Calculation method for hot-rolled coil post-cooling temperature distribution

Abstract

According to the hot-rolled coil post-cooling temperature distribution calculation method, a heat conduction differential equation is obtained through the Fourier law, the heat conduction differential equation is solved through a variable separation method and an intrinsic function method, a temperature field analytic expression of series expansion is obtained, then boundary conditions are substituted, and under the condition that temperature distribution at any initial moment is drawn up, the temperature distribution of the hot-rolled coil is calculated. And solving an undetermined coefficient in the temperature field expression according to the initial condition and the boundary condition to obtain a complete solution. Compared with traditional finite element modeling, the method based on a mathematical physical method has remarkable advantages. According to the method, the temperature distribution result in the post-cooling process of the hot-rolled coil can be obtained at a higher speed, the result precision is relatively accurate, and an efficient and reliable technical means is provided for temperature control and process optimization in the post-cooling process of the hot-rolled coil.

Description

Technical Field

[0001] The present invention relates to the field of hot rolling technology, and in particular to a method for calculating the temperature distribution of a hot rolled coil after cooling. Background Art

[0002] Heat transfer analysis is the study of how heat is transferred within an object. It usually involves considering factors such as the thermal conductivity of the material, temperature distribution, and time. We generally use the heat conduction differential equation to describe the heat conduction process. This partial differential equation takes into account the temperature changes in time and space. In the three-dimensional case, the temperature field can be defined as , corresponding to the cylindrical coordinate system is defined as If we assume that the temperature changes uniformly along the radial direction and does not change with the change of angle, the three-dimensional problem can be simplified to a one-dimensional problem. or to solve, and use the thermal conductivity coefficient to describe the thermal conductivity properties of the material.

[0003] In actual production, the analytical solutions of various nonlinear heat conduction differential equations are of great significance in heat transfer theory, and they are also helpful in calculating heat transfer and verifying numerical solutions. However, few people have explained the analytical solutions of heat conduction differential equations, so it is very meaningful to study this part. The study of temperature field in post-coiling cooling aims to gain an in-depth understanding of the temperature change distribution of metal during the cooling process after coiling. This kind of research can help study the occurrence of phase change, optimize metal processing technology, improve product quality, reduce energy consumption, reduce residual stress, and increase production efficiency.

[0004] At present, one of the commonly used methods for calculating the temperature distribution of hot-rolled steel coils during the post-coil cooling process is the calculation method based on numerical simulation, which uses the finite element method (FEM) or finite difference method (FDM) to discretize and solve the heat conduction equation to simulate the cooling process of the steel coil; or uses machine learning or deep learning algorithms to train models based on historical data to predict the cooling temperature of the steel coil. However, the above two methods are not very interpretable, and require high computing equipment and long calculation time. Summary of the invention

[0005] In order to solve the above technical problems, the purpose of the present invention is to provide a method for calculating the temperature distribution of hot rolled coil after cooling, which can significantly reduce the calculation time, improve the calculation efficiency, and avoid problems such as finite element simulation calculation results that cannot be explained.

[0006] A method for calculating the temperature distribution of a hot rolled coil after cooling according to the present invention comprises: Step 1: Using Fourier's law, derive the heat conduction differential equation in cylindrical coordinates; Step 2: Solve the homogeneous heat conduction differential equation by separation of variables method, and obtain the analytical solution expression of the temperature field with parameters through series expansion; Step 3: Determine the initial temperature field, and use the boundary conditions to construct a matrix to obtain the unknown coefficients in the homogeneous partial temperature field expression; Step 4: Considering the effect of phase change on temperature during cooling, internal heat source is introduced and the eigenfunction method is used to solve the non-homogeneous heat conduction differential equation; Step 5: Use boundary conditions to construct a matrix to obtain the unknown coefficients in the temperature field expression of the non-homogeneous part; Step 6: Take the superposition of the temperature expressions obtained from the homogeneous part and the inhomogeneous part as the complete solution of the temperature field and derive the temperature distribution law.

[0007] The present invention provides a method for calculating the temperature distribution of a hot rolled coil after cooling. The heat conduction differential equation is obtained by Fourier's law. The heat conduction differential equation is solved by separation of variables method and eigenfunction method, and the temperature field analytical expression of the series expansion is obtained. Then, the boundary conditions are substituted. When the temperature distribution at any initial time is proposed, the undetermined coefficients in the temperature field expression are solved by the initial conditions and boundary conditions to obtain a complete solution. The calculation method starts from the basic principles of heat transfer, ensures its rationality, and has significant advantages over traditional finite element modeling. BRIEF DESCRIPTION OF THE DRAWINGS

[0008] Figure 1 A flowchart of an online resource allocation optimization method for a UAV communication system of the present invention; Figure 2 The 48h cooling result diagram of 20 groups of characteristic values ​​is selected under the initial Gaussian distribution temperature field proposed for the present invention; Figure 3 A temperature drop comparison diagram of the center layer and the outer layer of the steel coil of 20 sets of characteristic values ​​is selected under the initial Gaussian distribution temperature field proposed for the present invention. DETAILED DESCRIPTION

[0009] The present invention provides a method for calculating the temperature distribution of a hot rolled coil after cooling, which is described below in conjunction with a specific example.

[0010] The temperature distribution of steel grade DP590 produced by a 2300mm hot rolling production line of a steel enterprise is calculated during the cooling process. The initial temperature distribution is proposed as: like Figure 1 As shown, a method for calculating the temperature distribution of a hot rolled coil after cooling in this embodiment comprises the following steps: Step 1: Using Fourier's law, derive the heat conduction differential equation in cylindrical coordinates, including: It is derived by establishing a temperature transfer model in a cylindrical coordinate system. The derivation process includes: A small part of the steel coil is selected as a microelement, and the heat conduction law satisfied by the entire steel coil is derived using the microelement method. According to Fourier's law: in, ϕ is the total heat flow of the microelement, λ is the thermal conductivity, A is the microelement area, T is the temperature, x is the microelement radius.

[0011] Corresponding to the cylindrical coordinate system: in, 、 They represent the total heat flux flowing radially into the microelement and the total heat flux flowing radially out of the microelement, respectively. φ represents the microelement angle, r is the inner diameter of the microelement, dr is the radial length of the microelement.

[0012] According to the total heat flow into the micro-element radially and the total heat flow out of the micro-element radially, the formula is satisfied: The homogeneous heat conduction differential equation is obtained by rearranging: If the influence of internal heat source caused by phase change is considered, the non-homogeneous heat conduction differential equation is obtained: In the formula, ρ is the material density, c is the specific heat capacity of the material, D is the thermal diffusion coefficient; t Represents time, T and represent the temperature of the homogeneous part and the temperature of the inhomogeneous part, respectively. N p is the number of phases, ΔH p , X p Represents the enthalpy change caused by each phase change and the corresponding phase volume fraction.

[0013] Step 2: Solve the homogeneous heat conduction differential equation by separation of variables method, and obtain the analytical solution expression of the temperature field with parameters through series expansion, which is: For a non-homogeneous partial differential equation, its general solution is the sum of the general solution of the corresponding homogeneous partial differential equation and the special solution of the non-homogeneous partial differential equation. The heat conduction differential equation is essentially a partial differential equation, so the form of this solution is still applicable.

[0014] For the homogeneous heat conduction differential equation .

[0015] In mathematical physics, the separation of variables method is a common method for solving homogeneous differential equations. Its main idea is to separate the variables of the function containing multiple variables in the equation through appropriate transformation, so that the equation can be transformed into several ordinary differential equations containing only one variable. T ( r,t )= R ( r ) G ( t ), here T ( r,t ) is used to indicate T and r , t related, R ( r )express R and r related, G ( t )express G and t Regarding: Will , , Substituting back the homogeneous heat conduction differential equation, we obtain: Because when hour, Established, so there are: β k = 0 o'clock, ; β k ≠ 0 o'clock, ;also .

[0016] Solve separately , , ,get R ( r ), G (t ), if the steel coil is regarded as i layer, the temperature field expression of the homogeneous part is: Assume that there are n+1 layers of steel coils. , in the expression T (i) ( r,t ) represents the temperature of the first layer of the homogeneous part of the steel coil, r (i) For the i The inner diameter length of the layer steel coil, represents the zero-order Bessel function of the first kind and the zero-order Bessel function of the second kind, D (i) Representative i The thermal diffusivity of the layer, All are homogeneous parts. i The undetermined coefficients of the layer steel coil exist. β k is the eigenvalue of the homogeneous partial differential equation, N k represent β k The number of .

[0017] Step 3: Determine the initial temperature field , the thermal conductivity of DP590 steel is calculated by Jmatpro software, and the matrix is ​​constructed using boundary conditions to obtain the unknown coefficients in the homogeneous partial temperature field expression, including: Step 3.1: For the homogeneous heat conduction differential equation, establish the following boundary conditions: in, λ (0) is the thermal conductivity of the 0th layer of steel coil, i.e. the innermost layer of steel coil; H (0) is the heat transfer coefficient of the 0th layer of steel coil; T (0) is the temperature of the 0th layer of the homogeneous part; T (i) For the homogeneous part i The temperature of the layer steel coil; r (i) For the i The inner diameter of the layer steel coil; λ (i) For the i Thermal conductivity of the layer steel coil; λ (n) For the n Layer steel coil, that is, the thermal conductivity of the outermost steel coil; T intis the ambient temperature of the innermost layer of the steel coil, T ext The ambient temperature of the outermost layer of the steel coil; H (n+1) It is the environmental heat transfer coefficient of the outermost layer of the steel coil.

[0018] Substituting the boundary conditions into the homogeneous partial temperature expressions, the two sets of equations obtained are: is the undetermined coefficient of the 0th layer of the homogeneous part; represents the thermal diffusion coefficient of layer 0; is the undetermined coefficient of the nth layer of steel coil in the homogeneous part; r (n+1) is the outer diameter length of the nth layer of steel coil. Assume that the number of steel coil layers is 200, the thickness of each layer is 2mm, the inner diameter of the steel coil is 200mm, and the outer diameter is 600mm. The relationship between the change of thermal physical parameters and temperature is shown in the following table: Table 1 Thermal properties of materials Step 3.2: Rewrite the two sets of equations obtained into matrix form and obtain ,in Step 3.3: ,when When , this formula has a non-trivial solution, and the matrix The rank of is 2n+1, so using The vector obtained Elements in satisfy: ,in, is the unknown coefficient determined by the initial temperature field and is given by Solve for the eigenvalue β k ,and β k The more values ​​there are, the more series expansion items there are, and the better the temperature field calculation effect is; Can be solved directly.

[0019] Step 3.4: By Solve for the unknown coefficients ; By determining the appropriate initial temperature field, the unknown coefficients are solved , thus solving for the coefficient .

[0020] Some of the calculation results are: Step 3.5: Obtain the temperature distribution of the homogeneous part.

[0021] Step 4: Considering the effect of phase change on temperature during cooling, introduce internal heat source and use eigenfunction method to solve the non-homogeneous heat conduction differential equation, including: For the inhomogeneous heat conduction differential equation .set up , substitute the given expression into the nonhomogeneous heat conduction differential equation. For the steel coil, if we regard it as i layer, the temperature field expression of the non-homogeneous part is: In the expression Represents the non-homogeneous part i The temperature of the layer steel coil, represents the zero-order Bessel function of the first kind and the zero-order Bessel function of the second kind, All are non-homogeneous parts. i Undetermined coefficients of layer steel coils, , is the eigenvalue of the inhomogeneous partial differential equation, represent The number of .

[0022] Step 5: Use boundary conditions to construct a matrix to obtain the unknown coefficients in the non-homogeneous temperature field expression, including: Step 5.1: Establish the following boundary conditions for the nonhomogeneous heat conduction differential equation: is the temperature of the 0th layer of the non-homogeneous part; For non-homogeneous parts i The temperature of the layer steel coil; Non-homogeneous part n The temperature of the outermost layer of steel coil.

[0023] Substituting the boundary conditions into the non-homogeneous partial temperature expressions, the two sets of equations obtained are: is the undetermined coefficient of the 0th layer of steel coil in the non-homogeneous part.

[0024] Step 5.2: Substitute the obtained non-homogeneous partial temperature field expressions into the corresponding boundary conditions to obtain two sets of equations, which are organized into matrix form to obtain ,in: Step 5.3: Directly solve for the unknown coefficients .

[0025] Step 5.4: Obtain the temperature distribution of the non-homogeneous part.

[0026] Step 6: The superposition of the temperature expressions obtained from the homogeneous part and the non-homogeneous part is used as the complete solution of the temperature field to obtain the temperature distribution law. The superposition process of the temperature field in step 6 includes: The complete solution of the temperature field of the steel coil is: .

[0027] Figure 2 The 48h cooling result diagram of 20 groups of characteristic values ​​is selected under the initial Gaussian distribution temperature field proposed for the present invention.

[0028] Figure 3 A temperature drop comparison diagram of the center layer and the outer layer of the steel coil of 20 sets of characteristic values ​​is selected under the initial Gaussian distribution temperature field proposed for the present invention.

[0029] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. For those skilled in the art, the present invention may have various modifications and variations. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A method for calculating the temperature distribution of hot rolled coil after cooling, characterized in that: include: Step 1: Using Fourier's law, derive the heat conduction differential equation in cylindrical coordinates; Step 2: Solve the homogeneous heat conduction differential equation by separation of variables method, and obtain the analytical solution expression of the temperature field with parameters through series expansion; Step 3: Determine the initial temperature field, and use the boundary conditions to construct a matrix to obtain the unknown coefficients in the homogeneous partial temperature field expression; Step 4: Considering the effect of phase change on temperature during cooling, internal heat source is introduced and the eigenfunction method is used to solve the non-homogeneous heat conduction differential equation; Step 5: Use boundary conditions to construct a matrix to obtain the unknown coefficients in the temperature field expression of the non-homogeneous part; Step 6: Take the superposition of the temperature expressions obtained from the homogeneous part and the inhomogeneous part as the complete solution of the temperature field and derive the temperature distribution law.

2. The method for calculating the temperature distribution of hot rolled coil after cooling according to claim 1, characterized in that: The heat conduction differential equation in step 1 is derived by establishing a temperature transfer model in a cylindrical coordinate system. The derivation process includes: A small part of the steel coil is selected as a microelement, and the heat conduction law satisfied by the entire steel coil is derived using the microelement method. According to Fourier's law: in, ϕ is the total heat flow of the microelement, λ is the thermal conductivity, A is the microelement area, T is the temperature, x is the radius of the microelement; Corresponding to the cylindrical coordinate system: in, 、 They represent the total heat flux flowing radially into the microelement and the total heat flux flowing radially out of the microelement, respectively. φ represents the microelement angle, r is the inner diameter of the microelement, dr is the radial length of the microelement; According to the total heat flow into the micro-element radially and the total heat flow out of the micro-element radially, the formula is satisfied: The homogeneous heat conduction differential equation is obtained by rearranging: If the influence of internal heat source caused by phase change is considered, the non-homogeneous heat conduction differential equation is obtained: In the formula, ρ is the material density, c is the specific heat capacity of the material, D is the thermal diffusion coefficient; t Represents time, T and represent the temperature of the homogeneous part and the temperature of the inhomogeneous part, respectively. N p is the number of phases, ΔH p , X p Represents the enthalpy change caused by each phase change and the corresponding phase volume fraction.

3. The method for calculating the temperature distribution of hot rolled coil after cooling according to claim 2, characterized in that: The calculation process of the parameter-containing temperature field analytical solution expression in step 2 includes: For the homogeneous heat conduction differential equation, let T ( r,t )= R ( r ) G ( t ), here T ( r,t ) is used to indicate T and r , t related, R ( r )express R and r related, G ( t )express G and t Regarding: Will , , Substituting back into the homogeneous heat conduction differential equation, we get: Because when hour, Established, so there are: β k = 0 o'clock, ; β k ≠ 0 o'clock, ;also ; Solve separately , , ,get R ( r ), G ( t ), if the steel coil is regarded as i layer, the temperature field expression of the homogeneous part is: Assume that there are n+1 layers of steel coils. , in the expression T (i) ( r,t ) represents the homogeneous part i The temperature of the layer steel coil, r (i) For the i The inner diameter length of the layer steel coil, represents the zero-order Bessel function of the first kind and the zero-order Bessel function of the second kind, D (i) Representative i The thermal diffusivity of the layer, All are homogeneous parts. i The undetermined coefficients of the layer steel coil exist. β k is the eigenvalue of the homogeneous partial differential equation, N k represent β k The number of .

4. The method for calculating the temperature distribution of hot rolled coil after cooling according to claim 3, characterized in that: The step 3 comprises: Step 3.1: For the homogeneous heat conduction differential equation, establish the following boundary conditions: in, λ (0) is the thermal conductivity of the 0th layer of steel coil, i.e. the innermost layer of steel coil; H (0) is the heat transfer coefficient of the 0th layer of steel coil; T (i) For the homogeneous part i The temperature of the layer steel coil; r (i) For the i The inner diameter length of the layer steel coil; λ (i) For the i Thermal conductivity of the layer steel coil; λ (n) For the n Layer steel coil, that is, the thermal conductivity of the outermost steel coil; T int is the ambient temperature of the innermost layer of the steel coil, T ext The ambient temperature of the outermost layer of the steel coil; H (n+1) is the environmental heat transfer coefficient of the outermost layer of the steel coil; Substituting the boundary conditions into the homogeneous partial temperature field expression, the two sets of equations obtained are: in, r (n+1) For the n The outer diameter length of the layer steel coil; Step 3.2: Rewrite the two sets of equations obtained into matrix form and obtain ,in: Step 3.3: ,when When , this formula has a non-trivial solution, and the matrix The rank of is 2n+1, so using The vector obtained Elements in satisfy: ,in, is the unknown coefficient determined by the initial temperature field and is given by Solve for the eigenvalue β k ,and β k The more values ​​there are, the more series expansion items there are, and the better the temperature field calculation effect is; Can be solved directly; Step 3.4: By Solve for the unknown coefficients ; By determining the appropriate initial temperature field, the unknown coefficients are solved , thus solving for the coefficient ; Step 3.5: Obtain the temperature distribution of the homogeneous part.

5. The method for calculating the temperature distribution of hot rolled coil after cooling according to claim 1, characterized in that: The calculation process of solving the non-homogeneous heat conduction differential equation using the eigenfunction method in step 4 includes: For the inhomogeneous heat conduction differential equation, let , substitute the given expression into the nonhomogeneous heat conduction differential equation. For the steel coil, if we regard it as i layer, the temperature field expression of the non-homogeneous part is: In the expression Represents the non-homogeneous part i The temperature of the layer steel coil, represents the zero-order Bessel function of the first kind and the zero-order Bessel function of the second kind, All are non-homogeneous parts. i Undetermined coefficients of layer steel coils, , is the eigenvalue of the inhomogeneous partial differential equation, represent The number of .

6. The method for calculating the temperature distribution of hot rolled coil after cooling according to claim 1, characterized in that: The step 5 comprises: Step 5.1: Establish the following boundary conditions for the nonhomogeneous heat conduction differential equation: is the temperature of the 0th layer of the non-homogeneous part; For non-homogeneous parts i The temperature of the layer steel coil; Non-homogeneous part n Layer steel coil, i.e. the temperature of the outermost steel coil; Substituting the boundary conditions into the non-homogeneous partial temperature expressions, the two sets of equations obtained are: Step 5.2: Substitute the obtained non-homogeneous partial temperature field expressions into the corresponding boundary conditions to obtain two sets of equations, which are organized into matrix form to obtain ,in: Step 5.3: Directly solve for the unknown coefficients ; Step 5.4: Obtain the temperature distribution of the non-homogeneous part.

7. The method for calculating the temperature distribution of hot rolled coil after cooling according to claim 1, characterized in that: The superposition process of the temperature field in step 6 includes: The complete solution of the temperature field is: .

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