Optimization Analysis of Mechanical Performance of Copper Stave with Special-shaped Tubes in the Blast Furnace Bosh

Abstract

Based on the theory of heat transfer, parametric modeling is established for the heat transfer model of copper cooling stave, which appears in recent years, with special-shaped tubes (elliptical, rectangular, double circular, three circular and ortho hexagonal) in a blast furnace (BF) bosh and the optimal tube for the cooling pipe is selected on the basis of the heat transfer characteristics of the stave. The heat transfer model of the hot end of stave embedded bricks which are not covered by slag, is analyzed using thermal-structural coupling method at the initial stage of blow-in under the normal working condition. The mutual influence of various parameters on the mechanical properties of copper stave is obtained using the response surface method. This method is combined with NSGA-II genetic algorithm to optimize the structure parameters and longevity technology of the bosh. The optimized structure of the furnace bosh is improved in heat transfer characteristics and mechanical properties, which proves the model and parameterized calculation program can be used as an optimized design and evaluation of the longevity technology of the bosh structure.

1. Introduction

The security and stability operation of stave is the basis for a long-life operation of a BF. A good cooling system and process conditions, especially in the high heat load area of the waist and the bosh, can form a stable slag on the surface of stave, which can protect the stave and prolong the service life of BF. In recent years, domestic and foreign scholars have studied the heat transfer characteristics and slagging capacity of the copper stave thermal structure in the bosh region from the perspective of heat transfer.^1,2,3,4,5,6) Y. Zhong et al.⁷⁾ conducted a systematic feasibility analysis on the buried pure copper tubular cast copper stave. G. S. Guo et al.⁸⁾ studied the heat transfer effect of cooling specific surface area on the stave based on the finite element method. S. Lin et al.⁹⁾ conducted a thermal test on a buried copper tubular stave, and concluded that the gas temperature has a significant effect on the stress distribution of the copper stave body. Y. Deng et al.¹⁰⁾ discussed the fracture mechanism of the rolled copper stave pipe and proposed a repairing method, K. Kawaoka et al.¹¹⁾ conducted a high temperature wear test on copper cooling stave copper material, and obtained that the strength of the copper cooling stave material would be significantly reduced in the environment exceeding 200°C. Recently, buried copper tubular copper stave has been successfully applied in many large BFs at home and abroad.¹²⁾ However, water cracking and hot surface damage have occurred in copper stave, and the mechanism of this phenomenon lacks quantitative analysis. In addition, the previous research on copper stave is limited to circular cooling water pipes or a single factor while lacks research on the mutual influence of various factors. Therefore, in this paper, based on the heat transfer theory, the heat transfer characteristics and mechanical properties of the copper stave that are commonly used in the BF bosh are analyzed. On the basis of optimizing the cooling water pipe type, the thickness of the copper stave body, the cooling specific surface area, the cooling water speed and the cooling water temperature are compared. The maximum thermal stress of the stave body under the normal working condition of the BF is used as an index to quantitatively calculate the correlation between each parameter and the index parameters. Taking the maximum thermal stress of the copper stave body as the optimization target, the structural parameters and longevity technology of the thermal structure of the bosh area are optimized, and the heat transfer model of the bosh structure before and after optimization is compared and analyzed using a combination of response surface model and NSGA-II genetic algorithm method, which verifies the effectiveness of this method to optimize the thermal structure of BF.

2. Boundary Conditions and Finite Element Model

The analysis is carried out for the structure of the bosh area of a BF,¹³⁾ which consists of the furnace shell (including bolts, locating pins, etc.), the packing layer, copper stave (including the cooling water pipes, the stave of the BF bosh is a circular water pipe in this paper) and embedded bricks, from the outside to the inside. The gap between adjacent stave walls and between the stave and the shell is filled with filler. The stave is a “four in and four out” buried pure copper tube cast copper stave. Considering the fact that the working environment of each stave is basically the same and for the purpose of reducing the calculation amount, a single stave + side gap filler + packing layer + furnace shell are taken as the calculating model. The upper and lower end faces of the stave have the same displacement and the circumferential displacement of the left and right sides is zero. The monolithic stave will have periodic mechanical characteristics when the heat transfer between the upper and lower sides is neglected. The temperature gradient of the upstream and downstream of the stave caused by the water temperature difference under working conditions has little influence on the coupling characteristics thus the average temperature is taken to reduce the calculation amount. The sector structure model of the single block stave in the cylindrical coordinate system is shown in Fig. 1.

Fig. 1.

Three-dimensional physical model of the BF bosh. (Online version in color.)

The expression of the integrated convective heat transfer coefficient h_s between the furnace shell and the surrounding air¹⁴⁾ is:   

$$h_s=9.3+0.058\cdot T_1$$(1)

Where T₁ is the ambient temperature around the furnace shell, °C

According to the research data of L. J. Wu et al.¹⁵⁾ based on the boundary condition substitution method, which determined that the comprehensive convective heat transfer coefficient h_x between the hot furnace gas and the tile hot surface in the range of 500–1248°C varies with temperature as shown in Eq. (2).   

$$h_x=-5.606+0.207T_2+8.414\times {10}^{-5}{T}_2{}^2$$(2)

Where T₂ represents the hot furnace gas temperature, °C.

The copper stave has the advantages of high thermal conductivity, stable dross, high load and long life. The buried copper tube cast copper stave avoids the welding process and completely eliminates the air gap layer. The structural size parameters of each layer in the bosh area of a blast furnace are shown in Table 1.

Table 1. Dimension parameters of stave.

Parameter name	Numerical value/mm	Parameter name	Numerical value/mm
Thickness of furnace shell	40	Distance between pipe and hot end (a)	140
Packing layer thickness	50	Distance of pipe end face (b)	80
Thickness of cooling stave (f)	180	Distance between pipes (e)	179.25
Height of cooling stave	1385	Side clearance width	20
Width of cooling stave	717	Depth of dovetail slot (c)	60
Width of brick (d)	100	Width of dovetail slot	80×100
Pipe diameter	45	Cooling plate inner radius	4000

3. Heat Transfer Characteristics Analysis of Various Tubular Copper Stave

In order to better compare the influence of each tube type on the heat transfer characteristics of the stave, the circular tube is taken as the reference standard, and the heat transfer characteristics of the stud tube stave in the two cases are considered, that is, the same cross-sectional area and the same circumference as the circular tube.

3.1. Heat Transfer Characteristics Analysis of the Stave with Same Cross-sectional Area

The equivalent diameter D_G of each shaped tube is   

$$D_{\text{G}}=4A/S$$(3)

Where A represents the cross-sectional area; S represents the wet perimeter length.

The equivalent diameter of each shaped tube is shown in Table 2. The ratio of the short axis of the rectangular tube to the elliptical tube b/a is calculated as 0.6.

Table 2. Equivalent diameter of section tube.

Name	Circular tube	Rectangular tube	Elliptical tube
DG/mm	45.000	38.614	42.270
Name	Double circular tube	Three circular tube	Ortho hexagonal tube
DG/mm	33.750	30.000	42.854

The relationship between the inertial force and the viscous force of a fluid in a given flow field can be characterized by the Reynolds number Re, the expression is   

$$Re=\rho \nu l/\mu =\nu l/\upsilon$$(4)

Where v is the fluid velocity; υ is the fluid kinematic viscosity; ρ is the fluid density; μ is the hydrodynamic viscosity; and l is the characteristic length.

The Reynolds number is the basis for judging the fluid flow state. For a circular tube, the fluid flow state will be laminar flow when Re≤2300; it will be turbulent flow when Re≥10⁴; and it will be transition flow when 2300<Re<10⁴. Taking a circular tube as an example, the convective heat transfer coefficient of the turbulent cooling water can be expressed by the Dietes and Bolt characteristic correlation¹⁶⁾ for a smooth or vertical long tube with a constant wall temperature and a small temperature difference between the fluid and the wall.   

$$Nu=0.023R{e}^{0.8}P{r}^n$$(5)

Where n is the index, n = 0.4 when the fluid is heated; Pr is the Prandtl number.

Combining Eqs. (4), (5) and Nusselt number Nu=h·D_G/k, the convective heat transfer coefficient expression of cooling water in a circular tube under turbulent flow conditions is obtained:   

$$h=3\mathrm{\hspace{0.17em} }899.750v^{0.8}$$(6)

According to this, the expressions of the convective heat transfer coefficient h of each type of internal cooling water are obtained, as shown in Table 3.

Table 3. Expression of convective heat transfer coefficient.

Name	Circular tube	Rectangular tube	Elliptical tube
H	h=3899.75v0.8	h=4020.97v0.8	h=3948.87v0.8
Name	Double circular tube	Three circular tube	Ortho hexagonal tube
H	h=4127.81v0.8	h=4154.17v0.8	h=3938.05v0.8

When the cooling water flows in each tube type, the pump power and the water flow velocity are the same. The enhanced heat transfer evaluation criterion can be used as the evaluation standard for each type of enhanced heat transfer. The calculation formula is   

$$\eta =\left(N{u}_i/N{u}_0\right)/{\left(f_i/f_0\right)}^{1/3}$$(7)

Where Nu_i and Nu₀ are the Nusselt number of the shaped tube and the standard tube respectively; f_i and f₀ are the resistance coefficients of the shaped tube and the standard tube respectively. The coefficient of resistance of the tube can be obtained by the Blasius formula.¹⁷⁾

The hot furnace gas temperature is 1200°C. Figure 2 is a graph showing the maximum temperature T_max of each tubular copper stave body in the bosh region of the BF under the same cross-sectional area and cooling water flow rate of 3 m/s.

Fig. 2.

T_max (v=3 m/s) for each tube type under the same cross-sectional area.

It can be seen from the calculation results in the figure that the T_max of each shaped tube has a slight decrease compared with that of the circular tube with the same cross-sectional area, and the temperature of cooling stave with rectangular tubes (b/a=0.6) and cooling stave with three circular tubes have the most obvious decrease. Figure 3 shows the variation of T_max and η in different b/a of rectangular tubes. As b/a increases, T_max increases continuously, and η decreases continuously. When b/a is greater than 0.6, T_max increases sharply while η drop tends to be flat. Once the copper cooling stave is damaged, the only way to replace it with a new one is to shut down the furnace, which can cause a significant productivity reducing of the enterprise. The main reason for the damage of the copper cooling stave is the high temperature and repeated deformation (mainly caused by the repeated change of temperature difference). Therefore, the pursuit of lower temperature is the primary consideration for the selection of cooling water tube parameters. From the calculation of the enhanced heat transfer evaluation criteria and the flow resistance characteristics of fluid, it can be seen that the flatter the cooling water tube is, the greater the resistance of the cooling water tube to the cooling water will be and thus under the same cooling water flow speed, the pressure at the inlet end of the water tube needs to be increased to improve the power of the water pump. At the same time, the flatter the cooling water tube means that the smaller the radial bearing capacity of the copper cooling wall is, and the more difficult the processing is. Therefore, in pursuit of a lower temperature, the evaluation criterion η of enhanced heat transfer can be appropriately reduced, which not only saves the cost of the enterprise, but also increases the deformation resistance of the cooling stave. According to the analysis results of heat transfer characteristics of copper cooling stave, the intersection of the maximum temperature T_max of copper cooling stave with rectangular tube and the evaluation criterion η of enhanced heat transfer should be the optimal tube type size. At this time, the rectangular tube b/a=0.62, as shown in Fig. 3. Figure 4 is a temperature field cloud diagram of the thermal structure of the rectangular section of the rectangular tube (b/a = 0.6).

Fig. 3.

T_max and η distribution of Rectangular tubes with different b/a.

Fig. 4.

Temperature field of each structure of the Copper Stave with rectangular tubes (b/a=0.6).

3.2. Heat Transfer Characteristics Analysis of Stave on the Same Perimeter

Figure 5 shows the calculation results of T_max for each tube type with the same circumference and cooling water speed of 3 m/s. It can be seen from it that the temperature of the stave with elliptical tubes decreases the most with the same circumference. Figure 6 shows the distribution of T_max and η under different b/a of elliptical tubes. From the perspective of Comprehensive consideration from the comprehensive evaluation index of heat transfer enhancement, the temperature of cooling stave, enterprise cost, etc., the optimal tube type for each tube type with the same circumference should be an elliptical tube with b/a=0.57.

Fig. 5.

T_max (v=3 m/s) for each tube type under the same circumference.

Fig. 6.

T_max and η distribution of elliptical tubes with different b/a.

Analyzing the calculation results of Figs. 2 and 5, it can be obtained that the copper stave of the rectangular tube has a small difference in heat transfer characteristics compared with the copper stave with elliptical tubes in the same cross-sectional area while has a slightly lower value in terms of enhanced heat transfer evaluation indicators. Considering the heat transfer, flow resistance characteristics and the stress concentration point of the rectangular tube during working process, the cooling water pipe of copper stave in the BF bosh area will be most suitable to change to an elliptical tube with b/a=0.57. At the same time, compared with the original stave with circular pipes, it can save a large amount of cooling water and improve efficiency while achieving the same cooling effect.

4. Optimization of Structure Longevity Technology in Blast Furnace Bosh Area

4.1. Introduction to Response Surface Method

The basic idea of response surface methodology is to express implicit functional functions by constructing polynomials with well-defined forms. Assume that the relationship between the performance function Z and the random variable Q=[Q₁, Q₂, ..., Q_R] is as shown in Eq. (8). The N samples of the random variable are randomly sampled, and the performance function values (z₁, z₂, ..., z_N) are statistically analyzed, and the system function is obtained by least squares theory fitting.¹⁸⁾   

$$\mathrm{Z}=a_0+\sum _{i=1}^R{a}_i{Q}_i+\sum _{i=1}^R\sum _{j=i}^R{a}_{ij}{Q}_i{Q}_j$$(8)

Where a₀, a_i, a_ij (i=1,..., R; j=i, ..., R) are the undetermined coefficients of the functional equation, for a total of 1+R+R(R+1)/2.

The matrix method is used to take three horizontal points for each random variable and then according to the Box-Behnken (BBD) sampling method, the center and the middle midpoint are taken as the sample value points. When the random variable distribution conforms to a certain law, the variable level q_s can be determined by Eq. (9).   

$${\int }_{-\infty }^{q_s}f(q)dq=p_n,\mathrm{\hspace{0.17em} }n=1,2,3$$(9)

Where f(q) represents a random variable probability density function. p_n represents a horizontal point, taking p₁=0.01, p₂=0.5, and p₃=0.99.

Proceeding numerical simulation of the sample values of S random variables yields S output points (z₁, z₂, ..., z_S). Then the Eq. (10) can be obtained using regression analysis, and the functional relationship of the performance function (8) can be determined. The design method used is the BBD method.   

$$S={\sum _{i=1}^S\left[z_i-\left(a_0+\sum _{i=1}^R{a}_i{q}_i+\sum _{i=1}^R\sum _{j=i}^R{a}_{ij}{q}_i{q}_j\right)\right]}^2$$(10)

4.2. Maximum Thermal Stress Response Surface Model of Copper Stave Body

Thermal stress is one of the key factors affecting the service life of the BF stave. From the numerical calculation of the thermal stress of the stave body, the influence law is analyzed when each parameter varies, which has far-reaching significance for the longevity of the BF. Based on the structure of blast furnace bosh in Table 1, the maximum thermal stress S_max (ie Von Mises equivalent stress) of the copper cooling stave with elliptical tube which has the same circumference with circular tube and a short-axis ratio of 0.6 is taken as research object, and the thickness of the copper stave body, the cooling specific surface area, the cooling water flow rate and the cooling water temperature as comparative sequences. The parameters are shown in Table 4.

Table 4. Levels of each parameter.

Random parameters	Horizontal encoding value
	−1	0	1
x1/mm	155	180	205
x2	0.7	1.0	1.3
x3/m·s−1	2	3	4
x4/°C	20	30	40

Note: x₁, x₂, x₃ and x₄ are the copper stave body thickness, cooling specific surface area, cooling water flow rate and cooling water temperature, respectively.

Multivariate regression fitting analysis is performed on the S_max calculation results, and Table 5 is the results. The model determines the coefficient R₂=0.9463 and the adjustment decision coefficient R_2adj=0.8925, which indicates that the calculation result of the model is well fitted to the experimental data, and can be used for the theoretical prediction of the maximum thermal stress S_max of the copper stave body with various parameters. The significance of each parameter to S_max is determined by the F test: the smaller the P value and the larger the F value, which means the higher the degree of significance. It can be seen from Table 5 that the influence of x₂ and x₄ on S_max is particularly prominent, and the interaction of x₁ and x₂ in the interaction term has a greater influence on S_max. Due to space limitations, results of the influence rule of partial interaction parameters on S_max are shown in Fig. 7.

Table 5. Calculations of response surface methodology.

Random variables	Quadratic sum	Degree of freedom	Mean square	F	P (P>F)
x1	432.55	1	432.55	5.25	0.0379
x2	2646.89	1	2646.89	32.15	0.0001
x3	732.44	1	732.44	8.90	0.0099
x4	1022.39	1	1022.39	12.42	0.0034
x1x2	862.74	1	862.74	10.48	0.0060
x1x3	1.26	1	1.26	0.02	0.9033
x1x4	0.75	1	0.75	9.08×10−3	0.9254
x2x3	12.33	1	12.33	0.15	0.7046
x2x4	2.81×10−3	1	2.81×10−3	3.41×10−5	0.9954
x3x4	8.19×10−3	1	8.19×10−3	9.95×10−5	0.9922
x12	440.18	1	440.18	5.35	0.0365
x22	0.29	1	0.29	3.48×10−3	0.9538
x32	74.01	1	74.01	0.90	0.3592
x42	8.45	1	8.45	0.10	0.7534
Residual error	1152.64	14	82.33
Sum	7497.06	28

Fig. 7.

The influence results of various parameters on S_max.

The degree of influence of each parameter on S_max ranges from largest to smallest: x₂, x₄, x₃, x₁, and the interaction between x₁ and x₂ is relatively large. According to the response surface theory, the greater the degree of influence is, the greater the influence of this factor on the target sequence will be. S_max increases firstly and then decreases with the increase of x₁ and based on this, when the value of x₂is small, the thickness of the stave should be moderately reduced. S_max decreases rapidly with the increase of x₂ and then becomes gentle. When the specific surface area increases from 0.7 to 1.1, S_max drops sharply, and based on this, the specific surface area can be appropriately increased to reduce the cooling wall stress. When x₃ is increased from 2 m/s to 4 m/s, the S_max decreases slightly because the heat transfer resistance of heat convection of cooling water is not the limiting element of the heat transfer of the stave any more. Therefore, an improvement in the cooling water velocity means an increase in the water flow rate, and at the same time it will lead to a significant increase in the water flow resistance and the running cost. When the value of x₄ is relatively small, the water speed can be reduced to reduce the cost but increased to reduce the cooling wall stress when the water temperature is high. In the actual operation, attention should also be paid to the repeated thermal stress of the body caused by repeated changes in water temperature which is easy to cause fatigue damage, so the water temperature pursuit is mainly based on stability.

According to formula (10), the response surface equation of the maximum thermal stress S_max of the copper stave body during normal operation of the blast furnace is   

$$\begin{array}{l}{S}_{max}=-421.8408+6.44753x_1+280.47565x_2-38.1062x_3\\ \hspace{1em}\hspace{1em}\hspace{1em}+0.52697x_4-1.95817x_1{x}_2+0.02244x_1{x}_3-1.729\\ \hspace{1em}\hspace{1em}\hspace{1em}\times {10}^{-3}{x}_1{x}_4+5.85167x_2{x}_3+8.83333\times {10}^{-3}{x}_2{x}_4\\ \hspace{1em}\hspace{1em}\hspace{1em}+4.525\times {10}^{-3}{x}_3{x}_4-0.013591x_1{}^2+2.33426x_2{}^2\\ \hspace{1em}\hspace{1em}\hspace{1em}+3.37783x_3{}^2+0.01268x_4{}^2\end{array}$$(11)

4.3. Target Optimization Based on Genetic Algorithm

Based on the response surface model of the copper stave bosh of the furnace, the genetic algorithm is used to optimize the maximum thermal stress of the body under multi-parameters condition.¹⁹⁾ Take the optimized overall mass m_T of the copper stave no higher than 95% of the initial mass and the water consumption L_s per unit time no higher than 95% of the water consumption in the initial unit time as the constraint condition.

The mathematical model of the optimized design can be expressed as   

$$\begin{array}{l}{S}_y=min(S_{max}(x_1,x_2,x_3,x_4))\\ \begin{array}{c}\text{s}\text{.t}\text{. }{x}_1\in (155,205),x_2\in (0.7,1.3),\\ x_3\in (2,4),x_4\in (20,40)\\ m_{\text{T}}\le 0.95m_0,L_s\le 0.95L_0\end{array}\end{array}$$(12)

Where ρ represents the copper stave material density; m₀ represents the initial mass of the elliptical tube copper stave and L₀ represents the initial unit time water consumption.

The genetic algorithm has a crossover probability of 0.6 and a mutation probability of 0.01. The calculation results of each random variable and objective function before and after optimization are shown in Table 6.

Table 6. Calculation results before and after optimization.

Name	x1/mm	x2	x3/m·s−1	x4/°C	Smax/MPa	mT /kg	Ls/m3
Initial value	180	1.0	3.0	30	205.2	0.172ρ	0.012
Optimal value	171	1.1	2.3	30	197.2	0.162ρ	0.011

In order to verify the accuracy of the optimized results, the optimized parameter values are brought into the finite element model, and the results are shown in Fig. 8. The comparison results of the response surface equation are 199.09818 MPa, and the approximation error is 0.9549%, which indicates that the response surface model and the NSGA-II genetic algorithm have higher precision for optimizing the structural parameters and longevity technology of the BF bosh area.

Fig. 8.

The optimized stress field of copper stave.

5. Conclusion

(1) From the perspective of the temperature of cooling stave, the comprehensive evaluation index of heat transfer enhancement, enterprise cost, machining difficulty, radial bearing capacity of cooling stave, etc., the optimal tube type for each tube type should be an elliptical tube with b/a=0.57. After the cooling water pipe of the copper stave in the bosh area is changed to the elliptical tube, the cooling effect can be ensured while the cooling water consumption is saved, which can help to effectively improve the enterprise efficiency.

(2) The degree of influence of each parameter on S_max is as follows: cooling specific surface area, cooling water temperature, cooling water velocity and thickness of the stave body. Interaction between the thickness of the stave body and the cooling specific surface area is relatively large in terms of interaction. When designing the structure of the bosh area, the influence of various factors on the distribution of the stress field of the stave can be referenced to ensure the safe operation of the stave while optimizing its structure.

(3) The objective function is optimized by the combination of response surface method and NSGA-II genetic algorithm. The optimized stave is improved in heat transfer characteristics, and the mass of the body and the amount of cooling water are reduced. The effectiveness of the optimization method of the thermal structure parameters of the BF bosh area combined with the response surface method and the genetic algorithm is verified.

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