Hybrid modeling of an industrial grinding-classification process

Powder Technology 279 (2015) 75–85

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Powder Technology journal homepage: www.elsevier.com/locate/powtec

Hybrid modeling of an industrial grinding-classification process Xiaoli Wang a,b, Yalin Wang a, Chunhua Yang a,⁎, Degang Xu a, Weihua Gui a a b

School of Information Science & Engineering, Central South University, Changsha, 410083, China Postdoctoral workstation, Aluminum Corporation of China Limited Zhongzhou Branch, Jiaozuo, 454171, China

a r t i c l e

i n f o

Article history: Received 16 November 2014 Received in revised form 2 March 2015 Accepted 22 March 2015 Available online 1 April 2015 Keywords: Grinding-classification process Diasporic bauxite Hybrid modeling Phenomenological method LSSVM

a b s t r a c t An industrial grinding-classification process of diasporic bauxite is modeled based on the integration of phenomenological and statistical learning methods. The breakage characteristics of the ore and running status of the whole process are first investigated by laboratory testing and process sampling, respectively. Based on the population balance model (PBM) framework, the breakage distribution function is estimated from laboratory test data. The breakage rates are back-calculated directly from the industrial data, where a nonlinear breakage rate function is proposed for coarse particles. They are then correlated to the operating variables (including the water flow rate and feed flow rate), ball characteristics and material properties using the least squares support vector machine (LSSVM) method so that the model is suitable to various grinding conditions. Material transportation through the mill was treated as two equal smaller fully mixed reactors followed by a large one. The particle size distribution (PSD) of the mill product is then predicted by sequentially solving the reactors in series, considering the nonlinear breakage kinetics. A spiral classifier model is obtained with the Rosin–Rammer curve, where the bypass, real classification effect and operating conditions are included. The simulation results of the whole process by using the sequential module approach (SMA) demonstrate reasonable agreement between the predicted and measured industrial process data. The models are finally applied to the process for the prediction of particle size indices and to provide valuable information for the operation and further optimization of the process. © 2015 Elsevier B.V. All rights reserved.

1. Introduction Grinding is a common size reduction process in mineral processing, the pharmaceutical and cement industries, and other fields. In mineral processing, it is usually followed by a classification process to produce a slurry with a desired particle size and solid concentration for flotation. The performance of the grinding-classification process plays a determinative role on the financial and technical indices of the whole mineral processing plant. Diasporic bauxite is the main raw ore for alumina production in China, where a large percentage of the world's alumina is produced. Due to the low grade of the ores, mineral processing, which is a potential method of using high silica bauxite [1], is applied before the Bayer process in China to greatly reduce the cost of alumina production and environmental pollution. In this process, diasporic ores excavated from several deposits are crushed and homogenized in a heap using a complex procedure to prepare the feed for grinding. The deposits are

⁎ Corresponding author at: Room 309, Minzhu Building, School of Information Science and Engineering, Central South University, Changsha, 410083, China. Tel./fax: +86 731 88836876. E-mail addresses: [email protected] (X. Wang), [email protected] (Y. Wang), [email protected] (C. Yang), [email protected] (D. Xu), [email protected] (W. Gui).

http://dx.doi.org/10.1016/j.powtec.2015.03.031 0032-5910/© 2015 Elsevier B.V. All rights reserved.

not the same for different heaps, and the grades of the ores steadily decrease, which has a strong effects on the grindability and floatability when a changeover between the heaps occurs. In addition, the process indices are manually measured and time-delayed and are then controlled by operators, depending on their experience. As a result, many problems, including a large fluctuation of product quality and throughput, waste of flotation agents, and low recovery of useful minerals, increase the cost. Therefore, modeling and process simulation are carried out to improve the process indices, including the particle size and solid concentration of the slurry flows. The population balance model (PBM) introduced by Refs. [2,3] is a popular way to model the grinding process. It predicts the product size distribution from the feed size distribution by dividing the process into two parts, material transport through the mill (described using the residence time distribution (RTD)) and breakage within the mill, including the selection of the particles for breakage (described using a breakage rate function) and the actual breakage resulting in a particular distribution of fragment sizes after the particle is selected (described using a breakage distribution function). Based on this mass balance framework, different PBMs have been proposed and used for design, simulation, control and optimization [e.g., 4–12] to improve the grinding effectiveness. The main difference between these models lies in the form of the breakage rate function and the material transport description.

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X. Wang et al. / Powder Technology 279 (2015) 75–85

PBM can also be combined with other methods to integrate the advantages of different methods. A combination of PBM with the discrete element method (DEM) is popular, where DEM is used to simulate different types of breakage, such as impact breakage and abrasion/ chipping breakage [13,14]. Akkisetty et al. [15] proposed a model combining PBM with a neural network, where the neural network is used for the prediction of the breakage rate and breakage distribution. The model is only validated by laboratory data. Models using solely data-based methods, such as neural networks, support vector regression, or fuzzy inference, have also been developed for soft sensing the particle size indices in the grinding-classification process [e.g., 16–18]. One key problem in predicting the particle size distribution is the lack of knowledge of the breakage rate of the material being ground. The properties of different materials are different, and each grindingclassification process also has its own features. For an industrial continuous grinding process, it is especially difficult to obtain accurate prediction values due to the complicated running conditions. Therefore, much work has been conducted to develop a ball mill model and a spiral classifier model for the processing of bauxite. In this work, we present a hybrid ball mill model and a spiral classifier model for the grinding-classification process of bauxite, where the material characteristics and operation conditions are all considered. The models are useful for particle size distribution (PSD) prediction, process simulation and further optimization. The work is organized as follows: In the next section, the process knowledge and sampling campaign are introduced. The laboratory test method and findings are provided in Section 3, and the breakage distribution is estimated. In Section 4, the continuous ball milling process is modeled, including the residence time distribution and the breakage rate function. Section 5 discusses the modeling of the two spiral classifiers. The simulation and application results using the sequential module approach (SMA) are shown and analyzed in Section 6. Finally, the work is concluded.

12 mm, but it usually contains more than 30% of + 12 mm particles because of the low efficiency of the crushing process, so that the performance of the milling process is reduced. The maximum diameter of the two spiral classifiers is 3 m. Spiral classifier 1 is of the high-weir type so that it is suited for classifying coarse particles, while classifier 2 is of the sinking type, which is suited for fine particles. The underflow of classifier 2 is directly taken as the concentrate due to the obvious selective breakage of the bauxite ore, which is different from the usual grinding process of many other ores.

2. Process description and sampling campaign

2.3. Process data analysis

2.1. Process description

The process data show a large fluctuation of the grindingclassification circuit. The maximum and minimum circulating loads are 142.2% and 495.1%, respectively, in all of the samples. The process indices of the greatest significance (including the − 0.075-mm particle size fraction and the solid concentration in the two overflows) are also unstable, as illustrated in Table 1. The grindability of the ore is a non-negligible disturbance. This may also be related to the great variation of the amount of coarse particles in the fresh feed, as shown in Fig. 2, where the mean value of the 30 samples and the max and the min values of each size interval are given. It can be seen that, in addition to the coarse particles, the fraction of the fine particles in the feed is large as well.

The industrial grinding-classification process consists of one ball mill and two spiral classifiers, as the schematic diagram shows in Fig. 1. The ball mill has an inner diameter of 3.6 m and a length of 5.0 m. The initial media charge of the ball mill is 80 tons of cast steel balls with a make-up diameter of 70:50 mm and a ratio of 2:1. A certain amount of 90-mm diameter fresh balls are replenished to the mill once a week. Diasporic ores excavated from several deposits are crushed and homogenized into heaps through a complex procedure and then used as the feed of the ball mill. The designed maximum size in the feed is

2.2. Sampling campaigns Several sampling campaigns were carried out on the whole grinding-classification process to assess the mill-classifier operation and to collect data for modeling. It is difficult to keep the samples representative because of the high flow rates of some streams and the rapid coarse particle sinking. Therefore, each flow was sampled four times in 2 h to obtain one blended sample. The PSD and concentration of each sample were carefully analyzed. Meanwhile, the operating conditions, including the flow rate of the fresh ore, water flow rate, and currents of mill and classifiers, were recorded from the on-line monitoring system. The sampling campaign was carried out twice a day, and 30 groups of data were finally obtained. The chemical compositions of all of the flows and each size interval of these flows were analyzed. When the sampling campaign was started, the balls were used for a long period of time so that their make-up was not clear. Therefore, the ball distribution and the ball load were sampled and estimated when the ball mill was stopped after the sampling campaign. Data reconciliation was then conducted using the commercial package Bilmat® to minimize the sampling and analysis errors and estimate the immeasurable slurry flow rates. The adjusted data were then used to investigate the process performance and perform the modeling work.

Fig. 1. Schematic diagram of the grinding-classification process.

X. Wang et al. / Powder Technology 279 (2015) 75–85 Table 1 Coarse and fine particle fractions in different flows. Flow name

Particle size (mm)

Max fraction (%)

Min fraction (%)

Underflow 1 Overflow 1 Overflow 2

+3.35 −0.075 −0.075

20.91 73.64 94.98

0.47 57.94 87.36

3. Experimental method and results Wet grinding tests of diasporic ores were performed in a laboratory ball mill to investigate the grindability of the ores and to estimate their breakage distribution parameters. The ball mill has a 240-mm diameter and 300-mm length. Cast steel balls with diameters of 32, 22, and 19 mm were used, and were numbered 32, 115 and 350. The ore was crushed to − 3.35 mm for tests using a laboratory jaw crusher. The crushed material was then sampled and sieved into eight mono-sized intervals, which are represented by their upper limits as 3.35, 2.0, 0.85, 0.5, 0.25, 0.15, 0.075, and 0.045 mm. Then, several make-up feeds, all of the mono-sized intervals and the − 3.35-mm material with natural size distributions were ground several times for different grinding times. More details of the tests can be found in Ref. [19]. Diasporic ores collected from several heaps of the feed to the industrial ball mill were tested. The reproducibility of the tests was first confirmed, as illustrated in Fig. 3, showing each feed ground three 25

3.5

3

2.5 −1

0

10 Size (mm)

1

10

Fig. 2. PSD and A/S of each size interval in the fresh feed.

2

10

n X

bk j :

ð2Þ

k¼i

In batch grinding, if the breakage rate is constant, the breakage rate of the top-size interval can be calculated as follows ln ðw1 ðt Þ=w1 ð0ÞÞ ¼ −S1 t

ð3Þ

where w1(0) is the initial mass fraction of the top-size interval. On semilog coordinates, the slope is −S1. If S1 does not vary with grinding time, the breakage of the top-size interval is called first-order or linear. Otherwise, it is called non-first order or nonlinear. From the mono-sized grinding data, it is found that the fine and the coarse fractions of the tested ores follow non-first order breakage and the deceleration rates of the breakage are non-uniform for different size intervals, as shown in Fig. 4. The test results of different ores have the same grindability trend but different breakage rates. Due to the non-uniform deceleration of the breakage rates, a back-calculation method based on the piecewise linearization and interpolation of the breakage rates was proposed to estimate the non-normalizable breakage distribution function of diasporic bauxite [21]. The empirical function used for the breakage distribution is [22] Bi; j ¼ b1

1 xj

!b

4

xi xj

!b

2

þ ð1−b1 Þ

1 xj

!b

4

Þ

xi xj

!b

3

:

ð4Þ

4. Model development for the industrial ball milling process

5

10

Bi j ¼

4.5

4

10

where i and j are the size-interval indices running up to n; i = 1 represents the coarsest particles; and i = n represents the finest particles. t, w, S and b denote grinding time, mass fraction, breakage rate function, and breakage distribution function, respectively. bij denotes the fraction of size j material, which appears with size i upon primary breakage. The first term and the second term on the right of Eq. (1) are the disappearance rate and the appearance rate of size i, respectively. Usually, the breakage distribution function is presented in cumulative form as

5

5.5

A/S

Particle size distribution (% )

Mean PSD Mean A/S Max value of each size interval Min value of each size interval

ð1Þ

The obtained breakage distribution is plotted in Fig. 5, with parameters of b1 = 0.406, b2 = 0.3038, b3 = 2.4388, b4 = 0.0134. The simulation results using the obtained breakage distribution are in good agreement with the experimental data (see Fig. 6). The determined breakage distribution function was then directly used to model the industrial mill for the sake of simplicity.

6

15

0 −2 10

times. It can be seen that the size distributions of the products are in very good agreement with each other. The size-discrete, time continuous PBM for batch grinding is [20] i−1 X dwi ðt Þ bi j S j ðt Þwi ðt Þ; n ≥ i ≥ j ≥ 1 ¼ −Si ðt Þwi ðt Þ þ dt j¼1 i N1

In the fresh feed, the grade (i.e., A/S, which is the ratio of the contents of Al2O3 and SiO2, a low A/S usually means less diaspore mineral and more silicate minerals, while a high A/S means more diaspore mineral and less silicate minerals) of the − 0.045-mm interval is the lowest, but it becomes much higher when the particle size increases (see Fig. 2). This may be because the ores have strong heterogeneity and selective breakage properties. The silicate minerals are easy to overgrind and produce finer particles, while the diaspore mineral is difficult to break. This is consistent with the findings of the laboratory tests, where nonlinear breakage was obvious [19]. Hence, there is a need for the careful operation of the grinding process to obtain diaspore minerals with the desired size while avoiding overgrinding the gangue minerals. Among all of the flows, the A/S of underflow 1 is the highest. The A/S of underflow 2 is not stable. The A/S of the coarsest and finest size intervals are both very low. As such, with the decrease of the grade of the fresh ore, the grade of underflow 2, which is taken as the concentrate, becomes unqualified more and more frequently. Therefore, the optimization of the process is extremely urgent.

20

77

For the continuous ball milling process, the breakage rate, breakage distribution and RTD of materials in the mill are the three main parts of the PBM. The model framework can be found in Fig. 7. The breakage distribution function is determined from the laboratory test (see the laboratory results in Section 3). The residence time distribution (RTD) is modeled as several fully mixers in series; the overall mean residence time is calculated from the industrial data. The breakage rate are fitted from industrial data; and to achieve higher precision and make the model more adaptive to variations in the process, LSSVM models are established between the milling conditions and the breakage rate parameters. In Fig. 7, N is the number of equivalent mixers for the mill, τ

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X. Wang et al. / Powder Technology 279 (2015) 75–85 2

2

10

10

Weight Fraction Finer Than Size (%)

b

Weight Fraction Finer Than Size (%)

a

feed 0.5min−1st grinding 0.5min−2nd 0.5min−3rd 1min−1st 1min−2nd 1min−3rd

1

10 −2 10

−1

0

10

1st grinding 2nd grinding 3rd grinding

0

1

10

1

10

10

10 −2 10

−1

0

10

Size (mm)

1

10

10

Size (mm)

Fig. 3. Reproducibility of the tests. a) Feed (−3.35 mm) with a natural size distribution ground for 0.5 and 1.0 min. b) Feed (−3.35 + 2.0 mm) ground for 1.0 min.

is the overall mean residence time, τl is the mean residence time of the lth mixer, and aT, α, μ, Λ are the breakage rate parameters. The details of obtaining the RTD model and the breakage rate parameters are as follows.

4.1. Residence time distribution (RTD) The RTD function describes the transport of materials through the mill, which is a significant factor that affects the fineness of the product. Based on radioactive tracing studies, several RTD models have been proposed, which can be found in Refs. [20,23–25]. RTD functions for ball mills can be adequately described by three unequal perfectly mixed segments, and when the last segment is significantly larger than the other two, it is consistent with post classification in the ball mill being considered [26]. Therefore, a model of two small and one large fully mixed reactor in series is used as the RTD model. Then, the overall mean residence time is τ = τ1 + τ2 + τ3 with τ1 = τ2 b τ3, where τ3 is the mean residence time for the larger reactor and τ1, τ2 are the mean residence times for the two smaller ones. The overall mean residence time can be calculated as follows τ ¼ W= F

ð5Þ

where W is the slurry hold-up in the mill (t) and F is the feed flow rate (t/h), including the circulating load. When the mill is running at a certain speed, W varies with the ball load, the slurry density, the feed rate, and other parameters [27]. As it is difficult to measure W in the industrial process, we assume that the slurry hold-up volume is constant during the sampling period because the ball load does not vary significantly. Then, the hold-up volume was measured at a sudden stop of the mill, and the slurry hold-up can then be calculated as follows W¼

C s V 0 ρw ρ ρm þ C s ρw −C s ρm m

ð6Þ

where V0 is the volume of the hold-up (m3); Cs is the slurry concentration (% wt), which is assumed to be equal to the discharge concentration; and ρm and ρw are the densities (t/m3) of the ore and water, respectively. From the measured data, τ is calculated using Eqs. (5) and (6) and plotted in Fig. 8. Let τ3 = γτ, then τ1 = (1 − γ)τ/2. Hence, only the parameter γ needs to be determined. According to the measured values of RTD, γ varies with the geometry of the mill (L/D for ball mill), feed flow rate, slurry concentration in 0

10 0

1

1

ln(w (t)/w (0))

Cumulative Breakage Distribution

10

−3.35+2.0 mm −2.0+0.85 −0.85+0.5 −0.5+0.25 −0.25+0.15

−1

10

0

1

2

3

4

5

6

Grinding time (min) Fig. 4. Breakage kinetics of mono-sized intervals.

7

8

Bi1 Bi2 Bi3 Bi4 Bi5 Bi6 Bi7 −2

10

−1

0

10

10

Size (mm) Fig. 5. The back-calculated breakage distribution.

1

10

X. Wang et al. / Powder Technology 279 (2015) 75–85

90

5.5 Overall mean residence time (min)

6

Weight Fraction Finer Than Size (%)

100

80 70

Time (min) 8.0

60 50 40 30

4.0 2.0 1.0

Feed Experimental Simulated

0.5

20 −2 10

−1

0

10

5 4.5 4 3.5 3 2.5 2

1

10

79

10

0

5

10

Size (mm)

8 j−1 X > > > cik c jk >− > > < k¼i ci j ¼ 1 > i−1 > > 1 X > > Sk bik ck j > : S −S i j k¼ j

the mill and mill speed [20,24,28]. For example, in the measured values, γ varies from 0.38 to 0.67 according to Ref. [20] for a wet overflow ball mill, from 0.73 to 0.76 in Ref. [24] for a ball mill and from 0.38 to 0.95 for a centrifugal mill in Ref. [28]. The direct measure of the RTD is not available in this study. Meanwhile, the breakage rate parameters are fitted from the industrial data. If γ is fitted simultaneously, they must influence each other during the fitting procedure. Therefore, γ was fixed at a certain value. In Ref. [20], γ = 0.45 when L/D = 1.34 and F = 105 t/h, and γ = 0.57 when L/D = 2.0 and F = 114 t/h, so γ was fixed at 0.5 for our mill that had a L/D of 5/3.6 = 1.38. The ball mill model is then solved for one mixer by another. That is, the product size distribution of the first mixer is calculated first and then used as the feed of the second mixer, and so on. For the lth mixer, i X

di j f i ; n ≥ i ≥ 1

  e j ¼ 1= 1 þ S j τ l :

25

30

i¼ j

ð9Þ

iN j

ð10Þ

The breakage rate function is dependent on many factors, including the ore properties, slurry concentration, and media characteristics [20, 29,30]. The breakage rate of the industrial grinding process is very different from that of the batch testing process, and it is impossible to measure directly. Usually, S is scaled up from laboratory batch or closed-loop test data or estimated directly from real plant data [4,10]. The scale-up procedure is not suitable when some of the detailed information of the process, such as the ball size distribution, cannot be obtained. The latter is a black-box procedure that cannot discriminate the influence of each factor. Therefore, the breakage rate function is first identified from the process data and then modeled as a function

ð7Þ

8 i¼ j e > < j i−1 X di j ¼ cik c jk ðek −ei Þ i N j > :

ib j

4.2. Breakage rate function

j¼1

ð8Þ

k¼ j

aT

N l l 1

l 1, 2,..., N

1

20

Fig. 8. The overall mean residence time of the industrial milling process.

Fig. 6. Simulation results of the test data using the back-calculated B.

pi ¼

15 Sampling point

2

N

Fig. 7. Model structure of the continuous ball milling process.

80

X. Wang et al. / Powder Technology 279 (2015) 75–85

of the ore properties and operating conditions using the least squares support vector machine (LSSVM). S has been empirically proposed to be a function of particle size for first-order breakage [20], Si ¼ aT

 α  α xi x 1 Q i ¼ aT i ; Λ≥0 x0 x0 1 þ ðxi =μ ÞΛ

ð11Þ

where xi is the upper limit of the size interval indexed by i, x0 is 1 mm, and aT and a are model parameters depending on the properties of the material and the grinding conditions. Qi is a correction factor, which is 1 for smaller sizes (normal breakage) and smaller than 1 for particles that are too large (Nxm) to be nipped and fractured properly by the ball size in the mill (abnormal breakage). μ is the particle size at which the correction factor is 0.5. Λ indicates how rapidly the breakage rates decrease as the particle size increases. It is difficult to determine whether the diasporic ore follows firstorder or non-first order breakage in the industrial ball mill. However, the heterogeneity of the ore is the most probable reason for non-first order behavior in the laboratory test [19], and there is a considerable fraction of coarse particles that are too large to be nipped by the balls. Therefore, particles larger than 12 mm are considered to follow nonfirst order breakage in the industrial milling process because 12 mm is an optimal size to trade-off between the power draw of the milling process and the crushing process, and it is also the prescribed top size in the fresh feed but cannot be satisfied. Then, the breakage rate is modeled as follows  Si ðt Þ ¼

Si ð0Þki ðt Þ; Si ð0Þ;

xi ≥ 12 mm xi b 12 mm

ð12Þ

where ki(t) is a slowing down factor and Si(0) is the first-order breakage expressed in Eq. (11). According to the test results for the size intervals −3.35 + 2.0 mm and −2.0 + 0.85 mm, the slowing down factor of the breakage rate is fitted by Eq. (13) and plotted in Fig. 9.  ki ðt Þ ¼ a1 exp a2

    t t 2 þ a3 exp a4 R ¼ 0:896 Si ð0Þ Si ð0Þ

ð13Þ

where a1 = 0.9528, a2 = − 1.409, a3 = 0.1866, a4 = 0.04257, and Si(0) are the breakages rate with first-order breakage. The reconciled process data were used in groups to estimate the breakage rate parameters. As Fig. 10 shows, the four estimated parameters are not the same for different data groups subjected to different operating conditions. 1.1 Experiment value Fitted curve

1

Investigation of the ball milling processes of quartz, hematite, and other minerals demonstrated that [4,20] the breakage rate parameters a and α are dependent on ore properties and milling conditions; μ varies with milling conditions, and Λ is mainly a function of ore properties. To make the model more suitable to the variation of the ore properties and the operating conditions, the LSSVM method [31] is adopted to find the correlations in Eqs. (14) to (17) because LSSVM is suitable for small sample data and can effectively overcome the disadvantages of overfitting and local optima [32]. aT ¼ Lssvm1 ðAs ; A Fe ; Q ; Q w ; I m ; Ic ; db max Þ

ð14Þ

α ¼ Lssvm2 ðAs ; A Fe ; Q ; Q w ; Im ; Ic Þ

ð15Þ

μ ¼ Lssvm3 ðQ ; Q w ; Im ; Ic ; db max Þ

ð16Þ

Λ ¼ Lssvm4 ðAs ; A Fe ; db max Þ

ð17Þ

where As is the ore grade (i.e., A/S), AFe is the % wt of ferrum (Fe), dbmax is the maximum ball size (mm), Q is the fresh feed rate (t/h), Q w is the water feed rate (t/h) to the mill, Im is the mill current (A) and Ic is the classifier current (A). The modeling process using the LSSVM method is simply described as follows. In the SVM method, for a given training sample set (xi, yi) (i = 1, 2, ⋯, M) with m inputs (namely, xi ∈ Rm) and one output and where M is the sample number, the training data in the original input space is mapped to a high-dimensional feature space by using the nonlinear function φ(x), so that the nonlinear regression problem in the original sample space is transformed into a linear regression problem in the high-dimensional feature space. The linear regression can be expressed as follows T

yi ðxi Þ ¼ w φðxi Þ þ b

where w and b are the weight vector and bias vector, respectively. According to the principle of structural risk minimization, the regression problem in Eq. (18) is transformed into the following optimization problem min J ðw; eÞ ¼

slowing down factor k(t)

M 1 T CX 2 e ; w wþ 2 2 i¼1 i

T

s:t: yi ¼ w φðxi Þ þ b þ ei ;

ð19Þ

ði ¼ 1; 2; ⋯; MÞ:

where ei is the error vector and C N 0 is a penalizing factor to control the trade-off between the model complexity and the training error. Solve this optimization problem by using the Lagrange multiplier method, the optimal ai and b can thereby be found, and the relationship can be represented as follows

0.9

yi ðxi Þ ¼

0.8

ð18Þ

M X

  α i K xi ; x j þ b

ð20Þ

i¼1

0.7 0.6 0.5 0.4 0.3 0.2 0.1

0

1

2 3 grinding time /Si(0)

4

Fig. 9. Slowing down factor of the large particles.

5

where ai is the Lagrange multiplier and K(xi, xj) is the kernel function that needs to satisfy the Mercer condition [33]. The kernel function is important to the performance of SVM models. The common kernel functions for nonlinear problems are RBF, polynomial, multi-layer perceptron, and sigmoid. Thus far, no analytical or empirical study has conclusively established the superiority of one kernel over another; thus, the kernel is usually selected by experiment or experience. For different applications, different kernels have been found to be the most suitable, or sometimes several kernels show almost identical performance [e.g., 34–36]. In this work, RBF, polynomial and spline kernels are compared. Polynomial is the worst of the three

X. Wang et al. / Powder Technology 279 (2015) 75–85

10 Value of breakage rate parameters

Value of breakage rate parameters

2.5 a α

2 1.5 1 0.5 0

81

0

10 20 Smaple number

30

μ Λ

8 6 4 2 0

0

10 20 Smaple number

30

Fig. 10. The estimated breakage rate parameters for the industrial milling process.

kernels, and RBF is slightly better than the spline kernel. Therefore, RBF is used as the kernel function, 

K xi ; x j



0

2 1    x −x j  B  i C ¼ exp@− A: 2σ 2

ð21Þ

Using the back-calculated breakage rates and the corresponding operating conditions, the model was developed mainly based on the MATLAB LS-SVM toolbox [37]. A simplex algorithm and crossvalidation were used to optimize the kernel and the penalizing parameters of the four LSSVM models. The parameters are obtained as σ1 = 0.3644, C1 = 99.8705; σ2 = 0.3561, C2 = 100.2673; σ3 = 0.8853, C3 = 56.3867; and σ4 = 0.5837, C4 = 159.2004. More details about how to find the optimal model parameters can be found in Ref. [38]. 5. Model development for the spiral classifiers The performance of the classification process is usually evaluated by an efficiency curve showing the mass fraction of a particular size interval d in the classifier feed that is reported to be in the underflow. A spiral classifier model has been developed by Lynch and Rao [39] in which the classification product is considered to be the result of real classification and bypass, which does not undergo classification, caused by the splitting of water. The effect of the bypass is equal to the fraction of feed water splitting to the underflow. Xie and Li [40] improved the expression of the bypass by considering that the bypass effect is not equal to

the fraction of feed water splitting to the underflow. The improved model is described in two parts as follows. EaðiÞ ¼ EcðiÞ þ aðiÞ

ð22Þ

   di −dmin m EcðiÞ ¼ 1− exp −0:693 d50c −dmin

ð23Þ

" aðiÞ ¼ Ea min

  # di −dmin k 1− dmax

ð24Þ

where Ea(i) and Ec(i) are the classification efficiency (CE, %) and the corrected classification efficiency (%) of the ith size interval, respectively; a(i) is the bypass (%) of the ith size interval; dmax and dmin are the maximum and minimum particle sizes (mm), respectively; d(i) is the particle size (mm) of the ith interval; d50c is the corrected cut size (d50, mm), which has equal (50%) probability of reporting to either product of the classifier; m serves as a direct measure of the sharpness of classification; Eamin is the classification efficiency of the finest size interval; and k is defined as the bypass number. The separation process is inherently complex and strongly nonlinear. Furthermore, the feed of classifier 1 contains a wide size range of particles, which makes it even more complex to describe using simple relationships. To characterize the feed size distribution, it is expressed as follows   B F i ¼ 1− exp −Adi

ð25Þ

Table 2 Summary of the model parameters and their fitting data source. Equations

Parameters

Fitting data source

Breakage distribution

Eq. (4)

b1 = 0.406, b2 = 0.3038, b3 = 2.4388, b4 = 0.0134

Laboratory test data

RTD model Slowing down factor for S

τ3 = γτ, τ1 = (1 − γ)τ/2 Eq. (13)

Industrial process data Experience value Laboratory test data

LSSVM model for breakage rate parameters

Eqs. (14) to (17)

LSSVM model for spiral classifier 1

Eqs. (26) to (29)

τ γ = 0.5 a1 = 0.9528, a2 = −1.409, a3 = 0.1866, a4 = 0.04257 σ1 = 0.3644, C1 = 99.8705; σ2 = 0.3561, C2 = 100.2673; σ3 = 0.8853, C3 = 56.3867; σ4 = 0.5837, C4 = 159.2004 σ1 = 0.1513, C1 = 170.125; σ2 = 0.7457, C2 = 50; σ3 = 0.4254, C3 = 50; σ4 = 2.1522, C4 = 50

Industrial process data

Industrial process data

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X. Wang et al. / Powder Technology 279 (2015) 75–85

Table 3 Statistics of the simulation results of the unit models. PSD of mill discharge

PSD of underflow 1

PSD of overflow 1

CE of classifier 1

PSD of underflow 2

PSD of overflow 2

CE of classifier 2

90.3% 79.2% 81.94% 66.67% 10.46% 6.10%

100.00% 85.19% 100.00% 98.15% 3.59% 2.20%

100.00% 100.00% 97.22% 80.56% 5.74% 5.74%

92.59% 81.48% 77.78% 68.52% 8.16% 5.54%

100.00% 94.44% 100% 100% 1.79% 1.79%

100.00% 100.00% 100% 100% 1.35% 1.11%

97.22% 88.89% 97.22% 91.67% 5.50% 1.47%

Weight Fraction Finer Than Size (%)

where A and B are called the characteristic parameters of the feed size distribution. A large value of A means the feed contains more fine particles, while a large value of B means the feed contains more coarse particles. The LSSVM method, as described in Section 4.2, is then employed to develop the correlations between the classification efficiency model parameters and the operating variables, including the feed flow rate Qc1 (t/h), solid concentration Cc1 (% wt), and feed size distribution characteristics A and B. The design variables are ignored because they

are rarely regulated in the industrial process. The LSSVM models are as follows

100 80

a

60 40 20 0 −2 10

0

10 Size (mm)

2

10

Ea min1 ¼ lssvm1 ðQ c1 ; C c1 ; A; BÞ

ð26Þ

d50c1 ¼ lssvm2 ðQ c1 ; C c1 ; A; BÞ

ð27Þ

m1 ¼ lssvm3 ðQ c1 ; C c1 ; A; BÞ

ð28Þ

Weight Fraction Finer Than Size (%)

RE ±20% RE ±10% AE ±5% AE ±3% Max AE Max AE of −0.075 mm

60 40 20 −2 10

CE of Classifier 2 (%)

CE of Classifier 1 (%)

80

0

10 Size (mm)

d

60 40 20 0 −2 10

0

10 Size (mm)

2

10

e 80 60 40 20 −2 10

2

10

100 80

80

100

b

0

10 Size (mm)

2

10

100 CE of Classifier 2 (%)

CE of Classifier 1 (%)

100

100

c

60 40 20 0 −2 10

−1

10 Size (mm)

0

10

80

f

60 40 20 0 −2 10

−1

10 Size (mm)

0

10

Fig. 11. Simulation and measured results of the grinding-classification process, (a), (b) and (c) correspond to sample 1, and (d), (e) and (f) correspond to sample 2. ( the mill, o is measured value, and ––– is simulated value).

is fresh feed to

X. Wang et al. / Powder Technology 279 (2015) 75–85

83

Table 4 Statistics of the simulation results of the process using SMM.

RE ±20% RE ±10% AE ±5% AE ±3% Max AE Max AE of −0.075 mm

PSD of mill Discharge

PSD of Underflow 1

PSD of overflow 1

CE of classifier 1

PSD of Underflow 2

PSD of overflow 2

CE of classifier 2

88% 68% 76.39% 54.16% 11.87% 6.7%

77.78% 64.81% 74% 61.1% 11.2% 3.19%

100% 94.44% 86.1% 75% 8.16% 3.68%

83.3% 66.7% 70.37% 57.41% 12.51% 3.32%

100% 88.89% 89% 86% 8.22% 2.04%

100% 100% 100% 92% 4.1% 2.15%

88.89% 75% 80.56% 69.44% 9.89% 3.19%

k1 ¼ lssvm4 ðQ c1 ; C c1 ; A; BÞ

ð29Þ

where the kernel and the penalizing parameters are σ1 = 0.1513, C1 = 170.125; σ2 = 0.7457, C2 = 50; σ3 = 0.4254, C3 = 50; and σ4 = 2.1522, C4 = 50, respectively. For classifier 2, because the feed contains mainly fine particles so that it cannot accurately be described by using A and B parameters, the fractions of +0.15 mm and −0.075 mm, denoted by A2 and B2, respectively, are used to characterize the feed size distribution. The solid flow rate Q c2 (t/h) and the solid concentration Cc2 (% wt) of the feed are also considered. A stepwise multiple-linear regression method was used to simplify the model structure. The individual variables, their multiplier, their logistic and the power of e were tested. Finally, the correlations are obtained as follows   2 R ¼ 0:759

Eamin ¼ −0:911B2 −0:001Q c2 A2 þ 30:659 lnB2 −59:255

ð30Þ d50c ¼ −21:401C c2 þ 57:565A2 þ 23:317B2 þ 53:22Q c2 C c2 −0:027Q c2 B2 −39:879C c2 B2 −123:727C c2 A2 −346:218

  2 R ¼ 0:813

ð31Þ m ¼ −0:471B2 þ 31:998 ln B2 −101:586

  2 R ¼ 0:772

ð32Þ

k ¼ 0:266Q c2 C c2 −0:002Q c2 B2 þ 0:05C c2 A2 þ 0:311C c2 B2   C 2 −28:016e c2 þ 34:75 R ¼ 0:877 :

ð33Þ

6. Results and discussion In this section, the off-line simulation results of individual unit models and the whole process and the application results of PSD index measurement are shown to illustrate the effectiveness of the

models. Firstly, Table 2 is given to summarize the model parameters and their corresponding data sources. 6.1. Simulation results In all of the 30 groups of data collected by the full-scale process sampling campaign described in Section 3, six groups with a wide range of fresh feed flow rates were selected for model validation, while the other 24 groups were used for model training. The ball mill model and the classifier models are first individually used and then put in series as the actual process to predict the PSD of the ball mill discharge and the overflow and underflow of the classifiers. The statistics of the simulation results for the individual models are given in Table 3, where AE and RE are the absolute error and relative error, respectively, between the simulation and the measured values. The AE and RE are calculated using Eqs. (34) and (35), respectively. RE ± 20% means that RE is within ± 20%; for example, for the PSD of the mill discharge, the number of samples that have a RE within ± 20% divided by the total number of the samples is 90.3%, that is, 90.3% of the samples' RE are within ± 20%. Max AE is the maximum AE in all of the samples. The Max AE of − 0.075 mm is the maximum AE in all of the −0.075 mm-size-interval samples. ^i AE ¼ yi −y

ð34Þ

^i Þ=yi  100% RE ¼ ðyi −y

ð35Þ

where yi and ŷi are the measured and simulated values, respectively. The unit models are then built in series to simulate the whole grinding-classification process by using the SMA. The convergence criterion is to iteratively minimize the simulated and set values of the size distribution of underflow 1. If the criterion cannot be satisfied, the set values will be replaced by the simulated values and re-calculated. The best and the worst simulated mill discharge size distributions with the corresponding classification efficiencies for the two classifiers are plotted in Fig. 11 and labeled as Sample 1 and Sample 2, respectively.

90 85

100

a

98

b

96 80 94 Predicted

Predicted

75 70 65

92 90 88 86

60 84 55 50 50

82 60

70 Measured

80

90

80 80

85

90 Measured

95

Fig. 12. Comparison of the measured and predicted PSD indices. (a) −0.075 mm of overflow 1. (b) −0.075 mm of overflow 2.

100

84

X. Wang et al. / Powder Technology 279 (2015) 75–85

The statistics of all of the simulation results of PSD are shown in Table 4, with the same notations as in Table 3. Meanwhile, for the solid flow rate of the circulating load, the relative errors between its simulated values and reconciled values are all within −9.66% and 4.57%, with a RRMSE equal to 4.56%. The RRMSE is the relative root mean square error, calculated as follows vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n   u1 X ^i Þ2 =yi : ðyi −y RRMSE ¼ t n i¼1

Table 5 Statistics of the qualitative accuracy of the application results.

Overflow 1 Overflow 2

ð36Þ

As the validation results of the unit models shown in Table 3, for all of the data points in each slurry flow, more than 90% of the relative errors are within ± 20%, and more than 80% of the absolute errors are within ±5%. The simulation results of the whole process (see Table 4) by using SMA are less accurate than those of the individual unit models (see Table 3). This may be caused by the error accumulation of the models when they are connected to each other. Both in Tables 3 and 4, some relative errors are large because the mass fraction of that size-interval is itself small. Therefore, although the corresponding absolute error is very small, the relative error is large. For example, when the measured value is 9.95% and the simulated value is 8.16%, the absolute error is 1.79%, which is within the allowed range, but the relative error is large. This situation mostly occurs for the −0.045 mm size-interval in underflow 1. For the ball mill, one group of the simulation results is quite poor, as shown in Fig. 11(d). This may be caused by abnormal conditions in the process because the circulating load was extremely high at that shift. The model precision becomes higher from the ball mill to classifier 2. The simulation results for classifier 2 are always of high accuracy because its feed has a narrow size range with mainly fine particles. The predicted size fractions of − 0.075 mm, the most significant process index, are reasonable for all of the slurry flows. 6.2. Application results The proposed models are then applied to the process for the measurement of the PSD indices. Some of the running results are shown here. Considering that the sieving work is time-consuming, only size intervals + 13.2, − 13.2 + 9.5, − 9.5 + 5.6, − 5.6 + 3.35, −3.35 + 0.075, and −0.075 mm are sieved for the fresh feed and the circulation flow every 8 h. They are then interpolated and used to predict the PSD once an hour. In practice, because the PSD is measured manually, the mass fraction of − 0.075 mm in the two overflows are mainly concerned. Therefore, the predicted and measured values of only − 0.075 mm are compared to illustrate the effectiveness of the

Measured and predicted value are not consistent

12.4%

80.5%

7.1%

72.1%

8.7%

19%

7. Conclusions and future work Mineral processing of bauxite is significant for the sustainable development of aluminate production in China. Hybrid models using the phenomenological and statistical learning methods are developed for the ball milling and spiral classification processes of bauxite based on a large amount of test and industrial running data. Operating conditions, ore properties and ball characteristics are considered in the models so that they are adaptable to variations in the running conditions. The simulation results using the individual unit models and the SMA both show 25

a

b 20

14 12

Frequency (%)

Frequency (%)

Measured and predicted values are consistent (the product is not qualified)

models, with 298 samples obtained under normal operating conditions plotted in Fig. 12. The prediction error distribution is shown in Fig. 13. It can be seen that 63.1% and 70.1% of the absolute errors are within ±2% for overflow 1 and overflow 2, respectively. This is worse than the simulation results of the −0.075-mm particles in Tables 3 and 4. This may be because the ore properties and operating conditions varied over a wider range during the application period. In the ball mill model, the ore properties only include the ore grade and the iron ore content. This is not enough because the mineralogy characteristics of the ore are complicated and difficult to describe by only one or two characteristics. Hence, material characterization is still a challenge for the future. To improve the model in this study, parameter updates according to the milling conditions are necessary. Fortunately, the precision is acceptable in the real application because the mass fraction of − 0.075 mm in the two overflows is subscribed to be ≥75% and ≥90%, respectively. Usually, only when the mass fraction of − 0.075 mm in the product cannot achieve the subscribed value do we need to determine the regulation degrees of the operating variables by quantifying the deviation of the process. Otherwise, the process variables are usually not regulated. As the qualitative statistics of the samples show in Table 5, more than 80% of the predicted values can accurately be predicted, whether the subscribed mass fraction in the product is achieved or not, which provides valuable information to the operators.

18 16

Measured and predicted values are consistent (the product is qualified)

10 8 6 4

15

10

5

2 0 −5

0 Absolute error

5

0 −5

0 Absolute error

Fig. 13. Distribution of the prediction error of the application results. (a) −0.075 mm of overflow 1. (b) −0.075 mm of overflow 2.

5

X. Wang et al. / Powder Technology 279 (2015) 75–85

reasonable accuracy. Notably, in more than 80% of the cases, it can be determined whether the subscribed mass fraction in the product is achieved or not using the prediction results, although the quantitative accuracy still needs to be improved, which is significant to the process operation. Optimization is being conducted to improve the performance of the grinding-classification process based on the developed models. Acknowledgments The authors are grateful to the workers in Zhongzhou Branch, China Aluminum Corporation and the other members of the project for their support of the experiments and the plant sampling campaigns. The authors acknowledge the financial support from the National Nature Science Foundation of China (61304126, 61273187), the Ph.D. Programs Foundation of the Ministry of Education of China (No. 20120162120022), the Postdoctoral Science Foundation of China (2012M521413), and the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (61321003). References [1] P. Smith, The processing of high silica bauxites—review of existing and potential processes, Hydrometallurgy 98 (2009) 162–176. [2] B. Epstein, The mathematical description of certain breakage mechanisms leading to the logarithmico-normal distribution, J. Franklin Inst. 244 (1947) 471–477. [3] B. Epstein, Logarithmico-normal distribution in breakage of solids, Ind. Eng. Chem. 40 (1948) 2289–2291. [4] L.G. Austin, K. Julianelli, A.S. Souza de, C.L. Schneider, Simulation of wet ball milling of iron ore at Carajas, Brazil, Int. J. Miner. Process. 84 (2007) 157–171. [5] E. Bilgili, J. Yepes, B. Scarlett, Formulation of a non-linear framework for population balance modeling of batch grinding: beyond first-order kinetics, Chem. Eng. Sci. 61 (2006) 33–34. [6] H. Dundar, H. Benzer, N.A. Aydogan, O. Altun, N.A. Toprak, O. Ozcan, D. Eksi, A. Sargın, Simulation assisted capacity improvement of cement grinding circuit: case study cement plant, Miner. Eng. 24 (2011) 205–210. [7] H. Dundar, H. Benzer, N. Aydogan, Application of population balance model to HPGR crushing, Miner. Eng. 50–51 (2013) 114–120. [8] J.A. Herbst, D.W. Fuerstenau, Scale-up procedure for continuous grinding mill design using population balance models, Int. J. Miner. Process. 7 (1980) 1–31. [9] K. Mitra, Multiobjective optimization of an industrial grinding operation under uncertainty, Chem. Eng. Sci. 64 (2009) 5043–5056. [10] S. Morrell, Y.T. Man, Using modelling and simulation for the design of full scale ball mill circuits, Miner. Eng. 10 (1997) 1311–1327. [11] M. Pieper, Z. Kutelova, S. Aman, J. Tomas, Modeling of baryte batch grinding in a vibratory disc mill, Adv. Powder Technol. 24 (2013) 229–234. [12] D.M. Weedon, A perfect mixing matrix model for ball mills, Miner. Eng. 14 (2001) 1225–1236. [13] L.M. Tavares, R.M. Carvalho de, Modeling breakage rates of coarse particles in ball mills, Miner. Eng. 22 (2009) 650–659. [14] E.T. Tuzcu, R.K. Rajamani, Modeling breakage rates in mills with impact energy spectra and ultra fast load cell data, Miner. Eng. 24 (2011) 252–260. [15] P.K. Akkisetty, U. Lee, G.V. Reklaitis, V. Venkatasubramanian, Population balance model-based hybrid neural network for a pharmaceutical milling process, J. Pharm. Innov. 5 (2010) 161–168. [16] K. Mitra, M. Ghivari, Modeling of an industrial wet grinding operation using datadriven techniques, Comput. Chem. Eng. 30 (2006) 508–520.

85

[17] A.K. Pani, H.K. Mohanta, Soft sensing of particle size in a grinding process: application of support vector regression, fuzzy inference and adaptive neuro fuzzy inference techniques for online monitoring of cement fineness, Powder Technol. 264 (2014) 484–497. [18] D. Sharbaro, P. Ascencio, P. Espinoza, F. Mujica, G. Cortes, Adaptive soft-sensors for on-line particle size estimation in wet grinding circuits, Control. Eng. Pract. 16 (2008) 171–178. [19] X. Wang, W. Gui, C. Yang, Y. Wang, Wet grindability of an industrial ore and its breakage parameters estimation using population balances, Int. J. Miner. Process. 98 (2011) 113–117. [20] L.G. Austin, R.R. Klimpel, P.T. Luckie, Process Engineering of Size Reduction: Ball Milling, Society of Mining Engineering, New York, 1984. [21] X. Wang, W. Gui, C. Yang, Y. Wang, Breakage distribution estimation of bauxite based on piecewise linearized breakage rate, Chin. J. Chem. Eng. 20 (2012) 1198–1205. [22] L.G. Austin, K. Shoji, V. Bhatia, V. Jindal, K. Savage, R. Klimpel, Some results on the description of size reduction as a rate process in various mills, Ind. Eng. Chem. Process. Des. Dev. 15 (1976) 187–196. [23] H. Cho, L.G. Austin, The equivalence between different residence time distribution models in ball milling, Powder Technol. 124 (2002) 112–118. [24] A.B. Makokha, M.H. Moys, M.M. Bwalya, Modeling the RTD of an industrial overflow ball mill as a function of load volume and slurry concentration, Miner. Eng. 24 (2011) 335–340. [25] G.B. Tupper, I. Govender, A.N. Mainza, N. Plint, A mechanistic model for slurry transport in tumbling mills, Miner. Eng. 43–44 (2013) 102–104. [26] R.P. King, Modelling and Simulation of Mineral Processing Systems, 1st ed. Butterworth-Heinemann Publishers, New York, 2001. [27] P. Songfack, R. Rajamani, Hold-up studies in a pilot scale continuous ball mill: dynamic variations due to changes in operating variables, Int. J. Miner. Process. 57 (1999) 105–123. [28] H.C. Cho, K.H. Kim, H. Lee, D.J. Kim, Study of residence time distribution and mill hold-up for a continuous centrifugal mill with various G/D ratios in a dry-grinding environment, Miner. Eng. 24 (2011) 77–81. [29] V. Deniz, The effects of ball filling and ball diameter on kinetic breakage parameters of barite powder, Adv. Powder Technol. 23 (2012) 640–646. [30] C. Tangsathitkulchai, Effects of slurry concentration and powder filling on the net mill power of a laboratory ball mill, Powder Technol. 137 (2003) 131–138. [31] J.A.K. Suykens, Least squares support vectors machines for classification and nonlinear modeling, Neural Netw. World 10 (2000) 29–47. [32] X.G. Zhang, Introduction to statistical learning theory and support vector machines, Acta Autom. Sin. 26 (2000) 32–42. [33] C. Cortes, V. Vapnik, Support vector networks, Mach. Learn. 20 (1995) 273–297. [34] R. Begg, J. Kamruzzaman, A machine learning approach for automated recognition of movement patterns using basic, kinetic and kinematic gait data, J. Biomech. 38 (2005) 401–408. [35] H. Spratt, H. Ju, A.R. Brasier, A structured approach to predictive modeling of a twoclass problem using multidimensional data sets, Methods 61 (2013) 73–85. [36] Z. Bao, D. Pi, Y. Sun, Nonlinear model predictive control based on support vector machine with multi-kernel, Chin. J. Chem. Eng. 15 (2007) 691–697. [37] K. De Brabanter, P. Karsmakers, F. Ojeda, C. Alzate, J. De Brabanter, K. Pelckmans, B. De Moor, J. Vandewalle, J.A.K. Suykens, Least square support vector machine MATLAB tool box, http://www.esat.kuleuven.be/sista/lssvmlab/ (available online at:). [38] A.K. Pani, H.K. Mohanta, Online monitoring and control of particle size in the grinding process using least square support vector regression and resilient back propagation neural network, ISA Trans. (2014)http://dx.doi.org/10.1016/j.isatra (in press). [39] A.J. Lynch, T.C. Rao, 11th International Mineral Processing Congress, Cagliar 1975, p. 245. [40] H. Xie, S. Li, An improvement of calculation method on classification efficiency, Nonferrous Met. 42 (1990) 39–45 (in Chinese).