^{1}

^{1}

^{2, 3}

^{1}

^{1}

^{2}

^{3}

A series of smelting reduction experiments has been carried out with high-phosphorus iron ore of the different bases and heating rates by thermogravimetric analyzer. The derivative thermo gravimetric (DTG) data have been obtained from the experiments. After analyzing its phase space reconstruction, it is found that DTG phase portrait contains with a clear double “

Chaos theory suggests that the randomness within deterministic system is often caused by the nonlinear dynamic factors instead of random factors from the outside. There are complex nonlinear dynamical systems in practical problems, because of the limit of measuring means; and the unknown information of complex systems and absence of physical quantities contained make it difficult to establish the directly analytical form of the mathematical model. Therefore, the reconstruction of nonlinear dynamical systems uses the time series of a physical quantity which was got by experiments to obtain the internal and inherent characteristics. Such research is an important part of nonlinear analysis [

The rest of the paper is organized as follows. The phase space reconstruction method and its analysis and discussion of the experimental data are shown in Section

Phase space reconstruction of the dynamic system is a new technology of time series quantitative analysis [

Let

Phase space reconstruction theory proposed by Takens proves that there is a suitable embedding dimension by which a regular trajectory (attractor) would be restored. In other words, it is a diffeomorphism from its trajectory to the original dynamic system in the reconstructed

In addition to this, if time series is long enough without the noise, the delay time

The average mutual information method is employed to obtain a delay time. It is a modification of the classical autocorrelation function method from the perspective of information theory and could distinguish effectively folded and unfolded orbit.

The calculated results of the autocorrelation function are tended to lag, and the emphasis is a linear correlation part. The average mutual information function includes nonlinear components; the information theory is employed to select

The vector sequence was reconstructed by time delay for time series

Here false neighbor is based on the above definition

The mean of all

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, flowing at a rate of 10 mL/min.

Raw materials employed in the experiment are taken from the Huimin high-phosphorus iron ore in Yunnan province. The composition of Huimin ore and auxiliary raw materials such as coal and CaO is shown in Tables

Mineral composition of Huimin ore (%).

TFe | FeO | SiO_{2} |
Al_{2}O_{3} |
MgO | CaO | S | P | TiO_{2} |
---|---|---|---|---|---|---|---|---|

41.02 | 1.36 | 17.98 | 5.60 | 0.21 | 0.35 | 0.06 | 0.9 | 0.28 |

The ingredients of solvent (%).

CaO | Alkali-metal and MgO | Fe | Ammonia precipitate | Sulfate | Loss On ignition |
---|---|---|---|---|---|

96.3 | 0.5 | 0.015 | 0.2 | 0.1 | 2 |

The composition of anthracite (%).

C | CaO |
SiO_{2} |
S | P | Ash | Volatile |
---|---|---|---|---|---|---|

76.43 | 1.59 | 7.80 | 0.35 | 0.05 | 15.29 | 7.78 |

As shown in Figure _{2}. Huimin limonite ore is a kind of water goethite.

XRD collection of illustrative plates of Huimin iron ore.

The particle size ranged from 0.001 to 0.1 mm. There are two genetic types for the limonite ore, one is the sedimentary limonite, which was implicit and crystalline aggregate; the other is the limonite aggregate formed by exogenous effect of oxidation and hydrolysis.

Derivative thermogravimetric data of Huimin ore reflect the change of the apparent reaction rate in the smelting reduction process. It is shown in Figure

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 time series of DTG.

Firstly, the features of DTG are analyzed, and then 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 indicates that no matter how the degree of the embedded noise is, it will evenly fill that space. Chaotic (or periodic) signals have a distinct spatial structure, and their correlation dimension will saturate at some point while embedding dimension is increasing.

By calculating the correlation dimension of DTG data to ensure that the correlation dimension will reach saturation status as shown in Figure

Evolution diagram of correlation dimension.

As can be seen from the calculated results in Figures

Calculated process of optimal embedding dimension.

Calculated process of optimal time delay.

(a) The phase diagram for

The DTG phase space reconstruction of Huimin ore with alkalinity ratio of five (heating rate of 10 K/min) was obtained; the calculated results in VRA (Visual Recurrence analysis) software [

(a) Heating rate of 10 K/min,

It can be seen from Figure

The condition with better smelting reduction (alkalinity

(a)

As can be seen from Figure

As can be seen from Figure

The relation of opposite angles length of graph and heating rate.

The recurrence plot proposed by Belaire-Franch and Contreras [

The recurrence plot was depicted with reconstructed phase space by the delay time

The recurrence plot of the DTG.

The phase portrait of the smelting reduction process reconstructed by the experimental data DTG has the statistical characteristics of the attractor that is double “

Considering that higher degree agreement of the most probable mechanism functions with the experimental data [

Comparison between the measured and model calculated results on reaction fraction of the reduction phase (prereduction and final reduction).

It can be seen from the Figure

Taken prereduction stage as an example, the reaction mechanism function

Therefore, the above-mentioned method though reasonable could not correctly express DTG data. The main reason is the complexity and dynamic randomness of the reaction in the smelting reduction process of high-phosphorus iron ore. Thus, dynamic system method by time series has an advantage of characterizing dynamic behavior.

The diversity of ore raw material composition and complexity of the chemical reaction combined with the recurrence plot analysis have showed that the system is stochastic dynamical systems; in the meanwhile it is the discrete dynamical system determined by characteristics of the attractor and discrete point, So it can be concluded that it belongs to the random discrete dynamic system [

(a) The initial low temperature segment, (b) the first intermediate process, (c) the second intermediate process, and (d) the third intermediate process. (The horizontal axis and vertical axis are, respectively,

It can be inferred from Figure

TGA experiment on smelting reduction of the Huimin high-phosphorus iron ore was made with five different alkalinities and four kinds of different heating rates at the same alkalinity, the derivative thermogravimetric (DTG) data have been obtained from the experiments. Then the phase space reconstruction analysis is found as follows.

The equations of mathematical physics for the attractor has not established accurately, every exaction cannot be reflect in the attractor, they are the faultiness of our method and we will develop future work combined these two aspects.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work This work was supported by the Startup Foundation for Doctors (no. PXY-BSQD-2014001) and The Youth Foundation of Ping Ding Shan University (No.PXY-QNJJ-2014008) to which the authors are greatly obliged.