Analysis and Prediction of Influencing Parameters on the Coal Classification Performance of a Novel Three Products Hydrocyclone Screen (TPHS) Based on Grey System Theory

Abstract

A novel hydrocyclone including a cylindrical screen embedded in a conventional hydrocyclone (CH), named three products hydrocyclone screen (TPHS), has been successfully designed. In TPHS, the combination of centrifugal classification and screening was employed to separate particles. In this paper, Grey theory, as an effective means to the laws of both complex and uncertainty system’s behavior with small samples, was used to investigate the operational (feed concentration and feed pressure) and structural (aperture size, spigot diameter, and vortex finder diameter) parameters on performance evaluation Hancock classification efficiency (HE), imperfection (I), and cut size (d_50c). The experiments of coal sample (0–1 mm) show that TPHS with coarser particles in underflow exhibited the absent “fish-hook”. The closeness calculated using the Grey System algorithm indicates that the performance of TPHS was closely related to the operation and structure parameters. Further, the order of grey incidence degree between different parameters and HE (or I or d_50c) is the spigot diameter and aperture size with the highest value, the feed pressure and vortex finder diameter with the middle value, and the feed concentration with the lowest value. The prediction using the GM(1, N) algorithm implies that the dynamic prediction model for the performance evaluation can be created depending on the operation, structure and previous performance value. The mean relative errors between the predicted and actual HE, I, and d₅₀ were 2.84%, 5.83%, and 3.57%, respectively, which exhibit the accurate prediction.

1. Introduction

Hydrocyclones are well known as significant devices to separate fine particles based on its size in a centrifugal force field. According to the high throughput, low cost, flexible operation and so on, hydrocyclones are now widely used in mineral, chemical, environmental, and some other industrial processes [1,2,3]. The conventional hydrocyclone (CH) includes one tangential feed inlet and two axial product outlets, i.e., smaller underflow at the bottom and bigger overflow at the top to discharge coarser and fine particles, respectively [4,5]. Nevertheless, on the one hand due to the inherent disadvantages (like fish-hook effect or unsatisfactory efficiency [6,7]), on the other hand owing to the new trendies (like the reducing processed particle size [8,9]), the hydrocyclone performance is partly weak, even the device cannot be adopted in some industrial process cases. Therefore, the device performance is tried to be improved by the structural modifications, such as (1) introducing new structures like filtering hydrocyclone [10,11], JK three-product cyclone [12], and two-inlet hydrocyclones [13,14], and (2) changing original structures including the feed pipe [15], the cylindrical section [16], the vortex finder [17], and the conical vessel [16], among others.

In view of the above-mentioned types of hydrocyclone, a new apparatus called three products hydrocyclone screen (TPHS) shown in Figure 1 has been developed [18,19]. Basically, TPHS can be considered as a CH embedded by a cylindrical screen to combine both centrifugal classification and screening. Thus, cylinder stratification, cylinder classification, and cone classification are generated in TPHS to separate particle depending on size. The working process between TPHS and CH are similar; however, besides the overflow and underflow, the present screen results in a new screen underflow product in TPHS. The previous experiment for TPHS and CH expose that the new cyclone could remove the “fish-hook” effect for better performances [18]. In addition, the exited computational fluid dynamics simulation and particle image velocimetry test reveal that TPHS, compared to CH, could eliminate the air column and short circuit flow for the more reasonable flow field distribution [20].

The previous researches demonstrate that some general understanding for TPHS is available. However, the previous research focused on the performance contrast of TPHS and CH; the details of effects of structural and operational parameters on TPHS are not yet reported and remains unexplored. To cyclone, Hancock classification efficiency, imperfection, and cut size are the important indexes to generally evaluate the device performance of cyclone, which are closely related to both operation parameters (including feed concentration and feed pressure) and structure parameters (cyclone diameter, vortex finder diameter, and spigot diameter) [21]. Since, TPHS is a derivative of hydrocyclones, the aforesaid parameters can be considered for the performance evaluation of the new cyclone. However, due to the particular cylindrical screen, the aperture size should be taken as one vital factor for TPHS performance. Usually, the experiments under different factors are adopted to explore the influence of conditions on cyclone performance [4], wherein the experiment result should be analyzed through the suitable data processing method.

Recently, with the development of data processing algorithm, Grey System theory [22,23,24] has been considered as an effective means to the laws of both complex and uncertainty systems’ behavior with small samples. As this algorithm explores the objective laws through information coverage and through the works of sequence operators, it is little constrained by experimental conditions. Thus, Grey System theory has been widely used in many fields including hydrocyclone [25,26,27]. Therefore, this method can be as well applied to the performance of TPHS.

Thus, based on the above discussion, the objective of this new investigation was to explore the device performance involving different operational and structural parameters for the aforementioned knowledge gap. To attain this purpose, Grey System theory has been used, considering a pilot-scale TPHS (φ 150 mm). Specifically, the objectives of the present study were to:

(1)

Comprehensively verify the separation characteristic including the size distribution and the partition curve;

(2)

systematically explore the closeness between the performance and the relevant factors (operational and structural); and

(3)

critically analyze and verify the prediction model of the performance evaluation based on the parameters of operation and structure.

2. Experimental

2.1. Experiment Rig

The diagram of experiment rig is exhibited in Figure 2a schematic diagram and (b) photograph. From this figure, it can be seen that the main devices adopted in the present study included a feed tank of 0.5 m³, a variable feed pump of 0–30 m³/h, a frequency converter of 0–50 Hz, a product sampler, some valves, a stirrer and a TPHS. An electromagnetic flow meter and a mechanical manometer were used to measure the feed mass flow and pressure, respectively. It is worth pointing out that the TPHS size and the target cut size were considered as 150 mm and 0.25 mm, respectively in terms of our previous work [18]. The details of structure parameters for TPHS are shown in

Table 1. Structures of TPHS in pilot-scale experiment.

Items	Value
Diameter of hydrocyclone	150 mm
Aperture size of cylindrical screen	0.5 mm, 0.6 mm and 0.7 mm
Length of inlet column	130 mm
Length of cylindrical envelope	370 mm
Length of vortex finder	300–400 mm
Angle of cone	20°
Size of inlet	58 mm × 16 mm
Diameter of screen underflow outlet	56 mm
Diameter of vortex finder	30 mm, 40 mm, 50 mm, 60 mm
Diameter of spigot	18 mm, 24 mm

. The aperture size of the cylindrical screen was in the range of ~0.5 mm to 0.7 mm (~2 to 3 times bigger than the target cut size). The length of inlet column and cylindrical envelope were 130 mm and 370 mm, respectively. The length of vortex finder was 300–400 mm. The size of inlet was 58 mm × 26 mm for feed stream. The spigot diameter of 18 mm and 24 mm and the cone angle of 20° were used to generate the crowding and hindered settling conditions in the cone classification for underflow stream. The diameter of vortex finder was considered ~30 mm to 60 mm for overflow stream.

2.2. Experiment Procedure

In this research, the coal samples (diameter ~0–1 mm and density ~1.45 g/cm³) from Guobei coal preparation plant, China, were used as the mineral particles. The experiment procedure can be described as follows.

(1)

The process cycle starts with a coal slurry being stirred ~10 min. Subsequently the homogeneous slurry was reported to TPHS by the variable speed pump with different experiment conditions (see

Table 2. Experimental conditions.

Experiment Number	1	2	3	4	5	6	7	8	9	10
Feed concentration (g/L)	60.00	100.00	140.00	180.00	220.00	60.00	100.00	140.00	180.00	220.00
Feed pressure (MPa)	0.04	0.05	0.06	0.04	0.05	0.06	0.04	0.05	0.06	0.04
Aperture size (mm)	0.60	0.70	0.80	0.60	0.70	0.80	0.60	0.70	0.80	0.60
Diameter of spigot (mm)	18.00	24.00	18.00	24.00	18.00	24.00	18.00	24.00	18.00	24.00
Diameter of vortex finder (mm)	30.00	40.00	50.00	60.00	30.00	40.00	50.00	60.00	30.00	40.00

for the details). Based on the previous works [18,19], the operational parameters feed concentration and feed pressure were considered as 60–220 g/L and 0.04–0.06 MPa, respectively, while the structural parameters aperture size, spigot diameter and vortex finder diameter were 0.60–0.80 mm, 18.00–24.00 mm, and 30.00–60.00 mm, respectively. As mentioned, Grey System theory is less limited to the experimental conditions., thus different parameters were combined from large to small in the present research.

(2)

Then, every experiment was run about 30 min to get the steady particle separation. Afterwards, the inlet and outlet streams were sampled simultaneously, where three sets of each stream sample were taken with the intervals of 5 min, i.e., at 40, 45, and 50 min for the sample representativeness.

(3)

Finally, all the samples were processed by the weighing, filtration, and drying for the granularity test. It is noted that wet sieving [28] was considered for the analysis of particle size distribution to avoid the particle aggregation.

2.3. Data Analysis

In this research, the mass flow ratio of solids, i.e., solid recovery, for each product stream was calculated by Equations (1)–(4) according to the least-squares method (LSM) [29]. So as to eliminate the effect of the bypass to underflow [30,31], the corrected partition curve was conducted using Equations (5) and (6). To evaluate the performance of TPHS, Hancock classification efficiency (HE) and the imperfection (I) was obtained by Equations (7) and (8), respectively [32,33,34], and the corrected cut size of the partition curve (d_50c) was gotten according to its definition (see the details in the annotation after Equation (8)). The Equations (1)–(8) are shown as follows:

$${\gamma }_1=\frac{g_{01}{g}_{22}-g_{02}{g}_{12}}{g_{11}{g}_{22}-g_{12}{g}_{12}}g\times 100\%$$

(1)

$${\gamma }_2=\frac{g_{02}{g}_{11}-g_{01}{g}_{12}}{g_{11}{g}_{22}-g_{12}{g}_{12}}g\times 100\%$$

(2)

$${\gamma }_3=100\%-{\gamma }_1-{\gamma }_2$$

(3)

$$\begin{array}{l}{g}_{01}={\displaystyle \sum _{j=1}^N\left(G_{0j}-G_{3j}\right)}\left(G_{1j}-G_{3j}\right)\\ g_{11}={\displaystyle \sum _{j=1}^N{\left(G_{0j}-G_{3j}\right)}^2}\\ g_{02}={\displaystyle \sum _{j=1}^N\left(G_{0j}-G_{3j}\right)}\left(G_{2j}-G_{3j}\right)\\ g_{12}={\displaystyle \sum _{j=1}^N\left(G_{1j}-G_{3j}\right)}\left(G_{2j}-G_{3j}\right)\\ g_{22}={\displaystyle \sum _{j=1}^N{\left(G_{2j}-G_{3j}\right)}^2}\end{array}\}$$

(4)

$$P_{tj}=\frac{{\gamma }_3{G}_{3j}}{G_{0j}}$$

(5)

$$P_{cj}=\frac{P_{tj}-R_f}{100\%-R_f}$$

(6)

$$HE=[\frac{{\gamma }_3\times (-d_{t3})}{-d_{t0}}+\frac{{\gamma }_3\times (+d_{t3})}{+d_{t0}}]-100\%$$

(7)

$$I=\frac{d_{75c}-d_{25c}}{2d_{50c}}$$

(8)

where,${\gamma }_1$, ${\gamma }_2$, and ${\gamma }_3$ represent the solid recovery in the overflow, screen underflow and underflow, respectively. G_0j, G_1j, G_2j, and G_3j are the mass percentage of the particle size interval (j) in the feed, overflow, screen underflow and underflow, respectively. P_tj means the partition number of the raw partition curve for the particle size interval (j). P_cj means the partition number of the corrected partition curve for the particle size interval (j), R_f indicates the percentage of water from the feed to the underflow. +d_t3 and −d_t3 is the cumulative weight percentage in underflow which size is coarser and finer than the target cut size, respectively. +d_t0 and −d_t0 are the cumulative weight percentage in the feed which size is coarser and finer than the target cut size, respectively. d_25c, d_50c, and d_75c are the particle size corresponding to the partition numbers 25%, 50%, and 75% from the corrected partition curve, respectively. d_50c is usually named corrected cut size of the partition curve.

In the present study, the grey system theory was used to explore the action law of different operational and structural factors on TPHS. Wherein, the grey system algorithm, given in Equations (9) and (10), was assumed to analyze the closeness between the performance and different operational and structural parameters, while the GM(1, N) algorithm [23,24] in Equations (11)–(14) was used to predict the performance evaluation for TPHS. The Equations (9)–(14) are shown as follows:

$$r(\overline{X_0},\overline{X_l})=\frac{1}{n}{\displaystyle \sum _{k=1}^{n}r\left(\overline{x_0}\left(k\right),\overline{x_l}\left(k\right)\right)}$$

(9)

$$\begin{array}{l}\overline{X_0}=\left(x_0\left(1\right),x_0\left(2\right),\dots ,x_0\left(m\right)\right)x_0{\left(1\right)}^{-1}=\left(\frac{x_0\left(1\right)}{x_0\left(1\right)},\frac{x_0\left(2\right)}{x_0\left(1\right)},\dots ,\frac{x_0\left(m\right)}{x_0\left(1\right)}\right)x_0\left(1\right)\ne 0\\ \overline{X_l}=\left(x_l\left(1\right),x_l\left(2\right),\dots ,x_l\left(n\right)\right)x_l{\left(1\right)}^{-1}=\left(\frac{x_l\left(1\right)}{x_l\left(1\right)},\frac{x_l\left(2\right)}{x_l\left(1\right)},\dots ,\frac{x_l\left(n\right)}{x_l\left(1\right)}\right)x_l\left(1\right)\ne 0\mathrm{and}l=1,2,\dots n\\ r\left(\overline{x_0}\left(k\right),\overline{x_l}\left(k\right)\right)=\frac{\underset{l}{\min }\underset{k}{\min }\left|\overline{x_0}\left(k\right)-\overline{x_l}\left(k\right)\right|+\xi \underset{l}{\max }\underset{k}{\max }\left|\overline{x_0}\left(k\right)-\overline{x_l}\left(k\right)\right|}{\left|\overline{x_0}\left(k\right)-\overline{x_l}\left(k\right)\right|+\xi \underset{l}{\max }\underset{k}{\max }\left|\overline{x_0}\left(k\right)-\overline{x_l}\left(k\right)\right|}k=1,2,\dots m\end{array}\}$$

(10)

$$\overline{x_0}\left(k\right)=a{z}_0{}^{\left(1\right)}\left(k\right)+{\displaystyle \sum _{l=1}^n{b}_l{\overline{x_l}}^{\left(1\right)}\left(k\right)}$$

(11)

$$\begin{array}{l}{\overline{x_l}}^{\left(1\right)}\left(k\right)={\displaystyle \sum \overline{x_l}\left(k\right)}k=1,2,\dots m\\ {\overline{x_0}}^{\left(1\right)}\left(k\right)={\displaystyle \sum \overline{x_0}\left(k\right)}k=1,2,\dots m\\ z_0{}^{\left(1\right)}\left(k\right)=\frac{{\overline{x_0}}^{\left(1\right)}\left(k\right)+{\overline{x_0}}^{\left(1\right)}\left(k-1\right)}{2}k=2,\dots m\end{array}\}$$

(12)

$$\stackrel{\wedge }{a}=\left[a,b_1,b_2,\dots ,b_n\right]={\left(B^{T}B\right)}^{-1}{B}^{T}y$$

(13)

$$B=\left[\begin{array}{l}{z}_0{}^{\left(1\right)}\left(2\right){\overline{x_1}}^{\left(1\right)}\left(2\right)\cdots {\overline{x_n}}^{\left(1\right)}\left(2\right)\\ \cdots \cdots \cdots \\ z_0{}^{\left(1\right)}\left(m\right){\overline{x_1}}^{\left(1\right)}\left(m\right)\cdots {\overline{x_n}}^{\left(1\right)}\left(m\right)\end{array}\right],y=\left[\begin{array}{l}{\overline{x_1}}^{\left(1\right)}\left(2\right)\\ ⋮\\ {\overline{x_1}}^{\left(1\right)}\left(m\right)\end{array}\right]$$

(14)

where, $X_0$ and $X_l$ stand the actual performance sequence (i.e., HE, etc.) and relevant factor sequence (i.e., feed concentration, etc.) of the system, respectively. $\overline{X_0}$ and $\overline{X_l}$ are the dimensionless values of $X_0$ and $X_l$, respectively. $\overline{x_0}\left(k\right)$ and $\overline{x_l}\left(k\right)$ are referred to as the value of the sequence $\overline{X_0}$ and $\overline{X_l}$, respectively, at point $k$. $r(\overline{X_0},\overline{X_l})$ represents the relative degree of grey incidence between $\overline{X_0}$ and $\overline{X_l}$ at point $k$. $\xi \in \left(0,1\right)$ means the distinguishing coefficient (equaled 0.5 in this study). The subscript l represents the l row in sequence. m and n are the number of the experiment and system’s relevant factor, respectively. the constant $a$ related to the performance evaluation is known as the development coefficient of the system. $b_l{\overline{x_l}}^{\left(1\right)}\left(k\right)$ and $b_l$ are the driving term and driving coefficient, respectively, which are related to the relevant factor of system. ${\overline{x_0}}^{\left(1\right)}\left(k\right)$ and ${\overline{x_l}}^{\left(1\right)}\left(k\right)$ are the accumulated generation operation (AGO) of $\overline{x_0}\left(k\right)$ and $\overline{x_l}\left(k\right)$, respectively. $z_0{}^{\left(1\right)}\left(k\right)$ is the mean generation of $\overline{x_0}\left(k\right)$.

3. Results and Discussions

Prior to the investigation for the effects of operational and structural parameters on the device performance, the separation characteristic of TPHS was verified compared to previous works. Following this, the closeness between the performance and the relevant factors (operation and structure parameters) was systematically explored using the Grey System theory. Then, the prediction and verification for the performance evaluation of TPHS were obtained by the GM(1, N) model. The details are described as follows.

3.1. Separation Characteristic

Figure 3a–c describes the particle size distribution for (a) overflow (b) screen underflow and (c) underflow in the TPHS at different experiments (experiment factors shown in Table 2). The particle size distribution for feed corresponding to each experiment is shown in Figure A1 in the Appendix C. It is apparent that the size distribution of TPHS in each product stream varied with different operational and structural parameters, however, the overall trends remained similar. In general, the size distribution in overflow and screen underflow were finer than that in underflow. Particularly, the dominant size distribution in screen underflow was in the range of ~0 mm to 0.25 mm, which was similar to that in overflow (~0 to 0.25 mm), relative to that in underflow (0.25 to 1 mm). It can be attributed to the adding cylindrical screen which combined centrifugal separation and screening [18].

The solid recovery and water recovery of each product stream for TPHS are exhibited in

Table 3. Solid recovery of each product stream and water recovery of underflow.

Experiment Number	Solid Recovery in Overflow ${\mathit{\gamma }}_1$ (%)	Solid Recovery in Screen Underflow ${\mathit{\gamma }}_2$ (%)	Solid Recovery in Underflow ${\mathit{\gamma }}_3$ (%)	Water Recovery in Underflow Rf (%)
1	12.76	32.20	55.04	10.26
2	13.20	40.30	46.48	6.87
3	17.77	39.04	43.19	4.89
4	9.43	46.85	43.72	5.39
5	13.05	51.38	35.57	7.41
6	5.69	42.74	51.57	6.01
7	27.30	34.45	38.28	3.35
8	12.49	47.91	39.60	3.95
9	6.45	52.79	40.76	6.62
10	10.40	52.54	37.07	8.16

, which were estimated using the formulas above (Equations (1)–(4)) and the definition, respectively. It is evident that the solid recovery in both underflow and screen underflow was more than that in overflow, which demonstrates that fewer particles discharged via overflow. In addition, the water recovery in underflow was less than that in overflow and screen underflow, which indicates that less water (i.e., lower bypass flow fraction [31]) flowed through underflow stream. In Figure 4, the absent “fish-hook” can be observed distinctly in all corrected partition curves (plotted by Equations (5) and (6)) of TPHS, which leads to the fewer fines sent to underflow and the increasing device performance. This result can be ascribed to the presence of cylindrical screen that altered the size distribution in each product stream [18]. Both the above size distribution and the corrected partition curves of TPHS shows the similar trends to the literature [18], which also verifies the reasonable results in this work.

3.2. Closeness Analysis

Table 4. Performance evaluation under different relevant factors for TPHS.

Items			Experiment Number
			1	2	3	4	5	6	7	8	9	10
Performance $X_0$	$X_0$	HE (%)	61.160	60.854	63.515	67.908	57.911	63.948	64.665	65.155	64.626	59.310
	$X_0$	I	0.370	0.380	0.383	0.347	0.307	0.380	0.396	0.360	0.340	0.365
	$X_0$	d50c (mm)	0.230	0.250	0.235	0.245	0.285	0.230	0.240	0.250	0.250	0.260
Relevant factors $X_1$	$X_1$	Feed concentration (g/L)	60.00	100.00	140.00	180.00	220.00	60.00	100.00	140.00	180.00	220.00
	$X_2$	Feed pressure (MPa)	0.04	0.05	0.06	0.04	0.05	0.06	0.04	0.05	0.06	0.04
	$X_3$	Aperture size (mm)	0.60	0.70	0.80	0.60	0.70	0.80	0.60	0.70	0.80	0.60
	$X_4$	Diameter of spigot (mm)	18.00	24.00	18.00	24.00	18.00	24.00	18.00	24.00	18.00	24.00
	$X_5$	Diameter of vortex finder (mm)	30.00	40.00	50.00	60.00	30.00	40.00	50.00	60.00	30.00	40.00

shows the performance evaluation (HE, I, and d_50c) under different operation and structure factors (feed concentration, feed pressure, aperture size, spigot diameter, and vortex finder diameter) for TPHS in the present study. The HE and I were calculated using Equations (7) and (8), respectively, while d_50c was estimated according to its definition. It is distinct that the performance evaluation HE, I and d_50c of TPHS fluctuates around 63%, 0.363 and 0.248 mm, respectively, with different factors.

To explore the effect of the operation and structure parameters on the performance of TPHS, the relative degree of grey incidence was obtained using Equations (9) and (10), which is shown in Figure 5. It is noted that the relative degree of grey incidence can be considered to reflect the closeness of the relationship between parameters and performance evaluation. The bigger relative degree of grey incidence reveals the higher closeness, vice versa. The details for the calculation are shown in Appendix A. From Figure 5, it is found that both the operational and structural parameters mainly influenced the device performance, in which the relative grey incidence degree of each factor was more than 0.5. In detail, the diameter of spigot shows the highest grey incidence degree of 0.895 and 0.878 for the HE and I, respectively, while the aperture size presents the highest grey incidence degree of 0.895 for the d_50c. Further, the feed pressure and vortex finder diameter display the smaller grey incidence degree which were ~97% and ~88% of the highest degree, respectively. Besides, the feed concentration exhibits the smallest grey incidence degree to each performance evaluation, which was only ~65% of the highest degree. Thus, the order of the grey incidence degree for the performance of TPHS can be summarized as follows: Spigot diameter and aperture size the highest, followed by feed pressure and vortex finder diameter, feed concentration the lowest. This trend reveals that the structure consisting of the spigot diameter and aperture size plays significant roles in the device performance, compared to the operation including feed pressure and feed concentration. To a certain extent, this trait also indicates that TPHS shows good adaptability to the operation conditions.

3.3. Prediction and Verification

To predict the performance of TPHS, the dimensionless sequence $\overline{X_0}$ and $\overline{X_l}$ (see

Table A1. Dimensionless of Grey System theory for TPHS.

Items		Experiment_
		1	2	3	4	5	6	7	8	9	10
HE	$\overline{X_0}$	1.000	0.995	1.039	1.110	0.947	1.046	1.057	1.065	1.057	0.864
$\overline{X_l}$
Feed concentration	$\overline{X_1}$	1.000	1.741	2.332	3.068	3.627	1.099	1.726	2.422	2.987	3.620
Feed pressure	$\overline{X_2}$	1.000	1.250	1.500	1.000	1.250	1.500	1.000	1.250	1.500	1.000
Aperture size	$\overline{X_3}$	1.000	1.200	1.400	1.000	1.200	1.400	1.000	1.200	1.400	1.000
Diameter of spigot	$\overline{X_4}$	1.000	1.333	1.000	1.333	1.000	1.333	1.000	1.333	1.000	1.333
Diameter of vortex finder	$\overline{X_5}$	1.000	1.333	1.667	2.000	1.000	1.333	1.667	2.000	1.000	1.333

in Appendix A) were conducted using the aforesaid GM(1, N) model (see Appendix B for the details of calculation). The sequence of parameter $\stackrel{\wedge }{a}$ given by Equations (13) and (14) are described in

Table 5. Parameter $\stackrel{\wedge }{a}$ for the dimensionless performance evaluation in TPHS.

Items	Parameter $\stackrel{\wedge }{\mathit{a}}$	HE (%)	I	d50c (mm)
Development coefficient of system	a	−1.842	−1.604	−1.690
Driving coefficient bl	b1	−0.019	−0.129	0.065
	b2	−6.227	−9.021	−9.541
	b3	7.902	10.994	11.632
	b4	−0.001	0.016	−0.127
	b5	0.175	−0.035	−0.142

, while the prediction models were obtained using Equations (11) and (12) for the dimensionless value of HE, I and d_50c are shown in Equation (15). Note that the coefficient a, b₁, b₂, b₃, b₄, and b₅ in Equation (13) were corresponding to the HE (or I or d_50c), feed concentration, feed pressure, aperture size, spigot size, and vortex finder diameter, respectively.

From Table 5, it is apparent that the systematic development coefficient a was negative to the prediction of HE, I and d_50c, while the driving coefficient b_l~b₅ fluctuated with different performance evaluations. In detail, the b₁~b₂ and b₃ were negative and positive, respectively, to all the prediction value. The b₄ was negative and positive to the predicted HE (or d_50c) and I, respectively, while the b₅ was negative and positive to the predicted I (or d_50c) and HE, respectively. The present prediction model indicates that the predicted dimensionless HE (I or d_50c) reduced with the decreasing accumulated generation operation (AGO) of the aperture size or the increasing AGO of the previous HE (I or d_50c), feed concentration and feed pressure. Further, with the increasing AGO of the spigot size, the predicted HE and d_50c decreased but the predicted I increased. Conversely, as the AGO of the vortex finder diameter rose, the predicted HE went up yet the I and d_50c went down. Note that the systematic development coefficient a related to the AGO of the HE (I or d_50c) was used in the GM(1, N) model. Thus, the prediction model (Equation (15)) for the performance evaluation of TPHS can be dynamically updated according to the actual operation, which improves the applicability and accuracy of the prediction model.

$$\left[\begin{array}{l}\overline{x_0}(k)\_HE\\ \overline{x_0}(k)\_I\\ \overline{x_0}(k)\_d_{50c}\end{array}\right]=\left[\begin{array}{l}-0.959-0.010-3.2424.113-0.00040.091\\ -1.604-0.129-9.02110.9940.016-0.035\\ -0.916-0.035-6.1716.305-0.0687-0.077\end{array}\right]\left[\begin{array}{lll}{\overline{x_0}}^{\left(1\right)}(k-1)\_HE& {\overline{x_0}}^{\left(1\right)}(k-1)\_I& {\overline{x_0}}^{\left(1\right)}(k-1)\_d_{50c}\\ {\overline{x_1}}^{\left(1\right)}\left(k\right)& {\overline{x_1}}^{\left(1\right)}\left(k\right)& {\overline{x_1}}^{\left(1\right)}\left(k\right)\\ {\overline{x_2}}^{\left(1\right)}\left(k\right)& {\overline{x_2}}^{\left(1\right)}\left(k\right)& {\overline{x_2}}^{\left(1\right)}\left(k\right)\\ {\overline{x_3}}^{\left(1\right)}\left(k\right)& {\overline{x_3}}^{\left(1\right)}\left(k\right)& {\overline{x_3}}^{\left(1\right)}\left(k\right)\\ {\overline{x_4}}^{\left(1\right)}\left(k\right)& {\overline{x_4}}^{\left(1\right)}\left(k\right)& {\overline{x_4}}^{\left(1\right)}\left(k\right)\\ {\overline{x_5}}^{\left(1\right)}\left(k\right)& {\overline{x_5}}^{\left(1\right)}\left(k\right)& {\overline{x_5}}^{\left(1\right)}\left(k\right)\end{array}\right]$$

(15)

The comparison between the predicted and actual value for HE, I and d_50c under the above experiment 1–10 (see Table 2 for experiment details) was used to verify the accuracy of perdition model. From the dimensionless performance evaluation (Equation (10)) for TPHS, the predicted dimensionless values which times the corresponding $x_0\left(1\right)$ can be converted to the prediction of HE, I, and d_50c, respectively. Figure 6 shows the comparison of the actual and predicted performance evaluation (a) HE (b) I and (c) d_50c, while the relative error and mean relative error are presented in

Table 6. Error checks.

Items		Relative Error = (Predicted-Actual)/Predicted*100%
		HE	I	d50c
Experiment	2	−2.49%	3.35%	−1.76%
	3	2.93%	−10.86%	2.50%
	4	−4.88%	−7.58%	5.42%
	5	2.22%	5.71%	−5.09%
	6	2.11%	4.81%	−2.96%
	7	0.62%	2.91%	3.86%
	8	4.44%	4.36%	−6.00%
	9	−5.44%	3.64%	−0.27%
	10	0.37%	−8.52%	4.26%
Mean relative error, %		2.83%	5.75%	3.57%

. In Figure 6, it is evident that the performance predicted by the Equations (14)–(16) shows good agreement with that obtained by the actual experiment (from experiment 1–10). From Table 6, it can be seen that although the relative error fluctuated, the mean relative error between the actual and predicted HE, I, and d_50c were 2.84%, 5.83%, and 3.57%, respectively. This result demonstrates that the prediction models obtained using the GM(1, N) model can well predict the performance evaluation HE, I, and d_50c for TPHS.

4. Conclusions

In the present study, a new three products hydrocyclone screen (TPHS) which combine both centrifugal classification and screening to separate particle depending on size has been successfully developed. The pilot-scale experiments of coal samples (0–1 mm) were widely carried out conducting a 150-mm TPHS aiming to explore the effects of operation (feed concentration and feed pressure) and structure (aperture size, spigot diameter, and vortex finder diameter) parameters on the device performance (Hancock classification efficiency (HE), imperfection (I), and the cut size (d_50c). The obtained results were verified firstly, and then critically analyzed using the Grey System algorithm and GM(1, N) algorithm. Based on the presented results, the following conclusions can be obtained:

(1)

In TPHS, the size distribution in overflow and screen underflow was finer than that in underflow, further, the absent “fish-hook” can be observed due to the presence of cylindrical screen. This is consistent with the previous literature.

(2)

The closeness analysis using the Grey System theory reveals that the operational and structural parameters greatly influenced the performance of TPHS with the relative grey incidence degree of ~0.5 to 1. Further, the order of grey incidence degree between different parameters and HE (or I or d_50c) is that: The spigot diameter and aperture size present the highest incidence degree; the feed pressure and vortex finder diameter exhibit the middle incidence degree; and the feed concentration shows the lowest incidence degree.

(3)

The dynamic performance prediction model using the GM(1, N) can be generated according to the operation, structure, and previous performance value. As the accumulated generation operation (AGO) of the aperture size decreased or the AGO of the previous HE (I or d_50c), feed concentration and feed pressure increased, the predicted HE (I or d_50c) decreased. Besides, the predicted HE and d_50c reduced but the predicted I improved with the increasing AGO of the spigot size, while the predicted HE declined yet the I and d_50c increased with the growing AGO of the vortex finder diameter.

(4)

The performance evaluations estimated using the prediction model present excellent agreement with those gotten using the actual pilot-scale experiment, wherein the mean relative errors were 2.84%, 5.83%, and 3.57%, respectively.