A new method for evaluating packing efficiency of particle size distribution during coal water slurry preparation

： Particle size distribution is a key problem in preparation of coal water slurry . Due to the large number of particles and irregular accumulation of particles, how to evaluate and calculate CWS particles has become an urgent need. In this paper, the volume accumulation of particles is simplified as plane accumulation, and the particle packing gradation model based on the three-particle packing gradation and the four-particle packing gradation is constructed. Based on the classical mathematical principle, the packing voids and packing probability of 5 typical CWS particle sizes were calculated. The packing styles of N 1 N 2 N 3 , N 2 N 3 N 4 and N 3 N 4 N 5 were evaluated and analyzed. The evaluation method of packing index based on ΣP×S min and ΣP×S max was established. The results show that the packing efficiency of particles can be quantitatively evaluated by using of cumulative calculation of voids areas and packing probabilities under different packing modes. The evaluation results are consistent to the tendency of CWS concentration. By this method, the particle size distribution of CWS can be controlled and adjusted during both wet milling processes and experimental investigations. Furthermore, the method may be used to other particle packing cases.


Introduction
The packing property of particles refers to the inner part of particles, the arrangement of particles in space or the structural characteristics of particles. The preparation of high concentration of coal water slurry requires higher packing density. In the study of dense packing of particles, there are two major theories: continuous and discontinuous packing of particles. Continuously distributed particles are composed of particles of all sizes within a certain range of particle size, the discontinuous distribution is composed of particles representing the finite size of the range. Studies on accumulation of discrete-size particles. (Furnas 1928;Westman et al 1930;Westman 1936；Suzuki 1985 The initiator of the classical continuous theory was Andreason (Andreason 1928；Andreason 1929； Andreason 1930), who described the actual particle distribution with equation of Gaudin-Schuhmann Golden-schutzmann .Fuller (Fuller 1906) obtained the experience curve of the most compact packing of continuous grain-size system by experiment. It is considered that the equation modulus m in the range of 0.33 ~ 0.5 has the minimum void fraction. Based on Andreason's theory, the Alfred equation is proposed by Dinger and Funk (Funk 1980；Funk1987), and it is found that the highest accumulation rate occurs when n is 0.37.Some researchers put forward the theory of inter-stage packing by analytic method, and established a new relation between the modulus n and the void fraction of Gaudin and Alfred equation, and calculated the packing efficiency of any particle size distribution (Zhang et al 2002).
With the rapid development of China's coal chemical industry, the coal production process of methanol, synthetic ammonia, olefins by using of coal water slurry as gasification media is expected to use 300 million tons in the future, the amount of CWS used for combustion is about 30-60million tons annually. Therefore, coal-water slurry gasification has become the focus of chemical industry and the preparation of high concentration coal-water slurry has become an important factor in CWS applications. The minimum void accumulation model and evaluation index E provides a good method to testify the coarse particle and fine particle distribution (Tu 2013；Tu et al 2013；Tu et al 2015；Yang   et al 2016). Increasing the size range of coal water slurry particles is beneficial to improving its packing efficiency, the packing efficiency is related to the particle size composition of raw materials However, sometimes it is difficult for engineers in CWS factories to understand these formulas. A simplified description and method for packing efficiency based on particle size distribution of CWS is needed. Fig.1 is the experimental diagram for particles evaluation. The size composition of CWS consists of 8-mesh, 14-mesh, 40-mesh, 80-mesh and 200-mesh. Most of the size composition of qualified CWS from 8 to 14-mesh is less than 1% . Therefore, five particle sizes of 14-40 mesh, 40-80 mesh, 80-200 mesh, 200-325 mesh, <325 Mesh were selected as the research objects. According to the packing model in Fig. 2, the packing voids of particles in CWS can be calculated simply by the distribution of the above five kinds of particle sizes. Figure 2 is the model of particle packing gradation. In Figure 2, R1R1R1, R1R1R2, R1R2R2, R2R2R2, R1R1R3, R1R2R3, R1R3R3, R2R3R3 and R3R3R3 are packed in a steady state with a radius of R1, R2, R3.

Experimental methods and procedures
R3R3R3R1，R3R3R3R2，R3R3R3R3 are packed in a non-steady state with a radius of R1, R2, R3.
The packing voids of particles in a unit volume is calculated and analyzed by the voids formed by particles in its cross section. The calculation procedure is as follows: 1) the particle volumes of the above five regions are defined as V1, V2, V3, V4, V5. The mass of a single particle in the five size range is defined as M1, M2, M3, M4, and M5. The number of particles per unit mass in the five size ranges is defined as W1, W2, W3, W4, W5. The number of particles per unit mass cross section is defined as N1, N2, N3, N4, N5 for each of the five dimensions.
2) the probability distribution, void area distribution and cumulative void area of 10 particle grading modes under N1N2N3 and N3N4N5 packing modes were calculated according to the three-particle steady state mechanism of graded packing.
3) the probability distribution, void area distribution and cumulative void area of 21 kinds of particle grading modes under N1N2N3 and N3N4N5 packing modes are calculated according to the four-particle unsteady mechanism of graded packing.
The median of the granularity range is used as the value of the granularity range. The values of D1, D2, D3, D4 and D5 are 940μm, 325μm, 138μm, 60μm and 22μm respectively. As coal particles are non-spherical particles, the spherical degree of broken particles is 0.75, so the diameter of the non-spherical particles is about 0.75 times diameter of the spherical particles. The particle diameters of the samples were 705μm, 244μm, 104μm, 45μm and 17μm respectively. The calculation is as follows: For the purposes of the study, the particle is approximately considered to be a sphere with a volume of 1mm 3 . The density of coal particles is 1.35g/cm 3 . The volume and mass of the coal particles in each size range are calculated as follows: The particle size distribution of 4 CWS samples are listed in Table 1 .
Samples A, B and C are coal water slurry samples from a methanol plant in Ningxia, and Samples D are coal water slurry samples from a methanol plant in Inner Mongolia. The particle size of each group was the average value of 10 CWS samples. The particle distribution parameters of samples A, B, C and D are calculated in Table 2.

Average size
According to the particle size distribution in Table 1, the average particle size values of A, B, C and D CWS samples are 170μm, 175μm, 170μm and 186μm respectively. The calculation is as follows: Comparing the concentration values of A, B, C and D samples, it is found that the concentration values of D samples are 55.07% , and the average size values of slurry are 186μm, and the average size difference is larger than that of A, B and C. The concentration difference of A, B and C samples was 1-2wt% , and the average particle size difference was 5μm. It is concluded that the larger the average size of CWS, the larger the number of coarse particles in CWS. The reason for this result lies in the incomplete grinding of CWS particles by steel rods in rod mills during wet overflow milling, causing more coarse particles to overflow into the finished slurry. The 14-40 mesh particle ratio of D sample in Table 1 is obviously higher than that of A, B and C sample.

Three-particle packing gradation 3.2.1 Heron Formula
The size of the void formed between particles can be expressed by the area of the triangle formed by the three central points of particles. As shown in Figure 3 There are 10 packing modes according to the three-particle packing gradation, which are R1R1R1,R1R1R2,R1R2R2,R2R2R2,R1R1R3,R1R2R3,R2R2R3,R1R3R3,R2R3R3 and R3R3R3, respectively.
According to the Formula (2) , the calculation values of the void areas of the three-particle packing in N1N2N3 and N3N4N5 packing modes are shown in Table 3.

Probability distribution of three-particle packing gradation
According to the Formula (3) , the probability distribution value P of each packing mode can be calculated. The probability distribution of coal water slurry samples A, B, C and D is shown in Fig.4.

Distribution of void areas in three-particle packing gradation
According to the calculation value Smin in Formula (2) and the probability value P in formula (3) , the value of P×Smin in each packing mode can be calculated. The probability distribution of three-particle packing model and the distribution of packing void areas of CWS samples A, B, C and D are shown in Fig. 5.
It can be seen from Fig.6

Probability distribution of four-particle packing gradation
According to the Formula (6) , the probability distribution value P of each packing mode can be calculated. The probability distribution of coal water slurry samples A, B, C and D is shown in Fig.7.
As can be seen from the Fig.7, the probability of four particles forming R3R3R3R3 is the highest in the N1N2N3 grading model, which are 57.9%，56.7%，56.6%，55.4% respectively, followed by R2R3R3R3 packing. Under the N2N3N4 packing gradation mode, the probability of four particles forming R2R3R3R3 packing is the highest which are 31.5%，30.6%，33.6%，35.5% respectively， followed by R3R3R3R3,R2R2R3R3 and R2R3R2R3.Under the N3N4N5 packing gradation mode, the probability of four particles forming R3R3R3R3 packing is the highest which are 80.2%，80.0%， 77.8%，76.6% respectively，followed by R2R3R3R3 and R1R3R3R3.

Cumulative void area of four-particle packing gradation
Based on the calculation of the probability distribution of packing gradation and the area distribution , the cumulative void area of packing gradation ∑P×Smax is calculated in order to further analyze the difference of packing void in different granular packing states, see Formula (7) . (7) The cumulative packing void areas of CWS samples A, B, C and D is shown in Fig.9. It can be seen from Fig. 9 that under the condition of N1N2N3 packing gradation, the accumulative packing voids area of sample A, B, C and D are 14470μm 2 ,14606μm 2 ,14610 μm 2 and 14886μm 2 respectively. The accumulated accumulation void area of D sample is the highest. Under the condition of N3N4N5 packing gradation, the accumulative packing voids area of sample A, B, C and D are 351μm 2 ,352μm 2 ,357μm 2 ,359μm 2 respectively. The accumulative packing voids area of sample D are the highest. As can be seen from Table 1, the concentration of sample D is the lowest. The results are consistent with the conclusion that the higher the packing efficiency the higher the CWS concentration.
Under the condition of N2N3N4 packing gradation, the accumulative packing voids area of sample A, B, C and D are 4761μm 2 ,4881μm 2 ,4457μm 2 ,4094μm 2 respectively. The accumulated void area of D sample is the lowest. The reason is the same as the situation of three-particle packing gradation and it is also that the fine coal powder can not fill the voids formed by medium-size coal powder effectively, which results in the decrease of packing voids and the decrease of slurry concentration. Therefore, it is feasible to use the larger value of the cumulative probability void area ∑P×Smax under N1N2N3 and N3N4N5 packing modes and the smaller value of the cumulative probability void area ∑P×Smax under N2N3N4 packing modes as the evaluation indexes of the CWS concentration reduction. Moreover, the results are basically consistent with the results of comparative analysis of packing voids of three-particle packing gradation. The consistency of the method is further proved.

Prediction evaluation procedure for packing efficiency
A process for predicting packing efficiency and slurry concentration based on packing gradation model is as follows: First step: Calculate the packing void area Smin or Smax.
Second step: Calculate the numbers of packing particles per unit volume according to the test value or designed values of particle size distribution.
Third step: Calculate probability of different packing modes by using of Excel softwar according to the formula of permutation and combination.
Fourth step: Calculate the values of ∑P×Smin or ∑P×Smax under the packing gradation of N1N2N3,N2N3N4 or N3N4N5 . Fifth step: Adjust particle size distribution of CWS according to the value of ∑P×Smin or ∑P×Smax. (1) The packing efficiency of particles can be quantitatively evaluated by using of cumulative calculation of voids areas and packing probabilities under different packing modes. Correspondently, the CWS concentration can also be predicted by this method.

Conclusions
(2) The slurry concentration of CWS is closely related to its particle distribution. The size of CWS particles determines the size of the voids formed by the packing gradation of particles, and the composition of particles determines the probability of the packing gradation of CWS particles.
Under the condition of wet overflow milling, the average size of slurry calculated by weighted average can reflect the state of packing gradation of slurry particles, which is un-proportionally with the slurry concentration. The larger the average size of slurry, the lower the concentration of CWS slurry, the smaller the average size of slurry, the higher the concentration of CWS slurry.