Continuous casting slab solidification process phase field simulation method and system based on dynamic temperature boundary conditions

Abstract

The invention relates to the technical field of metal solidification numerical simulation and calculation material science, and provides a continuous casting billet solidification process phase field simulation method and system based on dynamic temperature boundary conditions. The method comprises the following steps: S1, collecting physical property parameters and component data of steel, continuous casting machine crystallizer parameters and secondary cooling working condition parameters; s2, selecting a phase field method model, and determining initial parameters; s3, solving a phase field control equation, a solute field control equation and a temperature field control equation to obtain phase field variables evolved along with time, solute distribution data and temperature field data; when the temperature field control equation is solved, continuous casting process parameters are introduced into the phase field model, and the actual boundary heat transfer process is simulated by considering the combination of D, N and R thermal boundary conditions. According to the method, any combination and stable solution of the three types of thermal boundary conditions are realized, the adaptability and calculation efficiency of phase field simulation to the complex thermal boundary conditions in the solidification process are remarkably improved, and the analysis precision, the calculation efficiency and the engineering adaptability are considered.

Description

Technical Field

[0001] The present invention relates to the technical field of numerical simulation of metal solidification and computational materials science, and particularly relates to a phase-field simulation method and system for the solidification process of continuous casting billets based on dynamic temperature boundary conditions. Background Art

[0002] During the solidification process of continuous casting billets, the setting of the temperature field boundary conditions has a decisive influence on the dendritic growth morphology and segregation process. The thermal diffusivity is generally 3 - 4 orders of magnitude larger than the liquid-phase solute diffusivity. Traditional simulation methods often regard dendritic growth as an isothermal solidification process (such as constant undercooling or constant temperature gradient), ignoring the release of latent heat of solidification during dendritic growth. The temperature field generally adopts the Zero-Neumann adiabatic boundary condition. However, in actual continuous casting processes, the cooling conditions between the edge and the central region of the billet are significantly different, which is a non-isothermal solidification process, and process parameters such as the water flow rate in the secondary cooling zone and the drawing speed of the continuous casting machine change dynamically, resulting in a single temperature boundary condition being unable to truly reflect the non-uniform cooling process. For example, the existing patent (CN 117828950 A) uses the Zero-Neumann condition. Although multi-field coupling is achieved, the dynamic adjustment of the temperature field boundary with process parameters is not considered, making it difficult to achieve corresponding temperature field adjustment when the drawing speed is high or local cooling is uneven, resulting in a large simulation error.

[0003] In addition, the existing technology has insufficient research on the type switching mechanism of temperature field boundary conditions. For example, the rapid solidification of continuous casting thin slabs is often accompanied by sudden changes in temperature gradients, and Dirichlet, Neumann, or Robin boundary conditions need to be switched according to the local solid fraction or interface moving speed. However, existing models (such as CN 115171818 A) only use the temperature freezing approximation and fail to achieve dynamic adaptation of boundary conditions.

[0004] Therefore, there is an urgent need for a phase-field simulation method that supports multiple types of boundary conditions for the temperature field to improve the simulation calculation accuracy of the solidification structure formation of continuous casting billets. Summary of the Invention

[0005] The purpose of the present invention is to overcome the deficiencies of the prior art and provide a phase-field simulation method and system for the solidification process of continuous casting billets based on dynamic temperature boundary conditions, which is applicable to the accurate calculation and process optimization of the microstructure evolution of billets in the steel continuous casting process.

[0006] The present invention adopts the following technical solutions: On the one hand, the present invention provides a phase-field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions, including: S1. Collect the physical property parameters and composition data of the steel to be studied, the crystallizer parameters of the continuous casting machine, and the secondary cooling condition parameters; S2. Select the phase field method model and determine the initial parameters required for simulation calculation; S3. Based on the physical properties of the steel to be studied and the phase field model, the phase field, solute field and temperature field control equations are solved to obtain the phase field variables, solute distribution and temperature field data that evolve over time; wherein, when solving the temperature field control equation, the continuous casting machine crystallizer parameters and secondary cooling operating condition parameters are introduced into the phase field model, and a combination of three types of thermal boundary conditions, namely Dirichlet, Neumann and Robin, is used to simulate the heat transfer process of the actual boundary.

[0007] Any of the possible implementations described above further provides an implementation, in step S1, the physical property parameters include the solute concentration c of the molten steel ∞ , steel liquid density ρ, solute equilibrium distribution coefficient k, liquidus slope m, liquid phase solute diffusion rate D, Lewis number Le, Gibbs-Thomson coefficient, pure iron melting point temperature T M , anisotropic strength ε, thermal conductivity λ, latent heat of solidification L, specific heat capacity Cp, thermal capillary length d0, thermal diffusion coefficient α; the continuous casting machine crystallizer parameters include casting speed, crystallizer shape and size; the secondary cooling working condition parameters include the water volume of each section of the secondary cooling zone, the length of each section of the secondary cooling zone, and the cross-sectional size of the billet.

[0008] Any possible implementation as described above, further provides an implementation, in step S2, in step S2, the phase field method model selects the Karma model, and the initial parameters include: the dimensionless undercooling θ of dendrite growth, the nucleation size R and position, the environmental noise ξ, the time step Δt, the iterative time step number nstep, the spatial step Δx, and the number of grids N.

[0009] The dimensionless undercooling θ of dendrite growth described above is calculated, and the nucleation size R is specified as 2 to 4 times the width of the solid-liquid interface. When simulating columnar crystal growth, the nucleation position is the bottom of the computational domain, the ambient noise ξ is the global Gaussian noise of the phase field, and the time step Δt, the number of iterative time steps nstep, the spatial step Δx, and the number of grids N are all specified by the time and space of the actual simulated physical process.

[0010] For any of the possible implementations described above, a further implementation is provided. In step S3, the phase field, solute field, and temperature field control equations and calculation process are as follows: S31. A phase field order parameter φ is introduced to describe the solidification state of the system. When φ = -1, it is liquid phase, φ = 1, it is solid phase, and when φ changes from -1 to 1, it is solid-liquid interface. Based on the finite difference nine-point discretization format, the control equation containing the phase field parameter φ is solved as shown in formula (1): (1) where φ is the phase-field order parameter, U is the dimensionless concentration, θ is the dimensionless supercooling degree, Le is the Lewis number, λ is the coupling coefficient, k is the solute equilibrium distribution coefficient, ξ is the environmental noise, and c ∞ is the solute concentration of the molten steel; A(ψ) is the anisotropic function of the solid-liquid interface energy, M is the dimensionless liquidus slope, and its specific form is shown in Equation (2): , (2) where ε is the anisotropy strength, ω is 4, representing that the molten steel is a cubic crystal system metal structure, , is the initial grain orientation, m is the liquidus slope, L is the latent heat of solidification, and Cp is the specific heat capacity; S32. The solute field control equation is shown in Equation (3): (3) where is the anti-solute rejection term to eliminate the non-equilibrium effect of solute diffusion, is the dimensionless solute diffusion coefficient; The finite volume method is used to solve the solute field control equation (3). The computational domain is discretized into control volumes, and the solute field control equation (3) is integrated within the element and transformed into an interface flux balance by applying the divergence theorem; S33. The temperature field control equation is shown in Equation (4): (4) where is the dimensionless thermal diffusion coefficient; According to the operating parameters of the continuous casting mold and the secondary cooling zone, the type and specific values of the temperature boundary conditions are determined; the alternating direction implicit iteration method in the finite difference method is used to solve the temperature field control equation (4).

[0011] In traditional phase-field simulations, the Zero-Neumann boundary condition, i.e., the adiabatic boundary condition, is usually adopted for the temperature boundary. To couple the continuous casting process parameters and further consider the influence of the drawing speed and the water flow rate in the secondary cooling zone on the growth of columnar crystals at the edge of the slab, the following temperature field boundary condition assignment method can be adopted in the phase-field simulation to improve the simulation calculation accuracy.

[0012] For any of the possible implementation methods described above, a further implementation method is provided. In step S33, the method for determining the temperature boundary conditions is as follows: S331. Judge the formation position of the solidification structure of the continuous casting slab: The square root law of solidification is adopted in the mold area, and the macroscopic heat transfer mathematical model is adopted in the secondary cooling section to establish the fitting relationship between the shell thickness δ and the solidification time t, so as to judge whether the solidification position of the slab is inside the mold or enters the secondary cooling zone and its specific position; S332. When the solidification position of the ingot is in the crystallizer, the boundary conditions are determined by the following method: When the molten steel contacts the mold wall in the continuous casting process, the molten steel solidifies into a non-zero Neumann boundary condition. The specific heat flux forms are shown in Equations (5) and (6): Heat flux in the crystallizer: (5) , (6) Where, L d L is the distance between the shell and the meniscus when the molten steel solidifies. m V is the effective liquid level of molten steel in the crystallizer, c is the casting speed of the continuous casting machine, β is the shape coefficient of the crystallizer, is the instantaneous heat flux density of the crystallizer, is the actual measured average heat flux density of the crystallizer, C w is the specific heat capacity of the crystallizer cooling water, W m is the water flow rate of the crystallizer, ΔT is the temperature difference of the cooling water in and out of the crystallizer, S eff is the effective heating area of the copper plate of the crystallizer; The above temperature field boundary conditions are non-dimensionalized to be consistent with the dimensionless undercooling θ in the phase field. The dimensionless process of temperature in the phase field model is: (7) In formula (7), T is the temperature of the molten steel in the calculation domain, T M is the melting point of pure iron; the non-zero Neumann temperature field boundary condition in equation (5) is dimensionlessly transformed using equation (7) to derive equation (8), which is then coupled to the temperature field control equation in the phase field model for solution. The specific dimensionless form of the heat flux boundary condition in the crystallizer is: (8) Calculate the dimensionless undercooling θ according to formula (8): m Value, as the specific value of the boundary condition under the temperature field; The temperature field has upper, lower, left, and right boundaries. The heat flux values of the upper, left, and right boundary conditions are all 0, indicating that they are all adiabatic boundary conditions. The lower boundary condition is the calculated θ m The value is the temperature boundary condition coupled to the casting speed input into the phase field model; S333. When the solidification position of the ingot is in the secondary cooling section, the boundary conditions are determined by the following method: In the secondary cooling section of the continuous casting process, the temperature boundary condition during the solidification of the molten steel adopts the Robin boundary condition, and the specific heat flux forms are respectively expressed as Equation (9) and Equation (10): Heat flux density in the full-roll section of the secondary cooling zone , (9) Heat flux density in the movable section and segment section of the secondary cooling zone , (10); In the formula, q f is the heat flux density in the full-roll section of the secondary cooling zone, h f is the heat transfer coefficient in the full-roll section of the secondary cooling zone, T f is the cooling water temperature in the full-roll section of the secondary cooling zone; q k is the heat flux density in the movable section and segment section of the secondary cooling zone, h k is the heat transfer coefficient in the movable section and segment section of the secondary cooling zone, T w is the cooling water temperature in the movable section and segment section of the secondary cooling zone, and w is the water flow density; Convert the temperature boundary conditions of formula (9) - formula (10) into the phase-field temperature boundary conditions of formula (11) - formula (12) that couple the water flow in the secondary cooling zone to ensure the accuracy of the temperature field boundary conditions during the simulation of the billet solidification process and improve the calculation accuracy; Dimensionless heat flux density in the full-roll section of the secondary cooling zone (11) Dimensionless heat flux density in the movable section and segment section of the secondary cooling zone (12); In formula (11) - (12), θs1 and θs2 are the dimensionless cooling water temperatures obtained by calculating T f and T w according to formula (7); The specific process of assigning values to the lower boundary condition of the temperature field is as follows: Calculate the dimensionless heat flux densities θ f and θ w of the full-roll section and the movable section and segment section in the secondary cooling zone according to formula (11) - (12). When the billet solidification position is in the full-roll section, use θ f , and when it is in the movable section and segment section, use θ w value; The left, right, and upper boundaries of the temperature field are of Neumann type, with adiabatic boundary conditions, and the value is 0; When the temperature boundary conditions of the billet solidification process are obtained through the infrared temperature measurement method, the left, right, and upper boundaries of the temperature field are of Neumann type, with adiabatic boundary conditions, and the value is 0; the lower boundary adopts Dirichlet type boundary conditions, and convert the measured billet surface temperature T d into the dimensionless temperature θ d according to formula (7); S334. The two-dimensional temperature field is solved by using the alternating implicit iteration method, which is decomposed into one-dimensional implicit solutions in the alternating x and y directions. The order of the solution directions is swapped in odd and even steps to ensure the symmetry of the solution results. At the same time, a one-dimensional implicit solver is constructed, and the Thomas algorithm (or the TDMA algorithm) is used to solve the tridiagonal matrix, improving the efficiency of solving the temperature field control equation.

[0013] In any of the possible implementation manners described above, a further implementation manner is provided. In step S3, for the Dirichlet boundary condition, the temperature of the boundary nodes for grid initialization is set to a fixed value. ; for the Neumann boundary condition, the gradient term is corrected by one-sided difference, and the right-hand side term θ m value is set. , where θ1 and θ2 are the dimensionless temperature values of the first and second layers at the grid boundary, and Δx is the grid spacing; for the Robin boundary condition, a linear combination of the temperature and the gradient term is combined, and the convection parameter is introduced to implement it. The original formula is replaced by , where λ is the thermal conductivity of steel, h f is the heat transfer coefficient, and θ f is the dimensionless temperature of the cooling water, and after arrangement, it is used for correcting the boundary coefficient matrix of the implicit method.

[0014] In any of the possible implementation manners described above, a further implementation manner is provided. In step S3, the control equations of the phase field, solute field, and temperature field are solved, the calculation results at that time are displayed in real time, and the phase field data, solute field data, and temperature field data at different times after the calculation is completed are recorded to form a data file; the data file is imported into post-processing software for visualizing the calculation results.

[0015] In any of the possible implementation manners described above, a further implementation manner is provided. In step S3, the control equations of the phase field, solute field, and temperature field are solved by Matlab software, and a series of VTK format files with specified time steps are output; the VTK files are imported into ParaView software for further visualizing the calculation results.

[0016] In any of the possible implementation manners described above, a further implementation manner is provided. In step S331, the determination of the solidification position is corrected by combining the nail shooting test and the infrared temperature measurement method.

[0017] On the other hand, the present invention also provides a phase field simulation system for the continuous casting billet solidification process based on dynamic temperature boundary conditions. The system is used to implement the above method, and the system includes: A material parameter input unit for inputting the physical property parameters and composition data of the steel to be studied, the continuous caster mold parameters, and the secondary cooling condition parameters; A phase field model parameter input unit, which is used to determine the initial parameters for the calculation of the phase field model; A temperature boundary condition determination unit, which is used to determine the temperature boundary conditions of the mold and the secondary cooling zone in the continuous casting process; An equation solving unit, which solves the control equations of the phase field, solute field and temperature field according to the determined temperature boundary conditions, and obtains the phase field variables, solute distribution and temperature field data that evolve with time.

[0018] A data processing unit inputs the phase field, solute and temperature field data files into post-processing software, and extracts and analyzes the data of the dendrite growth process.

[0019] On the other hand, the present invention also provides an information data processing terminal, which includes a processor and a memory for storing executable instructions; the processor is configured to execute the executable instructions to execute the above-mentioned phase field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions.

[0020] On the other hand, the present invention also provides a computer-readable storage medium, on which a computer program is stored, and when the computer program is executed by a processor, the above-mentioned phase field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions is realized.

[0021] The beneficial effects of the present invention are as follows: 1. By introducing the switching of the temperature field boundary type and combining the alternating direction implicit method, the method provided by the present invention successfully realizes the arbitrary combination and stable solution of the three types of thermal boundary conditions of Dirichlet, Neumann, and Robin, significantly improving the adaptability and calculation efficiency of the phase field simulation to the complex thermal boundary conditions in the solidification process.

[0022] 2. The present invention introduces continuous casting process parameters such as the water flow rate in the secondary cooling zone and the drawing speed of the continuous casting machine into the phase field model, realizes the coupling of the thermal boundary conditions and the actual production process, and expands the simulation ability of the method for dendrite growth behavior under different working conditions. It provides a reliable basis for the cooling optimization design and defect prediction in the continuous casting process.

[0023] 3. The present invention performs non-dimensionalization processing consistent with the temperature field control equation for the typical heat flux boundaries of the mold and the secondary cooling zone, improves the physical consistency and numerical stability of the model, and takes into account the analytical accuracy, calculation efficiency and engineering adaptability. Description of the Drawings

[0024] Figure 1 The figures show the phase field, solute field and temperature field diagrams in the prior art where the upper, lower, left and right boundary conditions of the temperature field are all Zero-Neumann conditions.

[0025] Figure 2Shown are the phase field, solute field, and temperature field diagrams in Embodiment 1 where the upper, left, and right boundary conditions of the temperature field are Zero-Neumann conditions, and the lower boundary condition is a non-zero Neumann condition.

[0026] Figure 3 Shown are the phase field, solute field, and temperature field diagrams in Embodiment 2 where the left, right, and upper boundaries of the temperature field are set to the "N" type, and the lower boundary condition is a Dirichlet condition.

[0027] Figure 4 Shown are the phase field, solute field, and temperature field diagrams in Embodiment 3 where the lower boundary condition of the temperature field is a Robin condition.

[0028] Figure 5 Shown is the schematic flow diagram of a phase field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions according to an embodiment of the present invention. Detailed implementation manners

[0029] The specific embodiments of the present invention will be described in detail below with reference to specific drawings. It should be noted that the technical features described in the following embodiments or the combinations of technical features should not be considered isolated, and they can be combined with each other to achieve better technical effects.

[0030] As Figure 5 shown, a phase field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions according to an embodiment of the present invention includes: S1. Collect the physical property parameters and composition data of the steel to be studied, the continuous caster mold parameters, and the secondary cooling condition parameters; S2. Select a phase field method model and determine the initial parameters required for simulation calculation; S3. Solve the phase field, solute field, and temperature field control equations according to the physical property parameters of the steel to be studied and the phase field method model to obtain the phase field variables, solute distribution, and temperature field data evolving with time; wherein, when solving the temperature field control equation, the continuous caster mold parameters and the secondary cooling condition parameters are introduced into the phase field method model, and the heat transfer process of the actual boundary is simulated using a combination of Dirichlet, Neumann, and Robin types of thermal boundary conditions.

[0031] In a specific embodiment, in step S1, the physical property parameters include the solute concentration c of the molten steel ∞ , the density ρ of the molten steel, the solute equilibrium distribution coefficient k, the liquidus slope m, the solute diffusion rate D in the liquid phase, the Lewis number Le, the Gibbs-Thomson coefficient, and the melting point temperature T of pure iron M, anisotropy strength ε, thermal conductivity λ, latent heat of solidification L, specific heat capacity Cp, thermocapillary length d0, thermal diffusivity α; the continuous casting mold parameters include casting speed and mold shape and size; the secondary cooling condition parameters include the water volume of each section in the secondary cooling zone, the length of each section in the secondary cooling zone, and the billet cross-section size.

[0032] In a specific embodiment, in step S2, the phase field method model selects the Karma model, and the initial parameters include: dimensionless supercooling θ of dendrite growth, nucleation size R and position, environmental noise ξ, time step Δt, number of iterative time steps nstep, spatial step Δx, and number of grids N.

[0033] In a specific embodiment, in step S3, the control equations and calculation processes of the phase field, solute field, and temperature field are as follows: S31. Introduce a phase field order parameter φ to describe the solidification state of the system. When φ = -1, it is the liquid phase; when φ = 1, it is the solid phase; when φ changes from -1 to 1, it is the solid-liquid interface. Solve the control equation containing the phase field parameter φ based on the finite difference nine-point discretization format as shown in Equation (1): (1) In the formula, φ is the phase field order parameter, U is the dimensionless concentration, θ is the dimensionless supercooling, Le is the Lewis number, λ is the coupling coefficient, k is the solute equilibrium distribution coefficient, ξ is the environmental noise, c ∞ is the solute concentration of the molten steel; A(ψ) is the anisotropy function of the solid-liquid interface energy, M is the dimensionless liquidus slope, and the specific form is as shown in Equation (2): , (2) In the formula, ε is the anisotropy strength, ω is 4, representing that the molten steel is a cubic crystal system metal structure, , is the initial grain orientation, m is the liquidus slope, L is the latent heat of solidification, and Cp is the specific heat capacity; S32. The solute field control equation is as shown in Equation (3): (3) In the formula is the anti-solute trapping term to eliminate the non-equilibrium effect of solute diffusion, is the dimensionless solute diffusion coefficient; Use the finite volume method to solve the solute field control equation (3). Discretize the computational domain into control volumes, integrate the solute field control equation (3) within the element, and apply the divergence theorem to convert it into an interface flux balance; S33. The temperature field control equation is as shown in Equation (4): (4) In the formula, is the dimensionless thermal diffusivity; According to the operating parameters of the continuous casting machine crystallizer and the secondary cooling zone, the type and specific value of the temperature boundary condition are determined; the alternating direction implicit iteration method in the finite difference method is used to solve the temperature field control equation (4).

[0034] The main boundary conditions for solving the temperature field are divided into three categories: Dirichlet, Neumann, and Robin boundary conditions.

[0035] In the Dirichlet boundary condition, the temperature boundary is set to a constant value; in the Neumann boundary condition, the temperature boundary is set to a constant heat flux (heat flux density or temperature gradient); the Robin boundary condition is a convection heat transfer boundary condition, which constrains the boundary heat exchange through the heat transfer coefficient and the ambient temperature, and the temperature and heat flux on the boundary satisfy a linear combination.

[0036] In traditional phase-field simulations, Zero-Neumann boundary conditions (also known as adiabatic boundary conditions) are typically used for temperature boundaries. To couple continuous casting process parameters and further consider the effects of casting speed and secondary cooling zone water flow on columnar grain growth at the edges of the strand, phase-field simulations can employ the following temperature field boundary condition assignment method to improve simulation accuracy.

[0037] In a specific embodiment, in step S33, the method for determining the temperature boundary condition is as follows: In step S33, the method for determining the temperature boundary condition is as follows: S331. Determine the location of the solidification structure of the continuous casting slab: Use the solidification square root law in the mold area and the macroscopic heat transfer mathematical model in the secondary cooling section to establish a fitting relationship between the shell thickness δ and the solidification time t. This allows determination of whether the slab solidifies within the mold or enters the secondary cooling section, as well as its specific location. Furthermore, the model parameters can be calibrated by combining nail shooting tests and infrared temperature measurement.

[0038] This model is mainly applied to the growth process of columnar crystals at the boundary of the ingot. The Zero-Neumann adiabatic boundary is adopted in the casting direction. In the direction perpendicular to the casting direction, the temperature boundary is close to the side of the crystallizer wall (or secondary cooling water) with a non-zero Neumann boundary condition (or Robin boundary condition), and the other side close to the dendrite front of the molten steel is set to a Neumann boundary condition or a Dirichlet boundary condition.

[0039] S332. When the solidification position of the ingot is in the crystallizer, the boundary conditions are determined by the following method: When the molten steel contacts the mold wall in the mold during the continuous casting process, the solidification of the molten steel is a non-zero Neumann boundary condition, and the specific heat flux form is shown in Equations (5) and (6): Heat flux density in the mold: (5) , (6) In the formula, L d is the distance from the position of the billet shell to the meniscus when the molten steel solidifies, L m is the effective liquid level height of the molten steel in the mold, V c is the drawing speed of the continuous casting machine, β is the mold shape coefficient (obtained by calculating the mold size), is the instantaneous heat flux density of the mold, is the average heat flux density of the mold measured actually, C w is the specific heat capacity of the mold cooling water, W m is the water flow rate of the mold, ΔT is the temperature difference between the inlet and outlet cooling water of the mold, S eff is the effective heated area of the mold copper plate; The above temperature field boundary conditions are made dimensionless to be consistent with the dimensionless supercooling degree θ in the phase field. The dimensionless process of temperature in the phase field model is: (7) In Equation (7), T is the temperature of the molten steel in the calculation domain, T M is the melting point temperature of pure iron; after the non-zero Neumann temperature field boundary condition in Equation (5) is made dimensionless in the same way as Equation (7), Equation (8) is derived and coupled to the temperature field control equation in the phase field model for solution. The specific dimensionless form of the heat flux density boundary condition in the mold is: (8) Calculate the θ m value of the dimensionless supercooling degree as the specific value of the lower boundary condition of the temperature field; The temperature field has upper, lower, left, and right boundaries. Among them, the heat flux density values of the upper, left, and right boundary conditions are all 0, indicating adiabatic boundary conditions, and the lower boundary condition is the calculated θ m value, which is the temperature boundary condition coupled with the drawing speed and input into the phase field model; S333. When the solidification position of the billet is in the secondary cooling zone, the following method is used to determine the boundary condition: In the secondary cooling zone of the continuous casting process, when the molten steel solidifies, the temperature boundary condition adopts the Robin boundary condition, and the specific heat flux forms are respectively shown in Equations (9) and (10): Heat flux density in the full roll section of the secondary cooling zone, , (9) Heat flux density of the movable section and segment section in the secondary cooling zone , (10); In the formula, q f is the heat flux density of the dummy bar section in the secondary cooling zone, h f is the heat transfer coefficient of the dummy bar section in the secondary cooling zone, T f is the cooling water temperature of the dummy bar section in the secondary cooling zone; q k is the heat flux density of the movable section and segment section in the secondary cooling zone, h k is the heat transfer coefficient of the movable section and segment section in the secondary cooling zone, T w is the cooling water temperature of the movable section and segment section in the secondary cooling zone, and w is the water flow density; Convert the temperature boundary conditions of formula (9) - formula (10) into the phase field temperature boundary conditions of formula (11) - formula (12) that couple the water flow in the secondary cooling zone to ensure the accuracy of the temperature field boundary conditions during the simulation of the solidification process of the continuous casting billet and improve the calculation accuracy; Dimensionless heat flux density of the dummy bar section in the secondary cooling zone (11) Dimensionless heat flux density of the movable section and segment section in the secondary cooling zone (12); In formula (11) - (12), θs1 and θs2 are the dimensionless cooling water temperatures obtained by calculating according to formula (7) for T f and T w respectively; The specific process of assigning values to the lower boundary condition of the temperature field is as follows: Calculate the dimensionless heat flux densities θ f and θ w of the dummy bar section and the movable section and segment section in the secondary cooling zone according to formula (11) - (12). When the solidification position of the continuous casting billet is in the dummy bar section, use θ f , and when it is in the movable section and segment section, use θ w value; The left, right, and upper boundaries of the temperature field are of Neumann type, with adiabatic boundary conditions, and the heat flux density value is taken as 0; When the temperature boundary conditions of the solidification process of the continuous casting billet are obtained through the infrared temperature measurement method, the left, right, and upper boundaries of the temperature field are of Neumann type, with adiabatic boundary conditions, and the heat flux density value is taken as 0; the lower boundary adopts Dirichlet type boundary condition, and the measured surface temperature T d of the continuous casting billet is converted into the dimensionless temperature θ d according to formula (7); S334. Use the alternating implicit iterative method to solve the two-dimensional temperature field, decompose it into a one-dimensional implicit solution in alternating x and y directions, swap the order of the solution direction in odd and even steps to ensure the symmetry of the solution results, and construct a one-dimensional implicit solver at the same time. Use the Thomas algorithm to solve the tridiagonal matrix to improve the efficiency of solving the temperature field control equation.

[0040] For Dirichlet boundary conditions, the boundary node temperatures of the mesh are initialized to fixed values. ; For the Neumann boundary condition, the gradient term is modified by unilateral difference, and the right-hand term θ is set m value, ; For the Robin boundary condition, the convection parameter is introduced to realize the linear combination of temperature and gradient terms. After sorting, it is used to correct the boundary coefficient matrix of the implicit method; In a specific embodiment, the phase field, solute field and temperature field control equations are solved by Matlab software, the calculation results are displayed in real time, and the phase field data, solute field data and temperature field data at different times after the calculation is completed are recorded to form a series of VTK format file outputs with specified time steps; Furthermore, the VTK file is imported into ParaView software to further visualize the calculation results.

[0041] An embodiment of the present invention provides a phase field simulation system for a continuous casting solidification process based on dynamic temperature boundary conditions, the system being used to implement the above method, and comprising: Material parameter input unit, used to input the physical parameters and composition data of the steel to be studied, continuous casting machine mold parameters and secondary cooling condition parameters; Phase field model parameter input unit, used to determine the initial parameters used in phase field model calculation; Temperature boundary condition determination unit, used to determine the temperature boundary conditions of the crystallizer and secondary cooling zone in the continuous casting process; The equation solving unit solves the phase field, solute field and temperature field control equations according to the determined temperature boundary conditions to obtain the phase field variables, solute distribution and temperature field data evolving with time.

[0042] The data processing unit inputs the phase field, solute and temperature field data files into the post-processing software to extract and analyze the data of the dendrite growth process.

[0043] An embodiment of the present invention provides an information data processing terminal, comprising a processor and a memory for storing executable instructions; the processor is configured to execute the executable instructions to perform the above-mentioned phase field simulation method for the continuous casting solidification process based on dynamic temperature boundary conditions.

[0044] An embodiment of the present invention provides a computer-readable storage medium, on which a computer program is stored. When the computer program is executed by a processor, the above-mentioned phase-field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions is implemented. In the present invention, to improve the accuracy and physical rationality of the numerical simulation of the solidification process of the casting billet, the setting method of the temperature field boundary conditions has been improved. Compared with the Zero-Neumann type boundary condition where the heat fluxes at the upper, lower, left, and right boundaries are all zero in traditional phase-field simulations (as Figure 1 shown), the present invention adopts a boundary treatment method that is more in line with the actual working conditions: In the first embodiment ( Figure 2 ), without changing other calculation parameters (the same below), the lower boundary of the temperature field is set to a non-zero Neumann type boundary condition, that is, a heat flux density directed outward in a specified direction is assigned to the lower boundary to simulate the intense heat dissipation process that occurs in the initial stage when the molten steel enters the mold. This method can effectively reflect the true thermal state when the molten steel enters the mold, form a temperature gradient at the solidification interface, and promote the upward advancement of the solidification front, thus being closer to the characteristic of surface solidification first in the actual continuous casting process.

[0045] In the second embodiment ( Figure 3 ), the temperature data on the surface of the casting billet in the secondary cooling zone is obtained by infrared thermometry and used as the boundary input condition for the temperature field simulation. The left, right, and upper boundaries of the temperature field are set to the "N" type (adiabatic boundary), and the heat flux density is taken as 0; the lower boundary is set to the "D" type (Dirichlet boundary), and the measured surface temperature of the casting billet is directly assigned to the boundary after being converted into a dimensionless temperature. This setting method can accurately reflect the true heat dissipation state of the casting billet in the secondary cooling zone and effectively improve the simulation accuracy of the solute field distribution and interface evolution during the solidification process.

[0046] In the third embodiment ( Figure 4 ), the lower boundary of the temperature field is further set to the Robin type boundary condition, and the heat transfer coefficient and the cooling water temperature are specified in a structural form to achieve the co-control of the boundary heat flux and temperature difference. Compared with the Dirichlet condition, this method more fully considers the convective heat transfer mechanism between the casting billet and the spray cooling in each section of the secondary cooling zone, and has greater universality and engineering applicability.

[0047] The use of the multi-type boundary setting method described in the present invention not only significantly improves the accuracy of the solidification interface simulation calculation, but also makes the temperature field simulation results have higher physical authenticity, providing a reliable basis for cooling optimization design and defect prediction in the continuous casting process. The present invention solves the partial differential equations of the phase field, solute field and temperature field coupling through finite difference and finite volume methods and introduces a temperature field boundary type switching module. It further uses the alternating direction implicit iteration (ADI) method to solve the temperature field, realizing any combination of the three types of temperature field boundary conditions: Dirichlet, Neumann, and Robin. It is suitable for the calculation process of complex thermal boundary conditions in solidification, while improving the solution efficiency and ensuring numerical convergence. The typical heat flux density boundary conditions of the crystallizer and the secondary cooling zone are dimensionlessly normalized to be consistent with the temperature field control equation. The operating parameters such as the water flow rate in the secondary cooling zone and the continuous casting machine casting speed are introduced into the phase field model, expanding the application of phase field simulation to dendrite growth under different operating parameters of continuous casting solidification.

[0048] Although several embodiments of the present invention have been described herein, those skilled in the art will appreciate that modifications may be made to the embodiments herein without departing from the spirit of the present invention. The above embodiments are merely exemplary and should not be used as limitations on the scope of the present invention.

Claims

1. A phase field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions, characterized in that The method includes the following steps: S1. Collect the physical property parameters and composition data of the steel to be studied, the continuous caster mold parameters, and the secondary cooling condition parameters; S2. Select a phase-field method model and determine the initial parameters required for simulation calculation; S3. According to the physical property parameters of the steel to be studied and the phase-field method model, solve the control equations of the phase field, solute field, and temperature field to obtain the phase-field variables, solute distribution, and temperature field data evolving with time. Among them, when solving the temperature field control equation, the continuous caster mold parameters and the secondary cooling condition parameters are introduced into the phase-field method model, and a combination of Dirichlet, Neumann, and Robin types of thermal boundary conditions is used to simulate the heat transfer process of the actual boundary.

2. The phase-field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions according to claim 1, characterized in that In step S1, the physical property parameters include the solute concentration c of the molten steel ∞ , the density ρ of the molten steel, the solute equilibrium distribution coefficient k, the liquidus slope m, the liquid-phase solute diffusion rate D, the Lewis number Le, the Gibbs-Thomson coefficient, the melting point temperature T of pure iron M , the anisotropy strength ε, the thermal conductivity λ, the latent heat of solidification L, the specific heat capacity Cp, the thermocapillary length d0, the thermal diffusivity α; the continuous casting mold parameters include the casting speed and the shape and size of the mold; the secondary cooling condition parameters include the water volume of each section in the secondary cooling zone, the length of each section in the secondary cooling zone, and the cross-sectional size of the billet.

3. The phase field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions according to claim 1, characterized in that In step S2, the Karma model is selected as the phase-field method model, and the initial parameters include: the dimensionless undercooling θ of dendrite growth, the nucleation size R and position, the environmental noise ξ, the time step Δt, the number of iterative time steps nstep, the spatial step Δx, and the number of grids N.

4. The phase field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions according to claim 1, characterized in that, In step S3, the control equations and calculation processes of the phase field, solute field, and temperature field are as follows: S31. Introduce a phase-field order parameter φ to describe the solidification state of the system. When φ = -1, it represents the liquid phase; when φ = 1, it represents the solid phase; when φ varies from -1 to 1, it represents the solid-liquid interface. Solve the control equation containing the phase-field parameter φ based on the finite difference nine-point discretization format as shown in Equation (1): （1） where φ is the phase-field order parameter, U is the dimensionless concentration, θ is the dimensionless undercooling, Le is the Lewis number, λ is the coupling coefficient, k is the solute equilibrium partition coefficient, ξ is the environmental noise, and c ∞ is the solute concentration of the molten steel; A(ψ) is the anisotropy function of the solid-liquid interface energy, M is the dimensionless liquidus slope, and the specific form is shown in Equation (2): ， （2） In the formula, ε is the anisotropy strength, ω is 4, representing that the molten steel is a cubic crystal system metal structure, , is the initial grain orientation, m is the liquidus slope, L is the latent heat of solidification, and Cp is the specific heat capacity; S32. The solute field control equation is as shown in Equation (3): （3） In the formula is the reverse solute rejection term, which eliminates the non-equilibrium effect of solute diffusion, is the dimensionless solute diffusion coefficient; Use the finite volume method to solve the solute field control equation (3). Discretize the computational domain into control volumes, integrate the solute field control equation (3) within the element, and apply the divergence theorem to convert it into an interface flux balance; S33. The temperature field control equation is as shown in Equation (4): （4） wherein, is the dimensionless thermal diffusivity; According to the continuous caster mold parameters and the secondary cooling condition parameters, determine the type and specific values of the temperature boundary conditions; use the alternating direction implicit iteration method in the finite difference method to solve the temperature field control equation (4).

5. The phase field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions according to claim 4, wherein In step S33, the method for determining the temperature boundary conditions is as follows: S331. Judge the formation position of the solidification structure of the continuous casting billet: In the mold region, use the solidification square root law, and in the secondary cooling section, use the macroscopic heat transfer mathematical model to establish the fitting relationship between the shell thickness δ and the solidification time t, so as to judge whether the solidification position of the casting billet is within the mold or enters the secondary cooling zone and its specific position; S332. When the solidification position of the casting billet is within the mold, the following method is used to determine the boundary conditions: When the molten steel in the continuous casting process contacts the mold wall in the mold, the solidification of the molten steel is a non-zero Neumann boundary condition, and the specific heat flux forms are as shown in Equations (5) and (6): Heat flux density in the mold: (5) ， （6） Where, L d is the distance from the meniscus to the position of the shell during solidification of the molten steel, L m is the effective liquid level height of the molten steel in the mold, V c is the casting speed of the continuous caster, β is the mold shape coefficient, is the instantaneous heat flux density of the mold, is the average heat flux density of the mold measured actually, C w is the specific heat capacity of the mold cooling water, W m is the water flow rate of the mold, ΔT is the temperature difference of the cooling water entering and leaving the mold, S eff is the effective heated area of the mold copper plate; Nondimensionalize the above temperature field boundary conditions to be consistent with the dimensionless undercooling θ in the phase field. The nondimensional process of temperature in the phase field model is: （7） In Equation (7), T is the molten steel temperature in the calculation domain, and T M is the melting point temperature of pure iron; the non-zero Neumann temperature field boundary conditions in Equation (5) are dimensionless in the same way as Equation (7) to derive Equation (8), which is coupled to the temperature field control equation in the phase field model for solution. The specific dimensionless form of the heat flux density boundary condition in the mold is as follows: （8） Calculate the dimensionless degree of supercooling θ according to Equation (8). m value; The temperature field has upper, lower, left, and right boundaries. The heat flux density values of the upper, left, and right boundary conditions are all 0, indicating adiabatic boundary conditions, and the lower boundary condition is the calculated θ m value, which is the temperature boundary condition for coupling the casting speed input into the phase field model; S333. When the solidification position of the casting billet is in the secondary cooling section, the following method is used to determine the boundary conditions: In the secondary cooling section of the continuous casting process, the temperature boundary condition for the solidification of the molten steel adopts the Robin boundary condition, and the specific heat flux forms are respectively shown in Equations (9) and (10): Heat flux density in the full-roll section of the secondary cooling zone , (9) Heat flux density of the active section and segment in the secondary cooling zone , (10); where q f is the heat flux density in the full-roll section of the secondary cooling zone, h f is the heat transfer coefficient in the full-roll section of the secondary cooling zone, T f is the cooling water temperature in the full-roll section of the secondary cooling zone; q k is the heat flux density in the movable section and segment section of the secondary cooling zone, h k is the heat transfer coefficient in the movable section and segment section of the secondary cooling zone, T w is the cooling water temperature in the movable section and segment section of the secondary cooling zone, and w is the water flow density; The temperature boundary conditions in Equations (9) - (10) are transformed into the phase-field temperature boundary conditions in Equations (11) - (12) that are coupled with the water flow rate in the secondary cooling zone; Dimensionless heat flux density in the full-roll section of the secondary cooling zone (11) Dimensionless heat flux density of the active section and segment in the secondary cooling zone, (12); In formulas (11)-(12), θs1 and θs2 are respectively the dimensionless cooling water temperatures obtained by calculating T f and T w according to formula (7); The specific process of giving the boundary conditions of the temperature field is as follows: Calculate the dimensionless heat flux densities θ of the full-roll section, the movable section, and the segment section in the secondary cooling zone according to Equations (11)-(12). f and θ w When the solidification position of the continuous casting billet is in the full-roll section, use the value of θ f ; when it is in the movable section and the segment section, use the value of θ w value; The left, right, and upper boundaries of the temperature field are of the Neumann type, and the heat flux density value is taken as 0, that is, the adiabatic boundary condition; When the temperature boundary conditions of the continuous casting slab solidification process are obtained through the infrared temperature measurement method, the left, right, and upper boundaries of the temperature field are of Neumann type, and the heat flux density value is taken as 0, that is, the adiabatic boundary condition; the lower boundary adopts the Dirichlet type boundary condition, and the measured surface temperature T of the continuous casting slab d is converted into the dimensionless temperature θ according to Equation (7) d ; S334. The two-dimensional temperature field is solved by using the alternating implicit iteration method, which is decomposed into one-dimensional implicit solutions in the alternating x and y directions. The order of the solution directions is swapped in odd and even steps to ensure the symmetry of the solution results. At the same time, a one-dimensional implicit solver is constructed, and the Thomas algorithm is used to solve the tridiagonal matrix to improve the efficiency of solving the temperature field control equation.

6. The phase field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions according to claim 1, wherein In step S3, the control equations of the phase field, solute field, and temperature field are solved, the calculation results at that time are displayed in real time, and the phase field data, solute field data, and temperature field data at different times after the calculation is completed are recorded to form a data file; The data file is imported into the post-processing software to perform visualization processing on the calculation results.

7. The phase field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions according to claim 5, characterized in that In step S331, the determination of the solidification position is corrected by combining the nail shooting test and the infrared temperature measurement method.

8. A phase-field simulation system for the solidification process of continuous casting billets based on dynamic temperature boundary conditions, characterized in that, The system is used to implement the method according to any one of claims 1 - 7. The system includes: A material parameter input unit for inputting the physical property parameters and composition data of the steel to be studied, the continuous caster mold parameters, and the secondary cooling condition parameters; A phase-field model parameter input unit for determining the initial parameters used in the phase-field model calculation; A temperature boundary condition determination unit for determining the temperature boundary conditions of the mold and the secondary cooling zone in the continuous casting process; An equation solving unit for solving the control equations of the phase field, solute field, and temperature field according to the determined temperature boundary conditions to obtain the phase field variables, solute distribution, and temperature field data evolving with time; A data processing unit for inputting the phase field, solute, and temperature field data files into the post-processing software to extract and analyze the data of the dendrite growth process.

9. An information data processing terminal, comprising a processor and a memory for storing executable instructions; characterized in that, The processor is configured to execute the executable instructions to execute the phase-field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions according to any one of claims 1 - 7.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by the processor, it implements the phase-field simulation method for the solidification process of continuous casting billets based on dynamic temperature boundary conditions according to any one of claims 1 - 7.

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