Mathematical model of the blast furnace main trough lining condition

Introduction

The blast furnace runner system usually includes a main trough and pouring troughs, with the main trough serving as the location where molten cast iron is separated from slag. Consequently, it operates under the most severe conditions. For this reason, the improvement of its lining receives the greatest attention from both refractory manufacturers and end users.

Available research on heat transfer in the lining of the blast furnace main trough is far more limited compared to studies on the furnace’s internal lining. However, in recent years, information from international publications has made it possible to assess the efforts made to study this issue.

In scientific studies addressing molten cast iron – slag separation in blast furnace main troughs, a wide range of methods has been employed, combining experimental and numerical approaches [1 – 3]. Experimental investigations, such as those reported in [4; 5], relied on physical modeling in which analog fluids (e.g., oil and water to simulate cast iron and slag) were used to examine the influence of trough geometry (inclination angle, cross sectional shape) and tapping velocity on phase separation efficiency. In [5], in particular, a 1:10 scale model was utilized to validate the results of numerical simulations.

Numerical modeling plays a central role in analyzing the complex heat and mass transfer processes in the trough [6 – 9]. Researchers have adopted different numerical approaches, including the finite volume method [7] and the finite element method [8 – 11]), solving the Navier–Stokes equations to describe flow hydrodynamics and the heat conduction equation (with allowance for radiation [11 – 13]) to simulate the temperature field. Key factors influencing separation efficiency and lining service life have been considered, such as flow turbulence [7; 9], heat transfer between the melt and the refractory [14 – 17], thermal radiation [10; 12; 13], and refractory wear [10]. For example, in [7] a k–ε turbulence model was applied, while in [10; 13] nonlocal boundary conditions were introduced to account for thermal radiation. In [8], the authors focused on identifying critical isotherms to extend lining service life, employing a two dimensional heat transfer model and comparing simulation results with experimental data. In [10], it was demonstrated that an adaptive time step regulator could be developed to improve the efficiency of long cycle blast furnace simulations.

In most cases, numerical modeling results showed good agreement with experimental data, making it possible to identify the regions of maximum temperature and stress in the lining [14 – 17] – most frequently in the sidewalls – and to predict its wear. Nevertheless, some uncertainties remain, particularly concerning the precise placement of thermocouples for temperature measurement in an operating trough [18 – 20], which necessitates the use of additional data processing techniques (e.g., the GRSA hybrid algorithm [10]) to refine the results. Overall, the combination of experimental and numerical methods has yielded a more comprehensive understanding of the complex processes occurring in blast furnace main troughs and has provided a foundation for developing recommendations to optimize their design and operation.

Input data

This study is devoted to the development of a mathematical model of the refractory lining of the blast furnace main trough, carried out at the Institute of Metallurgy of the Ural Branch of the Russian Academy of Sciences. An algorithm is presented for calculating the temperature field in the refractory lining of the trough based on thermocouple readings obtained from the outer surface of the trough’s metal casing.

Fig. 1 shows a general (schematic) top view of two blast furnace main troughs.

The cross section of the refractory lining of the blast furnace main trough is shown in Fig. 2.

Calculation of heat conduction through a multilayer flat wall

The solution of this problem reduces to steady state heat conduction in a multilayer flat wall, each layer of which is homogeneous. It is assumed that the total thickness of the multilayer wall, equal to the sum of the thicknesses of the individual layers, is much smaller than the wall’s height and width. In this case, the isothermal surfaces are planes parallel to the boundary planes, including the planes of layer interfaces. The individual layers of the wall are assumed to have smooth boundary surfaces that fit tightly together, so that the temperatures of the contacting surfaces are equal (Fig. 3).

When considering heat conduction in a single layer wall, it is observed that the heat flux density does not change when moving from one isothermal surface to another along the x axis, i.e., from left to right.

The plane of the interface between the first and second layers likewise represents an isothermal surface with the same value of heat flux density as in the first layer. However, this plane serves as the “initial” surface for the second layer, in which a constant heat flux density q, equal to that in the first layer, is also established across the thickness δ₂. The same reasoning applies to all subsequent layers (δ₃, etc.).

The total heat flux, and hence its density, does not vary across the thickness of the multilayer flat wall (Q ≠ f (x) and q ≠ f (x)). Therefore, for any i-th layer of the multilayer flat wall, the following relation holds

This relation can be expressed sequentially for all layers, beginning with the first:

We then transform the obtained expressions into \(q\frac{1}{{{\alpha _1}}} = {T_{{f_1}}} - {T_2}\) and sum them (combining left-hand sides with left-hand sides and right-hand sides with right-hand sides):

\[q\left( {\frac{1}{{{\alpha _1}}} + \frac{{{\delta _1}}}{{{\lambda _1}}} + \frac{{{\delta _2}}}{{{\lambda _2}}} + \frac{{{\delta _3}}}{{{\lambda _3}}} + \frac{1}{{{\alpha _2}}}} \right) = {T_{{f_1}}} - {T_{{f_2}}}.\]

These derivations remain valid for an arbitrary number of layers. Thus, in the general case, the expression for the surface heat flux density (q_s) is written as:

where α₁ and α₂ are the heat transfer coefficients from the hot fluid to the wall and from the wall to the cold fluid, respectively, W/(m²·K); λ is the thermal conductivity of the material, W/(m·K).

Fig. 4 shows a scheme of heat transfer in the blast furnace main trough.

Temperature monitoring is performed using thermocouples installed on the casing of the main trough. The lining of the main trough consists of three refractory layers and is enclosed by a steel casing; therefore, in this case, the expression for the surface heat flux density takes the form:

where \({T_{{f_1}}}\) is the temperature of the hot fluid (cast iron, slag), K; Т₅ is the casing temperature, measured by a thermocouple (heat removal can be recorded with a sampling interval from 10 s to 24 h), K; α₁ is the heat transfer coefficient from the hot fluid to the inner wall of the trough, W/(m²·K); δ₁, δ₂, δ₃, δ₄ are the thicknesses of the refractory layers from the innermost to the outermost, m; λ₁, λ₂, λ₃, λ₄ are the thermal conductivities of the refractory materials from the innermost to the outermost, W/(m·K).

In the interfacial regions, the thermophysical properties of the materials are averaged.

Block diagram of the calculation algorithm

A block diagram of the algorithm for calculating the temperature variation across the lining layers is shown in Fig. 5.

Conclusions

The mathematical model developed for the blast furnace main trough lining, based on the solution of the steady state heat conduction problem in a multilayer wall, allows for an efficient evaluation of the thermal load on each refractory layer and the casing in real time (according to the configured heat removal sampling interval). This ensures continuous monitoring of the lining condition, enables prediction of its residual service life (specifically, the thickness of the inner layer), and supports timely decision making regarding repair or replacement. Such capabilities directly enhance the efficiency and safety of blast furnace operation, reduce the risk of emergency situations, and extend the service life of the lining. Future research will aim to further refine the model, for instance by incorporating transient thermal processes (introducing time dependence into the calculations) and by accounting for more complex geometric configurations of the trough.