Numerical Analysis on Erosion and Optimization of a Blast Furnace Main Trough

The main trough of a blast furnace (BF) is a main passage for hot metal and molten slag transportation from the taphole to the torpedo and the slag handling. Its appropriate working status and controlled erosion ensure a safe, stable, high-efficiency and low-cost continuous production of hot metal. In this work, the tapping process of a main trough of a BF in the east of China was numerically studied with the help of a CFD library written in C++, called OpenFOAM, based on the use of the Finite Volume Method (FVM). The results show that turbulence intensity downstream of the hot metal impact position becomes weaker and the turbulence area becomes larger in the main trough. During the tapping, thermal stress of wall refractory reaches the maximum value of 1.7 × 107 Pa at the 4 m position in the main trough. Furthermore, baffles in the main trough placed between 5.8 m and 6.2 m were found to control and reduce the impact of the turbulence on the refractory life. The metal flowrate upstream of the baffles can be decreased by 6%, and the flow velocity on the upper sidewall and bottom wall decrease by 9% and 7%, respectively, compared with the base model. By using baffles, the minimum fatigue life of the refractory in the main trough increases by 15 tappings compared with the base model, so the period between the maintenance stops can be prolonged by about 2 days.


Introduction
The main trough of a blast furnace (BF) is the only runner for hot metal and slag leaving the taphole and reaching the torpedo and slag handling unit, and an important place to separate slag from hot metal and desulfurize hot metal as well. The production rate and efficiency of the BF, hot metal quality and cost are affected directly by its working condition. With the large scale and long life length of the BF, the working intensity of the taphole is increasing [1]. About 4-7.5 t/min of hot metal and molten slag at 1730-1800 K [2] pass through the main trough. Tapping lasts for 70-120 min and there are typically 10-14 taps per day. In high-intensity tapping operation, the main trough consumes most refractory materials in the BF system [3,4]. At the end of a BF's campaign, the accumulated cost of lining and repairing of the trough system exceeds the sum of new lining and all auxiliary equipment costs of the BF. Therefore, studying the working status of the main trough and clarifying the erosion mechanisms of the refractory can help prolong the life of the main trough lining and save costs in hot metal production.
At present, the primary research methods of the main trough can be classified as hydraulic model experiments, industrial tests and numerical calculations, while the main research has been focused on tapping parameters, properties and structure of the main trough The governing equations solved in the hot metal include the continuity equation, momentum equation and energy equation. Assuming hot metal to be an incompressible Newtonian fluid, its volume expansion rate equals zero [14] so where x i is the three-dimensional coordinate of a point (m) (where x 1 , x 2 and x 3 represent the three orthogonal directions) and u i is the velocity corresponding to the coordinate point (m·s −1 ).

Momentum Conservation
Substituting the constitutive equation of a Newtonian fluid into the dynamic equation, the momentum conservation equation of the incompressible Newtonian fluid is obtained [14] as where p is the pressure (kg·m −1 ·s −2 ), t is the time (s), ρ expresses the density (kg·m −3 ), and ν is the dynamic viscosity of the fluid (m 2 ·s −1 ). The standard k-ε turbulence model has few empirical parameters, a simple form of calculation, usually gives accurate results and has a wide application range. Therefore, turbulence is calculated by the standard k-ε turbulence equations

Energy Conservation
The energy conservation is expressed by [14]: where K f is the thermal conductivity of hot metal (W·m −1 ·K −1 ), and C p f is the specific heat capacity of hot metal (J·kg −1 ·K −1 ). For a further study on the mechanical erosion, wall shear stress is introduced to describe the erosion caused by hot metal on the wall. According to Newton's law of viscosity [11,15,16], the wall shear stress of turbulence is where → u → n expresses the velocity gradient perpendicular to the wall, µ and ρ define the viscosity and density of hot metal, respectively, and µ t is the turbulent viscosity.

Governing Equations of Refractory 2.2.1. Fourier Equation
The heat conduction in the working layer of the refractories is described by the Fourier equation [17] ∂T f ∂t where K s and C ps define the thermal conductivity and specific heat capacity of the refractory material of the working layer, respectively (W·m −1 ·K −1 ) and (J·kg −1 ·K −1 ). The equations are solved by OpenFOAM with a displacement method. However, the displacement nodes are considered unknown, by which stress, strain and node force are expressed. The theory of thermal elasticity is similar to the elasticity theory, which establishes the basic equation of the elastomer problem with three aspects of dynamics, physics and geometry [18]. The following are thermal stress calculation equations of the main trough applied in this study.

Momentum balance
The momentum balance is given by where D is the displacement vector (m), ρ s is the density (kg·m −3 ) and σ is the stress tensor (Pa) of the refractory material, respectively. σ has nine components, where σ x , σ y , σ z express the normal stress (Pa) in the three directions in rectangular coordinates, τ xy , τ yx , τ yz , τ zy , τ xz and τ zx define the shear stress in six directions (Pa) and ∂ 2 (ρD) ∂t 2 denotes the volume force.

Strain Displacement
The strain displacement is expressed by where ε with three normal strains and six shear strains is the strain tensor (m), which is the same as the stress tensor.

Elasticity
The elasticity is described by where I is the unit tensor, G denotes the modulus of elasticity, and K s expresses the thermal conductivity of the refractory material. Combining Equations (10) and (11) with Equation (12), the governing equation of the elastic body is expressed by the displacement.
Adding the influence of temperature on the displacement to the elastic body described by Equation (10), the stress equation affected by temperature and the displacement is obtained from where a is the thermal expansion coefficient of the refractory material.
In order to improve the convergence, the equation is written in a new form

Conjugated Heat Transfer
Refractory materials and hot metal exchange energy, and the increased in heat in one phase is equal to the decrease in heat in the other phase. Therefore, the heat transfer formula of the solid-liquid interface is where → n represents the normal direction on the solid-liquid interface.

Mathematical Model
The geometry of the mathematical model in the simulations is the same as that of the trough of No. 2 BF in Lai Steel, as shown in Figure 1 [19]. During tapping, hot metal flows out of the taphole with an angle of 10 • , and then falls into the main trough. The height of hot metal from the upper surface of the main trough is 300 mm. The dropping position of hot metal is defined as the inlet of the main trough (the initial position is 4 m away from the taphole). As the tapping progresses, the speed of inlet metal flow gradually decreases, the inlet position moves toward the taphole as the tapping progresses. The refractory of working layer is the most severely eroded part of the main trough; therefore, only this part is studied in this paper, and the thickness of the refractory material on both sides is 550 mm. Based on previous studies [6,10] and plant data, the physical properties of the fluid and the refractory in the study are as presented in Table 1. The main component of the refractory castable is Al 2 O 3 -SiC-C and its thickness is shown in Figure 1b. inlet position moves toward the taphole as the tapping progresses. The refractory of working layer is the most severely eroded part of the main trough; therefore, only this part is studied in this paper, and the thickness of the refractory material on both sides is 550 mm. Based on previous studies [6,10] and plant data, the physical properties of the fluid and the refractory in the study are as presented in Table 1. The main component of the refractory castable is Al2O3-SiC-C and its thickness is shown in Figure 1b.
16.5 Specific heat capacity of hot metal (J × kg −1 × K −1 ) 850 Hot metal production rate (t × d −1 ) 3400 Density of the refractory (kg × m −3 ) 2850 Thermal conductivity of the refractory (W × m −1 × K −1 ) 2.6 Specific heat capacity of the refractory (J × kg −1 × K −1 ) 750 The computational grid of the main trough model used in this article, as shown in Figure 2, has been proven to give mesh-independent results for an average mesh size of 20 mm or smaller. Therefore, the model with 20 mm grids was used in the simulations. This model contains 6433548 hexahedral cells. The mesh size was maintained as 20 mm for the different cases even though the trough geometry was changed in some of the cases studied.  2850 Thermal conductivity of the refractory (W × m −1 × K −1 ) 2.6 Specific heat capacity of the refractory (J × kg −1 × K −1 ) 750 The computational grid of the main trough model used in this article, as shown in Figure 2, has been proven to give mesh-independent results for an average mesh size of 20 mm or smaller. Therefore, the model with 20 mm grids was used in the simulations. This model contains 6433548 hexahedral cells. The mesh size was maintained as 20 mm for the different cases even though the trough geometry was changed in some of the cases studied.

Assumptions
The following assumptions are made to simplify the model: (1) Hot metal is an incompressible Newtonian fluids; (2) Thermophysical parameters are constant; (3) The Marangoni effect and chemical reactions are ignorable; (4) Only hot metal is considered in the main trough; (5) The main trough lining is intact and experiences no corrosion.

Assumptions
The following assumptions are made to simplify the model: (1) Hot metal is an incompressible Newtonian fluids; (2) Thermophysical parameters are constant; (3) The Marangoni effect and chemical reactions are ignorable; (4) Only hot metal is considered in the main trough; (5) The main trough lining is intact and experiences no corrosion.

Inlet Boundary Conditions
The injection process of hot metal from the taphole to the dropping position is regarded as an oblique throwing motion [5] described by where α denotes the taphole angle, u 0 is the metal flow velocity at the taphole, while x and y are the coordinates of the dropping position of hot metal.
The tapping parameters at the reference plant are given in Table 2 and Figure 3. The taphole angle, the location of the taphole, the falling height of the hot metal and the position of the farthest drop point are used in Equation (17), and the maximum tapping speed is calculated. Combining the maximum tapping speed at the mouth with the Equation (17), the hot metal velocity at the entrance to the upper surface of the hot metal is obtained. As the flow rate at the inlet of the falling hot metal is equal to the one at the taphole, the entrance size is realized. The inlet temperature is 1773 K and the pressure gradient is zero. metal velocity at the entrance to the upper surface of the hot metal is obtained. As the flow rate at the inlet of the falling hot metal is equal to the one at the taphole, the entrance size is realized. The inlet temperature is 1773 K and the pressure gradient is zero.

Outlet Boundary Conditions
An outlet pressure of p = 100, 000 Pa (1 atm) is adopted as the boundary condition. Velocity, temperature, turbulent kinetic energy and turbulent energy dissipation rate are all set to have no gradients.

Wall Boundary Conditions
The wall surface of the refractory material satisfies the non-slip condition, so the wall surface applies the wall function boundary conditions to define the flow near the wall surface. The energy transfer between the refractory material and the main trough is described by a conjugate heat transfer expression [20]. The heat transfer coefficient of natural convection between the upper surface of the main trough and the air is 10 W × (K·m 2 ) −1 [21]. The initial temperature of the hot metal in the main trough is 1723 K. The initial temperature of the ramming material of the main trough working layer and the outer refractory material is set to 1273 K.

Internal Field Boundary Conditions
The initial value is the same as the initial value at the entrance and the pressure gradient is set to zero.

Time Step and Sub-Iterations
The time step size of the calculation is set to 0.001 s. In order to ensure the achievement of steady state, the number of sub-iterations was set to 10,000. Tapping time of No. 2 BF in Lai Steel is 90 min. A 90-min simulation may lead to a waste of computing resources, hence the time of hot metal tapping in the model was set to 90 s. The calculation time is about 90 s in the case where the accuracy of the calculation results is guaranteed.

Velocity Distribution of the Fluid in the Main Trough
The inspection sections and lines in the main trough are presented for the overall description of physical fields in Figure 4.

Velocity Distribution of the Fluid in the Main Trough
The inspection sections and lines in the main trough are presented for the overall description of physical fields in Figure 4.   Figure 4). In the early stage of tapping, the drop point is located 3.5 m from the left side of the main trough (the blue point). At this moment, there is obvious turbulence downstream, and the turbulence region falls within the range of 4.5-6 m. In the middle of hot metal tapping, the hot metal drop point moves towards the taphole. At this moment, turbulence still exists, and the range of action is expanded to 3.5~6.5 m. It is due to the increase in metal flow angle at the drop point of the hot metal (37°-50°). Therefore, the direction angle of metal flow becomes larger after metal flow is blocked by the bottom of the main trough, and turbulence is more likely to occur. Although the flow velocity of the falling metal is reduced, turbulence is fully developed. As tapping continues, the drop point moves further towards the taphole and the flow velocity of the falling metal flow decreases. However, the increase in the angle of the drop point contributes to a more intense scour effect of the hot metal flow on the bottom of the main trough. The shape of the turbulence region becomes narrower and longer, with an influ-   Figure 4). In the early stage of tapping, the drop point is located 3.5 m from the left side of the main trough (the blue point). At this moment, there is obvious turbulence downstream, and the turbulence region falls within the range of 4.5~6 m. In the middle of hot metal tapping, the hot metal drop point moves towards the taphole. At this moment, turbulence still exists, and the range of action is expanded to 3.5~6.5 m. It is due to the increase in metal flow angle at the drop point of the hot metal (37 • -50 • ). Therefore, the direction angle of metal flow becomes larger after metal flow is blocked by the bottom of the main trough, and turbulence is more likely to occur. Although the flow velocity of the falling metal is reduced, turbulence is fully developed. As tapping continues, the drop point moves further towards the taphole and the flow velocity of the falling metal flow decreases. However, the increase in the angle of the drop point contributes to a more intense scour effect of the hot metal flow on the bottom of the main trough. The shape of the turbulence region becomes narrower and longer, with an influence range of 2~6.5 m.  Figure 4) at three different time points. Figure 6 shows the temperature distribution of the central section of the main trough at different time points. Three lines are presented for the overall description of temperature. As the tapping progresses, the temperature increases significantly and the temperature changes greatly in the early and interim tapping stages. For Line 1, the extreme temperature of the initial time is increased by 11 K higher than the end time. For Line 2, the extreme temperature of the initial time is increased 10 K higher than the end time. During the interim to final tapping, the temperature experiences a small change, indicating that the temperature in the main trough tends to stabilize after 45 min of tapping at about 1740 K. Han et al. [22] found that the hot metal temperature gradually stabilized after 40 min of tapping in the tapping trough, so the findings agree well.    Figure 6 shows the temperature distribution of the central section of the main trough at different time points. Three lines are presented for the overall description of temperature. As the tapping progresses, the temperature increases significantly and the temperature changes greatly in the early and interim tapping stages. For Line 1, the extreme temperature of the initial time is increased by 11 K higher than the end time. For Line 2, the extreme temperature of the initial time is increased 10 K higher than the end time. During the interim to final tapping, the temperature experiences a small change, indicating that the temperature in the main trough tends to stabilize after 45 min of tapping at about 1740 K. Han et al. [22] found that the hot metal temperature gradually stabilized after 40 min of tapping in the tapping trough, so the findings agree well.  Figure 4) at three different time points. Figure 6 shows the temperature distribution of the central section of the main trough at different time points. Three lines are presented for the overall description of temperature. As the tapping progresses, the temperature increases significantly and the temperature changes greatly in the early and interim tapping stages. For Line 1, the extreme temperature of the initial time is increased by 11 K higher than the end time. For Line 2, the extreme temperature of the initial time is increased 10 K higher than the end time. During the interim to final tapping, the temperature experiences a small change, indicating that the temperature in the main trough tends to stabilize after 45 min of tapping at about 1740 K. Han et al. [22] found that the hot metal temperature gradually stabilized after 40 min of tapping in the tapping trough, so the findings agree well.  The temperature change curves of three different tapping times are drawn along the bottom and side of the main trough refractory material as shown in Figure 7. The temperature of the main trough sidewall and the main trough working layer increase as the hot metal is tapped. The heat preservation of the refractory material is good and thermal conductivity of it is much less than that of hot metal, so the temperature rise is small with a change range of about 17 K. At the end of tapping, the temperature shows the highest value. The center of the sidewall and the center of the bottom of the main trough has a better heat transfer where the temperature change is the largest, and the temperature change in the edge area is minor. The temperature change curves of three different tapping times are drawn along the bottom and side of the main trough refractory material as shown in Figure 7. The temperature of the main trough sidewall and the main trough working layer increase as the hot metal is tapped. The heat preservation of the refractory material is good and thermal conductivity of it is much less than that of hot metal, so the temperature rise is small with a change range of about 17 K. At the end of tapping, the temperature shows the highest value. The center of the sidewall and the center of the bottom of the main trough has a better heat transfer where the temperature change is the largest, and the temperature change in the edge area is minor. Thermocouples are installed in the main trough to record temperature values every 30 minutes and the tapping time of a BF is about 90 min. Therefore, there are about four temperature records for a tapping. The calculation time is 90 s in our cases. Since there is a fluctuation of the monitoring temperature, an averaging temperature of the four-time segments in hot metal is necessary. Comparisons between the simulated and measured temperatures in the plant are shown in Figure 8. During the tapping, the temperature of on-site monitoring and temperature of the numerical calculation fluctuate with a similar trend and the latter changes in a small range. The average temperature error is 1.3% so the temperature distributions agree reasonably well. Thus, the simulations are consistent with plant measurements. Thermocouples are installed in the main trough to record temperature values every 30 min and the tapping time of a BF is about 90 min. Therefore, there are about four temperature records for a tapping. The calculation time is 90 s in our cases. Since there is a fluctuation of the monitoring temperature, an averaging temperature of the four-time segments in hot metal is necessary. Comparisons between the simulated and measured temperatures in the plant are shown in Figure 8. During the tapping, the temperature of on-site monitoring and temperature of the numerical calculation fluctuate with a similar trend and the latter changes in a small range. The average temperature error is 1.3% so the temperature distributions agree reasonably well. Thus, the simulations are consistent with plant measurements.

Temperature Distribution of the Main Trough
The temperature change curves of three different tapping times are drawn alon bottom and side of the main trough refractory material as shown in Figure 7. The tem ature of the main trough sidewall and the main trough working layer increase as th metal is tapped. The heat preservation of the refractory material is good and thermal ductivity of it is much less than that of hot metal, so the temperature rise is small w change range of about 17 K. At the end of tapping, the temperature shows the hig value. The center of the sidewall and the center of the bottom of the main trough h better heat transfer where the temperature change is the largest, and the tempera change in the edge area is minor. Thermocouples are installed in the main trough to record temperature values e 30 minutes and the tapping time of a BF is about 90 min. Therefore, there are about temperature records for a tapping. The calculation time is 90 s in our cases. Since the a fluctuation of the monitoring temperature, an averaging temperature of the foursegments in hot metal is necessary. Comparisons between the simulated and meas temperatures in the plant are shown in Figure 8. During the tapping, the temperatu on-site monitoring and temperature of the numerical calculation fluctuate with a sim trend and the latter changes in a small range. The average temperature error is 1.3 the temperature distributions agree reasonably well. Thus, the simulations are consi with plant measurements.  Figure 9 shows the wall shear stress distribution of the main trough at the in middle and final stages of the tapping, respectively, and the shear force distributio the straight line is obtained. The shear stress on the bottom surface of the main trou seen to be greater than on the sidewall, while the shear stress near the hot metal d point is greater than at other locations. During the tapping process, the shear stress a upper part of the sidewall gradually decreases, and the shear stress concentration ar found between 4 m and 5 m. The shear stress between the lower part of the sidewall the bottom surface gradually increases, and the shear stress concentration area of the is approximately the same, located at 3-7 m.  Figure 9 shows the wall shear stress distribution of the main trough at the initial, middle and final stages of the tapping, respectively, and the shear force distribution on the straight line is obtained. The shear stress on the bottom surface of the main trough is seen to be greater than on the sidewall, while the shear stress near the hot metal drop point is greater than at other locations. During the tapping process, the shear stress at the upper part of the sidewall gradually decreases, and the shear stress concentration area is found between 4 m and 5 m. The shear stress between the lower part of the sidewall and the bottom surface gradually increases, and the shear stress concentration area of the two is approximately the same, located at 3~7 m.  Figure 9 shows the wall shear stress distribution of the main trough a middle and final stages of the tapping, respectively, and the shear force dis the straight line is obtained. The shear stress on the bottom surface of the ma seen to be greater than on the sidewall, while the shear stress near the hot point is greater than at other locations. During the tapping process, the shear upper part of the sidewall gradually decreases, and the shear stress concentr found between 4 m and 5 m. The shear stress between the lower part of the s the bottom surface gradually increases, and the shear stress concentration are is approximately the same, located at 3-7 m.

Wall Thermal Axial Stress
The equivalent stress of the main trough as a whole is shown in Figure  demonstrates that the location of the stress concentration is related to the loca falling metal flow. Due to the injection of high-temperature hot metal, the te gradient increases rapidly in the early stage of the tapping. The average stress v refractory material downstream of the main trough is less than at other tapping the stress at the entrance of the iron dam and the iron trough is also greater. In of casting, there is no turbulence, and the temperature gradient is reduced, s value is reduced compared with the initial stage of the tapping. Therefore, the the hot metal drop point and downstream of it still remains high. At the end due to the increase in the angle at the hot metal drop point, the falling hot met flow directly downstream when it reaches the bottom surface of the trough, flows to the sidewall. Under the effect of the high-temperature hot metal, the te gradient at the lower part of the sidewall and the stress value increase. At the

Wall Thermal Axial Stress
The equivalent stress of the main trough as a whole is shown in Figure 10, which demonstrates that the location of the stress concentration is related to the location of the falling metal flow. Due to the injection of high-temperature hot metal, the temperature gradient increases rapidly in the early stage of the tapping. The average stress value of the refractory material downstream of the main trough is less than at other tapping times, and the stress at the entrance of the iron dam and the iron trough is also greater. In the middle of casting, there is no turbulence, and the temperature gradient is reduced, so the stress value is reduced compared with the initial stage of the tapping. Therefore, the stress near the hot metal drop point and downstream of it still remains high. At the end of tapping, due to the increase in the angle at the hot metal drop point, the falling hot metal does not flow directly downstream when it reaches the bottom surface of the trough, but instead flows to the sidewall. Under the effect of the high-temperature hot metal, the temperature gradient at the lower part of the sidewall and the stress value increase. At the same time, due to the edges and corners of the lower part of the sidewall, the stress concentration is aggravated, so the thermal stress in the lower part of the sidewall becomes larger. The stress value near the hot metal drop point is larger than at other positions. T cross-sections at 3.5 m, 4 m and 4.5 m were selected for analysis and the results are sho in Figure 11. The temperature gradient on the surface of the lining is large, forming a stress-concentration layer. The temperature gradient in the early stage of the tappin relatively large, so the range of the high-stress distribution is larger than that in the mid of the tapping. However, as the tapping continues, the turbulence in the main tro changes, the temperature gradient in the refractory material increases, and the stres fected area expands again.  Figure 11. The temperature gradient on the surface of the lining is large, forming a thin stress-concentration layer. The temperature gradient in the early stage of the tapping is relatively large, so the range of the high-stress distribution is larger than that in the middle of the tapping. However, as the tapping continues, the turbulence in the main trough changes, the temperature gradient in the refractory material increases, and the stress-affected area expands again.  Figure 11. The temperature gradient on the surface of the lining is large, forming a t stress-concentration layer. The temperature gradient in the early stage of the tappin relatively large, so the range of the high-stress distribution is larger than that in the mid of the tapping. However, as the tapping continues, the turbulence in the main trou changes, the temperature gradient in the refractory material increases, and the stress fected area expands again. Figure 11. Equivalent stress distribution on the cross-section of the main trough at different tim Eight straight inspection lines were selected on the bottom of the main trough, as picted in Figure 12, to study the stress distribution. It can be seen from Figure 13 that fr Eight straight inspection lines were selected on the bottom of the main trough, as depicted in Figure 12, to study the stress distribution. It can be seen from Figure 13 that from the center to the sidewall, the thermal stress at different positions is not much different. In the early stage of the tapping, the thermal stress downstream is very large and the stress upstream and downstream in the main trough are significantly different. In the middle of the tapping, the thermal stress value decreases significantly, because of the relatively stable temperature and gentle rise of the hot metal temperature at this time. The temperature gradient is also much smaller than at the initial stage of the tapping. At the end of the tapping, the stress value increases again, but the peak value is smaller than at the beginning of the tapping. the center to the sidewall, the thermal stress at different positions is not much different. In the early stage of the tapping, the thermal stress downstream is very large and the stress upstream and downstream in the main trough are significantly different. In the middle of the tapping, the thermal stress value decreases significantly, because of the relatively stable temperature and gentle rise of the hot metal temperature at this time. The temperature gradient is also much smaller than at the initial stage of the tapping. At the end of the tapping, the stress value increases again, but the peak value is smaller than at the beginning of the tapping. Based on the analysis of the shear force and thermal stress, it is concluded that the refractory material on the bottom of the main trough experiences greater thermal stress during the tapping process, and the maximum thermal stress occurs at 4 m on the bottom of the main trough. Therefore, key points for further analysis are selected at this position and in nearby areas in order to observe the stress changes in a working cycle. After calculating the tapping process (which lasted 90 s in the simulations), an equally long period ("tapping gap") without hot metal supply was simulated to study the working state of the main trough during the tap cycle. This gave the stress curve depicted in Figure 13.  the early stage of the tapping, the thermal stress downstream is very large and the stress upstream and downstream in the main trough are significantly different. In the middle of the tapping, the thermal stress value decreases significantly, because of the relatively stable temperature and gentle rise of the hot metal temperature at this time. The temperature gradient is also much smaller than at the initial stage of the tapping. At the end of the tapping, the stress value increases again, but the peak value is smaller than at the beginning of the tapping. Based on the analysis of the shear force and thermal stress, it is concluded that the refractory material on the bottom of the main trough experiences greater thermal stress during the tapping process, and the maximum thermal stress occurs at 4 m on the bottom of the main trough. Therefore, key points for further analysis are selected at this position and in nearby areas in order to observe the stress changes in a working cycle. After calculating the tapping process (which lasted 90 s in the simulations), an equally long period ("tapping gap") without hot metal supply was simulated to study the working state of the main trough during the tap cycle. This gave the stress curve depicted in Figure 13.   Based on the analysis of the shear force and thermal stress, it is concluded that the refractory material on the bottom of the main trough experiences greater thermal stress during the tapping process, and the maximum thermal stress occurs at 4 m on the bottom of the main trough. Therefore, key points for further analysis are selected at this position and in nearby areas in order to observe the stress changes in a working cycle. After calculating the tapping process (which lasted 90 s in the simulations), an equally long period ("tapping gap") without hot metal supply was simulated to study the working state of the main trough during the tap cycle. This gave the stress curve depicted in Figure 13.
Since R = −1 (R is the stress ratio) for the stress curve, an average stress correction is required. The Goodman average stress correction method was applied, using the formulas where S max , S min and S a−1) represent the maximum stress, the minimum stress and the corrected stress amplitude (Pa), respectively. S u = 1.92 × 10 7 Pa is the ultimate tensile strength of refractory material [23]. Substituting the corrected result into the S-N stress fatigue life curve, shown in Figure 14  Since R ≠ − 1 (R is the stress ratio) for the stress curve, an average stress correction is required. The Goodman average stress correction method was applied, using the formulas where S max , S min and S a−1) represent the maximum stress, the minimum stress and the corrected stress amplitude (Pa), respectively. = 1.92 × 10 7 Pa is the ultimate tensile strength of refractory material [23].
Substituting the corrected result into the S-N stress fatigue life curve, shown in Figure  14

Effect of Baffle Size
An optimization of the main trough structure can be undertaken to suppress the formation of turbulence and reduce the impact of backflow. Suppressing the formation of turbulence in the main trough is here the top priority. On the basis of Section 4.1, it can be seen that the counter-clockwise turbulence in the main trough recirculates between 5.8 m and 6.2 m, so baffles were set at this position in the model, as reported in Table 3. Table 3. Size of the baffles that suppress turbulence.

Effect of Baffle Size
An optimization of the main trough structure can be undertaken to suppress the formation of turbulence and reduce the impact of backflow. Suppressing the formation of turbulence in the main trough is here the top priority. On the basis of Section 4.1, it can be seen that the counter-clockwise turbulence in the main trough recirculates between 5.8 m and 6.2 m, so baffles were set at this position in the model, as reported in Table 3. Table 3. Size of the baffles that suppress turbulence.

Main Channel Flow Field
When the tapping has lasted 30 min, the simulated velocity distribution of each cross-section of the main trough is presented in Figure 15. When the tapping has lasted 30 min, the simulated velocity distribution of each cro section of the main trough is presented in Figure 15. The figure shows that the installation of baffles changes the distribution of the fl field. At the 5 m cross-section, the area is affected by the return flow on the upper side the main trough that becomes larger because of the downstream baffles. At the 5.5 m a 5.7 m sections, the velocity concentration area in the lower part of the trough is far fr the bottom. On the sidewall, due to the blocking of the baffles, more hot metal flo downstream and the intensity of the turbulence is weakened. At the 6 m position, the metal at Baffle 1 and Baffle 2 is affected by the baffles, which causes the lateral width decrease, and the flow cannot be fully diffused. Therefore, more hot metal flows from middle of the baffles downstream without obvious speed concentration. Figure 16a presents the velocity distribution near the sidewall (Line 5) at 5.7 m in metal flow direction of the main trough. Due to the influence of baffles, the extreme spe on the upper sidewall for the Baffle 1 and Baffle 2 cases is decreased by 4.8% and 9.2 respectively, compared with the base model, which shows that adding baffles here suppress turbulence, and the turbulence speed on the sidewall is significantly reduced order to verify this conclusion, the conditions along Line 1 on the upper side of the m trough is studied. Figure 16b shows that the extreme speed of the Baffle 1 case is redu by 4.8% and for the Baffle 2 case by 4.6%. The velocity distribution at the sidewall is sho in Figure 16c. Affected by turbulence, there are more peaks at the sidewall, but the extre velocity of each model has decreased. The middle channel flows away from the lower p of the sidewall. From Figure 16d, showing the velocity distribution on the bottom surf of the main trough, it can be seen that the speed at the bottom surface has dropped sig icantly, especially for the case with Baffle 2, and is decreased by 6.8% compared with base model. The figure shows that the installation of baffles changes the distribution of the flow field. At the 5 m cross-section, the area is affected by the return flow on the upper side of the main trough that becomes larger because of the downstream baffles. At the 5.5 m and 5.7 m sections, the velocity concentration area in the lower part of the trough is far from the bottom. On the sidewall, due to the blocking of the baffles, more hot metal flows downstream and the intensity of the turbulence is weakened. At the 6 m position, the hot metal at Baffle 1 and Baffle 2 is affected by the baffles, which causes the lateral width to decrease, and the flow cannot be fully diffused. Therefore, more hot metal flows from the middle of the baffles downstream without obvious speed concentration. Figure 16a presents the velocity distribution near the sidewall (Line 5) at 5.7 m in the metal flow direction of the main trough. Due to the influence of baffles, the extreme speed on the upper sidewall for the Baffle 1 and Baffle 2 cases is decreased by 4.8% and 9.2%, respectively, compared with the base model, which shows that adding baffles here can suppress turbulence, and the turbulence speed on the sidewall is significantly reduced. In order to verify this conclusion, the conditions along Line 1 on the upper side of the main trough is studied. Figure 16b shows that the extreme speed of the Baffle 1 case is reduced by 4.8% and for the Baffle 2 case by 4.6%. The velocity distribution at the sidewall is shown in Figure 16c. Affected by turbulence, there are more peaks at the sidewall, but the extreme velocity of each model has decreased. The middle channel flows away from the lower part of the sidewall. From Figure 16d, showing the velocity distribution on the bottom surface of the main trough, it can be seen that the speed at the bottom surface has dropped significantly, especially for the case with Baffle 2, and is decreased by 6.8% compared with the base model. The flow at the two cross-sections 5.5 m upstream and 6.6 m downstream are selected for analysis, and the results are shown in Figure 17. The volume flow of the Baffle 1 and Baffle 2 models has decreased compared with the base model: the volume flow of Baffle 1 has dropped by 5.0% and for Baffle 2 by 6.4%. In Figure 17b, the addition of baffles effectively increases the downstream flow rate. It is due to the reduction in the return metal flow rate and direct downstream of much more hot metal, resulting in an increase in the downstream surface flow rate by 11%.

Stress analysis
Line 1 on the sidewall of the main trough and Line 2 at the bottom of the main trough are analyzed as shown in Figure 18. The shear stress of Baffle 2 is the smallest, and the decrease is about 26%. The addition of baffles is seen to reduce the time-averaged shear stress on the main trough wall, the hot metal to the main trough and the erosion of the sidewall. However, whether the addition of baffles reduce the time-averaged shear stress at the bottom of the main trough is unknown. The flow at the two cross-sections 5.5 m upstream and 6.6 m downstream are selected for analysis, and the results are shown in Figure 17. The volume flow of the Baffle 1 and Baffle 2 models has decreased compared with the base model: the volume flow of Baffle 1 has dropped by 5.0% and for Baffle 2 by 6.4%. In Figure 17b, the addition of baffles effectively increases the downstream flow rate. It is due to the reduction in the return metal flow rate and direct downstream of much more hot metal, resulting in an increase in the downstream surface flow rate by 11%. The flow at the two cross-sections 5.5 m upstream and 6.6 m downstream are selecte for analysis, and the results are shown in Figure 17. The volume flow of the Baffle 1 an Baffle 2 models has decreased compared with the base model: the volume flow of Baffle has dropped by 5.0% and for Baffle 2 by 6.4%. In Figure 17b, the addition of baffles effe tively increases the downstream flow rate. It is due to the reduction in the return met flow rate and direct downstream of much more hot metal, resulting in an increase in th downstream surface flow rate by 11%.

Stress analysis
Line 1 on the sidewall of the main trough and Line 2 at the bottom of the main troug are analyzed as shown in Figure 18. The shear stress of Baffle 2 is the smallest, and the d crease is about 26%. The addition of baffles is seen to reduce the time-averaged shear stre on the main trough wall, the hot metal to the main trough and the erosion of the sidewa However, whether the addition of baffles reduce the time-averaged shear stress at the bo tom of the main trough is unknown.

Effect of Baffle Width and Length
The second research point in this study lies in the inhibition backflow. In this sec baffles are set at the position where the recirculation is strong, aiming to suppres backflow and reduce the erosion on the sidewall. The study is divided into two parts influence of baffle width on the suppression of reflux is studied by the Baffle 3, 4 a models, while the influence of baffle height is analyzed by the Baffle 5, 6 and 7 mo Detailed parameters of these setups are shown in Table 4.

Effect of Baffle Width and Length
The second research point in this study lies in the inhibition backflow. In this section, baffles are set at the position where the recirculation is strong, aiming to suppress the backflow and reduce the erosion on the sidewall. The study is divided into two parts. The influence of baffle width on the suppression of reflux is studied by the Baffle 3, 4 and 5 models, while the influence of baffle height is analyzed by the Baffle 5, 6 and 7 models. Detailed parameters of these setups are shown in Table 4.

Effect of Baffle Width and Length
The second research point in this study lies in the inhibition backflow. In this section, baffles are set at the position where the recirculation is strong, aiming to suppress the backflow and reduce the erosion on the sidewall. The study is divided into two parts. The influence of baffle width on the suppression of reflux is studied by the Baffle 3, 4 and 5 models, while the influence of baffle height is analyzed by the Baffle 5, 6 and 7 models. Detailed parameters of these setups are shown in Table 4.

Flow Field Analysis
The velocity distribution of each cross-section of the main trough is presented in Figure 20 for the case where hot metal has been tapped for 30 min. Figure 20a shows that the velocity distribution of each model in the 3.5 m cross-section is quite different. As the baffles are located at 4~5 m in the main trough block the return metal flow, only a small amount of metal flow reaches the side wall of the main trough before the 4 m location. The figures present a general decline in velocity as the width of the baffles (in the z direction) increases, as the blocking effect becomes more significant. It can be seen from the 3.5 m cross-section of Figure 20b that as the height of the baffles increases, the metal flow rate upstream of the baffles is significantly reduced. Baffle 7 with the lowest height does not inhibit the backflow very well, so return hot metal flows from the upper end of the baffles.

Flow Field Analysis
The velocity distribution of each cross-section of the main trough is presented in Figure  20 for the case where hot metal has been tapped for 30 min. Figure 20a shows that the velocity distribution of each model in the 3.5 m cross-section is quite different. As the baffles are located at 4~5 m in the main trough block the return metal flow, only a small amount of metal flow reaches the side wall of the main trough before the 4 m location. The figures present a general decline in velocity as the width of the baffles (in the z direction) increases, as the blocking effect becomes more significant. It can be seen from the 3.5 m cross-section of Figure 20b that as the height of the baffles increases, the metal flow rate upstream of the baffles is significantly reduced. Baffle 7 with the lowest height does not inhibit the backflow very well, so return hot metal flows from the upper end of the baffles.

Flow Field Analysis
The velocity distribution of each cross-section of the main trough is presented in Figu  20 for the case where hot metal has been tapped for 30 min. Figure 20a shows that the locity distribution of each model in the 3.5 m cross-section is quite different. As the baff are located at 4~5 m in the main trough block the return metal flow, only a small amount metal flow reaches the side wall of the main trough before the 4 m location. The figu present a general decline in velocity as the width of the baffles (in the z direction) increas as the blocking effect becomes more significant. It can be seen from the 3.5 m cross-sect of Figure 20b that as the height of the baffles increases, the metal flow rate upstream of baffles is significantly reduced. Baffle 7 with the lowest height does not inhibit the backfl very well, so return hot metal flows from the upper end of the baffles.      Figure 22b was obtained. At the center of the main trough, the metal flow velocity basically does not change. At the edge of the trough, the speeds of the Baffle 5 and Baffle 7 cases are significantly lower than those of the base model; the largest decrease is about 10%. Baffle 6 is a too low design to prevent hot metal from flowing near the wall. Figure 22c shows the velocity distribution along Line 5 near the sidewall of the trough. It is seen that the velocity of the model with a baffle near the sidewall is less than in the base model, the largest decrease about 33%. Due to the blocking effect of the baffle, the metal flow velocity near the upper and lower sides is lower than in the base model. In the vicinity of the baffle wall, since the baffles block the backflow, the speed of the hot metal flow is greater than in the base model.  Figure 21b: at the same distance from the wall, the metal flow velocity of the baffle models is less than that of the base model. The extreme velocity of Baffle 5 is 0.233 m/s, and the extreme velocity of the base model is 0.246 m/s, so a decrease of about 5% is obtained by the design. Figure 22a depicts the velocity distribution along the white line on the sidewall of the main trough (20 mm from the baffle's wall). The extreme velocity of Baffle 7 shows the largest decrease 11.7%, because of the different analysis positions. The recirculation velocity near the upper surface is overall larger, while the metal flow velocity near the upper surface is smaller. After analyzing the results along Line 4 on the cross-section of the main trough, the result shown in Figure 22b was obtained. At the center of the main trough, the metal flow velocity basically does not change. At the edge of the trough, the speeds of the Baffle 5 and Baffle 7 cases are significantly lower than those of the base model; the largest decrease is about 10%. Baffle 6 is a too low design to prevent hot metal from flowing near the wall. Figure 22c shows the velocity distribution along Line 5 near the sidewall of the trough. It is seen that the velocity of the model with a baffle near the sidewall is less than in the base model, the largest decrease about 33%. Due to the blocking effect of the baffle, the metal flow velocity near the upper and lower sides is lower than in the base model. In the vicinity of the baffle wall, since the baffles block the backflow, the speed of the hot metal flow is greater than in the base model. Since baffles are set at a position of 4~5 m from the taphole, a cross-section at a distance of 4.5 m from the taphole is selected to study on the effect of baffles on reflux suppression. For main troughs of different widths, as shown in Figure 23a, the flow rate of the Baffle 3 case is the smallest among the four models. Compared with the cross-sectional flow of the base model, the flow rate drops by about 3.2%, and the flow rate of the Baffle 5 model drops by about 3.1%. The decrease in flow rate is inversely proportional to the width of the baffles, that is, the larger the width, the better the suppression effect on the backflow in the main trough. Figure 23b shows the flow rate of each model cross-section downstream of the baffles. Since the reflux metal flow is blocked by the baffles, the flow rate of the downstream cross-section increases, and the flow rate is inversely proportional Since baffles are set at a position of 4~5 m from the taphole, a cross-section at a distance of 4.5 m from the taphole is selected to study on the effect of baffles on reflux suppression. For main troughs of different widths, as shown in Figure 23a, the flow rate of the Baffle 3 case is the smallest among the four models. Compared with the cross-sectional flow of the base model, the flow rate drops by about 3.2%, and the flow rate of the Baffle 5 model drops by about 3.1%. The decrease in flow rate is inversely proportional to the width of the baffles, that is, the larger the width, the better the suppression effect on the backflow in the main trough. Figure 23b shows the flow rate of each model cross-section downstream of the baffles. Since the reflux metal flow is blocked by the baffles, the flow rate of the downstream cross-section increases, and the flow rate is inversely proportional to the upstream flow rate. It is shown that the hot metal flows downstream after being blocked by the baffles. For main troughs of different heights, the cross-sectional flow of the main trough at 4.5 m (the middle position of baffles) and 5.2 m (downstream of baffles) was calculated. The results shown in Figure 24 indicate that as the height of the baffles increases, the flow rate at the 4.5 m section increases significantly.

Stress Analysis
The influence of the addition of baffles on the wall was studied and the time-averaged shear stress of the main trough wall was analyzed as shown in Figures 25 and 26. to the upstream flow rate. It is shown that the hot metal flows downstream after being blocked by the baffles. For main troughs of different heights, the cross-sectional flow of the main trough at 4.5 m (the middle position of baffles) and 5.2 m (downstream of baffles) was calculated. The results shown in Figure 24 indicate that as the height of the baffles increases, the flow rate at the 4.5 m section increases significantly.    Figure 25a shows that since the baffles added in this part are on the sidewalls of the trough, they have little effect on the metal flow at the bottom of it. Therefore, all models with baffles added decrease the shear stress slightly compared with the base model. Figure 25b reveals that the maximum time-averaged shear stress of the model with baffles is greater than that of the base model. It is inconsistent with the analysis result of the flow field. The metal flow near the wall decreases because the upstream flow rate decreases, and the metal flow velocity slows down. The increase in wall shear occurs as backflow is partially obstructed by the baffles. It can be seen from Figure 26 that the shear stress on the wall of the model with baffles has been reduced, and the effects of Baffle 5 and Baffle 6 are better. As the position of Baffle 6 is relatively high, the effect of restraining the flow in this part is better. The shear stress on the wall of the baffles is greater than for other models, but the position of Baffle 7 is too low, and the backflow appears at the top of the baffles, leading to greater shear stress.
For baffles of different widths, the inhibitory effect of Baffle 5 on the reflow is significantly different from the case with baffles of maximum width, and the shear stress for Baffle   Figure 25a shows that since the baffles added in this part are on the sidewalls of the trough, they have little effect on the metal flow at the bottom of it. Therefore, all models with baffles added decrease the shear stress slightly compared with the base model. Figure 25b reveals that the maximum time-averaged shear stress of the model with baffles is greater than that of the base model. It is inconsistent with the analysis result of the flow field. The metal flow near the wall decreases because the upstream flow rate decreases, and the metal flow velocity slows down. The increase in wall shear occurs as backflow is partially obstructed by the baffles. It can be seen from Figure 26 that the shear stress on the wall of the model with baffles has been reduced, and the effects of Baffle 5 and Baffle 6 are better. As the position of Baffle 6 is relatively high, the effect of restraining the flow in this part is better. The shear stress on the wall of the baffles is greater than for other models, but the position of Baffle 7 is too low, and the backflow appears at the top of the baffles, leading to greater shear stress.
For baffles of different widths, the inhibitory effect of Baffle 5 on the reflow is significantly different from the case with baffles of maximum width, and the shear stress for Baffle 5 is smaller than for Baffle 3. As for the study of baffles of different heights, Baffle 7 has the best effect on reducing the reflow on the sidewall, but it is not as good as Baffle 6 and Baffle  Figure 25a shows that since the baffles added in this part are on the sidewalls of the trough, they have little effect on the metal flow at the bottom of it. Therefore, all models with baffles added decrease the shear stress slightly compared with the base model. Figure 25b reveals that the maximum time-averaged shear stress of the model with baffles is greater than that of the base model. It is inconsistent with the analysis result of the flow field. The metal flow near the wall decreases because the upstream flow rate decreases, and the metal flow velocity slows down. The increase in wall shear occurs as backflow is partially obstructed by the baffles. It can be seen from Figure 26 that the shear stress on the wall of the model with baffles has been reduced, and the effects of Baffle 5 and Baffle 6 are better. As the position of Baffle 6 is relatively high, the effect of restraining the flow in this part is better. The shear stress on the wall of the baffles is greater than for other models, but the position of Baffle 7 is too low, and the backflow appears at the top of the baffles, leading to greater shear stress.
For baffles of different widths, the inhibitory effect of Baffle 5 on the reflow is significantly different from the case with baffles of maximum width, and the shear stress for Baffle 5 is smaller than for Baffle 3. As for the study of baffles of different heights, Baffle 7 has the best effect on reducing the reflow on the sidewall, but it is not as good as Baffle 6 and Baffle 5 in suppressing the time-averaged shear stress on the sidewall. Comprehensively, Baffle 5 can not only effectively reduce the upstream flow, but also concentrates most of the shear force on the baffles, which is the best solution to reduce the detrimental effect of backflow on the sidewall.

Conclusions
The main trough of the blast furnace is the only passage for hot metal and slag leaving the taphole for the torpedo and the slag granulation unit, and an important place to separate slag from hot metal and desulfurize hot metal as well. Studying the working status of the main trough and the erosion mechanisms of the refractory can provide useful information that can be used to improve the life length of the main trough lining and save costs of hot metal production. A mathematical model based on OpenFOAM was applied to analyze the flow temperature, wall shear stress, wall thermal stress in the main trough and the influence of adding baffles. The main conclusions are as follows: (1) Turbulence intensity downstream of the hot metal dropping position becomes weaker and the turbulence ranges become larger (2~6.5 m from the beginning of the trough);