Mathematical model of blast furnace hearth condition based on data from thermocouples in refrigerator belts

Introduction

Effective control of hearth lining burnout in a blast furnace (BF) is an integral part of the comprehensive blast furnace process control system aimed at achieving high technical and economic performance.

There are several methods for monitoring hearth lining burnout in blast furnaces:

– application of systems analysis principles and solving direct and inverse problems of blast furnace heat performance control [1 – 3];

– systematic diagnostics of hearth lining condition using the reflected acoustic-ultrasonic signal (AU-E) method developed by the company [4];

– the method of group argument accounting, where the silicon content in hot metal is taken as the primary parameter for assessing the thermal state of the BF hearth [5];

– determination of the average heat flux of cooling water in the refrigerators [6; 7];

– analysis of the relationship between hearth lining design and lining wear [8 – 10].

One of the most reliable methods involves the installation of special thermocouples at specific points and in a defined number during a major overhaul. This subsequently allows for the monitoring of the thickness of the lining layers and assessment of the temperature gradient within the masonry [11 – 14]. The remaining thickness of the hearth and bottom lining, along with their two- and three-dimensional visualization, is calculated using both commercial software packages (e.g., Matlab [15]) and proprietary software solutions [16 – 18]. Among the available approaches, the finite element method is considered the most illustrative for modeling heat transfer [19; 20].

Monitoring the condition of the refractory hearth lining using a mathematical model [21] is feasible after the furnace blow-in and stabilization of its thermal regime. The mathematical model presented in this paper can be applied at any time after the overhaul of the first discharge category. If a significant amount of time has passed since the furnace was blown in, burnout or skull formation may occur in the hearth lining. In this case, it becomes necessary to use the results of ultrasonic control (USC) of the lower part of the blast furnace.

Input data

Fig. 1 shows a vertical section of the hearth of a 2000 m³ blast furnace. It can be seen that the first refrigerator belt covers most of the bottom, while the second and third belts are located in the hearth crucible zone.

Thermocouples are installed in front of the three refrigerator belts in the hearth and bottom zones (20 thermocouples per belt). The data from these thermocouples, stored in the blast furnace automated process control system (APCS) database, can be used to assess hearth burnout. The data sampling interval may be as short as one minute. The concept of reference thermocouple temperatures (RTT) is introduced – these are the thermocouple readings taken at the moment of ultrasonic control (USC) for hearth burnout. The USC results provide information on the thickness of all refractory lining layers at four vertical levels. According to the results of the USC, wear of the ceramic blocks and carbon blocks, as well as the presence of skull formation, is observed in the hearth belt. The horizontal sections were taken at elevations of 6.5, 7.3, 7.9, and 9.9 m.

Fig. 2 presents a horizontal section of the hearth referenced to the tuyeres and tapholes. The origin of the cylindrical coordinate system is located at the center of the bottom at elevation 0. The zero-angle reference is aligned with the axis of tuyere 1 (Fig. 2), followed clockwise with equal angular intervals. In this system, the height remains constant and corresponds to the elevation of the slicing plane.

Thus, four slicing planes define the geometry of the skull formation. Using CAD software, a highly accurate 3D model of the hearth skull can be generated.

The computational domain includes the sidewall zone of the hearth and the bottom zone, where heat transfer is described by the heat conduction equation for solid materials. The initial conditions are the known data from the latest USC (reference temperatures). The internal boundary is the contact surface between the molten metal and the lining (skull). The temperature of the inner surface of the lining is assumed to be equal to the current temperature of the hot metal, measured by thermocouples installed in the blast furnace taphole. If such thermocouples are absent, the temperature is assigned by the operator (e.g., 1450 °C). The external boundary condition is defined by the thermocouple readings in the refrigerator belts. It is assumed that the temperature of the cooling agent in the refrigerators remains constant. For simplification, an ideal thermal contact between lining layers is assumed. In the interface zones, the thermophysical properties of the materials are averaged.

Heat transfer through a multilayer cylindrical wall

The linear heat flux density through a cylindrical wall was calculated using formulas (1) – (3)

where T₁ and T₂ are the temperatures of the hot and cold fluids, °C (K); α₁ and α₂ are the heat transfer coefficients from the hot fluid to the wall and from the wall to the cold fluid, W/(m²·K); d₁ and d₂ are the inner and outer diameters of the cylindrical wall, m; λ is the thermal conductivity of the wall material, W/(m·K); k_l is the linear heat transfer coefficient through the cylindrical wall, W/(m·K); R₁ is the linear thermal resistance of the cylindrical wall, (m·K)/W.

When heat is transferred through a cylindrical wall composed of n layers with different thicknesses and thermal properties, the linear heat flux density is calculated using the following formula (4)

where λ_i is the thermal conductivity of the i-th layer, W/(m·K); d_i and d_i + 1 are the inner and outer diameters of the i-th layer, m.

In this case, the linear thermal resistance of the wall is calculated as

Heat transfer through the five-layer wall was modeled at four specified heights, with known layer thicknesses (e.g., in the region covered by the second refrigerator belt, the layers are shown in Fig. 3).

Due to the degradation of the ceramic cup and insulating material, their actual thicknesses in the temperature control sector (TCS) differ from the design values. The height of the TCS corresponds to the height of the refrigerator. In each horizontal section along the TCS, the temperature of the refractory lining materials varies from point to point, resulting in different degrees of lining erosion (or skull thickness). Each TCS is subdivided into equal-height segments. For each sectional plane, the thickness of all refractory layers is determined, and the temperature distribution across the plane is calculated. Based on the known temperature, the behavior of each layer under increasing or decreasing overall temperature is proportionally estimated.

Let us consider the cross-section at a height of 6.5 m (Fig. 4). The thickness of the skull varies in the radial direction (from S_(2–4) to S_(1–3)). The coordinates of points T₁ and Т₂ shift depending on the following conditions:

– if the skull thickness increases, T₁ and Т₂ move toward the center of the furnace, and the temperature in the sector decreases;

– if the skull thickness decreases, T₁ and Т₂ shift toward T₃ and Т₄, respectively, resulting in a temperature increase.

Based on the thermocouple temperatures recorded on the day of the ultrasonic control (USC), reference temperature distributions within the hearth masonry can be determined for all levels.

The thermal conductivity of the skull formation depends on various factors, including its density [22]. In the present calculations, the value of λ_s = 0.36 W/(m·°C) is assumed.

The Table above presents the input data used for the heat transfer calculations.

Heat conduction through a multilayer planar wall

The bottom of the analyzed blast furnace is composed of seven tightly bonded layers. In the area covered by the first refrigerator belt, the layers (from the inner to the outer surface) are arranged as follows: skull, ceramic layer, super-microporous carbon blocks, carbon blocks, graphite blocks, refractory packing mass, and an air-cooled zone at the bottom, where thermocouples are installed).

The first refrigerator belt is divided by 20 cylindrical cutting planes (Fig. 5, a). The boundary points of skull formation in two refrigerator belts are shown in Fig. 5, b.

Heat transfer through the planar wall is calculated using the surface heat flux density, which is related to the total heat flux by the equation q = Q/F, where F is the surface area of heat exchange.

For a wall composed of n layers, the formula for calculating heat transfer through the planar wall is given by

where δ_i and λ_i are the thickness and thermal conductivity of the i-th layer of the wall, and R_t is the overall thermal resistance of the multilayer wall, (m²·K)/W.

Then

where ΔT_i is the temperature drop across the i-th heat transfer section, °C (K); R_ti is the thermal resistance of the i-th heat transfer section, (m²·K)/W; k is the number of heat transfer sections.

Visualization of the BF hearth and bottom

Two- and three-dimensional models of the blast furnace hearth are visualized using the Visualization Toolkit (VTK) – an open-source C++ library for modeling, image processing, and applied visualization.

Three-dimensional objects are defined by sets of vertices and faces connecting those vertices. Since these models represent quasi-solids of revolution, the vertex coordinates are specified in a cylindrical coordinate system. The origin of this system is located on an imaginary vertical axis passing through the center of the furnace, starting at elevation 0. However, VTK requires vertex coordinates to be provided in Cartesian coordinates X, Y, Z. Coordinate transformation is performed using the following formulas: X = R cos(A); Y = R sin(A); Z = H.

The coordinates of the vertices on the outer surface of the furnace are constant and defined at the model design stage. The inner surface vertex coordinates vary depending on the lining thickness calculated by the hearth burnout mathematical model. The color rendering of the masonry is displayed as a color gradient corresponding to the lining temperature (Fig. 6).

To enhance clarity in the 3D representation of the hearth, any sector can be cut out. This allows for visualization of the internal condition of the lining, with azimuthal section planes freely defined, enabling comprehensive inspection of the entire masonry volume. The cross-section ends are not generated directly by VTK during object slicing; instead, they represent a separate, specially calculated set of points. As a result, each visualization is constructed from four surfaces: the outer surface (constant), the inner surface (model-calculated), and two azimuthal cross-sections (model-calculated).

The accuracy of predicting the remaining thickness of the blast furnace hearth lining is determined by comparing the model results with the findings obtained from ultrasonic control (USC) inspections.

Conclusions

The application of the developed mathematical model enables optimization of the blast furnace smelting process by providing real-time control of the hearth and bottom lining thickness through visual representation. In turn, this opens the way to significant cost reduction through timely decision-making regarding process parameters.