Evaluation strategy and mass balance for making decision about the amount of aluminum fluoride addition based on superheat degree

Weichao Yue; Weihua Gui; Xiaofang Chen; Zhaohui Zeng; Yongfang Xie

doi:10.3934/jimo.2018169

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March 2020, 16(2): 601-622. doi: 10.3934/jimo.2018169

Evaluation strategy and mass balance for making decision about the amount of aluminum fluoride addition based on superheat degree

Weichao Yue , Weihua Gui , Xiaofang Chen ^, , Zhaohui Zeng and Yongfang Xie

School of Information Science and Engineering, Central South University, Changsha 410083, China

*Corresponding author: Xiaofang Chen (e-mail: xiaofangchen@csu.edu.cn)

Received June 2018 Revised August 2018 Published March 2020 Early access October 2018

Fund Project: This project was supported by the National Natural Science Foundation of China (61773405, 61533020, 61621062 and 61725306); and the innovation project of Central South University (502390003)

The purpose of aluminum fluoride (AlF₃) addition is to adjust the superheat degree (SD) in the aluminum reduction process. Determining the appropriate amount of AlF₃ to add has long been a challenging industrial issue as a result of its inherent complexity. Because of the decreasing number of experienced technicians, the manual addition of AlF₃ is usually inexact, which easily leads to an unstable cell condition. In this paper, an evaluation strategy based on the SD for AlF₃ addition is proposed. An extended naïve Bayesian classifier (ENBC) is designed to estimate the states of SD and its trends that represent the current and potential cell condition respectively, and then the process is graded by evaluating the estimated results based on fuzzy theory. The reduction process is divided into a few situations based on the evaluation grades, and mass balance is introduced to determine the amount of AlF₃ addition in each situation. The results of experiments show that the proposed strategy is feasible, and the effectiveness of AlF₃ addition is improved compared to the existing method. Moreover, automatic AlF₃ addition is promising based on the proposed strategy.

Keywords: Aluminum reduction, AlF₃ addition amount, extended naïve Bayesian classifier, superheat degree, evaluation.

Mathematics Subject Classification: Primary: 93C95, 93C35; Secondary: 93C42, 93C10.

Citation: Weichao Yue, Weihua Gui, Xiaofang Chen, Zhaohui Zeng, Yongfang Xie. Evaluation strategy and mass balance for making decision about the amount of aluminum fluoride addition based on superheat degree. Journal of Industrial & Management Optimization, 2020, 16 (2) : 601-622. doi: 10.3934/jimo.2018169

References:

[1]	I. Barbeito and R. Cao, Smoothed stationary bootstrap bandwidth selection for density estimation with dependent data, Computational Statistics & Data Analysis, 104 (2016), 130-147. doi: 10.1016/j.csda.2016.06.015. Google Scholar
[2]	K. Barbé, L. G. Fuentes, L. Barford and L. Lauwers, A guaranteed blind and automatic probability density estimation of raw measurements, IEEE Transactions on Instrumentation & Measurement, 63 (2014), 2120-2128. Google Scholar
[3]	Z. G. Chen, Y. G. Li, X. F. Chen and W. H. Gui, Semantic Network Based on Intuitionistic Fuzzy Directed Hyper-Graphs and Application to Aluminum Electrolysis Cell Condition Identification, IEEE Access, 5 (2017), 20145-20156. doi: 10.1109/ACCESS.2017.2752200. Google Scholar
[4]	Y. Chien, Pattern classification and scene analysis, IEEE Transactions Automatic Control, 19 (1974), 462-463. doi: 10.1109/TAC.1974.1100577. Google Scholar
[5]	K. S. Chuang, H. L. Tzeng and S. Chen, Fuzzy c-means clustering with spatial information for image segmentation, Computerized Medical Imaging and Graphics, 30 (2006), 9-15. doi: 10.1016/j.compmedimag.2005.10.001. Google Scholar
[6]	P. Desclaux, AlF₃ additions based on bath temperature measurements, Light Metals, (1987), 309-313. Google Scholar
[7]	T. Drengstig, D. Ljungquist and B. A. Foss, On the AlF₃ and temperature control of an aluminum electrolysis cell, IEEE Transactions on Control Systems Technology, 6 (1998), 157-171. Google Scholar
[8]	M. Dupuis and I. GéniSim, Excess AlF₃ concentration in bath control logic, National Conference on Advancements in Aluminium Electrolysis, Indian Institute of Metals, Angul, (2006), 309-313. Google Scholar
[9]	P. M. Entner and G. A. Gudmundsson, Further development of the temperature model, Light Metals, (1996), 445-449. Google Scholar
[10]	W. Haupin and H. Kvande, Mathematical model of fluoride evolution from Hall-Héroult Cells, Essential Readings in Light Metals, (2016), 903-909. Google Scholar
[11]	Y. J. He, Y. Mao, W. L. Chen and Y. X. Chen, Nonlinear metric learning with kernel density estimation, IEEE Transactions Knowledge and Data Engineering, 27 (2015), 1602-1614. doi: 10.1109/TKDE.2014.2384522. Google Scholar
[12]	Y. L. He, R. Wang, S. Kwong and X. Z. Wang, Bayesian classifiers based on probability density estimation and their applications to simultaneous fault diagnosis, Information Sciences, 259 (2014), 252-268. doi: 10.1016/j.ins.2013.09.003. Google Scholar
[13]	Y. B. Huang, X. D. Qu and J. M. Zhou, Coupled heat/mass balance model for analyzing correlation between excess AlF₃ concentration and aluminum electrolyte temperature, Transactions of Nonferrous Metals Society of China, 19 (2009), 724-729. Google Scholar
[14]	M. M. Hyland, E. C. Patterson, F. S. Mcfadden and B. J. Welch, Aluminium fluoride consumption and control in smelting cells, Scand. J. Metall., 30 (2001), 404-414. Google Scholar
[15]	H. J. Kim, S. N. MacEachern and Y. Jung, Bandwidth selection for kernel density estimation with a markov chain monte carlo sample, arXiv. Preprint. arXiv., 1607.08274 (2016), 1-16. Google Scholar
[16]	K. R. Kloetstra, S. Benninghoff, M. A. Stam and B. W. Toebes, Optimisation of Aluminium Fluoride Control at Aluminium Delfzijl, Proceedings of the 7th Australasian Aluminium Smelting Workshop, (2001), 506-514. Google Scholar
[17]	M. Köhler, A. Schindler and S. Sperlich, A review and comparison of bandwidth selection methods for kernel regression, International Statistical Review, 82 (2014), 243-274. doi: 10.1111/insr.12039. Google Scholar
[18]	S. Kolås, Defining and verifying the 'correlation line' in aluminum electrolysis, JOM, 59 (2007), 55-60. Google Scholar
[19]	S Kolås and T. Støre, Bath temperature and AlF₃ control of an aluminium electrolysis cell, Control and Engineering Practice, 17 (2009), 1035-1043. Google Scholar
[20]	M. E. Maron and J. L. Kuhns, On relevance, probabilistic indexing and information retrieval, Journal of the ACM, 7 (1960), 216-244. doi: 10.1145/321033.321035. Google Scholar
[21]	A. Meghlaoui, Y. A. A. Farsi and N. H. Aljabri, Analytical and experimental study of fluoride evolution, Light Metals-Warrendale-Proceedings. TMS, 2001, 283-288. Google Scholar
[22]	A. R. Mugdadi and I. A. Ahmad, A bandwidth selection for kernel density estimation of functions of random variables, Computational Statistics & Data Analysis, 47 (2004), 49-62. doi: 10.1016/j.csda.2003.10.013. Google Scholar
[23]	R. J. Pak, The influence function of the optimal bandwidth for kernel density estimation, Communications in Statistics, 46 (2017), 602-608. doi: 10.1080/03610926.2014.1000501. Google Scholar
[24]	D. J. Salt, Bath chemistry control system, Essential Readings in Light Metals, (2016), 798-803. Google Scholar
[25]	A. T. Tabereaux, T. R. Alcorn and L Trembley, Lithium-Modified Low Ratio Electrolyte Chemistry for Improved Performance in Modern Reduction Cells, Essential Readings in Light Metals, (2016), 83-88. Google Scholar
[26]	J. Thonstand and S. Roselth, Equilibrium between bath and side ledge in aluminum cells-Basic principles, Light Metals, (1983), 414-424. Google Scholar
[27]	L. Y. Wang, W. H. Gui, K. L. Teo, R. C. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications, Journal of Industrial & Management Optimization, 5 (2009), 705-718. doi: 10.3934/jimo.2009.5.705. Google Scholar
[28]	X. Z. Wang, Y. L. He and D. D. Wang, Non-naive bayesian classifiers for classification problems with continuous attributes, IEEE Transactions on Cybernetics, 44 (2013), 21-39. doi: 10.1109/TCYB.2013.2245891. Google Scholar
[29]	M. J. Wilson, Practical considerations used in the development of a method for calculating aluminum fluoride additions based on cell temperatures, Light Metals (1992), 37-5-378. Google Scholar
[30]	J. Ye, H. Xu, E. Feng and Z. Xiu, Optimization of a fed-batch bioreactor for 1, 3-propanediol production using hybrid nonlinear optimal control, Journal of Process Control, 24 (2014), 1556-1569. Google Scholar
[31]	J. Yi, D. Huang, S. Fu and T. Li, Optimized relative transformation matrix using bacterial foraging algorithm for process fault detection, IEEE Transactions on Industrial Electronics, 63 (2016), 2595-2605. doi: 10.1109/TIE.2016.2515057. Google Scholar
[32]	W. C. Yue, X. F. Chen, W. H. Gui and H. L. Zhang, A knowledge reasoning Fuzzy-Bayesian network for root cause analysis of abnormal aluminum electrolysis cell condition, Fronters of Chemical Science and Engineering, 11 (2017), 414-428. doi: 10.1007/s11705-017-1663-x. Google Scholar
[33]	S. P. Zeng and F. W. Cui, Dynamic decision model for amount of AlF₃ addition in industrial aluminum electrolysis, International Conference on Mechatronics, Robotics and Automation, (2015), 307-318. Google Scholar
[34]	S. P. Zeng, S. S. Wang and Y. X. Qu, Control of temperature and aluminum fluoride concentration based on model prediction in aluminum electrolysis, Advances in Materials Science & Engineering, (2014), 1-5. Google Scholar
[35]	E. Zenteno, Z. A. Khan, M. Isaksson and P. Handel, Finding structural information about RF power amplifiers using an orthogonal nonparametric kernel smoothing estimator, IEEE Transactions on Vehicular Technology, 65 (2016), 2883-2889. doi: 10.1109/TVT.2015.2434497. Google Scholar
[36]	S. Q. Zhan, M. Li, J. M. Zhou and Y. W. Zhou, CFD simulation of dissolution process of alumina in an aluminum reduction cell with two-particle phase population balance model, Applied Thermal Engineering, 73 (2014), 805-818. doi: 10.1016/j.applthermaleng.2014.08.040. Google Scholar

show all references

References:

[1]	I. Barbeito and R. Cao, Smoothed stationary bootstrap bandwidth selection for density estimation with dependent data, Computational Statistics & Data Analysis, 104 (2016), 130-147. doi: 10.1016/j.csda.2016.06.015. Google Scholar
[2]	K. Barbé, L. G. Fuentes, L. Barford and L. Lauwers, A guaranteed blind and automatic probability density estimation of raw measurements, IEEE Transactions on Instrumentation & Measurement, 63 (2014), 2120-2128. Google Scholar
[3]	Z. G. Chen, Y. G. Li, X. F. Chen and W. H. Gui, Semantic Network Based on Intuitionistic Fuzzy Directed Hyper-Graphs and Application to Aluminum Electrolysis Cell Condition Identification, IEEE Access, 5 (2017), 20145-20156. doi: 10.1109/ACCESS.2017.2752200. Google Scholar
[4]	Y. Chien, Pattern classification and scene analysis, IEEE Transactions Automatic Control, 19 (1974), 462-463. doi: 10.1109/TAC.1974.1100577. Google Scholar
[5]	K. S. Chuang, H. L. Tzeng and S. Chen, Fuzzy c-means clustering with spatial information for image segmentation, Computerized Medical Imaging and Graphics, 30 (2006), 9-15. doi: 10.1016/j.compmedimag.2005.10.001. Google Scholar
[6]	P. Desclaux, AlF₃ additions based on bath temperature measurements, Light Metals, (1987), 309-313. Google Scholar
[7]	T. Drengstig, D. Ljungquist and B. A. Foss, On the AlF₃ and temperature control of an aluminum electrolysis cell, IEEE Transactions on Control Systems Technology, 6 (1998), 157-171. Google Scholar
[8]	M. Dupuis and I. GéniSim, Excess AlF₃ concentration in bath control logic, National Conference on Advancements in Aluminium Electrolysis, Indian Institute of Metals, Angul, (2006), 309-313. Google Scholar
[9]	P. M. Entner and G. A. Gudmundsson, Further development of the temperature model, Light Metals, (1996), 445-449. Google Scholar
[10]	W. Haupin and H. Kvande, Mathematical model of fluoride evolution from Hall-Héroult Cells, Essential Readings in Light Metals, (2016), 903-909. Google Scholar
[11]	Y. J. He, Y. Mao, W. L. Chen and Y. X. Chen, Nonlinear metric learning with kernel density estimation, IEEE Transactions Knowledge and Data Engineering, 27 (2015), 1602-1614. doi: 10.1109/TKDE.2014.2384522. Google Scholar
[12]	Y. L. He, R. Wang, S. Kwong and X. Z. Wang, Bayesian classifiers based on probability density estimation and their applications to simultaneous fault diagnosis, Information Sciences, 259 (2014), 252-268. doi: 10.1016/j.ins.2013.09.003. Google Scholar
[13]	Y. B. Huang, X. D. Qu and J. M. Zhou, Coupled heat/mass balance model for analyzing correlation between excess AlF₃ concentration and aluminum electrolyte temperature, Transactions of Nonferrous Metals Society of China, 19 (2009), 724-729. Google Scholar
[14]	M. M. Hyland, E. C. Patterson, F. S. Mcfadden and B. J. Welch, Aluminium fluoride consumption and control in smelting cells, Scand. J. Metall., 30 (2001), 404-414. Google Scholar
[15]	H. J. Kim, S. N. MacEachern and Y. Jung, Bandwidth selection for kernel density estimation with a markov chain monte carlo sample, arXiv. Preprint. arXiv., 1607.08274 (2016), 1-16. Google Scholar
[16]	K. R. Kloetstra, S. Benninghoff, M. A. Stam and B. W. Toebes, Optimisation of Aluminium Fluoride Control at Aluminium Delfzijl, Proceedings of the 7th Australasian Aluminium Smelting Workshop, (2001), 506-514. Google Scholar
[17]	M. Köhler, A. Schindler and S. Sperlich, A review and comparison of bandwidth selection methods for kernel regression, International Statistical Review, 82 (2014), 243-274. doi: 10.1111/insr.12039. Google Scholar
[18]	S. Kolås, Defining and verifying the 'correlation line' in aluminum electrolysis, JOM, 59 (2007), 55-60. Google Scholar
[19]	S Kolås and T. Støre, Bath temperature and AlF₃ control of an aluminium electrolysis cell, Control and Engineering Practice, 17 (2009), 1035-1043. Google Scholar
[20]	M. E. Maron and J. L. Kuhns, On relevance, probabilistic indexing and information retrieval, Journal of the ACM, 7 (1960), 216-244. doi: 10.1145/321033.321035. Google Scholar
[21]	A. Meghlaoui, Y. A. A. Farsi and N. H. Aljabri, Analytical and experimental study of fluoride evolution, Light Metals-Warrendale-Proceedings. TMS, 2001, 283-288. Google Scholar
[22]	A. R. Mugdadi and I. A. Ahmad, A bandwidth selection for kernel density estimation of functions of random variables, Computational Statistics & Data Analysis, 47 (2004), 49-62. doi: 10.1016/j.csda.2003.10.013. Google Scholar
[23]	R. J. Pak, The influence function of the optimal bandwidth for kernel density estimation, Communications in Statistics, 46 (2017), 602-608. doi: 10.1080/03610926.2014.1000501. Google Scholar
[24]	D. J. Salt, Bath chemistry control system, Essential Readings in Light Metals, (2016), 798-803. Google Scholar
[25]	A. T. Tabereaux, T. R. Alcorn and L Trembley, Lithium-Modified Low Ratio Electrolyte Chemistry for Improved Performance in Modern Reduction Cells, Essential Readings in Light Metals, (2016), 83-88. Google Scholar
[26]	J. Thonstand and S. Roselth, Equilibrium between bath and side ledge in aluminum cells-Basic principles, Light Metals, (1983), 414-424. Google Scholar
[27]	L. Y. Wang, W. H. Gui, K. L. Teo, R. C. Loxton and C. H. Yang, Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications, Journal of Industrial & Management Optimization, 5 (2009), 705-718. doi: 10.3934/jimo.2009.5.705. Google Scholar
[28]	X. Z. Wang, Y. L. He and D. D. Wang, Non-naive bayesian classifiers for classification problems with continuous attributes, IEEE Transactions on Cybernetics, 44 (2013), 21-39. doi: 10.1109/TCYB.2013.2245891. Google Scholar
[29]	M. J. Wilson, Practical considerations used in the development of a method for calculating aluminum fluoride additions based on cell temperatures, Light Metals (1992), 37-5-378. Google Scholar
[30]	J. Ye, H. Xu, E. Feng and Z. Xiu, Optimization of a fed-batch bioreactor for 1, 3-propanediol production using hybrid nonlinear optimal control, Journal of Process Control, 24 (2014), 1556-1569. Google Scholar
[31]	J. Yi, D. Huang, S. Fu and T. Li, Optimized relative transformation matrix using bacterial foraging algorithm for process fault detection, IEEE Transactions on Industrial Electronics, 63 (2016), 2595-2605. doi: 10.1109/TIE.2016.2515057. Google Scholar
[32]	W. C. Yue, X. F. Chen, W. H. Gui and H. L. Zhang, A knowledge reasoning Fuzzy-Bayesian network for root cause analysis of abnormal aluminum electrolysis cell condition, Fronters of Chemical Science and Engineering, 11 (2017), 414-428. doi: 10.1007/s11705-017-1663-x. Google Scholar
[33]	S. P. Zeng and F. W. Cui, Dynamic decision model for amount of AlF₃ addition in industrial aluminum electrolysis, International Conference on Mechatronics, Robotics and Automation, (2015), 307-318. Google Scholar
[34]	S. P. Zeng, S. S. Wang and Y. X. Qu, Control of temperature and aluminum fluoride concentration based on model prediction in aluminum electrolysis, Advances in Materials Science & Engineering, (2014), 1-5. Google Scholar
[35]	E. Zenteno, Z. A. Khan, M. Isaksson and P. Handel, Finding structural information about RF power amplifiers using an orthogonal nonparametric kernel smoothing estimator, IEEE Transactions on Vehicular Technology, 65 (2016), 2883-2889. doi: 10.1109/TVT.2015.2434497. Google Scholar
[36]	S. Q. Zhan, M. Li, J. M. Zhou and Y. W. Zhou, CFD simulation of dissolution process of alumina in an aluminum reduction cell with two-particle phase population balance model, Applied Thermal Engineering, 73 (2014), 805-818. doi: 10.1016/j.applthermaleng.2014.08.040. Google Scholar

Figure 1. Sketch of aluminum reduction cell

Parameter	Ab.	Value	Role analysis
Aluminum level	AL	20-23 cm	The height of the molten aluminum. A higher AL leads to greater heat loss, and vice versa. A suitable AL can stabilize the cell voltage.
Molecular ratio	MR	2.64-3.0	This affects the dissolution of the alumina in the electrolyte, with a higher MR leading to a lower SD, and vice versa.
Electrolyte level	EL	23-28 cm	This stabilize the thermal balance of the cell. Thus, the thermal balance is robust with a suitable EL.
Waving	WA	0-20 mv	A strong low-frequency noise may be due to insufficient energy intake for the cell.
Vibration	VI	0-50 mv	VI is an indicator of the stability of the cell. A greater VI is more likely for a cold cell.
Under/over number ratio	UO	0.75-1	The UO is the ratio between the under and over feeding times. A smaller UO is more likely for a cold cell, and vice versa.
Tapping amount	TA	2.9-3.05 ton	The TA has a great influence on the energy balance. A greater TA is more likely for a hot cell, and vice versa.
Electrolyte temperature	ET	955-965℃	This affects the entire operation condition of the cell. A higher temperature is more likely for a hot cell, and vice versa.

Definitions for SD			Definitions for dSD
Label	Meaning	Membership	Label	Meaning	Membership
VL	Very low	$\mu \left( VL \right)=P\left( VL\left\| {{{\bf{x}}}_{i}} \right. \right)$	HN	High negative	$\mu \left( HN \right)=P\left( HN\left\| \Delta {{{\bf{x}}}_{i}} \right. \right)$
LL	Little low	$\mu \left( LL \right)=P\left( LL\left\| {{{\bf{x}}}_{i}} \right. \right)$	LN	Low negative	$\mu \left( LN \right)=P\left( LN\left\| \Delta {{{\bf{x}}}_{i}} \right. \right)$
N	Normal	$\mu \left( N\right)=P\left( N\left\| {{{\bf{x}}}_{i}} \right. \right)$	Z	zero	$\mu \left( N \right)=P\left( N\left\| \Delta {{{\bf{x}}}_{i}} \right. \right)$
LH	Little high	$\mu \left( LP \right)=P\left( LP\left\| {{{\bf{x}}}_{i}} \right. \right)$	LP	Low positive	$\mu \left( LH \right)=P\left( LH\left\| \Delta {{{\bf{x}}}_{i}} \right. \right)$
VH	Very high	$\mu \left( LH \right)=P\left( LH\left\| {{{\bf{x}}}_{i}} \right. \right)$	HP	High positive	$\mu \left( HP \right)=P\left( HP\left\| \Delta {{{\bf{x}}}_{i}} \right. \right)$

Data set names	Number of instances			Accuracy rate
Data set names	training	test	attributes	ENBC	NBC
hline seeds	135	75	7	0.9733	0.8933
banknote	1297	75	5	0.9467	0.8267

Labels	Index	Parameters
Labels	Index	MR	EL (cm)	WA (mv)	VI (mv)	UO	TA kg	ET (℃)	AL (cm)
VL	1	3.05	33	537	799	0.76	3023	959	22.0
LL	2	2.98	32	89	114	0.52	2988	960	23.0
N	3	2.79	23	6	10	0.40	2811	974	25.5
LH	4	2.87	31	2	6	1.18	2787	976	21.0
VH	5	2.52	25	3	5	0.33	2896	985	24.0

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American Institute of Mathematical Sciences

Evaluation strategy and mass balance for making decision about the amount of aluminum fluoride addition based on superheat degree

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