A finite element calculation method for solidification heat transfer during round billet continuous casting

Abstract

The present invention discloses a finite element calculation method for solidification heat transfer during continuous casting of round billets, and relates to the technical field of steel continuous casting. The method comprises establishing a two-dimensional geometric model of the round billet according to the cross-sectional dimensions of the round billet; meshing the two-dimensional geometric model of the round billet to construct a two-dimensional finite element model of the round billet; establishing a two-dimensional finite element solidification heat transfer model of the round billet in view of the problem of solidification heat transfer during continuous casting of the round billet; and calculating the temperature field change during continuous casting using the two-dimensional finite element solidification heat transfer model of the round billet. The method fully considers the arc-shaped boundary of the round billet, and adopts general quadrilateral units. When the grid is dense enough, the arbitrary curve of the boundary of the calculation area can be simulated by a segmented straight line on the boundary, while avoiding the serrated boundary caused by the use of rectangular units; the calculation accuracy of the calculation using the finite element method is improved compared with the calculation accuracy of the finite difference method and the like; the appropriate boundary conditions are selected according to the different cooling conditions of the crystallizer and the secondary cooling zone, so as to more accurately simulate the heat transfer phenomenon during the actual continuous casting production process of the round billet.

Description

Technical Field

[0001] The invention relates to the technical field of steel continuous casting, and in particular to a finite element calculation method for round billet continuous casting solidification heat transfer. Background Art

[0002] Round billet continuous casting is a continuous steel casting technology that mainly produces billets for seamless steel pipes. It can also be used to produce hub steel, gear steel, bearing steel, etc. Various specifications of round billets of different materials are widely used in the fields of railways, chemicals, petroleum, etc. Round billet continuous casting has the advantages of high billet casting precision, good quality, low energy consumption and high metal recovery rate. Compared with the traditional process of first rolling into billets and then rolling steel pipes with ingots and continuous casting billets, directly using continuous casting round billets to perforate and roll steel pipes can not only greatly save metal and improve the yield rate, but also is of great benefit to ensuring the quality of billets. Therefore, round billet continuous casting technology is booming today. However, there are still many quality defects in continuously cast round billets, mainly internal quality defects and surface quality defects.

[0003] The internal quality defects of continuous casting round billets mainly include cracks between dendrites and central shrinkage holes. It is generally believed that cracks between dendrites are mainly caused by uneven secondary cooling and thermal stress. Surface defects are mainly surface cracks, which are roughly longitudinal cracks, star-shaped cracks, surface scars, hairline cracks, etc. Among them, longitudinal cracks are mainly caused by the uneven growth of the billet shell in the crystallizer, and are formed after the unevenness is aggravated in the secondary cooling zone. The through-length cracks in the longitudinal cracks are relatively long, often running through the entire body of the billet, with a depth ranging from 30mm to 100mm, which has a great impact on the rear sharing. The main reason for the through-length cracks is that during the casting process, the billet is cooled by the secondary cooling water spray in the secondary cooling zone, which accelerates the heat transfer and makes the billet cool rapidly. If the secondary cooling water is sprayed unevenly, it will lead to an increase in internal tissue stress and cracking of the surface. The formation of transverse cracks on the surface of the continuous casting round billet is the result of the interaction between heat transfer, mass transfer and stress. During the solidification process of the ingot, if the intensity of the primary cooling and secondary cooling is too high, the columnar crystals will be more developed, and the developed columnar crystals are the origin of the initiation of small cracks; in addition, the uneven cooling of the ingot surface will cause uneven solidification structure, thereby generating internal stress between the tissues. If the ingot straightening temperature is just in the brittle temperature range of the cast steel, the plastic deformation ability of the material is poor during straightening in this range, especially when the inner arc side is subjected to tensile stress, it is easy to produce transverse cracks.

[0004] Therefore, selecting a suitable cooling system for the billet in the crystallizer and the secondary cooling zone and ensuring uniform cooling are crucial to reducing the quality defects of the continuous casting round billet and improving the quality of the billet. In order to obtain a suitable cooling system and achieve uniform cooling during the continuous casting process, it is necessary to establish an online control system for the continuous casting billet, quickly and accurately predict the temperature changes of the round billet at each moment, and adjust the cooling water volume in real time accordingly. The key to building an online control system for round billet continuous casting lies in the determination of the core algorithm for round billet solidification heat transfer, that is, the round billet solidification heat transfer model. In the prior art, the Chinese patent "Process Simulation and Reproduction System for Continuous Casting Production Process of Large Square Billet" with document number CN 101559480B uses rectangular grids to divide the tracking unit. The boundary of the round billet is arc-shaped. If the rectangular grid is used for approximate simulation, the boundary will be jagged, the calculation accuracy will be greatly reduced, and the calculated temperature will have a certain gap with the actual production process; the Chinese patent application "A Method for Simulating the Solidification Process of Continuously Casting Round Billet" with document number CN 109446748 A simulates the solidification process of the continuously cast round billet by establishing a mathematical model of the solidification heat transfer of the round billet, but the patent uses the finite difference method for calculation, which is not suitable for processing complex boundaries and the processing of irregular areas is also cumbersome, so it is not suitable for calculating the solidification heat transfer of the round billet; the document number is CN 110941889 A's Chinese patent application "Research Method for the Initiation and Extension of Micro and Macro Cracks in Continuously Casting Special-shaped Billets" simulates the temperature field and stress field of the special-shaped billet in the crystallizer through ANSYS to find out the positions where cracks in the billet are most likely to initiate and extend. However, the simulation of the temperature field in this patent is done with the help of ANSYS finite element software, and its principle is not explored, and the specific calculation method is unknown. Therefore, it cannot be applied to the online control system for continuous casting of round billets.

[0005] In summary, establishing a round billet solidification heat transfer model and calculating the temperature field at each moment of the round billet continuous casting process are of great significance for real-time control of cooling water, ensuring a reasonable cooling system, achieving uniform cooling of the billet, reducing quality defects, and improving the quality of the billet. However, there are currently few processes and methods that can provide a basis for round billet continuous casting parameter optimization. The existing methods are relatively simple, and the calculation of round billet solidification heat transfer is relatively rough, which has a certain gap with the actual production situation. Summary of the invention

[0006] In view of the deficiencies in the above-mentioned prior art, the present invention provides a finite element calculation method for solidification heat transfer of round billet continuous casting, aiming to predict the temperature changes at various moments in the round billet continuous casting solidification process, provide a basis for the design optimization of round billet continuous casting process parameters, and provide an algorithm and theoretical basis for establishing an online solidification heat transfer model of round billets, optimizing the cooling water volume, reducing billet quality defects, and improving billet quality.

[0007] The technical solution of the present invention is:

[0008] A finite element calculation method for round billet continuous casting solidification heat transfer, the method comprising the following steps:

[0009] Step 1: According to the cross-sectional dimensions of the round billet, a two-dimensional geometric model of the round billet is established using finite element analysis software;

[0010] Step 2: Mesh the two-dimensional geometric model of the round billet, construct a two-dimensional finite element model of the round billet and save the mesh information;

[0011] Step 3: Aiming at the solidification heat transfer problem of round billet continuous casting, a two-dimensional finite element solidification heat transfer model of round billet is established;

[0012] Step 4: Use the round billet two-dimensional finite element solidification heat transfer model to calculate the temperature field changes during the continuous casting process.

[0013] Further, according to the finite element calculation method of round billet continuous casting solidification heat transfer, the grid information includes unit information, node information, and boundary information; the unit information includes the unit number and the nodes contained in the unit; the node information includes the node number and the node coordinates; the boundary refers to the boundary of the round billet two-dimensional finite element model; the boundary information includes the unit information and node information on the boundary.

[0014] Further, according to the finite element calculation method for round billet continuous casting solidification heat transfer, step 3 includes the following steps:

[0015] Step 3.1: Construct the basic equation of finite element heat transfer;

[0016] Step 3.2: Construct a finite element calculation model for transient heat transfer problems;

[0017] Step 3.3: Construct a planar four-node quadrilateral element and perform heat transfer analysis based on the element for different heat transfer boundary conditions.

[0018] Further, according to the finite element calculation method of round billet continuous casting solidification heat transfer, the method of constructing a planar four-node quadrilateral unit is: select four vertices as interpolation nodes, and encode the nodes as i, j, k and l in a counterclockwise direction to construct a typical planar four-node quadrilateral unit.

[0019] Further, according to the finite element calculation method for round billet continuous casting solidification heat transfer, the heat transfer analysis based on the plane four-node quadrilateral unit for different heat transfer boundary conditions includes:

[0020] (1) When there is no heat transfer boundary and the unit is completely internal: the calculation formula of the unit temperature stiffness matrix element is formula (18), the calculation formula of the unit heat capacity matrix element is formula (19), and the calculation formula of the node load is formula (20);

[0021] (2) If one edge of the unit is a heat transfer boundary that meets the second type of heat transfer boundary condition S2: the calculation formula of the unit temperature stiffness matrix element is formula (18), the calculation formula of the unit heat capacity matrix element is formula (19), and the calculation formula of the node load is formula (24);

[0022] (3) If one edge of the unit is a heat transfer boundary that meets the third type of heat transfer boundary condition S3: the calculation formula of the unit temperature stiffness matrix element is formula (25) and formula (26), the calculation formula of the unit heat capacity matrix element is formula (19), and the calculation formula of the node load is formula (27);

[0023]

[0024]

[0025]

[0026]

[0027]

[0028]

[0029]

[0030] In the above formula, k mn and k mm are all element temperature stiffness matrix elements; k is the heat transfer coefficient; N m and N n are all shape functions; a is the upper limit of the integration in the parameter space; b is the lower limit of the integration in the parameter space; J is the Jacobian matrix; |J| is the determinant of the Jacobian matrix; J * is the adjoint matrix of the Jacobian matrix; x, y are Cartesian coordinate systems; ξ, η are coordinate systems in the parameter space; c mn represents the unit heat capacity matrix element; ρ is the material density; c T is the material specific heat; p m represents the node load; q v is the heat generated in the unit volume of the material inside the unit volume, which is called the internal heat source density; L is the heat transfer boundary length of the unit; is the given solidification heat flux density on the boundary that meets the boundary condition S2; is the convective heat transfer coefficient between the object and the surrounding medium; T ∞ is the ambient temperature; t is the time.

[0031] Further, according to the finite element calculation method for round billet continuous casting solidification heat transfer, step 4 includes the following steps:

[0032] Step 4.1: Read the round billet steel grade information, continuous casting machine structural parameters and continuous casting process parameters, and calculate the solidus temperature and liquidus temperature;

[0033] Step 4.2: Import the mesh information of the round billet 2D finite element model;

[0034] Step 4.3: Determine the time step; the time step is the time for calculating the movement of each slice; the slice represents the two-dimensional finite element model of the round blank;

[0035] Step 4.4: Determine the unit phase region and calculate the physical property parameters of the unit;

[0036] Step 4.5: Calculate the unit heat transfer matrix, node load, unit heat capacity matrix and integrate the total heat transfer matrix, total temperature load, and total heat capacity matrix; the integrated total heat transfer matrix is ​​to decompose each unit heat transfer matrix into sub-matrices, and then "fill in" the block sub-matrices determined by each node sequence into the total heat transfer matrix according to the unit node number; the integrated total temperature load and total heat capacity matrix are the same as the integrated total heat transfer matrix;

[0037] Step 4.6: Solve the temperature field, that is, solve the finite element two-dimensional unsteady heat conduction control equation At the same time, determine whether the slice position exceeds the air cooling zone. If so, the calculation ends. If not, the slice continues to move and go to step 4.4.

[0038] Further, according to the finite element calculation method of round billet continuous casting solidification heat transfer, the steel type information includes steel grade and chemical composition; the continuous casting machine structural parameters include crystallizer height and secondary cooling zone structural parameters; the secondary cooling zone structural parameters include the number, length, inlet position and outlet position of secondary cooling zones; the continuous casting process parameters include casting temperature, pulling speed, crystallizer heat flux density, water flow and water temperature in each secondary cooling zone, ambient temperature and round billet cross-sectional dimensions; the round billet cross-sectional dimensions include round billet radius.

[0039] Furthermore, according to the finite element calculation method of round billet continuous casting solidification heat transfer, the phase region refers to the liquid phase region, the solid phase region and the solid-liquid two-phase region; the physical property parameters include unit density, unit specific heat, unit solid phase ratio and unit thermal conductivity coefficient.

[0040] Furthermore, according to the finite element calculation method of round billet continuous casting solidification heat transfer, the method for determining the unit phase zone is: firstly, the center temperature of the planar four-node quadrilateral unit is calculated by the inverse distance weighted interpolation method, and finally, the phase zone of the unit is determined according to the unit center temperature.

[0041] Further, according to the finite element calculation method for solidification heat transfer of round billet continuous casting, the Gauss-Seidel iteration method is used to solve the matrix equation Calculate the temperature at each node.

[0042] In general, the above technical scheme conceived by the present invention has the following beneficial effects compared with the prior art: the finite element calculation method for solidification heat transfer of round billet continuous casting provided by the present invention fully considers the arc boundary of the round billet, adopts general quadrilateral units, and when the grid is dense enough, the arbitrary curve of the boundary of the calculation area can be simulated by segmented straight lines on the boundary, while avoiding the serrated boundary caused by the use of rectangular units, and the calculation accuracy is guaranteed to a certain extent; the finite element method is used for calculation, and the calculation accuracy is improved to a certain extent compared with the finite difference method; according to the different cooling conditions of the crystallizer and the secondary cooling zone, the appropriate boundary conditions are selected to more accurately simulate the heat transfer phenomenon in the actual continuous casting production process of the round billet. In short, the actual situation in the continuous casting production process of the round billet is fully considered, and the solidification heat transfer phenomenon and temperature change of the round billet continuous casting are described, so as to provide support and guidance for determining the appropriate cooling system, optimizing the secondary cooling water distribution, ensuring uniform cooling of the billet, and improving the quality of the billet. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 A schematic diagram of the process of a finite element method for solidification heat transfer during continuous casting of round billets provided in an embodiment of the present invention;

[0044] Figure 2 A schematic diagram of a two-dimensional finite element model of a round billet provided in an embodiment of the present invention;

[0045] Figure 3 The temperature distribution cloud diagrams of the round billet at different positions of the continuous casting machine provided by the embodiment of the present invention, wherein Figure (a) is the temperature distribution cloud diagram of the billet at the outlet of the crystallizer; Figure (b) is the temperature distribution cloud diagram of the billet at the outlet of the secondary cooling zone 1; Figure (c) is the temperature distribution cloud diagram of the billet at the outlet of the secondary cooling zone 2; Figure (d) is the temperature distribution cloud diagram of the billet at the outlet of the secondary cooling zone 3;

[0046] Figure 4 A comparison chart of the calculated surface temperature and the measured surface temperature of a round billet provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0047] In order to facilitate the understanding of the present application, the present application will be described more fully below with reference to the relevant drawings. The preferred embodiments of the present application are given in the drawings. However, the present application can be implemented in many different forms and is not limited to the embodiments described herein. On the contrary, the purpose of providing these embodiments is to make the disclosure of the present application more thoroughly and comprehensively understood.

[0048] This embodiment takes the continuous casting of 500mm 42CrMo round billets in a domestic steel plant as an example, and adopts the finite element calculation method of the round billet continuous casting solidification heat transfer of the present invention to simulate the temperature change during the continuous casting of the 42CrMo round billet.

[0049] A finite element calculation method for solidification heat transfer of round billet continuous casting in this embodiment is as follows: Figure 1 As shown, the specific steps include:

[0050] Step 1: According to the cross-sectional dimensions of the round billet, a two-dimensional geometric model of the round billet is established using finite element analysis software;

[0051] According to the cross-sectional dimensions of the round billet, a circular surface is generated in ANSYS or other finite element software with the coordinate origin as the center and the round billet radius as the radius, and a two-dimensional geometric model of the round billet is established. The cross-sectional dimensions of the round billet include the round billet radius.

[0052] In this embodiment, the diameter of the round billet is 500 mm.

[0053] Step 2: Mesh the two-dimensional geometric model of the round billet, construct a two-dimensional finite element model of the round billet and save the mesh information;

[0054] In finite element analysis software such as ANSYS, select the unit type as PLANE55, the unit shape as quadrilateral unit, and determine the unit size, then mesh the two-dimensional geometric model of the round billet to establish a two-dimensional finite element model of the round billet. Output the mesh information as a DAT format file and save it. The mesh information includes unit information, node information, and boundary information. The unit information includes the unit number and which nodes are included in the unit; the node information includes the node number and the node coordinates; the boundary refers to the model boundary. Due to the special shape of the round billet, the inner and outer arcs and side arcs are not distinguished; the boundary information includes which units and nodes are on the boundary.

[0055] In this embodiment, the unit size of the central area is 2 mm, and the unit division method is mapped division, and the unit area is automatically divided by ANSYS finite element software. Figure 2 shown.

[0056] Step 3: Aiming at the solidification heat transfer problem of round billet continuous casting, a two-dimensional finite element solidification heat transfer model of round billet is established;

[0057] Step 3.1: Construct the basic equation of finite element heat transfer;

[0058] The fundamental variable in a heat transfer process is temperature, which is a function of the geometric position in the object and time.

[0059] According to Fourier's heat transfer law and the law of conservation of energy, the control equation of the heat conduction problem can be established, that is, the transient temperature field T(x, y, z, t) of the object should satisfy the following equation (1).

[0060]

[0061] Where ρ is the material density, kg / m 3 ;c T is the specific heat of the material, J / (kg·K); κ x , κ y , κ z The thermal conductivity coefficients along the x, y, and z directions are W / (m·K) respectively; Q(x, y, z, t) is the heat source intensity inside the object, W / kg.

[0062] There are three types of heat transfer boundary conditions, namely

[0063] The first type of boundary condition S1 is the Dirichlet condition, which gives a temperature value on the boundary

[0064]

[0065] The second type of boundary condition S2 is the Neumann condition for a given heat flux density.

[0066]

[0067] The third type of boundary condition S3 is the Neumann condition for given convective heat transfer

[0068]

[0069] Where n x ,n y ,n z is the direction cosine of the normal outside the boundary; is the temperature given on the boundary that meets boundary condition S1, °C; is the given solidification heat flux density on the boundary that meets boundary condition S2, W / m 2 ; is the convection heat transfer coefficient between the object and the surrounding medium, W / (m 2 ·K); T ∞ is the ambient temperature; t is the time, s; and the boundary of the object Ω is

[0070] If the initial condition IC of the problem is

[0071]

[0072] in, is the initial temperature, ℃.

[0073] The corresponding variational formulation is that in the permissible temperature field that satisfies the boundary conditions S1, S2, S3 and the initial condition IC, the real temperature field makes the following functional I take the minimum value, that is,

[0074]

[0075] In the process of dealing with practical problems, boundary conditions S2 and S3 are difficult to meet in advance. Therefore, these two conditions can be coupled into the functional, that is,

[0076]

[0077] Step 3.2: Construct a finite element calculation model for transient heat transfer problems;

[0078] In transient heat transfer problems, the temperature field of the element will change with time, that is,

[0079]

[0080] Where N(x,y,z) is the shape function, and the node temperature here is is time-varying, i.e.

[0081]

[0082] Substitute (8) into (7) and Finding the extreme value of variation, we can get

[0083]

[0084] in

[0085]

[0086]

[0087]

[0088]

[0089] Equation (10) is called the unit heat transfer equation; It is called the unit heat transfer matrix; is the unit node temperature array; is the unit heat capacity matrix; is the equivalent temperature load array of the unit node.

[0090] Step 3.3: Construct a planar four-node quadrilateral element and perform heat transfer analysis based on the element for different heat transfer boundary conditions.

[0091] Select four vertices as interpolation nodes, and encode the nodes as i, j, k and l in a counterclockwise direction to construct a typical planar four-node quadrilateral unit; the temperature field inside the unit is calculated using a bilinear interpolation function:

[0092] T(x,y)=a1+a2x+a3y+a4xy (15)

[0093] In the formula, a1, a2, a3, and a4 are unknown coefficients.

[0094] The calculation of the coefficients of the bilinear interpolation function in any plane four-node quadrilateral element is very difficult, and some integral terms in the heat transfer matrix, heat capacity matrix and load vector expressions cannot be obtained by analytical primitive functions. Therefore, the finite element method does not directly calculate the quadrilateral element model of arbitrary shape, but uses the isoparametric mapping method to indirectly calculate the quadrilateral element model.

[0095] The present invention selects the square [-1,1]×[-1,1] as the reference square unit in the parameter space. That is, the coordinates of the vertices of the square in the parameter space are i(-1,-1), j(1,-1), k(1,1) and l(-1,1). The geometric mapping function of the square [-1,1]×[-1,1] in the parameter space to the quadrilateral in the physical space is written as a combination of node shape functions, that is,

[0096]

[0097] The square node shape function is

[0098]

[0099] The finite element method converts the calculation method of the temperature stiffness matrix elements of any quadrilateral unit in physical space to the global coordinate function into the calculation method of the square unit to the local coordinate in parameter space, that is,

[0100]

[0101] The upper and lower limits of the integration in the parameter space are the coordinate variation range of the square. For the [-1,1]×[-1,1] square, a=-1, b=1.

[0102] The heat capacity matrix elements of any quadrilateral unit in physical space can also be calculated in square units in parameter space, that is,

[0103]

[0104] The nodal loads generated by the heat source in any quadrilateral element in the physical space are also calculated in the square element in the parameter space, that is,

[0105]

[0106] Among them, q v The heat generated per unit volume inside the material is called the internal heat source density, W / m 2; J is called the Jacobian matrix

[0107]

[0108] |J| is called the Jacobian matrix determinant

[0109]

[0110] J * The adjoint matrix of the Jacobian matrix is ​​called

[0111]

[0112] Regarding the treatment of heat transfer boundaries, a four-node quadrilateral unit consisting of four nodes i, j, k, and l is used to perform heat transfer analysis for different heat transfer boundary conditions, and the unit matrices for the following three situations are derived:

[0113] (1) No heat transfer boundary, that is, completely internal unit;

[0114] (2) If a certain edge of the unit, for example, the ij edge, is a heat transfer boundary that meets the second type of heat transfer boundary condition S2: Constant to describe;

[0115] (3) If a certain edge of the unit, for example, the ij edge, is a heat transfer boundary that meets the third type of heat transfer boundary condition S3: Constant to describe.

[0116] (1) No heat transfer boundary, that is, completely internal unit;

[0117] When there is no heat transfer boundary and the unit is completely internal, the calculation formula of the unit temperature stiffness matrix element is formula (18), the calculation formula of the unit heat capacity matrix element is formula (19), and the calculation formula of the node load is formula (20).

[0118] (2) If one edge of the unit, for example, the ij edge, meets the second type of heat transfer boundary condition S2: Constant to describe;

[0119] The calculation formula of the unit temperature stiffness matrix element is formula (18), the calculation formula of the unit heat capacity matrix element is formula (19), and the calculation formula of the node load is as follows:

[0120]

[0121] Where L is the length of the ij side, m.

[0122] (3) If one edge of the unit, for example, the ij edge, meets the third type of heat transfer boundary condition S3: Constant to describe.

[0123] The calculation formula of the unit heat capacity matrix element is formula (19), and the calculation formula of the unit temperature stiffness matrix element and the node load is as follows:

[0124]

[0125]

[0126]

[0127] In the above formula, k mn and k mm are all element temperature stiffness matrix elements; k is the heat transfer coefficient; N m and N n are all shape functions; a is the upper limit of the integration in the parameter space; b is the lower limit of the integration in the parameter space; J is the Jacobian matrix; |J| is the determinant of the Jacobian matrix; J * is the adjoint matrix of the Jacobian matrix; x, y are Cartesian coordinate systems; ξ, η are coordinate systems in the parameter space; c mn represents the unit heat capacity matrix element; ρ is the material density; c T is the material specific heat; p m represents the node load; q v is the heat generated in the unit volume of the material inside the unit volume, called the internal heat source density; L is the heat transfer boundary length of the unit, m; is the given solidification heat flux density on the boundary that meets the boundary condition S2; is the convective heat transfer coefficient between the object and the surrounding medium; T ∞ is the ambient temperature; t is the time.

[0128] Step 4: Use the round billet two-dimensional finite element solidification heat transfer model to calculate the temperature field changes during the continuous casting process.

[0129] Step 4.1: Read the round billet steel grade information, continuous casting machine structural parameters and continuous casting process parameters, and calculate the solidus temperature and liquidus temperature;

[0130] The steel type information includes steel grade and chemical composition; the continuous casting machine structural parameters include crystallizer height and secondary cooling zone structural parameters; the secondary cooling zone structural parameters include the number, length, inlet position and outlet position of secondary cooling zones; the continuous casting process parameters include casting temperature, drawing speed, crystallizer heat flux density, water flow and water temperature of each secondary cooling zone, ambient temperature and round billet cross-sectional dimensions; the round billet cross-sectional dimensions include round billet radius.

[0131] In this embodiment, the chemical composition of the steel is shown in Table 1.

[0132] Table 1 Chemical composition of Φ500mm42CrMo round billet from a domestic steel plant

[0133] chemical composition C Si Mn P S Cr Mo Mass fraction (%) 0.42 0.21 0.60 0.015 0.015 1.0 0.20

[0134] In this embodiment, the round billet continuous casting machine includes three secondary cooling zones, wherein the length of the secondary cooling zone 1 is 0.299m, and the inlet and outlet are 0.82m and 1.119m away from the meniscus respectively; the length of zone 2 is 0.980m, and the inlet and outlet are 1.119m and 2.099m away from the meniscus respectively; the length of zone 3 is 1.160m, and the inlet and outlet are 2.099m and 3.259m away from the meniscus respectively. The solidus and liquidus temperatures of the 42CrMo round billet are 1446℃ and 1495℃ respectively, the casting temperature is 1522℃, the pulling speed is 0.33m / min, the cooling water volume of the crystallizer is L / min, the inlet and outlet temperature difference is 2.8℃, the water temperature of the secondary cooling zone is 25℃, the ambient temperature is 75℃, and the cooling conditions of the secondary cooling zone are shown in Table 2.

[0135] Table 2 Cooling conditions of Φ500mm42CrMo round billet in the secondary cooling zone of a domestic steel plant

[0136] Second cooling zone 1 2 3 Cooling water volume (L / min) 37 18 12

[0137] Step 4.2: Import the mesh information of the round billet 2D finite element model;

[0138] The mesh information obtained after meshing the two-dimensional geometric model of the round billet in step 2 is imported into the calculation program. The mesh information includes unit information, node information, and boundary information. The unit information includes the unit number and which nodes are included in the unit; the node information includes the node number and the node coordinates; and the boundary information includes which units and nodes are on the boundary.

[0139] Step 4.3: Determine the time step;

[0140] The time step is the time for each calculation of the slice (called slice because the two-dimensional model has no thickness). The time step multiplied by the pulling speed is the distance the slice moves each time, and the accumulated moving distance is the distance between the slice and the meniscus, that is, the position of the slice at this time.

[0141] In this embodiment, the time step is 0.1 s and the pulling speed is 0.33 m / min.

[0142] Step 4.4: Determine the unit phase region and calculate the physical property parameters of the unit;

[0143] The quadrilateral element calculates the unit center temperature through the inverse distance weighted interpolation method, and the phase region where the unit is located is determined according to the unit center temperature and the unit physical property parameters are calculated.

[0144] The phase region refers to the liquid phase region, the solid phase region and the solid-liquid two-phase region;

[0145] The physical property parameters include unit density, unit specific heat, unit solid phase ratio and unit thermal conductivity.

[0146] In this embodiment, the solidus temperature is 1446° C., the liquidus temperature is 1495° C., the unit phase region is determined according to the unit temperature and the corresponding calculation formula is selected to calculate the physical property parameters of the unit.

[0147] Step 4.5: Calculate the unit heat transfer matrix, node load, unit heat capacity matrix and integrate the total heat transfer matrix, total temperature load, and total heat capacity matrix;

[0148] According to the formula in step 3.2, the unit heat transfer matrix, node load, and unit heat capacity matrix of each unit are calculated respectively, and boundary conditions are added to the units on the boundary. The boundary conditions are calculated according to the different positions of the slice in the continuous casting machine and the cooling conditions of the crystallizer and the secondary cooling zone read in step 4.1.

[0149] The integrated total heat transfer matrix is ​​to decompose each unit heat transfer matrix into sub-matrices, and then "fill in" the block sub-matrices determined by each node sequence into the total heat transfer matrix according to the unit node number. The integrated total temperature load and total heat capacity matrices are the same as the integrated total heat transfer matrix.

[0150] Step 4.6: Solve the temperature field, that is, solve the finite element two-dimensional unsteady heat conduction control equation At the same time, determine whether the slice position exceeds the air cooling zone. If yes, the calculation ends. If not, the slice continues to move and go to step 4.4;

[0151] Solve the matrix equation using Gauss-Seidel iteration method Calculate the temperature of each node. After the solution is completed, determine whether the slice position exceeds the air cooling zone. If it has exceeded the air cooling zone, the calculation ends and the round billet continuous casting solidification heat transfer calculation is completed. If it has not exceeded the air cooling zone, the slice moves to the next position and goes to step 4.4.

[0152] This embodiment uses the Gauss-Seidel iteration method to solve the matrix equation. The error is the square root of the sum of the squares of the differences between two iterations of all nodes. The error tolerance is 0.001. In order to improve the calculation efficiency, the parallel calculation function parallel_for in OpenCV is used for CPU parallel calculation.

[0153] Step 5: Visualize and post-process the calculation results of round billet continuous casting solidification heat transfer;

[0154] After the solidification heat transfer calculation of round billet continuous casting is completed, the calculation results at different positions of the continuous casting machine are extracted and imported into the Tecplot software for visualization to form a temperature cloud map; the temperature change on the surface of the round billet is extracted and imported into the Origin software for visualization to form a temperature change curve.

[0155] This embodiment provides a temperature distribution cloud diagram of a Φ500mm42CrMo round billet at different positions of a continuous casting machine in a domestic steel plant, such as Figure 3 As shown; the calculated surface temperature of the round billet is compared with the measured temperature, as shown Figure 4 As shown. Figure 3 It can be seen that in the presence of a liquid core, the temperature gradient gradually increases from the surface of the billet to the inner center, reaches the maximum at the solidification front, and gradually decreases to zero after reaching the liquid phase zone. The liquid core of the round billet exists for a long time and still exists at the end of the secondary cooling zone, so a longer air cooling zone is required for cooling. Figure 4 It can be seen that by tracking the round billet continuous casting production process, the calculated results are basically consistent with the measured billet surface temperature, the surface temperature of the secondary cooling zone basically shows a gentle downward trend, and the billet temperature is well controlled.

[0156] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some or all of the technical features therein. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope defined by the claims of the present invention.

Claims

1. A finite element calculation method for solidification heat transfer of round billet continuous casting, characterized in that: The method comprises the following steps: Step 1: According to the cross-sectional dimensions of the round billet, a two-dimensional geometric model of the round billet is established using finite element analysis software; Step 2: Mesh the two-dimensional geometric model of the round billet, construct a two-dimensional finite element model of the round billet and save the mesh information; Step 3: Aiming at the solidification heat transfer problem of round billet continuous casting, a two-dimensional finite element solidification heat transfer model of round billet is established; Step 4: Use the round billet two-dimensional finite element solidification heat transfer model to calculate the temperature field change during the continuous casting process; The step 3 comprises the following steps: Step 3.1: Construct the basic equation of finite element heat transfer; Step 3.2: Construct a finite element calculation model for transient heat transfer problems; Step 3.3: Construct a planar four-node quadrilateral element and perform heat transfer analysis based on the element for different heat transfer boundary conditions; The heat transfer analysis based on the plane four-node quadrilateral unit for different heat transfer boundary conditions includes: (1) When there is no heat transfer boundary and the unit is completely internal: the calculation formula of the unit temperature stiffness matrix element is formula (18), the calculation formula of the unit heat capacity matrix element is formula (19), and the calculation formula of the node load is formula (20); (2) If one edge of the unit is a heat transfer boundary that meets the second type of heat transfer boundary condition S2: the calculation formula of the unit temperature stiffness matrix element is formula (18), the calculation formula of the unit heat capacity matrix element is formula (19), and the calculation formula of the node load is formula (24); (3) If one edge of the unit is a heat transfer boundary that meets the third type of heat transfer boundary condition S3: the calculation formula of the unit temperature stiffness matrix element is formula (25) and formula (26), the calculation formula of the unit heat capacity matrix element is formula (19), and the calculation formula of the node load is formula (27); In the above formula, k mn and k mm are all element temperature stiffness matrix elements; k is the heat transfer coefficient; N m and N n are all shape functions; a is the upper limit of the integration in the parameter space; b is the lower limit of the integration in parameter space; J is the Jacobian matrix; |J| is the determinant of the Jacobian matrix; J * is the adjoint matrix of the Jacobian matrix; x, y are Cartesian coordinate systems; ξ, η are coordinate systems in the parameter space; c mn represents the unit heat capacity matrix element; ρ is the material density; c T is the specific heat of the material; p m represents the node load; q v is the heat generated in the unit volume of the material inside the unit volume, which is called the internal heat source density; L is the heat transfer boundary length of the unit; is the given solidification heat flux density on the boundary that meets boundary condition S2; is the convective heat transfer coefficient between the object and the surrounding medium; T ∞ is the ambient temperature; t is the time.

2. The finite element calculation method for round billet continuous casting solidification heat transfer according to claim 1 is characterized in that: The mesh information includes unit information, node information, and boundary information; the unit information includes the unit number and the nodes contained in the unit; the node information includes the node number and the node coordinates; the boundary refers to the boundary of the two-dimensional finite element model of the round billet; the boundary information includes the unit information and node information on the boundary.

3. The finite element calculation method for round billet continuous casting solidification heat transfer according to claim 2, characterized in that: The method for constructing a planar four-node quadrilateral unit is: selecting four vertices as interpolation nodes, and encoding the nodes as i, j, k and l in a counterclockwise direction to construct a typical planar four-node quadrilateral unit.

4. The finite element calculation method for round billet continuous casting solidification heat transfer according to claim 1, characterized in that: The step 4 comprises the following steps: Step 4.1: Read the round billet steel grade information, continuous casting machine structural parameters and continuous casting process parameters, and calculate the solidus temperature and liquidus temperature; Step 4.2: Import the mesh information of the round billet 2D finite element model; Step 4.3: Determine the time step; the time step is the time for calculating the slice movement each time; The slice represents a two-dimensional finite element model of the round blank; Step 4.4: Determine the unit phase region and calculate the physical property parameters of the unit; Step 4.5: Calculate the unit heat transfer matrix, node load, unit heat capacity matrix and integrate the total heat transfer matrix, total temperature load, and total heat capacity matrix; the integrated total heat transfer matrix is ​​to decompose each unit heat transfer matrix into sub-matrices, and then "fill in" the block sub-matrices determined by each node sequence into the total heat transfer matrix according to the unit node number; the integrated total temperature load and total heat capacity matrix are the same as the integrated total heat transfer matrix; Step 4.6: Solve the temperature field, that is, solve the finite element two-dimensional unsteady heat conduction control equation At the same time, determine whether the slice position exceeds the air cooling zone. If so, the calculation ends. If not, the slice continues to move and go to step 4.

4.

5. The finite element calculation method for round billet continuous casting solidification heat transfer according to claim 4, characterized in that: The information includes steel grade and chemical composition; the continuous casting machine structural parameters include crystallizer height and secondary cooling zone structural parameters; the secondary cooling zone structural parameters include the number, length, inlet position and outlet position of secondary cooling zones; the continuous casting process parameters include casting temperature, pulling speed, crystallizer heat flux density, water flow and water temperature of each secondary cooling zone, ambient temperature and round billet cross-sectional dimensions; the round billet cross-sectional dimensions include round billet radius.

6. The finite element calculation method for round billet continuous casting solidification heat transfer according to claim 4, characterized in that: The phase region refers to the liquid phase region, the solid phase region and the solid-liquid two-phase region; the physical property parameters include unit density, unit specific heat, unit solid phase ratio and unit thermal conductivity coefficient.

7. The finite element calculation method for round billet continuous casting solidification heat transfer according to claim 6, characterized in that: The method for determining the unit phase zone is: firstly, calculating the center temperature of the plane four-node quadrilateral unit by an inverse distance weighted interpolation method, and finally determining the phase zone where the unit is located according to the unit center temperature.

8. The finite element calculation method for round billet continuous casting solidification heat transfer according to claim 6, characterized in that: Solve the matrix equation using Gauss-Seidel iteration method Calculate the temperature at each node.

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