CN116052806A - Finite volume calculation method for solidification heat transfer of irregular section continuous casting billet - Google Patents

Abstract

The invention designs a finite volume calculation method for solidification heat transfer of a continuous casting billet with an irregular section; firstly, constructing a 1/2 continuous casting blank simulation calculation domain according to the cross section size of the irregular cross section continuous casting blank; dividing unstructured grids in the established 1/2 continuous casting blank simulation calculation domain, and storing grid information; then, establishing an unstructured limited volume heat transfer model aiming at the continuous casting blank continuous casting solidification heat transfer problem; finally, simulating solidification heat transfer phenomenon of irregular continuous casting blank in continuous casting process by using the established non-structural limited volume heat transfer model; the calculation is carried out by adopting a volumetric method, and compared with a finite element method, the calculation rate is improved; selecting proper boundary conditions according to different cold areas, and more accurately simulating the solidification heat transfer of the whole continuous casting production process of the irregular continuous casting blank; the actual situation of continuous casting of the irregular continuous casting billet is fully considered, and the temperature field change in the continuous casting process of the irregular continuous casting billet is described, so that support is provided for uniformly cooling the irregular continuous casting billet and improving the quality of the irregular continuous casting billet.

Description

Finite volume calculation method for solidification heat transfer of irregular section continuous casting billet

Technical Field

The invention relates to the technical field of steel continuous casting, in particular to a finite volume calculation method for solidification heat transfer of a continuous casting billet with an irregular section.

Background

The continuous casting process is a process of forming a cast slab having a certain shape and size after the molten steel is refined. In the continuous casting process, after flowing into a crystallizer from a tundish, molten steel is subjected to heat exchange with a copper wall, and a part of molten steel is solidified to form a primary green shell. Under the action of the withdrawal and straightening machine, the casting blank continuously moves down to enter a secondary cooling area to finish cooling finally.

The square billet is a square-section billet produced by a continuous casting machine. The bloom continuous casting machine mainly produces medium and high carbon alloy steel and is used for rolling high-strength sectional materials, wires, channel steel and other steel types with high requirements on internal quality and compression ratio. The reasonable secondary cooling system can ensure the uniformity and stability of the casting blank cooling process, prevent the formation of quality defects such as cracks, bulging and the like, and has very important influence on the shape and depth of liquid cavities in the casting blank.

The round billet continuous casting has the advantages of high casting blank precision, good quality, low energy consumption and high metal yield. However, continuous casting round billets still have a number of quality defects, mainly internal quality defects and surface quality defects. The internal quality defects of the continuous casting round billet mainly comprise cracks among dendrites, central shrinkage cavities and the like. The surface defects are mainly surface cracks, and the forms of the surface defects are approximately longitudinal cracks, star-shaped cracks, surface pits, hairlines and the like.

The shaped slab is a continuous casting slab having a complicated cross section except for square slabs, round slabs and rectangular slabs. The beam blank is widely applied to the fields of traffic, construction, heavy equipment manufacture and the like by virtue of the excellent bearing capacity, the excellent section stability, the structural weight saving and the like. However, due to the complex cross-sectional shape, various quality defects are more likely to occur in the special-shaped blank compared with the casting blank with the conventional shape, the current research on the special-shaped blank continuous casting technology is difficult to meet the development requirement of the modern special-shaped blank continuous casting, and the continuous casting control level is far from reaching the expectations of people.

The square billet is often rounded at the corner in the actual production process, the round billet boundary is arc-shaped, the section of the special-shaped billet is complex and also comprises a plurality of sections of arc-shaped, but the prior art is generally simulated by dividing orthogonal grid units, saw-tooth shapes can be shown at the boundary, and the calculation accuracy is greatly reduced.

At present, three methods, namely a finite difference method, a finite element method and a finite volume method, are mainly used for computer numerical simulation. The finite difference method is the earliest method adopted by computer numerical simulation. The method divides the solution domain into differential grids, and replaces the continuous solution domain with a limited number of grid nodes. The finite difference method uses Taylor series expansion and other methods to replace the derivative in the control equation with the difference quotient of the function values on the grid nodes to carry out the discretization, thereby establishing an algebraic equation set taking the values on the grid nodes as unknowns. The method is an approximate numerical solution that directly changes the differential problem into an algebraic problem. The method has the advantages of visual mathematical concept, simple expression, simple mathematical modeling, easy programming and parallelism, is not suitable for processing complex boundaries, and is very complicated for processing irregular areas.

The basic solution idea of the finite element method is to divide the calculation domain into a finite number of units which are not overlapped with each other, select some proper nodes as interpolation points of the solution function in each unit, rewrite the variables in the differential equation into linear expression composed of node values of each variable or its derivative and the selected interpolation function, and discretely solve the differential equation by means of the variational principle or the weighted allowance method. The prior art discloses geometric discretization of a two-dimensional cross section of a continuous casting billet by establishing a two-dimensional finite element model; then, constructing a continuous casting billet two-dimensional finite element solidification heat transfer model aiming at the problem of continuous casting solidification heat transfer; finally, introducing structural parameters and other information of the continuous casting machine, and calculating the temperature field change in the continuous casting process by using the model; the mixed triangle unit and the quadrilateral unit can better simulate arc boundaries and can adapt to various complex shapes. However, the formed finite element matrix equation is complex, when n unit nodes exist, an n multiplied by n matrix equation is formed, the matrix is irregular, the solving difficulty is high, the convergence speed is low during solving, and the calculation efficiency is low; secondly, the finite element method has higher requirements on the quality of the grid, and the condition of no solution is easy to appear when the quality of the grid is poor.

Disclosure of Invention

Aiming at the defects existing in the prior art, the invention provides a finite volume calculation method for solidification and heat transfer of irregular section continuous casting billets, which aims at calculating the temperature change conditions at all moments in the continuous casting solidification process of the irregular section continuous casting billets, pushing the simulation and heat transfer analysis of the irregular section continuous casting billets, establishing an on-line solidification and heat transfer model, optimizing cooling water distribution in a secondary cooling area, reducing quality defects of the continuous casting billets and providing a theoretical basis for improving the quality of the continuous casting billets.

The finite volume calculation method for solidifying and transferring heat of irregular section continuous casting billet specifically comprises the following steps:

step 1: constructing a 1/2 continuous casting blank simulation calculation domain according to the cross section size of the irregular cross section continuous casting blank; the irregular section continuous casting billet comprises a special-shaped billet, a round square billet and a round billet;

determining coordinates of key points according to the section size, importing the coordinates into ANSYS software, connecting lines of the key points in the ANSYS and generating a geometric surface, and constructing a 1/2 simulation calculation domain; the section size comprises wing tip thickness, wide face width, narrow face width, web length, web thickness, inner edge arc radius, flange radius and fillet radius for the special-shaped blank; the fillet radius comprises a wide face width, a narrow face width and a fillet radius for the fillet square billet; the round billet comprises a round billet radius; the key points comprise web end points, wing tip end points, inner edge arc end points, flange end points and fillet end points for the special-shaped blank; the round-corner square billet comprises round-corner end points, wide-face end points and narrow-face end points; the circle blank comprises a circle center;

step 2: dividing unstructured grids in the 1/2 continuous casting blank simulation calculation domain established in the step 1, and storing grid information; the grid information comprises unit information, node information and unit plane information;

step 2.1: selecting a cell type and a cell shape in ANSYS, determining a cell size, and then dividing a non-mechanical grid in a continuous casting billet simulation calculation domain;

step 2.2: outputting the unit information and the node information into a DAT format file and storing the DAT format file; the unit information comprises a unit number and nodes contained in the unit; the node information comprises a node number and a node coordinate;

step 2.3: extracting unit face information according to the unit information and the node information and storing the unit face information;

because the cross section shape of the irregular cross section continuous casting billet is complex, the square units cannot be completely used for grid division when ANSYS performs regional division, and partial triangular units exist, so that whether the unit shapes are square units or not needs to be judged, and if not, the grids need to be re-divided, and the step is transferred to 2.1;

extracting unit surface information through the relationship among the units, the nodes and the surfaces, outputting the unit surface information into a DAT format file, and storing the DAT format file; the relation among the units, the nodes and the planes is that one unit consists of four nodes, four unit planes are arranged, and each unit plane consists of two nodes;

step 3: aiming at the problem of continuous casting solidification heat transfer of a continuous casting blank, an unstructured limited volume heat transfer model is established;

step 3.1: a finite volume calculation model of a steady-state heat transfer problem is constructed by a discrete diffusion equation under a regular Cartesian coordinate system;

step 3.1.1: discrete steady-state heat transfer diffusion equations;

the basic variable of the heat transfer process is temperature, which is a function of geometric position in the object and time; firstly, consider a regular Cartesian grid on a simple rectangular domain; based on the grid, the steady-state diffusion equation is:

wherein phi represents a scalar quantity, which is the temperature in the heat transfer problem; Γ -shaped structure ^φ Is a diffusion coefficient, i.e., a thermal conductivity coefficient, W/(mK) in the case of heat transfer; q (Q) ^φ Representing the generation rate of phi in unit volume in the calculation domain, namely the source term, namely the internal heat source in the heat transfer problem, W/m ³ The method comprises the steps of carrying out a first treatment on the surface of the The equation is written as generalThe form is:

▽·J ^φ,D ＝Q ^φ (2)

wherein J ^φ,D Is the diffusion flux;

taking unit C as an example, discretizing the unit C with the formula (1) converts it into:

where the subscript f represents the individual cell faces on cell C, S represents the surface vector,

representing the rate of generation of phi within cell C, i.e., source term, V _C Representing the control volume, m, of unit C ³ The method comprises the steps of carrying out a first treatment on the surface of the Expanding the above method to obtain:

wherein subscript e, w, s, n represents the four north-south faces of element C; for a uniform Cartesian grid, the surface vector of the cell face is calculated from the following equation:

S _e ＝+(Δy) _e i＝||S _e ||i＝S _e i S _w ＝-(Δy) _w i＝-||S _w ||i＝-S _w i

S _n ＝+(Δx) _n j＝||S _n ||j＝S _n j S _s ＝-(Δx) _s j＝-||S _s ||j＝-S _s j (5)

wherein Δx represents an increment in the x-axis direction, Δy represents an increment in the y-axis direction, i represents a unit vector in the x-axis direction, and j represents a unit vector in the y-axis direction;

thus, by way of example, the diffusion flux of the eastern cell surface is:

in the discrete of the conservation equation, if a single integral point pattern is used, the discrete form of the diffusion flux is:

assuming that phi varies linearly between cell C-shaped centers, the gradient along direction i across cell plane e is:

wherein subscript E represents the unit to the eastern side of unit C; substituting the above formula into formula (7) yields a discrete form of the diffusion flux of the unit face e, namely:

the coefficients are therefore:

wherein d _CE Representing the vector between the cell C and the cell E, d _CE Representing the distance between the cell C and the cell E centroid;

obtaining each coefficient based on the above process processing unit surface w, n, s, and substituting the coefficient into formula (4) to obtain algebraic form of diffusion equation, namely:

a _C φ _C +a _E φ _E +a _W φ _W +a _N φ _N +a _S φ _S ＝b _C (11)

wherein:

a _C ＝FluxC _e +FluxC _w +FluxC _n +FluxC _s

the above method is simplified to obtain:

wherein:

subscript F represents adjacent cells (E, W, N, S) of cell C and subscript F represents cell face (e, w, n, s) of cell C;

step 3.1.2: performing heat transfer analysis for different heat transfer boundary conditions;

discretization around cell C yields the following relationship:

the flux dispersion of the inner cell surface has been completed, while the dispersion of the boundary flux is to construct the information about phi _C And therefore:

wherein the subscript b represents the boundary cell face;

for heat transfer problems, there are dirichlet, noerman, mixed boundary condition types, consider the following three boundary conditions for a certain cell face of the cell C:

(1) For a certain cell surface of the cell, for example, when the cell surface e is a dirichlet boundary condition; the value of the boundary unknowns phi given the dirichlet boundary conditions, namely:

φ _b ＝φ _specified (17)

in this case:

wherein:

d _Cb representing the vector between the centroid of the cell C and the face of the boundary cell；

(2) For a certain cell surface of the cell, for example, cell surface e is a Neiman boundary condition; given the flux of phi at the boundary or the normal phase gradient of the boundary cell surface, the boundary condition is called the Neiman boundary condition; under this condition, the given flux expression is:

q _b flux for phi at a given boundary;

the above is flux

The reason is that:

wherein:

(3) When a certain unit surface of the unit, for example, a unit surface e, is a mixed boundary condition;

if the given boundary condition includes a convective transport coefficient (h _∞ ) And a surrounding reference value phi (phi) _∞ ) This condition is called a mixing boundary condition, namely:

the above is rewritten as:

from which phi is obtained _b ：

Will phi _b In expression (23), the flux equation is converted into:

wherein:

step 3.2: constructing an unstructured finite volume calculation model of a steady-state heat transfer problem in a non-orthogonal unstructured grid;

typically, the unstructured grid is non-orthogonal; thus, the surface vector S _f And the vector CF connecting the centroids of the coplanar units are not collinear; in this case, the gradient in the interface normal direction cannot be written as phi _F And phi _C As it has a component perpendicular to the CF direction;

in the case of an orthogonal grid, since CF and interfacial unit normal vector n are collinear, the gradient expression in the direction perpendicular to the interface is:

wherein r is _C 、r _F Is the vector between the centroid and the origin of coordinates of C, F, d _CF Is the distance between the centers of the C shape and the F shape;

for non-orthogonal grids, include _F And phi _C The gradient direction of the expression of (C) must be along the line of the centers of the two C and F;

if e represents a unit vector along the direction of the centroid of both C and F:

wherein d is _CF Is the vector between the centers of the C shape and the F shape;

thus, the gradient in the e-direction is written as:

thus, to achieve flux linearization in a non-orthogonal grid, the surface vector is written as two vectors E _f And T _f And, namely:

S _f ＝E _f +T _f (31)

wherein E is _f In line with the CF direction, in order to be able to write a part of the diffusion flux as phi _F And phi _C Such that:

decomposing S by using a minimum correction method _f Make E _f And T _f Vertical, thereby minimizing the non-orthogonal correction portion in equation (32) as much as possible; phi with increasing non-orthogonality _F And phi _C The contribution to the diffusion flux will be reduced; e (E) _f The formula of (2) is as follows:

E _f ＝(e·S _f )e＝(S _f cosθ)e (33)

wherein e represents a unit vector;

wherein θ is E _f And S is equal to _f Is included in the plane of the first part; substituting the respective expressions into a semi-discrete equation of the diffusion flux and expanding to obtain a final form of a discrete equation based on the unstructured grid:

wherein:

wherein E is _f Is vector E _f Is a mold of (2);

the processing of the non-orthogonal grid boundary conditions is similar to the case of orthogonal grids, but there is still a slight difference between the two, which is related to the presence of non-orthogonal spreading amounts; also three boundary conditions are distinguished:

(1) Dirichlet boundary conditions:

wherein:

(2) A Neiman boundary condition; the processing of the noerman boundary condition for a non-orthogonal grid is the same as in an orthogonal grid, i.e. the boundary fluxes given are added directly to the equation only as source terms;

(3) Mixing boundary conditions:

wherein:

step 3.3: constructing an unstructured finite volume calculation model of the transient heat transfer problem;

for transient simulation, the control equation needs to be discretized in space and time; the spatial dispersion is performed on the spatial domain, just as the steady state problem is processed, and the time dispersion needs to establish a time coordinate, and the derivative or integral of the transient item is calculated based on the time coordinate;

in general, the control equation for the transient behavior of the variable φ is:

wherein the function is

Representing a spatial operator comprising all non-transient items,/->

Representing a transient operator;

integrating and spatially dispersing the formula (40) over unit C yields:

wherein the method comprises the steps of

Is a spatially discrete operator at reference time t, which is written in the algebraic form:

derivative using Taylor series expansion in backward Euler format

Expressed as a function of discrete grid point values;

the value of the function T at the time T- Δt is written as a function of the T value at the time T and its derivative using the taylor formula, namely:

and (3) obtaining an expression of the first derivative through arrangement:

substituting T in equation (44) with (ρΦ) and substituting the expression of the derivative in equation (41), the discrete equation becomes:

algebraic relation of the reference space operator, and complete algebraic form of transient equation is:

wherein:

wherein the superscript °denotes the variable value of the previous time step, and the superscript · denotes the multiplication by · phi _C Is a non-steady state term coefficient of (2);

step 4: simulating solidification heat transfer phenomenon of irregular continuous casting blank in the continuous casting process by using the non-structural limited volume heat transfer model established in the step 3;

step 4.1: because the units are more, the required data storage space is larger, the data application storage space is required to be in advance, and the data overflow in the subsequent calculation process is prevented;

step 4.2: reading parameter information of irregular continuous casting blanks;

reading steel grade information, structural parameters of a continuous casting machine and continuous casting process parameters, and calculating solidus temperature and liquidus temperature; the steel grade information comprises steel marks and chemical components; the structural parameters of the continuous casting machine comprise the height of a crystallizer and the structural parameters of a secondary cooling zone; the secondary cooling zone structure parameters comprise the number, the length, the inlet position and the outlet position of secondary cooling zones; the continuous casting process parameters comprise casting temperature, drawing speed, heat flux density of a crystallizer, water flow rate and water temperature of each secondary cooling area, ambient temperature and section size;

step 4.3: importing the DAT format file exported in the step 2 into an unstructured finite volume heat transfer model, and storing the model; the grid information comprises unit information, node information and unit plane information; the unit information comprises a unit number and nodes contained in the unit; the node information comprises a node number and a node coordinate; the unit face information comprises unit face numbers and nodes forming the unit face;

step 4.4: calculating grid related parameters; the related parameters comprise a unit area, a unit centroid, a unit surface length and a unit surface vector;

step 4.4: determining a time step; the time step is the time of each calculation of the slice movement; the time step length multiplied by the pulling speed is the distance of each movement of the slice, and the accumulated movement distance is the position of the slice;

step 4.5: calculating the temperature gradient of the unit by using the green-Gaussian or gradient theorem according to the grid related parameters obtained by calculation in the step 4.3;

step 4.6: judging the phase region of the unit according to the temperature of the unit surface and calculating physical parameters of the unit; the phase region refers to a liquid phase region, a solid phase region and a solid-liquid two-phase region; the physical parameters comprise unit surface density, unit surface specific heat, unit surface solid phase rate and unit surface heat conduction coefficient;

step 4.7: calculating the coefficient a according to the formulas (35) and (47) in step 3 _C 、a _F 、b _C 、

And substituting into the discrete equation (46);

step 4.8: summarizing the discrete equations in the step 4.7, and solving the discrete equation set by adopting a Gaussian-Saidel iteration method to obtain the temperature of the unit surface; after the solving is finished, judging whether the position of the slice exceeds the air cooling area in sequence, if so, finishing calculation, and if not, moving the slice to the next position, and turning to the step 4.5;

step 5: carrying out visualization and result post-treatment on the calculation result of the continuous casting solidification heat transfer of the irregular continuous casting billet;

extracting calculation results at different positions of a continuous casting machine after the irregular continuous casting blank continuous casting solidification heat transfer calculation is completed, and introducing the calculation results into Tecplot software for visualization treatment to form a temperature cloud picture; respectively extracting the temperature change conditions of the characteristic points, and importing the characteristic points into Origin software for visualization processing to form a temperature change curve; the characteristic points comprise web surface centers, inner edge surface centers, narrow-face film mounting centers and round corner surface centers for the special-shaped billets, wide-face surface centers, narrow-face surface centers, casting blank centers and round corner centers for the round corner billets, and casting blank centers and casting blank surfaces for the round billets.

The invention has the beneficial technical effects that:

the basic idea of the finite volume method is that a calculation area is divided into a series of non-repeated control volumes, and one control volume is arranged around each grid point; integrating the differential equation to be solved for each control volume results in a set of discrete equations where the unknowns are the values of the dependent variables at the grid points. The basic thought of the method is easy to understand, direct physical interpretation can be obtained, the method can be suitable for various complex areas, meanwhile, the finally formed matrix equation is a diagonal matrix, the solving difficulty is low, the convergence speed is high, and the solving speed is far faster than that of a finite element method.

The finite volume calculation method for solidification heat transfer of the irregular section continuous casting billet fully considers the complex section shape of the special-shaped billet, and can better simulate the cross section of an actual special-shaped billet by adopting a general quadrilateral unit; the special condition that square billets are usually provided with round corners in the actual production process is considered, unstructured grids are adopted, when the grids are sufficiently dense, arbitrary curves of regional boundaries can be calculated by piecewise straight line simulation on the boundaries, and the calculation accuracy is guaranteed to a certain extent; the calculation is carried out by adopting a volumetric method, and compared with a finite element method, the calculation rate is improved; the arc boundary of the round billet is fully considered, an unstructured grid is adopted, when the grid is sufficiently dense, the arbitrary curve of the boundary of the area can be calculated by using piecewise straight line simulation on the boundary, and the calculation precision is ensured to a certain extent; the calculation is carried out by adopting a volumetric method, and compared with a finite element method, the calculation rate is improved; and proper boundary conditions are selected according to different cold areas, so that the full-flow solidification heat transfer of irregular continuous casting blank continuous casting production is more accurately simulated. The actual situation of continuous casting of the irregular continuous casting billet is fully considered, and the temperature field change in the continuous casting process of the irregular continuous casting billet is described, so that support is provided for uniformly cooling the irregular continuous casting billet and improving the quality of the irregular continuous casting billet.

Drawings

FIG. 1 is a schematic flow chart of a finite volume calculation method for solidifying and transferring heat of a continuous casting billet with an irregular section according to the embodiment of the invention;

FIG. 2 is a schematic diagram of the cross-sectional dimensions of a preform according to an embodiment of the present invention;

FIG. 3 is a schematic diagram of 1/2 special-shaped blank grid division provided by the embodiment of the invention;

fig. 4 is a cloud chart of temperature distribution of a special-shaped blank at different positions of a continuous casting machine according to an embodiment of the invention; wherein the graph (a) is a temperature distribution cloud graph of casting blanks at the outlet of a crystallizer; the diagram (b) is a temperature distribution cloud diagram of a casting blank at the outlet of the secondary cooling zone 1; the diagram (c) is a temperature distribution cloud diagram of a casting blank at the outlet of the secondary cooling zone 2; the diagram (d) is a temperature distribution cloud diagram of a casting blank at the outlet of the secondary cooling 3 region; the diagram (e) is a temperature distribution cloud diagram of a casting blank at the outlet of the secondary cooling 4 region; the diagram (f) is a temperature distribution cloud diagram of a casting blank at the outlet of the secondary cooling 5 region;

FIG. 5 is a graph showing temperature change at each characteristic point of a preform according to an embodiment of the present invention;

fig. 6 is a graph showing a comparison between calculated temperature and measured temperature of a flange surface center of a beam blank according to an embodiment of the present invention.

Detailed Description

The invention is further described below with reference to the drawings and examples;

in the embodiment, the continuous casting process of 450mm multiplied by 350mm multiplied by 90mm Q235 steel special-shaped blank of a certain steel factory in China is taken as an example, and the temperature change condition of each moment in the continuous casting process of the Q235 steel special-shaped blank is calculated by adopting the finite volume calculation method for solidifying and transferring heat of the irregular section continuous casting blank.

An unstructured finite volume method for simulating continuous casting solidification heat transfer of a special-shaped blank, as shown in figure 1, specifically comprises the following steps:

step 1: constructing a 1/2 special-shaped blank simulation calculation domain according to the section size of the special-shaped blank;

determining key point coordinates according to the section size of the special-shaped blank, importing the key point coordinates into ANSYS software, connecting lines by the key points in the ANSYS, generating a geometric surface, and constructing a 1/2 special-shaped blank simulation calculation domain; the section size of the special-shaped blank comprises wing tip thickness, wide surface width, narrow surface width, web length, web thickness, inner edge arc radius, flange radius and fillet radius; the key points comprise web endpoints, wing tip endpoints, inner edge arc endpoints, flange endpoints and fillet endpoints;

in this embodiment, the width of the wide face of the profiled blank is 0.435m, the width of the narrow face is 0.320m, the thickness of the wing tip is 0.045m, the length of the web is 0.229m, the thickness of the web is 0.090m, the radius of the flange is 0.020m, the radius of the inner arc is 0.100m, and the radius of the fillet is 0.010m. The schematic cross-sectional dimensions of the preform of this embodiment are shown in fig. 2.

Step 2: dividing unstructured grids in a simulation calculation domain, and storing grid information; the grid information comprises unit information, node information and unit plane information;

step 2.1: selecting a cell type and a cell shape in ANSYS, determining a cell size, and then dividing a non-mechanical grid in a simulation calculation domain;

in this embodiment, the cell type is selected from PLANE55, the cell shape is selected from quadrilateral cell, the cell size is selected from 5mm, and the area cell subdivision is automatically performed by ANSYS software. A schematic diagram of the mesh division of the beam blank in this embodiment 1/2 is shown in fig. 3.

Step 2.2: outputting the unit information and the node information into a DAT format file and storing the DAT format file; the unit information comprises a unit number and nodes contained in the unit; the node information comprises a node number and a node coordinate;

step 2.3: because the section shape of the special-shaped blank is complex, the square units cannot be completely used for grid division when ANSYS performs regional division, and partial triangular units exist, so that whether the unit shapes are square units or not needs to be judged, and if not, the grids need to be divided again, and the step is shifted to 2.1;

step 2.4: extracting unit face information according to the unit information and the node information and storing the unit face information;

extracting unit surface information through the relationship among the units, the nodes and the surfaces, outputting the unit surface information into a DAT format file, and storing the DAT format file; the relation among the units, the nodes and the planes is that one unit consists of four nodes, four unit planes are arranged, and each unit plane consists of two nodes;

step 3: aiming at the problem of continuous casting solidification heat transfer of the special-shaped blank, an unstructured limited volume heat transfer model is established;

step 3.1: a finite volume calculation model of a steady-state heat transfer problem is constructed by a discrete diffusion equation under a regular Cartesian coordinate system;

step 3.1.1: discrete steady-state heat transfer diffusion equations;

step 3.1.2: performing heat transfer analysis for different heat transfer boundary conditions;

the analytical solution of any ordinary differential equation or partial differential equation depends on a constant determined by the boundary conditions. For heat transfer problems, dirichlet, noerman, mixed boundary condition types exist. Boundary conditions apply to boundary cells that have one or more cell faces on the boundary.

Step 3.2: constructing an unstructured finite volume calculation model of a steady-state heat transfer problem in a non-orthogonal unstructured grid;

step 3.3: constructing an unstructured finite volume calculation model of the transient heat transfer problem;

step 4: simulating solidification heat transfer phenomenon of the special-shaped blank in the continuous casting process by using the non-structural limited volume heat transfer model established in the step 3;

step 4.1: reading special-shaped blank steel type information, structural parameters of a continuous casting machine and continuous casting process parameters, and calculating solidus temperature and liquidus temperature; the steel grade information comprises steel marks and chemical components; the structural parameters of the continuous casting machine comprise the height of a crystallizer and the structural parameters of a secondary cooling zone; the secondary cooling zone structure parameters comprise the number, the length, the inlet position and the outlet position of secondary cooling zones; the continuous casting process parameters comprise casting temperature, drawing speed, heat flux density of a crystallizer, water flow rate and water temperature of each secondary cooling area, ambient temperature and section size of the special-shaped blank. The section size of the special-shaped blank comprises wing tip thickness, wide surface width, narrow surface width, web length, web thickness, inner edge arc radius, flange radius and fillet radius;

in this example, the steel grade is Q235 and the chemical composition of the steel grade is shown in Table 1.

Table 1 chemical composition of 450mm x 350mm x 90mm Q235 beam blank in certain steel works in China;

chemical composition	C	Si	Mn	P	S
						Mass fraction (%)	0.19	0.21	0.47	0.025	0.027

In the embodiment, the height of the crystallizer of the beam blank continuous casting machine is 0.7m; a total of 5 secondary cooling zones, wherein the secondary cooling zone 1 has a length of 0.647m, and the inlet and outlet are spaced 0.7m and 1.347m from the meniscus, respectively; zone 2 has a length of 1.486m and inlet and outlet are spaced from the meniscus 1.347m and 2.833m, respectively; zone 3 has a length of 2.337m and inlet and outlet are spaced from the meniscus 2.833m and 5.170m respectively; zone 4 is 2.378m long with the inlet and outlet being 5.170m and 7.548m from the meniscus respectively; zone 5 is 2.224m in length with the inlet and outlet being spaced 7.548m and 9.772m from the meniscus, respectively. The casting temperature of the Q235 special-shaped blank is 1545 ℃, and the pulling speed is 0.9m/min; the wide-surface heat flux density of the crystallizer is 247w/m2, and the narrow-surface heat flux density is 243.677w/m2. The cooling conditions in the secondary cooling zone are shown in table 2. The secondary cooling water temperature is 25 ℃, and the ambient temperature of the air cooling area is 90 ℃.

Table 2 two cold cooling conditions of 450mm x 350mm x 90mm Q235 special-shaped blanks of a certain steel mill in China;

two cold areas	1	2	3	4	5
						Inner arc water volume (L/min)	36	33	21	13	13
Outer arc water volume (L/min)	36	33	21	13	13
						Side arc water volume (L/min)	40	37	19	8.4	8.4
Total water (L/min)	112	103	60	34..4	34.4

In this embodiment, a calculation formula of the corresponding solidus-liquidus temperature is selected according to the C content of the Q235 steel:

T _l ＝1537-{73.1[C]+14[Si]+4[Mn]+30[P]+45[S]+1.5[Cr]+2.5[Al]+3.5[Ni]+4[V]+5[Mo]}

＝1516℃ (48)

T _s ＝1527.0-{187.5[C]+20.5[Si]+6.5[Mn]+500[P]+700[S]+2.0[Cr]+11.5[Ni]+5.5[Al]}

＝1452℃ (49)

wherein T is _l Is the liquidus temperature; ts is solidus temperature; [ C]、[Si]、[Mn]、[P]、[S]、[Cr]、[Al]、[Ni]、[V]、[Mo]The mass fractions of the corresponding elements are respectively;

step 4.2: importing grid information of a 1/2 special blank two-dimensional geometric model;

importing the DAT format file exported in the step 2 into a heat transfer model, and storing the DAT format file; the grid information comprises unit information, node information and unit plane information; the unit information comprises a unit number and nodes contained in the unit; the node information comprises a node number and a node coordinate; the unit face information comprises unit face numbers and nodes forming the unit face;

step 4.3: calculating grid related parameters; the related parameters comprise a unit area, a unit centroid, a unit surface length and a unit surface vector;

step 4.4: determining a time step; the time step is the time of each calculation of the movement of the slice (called slice because the two-dimensional model has no thickness); the time step length multiplied by the pulling speed is the distance of each movement of the slice, and the accumulated movement distance is the position of the slice;

in this example, the time step is 0.1s and the pull rate is 0.9m/min.

Step 4.5: calculating the temperature gradient of the unit by using the green-Gaussian or gradient theorem according to the grid related parameters obtained by calculation in the step 4.3;

step 4.6: judging the phase region of the unit according to the temperature of the unit surface and calculating physical parameters of the unit; the phase region refers to a liquid phase region, a solid phase region and a solid-liquid two-phase region; the physical parameters comprise unit surface density, unit surface specific heat, unit surface solid phase rate and unit surface heat conduction coefficient;

in this example, the solidus temperature was 1485℃and the liquidus temperature was 1515℃as known from step 4.2. And judging a unit surface phase region according to the unit surface temperature, and calculating physical property parameters of the unit by adopting a proper formula.

Step 4.7: calculating the coefficient a according to the formulas (35) and (47) in step 3 _C 、a _F 、b _C 、

And substituting into the discrete equation (46);

step 4.8: summarizing the discrete equations in the step 4.7, and solving the discrete equation set by adopting a Gaussian-Saidel iteration method to obtain the temperature of the unit surface; judging whether the position of the slice exceeds the air cooling area after the solving is finished, if so, finishing calculation, finishing the calculation of the continuous casting solidification heat transfer of the special-shaped blank, and if not, moving the slice to the next position, and turning to the step 4.5;

in the embodiment, a Gaussian-Saidel iteration method is adopted to solve a discrete equation set, the error is the square sum of the difference between two iterations of all nodes, the error margin is 0.001, and in order to improve the calculation efficiency, a parallel calculation function parallel_for in OpenCV is adopted to carry out CPU parallel calculation.

Step 5: carrying out visualization and result post-treatment on the calculation result of the continuous casting solidification heat transfer of the special-shaped blank;

extracting calculation results at different positions of a continuous casting machine after the calculation of the continuous casting solidification heat transfer of the special-shaped blank is completed, and introducing the calculation results into Tecplot software for visualization processing to form a temperature cloud picture; and respectively extracting the temperature change conditions of the web center, the inner edge center, the narrow surface center and the round corner center, and introducing the temperature change conditions into Origin software for visualization processing to form a temperature change curve.

The embodiment provides a temperature distribution cloud picture of 450mm multiplied by 350mm multiplied by 90mm Q235 special-shaped blanks of a certain steel mill at different positions of a continuous casting machine in China, as shown in fig. 4, wherein the picture (a) is a temperature distribution cloud picture of a casting blank at an outlet of a crystallizer; the diagram (b) is a temperature distribution cloud diagram of a casting blank at the outlet of the secondary cooling zone 1; the diagram (c) is a temperature distribution cloud diagram of a casting blank at the outlet of the secondary cooling zone 2; the diagram (d) is a temperature distribution cloud diagram of a casting blank at the outlet of the secondary cooling 3 region; the diagram (e) is a temperature distribution cloud diagram of a casting blank at the outlet of the secondary cooling 4 region; and the graph (f) is a temperature distribution cloud graph of the casting blank at the outlet of the secondary cooling 5 region. The corresponding characteristic point temperature change curves are shown in fig. 5. The calculated temperature at the center of the flange surface is compared with the measured temperature as shown in fig. 6. It can be seen from fig. 6 that the simulation calculation result substantially coincides with the actual temperature. Due to the shape and cooling characteristics of the special-shaped blank, the central high-temperature area of the special-shaped blank gradually decreases and gradually moves to the inner edge side along with the cooling. Because the inner edge is concave, the heat transfer effect is poor, and the temperature of the surface of the inner edge is higher than that of other positions of the surface. The heat dissipation is faster in the two-dimensional heat transfer mode at the corner of the special-shaped blank, so that the temperature at the corner is the lowest.

Claims (7)

1. The finite volume calculation method for solidifying and transferring heat of irregular section continuous casting blank is characterized by comprising the following steps:

step 1: constructing a 1/2 continuous casting blank simulation calculation domain according to the cross section size of the irregular cross section continuous casting blank; the irregular section continuous casting billet comprises a special-shaped billet, a round square billet and a round billet;

step 2: dividing unstructured grids in the 1/2 continuous casting blank simulation calculation domain established in the step 1, and storing grid information; the grid information comprises unit information, node information and unit plane information;

step 3: aiming at the problem of continuous casting solidification heat transfer of a continuous casting blank, an unstructured limited volume heat transfer model is established;

step 4: simulating solidification heat transfer phenomenon of irregular continuous casting blank in the continuous casting process by using the non-structural limited volume heat transfer model established in the step 3;

step 5: and (5) visualizing and performing result post-processing on the irregular continuous casting billet continuous casting solidification heat transfer calculation result.

2. The method for calculating the finite volume of solidification heat transfer of continuous casting billets with irregular sections according to claim 1, wherein the constructing 1/2 simulation calculation domain of continuous casting billets in step 1 is specifically as follows:

determining coordinates of key points according to the section size, importing the coordinates into ANSYS software, connecting lines of the key points in the ANSYS and generating a geometric surface, and constructing a 1/2 simulation calculation domain; the section size comprises wing tip thickness, wide face width, narrow face width, web length, web thickness, inner edge arc radius, flange radius and fillet radius for the special-shaped blank; the fillet radius comprises a wide face width, a narrow face width and a fillet radius for the fillet square billet; the round billet comprises a round billet radius; the key points comprise web end points, wing tip end points, inner edge arc end points, flange end points and fillet end points for the special-shaped blank; the round-corner square billet comprises round-corner end points, wide-face end points and narrow-face end points; the circle center is included for the round billet.

3. The method for calculating the finite volume of the solidification heat transfer of the irregular section continuous casting billet according to claim 1, wherein the step 2 is specifically:

step 2.1: selecting a cell type and a cell shape in ANSYS, determining a cell size, and then dividing a non-mechanical grid in a continuous casting billet simulation calculation domain;

step 2.2: outputting the unit information and the node information into a DAT format file and storing the DAT format file; the unit information comprises a unit number and nodes contained in the unit; the node information comprises a node number and a node coordinate;

step 2.3: extracting unit face information according to the unit information and the node information and storing the unit face information;

because the cross section shape of the irregular cross section continuous casting billet is complex, the square units cannot be completely used for grid division when ANSYS performs regional division, and partial triangular units exist, so that whether the unit shapes are square units or not needs to be judged, and if not, the grids need to be re-divided, and the step is transferred to 2.1;

extracting unit surface information through the relationship among the units, the nodes and the surfaces, outputting the unit surface information into a DAT format file, and storing the DAT format file; the relation among the units, the nodes and the planes is that one unit consists of four nodes, and four unit planes are provided, and each unit plane consists of two nodes.

4. The method for calculating the finite volume of solidification heat transfer of irregular section continuous casting billet according to claim 1, wherein the step 3 is specifically:

step 3.1: a finite volume calculation model of a steady-state heat transfer problem is constructed by a discrete diffusion equation under a regular Cartesian coordinate system;

step 3.2: constructing an unstructured finite volume calculation model of a steady-state heat transfer problem in a non-orthogonal unstructured grid;

typically, the unstructured grid is non-orthogonal; thus, the surface vector S _f And the vector CF connecting the centroids of the coplanar units are not collinear; in this case, the gradient in the interface normal direction cannot be written as phi _F And phi _C As it has a component perpendicular to the CF direction;

in the case of an orthogonal grid, since CF and interfacial unit normal vector n are collinear, the gradient expression in the direction perpendicular to the interface is:

wherein r is _C 、r _F Is the vector between the centroid and the origin of coordinates of C, F, d _CF Is the distance between the centers of the C shape and the F shape;

for non-orthogonal grids, include _F And phi _C The gradient direction of the expression of (C) must be along the line of the centers of the two C and F;

if e represents a unit vector along the direction of the centroid of both C and F:

wherein d is _CF Is the vector between the centers of the C shape and the F shape;

thus, the gradient in the e-direction is written as:

thus, to achieve flux linearization in a non-orthogonal grid, the surface vector is written as two vectors E _f And T _f And, namely:

S _f ＝E _f +T _f (31)

wherein E is _f In line with the CF direction, in order to be able to write a part of the diffusion flux as phi _F And phi _C Such that:

decomposing S by using a minimum correction method _f Make E _f And T _f Vertical, thereby minimizing the non-orthogonal correction portion in equation (32) as much as possible; phi with increasing non-orthogonality _F And phi _C The contribution to the diffusion flux will be reduced; e (E) _f The formula of (2) is as follows:

E _f ＝(e·S _f )e＝(S _f cosθ)e (33)

wherein e represents a unit vector;

wherein θ is E _f And S is equal to _f Is included in the plane of the first part; substituting the respective expressions into a semi-discrete equation of the diffusion flux and expanding to obtain a final form of a discrete equation based on the unstructured grid:

wherein:

wherein E is _f Is vector E _f Is a mold of (2);

the processing of the non-orthogonal grid boundary conditions is similar to the case of orthogonal grids, but there is still a slight difference between the two, which is related to the presence of non-orthogonal spreading amounts; also three boundary conditions are distinguished:

(1) Dirichlet boundary conditions:

wherein:

(2) A Neiman boundary condition; the processing of the noerman boundary condition for a non-orthogonal grid is the same as in an orthogonal grid, i.e. the boundary fluxes given are added directly to the equation only as source terms;

(3) Mixing boundary conditions:

wherein:

step 3.3: constructing an unstructured finite volume calculation model of the transient heat transfer problem;

for transient simulation, the control equation needs to be discretized in space and time; the spatial dispersion is performed on the spatial domain, just as the steady state problem is processed, and the time dispersion needs to establish a time coordinate, and the derivative or integral of the transient item is calculated based on the time coordinate;

in general, the control equation for the transient behavior of the variable φ is:

/>

where the function i (phi) represents a spatial operator, which contains all non-transient terms,

representing a transient operator;

integrating and spatially dispersing the formula (40) over unit C yields:

wherein the method comprises the steps of

Is a spatially discrete operator at reference time t, which is written in the algebraic form:

derivative using Taylor series expansion in backward Euler format

Expressed as a function of discrete grid point values;

the value of the function T at the time T- Δt is written as a function of the T value at the time T and its derivative using the taylor formula, namely:

and (3) obtaining an expression of the first derivative through arrangement:

substituting T in equation (44) with (ρΦ) and substituting the expression of the derivative in equation (41), the discrete equation becomes:

algebraic relation of the reference space operator, and complete algebraic form of transient equation is:

wherein:

wherein the superscript °denotes the variable value of the previous time step, and the superscript · denotes the multiplication by · phi _C Is a non-steady state term coefficient of (c).

5. The method for calculating the finite volume of the solidification heat transfer of the irregular section continuous casting billet according to claim 4, wherein the step 3.1 is specifically:

step 3.1.1: discrete steady-state heat transfer diffusion equations;

the basic variable of the heat transfer process is temperature, which is a function of geometric position in the object and time; firstly, consider a regular Cartesian grid on a simple rectangular domain; based on the grid, the steady-state diffusion equation is:

wherein phi represents a scalar quantity, which is the temperature in the heat transfer problem; Γ -shaped structure ^φ Is a diffusion coefficient, i.e., a thermal conductivity coefficient, W/(mK) in the case of heat transfer; q (Q) ^φ Representing the generation rate of phi in unit volume in the calculation domain, namely the source term, namely the internal heat source in the heat transfer problem, W/m ³ The method comprises the steps of carrying out a first treatment on the surface of the This equation is written in a general form, namely:

wherein J ^φ,D Is the diffusion flux;

taking unit C as an example, discretizing the unit C with the formula (1) converts it into:

where the subscript f represents the individual cell faces on cell C, S represents the surface vector,

representing the rate of generation of phi within cell C, i.e., source term, V _C Representing the control volume, m, of unit C ³ The method comprises the steps of carrying out a first treatment on the surface of the Expanding the above method to obtain:

wherein subscript e, w, s, n represents the four north-south faces of element C; for a uniform Cartesian grid, the surface vector of the cell face is calculated from the following equation:

S _e ＝+(Δy) _e i＝||S _e ||i＝S _e i S _w ＝-(Δy) _w i＝-||S _w ||i＝-S _w i

S _n ＝+(Δx) _n j＝||S _n ||j＝S _n j S _s ＝-(Δx) _s j＝-||S _s ||j＝-S _s j (5)

wherein Δx represents an increment in the x-axis direction, Δy represents an increment in the y-axis direction, i represents a unit vector in the x-axis direction, and j represents a unit vector in the y-axis direction;

thus, by way of example, the diffusion flux of the eastern cell surface is:

in the discrete of the conservation equation, if a single integral point pattern is used, the discrete form of the diffusion flux is:

assuming that phi varies linearly between cell C-shaped centers, the gradient along direction i across cell plane e is:

wherein subscript E represents the unit to the eastern side of unit C; substituting the above formula into formula (7) yields a discrete form of the diffusion flux of the unit face e, namely:

the coefficients are therefore:

wherein d _CE Representing the vector between the cell C and the cell E, d _CE Representing the distance between the cell C and the cell E centroid;

obtaining each coefficient based on the above process processing unit surface w, n, s, and substituting the coefficient into formula (4) to obtain algebraic form of diffusion equation, namely:

a _C φ _C +a _E φ _E +a _W φ _W +a _N φ _N +a _S φ _S ＝b _C (11)

wherein:

a _C ＝FluxC _e +FluxC _w +FluxC _n +FluxC _s

the above method is simplified to obtain:

wherein:

subscript F represents adjacent cells (E, W, N, S) of cell C and subscript F represents cell face (e, w, n, s) of cell C;

step 3.1.2: performing heat transfer analysis for different heat transfer boundary conditions;

discretization around cell C yields the following relationship:

the flux dispersion of the inner cell surface has been completed, while the dispersion of the boundary flux is to construct the information about phi _C And therefore:

wherein the subscript b represents the boundary cell face;

for heat transfer problems, there are dirichlet, noerman, mixed boundary condition types, consider the following three boundary conditions for a certain cell face of the cell C:

(1) For a certain cell surface of the cell, for example, when the cell surface e is a dirichlet boundary condition; the value of the boundary unknowns phi given the dirichlet boundary conditions, namely:

φ _b ＝φ _specified (17)

in this case:

/>

wherein:

d _Cb a vector representing the orientation between the centroid of cell C and the boundary cell face;

(2) For a certain cell surface of the cell, for example, cell surface e is a Neiman boundary condition; given the flux of phi at the boundary or the normal phase gradient of the boundary cell surface, the boundary condition is called the Neiman boundary condition; under this condition, the given flux expression is:

q _b flux for phi at a given boundary;

the above is flux

The reason is that:

wherein:

(3) When a certain unit surface of the unit, for example, a unit surface e, is a mixed boundary condition;

if the given boundary condition includes a convective transport coefficient (h _∞ ) And a surrounding reference value phi (phi) _∞ ) This condition is called a mixing boundary condition, namely:

the above is rewritten as:

from which phi is obtained _b ：

Will phi _b In expression (23), the flux equation is converted into:

wherein:

6. the method for calculating the finite volume of solidification heat transfer of irregular section continuous casting billet according to claim 1, wherein the step 4 is specifically:

step 4.1: because the units are more, the required data storage space is larger, the data application storage space is required to be in advance, and the data overflow in the subsequent calculation process is prevented;

step 4.2: reading parameter information of irregular continuous casting blanks;

reading steel grade information, structural parameters of a continuous casting machine and continuous casting process parameters, and calculating solidus temperature and liquidus temperature; the steel grade information comprises steel marks and chemical components; the structural parameters of the continuous casting machine comprise the height of a crystallizer and the structural parameters of a secondary cooling zone; the secondary cooling zone structure parameters comprise the number, the length, the inlet position and the outlet position of secondary cooling zones; the continuous casting process parameters comprise casting temperature, drawing speed, heat flux density of a crystallizer, water flow rate and water temperature of each secondary cooling area, ambient temperature and section size;

step 4.3: importing the DAT format file exported in the step 2 into an unstructured finite volume heat transfer model, and storing the model; the grid information comprises unit information, node information and unit plane information; the unit information comprises a unit number and nodes contained in the unit; the node information comprises a node number and a node coordinate; the unit face information comprises unit face numbers and nodes forming the unit face;

step 4.4: calculating grid related parameters; the related parameters comprise a unit area, a unit centroid, a unit surface length and a unit surface vector;

step 4.4: determining a time step; the time step is the time of each calculation of the slice movement; the time step length multiplied by the pulling speed is the distance of each movement of the slice, and the accumulated movement distance is the position of the slice;

step 4.5: calculating the temperature gradient of the unit by using the green-Gaussian or gradient theorem according to the grid related parameters obtained by calculation in the step 4.3;

step 4.6: judging the phase region of the unit according to the temperature of the unit surface and calculating physical parameters of the unit; the phase region refers to a liquid phase region, a solid phase region and a solid-liquid two-phase region; the physical parameters comprise unit surface density, unit surface specific heat, unit surface solid phase rate and unit surface heat conduction coefficient;

step 4.7: calculating the coefficient a according to the formulas (35) and (47) in step 3 _C 、a _F 、b _C 、

And substituting into the discrete equation (46);

step 4.8: summarizing the discrete equations in the step 4.7, and solving the discrete equation set by adopting a Gaussian-Saidel iteration method to obtain the temperature of the unit surface; and after the solving is finished, judging whether the position of the slice exceeds the air cooling area in sequence, if so, finishing calculation, and if not, moving the slice to the next position, and turning to the step 4.5.

7. The method for calculating the finite volume of the solidification heat transfer of the irregular section continuous casting billet according to claim 1, wherein the step 5 is characterized in that the calculation result of the solidification heat transfer of the irregular continuous casting billet is visualized and the result is post-processed as follows:

extracting calculation results at different positions of a continuous casting machine after the irregular continuous casting blank continuous casting solidification heat transfer calculation is completed, and introducing the calculation results into Tecplot software for visualization treatment to form a temperature cloud picture; respectively extracting the temperature change conditions of the characteristic points, and importing the characteristic points into Origin software for visualization processing to form a temperature change curve; the characteristic points comprise web surface centers, inner edge surface centers, narrow-face film mounting centers and round corner surface centers for the special-shaped billets, wide-face surface centers, narrow-face surface centers, casting blank centers and round corner centers for the round corner billets, and casting blank centers and casting blank surfaces for the round billets.

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