Hot Metal Temperature Prediction During Desulfurization in the Ladle

Abstract

Hot metal is loaded from the torpedo car to the ladle, where desulfurization is realized. The main aim of the desulfurization process is to reduce the amount of sulfur in hot metal. During this process, the temperature of desulfurized hot metal is changed. The continuous measurement of this temperature during the desulfurization process is not implemented due to the device’s construction and the increasing costs. The temperature is measured only at the beginning and end of this process. For this reason, the paper focuses on the mathematical model proposal for the continuous measurement and the prediction of the desulfurized hot metal temperature. Information about this temperature will provide insight into the course of thermal processes and allow us to predict the temperature before pouring from the ladle into the oxygen converter.

1. Introduction

Hot metal (i.e., pig iron in a liquid state) is loaded from the torpedo car to the ladle after its production in the blast furnace and transportation using the torpedo car. After pouring into the ladle, the desulfurization process occurs to decrease the sulfur content in the hot metal. When the required sulfur content in the hot metal is reached, the hot metal is poured from the ladle into the oxygen converter for further processing (see Figure 1). Hot metal processing in heat aggregates is characterized by high energy consumption. Information about the temperature of hot metal in the ladle is essential for process control and energy savings.

The hot metal produced in the blast furnace is an important steelmaking material. It contains iron, carbon, manganese, silicon, sulfurs, and phosphorus. Sulfur and phosphorus are considered undesirable impurities. The sulfur content negatively affects the surface and internal quality of steel products, increases the steel brittleness at higher temperatures, deteriorates mechanical properties, and lowers steel’s intergranular strength and melting point (Cao and Nastac [1]). Pretreatment of the hot metal by desulfurization is used as a remedial measure. It is about processing hot metal by the desulfurization process before the steelmaking process in the oxygen converter. Hot metal pretreatment by desulfurization is the essential process for low (i.e., sulfur content < 0.01%) and ultra-low-sulfur steel (i.e., sulfur content < 0.005%) production (Zhou et al.) [2]. Nowadays, hot metal desulfurization is realized by desulfurization mixtures injection into torpedo ladles or open ladles, adding desulfurization mixtures into open ladles using the Kanbara reactor impeller stirring system, and desulfurization in a basic oxygen furnace or in the steel ladle. Injection technology for hot metal desulfurization is realized by the deep injection of calcium carbide, lime, magnesium, soda ash, or their mixtures into ladles. Mainly, nitrogen is used as a high-speed transport gas injected through refractory lined lances immersed into the hot metal (Hüsken) [3].

The improvement of the hot metal desulfurization process is currently being carried out using experimental approaches and developing mathematical and simulation models. Brodrick [4] described the desulfurization process of the hot metal iron. It has been described that the injection depth in the ladle construction is important for maximizing the reaction time, and the higher isolation of the wall layers is needed to minimize heat losses to the environment.

Kalling et al. [5] realized the experimental examination of the hot metal desulfurization with lime in a rotary furnace. This method requires maintaining the lime in a fine powdered form for rapid and complete desulfurization. Gurov et al. [6] presented the method of hot metal desulfurization in the ladle based on a bell evaporator used for the controlled entry of magnesium ingots into the hot metal. Shevchenko et al. [7] analyzed the efficiency of the methods for hot metal desulfurization based on magnesium injection. Mahendra et al. [8] described an analysis of the influence of residual mixtures, i.e., lime, aluminum dross, and fluorspar, for the cost reduction of desulfurization. The results showed that the mixture fusion’s point and the formed slag’s viscosity are reduced using CaF₂.

Experimental measurements serve to obtain industrial data, which are used to verify the proposed mathematical models. The aim of the paper described by Ochoterena et al. [9] was a study of setting operational parameters and reaction mechanisms affecting the efficiency of the hot metal desulfurization process. The accuracy of a mass transfer model proposed in this study was verified by comparing the sulfur content between model results and industrial experiments. The lance-injected desulfurization mathematical method for single-hole or double-hole injection was investigated by Ma et al. [10]. The mathematical model includes parameters such as hot metal, atmosphere pressure, and nitrogen.

Barron et al. [11] derived a kinetic model for the desulfurization process realized in the ladle. The results showed that a hot metal temperature belongs to several factors affecting the selection of the desulfurization mixture. Zhao et al. [12] proposed a three-dimensional mathematical model to determine the multiphase flow, motion, and dispersion of desulfurized particles in the hot metal desulfurization process. The dependence between the turbulent energy dissipation rate and the impeller rotation speed was described using the regression formula. Rodriguez et al. [13] described a model that includes thermodynamics, kinetics, and transport processes for the sulfur level measurement during the calcium carbide injection-based desulfurization process. The effect of the reactor shape (i.e., cylindrical and spherical) was investigated. Ashhab [14] described the desulfurization of a hot metal process based on injecting powdered calcium diamide using artificial neural net modeling and an optimization algorithm.

Currently, mathematical models are transformed on computational tools in the form of software products used to, for example, predict and understand material properties, fluid mechanics, etc. Grillo et al. [15] described the study of hot metal desulfurization by CaO–sodalite and CaO–fluorspar mixtures using the software THERMOCALC TCW v.5 whose database used it was Slag3. Computational fluid dynamics (i.e., CFD) simulations were used for the assumption verification of the mono-injection model derived from the continuity equation for hot metal desulfurization by lime powder mono-injection [16] and the study of refractories for a hot metal ladle during desulfurization [17].

A multiphase numerical simulation (i.e., steel–slag–argon) using ANSYS FLUENT to model the heat losses of the hot metal into the ladle wall during the argon injection in the secondary refining process was described by [18]. The results showed that the alumina (i.e., Al₂O₃) refractories keep the hot metal temperature better than magnesia-carbon (i.e., MgO-C) refractories. The liquid steel temperature prediction in the ladle furnace refining process by the random forest method was examined in [19]. The maximum error of the temperature prediction was 8.9 °C.

The literature review shows the desulfurization process research through experimental measurements: mathematical and software modeling, respectively. This research is oriented toward improvements in desulfurization devices’ construction parameters [2,10]; operating temperatures values [4]; desulfurization mixtures’ mass and composition [5,11,15]; magnesium consumption [6]; magnesium injection [7]; desulfurization costs [8]; operational parameters and reaction mechanisms [9,12]; the constructions of models in the mass transfer, kinetics, thermodynamics, or neural net forms [9,11,13,14]; commercial software use [15,16,17]; measurement sensors use [17]; modeling heat losses into the ladle’s wall [18]; and liquid steel temperature prediction [19] in the ladle furnace refining process.

The realized investigations aimed to track the sulfur content in the hot metal, but the hot metal temperature drop was not tracked and modeled during the desulfurization process. The model described in this paper, compared to the referenced studies, integrates theoretical knowledge of the heat transfer, such as heat radiation, convection, conduction, heat accumulation, and heat consumed by chemical reactions, to monitor the temperature change of not only the hot metal but also the thermal state of the ladle. Information about the thermal state of the ladle allows for the determination of heat losses to the ladle walls during the desulfurization process.

The temperature drop in the hot metal transported and processed in aggregates between the blast furnace and the oxygen converter is significant for controlling processes in the steel plant (see Figure 1). Measuring this temperature drop can lead to better control of processes by improving the input parameter values (i.e., dependent on the temperature) of industrial aggregates and thus lower energy consumption, higher quality, and greater productivity. The temperature of the hot metal is not continuously measured during its desulfurization process. It is measured only in two time steps, i.e., in the time before desulfurization, and in the time after desulfurization, when the slag has already been removed. It is too late from the operator’s perspective to determine the optimal setting of temperature-dependent inputs for subsequent operations (e.g., for the steelmaking process). For this reason, this paper focuses on creating a model for predicting the hot metal temperature drop in the ladle during the desulfurization process. The proposed model is based on the heat losses calculation through heat conduction (i.e., polar coordinates for the ladle’s vertical wall and Cartesian coordinates for the ladle’s bottom wall), heat convection, heat radiation, heat accumulation, and the heat calculation released and consumed by chemical reactions. The proposed model is used to the influence analysis of the measured variables and process parameters on the hot metal temperature change in the ladle. The model’s accuracy is verified using data obtained from operational processes.

2. Materials and Methods

The problem of predicting the hot metal temperature drop during the desulfurization process in the ladle is solved by a mathematical model proposal in the form of cooperating thermal processes. The individual parts of the proposed model are implemented in the MATLAB R2023b environment into the simulation model form.

2.1. Ladle, Hot Metal, and Desulfurization Process Description

Two types of layer distribution in the selected ladle (see Figure 2) are considered, the first for the bottom wall in the sequence from the ladle’s bottom wall inner surface and the second for the vertical wall in the sequence from the ladle’s vertical wall inner surface (see Figure 3). The vertical wall includes four layers, and the bottom wall includes three. The bottom ladle’s wall does not contain a layer marked number 3 (see Figure 3).

The thermophysical properties (i.e., density $\rho$, thermal conductivity $\lambda$, and heat capacity c) and dimensions (i.e., thickness d) of the ladle’s individual wall layers are shown in

Table 1. Thermophysical properties and dimensions of the ladle’s wall layers.

		Layer	1	2	3	4
d, mm	vertical wall		150	72	10	8
d, mm	bottom wall		200	201	0	8
$\rho$, kg·m−3			2440	2100	390	7846
		25	-	1.55	0.13	60.2
		250	2.4	1.47	-
		400	2.3	1.5	-
$\lambda$,	t, °C	600	-	-	0.13
W·m−1·K−1		800	2.1	1.57	0.16
		1000	2.1	1.6	0.19
		1200	2	-	-
		1250	-	1.61	-
$c_p$,	$c_p=a+b·t$	a	844	960	969	494
J·kg−1·K−1		b	0.42	0.13	0	0

. The manufacturer provided these values [20,21,22].

The thermophysical properties of the hot metal (i.e., pig iron in a liquid state) before the desulfurization process are obtained from the literature [23,24].

The desulfurization process occurs according to the workflow of the individual operations listed in

Table 2. The desulfurization process individual operations.

Operation	Time (min)
Ejection of crane hooks	0.25
Lifting of the hooks	1
Inserting the mobile stand into the box	1
Taking a sample and temperature measurement	2
Sample analysis	2
Calculation of the mixture amount	0.5
Nozzle lowering	0.25
Dosage of the mixture	12
Lifting the nozzle	0.25
Ladle tilting	1
Ejection of the slag bubbler nozzle	0.25
Slag withdrawal	7
Pulling in the slag bubbler nozzle	0.25
Ladle alignment	0.5
Taking a sample and temperature measurement	2
Sample analysis	0.5
Removing the mobile stand from the box	1
Hooks lowering	1
Pulling in the hooks	0.25
Total time	33

. The duration of the individual operations is approximate.

2.2. Mathematical Model

The hot metal desulfurization process for the sulfur content reduction is carried out in the ladle. This ladle can work in three stages:

• Ladle preheating—before entering the cycle, a ladle is preheated to a required temperature according to the technological process diagram.
• Empty ladle—after preheating or emptying, the ladle waits until it is moved to the torpedo car and filled with hot metal.
• Full ladle—the desulfurization process occurs if the ladle is filled with hot metal. After desulfurizing, the ladle is moved and emptied into the oxygen converter. Subsequently, it continues with step 2 (i.e., if the ladle lining repair is unnecessary) or 1 (i.e., after the ladle lining repair).

The mathematical model proposal is based on ongoing thermal processes during the desulfurization process, i.e., heat conduction, heat convection, heat radiation, heat accumulation, and chemical reactions.

2.2.1. Heat Conduction

It is a heat energy exchange process from high- to low-temperature areas in solids caused by a temperature gradient [25,26]. In the investigated case, it is the heat conduction in the vertical and bottom walls of the ladle.

The Fourier heat conduction differential equation expressed for the one-dimensional heat transfer was used for the heat conduction process in the ladle’s wall. This equation represents temperature T (K) change over time $\tau$ (s) in direction r (m) axis:

$$\begin{array}{c}\hfill \varrho c_p{\displaystyle \frac{\partial T(r,\tau )}{\partial \tau }}={\displaystyle \frac{1}{r^{\gamma }}}{\displaystyle \frac{\partial }{\partial r}}\left(r^{\gamma }\lambda {\displaystyle \frac{\partial T(r,\tau )}{\partial r}}\right),\end{array}$$

(1)

A value of $\gamma$ equal to 0 represents a Cartesian coordinates system that is considered for the solution of heat conduction in the bottom wall (see, Figure 3). A value of $\gamma$ equal to 1 represents a polar coordinates system that is considered for the solution of heat conduction in the vertical wall (see Figure 3).

The initial (i.e., the ladle temperatures at time $\tau =0$) and boundary conditions (i.e., Dirichlet, Neumann, and Robin) has to be determined for the solution of the Fourier heat conduction differential Equation (1) [27]. The Dirichlet boundary condition assumes that the temperature on the ladle’s wall surface determined in time $\tau$ is given (i.e., $T_a\left(\tau \right)$ and $T_b\left(\tau \right)$). The Neumann boundary condition assumes that the heat flow at the ladle’s wall surface determined in time t is given (i.e., $i_a\left(\tau \right)$ and $i_b\left(\tau \right)$). The Robin boundary condition assumes that the surrounding temperature at the ladle’s wall surface determined in time $\tau$ is given (i.e., $T_{sa}\left(\tau \right)$ and $T_{sb}\left(\tau \right)$) [27].

In the case of polar coordinates, variable a represents the coordinate on the ladle’s vertical wall inner surface, and variable b represents the coordinate on the ladle’s vertical wall outer surface. The r axis is oriented from the ladle’s center axis (i.e., the coordinate is equal to 0) to the ladle’s vertical wall inner surface (see Figure 3).

In the case of Cartesian coordinates, variable a represents the coordinate on the ladle’s bottom wall inner surface, and variable b represents the coordinate on the ladle’s bottom wall outer surface. The r axis is oriented from the ladle’s bottom wall inner surface (i.e., the coordinate is to equal 0) to the ladle’s bottom wall outer surface (see Figure 3).

The spatial discretization (i.e., using index i), temporal discretization (i.e., using index k), and the replacement of the derivative with a difference are applied to Equation (1).

An equation for the polar coordinates (i.e., for the ladle’s vertical wall) is in the following form

$$\begin{array}{cc}\hfill T_{i,k+1}& =\left({\displaystyle \frac{a_{r_{i+1,i}}·r_{i+1,i}·\frac{T_{i+1,k}-T_{i,k}}{\Delta r}-a_{r_{i,i-1}}·r_{i,i-1}·\frac{T_{i,k}-T_{i-1,k}}{\Delta r}}{r_i\Delta r}}\right)·\Delta \tau +T_{i,k},\end{array}$$

(2)

where $\Delta r$ (m) is the element size, $\Delta \tau$ (s) is the time step, r (m) is the distance, T (K) is the temperature, and a (m²s⁻¹) is the thermal diffusivity coefficient.

An equation for the Cartesian coordinates (i.e., for the ladle’s bottom wall) is in the following form:

$$\begin{array}{cc}\hfill T_{i,k+1}& =\left({\displaystyle \frac{a_{r_{i+1,i}}·(T_{i+1,k}-T_{i,k})-a_{r_{i,i-1}}·(T_{i,k}-T_{i-1,k})}{\Delta r^2}}\right)·\Delta \tau +T_{i,k}.\end{array}$$

(3)

The thermal diffusivity coefficient is calculated according to the following equation:

$$\begin{array}{c}\hfill a={\displaystyle \frac{\lambda }{\varrho c_p}}.\end{array}$$

(4)

2.2.2. Heat Convection

Convection is the transfer of heat through the flow of a heat-carrying fluid (i.e., liquid or gas) from the body’s surface to the surroundings. In the case of a ladle, free convection is assumed between the hot metal and the surrounding air, the hot metal and the ladle’s wall, and the ladle’s wall and the surrounding air. The heat flow to the wall of the ladle depends on the value of the heat transfer coefficient by convection ($\alpha$, W·m⁻²·K⁻¹), on the size of the surface area (A, m²), and on the difference between the temperature of the hot metal ($T_{HM}$, K) and the temperature of the surrounding air ($T_s$, K) according to the following equation [28,29]:

$$\begin{array}{c}\hfill I_{Q_c}\left(\tau \right)=\alpha A(T_{HM}\left(\tau \right)-T_s).\end{array}$$

(5)

The heat transfer coefficient by convection depends on the shape and size of the walls of the body, the temperature and temperature difference, the physical properties of the fluid, and on the flow character.

2.2.3. Heat Radiation

A body whose temperature is higher than 0 K radiates thermal energy through its surface. Heat transfer by radiation has a different character than heat transfer by conduction and flow. In conduction and flow, it is a molecular transfer of heat energy in solid, liquid, and gaseous substances, dependent on the temperature difference. In the case of radiation, the intensity of the heat flow depends on the power of the temperature, which in the case of radiation from a perfectly black body represents a value of four. Since the heat flux density during radiation is applied in an exponential temperature dependence, radiation begins to be applied more significantly from a temperature of 1000 K, which is also the case of cooling hot metal in the ladle. The heat flow by radiation between two gray bodies with temperatures $T_{HM}$ and $T_s$ with the resulting emissivity ${ϵ}_{12}$ through surface A is given by the equation [30,31,32]

$$\begin{array}{c}\hfill I_{Q_r}\left(\tau \right)={ϵ}_{12}\sigma A(T_{HM}^4\left(\tau \right)-T_s^4),\end{array}$$

(6)

2.2.4. Heat Accumulation

In general, the accumulation process represents a change in an extensive quantity proportional to a change in an intensive quantity per unit of time. In the case of the heat accumulation process, these quantities are heat and temperature, and the process can be described according to the equation

$$\begin{array}{c}\hfill {\displaystyle \frac{dQ\left(\tau \right)}{d\tau }}=C{\displaystyle \frac{dT\left(\tau \right)}{d\tau }},\end{array}$$

(7)

where C is the heat capacity (J·K⁻¹). If we consider heat capacity as a product of mass and specific heat capacity, then relation (7) has the form

$$\begin{array}{c}\hfill {\displaystyle \frac{dQ\left(\tau \right)}{d\tau }}=m{c}_p{\displaystyle \frac{dT\left(\tau \right)}{d\tau }}.\end{array}$$

(8)

In the case of cooling hot metal in a ladle, we consider the weight and specific heat capacity of hot metal.

2.2.5. Heat of Reaction

The heat released during a chemical reaction is expressed as the change in the enthalpy at a constant pressure. The desulfurization process of hot metal is carried out in a ladle to reduce the sulfur content using a desulfurization mixture injection. The used desulfurization mixture consists of magnesium, fluorspar, and lime and is transported by injected nitrogen gas. The following chemical reaction is considered for the chemical heat release in this process:

$$\begin{array}{c}\hfill {\mathrm{Mg}}_{\left(\mathrm{g}\right)}+{\mathrm{S}}_{\left(s\right)}={\mathrm{MgS}}_{\left(\mathrm{s}\right)}+{\Delta }_r{H}^{\circ },\end{array}$$

(9)

where ${\Delta }_r{H}^{\circ }\left(T\right)$ is the reaction enthalpy of a chemical reaction at temperature T, which is calculated according to the equation

$$\begin{array}{c}\hfill {\Delta }_r{H}^{\circ }\left(T\right)={\Delta }_r{H}^{\circ }\left(T_0\right)+{\int }_{T_0}^T{C}_p^{\circ }\left(T\right)dT,\end{array}$$

(10)

where ${\Delta }_r{H}^{\circ }\left(T_0\right)$ is the reaction enthalpy for a specified chemical reaction at temperature $T_0$ = 298.15 K and is calculated according to the formula

$$\begin{array}{c}\hfill {\Delta }_r{H}^{\circ }\left(T_0\right)={\Delta }_r{H}_{\mathrm{MgS}}^{\circ }\left(T_0\right)-{\Delta }_r{H}_{\mathrm{Mg}}^{\circ }\left(T_0\right)-{\Delta }_r{H}_{\mathrm{S}}^{\circ }\left(T_0\right).\end{array}$$

(11)

Molar heat capacity is determined according to

$$\begin{array}{c}\hfill C_p^{\circ }\left(T\right)=R\sum _{i=1}^7\Delta a_i{T}^{i-3},\end{array}$$

(12)

$$\begin{array}{c}\hfill \Delta a_i=a_{i,\mathrm{MgS}}-a_{i,\mathrm{Mg}}-a_{i,\mathrm{MgS}},\end{array}$$

(13)

where $a_{i,j}$ is the ith coefficient of the NASA polynomial for the jth chemical reaction substance [33].

The total heat flow during the desulfurization process consists of the reaction enthalpy of the chemical reaction (9) and the heat required to heat the other components of the desulfurization mixture and the nitrogen gas, i.e.,

$$\begin{array}{c}\hfill I_{Q_d}=-{\displaystyle \frac{{\Delta }_r{H}^{\circ }\left(T\right)}{M{h}_S}}{m}_S+\sum _{l=1}^n{\displaystyle \frac{C_{p,l}^{\circ }\left(T\right)}{M{h}_l}}{m}_l\Delta T,\end{array}$$

(14)

where $m_S$ is the decrease in sulfur during desulfurization (kg·s⁻¹), $m_l$ is the mass flow of the lth component (i.e., substance) of the desulfurization mixture (kg·s⁻¹), $M{h}_l$ is the molar mass of the lth component (i.e., substance) of the desulfurization mixture (kg·kmol⁻¹), $\Delta T$ is the temperature difference between the desulfurization mixture temperature and the hot metal temperature (K), and n is the desulfurization mixture components number.

It is needed to consider the consumed heat of these components in the desulfurization mixture, i.e., CaF₂, P₂O₅, SiO₂, Fe₂O₃, Al₂O₃, CaCO₃ in the fluorspar; CaO, MgO, SiO₂, Fe₂O₃, Al₂O₃, CaCO₃ in the lime; and Mg in the magnesium. Maximal and minimal yields of the desulfurization mixture are determined using the desulfurization process of the hot metal realized at various input (i.e., 0.01%, 0.011%, ⋯) and output (i.e., 0.001%, 0.002%, ⋯) sulfur concentrations.

2.2.6. Model of Hot Metal Cooling

The hot metal cooling in the ladle is based on a synthesis of elementary models in the heat balance form

$$\begin{array}{c}\hfill {\displaystyle \frac{d{Q}_{HM\left(\tau \right)}}{d\tau }}=-I_{Q_w}\left(\tau \right)-I_{Q_c}\left(\tau \right)-I_{Q_r}\left(\tau \right)-I_{Q_d}\left(\tau \right),\end{array}$$

(15)

or

$$\begin{array}{c}\hfill m_{HM}{c}_{HM}{\displaystyle \frac{d{T}_{HM}\left(\tau \right)}{d\tau }}=-I_{Q_w}\left(\tau \right)-I_{Q_c}\left(\tau \right)-I_{Q_r}\left(\tau \right)-I_{Q_d}\left(\tau \right),\end{array}$$

(16)

where $Q_{HM}$ (W) is the accumulated heat in hot metal that is reduced by heat losses into the ladle wall $I_{Q_w}$ (W), heat losses through the ladle opening at convection $I_{Q_c}$ (W) and radiation $I_{Q_r}$ (W) heat transfer, and the heat of the chemical reaction from the desulfurization process $I_{Q_d}$ (W). The hot metal condition is defined by mass $m_{HM}$ (kg), the temperature $T_{HM}$ (K), and the specific heat capacity $c_{HM}$ (J·kg⁻¹·K⁻¹).

Euler’s method is used for the ordinary differential Equation (16) solution. The equation obtained using time discretization and replacing the derivative with a difference is the following form:

$$\begin{array}{c}\hfill T_{HM,k+1}=T_{HM,k}+{\displaystyle \frac{\Delta \tau }{m_{HM}{c}_{HM}}}\left(-I_{Q_w,k}-I_{Q_c,k}-I_{Q_r,k}-I_{Q_d,k}\right).\end{array}$$

(17)

2.2.7. Model of Ladle Preheating

This model assumes that a steady state occurred in the ladle’s wall during the preheating process. Robin’s boundary conditions are used, i.e., the flue gas temperature (i.e., $T_{sa}\left(\tau \right)$) surrounding the ladle’s inner wall surface and the environment temperature (i.e., $T_{sb}\left(\tau \right)$) surrounding the ladle’s outer wall surface.

2.2.8. Model of Ladle Cooling

The empty ladle’s cooling model is based on cooling the ladle’s wall. This model is constructed using the heat conduction model with the following boundary conditions, i.e., Neumann’s condition (i.e., $i_a\left(\tau \right)$—the convection and radiation heat flows) on the ladle’s wall inner surface and Robin’s condition (i.e., $T_{sb}\left(\tau \right)$—the environment temperature) on the ladle’s wall outer surface.

2.3. Implementation

The synthesis process is used to implement the presented thermal and thermochemical mathematical models into the proposed model. The algorithm of the ladle’s desulfurization process (see Figure 4) results from this implementation process. This algorithm has three essential steps:

• Preheating: The ladle is heated to the required temperature after the new ladle enters the process or the ladle’s walls are repaired. The ladle wall’s temperatures are calculated using this step.
• Hot metal cooling: The hot metal temperature and ladle wall’s temperatures are tracked if the hot metal stored in the ladle is desulfurized and transported between the torpedo car and the oxygen converter. The tracked temperatures are calculated in this step.
• Ladle’s walls cooling: The empty ladle cooling (i.e., the ladle’s vertical and bottom walls cooling) is started after realizing the previous steps. The ladle wall’s temperatures are calculated in this step.

The models are interconnected to determine the change in the thermal state of the ladle (i.e., heat accumulation in its walls), namely, the preheating model and the ladle wall cooling model. The thermal state of the ladle determines the heat losses into the ladle walls during cooling the hot metal (i.e., the hot metal cooling’s model). The contact between the hot metal and the ladle’s wall affects the ladle thermal state.

At the beginning of the simulation, a decision must be made whether the ladle is in a state of after preheating repair or in operation. In the case of a new or repaired ladle, the preheating function is used. The ladle wall’s temperatures (i.e., the temperature data history) are loaded if it is the case of the ladle being in operation. The ladle wall’s layers parameters must be set in the mathematical model if the ladle is repaired or new. Heat transfer calculations determine the temperature distribution in the ladle wall’s lining during preheating. The hot metal desulfurization and cooling process is simulated if the ladle is filled with hot metal. The ladle’s wall cooling process is simulated if the ladle is empty.

After preheating the ladle or loading the ladle’s stored temperatures from the previous desulfurization cycle performed in the ladle (i.e., the time of pouring hot metal into the oxygen converter), the process of ladle wall cooling must follow (i.e., the ladle wall cooling process). The time interval for cooling the ladle walls is equal to the period from the end of preheating to the time of pouring the hot metal into the ladle, or from the time when the hot metal is poured into the converter to the time of pouring the hot metal into the ladle. The proposed model computes the temperature drop in the ladle walls for this interval.

After pouring the hot metal charge into the ladle, the cooling process of the hot metal (i.e., the hot metal cooling model is used) follows. This is due to heat losses through the ladle pouring hole (i.e., radiation and convection process) and heat losses to the ladle walls (i.e., convection process). This results in a decrease in the hot metal temperature and an increase in the temperature of the ladle walls. The model calculates the temperature change in the hot metal and the ladle walls for the time period from the hot metal pouring into the ladle to the hot metal pouring into the oxygen converter.

After the hot metal is poured into the oxygen converter, the temperatures in the ladle walls are saved in a file. Subsequently, it is checked whether a repair of the ladle lining is required or the desulfurization process of a new charge of hot metal can follow. If lining repair is needed, the ladle will undergo a preheating process after the repair, during which the temperatures in the ladle walls are calculated by the preheating model. After preheating, or if no repair is needed, the ladle is transferred to the torpedo car, where it waits to pour a new charge (i.e., the ladle wall cooling model is used).

After the simulation, the output data (i.e., temperature values) are displayed in graphs and saved in files.

The MATLAB programming environment is used for the mathematical model implementation (i.e., in the m-functions form), where the connection of individual mathematical models (i.e., listed in the previous chapter) represents the structure of the mathematical model.

3. Results and Discussion

The proposed mathematical model of the hot metal desulfurization process in the ladle was verified using measured variables and the computing simulation with the MATLAB model. Subsequently, the sensitivity analysis of the impact of the measured variables and selected process parameters on the hot metal temperature change was evaluated.

3.1. Input Data

The three ladles created for the desulfurization process of 150 t hot metal were used for the simulations. The ladles parameters and the thermophysical properties of the ladle walls layer’s materials were determined by the manufacturer [20,21,22], and the input values of measured variables were determined using measurements realized in the steel mill.

The measured data used in the simulation process were as follows: hot metal temperature before desulfurization—$T_1$, hot metal temperature after desulfurization—$T_2$, hot metal weight before desulfurization—$m_1$, hot metal weight after desulfurization—$m_2$, desulfurization mixture mass—$m_z$, nitrogen volume—$V_{N_2}$, sulfur concentration in hot metal before desulfurization—$S_0$, sulfur concentration in hot metal after desulfurization—$S_k$, full ladle time—${\tau }_F$, and empty ladle time—${\tau }_E$.

Table 3. Selected parameters of the measured variables.

Variable	Unit	Minimal	Maxima	Average
$T_1$	°C	1298	1419	1359.18
$T_2$	°C	1280	1413	1340.67
$m_1$	t	142.2	162.5	153.28
$m_2$	t	140	160.8	151.58
$m_z$	kg	379	1303	622.87
$V_{N_2}$	dm3	3540	19,923	9985.24
$S_0$	%	0.021	0.126	0.052
$S_k$	%	0.001	0.0037	0.002
${\tau }_E$	min	7.32	116	45.97
${\tau }_F$	min	33.45	133	64.27

shows the measured data’s minimal, maximal, and average values based on the three realized sets of cycles from the three ladles (the 1st ladle—17 cycles, the 2nd ladle—21 cycles, and the 3rd ladle—17 cycles), a sample of a total of 55 cycles. These values for the 1st ladle are shown in

Table 4. Selected parameters of the measured variables, the 1st ladle.

Variable	Unit	Minimal	Maximal	Average
$T_1$	°C	1317	1400	1342.53
$T_2$	°C	1294	1384	1322.53
$m_1$	t	155	162.5	158.04
$m_2$	t	153	160.8	156.51
$m_z$	kg	493	1303	686.35
$V_{N_2}$	dm3	7834	19,923	11,323.35
$S_0$	%	0.04	0.126	0.059
$S_k$	%	0.002	0.003	0.002
${\tau }_E$	min	16	116	54.82
${\tau }_F$	min	40	133	68.41

, those for the 2nd ladle are shown in

Table 5. Selected parameters of the measured variables, the 2nd ladle.

Variable	Unit	Minimal	Maximal	Average
$T_1$	°C	1321	1395	1358.86
$T_2$	°C	1302	1378	1339.19
$m_1$	t	148.7	161.2	152.74
$m_2$	t	147	156.4	151.12
$m_z$	kg	379	923	618.38
$V_{N_2}$	dm3	5793	14,406	10,113.86
$S_0$	%	0.021	0.09	0.051
$S_k$	%	0.001	0.002	0.002
${\tau }_E$	min	7.32	102.33	45.4
${\tau }_F$	min	33.45	113.45	64.98

, and those for the 3rd ladle are shown in

Table 6. Selected parameters of the measured variables, the 3rd ladle.

Variable	Unit	Minimal	Maximal	Average
$T_1$	°C	1298	1419	1376.24
$T_2$	°C	1280	1413	1360.65
$m_1$	t	142.2	158.2	149.2
$m_2$	t	140	156.1	147.22
$m_z$	kg	413	734	564.94
$V_{N_2}$	dm3	3540	13,740	8488.24
$S_0$	%	0.03	0.067	0.048
$S_k$	%	0.0013	0.0037	0.002
${\tau }_E$	min	10.72	109.38	37.81
${\tau }_F$	min	1	10	8.94

.

The figures (see Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9) show the frequent occurrence of the selected variables within 55 cycles, i.e., the hot metal temperature before and after desulfurization, the hot metal weight, the desulfurization mixture weight, the sulfur concentration in hot metal before desulfurization, the sulfur concentration in hot metal after desulfurization, the hot metal temperature difference during desulfurization, the nitrogen volume, the empty ladle time, and the full ladle time.

These figures show that the highest occurrence of the selected variables is in these ranges: 1310–1370 °C for the $T_1$ temperature, 1310–1340 °C for the $T_2$ temperature, 148–156 t for the $m_1$ weight, 510–670 kg for the $m_z$ weight, 0.04–0.06% for the $S_0$ sulfur concentration, 0.0015–0.002% for the $S_k$ sulfur concentration, 18–25 °C for the change in the temperature during the desulfurization process, 9000–11,750 dm³ for the nitrogen volume $V_{N_2}$, 30–55 min for the ${\tau }_E$ ladle time, and 35–60 min for the ${\tau }_F$ ladle time.

The mixture consisting of 20% magnesium, 5% fluorspar, and 75% lime was used for the hot metal desulfurization during experimental measurements. This desulfurization mixture was dosed in the nitrogen gas into the hot metal placed in the ladle.

3.2. Model Verification

The input temperature of the hot metal poured into the ladle was not measured. As a result, during the time interval between pouring the hot metal into the ladle and measuring the temperature $T_1$, it was impossible to determine the hot metal’s temperature drop or the ladle’s thermal state. It was necessary to propose an algorithm to determine the hot metal temperature after pouring it into the ladle because the first temperature measurement (i.e., $T_1$) was realized approximately after three minutes (see Table 2). The determination of the initial temperature of the hot metal was essential for the ladle’s thermal state sets. This ladle’s thermal state was needed for the heat loss into ladle calculation in the subsequent hot metal cooling process. The modified Fibonacci algorithm was used to solve this problem.

3.2.1. Fibonacci Method

In the Fibonacci method, two conditions are assumed. The first condition assumes that the temperature drop of the hot metal cannot exceed 100 °C during the given time interval. The second condition assumes that the temperature of the poured hot metal cannot be lower than its measured temperature $T_1$, after being poured and cooled in the ladle. For this reason, the interval in which we search for the input temperature of the hot metal was set within the range from $T_1$ to $T_1$ + 100 °C. The Fibonacci method identified the initial hot metal temperature within this interval. A set of iterations was performed, differing by the temperature value of the poured hot metal (i.e., determined by the Fibonacci method). These iterations were calculated for the time period between pouring the hot metal into the ladle and measuring the temperature $T_1$. The temperature value of the poured hot metal was selected such that the calculated temperature $T_1$ at the time of measurement showed the smallest difference between the calculated and measured temperature $T_1$. The chosen value of the poured hot metal temperature was then used as input for the simulation of the entire current cycle.

In this method, the number of steps must be specified. The number of steps can be determined by the accuracy ${\epsilon }_F$, the input interval $⟨a,b⟩$, and the Fibonacci sequence of numbers. The eleven values of this sequence are shown in

Table 7. Fibonacci sequence of numbers.

n	0	1	2	3	4	5	6	7	8	9	10	11
$F_n$	1	1	2	3	5	8	13	21	34	55	89	144

.

In the row $F_n$ of Table 7, the first two values are equal to one (i.e., $F_0=F_1=1$), and the remaining values are calculated according to $F_{n+2}=F_{n+1}+F_n$ ($n=0,1,2,\cdots$).

The interval $⟨a,b⟩$ determines the range of values where the initial temperature of the hot metal is located. The Fibonacci method divides this interval into sections (see Figure 10). We assumed that $a=T_1$ and $b=(T_1+100)$ because only cooling of the hot metal in the ladle could have occurred (i.e., maximal cooling by 100 °C in the solved problem). The values $T_{initial,1}$ and $T_{initial,2}$ are calculated according to

$$\begin{array}{c}\hfill T_{initial,1}=a+{\displaystyle \frac{F_n}{F_{n+2}}}·(b-a)\end{array}$$

(18)

$$\begin{array}{c}\hfill T_{initial,2}=a+{\displaystyle \frac{F_{n+1}}{F_{n+2}}}·(b-a)\end{array}$$

(19)

The Fibonacci number $F_{n+2}$ must be calculated for the $T_{initial,1}$ and $T_{initial,2}$ determination

$$\begin{array}{c}\hfill F_{n+2}={\displaystyle \frac{2·(b-a)}{{\epsilon }_F}}\end{array}$$

(20)

After the number $F_{n+2}$ calculation, finding the nearest higher number in column $F_n$ of the Table 7 is necessary. The calculated $F_{n+2}$ number has to be replaced by this higher number. The numbers $F_{n+1}$ and $F_n$ are subsequently determined using values in Table 7. Two simulations by the hot metal cooling model are realized if the $T_{initial,1}$ and $T_{initial,2}$ are calculated. The first is for the $T_{initial,1}$ temperature and the second is for the $T_{initial,2}$ temperature in the time range between the pouring of the hot metal into the ladle and the $T_1$ temperature measurement. The values $T_1\left(T_{initial,1}\right)$ and $T_1\left(T_{initial,2}\right)$ (i.e., the hot metal temperature in the time of the $T_1$ temperature measurement) are determined using these simulations. These values are used for the objective function calculations

$$\begin{array}{c}\hfill F_1=|T_1-T_1\left(T_{initial,1}\right)|\end{array}$$

(21)

$$\begin{array}{c}\hfill F_2=|T_1-T_1\left(T_{initial,2}\right)|\end{array}$$

(22)

The interval $⟨a,b⟩$ is shortened using these values, i.e., $⟨a,b=T_{initial,2}⟩$ if $F_1<F_2$; otherwise, $⟨a=T_{initial,1},b⟩$. Subsequently, $T_{initial,1}$ and $T_{initial,2}$ values have to be calculated with a new value of a or b. The presented procedure of interval shortening is repeated n times, where n is equal to the value from column n at the determined value of $F_n$. After the end of this procedure, the ladle’s thermal state is set using the ladle’s temperature values obtained at a lower value between $F_1$ and $F_2$.

3.2.2. Relative Error and Correlation Coefficient Calculation

The verification of the proposed model was realized by determining the average relative error (i.e., AvgE), the minimal relative error (i.e., MinE), the maximal relative error (i.e., MaxE), and the correlation coefficient, and comparing the measured (i.e., after desulfurization) and modeled (i.e., Tmod) temperatures. The relative error between the measured and modeled hot metal temperature was estimated according to

$$\begin{array}{c}\hfill E_i={\displaystyle \frac{|T_i^{measured}-T_i^{modeled}|}{T_i^{measured}}}·100,\end{array}$$

(23)

$$\begin{array}{c}\hfill \mathrm{AvgE}={\displaystyle \frac{{\sum }_{i=1}^n{E}_i}{n}},\end{array}$$

(24)

$$\begin{array}{c}\hfill \mathrm{MinE}=\underset{i}{min}\left\{E_i\right\},\end{array}$$

(25)

$$\begin{array}{c}\hfill \mathrm{MaxE}=\underset{i}{max}\left\{E_i\right\},\end{array}$$

(26)

where $E_i$ is the relative error for i-$\mathrm{th}$ cycle (%), $T_{measured}$ is the measured hot metal temperature after desulfurization (°C), $T_{modeled}$ is the modeled hot metal temperature after desulfurization (°C), and AvgE, MinE, and MaxE are the average, minimal, and maximal relative errors (%).

The first simulations were carried out for the first ladle. Figure 11a shows the behavior of measured hot metal temperatures before desulfurization, after desulfurization, and after desulfurization from the model (i.e., Tmod) for 17 cycles. From the verification perspective, we are interested in comparing the measured and modeled temperature profiles after the desulfurization process. The average relative error between the model and the measurement is 0.4262%, while the maximum relative error is 0.9141%. The correlation between the measured hot metal temperatures after desulfurization and the modeled hot metal temperatures after desulfurization is shown in Figure 11b, and it is represented by a correlation coefficient of 0.95737. This close correlation can be visually seen in Figure 12a, where the temperature profiles after desulfurization—i.e., both the measured and modeled temperatures—overlap.

Subsequent simulations were carried out for the second ladle (see Figure 12) and the third ladle (see Figure 13). The results confirmed the low value of the average relative error (i.e., 0.4288% and 0.4272%) and the maximum relative error (i.e., 0.855% and 1.3725%), and showed the high value of the correlation of the temperature after desulfurization between the measurement and the model (i.e., 0.97673 and 0.95732).

Data obtained from an operational experiment were used to determine the heat transfer coefficient, while the measured and modeled (i.e., obtained by the proposed model) surface temperatures were compared.

The simulations presented in this chapter were carried out using the MATLAB simulation tool on a PC with the following configuration:

• Processor: 11th Gen Intel(R) Core(TM) i5-11400H @ 2.70GHz 2.69 GHz.
• RAM: 16.0 GB.
• 64-bit operating system, x64 processor type.

Table 8. Durations of cycles for the 1st ladle.

Cycle	Simulation	Relative	Empty Ladle	Full Ladle
	Duration (s)	Error (%)	Time (min)	Time (min)
1	3.7926	0.4696	116	95
2	2.9724	0.3066	76	55
3	2.5231	0.2191	24	59
4	2.5165	0.1541	38	44
5	2.3177	0.2209	16	44
6	2.9268	0.0639	76	49
7	3.2185	0.9141	67	85
8	3.2924	0.1463	23	133
9	3.0518	0.0259	51	85
10	2.761	0.6976	45	62
11	3.3287	0.6071	90	76
12	2.8865	0.7028	48	72
13	2.685	0.6668	20	78
14	2.7151	0.5212	42	58
15	2.4111	0.4732	28	40
16	3.5645	0.4409	115	71
17	2.8391	0.6273	57	57

,

Table 9. Durations of cycles for the 2nd ladle.

Cycle	Simulation	Relative	Empty Ladle	Full Ladle
	Duration (s)	Error (%)	Time (min)	Time (min)
1	2.1755	0.5563	7	45
2	2.3317	0.0468	10	33
3	3.0239	0.4914	44	61
4	2.7815	0.855	37	47
5	3.4225	0.3183	86	62
6	2.823	0.4927	44	50
7	2.9345	0.4078	17	89
8	3.4222	0.7313	102	53
9	2.7008	0.1417	36	49
10	2.7478	0.7285	36	60
11	3.4721	0.4111	90	69
12	3.1291	0.2572	47	79
13	3.5853	0.8097	47	113
14	3.8193	0.1325	94	98
15	2.8083	0.3454	50	49
16	2.5554	0.487	29	44
17	3.2363	0.229	26	108
18	2.8041	0.2115	16	73
19	2.8287	0.5426	40	65
20	2.8584	0.2884	38	67
21	2.9587	0.5203	57	50

and

Table 10. Durations of cycles for the 3th ladle.

Cycle	Simulation	Relative	Empty Ladle	Full Ladle
	Duration (s)	Error (%)	Time (min)	Time (min)
1	2.6978	0.4063	30	52
2	3.0312	1.0818	60	60
3	2.736	0.0612	35	51
4	3.0763	0.5079	62	47
5	2.3839	0.2893	18	41
6	2.4643	0.2471	11	55
7	2.8469	0.3173	43	50
8	3.7985	1.3725	109	75
9	1.3455	0.6352	43	64
10	3.1876	0.2673	37	73
11	3.5539	0.0787	90	59
12	2.5923	0.1517	11	54
13	2.7904	0.1082	16	63
14	2.3076	0.1372	11	89
15	2.6663	0.3727	25	51
16	2.0798	0.5089	15	63
17	2.1872	0.7185	26	59

show the simulation durations for the individual cycles within the time interval that includes the empty ladle time duration (i.e., ladle wall cooling model) and the full ladle time duration (i.e., hot metal cooling model). These tables also show the relative error for each cycle. From the tables, it can be observed that the simulation time ranges from 1.3455 s to 3.8193 s. Despite the complexity of the model and the numerical methods used, the cycle simulation time is in the order of a few seconds. The changes in simulation durations correlate with changes in the actual cycle times. When the duration of the real cycle, including both the empty and full ladle times, is increased, the simulation time also increases. We can conclude that the simulation duration for one cycle is a maximum of approximately 4 s, which is negligible compared to the desulfurization process duration of around 30 min. Given the maximum simulation time of 3.8193 s and a maximum deviation of 1.3725%, we can conclude that the proposed model is suitable for predicting temperatures in the examined ladle with sufficient accuracy.

The operator uses only one type of ladle, whose construction and dimensions correspond to the layer configuration shown in Figure 3 and the values listed in Table 1. The proposed model can be extended to other types of ladles by including their parameters, dimensions, and properties as input to the proposed model.

If a ladle with different dimensions from those described in Table 1 and with different layers from those shown in Figure 3 is used, the following parameters of the ladle need to be adjusted in the proposed model:

• The size of the ladle opening which affects the hot metal temperature drop due to heat losses by radiation and convection to the surroundings.
• The inner and outer diameters of the ladle, as well as the height of the ladle, influence the mass of hot metal charged into it.
• The composition and dimensions of the individual layers of the furnace walls affect the hot metal’s temperature drop due to heat losses by convection to the furnace walls and, subsequently, the temperature distribution in the furnace walls.

The proposed model uses a desulfurization mixture containing magnesium, which reacts with sulfur. If a different desulfurization mixture is used, the chemical reactions and corresponding heat calculations need to be analyzed and adjusted accordingly.

3.3. Sensitivity Analysis

This analysis evaluated the influence of the selected measured variables and the heat transfer and accumulation parameters on the temperature change in the hot metal in the ladle (i.e., the output value of the proposed model) after stabilizing this temperature (i.e., after 20 identical cycles of the same ladle). The temperature behaviors (i.e., time dependent) in the selected places, i.e., in the hot metal, the ladle’s vertical wall working layer, and the ladle’s vertical wall insulating layer, are shown in Figure 14.

The analysis was carried out for the cycle chosen from the cycles carried out on the first ladle. The selected cycle consisted of the following measured values: the hot metal temperature before desulfurization (i.e., 1370 °C), the hot metal temperature after desulfurization (i.e., 1349 °C), the empty ladle time (i.e., 42 min), the full ladle time (i.e., 58 min), the hot metal weight (i.e., 155 t), the desulfurization mixture weight (i.e., 493 kg), the sulfur concentration before desulfurization (i.e., 0.049%), the sulfur concentration after desulfurization (i.e., 0.003%), and the nitrogen volume (i.e., 7920 dm³). The individual measured variables and the heat transfer and accumulation parameters (i.e., model parameters) were changed in the range from −5 to +5%.

The group of measured variables included the data of the following variables: $T_1$, $m_1$, $m_z$, $S_0$, ${\tau }_E$, ${\tau }_F$, and $V_{N_2}$. The impact of the measured variables change (i.e., in the range from −5% to 5%) on the final value of the hot metal temperature (i.e., the end hot metal temperature change $\Delta T_{HM}$) after the simulation of the 20 identical cycles is shown in Figure 15. Given this change, this figure shows that the highly significant impact have variables $T_1$ (i.e., circa 64 °C), and, subsequently, ${\tau }_F$ (i.e., circa 1.5 °C) and $m_1$ (i.e., circa 1.2 °C). The least significant impact has variables $V_{N_2}$ (i.e., circa 0.005 °C), ${\tau }_E$ (i.e., circa 0.37 °C), and $m_z$ (i.e., circa 0.43 °C).

The group of the heat transfer and accumulation parameters includes the following data: the hot metal specific heat capacity $c_{HM}$, the hot metal emissivity ${ϵ}_{HM}$, the air emissivity ${ϵ}_{air}$, the heat transfer coefficient by flow into hot metal ${\alpha }_{HM}$, the heat transfer coefficient by flow into air ${\alpha }_{air}$, and the working layer thermal diffusivity $a_w$. The impact of the heat transfer and accumulation parameters change (i.e., in the range from −5% to 5%) on the final value of the hot metal temperature after the simulation of the 20 identical cycles is shown in Figure 16. Given this change, this figure shows that the highly significant impact has $c_{HM}$ (i.e., circa 2 °C), and subsequently, ${ϵ}_{air}$ (i.e., circa 0.87 °C) and ${ϵ}_{HM}$ (i.e., circa 0.65 °C). The least significant impact has ${\alpha }_{HM}$ (i.e., circa 0.01 °C), $a_w$ (i.e., circa 0.34 °C), and ${\alpha }_{air}$ (i.e., circa 0.41 °C).

The impact of individual changes in measured variables (i.e., the temperature of hot metal by 10 °C, hot metal weight by 10 t, desulfurization mixture weight by 10 kg, empty ladle time by 1 min, full ladle time by 1 min, initial sulfur concentration by 0.01%, and volume of nitrogen by 1 m³) is shown in Figure 17a. The impact of individual changes in the heat transfer and accumulation parameters (i.e., hot metal specific heat capacity by 100 J·kg⁻¹·K⁻¹, hot metal emissivity by 0.1, air emissivity by 0.1, heat transfer coefficient into the hot metal by 100 W·m⁻²·K⁻¹, heat transfer coefficient into the air by 1 W·m⁻²·K⁻¹, and working layer thermal diffusivity by 10⁻⁷ m²·s⁻¹) is shown in Figure 17b.

Figure 17a shows that the highly significant impact on the end value of the hot metal temperature among the measured variables has the input hot metal temperature and, subsequently, the initial sulfur concentration (i.e., before desulfurization). A change in the initial sulfur concentration of hot metal by 0.01% corresponds to a change in the resulting temperature of hot metal by 2.32 °C. This is a significant influence (i.e., depending on the accuracy of the measuring devices), so the lowest possible uncertainty of the sulfur concentration measurement is required.

An increase in the initial hot metal temperature by 10 °C leads to an increase in the final hot metal temperature by 9.27 °C; similarly, a change in the weight of hot metal by 10 t causes a change in the temperature by 1.46 °C, and an increase in the time of a full ladle by 1 min causes a decrease in temperature by 0.488 °C. If the time of the full ladle is increased, then heat losses to the ambient and ladle wall are increased, and thus the end hot metal temperature is decreased.

Figure 17b shows that the hot metal specific heat capacity has a highly significant impact among the heat transfer and accumulation parameters and, subsequently, the air emissivity and the hot metal emissivity that affect heat losses by radiation to the surrounding.

The sensitivity analysis shows that the most significant factors influencing the model’s accuracy and reliability from the measured data are the input temperature of the hot metal, sulfur concentration, and the hot metal mass. Operators must ensure accurate operational measurements of these inputs using appropriate devices tailored to the specific operational needs.

Among the heat transfer and accumulation parameters, the most significant factors influencing the model’s accuracy are the heat capacity and emissivity of the hot metal and the emissivity of the surrounding air. The heat capacity of the hot metal is a crucial parameter for heat accumulation. The desulfurization process of the hot metal occurs at temperatures much higher than 1000 K, and at these values, the radiative heat transfer plays a significant role. The heat transfer by radiation depends on the emissivity of the environment, and this fact explains the substantial impact of emissivity on the final temperature of the hot metal.

4. Conclusions

A mathematical and simulation model for hot metal temperature prediction in the ladle is described in this article. The proposed approach’s novelty is based on applying thermal and thermochemical mathematical models to modeling the hot metal temperature drop during the desulfurization process in the ladle. These models were realized and verified by simulations on three ladles.

The hot metal temperature drop when the hot metal is placed in the ladle, as well as the change in the ladle’s wall temperature when the ladle is empty, were determined by this model. The accuracy was evaluated by comparing the measured and modeled hot metal temperatures (see Figure 11, Figure 12 and Figure 13). Subsequently, the correlation coefficient, and the average, maximal, and minimal relative errors were calculated. The relative error values (i.e., maximal average error 0.43%) and the correlation coefficient values (i.e., minimal coefficient 0.95732) showed a strong correlation of the measured and modeled temperatures.

After verification of the proposed model, the sensitive analysis was performed. This analysis evaluated the impact of the measured variables and the heat transfer and accumulation parameters on the hot metal temperature change in the ladle. Among the measured variables, the input hot metal temperature, the hot metal weight, and the initial sulfur concentration had the most significant impact on the end hot metal temperature. The measurement of these variables needs to be refined and stabilized because of the reduction in measurement uncertainty. Among the heat transfer and accumulation parameters, the hot metal specific heat capacity, the air emissivity, and the hot metal emissivity had the most significant impact on the end hot metal temperature. The obtaining of the measured data and the determination of the heat transfer and accumulation parameters require high precision to enhance the model’s reliability.

The proposed model enables determining the ladle’s thermal state and assessing whether the ladle preheating is necessary. A low thermal state of the ladle can lead to a significant drop in the hot metal temperature during the desulfurization process due to increased heat losses into the ladle’s wall. The temperature of the hot metal at the outlet of the ladle is equal to the temperature of the hot metal poured into the oxygen converter. This temperature affects the steelmaking process within the oxygen converter. This includes the progression of chemical reactions and the control of the scrap charge mass concerning its impact on the temperature drop. In steelmaking, alongside monitoring the carbon content in steel, the end hot metal temperature is critical. The low temperatures can obstruct subsequent operations such as transportation, casting, and rolling. For this reason, continuous measurement of the hot metal temperature drop in the ladle and determining the ladle’s thermal state are highly significant.