**5. Conclusions and Future Work**

Not only warehouses and manufacturing facilities but metallurgical plants such as a steel plant or copper smelter—which are more difficult to automate—will be transformed in the future with use of new machine capabilities, automation, and improved sensors and controls. Steel is, by far, the major metal produced in the world, representing about 95% by tonnage of all metal production, with the world copper smelter output some 80 to 90 times less than world steel output. This, in part, helps understand that certainly more developments are attained in the iron and steel industry relative to copper. However, the computational framework described in this paper will help close the gap, regarding HI and LIBS and other radiometric sensors, as it enables the implementation of these technologies and justifies their further development within the copper industry. In particular, LIBS is expected to have an increasing importance in handling problematic feeds as existing copper smelters are confronted with increasing amounts of arsenic, bismuth, and antimony [42].

Modern sensors will be vital in supporting smelter-wide responses to increasingly challenging feeds that are being confronted throughout the world, as described in the previous section. It should be noted in particular that smelting and converting operations are central within copper and nickel–copper smelting and are linked to supporting operations throughout the smelter. High-quality and reliable process instrumentation and controls are therefore important in maximizing the global operating efficiency. Additionally, the monitoring of the furnace integrity, refractory wear, preventative maintenance, and plant safety are also key aspects that constantly need attention at the plant. Many high-performance smelting furnaces today include water-cooled copper blocks generally externally mounted on a furnace sidewall to protect the refractory lining at the hot face. The Peirce-Smith converter operates with a converter hood that includes water-cooled panels on the cold face in order to protect the steel wall at the hot face. Recent developments in the instrumentations for detecting and measuring the presence of small levels of water vapor in furnace offgas can signal a water leak and lead to improved furnace monitoring [43]. However, the implementation of such measures often requires quantitative justification.

This paper showed that combining thermochemical equilibrium data with a knowledge of smelting and converting dynamics provides a powerful tool for advancing smelting operations in the form of DES-TAFD hybrid simulations. The specialized use of Newton's Method, Runge-Kutta-Fehlberg, and Hermite interpolation within a DES are, in fact, an advancement within the industrial system analysis, which can be adapted to other industrial contexts, supporting modernization projects that include novel sensors and other technology.

**Author Contributions:** Conceptualization, N.T.; methodology, A.N.; software, A.N.; validation, A.N., N.T., and A.R.; formal analysis, A.N., A.R., and J.-C.N.; investigation, A.N., R.P., N.T., A.R., and P.J.M.; data curation, A.N.; writing—original draft preparation, A.N., R.W., R.P., A.R., and P.J.M.; writing—review and editing, A.N., R.W., J.N., and P.M.; and funding acquisition, A.N. and R.P. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Conicyt, Anillo Minería ACM 170008, supported by the Chilean government, and NSERC, grant number 2020-04605, supported by the Canadian government.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A Proxy Function for Gibbs Free Energy Balance Based on Goto's Model**

The approach of Navarra et al. [18,19] to obtain a viable proxy function *f* <sup>G</sup> considers that each mole of wustite FeO contains one mole of ferrous, that each mole of magnetite FeO·Fe2O3 contains one ferrous and two ferric, and that all other iron-bearing slag compounds are negligible. It follows that α = Fe3+/Fe2<sup>+</sup> can be taken as

$$\alpha = \frac{2n\_{\text{FeSO4,Slag}}}{n\_{\text{FeO,Slag}} + n\_{\text{FeSO4,Slag}}} \tag{A1}$$

in which *nij* generally denotes the number of moles of *i* within phase *j*.

When setting Δ*G* = 0, Equation (9) can be reorganized:

$$0 = a\_{\mathrm{FeS,Matter}} \left( a\_{\mathrm{Fe\ominus O4,Slag}} \right)^3 - \left( a\_{\mathrm{FeO,Slag}} \right)^{10} p\_{\mathrm{SO2,Oftgas}} e^{\left(\frac{\Delta S\_0}{R} - \frac{\Delta H\_0}{RT}\right)}$$

Using the expressions from Goto [30] and Kemori et al. [31] for the activity coefficients of (*a*FeS,Matte, *a*FeO,Slag, and *a*Fe3O4,Slag), a series of algebraic manipulations were performed by Navarra et al. [18,19] to obtain the following form that explicitly features *T* and α:

$$0 = \prod\_{l=1}^{3} (A\_l + B\_l \alpha)^{C\_l + D\_l/T} - \prod\_{l=4}^{9} (A\_l + B\_l \alpha)^{C\_l + D\_l/T}$$

In which the coefficients (*Al*, *Bl*, *Cl*, and *Dl*) are given in Table A1, from which a viable proxy function is obtained:

$$f\_{\mathcal{G}}(T,\alpha) = \prod\_{l=1}^{3} (A\_l + B\_l \alpha)^{\mathbb{C}\_l + D\_l/T} - \prod\_{l=4}^{9} (A\_l + B\_l \alpha)^{\mathbb{C}\_l + D\_l/T} \tag{A2}$$

Indeed, Δ*G* = 0 if and only if *f* <sup>G</sup> = 0. Moreover, the partial derivates of *f* <sup>G</sup> can be obtained with respect to α and *T*, so to complete the Newton iterations described by Equations (10) and (11). To obtain the expression for <sup>∂</sup> *fG* <sup>∂</sup>*<sup>T</sup>* , it is helpful to notice that *D*<sup>l</sup> is zero for all factors except for the third and ninth.

$$\begin{aligned} f\_{\mathbb{G}}(T,\alpha) &= \binom{2}{l-1} (A\_l + B\_l \alpha)^{\mathbb{C}\_l} \Big| (A\_3 + B\_3 \alpha)^{\mathbb{C}\_3 + D\_3/T} \\ &- \binom{8}{l-4} (A\_l + B\_l \alpha)^{\mathbb{C}\_l} \Big| (A\_9 + B\_9 \alpha)^{\mathbb{C}\_9 + D\_9/T} \end{aligned} \tag{A3}$$

However, to obtain an expression for <sup>∂</sup> *fG* ∂α , it is more effective to work directly with Equation (A2).

Within Table A1, ϕ denotes the volume fraction of oxygen within the blast, which can be related to *p*SO2,offgas. Additionally, the mole fraction of FeS within the matte is taken to be

$$X\_{\text{FeS,Matter}} = \frac{n\_{\text{FeS,Matter}}}{n\_{\text{FeS,Matter}} + n\_{\text{NiS,Matter}} + n\_{\text{Cu2S,Matter}} + n\_{\text{CoS,Matter}}} \tag{A4}$$

which supports the modeling of nickel–copper smelters, as well as copper smelters in which *n*NiS,Matte and *n*CoS,Matte are set to zero. Moreover, Table A1 has several instances of the silica-to-iron mole ratio (*n*SiO2,Slag/*n*Fe,Slag), which can be related to the silica-to-iron mass ratio *r* within the slag:

$$r = \left(\frac{M\_{\rm SiO2}}{M\_{\rm Fe}}\right) \left(\frac{n\_{\rm SiO2, Slag}}{n\_{\rm Fe, Slag}}\right) \tag{A5}$$

in which *M*SiO2 and *M*Fe are the molar masses of silica and iron, respectively; *r* is a common operational parameter used to control the flux additions.


**Table A1.** Coefficients for Equation (A2) (adapted from [19]).

\* In which *K* = *e*−Δ*H*0/15,430*R*(0.54 + 0.52*X*FeS,Matte + 1.4*X*FeS,Matte ln *X*FeS,Matte) 1458/15,430.

#### **References**


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