surface fractal dimension based on SIM can be inferred according to eq 7, and the results are shown in Figures 8 and 9. The Ds for each coal sample was obtained by

1=D

½PðηÞ RðηÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ AðηÞ

ð5Þ

where R(η) is a constant depending on the choice of the yardstick length, η is used to measure the length along the walking path, and P(η) and A(η) represent the perimeter and the area of islands separately. When islands are derived from an initial self-affined fractal surface of dimension Ds by sectioning with a plane, their coastlines are self-similar fractals with dimension D = Ds - 1. Equation 5 shows that there exists a power law relationship between the perimeter and the area of the island: ½PðηÞ1=D µ ½AðηÞ1=2

ð6Þ

Logarithmic transformation of the perimeter area relation, eq 6, could be expressed in the following form: logðPÞ ¼ k logðAÞ þ β

ð7Þ

where β is a constant. When the graph of log(P) versus log(A) is rectilinear, the fractal dimension can be deduced from the slope: ð8Þ Ds ¼ 2k þ 1 The Hitachi S-2150 SEM was used to analyze the microstructure of the coal samples. The surface structures of the eight bituminous coal samples are illustrated in Figures 6 and 7. It is obvious that the larger the mean diameter of the coals is, the more large particles there are.

(35) Gosiewska, A.; Drelich, J.; Laskowski, J. S.; Pawlik, M. J. Colloid Interface Sci. 2002, 247, 107–116.

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Figure 9. Fractal dimension of NMG bituminous coals using the SIM method.

determining the slope of the best-fit line through the data points using linear regression. It shows that the correlation coefficients of the lines are all larger than 0.95, which means that the goodness-of-fit can be considered excellent.36 In each figure, linear equations obtained by eq 7 were given for every coal sample, and from the slopes of these equations, the corresponding fractal dimensions are calculated according to eq 8. To minimize the artificial errors, four images under the same 1000 magnification were taken for each coal sample at different locations. Then, the average of the four values calculated from the images using the same method was taken as the fractal dimension of that coal sample. The standard deviation values were also recorded to evaluate the errors, of which the largest was only 0.012. Hence, the deviations of the fractal dimensions between the four images were very small. The surface fractal dimensions obtained are summarized in Table 4 with their R2 and standard deviation values. Figure 5 shows that the fractal dimensions decrease obviously with the increase of the pulverized coal particle size. Therefore, from the above SIM analysis, the same conclusion will be drawn. Furthermore, compared with that utilizing single power-law theory, the higher R2 values indicate that SIM is a better method to describe particle size distributions of coals. 3.3. Grey Relational Analysis of the Factors Influencing the Fractal Dimension. The grey system theory proposed by Deng37

Table 4. Fractal Dimensions Calculated from SIM correlation fractal dimensions (Ds) coefficients (R2) SH_14.705 SH_17.439 SH_21.3 SH_44.264 NMG_12.561 NMG_14.999 NMG_25.862 NMG_52.778

2.484 2.457 2.244 2.2125 2.492667 2.456 2.243 2.232

0.961 0.962 0.977 0.97 0.963 0.955 0.973 0.974

standard deviation (std dev) 0.011269 0.003536 0.002828 0.003202 0.01159 0.007071 0.00495 0.002828

has been widely applied to various fields.38,39 It has been proven to be useful for dealing with poor, incomplete and uncertain information. GRA is part of grey system theory, which is suitable for solving problems with complicated interrelationships between multiple factors and variables.40 Grey theory makes use of relatively small data sets and does not demand strict compliance to certain statistical laws, with simple or linear relationships among the observables. It is suitable to apply grey system theory to the identification of influencing factors of fractal dimensions. GRA is a quantitative analysis to explore the similarity and dissimilarity among factors in developing dynamic processes. The theory proposes a dependence to measure the correlation degree of factors; the more similarities develop, (38) Caydas, U.; Hascalik, A. Optics Laser Technol. 2008, 40, 987– 994. (39) Chang, T. C.; Lin, S. J. J. Environ. Manage. 1999, 56, 247–257. (40) Kuo, Y.; Yang, T.; Huang, G. W. Comput. Ind. Eng. 2008, 55 80–93.

(36) Millan, H.; Posada, M. G. Geoderma 2005, 125, 25–38. (37) Deng, J. Syst. Control Lett. 1982, 1, 288–294.

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Table 5. Influencing Factors of Fractal Dimensions

power law theory

SH

NMG

slit island method

SH

NMG

Table 6. Results of Grey Relational Analysis on Fractal Dimensions

fixed carbon

volatile

mean diameter

fractal dimensions

0.5358

0.2422

44.264

2.12553

0.5358 0.5358 0.5358 0.3895 0.3895 0.3895 0.3895 0.5358

0.2422 0.2422 0.2422 0.3569 0.3569 0.3569 0.3569 0.2422

21.3 17.439 14.705 52.778 25.862 14.999 12.561 44.264

2.16622 2.24008 2.25946 2.0364 2.09861 2.14945 2.23868 2.2125

0.5358 0.5358 0.5358 0.3895 0.3895 0.3895 0.3895

0.2422 0.2422 0.2422 0.3569 0.3569 0.3569 0.3569

21.3 17.439 14.705 52.778 25.862 14.999 12.561

2.244 2.457 2.484 2.232 2.243 2.456 2.49267

power law theory slit island method

volatile (X2)

mean diameter (X3)

sequence

0.8636

0.7914

0.6369

X1 >X2 >X3

0.8680

0.8324

0.6478

X1 >X2 >X3

where k1 and k2 are the slopes of the first and the second domains, respectively, and log dc is the cutoff of the whole domain. The fractal dimensions of the piecewise model can be expressed as D1 ¼ 3 - k1 D2 ¼ 3 - k2

ð10Þ

3.4.2. Quantification of PSD Curves Applying Piecewise Fractal Model. From the geometrical point of view, four quantities k1, k2, Yc, and dc are required to quantify a PSD curve, which correspond respectively to the slopes of the two lines and the coordinates of the intersection. The slopes and intersection are automatically determined through the leastsquares technique and solving equations for the two lines. On the basis of eq 10, the slopes k1 and k2 are replaced respectively by the fractal dimensions D1 (the primary fractal fragmentation dimensions) and D2 (the secondary fractal fragmentation dimensions). The PSD curves of all of the coal samples applying the piecewise fractal model are shown in Figures 10 and 11. In these figures, the piecewise linear regression plots clearly show there are two sets of powerlaw functions which can be fitted across data points, indicating two fractal dimensions calculated from the two separate regression equations for different ranges of particle sizes. Table 7 summarizes the results of fitting eq 9 to the piecewise fractal model in which Δ represents D1 - D2. 3.4.3. Grey Relational Analysis of the Factors Influencing Piecewise Fractal Dimensions. The influencing factors of the piecewise fractal dimensions were analyzed by the grey relational analysis method. The parameters of fixed carbon and volatility were also chosen to represent the quality of coal species. The fractal dimensions obtained by the piecewise fractal model D1 and D2 were affirmed as a reference sequence, and fixed carbon, volatility, mean diameter of the coal particles, and the values of breaking points Yc and dc were affirmed as comparison sequences. An equalization approach was used in processing the data to obtain a dimensionless eigenvector matrix. The index for distinguishability was set to 0.5, and grey relational sequences considering the influence of fixed carbon, volatility, and mean diameter of the coal particles and the values of breaking points on the piecewise fractal dimensions were obtained, see Table 8. From the order, the degree of the effect of comparison sequences on the reference sequence can be worked out. On the basis of the above calculation results, Yc variables markedly influence both fractal dimensions for the whole ranges of particle sizes. On the other hand, the influence of the coal particle size is not negligible. As can be seen, the fractal dimensions of the piecewise fractal model are also relatively markedly related with coal qualities such as fixed carbon and volatility, just as single power law and SIM fractal dimensions are. So it is concluded that all of the fractal dimensions about the PSD of coal particles are largely dependent upon the components of the coals.

the more factors correlate. It uses the grey relational grade to measure the relational degree of factors.41 The analysis procedure was described in detail in our previous papers.42,43 The influencing factors of fractal dimensions were analyzed by the grey relational analysis, as listed in Table 5. The parameters of fixed carbon and volatility were chosen to represent the quality of coal species. The fractal dimensions obtained by the power law theory and SIM were affirmed as a reference sequence, and fixed carbon, volatility, and mean diameter of the coal particles were affirmed as comparison sequences. An equalization approach was used in processing the data mentioned in Table 5 to obtain a dimensionless eigenvector matrix. The index for distinguishability was set to 0.5, and the grey relational sequences considering the influence of fixed carbon, volatility, and mean diameter of the coal particles on the fractal dimensions of different methods were obtained, see Table 6. From the order, the degree of the effect of comparison sequences on the reference sequence can be worked out. Generally, grey relational-grade X > 0.9 indicates a marked influence, X > 0.8 a relatively marked influence, X > 0.7 a noticeable influence, and X < 0.6 a negligible influence.44 On the basis of above calculation results, fixed carbon variables relatively markedly influence fractal dimensions obtained by both power law theory and SIM. On the other hand, the influence of the coal particle size is not negligible. 3.4. Fractal Dimension of Coals Applying the Piecewise Approach. 3.4.1. Piecewise Fractal Model. On the basis of the natural organization of PSD, it can be assumed that there could be two intervals of particle size, over which the values of Ds could be different and a critical value dc could also exist.25 The followingequation, eq 9, suggests the existence of two power laws separated by a breaking point, log dc. It might suggest that log-transformed data would yield two straight lines separated by a cutoff, log dc. log½Ms ðdedi Þ ¼ ½k1 log di þ log m1 ðlog di glog dc Þ þ ½k2 log di þ log m2 ðlog di elog dc Þ

fixed carbon (þX1)

ð9Þ

(41) Kung, C. Y.; Wen, K. L. Decis. Support Syst. 2007, 43, 842–852. (42) Han, X. X.; Jiang, X. M.; Liu, J. G.; Wang, H. Oil Shale 2006, 23, 99–109. (43) Wang, H.; Jiang, X. M.; Liu, J. G.; Lin, W. G. Energy Fuels 2007, 21, 1924–1930. (44) Fu, C. Y.; Zheng, J. S.; Zhao, J. M.; Xu, W. D. Corros. Sci. 2001, 43, 881–889.

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Figure 10. Fractal dimension of SH bituminous coals applying a piecewise approach.

1.88 and close to 2. These results are consistent with the findings of former studies.33,45 It means that the energy dissipation in the comminution process occurs in a fractal domain intermediate between a surface and a volume: For finer particles created by milling, the fractal exponent D is close to 2, while for larger particles created by cutting, it is close to 3. This may suggest that two different breakage mechanisms are at work in the fragmentation process for producing fine and coarse particles.32 We will discuss this part about the breakage mechanisms for the superfine pulverized coal particles in detail in another paper. Relationships between coal particle size and piecewise fractal dimensions are presented in Figure 13, which is different from the trends in Figure 5. This is because of the difference of mechanism between the single and piecewise fractal dimensions. Figure 13 clearly shows that, with the increase of the particle size, the primary fractal fragmentation dimensions decrease while the secondary fractal fragmentation dimensions increase, this trend being opposite that of single fractal dimensions. It has been known that the energy dissipation for larger particles is substantially in the volume (D1 close to 3), and for smaller particles it is substantially on the surface (D2 close to 2). With the increase of the mean particle size, there are more large-scale particles in the secondary fractal dimension dominant regions, which results in a D2 closer to 2. In other words, the trend of D2 is going up. In the comminution process, more energy dissipation takes place on the surfaces of the particles with the

The grey relational analysis results suggest that the breaking point Yc has a great influence on the piecewise fractal dimension. The relationship between the secondary fractal dimensions and the breaking points are shown in Figure 12. It reveals that there is a quadratic polynomial relationship between D2 and Yc for both SH and NMG coals. The correlation coefficients of the fitted lines are all larger than 0.95, and the goodness of fit can be considered excellent. However, it is not known, from a mechanical point of view, why there exists this kind of relationship between Ds and Yc, which demands further investigations. 3.4.4. Interpretation of the Piecewise Fractal Dimensions. A quantitative understanding of the evolution of particle size distribution is a necessary step for appropriate process control and optimization of the comminution operation.45 According to the piecewise fractal model, it is possible to partition the PSD into two subsets such that each subset could be fractal with its own scaling exponent. This bilinearity emphasizes the two distinct comminution mechanisms and provides two fractal exponents, D1 (for larger particles, primary comminution) and D2 (for smaller particles, secondary comminution).32 Table 7 shows that the primary fractal fragmentation dimensions (D1) of the coal particles are greater than 2.5 (between 2.70 and 2.89) and close to 3, while the secondary fractal fragmentation dimensions (D2) are between 1.61 and (45) Tasdemir, A. Miner. Eng. 2009, 22, 156–167.

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Figure 11. Fractal dimension of NMG bituminous coals applying a piecewise approach. Table 7. Regression Analyses of PSD for Piecewise Fractal Methods SH

NMG

mean diameter

D1

D2

D1 - D2

dc

Yc

R1 2

R22

14.705 17.439 21.3 44.264 12.561 14.999 25.862 52.778

2.84123 2.88645 2.72265 2.70294 2.87728 2.698 2.79027 2.79362

1.60997 1.67856 1.7863 1.84383 1.69433 1.71734 1.8263 1.88356

1.23126 1.20789 0.93635 0.85911 1.18295 0.98066 0.96397 0.91006

10.67271 16.33554 26.27534 29.13141 12.45098 20.06869 36.45806 63.66901

0.700733 0.794417 0.77744 0.583411 0.790892 0.808441 0.847128 0.8

0.98243 0.95133 0.93512 0.97387 0.94655 0.93683 0.92873 0.95371

0.99616 0.99812 0.99777 0.99394 0.99755 0.99935 0.99876 0.99275

Table 8. Results of Grey Relational Analysis on Piecewise Fractal Dimensions D1 D2

fixed carbon (X1)

volatile (X2)

mean diameter (X3)

dc (X4)

Yc (X5)

sequence

0.8140 0.7939

0.7834 0.7896

0.6437 0.6491

0.6620 0.6766

0.9071 0.9054

X5 > X1 > X2 > X4 > X3 X5 > X1 > X2 > X4 > X3

increase of D2. Therefore, less energy dissipation occurs in the primary fractal dimension dominant regions; that is, the fractal dimension D1 will deviate from 3. In other words, the trend of D1 is going down. Considering the trends of D1 and D2, it is apparent that Δ (D1 - D2) decreases with the increase of the particle size. 3.5. Analysis of the Economic Granule Size of Pulverized Coal. It has a certain guidance value for safe and economical operation and low-energy consumption for coal-fired power plants to confirm the economic granule size of pulverized coal, that is, the economic fineness of pulverized coal. 3.5.1. New Definition of the Economic Fineness of Pulverized Coal Particles. Assume the expenses of heat loss due to

unburned fuels of boilers are E1, the power consumptions of coal mills are E2, and the abrasions of metallic materials are E3. Then, eq 11 is defined as f ðxÞ ¼ E1 þ E2 þ E3

ð11Þ

The traditional definition: If there exists a particle size x, making f(x) the smallest, then the corresponding x is defined as the economic fineness of pulverized coal. But in the definition, the damage caused by pollutants such as dusts, SO2, and NOx and the problems of stability, ash deposition, slagging, abrasion, and corrosion in the combustion process are not taken into account. In addition, with the explosion of oil prices, the expense of oil utilized for ignition and the 852

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account to evaluate the economic granule size of pulverized coal, which is defined as the economic fineness of pulverized coal based on E2. Further investigations will be carried out considering more influencing factors in the future. As it is known, the mean particle size of the coal sample is proportional to the power consumption of coal mills, E2; that is, the smaller the mean particle size is, the more power consumption of coal mills there will be. Furthermore, it is more convenient to confirm the economic granule size of pulverized coal by applying mean particle size. Otherwise, a further step will be needed to convert E2 to the corresponding mean particle size. So, in this work, the mean particle size is used to represent the power consumption of coal mills. 3.5.3. Identification of the Economic Fineness of Pulverized Coal Particles. In this part, we utilize the piecewise fractal dimension combined with a neural network method to identify the economic fineness of pulverized coal particles. Artificial neural networks are a new information processing system which models and spreads the intellectual functionality of humankind that has been widely accepted as a technology that offers an alternative way to simulate complex and ill-defined problems.46 At present, the most extensively adopted algorithm for the learning phase is the back propagation (BP) one, which is a generalization of the steepest descent method.47 The typical BP network consists of an input layer, an output layer, and at least one hidden layer. Each layer contains neurons, and each neuron is a simple microprocessing element which receives and combines signals from many neurons via weighted connections and is connected to all of the neurons in the next layer. In this work, a BP network model was used to analyze the economic fineness of pulverized coal particles. The dimensions of the input layer and the output layer in the BP network structure are designed according to the main purpose that the breaking points Yc of piecewise fractal dimensions can be predicted at random mean particle sizes. That is to say, the breaking point Yc is the output part, and the mean particle size is the input part. Thus, the input layer consists of one input neuron corresponding to the mean particle size, the transfer function of which is a tangent sigmoid transfer function (tansig). The output layer is composed of eight output neurons corresponding to eight kinds of coal samples, of which the transfer function is a linear transfer function (purelin). An important step in the modeling process is a determination of the adequate number of neurons in the hidden layer. The optimal number of neurons in the hidden layer was determined by varying the number of hidden neurons and observing the root-mean-square error between the experimental results and the calculated output of the BP network.48 The number of neurons used for the hidden layer is optimized by trial-and-error training assays, and it is confirmed that choosing 11 hidden neurons can make the network model move toward convergence in a short time. The transfer function of the hidden neurons is tansig. A three-layer network model for the primary fragmentation of oil shale particles has been set up. The input layer of the model has one neuron, the hidden layer has 11 neurons, and the output layer has eight neurons. The learning rate of the network model is 10-4. Training is carried out by

Figure 12. Relationship between the secondary fractal dimensions and the breaking points.

Figure 13. Relationships between coal particle size and piecewise fractal dimensions.

combustion stability of the boilers cannot be neglected. Assume the expenses of desulfurization and denitrification are E4; the oil consumptions are E5; and ash deposition, slagging, abrasion, and corrosion are E6. Then f(x) is defined as f ðxÞ ¼ E1 þ E2 þ E3 þ E4 þ E5 þ E6

ð12Þ

If there is x, making f(x) the smallest, then x is defined as the economic fineness of pulverized coal. This is the new definition of the economic granule size of pulverized coal.31 3.5.2. Analysis of the Economic Fineness of Pulverized Coal Particles. Superfine pulverized coal combustion has better stability, higher combustion efficiency, lower NOx and SO2 emission, and higher comprehensive efficiency. At the same time, it can reduce the cost of oil utilized for ignition and combustion stability. Furthermore, with a decrease of the coal particle size, the ability for pulverized coal to flow becomes better, which is propitious in preventing the situations of ash deposition, slagging, abrasion, corrosion, and so forth. From the above analysis, it indicates that the technology of superfine pulverized coal combustion can lower the expenses of E1, E3, E4, E5, and E6 at the cost of increasing E2. Consequently, the power consumption of coal mills, E2, is the key point of identifying the economic fineness of pulverized coal. Therefore, in this work, only E2 was taken into

(46) Guo, B.; Shen, Y.; Li, D.; Zhao, F. Fuel 1997, 76, 1159–64. (47) Piron, E.; Latrille, E.; Rene, F. Comput. Chem. Eng. 1997, 21, 1021–1030. (48) Cui, Z. G.; Han, X. X.; Jiang, X. M.; Liu, J. G. Oil Shale 2009, 26, 114–124.

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Table 9. Comparison between Experimental Data and Network Prediction Results SH_14.705 SH_17.439 SH_21.3 SH_44.264 NMG_12.561 NMG_14.999 NMG_25.862 NMG_52.778

experiment data

prediction data

error

0.700733 0.794417 0.77744 0.583411 0.790892 0.808441 0.847128 0.8

0.7014 0.7855 0.7808 0.5838 0.7917 0.8086 0.845 0.8003

0.000952 -0.01122 0.004322 0.000667 0.001022 0.000197 -0.00251 0.000375

Figure 15. Identification of the economic fineness of NMG pulverized coal particles.

particles created by milling and the least energy consumption of the secondary comminution. The greatest Yc could be calculated from the fitted lines, and the corresponding mean particle size is identified as the economic granule size of the pulverized coal. The final results show that the economic fineness of SH pulverized coal particles is about 18.94, while that of NMG pulverized coal particles is about 36.13. Therefore, we can draw the conclusion that the economic fineness of pulverized coal particles which is related to the coal quantity varies with different coals.

Figure 14. Identification of the economic fineness of SH pulverized coal particles.

repeatedly presenting the entire set of training patterns until all error signals between the desired and actual outputs over all of the training patterns are minimized and within the preset training precision (10-9 in this work). In order to verify the validity of the network model, the experimental data of eight different mean particle sizes of both SH and NMG coals are used as check-up samples to compare with the prediction results. The values and errors between them are listed in Table 9, showing that the prediction data agree with the experimental data very well. Then, this trained model is used to predict more breaking points with different mean particle sizes, which are illustrated in Figures 14 and 15. Attention should be drawn to there being quadratic polynomial relationships between mean particle sizes and Yc for both SH and NMG coals. The correlation coefficients of the fitted lines are all larger than 0.90, which means the goodness of fit is good. It is known that Yc log Ms (d e dc) represents the cumulative mass of particles which are smaller than dc. The higher the value of Yc is, the larger the mass fraction of particles smaller than dc is, and thus more fine particles are produced. Therefore, a larger Yc means that there are more small particles created by milling, which is propitious in the combustion of the superfine pulverized coal. On the basis of the above discussions, we proposed that the main comminution process of producing finer particles is milling, in which the energy dissipation is proportional to the surface. However, the energy dissipation in the larger particle comminution process (mainly cutting) is proportional to the volume, which is smaller than that of milling. The finer particles created by milling, that is, the lower the value of Yc is, the larger the energy consumption that will have to be exerted in the comminution process. So it is inferred that the maximum value of Yc corresponds to the largest mass fraction of fine

4. Conclusions On the basis of the experiments and analysis, the following conclusions can be drawn: 1. The single power law fractal dimensions range from 2.04 to 2.26, while the fractal dimensions calculated by SIM range from 2.21 to 2.49. There is a similar trend that the fractal dimensions decrease with the increase of the pulverized coal particle size, which means the more fine particles there are, the larger the fractal dimensions and the higher the degree of particle fragmentation will be. 2. On the basis of grey relational analysis, fixed carbon variables relatively markedly influence fractal dimensions obtained by both power law theory and SIM. And, the influence of the coal particle size is not negligible. 3. A piecewise fractal model is successfully applied to quantify the PSD curves. The piecewise fractal dimensions about PSD of coal particles are largely dependent upon the components of the coals. From the grey relational analysis results, it is known the breaking points have great influence on the piecewise fractal dimensions. A quadratic polynomial relationship exists between the secondary fractal dimensions and the breaking points. 4. The primary fractal fragmentation dimensions of the coal particles are close to 3, while the secondary fractal fragmentation dimensions are close to 2. This may suggest that two different breakage mechanisms are at work in the fragmentation process for producing finer and coarser particles, which is consistent with the findings of former studies. Furthermore, with the increase of the particle size, the primary fractal fragmentation dimensions decrease, while the secondary fractal fragmentation dimensions increase. 854

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5. The piecewise fractal dimensions combined with a neural network method are utilized to identify the economic fineness based on the analysis of power consumptions of coal mills, E2. Final results indicate that the economic fineness of SH pulverized coal particles based on E2 is about 18.94, while that of NMG pulverized coal particles

is about 36.13. Furthermore, a conclusion can be drawn that the economic fineness of pulverized coal particles related to the coal quantity varies with different coals. Acknowledgment. This work was supported by the National Natural Science Foundation of China (50876060).

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