Research On The Fractal and Percolation Characteristics of Coal-Based Porous Media For Filtration and Impregnation

: In order to distinguish the difference in the heterogeneous fractal structure of porous graphite used for filtration and impregnation, the fractal dimensions obtained through the mercury intrusion porosimetry (MIP) along with the fractal theory were used to calculate the volumetric FD of the graphite samples. The FD expression of the tortuosity along with all parameters from MIP test was optimized to simplify the calculation. In addition, the percolation evolution process of mercury in the porous media was analyzed in combination with the experimental data. As indicated in the analysis, the FDs in the backbone formation regions of sample vary from 2.695 to 2.984, with 2.923 to 2.991 in the percolation regions and 1.224 to 1.544 in the tortuosity. According to the MIP test, the mercury distribution in porous graphite manifested a transitional process from local aggregation, gradual expansion, and infinite cluster connection to global connection.


Introduction
Over years, the study of fractal porous media has caught intensive attentions of many scholars and researchers. The structural characterization and description have been considered as the foundation for the study of thermophysical phenomena, such as the flow, phase transition, heat and mass transfer in porous media [1][2][3]. Due to the randomness and heterogeneity of the structures in porous media, the traditional Euclidean geometry fails to describe the structural characterization accurately. However, the fractal geometry has gradually adopted by many researchers and proven to be valid and fruitful.
The fractal theory has been considered as a powerful tool in studying fractal geometry and fractal phenomena in nature [4]. Some researchers, including Katz, Winslow, Krohn and Tyler, et al., have successfully described the structural characteristics of porous media such as sandstone, rock, coal char, cement and soil based on the fractal method, derived the expressions of fractal dimension (FD), proposed the physical meaning and obtained the parameters [5][6][7][8][9][10].
Measurement and description are two key steps in the structural characterization of fractal media.
Image processing methods are often used to characterize the fractal features of 2-dimensional (2-D) planes of a porous media. For example, a 2-D image of a porous media can be obtained through the scanning electron microscopy (SEM). The FD of the 2-D fractal structure can be developed by boxcounting method, the area-circumference method or the area-radius method [11][12][13].
Regarding the 3-dimensional (3-D) fractal media, the experimental measurement and the image processing method are combined to obtain the structural parameters. Some structural parameters are obtained through experiments with the adoption of mercury intrusion porosimetry (MIP) and nitrogen (N2) low pressure gas adsorption (LPGA). Some parameters are obtained through the image processing method, such as the micro-CT and SEM. The volumetric of the 3-D porous media can be obtained through the fractal method with the characteristic parameters. Relevant productive researches have been conducted by scholars such as Go´mez-Carracedo et al., Yao et al., and Wu et al. [14][15][16]. The physical and chemical stability of the porous graphite has contributed to the wide adoption in various fields such as materials, chemicals, biology and environment [17]. In industrial applications such as impregnation and filtration, the effective description of the heterogeneous structure plays a crucial role to study the fluid flow phenomena in porous graphite [18]. Many studies have indicated that the structure of porous graphite demonstrates some typical fractal characteristics, which can be described through the fractal method [19,20]. In this study, the fractal characteristics of the porous graphite, which was prepared from coal-based raw materials, were described with an optimized method developed to describe the tortuosity fractal dimension. In order to minimize the errors caused by the measurement method and instrument accuracy, only the parameters through the MIP were used to obtain the FDs of volume. Moreover, the evolution process and percolation characteristics of the mercury occupation in porous graphite were analyzed based on the data obtained from the MIP.

Experimental
As an effective method for determining the distribution of some mesopores and macropores in porous medium, the MIP is developed based on the Young-Laplace law and Washburn equation to measure complex pore structures [21,22]. Many studies were conducted to measure the pore structure and permeability of porous media through the MIP [23][24][25][26].
Porous graphite samples were prepared using coal coke particles as raw materials and mediumtemperature coal tar pitch as binder by following different aggregate formulas. Four samples were selected for filtration (No.1-4#) and four for impregnation (No.5-8#). The MIP test was conducted by Micromeritics AutoPore IV 9520, with a maximum pressure of 60,000 psia, a pressure accuracy of 0.01 psia, a pore size measurement range of 3 nm to 1000 μm, and a volumetric accuracy of 0.1 μL.
The main steps of the MIP test are provided as follows: a. The samples were firstly processed into a cylinder shape with a diameter of 8 mm and a height of 10 mm. Sample cylinders were cleaned and dried in the oven for 10 hours at 100℃.
b. The sample was weighed and placed in the sample tube. Tube was sealed with think resins.
c. Measurement in the low-pressure station: Low-pressure tests were performed under the vacuum of less than 50 μmHg. Mercury was removed and the samples were scaled.
d. Measurement in the high-pressure station: Sample was pressurized in the station where the pressure was increased to 30,000 psia.
e. Data collection and process were conducted through the software connected to the test system.

Results and discussion
3.1 The Pore structural parameters of porous graphite Where V is the volume of the 3-D fractal media and v D standards for the volumetric of the fractal media.
The box-counting method was widely used to calculate the FD of 2-D fractal images effectively by many researchers. [27,28]. However, applying the box-counting method to obtain the volumetric of the 3-D fractal media is challenging. To calculate volumetric FD of the 3-D fractal media through the box-counting method, various units of 3-D under different scales are needed to conduct multiple boxcounting and calculation, which can be difficult and time-consuming [28].
Angulo et al. established an effective method to calculate the FD of a 3-D fractal media [29]. According to the Yong-Laplace theory, when fluid flows through a pore, the capillary pressure depends on the pore size which makes the replacement of geometric scale with the pressure scale possible to measure the intrusion volume and the corresponding pore size under a certain capillary pressure, which has been proven to be effective When the capillary pressure ( c P ) is above the threshold pressure ( t P ), the intrusion volume tends to rise rapidly. The intrusion volume is related to the pressure and the FD of the media. The relationship among FD, pressure and intrusion volume is shown as follows [29].
(3 ) Taking the logarithm of both sides of Eq. (6) and setting ct P P P    , the following expression can be obtained as Eq. (4).
Where v D is the volumetric of fractal media, c P stands for the capillary pressure, t P refers to the threshold pressure, V indicates the intrusion volume, respectively. Eq.8 shows a close linear By measuring the different parameters of log( ) V and log( ) P  experimentally, the volumetric of the fractal medium can be calculated. In MIP test, the boundary between the backbone formation region and the percolation region should be closely monitored, which shows different characteristics.
Therefore, the FDs of different regions can be obtained by analyzing the data corresponding to different regions. The double logarithmic curves of -VP  for eight porous graphite samples are developed and shown in Fig.1, and the results of FDs and relationship equations are shown in Table 2. Meanwhile the percolation region demonstrates better linearity and fewer RMS errors in corresponding curves than the backbone region. Generally, the results included in Table 2 show an excellent linear correlation, with 2 R from 0.906 to 0.999. to the irregularity and heterogeneity of the geometry than those in the impregnation samples. However, Table 1 and Table 2indicate that no single direct correlation is observed between the FD and the media porosity. The FD characterizes the heterogeneity and complexity of the internal structure of the fractal media without a direct relation to the porosity of the media. In other words, the FD solely characterizes the proportion of pores in the porous media. Often the higher FDs indicate a greater heterogeneity of pore structure in porous media [30,31]. Among four porous graphite samples from 1# to 4# for filtration, sample 4# demonstrates the highest FD in percolation region with a medium porosity.
Sample 2# has the lowest FD in the backbone formation region with a highest porosity. Such phenomenon explains the easy mistakes by simply predicting the level of FD based on the porosity of porous media.
Many researchers have shown that different expressions of FDs highly depend on different pore structural parameters of fractal media, such as porosity, minimum pore size, maximum pore size, and even the throat size of the pores [32,33,44]. The pore size parameters obtaining is often affected by the accuracy of the measuring equipment and the measurement method, especially for the measurement in the nanometer scale. The MIP has shown some advantages to obtain the FDs and structural parameters. The intrusion volume ( V ) and pressure difference ( P  ) can be obtained with less complexity and higher accuracy than most of other methods. Moreover, the aforementioned structural parameters needed and errors caused by the measurement methods and instrument can be minimized.

FD of tortuosity by MIP test
The  can be obtained through the MIP tests, without considering the distribution of the capillary or whether the capillary channel runs through the whole media. Therefore, the characteristic length of the capillary ( c L ) is more accurate as the representative length of the capillary ( 0 L ) than the sample length. Eq. (7) can be modified and shown follows. In the MIP test, the obtained av  ， av  and c L are featured with a high reliability. Therefore, calculating the FD of tortuosity of the capillary structure in porous media by following Eq. (9) offers more advantages.

Percolation evolution and characteristics in MIP test
Percolation theory was first proposed by Broadbent and Hammersley and has become a powerful tool for studying disorder and random phenomena in nature, and is widely used in the fields of physics, chemistry, biology, environment, finance, social problems and so on [39,40].During the MIP test of porous graphite, an increasing amount of mercury enters the pores under the pressure gradient, and overcomes the capillary resistance to slowly intrude into the interior of the porous graphite. When the pressure reaches a certain critical point, also known as the thresholds, the intruded mercury begins to expand and fill the whole capillary network. At this point despite that the pressure continues to rise, the cumulative mercury intrusion volume no longer increases significantly. The mercury in the porous graphite has achieved a percolation distribution. Fig.2 shows the relationship between cumulative intrusion volume and pressure for samples 1-4# during the MIP test. At the beginning of the test, only traces of (<0.03 mL/g) mercury intruded into the porous graphite, which was almost negligible. When the pressure was increased to reach the threshold value, the volume of the intruded mercury rises rapidly, and the mercury began to occupy the vacant space of the percolation region in the porous medium. According to the test, the pressures thresholds of the samples 1-4# were 8.49, 4.67, 13.05 and 11.19 psia, respectively. When the pressure rose above the threshold, the increasing of the cumulative mercury immersion volume in each sample started to slow down which corresponded to the relatively flat region in the curve, indicating that the mercury had completely intruded into the percolation network structure. When the pressure continued to increase, some new mercury continuously entered into the pores. Meanwhile an equivalent volume of mercury was pushed out of the capillary network, suggesting that the total volume within the capillary remained unchanged.
Even when the pressure was increased to 30,000 psia, the cumulative intrusion volume of mercury changed little. The process of pressurized mercury intrusion in porous graphite can be described as a process in which the fluid gradually occupies the pores, with the percolation evolution through local aggregation/gradual expansion/infinite cluster connection/global connection. In the extrusion cycle, the mercury volume demonstrated little changes, and the curve stayed flat, indicating that the antiintrusion force was limited with a stable percolation network structure.  According to Fig.4, the incremental intrusion volume increased sharply when the capillary pressure got close to the threshold value. When the pressure rose slightly higher than the threshold, the incremental intrusion volumes of samples 1-4# reached the peak, which were 0.144, 0.108, 0.137, 0.172mL/g, respectively. The incremental intrusion volume rapidly dropped to near zero after the peak.
In addition, no direct relation was observed between the incremental intrusion volume and the sample porosity. For example, sample 4# had the largest incremental intrusion with porosity lower than sample 1# and sample 2#.Similarly, in Fig.5, sample 6# had the highest incremental intrusion (0.026mL/g)at the threshold, with the lowest porosity (9.53%).In spite that Sample 7# had a highest porosity (15.75%),the incremental intrusion of sample 7# (0.014mL/g)at the threshold point only the ranked number three among all samples, just a little higher than that of sample 8# (0.011mL/g).  Table 4.

Conclusions
The pore structural parameters of porous graphite were effectively measured and described through the MIP test. Samples for filtration have a higher porosity and average pore diameter than the samples for impregnation with a much lower tortuosity. The expression of FD for 3-D fractal media is proven to be valid to minimize the errors caused by the accuracy of measuring methods and instruments. In addition, the expression is more concise with more reliable data obtained from MIP. For all the samples, comparing with the percolation regions, the FDs in backbone formation regions are lower. The parameters of average tortuosity, characteristic length of capillary and average diameter of pores can be effectively adopted to develop the FD expression of the tortuosity of porous graphite. The parameters obtained only from MIP test offer some advantages including simplified relevant parameters with clear physical meanings, accessible measuring methods and the retrieval of reliable data, which all together contribute to the minimization of errors caused by the measurement methods and the instrument accuracy.
The microstructure of porous graphite and the intrusion evolution of mercury in pores indicate some conspicuous percolation characteristics in MIP test. The cumulative intrusion volume increases sharply as the pressure reaches the pressure threshold. After the percolation distribution of the mercury liquid in the pores, the cumulative intrusion volume demonstrates little changes even when the pressure is increased to the maximum limit. When the percolation distribution of the mercury is formed, the samples for impregnation demonstrate a much higher threshold pressure than the samples for filtration.