Phase field simulation method and system for continuous casting billet solidification process based on dynamic temperature boundary condition

Abstract

The application relates to the technical field of metal solidification numerical simulation and computational material science, and provides a continuous casting billet solidification process phase field simulation method and system based on a dynamic temperature boundary condition; the method comprises the following steps: S1, collecting physical property parameters and component data of steel, and collecting continuous casting machine crystallizer parameters and secondary cooling working condition parameters; S2, selecting a phase field method model and determining initial parameters; S3, solving phase field, solute field and temperature field control equations to obtain phase field variables, solute distribution and temperature field data evolving with time; when the temperature field control equation is solved, the continuous casting process parameters are introduced into the phase field model, and the combination of three types of thermal boundary conditions, namely D, N and R, is considered to simulate the actual boundary heat transfer process. The application realizes the arbitrary combination and stable solution of the three types of thermal boundary conditions, significantly improves the adaptability and calculation efficiency of the phase field simulation to the complex thermal boundary conditions of the solidification process, and considers the analysis accuracy, calculation efficiency and engineering adaptability.

Description

Technical Field

[0001] The present invention relates to the field of metal solidification numerical simulation and computational material science technology, and in particular to a phase field simulation method and system for a continuous casting billet solidification process based on dynamic temperature boundary conditions. Background Art

[0002] During the solidification process of a continuous casting billet, the temperature field boundary conditions have a crucial influence on the dendrite growth morphology and segregation process. The thermal diffusivity is generally 3–4 orders of magnitude greater than the diffusion coefficient of the liquid phase solute. Traditional simulation methods often treat dendrite growth as an isothermal solidification process (e.g., with a constant undercooling or constant temperature gradient), ignoring the release of latent heat of solidification during dendrite growth. Zero-Neumann adiabatic boundary conditions are generally used for the temperature field. However, in actual continuous casting processes, the cooling conditions at the edges and center of the billet differ significantly, resulting in a non-isothermal solidification process. Furthermore, process parameters such as the water flow rate in the secondary cooling zone and the casting speed of the continuous casting machine vary dynamically, making a single temperature boundary condition incapable of accurately reflecting the non-uniform cooling process. For example, the existing patent (CN 117828950 A) uses zero-Neumann conditions. While achieving multi-field coupling, it fails to consider the dynamic adjustment of the temperature field boundary with process parameters. This makes it difficult to achieve corresponding temperature field adjustment at high casting speeds or when cooling is unevenly distributed, resulting in significant simulation errors.

[0003] Furthermore, existing technologies lack research on the switching mechanisms of temperature field boundary conditions. For example, the rapid solidification of thin continuous slabs is often accompanied by abrupt temperature gradient changes, requiring switching between Dirichlet, Neumann, or Robin boundary conditions based on the local solid fraction or interface migration velocity. 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 in 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 shortcomings of the existing technology and provide a phase field simulation method and system for the continuous casting slab solidification process based on dynamic temperature boundary conditions, which is suitable for the precise calculation and process optimization of the microstructure evolution of the cast slab in the steel continuous casting process.

[0006] The present invention adopts the following technical solutions:

[0007] In one aspect, the present invention provides a phase field simulation method for a continuous casting slab solidification process based on dynamic temperature boundary conditions, comprising:

[0008] S1. Collect the physical properties and composition data of the steel to be studied, the continuous casting machine mold parameters and the secondary cooling condition parameters;

[0009] S2. Select the phase field method model and determine the initial parameters required for simulation calculation;

[0010] 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.

[0011] 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.

[0012] 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.

[0013] 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.

[0014] 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:

[0015] 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):

[0016] (1)

[0017] Where 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 ambient noise, and c ∞ is the solute concentration of the molten steel; A(ψ) is the anisotropy function of the solid-liquid interface energy, and M is the dimensionless liquidus slope. The specific form is shown in formula (2):

[0018] , (2)

[0019] In the formula, ε is the anisotropic strength, ω is 4, which means that the molten steel is a cubic metal structure. , is the initial orientation of the grain, m is the liquidus slope, L is the latent heat of solidification, and Cp is the specific heat capacity;

[0020] S32, the solute field control equation is shown in formula (3):

[0021] (3)

[0022] In the formula is the antisolute retention term, which eliminates the non-equilibrium effect of solute diffusion. is the dimensionless solute diffusion coefficient;

[0023] The finite volume method is used to solve the solute field control equation (3). The computational domain is discretized into a control volume. The solute field control equation (3) is integrated within the unit and converted into an interface flux balance by applying the divergence theorem.

[0024] S33, the temperature field control equation is shown in formula (4):

[0025] (4)

[0026] Where, is the dimensionless thermal diffusivity;

[0027] 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).

[0028] 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.

[0029] In any of the possible implementations described above, a further implementation is provided, in step S33, the method for determining the temperature boundary condition is as follows:

[0030] 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.

[0031] S332. When the solidification position of the ingot is in the crystallizer, the boundary conditions are determined by the following method:

[0032] 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):

[0033] Heat flux in the crystallizer: (5)

[0034] , (6)

[0035] 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 crystallizer copper plate;

[0036] 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:

[0037] (7)

[0038] In formula (7), T is the temperature of the molten steel in the calculation domain, TM 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:

[0039] (8)

[0040] Calculate the dimensionless undercooling θ according to formula (8): m Value, as the specific value of the boundary condition under the temperature field;

[0041] 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;

[0042] S333. When the solidification position of the ingot is in the secondary cooling section, the boundary conditions are determined by the following method:

[0043] 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):

[0044] Heat flux density of the second cooling zone full roller section, , (9)

[0045] Heat flux density of active section and fan section in the second cooling zone, , (10);

[0046] Where q f is the heat flux density of the second cooling zone foot roll section, h f is the heat transfer coefficient of the second cooling zone roller section, T f is the cooling water temperature of the second cooling zone roller section; q k is the heat flux density of the active section and fan section of the secondary cooling zone, h k is the heat transfer coefficient of the active section and fan section of the secondary cooling zone, T w is the cooling water temperature of the active section and fan section of the secondary cooling zone, and w is the water flow density;

[0047] The temperature boundary conditions (9)-(10) are transformed into the phase field temperature boundary conditions (11)-(12) coupled with the water flow in the secondary cooling zone to ensure the accuracy of the temperature field boundary conditions in the simulation of the solidification process of the casting and improve the calculation accuracy;

[0048] The dimensionless heat flux density of the second cooling zone foot roller segment, (11)

[0049] The dimensionless heat flux density of the active section and fan-shaped section of the second cooling zone, (12);

[0050] In formulas (11) and (12), θs1 and θs2 are T f 、T w The dimensionless cooling water temperature calculated according to formula (7);

[0051] The specific process of giving the boundary conditions under the temperature field is as follows: Calculate the dimensionless heat flux density θ of the foot roller segment, active segment, and fan segment in the secondary cooling zone according to equations (11)-(12): f and θ w , when the solidification position of the billet is at the full roller section, use θ f , use θ in active segment and fan segment w value;

[0052] The left, right and upper boundaries of the temperature field are of Neumann type, with adiabatic boundary conditions and a value of 0;

[0053] When the infrared temperature measurement method is used to obtain clear temperature boundary conditions during the solidification process of the billet, the left, right and upper boundaries of the temperature field are Neumann type, adiabatic boundary conditions, and the value is 0; the lower boundary adopts the Dirichlet type boundary condition, and the measured billet surface temperature T d Convert to dimensionless temperature θ according to formula (7) d ;

[0054] 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 (or TDMA algorithm) to solve the tridiagonal matrix to improve the efficiency of solving the temperature field control equation.

[0055] Any possible implementation described above further provides an implementation, in step S3, for the Dirichlet boundary condition, the boundary node temperature of the grid initialization is set to a fixed value, ; For the Neumann boundary condition, the gradient term is modified by unilateral difference, and the right-hand term θ is set m value, , θ1 and θ2 are the dimensionless temperature values ​​of the first and second layers of the grid boundary, and Δx is the grid spacing; for the Robin boundary condition, the convection parameter is introduced to realize the linear combination of temperature and gradient terms, and the original formula is replaced by , λ is the thermal conductivity of steel, h f is the heat transfer coefficient, θ fis the dimensionless temperature of cooling water, which is used to correct the boundary coefficient matrix of the implicit method after sorting.

[0056] Any possible implementation as described above, further provides an implementation method, in step S3, solving the phase field, solute field and temperature field control equations, displaying the calculation results in real time, and recording the phase field data, solute field data, and temperature field data at different times after the calculation is completed to form a data file; importing the data file into post-processing software to visualize the calculation results.

[0057] Any possible implementation method as described above is further provided. In step S3, the phase field, solute field and temperature field control equations are solved by Matlab software to form a series of VTK format file outputs with specified time steps; the VTK file is imported into ParaView software to further visualize the calculation results.

[0058] In any of the possible implementations described above, a further implementation is provided, in which, in step S331, the solidification position determination is calibrated by combining a nail shooting test and an infrared temperature measurement method.

[0059] On the other hand, the present invention also provides a phase field simulation system for the continuous casting solidification process based on dynamic temperature boundary conditions, the system is used to implement the above method, and the system includes:

[0060] 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;

[0061] Phase field model parameter input unit, used to determine the initial parameters used in phase field model calculation;

[0062] Temperature boundary condition determination unit, used to determine the temperature boundary conditions of the crystallizer and secondary cooling zone in the continuous casting process;

[0063] 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.

[0064] 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.

[0065] On the other hand, the present invention also 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 execute the above-mentioned phase field simulation method of the continuous casting billet solidification process based on dynamic temperature boundary conditions.

[0066] In another aspect, the application also provides a computer readable storage medium, which stores a computer program, and the computer program is executed by a processor to implement the phase field simulation method for continuous casting billet solidification process based on dynamic temperature boundary conditions.

[0067] The application has the following beneficial effects:

[0068] 1. The method provided by the application successfully realizes the stable solution of any combination of Dirichlet, Neumann and Robin three types of thermal boundary conditions by introducing temperature field boundary type switching and combining the alternating direction implicit method, and significantly improves the adaptability and calculation efficiency of the phase field simulation to complex thermal boundary conditions in the solidification process.

[0069] 2. The application introduces the continuous casting process parameters such as water flow in the secondary cooling zone and casting speed into the phase field model, realizes the coupling of the thermal boundary condition and the actual production process, and expands the simulation capability of the method to dendrite growth behavior under different working conditions. It provides a reliable basis for cooling optimization design and defect prediction in the continuous casting process.

[0070] 3. The application performs dimensionless processing consistent with the temperature field control equation for the typical heat flow boundary of the crystallizer and the secondary cooling zone, improves the physical consistency and numerical stability of the model, and balances the analytical precision, calculation efficiency and engineering adaptability. BRIEF DESCRIPTION OF DRAWINGS

[0071] Figure 1 Fig. 1 shows the phase field, solute field and temperature field diagrams of the prior art in which the upper, lower, left and right boundary conditions of the temperature field are all Zero-Neumann conditions.

[0072] Figure 2 Fig. 2 shows the phase field, solute field and temperature field diagrams of Example 1 in which the upper, left and right boundary conditions of the temperature field are all Zero-Neumann conditions, and the lower boundary condition is a non-zero Neumann condition.

[0073] Figure 3 Fig. 3 shows the phase field, solute field and temperature field diagrams of Example 2 in which the left, right and upper boundary conditions of the temperature field are set to "N" type, and the lower boundary condition is a Dirichlet condition.

[0074] Figure 4 Fig. 4 shows the phase field, solute field and temperature field diagrams of Example 3 in which the lower boundary condition of the temperature field is a Robin condition.

[0075] Figure 5 Fig. 5 shows the flowchart of the phase field simulation method for continuous casting billet solidification process based on dynamic temperature boundary conditions according to an embodiment of the application. DETAILED DESCRIPTION

[0076] The following will describe in detail specific embodiments of the present invention with reference to the accompanying drawings. It should be noted that the technical features or combinations of technical features described in the following embodiments should not be considered isolated, and they can be combined with each other to achieve better technical effects.

[0077] like Figure 5 As shown, a phase field simulation method for a continuous casting slab solidification process based on dynamic temperature boundary conditions according to an embodiment of the present invention includes:

[0078] S1. Collect the physical properties and composition data of the steel to be studied, the continuous casting machine mold parameters and the secondary cooling condition parameters;

[0079] S2. Select the phase field method model and determine the initial parameters required for simulation calculation;

[0080] 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.

[0081] In a specific embodiment, 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.

[0082] In a specific embodiment, 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.

[0083] In a specific embodiment, in step S3, the phase field, solute field and temperature field control equations and calculation process are as follows:

[0084] 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):

[0085] (1)

[0086] Where 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 ambient noise, and c ∞ is the solute concentration of the molten steel; A(ψ) is the solid-liquid interface energy anisotropy function, and M is the dimensionless liquidus slope. The specific form is shown in formula (2):

[0087] , (2)

[0088] In the formula, ε is the anisotropic strength, ω is 4, which means that the molten steel is a cubic metal structure. , is the initial orientation of the grain, m is the liquidus slope, L is the latent heat of solidification, and Cp is the specific heat capacity;

[0089] S32, the solute field control equation is shown in formula (3):

[0090] (3)

[0091] In the formula is the antisolute retention term, which eliminates the non-equilibrium effect of solute diffusion. is the dimensionless solute diffusion coefficient;

[0092] The finite volume method is used to solve the solute field control equation (3). The computational domain is discretized into a control volume. The solute field control equation (3) is integrated within the unit and converted into an interface flux balance by applying the divergence theorem.

[0093] S33, the temperature field control equation is shown in formula (4):

[0094] (4)

[0095] Where, is the dimensionless thermal diffusivity;

[0096] 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).

[0097] The main boundary conditions for solving the temperature field are divided into three categories: Dirichlet, Neumann, and Robin boundary conditions.

[0098] 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.

[0099] 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.

[0100] In a specific embodiment, in step S33, the method for determining the temperature boundary condition is as follows:

[0101] In step S33, the method for determining the temperature boundary condition is as follows:

[0102] 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.

[0103] Furthermore, the model parameters can be calibrated by combining nail shooting tests and infrared temperature measurement.

[0104] 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.

[0105] S332. When the solidification position of the ingot is in the crystallizer, the boundary conditions are determined by the following method:

[0106] 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):

[0107] Heat flux in the crystallizer: (5)

[0108] , (6)

[0109] where L d is the distance from the meniscus to the shell position during the solidification of the liquid steel, L m is the effective liquid level in the mould, V c is the casting speed, β is the mould shape coefficient (calculated by the mould size), is the instantaneous heat flux density of the mould, is the average heat flux density of the mould actually measured, C w is the specific heat capacity of the cooling water of the mould, W m is the water flow of the mould, ΔT is the temperature difference of the cooling water entering and leaving the mould, S eff is the effective heating area of the copper plate of the mould;

[0110] The above temperature field boundary conditions are dimensionless in accordance with the dimensionless supercooling degree θ in the phase field, and the dimensionless process of temperature in the phase field model is:

[0111] (7)

[0112] In formula (7), T is the temperature of the calculated domain liquid steel, T M is the melting point temperature of pure iron; the non-zero Neumann temperature field boundary condition in formula (5) is dimensionless in formula (7), and formula (8) is derived after the same dimensionless is carried out, which is coupled to the temperature field control equation of the phase field model to solve, and the specific dimensionless form of the heat flux density boundary condition in the mould is:

[0113] (8)

[0114] According to formula (8), the value of the dimensionless supercooling degree θ m is calculated, which is the specific value of the lower boundary condition of the temperature field;

[0115] The temperature field has upper, lower, left and right boundaries, wherein the heat flux density values of the upper, left and right boundary conditions are all 0, indicating that they are all adiabatic boundary conditions, and the lower boundary condition is the calculated θ m value, that is, the temperature boundary condition of the coupled casting speed input to the phase field model;

[0116] S333、When the solidification position of the casting blank is in the secondary cooling section, the following method is used to determine the boundary condition:

[0117] In the secondary cooling section of the continuous casting process, the temperature boundary condition during the solidification of the liquid steel adopts the Robin boundary condition, and the specific heat flux forms of formula (9) and formula (10) are respectively:

[0118] Heat flux density of the second cooling zone full roller section, , (9)

[0119] Heat flux density of active section and fan section in the second cooling zone, , (10);

[0120] Where q f is the heat flux density of the second cooling zone foot roll section, h f is the heat transfer coefficient of the second cooling zone roller section, T f is the cooling water temperature of the second cooling zone roller section; q k is the heat flux density of the active section and fan section of the secondary cooling zone, h k is the heat transfer coefficient of the active section and fan section of the secondary cooling zone, T w is the cooling water temperature of the active section and fan section of the secondary cooling zone, and w is the water flow density;

[0121] The temperature boundary conditions (9)-(10) are transformed into the phase field temperature boundary conditions (11)-(12) coupled with the water flow in the secondary cooling zone to ensure the accuracy of the temperature field boundary conditions in the simulation of the solidification process of the casting and improve the calculation accuracy;

[0122] The dimensionless heat flux density of the second cooling zone foot roller segment, (11)

[0123] The dimensionless heat flux density of the active section and fan-shaped section of the second cooling zone, (12);

[0124] In formulas (11) and (12), θs1 and θs2 are T f 、T w The dimensionless cooling water temperature calculated according to formula (7);

[0125] The specific process of giving the boundary conditions under the temperature field is as follows: Calculate the dimensionless heat flux density θ of the foot roller segment, active segment, and fan segment in the secondary cooling zone according to equations (11)-(12): f and θ w , when the solidification position of the billet is at the full roller section, use θ f , use θ in active segment and fan segment w value;

[0126] The left, right and upper boundaries of the temperature field are of Neumann type, with adiabatic boundary conditions and heat flux density value of 0;

[0127] When the infrared temperature measurement method is used to obtain clear temperature boundary conditions during the solidification process of the billet, the left, right and upper boundaries of the temperature field are Neumann type, adiabatic boundary conditions, and the heat flux value is 0; the lower boundary adopts the Dirichlet type boundary condition, and the measured billet surface temperature Td Convert to dimensionless temperature θ according to formula (7) d ;

[0128] 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.

[0129] 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;

[0130] 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;

[0131] Furthermore, the VTK file is imported into ParaView software to further visualize the calculation results.

[0132] 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:

[0133] 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;

[0134] Phase field model parameter input unit, used to determine the initial parameters used in phase field model calculation;

[0135] Temperature boundary condition determination unit, used to determine the temperature boundary conditions of the crystallizer and secondary cooling zone in the continuous casting process;

[0136] 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.

[0137] 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.

[0138] 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.

[0139] The embodiment of the present invention provides a computer-readable storage medium having a computer program stored thereon, wherein the computer program, when executed by a processor, implements the above-mentioned phase field simulation method for the solidification process of a continuous casting billet based on dynamic temperature boundary conditions.

[0140] In the present invention, in order 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 condition is improved. Compared with the Zero-Neumann type boundary condition (such as Figure 1 As shown), the present invention adopts a boundary processing method that is more in line with actual working conditions:

[0141] In Example 1 ( 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. This specifies an outward heat flux at the lower boundary to simulate the intense heat dissipation that occurs when the molten steel initially enters the mold. This approach effectively reflects the actual thermal state of the molten steel upon entering the mold, forming a temperature gradient at the solidification interface and promoting the upward advancement of the solidification front, thus more closely resembling the surface-first solidification characteristic of the actual continuous casting process.

[0142] In Example 2 ( Figure 3 ), infrared temperature measurement is used to obtain surface temperature data in the secondary cooling zone of the ingot, which is used as the boundary input for the temperature field simulation. The left, right, and upper boundaries of the temperature field are set to "N" type (adiabatic boundaries), with a heat flux of 0. The lower boundary is set to "D" type (Dirichlet boundary), converting the measured surface temperature of the ingot to a dimensionless temperature and assigning it directly to the boundary. This setting accurately reflects the actual heat dissipation state of the ingot in the secondary cooling zone, effectively improving the simulation accuracy of the solute field distribution and interface evolution during the solidification process.

[0143] In Example 3 ( Figure 4), and the lower boundary of the temperature field is set as a Robin type boundary condition, and the heat transfer coefficient and cooling water temperature are specified in the form of a structure, so as to realize the joint control of the boundary heat flow and temperature difference. Compared with the Dirichlet condition, the method more fully considers the convective heat transfer mechanism between the casting blank and the spray cooling of each section of the secondary cooling zone, and is more universal and engineering applicable.

[0144] The multi-type boundary setting method of the application not only significantly improves the accuracy of the simulation calculation of the solidification interface, but also makes the simulation result of the temperature field have higher physical authenticity, and provides a reliable basis for the cooling optimization design and defect prediction in the continuous casting process. The application solves the coupled partial differential equations of the phase field, the solute field and the temperature field by the finite difference method and the finite volume method, and further uses the alternating direction implicit iteration (ADI) method to solve the temperature field, realizes the arbitrary combination of the Dirichlet, Neumann and Robin three types of boundary conditions of the temperature field, and is suitable for the calculation process of the complex thermal boundary condition in solidification, improves the solving efficiency, and guarantees the numerical convergence. The typical heat flux density boundary conditions of the crystallizer and the secondary cooling zone are subjected to dimensionless normalization processing consistent with the temperature field control equation, the working condition parameters such as the water flow of the secondary cooling zone and the casting speed of the continuous casting machine are introduced into the phase field model, and the application of the phase field simulation in the dendrite growth under different working condition parameters of the continuous casting blank is expanded.

[0145] Although several embodiments of the application have been described in this paper, those skilled in the art should understand that the embodiments in this paper can be changed without departing from the spirit of the application. The above-mentioned embodiments are only exemplary, and the embodiments in this paper should not be used as a limitation of the scope of the application.

Claims

1. A phase field simulation method for continuous casting solidification process based on dynamic temperature boundary conditions, characterized in that: The method comprises: S1. Collect the physical properties and composition data of the steel to be studied, the continuous casting machine mold parameters and the secondary cooling condition parameters; S2. Select the 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 based on the physical properties of the steel to be studied and the phase field model to obtain phase field variables, solute distribution, and temperature field data that evolve over time. When solving the temperature field control equation, the continuous casting machine mold parameters and secondary cooling condition parameters are introduced into the phase field model, and a combination of three types of thermal boundary conditions (Dirichlet, Neumann, and Robin) is used to simulate the heat transfer process at the actual boundary. The method for determining the temperature boundary conditions is as follows: Step 1: Determine the location of the continuous casting solidification structure formation: Use the solidification square root law in the crystallizer area and the macro heat transfer mathematical model in the secondary cooling section to establish a fitting relationship between the shell thickness δ and the solidification time t, thereby determining whether the solidification position of the continuous casting is located in the crystallizer or enters the secondary cooling zone, and its specific location; Step 2: 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 dimensionless form of the heat flux boundary condition in the mold is: ； Calculate θ according to the above formula m Value; where: L is the latent heat of solidification, Cp is the specific heat capacity, λ is the coupling coefficient, β is the crystallizer shape coefficient, L d V is the distance between the shell and the meniscus when the molten steel solidifies. c is the casting speed of the continuous casting machine; 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; Step 3: When the solidification position of the ingot is in the secondary cooling section, the boundary conditions are determined by the following method: The temperature boundary condition during the solidification of molten steel adopts the Robin boundary condition. The temperature boundary condition is converted into the phase field temperature boundary condition coupled with the water flow in the secondary cooling zone as follows: The dimensionless heat flux density of the second cooling zone foot roller segment, ; The dimensionless heat flux density of the active section and fan-shaped section of the second cooling zone, ; Where: θ is the dimensionless undercooling, h f is the heat transfer coefficient of the second cooling zone roller section, h k is the heat transfer coefficient of the active section and fan section of the secondary cooling zone, θs1 is the dimensionless cooling water temperature of the foot roller section of the secondary cooling zone, and θs2 is the dimensionless cooling water temperature of the active section and fan section of the secondary cooling zone; The specific process of giving the boundary condition value under the temperature field is as follows: when the solidification position of the billet is in the full roller section, use θ f Value, use θ in active segment and fan segment w The left, right and upper boundaries of the temperature field are of Neumann type, and the heat flux value is 0, that is, the adiabatic boundary condition; Step 4. 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.

2. The phase field simulation method for the continuous casting solidification process 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 ∞ , 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.

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

4. The phase field simulation method for the continuous casting solidification process based on dynamic temperature boundary conditions according to claim 1, characterized in that: 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, Le is the Lewis number, k is the solute equilibrium distribution coefficient, ξ is the ambient noise, and c ∞ is the solute concentration of the molten steel; A(ψ) is the anisotropy function of the solid-liquid interface energy, and M is the dimensionless liquidus slope. The specific form is shown in formula (2): ， （2） In the formula, ε is the anisotropic strength, ω is 4, which means that the molten steel is a cubic metal structure. , is the initial orientation of the grain, m is the liquidus slope; S32, the solute field control equation is shown in formula (3): （3） In the formula is the antisolute retention term, which eliminates 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 a control volume. The solute field control equation (3) is integrated within the unit and converted into an interface flux balance by applying the divergence theorem. S33, the temperature field control equation is shown in formula (4): （4） Where, is the dimensionless thermal diffusivity.

5. The phase field simulation method for the continuous casting solidification process based on dynamic temperature boundary conditions according to claim 1, characterized in that: In step S3, the phase field, solute field and temperature field control equations are solved, 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 data file; The data files are imported into post-processing software to visualize the calculation results.

6. The phase field simulation method for continuous casting solidification process based on dynamic temperature boundary conditions according to claim 1, characterized in that: In step 1, the solidification position determination is calibrated by combining the nail shooting test and infrared temperature measurement.

7. A phase field simulation system for continuous casting solidification process based on dynamic temperature boundary conditions, characterized in that: The system is used to implement the method according to any one of claims 1 to 6, and the system includes: 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; An equation solving unit solves the phase field, solute field and temperature field control equations according to the determined temperature boundary conditions to obtain phase field variables, solute distribution and temperature field data that evolve over time; 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.

8. 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 perform the phase field simulation method for the continuous casting solidification process based on dynamic temperature boundary conditions according to any one of claims 1 to 6.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by a processor, the phase field simulation method for the continuous casting slab solidification process based on dynamic temperature boundary conditions is implemented as described in any one of claims 1 to 6.

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