*3.4. Estimation of Threshold Crossing Times Using Time-Adaptive Finite Di*ff*erences*

Time-adaptive finite differences are an important feature of our smelter DES frameworks, allowing the correct placement of threshold-crossing events within the future event list (Figure 8). In the approach of Navarra et al. [18,19], the Newton iterations of Equations (10) and (11) are nested within the Runge-Kutta-Fehlberg (RKF) method, which is a well-known time-adaptive finite difference (TAFD) scheme [34,35]. This method is a combination of finite difference schemes based on fourth and fifth-order Taylor expansions [34]; an attempted timestep is only accepted if the fourth and fifth order estimates of *T* and α are within an acceptable error tolerance; otherwise, a smaller timestep is attempted. Moreover, each attempted timestep uses the previous results of *T* and α as a starting point for the Newton iterations, so that convergence is faster, thus making the computation more efficient.

**Figure 8.** Threshold-crossing event in relation to another event *e*. Time-adaptive finite differences determine if the threshold is crossed (**a**) before *e* or (**b**) after *e*.

The efficiency of DES depends on this time-adaptive aspect to limit the computational effort devoted to dynamics that occur between the discrete events (Figure 7). Alternatives to the approach of Navarra et al. [18,19] include the use of cubic Hermite spline interpolation [36,37] and Richardson extrapolation [38]. These techniques can be adapted to isolate the threshold crossing event of a state variable and improve the accuracy with which we can find the time at which it crosses. In fact, it is important to recognize that both the efficiency and accuracy of the method rely on an accurate time estimate for threshold crossing. In our case, the threshold is first contained in the interval bounded by the two timesteps bracketing an evaluation of the state variable below and above the threshold. This precisely guarantees we have access to values already computed at both of the upper and lower estimates (as a result of the RKF attempt), which includes state variable values, as well as their time derivatives (by direct evaluation if the right-hand side at the state variable values). With this data, we constructed a cubic Hermite spline, which is, therefore, a valid fourth-order interpolation of the solution over this interval. Since the Hermite spline is, in fact, a cubic polynomial, we can compute analytically the crossing time as a simple root-finding problem. We also note that this approach is robust, since the data at the end of the interval provided need not be exact. In fact, if the state variable data has a fourth-order accurate truncation error (e.g., by only using a third order Runge-Kutta integrator) and the derivatives are third-order accurate, the interpolant is guaranteed to be fourth-order accurate [37]. Further, while this approach is tailored to the RKF used here, it may be expanded by considering Hermite quintics if a further need for accuracy is required.

While conventional DES frameworks support basic representations of smelter logistics, a hybridization with time-adaptive finite differences supports a detailed representation of the kinetics that occur within the individual unit operations. In particular, the detailed representation of individual slag-blow segments may require a dynamic simulation of the evolving thermochemical states of matte and slag. Moreover, the DES-TAFD hybridization allows the simulation of several simultaneous converting cycles, in conjunction with the intervening actions involving cranes and ladles, the delivery of a cold charge, and other phenomena. Nonetheless, the explicit representation of slag-blow segments should only be implemented if it is beneficial to the particular engineering project. Beyond the internal dynamics of smelter, a project may require an explicit representation of market-related or environmental phenomena (Figure 6). The hybridization of DES and TAFD is indeed capable of linking detailed representations of diverse aspects throughout the smelter and beyond, whose coordination may be critical to the sustainability of the smelter. Furthermore, aging smelters will not be sustainable unless they can successfully modernize their operational practices, benefitting from sensors and other novel technology.

#### **4. Sample Computations and Context**

The sample calculations presented in this section are typical of an aging copper smelter that:


Regarding the third point, there may be operator experiences in treating marginal feeds that had a manageable amount of the trace elements; such feeds were treated by driving the individual slag-blow segments to a relatively low iron content (e.g., ~3% Fe), in combination with particular flux compositions. Indeed, a more forceful elimination of the undesirable elements could be induced at an even lower iron content, but this would increase the level of entrained into the slag, under imperfect mixing, as blister copper would be precipitated heterogeneously in certain regions of the vessel. Better endpoint control on the individual segments would allow a more careful advance toward 0% Fe with a limited risk of copper entrainment and could be attained by adapting the endpoint approach at Boliden's Rönnskär Smelter [5], mentioned earlier; this approach depended on customized sensor development [10–12].

From the authors' experience in copper smelter projects, it is incumbent on outside experts to integrate their general understanding with the detailed understanding of in-house experts at the given smelter. There may be issues that were already confronted with some amount of success, but these issues may gradually become critical. For instance, the increasing presence of penalty elements is especially common in custom smelters that receive concentrates from a combination of regional mines, which are themselves confronted with ore blending challenges [29]. The experiences of the smelter should be cross-checked with thermochemical models and a literature review [22,23] in proposing new operational modes that may require a technological upgrade. In effect, this can be a standardization and optimization of approaches that the in-house experts were already considering.

The DES framework described in Sections 3.1–3.4 was implemented using Rockwell ArenaTM software and replicated the general aspects of conventional smelters while incorporating smelter-specific data. Tables 1–3 contain sample data that are loosely based on published values from [2,5,6,13,39,40]. As stated in Section 3.1, conventional smelters have smelting capacities that normally exceed the downstream converting capacity; indeed, the smelting furnace should not usually produce matte at full capacity, nor should it function in fits and starts. The definition of so-called short and long converter cycles (Figure 6) provides the operational flexibility to set a fixed smelting rate. Moreover, some smelters may have several cycle types under consideration to respond to build-ups of cold charges or compositional imbalances in the feeds that are received from the supplying mines. For simplicity, the current computations consider only two types of cycles.


**Table 1.** Feed compositions entering the smelting furnace.


**Table 2.** Examples of smelting furnace operational parameters.


**Table 3.** Examples of Peirce–Smith (PS) converting data describing short and long cycles.


\* Assume that ladles are added at the beginning of the segment, each carrying 30 t of matte.

To maintain stable throughput, the blending strategy ensures comparatively stable portions of the main elements (Table 1) but with unavoidable fluctuations in the undesirable trace elements. Additionally, to maintain stable throughput, the smelting parameters are kept constant (Table 2),

including the grade of the output furnace matte; given that the matte is a mixture of FeS and Cu2S, an elemental mass balance determines that a 60% Cu grade corresponds to roughly 24% S and 16% Fe.

Effectively, the smelting furnace blasts 550 Nm3/min of oxygen into a bath in order to decrease the iron content from 39% to 16% as it burns the FeS. The smelting blast also includes 550 Nm3/min of nitrogen, which carries away a portion of the heat. The control of the temperature at 1275 ◦C depends on this nitrogen flow. Given the stable temperature *T* = 1275 ◦C = 1548 K, oxygen enrichment ϕ = 0.5, and silica-iron ratio *r* = 0.7, the Newton iterations of Equation (10) are used to obtain the degree of oxidation α = 0.166, with a corresponding throughput of 36.928 t/h of matte or equivalently 1.231 ladles/h. At a matte grade of 60% Cu, this corresponds to 22.157 t Cu/h.

Figure 9 shows graphs of the matte content using the data from Tables 1–3. The largest declines correspond to the initiation of long cycles (four ladles = 120 t) and short cycles (three ladles = 90 t), and the other declines correspond to the recharging actions that occur during the cycles at the second and third slag-blow segments. As is common in larger smelters, the offgas handling capacity allows for the execution of two simultaneous converter cycles.

**Figure 9.** Matte levels of the smelting furnace with data from Tables 1–3, in which (**a**) exceeding 800 t triggers two long cycles and (**b**) exceeding 700 triggers three long cycles. The second case is safely below the maximum holding capacity of 900 t.

Within Table 3, the times for the initial slag-blow segments are fixed as operational parameters. However, the durations of the final slag-blow segments depend on the exact nature of the charge (including the cold dope), as well as variable operator behavior and, potentially, other factors. By extension, the copper-blow durations are also variable. For the purposes of Figure 9, these fluctuating values are used to define uniform distributions, e.g., the copper-blow durations are uniformly distributed between 3.8 and 5.2 h for the short cycles and between 3.8 and 6.2 h for the long cycles. Uniform distributions, triangular distributions (Figure 5a), and other simple forms are typical of the early phases of a smelter project. More advanced phases can include more elaborate distributions that are supported by goodness-of-fit testing of the plant data [41].

An analysis of Table 3 reveals that the average conversion rates are 0.571 ladles/h and 0.737 ladles/h for the short and long cycles, respectively. Considering that the smelting furnace produces 1.231 ladles/h that is fed into two simultaneous converter cycles (i.e., 0.615 ladles/h per converter), the balancing of the smelting and converting throughput will require a combination of the long and short cycles. (Incidentally, the long cycle copper-blow is more productive than that of the short cycle; this may be because it is loaded with cold copper scrap, which makes the heat balance less dependent on the nitrogen convection, thus supporting a more intense higher-oxygen blast).

Figure 9 considers a threshold criterion that, whenever the matte level surpasses a critical value, the next several cycles are set to a long cycle. Poor adjustment of this threshold can diminish the overall throughput, as the smelting furnace is deactivated when it reaches the 900-t capacity given in Table 1; this is depicted in Figure 9a, in which the threshold is set to 800 t, which triggers two long cycles, which is insufficient to prevent the approach to 900 t observed at time 140 h. A balanced production is

shown in Figure 9b, which applies a threshold of 700 t, which triggers three long cycles. In general, a DES-TAFD hybrid can be adapted to incorporate the operational decision-making of the smelter that is being studied; this flexibility is vital in the context of reengineering projects [20].

Table 4 and Figure 10 describe modified dynamics in which an array of sensors track the approach to 0% Fe and the presence of the undesirable trace elements. The data-driven approach to removing the iron is reflected by the (uniform) variation ascribed to all of the slag-blow segments, whereas the previous configuration (Table 3) fixed predetermined blow times for the initial segments. Moreover, Table 4 follows the action graph of Figure 5b, in which the commitment to an extended cycle is made only after the cycle begins. Indeed, the regular short cycles correspond to 2-5-13, the extended short cycles to 2-5-18-12, the regular long cycles to 1-4-7-10, and the extended long cycles to 1-3-6-9-11. For the sake of Figure 10, it is assumed that 30% of the short cycles are extended and 20% of the long cycles are extended, although different values could be tested and should be driven by actual forecasts of what the incoming feed might be.

**Table 4.** Example of PS converting data, including short and long cycles that can be extended.


\* Assume that ladles are added at the beginning of the segment, each carrying 30 t of matte.

**Figure 10.** Matte level of smelting furnace with data from Table 1, Table 2, and Table 4, in which exceeding 700 triggers three long cycles, 30% of short cycles are extended, and 20% of long cycles are extended.

A comparison between Figures 9 and 10 shows the effect that responsive converter cycles (Table 4) can have on the matte levels; although the matte levels of Figure 10 remain mostly between 600 and 700 t, the pattern of peaks and dips is far less regular. These fluctuations may be acceptable, considering that it would allow the smelter to handle problematic feeds. Otherwise, the feed rate into the smelting furnace could be controlled in coordination with a variable blast rate, which may be the subject of a following stage of engineering.

To the authors' knowledge, the computations of Figure 10 are the first published instance in which several converters are simultaneously drawing upon a smelting furnace and operational decisions are finalized after the cycles begin, i.e., in response to incoming sensor data. This level of detail is necessary to evaluate the utility of sensors in supporting smelter-wide responses to problematic feeds. The computations described by Table 4 and Figure 10 rely on the representation of individual slag-blow

segments (Figure 5b), although other approaches might require the detailed representation of additional aspects within or surrounding the smelter (Figure 6).
