Study on the Kinetics of Carbothermic Reduction of Stainless Steel Dust by Walnut Shell Biochar

Abstract

Stainless steel dust (SSD) is a by-product generated during the smelting process of stainless steel, which is rich in valuable metals such as Fe, Cr, Ni, and Mn. To optimize the carbothermic reduction process of SSD, this study first conducted the thermodynamic analysis of the carbothermic reduction of SSD and then employed walnut shell biochar as a reductant with non-isothermal thermogravimetric analysis with linear heating rates of 5 °C/min, 10 °C/min, 15 °C/min, and 20 °C/min. The activation energies of the carbothermic reduction reactions were calculated using the FWO method, KAS method, and Friedman method, respectively. Subsequently, the corresponding kinetic models were fitted and matched using the Málek method. The results indicate that before 600 °C, the direct reduction of SSD by carbon plays a dominant role. As the temperature increases, the indirect reduction becomes the main reduction reaction for SSD due to the generation of CO. The activation energies calculated by the Flynn–Wall–Ozawa (FWO) method, Kissinger–Akahira–Sunose (KAS) method, and Friedman method are 412.120 kJ/mol, 416.930 kJ/mol, and 411.778 kJ/mol, respectively, showing close values and a general trend of increasing activation energy as the conversion rate increased from 10% to 90%. Moreover, the reduction reaction is staged. In the conversion range of 10% to 50%, the carbothermic reduction reaction conforms to the shrinking core model within phase boundary reactions, coded as R1/4. In the conversion range of 50% to 60%, it conforms to the shrinking core model within phase boundary reactions, coded as R1/2; in the conversion range of 60% to 90%, the carbothermic reduction reaction follows the second-order chemical reaction model, coded as F2.

1. Introduction

Stainless steel dust (SSD) is a by-product generated during the stainless steel smelting process [1,2,3]. In 2024, global crude stainless steel production is estimated to be approximately 60 million tons. During the smelting of each ton of crude stainless steel, 18 to 33 kg of dust are emitted [4,5,6]. The accumulation of large amounts of SSD not only imposes significant pressure on the environment but also poses considerable health risks to humans. Given that SSD is rich in valuable metallic elements such as Fe, Cr, Ni, and Mn, it represents a promising resource for recycling [7]. Therefore, the circular utilization of SSD holds substantial significance [8,9,10].

Currently, the mainstream processes for handling SSD primarily include landfilling, pyrometallurgical treatment, and hydrometallurgical treatment [11,12,13,14]. The landfill method involves solidifying SSD and burying it underground. Although this process is relatively simple, it is gradually being phased out, as it fails to effectively recover valuable metal elements within the dust. Currently, the most common hydrometallurgical treatment methods for stainless steel dust are acid washing, solvent extraction, and chemical precipitation [15]. Acid washing is the process of leaching dissolved metal oxides with inorganic acids (e.g., HNO₃, 15~25%vol) to create soluble salts. Solvent extraction is the process of using organic solvents to selectively separate target metals from the leach solution with high recovery rates [16], for example, the use of amine extractants to extract Cr⁶⁺ at pH = 1~2. Chemical precipitation alters the pH or adds precipitant to precipitate metal ions; further treatment is required to improve purity [17], for example, precipitating Fe³⁺ at pH = 4~5 and then adjusting pH = 8~9 to precipitate Cr³⁺/Ni²⁺. Hydrometallurgy suffers from high costs and a tendency to generate secondary pollution, which limits its widespread adoption [18,19,20]. In contrast, pyrometallurgical treatment focuses on recovering valuable metals from SSD through carbothermic reduction processes. Representative technologies in this category include the Inmetco process, Fastmet/Fastmelt process, and OxyCup process. The Fastmet and Inmetco processes use a rotating hearth furnace for carbothermal reduction at 1300 °C~1350 °C [21], and the Inmetco process then uses an electric arc furnace for slag–iron separation [22,23]. The OxyCup processes use a shaft furnace for carbothermal reduction with total melting and reduction at 1400 °C [24]. The reduction products of these pyrometallurgical treatments are alloys that can be employed directly as steelmaking raw materials or alloy additives. These technologies feature broad applicability, enabling the efficient recovery of various metals with relatively lower environmental impact and energy consumption. Consequently, they have become the predominant methods employed by steel mills for treating SSD [25].

As a renewable energy source, biomass has gained considerable attention from researchers due to its widespread distribution, diverse types, and low pollution characteristics [26,27,28]. However, biomass also poses several challenges, including low fixed carbon content, high volatile matter content, lower calorific value, and seasonal limitations [29]. Consequently, carbonizing biomass in an anoxic or low-oxygen environment to produce biochar offers advantages such as ease of storage, higher fixed carbon content, greater calorific value, and pollution-free characteristics, showcasing a broad application potential [30,31,32]. In recent years, scholars both domestically and internationally have conducted extensive research on using biochar as a substitute for traditional fossil fuels, applying it in energy development and smelting sectors. Net emissions of 0.5 to 0.8 tons of CO₂ can be decreased by melting one ton of iron with biochar. Utilizing biochar as a clean energy source can not only significantly reduce greenhouse gas emissions but also promote a green transformation of the energy structure [33,34,35].

The focus of this study is to conduct thermodynamic analysis and non-isothermal kinetic studies of the carbothermic reduction process of SSD with walnut shell biochar as a reductant using thermal gravimetric analysis under various linear heating rates. The objectives include establishing reduction reaction models, exploring the patterns of activation energy changes, and examining the impact of different heating rates on the reduction process. The study aims to uncover the kinetic behavior of the carbothermic reduction of SSD, thereby providing experimental and theoretical bases for the resource utilization of SSD.

2. Thermodynamics Analysis of the Carbothermic Reduction Process of SSD

To investigate the likelihood of the main components in stainless steel dust undergoing carbothermic reduction reactions and to analyze the trend of the standard Gibbs free energy change of these reactions with temperature, thermodynamic calculations were performed on metal oxides such as Fe, Cr, Ni, and Mn in stainless steel dust using Factsage 8.1 software. The results shown in Figure 1 encompass both direct reduction reactions of metal oxides with carbon and indirect reduction reactions of metal oxides with CO.

As shown in Figure 1, the reduction of iron oxides and nickel oxides, such as Fe₂O₃, Fe₃O₄, and NiO, in stainless steel dust occurs in stages. The standard Gibbs free energy change (ΔG⁰) for direct reduction reactions decreases with increasing temperature, indicating that higher temperatures favor these reduction processes. The initial temperatures for the direct reduction of Fe₂O₃, Fe₃O₄, and NiO are 317 °C, 664 °C, and 425 °C, respectively, leading to the formation of lower-valent iron oxides and metal Ni. When the reduction temperature reaches 719 °C, FeO undergoes reduction to metal iron, releasing significant amounts of reductive CO gas.

FeCr₂O₄ is the primary chromium-containing phase in SSD. It undergoes stepwise reduction at temperatures reaching 1045 °C, 1218 °C, and 1269 °C until it is ultimately reduced to metal chromium. In contrast, the direct reduction of MnO requires a much higher temperature of up to 1421 °C. Moreover, the ΔG⁰ values for the indirect reduction reactions of MnO and FeCr₂O₄ are greater than zero (>0), indicating that these reactions are not spontaneous under these conditions. However, the indirect reduction of Fe₂O₃ to Fe₃O₄ has a ΔG⁰ less than zero (<0), suggesting that this reaction can proceed spontaneously. Subsequent indirect reduction of Fe₃O₄ also starts at a relatively low temperature of only 576 °C. Additionally, the indirect reduction of NiO has a ΔG⁰ < 0, indicating that it is easy to occur spontaneously. Therefore, selecting an appropriate temperature range can achieve efficient reduction of SSD and effective control of the reduction products.

3. Experimental

3.1. Characterization of Raw Materials

The raw materials used in the experiment consist of two components: stainless steel dust (SSD) sourced from a stainless steel smelting plant in China, and walnut shell biochar obtained from a crop residue processing plant in China. The particle size of the SSD and the walnut shell biochar were measured using a laser particle size analyzer (Matersizer 2000, Malvern Panalytical, Malven, UK), with the particle size distribution shown in Figure 2. As indicated in Figure 2, the particle size of the SSD generally follows a normal distribution, predominantly falling within the range of 1–10 μm, which accounts for 77.72% of the total distribution. The fraction of particles within the 10–100 μm range constitutes 19.29% of the distribution. Overall, the particle sizes are relatively fine, with an average particle size of 14.86 ± 0.57 μm and a median particle size of 3.31 μm. The average particle size of walnut shell biochar was 144.40 ± 0.77 μm.

The chemical composition analysis of the SSD was conducted using Inductively Coupled Plasma–Atomic Emission Spectroscopy (ICP-AES, IRIS Advantage ER/S, Thermo Elemental, Waltham, MA, USA).

Table 1. Chemical composition and content of SSD (wt%).

TFe	Fe2O3	CaO	MgO	ZnO	MnO	Cr2O3	NiO	Al2O3	SiO2	LOI
48.41	69.60	5.21	0.55	0.18	6.45	11.22	1.03	0.20	3.38	2.72

presents the specific chemical composition and content of the SSD. As shown in Table 1, the SSD is primarily composed of Fe₂O₃, Cr₂O₃, MnO, and NiO, with their respective weight contents being 69.60 wt%, 11.22 wt%, 6.45 wt%, and 1.03 wt%. The total iron content reaches 48.41 wt%, indicating a high potential for recovery and utilization. The impurity components mainly include CaO, SiO₂, MgO, and Al₂O₃, with their contents at 5.21 wt%, 3.38 wt%, 0.55 wt%, and 0.20 wt%, respectively.

The phase composition of the SSD was analyzed using X-ray diffraction (XRD, X’Pert PRO MPD, PANalytical, Almelo, The Netherlands). The operational conditions were as follows: Cu Kα radiation, a scanning range of 20° to 70°, with a current of 40 mA and tube voltage of 40 kW. The XRD pattern of the SSD is shown in Figure 3. The main phase composition of the SSD includes Fe₂O₃, Fe₃O₄, FeCr₂O₄, and SiO₂.

The microscopic morphology of SSD and walnut shell biochar was observed using a field emission scanning electron microscope (SEM-EDS, Nova NanoSEM400, FEI, Hillsboro, OR, USA), as shown in Figure 4 and Figure 5. As indicated by Figure 4, the SSD consists of particles of various sizes, exhibiting spherical and irregular polyhedral shapes. Moreover, there is a certain degree of agglomeration observed among some of the particles. In contrast, the surface of the walnut shell biochar displays a honeycomb-like porous structure, with an interconnected network of channels inside. This porous structure provides a larger contact area for carbothermic reduction reactions.

The industrial analysis and elemental analysis results of the walnut shell biochar are shown in

Table 2. Industrial analysis of walnut shell biochar (wt%).

Sample	Fixed Carbon	Volatile Matter on Air-Dried Basis	Ash Content on Air-Dried Basis	Moisture Content on Air-Dried Basis	High Heat Value
Walnut shell biochar	86.48	11.10	1.85	0.56	7891 cal/g

and

Table 3. Elemental analysis of walnut shell biochar and graphite (wt%).

Sample	C	H	O	N	S
Walnut shell biochar	74.31	2.35	8.36	1.33	0.44

, respectively. As indicated by Table 2 and Table 3, the fixed carbon content of the walnut shell biochar is 86.48%, with an ash content of merely 1.85%. This is even higher than that of commonly used bituminous coal and anthracite, which have fixed carbon contents ranging from 63% to 75%. Furthermore, the nitrogen and sulfur contents are relatively low at 1.33% and 0.44%, respectively, and the high calorific value is 7891 cal/g, which is beneficial for reducing the emission of harmful gases such as SO₂.

3.2. Thermogravimetric Analysis

The non-isothermal thermogravimetric experiments were conducted using a simultaneous thermal analyzer (STA449C, NETZSCH, Selb, Germany). The carbothermic reduction experiments were performed with a mixture of walnut shell biochar and the SSD at a C/O molar ratio of 1.2 (the molar ratio of fixed carbon in the walnut shell biochar to the main metal oxides in the SSD). Each experiment used a sample mass of 20 ± 3 mg, which was heated from room temperature to 1400 °C at heating rates of 5 °C/min, 10 °C/min, 15 °C/min, and 20 °C/min, respectively. All experiments were carried out under an Ar atmosphere, continuously recording the mass change of the samples. Subsequently, based on the thermogravimetric experimental data, the conversion rate of the carbothermic reduction of the SSD was calculated. The activation energy for the carbothermic reduction reaction was then determined by fitting the data using the Flynn–Wall–Ozawa (FWO) method, Kissinger–Akahira–Sunose (KAS) method, and Friedman method [36]. Finally, the kinetic model of the carbothermic reduction of the SSD was fitted using the Málek method.

The conversion rate during the reaction process is denoted by α, as shown in Equation (1). In Equation (1), m₀, m_t, and m_∞ denote the mass of the sample at the initiation of the reaction (g), the mass of the sample at time t (g), and the mass of the sample after the reaction (g), respectively [37].

$$\alpha ={\displaystyle \frac{m_0-m_t}{m_0-m_{\infty }}}$$

(1)

In Equation (2), β represents the heating rate (K/min), which is maintained as a constant value during thermogravimetric experiments.

$$\beta ={\displaystyle \frac{dT}{dt}}$$

(2)

Assuming that the reaction process of a substance depends solely on the conversion rate and temperature, and these two parameters are independent of each other, the kinetic equation of differential form and integral form for non-isothermal reactions can be expressed as shown in Equation (3) and Equation (4), respectively.

$${\displaystyle \frac{d\alpha }{dt}}=f(\alpha )K(T)$$

(3)

$$G(\alpha )={\displaystyle {\int }_0^{\alpha }\frac{d\alpha }{f(\alpha )}}=K(T)t$$

(4)

${\displaystyle \frac{d\alpha }{dt}}$ denotes the derivative of the conversion rate, expressed in units of percent per minute (%/min). f(α) represents the differential mechanism function, and G(α) represents the integral mechanism function. K(T) represents the relationship between the rate constant and temperature, expressed by the Arrhenius equation, as illustrated in Equation (5).

$$K(T)=A\exp (-{\displaystyle \frac{E}{\mathrm{R}T}})$$

(5)

In Equation (5), A is the frequency factor (s⁻¹), E is the activation energy (kJ/mol), R is the universal gas constant (8.314 J/(mol·K)), and T is the reaction temperature (K).

By inserting Equation (5) into Equation (3), it can be transformed into Equation (6) [38].

$${\displaystyle \frac{d\alpha }{dt}}=({\displaystyle \frac{A}{\beta }})\exp (-{\displaystyle \frac{E}{\mathrm{R}T}})f(\alpha )$$

(6)

By combining Equations (4) and (6), the integral form of the kinetic equation is obtained, as shown in Equation (7).

$$G(\alpha )={\displaystyle {\int }_0^{\alpha }\frac{d\alpha }{f(\alpha )}}={\displaystyle \frac{A}{\beta }}{\displaystyle {\int }_{T_0}^T\exp (-\frac{E}{\mathrm{R}T})}dT$$

(7)

In Equation (7), T₀ represents the initial reaction temperature (K), and T denotes the instantaneous temperature at a given time (K).

Model-free kinetic analysis, known as iso-conversional methods, refers to techniques that directly determine the apparent activation energy without assuming a specific reaction mechanism function. This approach avoids errors that may arise from incorrect assumptions about different reaction mechanisms, making it widely applicable in kinetic analysis. Commonly used model-free kinetic methods include the FWO method, KAS method, and Friedman method [39,40,41].

In the FWO method, the kinetic equation can be expressed as Equation (8):

$$\lg \beta =\lg ({\displaystyle \frac{AE}{\mathrm{R}G(\alpha )}})-2.315-0.4567\times {\displaystyle \frac{E}{\mathrm{R}T}}$$

(8)

In the KAS method, the thermokinetic equation can be expressed as Equation (9):

$$\ln {\displaystyle \frac{\beta }{T^2}}=\ln ({\displaystyle \frac{AE}{\mathrm{R}G(\alpha )}})-{\displaystyle \frac{E}{\mathrm{R}T}}$$

(9)

The Friedman method is a model-free analysis approach based on the differential form, and its thermokinetic equation can be expressed as shown in Equation (10):

$$\ln (\beta {\displaystyle \frac{d\alpha }{dT}})=\ln A+\ln f(\alpha )-{\displaystyle \frac{E}{\mathrm{R}T}}$$

(10)

The Málek method is an advanced kinetic calculation approach. It first determines the activation energy E of the reaction using iso-conversional methods such as the FWO method. Then, it compares experimental data points with graphs generated based on various model functions to identify the appropriate reaction kinetic model function f(α). The advantage of this method starts from the calculation of activation energy using iso-conversional methods and progressively obtaining comprehensive kinetic results, thus avoiding the hassle and subjective errors associated with individually testing each f(α). However, when applying the Málek method, it is crucial to have a sufficient number of experimental data points to avoid difficulties in determining the reaction model. Equations (11) and (12) represent the specific mathematical expressions for the Málek method [42], G(0.5) denotes the value of G(α) for a conversion rate of 0.5, and f(0.5) denotes the value of f(α) for a conversion rate of 0.5:

$$y(\alpha )={\displaystyle \frac{G(\alpha )f(\alpha )}{G(0.5)f(0.5)}}$$

(11)

$$y(\alpha )={\displaystyle \frac{{(\frac{T}{T_{0.5}})}^2\frac{d\alpha }{dt}}{{(\frac{d\alpha }{dt})}_{0.5}}}$$

(12)

4. Results and Discussion

4.1. Thermogravimetric Analysis Results

Figure 6 shows the thermogravimetric analysis and conversion rate curves of a mixture of SSD and walnut shell biochar at a C/O molar ratio of 1.2, recorded during heating from room temperature to 1400 °C at different heating rates (5 °C/min, 10 °C/min, 15 °C/min, and 20 °C/min). Combining the thermodynamic calculation results from Figure 1, it can be observed that when the reaction temperature is below 600 °C, the thermogravimetric curves are relatively flat, indicating slow reaction progression. This stage corresponds to the reduction processes of 3Fe₂O₃ + C→2Fe₃O₄ + CO, along with minor reactions such as NiO + C→Ni + CO. As the temperature gradually increases to 1200 °C, the mass loss of the sample sharply increases, indicating more vigorous reactions. This stage corresponds to the stepwise reduction of iron oxides and the reduction of ferric chromite, represented by the following reactions: Fe₃O₄ + C→3FeO + CO, FeO + CO→Fe + CO₂, Fe₃O₄ + CO→3FeO + CO₂, and FeCr₂O₄ + C→Fe + Cr₂O₃ + CO. Within this temperature range, the iron oxides in the SSD are largely reduced to metallic iron. Between 1200 °C and 1400 °C, the rate of mass loss gradually decreases, indicating that the primary reactions occurring are the stepwise reduction of chromium oxides, mainly through the reactions Cr₃O₄ + 4C→3Cr + 4CO and Cr₂O₃ + 3C→2Cr + 3CO.

As the heating rate increases, the onset reaction temperature of the sample rises, and the residual mass also increases. Specifically, at a heating rate of 5 °C/min, the onset reaction temperature is 524.12 °C, with a residual mass of 65.26%. When the heating rate is increased to 20 °C/min, the onset reaction temperature rises to 541.04 °C, and the residual mass increases to 67.93%. This indicates that while increasing the heating rate accelerates the thermal motion of reactant molecules and promotes the reaction, it can also lead to discrepancies between the diffusion rate of the reactants and the reaction rate, potentially resulting in incomplete reactions. Therefore, a relatively lower heating rate may facilitate a more thorough diffusion of the reactants, ensuring a more complete reaction [43].

4.2. Calculation of Activation Energy for the Carbothermic Reduction Process of SSD

Based on the Flynn–Wall–Ozawa (FWO) equation, a relationship can be established between the logarithm of the heating rate (lgβ) and the reciprocal of the reaction temperature (1000/T), as illustrated in Figure 7. In the process of calculating the activation energy, only the conversion range from 10% to 90% is considered to mitigate potential biases that could be introduced by experimental data from the initial and final stages of the reaction. The thermogravimetric experimental data were subjected to linear fitting using the data points corresponding to the same conversion rate but under four different heating rates. It was observed that the correlation coefficients (R²) of the fitted curves were consistently high. Consequently, the activation energy for the carbothermic reduction reaction between SSD and walnut shell biochar can be determined by calculating the slopes of these fitted curves. Furthermore, as the conversion rate increases, the slope of the fitted line gradually becomes steeper, indicating that the activation energy increases with the rise in conversion rate. This suggests that the reaction becomes progressively more difficult to proceed as the conversion rate increases.

Similarly, the KAS method also focuses on the conversion range from 10% to 90%. It examines the relationship between ln(β/T²) and the reciprocal of the reaction temperature (1000/T). Thermogravimetric experimental data were subjected to linear fitting using data points corresponding to the same conversion rate but under four different heating rates. The outcomes are presented in Figure 8. The correlation coefficients (R²) of the fitted curves are relatively high. Consequently, the activation energy for the carbothermic reduction reaction between the SSD and walnut shell biochar can be determined by calculating the slopes of these fitted curves. From the fitting results, it can be observed that, similar to the FWO method, the slope of the fitted line increases as the conversion rate rises. This indicates that the required activation energy becomes higher as the reaction progresses, suggesting that the reaction becomes progressively less active.

The Friedman method similarly focuses on the conversion range from 10% to 90%. It examines the relationship between ln(β·dα/dT) and the reciprocal of the reaction temperature (1000/T) based on thermogravimetric experimental data. This relationship is depicted in Figure 9. The fitting results obtained using this method are consistent with those from the preceding two methods. It is observed that as the conversion rate α increases, the slope of the fitted line gradually becomes steeper, indicating a progressive increase in the activation energy.

The variation trends of the activation energy calculated by the three methods with respect to the conversion rate, along with the specific fitting parameters, are shown in Figure 10 and

Table 4. Fitting parameters of FWO, KAS, and Friedman methods for mixture at different conversion rates.

α	FWO	R2	KAS	R2	Friedman	R2
	E/(kJ·mol−1)		E/(kJ·mol−1)		E/(kJ·mol−1)
10%	94.282	0.9919	92.147	0.9882	114.511	0.9697
20%	123.206	0.9910	121.625	0.9884	129.265	0.9897
30%	150.169	0.9894	148.042	0.9866	154.288	0.9823
40%	223.950	0.9924	216.789	0.9978	322.210	0.9985
50%	396.592	0.9892	397.554	0.9878	344.382	0.9645
60%	472.770	0.9654	477.024	0.9517	445.184	0.9487
70%	619.718	0.9722	631.145	0.9456	574.358	0.9483
80%	813.819	0.9384	834.552	0.9584	745.666	0.9411
90%	814.574	0.9480	833.485	0.9645	876.140	0.9813
Mean value	412.120	-	416.930	-	411.778	-

, respectively. From Figure 10 and Table 4, it can be observed that the activation energy derived from the FWO method ranges from 94.282 to 814.574 kJ/mol, with an average value of 412.120 kJ/mol. The activation energy derived from the KAS method ranges from 92.147 kJ/mol to 834.552 kJ/mol, with an average value of 416.930 kJ/mol, whereas the activation energy obtained via the Friedman method spans from 114.511 kJ/mol to 876.140 kJ/mol, with an average value of 411.778 kJ/mol.

It is noted that the activation energy increases as the conversion rate increases. This phenomenon can be explained by the initial stage of the reaction, where the reactants are in close contact, and reactions predominantly take place on the surface of the powders or particles. During this phase, reactions are relatively facile, necessitating lower activation energy. However, as the conversion rate increases, the quantity of unreacted solid material diminishes, and the remaining reactive sites may become less active or increasingly inaccessible. Thus, higher activation energy is required for subsequent reactions to overcome these challenges.

4.3. Fitting Results of Kinetic Models for the Carbothermic Reduction Process of SSD

Table 5. Differential and integral expressions of common reaction conversion degree function. Adapted from Refs. [36,44].

Model	Mechanism	Code	Differential Model f(α)	Integral Model G(α)
Order of reaction (n)	n = 0	F0	1	α
	n = 1	F1	1 − α	−ln(1 − α)
	n = 2	F2	(1 − α)2	(1 − α)−1 − 1
Phase boundary reaction	Shrinking core, m = 1/4	R1/4	1/4(1 − α)−3	1 − (1 − α)4
	Shrinking core, m = 1/3	R1/3	1/3(1 − α)−2	1 − (1 − α)3
	Shrinking core, m = 1/2	R1/2	1/2(1 − α)−3	1 − (1 − α)2
	Shrinking core, m = 2	R2	2(1 − α)1/2	1 − (1 − α)1/2
	Shrinking core, m = 3	R3	3(1 − α)2/3	1 − (1 − α)1/3
Diffusion	The one-dimensional diffusion	D1	1/2α−1	α2
	The two-dimensional diffusion	D2	[−ln(1 − α)]−1	α + (1 − α) ln(1 − α)
	The three-dimensional diffusion	D3	(1 − α)1/2 [1 − (1 − α)1/2] − 1	[1 − (1 − α)1/2]2
Random nucleation and nuclei growth	Two dimension	A2	2(1 − α) [−ln(1 − α)]1/2	−ln(1 − α)1/2
	Three dimension	A3	3(1 − α) [−ln(1 − α)]1/3	−ln(1 − α)1/3
Exponential nucleation	Power series law, n = 1/2	P2	2α1/2	α1/2
	Power series law, n = 1/3	P3	3α2/3	α1/3
	Power series law, n = 1/4	P4	4α3/4	α1/4

lists sixteen frequently encountered kinetic models, detailing their model code, corresponding differential equations, and integral equations.

Figure 11 shows the fitting results obtained using the Málek method. As indicated by Figure 11, the carbothermic reduction of SSD is a complex process. Specifically, within the conversion range of 10% to 50%, the reaction process conforms to R1/4. Within the conversion range of 50% to 60%, the reaction process conforms to R1/2. Within the conversion range of 70% to 90%, the reaction process conforms to F2.

5. Conclusions

(1)

Based on the thermogravimetric analysis results at different heating rates, the carbothermic reduction of stainless steel dust with biochar can be primarily divided into three stages: when the reaction temperature is below 600 °C, this stage is characterized by the direct reduction of Fe₂O₃ and partial direct reduction of NiO; when the reaction temperature is between 600 °C and 1200 °C, the reduction process is mainly dominated by the indirect reduction of Fe₃O₄ and FeO, accompanied by the direct reduction of FeCr₂O₄; and when the reaction temperature is between 1200 °C and 1400 °C, the primary reactions involve the stepwise reduction of chromium oxides.

(2)

The activation energy for the carbothermic reduction of stainless steel dust, as determined by the FWO method, ranges from 94.282 to 814.574 kJ/mol, with an average value of 412.120 kJ/mol. Utilizing the KAS method, the activation energy range is from 92.147 to 833.485 kJ/mol, with an average value of 416.930 kJ/mol. The activation energy computed using the Friedman method spans from 114.511 to 876.140 kJ/mol, with an average value of 411.778 kJ/mol. Furthermore, it is noted that the activation energy increases as the conversion rate increases.

(3)

Using walnut shell biochar as the reductant, SSD’s carbothermic reduction reaction is staged. In the conversion range of 10% to 50%, the carbothermic reduction reaction conforms to the shrinking core model within phase boundary reactions, coded as R1/4. In the conversion range of 50% to 60%, it conforms to the shrinking core model within phase boundary reactions, coded as R1/2, and in the conversion range of 60% to 90%, the carbothermic reduction reaction follows the second-order chemical reaction model, coded as F2.