Research on the Thermodynamic Simulation Model of Antimony–Lead Synergistic Side-Blown Oxidation Smelting Process Based on MetCal

Abstract

On the basis of the theory of polyphase equilibrium and the utilization of the MetCal software platform (MetCal v7.81), we adopted the chemical equilibrium constant method and successfully constructed a multiphase equilibrium model and simulation system for the antimony–lead synergistic side-blown oxidation smelting process. In typical production conditions, which encompass factors such as the composition of raw material, the ratio of oxygen to material, and oxygen-enriched concentration, the equilibrium product composition and pivotal technical indices are modeled and computed. Calculation results indicated that, other than the trace elements in the smelting slag, the relative errors of the calculated values for the content of major elements in the antimony-rich crude lead and smelting slag were less than 10% of the measured value after average treatment in production. Therefore, our results showed that the developed model and system preferably embodied the practical production condition of the antimony–lead synergistic side-blown oxidation smelting process, which is capable of precisely forecasting the smelting outcomes and optimizing the process parameters, thereby offering effective guidance for the practical execution of the antimony–lead synergistic side-blown oxidation smelting process.

1. Introduction

Antimony is a vital strategic metal and has a wide range of uses in industry [1,2]. China is rich in antimony resources, with proven reserves of 640,000 tons of antimony in 2024, accounting for more than 30% of the global total [3]. At the same time, in 2023, the annual consumption of antimony resources in China reached approximately 150,000 tons, ranking it among the top in the world [4]. Antimony sulfide concentrate is one of the main raw materials for the production of metal antimony [5]; however, with increasing demands for antimony resources and the continuous shortage of single antimony ore resources, the extraction of antimony from complex resources containing antimony is the only means possible to sustainably develop the antimony industry [6,7]. Currently, more than 10 types of antimony-containing complex resources can meet the requirements of industrial production, including stibnite (Sb₂S₃), jamesonite (Pb₄FeSb₆S₁₄), valentinite (Sb₂O₃, Sb 83.3%), senarmontite (Sb₂O₃, Sb 83.54%), and antimony–gold ore (Sb₂S₃, Sb 59.92%) [8,9,10]. Among these resources, antimony–gold symbiotic ore is a compound antimony ore with more economic value than single antimony ore [11]. As a result, this symbiotic ore is gradually replacing antimony sulfide concentrate and becoming one of the important raw materials for the production of metal antimony [12].

Currently, the treatment methods of antimony–gold symbiotic concentrate include the hydrometallurgical process and the pyrometallurgical process, with the latter method primarily used in enterprise production [13,14]. Volatilization smelting in a blast furnace is a typical pyrometallurgical process used for antimony-containing concentrates [15,16]. It faces issues, however, such as high energy consumption, difficulty in controlling environmental pollution, high production costs, and challenges in recovering precious metals [9]. The hydrometallurgical processes include the alkaline sulfide leach process [17,18], the acidic chloride leach process [19,20], and the pulp electrolysis process [21,22]. Although these processes do not produce sulfur dioxide (SO₂) and have low energy consumption, they do encounter issues such as high consumption of water, large wastewater volumes, chlorine corrosion, and power consumption [23]. The oxygen-rich side-blown smelting process [24,25,26] has emerged as a novel technology in antimony smelting, characterized by its robust adaptability to raw materials, environmentally friendly approach, long furnace body life, high comprehensive metal recovery rate, and cost-effectiveness [27]. To address the issues of low metal recovery, high energy consumption, and high pollution associated with the existing treatment of antimony–gold concentrate, some scholars have proposed the adoption of a new synergistic oxygen-rich side-blown smelting process for lead concentrate and antimony–gold concentrate based on similar properties and its easy association with antimony and lead [28]. These scholars have initiated production practices. Zhang et al. [29] used the thermodynamic software FactSage 7.3 to calculate the reaction trend of various metal sulfides, the dominant region map of Me-S-O diagrams, and the distribution rule of phase equilibrium in the antimony–gold concentrate and lead concentrate synergistic smelting process and a verification test was carried out. However, the results were not quantitatively described in this research, and the yield of each product and the distribution behavior of each element in the product were not clear.

As advancements in computer hardware and software have been achieved, the quantitative analysis of the pyrometallurgical process has been performed using the multiphase equilibrium model [30] and leveraging computer-aided simulation techniques, which offer such benefits as high research efficiency, good reproducibility, low cost, and good security. Currently, scholars have performed thermodynamic simulation analysis and have conducted research on metallurgical processes, such as flash smelting [31,32], bottom-blown smelting [33,34], and side-blown smelting [35,36] of copper, lead, and other raw materials. These processes can better calculate and predict production output and reveal the distribution law of elements, providing theoretical guidance to optimize the production process.

Given these findings, in this study, we adopted the multiphase chemical equilibrium constant method [37,38], which was grounded in the mechanism of process and production practice of the antimony–lead synergistic side-blown oxidation smelting process and was based on the MetCal software platform (MetCal v7.81) [39]. We also researched and established a multiphase thermodynamic simulation model for the synergistic oxidation smelting of antimony and lead. Under typical production conditions, we conducted a thermodynamic simulation calculation for the antimony–lead synergistic side-blown oxidation smelting process while establishing a solid numerical modeling foundation for subsequent quantitative control and optimization of process parameters in this synergistic smelting process.

2. Process Mechanism and Model Establishment

2.1. Process Mechanism

The antimony–lead synergistic side-blown smelting process is based on a three-furnace segmented reaction design, including three stages of oxidation smelting, reduction smelting, and slag smelting [24]. In this study, we focused on the antimony–lead synergistic side-blown oxidation smelting process, which was completed in a side-blown oxidation furnace, as shown in Figure 1.

We added lead concentrate, antimony–gold concentrate, quartz sand, limestone, and other materials produced by belt ingredient mixing and granulation to a side-blown oxidation furnace. We added rich oxygen from both sides of the furnace body drum to the slag layer while strongly stirring the furnace high-temperature melt (1100–1300 °C). The antimony-containing and lead-containing materials oxidized rapidly, melted, and generated the primary antimony-rich crude lead, smelting slag, flue gas, and flue dust as well as other products in the furnace. The distribution rate of antimony and lead in the flue gas was low, while the distribution rate in the primary antimony-rich crude lead, smelting slag, and flue dust was high. Antimony and lead in the primary antimony-rich crude lead existed primarily in the form of monomers, and antimony and lead in the smelting slag and flue dust existed primarily in the form of oxides. Because of the large phase boundary area, the gas gave the molten pool a high stirring kinetic energy, the mass and heat transfer rate in the furnace was fast, and the phase composition of each product tended toward thermodynamic equilibrium quickly.

The products were precipitated and separated in the cylinder area of the furnace, the antimony-rich crude lead was discharged into the refining furnace from the siphon, and the smelting slag (containing approximately 30 wt% Pb, as analyzed using X-ray fluorescence (XRF)) was sent to the side-blown reduction furnace through the chute for further reduction smelting, which produced the secondary antimony-rich crude lead, reduction slag, flue gas, and flue dust. The gas generated during the oxidation and reduction smelting process was cooled in a waste heat boiler, and the flue dust was collected in an electric precipitator and then sent to the sulfuric acid system for acid production.

2.2. Model Assumption

In keeping with the mechanism of the antimony–lead synergistic side-blown oxidation smelting process, the products of this oxidative smelting process included antimony-rich crude lead, smelting slag, flue gas, and flue dust. In modeling the multiphase equilibrium model for this process, we assume that the equilibrium product phases encompassed three distinct phases: antimony-rich crude lead, smelting slag, and flue gas. The flue dust was composed of mixed mineral dust and product dust. Based on the reaction mechanism of the antimony–lead synergistic side-blown oxidation smelting process and literature reports, the composition of each equilibrium product is assumed to be as follows:

(1)

antimony-rich crude lead (ALd): Pb, PbS, Bi, Sb, Zn, Cu₂S, FeS, As, Ag, and Other1;

(2)

smelting slag (Sl): PbO, PbSO₄, PbS, ZnO, Cu₂O, Cu₂S, As₂O₃, Bi₂O₃, Sb₂O₃, Sb₂O₅, FeO, Fe₃O₄, FeS, CaO, Al₂O₃, MgO, SiO₂, Ag, and Other2;

(3)

flue gas (gas): O₂, Sb₂O₃, Sb₂S₃, Pb, PbO, PbS, As₂O₃, As₂S₃, Zn, ZnO, ZnS, SO₂, S₂, CO, CO₂, N₂, and H₂O;

(4)

flue dust (Dt): Bi₂S₃, Sb₂S₃, As₂S₃, FeS, SiO₂, Ag, Al₂O₃, H₂O, FeS₂, PbS, ZnS, Cu₂S, Fe₂O₃, CaCO₃, MgO, CaO, Pb, Bi, Zn, As, Sb, PbO, PbSO₄, ZnO, Cu₂O, Bi₂O₃, Sb₂O₃, Sb₂O₅, FeO, Fe₃O₄, As₂O₃, and Other3.

Among these products, “Other” denoted impurity elements that were not involved in the reactions but had the potential to influence the modeling of mass balance relationships.

2.3. Modeling Principles

We characterized the antimony–lead collaborative side-blown oxidation smelting process as a multiphase and multicomponent reactive system. The mathematical model for this process was developed using the chemical equilibrium constant method. In this method, at a constant temperature and pressure, we established a mathematical model based on the total moles of each element present and the independent chemical reactions occurring within the system when the system was at equilibrium. This model was expressed as a set of matrix nonlinear equations. By solving the mathematical model, we calculated the molar number of each phase and component in the system.

The multiphase multicomponent reaction system featured a set of linear independent molecular equation vectors, which were defined as independent components, and the rest were defined as subordinate components. Assuming that $N_e$ and $N_c$, respectively, denoted the number of elements and the chemical composition in the antimony–lead synergistic side-blown oxidation smelting process, we determined the number of all components in the system by the reaction between the components, with the count of independent reactions denoted as $N_b$, which was equal to $N_c-N_e$. These $N_b$ independent reactions were represented by the following matrix equation:

$$\left(V_{j,i}\right)\left(A_{i,k}\right)=\left(B_{j,k}\right)$$

(1)

where $V_{j,i}$ denotes the matrix of stoichiometric coefficients, $A_{i,k}$ denotes the molecular equation matrix of independent components, $B_{j,k}$ denotes the molecular equation matrix of subordinate components, and $i,j,\mathrm{and}k$ denote, respectively, the independent component number, subordinate component number, and element type number.

According to the rules of matrix operation, $V_{j,i}$ can be calculated by the following equation:

$$\left(V_{j,i}\right)=\left(B_{j,k}\right)\left(U_{k,i}\right)$$

(2)

where $\left(U_{k,i}\right)$ is the computable inverse matrix of $V_{j,i}$.

For the antimony–lead synergistic side-blown oxidation smelting process, we assumed that 0.2% of the inert constituent “Other” was allocated to the antimony-rich crude lead, while 99.8% was directed toward the smelting slag. On the basis of product assumptions and the principle of phase equilibrium, the composition of the multiphase equilibrium products included 17 elements and 46 compounds. The 17 elements included 15 individual elements (Pb, Zn, Cu, Fe, S, As, Sb, Ag, Bi, Al, Mg, O, H, N, and C) and 2 pseudo-elements (SiO₂ and CaO). The total number of compound types corresponded to the sum of the types of compounds present in the antimony-rich crude lead, smelting slag, and flue gas. Therefore, in Equations (1) and (2), i and k ranged from 1 to 17, and j ranged from 1 to 27. There were 17 independent components and 27 subordinate components, excluding “Other,” which were represented by the stoichiometric coefficient matrices shown in

Table A1. Stoichiometry matrix for 17 independent species.

Component	Phase	Pb	Zn	Cu	Fe	S	As	Sb	Mg	Al	Bi	Ag	SiO2	CaO	O	C	H	N
Pb	Ld	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
PbS	Ld	1	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0
Zn	Ld	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
Cu2S	Ld	0	0	2	0	1	0	0	0	0	0	0	0	0	0	0	0	0
FeS	Ld	0	0	0	1	1	0	0	0	0	0	0	0	0	0	0	0	0
As	Ld	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0
Sb	Ld	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0
Bi	Ld	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0
Ag	Ld	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0
Al2O3	Sl	0	0	0	0	0	0	0	0	2	0	0	0	0	3	0	0	0
MgO	Sl	0	0	0	0	0	0	0	1	0	0	0	0	0	1	0	0	0
SiO2	Sl	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0
CaO	Sl	0	0	0	0	0	0	0	0	0	0	0	0	1	0	0	0	0
O2	Gas	0	0	0	0	0	0	0	0	0	0	0	0	0	2	0	0	0
CO	Gas	0	0	0	0	0	0	0	0	0	0	0	0	0	1	1	0	0
H2O	Gas	0	0	0	0	0	0	0	0	0	0	0	0	0	1	0	2	0
N2	Gas	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	2

and

Table A2. Stoichiometry matrix for 27 subordinate species.

Component	Phase	Pb	Zn	Cu	Fe	S	As	Sb	Bi	Ag	SiO2	CaO	O	C	H	N
PbO	Sl	1	0	0	0	0	0	0	0	0	0	0	1	0	0	0
PbSO4	Sl	1	0	0	0	1	0	0	0	0	0	0	4	0	0	0
PbS	Sl	1	0	0	0	1	0	0	0	0	0	0	0	0	0	0
ZnO	Sl	0	1	0	0	0	0	0	0	0	0	0	1	0	0	0
Cu2O	Sl	0	0	2	0	0	0	0	0	0	0	0	1	0	0	0
Cu2S	Sl	0	0	2	0	1	0	0	0	0	0	0	0	0	0	0
FeO	Sl	0	0	0	1	0	0	0	0	0	0	0	1	0	0	0
Fe3O4	Sl	0	0	0	3	0	0	0	0	0	0	0	4	0	0	0
FeS	Sl	0	0	0	1	1	0	0	0	0	0	0	0	0	0	0
As2O3	Sl	0	0	0	0	0	2	0	0	0	0	0	3	0	0	0
Sb2O3	Sl	0	0	0	0	0	0	2	0	0	0	0	3	0	0	0
Bi2O3	Sl	0	0	0	0	0	0	0	2	0	0	0	3	0	0	0
Sb2O5	Sl	0	0	0	0	0	0	2	0	0	0	0	5	0	0	0
Ag	Sl	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0
Pb	Gas	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0
PbO	Gas	1	0	0	0	0	0	0	0	0	0	0	1	0	0	0
PbS	Gas	1	0	0	0	1	0	0	0	0	0	0	0	0	0	0
Zn	Gas	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0
ZnO	Gas	0	1	0	0	0	0	0	0	0	0	0	1	0	0	0
ZnS	Gas	0	1	0	0	1	0	0	0	0	0	0	0	0	0	0
As2O3	Gas	0	0	0	0	0	2	0	0	0	0	0	3	0	0	0
As2S3	Gas	0	0	0	0	3	2	0	0	0	0	0	0	0	0	0
Sb2O3	Gas	0	0	0	0	0	0	2	0	0	0	0	3	0	0	0
Sb2S3	Gas	0	0	0	0	3	0	2	0	0	0	0	0	0	0	0
SO2	Gas	0	0	0	0	1	0	0	0	0	0	0	2	0	0	0
S2	Gas	0	0	0	0	2	0	0	0	0	0	0	0	0	0	0
CO2	Gas	0	0	0	0	0	0	0	0	0	0	0	2	1	0	0

of Appendix A, respectively.

Table 1. Equilibrium reactions and equilibrium constants of the antimony–lead synergistic side-blown oxidation melting system.

Equilibrium Reaction	Kj	Equilibrium Reaction	Kj
2Pb(ALd) + O2(Gas) = 2PbO(Sl)	K1	As2O3(Sl) = As2O3(Gas)	K15
PbS(ALd) + O2(Gas) = Pb(Sl) + SO2(Gas)	K2	Sb2O3(Sl) = Sb2O3(Gas)	K16
2Zn(ALd) + O2(Gas) = 2ZnO(Sl)	K3	6FeO(Sl) + O2(Gas) = 2Fe3O4(Sl)	K17
4Sb(ALd) + 3O2(Gas) = 2Sb2O3(Sl)	K4	2FeS(Sl) + 3O2(Gas) = 2FeO(Sl) + 2SO2(Gas)	K18
4As(ALd) + 3O2(Gas) = 2As2O3(Sl)	K5	2Cu2S(Sl) + 3O2(Gas) = 2Cu2O(Sl) + 2SO2(Gas)	K19
Pb(ALd) = Pb(Gas)	K6	2PbO(Sl) + O2(Gas) + 2SO2(Gas) = 2PbSO4(Sl)	K20
Ag(ALd) = Ag(Sl)	K7	2ZnS(Gas) + 3O2(Gas) = 2ZnO(Gas) + 2SO2(Gas)	K21
2Cu2S(Sl) + 3O2(Gas) = 2Cu2O(Sl) + 2SO2(Gas)	K8	2Zn(Gas) + O2(Gas) = 2ZnO(Gas)	K22
2FeS(ALd) + 3O2(Gas) = 2FeO(Sl) + 2SO2(Gas)	K9	2As2S3(Gas) + 9O2(Gas) = 2As2O3(Gas) + 6SO2(Gas)	K23
2Pb(Gas) + O2(Gas) = 2PbO(Gas)	K10	2Sb2S3(Gas) + 9O2(Gas) = 2Sb2O3(Gas) + 6SO2(Gas)	K24
PbS(Sl) + 2PbO(Sl) = 3Pb(ALd) + SO2(Gas)	K11	S2(Gas) + 2O2(Gas) = 2SO2(Gas)	K25
PbS(Sl) = PbS(Gas)	K12	2CO(Gas) + O2(Gas) = 2CO2(Gas)	K26
ZnO(Sl) = ZnO(Gas)	K13	4Sb(ALd) + 5O2(Gas) = 2Sb2O5(Sl)	K27
4Bi(ALd) + 3O2(Gas) = 2Bi2O3(Sl)	K14

shows the equilibrium reactions of 27 subordinate components and their equilibrium constant $K_j$. The equilibrium constant $K_j$ of the independent reaction can be expressed by the following equation:

$$K_j=\exp \left(-{\displaystyle \frac{\left(\Delta G_{bj}^0-{\displaystyle \sum V_{ji}\Delta G_{ai}^0}\right)}{RT}}\right)$$

(3)

where R denotes the gas constant, T denotes the equilibrium temperature of the system, $\mathsf{\Delta }{G}_{\mathit{ai}}^0$ denotes the standard Gibbs free energy of formation of i independent components, and $\mathsf{\Delta }{G}_{\mathit{bi}}^0$ denotes the standard Gibbs free energy of formation of j subordinate components.

When the multiphase reaction system of antimony–lead synergistic side-blown oxidation smelting reached the chemistry balance state, we determined the connection between 17 independent components and 27 subordinate components as shown in the following equation:

$$Y_j=\left({\displaystyle \frac{Z_{m,j}}{{\gamma }_j}}\right)\left(K_j\right){\displaystyle \prod _i\left(\frac{{\gamma }_i{X}_i}{Z_{m,i}}\right)}$$

(4)

where $X_i$ denotes the molar number of i independent components, ${\mathsf{\gamma }}_i$ denotes the activity coefficient of i independent components, $Y_j$ denotes the molar number of $j$ subordinate components, ${\mathsf{\gamma }}_j$ denotes the activity coefficient of $j$ subordinate components, $Z_{m,i}$ denotes the molar number of i independent components, $Z_{m,j}$ denotes the molar number of i subordinate components, and $m$ denotes the product phase.

The total molar number $Z_m$ of each component in the $m$ product phase in Equation (4) can be calculated by the following equation:

$$Z_m={\displaystyle \sum _{i\left(m\right)}{X}_i+{\displaystyle \sum _{j\left(m\right)}{Y}_j}}$$

(5)

where i(m) denotes that the sum is obtained only when the i independent component belongs to the $m$ product phase, and j(m) denotes that the sum is obtained only when the $j$ subordinate component pertains to the $m$ product phase.

According to the law of conservation of mass, the mass of each element can be calculated by the following equation:

$$Q_k={\displaystyle \sum _i{A}_{i,k}{X}_f+{\displaystyle \sum _j{B}_{j,k}{Y}_f}}$$

(6)

where $Q_k$ denotes the molar number of the $k$ element.

For a closed pyrometallurgical system, we assumed that the number of product phases was defined as $N_p$ when the temperature, pressure, and number of elements of the system were given and had reached equilibrium. From Equations (3), (4) and (6), we observed that there were $\left(N_c+N_p\right)$ equations in the multiphase reaction system and that the number of variables to be solved was $\left(X_i+Y_j+Z_m\right)$. The equations were matched in quantity to the variables that had to be resolved. The amount of each species in each phase at the equilibrium of this system can be obtained by solving the system of nonlinear equations consisting of Equations (4)–(6) according to the Newton–Raphson algorithm.

2.4. Mathematical Models and Computing Systems

On the basis of the mechanism of the antimony–lead synergistic side-blown oxidation smelting process and the modeling principle described in Section 2.3, we constructed the multiphase equilibrium model for this process using the chemical equilibrium constant method. The model can be solved by the calculation flowchart shown in Figure 2. After considering the heat balance connection between the input material and the output material within the smelting system, we developed the multiphase equilibrium calculation system in Figure 3 using the following equation, which was based on the MetCal software development platform (MetCal v7.81):

$${\displaystyle \sum _i^{n_{\mathrm{A}}}\Delta H_{298,{\mathrm{A}}_i}}+{\displaystyle \sum _i^{n_{\mathrm{A}}}{\displaystyle {\int }_{298}^{T_i}C{p}_{{\mathrm{A}}_i}}\mathrm{d}T}={\displaystyle \sum _j^{n_{\mathrm{B}}}\Delta H_{298,{\mathrm{B}}_j}}+{\displaystyle \sum _j^{n_{\mathrm{B}}}{\displaystyle {\int }_{298}^{T_j}{C}_{p{\mathrm{B}}_j}}\mathrm{d}T}+Q_{\mathrm{Loss}}$$

(7)

where A_i denotes the reactant; T_i denotes the initial temperature of the reactant A_i; B_j denotes the product; T_j denotes the temperature of the product B_j; n_A denotes the number of reactants; n_B denotes the number of products; H denotes the enthalpy value; C_p denotes the heat capacity; and Q_Loss denotes the amount of heat loss.

3. Materials and Methods

3.1. Raw Material Composition

The raw materials for the side-blowing oxidation smelting process of an antimony and lead smelting enterprise in China included lead concentrate, antimony–gold concentrate, quartz sand, limestone, air, and industrial oxygen (oxygen volume concentration of 95%). We conducted XRF analysis, and the technical parameters of the equipment were as follows: in the linear range of less than 1%, the streaming gas detector counted 3000 kcps per second, and the scintillation detector counted 1500 kcps per second; the angular positioning accuracy of the scanning channel was less than 0.0001 degrees; and the range of analyzed elemental concentration was xppm—100%. We also used a spectrometer to analyze its elemental composition. According to the XRF analysis, we obtained the average value of the elemental composition of the raw materials used by the enterprise in December 2023, as shown in

Table 2. Chemical composition of lead concentrate (wt.%).

Pb	Zn	Bi	Cu	Fe	Ca	SiO2
40.76	5.90	0.54	0.65	14.80	2.1	8.58
S	Mg	Al	C	O	Other
16.30	1.14	0.26	0.63	7.31	1.03

,

Table 3. Chemical composition of antimony–gold concentrate (wt.%).

Sb	Fe	S	SiO2	As	Al	Ag	O	Other
40.55	3.50	27.00	12.34	13.82	0.36	1.22	0.32	0.90

,

Table 4. Chemical composition of quartz (wt.%).

SiO2	CaO	FeO	O	Other
85.00	5.00	2.64	0.29	7.07

and

Table 5. Chemical composition of limestone (wt.%).

FeO	CaO	SiO2	O	Other
0.49	53.00	1.04	0.43	45.04

.

3.2. Calculation Condition and Mixed Ore Composition

Based on typical production data from a domestic antimony–lead smelting enterprise’s side-blown oxidative smelting process, the total raw material input was 50 t/h, with lead concentrate being 85.2%, antimony–gold concentrate 10.6%, quartz 0.2%, and limestone 4%. The oxygen–material ratio was set at 130 Nm³/t, and the oxygen enrichment concentration was 90%. We calculated the melting temperature according to the heat balance. We assumed that the temperature of the antimony-rich crude lead was 300 °C lower than that of the smelting slag and that the temperature of the flue gas and dust was 20 °C higher than that of the smelting slag. The elemental composition of each raw material is detailed in Table 2, Table 3, Table 4 and Table 5. The outer wall area was 400 m², the furnace mouth area was 8 m², and the furnace lining thickness was 1 m. The ambient temperature was 30 °C, and the liner material was made of magnesio-chrome tiles with a coefficient of blackness of 0.8 and a convective heat transfer coefficient of 3.5 W/(m²·K).

The mixed ore is obtained by batching and physical mixing of lead concentrate, antimony–gold concentrate, quartz, and lime melt, and its physical phase composition is shown in

Table 6. Phase composition of mixed ore (wt.%).

PbS	ZnS	Cu2S	Fe2O3	FeS	CaCO3	SiO2	Ag	Al2O3
37.23	6.96	0.64	10.05	0.53	4.15	8.18	0.12	0.46
As2S3	Sb2S3	Bi2S3	MgO	H2O	FeS2	CaO	Other
2.30	5.73	0.53	4.15	6.95	10.12	1.98	2.60

. The physical phase composition of the mixed ore was obtained by summing the physical phase compositions of the four solid raw materials. Among them, the physical phase compositions of antimony–gold concentrate and lead concentrate were obtained from the elemental analysis results obtained from XRF analysis of samples taken under typical working conditions and, based on the characteristics of the two types of concentrates belonging to sulfide ores and the principle of the minimum free energy [42], they were calculated by the constructed physical phase computation model assuming their physical phase types (i.e., compound compositions).

3.3. Thermodynamic Data

By combining Equation (8), the Kirchhoff formula, with Equation (9), which details the connection between the temperature and the standard molar entropy change in reactions, we computed the Gibbs free energy of the components in the antimony–lead synergistic side-blown oxidation smelting process using Equation (10):

$$\Delta H_T^{\theta }=\Delta H_{298}^{\theta }+{\displaystyle {\int }_{298}^T{C}_p} \mathrm{d}T$$

(8)

$$\Delta S_T^{\theta }=\Delta S_{298}^{\theta }+{\displaystyle {\int }_{298}^T\frac{C_p}{T}} \mathrm{d}T$$

(9)

$$\Delta G_T^{\theta }=\Delta H_{298}^{\theta }-T·\Delta S_{298}^{\theta }+{\displaystyle {\int }_{298}^T{C}_p} \mathrm{d}T-T{\displaystyle {\int }_{298}^T\frac{C_p}{T}}\mathrm{d}T$$

(10)

We extracted the standard thermodynamic data for the products from the MetCal software (MetCal v7.81)’s database, as shown in Appendix A,

Table A3. Standard thermodynamic parameters of product components.

Component	State	$\mathit{\Delta }{\mathit{H}}_{298}^{\mathbf{\Theta }}$/ (kJ·mol−1)	$\mathit{\Delta }{\mathit{S}}_{298}^{\mathbf{\Theta }}$ (J·K−1·mol−1)	cp = a + b × 10−3T + c × 105T−2 + d × 10−6T2
				a	b	c	d
Pb	Liquid	3.873	70.506	27.159	1.029	0	0
PbS	Liquid	−93.143	84.129	66.946	0	0	0
Bi	Liquid	9.271	71.980	27.197	0	0	0
Sb	Liquid	17.531	62.712	31.381	0	0	0
Zn	Liquid	5.727	48.549	31.381	0	0	0
Cu2S	Liquid	−68.100	132.462	89.665	0	0	0
FeS	Liquid	−64.631	91.208	62.552	0	0	0
As	Liquid	21.568	53.284	28.833	0	0	0
Ag	Liquid	6.393	43.220	33.473	0	0	0
PbO	Liquid	−202.249	73.379	65.000	0	0	0
PbSO4	Liquid	−923.159	148.494	186.004	0	0	0
ZnO	Liquid	−309.542	47.920	60.669	0	0	0
Cu2O	Liquid	−130.224	96.402	99.916	0	0	0
As2O3	Liquid	−643.439	128.125	152.720	0	0	0
Sb2O3	Liquid	−675.490	143.628	156.904	0	0	0
Sb2O5	Liquid	−971.925	125.105	141.331	−3.732	−20.113	0
Bi2O3	Liquid	−578.024	149.814	202.005	0	0	0
FeO	Liquid	−257.276	57.591	68.201	0	0	0
Fe3O4	Liquid	−993.334	198.385	213.389	0	0	0
SiO2	Liquid	−927.548	9.310	85.774	0.000	0.000	0.000
CaO	Liquid	−572.908	40.980	62.762	0.000	0.000	0.000
MgO	Liquid	−561.018	12.833	66.946	0.000	0.000	0.000
Al2O3	Liquid	−595.568	45.145	144.866	0.000	0.000	0.000
Pb	Gas	195.205	175.377	28.063	−11.029	−9.310	4.728
PbO	Gas	68.139	240.048	41.612	−3.526	−20.136	1.014
PbS	Gas	127.959	251.416	37.350	0.194	−2.096	0.140
Zn	Gas	130.403	16.992	20.898	−0.133	−0.067	0.034
ZnO	Gas	136.518	242.811	37.671	−0.286	−1.985	0.735
ZnS	Gas	204.322	236.404	166.350	−85.742	−166.125	21.952
As2O3	Gas	−322.845	371.925	82.134	6.444	−5.356	0
As2S3	Gas	27.042	314.289	96.201	1.071	−8.213	0
Sb2O3	Gas	−708.564	129.903	180.004	0	0	0
Sb2S3	Gas	119.661	409.820	107.636	0.209	−7.255	0
O2	Gas	0	205.154	34.860	1.312	−14.141	0.163
SO2	Gas	−296.820	248.226	54.781	3.350	−24.745	−0.241
S2	Gas	128.603	228.169	34.672	3.286	−2.816	−0.312
CO	Gas	−110.544	197.665	29.932	5.415	−10.814	−1.054
CO2	Gas	−393.515	213.774	54.437	5.116	−43.579	−0.806
N2	Gas	0	191.615	23.529	12.117	1.210	−3.076
H2O	Gas	−241.832	188.837	31.438	14.106	−24.952	−1.832

. To negate the effects of reaction kinetics, we considered both production analysis and literature references [43,44,45,46] and adjusted the activity coefficients for certain products, as noted in Appendix A,

Table A4. Activity coefficients of product components.

Component	Phase	Activity Coefficient
PbO	Sl	1
PbSO4	Sl	0.8
PbS	Sl	0.5
ZnO	Sl	0.1
Cu2O	Sl	0.002
Cu2S	Sl	50
FeO	Sl	0.0001
Fe3O4	Sl	0.1
FeS	Sl	0.0001
As2O3	Sl	0.003
Sb2O3	Sl	0.002
Sb2O5	Sl	MQC
Bi2O3	Sl	1.64
CaO	Sl	0.1
MgO	Sl	1
Al2O3	Sl	1
SiO2	Sl	0.1
Ag	Sl	1.351
Pb	ALd	0.35
PbS	ALd	20
Sb	ALd	0.0078
CuS	ALd	0.028
FeS	ALd	100
Zn	ALd	0.066
As	ALd	0.058
Bi	ALd	18
Ag	ALd	0.045

. MQC indicates that the activity of the component is determined using MetCal v7.81’s modified quasi-chemical solution model for activity calculations. We treated the flue gas as an ideal gas, which suggested that its component activity coefficient should be set to 1.

4. Result and Discussion

4.1. Calculated Result

According to the calculation conditions and the phase composition of mixed ore of Section 3.2, we used the constructed multiphase equilibrium calculation system for the antimony–lead synergistic side-blown oxidation smelting process to calculate the composition and yield of equilibrium products.

Table 7. Calculation results of the main technical index.

Name	Unit	Value	Name	Unit	Value
Pb content of ALd	%	89.29	Pb content of dust	%	38.03
Sb content of ALd	%	3.24	Sb content of dust	%	5.05
Direct yield of Pb	%	20.62	Temperature of ALd	°C	829
Direct yield of Sb	%	5.87	Temperature of slag	°C	1129
FeO/SiO2 in slag		2.30	Temperature of gas	°C	1149
CaO/SiO2 in slag		0.53	Yield of dust	%	13.88
CaO/FeO in slag		0.28	Yield of ALd	%	6.18
Pb content of slag	%	32.88	Yield of slag	%	48.60
Sb content of slag	%	5.34	Yield of gas	%	33.69

and

Table 8. Calculation results of heat balance.

Type	Heat Type	Material Name	Temperature (°C)	Heat Quantity (MJ/h)	Ratio of Heat (%)
Heat income	Physical heat	Mixed ore	25	0.00	0.00
		Industrial oxygen	25	0.00	0.00
		Air	25	0.00	0.00
	Chemical heat		25	59,250.37	100.00
	Exchange heat	Cooling inlet water	37
	Total			55,250.37	100.00
Heat outcome	Physical heat	Antimony-rich crude lead	829	480.12	0.81
		Smelting slag	1129	23,199.92	39.16
		Flue gas	1149	24,207.03	40.86
		Dust	1149	5186.27	8.75
	Exchange heat	Cooling outlet water	38	836.39	1.41
	Natural heat dissipation		60	5340.64	9.01
	Total			59,250.37	100.00

, respectively, show the main technical indices and heat balance calculation results of the antimony–lead synergistic oxidation smelting process. And the allocation behavior of the accessory elements in the products is illustrated in Figure 4.

4.2. Results Comparison

On the basis of production report data from a domestic antimony–lead smelting enterprise’s side-blown oxidation smelting section released in mid-December 2023, we compared the average analysis results of samples of the antimony-rich crude lead and smelting slag with the values calculated using our model, as shown in

Table 9. Calculation results and industrial data (wt.%).

Value	Phase	Pb	Sb	Zn	Cu	Fe	CaO	SiO2	S	As	Bi	Ag
Calculated	ALd	89.292	3.238	0.002	1.319	0	-	-	0.348	0.555	4.284	0.903
Measured		85.176	3.370	-	1.318	-	-	-	-	0.553	4.366	0.892
Calculated	Sl	32.875	5.275	6.590	0.562	17.062	6.083	11.558	1.282	2.06	0.07	0.061
Measured		33.351	5.815	7.160	0.519	17.525	6.050	11.797	1.226	2.064	0.070	0.058

.

4.3. Discussion

From the results of the data presented in Figure 3, it is evident that elements such as Pb, Sb, Zn, As, Cu, and Fe were mainly enriched in the smelting slag during the antimony–lead synergistic side-blown oxidation smelting process, whereas Ag and Bi were mainly distributed in the antimony-rich crude lead. With the exception of S, the distribution rate of the other associated elements in the flue gas was not significant. The results show that the elements Pb, Sb, Zn, As, Cu, and Fe are more easily oxidized into slag, while Ag and Bi are less easily oxidized, so they are more distributed in antimony-rich crude lead.

Reviewing the data presented in Table 9, we observed that the calculated values for most elements in the products closely matched the average production measurements, except for certain elements that were not detected during production. Specifically, the relative errors for Pb, Sb, Cu, As, Bi, and Ag in the antimony-rich crude lead were 4.83%, 3.92%, 0.08%, 0.36%, 1.88%, and 1.23%, respectively. For the smelting slag, the relative errors for Pb, Sb, Zn, Cu, Fe, CaO, SiO₂, S, As, Bi, and Ag were 1.43%, 9.29%, 7.96%, 8.29%, 2.64%, 0.55%, 2.03%, 4.57%, 0.10%, 0%, and 5.17%, respectively. Notably, the discrepancies between simulated and measured values for Zn, Cu, Sb, and As in the smelting slag were substantial, which we attributed to detection errors associated with the quantification of trace elements in the production analysis, as well as inherent errors in the model itself.

In order to further improve the accuracy and reliability, it is necessary to continue to optimize the model and to carry out more precise detection in the future. Although there are certain discrepancies between the production data and the calculated values of the model, these errors are currently in a controllable range, with relative errors remaining below 10%. These comparison results demonstrated that the model developed in this research was able to capture the multiphase reaction characteristics of the antimony–lead synergistic side-blown oxidation smelting process. This process had the capacity to precisely forecast the smelting production process and to refine process parameters, thus serving as an effective instrument for the subsequent systematic thermodynamic analysis of the process.

5. Conclusions

• On the basis of the multiphase reaction mechanism and features of the antimony–lead synergistic side-blown oxidation smelting process, we constructed a thermodynamic simulation model and calculation system of the antimony–lead synergistic side-blown oxidation smelting process using the chemical equilibrium constant approach and MetCal software platform (MetCal v7.81), which provided a software-based tool for the thermodynamic calculation and analysis of the smelting process. The model is based on the MetCal platform (MetCal v7.81), which has high accuracy, low error predictions and fast computation speed.
• Using the established calculation system, we conducted an instance validation under the typical production conditions of a domestic enterprise. The outcomes from the product composition closely matched the actual manufacturing outcomes, demonstrating that the constructed model effectively embodied the multiphase reaction features of the antimony–lead synergistic side-blown oxidation smelting process and possessed the ability to accurately forecast the output of this process.
• Through verification and contrast experiments, we found that the calculated values of the main technical index for the antimony–lead synergistic side-blown oxidation smelting process had a small relative error compared with the average measured values from industrial production during the same period. The relative errors of the calculated mass fractions of Pb, Sb, Zn, Cu, Fe, CaO, SiO₂, S, As, Bi, and Ag in antimony-rich crude lead and smelting slag are less than 10%. Although there is an error margin, it is acceptable. In the future, the constructed model and calculation system can be used to carry out conditional experiments to optimize and regulate different process parameters, so as to provide a model basis for the subsequent on-line optimization and regulation of the antimony and lead synergistic side-blowing oxidation smelting process.