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doi: 10.3934/jimo.2022003
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Hybrid modeling and distributed optimization control method for the iron removal process

1. 

School of Automation, Central South University, Changsha Hunan 410083, China

2. 

School of Electrical Engineering, Guangxi University, Nanning Guangxi 53004, China

* Corresponding author: Ning Chen

Received  July 2021 Revised  November 2021 Early access January 2022

Fund Project: The research is supported in part by the Program of National Natural Science Foundation of China (61673399), and in part by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (61621062)

Iron removal by goethite is a key procedure in zinc hydrometallurgy. Due to its complex chemical reaction mechanism, multiple reactors cascade, and high uncertainty, it is difficult to optimize and control in industrial process. In this paper, a distributed optimization control method which is based on a novel hybrid model of the iron removal process is proposed. By combining the mechanism model and a data-driven oxygen mass transfer coefficient model, a hybrid model is first established. Then, to overcome the influence of the former reactor on the latter reactor, the ratio of the status in each subsystem to the set point is taken as a new status, and a distributed optimization control problem is constructed. Considering the high dimensionality of this problem, it is necessary to reconstruct it by Virtual Motion Camouflage (VMC), so that the optimal control problem is transformed into a nonlinear constrained optimal trajectory planning problem. And a Legendre pseudo-spectral method is used to solve the problem accurately to obtain the optimal trajectory of the ion concentration. Finally, simulation results show that the proposed method can effectively reflect the industrial process, and track the fluctuation of inlet ion concentrations with a nice real-time performance.

Citation: Ning Chen, Yan Xia Zhao, Jia Yang Dai, Yu Qian Guo, Wei Hua Gui, Jun Jie Peng. Hybrid modeling and distributed optimization control method for the iron removal process. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022003
References:
[1]

C. B. JulioL. D. PolliL and S. M. Tania, Importance of roasted sulphide concentrates characterization in the hydrometallurgical extraction of zinc, Minerals Engineering, 21 (2008), 100-110. 

[2]

M. LoanO. M. G. Newman and R. M. G. Cooper, Defining the Paragoethite process for iron removal in zinc hydrometallurgy, Hydrometallurgy, 81 (2006), 104-129.  doi: 10.1016/j.hydromet.2005.11.002.

[3]

M. R. C. Ismael, Iron recovery from sulphate leach liquors in zinc hydrometallurgy, Minerals Engineering, 16 (2003), 31-39.  doi: 10.1016/S0892-6875(02)00310-2.

[4]

N. Chen, S. Yang and W. H. Gui, Fuzzy cognitive network control of iron sinking process by goethite method, The 35th China Control Conference, Chengdu, (in Chinese), 2016.

[5]

Y. F. XieS. W. Xie and Y. G. Li, Dynamic modeling and optimal control of goethite process based on the rate-controlling step, Control Engineering Practice, 58 (2017), 54-65.  doi: 10.1016/j.conengprac.2016.10.001.

[6]

N. ChenJ. Q. Zhou and J. J. Peng, Modeling of goethite iron precipitation process based on time-delay fuzzy gray cognitive network, Journal of Central South University, 26 (2019), 63-74.  doi: 10.1007/s11771-019-3982-1.

[7]

F. Q. Xiong, W. H. Gui and C. H. Yang, Dynamic modeling of iron removal process based on goethite method, Journal of Central South University, (Natural Science Edition), 43 (2012), 541–547.

[8]

T. HaakanaM. Lahtinen and H. Takala, Development and modelling of a novel reactor for direct leaching of zinc sulphide concentrates, Chemical Engineering Science, 62 (2007), 5648-5654.  doi: 10.1016/j.ces.2006.12.075.

[9]

Y. F. XieS. W. Xie and X. F. Chen, An integrated predictive model with an on-line updating strategy for iron precipitation in zinc hydrometallurgy, Hydrometallurgy, 151 (2015), 62-72.  doi: 10.1016/j.hydromet.2014.11.004.

[10]

Y. F. Xie and S. W. Xie, Coordinated optimization for the descent gradient of technical index in the iron removal process, IEEE Transactions on Cybernetics, 48 (2018), 3313-3322.  doi: 10.1109/TCYB.2018.2833805.

[11]

N. ChenJ. Y. Dai and X. J. Zhou, Distributed model predictive control of iron precipitation process by goethite based on dual iterative method, International Journal of Control, Automation and Systems, 17 (2019), 1233-1245.  doi: 10.1007/s12555-017-0742-6.

[12]

A. MujahedS. Alsabbah and I. Mujtaba, A predictive neural network-based cascade control for ph reactors, Mathematical Problems in Engineering: Theory, Methods and Applications, 5638632 (2016), 1-7. 

[13]

I. F. Nusyirwan and C. Bil, Effect of uncertainties on UCAV trajectory optimisation using evolutionary programming, 2007 Information, Decision and Control, Australia, 219–223. doi: 10.1109/IDC.2007.374553.

[14]

R. Dai, B-splines based optimal control solution, Aiaa Guidance, Navigation & Control Conference, California, 2010. doi: 10.2514/6.2010-7888.

[15]

Y. Xu and N. Li, Bio-inspired varying subspace based computational framework for a class of nonlinear constrained optimal trajectory planning problems, Bioinspiration & Biomimetics, 9 (2014), 036010. 

[16]

Y. Xu and G. Basset, Virtual motion camouflage based phantom track generation through cooperative electronic combat air vehicles, Automatica, 46 (2010), 1454-1461.  doi: 10.1016/j.automatica.2010.05.027.

[17]

R. Strydom and M. Srinivasan, UAS stealth: Target pursuit at constant distance using a bio-inspired motion camouflage guidance law, Bioinspiration & Biomimetics, 12 (2017), 055002.  doi: 10.1088/1748-3190/aa7d65.

[18]

S. G. Defterli and Y. Xu, Virtual motion camouflage based visual servo control of a leaf picking mechanism, 11th Annual Dynamic Systems and Control Conference (DSCC 2018), Atlanta, (2018).  doi: 10.1115/DSCC2018-9042.

[19]

M. Develle and Y. Xu, Optimal attitude control allocation via the B-spline augmented virtual motion camouflage method, IEEE Transactions on Aerospace and Electronic Systems, 51 (2015), 1774-1780. 

[20]

D. J. KwakB. Choi and D. Cho, Decentralized trajectory optimization using virtual motion camouflage and particle swarm optimization, Autonomous Robots, 38 (2015), 161-177.  doi: 10.1007/s10514-014-9399-7.

[21]

N. ChenJ. Y. Dai and W. H. Gui, A hybrid prediction model with a selectively updating strategy for iron removal process in zinc hydrometallurgy, Science China-Information Sciences, 63 (2020), 119205:1-119205:3.  doi: 10.1007/s10514-014-9399-7.

[22]

Y. J. XuC. Remeikas and K. Pham, Local pursuit strategy-inspired cooperative trajectory planning algorithm for a class of nonlinear constrained dynamical systems, Internat. J. Control, 87 (2014), 506-523.  doi: 10.1080/00207179.2013.845911.

[23]

Y. W. LiuZ. Y. LinK. G. ZhaoJ. Ye and X. D. Huang, Multiobjective gearshift optimization with Legendre pseudospectral method for seamless two-speed transmission, Mechanism and Machine Theory, 145 (2020), 103682.  doi: 10.1016/j.mechmachtheory.2019.103682.

[24]

I. Matychyn, Pursuit strategy of motion camouflage in dynamic games, Dyn. Games Appl., 10 (2020), 145-156.  doi: 10.1007/s13235-019-00316-0.

[25]

G. BassetY. J. Xu and N. Li, Fast trajectory planning via the B-spline augmented virtual motion camouflage approach, 2011 50th IEEE Conference on Decision and Control and European Control Conference, (2011).  doi: 10.1109/CDC.2011.6160835.

[26]

S. P. Kim and R. G. Melton, Constrained station relocation in geostationary equatorial orbit using a legendre pseudospectral method, Journal of Guidance, Control, and Dynamics, 38 (2014), 711-719.  doi: 10.2514/1.G000114.

[27]

S. B. Xu and S. B. Li, Legendre pseudospectral method for optimal control problem and its application, Jounal of Control and Decision, 29 (2014), 2113-2120. 

[28]

S. W. XieY. F. Xie and F. B. Li, Hybrid fuzzy control for the goethite process in zinc production plant combining type-1 and type-2 fuzzy logics, Neurocomputing, 366 (2019), 170-177.  doi: 10.1016/j.neucom.2019.06.089.

[29]

H. ZhangW. N. Wang and G. Y. Xu, Optimal control of formation reconfiguration for multiple UAVs based on Legendre Pseudospectral Method, Chinese Control and Decision Conference, Chongqing, (2017), 6230-6235.  doi: 10.1109/CCDC.2017.7978292.

[30]

P. D. Laurie, Computation of Gauss-type quadrature formulas, J. Comput. Appl. Math., 127 (2001), 201-217.  doi: 10.1016/S0377-0427(00)00506-9.

show all references

References:
[1]

C. B. JulioL. D. PolliL and S. M. Tania, Importance of roasted sulphide concentrates characterization in the hydrometallurgical extraction of zinc, Minerals Engineering, 21 (2008), 100-110. 

[2]

M. LoanO. M. G. Newman and R. M. G. Cooper, Defining the Paragoethite process for iron removal in zinc hydrometallurgy, Hydrometallurgy, 81 (2006), 104-129.  doi: 10.1016/j.hydromet.2005.11.002.

[3]

M. R. C. Ismael, Iron recovery from sulphate leach liquors in zinc hydrometallurgy, Minerals Engineering, 16 (2003), 31-39.  doi: 10.1016/S0892-6875(02)00310-2.

[4]

N. Chen, S. Yang and W. H. Gui, Fuzzy cognitive network control of iron sinking process by goethite method, The 35th China Control Conference, Chengdu, (in Chinese), 2016.

[5]

Y. F. XieS. W. Xie and Y. G. Li, Dynamic modeling and optimal control of goethite process based on the rate-controlling step, Control Engineering Practice, 58 (2017), 54-65.  doi: 10.1016/j.conengprac.2016.10.001.

[6]

N. ChenJ. Q. Zhou and J. J. Peng, Modeling of goethite iron precipitation process based on time-delay fuzzy gray cognitive network, Journal of Central South University, 26 (2019), 63-74.  doi: 10.1007/s11771-019-3982-1.

[7]

F. Q. Xiong, W. H. Gui and C. H. Yang, Dynamic modeling of iron removal process based on goethite method, Journal of Central South University, (Natural Science Edition), 43 (2012), 541–547.

[8]

T. HaakanaM. Lahtinen and H. Takala, Development and modelling of a novel reactor for direct leaching of zinc sulphide concentrates, Chemical Engineering Science, 62 (2007), 5648-5654.  doi: 10.1016/j.ces.2006.12.075.

[9]

Y. F. XieS. W. Xie and X. F. Chen, An integrated predictive model with an on-line updating strategy for iron precipitation in zinc hydrometallurgy, Hydrometallurgy, 151 (2015), 62-72.  doi: 10.1016/j.hydromet.2014.11.004.

[10]

Y. F. Xie and S. W. Xie, Coordinated optimization for the descent gradient of technical index in the iron removal process, IEEE Transactions on Cybernetics, 48 (2018), 3313-3322.  doi: 10.1109/TCYB.2018.2833805.

[11]

N. ChenJ. Y. Dai and X. J. Zhou, Distributed model predictive control of iron precipitation process by goethite based on dual iterative method, International Journal of Control, Automation and Systems, 17 (2019), 1233-1245.  doi: 10.1007/s12555-017-0742-6.

[12]

A. MujahedS. Alsabbah and I. Mujtaba, A predictive neural network-based cascade control for ph reactors, Mathematical Problems in Engineering: Theory, Methods and Applications, 5638632 (2016), 1-7. 

[13]

I. F. Nusyirwan and C. Bil, Effect of uncertainties on UCAV trajectory optimisation using evolutionary programming, 2007 Information, Decision and Control, Australia, 219–223. doi: 10.1109/IDC.2007.374553.

[14]

R. Dai, B-splines based optimal control solution, Aiaa Guidance, Navigation & Control Conference, California, 2010. doi: 10.2514/6.2010-7888.

[15]

Y. Xu and N. Li, Bio-inspired varying subspace based computational framework for a class of nonlinear constrained optimal trajectory planning problems, Bioinspiration & Biomimetics, 9 (2014), 036010. 

[16]

Y. Xu and G. Basset, Virtual motion camouflage based phantom track generation through cooperative electronic combat air vehicles, Automatica, 46 (2010), 1454-1461.  doi: 10.1016/j.automatica.2010.05.027.

[17]

R. Strydom and M. Srinivasan, UAS stealth: Target pursuit at constant distance using a bio-inspired motion camouflage guidance law, Bioinspiration & Biomimetics, 12 (2017), 055002.  doi: 10.1088/1748-3190/aa7d65.

[18]

S. G. Defterli and Y. Xu, Virtual motion camouflage based visual servo control of a leaf picking mechanism, 11th Annual Dynamic Systems and Control Conference (DSCC 2018), Atlanta, (2018).  doi: 10.1115/DSCC2018-9042.

[19]

M. Develle and Y. Xu, Optimal attitude control allocation via the B-spline augmented virtual motion camouflage method, IEEE Transactions on Aerospace and Electronic Systems, 51 (2015), 1774-1780. 

[20]

D. J. KwakB. Choi and D. Cho, Decentralized trajectory optimization using virtual motion camouflage and particle swarm optimization, Autonomous Robots, 38 (2015), 161-177.  doi: 10.1007/s10514-014-9399-7.

[21]

N. ChenJ. Y. Dai and W. H. Gui, A hybrid prediction model with a selectively updating strategy for iron removal process in zinc hydrometallurgy, Science China-Information Sciences, 63 (2020), 119205:1-119205:3.  doi: 10.1007/s10514-014-9399-7.

[22]

Y. J. XuC. Remeikas and K. Pham, Local pursuit strategy-inspired cooperative trajectory planning algorithm for a class of nonlinear constrained dynamical systems, Internat. J. Control, 87 (2014), 506-523.  doi: 10.1080/00207179.2013.845911.

[23]

Y. W. LiuZ. Y. LinK. G. ZhaoJ. Ye and X. D. Huang, Multiobjective gearshift optimization with Legendre pseudospectral method for seamless two-speed transmission, Mechanism and Machine Theory, 145 (2020), 103682.  doi: 10.1016/j.mechmachtheory.2019.103682.

[24]

I. Matychyn, Pursuit strategy of motion camouflage in dynamic games, Dyn. Games Appl., 10 (2020), 145-156.  doi: 10.1007/s13235-019-00316-0.

[25]

G. BassetY. J. Xu and N. Li, Fast trajectory planning via the B-spline augmented virtual motion camouflage approach, 2011 50th IEEE Conference on Decision and Control and European Control Conference, (2011).  doi: 10.1109/CDC.2011.6160835.

[26]

S. P. Kim and R. G. Melton, Constrained station relocation in geostationary equatorial orbit using a legendre pseudospectral method, Journal of Guidance, Control, and Dynamics, 38 (2014), 711-719.  doi: 10.2514/1.G000114.

[27]

S. B. Xu and S. B. Li, Legendre pseudospectral method for optimal control problem and its application, Jounal of Control and Decision, 29 (2014), 2113-2120. 

[28]

S. W. XieY. F. Xie and F. B. Li, Hybrid fuzzy control for the goethite process in zinc production plant combining type-1 and type-2 fuzzy logics, Neurocomputing, 366 (2019), 170-177.  doi: 10.1016/j.neucom.2019.06.089.

[29]

H. ZhangW. N. Wang and G. Y. Xu, Optimal control of formation reconfiguration for multiple UAVs based on Legendre Pseudospectral Method, Chinese Control and Decision Conference, Chongqing, (2017), 6230-6235.  doi: 10.1109/CCDC.2017.7978292.

[30]

P. D. Laurie, Computation of Gauss-type quadrature formulas, J. Comput. Appl. Math., 127 (2001), 201-217.  doi: 10.1016/S0377-0427(00)00506-9.

Figure 1.  Process diagram of the iron removal process
Figure 2.  The dissolution and reaction process of oxygen
Figure 3.  The hybrid modeling block diagram
Figure 4.  Block diagram of VMC based optimized control of double-layer structure
Figure 5.  The prey-predator relationship in MC
Figure 6.  Variation of ferrous ion concentration in #1 reactor
Figure 7.  Variation of ferric ion concentration in #1 reactor
Figure 8.  Variation of ferrous ion concentration in #2 reactor
Figure 9.  Variation of ferric ion concentration in #2 reactor
Figure 10.  Variation of ferrous ion concentration in #3 reactor
Figure 11.  Variation of ferric ion concentration in #3 reactor
Figure 12.  Variation of ferrous ion concentration in #4 reactor
Figure 13.  Variation of ferric ion concentration in #4 reactor
Figure 14.  Variation of ferrous ion concentration in #5 reactor
Figure 15.  Variation of ferric ion concentration in #5 reactor
Figure 16.  The concentration of ferrous ions at the outlet of #5 reactor within 80 hours
Figure 17.  The ferrous ion concentration curve at the outlet of the #1 reactor
Figure 18.  The ferrous ion concentration curve at the outlet of the #2 reactor
Table 1.  Optimal control of the iron removal process based on VMC
Initialization Step0: Determine the expected ion concentration decline curve from process history data. Define the initial time $ {t_0} $, the end time $ {t_f} $ and the number of LGL points $ N $. Determine the LGL points based on Quasi Newton method [27]. Calculate the differential matrix $ {D_{kj}} $.
Iteration Step1: Initialize path control parameter $ v $.
Step2: Disperse $ \mathit{\boldsymbol{{c_{pi}} }}$ at the LGL points to obtain $ \mathit{\boldsymbol{c_{pj}^i({\tau _k}) }}$.
Step3: Disperse the path control parameter $ v $ at the LGL points
to obtain $ v({\tau _k})(k = 1,2,...,N - 1) $.
Step4: Disperse the constraints and performance functions in Eq.
(52) at the LGL points.
Step5: Calculate the optimal path control parameter $ v({\tau _k}) $ by the
fmincon solution tool in MATLAB(the fmincon function uses the
sequential quadratic programming method, and solve the quadratic
programming sub-problem in each iteration).
Step6: Calculate $ \mathit{\boldsymbol{c_j^i({\tau _k}) }}$ in Eq.(48).
Step7: If the convergence criterion is satisfied or the maximum num-
ber of iterations has reached, the optimization is terminated. Other-
wise, recalculate path control parameter $ v $ and go back to Step1.
Initialization Step0: Determine the expected ion concentration decline curve from process history data. Define the initial time $ {t_0} $, the end time $ {t_f} $ and the number of LGL points $ N $. Determine the LGL points based on Quasi Newton method [27]. Calculate the differential matrix $ {D_{kj}} $.
Iteration Step1: Initialize path control parameter $ v $.
Step2: Disperse $ \mathit{\boldsymbol{{c_{pi}} }}$ at the LGL points to obtain $ \mathit{\boldsymbol{c_{pj}^i({\tau _k}) }}$.
Step3: Disperse the path control parameter $ v $ at the LGL points
to obtain $ v({\tau _k})(k = 1,2,...,N - 1) $.
Step4: Disperse the constraints and performance functions in Eq.
(52) at the LGL points.
Step5: Calculate the optimal path control parameter $ v({\tau _k}) $ by the
fmincon solution tool in MATLAB(the fmincon function uses the
sequential quadratic programming method, and solve the quadratic
programming sub-problem in each iteration).
Step6: Calculate $ \mathit{\boldsymbol{c_j^i({\tau _k}) }}$ in Eq.(48).
Step7: If the convergence criterion is satisfied or the maximum num-
ber of iterations has reached, the optimization is terminated. Other-
wise, recalculate path control parameter $ v $ and go back to Step1.
Table 2.  Comparison of accuracy and efficiency of D-LWKPCR when $ \bar N $ changes
$ \bar N $ RMSE(g/L) MAE(g/L) Time(s)
5 0.8303 0.6880 0.4840
6 0.6933 0.4977 0.5300
8 0.6378 0.4142 0.5770
10 0.6168 0.3732 0.6860
12 0.6269 0.3887 0.6870
15 0.6445 0.4352 0.7330
20 0.6694 0.4719 0.8110
$ \bar N $ RMSE(g/L) MAE(g/L) Time(s)
5 0.8303 0.6880 0.4840
6 0.6933 0.4977 0.5300
8 0.6378 0.4142 0.5770
10 0.6168 0.3732 0.6860
12 0.6269 0.3887 0.6870
15 0.6445 0.4352 0.7330
20 0.6694 0.4719 0.8110
Table 3.  Parameter identification results of the hybrid model based on D-LWKPCR
Parameter $ k_1^1 $ $ k_2^1 $ $ k_3^1 $ $ \alpha $ $ \beta $ $ \gamma $ $ \delta $ $ \sigma $
Identification results 1.5963 0.0013 20.0868 1.3592 1.2807 0.3764 4.4953 1.9341
Parameter $ k_1^1 $ $ k_2^1 $ $ k_3^1 $ $ \alpha $ $ \beta $ $ \gamma $ $ \delta $ $ \sigma $
Identification results 1.5963 0.0013 20.0868 1.3592 1.2807 0.3764 4.4953 1.9341
Table 4.  Parameter identification results of the hybrid model based on JITL-SWPPLS
Parameter $ k_1^1 $ $ k_2^1 $ $ k_3^1 $ $ \alpha $ $ \beta $ $ \gamma $ $ \bar \sigma $
Identification results 1.7113 0.3521 22.9975 1.5091 1.8809 0.3764 0.6421
Parameter $ k_1^1 $ $ k_2^1 $ $ k_3^1 $ $ \alpha $ $ \beta $ $ \gamma $ $ \bar \sigma $
Identification results 1.7113 0.3521 22.9975 1.5091 1.8809 0.3764 0.6421
Table 5.  The set-points at the outlet of #1-#5 reactors
Ion concentration #1 reactor #2 reactor #3 reactor #4 reactor #5 reactor
$ F{e^{2 + }} $ 9.61 5.89 3.66 1.57 0.68
$ F{e^{3 + }} $ 1.08 0.92 0.8 0.67 0.5
Ion concentration #1 reactor #2 reactor #3 reactor #4 reactor #5 reactor
$ F{e^{2 + }} $ 9.61 5.89 3.66 1.57 0.68
$ F{e^{3 + }} $ 1.08 0.92 0.8 0.67 0.5
Table 6.  The ranges of oxygen addition and pH value of #1-#5 reactors
Reactor #1 reactor #2 reactor #3 reactor #4 reactor #5 reactor
Oxygen addition($ {m^3}/h $) [10, 50] [50,100] [60,115] [70,135] [40, 90]
pH [3.0, 3.5] [3.0, 3.5] [3.0, 3.5] [3.0, 3.5] [3.0, 3.5]
Reactor #1 reactor #2 reactor #3 reactor #4 reactor #5 reactor
Oxygen addition($ {m^3}/h $) [10, 50] [50,100] [60,115] [70,135] [40, 90]
pH [3.0, 3.5] [3.0, 3.5] [3.0, 3.5] [3.0, 3.5] [3.0, 3.5]
Table 7.  Performance of the proposed method in #1 reactor
Performance Control error of $ F{e^{2 + }} $(%) Control error of $ F{e^{3 + }} $(%) Oxygen addition ($ {m^3}/h $) Time(s)
$ N = 4 $ 4.37 21.96 31.3 1.85
$ N = 10 $ 1.75 8.27 38.68 1.92
$ N = 20 $ 0.87 4.05 30.11 2.08
$ N = 30 $ 0.58 2.69 31.03 2.47
$ N = 40 $ 0.19 1.41 29.67 2.63
$ N = 50 $ 0.48 3.33 35.7 2.72
Performance Control error of $ F{e^{2 + }} $(%) Control error of $ F{e^{3 + }} $(%) Oxygen addition ($ {m^3}/h $) Time(s)
$ N = 4 $ 4.37 21.96 31.3 1.85
$ N = 10 $ 1.75 8.27 38.68 1.92
$ N = 20 $ 0.87 4.05 30.11 2.08
$ N = 30 $ 0.58 2.69 31.03 2.47
$ N = 40 $ 0.19 1.41 29.67 2.63
$ N = 50 $ 0.48 3.33 35.7 2.72
Table 8.  The performance of the proposed method in the overall system
Performance Control error of $ F{e^{2 + }} $(%) Control error of $ F{e^{3 + }} $(%) Oxygen addition ($ {m^3}/h $) Time(s)
$ N = 4 $ 20.12 7.6 340.04 9.7
$ N = 10 $ 8.72 2.96 338.62 10.29
$ N = 20 $ 4.27 1.48 330.45 11.03
$ N = 30 $ 3.63 1.38 350.24 12.7
$ N = 40 $ 1.31 0.34 322.16 13.4
$ N = 50 $ 3.6 0.94 350.34 13.96
Performance Control error of $ F{e^{2 + }} $(%) Control error of $ F{e^{3 + }} $(%) Oxygen addition ($ {m^3}/h $) Time(s)
$ N = 4 $ 20.12 7.6 340.04 9.7
$ N = 10 $ 8.72 2.96 338.62 10.29
$ N = 20 $ 4.27 1.48 330.45 11.03
$ N = 30 $ 3.63 1.38 350.24 12.7
$ N = 40 $ 1.31 0.34 322.16 13.4
$ N = 50 $ 3.6 0.94 350.34 13.96
Table 9.  The $ F{e^{2 + }} $ ion concentration values at the outlet of each reactor when $ N $ = 40
Reactor The first method The second method The set points of $ F{e^{2 + }} $ ion concentration
#1 9.60 9.50 9.61
#2 5.89 5.66 5.89
#3 3.66 3.52 3.66
#4 1.57 1.44 1.57
#5 0.68 0.63 0.68
Reactor The first method The second method The set points of $ F{e^{2 + }} $ ion concentration
#1 9.60 9.50 9.61
#2 5.89 5.66 5.89
#3 3.66 3.52 3.66
#4 1.57 1.44 1.57
#5 0.68 0.63 0.68
Table 10.  The $ F{e^{3 + }} $ ion concentration values at the outlet of each reactor when $ N $ = 40
Reactor The first method The second method The set points of $ F{e^{3 + }} $ion concentration
#1 1.08 1.00 1.08
#2 0.92 0.91 0.92
#3 0.8 0.79 0.8
#4 0.67 0.67 0.67
#5 0.5 0.49 0.5
Reactor The first method The second method The set points of $ F{e^{3 + }} $ion concentration
#1 1.08 1.00 1.08
#2 0.92 0.91 0.92
#3 0.8 0.79 0.8
#4 0.67 0.67 0.67
#5 0.5 0.49 0.5
Table 11.  System performance comparison of the two methods
Algorithm The first method The second method
Performance Oxygen addition Time(s) Oxygen addition Time(s)
(${m^3}/h$) (${m^3}/h$)
#1 38.07 3.39 29.67 2.63
#2 68.32 3.25 75.43 2.58
#3 90.64 3.78 95.43 2.44
#4 80.25 3.56 79.24 2.32
#5 46.25 3.12 45.42 2.10
Algorithm The first method The second method
Performance Oxygen addition Time(s) Oxygen addition Time(s)
(${m^3}/h$) (${m^3}/h$)
#1 38.07 3.39 29.67 2.63
#2 68.32 3.25 75.43 2.58
#3 90.64 3.78 95.43 2.44
#4 80.25 3.56 79.24 2.32
#5 46.25 3.12 45.42 2.10
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