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Percolation Research on Fractal Structure of Coal Char Hu Song,* Sun Xuexing, Xiong Youhui, Xiang Jun, Li Min, and Li Peisheng State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, WuHan, 430074, China Received January 30, 2002
Some parameters about pore network structure of coal char particle are calculated by nitrogen isotherm adsorption/desorption and SEM measured data. On the basis of the correlation between the fractal dimension and the porosity of a porous solid, fractal models are constructed to represent coal particle and char (39.53% burnoff level) particles, respectively. Using the Monte Carlo method, random numbers with the similar distribution as that of pore radii of coal/char particle are assigned to the lattice sites of fractal models. Transport process on the fractal structure of model is studied with the help of invasion percolation simulation. It is shown that D profiles in case of model (a) has two steps. Percolation threshold of fractal model (a) and (b) are 0.425 and 0.430, respectively. Some critical parameters of coal/char particles are calculated.
1. Introduction Martial transportation in particles is an important subject for coal utilization research projects. Most of the studies1,2 have suggested that pore structure relates to the transport character of coal char. As do randomly connected mediums when combined with the solid and porous features of coal, the inner pore structures of coal and char show complex and chaotic characteristics.3,4 It is hardly described by Euclidean geometry. Fractal theory5 is one of the methods of nonlinear mathematics and has become increasingly popular in the social and natural sciences as a means for characterizing intricate phenomena. It has been proved that the inner porous networks6 of coal and char have fractal characteristics, which are reflected in measured data.7,8 In a disordered porous medium such as a char particle, the morphology of the system and its geometry play fundamental roles in the transport and reaction in the system, which are often more important than the roles of other influencing factors.9 Louis and Pereira10 suggested the application of the fractal model (Menger sponge) concept to the porous catalysts to characterize the structural effects of the catalysts. Heping Xie et al.11 found the relationship between bulk fractal dimension and porosity. It is possible for us to construct the fractal model to represent particles of coal and char. * Corresponding author. (1) Sastry, P. U.; et al. Solid State Commun. 2000, 114, 329. (2) Clarkson, C. R.; et al. Fuel 1999, 78, 1333. (3) Nakagawa, T.; et al. Fuel 2000, 79, 1341. (4) Hongwei, Z.; Heping, X.; Kwasniewski, M. A. Nat. Sci. Prog. 2001, 7, 682. (5) Youzhong, R.; Jian, F.; Zhibo, C.; Yuanquan, C. J. Combust. Sci. Technol. 1996, 2, 8. (6) Wang, F. M. Ind. Eng. Chem. Res. 1997, 36, 1598. (7) Oleschko, K.; Fuentes, C.; Brambila, F.; Alvarez, R. Soil Technol. 1997, 10, 185. (8) Nirupamn, S.; Chaudhuri, B. B. Pattern Recognit. 1992, 25, 1035. (9) Sahimi, M.; Gavalas, G. R.; Tsotsis, T. T. Chem. Eng. Sci. 1990, 45, 1443. (10) Louis, L.; Pereira, C. J. Chem. Eng. Sci. 1990, 45, 2027. (11) Hongwei, Z.; Heping, X. Journal of Xi’An Mining Institute 1997, 172, 47.
Transport phenomena in porous medium can be modeled through percolation processes.12 Examples of this range from viscous flow of a liquid phase into a bed of glass beads to the problem of water leakage or oil recovery in a sedimentary rock. In studies of percolation model or microlevel displacement experiments, porous medium has been commonly represented as an ordinary network of pores generated by the random distribution of pore sizes. Invasion percolation (IP) is a dynamical process of growing a single percolated cluster. It was proposed to simulate the displacement of one fluid by the other in a random porous medium at very low and constant injection rates.13 In this regime, the viscous forces prevail over the capillary forces, and the invading fluid follows a path affected by some conditions causing the macropores to be invaded first. Invasion percolation with trapping assumes that the defending fluid is incompressible; therefore, it can be trapped in the porous medium if it is surrounded by the invading fluid. However, in some real problems, the presence of fields or topological features makes the description of the dynamics of the system more complicated than the one given by percolation simulations. Moreover, it was observed that the properties of natural porous medium percolation are highly correlated to fractal characters of porous medium.14 Mandelblort15 first proposed fractal percolation process, and several authors16 had studied it. But few papers about percolation research on fractal model were published. To make the modeling more realistic, some other factors, such as fractal structure and pore radii distribution, were considered in this paper. The aim of this paper is to study molecular transportation process in coal/char particles with the help of (12) Vidales, A. M. Phys. A 2000, 285, 259. (13) Felinto, D.; et al. Phys. A 2001, 293, 307. (14) Hewett, T. Proceedings of the Society of Petroleum Engineering, 58th Annual Technical Conference and Exhibition, October 5-8, 1983, San Franciso. (15) Mandelbrot, B. B. The Fractal Geometry of Nature; W. H. Freeman: New York, 1982. (16) Chayes, L. J. Phys. A 1995, 28, 295-301.
10.1021/ef020003s CCC: $22.00 © 2002 American Chemical Society Published on Web 07/20/2002
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Table 1. Sample Analyses of Huainan Coal
variables N and Nm is represented in the following form:
Proximate Analysis (wt % a.d)
ln(Nd - Nm) Df ) In N
moisture
volatile matter
ash
fixed carbon
2.3
25.7
14.7
57.3
Ultimate Analysis (wt % d.a.f) N
C
S
H
O
1.5
69.3
0.6
4.6
7.0
invasion percolation simulation on constructed fractal models. 2. Research Procedure and Theory of Fractal Model Construction 2.1. Sample Preparation. Huainan low metamorphism coal was used in the present study, and its properties are shown in Table 1. The devolatilization of the coals was processed in a drop tube furnace heated within the range of 1070-1076 K in an atmosphere designed (oxidant:nitrogen ) 1:4) to yield the char, and was then immediately cooled and protected with N2. The conversion degree of char particles varies with the diameters of coal particles. Coal samples were sizeadjusted to 40-60 µm with sieves to avoid any unevenness in the heat treatment caused by the difference in coal particle sizes. At the same time, a statistical method was used to diminish the difference conversation among char particles in data analysis process. Porosity is the average value of several calculated data of SEM photographs, and pore radii distribution represents the character of all char particles. There is a possibility that the pore characters of porous materials differ from each other depending on the moisture contents.17 To avoid this type of influence on pore structure, coal samples were dryer-packed immediately after heat treatment. The burnoff level of the combusted coal was determined by means of thermogravimetric analysis (TGA). It is defined as percentage of maximum possible loss weight. Burnoff level of char sample is 39.53%. 2.2. Research Procedure. Figure 1 shows the research procedure of fractal percolation simulation. The whole procedure can be divided into three parts: experiment, data analysis, and percolation simulation. In the experimental part, inner pore characters of coal and char particles were measured using nitrogen isotherm adsorption/desorption and SEM methods. Some parameters, such as fractal dimension, pore radii distribution, and two-dimensional porosity, were calculated in the data analysis part. The percolation process was simulated on the surface of fractal cube model in the last part. 2.3. Theory about Fractal Model Construction. To construct a porous fractal solid from a Euclidean solid, the solid is divided into N equal parts in each direction to form Nd small solids and then Nm small solid objects are dug out. Next, the same operation of digging out will be iterated at each of the remaining objects. Repeating this procedure, a porous fractal solid with a desired fractal dimension can be finally obtained. Hence, the capacity dimension equation with two integral (17) Kaneko, K.; Fujiwara, Y.; Nishikawa, K. J. Colloid Interface Sci. 1989, 127, 298.
(1)
where N is the number of one-dimensional objects, Nm can be the numbers of one (d ) l), two (d ) 2), or three (d ) 3) dimensional objects corresponding to lines, areas, or solids. Df is the resulting final fractal dimension of lines, areas or solids. The following equation generally is used to define the porosity of the porous solid:
f )
V0 VT
(2)
where f is the porosity, V0 is the pore volume of porous solid, and VT is the total volume of the solid. The volume of the pores in fractal structure can be represented as
( (
V0 ) 1 -
))
Nd - Nm Nd
i
VT
(3)
where i represents the number of iteration. Therefore, the porosity of the fracture structure can be written as
( (
f ) 1 -
))
N d - Nm Nd
i
(4)
it should be mentioned that N and Nm in eqs 1 and 4 are the same and that i, the number of iterations, has to be natural number. The iteration has to be stopped at some point where i is an intrinsic property of the porous solids. By eliminating Nm from eqs 1 and 4, we obtain eq 5. Equation 5 illustrates the correlation between the fractal dimension and the porosity of a porous solid.
ln(1 - f) ) i(Df - d)ln N
(5)
3. Percolation Algorithms The main feature of this algorithm is that occupation is performed in a locally restricted way. In invasion percolation standard algorithms, a site belonging to the interface of occupied-unoccupied elements is occupied if it is of the lowest value. The modified invasion algorithm is defined as follows: (1) To each site of a square lattice, we assign a random number r with the same distribution as that of pore radii of coal/char particles in the interval [0,1] to represent the resistance of the corresponding pore for invasion. (2) We start from a configuration in which all sites are occupied by defending fluid if random numbers generated are bigger than the preset probability P, except for those sites on the left-hand face which are occupied by invading fluid. (3) At each time step, all pores filled with defending fluid which are in contact with invading fluid are added in a list of accessible (boundary) pores. The invading (wetting) phase evolves by occupying the pore in the boundary abiding with the multiple occupation.12 Multiple occupation occurs when each already occupied site belonging to the interface is visited. Assume
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Figure 1. Process of fractal percolation research on coal char particle.
that it has a measure r. All of its unoccupied nearest neighbors are inspected: if a neighbor has a value that belongs to the interval r+δ (δ is a preset value belonging to the interval (0,1)), it is automatically occupied. Thus, multiple occupation can occur. Once all the neighbors are checked, the site is killed. Then, another site is visited until all the sites of the interface are inspected. To kill a site means that after all its neighbors have been inspected for the δ-condition to be fulfilled: it is no more useful to consider that site as a part of the active interface. (4) The invasion process ends when the displacing fluid reaches the opposite face of the lattice. The socalled breakthrough configuration, and a single invading percolated cluster has been set up. P changes from 0 to 1, and the procedure repeats from steps 2-4. It is important to analyze the relationship between fractal dimension D of lattice and probability P. Generally, the fractal body shows the following relation:
F(R) ∼ R(D-d)
(6)
where d is the dimension of the constructional space of the lattice, D is the fractal dimension, R is the radius of body, and F(R) is the correlation function of the graph. F(R,P) corresponds with P. Therefore, the fractal dimension D(P) of the percolated lattice can be calculated. 4. Result and Discussion 4.1. Fractal Model Construction. Fractal dimensions of coal and char particle have been calculated by N2 isotherm adsorption data.18 The porosity of the solids is obtained from SEM analysis. Material steps can be seen in the literature.19 Two-dimensional porosity, which was statistically calculated from the SEM photograph, represents porosity in one section. Other (18) Longjun, X.; Daijun, Z.; Xuefu, X. Coal Conversion 1995, 18, 31. (19) Song, H.; et al. Nat. Sci. Prog. 2002, 2.
Table 2. Fractal Cube Structure Parameter of Huainan Coal Measured Parameters average burnoff Vm/ pore BET SBET/ porosity level/% mL g-1 constant C sq m g-1 bulk/mL g-1 radius/nm 0 39.53
0.8381 1.3080
10.3189 3.4884
3.6922 5.6940
0.001411 0.002358
16.9306 18.0827
Calculated Parameters burnoff level/%
D
f
N
Nm
i
simulated D
simulated f
0 39.53
2.6248 2.8312
0.7210 0.5170
6 5
104 30
2 3
2.6334 2.8295
0.7312 0.5610
parameters about the porous structure of coal and char particles are shown in Table 2. Some parameters, such as N, Nm and i, must be determined with the help of eqs 1 and 5 before we construct the fractal model. The calculation algorithms are different from those in the literature.20 Three parameterssN, Nm, and iswere optimized using a least squares routine coupled with a multiparameter optimization algorithm in MATLAB. Relativaty errors of D and f between calculated and simulated data are limited in the little range. On the bases of these parameters, the fractal models were established following steps introduced in section 2.3. Black points represent unconnected solids; white points represent pores (see Figure 2) with the same characters. The line connected two white points represents the channel between pores, named bond or pore throat. 4.2. Percolation Process Simulation. Only surfaces of constructed fractal models are considered in this study. Using the Monte Carlo simulation, a random value of r assigned to each site on a square lattice has a similar distribution as that of the pore radii of coal and char particles. The difference between measured data and simulation result is little (see Figure 3). One (20) Huang, S. F.; et al. Phys. A 1999, 274, 419.
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Energy & Fuels, Vol. 16, No. 5, 2002 1131
Figure 2. Fractal models. Models a and b corresponding to coal particle and char particle with 39.53% burnoff level, respectively.
Figure 3. Pore radii distribution of coal and char and distribution of random numbers simulated by Monte Carlo simulation; (a) and (b) are pore radii distributions of coal particle and char particle with 39.53% burnoff level, respectively; (c) and (d) are simulation results.
must not confound space distribution and probability distribution of random r values. Random r values uniformly distribute in the two-dimensional lattice space. The invading fluid is injected from the left-hand face, and therefore it can only escape through the right-hand face of the lattice. Figure 4 shows the simulation result of two fractal models. Where white point represents site occupied with defending fluid, gray point represents coal char solid and black point represents site occupied with invasion fluid during invasion process. Simulations were run over lattices and results were averaged over 200 times. The clusters do not represent a homogeneously spread invasion; instead the growth
of cluster is governed by the local heterogeneity of the lattice. These findings also show that the transport properties are affected by the fractal character of the lattice and the percolation cluster properties can be correlated with the fractal dimension of the fractal lattice. The invasion percolation clusters corresponding to fractal models with different P are given in Figure 4. From over 200 repeated simulations, we found that no typical shapes and patterns of displacement or ramification are expected for this type of fractal model structures. The invasion cluster may reflect different types of distribution and the ramification is totally dependent on the random numbers distribution compared with
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Figure 4. Percolation graphs at different P values.
Figure 5. Relationship between D and P.
percolation simulations of model a and b. One can observed the significant changes of graphs (see Figure 4) near the percolation threshold (Pc) site. It is contrast to the other simulation21 that increased P value leads to a decrease of percentage of occupied sites. The interface between the fluids advances pore by pore, its dynamnic is determined by the rule: the site with r bigger than probability P could be invaded. The fractal dimension of simulated lattice against P value was plotted (see Figure 5). Similarly, each curve was obtained from an average over 200 realizations. Fractal dimension (D) of invaded models changes with different shapes when the P value gradually increases. It was shown that this quantity tends to a step function with a threshold at P ) Pc, where Pc is the lattice (21) Haixing, L.; Juping, T.; Boming, Y.; Kailun, Y. T J. Northwest Inst. Arch. Eng. 1996, 3, 118.
critical percolation threshold in the invasion percolation problem. The values of D of models a and b tend toward the original fractal dimension of lattice respectively when the value of P approaches 1. One can observe a novel and interesting feature that D profiles in case model a exhibits two steps: the first was located at the interval (P) [0.38,0.425] and the second was located at the interval (P) [0.68,0.78]. The fractal dimension of model a falls from 1.87 to 1.775 when P changes from 0.38 to 0.425. The explanation about this phenomenon is introduced in the next paragraph. It is relatively common to model b on only one step in the interval (P) [0.400,0.430]. From Figure 4 we can see that the breakthrough configurations of models a and b are established at P ) Pc ) 0.425 and P ) Pc ) 0.430, respectively. The different Pc values could be caused by two reasons: percolation property related with the fractal dimension of the original lattice22 and level iterational shifting of the threshold Pc values.23 Different percolation threshold values between models a and b mean that the transport action is not the same for the two models. The abnormity of the D profile of model a is caused by the size of lattice.24 R. A. Zara found that, in twodimensional lattices with large size, the walks could easily be blocked since hindrances to the growth are set everywhere. For the bigger size lattice, invasion breakthrough configuration attenuates quickly and the second invasion range cannot exit. It turned out that the percolation threshold Pc value is always decided by the first inversion range, no matter if the second inversion range exits or not. (22) Babadagli, T. Phys. A 2000, 285, 248. (23) Young, T.-F.; Fang, H.-j. Phys. A 2000, 281, 276. (24) Reginaldo, A. Z.; Onody, R. N. Phys. A 2000, 277, 1.
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On the bases of the percolation simulation answers, some conclusions about transport character in coal and char particles can be deduced. Because of similar distribution, the critical parameter of the pore structure corresponding to the percolation threshold Pc values of fractal models can be calculated by the following equation:
Dt - D0 Mt - M0 ) Dl - D0 Ml - M0
(7)
where Dt and Mt are distributions of measured and simulated critical sites, respectively (see Figure 3); D0 and M0 are distributions of measured and simulated original sites, respectively (see Figure 3); and Dl and Ml are distributions of measured and simulated terminal sites, respectively (see Figure 3). Critical measured sites represent critical pore radii, which decide the transport characters in realistic situations. Utilizing eq 7, Mt values of A and B are calculated and equal to 13.6 nm and 14.0 nm, respectively. The percentage of participated pores can be calculated using eq 8.
Np )
∫MM f(r) dr l
t
(8)
where Np is the percentage of participated pore and f(r) is the distribution function of pore radii. The lower
integral limit is the measured site Mt and the upper integral limit is the terminal site Ml. The percentage of the participated pore is about 89.56 and 91.10. 5. Conclusion Pore structure of coal and char particles influences transport characters. It can be simulated by percolation theory. Fractal model is established with the help of fractal theory and data measured by SEM and nitrogen isotherm adsorption/desorption methods. Combined with Monte Carlo simulation, fractal model is perfected with similar characters of pore network of coal and char particles. During simulation process, a novel and interesting feature emerges: D profiles in the case of model a arise in two steps. Research shows that the percolation threshold Pc is decided by the first step. The percolation threshold Pc values of fractal models a and b are 0.425 and 0.430, respectively. Measured critical sites of coal particle and char particle with 39.53% burnoff level are 13.6 nm and 14.0 nm, respectively. Percentages of participated pore of coal particle and char particle with 39.53% burnoff level are 89.56 and 91.10, respectively. Acknowledgment. This study was supported by China Natural Science Foundations, under Contract 50176014. EF020003S
