Method for calculating stress on nitride dendrite in forced convection during solidification of molten steel

Abstract

The application provides a method for calculating stress of nitride dendrites under forced convection in a steel liquid solidification process, and relates to the technical field of metal solidification. The composition of the steel grade and the involved thermodynamic and kinetic parameters and complex F-H boundary conditions are collected. The free boundary condition is added to the right side along the x direction velocity, and the stress growth model of Fe-C-M-N quaternary alloy steel in the flow field is established by coupling D2Q9 and cellular automaton model. Finally, the visual processing module is used to analyze the shape, size and flow field distribution of the nitride dendrites during solidification. The application optimizes the solidification technology, predicts the stress analysis of the nitride dendrites under forced convection, and provides theoretical guidance for improving the quality of the casting blank.

Description

Technical Field

[0001] This invention relates to the field of metal solidification technology, and in particular to a method for calculating the force exerted on nitride dendrites by forced convection during the solidification of molten steel. Background Technology

[0002] The steel industry is a vital pillar industry of the national economy. Predicting inclusion precipitation during the solidification process of molten steel is crucial for controlling cracks in cast billets and improving billet quality. Large inclusions affect the fatigue toughness of steel and can cause cracks, while the precipitation of small inclusions between grains can refine the grains, significantly improving the steel's strength, toughness, weldability, and yield stress. Therefore, the inclusion precipitation process is extremely important, and the continuously flowing molten steel further complicates this phenomenon.

[0003] In the paper "Thuswaldner J, Bernhard C. Polydimensional modelling of dendritic growth and microsegregation in multicomponent alloys[J]. Acta Materialsalia, 2010, 58(7):2738-2751," Michelic et al., based on the CA method, established a model for calculating the dendritic micromorphology and solute distribution during the solidification process of Fe-C-Si-Mn-PS alloys by dividing the computational domain in time and space and discretizing the mass transfer and heat transfer equations. The simulation objects cover the main solute elements of steel materials, but ignore the interactions between solutes. This model not only has a transition from the front of the plane to the dendritic solidification mode, but also reproduces the expected secondary arms.

[0004] Chinese patent "CN202111241849.4" describes a method for efficiently simulating dendrite growth using the phase-field method. This method establishes a phase-field model for microstructure simulation and sets initial conditions for the phase-field simulation. For columnar crystal simulation, the initial condition is a thin solid phase layer at the bottom of the computational domain; for single equiaxed crystal simulation, the initial condition is a small crystal nucleus at the center of the computational domain. The method determines the moving boundary position of the computational domain in the phase field, ensuring that the area above the boundary is a pure liquid phase region unaffected by diffusion interfaces. It also determines the moving boundary position of the computational domain in the solute field, ensuring that the area above the boundary is a liquid phase region unaffected by solute diffusion. The moving boundaries of the phase field and the concentration field are compared to determine the final computational domain boundary. The newly determined computational domain boundary is then used for phase-field simulation to obtain dendrite growth results. This invention only performs adaptive control on the computational domain, is easy to program, and can be used in conjunction with adaptive mesh methods, multi-mesh methods, etc., to improve computational efficiency.

[0005] The literature “Thuswaldner J, Bernhard C. Polydimensional modelling of dendritic growth and microsegregation in multicomponent alloys[J].Acta Materialia,2010,58(7):2738-2751” uses the CA method to discretize the computational domain, but in order to simplify the calculation, the interaction between solutes is not considered, and the calculation results have certain errors.

[0006] According to Chinese patent "CN202111241849.4", compared to other simulation methods, the phase-field method does not require tracking complex solid-liquid interfaces, can be coupled with other physical fields, and is closer to engineering reality, thus quantitatively simulating the microstructure growth process. Therefore, the phase-field method has received widespread attention and application since its inception, becoming an important tool for simulating material solidification and microstructure evolution. However, a drawback of the phase-field method is its large computational load and high computer performance requirements, which to some extent limits its development. In recent years, in order to reduce the limitation of computer performance on phase-field simulation, the optimization algorithms for solving phase-field models have been continuously improved and have made some progress. Currently, optimization algorithms applied to the phase-field method include adaptive mesh methods, multi-mesh methods, and parallel computing. Most of these methods optimize the mesh, which is complex to program and prone to non-convergence during the calculation process. Summary of the Invention

[0007] To address the shortcomings of existing technologies, this invention provides a method for calculating the stress on nitride dendrites during the solidification process of molten steel under forced convection. It predicts the stress on nitride dendrites under forced convection under different process conditions, optimizing solidification techniques and providing theoretical guidance for the stress-induced growth of dendrites under forced convection.

[0008] A method for calculating the force exerted on nitride dendrites by forced convection during the solidification of molten steel, specifically including the following steps:

[0009] Step 1: Collect the composition of the steel grade and the thermodynamic and kinetic parameters of the solidification process, as well as the flow field boundary conditions;

[0010] The thermodynamic and kinetic parameters of the solidification process specifically include density, relative atomic mass, pouring temperature, cooling rate, supercooling, liquidus slope, solute partition coefficient, Gibbs-Thomson coefficient, anisotropy parameter, growth kinetic parameters, solute diffusion coefficient, latent heat, density, thermal conductivity, and specific heat.

[0011] Step 2: Based on the theory of metal solidification, the cell growth direction, temperature field and solute field distribution at the nitride dendrite interface are calculated using a cellular automata model; at the same time, the D2Q9 model is used to calculate the flow field distribution.

[0012] Step 2.1: Introduce the cellular automata model of metal solidification theory to calculate the interface growth direction and temperature field distribution;

[0013] The cellular automaton model discretizes the computational domain in both time and space, with each discrete unit called a cell. It includes cell rules and a lattice grid; the space is discretized into a grid, and cells are distributed within the grid cells. It features discrete time steps; changes in cell states are defined between two time steps, and the cell state remains fixed within the same time step. Each cell has a finite number of states, meaning it can only switch between a fixed number of states. All cell state updates follow the same rules, and no cell enjoys special treatment. The rules of the cellular automaton are defined locally, meaning the cell state is only affected by the states of its surrounding cells.

[0014] First, assuming the solid-liquid interface is in thermodynamic equilibrium, the interface growth direction is calculated using the following formula based on the law of conservation of solute:

[0015]

[0016] in Indicates the direction of interface growth; f is the gradient operator; s Indicates the increase in solid fraction at the interface mesh; n x Indicates the scalar growth of the interface along the x-direction; n y This indicates that the interface grows a scalar along the y-direction. This represents the unit vector that grows along the x and y directions of the interface.

[0017] The interface curvature K is calculated using the following formula:

[0018]

[0019] in, and They represent the solid fraction f, respectively. s First-order partial derivatives on the x-axis and y-axis; solid fraction f s First, take the partial derivative with respect to the x-axis, then take the second partial derivative with respect to the y-axis; and These are the solid fractions f s The second-order partial derivatives on the x-axis and y-axis; the liquid phase temperature field distribution is calculated using the following formula:

[0020]

[0021] Where T represents temperature, λ represents heat transfer coefficient, ρ represents matrix density, and c p q represents the specific heat capacity of the matrix, L represents the heat released per unit mass of matrix during solidification, and q represents the heat released per unit mass of matrix during solidification. w This represents the heat flux density applied to the heat dissipation edge, where t is time and the unit is seconds; u x and u y These represent the fluid velocities along the x-axis and y-axis, respectively, in m·s. -1 , ρ and c p Different parameters are used in the solid, liquid, and two-phase regions. In the solid-liquid two-phase region, f is used. s Interpolation calculation, ρ = ρ S ·f s +ρ L ·(1-f s ), c p =c S,p ·f s +c L,p ·(1-f s );ρ S ρ represents the density of the solid phase; L c represents the density of the liquid phase. L,p Indicates the heat capacity of the liquid phase; c S,p Indicates the heat capacity of the solid phase;

[0022] Step 2.2: Probe the cell state, determine the fluid flow boundary, and calculate the velocity distribution of the flow field using the D2Q9 model;

[0023] The D2Q9 model, also known as the lattice Boltzmann model, is a 2D discrete velocity model with nine discrete velocity components. The specific model is established as follows:

[0024] Neglecting the effects of external forces, the velocity distribution of the flow field is calculated using the following formula:

[0025]

[0026] Where τ is the dimensionless single-step relaxation time, and f i () is the liquid phase particle distribution function, f i eq (x,t) is the liquid phase particle equilibrium distribution function; where c i Let c be the migration velocity of the liquid fluid particles along the i-th direction of the lattice, and Δt be the time step; where c i and weighting coefficient w i The following formula yields:

[0027]

[0028]

[0029] Where c is the lattice velocity; the matrix density ρ, macroscopic velocity u, and hydrodynamic viscosity v are calculated by the following formulas:

[0030]

[0031] Where f i Let c be the entry node distribution function, where i = 0, 1, 2, 3, 4, 5, 6, 7, 8 are the ordinal numbers of the corner nodes, and c is the indices of the corner nodes. s The velocity of sound in the lattice, the velocity of the fluid along the x-axis, and the velocity of the fluid along the y-axis are calculated using the following formulas:

[0032]

[0033]

[0034] Wherein, the inlet density ρ in Export density ρ out Configure it yourself;

[0035] Step 2.3: Combine the flow field velocity distribution with the calculation of the solute field distribution using a cellular automata model;

[0036] The heat transfer in the solute field of the hot phase is calculated using the following formula:

[0037]

[0038] The heat transfer in a solid solute field is calculated using the following formula:

[0039]

[0040] Among them, c L,i With c S,i These represent the concentrations of the i-th element in the solid and liquid phases, respectively, in wt.%; i = 1, 2, 3…n-1, where the n-th element represents the solvent; D s,i and This represents the matrix of diffusion coefficient of solute i in the solid and Darken coefficient in the liquid phase, in units of m. 2 ·s -1 ;▽ represents the gradient operator, This represents the Darken coefficient matrix in the liquid phase. To simplify calculations, it is assumed that the coefficients are located along the x-axis and y-axis. It is calculated using the following formula:

[0041]

[0042] Where, δ ki and Represents the Kronecker delta function and the single Darken coefficient matrix in the liquid phase; xm x i With x j Let m, i, and j represent the mole fractions of elements m, i, and j, respectively. and a m Let a and b represent the self-diffusion coefficient and activity of element m, respectively, where k is the element and a is the activity of element m. m Calculated using the following formula:

[0043]

[0044] Where [%n] and [%m] represent the concentrations of elements n and m based on 1% by mass, respectively. The solute interaction coefficient and the undercooling at the solid-liquid interface are calculated using the following formula:

[0045]

[0046] Where Γ is the Gibbs-Thomson coefficient, K·m; K is the interface curvature, a function of the solid-liquid interface normal phase and growth direction. Calculated using the following formula:

[0047]

[0048] in, θ and ε represent the angles between the growth direction and the positive x-axis, respectively, in rad, and represent the anisotropy parameters, where the angles are... The following formula is used for calculation:

[0049]

[0050] According to the law of conservation of solute at the interface, the growth rate of the interface cells under thermodynamic equilibrium is calculated using the following formula:

[0051]

[0052] in, For solute transport, only the effect of liquid-phase solute interactions on dendrite growth is considered; D n S,i and D nL,ij Let m represent the diffusion coefficient of solute i in the solid phase and the Darken coefficient matrix in the liquid phase, respectively. 2 ·s -1 ;c L,i Indicates the interfacial cellular fluid phase composition; c S,i Indicates the solid phase composition of the interface cell; Let i be the thermodynamic equilibrium concentration of element i at the solid-liquid interface. Let i be the solid-phase thermodynamic equilibrium concentration of element i at the solid-liquid interface; Growth rate at the solidification front of the interfacial cells; The direction of interface growth is indicated by the growth rate of the solidification front of the interface cell. The growth of the interface cell per unit time step is calculated by the increase in solid fraction, as shown in the following formula:

[0053]

[0054]

[0055]

[0056] in, and The cellular solid phase indices Δf of NbN at the previous time step and at this time step, respectively. s,NbN The increase in the solid fraction; Δt is the time step; Indicates along The direction passes through the center of the cell by a unit length; Δl = 1 μm is the length of the grid cell; θ represents the angle between the dendrite growth direction and the x-axis direction. The velocity of the cell front is expressed in m·s. -1 , Δf s Δl represents the increase in solid fraction at the interface mesh; Δl is the cell size.

[0057] The forced convection stress simulation for NbN dendrites makes the following assumptions: the NbN precipitated in the steel does not contain other phases; only NbN precipitation in the liquid phase is considered; the increase in interfacial energy during growth is ignored; the heat change during NbN precipitation is ignored; when a cell in the computational domain meets the NbN precipitation condition, the cell is marked and its mesh is refined to 3×3, and its growth is calculated based on dynamic chemical equilibrium, as shown in the following formula:

[0058]

[0059] Among them, M NbN Δx represents the relative molecular mass of NbN; Δf represents the amount of reaction; NbN ρ represents the increase in NbN solid fraction zen; Fe ρ represents the density of Fe; NbN Indicates the density of NbN;

[0060] Step 3: Using the model established above, perform visualization processing to display the shape, size, and flow field distribution of MN dendrites during solidification.

[0061] First, consider the solid boundary point x. b Suppose it has a virtual equilibrium distribution function as follows:

[0062]

[0063] Where u f It is the solid phase inside x f The fluid velocity at that location, u bf Let ρ(x) be the undetermined virtual velocity. f ,t) represents x f The matrix density at time t, in order to solve u bf Construct an interpolation factor α, and α is related to the solid boundary x. w Location-related, u bf It can be obtained from the following formula:

[0064]

[0065]

[0066]

[0067] Where β is used to calculate x f The interpolation factor, u, constructed from the bounce distribution w It is x w Interface movement speed; x b For solid-phase boundary nodes;

[0068] To reduce calculation errors caused by excessively large time steps when calculating cell growth based on dynamic chemical equilibrium, the cells are spatially refined into 3×3 subdivisions and simultaneously subdivided temporally. During the calculation of forced convection on NbN dendrite growth, the growth within one time step is calculated repeatedly in cycles to minimize errors caused by the time step. When the NbN volume in the matrix cell increases and comes into contact with adjacent liquid phase cells, the adjacent liquid phase cells are subdivided into 3×3 FH-format lower boundary NbN precipitate cells, allowing NbN growth to continue. Data analysis and visualization software are used to display the shape, size, and flow field distribution of the NbN dendrites during solidification.

[0069] The beneficial effects of adopting the above technical solution are as follows:

[0070] This invention provides a method for calculating the stress on nitride dendrites during the solidification of molten steel under forced convection. This invention considers the interaction between solutes during the dendrite stress analysis under the flow field and employs a modified FH scheme to improve the accuracy of the calculation. To reduce calculation errors caused by excessively large time steps, the cell is spatially refined to 3×3 and simultaneously temporally refined. When calculating the growth of MN dendrites under forced convection, the growth within one time step is calculated repeatedly in cycles to reduce calculation errors caused by the time step. Attached Figure Description

[0071] Figure 1A flowchart illustrating the method for calculating the force exerted on nitride dendrites by forced convection during the solidification process of molten steel, provided in this embodiment of the invention.

[0072] Figure 2 A flowchart illustrating the calculation procedure for the forced convection force on nitride dendrites provided in this embodiment of the invention;

[0073] Figure 3 This is a schematic diagram of the D2Q9 model according to an embodiment of the present invention;

[0074] Figure 4 This is a schematic diagram of the node distribution function provided in an embodiment of the present invention;

[0075] Figure 5 A schematic diagram of the complex FH format provided in the embodiments of the present invention;

[0076] Figure 6 The simulation diagram of the evolution of NbN dendrite morphology and Nb solute distribution in the flow field provided in the embodiment of the present invention. Detailed Implementation

[0077] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0078] A method for calculating the stress on nitride dendrites caused by forced convection during the solidification of molten steel. This example uses a Fe-C-Nb-N quaternary alloy system. Figure 1 As shown, the specific steps include:

[0079] Step 1: Collect the composition of the steel grade and the thermodynamic and kinetic parameters of the solidification process, as well as the flow field boundary conditions;

[0080] The thermodynamic and kinetic parameters of the solidification process specifically include density, relative atomic mass, pouring temperature, cooling rate, supercooling, liquidus slope, solute partition coefficient, Gibbs-Thomson coefficient, anisotropy parameter, growth kinetic parameters, solute diffusion coefficient, latent heat, density, thermal conductivity, and specific heat.

[0081] In this embodiment, the steel composition used is shown in Table 1;

[0082] Table 1 Main Components of Steel Grades

[0083]

[0084] Step 2: Based on the theory of metal solidification, the cell growth direction, temperature field and solute field distribution at the nitride dendrite interface are calculated using a cellular automata model; at the same time, the D2Q9 model is used to calculate the flow field distribution.

[0085] Step 2.1: Introduce the cellular automata model of metal solidification theory to calculate the interface growth direction and temperature field distribution;

[0086] The cellular automaton model discretizes the computational domain in both time and space, with each discrete unit called a cell. It includes cell rules and a lattice grid; the space is discretized into a grid, and cells are distributed within the grid cells. It features discrete time steps; changes in cell states are defined between two time steps, and the cell state remains fixed within the same time step. Each cell has a finite number of states, meaning it can only switch between a fixed number of states. All cell state updates follow the same rules, and no cell enjoys special treatment. The rules of the cellular automaton are defined locally, meaning the cell state is only affected by the states of its surrounding cells.

[0087] First, assuming the solid-liquid interface is in thermodynamic equilibrium, the interface growth direction is calculated using the following formula based on the law of conservation of solute:

[0088]

[0089] in Indicates the direction of interface growth; f is the gradient operator; s Indicates the increase in solid fraction at the interface mesh; n x Indicates the scalar growth of the interface along the x-direction; n y This indicates that the interface grows a scalar along the y-direction. This represents the unit vector that grows along the x and y directions of the interface.

[0090] The interface curvature K is calculated using the following formula:

[0091]

[0092] in, and They represent the solid fraction f, respectively. s First-order partial derivatives on the x-axis and y-axis; solid fraction f s First, take the partial derivative with respect to the x-axis, then take the second partial derivative with respect to the y-axis; and These are the solid fractions f s Second-order partial derivatives on the x-axis and y-axis;

[0093] The liquid phase temperature field distribution is calculated using the following formula:

[0094]

[0095] Where T represents temperature, λ represents heat transfer coefficient, ρ represents matrix density, and cp q represents the specific heat capacity of the matrix, L represents the heat released per unit mass of matrix during solidification, and q represents the heat released per unit mass of matrix during solidification. w This represents the heat flux density applied to the heat dissipation edge, where t is time and the unit is seconds; u x and u y These represent the fluid velocities along the x-axis and y-axis, respectively, in m·s. -1 , ρ and c p Different parameters are used in the solid, liquid, and two-phase regions. In the solid-liquid two-phase region, f is used. s Interpolation calculation, ρ = ρ S ·f s +ρ L ·(1-f s ), c p =c S,p ·f s +c L,p ·(1-f s );ρ S ρ represents the density of the solid phase; L c represents the density of the liquid phase. L,p Indicates the heat capacity of the liquid phase; c S,p Indicates the heat capacity of the solid phase;

[0096] Step 2.2: Probe the cell state, determine the fluid flow boundary, and calculate the velocity distribution of the flow field using the D2Q9 model, such as... Figure 3 As shown;

[0097] The D2Q9 model, also known as the lattice Boltzmann model, is a 2D discrete velocity model with nine discrete velocity components. The specific model is established as follows:

[0098] Neglecting the effects of external forces, the velocity distribution of the flow field is calculated using the following formula:

[0099]

[0100] Where τ is the dimensionless single-step relaxation time, and f i () is the liquid phase particle distribution function, f i eq (x,t) is the liquid phase particle equilibrium distribution function; where c i Let c be the migration velocity of the liquid fluid particles along the i-th direction of the lattice, and Δt be the time step; where c i and weighting coefficient w i The following formula yields:

[0101]

[0102]

[0103] Where c is the lattice velocity; the matrix density ρ, macroscopic velocity u, and hydrodynamic viscosity v are calculated by the following formulas:

[0104]

[0105] Where f i Let c be the entry node distribution function, where i = 0, 1, 2, 3, 4, 5, 6, 7, 8 are the ordinal numbers of the corner nodes, and c is the indices of the corner nodes. s The velocity of sound in the lattice, the velocity of the fluid along the x-axis, and the velocity of the fluid along the y-axis are calculated using the following formulas, with the inlet node distribution as shown. Figure 4 As shown:

[0106]

[0107]

[0108] Wherein, the inlet density ρ in Export density ρ out Configure it yourself;

[0109] Step 2.3: Combine the flow field velocity distribution with the calculation of the solute field distribution using a cellular automata model;

[0110] The calculation of the liquid phase solute field is a heat transfer process. The heat transfer of the hot phase solute field is calculated using the following formula:

[0111]

[0112] The heat transfer in a solid solute field is calculated using the following formula:

[0113]

[0114] Among them, c L,i With c S,i These represent the concentrations of the i-th element in the solid and liquid phases, respectively, in wt.%; i = 1, 2, 3…n-1, where the n-th element represents the solvent; D s,i and This represents the matrix of diffusion coefficient of solute i in the solid and Darken coefficient in the liquid phase, in units of m. 2 ·s -1 ;▽ represents the gradient operator, This represents the Darken coefficient matrix in the liquid phase. To simplify calculations, it is assumed that the coefficients are located along the x-axis and y-axis. It is calculated using the following formula:

[0115]

[0116] Where, δ ki and Represents the Kronecker delta function and the single Darken coefficient matrix in the liquid phase; x m x i With x j Let m, i, and j represent the mole fractions of elements m, i, and j, respectively. and a m Let a and b represent the self-diffusion coefficient and activity of element m, respectively, where k is the element and a is the activity of element m. m Calculated using the following formula:

[0117]

[0118] Where [%n] and [%m] represent the concentrations of elements n and m based on 1% by mass, respectively. Solute interaction coefficients. In this embodiment, the solute interaction coefficients are shown in Table 2:

[0119] Table 2 Solute interaction coefficients

[0120]

[0121] The undercooling at the solid-liquid interface is calculated using the following formula:

[0122]

[0123] Where Γ is the Gibbs-Thomson coefficient, K·m; K is the interface curvature, a function of the solid-liquid interface normal phase and growth direction. Calculated using the following formula:

[0124]

[0125] in, θ and ε represent the angles between the growth direction and the positive x-axis, respectively, in rad, and represent the anisotropy parameters, where the angles are... The following formula is used for calculation:

[0126]

[0127] According to the law of conservation of solute at the interface, the growth rate of the interface cells under thermodynamic equilibrium is calculated using the following formula:

[0128]

[0129] in, For solute transport, only the effect of liquid-phase solute interactions on dendrite growth is considered; D n S,i and D nL,ij Let m represent the diffusion coefficient of solute i in the solid phase and the Darken coefficient matrix in the liquid phase, respectively. 2·s -1 ;c L,i Indicates the interfacial cellular fluid phase composition; c S,i Indicates the solid phase composition of the interface cell; Let i be the thermodynamic equilibrium concentration of element i at the solid-liquid interface. Let i be the solid-phase thermodynamic equilibrium concentration of element i at the solid-liquid interface; Growth rate at the solidification front of the interfacial cells; The direction of interface growth is indicated by the growth rate of the solidification front of the interface cell. The growth of the interface cell per unit time step is calculated by the increase in solid fraction, as shown in the following formula:

[0130]

[0131]

[0132]

[0133] in, and The cellular solid phase indices Δf of NbN at the previous time step and at this time step, respectively. s,NbN The increase in the solid fraction; Δt is the time step; Indicates along The direction passes through the center of the cell by a unit length; Δl = 1 μm is the length of the grid cell; θ represents the angle between the dendrite growth direction and the x-axis direction. The velocity of the cell front is expressed in m·s. -1 , Δf s Δl represents the increase in solid fraction at the interface mesh; Δl is the cell size.

[0134] The forced convection stress simulation for NbN dendrites makes the following assumptions: the NbN precipitated in the steel does not contain other phases; only NbN precipitation in the liquid phase is considered; the increase in interfacial energy during growth is ignored; the heat change during NbN precipitation is ignored; when a cell in the computational domain meets the NbN precipitation condition, the cell is marked and its mesh is refined to 3×3, and its growth is calculated based on dynamic chemical equilibrium, as shown in the following formula:

[0135]

[0136] Among them, M NbN Δx represents the relative molecular mass of NbN; Δf represents the amount of reaction; NbN ρ represents the increase in NbN solid fraction zen; Fe ρ represents the density of Fe; NbN Indicates the density of NbN;

[0137] Step 3: Using the model established above, perform visualization processing to display the shape, size, and flow field distribution of MN dendrites during solidification.

[0138] Therefore, complex FH boundary conditions are as follows Figure 5 As shown, firstly, for the solid boundary point x b Suppose it has a virtual equilibrium distribution function as follows:

[0139]

[0140] Where u f It is the solid phase inside x f The fluid velocity at that location, u bf Let ρ(x) be the undetermined virtual velocity. f ,t) represents x f The matrix density at time t, in order to solve u bf Construct an interpolation factor α, and α is related to the solid boundary x. w Location-related, u bf It can be obtained from the following formula:

[0141]

[0142]

[0143]

[0144] Where β is used to calculate x f The interpolation factor, u, constructed from the bounce distribution w It is x w Interface movement speed; x b For solid-phase boundary nodes;

[0145] To reduce calculation errors caused by excessively large time steps when calculating cell growth based on dynamic chemical equilibrium, the cells are spatially refined into 3×3 subdivisions and simultaneously subdivided temporally. During the calculation of forced convection on NbN dendrite growth, the growth within one time step is calculated repeatedly in cycles to minimize errors caused by the time step. When the NbN volume in the matrix cell increases and comes into contact with adjacent liquid phase cells, the adjacent liquid phase cells are subdivided into 3×3 FH-format lower boundary NbN precipitate cells, allowing NbN growth to continue. Data analysis and visualization software are used to display the shape, size, and flow field distribution of the NbN dendrites during solidification.

[0146] In this embodiment, the model parameter values ​​involved in the calculation process are shown in Table 3:

[0147] Table 3 Model Parameters

[0148]

[0149] The embodiment uses the Visual Studio 2019 platform and the C++ language to write a mathematical model for calculating the force on MN dendrites under forced convection. Figure 2 The numerical simulation program shown is implemented, and a visualization module is developed to obtain the following results: Figure 6 The figure shows the evolution of Nb dendrite morphology and Nb solute distribution in the solidification flow field of molten steel. The numerical simulation yields a dendrite growth model under the flow field, which provides theoretical guidance for optimizing solidification technology, predicting the stress on dendrites under forced convection, and improving the quality of cast billets.

[0150] The above description is merely a preferred embodiment of this disclosure and an explanation of the technical principles employed. Those skilled in the art should understand that the scope of the invention involved in the embodiments of this disclosure is not limited to technical solutions formed by specific combinations of the above-described technical features, but should also cover other technical solutions formed by arbitrary combinations of the above-described technical features or their equivalents without departing from the above-described inventive concept. For example, technical solutions formed by substituting the above-described features with (but not limited to) technical features with similar functions disclosed in the embodiments of this disclosure.

Claims

1. A method for calculating the stress on nitride dendrites in a molten steel solidification process by forced convection, characterized in that, It comprises the following steps: Step 1: collect the composition of the steel grade and the thermodynamic and kinetic parameters of the solidification process, the flow field boundary conditions; Step 2: based on the metal solidification theory, the cellular automaton model is used to calculate the growth direction of the nitride dendrite interface cell, the temperature field and the solute field distribution; at the same time, the D2Q9 model is used to calculate the flow field distribution; The cellular automaton model in step 2 is to discretize the calculation domain in time and space, and each discrete unit is called a cell; including the rules of the cell and the lattice grid, the space is discretized into a grid, and the cells are distributed in each grid of the grid; The time step is discrete; the change of cell state is defined between two time steps, and the state of the cell is fixed within the same time step; the number of states of each cell is limited, that is, the cell can only switch among a few fixed states, and the update of all cell states follows the same rules, and there is no cell enjoying special treatment; the rules of cellular automata are defined locally, that is, the state of a cell is only affected by the states of its surrounding cells; Said step 2 specifically comprises the following steps: Step 2.1: introduce the cellular automaton model of metal solidification theory to calculate the interface growth direction and temperature field distribution; Firstly, it is assumed that the solid-liquid interface is in a state of thermodynamic equilibrium, and the interface growth direction is calculated according to the solute conservation law as follows: （1）； wherein represents the interface growth direction; is the gradient operator; f s represents the increase of the interface mesh solid fraction; n x represents the interface growth scalar along the x direction; n y represents the interface growth scalar along the y direction; , represents the interface growth unit vector along the x, y directions; The interface curvature K is calculated by the following formula: （2）； wherein, and respectively represent the solid phase rate f s the first-order partial derivative on the x-axis and the y-axis; is the solid phase rate f s the second-order partial derivative on the y-axis after the first-order partial derivative on the x-axis, and respectively represent the solid phase rate f s the second-order partial derivative on the x-axis and the y-axis; The calculation of the liquid phase temperature field distribution is calculated by the following formula: （3）； wherein T represents temperature, λ represents heat transfer coefficient, ρ represents matrix density, c p represents matrix specific heat, L represents heat released during solidification per unit mass of matrix, q w represents heat flux density loaded on the heat dissipation side, t is time, with unit of s; u x and u y are respectively the velocity of fluid in x-axis and y-axis, with unit of m·s -1 , ρ and c p adopt different parameters in solid, liquid phase and two-phase region respectively, f s interpolation calculation is adopted in solid-liquid two-phase region, ρ= ρ S ·f s + ρ L ·(1-f s ), c p = c S,p ·f s + c L,p ·(1-f s ); ρ S represents solid phase density; ρ L represents liquid phase density; c L,p represents liquid phase heat capacity; c S,p represents solid phase heat capacity; Step 2.2: detect the cell state, determine the fluid flow boundary, and calculate the flow field velocity distribution by using the D2Q9 model; The D2Q9 model in step 2.2 is a 2D space 9 discrete velocity model of lattice Boltzmann model, and the specific model is established as follows: Neglecting external force, the velocity distribution of the flow field is calculated by the following formula: ； wherein is the dimensionless single-step relaxation time, f i is the liquid phase particle distribution function, is the liquid phase particle equilibrium distribution function; wherein c i is the migration velocity of the liquid phase fluid particle in the direction of the lattice i, is the time step; wherein c i and the weighting factor w i is obtained from the following equation: （4）； （5）； Wherein, c is the lattice velocity; the base density ρ, the macroscopic velocity u and the fluid dynamics viscosity v are calculated by the following formula: （6）； where f i is the inlet node distribution function, where i = 0, 1, 2, 3, 4, 5, 6, 7, 8 is the corner node ordinal, c s is the lattice sound speed, the fluid x-axis direction velocity, and the fluid y-axis direction velocity are calculated from the following equations: （7）； （8）； Wherein, the inlet density p in , the outlet density p out is set automatically; Step 2.3: combined with the flow field velocity distribution, the solute field distribution is calculated by using the cellular automaton model; Step 2.3 specifically: the heat transfer of the hot phase solute field is calculated by the following formula: （9）； The heat transfer of the solid phase solute field is calculated by the following formula: （10）； where c L,i and c S,i respectively represent the concentration of the ith element in the solid and liquid phases, with units of wt.%; i = 1, 2, 3...n-1, and the nth element represents the solvent; D s,i and represents the diffusion coefficient of solute i in the solid and the Darken coefficient matrix in the liquid phase, with units of m 2 ·s -1 ; ▽ is the gradient operator, represents the Darken coefficient matrix in the liquid phase. To simplify the calculation, it is assumed that the is calculated by the following formula: （11）； where δ ki and denote the Kronecker delta function and a single Darken coefficient matrix in the liquid phase; x m , x i and x j denote the mole fraction of elements m, i, j, respectively; and a m denote the self-diffusion coefficient of element m and the activity of element m, respectively, k is the element, a m is calculated from the following equation: （12）； where [%n] and [%m] represent the concentration of element n and element m, respectively, based on 1% by mass, The solute interaction coefficient, the degree of supercooling at the solid-liquid interface is calculated from the following formula: （13）； Wherein, Γ is the Gibbs-Thomson coefficient, K·m; K is the interface curvature, and the function f(φ,θ) of the solid-liquid interface normal and growth direction is calculated by the following formula: （14）； where φ and θ are the angles between the growth direction and the positive x-axis, respectively, in rad, denotes the anisotropy parameter, where the angle φ is calculated as follows: （15）； According to the solute conservation law at the interface, the growth velocity of the interface cell under the thermodynamic equilibrium state is solved by the following formula: （16）； where, is the solute transport term, which only considers the effect of solute-solute interaction in the liquid phase on dendrite growth; D n S,i and DnL,ij represent the diffusion coefficient of solute i in the solid phase and the Darken coefficient matrix in the liquid phase, respectively, m 2 ·s -1 ; cL,i represents the interfacial cell liquid phase composition; cS,i represents the interfacial cell solid phase composition; c* L,i is the liquid phase thermodynamic equilibrium concentration of element i at the solid-liquid interface; c* S,i is the solid phase thermodynamic equilibrium concentration of element i at the solid-liquid interface; the interfacial cell solidification front growth velocity; represents the interfacial growth direction, and the growth of the interfacial cell within a time step is calculated by the increase of the solid fraction through the solved interfacial cell solidification front growth velocity, as shown in the following equation: （17）； （18）； （19）； where, and are the cellular solid fraction of NbN at the previous time and the current time, respectively is the increase of the solid fraction; Δt is the time step; L φ represents the unit length passing through the center of the cell along the direction; Δl = 1 μm is the grid cell length; θ represents the angle between the dendrite growth direction and the x-axis direction, represents the interface cell front velocity, with the unit of m·s -1 , Δf s represents the increase of the interface grid solid fraction; Δl is the cell size; The simulation calculation of forced convection on NbN dendrite stress calculation is based on the following assumptions: the NbN precipitated in the steel does not contain other phases; only the NbN precipitation in the liquid phase is considered; the increase of interface energy during growth is ignored; the heat change when NbN precipitates is ignored; when a cell in the calculation region meets the NbN precipitation condition, mark the cell and perform 3*3 grid refinement on the cell, and calculate its growth according to the dynamic chemical equilibrium, as shown in the following formula: （20）； wherein MNbN represents the relative molecular mass of NbN; Δx represents the reaction amount; Δf NbN represents the increase in the solid phase fraction of NbN, zEN; p Fe represents the density of Fe; p NbN represents the density of NbN; Step 3: through the above D2Q9 model, the shape, size and flow field distribution of MN dendrite during solidification are displayed by visual processing.

2. The method according to claim 1, wherein the method is characterized by: The thermodynamic and kinetic parameters of the solidification process include density, relative atomic mass, pouring temperature, cooling rate, supercooling degree, liquidus slope, solute partition coefficient, Gibbs-Thomson coefficient, anisotropy parameter, growth kinetics parameter, solute diffusion coefficient, latent heat, density, thermal conductivity, specific heat.

3. The method according to claim 1, wherein the method is characterized by: The step 3 is specifically: first, the solid phase boundary point x b Assuming it has a virtual state equilibrium distribution function as follows: （21）； where u f is the fluid velocity inside the solid phase at x f t, u bf is the virtual velocity to be determined, denotes the matrix density at x f at time t, in order to solve u bf , an interpolation factor a is constructed, a is related to the position of the solid phase boundary x w t, u bf is obtained from the following formula: （22）； （23）； （24）； where β is calculated as x f Interpolation factor for bounce distribution construction, u w is x w Interface velocity at x b is a solid phase boundary node; In order to reduce the calculation error caused by the large time step, the cell is spatially refined by 3*3 and temporally refined at the same time when the cell growth is calculated according to the dynamic chemical equilibrium; the growth in a time step is calculated multiple times to reduce the calculation error caused by the time step when the forced convection is calculated to calculate the stress of the NbN dendrite; when the volume of the NbN in the substrate cell increases and contacts the adjacent liquid phase cell, the adjacent liquid phase cell is finely divided into 3*3 F-H format lower boundary NbN precipitation cell, and the NbN growth is continued; the shape, size and flow field distribution of the NbN dendrite during solidification are displayed by using data analysis and visualization processing software.

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