Parameter optimal identification and dynamic behavior analysis of nonlinear model for the solution purification process of zinc hydrometallurgy

Qianqian Wang; Minan Tang; Aimin An; Jiawei Lu; Yingying Zhao

doi:10.3934/jimo.2021159

Article Contents

2022, Volume 18, Issue 1: 693-712. Doi: 10.3934/jimo.2021159

This issue Previous Article Parallel-machine scheduling in shared manufacturing Next Article

Parameter optimal identification and dynamic behavior analysis of nonlinear model for the solution purification process of zinc hydrometallurgy

1.
College of Electrical and Information Engineering, Lanzhou University of Technology, Lanzhou, 730050, China
2.
College of Automation and Electrical Engineering, Lanzhou Jiaotong University, Lanzhou, 730070, China
3.
College of Electrical and Information Engineering, Key Laboratory of Gansu Advanced Control for Industrial Processes, National Demonstration Center for Experimental Electrical and Control Engineering Education, Lanzhou University of Technology, Lanzhou, 730050, China

^* Corresponding author: Aimin An, Ph.D Professor; Email: anaiminll@163.com; ORCID:0000-0003-3607-6536
^* Corresponding author: Aimin An, Ph.D Professor; Email: anaiminll@163.com; ORCID:0000-0003-3607-6536

Received: February 28, 2021

Revised: May 31, 2021

Early access: September 2021

Published: January 2022

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Abstract

Impurity removal is a momentous part of zinc hydrometallurgy process, and the quality of products and the stability of the whole process are affected directly by its control effect. The application of dynamic model is of great significance to the prediction of key indexes and the optimization of process control. In this paper, considering the complex coupling relationship of stage II purification process, a hybrid modeling method of mechanism modeling and parameter identification modeling was proposed on the basis of not changing the actual production process of lead-zinc smeltery. Firstly, the overall nonlinear dynamic mechanism model was established, and then the deviation between the theoretical value and the actual detected outlet ion concentration was taken as the objective function to establish the parameter identification optimization model. Since the built model is nonlinear, it may pose implementation problems. On the premise of deriving the gradient vector and Hessian matrix of the objective function with respect to the parameter vector, an optimization algorithm based on the steepest descent method and Newton method is proposed. Finally, using the historical production data of a lead-zinc smeltery in China, the model parameters were accurately inversed. An intensive simulation validation and analysis of the dynamic characteristics about the whole model shows the accuracy and the potential of the model, also in the perspective of practical implementation, which provides the basis for the optimal control of system output and the guidance for the optimal control of zinc powder addition.

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Mathematics Subject Classification: Primary: 93-10.

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Figure 1. Flow chart of roasting, leaching and purification process of a lead-zinc smeltery

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Figure 2. Temperature profiles of stage II solution purification in a lead-zinc smeltery

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Figure 3. The equivalent CSTR model

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Figure 4. Flow chart of solution algorithm for parameter identification model

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Figure 5. Stage II purification reaction tank of zinc hydrometallurgy in a lead-zinc smeltery

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Figure 6. Deviation function profile and parameter increment profile in solving process.(a)Deviation function profile; (b)Parameter increment profile

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Figure 7. Nonlinear dynamic system model of the stage II purification process

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Figure 8. Response profiles of outlet ion concentration.(a)Response profile of cobalt ion concentration; (b)Response profile of cadmium ion concentration

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Figure 9. Variation profiles of outlet ion concentration when $ {u_{\rm{b}}} $ is constant and $ {u_{\rm{a}}} $ is variable.(a)Cobalt ion concentration at the outlet; (b)Cadmium ion concentration at the outlet

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Figure 10. Variation profiles of outlet ion concentration when $ {u_{\rm{a}}} $ is constant and $ {u_{\rm{b}}} $ is variable.(a)Cobalt ion concentration at the outlet; (b)Cadmium ion concentration at the outlet

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Figure 11. Variation profiles of outlet ion concentration when $ {u_{\rm{a}}} $ and $ {u_{\rm{b}}} $ change

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Figure 12. The influence of inlet flow $ Q $ on impurity ions concentration.(a) Cobalt ion concentration at the outlet; (b)Cadmium ion concentration at the outlet

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Figure 13. Model test results.(a)Comparison of cobalt ion concentration at the outlet; (b) Comparison of cadmium ion concentration at the outlet

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Table 1. Values of relevant parameters in the reaction process

Parameter	Symbol	Value
Solution flow rate/(${{\rm{m}}^3}/{\rm{h}}$)	$Q$	160
Volume of single reaction tank/${{\rm{m}}^3}$	${V_p}$	108
Volume utilization of reaction tank/$\% $	$\backslash$	80
Area coefficient/(${{\rm{m}}^2}/{\rm{kg}}$)	$p$	174

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Table 2. Characteristics of sample data

Data type	Symbol	Average value	Maximum value	Minimum value
Inlet cobalt ion concentration/(${\rm{mg/L}}$)	${x_{{\rm{a0}}}}$	35.251	51.190	13.382
Inlet cadmium ion concentration/(${\rm{mg/L}}$)	${x_{{\rm{b0}}}}$	298.278	433.148	113.229
Outlet cobalt ion concentration/(${\rm{mg/L}}$)	${\bar x_{\rm{a}}}$	0.439	0.945	0.257
Outlet cadmium ion concentration/(${\rm{mg/L}}$)	${\bar x_b}$	15.772	58.943	1.449

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Table 3. Inlet ion concentration values

Inlet ion	Symbol	Value
Cobalt ion/(${\rm{g/L}}$)	${x_{{\rm{a0}}}}$	0.035
Cadmium ion/(${\rm{g/L}}$)	$ {x_{{\rm{b0}}}} $	0.298

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Table 4. Model error results

Error	Model in this paper	Original model [8]	Reformulated model [8]
Maximum error (Cobalt ion) /${\rm{\% }}$	31.12	49.10	32.27
Maximum error (Cadmium ion)/${\rm{\% }}$	24.40	$\backslash$	$\backslash$
Average error (Cobalt ion)/${\rm{\% }}$	11.05	13.81	11.90
Average error (Cadmium ion)/${\rm{\% }}$	10.85	$\backslash$	$\backslash$

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