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A series of direct smelting reduction experiment has been carried out with high phosphorous iron ore of the different bases by thermogravimetric analyzer. The derivative thermogravimetric (DTG) data have been obtained from the experiments. One-step forward local weighted linear (LWL) method , one of the most suitable ways of predicting chaotic time-series methods which focus on the errors, is used to predict DTG. In the meanwhile, empirical mode decomposition-autoregressive (EMD-AR), a data mining technique in signal processing, is also used to predict DTG. The results show that (1) EMD-AR(4) is the most appropriate and its error is smaller than the former; (2) root mean square error (RMSE) has decreased about two-thirds; (3) standardized root mean square error (NMSE) has decreased in an order of magnitude. Finally in this paper, EMD-AR method has been improved by golden section weighting; its error would be smaller than before. Therefore, the improved EMD-AR model is a promising alternative for apparent reaction rate (DTG). The analytical results have been an important reference in the field of industrial control.

The complexities in smelting reduction process of iron and steel come from the interaction between the fluid dynamics and chemical kinetics mechanisms. The research direction has changed to how to establish hybrid kinetic model by combining fluid dynamics equations with chemical kinetics algebraic equations and taking factors at the level of microchemical reactions into consideration. Some macrokinetic parameters should be modeled, such as the volume of wind, differential pressure, and feed rate (obtained by dynamic and real-time detection in the process). However, micro-kinetic parameters in chemical reaction process cannot be got by detection and identification online. The apparent reaction rate changes with the chemical reaction and fluid. In addition, the significance of its kinetic analysis is pretty obvious. Thus, the majority of research focuses on mathematical and physical model based on kinetics [

The rest of the paper is organized as follows. In Section

First, the features of DTG are analyzed. Correlation dimension, an important evidence to establish the existence of chaotic continuous systems, is calculated with the reconstruction map of the fractal dimension. For the truly random signals, the correlation dimension graph will look like a 45-degree straight line. It is indicating that no matter how you embed the noise, it will evenly fill that space. Chaotic (or periodic) signals have a distinct spatial structure, and their correlation dimension will saturate as some point while embedding dimension is increased.

The correlation dimension reaches saturation by calculating the correlation dimension of DTG data, as shown in Figure

Evolution diagram of correlation dimension.

In VRA (visual recurrence analysis), we can construct such a model from a range of classes, such as nearest neighbor, locally constant, kernel regression, locally linear, locally weighted linear, radial basis models, and support vector regression [

With a given time-series

One-step estimate is generated by

Suppose

The weight of

The locally weighted linear methods can be expressed in

The situation that the embedding dimension is one (

Then partial derivatives of

Solutions are expressed in

The calculated performance is otherwise known as the root mean square error (RMSE) and the normalized mean squared error (NMSE). NMSE increased the role of large-value errors in the indicators, thereby improved the sensitivity of this indicator. Equations (

Based on (

Derivative thermogravimetric data of the Huimin ore reflect the change of the apparent reaction rate in the smelting reduction process. Experiment extent is confirmed with bases 0.8, 1.1, 1.4, 1.7, 2.0; respectively, the compositions of each type of materials required are shown in Table

Compositions of each type of materials required.

Bases ( | 0.8 | 1.1 | 1.4 | 1.7 | 2.0 |
---|---|---|---|---|---|

Mineral content (mg) | 7.48 | 6.85 | 6.82 | 6.53 | 6.37 |

Coal content (mg) | 1.40 | 1.17 | 1.16 | 1.11 | 1.09 |

Calcium oxide content (mg) | 1.12 | 1.98 | 2.02 | 2.36 | 2.54 |

The TGA experiment equipment is produced by Netzsch, modeling as STA 449 F3. It is protected with nitrogen gas flowing at 50 mL/min. And argon, acting as purge gas, flows at the rate of 10 mL/min. There are 200 points around 1400°C predicted. The performances of their prediction charts using one-step forward locally weighted linear method are shown in Figure

Prediction effect for different values of

From Figure

Errors analysis.

Bases ( | One-step LWL | |

RMSE | NMSE | |

0.8 | 0.1099 | 0.8771 |

1.1 | 0.1047 | 0.9404 |

1.4 | 0.1059 | 0.9841 |

1.7 | 0.1073 | 0.9704 |

2.0 | 0.1196 | 0.9031 |

Empirical mode decomposition (EMD) has been successfully applied to many fields, such as communication, society, economy, engineering, and achieved good effects [

EMD method is developed from the simple assumption that any signal consists of different simple intrinsic modes of oscillations. Each linear or nonlinear mode will have the same number of extreme and zero crossings. There is only one extreme between successive zerocrossings. Each mode should be independent of the others. In this way, each signal could be decomposed into a number of intrinsic mode functions (IMFs), each of which must satisfy the following definitions [

In the whole data set, the number of extreme and the number of zero-crossings must either equal or differ from each other at most by one.

At any point, the mean value of the envelope defined by local maxima and the envelope defined by the local minima is zero.

An IMF represents a simple oscillatory mode compared with the simple harmonic function.

With the definition, any signal

Identify all local extremes, and then connect all the local maxima by a cubic spline line as the upper envelope.

Repeat the procedure for the local minima to produce the lower envelope. The upper and lower envelopes should cover all the data among them.

The mean of upper and low envelope value is designated as

Generally speaking,

The value of standard deviation

When

The new sequence is treated as the original data, and repeat the above (

Repeat (

The process is expressed as follows

The rate trend characterized by five kinds of base after EMD decomposition is shown in Figure

EMD trend line.

Seen from Figure

Consider

Equation (

Huimin ore was analyzed with different base at sampling points of 8144, after the decomposition of EMD, the data can be divided into 13 groups, and the last group is a trend term.

Firstly, each component is tried out through the ADF stationary test, as can be seen from Table

ADF test table.

Prob.* | ||
---|---|---|

Augmented Dickey-Fuller test statistic | −32.84661 | 0.0000 |

Test critical values 1% level | −3.430977 | |

5% level | −2.861702 | |

10% level | −2.566898 |

*MacKinnon 1996 [

Augmented Dickey-Fuller test equation.

Dependent variable: D(SER01).

Date: 06/07/11 Time: 16: 10.

Sample (adjusted): 10 8144.

Included observations: 8135 after adjustments.

lists autocorrelation and partial autocorrelation.

Autocorrelation | Partial correlation | AC | PAC | Prob. | ||
---|---|---|---|---|---|---|

1 | 0.952 | 0.952 | 7384.9 | 0.0000 | ||

2 | 0.825 | −0.866 | 12936. | 0.0000 | ||

3 | 0.648 | 0.253 | 16355. | 0.0000 | ||

4 | 0.455 | 0.317 | 18040. | 0.0000 | ||

5 | 0.279 | 0.066 | 18673. | 0.0000 | ||

6 | 0.142 | −0.086 | 18836. | 0.0000 | ||

7 | 0.052 | −0.075 | 18858. | 0.0000 | ||

8 | 0.008 | −0.024 | 18859. | 0.0000 | ||

9 | −0.003 | −0.004 | 18859. | 0.0000 | ||

10 | 0.003 | −0.014 | 18859. | 0.0000 | ||

11 | 0.013 | −0.022 | 18861. | 0.0000 | ||

12 | 0.016 | −0.021 | 18863. | 0.0000 | ||

13 | 0.007 | −0.021 | 18863. | 0.0000 | ||

14 | −0.014 | −0.034 | 18865. | 0.0000 | ||

15 | −0.046 | −0.034 | 18882. | 0.0000 | ||

16 | −0.081 | −0.011 | 18935. | 0.0000 | ||

17 | −0.115 | −0.007 | 19042. | 0.0000 | ||

18 | −0.142 | −0.019 | 19208. | 0.0000 | ||

19 | −0.162 | −0.020 | 19421. | 0.0000 | ||

20 | −0.171 | 0.007 | 19659. | 0.0000 | ||

21 | −0.170 | 0.032 | 19895. | 0.0000 | ||

22 | −0.159 | 0.025 | 20100. | 0.0000 | ||

23 | −0.139 | 0.002 | 20258. | 0.0000 | ||

24 | −0.112 | 0.014 | 20360. | 0.0000 | ||

25 | −0.080 | 0.024 | 20411. | 0.0000 | ||

26 | −0.044 | 0.013 | 20428. | 0.0000 | ||

27 | −0.009 | 0.004 | 20428. | 0.0000 | ||

28 | 0.022 | −0.026 | 20432. | 0.0000 | ||

29 | 0.048 | −0.005 | 20451. | 0.0000 | ||

30 | 0.064 | 0.002 | 20484. | 0.0000 | ||

31 | 0.069 | −0.026 | 20523. | 0.0000 | ||

32 | 0.065 | −0.011 | 20557. | 0.0000 | ||

33 | 0.052 | 0.013 | 20579. | 0.0000 | ||

34 | 0.034 | −0.005 | 20589. | 0.0000 | ||

35 | 0.015 | −0.018 | 20591. | 0.0000 | ||

36 | −0.002 | −0.003 | 20591. | 0.0000 |

Then with the previous 7944 data points, each group of data is predicted to 8144 in the method of AR (

Calculation results are represented in

Equation (

The results calculated by using MATLAB programming are shown in Figure

EMD-AR prediction effects.

As can be seen from Figure

Error analysis.

Base ( | RMSE | NMSE | AE | ||||
---|---|---|---|---|---|---|---|

One-step LWL | EMD-AR | Improved EMD-AR | One-step LWL | EMD-AR | Improved EMD-AR | Improved EMD-AR | |

0.8 | 0.1099 | 0.0417 | 0.0306 | 0.8771 | 0.0751 | 0.0679 | −0.0017 |

1.1 | 0.1047 | 0.0312 | 0.0289 | 0.9404 | 0.0812 | 0.0778 | −7.53E (−5) |

1.4 | 0.1059 | 0.0389 | 0.0326 | 0.9841 | 0.0956 | 0.0935 | −6.31E (−4) |

1.7 | 0.1073 | 0.0527 | 0.0266 | 0.9704 | 0.0678 | 0.0596 | 0.0024 |

2.0 | 0.1196 | 0.0297 | 0.0278 | 0.9031 | 0.0534 | 0.0489 | 0.0016 |

Based on Table

In Figure

Characteristic of normal distribution for DTG data.

State space chart of DTG (

As can be seen, the predicted value was identical with the true value of the DTG using one-step forward locally weighted linear prediction method in a high degree, which shows that the method is better; this can also be seen by the errors’ data itself. The evolution model of the apparent rate is expressed much more clearly.

EMD-AR (

DTG trend item is parabolic curve, indicating there is an inner maximum of the apparent reaction rate existing in the unsteady-state process. And the temperature ranges from 600 to 800°C; when the temperature falls in this range, various crystalline phases of iron oxides changes, but the mass will not change much.

This work was supported by National Natural Science Foundation of China under Contract (no. 51064015), the Yunnan Provincial Foundation Key project under Contract (no. ZD2010001), and the National Science Council of Taiwan (NSC 100-2628-H-161-001-MY4), to which the authors are greatly obliged.

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