*3.2. Detailing of Smelter Dynamics Within Discrete Event Simulation*

In a previous work, Navarra et al. [18,19] incorporated thermochemical equilibria within a discrete event simulation (DES); it was suggested generally that the hybridization of time-adaptive finite differences (TAFD) and DES is a suitable paradigm for multiphase smelter reengineering projects [20]. Within copper and nickel–copper smelters, thermochemical equilibria determine the iron-speciation of

smelting and converting slags, as described in the following section. However, the early phases of a smelter reengineering project can assume fixed molar ratios of iron and oxygen.

Assuming that the smelter feed is composed mainly of iron, sulfur, and copper, a smelter accomplishes the following unbalanced reaction:

$$\text{(Cu,Fe,S)}\_{\text{(Fuel)}} + \text{O}\_{2(\text{Blast})} + \text{Flux} \rightarrow \text{(FeO}\_3,\text{Flux})\_{\text{(Slag)}} + \text{SO}\_{2(\text{Offgas})} + \text{Cu}\_{\text{(Bilster)}}\tag{1a}$$

In which FeOx represents a mixture of wustite FeO and magnetite Fe3O4, such that *x* = 1 and *x* = 1.25 corresponds to pure wustite and magnetite, respectively. For simplicity, *x* can be fixed to 1, although typical values can range between 1.0 and 1.1, depending on the nature and quantity of the flux and the monitoring and control of the process itself; in particular, a low level of magnetite in slag is desirable, which is associated with low slag viscosity. In practice, the flux is predominantly silica SiO2, but certain smelters include varying quantities of CaO and other stable oxides; CaO is especially common in continuous converting [21], which is an alternative to the conventional PS converting [16]. The SO2 is captured for sulfuric acid production, and the blister copper is subject to fire refining prior to being cast into anodes that undergo electrolytic refining. A similar reaction can describe nickel–copper smelters:

$$\begin{array}{l} \text{(NI, CU,CO,FE,S)}\_{\text{(FEED)}} + \text{O}\_{\text{2(Blast)}} + \text{Flux} \\ \rightarrow \text{(FeO}\_{\text{x}}, \text{Flux)}\_{\text{(Slag)}} + \text{SO}\_{\text{2(Offgas)}} + \text{(Ni,Cu,Co,S)}\_{\text{(Besselmer)}} \\ \end{array} \tag{1b}$$

The subsequent processing of Bessemer matte depends on the given nickel–copper plant. Equation (1a,b) provide more detail to Figure 3 for copper and nickel–copper smelters, respectively.

Depending on the scope and phase of the project, *x* can be regarded as a single global value for the entire smelter or as distinct values for the smelting furnace(s) and converters. Equation (1a) can thus be rewritten

$$\begin{array}{c} \text{(Cu,Fe,S)}\_{\text{(Feed)}} + \text{O}\_{2(\text{Blast})} + \text{SFlux} + \text{CFlux} \\ \rightarrow \text{(FeO}\_{\text{xy}}, \text{SFlux})\_{\text{(SSlag)}} + \text{(FeO}\_{\text{xC}}, \text{CFlux})\_{\text{(CSlag)}} + \text{SO}\_{2(\text{Offgas})} + \text{Cu}\_{\text{(Blister)}} \end{array} \tag{2}$$

For copper smelters, which decompose the global *x* into *x*<sup>S</sup> and *x*C, characterize the slag of the smelting furnace. A similar decomposition of the global *x* could be applied to Equation (1b) in the case of nickel–copper smelters.

Moreover, the individual slag-blow segments of PS cycles can each be assigned appropriate *x* values. Therefore, Equation (2) could be further detailed as:

$$\begin{array}{c} \text{(Cu,Fe,S)}\_{\text{(Fred)}} + \text{O}\_{2(\text{Blat})} + \text{SFlux} + \text{CFlux1} + \text{CFlux2} + \dots + \text{CFluxn} \\ \rightarrow \\ \text{(FeO}\_{\text{X}s}, \text{SFlux})\_{\text{(SSlag)}} + \left( \text{FeO}\_{\text{X}\_{\text{C}1}}, \text{CFlux1} \right)\_{\text{(CSlag1)}} + \left( \text{FeO}\_{\text{X}\_{\text{C}2}}, \text{CFlux2} \right)\_{\text{(CSlag2)}} + \dots \\ + \left( \text{FeO}\_{\text{x}\_{\text{C}n}}, \text{CFluxn} \right)\_{\text{(CSlagn)}} + \text{SO}\_{2}(\text{Offgas}) + \text{Cu}(\text{Bliter}) \end{array} \tag{3}$$

In which *x*<sup>S</sup> characterizes the slag of the smelting furnace, and, depending on the level of detail, xCi can characterize the types of converter cycles or can characterize the individual types of slag-blow segments, for *i* = 1 to *n*. For example, Figure 5a shows an action graph that occurs within a smelter that practices two kinds of converter cycles, long and short; hence, *n* = 2. Figure 5b shows a more detailed representation, which considers 13 kinds of blow segments. (The slag-blow segments are punctuated with charging and skimming actions, although these are not explicitly shown in Figure 5b). For a conventional copper smelter, actions 1–9 describe slag-blow segments (Figure 4a); hence, *n* = 9, and the remaining actions 10–13 represent copper-blow segments (Figure 4b) that complete the cycle as a batch of blister copper is discharged. In the case of a nickel–copper smelter, all of the arcs represent slag-blow segments; hence, *n* = 13, noting that the discharge is the so-called Bessemer matte (Ni, Cu, Co and S) that is described in Figure 3 and Equation (1b).

**Figure 5.** Examples of action graphs that represent Peirce-Smith converting cycles, which consider two types of cycles: long and short. (**a**) The low-detail representation shows the long and short cycles as single actions that are characterized by broad distributions of cycle times, whereas (**b**) a more detailed representation considers individual blow segments, from 1 to 13; each of the segments can be characterized by comparatively narrow time durations (which were omitted from the figure).

The decision to apply one segment versus another (e.g., segment 3 versus 4) would depend on the state of the plant, to the extent that the state variables can monitored with the available sensors. Even if the resulting slag compositions for the different cycles (Figure 5a) or blow segments (Figure 5b) are relatively consistent, it may be unclear how frequently each cycle or segment will be applied, e.g., depending on how often certain plant conditions occur. A global mass balance based on Equation (3) requires an estimation of how often each of the different cycles (Figure 5a) or segments (Figure 5b) are applied; such estimations are the result of DES computations, as described below.

The broad distributions of Figure 5a approximate the combined effects of narrower distributions that would characterize the individual segments of Figure 5b. A proposed technological change within the smelter (e.g., installation of new sensors) may require a rethinking of the slag-blow and the definition of new action graphs. Depending on the project, it may be necessary to further decompose the actions of Figure 5b into sub-actions and sub-sub-actions, possibly including thermochemical modeling [22,23] or computational fluid dynamics [24]. This decomposition may be essential in order to properly simulate the system with and without the technological change, thereby evaluating the benefit of the proposed change. In the case of sensors, it is necessary to simulate how the additional information will be incorporated into the decisions and operational actions of the smelter, thereby computing the value of these better-informed decisions and actions.

In many reengineering projects, the phenomena that occur within the smelter may be less important than the phenomena that occur outside of the smelter. For instance, the DES model of the Hernán Videla Lira (HVL) Smelter developed by Navarra et al. [25] focuses on the smelter-wide response to changing meteorological conditions and has a comparatively simple representation of converter cycles, similar to Figure 5a. The HVL Smelter considers distinct categories of meteorological conditions—normal, unfavorable, and extreme—to describe the potential for the surrounding atmosphere to disperse the SO2 effluent. If the smelter is running in its normal operational mode when the unfavorable meteorological conditions emerge, there is a so-called "environmental incident". The model of Navarra et al. [25] computes the trade-off between production and environmental risk. Moreover, this model quantifies the improved trade-off that can result from a more accurate array of meteorological sensors.

DES development is a means to extend the static mass balances, to detail the critical phenomena that are driving and/or constraining a particular phase of an engineering project. The simulated events can dynamically affect the mass balances that are detailed throughout the model. Figure 6 makes the distinction between the events that occur outside of the smelter and within the smelter, which constitute the external and internal logistics, respectively. There is a further distinction between the logistical coordination of smelter equipment (furnaces, ladles, cranes, etc.) and the kinetics that occur within the equipment. In general, the state variables that describe the system, and the events that would alter these variables, can be positioned within the concentric ellipses of Figure 6. The incorporation of variables and events within a DES model must be guided by the scope and the phase of the engineering project. For example, it is not recommended to detail individual crane movements, unless the particular project would benefit from a comprehension of this aspect [26]. Likewise, it is not recommended to detail particular equipment breakdown events, unless the project would benefit from a comprehension of this aspect [27].

**Figure 6.** Relationship between smelter kinetics, internal and external smelter logistics, and broader system dynamics.

The computational efficiency of DES is due to adaptive time stepping, as the virtual clock advances from one event to the next without explicitly representing the dynamics that occur between events (Figure 7a). The sequencing of events is governed by the future event list (Figure 7b). Within this scheme, a prolonged activity is represented by a sequence of events, including starting and ending the activity. For example, a basic representation of converter cycles described by Figure 5a may include only two events: the start and end of the cycle. A more detailed representation (e.g., Figure 5b) may include several intervening events to represent individual slag- and copper-blow segments, as well as the intervening skimming, charging, actions of the operators, etc. with the level of details that correspond to the given project. Incidentally, DES applies random number generation to determine the duration and outcome of the activities and is thus a form of Monte Carlo simulation [28]; the distributions and action paths illustrated in Figure 5 can be incorporated into the framework.

**Figure 7.** Fundamental components of a discrete event simulation (DES) framework, including (**a**) a virtual timeline that is subject to discrete steps and (**b**) a future event list.

Moreover, the simultaneous operation of several converters in unison with other logistical phenomena is integrated into one single future event list (Figure 7b); hence, a system-wide representation. Periods of time with relatively few events are computed relatively quickly, thereby focusing the computational efforts on periods of time that are more heavily packed with activity. This time-adaptive aspect of DES allows the simulation of thousands of operating days within minutes.

A DES framework can include operational criteria that determine the action pathway of converter cycles, allowing the computation of frequency confidence intervals. Following the example of Figure 5a, the average frequency of short cycles may be between 2.8 and 3.2 cycles/day with 95% confidence and that of long cycles may be between 0.9 and 1.2 cycles/day with 95% confidence; this result will allow a mass balance based on Equation (3), given the data about the matte that are charged within each cycle and the corresponding flux and oxygen requirements. In a slightly more detailed representation, the DES framework may include the criteria that would determine the more detailed action paths of Figure 5b.

Standard DES frameworks do not explicitly represent the dynamics that occur between events. However, a linearly dynamic state variable can be represented as a combination of discretely dynamic state variables. For instance, the mass of feed stockpile *k* may be computed at a time *t*, as

$$m\_k(t) = m\_k^{\text{Previous}} + \dot{m}\_k^{\text{Previous}} \cdot \left(t - t^{\text{Previous}}\right) \tag{4}$$

in which *m*Previous *<sup>k</sup>* and . *m*Previous *<sup>k</sup>* are the mass and rate change of *k* that were computed at the previous event, which occurred at time *t* Previous. Thus, each feed *k* would require two discretely dynamic variables (*m*Previous *<sup>k</sup>* and . *m*Previous *<sup>k</sup>* ), in addition to the *t* Previous variable that remembers the time of the previous event. Equation (4) can used in simulations that consider alternating modes of operation that control feed blends in response to imbalances in incoming concentrates [29].

Other linearly continuous variables can be implemented in a manner similar to Equation (4), representing each of these continuous variables as two discrete variables: level and rate (e.g., *m*Previous *k* and . *m*Previous *<sup>k</sup>* ). Considering the DES representation of time (Figure 7), this constitutes a time-adaptive finite difference (TAFD) scheme. However, a full representation of the continuous dynamics requires the detection of threshold-crossing events, as described in Section 3.4. These threshold-crossing events are especially important in assessing the installation of sensors whose role may be to signal the need for corrective actions precisely when critical thresholds are crossed.
