*3.3. Slag Iron Speciation and Other Thermochemical Considerations*

The iron oxide speciation, i.e., the balance between FeO and Fe3O4, can be quantified as the oxygen-to-iron ratio *x* presented in Equations (1)–(3). Indeed, *x* represents a degree of freedom that must be resolved in order to complete the mass balance. This degree of freedom can also be expressed as the ratio of ferric to ferrous ions within the slag, α = Fe<sup>3</sup>+/Fe2<sup>+</sup>, often called the degree of oxidation. The homeomorphic relationship between *x* and α is given by

$$\alpha = \frac{2 + 3\alpha}{2 + 2\alpha} \tag{5}$$

Equation (1a) can thus be rewritten as

$$(\text{Cu}, \text{Fe}, \text{S})\_{\text{(Food)}} + \text{O}\_{2(\text{Blast})} + \text{Flux} \rightarrow (\text{FeO}\_{(\frac{2+3\text{H}}{2+2a})}, \text{Flux})\_{\text{(Slag)}} + \text{SO}\_{2(\text{Offgas})} + \text{Cu}\_{(\text{Blister})} \tag{6}$$

similar to Equation (1b), in which the minimum α = 0 corresponds to pure wustite FeO and α = 2 corresponds to pure magnetite Fe3O4. Equation (6) can be further detailed in a similar manner as Equations (2) and (3) by assigning appropriate subscripts to α, as in [20].

In the modeling of slag chemistry, α is preferred over *x* to avoid ambiguity between the reactive oxygen of the blast and the inert oxygen that is strongly bonded within the flux (i.e., within the SiO2, CaO, etc.). For instance, the role of SiO2 flux is made more evident by expressing the wustite as a component within a fayalite matrix FeO·2SiO2; hence, the balance of FeO versus Fe3O4 is considered as FeO·2SiO2 and Fe3O4. In practice, SiO2 is added into the slag in proportions that surpass the

stoichiometry of fayalite and may be accompanied by other stable oxides. Under matte-processing conditions, the stable molecules SiO2, CaO, etc. can be regarded as if they were indivisible atoms. Most notably, the strongly bonded oxygen is not explicitly represented in Equations (1)–(3) and is not taken into account in *x*; these equations only explicitly consider the blast oxygen. The degree of oxidation α considers only the iron species isolated from any mention of the blast and flux oxygen.

Within Equation (6) and its nickel–copper equivalent, the slag-blow reaction can be isolated and balanced as:

$$\text{FeS}\_{\text{(Matter)}} + \left(\frac{2+3a}{4+4\alpha}\right) \text{O}\_{2\text{(Blast)}} \rightarrow \text{FeO}\_{\left(\frac{2+3a}{2+2a}\right)\left(\text{Slag}\right)} + \text{SO}\_{2\text{(Offgas)}}\tag{7}$$

which applies to both the smelting and converting furnaces for both copper and nickel–copper smelters. Indeed, the melting of the feed of Equation (6) results in molten matte that is a mixture of FeS and Cu2S; in the case of nickel–copper smelters, the matte will also contain nickel and cobalt sulfides [18], but Equation (7) is still correct. The incoming blast includes N2, as well as O2 (see Figure 4). As the N2 passes through the bath and is exhausted into the offgas, along with the SO2, it carries away sensible heat and is a critical consideration in controlling the bath temperature.

To resolve the degree of freedom α (or equivalently *x*), the equilibrium between iron oxide species can be expressed as

$$\text{FeS}\_{\text{(Matter)}} + 3\text{Fe3O}\_{\text{4(Slag)}} \leftrightarrow 10\text{FeO}\_{\text{(Slag)}} + \text{SO}\_{2(\text{Offgas})} \tag{8}$$

having enthalpy and entropy values Δ*H*<sup>0</sup> = 622,549 J/mol and Δ*S*<sup>0</sup> = 342.64 J/mol K, respectively, which can be obtained from HSC ChemistryTM. The corresponding Gibbs free energy balance is

$$
\Delta G = \Delta H\_0 - T\Delta S\_0 + RT\ln\left[\frac{\left(a\_{\text{FeO,Slag}}\right)^{10} p\_{\text{SO2,Ofgas}}}{a\_{\text{FeS,Matter}} \left(a\_{\text{FeSO4,Slag}}\right)^3}\right] \tag{9}
$$

which is set to zero to assume equilibrium. *R* is the ideal gas constant, *T* is the bath temperature, and *aij* is the activity of species *i* within phase *j*. The activity of SO2 in the offgas is taken to be the partial pressure *p*SO2,Offgas.

Within Equation (9), the activities (*a*FeS,Matte, *a*FeO,Slag, and *a*Fe3O4,Slag) can be re-expressed in terms of α, *T*, and the operational parameters. The usual parameters include the oxygen enrichment of the blast ϕ and the silica–iron mass ratio *r* = (*m*SiO2,Slag/*m*Fe,Slag), which are considered in Section 4. Empirical measurements relate the activities *aij* to their respective mole fractions *Xij*. In particular, the classic model of Goto [30,31] is validated for smelting and converting, in both the copper and nickel–copper contexts [32], and is the subject of Appendix A.

Iron speciation computations are simpler for smelting furnaces than for converting, since the smelting bath temperature can usually be treated as if it were at a steady state and is approximately uniform and constant. Under this simplification, α can be resolved through an application of Newton's Method [18,19]:

$$a^{(k)} = a^{(k-1)} - \frac{f\_G}{\frac{\partial f\_G}{\partial a}}\tag{10}$$

or, in case *T* is not constant,

$$
\begin{pmatrix} T^{(k)} \\ \alpha^{(k)} \end{pmatrix} = \begin{pmatrix} T^{(k-1)} \\ \alpha^{(k-1)} \end{pmatrix} - \frac{1}{\frac{\partial f\_H}{\partial T} \frac{\partial f\_G}{\partial \alpha} - \frac{\partial f\_H}{\partial \alpha} \frac{\partial f\_G}{\partial T}} \begin{bmatrix} \frac{\partial f\_G}{\partial \alpha} & -\frac{\partial f\_H}{\partial \alpha} \\ -\frac{\partial f\_G}{\partial T} & \frac{\partial f\_H}{\partial T} \end{bmatrix} \begin{bmatrix} f\_G \\ f\_H \end{bmatrix} \tag{11}
$$

which is a two-variable form of Newton's Method, in which (*T*(*k*) , α(k)) denote the results of the *k*th Newton iteration. The righthand sides of Equations (10) and (11) include proxy functions, *f* <sup>G</sup> and *f* H, and their derivatives, which are all evaluated at the preceding values (*T*(k−1), α(k−1)), considering (*T*(0), α(0)) = (1473 K, 0.15) as typical starting values. The proxy function *f* <sup>G</sup> must be formulated so that

*f* <sup>G</sup> = 0 when the Gibbs free energy balance of Equation (9) is satisfied, i.e., when Δ*G* = 0. Appendix A presents a formulation of *f* <sup>G</sup> that is based on the classic Goto model [30,31]. Likewise, *f* <sup>H</sup> is formulated such that *f* <sup>H</sup> = 0 when the heat balance is satisfied [19].

Following the results of Appendix A, it is relatively simple to program Equations (10) and (11) into a simulation platform, thereby relating slag chemistry to the wustite–magnetite balance and, indeed, to the overall mass balance. Depending on the project, Goto's model may be an appropriate starting point, although it does not consider olivine slags [21], nor does it consider the transport of minor elements. In practice, it is preferable to have more wustite than magnetite, since the latter increases the slag viscosity and the entrainment of matte into the slag [21,33]. The modification of slag chemistry through flux additions affects the migration of minor elements [22,23], possibly at the expense of having higher slag viscosity [21].

There is an interest to develop DES platforms that draw upon state-of-the-art thermochemical databases [19] to assist in the retrieval of valuable elements such as gold, silver, and platinum and the handling of deleterious elements such as arsenic, bismuth, and antimony. The authors suspect that the partition of trace elements can be efficiently computed as a function of the main elements a posteriori when Equations (10) and (11) converge. This is an area of future research, and the resulting platforms would support smelter-wide strategies for the processing increasingly problematic feeds. Yet, in reality, for similar apparent conditions like matte grade, temperature, etc., the balance of FeO versus Fe3O4 can depend on various parameters, including flux quality, refractory wear, and amount of charge (hence, affecting mixing), and so, an empirical approach to speciation may be more effective.
